Euclidean Alogrithms

Fourier theorem A Fourier series is an expansion of a periodic function into a sum of trigonometric functions. Read more Gaussian function Gaussian is a function of the base form with f(x) = exp(-x^2) and with parametric extension f(x) = aexp(-(x-b)^2)/2c^2) for arbitrary real constants a, b and non-zero c. Read more © 2025 bikashpandey.com. All Rights Reserved. Privacy Policy [email protected] Maths Fourier theorem A Fourier series is an expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series. By expressing a function as a sum of sines and cosines, may problems involving the function become easier to analyze because trigonometric functions are well understood. For example, Fourier series were first used by Josheph Fourier(1786-1830) to find solutions to the heat equation. This application is possible because the derivatives of trigonometric functions fall into simple patterns. Fourier series cannot be used to approximate arbitrary functions, because most functions fall into simple patterns. Fourier series cannot be used to approximate arbitrary functions, because most functions have infinitely many terms in their Fourier series, and the series do not always converge. Well-behaved functions, for example smooth functions, have Fourier series that converge to the original function. The coefficients of the Fourier series are determined by integrals of the function multiplied by trigonometric functions, described in Fourier series Definition. The study of the convergence of Fourier series focus on the behaviors of the partial sums, which means studying the behavior of the sum as more and more terms from the series are summed. Fourier series are closely related to the Fourier transform. a more general tool that can even find the frequency information for functions that are not periodic. Periodic functions are identified with functions on a circle; for this reason Fourier series are the subject of Fourier analysis on the circle group, denoted by T or S1. The Fourier transform is also part of Fourier analysis, but is defined for functions on Rn. Since Fourier's time, many different approaches to defining and understanding the concept of Fourier series have been discovered, all of which are consistent with one another, but each of which emphasizes different aspects of the topic. Some of the more powerful and elegant approaches are based on the mathematical ideas and tools that were not available in Fourier's time. Fourier originally defined the Fourier series for real-valued functions of real arguments, and used the sine and cosine functions in the decomposition. May other Fourier-related transforms have since been defined, extending his initial idea to many applications and birthing an area of mathematics called Fourier analysis. Fourier transform The Fourier Transform is an integral transform that makes a function as input and outputs another function that describes the extent to which various frequencies are present in the original function. The output of the transform is a complex-valued function of frequency. The output of the transform is a complex valued function of frequency. The term Fourier transform refers to both this complex-valued function and the mathematical operation. When a distinction needs to be made, the output of the operation is sometimes called the frequency domain representation of the original function. The Fourier transform is analogous to decomposing the sound of a musical chord into the intensities of its constituent pitches. Functions that are localized in the time domain have Fourier transforms that are spread out across the frequency domain and vice versa,a phenomenon known as the uncertainty principle. The critical case for this principle is the Gaussian function, of substantial importance in probability theory and statistics as well as in the study of physical phenomena exhibiting normal distribution(example: diffusion). The Fourier transform of a Gaussian function is another Gaussian function. Joseph Fourier introduced the sine and cosine transforms ( which correspond to the imaginary and real components of the modern Fourier transform) in his study of heat transfer, where Gaussian functions appear as solutions of the heat equation. Comments# frozen_string_literal: true def gcd(first, second) if first.zero? return second end gcd(second % first, first) end first = 36 second = 60 puts "GCD of #{first} and #{second} is #{gcd(first, second)}"[email protected] Gaussian function Gaussian is a function of the base form with f(x) = exp(-x^2) and with parametric extension f(x) = aexp(-(x-b)^2)/2c^2) for arbitrary real constants a, b and non-zero c. It is named after the mathematician Carl Friedrich Gauss. The graph of a Gaussian is a characteristic symmetric "bell curve" shape. The parameter is the height of the curve's peak, b is the position of the center of the peak, and c (the standard deviation, sometimes called the Gaussian RMS width) controls the width of the "bell". Gaussian functions are widely used in statistics to describe the normal distributions, in signal processing to define the Gaussian filters, in image processing where two-dimensional Gaussians are used for Gaussian blurs, and in mathematics to solve heat equations and diffusion equations and to define the Weierstrass transform. They are also abundantly used in quantum chemistry to form basic sets. The most common method for estimating the Gaussian parameters is to take the logarithm of data and fit a parabola to the resulting data set. While this provides a simple curve fitting procedure, the resulting algorithm may be biased by excessively weighting small data values, which can produce large errors in the profile estimate. One can partially compensate for this problem through weighted least squares estimation, reducing the weight of small data values, but this too can be biased by allowing the tail of the Gaussian to dominate the fit. In order to remove the bias, one can instead use an iteratively reweighted least squares procedure, in which the weights are updated at each iteration. It is also possible to perform non-linear regression directly on the data without involving the logarithmic data transformation. Parameter Precision Once one has an algorithm for estimating the Gaussian parameters, it is also important to know how precise those estimates are. Any least squares estimation algorithm can provide numerical estimates for the variance of each parameter ( i.e. the variance of estimated height, position, and width of the function). One can also use Cramer-Rao bound theory to obtain an analytical expression for the lower bound on the parameter variances, given certain assumptions about the data. 1. The noise in the measured profile is either independent and identically distributed Gaussian, or the noise is Poissin-distributed. 2. The spacing between each sampling ( i.e the distance between pixels measuring the data) is uniform. 3. The peak is "well-sampled", so that less than 10% of the area or volume under the peak(area if a 1D Gaussian, volume if a 2D Gaussian) lies outside the measurement region. 4. The width of the peak is much larger than the distance between sample locations( i.e. the detector pixels must be al least 5 time smaller than the Gaussian FWHM). Applications Gaussian functions appear in many context in natural sciences, the social sciences, mathematics and engineering. Some examples include: 1. In statistics and probability theory, Gaussian functions appear as the density function of the normal distribution, which is a limiting probability distribution of complicated sums, according to central limit theorem. 2. A Gaussian function is the wave function of the ground state of the quantum harmonic oscillator. 3. The molecular orbitals used in computational chemistry can be linear combinations of Gaussian functions called Gaussian orbitals. 3. Gaussian functions are also associated with the vacuum state in quantum field theory. 4. Gaussian beams are used in optical systems, microwave systems and lasers. 5. In scale space representation, Gaussian functions are used as smoothing kernels for generating multi-scale representations in computer vision and image processing. Specifically, derivatives of Gaussians ( Hermite functions ) are used as a basis for defining a large number of types of visual operations. 6. Gaussian functions are used to define some types of artificial neural networks. 7. In fluorescence microscopy a 2D Gaussian function is used to approximate the Airy disk, describing the intensity distribution produced by a point source. 8. In signal processing they serve to define Gaussian filter, such as in image processing where 2D Gaussian kernel, which may be approximated by the Binomial coefficient or sampling a Gaussina. 9. In geostatistics, they have been used for understanding the variability between the patterns of a complex training image. They are used with kernel methods to cluster the patterns in the feature space. CommentsBlog Open user menuuser photo Featured Ruby Maths Fourier theorem A Fourier series is an expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series. By expressing a function as a sum of sines and cosines, may problems involving the function become easier to analyze because trigonometric functions are well understood. For example, Fourier series were first used by Josheph Fourier(1786-1830) to find solutions to the heat equation. This application is possible because the derivatives of trigonometric functions fall into simple patterns. Fourier series cannot be used to approximate arbitrary functions, because most functions fall into simple patterns. Fourier series cannot be used to approximate arbitrary functions, because most functions have infinitely many terms in their Fourier series, and the series do not always converge. Well-behaved functions, for example smooth functions, have Fourier series that converge to the original function. The coefficients of the Fourier series are determined by integrals of the function multiplied by trigonometric functions, described in Fourier series Definition. The study of the convergence of Fourier series focus on the behaviors of the partial sums, which means studying the behavior of the sum as more and more terms from the series are summed. Fourier series are closely related to the Fourier transform. a more general tool that can even find the frequency information for functions that are not periodic. Periodic functions are identified with functions on a circle; for this reason Fourier series are the subject of Fourier analysis on the circle group, denoted by T or S1. The Fourier transform is also part of Fourier analysis, but is defined for functions on Rn. Since Fourier's time, many different approaches to defining and understanding the concept of Fourier series have been discovered, all of which are consistent with one another, but each of which emphasizes different aspects of the topic. Some of the more powerful and elegant approaches are based on the mathematical ideas and tools that were not available in Fourier's time. Fourier originally defined the Fourier series for real-valued functions of real arguments, and used the sine and cosine functions in the decomposition. May other Fourier-related transforms have since been defined, extending his initial idea to many applications and birthing an area of mathematics called Fourier analysis. Fourier transform The Fourier Transform is an integral transform that makes a function as input and outputs another function that describes the extent to which various frequencies are present in the original function. The output of the transform is a complex-valued function of frequency. The output of the transform is a complex valued function of frequency. The term Fourier transform refers to both this complex-valued function and the mathematical operation. When a distinction needs to be made, the output of the operation is sometimes called the frequency domain representation of the original function. The Fourier transform is analogous to decomposing the sound of a musical chord into the intensities of its constituent pitches. Functions that are localized in the time domain have Fourier transforms that are spread out across the frequency domain and vice versa,a phenomenon known as the uncertainty principle. The critical case for this principle is the Gaussian function, of substantial importance in probability theory and statistics as well as in the study of physical phenomena exhibiting normal distribution(example: diffusion). The Fourier transform of a Gaussian function is another Gaussian function. Joseph Fourier introduced the sine and cosine transforms ( which correspond to the imaginary and real components of the modern Fourier transform) in his study of heat transfer, where Gaussian functions appear as solutions of the heat equation. CommentsMaths Euclidean Alogrithms From geeksforgeeks. The Eucliean alogrithm is a way of find the greatest common divisor of two positive integers. GCD of two numbers is the largest number that divides both of them. A simple way to find GCD is to factorize both numbers and multiply common prime factors. 36 = 2 * 2 * 3 * 3 60 = 2 * 2 * 3 * 5 GCD = Multiplication of common factors = 2 * 2 * 3 = 12 Basic Euclidean Algorithm for GCD: The alogirithm is based on the below facts: - If we subtract a smaller number from a larger one ( we reduce a large number ), GCD does not change. So, if we keep subtracting repeatedly the larger of two, we end up with GCD. - Now instead of subtraction, if we divide the larger number, the algorithm stops when we find the reminder 0. # frozen_string_literal: true def gcd(first, second) if first.zero? return second end gcd(second % first, first) end first = 36 second = 60 puts "GCD of #{first} and #{second} is #{gcd(first, second)}" CommentsThe English definite article the, written þe in Middle English, derives from an Old English demonstrative, which, according to gender, was written se (masculine), seo (feminine) (þe and þeo in the Northumbrian dialect), or þæt (neuter). The neuter form þæt also gave rise to the modern demonstrative that. The ye occasionally seen in pseudo-archaic usage such as "Ye Olde Englishe Tea Shoppe" is actually a form of þe, where the letter thorn (þ) came to be written as a y. Multiple demonstratives can give rise to multiple definite articles. Macedonian, for example, in which the articles are suffixed, has столот (stolot), the chair; столов (stolov), this chair; and столон (stolon), that chair. These derive from the Proto-Slavic demonstratives *tъ "this, that", *ovъ "this here" and *onъ "that over there, yonder" respectively. Colognian prepositions articles such as in dat Auto, or et Auto, the car; the first being specifically selected, focused, newly introduced, while the latter is not selected, unfocused, already known, general, or generic. Standard Basque distinguishes between proximal and distal definite articles in the plural (dialectally, a proximal singular and an additional medial grade may also be present). The Basque distal form (with infix -a-, etymologically a suffixed and phonetically reduced form of the distal demonstrative har-/hai-) functions as the default definite article, whereas the proximal form (with infix -o-, derived from the proximal demonstrative hau-/hon-) is marked and indicates some kind of (spatial or otherwise) close relationship between the speaker and the referent (e.g., it may imply that the speaker is included in the referent): etxeak ("the houses") vs. etxeok ("these houses [of ours]"), euskaldunak ("the Basque speakers") vs. euskaldunok ("we, the Basque speakers"). Speakers of Assyrian Neo-Aramaic, a modern Aramaic language that lacks a definite article, may at times use demonstratives aha and aya (feminine) or awa (masculine) – which translate to "this" and "that", respectively – to give the sense of "the".[10] In Indonesian, the third person possessive suffix -nya could be also used as a definite article.[11] Indefinite articles edit Indefinite articles typically arise from adjectives meaning one. For example, the indefinite articles in the Romance languages—e.g., un, una, une—derive from the Latin adjective unus. Partitive articles, however, derive from Vulgar Latin de illo, meaning (some) of the. The English indefinite article an is derived from the same root as one. The -n came to be dropped before consonants, giving rise to the shortened form a. The existence of both forms has led to many cases of juncture loss, for example transforming the original a napron into the modern an apron. The Persian indefinite article is yek, meaning one. See also edit English articles Al- (definite article in Arabic) Definiteness Definite description False title References edit Recasens, Marta; Martí, M. Antònia; Taulé, Mariona (2009-06-16), Winkler, Susanne; Featherston, Sam (eds.), "First-mention definites:More than exceptional cases", The Fruits of Empirical Linguistics II, vol. 102, Berlin, New York: Mouton de Gruyter, pp. 217–238, doi:10.1515/9783110216158.217, ISBN 978-3-11-021347-8, retrieved 2023-01-16 New perspectives on Hispanic contact : linguistics in the Americas. Melvin González-Rivera, and Sandro Sessarego. Madrid: Iberoamericana. 2015. ISBN 978-3-95487-831-4. OCLC 969386958. Burchfield, R. W. (1996). The New Fowler's Modern English Usage (3rd ed.). Oxford University Press. p. 512. ISBN 978-0199690367. Argetsinger, Amy (1 September 2015). "Why does everyone call Donald Trump 'The Donald'? It's an interesting story". The Washington Post. Retrieved 3 October 2017. Lawrence, Erma (1977). Haida dictionary. Fairbanks: Alaska Native Language Center. p. 64. Master, Peter (1997). "The English article system: Acquisition, function, and pedagogy". System. 25 (2): 215–232. doi:10.1016/S0346-251X(97)00010-9. Kusmenko, J K. "The typology of the language contact on the Balkans and in Scandinavia. A case of the suffixed definite article" (PDF). Archived (PDF) from the original on 2021-10-04. See s.5 Summary. Retrieved 2 February 2012. Simona, Ropati (1986). Tokelau Dictionary. New Zealand: Office of Tokelau Affairs. p. Introduction. Greenberg, Joseph H. (2005). Genetic linguistics : essays on theory and method. William Croft. Oxford: Oxford University Press. ISBN 978-0-19-151452-4. OCLC 132691297. Solomon, Zomaya S. (1997). Functional and other exotic sentences in Assyrian Aramaic, Journal of Assyrian Academic Studies, XI/2:44-69. "20 Contoh Penggunaan Kata nya dalam Kalimat" [Twenty Examples of the Use of 'nya' in Sentences]. kumparan.com (in Indonesian). Retrieved 2024-05-26. External links edit Look up article, definite article, or indefinite article in Wiktionary, the free dictionary. Wikisource has the text of the 1921 Collier's Encyclopedia article Article. "The Definite Article, 'The': The Most Frequently Used Word in World's Englishes" Ing, John (September 17, 2019). "How to Use Articles Correctly in English (A, An & The)". Retrieved May 3, 2020. Last edited 20 days ago by Kaloan-koko Related articles French articles and determiners Definiteness Semantic feature of noun phrases in linguistics English determiners Determiners in the English language Wikipedia Wikimedia Foundation Powered by MediaWiki Content is available under CC BY-SA 4.0 unless otherwise noted. Privacy policy Contact Wikipedia Code of Conduct Developers Statistics Cookie statement Terms of Use DesktopThe English definite article the, written þe in Middle English, derives from an Old English demonstrative, which, according to gender, was written se (masculine), seo (feminine) (þe and þeo in the Northumbrian dialect), or þæt (neuter). The neuter form þæt also gave rise to the modern demonstrative that. The ye occasionally seen in pseudo-archaic usage such as "Ye Olde Englishe Tea Shoppe" is actually a form of þe, where the letter thorn (þ) came to be written as a y. Multiple demonstratives can give rise to multiple definite articles. Macedonian, for example, in which the articles are suffixed, has столот (stolot), the chair; столов (stolov), this chair; and столон (stolon), that chair. These derive from the Proto-Slavic demonstratives *tъ "this, that", *ovъ "this here" and *onъ "that over there, yonder" respectively. Colognian prepositions articles such as in dat Auto, or et Auto, the car; the first being specifically selected, focused, newly introduced, while the latter is not selected, unfocused, already known, general, or generic. Standard Basque distinguishes between proximal and distal definite articles in the plural (dialectally, a proximal singular and an additional medial grade may also be present). The Basque distal form (with infix -a-, etymologically a suffixed and phonetically reduced form of the distal demonstrative har-/hai-) functions as the default definite article, whereas the proximal form (with infix -o-, derived from the proximal demonstrative hau-/hon-) is marked and indicates some kind of (spatial or otherwise) close relationship between the speaker and the referent (e.g., it may imply that the speaker is included in the referent): etxeak ("the houses") vs. etxeok ("these houses [of ours]"), euskaldunak ("the Basque speakers") vs. euskaldunok ("we, the Basque speakers"). Speakers of Assyrian Neo-Aramaic, a modern Aramaic language that lacks a definite article, may at times use demonstratives aha and aya (feminine) or awa (masculine) – which translate to "this" and "that", respectively – to give the sense of "the".[10] In Indonesian, the third person possessive suffix -nya could be also used as a definite article.[11] Indefinite articles edit Indefinite articles typically arise from adjectives meaning one. For example, the indefinite articles in the Romance languages—e.g., un, una, une—derive from the Latin adjective unus. Partitive articles, however, derive from Vulgar Latin de illo, meaning (some) of the. The English indefinite article an is derived from the same root as one. The -n came to be dropped before consonants, giving rise to the shortened form a. The existence of both forms has led to many cases of juncture loss, for example transforming the original a napron into the modern an apron. The Persian indefinite article is yek, meaning one. See also edit English articles Al- (definite article in Arabic) Definiteness Definite description False title References edit Recasens, Marta; Martí, M. Antònia; Taulé, Mariona (2009-06-16), Winkler, Susanne; Featherston, Sam (eds.), "First-mention definites:More than exceptional cases", The Fruits of Empirical Linguistics II, vol. 102, Berlin, New York: Mouton de Gruyter, pp. 217–238, doi:10.1515/9783110216158.217, ISBN 978-3-11-021347-8, retrieved 2023-01-16 New perspectives on Hispanic contact : linguistics in the Americas. Melvin González-Rivera, and Sandro Sessarego. Madrid: Iberoamericana. 2015. ISBN 978-3-95487-831-4. OCLC 969386958. Burchfield, R. W. (1996). The New Fowler's Modern English Usage (3rd ed.). Oxford University Press. p. 512. ISBN 978-0199690367. Argetsinger, Amy (1 September 2015). "Why does everyone call Donald Trump 'The Donald'? It's an interesting story". The Washington Post. Retrieved 3 October 2017. Lawrence, Erma (1977). Haida dictionary. Fairbanks: Alaska Native Language Center. p. 64. Master, Peter (1997). "The English article system: Acquisition, function, and pedagogy". System. 25 (2): 215–232. doi:10.1016/S0346-251X(97)00010-9. Kusmenko, J K. "The typology of the language contact on the Balkans and in Scandinavia. A case of the suffixed definite article" (PDF). Archived (PDF) from the original on 2021-10-04. See s.5 Summary. Retrieved 2 February 2012. Simona, Ropati (1986). Tokelau Dictionary. New Zealand: Office of Tokelau Affairs. p. Introduction. Greenberg, Joseph H. (2005). Genetic linguistics : essays on theory and method. William Croft. Oxford: Oxford University Press. ISBN 978-0-19-151452-4. OCLC 132691297. Solomon, Zomaya S. (1997). Functional and other exotic sentences in Assyrian Aramaic, Journal of Assyrian Academic Studies, XI/2:44-69. "20 Contoh Penggunaan Kata nya dalam Kalimat" [Twenty Examples of the Use of 'nya' in Sentences]. kumparan.com (in Indonesian). Retrieved 2024-05-26. External links edit Look up article, definite article, or indefinite article in Wiktionary, the free dictionary. Wikisource has the text of the 1921 Collier's Encyclopedia article Article. "The Definite Article, 'The': The Most Frequently Used Word in World's Englishes" Ing, John (September 17, 2019). "How to Use Articles Correctly in English (A, An & The)". Retrieved May 3, 2020. Last edited 20 days ago by Kaloan-koko Related articles French articles and determiners Definiteness Semantic feature of noun phrases in linguistics English determiners Determiners in the English language Wikipedia Wikimedia Foundation Powered by MediaWiki Content is available under CC BY-SA 4.0 unless otherwise noted. Privacy policy Contact Wikipedia Code of Conduct Developers Statistics Cookie statement Terms of Use DesktopBlog Open user menuuser photo Featured Ruby Maths Fourier theorem A Fourier series is an expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series. By expressing a function as a sum of sines and cosines, may problems involving the function become easier to analyze because trigonometric functions are well understood. For example, Fourier series were first used by Josheph Fourier(1786-1830) to find solutions to the heat equation. This application is possible because the derivatives of trigonometric functions fall into simple patterns. Fourier series cannot be used to approximate arbitrary functions, because most functions fall into simple patterns. Fourier series cannot be used to approximate arbitrary functions, because most functions have infinitely many terms in their Fourier series, and the series do not always converge. Well-behaved functions, for example smooth functions, have Fourier series that converge to the original function. The coefficients of the Fourier series are determined by integrals of the function multiplied by trigonometric functions, described in Fourier series Definition. The study of the convergence of Fourier series focus on the behaviors of the partial sums, which means studying the behavior of the sum as more and more terms from the series are summed. Fourier series are closely related to the Fourier transform. a more general tool that can even find the frequency information for functions that are not periodic. Periodic functions are identified with functions on a circle; for this reason Fourier series are the subject of Fourier analysis on the circle group, denoted by T or S1. The Fourier transform is also part of Fourier analysis, but is defined for functions on Rn. Since Fourier's time, many different approaches to defining and understanding the concept of Fourier series have been discovered, all of which are consistent with one another, but each of which emphasizes different aspects of the topic. Some of the more powerful and elegant approaches are based on the mathematical ideas and tools that were not available in Fourier's time. Fourier originally defined the Fourier series for real-valued functions of real arguments, and used the sine and cosine functions in the decomposition. May other Fourier-related transforms have since been defined, extending his initial idea to many applications and birthing an area of mathematics called Fourier analysis. Fourier transform The Fourier Transform is an integral transform that makes a function as input and outputs another function that describes the extent to which various frequencies are present in the original function. The output of the transform is a complex-valued function of frequency. The output of the transform is a complex valued function of frequency. The term Fourier transform refers to both this complex-valued function and the mathematical operation. When a distinction needs to be made, the output of the operation is sometimes called the frequency domain representation of the original function. The Fourier transform is analogous to decomposing the sound of a musical chord into the intensities of its constituent pitches. Functions that are localized in the time domain have Fourier transforms that are spread out across the frequency domain and vice versa,a phenomenon known as the uncertainty principle. The critical case for this principle is the Gaussian function, of substantial importance in probability theory and statistics as well as in the study of physical phenomena exhibiting normal distribution(example: diffusion). The Fourier transform of a Gaussian function is another Gaussian function. Joseph Fourier introduced the sine and cosine transforms ( which correspond to the imaginary and real components of the modern Fourier transform) in his study of heat transfer, where Gaussian functions appear as solutions of the heat equation. [email protected]# frozen_string_literal: true def gcd(first, second) if first.zero? return second end gcd(second % first, first) end first = 36 second = 60 puts "GCD of #{first} and #{second} is #{gcd(first, second)}"Ruby Maths Fourier theorem A Fourier series is an expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series. By expressing a function as a sum of sines and cosines, may problems involving the function become easier to analyze because trigonometric functions are well understood. For example, Fourier series were first used by Josheph Fourier(1786-1830) to find solutions to the heat equation. This application is possible because the derivatives of trigonometric functions fall into simple patterns. Fourier series cannot be used to approximate arbitrary functions, because most functions fall into simple patterns. Fourier series cannot be used to approximate arbitrary functions, because most functions have infinitely many terms in their Fourier series, and the series do not always converge. Well-behaved functions, for example smooth functions, have Fourier series that converge to the original function. The coefficients of the Fourier series are determined by integrals of the function multiplied by trigonometric functions, described in Fourier series Definition. The study of the convergence of Fourier series focus on the behaviors of the partial sums, which means studying the behavior of the sum as more and more terms from the series are summed. Fourier series are closely related to the Fourier transform. a more general tool that can even find the frequency information for functions that are not periodic. Periodic functions are identified with functions on a circle; for this reason Fourier series are the subject of Fourier analysis on the circle group, denoted by T or S1. The Fourier transform is also part of Fourier analysis, but is defined for functions on Rn. Since Fourier's time, many different approaches to defining and understanding the concept of Fourier series have been discovered, all of which are consistent with one another, but each of which emphasizes different aspects of the topic. Some of the more powerful and elegant approaches are based on the mathematical ideas and tools that were not available in Fourier's time. Fourier originally defined the Fourier series for real-valued functions of real arguments, and used the sine and cosine functions in the decomposition. May other Fourier-related transforms have since been defined, extending his initial idea to many applications and birthing an area of mathematics called Fourier analysis. Fourier transform The Fourier Transform is an integral transform that makes a function as input and outputs another function that describes the extent to which various frequencies are present in the original function. The output of the transform is a complex-valued function of frequency. The output of the transform is a complex valued function of frequency. The term Fourier transform refers to both this complex-valued function and the mathematical operation. When a distinction needs to be made, the output of the operation is sometimes called the frequency domain representation of the original function. The Fourier transform is analogous to decomposing the sound of a musical chord into the intensities of its constituent pitches. Functions that are localized in the time domain have Fourier transforms that are spread out across the frequency domain and vice versa,a phenomenon known as the uncertainty principle. The critical case for this principle is the Gaussian function, of substantial importance in probability theory and statistics as well as in the study of physical phenomena exhibiting normal distribution(example: diffusion). The Fourier transform of a Gaussian function is another Gaussian function. Joseph Fourier introduced the sine and cosine transforms ( which correspond to the imaginary and real components of the modern Fourier transform) in his study of heat transfer, where Gaussian functions appear as solutions of the heat equation. CommentsFourier theorem A Fourier series is an expansion of a periodic function into a sum of trigonometric functions. Read more Gaussian function Gaussian is a function of the base form with f(x) = exp(-x^2) and with parametric extension f(x) = aexp(-(x-b)^2)/2c^2) for arbitrary real constants a, b and non-zero c. Read more © 2025 bikashpandey.com. All Rights Reserved. Privacy Policy [email protected] admin monkey bou 🐒 👉 bkp by Om Luitel 10 minutes agoFourier theorem A Fourier series is an expansion of a periodic function into a sum of trigonometric functions. Read more Gaussian function Gaussian is a function of the base form with f(x) = exp(-x^2) and with parametric extension f(x) = aexp(-(x-b)^2)/2c^2) for arbitrary real constants a, b and non-zero c. Read more © 2025 bikashpandey.com. All Rights Reserved. Privacy Policy [email protected] Maths Fourier theorem A Fourier series is an expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series. By expressing a function as a sum of sines and cosines, may problems involving the function become easier to analyze because trigonometric functions are well understood. For example, Fourier series were first used by Josheph Fourier(1786-1830) to find solutions to the heat equation. This application is possible because the derivatives of trigonometric functions fall into simple patterns. Fourier series cannot be used to approximate arbitrary functions, because most functions fall into simple patterns. Fourier series cannot be used to approximate arbitrary functions, because most functions have infinitely many terms in their Fourier series, and the series do not always converge. Well-behaved functions, for example smooth functions, have Fourier series that converge to the original function. The coefficients of the Fourier series are determined by integrals of the function multiplied by trigonometric functions, described in Fourier series Definition. The study of the convergence of Fourier series focus on the behaviors of the partial sums, which means studying the behavior of the sum as more and more terms from the series are summed. Fourier series are closely related to the Fourier transform. a more general tool that can even find the frequency information for functions that are not periodic. Periodic functions are identified with functions on a circle; for this reason Fourier series are the subject of Fourier analysis on the circle group, denoted by T or S1. The Fourier transform is also part of Fourier analysis, but is defined for functions on Rn. Since Fourier's time, many different approaches to defining and understanding the concept of Fourier series have been discovered, all of which are consistent with one another, but each of which emphasizes different aspects of the topic. Some of the more powerful and elegant approaches are based on the mathematical ideas and tools that were not available in Fourier's time. Fourier originally defined the Fourier series for real-valued functions of real arguments, and used the sine and cosine functions in the decomposition. May other Fourier-related transforms have since been defined, extending his initial idea to many applications and birthing an area of mathematics called Fourier analysis. Fourier transform The Fourier Transform is an integral transform that makes a function as input and outputs another function that describes the extent to which various frequencies are present in the original function. The output of the transform is a complex-valued function of frequency. The output of the transform is a complex valued function of frequency. The term Fourier transform refers to both this complex-valued function and the mathematical operation. When a distinction needs to be made, the output of the operation is sometimes called the frequency domain representation of the original function. The Fourier transform is analogous to decomposing the sound of a musical chord into the intensities of its constituent pitches. Functions that are localized in the time domain have Fourier transforms that are spread out across the frequency domain and vice versa,a phenomenon known as the uncertainty principle. The critical case for this principle is the Gaussian function, of substantial importance in probability theory and statistics as well as in the study of physical phenomena exhibiting normal distribution(example: diffusion). The Fourier transform of a Gaussian function is another Gaussian function. Joseph Fourier introduced the sine and cosine transforms ( which correspond to the imaginary and real components of the modern Fourier transform) in his study of heat transfer, where Gaussian functions appear as solutions of the heat equation. Comments# frozen_string_literal: true def gcd(first, second) if first.zero? return second end gcd(second % first, first) end first = 36 second = 60 puts "GCD of #{first} and #{second} is #{gcd(first, second)}"[email protected] Gaussian function Gaussian is a function of the base form with f(x) = exp(-x^2) and with parametric extension f(x) = aexp(-(x-b)^2)/2c^2) for arbitrary real constants a, b and non-zero c. It is named after the mathematician Carl Friedrich Gauss. The graph of a Gaussian is a characteristic symmetric "bell curve" shape. The parameter is the height of the curve's peak, b is the position of the center of the peak, and c (the standard deviation, sometimes called the Gaussian RMS width) controls the width of the "bell". Gaussian functions are widely used in statistics to describe the normal distributions, in signal processing to define the Gaussian filters, in image processing where two-dimensional Gaussians are used for Gaussian blurs, and in mathematics to solve heat equations and diffusion equations and to define the Weierstrass transform. They are also abundantly used in quantum chemistry to form basic sets. The most common method for estimating the Gaussian parameters is to take the logarithm of data and fit a parabola to the resulting data set. While this provides a simple curve fitting procedure, the resulting algorithm may be biased by excessively weighting small data values, which can produce large errors in the profile estimate. One can partially compensate for this problem through weighted least squares estimation, reducing the weight of small data values, but this too can be biased by allowing the tail of the Gaussian to dominate the fit. In order to remove the bias, one can instead use an iteratively reweighted least squares procedure, in which the weights are updated at each iteration. It is also possible to perform non-linear regression directly on the data without involving the logarithmic data transformation. Parameter Precision Once one has an algorithm for estimating the Gaussian parameters, it is also important to know how precise those estimates are. Any least squares estimation algorithm can provide numerical estimates for the variance of each parameter ( i.e. the variance of estimated height, position, and width of the function). One can also use Cramer-Rao bound theory to obtain an analytical expression for the lower bound on the parameter variances, given certain assumptions about the data. 1. The noise in the measured profile is either independent and identically distributed Gaussian, or the noise is Poissin-distributed. 2. The spacing between each sampling ( i.e the distance between pixels measuring the data) is uniform. 3. The peak is "well-sampled", so that less than 10% of the area or volume under the peak(area if a 1D Gaussian, volume if a 2D Gaussian) lies outside the measurement region. 4. The width of the peak is much larger than the distance between sample locations( i.e. the detector pixels must be al least 5 time smaller than the Gaussian FWHM). Applications Gaussian functions appear in many context in natural sciences, the social sciences, mathematics and engineering. Some examples include: 1. In statistics and probability theory, Gaussian functions appear as the density function of the normal distribution, which is a limiting probability distribution of complicated sums, according to central limit theorem. 2. A Gaussian function is the wave function of the ground state of the quantum harmonic oscillator. 3. The molecular orbitals used in computational chemistry can be linear combinations of Gaussian functions called Gaussian orbitals. 3. Gaussian functions are also associated with the vacuum state in quantum field theory. 4. Gaussian beams are used in optical systems, microwave systems and lasers. 5. In scale space representation, Gaussian functions are used as smoothing kernels for generating multi-scale representations in computer vision and image processing. Specifically, derivatives of Gaussians ( Hermite functions ) are used as a basis for defining a large number of types of visual operations. 6. Gaussian functions are used to define some types of artificial neural networks. 7. In fluorescence microscopy a 2D Gaussian function is used to approximate the Airy disk, describing the intensity distribution produced by a point source. 8. In signal processing they serve to define Gaussian filter, such as in image processing where 2D Gaussian kernel, which may be approximated by the Binomial coefficient or sampling a Gaussina. 9. In geostatistics, they have been used for understanding the variability between the patterns of a complex training image. They are used with kernel methods to cluster the patterns in the feature space. CommentsBlog Open user menuuser photo Featured Ruby Maths Fourier theorem A Fourier series is an expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series. By expressing a function as a sum of sines and cosines, may problems involving the function become easier to analyze because trigonometric functions are well understood. For example, Fourier series were first used by Josheph Fourier(1786-1830) to find solutions to the heat equation. This application is possible because the derivatives of trigonometric functions fall into simple patterns. Fourier series cannot be used to approximate arbitrary functions, because most functions fall into simple patterns. Fourier series cannot be used to approximate arbitrary functions, because most functions have infinitely many terms in their Fourier series, and the series do not always converge. Well-behaved functions, for example smooth functions, have Fourier series that converge to the original function. The coefficients of the Fourier series are determined by integrals of the function multiplied by trigonometric functions, described in Fourier series Definition. The study of the convergence of Fourier series focus on the behaviors of the partial sums, which means studying the behavior of the sum as more and more terms from the series are summed. Fourier series are closely related to the Fourier transform. a more general tool that can even find the frequency information for functions that are not periodic. Periodic functions are identified with functions on a circle; for this reason Fourier series are the subject of Fourier analysis on the circle group, denoted by T or S1. The Fourier transform is also part of Fourier analysis, but is defined for functions on Rn. Since Fourier's time, many different approaches to defining and understanding the concept of Fourier series have been discovered, all of which are consistent with one another, but each of which emphasizes different aspects of the topic. Some of the more powerful and elegant approaches are based on the mathematical ideas and tools that were not available in Fourier's time. Fourier originally defined the Fourier series for real-valued functions of real arguments, and used the sine and cosine functions in the decomposition. May other Fourier-related transforms have since been defined, extending his initial idea to many applications and birthing an area of mathematics called Fourier analysis. Fourier transform The Fourier Transform is an integral transform that makes a function as input and outputs another function that describes the extent to which various frequencies are present in the original function. The output of the transform is a complex-valued function of frequency. The output of the transform is a complex valued function of frequency. The term Fourier transform refers to both this complex-valued function and the mathematical operation. When a distinction needs to be made, the output of the operation is sometimes called the frequency domain representation of the original function. The Fourier transform is analogous to decomposing the sound of a musical chord into the intensities of its constituent pitches. Functions that are localized in the time domain have Fourier transforms that are spread out across the frequency domain and vice versa,a phenomenon known as the uncertainty principle. The critical case for this principle is the Gaussian function, of substantial importance in probability theory and statistics as well as in the study of physical phenomena exhibiting normal distribution(example: diffusion). The Fourier transform of a Gaussian function is another Gaussian function. Joseph Fourier introduced the sine and cosine transforms ( which correspond to the imaginary and real components of the modern Fourier transform) in his study of heat transfer, where Gaussian functions appear as solutions of the heat equation. CommentsMaths Euclidean Alogrithms From geeksforgeeks. The Eucliean alogrithm is a way of find the greatest common divisor of two positive integers. GCD of two numbers is the largest number that divides both of them. A simple way to find GCD is to factorize both numbers and multiply common prime factors. 36 = 2 * 2 * 3 * 3 60 = 2 * 2 * 3 * 5 GCD = Multiplication of common factors = 2 * 2 * 3 = 12 Basic Euclidean Algorithm for GCD: The alogirithm is based on the below facts: - If we subtract a smaller number from a larger one ( we reduce a large number ), GCD does not change. So, if we keep subtracting repeatedly the larger of two, we end up with GCD. - Now instead of subtraction, if we divide the larger number, the algorithm stops when we find the reminder 0. # frozen_string_literal: true def gcd(first, second) if first.zero? return second end gcd(second % first, first) end first = 36 second = 60 puts "GCD of #{first} and #{second} is #{gcd(first, second)}" CommentsThe English definite article the, written þe in Middle English, derives from an Old English demonstrative, which, according to gender, was written se (masculine), seo (feminine) (þe and þeo in the Northumbrian dialect), or þæt (neuter). The neuter form þæt also gave rise to the modern demonstrative that. The ye occasionally seen in pseudo-archaic usage such as "Ye Olde Englishe Tea Shoppe" is actually a form of þe, where the letter thorn (þ) came to be written as a y. Multiple demonstratives can give rise to multiple definite articles. Macedonian, for example, in which the articles are suffixed, has столот (stolot), the chair; столов (stolov), this chair; and столон (stolon), that chair. These derive from the Proto-Slavic demonstratives *tъ "this, that", *ovъ "this here" and *onъ "that over there, yonder" respectively. Colognian prepositions articles such as in dat Auto, or et Auto, the car; the first being specifically selected, focused, newly introduced, while the latter is not selected, unfocused, already known, general, or generic. Standard Basque distinguishes between proximal and distal definite articles in the plural (dialectally, a proximal singular and an additional medial grade may also be present). The Basque distal form (with infix -a-, etymologically a suffixed and phonetically reduced form of the distal demonstrative har-/hai-) functions as the default definite article, whereas the proximal form (with infix -o-, derived from the proximal demonstrative hau-/hon-) is marked and indicates some kind of (spatial or otherwise) close relationship between the speaker and the referent (e.g., it may imply that the speaker is included in the referent): etxeak ("the houses") vs. etxeok ("these houses [of ours]"), euskaldunak ("the Basque speakers") vs. euskaldunok ("we, the Basque speakers"). Speakers of Assyrian Neo-Aramaic, a modern Aramaic language that lacks a definite article, may at times use demonstratives aha and aya (feminine) or awa (masculine) – which translate to "this" and "that", respectively – to give the sense of "the".[10] In Indonesian, the third person possessive suffix -nya could be also used as a definite article.[11] Indefinite articles edit Indefinite articles typically arise from adjectives meaning one. For example, the indefinite articles in the Romance languages—e.g., un, una, une—derive from the Latin adjective unus. Partitive articles, however, derive from Vulgar Latin de illo, meaning (some) of the. The English indefinite article an is derived from the same root as one. The -n came to be dropped before consonants, giving rise to the shortened form a. The existence of both forms has led to many cases of juncture loss, for example transforming the original a napron into the modern an apron. The Persian indefinite article is yek, meaning one. See also edit English articles Al- (definite article in Arabic) Definiteness Definite description False title References edit Recasens, Marta; Martí, M. Antònia; Taulé, Mariona (2009-06-16), Winkler, Susanne; Featherston, Sam (eds.), "First-mention definites:More than exceptional cases", The Fruits of Empirical Linguistics II, vol. 102, Berlin, New York: Mouton de Gruyter, pp. 217–238, doi:10.1515/9783110216158.217, ISBN 978-3-11-021347-8, retrieved 2023-01-16 New perspectives on Hispanic contact : linguistics in the Americas. Melvin González-Rivera, and Sandro Sessarego. Madrid: Iberoamericana. 2015. ISBN 978-3-95487-831-4. OCLC 969386958. Burchfield, R. W. (1996). The New Fowler's Modern English Usage (3rd ed.). Oxford University Press. p. 512. ISBN 978-0199690367. Argetsinger, Amy (1 September 2015). "Why does everyone call Donald Trump 'The Donald'? It's an interesting story". The Washington Post. Retrieved 3 October 2017. Lawrence, Erma (1977). Haida dictionary. Fairbanks: Alaska Native Language Center. p. 64. Master, Peter (1997). "The English article system: Acquisition, function, and pedagogy". System. 25 (2): 215–232. doi:10.1016/S0346-251X(97)00010-9. Kusmenko, J K. "The typology of the language contact on the Balkans and in Scandinavia. A case of the suffixed definite article" (PDF). Archived (PDF) from the original on 2021-10-04. See s.5 Summary. Retrieved 2 February 2012. Simona, Ropati (1986). Tokelau Dictionary. New Zealand: Office of Tokelau Affairs. p. Introduction. Greenberg, Joseph H. (2005). Genetic linguistics : essays on theory and method. William Croft. Oxford: Oxford University Press. ISBN 978-0-19-151452-4. OCLC 132691297. Solomon, Zomaya S. (1997). Functional and other exotic sentences in Assyrian Aramaic, Journal of Assyrian Academic Studies, XI/2:44-69. "20 Contoh Penggunaan Kata nya dalam Kalimat" [Twenty Examples of the Use of 'nya' in Sentences]. kumparan.com (in Indonesian). Retrieved 2024-05-26. External links edit Look up article, definite article, or indefinite article in Wiktionary, the free dictionary. Wikisource has the text of the 1921 Collier's Encyclopedia article Article. "The Definite Article, 'The': The Most Frequently Used Word in World's Englishes" Ing, John (September 17, 2019). "How to Use Articles Correctly in English (A, An & The)". Retrieved May 3, 2020. Last edited 20 days ago by Kaloan-koko Related articles French articles and determiners Definiteness Semantic feature of noun phrases in linguistics English determiners Determiners in the English language Wikipedia Wikimedia Foundation Powered by MediaWiki Content is available under CC BY-SA 4.0 unless otherwise noted. Privacy policy Contact Wikipedia Code of Conduct Developers Statistics Cookie statement Terms of Use DesktopBlog Open user menuuser photo Featured Ruby Maths Fourier theorem A Fourier series is an expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series. By expressing a function as a sum of sines and cosines, may problems involving the function become easier to analyze because trigonometric functions are well understood. For example, Fourier series were first used by Josheph Fourier(1786-1830) to find solutions to the heat equation. This application is possible because the derivatives of trigonometric functions fall into simple patterns. Fourier series cannot be used to approximate arbitrary functions, because most functions fall into simple patterns. Fourier series cannot be used to approximate arbitrary functions, because most functions have infinitely many terms in their Fourier series, and the series do not always converge. Well-behaved functions, for example smooth functions, have Fourier series that converge to the original function. The coefficients of the Fourier series are determined by integrals of the function multiplied by trigonometric functions, described in Fourier series Definition. The study of the convergence of Fourier series focus on the behaviors of the partial sums, which means studying the behavior of the sum as more and more terms from the series are summed. Fourier series are closely related to the Fourier transform. a more general tool that can even find the frequency information for functions that are not periodic. Periodic functions are identified with functions on a circle; for this reason Fourier series are the subject of Fourier analysis on the circle group, denoted by T or S1. The Fourier transform is also part of Fourier analysis, but is defined for functions on Rn. Since Fourier's time, many different approaches to defining and understanding the concept of Fourier series have been discovered, all of which are consistent with one another, but each of which emphasizes different aspects of the topic. Some of the more powerful and elegant approaches are based on the mathematical ideas and tools that were not available in Fourier's time. Fourier originally defined the Fourier series for real-valued functions of real arguments, and used the sine and cosine functions in the decomposition. May other Fourier-related transforms have since been defined, extending his initial idea to many applications and birthing an area of mathematics called Fourier analysis. Fourier transform The Fourier Transform is an integral transform that makes a function as input and outputs another function that describes the extent to which various frequencies are present in the original function. The output of the transform is a complex-valued function of frequency. The output of the transform is a complex valued function of frequency. The term Fourier transform refers to both this complex-valued function and the mathematical operation. When a distinction needs to be made, the output of the operation is sometimes called the frequency domain representation of the original function. The Fourier transform is analogous to decomposing the sound of a musical chord into the intensities of its constituent pitches. Functions that are localized in the time domain have Fourier transforms that are spread out across the frequency domain and vice versa,a phenomenon known as the uncertainty principle. The critical case for this principle is the Gaussian function, of substantial importance in probability theory and statistics as well as in the study of physical phenomena exhibiting normal distribution(example: diffusion). The Fourier transform of a Gaussian function is another Gaussian function. Joseph Fourier introduced the sine and cosine transforms ( which correspond to the imaginary and real components of the modern Fourier transform) in his study of heat transfer, where Gaussian functions appear as solutions of the heat equation. CommentsMaths Euclidean Alogrithms From geeksforgeeks. The Eucliean alogrithm is a way of find the greatest common divisor of two positive integers. GCD of two numbers is the largest number that divides both of them. A simple way to find GCD is to factorize both numbers and multiply common prime factors. 36 = 2 * 2 * 3 * 3 60 = 2 * 2 * 3 * 5 GCD = Multiplication of common factors = 2 * 2 * 3 = 12 Basic Euclidean Algorithm for GCD: The alogirithm is based on the below facts: - If we subtract a smaller number from a larger one ( we reduce a large number ), GCD does not change. So, if we keep subtracting repeatedly the larger of two, we end up with GCD. - Now instead of subtraction, if we divide the larger number, the algorithm stops when we find the reminder 0. # frozen_string_literal: true def gcd(first, second) if first.zero? return second end gcd(second % first, first) end first = 36 second = 60 puts "GCD of #{first} and #{second} is #{gcd(first, second)}" CommentsThe English definite article the, written þe in Middle English, derives from an Old English demonstrative, which, according to gender, was written se (masculine), seo (feminine) (þe and þeo in the Northumbrian dialect), or þæt (neuter). The neuter form þæt also gave rise to the modern demonstrative that. The ye occasionally seen in pseudo-archaic usage such as "Ye Olde Englishe Tea Shoppe" is actually a form of þe, where the letter thorn (þ) came to be written as a y. Multiple demonstratives can give rise to multiple definite articles. Macedonian, for example, in which the articles are suffixed, has столот (stolot), the chair; столов (stolov), this chair; and столон (stolon), that chair. These derive from the Proto-Slavic demonstratives *tъ "this, that", *ovъ "this here" and *onъ "that over there, yonder" respectively. Colognian prepositions articles such as in dat Auto, or et Auto, the car; the first being specifically selected, focused, newly introduced, while the latter is not selected, unfocused, already known, general, or generic. Standard Basque distinguishes between proximal and distal definite articles in the plural (dialectally, a proximal singular and an additional medial grade may also be present). The Basque distal form (with infix -a-, etymologically a suffixed and phonetically reduced form of the distal demonstrative har-/hai-) functions as the default definite article, whereas the proximal form (with infix -o-, derived from the proximal demonstrative hau-/hon-) is marked and indicates some kind of (spatial or otherwise) close relationship between the speaker and the referent (e.g., it may imply that the speaker is included in the referent): etxeak ("the houses") vs. etxeok ("these houses [of ours]"), euskaldunak ("the Basque speakers") vs. euskaldunok ("we, the Basque speakers"). Speakers of Assyrian Neo-Aramaic, a modern Aramaic language that lacks a definite article, may at times use demonstratives aha and aya (feminine) or awa (masculine) – which translate to "this" and "that", respectively – to give the sense of "the".[10] In Indonesian, the third person possessive suffix -nya could be also used as a definite article.[11] Indefinite articles edit Indefinite articles typically arise from adjectives meaning one. For example, the indefinite articles in the Romance languages—e.g., un, una, une—derive from the Latin adjective unus. Partitive articles, however, derive from Vulgar Latin de illo, meaning (some) of the. The English indefinite article an is derived from the same root as one. The -n came to be dropped before consonants, giving rise to the shortened form a. The existence of both forms has led to many cases of juncture loss, for example transforming the original a napron into the modern an apron. The Persian indefinite article is yek, meaning one. See also edit English articles Al- (definite article in Arabic) Definiteness Definite description False title References edit Recasens, Marta; Martí, M. Antònia; Taulé, Mariona (2009-06-16), Winkler, Susanne; Featherston, Sam (eds.), "First-mention definites:More than exceptional cases", The Fruits of Empirical Linguistics II, vol. 102, Berlin, New York: Mouton de Gruyter, pp. 217–238, doi:10.1515/9783110216158.217, ISBN 978-3-11-021347-8, retrieved 2023-01-16 New perspectives on Hispanic contact : linguistics in the Americas. Melvin González-Rivera, and Sandro Sessarego. Madrid: Iberoamericana. 2015. ISBN 978-3-95487-831-4. OCLC 969386958. Burchfield, R. W. (1996). The New Fowler's Modern English Usage (3rd ed.). Oxford University Press. p. 512. ISBN 978-0199690367. Argetsinger, Amy (1 September 2015). "Why does everyone call Donald Trump 'The Donald'? It's an interesting story". The Washington Post. Retrieved 3 October 2017. Lawrence, Erma (1977). Haida dictionary. Fairbanks: Alaska Native Language Center. p. 64. Master, Peter (1997). "The English article system: Acquisition, function, and pedagogy". System. 25 (2): 215–232. doi:10.1016/S0346-251X(97)00010-9. Kusmenko, J K. "The typology of the language contact on the Balkans and in Scandinavia. A case of the suffixed definite article" (PDF). Archived (PDF) from the original on 2021-10-04. See s.5 Summary. Retrieved 2 February 2012. Simona, Ropati (1986). Tokelau Dictionary. New Zealand: Office of Tokelau Affairs. p. Introduction. Greenberg, Joseph H. (2005). Genetic linguistics : essays on theory and method. William Croft. Oxford: Oxford University Press. ISBN 978-0-19-151452-4. OCLC 132691297. Solomon, Zomaya S. (1997). Functional and other exotic sentences in Assyrian Aramaic, Journal of Assyrian Academic Studies, XI/2:44-69. "20 Contoh Penggunaan Kata nya dalam Kalimat" [Twenty Examples of the Use of 'nya' in Sentences]. kumparan.com (in Indonesian). Retrieved 2024-05-26. External links edit Look up article, definite article, or indefinite article in Wiktionary, the free dictionary. Wikisource has the text of the 1921 Collier's Encyclopedia article Article. "The Definite Article, 'The': The Most Frequently Used Word in World's Englishes" Ing, John (September 17, 2019). "How to Use Articles Correctly in English (A, An & The)". Retrieved May 3, 2020. Last edited 20 days ago by Kaloan-koko Related articles French articles and determiners Definiteness Semantic feature of noun phrases in linguistics English determiners Determiners in the English language Wikipedia Wikimedia Foundation Powered by MediaWiki Content is available under CC BY-SA 4.0 unless otherwise noted. Privacy policy Contact Wikipedia Code of Conduct Developers Statistics Cookie statement Terms of Use DesktopMaths Gaussian function Gaussian is a function of the base form with f(x) = exp(-x^2) and with parametric extension f(x) = aexp(-(x-b)^2)/2c^2) for arbitrary real constants a, b and non-zero c. It is named after the mathematician Carl Friedrich Gauss. The graph of a Gaussian is a characteristic symmetric "bell curve" shape. The parameter is the height of the curve's peak, b is the position of the center of the peak, and c (the standard deviation, sometimes called the Gaussian RMS width) controls the width of the "bell". Gaussian functions are widely used in statistics to describe the normal distributions, in signal processing to define the Gaussian filters, in image processing where two-dimensional Gaussians are used for Gaussian blurs, and in mathematics to solve heat equations and diffusion equations and to define the Weierstrass transform. They are also abundantly used in quantum chemistry to form basic sets. The most common method for estimating the Gaussian parameters is to take the logarithm of data and fit a parabola to the resulting data set. While this provides a simple curve fitting procedure, the resulting algorithm may be biased by excessively weighting small data values, which can produce large errors in the profile estimate. One can partially compensate for this problem through weighted least squares estimation, reducing the weight of small data values, but this too can be biased by allowing the tail of the Gaussian to dominate the fit. In order to remove the bias, one can instead use an iteratively reweighted least squares procedure, in which the weights are updated at each iteration. It is also possible to perform non-linear regression directly on the data without involving the logarithmic data transformation. Parameter Precision Once one has an algorithm for estimating the Gaussian parameters, it is also important to know how precise those estimates are. Any least squares estimation algorithm can provide numerical estimates for the variance of each parameter ( i.e. the variance of estimated height, position, and width of the function). One can also use Cramer-Rao bound theory to obtain an analytical expression for the lower bound on the parameter variances, given certain assumptions about the data. 1. The noise in the measured profile is either independent and identically distributed Gaussian, or the noise is Poissin-distributed. 2. The spacing between each sampling ( i.e the distance between pixels measuring the data) is uniform. 3. The peak is "well-sampled", so that less than 10% of the area or volume under the peak(area if a 1D Gaussian, volume if a 2D Gaussian) lies outside the measurement region. 4. The width of the peak is much larger than the distance between sample locations( i.e. the detector pixels must be al least 5 time smaller than the Gaussian FWHM). Applications Gaussian functions appear in many context in natural sciences, the social sciences, mathematics and engineering. Some examples include: 1. In statistics and probability theory, Gaussian functions appear as the density function of the normal distribution, which is a limiting probability distribution of complicated sums, according to central limit theorem. 2. A Gaussian function is the wave function of the ground state of the quantum harmonic oscillator. 3. The molecular orbitals used in computational chemistry can be linear combinations of Gaussian functions called Gaussian orbitals. 3. Gaussian functions are also associated with the vacuum state in quantum field theory. 4. Gaussian beams are used in optical systems, microwave systems and lasers. 5. In scale space representation, Gaussian functions are used as smoothing kernels for generating multi-scale representations in computer vision and image processing. Specifically, derivatives of Gaussians ( Hermite functions ) are used as a basis for defining a large number of types of visual operations. 6. Gaussian functions are used to define some types of artificial neural networks. 7. In fluorescence microscopy a 2D Gaussian function is used to approximate the Airy disk, describing the intensity distribution produced by a point source. 8. In signal processing they serve to define Gaussian filter, such as in image processing where 2D Gaussian kernel, which may be approximated by the Binomial coefficient or sampling a Gaussina. 9. In geostatistics, they have been used for understanding the variability between the patterns of a complex training image. They are used with kernel methods to cluster the patterns in the feature space. [email protected]# frozen_string_literal: true def gcd(first, second) if first.zero? return second end gcd(second % first, first) end first = 36 second = 60 puts "GCD of #{first} and #{second} is #{gcd(first, second)}"Ruby Maths Fourier theorem A Fourier series is an expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series. By expressing a function as a sum of sines and cosines, may problems involving the function become easier to analyze because trigonometric functions are well understood. For example, Fourier series were first used by Josheph Fourier(1786-1830) to find solutions to the heat equation. This application is possible because the derivatives of trigonometric functions fall into simple patterns. Fourier series cannot be used to approximate arbitrary functions, because most functions fall into simple patterns. Fourier series cannot be used to approximate arbitrary functions, because most functions have infinitely many terms in their Fourier series, and the series do not always converge. Well-behaved functions, for example smooth functions, have Fourier series that converge to the original function. The coefficients of the Fourier series are determined by integrals of the function multiplied by trigonometric functions, described in Fourier series Definition. The study of the convergence of Fourier series focus on the behaviors of the partial sums, which means studying the behavior of the sum as more and more terms from the series are summed. Fourier series are closely related to the Fourier transform. a more general tool that can even find the frequency information for functions that are not periodic. Periodic functions are identified with functions on a circle; for this reason Fourier series are the subject of Fourier analysis on the circle group, denoted by T or S1. The Fourier transform is also part of Fourier analysis, but is defined for functions on Rn. Since Fourier's time, many different approaches to defining and understanding the concept of Fourier series have been discovered, all of which are consistent with one another, but each of which emphasizes different aspects of the topic. Some of the more powerful and elegant approaches are based on the mathematical ideas and tools that were not available in Fourier's time. Fourier originally defined the Fourier series for real-valued functions of real arguments, and used the sine and cosine functions in the decomposition. May other Fourier-related transforms have since been defined, extending his initial idea to many applications and birthing an area of mathematics called Fourier analysis. Fourier transform The Fourier Transform is an integral transform that makes a function as input and outputs another function that describes the extent to which various frequencies are present in the original function. The output of the transform is a complex-valued function of frequency. The output of the transform is a complex valued function of frequency. The term Fourier transform refers to both this complex-valued function and the mathematical operation. When a distinction needs to be made, the output of the operation is sometimes called the frequency domain representation of the original function. The Fourier transform is analogous to decomposing the sound of a musical chord into the intensities of its constituent pitches. Functions that are localized in the time domain have Fourier transforms that are spread out across the frequency domain and vice versa,a phenomenon known as the uncertainty principle. The critical case for this principle is the Gaussian function, of substantial importance in probability theory and statistics as well as in the study of physical phenomena exhibiting normal distribution(example: diffusion). The Fourier transform of a Gaussian function is another Gaussian function. Joseph Fourier introduced the sine and cosine transforms ( which correspond to the imaginary and real components of the modern Fourier transform) in his study of heat transfer, where Gaussian functions appear as solutions of the heat equation. CommentsFourier theorem A Fourier series is an expansion of a periodic function into a sum of trigonometric functions. Read more Gaussian function Gaussian is a function of the base form with f(x) = exp(-x^2) and with parametric extension f(x) = aexp(-(x-b)^2)/2c^2) for arbitrary real constants a, b and non-zero c. Read more © 2025 bikashpandey.com. All Rights Reserved. Privacy Policy [email protected] admin monkey bou 🐒 👉 bkp by Om Luitel 10 minutes ago7278302866463966781https://youtube.com/watch?v=GUS_MnJQ5Uo&lc=UgxkvBdpKjTYrynRE2V4AaABAg&si=m2nw-jfZ1pQMq0Eg733569203908433022673356920390843302267407414467280806430-74074144672808064317407414467280806430-7407414467280806431BTJEZBZ3Q9BTJEZBZ3Q9Blog Open user menuuser photo Featured Ruby Maths Fourier theorem A Fourier series is an expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series. By expressing a function as a sum of sines and cosines, may problems involving the function become easier to analyze because trigonometric functions are well understood. For example, Fourier series were first used by Josheph Fourier(1786-1830) to find solutions to the heat equation. This application is possible because the derivatives of trigonometric functions fall into simple patterns. Fourier series cannot be used to approximate arbitrary functions, because most functions fall into simple patterns. Fourier series cannot be used to approximate arbitrary functions, because most functions have infinitely many terms in their Fourier series, and the series do not always converge. Well-behaved functions, for example smooth functions, have Fourier series that converge to the original function. The coefficients of the Fourier series are determined by integrals of the function multiplied by trigonometric functions, described in Fourier series Definition. The study of the convergence of Fourier series focus on the behaviors of the partial sums, which means studying the behavior of the sum as more and more terms from the series are summed. Fourier series are closely related to the Fourier transform. a more general tool that can even find the frequency information for functions that are not periodic. Periodic functions are identified with functions on a circle; for this reason Fourier series are the subject of Fourier analysis on the circle group, denoted by T or S1. The Fourier transform is also part of Fourier analysis, but is defined for functions on Rn. Since Fourier's time, many different approaches to defining and understanding the concept of Fourier series have been discovered, all of which are consistent with one another, but each of which emphasizes different aspects of the topic. Some of the more powerful and elegant approaches are based on the mathematical ideas and tools that were not available in Fourier's time. Fourier originally defined the Fourier series for real-valued functions of real arguments, and used the sine and cosine functions in the decomposition. May other Fourier-related transforms have since been defined, extending his initial idea to many applications and birthing an area of mathematics called Fourier analysis. Fourier transform The Fourier Transform is an integral transform that makes a function as input and outputs another function that describes the extent to which various frequencies are present in the original function. The output of the transform is a complex-valued function of frequency. The output of the transform is a complex valued function of frequency. The term Fourier transform refers to both this complex-valued function and the mathematical operation. When a distinction needs to be made, the output of the operation is sometimes called the frequency domain representation of the original function. The Fourier transform is analogous to decomposing the sound of a musical chord into the intensities of its constituent pitches. Functions that are localized in the time domain have Fourier transforms that are spread out across the frequency domain and vice versa,a phenomenon known as the uncertainty principle. The critical case for this principle is the Gaussian function, of substantial importance in probability theory and statistics as well as in the study of physical phenomena exhibiting normal distribution(example: diffusion). The Fourier transform of a Gaussian function is another Gaussian function. Joseph Fourier introduced the sine and cosine transforms ( which correspond to the imaginary and real components of the modern Fourier transform) in his study of heat transfer, where Gaussian functions appear as solutions of the heat equation. CommentsBlog Open user menuuser photo Featured Ruby Maths Fourier theorem A Fourier series is an expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series. By expressing a function as a sum of sines and cosines, may problems involving the function become easier to analyze because trigonometric functions are well understood. For example, Fourier series were first used by Josheph Fourier(1786-1830) to find solutions to the heat equation. This application is possible because the derivatives of trigonometric functions fall into simple patterns. Fourier series cannot be used to approximate arbitrary functions, because most functions fall into simple patterns. Fourier series cannot be used to approximate arbitrary functions, because most functions have infinitely many terms in their Fourier series, and the series do not always converge. Well-behaved functions, for example smooth functions, have Fourier series that converge to the original function. The coefficients of the Fourier series are determined by integrals of the function multiplied by trigonometric functions, described in Fourier series Definition. The study of the convergence of Fourier series focus on the behaviors of the partial sums, which means studying the behavior of the sum as more and more terms from the series are summed. Fourier series are closely related to the Fourier transform. a more general tool that can even find the frequency information for functions that are not periodic. Periodic functions are identified with functions on a circle; for this reason Fourier series are the subject of Fourier analysis on the circle group, denoted by T or S1. The Fourier transform is also part of Fourier analysis, but is defined for functions on Rn. Since Fourier's time, many different approaches to defining and understanding the concept of Fourier series have been discovered, all of which are consistent with one another, but each of which emphasizes different aspects of the topic. Some of the more powerful and elegant approaches are based on the mathematical ideas and tools that were not available in Fourier's time. Fourier originally defined the Fourier series for real-valued functions of real arguments, and used the sine and cosine functions in the decomposition. May other Fourier-related transforms have since been defined, extending his initial idea to many applications and birthing an area of mathematics called Fourier analysis. Fourier transform The Fourier Transform is an integral transform that makes a function as input and outputs another function that describes the extent to which various frequencies are present in the original function. The output of the transform is a complex-valued function of frequency. The output of the transform is a complex valued function of frequency. The term Fourier transform refers to both this complex-valued function and the mathematical operation. When a distinction needs to be made, the output of the operation is sometimes called the frequency domain representation of the original function. The Fourier transform is analogous to decomposing the sound of a musical chord into the intensities of its constituent pitches. Functions that are localized in the time domain have Fourier transforms that are spread out across the frequency domain and vice versa,a phenomenon known as the uncertainty principle. The critical case for this principle is the Gaussian function, of substantial importance in probability theory and statistics as well as in the study of physical phenomena exhibiting normal distribution(example: diffusion). The Fourier transform of a Gaussian function is another Gaussian function. Joseph Fourier introduced the sine and cosine transforms ( which correspond to the imaginary and real components of the modern Fourier transform) in his study of heat transfer, where Gaussian functions appear as solutions of the heat equation. CommentsMaths Euclidean Alogrithms From geeksforgeeks. The Eucliean alogrithm is a way of find the greatest common divisor of two positive integers. GCD of two numbers is the largest number that divides both of them. A simple way to find GCD is to factorize both numbers and multiply common prime factors. 36 = 2 * 2 * 3 * 3 60 = 2 * 2 * 3 * 5 GCD = Multiplication of common factors = 2 * 2 * 3 = 12 Basic Euclidean Algorithm for GCD: The alogirithm is based on the below facts: - If we subtract a smaller number from a larger one ( we reduce a large number ), GCD does not change. So, if we keep subtracting repeatedly the larger of two, we end up with GCD. - Now instead of subtraction, if we divide the larger number, the algorithm stops when we find the reminder 0. # frozen_string_literal: true def gcd(first, second) if first.zero? return second end gcd(second % first, first) end first = 36 second = 60 puts "GCD of #{first} and #{second} is #{gcd(first, second)}" CommentsThe English definite article the, written þe in Middle English, derives from an Old English demonstrative, which, according to gender, was written se (masculine), seo (feminine) (þe and þeo in the Northumbrian dialect), or þæt (neuter). The neuter form þæt also gave rise to the modern demonstrative that. The ye occasionally seen in pseudo-archaic usage such as "Ye Olde Englishe Tea Shoppe" is actually a form of þe, where the letter thorn (þ) came to be written as a y. Multiple demonstratives can give rise to multiple definite articles. Macedonian, for example, in which the articles are suffixed, has столот (stolot), the chair; столов (stolov), this chair; and столон (stolon), that chair. These derive from the Proto-Slavic demonstratives *tъ "this, that", *ovъ "this here" and *onъ "that over there, yonder" respectively. Colognian prepositions articles such as in dat Auto, or et Auto, the car; the first being specifically selected, focused, newly introduced, while the latter is not selected, unfocused, already known, general, or generic. Standard Basque distinguishes between proximal and distal definite articles in the plural (dialectally, a proximal singular and an additional medial grade may also be present). The Basque distal form (with infix -a-, etymologically a suffixed and phonetically reduced form of the distal demonstrative har-/hai-) functions as the default definite article, whereas the proximal form (with infix -o-, derived from the proximal demonstrative hau-/hon-) is marked and indicates some kind of (spatial or otherwise) close relationship between the speaker and the referent (e.g., it may imply that the speaker is included in the referent): etxeak ("the houses") vs. etxeok ("these houses [of ours]"), euskaldunak ("the Basque speakers") vs. euskaldunok ("we, the Basque speakers"). Speakers of Assyrian Neo-Aramaic, a modern Aramaic language that lacks a definite article, may at times use demonstratives aha and aya (feminine) or awa (masculine) – which translate to "this" and "that", respectively – to give the sense of "the".[10] In Indonesian, the third person possessive suffix -nya could be also used as a definite article.[11] Indefinite articles edit Indefinite articles typically arise from adjectives meaning one. For example, the indefinite articles in the Romance languages—e.g., un, una, une—derive from the Latin adjective unus. Partitive articles, however, derive from Vulgar Latin de illo, meaning (some) of the. The English indefinite article an is derived from the same root as one. The -n came to be dropped before consonants, giving rise to the shortened form a. The existence of both forms has led to many cases of juncture loss, for example transforming the original a napron into the modern an apron. The Persian indefinite article is yek, meaning one. See also edit English articles Al- (definite article in Arabic) Definiteness Definite description False title References edit Recasens, Marta; Martí, M. Antònia; Taulé, Mariona (2009-06-16), Winkler, Susanne; Featherston, Sam (eds.), "First-mention definites:More than exceptional cases", The Fruits of Empirical Linguistics II, vol. 102, Berlin, New York: Mouton de Gruyter, pp. 217–238, doi:10.1515/9783110216158.217, ISBN 978-3-11-021347-8, retrieved 2023-01-16 New perspectives on Hispanic contact : linguistics in the Americas. Melvin González-Rivera, and Sandro Sessarego. Madrid: Iberoamericana. 2015. ISBN 978-3-95487-831-4. OCLC 969386958. Burchfield, R. W. (1996). The New Fowler's Modern English Usage (3rd ed.). Oxford University Press. p. 512. ISBN 978-0199690367. Argetsinger, Amy (1 September 2015). "Why does everyone call Donald Trump 'The Donald'? It's an interesting story". The Washington Post. Retrieved 3 October 2017. Lawrence, Erma (1977). Haida dictionary. Fairbanks: Alaska Native Language Center. p. 64. Master, Peter (1997). "The English article system: Acquisition, function, and pedagogy". System. 25 (2): 215–232. doi:10.1016/S0346-251X(97)00010-9. Kusmenko, J K. "The typology of the language contact on the Balkans and in Scandinavia. A case of the suffixed definite article" (PDF). Archived (PDF) from the original on 2021-10-04. See s.5 Summary. Retrieved 2 February 2012. Simona, Ropati (1986). Tokelau Dictionary. New Zealand: Office of Tokelau Affairs. p. Introduction. Greenberg, Joseph H. (2005). Genetic linguistics : essays on theory and method. William Croft. Oxford: Oxford University Press. ISBN 978-0-19-151452-4. OCLC 132691297. Solomon, Zomaya S. (1997). Functional and other exotic sentences in Assyrian Aramaic, Journal of Assyrian Academic Studies, XI/2:44-69. "20 Contoh Penggunaan Kata nya dalam Kalimat" [Twenty Examples of the Use of 'nya' in Sentences]. kumparan.com (in Indonesian). Retrieved 2024-05-26. External links edit Look up article, definite article, or indefinite article in Wiktionary, the free dictionary. Wikisource has the text of the 1921 Collier's Encyclopedia article Article. "The Definite Article, 'The': The Most Frequently Used Word in World's Englishes" Ing, John (September 17, 2019). "How to Use Articles Correctly in English (A, An & The)". Retrieved May 3, 2020. Last edited 20 days ago by Kaloan-koko Related articles French articles and determiners Definiteness Semantic feature of noun phrases in linguistics English determiners Determiners in the English language Wikipedia Wikimedia Foundation Powered by MediaWiki Content is available under CC BY-SA 4.0 unless otherwise noted. Privacy policy Contact Wikipedia Code of Conduct Developers Statistics Cookie statement Terms of Use DesktopThe English definite article the, written þe in Middle English, derives from an Old English demonstrative, which, according to gender, was written se (masculine), seo (feminine) (þe and þeo in the Northumbrian dialect), or þæt (neuter). The neuter form þæt also gave rise to the modern demonstrative that. The ye occasionally seen in pseudo-archaic usage such as "Ye Olde Englishe Tea Shoppe" is actually a form of þe, where the letter thorn (þ) came to be written as a y. Multiple demonstratives can give rise to multiple definite articles. Macedonian, for example, in which the articles are suffixed, has столот (stolot), the chair; столов (stolov), this chair; and столон (stolon), that chair. These derive from the Proto-Slavic demonstratives *tъ "this, that", *ovъ "this here" and *onъ "that over there, yonder" respectively. Colognian prepositions articles such as in dat Auto, or et Auto, the car; the first being specifically selected, focused, newly introduced, while the latter is not selected, unfocused, already known, general, or generic. Standard Basque distinguishes between proximal and distal definite articles in the plural (dialectally, a proximal singular and an additional medial grade may also be present). The Basque distal form (with infix -a-, etymologically a suffixed and phonetically reduced form of the distal demonstrative har-/hai-) functions as the default definite article, whereas the proximal form (with infix -o-, derived from the proximal demonstrative hau-/hon-) is marked and indicates some kind of (spatial or otherwise) close relationship between the speaker and the referent (e.g., it may imply that the speaker is included in the referent): etxeak ("the houses") vs. etxeok ("these houses [of ours]"), euskaldunak ("the Basque speakers") vs. euskaldunok ("we, the Basque speakers"). Speakers of Assyrian Neo-Aramaic, a modern Aramaic language that lacks a definite article, may at times use demonstratives aha and aya (feminine) or awa (masculine) – which translate to "this" and "that", respectively – to give the sense of "the".[10] In Indonesian, the third person possessive suffix -nya could be also used as a definite article.[11] Indefinite articles edit Indefinite articles typically arise from adjectives meaning one. For example, the indefinite articles in the Romance languages—e.g., un, una, une—derive from the Latin adjective unus. Partitive articles, however, derive from Vulgar Latin de illo, meaning (some) of the. The English indefinite article an is derived from the same root as one. The -n came to be dropped before consonants, giving rise to the shortened form a. The existence of both forms has led to many cases of juncture loss, for example transforming the original a napron into the modern an apron. The Persian indefinite article is yek, meaning one. See also edit English articles Al- (definite article in Arabic) Definiteness Definite description False title References edit Recasens, Marta; Martí, M. Antònia; Taulé, Mariona (2009-06-16), Winkler, Susanne; Featherston, Sam (eds.), "First-mention definites:More than exceptional cases", The Fruits of Empirical Linguistics II, vol. 102, Berlin, New York: Mouton de Gruyter, pp. 217–238, doi:10.1515/9783110216158.217, ISBN 978-3-11-021347-8, retrieved 2023-01-16 New perspectives on Hispanic contact : linguistics in the Americas. Melvin González-Rivera, and Sandro Sessarego. Madrid: Iberoamericana. 2015. ISBN 978-3-95487-831-4. OCLC 969386958. Burchfield, R. W. (1996). The New Fowler's Modern English Usage (3rd ed.). Oxford University Press. p. 512. ISBN 978-0199690367. Argetsinger, Amy (1 September 2015). "Why does everyone call Donald Trump 'The Donald'? It's an interesting story". The Washington Post. Retrieved 3 October 2017. Lawrence, Erma (1977). Haida dictionary. Fairbanks: Alaska Native Language Center. p. 64. Master, Peter (1997). "The English article system: Acquisition, function, and pedagogy". System. 25 (2): 215–232. doi:10.1016/S0346-251X(97)00010-9. Kusmenko, J K. "The typology of the language contact on the Balkans and in Scandinavia. A case of the suffixed definite article" (PDF). Archived (PDF) from the original on 2021-10-04. See s.5 Summary. Retrieved 2 February 2012. Simona, Ropati (1986). Tokelau Dictionary. New Zealand: Office of Tokelau Affairs. p. Introduction. Greenberg, Joseph H. (2005). Genetic linguistics : essays on theory and method. William Croft. Oxford: Oxford University Press. ISBN 978-0-19-151452-4. OCLC 132691297. Solomon, Zomaya S. (1997). Functional and other exotic sentences in Assyrian Aramaic, Journal of Assyrian Academic Studies, XI/2:44-69. "20 Contoh Penggunaan Kata nya dalam Kalimat" [Twenty Examples of the Use of 'nya' in Sentences]. kumparan.com (in Indonesian). Retrieved 2024-05-26. External links edit Look up article, definite article, or indefinite article in Wiktionary, the free dictionary. Wikisource has the text of the 1921 Collier's Encyclopedia article Article. "The Definite Article, 'The': The Most Frequently Used Word in World's Englishes" Ing, John (September 17, 2019). "How to Use Articles Correctly in English (A, An & The)". Retrieved May 3, 2020. Last edited 20 days ago by Kaloan-koko Related articles French articles and determiners Definiteness Semantic feature of noun phrases in linguistics English determiners Determiners in the English language Wikipedia Wikimedia Foundation Powered by MediaWiki Content is available under CC BY-SA 4.0 unless otherwise noted. Privacy policy Contact Wikipedia Code of Conduct Developers Statistics Cookie statement Terms of Use DesktopMaths Euclidean Alogrithms From geeksforgeeks. The Eucliean alogrithm is a way of find the greatest common divisor of two positive integers. GCD of two numbers is the largest number that divides both of them. A simple way to find GCD is to factorize both numbers and multiply common prime factors. 36 = 2 * 2 * 3 * 3 60 = 2 * 2 * 3 * 5 GCD = Multiplication of common factors = 2 * 2 * 3 = 12 Basic Euclidean Algorithm for GCD: The alogirithm is based on the below facts: - If we subtract a smaller number from a larger one ( we reduce a large number ), GCD does not change. So, if we keep subtracting repeatedly the larger of two, we end up with GCD. - Now instead of subtraction, if we divide the larger number, the algorithm stops when we find the reminder 0. # frozen_string_literal: true def gcd(first, second) if first.zero? return second end gcd(second % first, first) end first = 36 second = 60 puts "GCD of #{first} and #{second} is #{gcd(first, second)}" CommentsBlog Open user menuuser photo Featured Ruby Maths Fourier theorem A Fourier series is an expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series. By expressing a function as a sum of sines and cosines, may problems involving the function become easier to analyze because trigonometric functions are well understood. For example, Fourier series were first used by Josheph Fourier(1786-1830) to find solutions to the heat equation. This application is possible because the derivatives of trigonometric functions fall into simple patterns. Fourier series cannot be used to approximate arbitrary functions, because most functions fall into simple patterns. Fourier series cannot be used to approximate arbitrary functions, because most functions have infinitely many terms in their Fourier series, and the series do not always converge. Well-behaved functions, for example smooth functions, have Fourier series that converge to the original function. The coefficients of the Fourier series are determined by integrals of the function multiplied by trigonometric functions, described in Fourier series Definition. The study of the convergence of Fourier series focus on the behaviors of the partial sums, which means studying the behavior of the sum as more and more terms from the series are summed. Fourier series are closely related to the Fourier transform. a more general tool that can even find the frequency information for functions that are not periodic. Periodic functions are identified with functions on a circle; for this reason Fourier series are the subject of Fourier analysis on the circle group, denoted by T or S1. The Fourier transform is also part of Fourier analysis, but is defined for functions on Rn. Since Fourier's time, many different approaches to defining and understanding the concept of Fourier series have been discovered, all of which are consistent with one another, but each of which emphasizes different aspects of the topic. Some of the more powerful and elegant approaches are based on the mathematical ideas and tools that were not available in Fourier's time. Fourier originally defined the Fourier series for real-valued functions of real arguments, and used the sine and cosine functions in the decomposition. May other Fourier-related transforms have since been defined, extending his initial idea to many applications and birthing an area of mathematics called Fourier analysis. Fourier transform The Fourier Transform is an integral transform that makes a function as input and outputs another function that describes the extent to which various frequencies are present in the original function. The output of the transform is a complex-valued function of frequency. The output of the transform is a complex valued function of frequency. The term Fourier transform refers to both this complex-valued function and the mathematical operation. When a distinction needs to be made, the output of the operation is sometimes called the frequency domain representation of the original function. The Fourier transform is analogous to decomposing the sound of a musical chord into the intensities of its constituent pitches. Functions that are localized in the time domain have Fourier transforms that are spread out across the frequency domain and vice versa,a phenomenon known as the uncertainty principle. The critical case for this principle is the Gaussian function, of substantial importance in probability theory and statistics as well as in the study of physical phenomena exhibiting normal distribution(example: diffusion). The Fourier transform of a Gaussian function is another Gaussian function. Joseph Fourier introduced the sine and cosine transforms ( which correspond to the imaginary and real components of the modern Fourier transform) in his study of heat transfer, where Gaussian functions appear as solutions of the heat equation. CommentsBlog Open user menuuser photo Featured Ruby Maths Fourier theorem A Fourier series is an expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series. By expressing a function as a sum of sines and cosines, may problems involving the function become easier to analyze because trigonometric functions are well understood. For example, Fourier series were first used by Josheph Fourier(1786-1830) to find solutions to the heat equation. This application is possible because the derivatives of trigonometric functions fall into simple patterns. Fourier series cannot be used to approximate arbitrary functions, because most functions fall into simple patterns. Fourier series cannot be used to approximate arbitrary functions, because most functions have infinitely many terms in their Fourier series, and the series do not always converge. Well-behaved functions, for example smooth functions, have Fourier series that converge to the original function. The coefficients of the Fourier series are determined by integrals of the function multiplied by trigonometric functions, described in Fourier series Definition. The study of the convergence of Fourier series focus on the behaviors of the partial sums, which means studying the behavior of the sum as more and more terms from the series are summed. Fourier series are closely related to the Fourier transform. a more general tool that can even find the frequency information for functions that are not periodic. Periodic functions are identified with functions on a circle; for this reason Fourier series are the subject of Fourier analysis on the circle group, denoted by T or S1. The Fourier transform is also part of Fourier analysis, but is defined for functions on Rn. Since Fourier's time, many different approaches to defining and understanding the concept of Fourier series have been discovered, all of which are consistent with one another, but each of which emphasizes different aspects of the topic. Some of the more powerful and elegant approaches are based on the mathematical ideas and tools that were not available in Fourier's time. Fourier originally defined the Fourier series for real-valued functions of real arguments, and used the sine and cosine functions in the decomposition. May other Fourier-related transforms have since been defined, extending his initial idea to many applications and birthing an area of mathematics called Fourier analysis. Fourier transform The Fourier Transform is an integral transform that makes a function as input and outputs another function that describes the extent to which various frequencies are present in the original function. The output of the transform is a complex-valued function of frequency. The output of the transform is a complex valued function of frequency. The term Fourier transform refers to both this complex-valued function and the mathematical operation. When a distinction needs to be made, the output of the operation is sometimes called the frequency domain representation of the original function. The Fourier transform is analogous to decomposing the sound of a musical chord into the intensities of its constituent pitches. Functions that are localized in the time domain have Fourier transforms that are spread out across the frequency domain and vice versa,a phenomenon known as the uncertainty principle. The critical case for this principle is the Gaussian function, of substantial importance in probability theory and statistics as well as in the study of physical phenomena exhibiting normal distribution(example: diffusion). The Fourier transform of a Gaussian function is another Gaussian function. Joseph Fourier introduced the sine and cosine transforms ( which correspond to the imaginary and real components of the modern Fourier transform) in his study of heat transfer, where Gaussian functions appear as solutions of the heat equation. CommentsMaths Gaussian function Gaussian is a function of the base form with f(x) = exp(-x^2) and with parametric extension f(x) = aexp(-(x-b)^2)/2c^2) for arbitrary real constants a, b and non-zero c. It is named after the mathematician Carl Friedrich Gauss. The graph of a Gaussian is a characteristic symmetric "bell curve" shape. The parameter is the height of the curve's peak, b is the position of the center of the peak, and c (the standard deviation, sometimes called the Gaussian RMS width) controls the width of the "bell". Gaussian functions are widely used in statistics to describe the normal distributions, in signal processing to define the Gaussian filters, in image processing where two-dimensional Gaussians are used for Gaussian blurs, and in mathematics to solve heat equations and diffusion equations and to define the Weierstrass transform. They are also abundantly used in quantum chemistry to form basic sets. The most common method for estimating the Gaussian parameters is to take the logarithm of data and fit a parabola to the resulting data set. While this provides a simple curve fitting procedure, the resulting algorithm may be biased by excessively weighting small data values, which can produce large errors in the profile estimate. One can partially compensate for this problem through weighted least squares estimation, reducing the weight of small data values, but this too can be biased by allowing the tail of the Gaussian to dominate the fit. In order to remove the bias, one can instead use an iteratively reweighted least squares procedure, in which the weights are updated at each iteration. It is also possible to perform non-linear regression directly on the data without involving the logarithmic data transformation. Parameter Precision Once one has an algorithm for estimating the Gaussian parameters, it is also important to know how precise those estimates are. Any least squares estimation algorithm can provide numerical estimates for the variance of each parameter ( i.e. the variance of estimated height, position, and width of the function). One can also use Cramer-Rao bound theory to obtain an analytical expression for the lower bound on the parameter variances, given certain assumptions about the data. 1. The noise in the measured profile is either independent and identically distributed Gaussian, or the noise is Poissin-distributed. 2. The spacing between each sampling ( i.e the distance between pixels measuring the data) is uniform. 3. The peak is "well-sampled", so that less than 10% of the area or volume under the peak(area if a 1D Gaussian, volume if a 2D Gaussian) lies outside the measurement region. 4. The width of the peak is much larger than the distance between sample locations( i.e. the detector pixels must be al least 5 time smaller than the Gaussian FWHM). Applications Gaussian functions appear in many context in natural sciences, the social sciences, mathematics and engineering. Some examples include: 1. In statistics and probability theory, Gaussian functions appear as the density function of the normal distribution, which is a limiting probability distribution of complicated sums, according to central limit theorem. 2. A Gaussian function is the wave function of the ground state of the quantum harmonic oscillator. 3. The molecular orbitals used in computational chemistry can be linear combinations of Gaussian functions called Gaussian orbitals. 3. Gaussian functions are also associated with the vacuum state in quantum field theory. 4. Gaussian beams are used in optical systems, microwave systems and lasers. 5. In scale space representation, Gaussian functions are used as smoothing kernels for generating multi-scale representations in computer vision and image processing. Specifically, derivatives of Gaussians ( Hermite functions ) are used as a basis for defining a large number of types of visual operations. 6. Gaussian functions are used to define some types of artificial neural networks. 7. In fluorescence microscopy a 2D Gaussian function is used to approximate the Airy disk, describing the intensity distribution produced by a point source. 8. In signal processing they serve to define Gaussian filter, such as in image processing where 2D Gaussian kernel, which may be approximated by the Binomial coefficient or sampling a Gaussina. 9. In geostatistics, they have been used for understanding the variability between the patterns of a complex training image. They are used with kernel methods to cluster the patterns in the feature space. CommentsMaths Gaussian function Gaussian is a function of the base form with f(x) = exp(-x^2) and with parametric extension f(x) = aexp(-(x-b)^2)/2c^2) for arbitrary real constants a, b and non-zero c. It is named after the mathematician Carl Friedrich Gauss. The graph of a Gaussian is a characteristic symmetric "bell curve" shape. The parameter is the height of the curve's peak, b is the position of the center of the peak, and c (the standard deviation, sometimes called the Gaussian RMS width) controls the width of the "bell". Gaussian functions are widely used in statistics to describe the normal distributions, in signal processing to define the Gaussian filters, in image processing where two-dimensional Gaussians are used for Gaussian blurs, and in mathematics to solve heat equations and diffusion equations and to define the Weierstrass transform. They are also abundantly used in quantum chemistry to form basic sets. The most common method for estimating the Gaussian parameters is to take the logarithm of data and fit a parabola to the resulting data set. While this provides a simple curve fitting procedure, the resulting algorithm may be biased by excessively weighting small data values, which can produce large errors in the profile estimate. One can partially compensate for this problem through weighted least squares estimation, reducing the weight of small data values, but this too can be biased by allowing the tail of the Gaussian to dominate the fit. In order to remove the bias, one can instead use an iteratively reweighted least squares procedure, in which the weights are updated at each iteration. It is also possible to perform non-linear regression directly on the data without involving the logarithmic data transformation. Parameter Precision Once one has an algorithm for estimating the Gaussian parameters, it is also important to know how precise those estimates are. Any least squares estimation algorithm can provide numerical estimates for the variance of each parameter ( i.e. the variance of estimated height, position, and width of the function). One can also use Cramer-Rao bound theory to obtain an analytical expression for the lower bound on the parameter variances, given certain assumptions about the data. 1. The noise in the measured profile is either independent and identically distributed Gaussian, or the noise is Poissin-distributed. 2. The spacing between each sampling ( i.e the distance between pixels measuring the data) is uniform. 3. The peak is "well-sampled", so that less than 10% of the area or volume under the peak(area if a 1D Gaussian, volume if a 2D Gaussian) lies outside the measurement region. 4. The width of the peak is much larger than the distance between sample locations( i.e. the detector pixels must be al least 5 time smaller than the Gaussian FWHM). Applications Gaussian functions appear in many context in natural sciences, the social sciences, mathematics and engineering. Some examples include: 1. In statistics and probability theory, Gaussian functions appear as the density function of the normal distribution, which is a limiting probability distribution of complicated sums, according to central limit theorem. 2. A Gaussian function is the wave function of the ground state of the quantum harmonic oscillator. 3. The molecular orbitals used in computational chemistry can be linear combinations of Gaussian functions called Gaussian orbitals. 3. Gaussian functions are also associated with the vacuum state in quantum field theory. 4. Gaussian beams are used in optical systems, microwave systems and lasers. 5. In scale space representation, Gaussian functions are used as smoothing kernels for generating multi-scale representations in computer vision and image processing. Specifically, derivatives of Gaussians ( Hermite functions ) are used as a basis for defining a large number of types of visual operations. 6. Gaussian functions are used to define some types of artificial neural networks. 7. In fluorescence microscopy a 2D Gaussian function is used to approximate the Airy disk, describing the intensity distribution produced by a point source. 8. In signal processing they serve to define Gaussian filter, such as in image processing where 2D Gaussian kernel, which may be approximated by the Binomial coefficient or sampling a Gaussina. 9. In geostatistics, they have been used for understanding the variability between the patterns of a complex training image. They are used with kernel methods to cluster the patterns in the feature space. [email protected]@bikashpandey.com# frozen_string_literal: true def gcd(first, second) if first.zero? return second end gcd(second % first, first) end first = 36 second = 60 puts "GCD of #{first} and #{second} is #{gcd(first, second)}"# frozen_string_literal: true def gcd(first, second) if first.zero? return second end gcd(second % first, first) end first = 36 second = 60 puts "GCD of #{first} and #{second} is #{gcd(first, second)}"# frozen_string_literal: true def gcd(first, second) if first.zero? return second end gcd(second % first, first) end first = 36 second = 60 puts "GCD of #{first} and #{second} is #{gcd(first, second)}"Ruby Maths Fourier theorem A Fourier series is an expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series. By expressing a function as a sum of sines and cosines, may problems involving the function become easier to analyze because trigonometric functions are well understood. For example, Fourier series were first used by Josheph Fourier(1786-1830) to find solutions to the heat equation. This application is possible because the derivatives of trigonometric functions fall into simple patterns. Fourier series cannot be used to approximate arbitrary functions, because most functions fall into simple patterns. Fourier series cannot be used to approximate arbitrary functions, because most functions have infinitely many terms in their Fourier series, and the series do not always converge. Well-behaved functions, for example smooth functions, have Fourier series that converge to the original function. The coefficients of the Fourier series are determined by integrals of the function multiplied by trigonometric functions, described in Fourier series Definition. The study of the convergence of Fourier series focus on the behaviors of the partial sums, which means studying the behavior of the sum as more and more terms from the series are summed. Fourier series are closely related to the Fourier transform. a more general tool that can even find the frequency information for functions that are not periodic. Periodic functions are identified with functions on a circle; for this reason Fourier series are the subject of Fourier analysis on the circle group, denoted by T or S1. The Fourier transform is also part of Fourier analysis, but is defined for functions on Rn. Since Fourier's time, many different approaches to defining and understanding the concept of Fourier series have been discovered, all of which are consistent with one another, but each of which emphasizes different aspects of the topic. Some of the more powerful and elegant approaches are based on the mathematical ideas and tools that were not available in Fourier's time. Fourier originally defined the Fourier series for real-valued functions of real arguments, and used the sine and cosine functions in the decomposition. May other Fourier-related transforms have since been defined, extending his initial idea to many applications and birthing an area of mathematics called Fourier analysis. Fourier transform The Fourier Transform is an integral transform that makes a function as input and outputs another function that describes the extent to which various frequencies are present in the original function. The output of the transform is a complex-valued function of frequency. The output of the transform is a complex valued function of frequency. The term Fourier transform refers to both this complex-valued function and the mathematical operation. When a distinction needs to be made, the output of the operation is sometimes called the frequency domain representation of the original function. The Fourier transform is analogous to decomposing the sound of a musical chord into the intensities of its constituent pitches. Functions that are localized in the time domain have Fourier transforms that are spread out across the frequency domain and vice versa,a phenomenon known as the uncertainty principle. The critical case for this principle is the Gaussian function, of substantial importance in probability theory and statistics as well as in the study of physical phenomena exhibiting normal distribution(example: diffusion). The Fourier transform of a Gaussian function is another Gaussian function. Joseph Fourier introduced the sine and cosine transforms ( which correspond to the imaginary and real components of the modern Fourier transform) in his study of heat transfer, where Gaussian functions appear as solutions of the heat equation. CommentsRuby Maths Fourier theorem A Fourier series is an expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series. By expressing a function as a sum of sines and cosines, may problems involving the function become easier to analyze because trigonometric functions are well understood. For example, Fourier series were first used by Josheph Fourier(1786-1830) to find solutions to the heat equation. This application is possible because the derivatives of trigonometric functions fall into simple patterns. Fourier series cannot be used to approximate arbitrary functions, because most functions fall into simple patterns. Fourier series cannot be used to approximate arbitrary functions, because most functions have infinitely many terms in their Fourier series, and the series do not always converge. Well-behaved functions, for example smooth functions, have Fourier series that converge to the original function. The coefficients of the Fourier series are determined by integrals of the function multiplied by trigonometric functions, described in Fourier series Definition. The study of the convergence of Fourier series focus on the behaviors of the partial sums, which means studying the behavior of the sum as more and more terms from the series are summed. Fourier series are closely related to the Fourier transform. a more general tool that can even find the frequency information for functions that are not periodic. Periodic functions are identified with functions on a circle; for this reason Fourier series are the subject of Fourier analysis on the circle group, denoted by T or S1. The Fourier transform is also part of Fourier analysis, but is defined for functions on Rn. Since Fourier's time, many different approaches to defining and understanding the concept of Fourier series have been discovered, all of which are consistent with one another, but each of which emphasizes different aspects of the topic. Some of the more powerful and elegant approaches are based on the mathematical ideas and tools that were not available in Fourier's time. Fourier originally defined the Fourier series for real-valued functions of real arguments, and used the sine and cosine functions in the decomposition. May other Fourier-related transforms have since been defined, extending his initial idea to many applications and birthing an area of mathematics called Fourier analysis. Fourier transform The Fourier Transform is an integral transform that makes a function as input and outputs another function that describes the extent to which various frequencies are present in the original function. The output of the transform is a complex-valued function of frequency. The output of the transform is a complex valued function of frequency. The term Fourier transform refers to both this complex-valued function and the mathematical operation. When a distinction needs to be made, the output of the operation is sometimes called the frequency domain representation of the original function. The Fourier transform is analogous to decomposing the sound of a musical chord into the intensities of its constituent pitches. Functions that are localized in the time domain have Fourier transforms that are spread out across the frequency domain and vice versa,a phenomenon known as the uncertainty principle. The critical case for this principle is the Gaussian function, of substantial importance in probability theory and statistics as well as in the study of physical phenomena exhibiting normal distribution(example: diffusion). The Fourier transform of a Gaussian function is another Gaussian function. Joseph Fourier introduced the sine and cosine transforms ( which correspond to the imaginary and real components of the modern Fourier transform) in his study of heat transfer, where Gaussian functions appear as solutions of the heat equation. CommentsRuby Maths Fourier theorem A Fourier series is an expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series. By expressing a function as a sum of sines and cosines, may problems involving the function become easier to analyze because trigonometric functions are well understood. For example, Fourier series were first used by Josheph Fourier(1786-1830) to find solutions to the heat equation. This application is possible because the derivatives of trigonometric functions fall into simple patterns. Fourier series cannot be used to approximate arbitrary functions, because most functions fall into simple patterns. Fourier series cannot be used to approximate arbitrary functions, because most functions have infinitely many terms in their Fourier series, and the series do not always converge. Well-behaved functions, for example smooth functions, have Fourier series that converge to the original function. The coefficients of the Fourier series are determined by integrals of the function multiplied by trigonometric functions, described in Fourier series Definition. The study of the convergence of Fourier series focus on the behaviors of the partial sums, which means studying the behavior of the sum as more and more terms from the series are summed. Fourier series are closely related to the Fourier transform. a more general tool that can even find the frequency information for functions that are not periodic. Periodic functions are identified with functions on a circle; for this reason Fourier series are the subject of Fourier analysis on the circle group, denoted by T or S1. The Fourier transform is also part of Fourier analysis, but is defined for functions on Rn. Since Fourier's time, many different approaches to defining and understanding the concept of Fourier series have been discovered, all of which are consistent with one another, but each of which emphasizes different aspects of the topic. Some of the more powerful and elegant approaches are based on the mathematical ideas and tools that were not available in Fourier's time. Fourier originally defined the Fourier series for real-valued functions of real arguments, and used the sine and cosine functions in the decomposition. May other Fourier-related transforms have since been defined, extending his initial idea to many applications and birthing an area of mathematics called Fourier analysis. Fourier transform The Fourier Transform is an integral transform that makes a function as input and outputs another function that describes the extent to which various frequencies are present in the original function. The output of the transform is a complex-valued function of frequency. The output of the transform is a complex valued function of frequency. The term Fourier transform refers to both this complex-valued function and the mathematical operation. When a distinction needs to be made, the output of the operation is sometimes called the frequency domain representation of the original function. The Fourier transform is analogous to decomposing the sound of a musical chord into the intensities of its constituent pitches. Functions that are localized in the time domain have Fourier transforms that are spread out across the frequency domain and vice versa,a phenomenon known as the uncertainty principle. The critical case for this principle is the Gaussian function, of substantial importance in probability theory and statistics as well as in the study of physical phenomena exhibiting normal distribution(example: diffusion). The Fourier transform of a Gaussian function is another Gaussian function. Joseph Fourier introduced the sine and cosine transforms ( which correspond to the imaginary and real components of the modern Fourier transform) in his study of heat transfer, where Gaussian functions appear as solutions of the heat equation. CommentsFourier theorem A Fourier series is an expansion of a periodic function into a sum of trigonometric functions. Read more Gaussian function Gaussian is a function of the base form with f(x) = exp(-x^2) and with parametric extension f(x) = aexp(-(x-b)^2)/2c^2) for arbitrary real constants a, b and non-zero c. Read more © 2025 bikashpandey.com. All Rights Reserved. Privacy Policy [email protected] theorem A Fourier series is an expansion of a periodic function into a sum of trigonometric functions. Read more Gaussian function Gaussian is a function of the base form with f(x) = exp(-x^2) and with parametric extension f(x) = aexp(-(x-b)^2)/2c^2) for arbitrary real constants a, b and non-zero c. Read more © 2025 bikashpandey.com. All Rights Reserved. Privacy Policy [email protected] theorem A Fourier series is an expansion of a periodic function into a sum of trigonometric functions. Read more Gaussian function Gaussian is a function of the base form with f(x) = exp(-x^2) and with parametric extension f(x) = aexp(-(x-b)^2)/2c^2) for arbitrary real constants a, b and non-zero c. Read more © 2025 bikashpandey.com. All Rights Reserved. Privacy Policy [email protected] admin monkey bou 🐒 👉 bkp by Om Luitel 10 minutes ago7407060620271091713740706062027109171374070606202710917137407060620271091713https://youtube.com/watch?v=GUS_MnJQ5Uo&lc=UgxkvBdpKjTYrynRE2V4AaABAg&si=m2nw-jfZ1pQMq0EgBlog Open user menuuser photo Featured Ruby Maths Fourier theorem A Fourier series is an expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series. By expressing a function as a sum of sines and cosines, may problems involving the function become easier to analyze because trigonometric functions are well understood. For example, Fourier series were first used by Josheph Fourier(1786-1830) to find solutions to the heat equation. This application is possible because the derivatives of trigonometric functions fall into simple patterns. Fourier series cannot be used to approximate arbitrary functions, because most functions fall into simple patterns. Fourier series cannot be used to approximate arbitrary functions, because most functions have infinitely many terms in their Fourier series, and the series do not always converge. Well-behaved functions, for example smooth functions, have Fourier series that converge to the original function. The coefficients of the Fourier series are determined by integrals of the function multiplied by trigonometric functions, described in Fourier series Definition. The study of the convergence of Fourier series focus on the behaviors of the partial sums, which means studying the behavior of the sum as more and more terms from the series are summed. Fourier series are closely related to the Fourier transform. a more general tool that can even find the frequency information for functions that are not periodic. Periodic functions are identified with functions on a circle; for this reason Fourier series are the subject of Fourier analysis on the circle group, denoted by T or S1. The Fourier transform is also part of Fourier analysis, but is defined for functions on Rn. Since Fourier's time, many different approaches to defining and understanding the concept of Fourier series have been discovered, all of which are consistent with one another, but each of which emphasizes different aspects of the topic. Some of the more powerful and elegant approaches are based on the mathematical ideas and tools that were not available in Fourier's time. Fourier originally defined the Fourier series for real-valued functions of real arguments, and used the sine and cosine functions in the decomposition. May other Fourier-related transforms have since been defined, extending his initial idea to many applications and birthing an area of mathematics called Fourier analysis. Fourier transform The Fourier Transform is an integral transform that makes a function as input and outputs another function that describes the extent to which various frequencies are present in the original function. The output of the transform is a complex-valued function of frequency. The output of the transform is a complex valued function of frequency. The term Fourier transform refers to both this complex-valued function and the mathematical operation. When a distinction needs to be made, the output of the operation is sometimes called the frequency domain representation of the original function. The Fourier transform is analogous to decomposing the sound of a musical chord into the intensities of its constituent pitches. Functions that are localized in the time domain have Fourier transforms that are spread out across the frequency domain and vice versa,a phenomenon known as the uncertainty principle. The critical case for this principle is the Gaussian function, of substantial importance in probability theory and statistics as well as in the study of physical phenomena exhibiting normal distribution(example: diffusion). The Fourier transform of a Gaussian function is another Gaussian function. Joseph Fourier introduced the sine and cosine transforms ( which correspond to the imaginary and real components of the modern Fourier transform) in his study of heat transfer, where Gaussian functions appear as solutions of the heat equation. CommentsMaths Euclidean Alogrithms From geeksforgeeks. The Eucliean alogrithm is a way of find the greatest common divisor of two positive integers. GCD of two numbers is the largest number that divides both of them. A simple way to find GCD is to factorize both numbers and multiply common prime factors. 36 = 2 * 2 * 3 * 3 60 = 2 * 2 * 3 * 5 GCD = Multiplication of common factors = 2 * 2 * 3 = 12 Basic Euclidean Algorithm for GCD: The alogirithm is based on the below facts: - If we subtract a smaller number from a larger one ( we reduce a large number ), GCD does not change. So, if we keep subtracting repeatedly the larger of two, we end up with GCD. - Now instead of subtraction, if we divide the larger number, the algorithm stops when we find the reminder 0. # frozen_string_literal: true def gcd(first, second) if first.zero? return second end gcd(second % first, first) end first = 36 second = 60 puts "GCD of #{first} and #{second} is #{gcd(first, second)}" CommentsThe English definite article the, written þe in Middle English, derives from an Old English demonstrative, which, according to gender, was written se (masculine), seo (feminine) (þe and þeo in the Northumbrian dialect), or þæt (neuter). The neuter form þæt also gave rise to the modern demonstrative that. The ye occasionally seen in pseudo-archaic usage such as "Ye Olde Englishe Tea Shoppe" is actually a form of þe, where the letter thorn (þ) came to be written as a y. Multiple demonstratives can give rise to multiple definite articles. Macedonian, for example, in which the articles are suffixed, has столот (stolot), the chair; столов (stolov), this chair; and столон (stolon), that chair. These derive from the Proto-Slavic demonstratives *tъ "this, that", *ovъ "this here" and *onъ "that over there, yonder" respectively. Colognian prepositions articles such as in dat Auto, or et Auto, the car; the first being specifically selected, focused, newly introduced, while the latter is not selected, unfocused, already known, general, or generic. Standard Basque distinguishes between proximal and distal definite articles in the plural (dialectally, a proximal singular and an additional medial grade may also be present). The Basque distal form (with infix -a-, etymologically a suffixed and phonetically reduced form of the distal demonstrative har-/hai-) functions as the default definite article, whereas the proximal form (with infix -o-, derived from the proximal demonstrative hau-/hon-) is marked and indicates some kind of (spatial or otherwise) close relationship between the speaker and the referent (e.g., it may imply that the speaker is included in the referent): etxeak ("the houses") vs. etxeok ("these houses [of ours]"), euskaldunak ("the Basque speakers") vs. euskaldunok ("we, the Basque speakers"). Speakers of Assyrian Neo-Aramaic, a modern Aramaic language that lacks a definite article, may at times use demonstratives aha and aya (feminine) or awa (masculine) – which translate to "this" and "that", respectively – to give the sense of "the".[10] In Indonesian, the third person possessive suffix -nya could be also used as a definite article.[11] Indefinite articles edit Indefinite articles typically arise from adjectives meaning one. For example, the indefinite articles in the Romance languages—e.g., un, una, une—derive from the Latin adjective unus. Partitive articles, however, derive from Vulgar Latin de illo, meaning (some) of the. The English indefinite article an is derived from the same root as one. The -n came to be dropped before consonants, giving rise to the shortened form a. The existence of both forms has led to many cases of juncture loss, for example transforming the original a napron into the modern an apron. The Persian indefinite article is yek, meaning one. See also edit English articles Al- (definite article in Arabic) Definiteness Definite description False title References edit Recasens, Marta; Martí, M. Antònia; Taulé, Mariona (2009-06-16), Winkler, Susanne; Featherston, Sam (eds.), "First-mention definites:More than exceptional cases", The Fruits of Empirical Linguistics II, vol. 102, Berlin, New York: Mouton de Gruyter, pp. 217–238, doi:10.1515/9783110216158.217, ISBN 978-3-11-021347-8, retrieved 2023-01-16 New perspectives on Hispanic contact : linguistics in the Americas. Melvin González-Rivera, and Sandro Sessarego. Madrid: Iberoamericana. 2015. ISBN 978-3-95487-831-4. OCLC 969386958. Burchfield, R. W. (1996). The New Fowler's Modern English Usage (3rd ed.). Oxford University Press. p. 512. ISBN 978-0199690367. Argetsinger, Amy (1 September 2015). "Why does everyone call Donald Trump 'The Donald'? It's an interesting story". The Washington Post. Retrieved 3 October 2017. Lawrence, Erma (1977). Haida dictionary. Fairbanks: Alaska Native Language Center. p. 64. Master, Peter (1997). "The English article system: Acquisition, function, and pedagogy". System. 25 (2): 215–232. doi:10.1016/S0346-251X(97)00010-9. Kusmenko, J K. "The typology of the language contact on the Balkans and in Scandinavia. A case of the suffixed definite article" (PDF). Archived (PDF) from the original on 2021-10-04. See s.5 Summary. Retrieved 2 February 2012. Simona, Ropati (1986). Tokelau Dictionary. New Zealand: Office of Tokelau Affairs. p. Introduction. Greenberg, Joseph H. (2005). Genetic linguistics : essays on theory and method. William Croft. Oxford: Oxford University Press. ISBN 978-0-19-151452-4. OCLC 132691297. Solomon, Zomaya S. (1997). Functional and other exotic sentences in Assyrian Aramaic, Journal of Assyrian Academic Studies, XI/2:44-69. "20 Contoh Penggunaan Kata nya dalam Kalimat" [Twenty Examples of the Use of 'nya' in Sentences]. kumparan.com (in Indonesian). Retrieved 2024-05-26. External links edit Look up article, definite article, or indefinite article in Wiktionary, the free dictionary. Wikisource has the text of the 1921 Collier's Encyclopedia article Article. "The Definite Article, 'The': The Most Frequently Used Word in World's Englishes" Ing, John (September 17, 2019). "How to Use Articles Correctly in English (A, An & The)". Retrieved May 3, 2020. Last edited 20 days ago by Kaloan-koko Related articles French articles and determiners Definiteness Semantic feature of noun phrases in linguistics English determiners Determiners in the English language Wikipedia Wikimedia Foundation Powered by MediaWiki Content is available under CC BY-SA 4.0 unless otherwise noted. Privacy policy Contact Wikipedia Code of Conduct Developers Statistics Cookie statement Terms of Use DesktopMaths Gaussian function Gaussian is a function of the base form with f(x) = exp(-x^2) and with parametric extension f(x) = aexp(-(x-b)^2)/2c^2) for arbitrary real constants a, b and non-zero c. It is named after the mathematician Carl Friedrich Gauss. The graph of a Gaussian is a characteristic symmetric "bell curve" shape. The parameter is the height of the curve's peak, b is the position of the center of the peak, and c (the standard deviation, sometimes called the Gaussian RMS width) controls the width of the "bell". Gaussian functions are widely used in statistics to describe the normal distributions, in signal processing to define the Gaussian filters, in image processing where two-dimensional Gaussians are used for Gaussian blurs, and in mathematics to solve heat equations and diffusion equations and to define the Weierstrass transform. They are also abundantly used in quantum chemistry to form basic sets. The most common method for estimating the Gaussian parameters is to take the logarithm of data and fit a parabola to the resulting data set. While this provides a simple curve fitting procedure, the resulting algorithm may be biased by excessively weighting small data values, which can produce large errors in the profile estimate. One can partially compensate for this problem through weighted least squares estimation, reducing the weight of small data values, but this too can be biased by allowing the tail of the Gaussian to dominate the fit. In order to remove the bias, one can instead use an iteratively reweighted least squares procedure, in which the weights are updated at each iteration. It is also possible to perform non-linear regression directly on the data without involving the logarithmic data transformation. Parameter Precision Once one has an algorithm for estimating the Gaussian parameters, it is also important to know how precise those estimates are. Any least squares estimation algorithm can provide numerical estimates for the variance of each parameter ( i.e. the variance of estimated height, position, and width of the function). One can also use Cramer-Rao bound theory to obtain an analytical expression for the lower bound on the parameter variances, given certain assumptions about the data. 1. The noise in the measured profile is either independent and identically distributed Gaussian, or the noise is Poissin-distributed. 2. The spacing between each sampling ( i.e the distance between pixels measuring the data) is uniform. 3. The peak is "well-sampled", so that less than 10% of the area or volume under the peak(area if a 1D Gaussian, volume if a 2D Gaussian) lies outside the measurement region. 4. The width of the peak is much larger than the distance between sample locations( i.e. the detector pixels must be al least 5 time smaller than the Gaussian FWHM). Applications Gaussian functions appear in many context in natural sciences, the social sciences, mathematics and engineering. Some examples include: 1. In statistics and probability theory, Gaussian functions appear as the density function of the normal distribution, which is a limiting probability distribution of complicated sums, according to central limit theorem. 2. A Gaussian function is the wave function of the ground state of the quantum harmonic oscillator. 3. The molecular orbitals used in computational chemistry can be linear combinations of Gaussian functions called Gaussian orbitals. 3. Gaussian functions are also associated with the vacuum state in quantum field theory. 4. Gaussian beams are used in optical systems, microwave systems and lasers. 5. In scale space representation, Gaussian functions are used as smoothing kernels for generating multi-scale representations in computer vision and image processing. Specifically, derivatives of Gaussians ( Hermite functions ) are used as a basis for defining a large number of types of visual operations. 6. Gaussian functions are used to define some types of artificial neural networks. 7. In fluorescence microscopy a 2D Gaussian function is used to approximate the Airy disk, describing the intensity distribution produced by a point source. 8. In signal processing they serve to define Gaussian filter, such as in image processing where 2D Gaussian kernel, which may be approximated by the Binomial coefficient or sampling a Gaussina. 9. In geostatistics, they have been used for understanding the variability between the patterns of a complex training image. They are used with kernel methods to cluster the patterns in the feature space. Commentshello@bikashpandey.com7407414467280806430-7407414467280806431BTJEZBZ3Q9# frozen_string_literal: true def gcd(first, second) if first.zero? return second end gcd(second % first, first) end first = 36 second = 60 puts "GCD of #{first} and #{second} is #{gcd(first, second)}"Ruby Maths Fourier theorem A Fourier series is an expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series. By expressing a function as a sum of sines and cosines, may problems involving the function become easier to analyze because trigonometric functions are well understood. For example, Fourier series were first used by Josheph Fourier(1786-1830) to find solutions to the heat equation. This application is possible because the derivatives of trigonometric functions fall into simple patterns. Fourier series cannot be used to approximate arbitrary functions, because most functions fall into simple patterns. Fourier series cannot be used to approximate arbitrary functions, because most functions have infinitely many terms in their Fourier series, and the series do not always converge. Well-behaved functions, for example smooth functions, have Fourier series that converge to the original function. The coefficients of the Fourier series are determined by integrals of the function multiplied by trigonometric functions, described in Fourier series Definition. The study of the convergence of Fourier series focus on the behaviors of the partial sums, which means studying the behavior of the sum as more and more terms from the series are summed. Fourier series are closely related to the Fourier transform. a more general tool that can even find the frequency information for functions that are not periodic. Periodic functions are identified with functions on a circle; for this reason Fourier series are the subject of Fourier analysis on the circle group, denoted by T or S1. The Fourier transform is also part of Fourier analysis, but is defined for functions on Rn. Since Fourier's time, many different approaches to defining and understanding the concept of Fourier series have been discovered, all of which are consistent with one another, but each of which emphasizes different aspects of the topic. Some of the more powerful and elegant approaches are based on the mathematical ideas and tools that were not available in Fourier's time. Fourier originally defined the Fourier series for real-valued functions of real arguments, and used the sine and cosine functions in the decomposition. May other Fourier-related transforms have since been defined, extending his initial idea to many applications and birthing an area of mathematics called Fourier analysis. Fourier transform The Fourier Transform is an integral transform that makes a function as input and outputs another function that describes the extent to which various frequencies are present in the original function. The output of the transform is a complex-valued function of frequency. The output of the transform is a complex valued function of frequency. The term Fourier transform refers to both this complex-valued function and the mathematical operation. When a distinction needs to be made, the output of the operation is sometimes called the frequency domain representation of the original function. The Fourier transform is analogous to decomposing the sound of a musical chord into the intensities of its constituent pitches. Functions that are localized in the time domain have Fourier transforms that are spread out across the frequency domain and vice versa,a phenomenon known as the uncertainty principle. The critical case for this principle is the Gaussian function, of substantial importance in probability theory and statistics as well as in the study of physical phenomena exhibiting normal distribution(example: diffusion). The Fourier transform of a Gaussian function is another Gaussian function. Joseph Fourier introduced the sine and cosine transforms ( which correspond to the imaginary and real components of the modern Fourier transform) in his study of heat transfer, where Gaussian functions appear as solutions of the heat equation. CommentsFourier theorem A Fourier series is an expansion of a periodic function into a sum of trigonometric functions. Read more Gaussian function Gaussian is a function of the base form with f(x) = exp(-x^2) and with parametric extension f(x) = aexp(-(x-b)^2)/2c^2) for arbitrary real constants a, b and non-zero c. Read more © 2025 bikashpandey.com. All Rights Reserved. Privacy Policy [email protected] admin monkey bou 🐒 👉 bkp by Om Luitel 10 minutes ago7407060620271091713https://youtu.be/oKcXSKEM2zc?si=v7KBWMXyXL-4-bYChttps://youtu.be/OEsYpH5X6Ow?si=IhBb_t0dLO7yHUhCThe English definite article the, written þe in Middle English, derives from an Old English demonstrative, which, according to gender, was written se (masculine), seo (feminine) (þe and þeo in the Northumbrian dialect), or þæt (neuter). The neuter form þæt also gave rise to the modern demonstrative that. The ye occasionally seen in pseudo-archaic usage such as "Ye Olde Englishe Tea Shoppe" is actually a form of þe, where the letter thorn (þ) came to be written as a y. Multiple demonstratives can give rise to multiple definite articles. Macedonian, for example, in which the articles are suffixed, has столот (stolot), the chair; столов (stolov), this chair; and столон (stolon), that chair. These derive from the Proto-Slavic demonstratives *tъ "this, that", *ovъ "this here" and *onъ "that over there, yonder" respectively. Colognian prepositions articles such as in dat Auto, or et Auto, the car; the first being specifically selected, focused, newly introduced, while the latter is not selected, unfocused, already known, general, or generic. Standard Basque distinguishes between proximal and distal definite articles in the plural (dialectally, a proximal singular and an additional medial grade may also be present). The Basque distal form (with infix -a-, etymologically a suffixed and phonetically reduced form of the distal demonstrative har-/hai-) functions as the default definite article, whereas the proximal form (with infix -o-, derived from the proximal demonstrative hau-/hon-) is marked and indicates some kind of (spatial or otherwise) close relationship between the speaker and the referent (e.g., it may imply that the speaker is included in the referent): etxeak ("the houses") vs. etxeok ("these houses [of ours]"), euskaldunak ("the Basque speakers") vs. euskaldunok ("we, the Basque speakers"). Speakers of Assyrian Neo-Aramaic, a modern Aramaic language that lacks a definite article, may at times use demonstratives aha and aya (feminine) or awa (masculine) – which translate to "this" and "that", respectively – to give the sense of "the".[10] In Indonesian, the third person possessive suffix -nya could be also used as a definite article.[11] Indefinite articles edit Indefinite articles typically arise from adjectives meaning one. For example, the indefinite articles in the Romance languages—e.g., un, una, une—derive from the Latin adjective unus. Partitive articles, however, derive from Vulgar Latin de illo, meaning (some) of the. The English indefinite article an is derived from the same root as one. The -n came to be dropped before consonants, giving rise to the shortened form a. The existence of both forms has led to many cases of juncture loss, for example transforming the original a napron into the modern an apron. The Persian indefinite article is yek, meaning one. See also edit English articles Al- (definite article in Arabic) Definiteness Definite description False title References edit Recasens, Marta; Martí, M. Antònia; Taulé, Mariona (2009-06-16), Winkler, Susanne; Featherston, Sam (eds.), "First-mention definites:More than exceptional cases", The Fruits of Empirical Linguistics II, vol. 102, Berlin, New York: Mouton de Gruyter, pp. 217–238, doi:10.1515/9783110216158.217, ISBN 978-3-11-021347-8, retrieved 2023-01-16 New perspectives on Hispanic contact : linguistics in the Americas. Melvin González-Rivera, and Sandro Sessarego. Madrid: Iberoamericana. 2015. ISBN 978-3-95487-831-4. OCLC 969386958. Burchfield, R. W. (1996). The New Fowler's Modern English Usage (3rd ed.). Oxford University Press. p. 512. ISBN 978-0199690367. Argetsinger, Amy (1 September 2015). "Why does everyone call Donald Trump 'The Donald'? It's an interesting story". The Washington Post. Retrieved 3 October 2017. Lawrence, Erma (1977). Haida dictionary. Fairbanks: Alaska Native Language Center. p. 64. Master, Peter (1997). "The English article system: Acquisition, function, and pedagogy". System. 25 (2): 215–232. doi:10.1016/S0346-251X(97)00010-9. Kusmenko, J K. "The typology of the language contact on the Balkans and in Scandinavia. A case of the suffixed definite article" (PDF). Archived (PDF) from the original on 2021-10-04. See s.5 Summary. Retrieved 2 February 2012. Simona, Ropati (1986). Tokelau Dictionary. New Zealand: Office of Tokelau Affairs. p. Introduction. Greenberg, Joseph H. (2005). Genetic linguistics : essays on theory and method. William Croft. Oxford: Oxford University Press. ISBN 978-0-19-151452-4. OCLC 132691297. Solomon, Zomaya S. (1997). Functional and other exotic sentences in Assyrian Aramaic, Journal of Assyrian Academic Studies, XI/2:44-69. "20 Contoh Penggunaan Kata nya dalam Kalimat" [Twenty Examples of the Use of 'nya' in Sentences]. kumparan.com (in Indonesian). Retrieved 2024-05-26. External links edit Look up article, definite article, or indefinite article in Wiktionary, the free dictionary. Wikisource has the text of the 1921 Collier's Encyclopedia article Article. "The Definite Article, 'The': The Most Frequently Used Word in World's Englishes" Ing, John (September 17, 2019). "How to Use Articles Correctly in English (A, An & The)". Retrieved May 3, 2020. Last edited 20 days ago by Kaloan-koko Related articles French articles and determiners Definiteness Semantic feature of noun phrases in linguistics English determiners Determiners in the English language Wikipedia Wikimedia Foundation Powered by MediaWiki Content is available under CC BY-SA 4.0 unless otherwise noted. Privacy policy Contact Wikipedia Code of Conduct Developers Statistics Cookie statement Terms of Use DesktopMaths Euclidean Alogrithms From geeksforgeeks. The Eucliean alogrithm is a way of find the greatest common divisor of two positive integers. GCD of two numbers is the largest number that divides both of them. A simple way to find GCD is to factorize both numbers and multiply common prime factors. 36 = 2 * 2 * 3 * 3 60 = 2 * 2 * 3 * 5 GCD = Multiplication of common factors = 2 * 2 * 3 = 12 Basic Euclidean Algorithm for GCD: The alogirithm is based on the below facts: - If we subtract a smaller number from a larger one ( we reduce a large number ), GCD does not change. So, if we keep subtracting repeatedly the larger of two, we end up with GCD. - Now instead of subtraction, if we divide the larger number, the algorithm stops when we find the reminder 0. # frozen_string_literal: true def gcd(first, second) if first.zero? return second end gcd(second % first, first) end first = 36 second = 60 puts "GCD of #{first} and #{second} is #{gcd(first, second)}" CommentsBlog Open user menuuser photo Featured Ruby Maths Fourier theorem A Fourier series is an expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series. By expressing a function as a sum of sines and cosines, may problems involving the function become easier to analyze because trigonometric functions are well understood. For example, Fourier series were first used by Josheph Fourier(1786-1830) to find solutions to the heat equation. This application is possible because the derivatives of trigonometric functions fall into simple patterns. Fourier series cannot be used to approximate arbitrary functions, because most functions fall into simple patterns. Fourier series cannot be used to approximate arbitrary functions, because most functions have infinitely many terms in their Fourier series, and the series do not always converge. Well-behaved functions, for example smooth functions, have Fourier series that converge to the original function. The coefficients of the Fourier series are determined by integrals of the function multiplied by trigonometric functions, described in Fourier series Definition. The study of the convergence of Fourier series focus on the behaviors of the partial sums, which means studying the behavior of the sum as more and more terms from the series are summed. Fourier series are closely related to the Fourier transform. a more general tool that can even find the frequency information for functions that are not periodic. Periodic functions are identified with functions on a circle; for this reason Fourier series are the subject of Fourier analysis on the circle group, denoted by T or S1. The Fourier transform is also part of Fourier analysis, but is defined for functions on Rn. Since Fourier's time, many different approaches to defining and understanding the concept of Fourier series have been discovered, all of which are consistent with one another, but each of which emphasizes different aspects of the topic. Some of the more powerful and elegant approaches are based on the mathematical ideas and tools that were not available in Fourier's time. Fourier originally defined the Fourier series for real-valued functions of real arguments, and used the sine and cosine functions in the decomposition. May other Fourier-related transforms have since been defined, extending his initial idea to many applications and birthing an area of mathematics called Fourier analysis. Fourier transform The Fourier Transform is an integral transform that makes a function as input and outputs another function that describes the extent to which various frequencies are present in the original function. The output of the transform is a complex-valued function of frequency. The output of the transform is a complex valued function of frequency. The term Fourier transform refers to both this complex-valued function and the mathematical operation. When a distinction needs to be made, the output of the operation is sometimes called the frequency domain representation of the original function. The Fourier transform is analogous to decomposing the sound of a musical chord into the intensities of its constituent pitches. Functions that are localized in the time domain have Fourier transforms that are spread out across the frequency domain and vice versa,a phenomenon known as the uncertainty principle. The critical case for this principle is the Gaussian function, of substantial importance in probability theory and statistics as well as in the study of physical phenomena exhibiting normal distribution(example: diffusion). The Fourier transform of a Gaussian function is another Gaussian function. Joseph Fourier introduced the sine and cosine transforms ( which correspond to the imaginary and real components of the modern Fourier transform) in his study of heat transfer, where Gaussian functions appear as solutions of the heat equation. CommentsBTJEZBZ3Q97407414467280806430-74074144672808064317335692039084330226https://youtube.com/watch?v=GUS_MnJQ5Uo&lc=UgxkvBdpKjTYrynRE2V4AaABAg&si=m2nw-jfZ1pQMq0Eg72783028664639667817407060620271091713https://youtu.be/UuXqv-fQ9PY?si=3wMkWfSSMexa8BZgThe English definite article the, written þe in Middle English, derives from an Old English demonstrative, which, according to gender, was written se (masculine), seo (feminine) (þe and þeo in the Northumbrian dialect), or þæt (neuter). The neuter form þæt also gave rise to the modern demonstrative that. The ye occasionally seen in pseudo-archaic usage such as "Ye Olde Englishe Tea Shoppe" is actually a form of þe, where the letter thorn (þ) came to be written as a y. Multiple demonstratives can give rise to multiple definite articles. Macedonian, for example, in which the articles are suffixed, has столот (stolot), the chair; столов (stolov), this chair; and столон (stolon), that chair. These derive from the Proto-Slavic demonstratives *tъ "this, that", *ovъ "this here" and *onъ "that over there, yonder" respectively. Colognian prepositions articles such as in dat Auto, or et Auto, the car; the first being specifically selected, focused, newly introduced, while the latter is not selected, unfocused, already known, general, or generic. Standard Basque distinguishes between proximal and distal definite articles in the plural (dialectally, a proximal singular and an additional medial grade may also be present). The Basque distal form (with infix -a-, etymologically a suffixed and phonetically reduced form of the distal demonstrative har-/hai-) functions as the default definite article, whereas the proximal form (with infix -o-, derived from the proximal demonstrative hau-/hon-) is marked and indicates some kind of (spatial or otherwise) close relationship between the speaker and the referent (e.g., it may imply that the speaker is included in the referent): etxeak ("the houses") vs. etxeok ("these houses [of ours]"), euskaldunak ("the Basque speakers") vs. euskaldunok ("we, the Basque speakers"). Speakers of Assyrian Neo-Aramaic, a modern Aramaic language that lacks a definite article, may at times use demonstratives aha and aya (feminine) or awa (masculine) – which translate to "this" and "that", respectively – to give the sense of "the".[10] In Indonesian, the third person possessive suffix -nya could be also used as a definite article.[11] Indefinite articles edit Indefinite articles typically arise from adjectives meaning one. For example, the indefinite articles in the Romance languages—e.g., un, una, une—derive from the Latin adjective unus. Partitive articles, however, derive from Vulgar Latin de illo, meaning (some) of the. The English indefinite article an is derived from the same root as one. The -n came to be dropped before consonants, giving rise to the shortened form a. The existence of both forms has led to many cases of juncture loss, for example transforming the original a napron into the modern an apron. The Persian indefinite article is yek, meaning one. See also edit English articles Al- (definite article in Arabic) Definiteness Definite description False title References edit Recasens, Marta; Martí, M. Antònia; Taulé, Mariona (2009-06-16), Winkler, Susanne; Featherston, Sam (eds.), "First-mention definites:More than exceptional cases", The Fruits of Empirical Linguistics II, vol. 102, Berlin, New York: Mouton de Gruyter, pp. 217–238, doi:10.1515/9783110216158.217, ISBN 978-3-11-021347-8, retrieved 2023-01-16 New perspectives on Hispanic contact : linguistics in the Americas. Melvin González-Rivera, and Sandro Sessarego. Madrid: Iberoamericana. 2015. ISBN 978-3-95487-831-4. OCLC 969386958. Burchfield, R. W. (1996). The New Fowler's Modern English Usage (3rd ed.). Oxford University Press. p. 512. ISBN 978-0199690367. Argetsinger, Amy (1 September 2015). "Why does everyone call Donald Trump 'The Donald'? It's an interesting story". The Washington Post. Retrieved 3 October 2017. Lawrence, Erma (1977). Haida dictionary. Fairbanks: Alaska Native Language Center. p. 64. Master, Peter (1997). "The English article system: Acquisition, function, and pedagogy". System. 25 (2): 215–232. doi:10.1016/S0346-251X(97)00010-9. Kusmenko, J K. "The typology of the language contact on the Balkans and in Scandinavia. A case of the suffixed definite article" (PDF). Archived (PDF) from the original on 2021-10-04. See s.5 Summary. Retrieved 2 February 2012. Simona, Ropati (1986). Tokelau Dictionary. New Zealand: Office of Tokelau Affairs. p. Introduction. Greenberg, Joseph H. (2005). Genetic linguistics : essays on theory and method. William Croft. Oxford: Oxford University Press. ISBN 978-0-19-151452-4. OCLC 132691297. Solomon, Zomaya S. (1997). Functional and other exotic sentences in Assyrian Aramaic, Journal of Assyrian Academic Studies, XI/2:44-69. "20 Contoh Penggunaan Kata nya dalam Kalimat" [Twenty Examples of the Use of 'nya' in Sentences]. kumparan.com (in Indonesian). Retrieved 2024-05-26. External links edit Look up article, definite article, or indefinite article in Wiktionary, the free dictionary. Wikisource has the text of the 1921 Collier's Encyclopedia article Article. "The Definite Article, 'The': The Most Frequently Used Word in World's Englishes" Ing, John (September 17, 2019). "How to Use Articles Correctly in English (A, An & The)". Retrieved May 3, 2020. Last edited 20 days ago by Kaloan-koko Related articles French articles and determiners Definiteness Semantic feature of noun phrases in linguistics English determiners Determiners in the English language Wikipedia Wikimedia Foundation Powered by MediaWiki Content is available under CC BY-SA 4.0 unless otherwise noted. Privacy policy Contact Wikipedia Code of Conduct Developers Statistics Cookie statement Terms of Use DesktopMaths Euclidean Alogrithms From geeksforgeeks. The Eucliean alogrithm is a way of find the greatest common divisor of two positive integers. GCD of two numbers is the largest number that divides both of them. A simple way to find GCD is to factorize both numbers and multiply common prime factors. 36 = 2 * 2 * 3 * 3 60 = 2 * 2 * 3 * 5 GCD = Multiplication of common factors = 2 * 2 * 3 = 12 Basic Euclidean Algorithm for GCD: The alogirithm is based on the below facts: - If we subtract a smaller number from a larger one ( we reduce a large number ), GCD does not change. So, if we keep subtracting repeatedly the larger of two, we end up with GCD. - Now instead of subtraction, if we divide the larger number, the algorithm stops when we find the reminder 0. # frozen_string_literal: true def gcd(first, second) if first.zero? return second end gcd(second % first, first) end first = 36 second = 60 puts "GCD of #{first} and #{second} is #{gcd(first, second)}" CommentsBlog Open user menuuser photo Featured Ruby Maths Fourier theorem A Fourier series is an expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series. By expressing a function as a sum of sines and cosines, may problems involving the function become easier to analyze because trigonometric functions are well understood. For example, Fourier series were first used by Josheph Fourier(1786-1830) to find solutions to the heat equation. This application is possible because the derivatives of trigonometric functions fall into simple patterns. Fourier series cannot be used to approximate arbitrary functions, because most functions fall into simple patterns. Fourier series cannot be used to approximate arbitrary functions, because most functions have infinitely many terms in their Fourier series, and the series do not always converge. Well-behaved functions, for example smooth functions, have Fourier series that converge to the original function. The coefficients of the Fourier series are determined by integrals of the function multiplied by trigonometric functions, described in Fourier series Definition. The study of the convergence of Fourier series focus on the behaviors of the partial sums, which means studying the behavior of the sum as more and more terms from the series are summed. Fourier series are closely related to the Fourier transform. a more general tool that can even find the frequency information for functions that are not periodic. Periodic functions are identified with functions on a circle; for this reason Fourier series are the subject of Fourier analysis on the circle group, denoted by T or S1. The Fourier transform is also part of Fourier analysis, but is defined for functions on Rn. Since Fourier's time, many different approaches to defining and understanding the concept of Fourier series have been discovered, all of which are consistent with one another, but each of which emphasizes different aspects of the topic. Some of the more powerful and elegant approaches are based on the mathematical ideas and tools that were not available in Fourier's time. Fourier originally defined the Fourier series for real-valued functions of real arguments, and used the sine and cosine functions in the decomposition. May other Fourier-related transforms have since been defined, extending his initial idea to many applications and birthing an area of mathematics called Fourier analysis. Fourier transform The Fourier Transform is an integral transform that makes a function as input and outputs another function that describes the extent to which various frequencies are present in the original function. The output of the transform is a complex-valued function of frequency. The output of the transform is a complex valued function of frequency. The term Fourier transform refers to both this complex-valued function and the mathematical operation. When a distinction needs to be made, the output of the operation is sometimes called the frequency domain representation of the original function. The Fourier transform is analogous to decomposing the sound of a musical chord into the intensities of its constituent pitches. Functions that are localized in the time domain have Fourier transforms that are spread out across the frequency domain and vice versa,a phenomenon known as the uncertainty principle. The critical case for this principle is the Gaussian function, of substantial importance in probability theory and statistics as well as in the study of physical phenomena exhibiting normal distribution(example: diffusion). The Fourier transform of a Gaussian function is another Gaussian function. Joseph Fourier introduced the sine and cosine transforms ( which correspond to the imaginary and real components of the modern Fourier transform) in his study of heat transfer, where Gaussian functions appear as solutions of the heat equation. CommentsBTJEZBZ3Q9Maths Gaussian function Gaussian is a function of the base form with f(x) = exp(-x^2) and with parametric extension f(x) = aexp(-(x-b)^2)/2c^2) for arbitrary real constants a, b and non-zero c. It is named after the mathematician Carl Friedrich Gauss. The graph of a Gaussian is a characteristic symmetric "bell curve" shape. The parameter is the height of the curve's peak, b is the position of the center of the peak, and c (the standard deviation, sometimes called the Gaussian RMS width) controls the width of the "bell". Gaussian functions are widely used in statistics to describe the normal distributions, in signal processing to define the Gaussian filters, in image processing where two-dimensional Gaussians are used for Gaussian blurs, and in mathematics to solve heat equations and diffusion equations and to define the Weierstrass transform. They are also abundantly used in quantum chemistry to form basic sets. The most common method for estimating the Gaussian parameters is to take the logarithm of data and fit a parabola to the resulting data set. While this provides a simple curve fitting procedure, the resulting algorithm may be biased by excessively weighting small data values, which can produce large errors in the profile estimate. One can partially compensate for this problem through weighted least squares estimation, reducing the weight of small data values, but this too can be biased by allowing the tail of the Gaussian to dominate the fit. In order to remove the bias, one can instead use an iteratively reweighted least squares procedure, in which the weights are updated at each iteration. It is also possible to perform non-linear regression directly on the data without involving the logarithmic data transformation. Parameter Precision Once one has an algorithm for estimating the Gaussian parameters, it is also important to know how precise those estimates are. Any least squares estimation algorithm can provide numerical estimates for the variance of each parameter ( i.e. the variance of estimated height, position, and width of the function). One can also use Cramer-Rao bound theory to obtain an analytical expression for the lower bound on the parameter variances, given certain assumptions about the data. 1. The noise in the measured profile is either independent and identically distributed Gaussian, or the noise is Poissin-distributed. 2. The spacing between each sampling ( i.e the distance between pixels measuring the data) is uniform. 3. The peak is "well-sampled", so that less than 10% of the area or volume under the peak(area if a 1D Gaussian, volume if a 2D Gaussian) lies outside the measurement region. 4. The width of the peak is much larger than the distance between sample locations( i.e. the detector pixels must be al least 5 time smaller than the Gaussian FWHM). Applications Gaussian functions appear in many context in natural sciences, the social sciences, mathematics and engineering. Some examples include: 1. In statistics and probability theory, Gaussian functions appear as the density function of the normal distribution, which is a limiting probability distribution of complicated sums, according to central limit theorem. 2. A Gaussian function is the wave function of the ground state of the quantum harmonic oscillator. 3. The molecular orbitals used in computational chemistry can be linear combinations of Gaussian functions called Gaussian orbitals. 3. Gaussian functions are also associated with the vacuum state in quantum field theory. 4. Gaussian beams are used in optical systems, microwave systems and lasers. 5. In scale space representation, Gaussian functions are used as smoothing kernels for generating multi-scale representations in computer vision and image processing. Specifically, derivatives of Gaussians ( Hermite functions ) are used as a basis for defining a large number of types of visual operations. 6. Gaussian functions are used to define some types of artificial neural networks. 7. In fluorescence microscopy a 2D Gaussian function is used to approximate the Airy disk, describing the intensity distribution produced by a point source. 8. In signal processing they serve to define Gaussian filter, such as in image processing where 2D Gaussian kernel, which may be approximated by the Binomial coefficient or sampling a Gaussina. 9. In geostatistics, they have been used for understanding the variability between the patterns of a complex training image. They are used with kernel methods to cluster the patterns in the feature space. Commentshello@bikashpandey.com7407414467280806430-7407414467280806431# frozen_string_literal: true def gcd(first, second) if first.zero? return second end gcd(second % first, first) end first = 36 second = 60 puts "GCD of #{first} and #{second} is #{gcd(first, second)}"Ruby Maths Fourier theorem A Fourier series is an expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series. By expressing a function as a sum of sines and cosines, may problems involving the function become easier to analyze because trigonometric functions are well understood. For example, Fourier series were first used by Josheph Fourier(1786-1830) to find solutions to the heat equation. This application is possible because the derivatives of trigonometric functions fall into simple patterns. Fourier series cannot be used to approximate arbitrary functions, because most functions fall into simple patterns. Fourier series cannot be used to approximate arbitrary functions, because most functions have infinitely many terms in their Fourier series, and the series do not always converge. Well-behaved functions, for example smooth functions, have Fourier series that converge to the original function. The coefficients of the Fourier series are determined by integrals of the function multiplied by trigonometric functions, described in Fourier series Definition. The study of the convergence of Fourier series focus on the behaviors of the partial sums, which means studying the behavior of the sum as more and more terms from the series are summed. Fourier series are closely related to the Fourier transform. a more general tool that can even find the frequency information for functions that are not periodic. Periodic functions are identified with functions on a circle; for this reason Fourier series are the subject of Fourier analysis on the circle group, denoted by T or S1. The Fourier transform is also part of Fourier analysis, but is defined for functions on Rn. Since Fourier's time, many different approaches to defining and understanding the concept of Fourier series have been discovered, all of which are consistent with one another, but each of which emphasizes different aspects of the topic. Some of the more powerful and elegant approaches are based on the mathematical ideas and tools that were not available in Fourier's time. Fourier originally defined the Fourier series for real-valued functions of real arguments, and used the sine and cosine functions in the decomposition. May other Fourier-related transforms have since been defined, extending his initial idea to many applications and birthing an area of mathematics called Fourier analysis. Fourier transform The Fourier Transform is an integral transform that makes a function as input and outputs another function that describes the extent to which various frequencies are present in the original function. The output of the transform is a complex-valued function of frequency. The output of the transform is a complex valued function of frequency. The term Fourier transform refers to both this complex-valued function and the mathematical operation. When a distinction needs to be made, the output of the operation is sometimes called the frequency domain representation of the original function. The Fourier transform is analogous to decomposing the sound of a musical chord into the intensities of its constituent pitches. Functions that are localized in the time domain have Fourier transforms that are spread out across the frequency domain and vice versa,a phenomenon known as the uncertainty principle. The critical case for this principle is the Gaussian function, of substantial importance in probability theory and statistics as well as in the study of physical phenomena exhibiting normal distribution(example: diffusion). The Fourier transform of a Gaussian function is another Gaussian function. Joseph Fourier introduced the sine and cosine transforms ( which correspond to the imaginary and real components of the modern Fourier transform) in his study of heat transfer, where Gaussian functions appear as solutions of the heat equation. CommentsFourier theorem A Fourier series is an expansion of a periodic function into a sum of trigonometric functions. Read more Gaussian function Gaussian is a function of the base form with f(x) = exp(-x^2) and with parametric extension f(x) = aexp(-(x-b)^2)/2c^2) for arbitrary real constants a, b and non-zero c. Read more © 2025 bikashpandey.com. All Rights Reserved. Privacy Policy [email protected] Euclidean Alogrithms From geeksforgeeks. The Eucliean alogrithm is a way of find the greatest common divisor of two positive integers. GCD of two numbers is the largest number that divides both of them. A simple way to find GCD is to factorize both numbers and multiply common prime factors. 36 = 2 * 2 * 3 * 3 60 = 2 * 2 * 3 * 5 GCD = Multiplication of common factors = 2 * 2 * 3 = 12 Basic Euclidean Algorithm for GCD: The alogirithm is based on the below facts: - If we subtract a smaller number from a larger one ( we reduce a large number ), GCD does not change. So, if we keep subtracting repeatedly the larger of two, we end up with GCD. - Now instead of subtraction, if we divide the larger number, the algorithm stops when we find the reminder 0. # frozen_string_literal: true def gcd(first, second) if first.zero? return second end gcd(second % first, first) end first = 36 second = 60 puts "GCD of #{first} and #{second} is #{gcd(first, second)}" CommentsThe English definite article the, written þe in Middle English, derives from an Old English demonstrative, which, according to gender, was written se (masculine), seo (feminine) (þe and þeo in the Northumbrian dialect), or þæt (neuter). The neuter form þæt also gave rise to the modern demonstrative that. The ye occasionally seen in pseudo-archaic usage such as "Ye Olde Englishe Tea Shoppe" is actually a form of þe, where the letter thorn (þ) came to be written as a y. Multiple demonstratives can give rise to multiple definite articles. Macedonian, for example, in which the articles are suffixed, has столот (stolot), the chair; столов (stolov), this chair; and столон (stolon), that chair. These derive from the Proto-Slavic demonstratives *tъ "this, that", *ovъ "this here" and *onъ "that over there, yonder" respectively. Colognian prepositions articles such as in dat Auto, or et Auto, the car; the first being specifically selected, focused, newly introduced, while the latter is not selected, unfocused, already known, general, or generic. Standard Basque distinguishes between proximal and distal definite articles in the plural (dialectally, a proximal singular and an additional medial grade may also be present). The Basque distal form (with infix -a-, etymologically a suffixed and phonetically reduced form of the distal demonstrative har-/hai-) functions as the default definite article, whereas the proximal form (with infix -o-, derived from the proximal demonstrative hau-/hon-) is marked and indicates some kind of (spatial or otherwise) close relationship between the speaker and the referent (e.g., it may imply that the speaker is included in the referent): etxeak ("the houses") vs. etxeok ("these houses [of ours]"), euskaldunak ("the Basque speakers") vs. euskaldunok ("we, the Basque speakers"). Speakers of Assyrian Neo-Aramaic, a modern Aramaic language that lacks a definite article, may at times use demonstratives aha and aya (feminine) or awa (masculine) – which translate to "this" and "that", respectively – to give the sense of "the".[10] In Indonesian, the third person possessive suffix -nya could be also used as a definite article.[11] Indefinite articles edit Indefinite articles typically arise from adjectives meaning one. For example, the indefinite articles in the Romance languages—e.g., un, una, une—derive from the Latin adjective unus. Partitive articles, however, derive from Vulgar Latin de illo, meaning (some) of the. The English indefinite article an is derived from the same root as one. The -n came to be dropped before consonants, giving rise to the shortened form a. The existence of both forms has led to many cases of juncture loss, for example transforming the original a napron into the modern an apron. The Persian indefinite article is yek, meaning one. See also edit English articles Al- (definite article in Arabic) Definiteness Definite description False title References edit Recasens, Marta; Martí, M. Antònia; Taulé, Mariona (2009-06-16), Winkler, Susanne; Featherston, Sam (eds.), "First-mention definites:More than exceptional cases", The Fruits of Empirical Linguistics II, vol. 102, Berlin, New York: Mouton de Gruyter, pp. 217–238, doi:10.1515/9783110216158.217, ISBN 978-3-11-021347-8, retrieved 2023-01-16 New perspectives on Hispanic contact : linguistics in the Americas. Melvin González-Rivera, and Sandro Sessarego. Madrid: Iberoamericana. 2015. ISBN 978-3-95487-831-4. OCLC 969386958. Burchfield, R. W. (1996). The New Fowler's Modern English Usage (3rd ed.). Oxford University Press. p. 512. ISBN 978-0199690367. Argetsinger, Amy (1 September 2015). "Why does everyone call Donald Trump 'The Donald'? It's an interesting story". The Washington Post. Retrieved 3 October 2017. Lawrence, Erma (1977). Haida dictionary. Fairbanks: Alaska Native Language Center. p. 64. Master, Peter (1997). "The English article system: Acquisition, function, and pedagogy". System. 25 (2): 215–232. doi:10.1016/S0346-251X(97)00010-9. Kusmenko, J K. "The typology of the language contact on the Balkans and in Scandinavia. A case of the suffixed definite article" (PDF). Archived (PDF) from the original on 2021-10-04. See s.5 Summary. Retrieved 2 February 2012. Simona, Ropati (1986). Tokelau Dictionary. New Zealand: Office of Tokelau Affairs. p. Introduction. Greenberg, Joseph H. (2005). Genetic linguistics : essays on theory and method. William Croft. Oxford: Oxford University Press. ISBN 978-0-19-151452-4. OCLC 132691297. Solomon, Zomaya S. (1997). Functional and other exotic sentences in Assyrian Aramaic, Journal of Assyrian Academic Studies, XI/2:44-69. "20 Contoh Penggunaan Kata nya dalam Kalimat" [Twenty Examples of the Use of 'nya' in Sentences]. kumparan.com (in Indonesian). Retrieved 2024-05-26. External links edit Look up article, definite article, or indefinite article in Wiktionary, the free dictionary. Wikisource has the text of the 1921 Collier's Encyclopedia article Article. "The Definite Article, 'The': The Most Frequently Used Word in World's Englishes" Ing, John (September 17, 2019). "How to Use Articles Correctly in English (A, An & The)". Retrieved May 3, 2020. Last edited 20 days ago by Kaloan-koko Related articles French articles and determiners Definiteness Semantic feature of noun phrases in linguistics English determiners Determiners in the English language Wikipedia Wikimedia Foundation Powered by MediaWiki Content is available under CC BY-SA 4.0 unless otherwise noted. Privacy policy Contact Wikipedia Code of Conduct Developers Statistics Cookie statement Terms of Use DesktopBlog Open user menuuser photo Featured Ruby Maths Fourier theorem A Fourier series is an expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series. By expressing a function as a sum of sines and cosines, may problems involving the function become easier to analyze because trigonometric functions are well understood. For example, Fourier series were first used by Josheph Fourier(1786-1830) to find solutions to the heat equation. This application is possible because the derivatives of trigonometric functions fall into simple patterns. Fourier series cannot be used to approximate arbitrary functions, because most functions fall into simple patterns. Fourier series cannot be used to approximate arbitrary functions, because most functions have infinitely many terms in their Fourier series, and the series do not always converge. Well-behaved functions, for example smooth functions, have Fourier series that converge to the original function. The coefficients of the Fourier series are determined by integrals of the function multiplied by trigonometric functions, described in Fourier series Definition. The study of the convergence of Fourier series focus on the behaviors of the partial sums, which means studying the behavior of the sum as more and more terms from the series are summed. Fourier series are closely related to the Fourier transform. a more general tool that can even find the frequency information for functions that are not periodic. Periodic functions are identified with functions on a circle; for this reason Fourier series are the subject of Fourier analysis on the circle group, denoted by T or S1. The Fourier transform is also part of Fourier analysis, but is defined for functions on Rn. Since Fourier's time, many different approaches to defining and understanding the concept of Fourier series have been discovered, all of which are consistent with one another, but each of which emphasizes different aspects of the topic. Some of the more powerful and elegant approaches are based on the mathematical ideas and tools that were not available in Fourier's time. Fourier originally defined the Fourier series for real-valued functions of real arguments, and used the sine and cosine functions in the decomposition. May other Fourier-related transforms have since been defined, extending his initial idea to many applications and birthing an area of mathematics called Fourier analysis. Fourier transform The Fourier Transform is an integral transform that makes a function as input and outputs another function that describes the extent to which various frequencies are present in the original function. The output of the transform is a complex-valued function of frequency. The output of the transform is a complex valued function of frequency. The term Fourier transform refers to both this complex-valued function and the mathematical operation. When a distinction needs to be made, the output of the operation is sometimes called the frequency domain representation of the original function. The Fourier transform is analogous to decomposing the sound of a musical chord into the intensities of its constituent pitches. Functions that are localized in the time domain have Fourier transforms that are spread out across the frequency domain and vice versa,a phenomenon known as the uncertainty principle. The critical case for this principle is the Gaussian function, of substantial importance in probability theory and statistics as well as in the study of physical phenomena exhibiting normal distribution(example: diffusion). The Fourier transform of a Gaussian function is another Gaussian function. Joseph Fourier introduced the sine and cosine transforms ( which correspond to the imaginary and real components of the modern Fourier transform) in his study of heat transfer, where Gaussian functions appear as solutions of the heat equation. CommentsMaths Gaussian function Gaussian is a function of the base form with f(x) = exp(-x^2) and with parametric extension f(x) = aexp(-(x-b)^2)/2c^2) for arbitrary real constants a, b and non-zero c. It is named after the mathematician Carl Friedrich Gauss. The graph of a Gaussian is a characteristic symmetric "bell curve" shape. The parameter is the height of the curve's peak, b is the position of the center of the peak, and c (the standard deviation, sometimes called the Gaussian RMS width) controls the width of the "bell". Gaussian functions are widely used in statistics to describe the normal distributions, in signal processing to define the Gaussian filters, in image processing where two-dimensional Gaussians are used for Gaussian blurs, and in mathematics to solve heat equations and diffusion equations and to define the Weierstrass transform. They are also abundantly used in quantum chemistry to form basic sets. The most common method for estimating the Gaussian parameters is to take the logarithm of data and fit a parabola to the resulting data set. While this provides a simple curve fitting procedure, the resulting algorithm may be biased by excessively weighting small data values, which can produce large errors in the profile estimate. One can partially compensate for this problem through weighted least squares estimation, reducing the weight of small data values, but this too can be biased by allowing the tail of the Gaussian to dominate the fit. In order to remove the bias, one can instead use an iteratively reweighted least squares procedure, in which the weights are updated at each iteration. It is also possible to perform non-linear regression directly on the data without involving the logarithmic data transformation. Parameter Precision Once one has an algorithm for estimating the Gaussian parameters, it is also important to know how precise those estimates are. Any least squares estimation algorithm can provide numerical estimates for the variance of each parameter ( i.e. the variance of estimated height, position, and width of the function). One can also use Cramer-Rao bound theory to obtain an analytical expression for the lower bound on the parameter variances, given certain assumptions about the data. 1. The noise in the measured profile is either independent and identically distributed Gaussian, or the noise is Poissin-distributed. 2. The spacing between each sampling ( i.e the distance between pixels measuring the data) is uniform. 3. The peak is "well-sampled", so that less than 10% of the area or volume under the peak(area if a 1D Gaussian, volume if a 2D Gaussian) lies outside the measurement region. 4. The width of the peak is much larger than the distance between sample locations( i.e. the detector pixels must be al least 5 time smaller than the Gaussian FWHM). Applications Gaussian functions appear in many context in natural sciences, the social sciences, mathematics and engineering. Some examples include: 1. In statistics and probability theory, Gaussian functions appear as the density function of the normal distribution, which is a limiting probability distribution of complicated sums, according to central limit theorem. 2. A Gaussian function is the wave function of the ground state of the quantum harmonic oscillator. 3. The molecular orbitals used in computational chemistry can be linear combinations of Gaussian functions called Gaussian orbitals. 3. Gaussian functions are also associated with the vacuum state in quantum field theory. 4. Gaussian beams are used in optical systems, microwave systems and lasers. 5. In scale space representation, Gaussian functions are used as smoothing kernels for generating multi-scale representations in computer vision and image processing. Specifically, derivatives of Gaussians ( Hermite functions ) are used as a basis for defining a large number of types of visual operations. 6. Gaussian functions are used to define some types of artificial neural networks. 7. In fluorescence microscopy a 2D Gaussian function is used to approximate the Airy disk, describing the intensity distribution produced by a point source. 8. In signal processing they serve to define Gaussian filter, such as in image processing where 2D Gaussian kernel, which may be approximated by the Binomial coefficient or sampling a Gaussina. 9. In geostatistics, they have been used for understanding the variability between the patterns of a complex training image. They are used with kernel methods to cluster the patterns in the feature space. CommentsBTJEZBZ3Q97407414467280806430-7407414467280806431hello@bikashpandey.com# frozen_string_literal: true def gcd(first, second) if first.zero? return second end gcd(second % first, first) end first = 36 second = 60 puts "GCD of #{first} and #{second} is #{gcd(first, second)}"7335692039084330226Ruby Maths Fourier theorem A Fourier series is an expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series. By expressing a function as a sum of sines and cosines, may problems involving the function become easier to analyze because trigonometric functions are well understood. For example, Fourier series were first used by Josheph Fourier(1786-1830) to find solutions to the heat equation. This application is possible because the derivatives of trigonometric functions fall into simple patterns. Fourier series cannot be used to approximate arbitrary functions, because most functions fall into simple patterns. Fourier series cannot be used to approximate arbitrary functions, because most functions have infinitely many terms in their Fourier series, and the series do not always converge. Well-behaved functions, for example smooth functions, have Fourier series that converge to the original function. The coefficients of the Fourier series are determined by integrals of the function multiplied by trigonometric functions, described in Fourier series Definition. The study of the convergence of Fourier series focus on the behaviors of the partial sums, which means studying the behavior of the sum as more and more terms from the series are summed. Fourier series are closely related to the Fourier transform. a more general tool that can even find the frequency information for functions that are not periodic. Periodic functions are identified with functions on a circle; for this reason Fourier series are the subject of Fourier analysis on the circle group, denoted by T or S1. The Fourier transform is also part of Fourier analysis, but is defined for functions on Rn. Since Fourier's time, many different approaches to defining and understanding the concept of Fourier series have been discovered, all of which are consistent with one another, but each of which emphasizes different aspects of the topic. Some of the more powerful and elegant approaches are based on the mathematical ideas and tools that were not available in Fourier's time. Fourier originally defined the Fourier series for real-valued functions of real arguments, and used the sine and cosine functions in the decomposition. May other Fourier-related transforms have since been defined, extending his initial idea to many applications and birthing an area of mathematics called Fourier analysis. Fourier transform The Fourier Transform is an integral transform that makes a function as input and outputs another function that describes the extent to which various frequencies are present in the original function. The output of the transform is a complex-valued function of frequency. The output of the transform is a complex valued function of frequency. The term Fourier transform refers to both this complex-valued function and the mathematical operation. When a distinction needs to be made, the output of the operation is sometimes called the frequency domain representation of the original function. The Fourier transform is analogous to decomposing the sound of a musical chord into the intensities of its constituent pitches. Functions that are localized in the time domain have Fourier transforms that are spread out across the frequency domain and vice versa,a phenomenon known as the uncertainty principle. The critical case for this principle is the Gaussian function, of substantial importance in probability theory and statistics as well as in the study of physical phenomena exhibiting normal distribution(example: diffusion). The Fourier transform of a Gaussian function is another Gaussian function. Joseph Fourier introduced the sine and cosine transforms ( which correspond to the imaginary and real components of the modern Fourier transform) in his study of heat transfer, where Gaussian functions appear as solutions of the heat equation. Commentshttps://youtube.com/watch?v=GUS_MnJQ5Uo&lc=UgxkvBdpKjTYrynRE2V4AaABAg&si=m2nw-jfZ1pQMq0EgFourier theorem A Fourier series is an expansion of a periodic function into a sum of trigonometric functions. Read more Gaussian function Gaussian is a function of the base form with f(x) = exp(-x^2) and with parametric extension f(x) = aexp(-(x-b)^2)/2c^2) for arbitrary real constants a, b and non-zero c. Read more © 2025 bikashpandey.com. All Rights Reserved. Privacy Policy [email protected] admin monkey bou 🐒 👉 bkp by Om Luitel 10 minutes ago7407060620271091713The English definite article the, written þe in Middle English, derives from an Old English demonstrative, which, according to gender, was written se (masculine), seo (feminine) (þe and þeo in the Northumbrian dialect), or þæt (neuter). The neuter form þæt also gave rise to the modern demonstrative that. The ye occasionally seen in pseudo-archaic usage such as "Ye Olde Englishe Tea Shoppe" is actually a form of þe, where the letter thorn (þ) came to be written as a y. Multiple demonstratives can give rise to multiple definite articles. Macedonian, for example, in which the articles are suffixed, has столот (stolot), the chair; столов (stolov), this chair; and столон (stolon), that chair. These derive from the Proto-Slavic demonstratives *tъ "this, that", *ovъ "this here" and *onъ "that over there, yonder" respectively. Colognian prepositions articles such as in dat Auto, or et Auto, the car; the first being specifically selected, focused, newly introduced, while the latter is not selected, unfocused, already known, general, or generic. Standard Basque distinguishes between proximal and distal definite articles in the plural (dialectally, a proximal singular and an additional medial grade may also be present). The Basque distal form (with infix -a-, etymologically a suffixed and phonetically reduced form of the distal demonstrative har-/hai-) functions as the default definite article, whereas the proximal form (with infix -o-, derived from the proximal demonstrative hau-/hon-) is marked and indicates some kind of (spatial or otherwise) close relationship between the speaker and the referent (e.g., it may imply that the speaker is included in the referent): etxeak ("the houses") vs. etxeok ("these houses [of ours]"), euskaldunak ("the Basque speakers") vs. euskaldunok ("we, the Basque speakers"). Speakers of Assyrian Neo-Aramaic, a modern Aramaic language that lacks a definite article, may at times use demonstratives aha and aya (feminine) or awa (masculine) – which translate to "this" and "that", respectively – to give the sense of "the".[10] In Indonesian, the third person possessive suffix -nya could be also used as a definite article.[11] Indefinite articles edit Indefinite articles typically arise from adjectives meaning one. For example, the indefinite articles in the Romance languages—e.g., un, una, une—derive from the Latin adjective unus. Partitive articles, however, derive from Vulgar Latin de illo, meaning (some) of the. The English indefinite article an is derived from the same root as one. The -n came to be dropped before consonants, giving rise to the shortened form a. The existence of both forms has led to many cases of juncture loss, for example transforming the original a napron into the modern an apron. The Persian indefinite article is yek, meaning one. See also edit English articles Al- (definite article in Arabic) Definiteness Definite description False title References edit Recasens, Marta; Martí, M. Antònia; Taulé, Mariona (2009-06-16), Winkler, Susanne; Featherston, Sam (eds.), "First-mention definites:More than exceptional cases", The Fruits of Empirical Linguistics II, vol. 102, Berlin, New York: Mouton de Gruyter, pp. 217–238, doi:10.1515/9783110216158.217, ISBN 978-3-11-021347-8, retrieved 2023-01-16 New perspectives on Hispanic contact : linguistics in the Americas. Melvin González-Rivera, and Sandro Sessarego. Madrid: Iberoamericana. 2015. ISBN 978-3-95487-831-4. OCLC 969386958. Burchfield, R. W. (1996). The New Fowler's Modern English Usage (3rd ed.). Oxford University Press. p. 512. ISBN 978-0199690367. Argetsinger, Amy (1 September 2015). "Why does everyone call Donald Trump 'The Donald'? It's an interesting story". The Washington Post. Retrieved 3 October 2017. Lawrence, Erma (1977). Haida dictionary. Fairbanks: Alaska Native Language Center. p. 64. Master, Peter (1997). "The English article system: Acquisition, function, and pedagogy". System. 25 (2): 215–232. doi:10.1016/S0346-251X(97)00010-9. Kusmenko, J K. "The typology of the language contact on the Balkans and in Scandinavia. A case of the suffixed definite article" (PDF). Archived (PDF) from the original on 2021-10-04. See s.5 Summary. Retrieved 2 February 2012. Simona, Ropati (1986). Tokelau Dictionary. New Zealand: Office of Tokelau Affairs. p. Introduction. Greenberg, Joseph H. (2005). Genetic linguistics : essays on theory and method. William Croft. Oxford: Oxford University Press. ISBN 978-0-19-151452-4. OCLC 132691297. Solomon, Zomaya S. (1997). Functional and other exotic sentences in Assyrian Aramaic, Journal of Assyrian Academic Studies, XI/2:44-69. "20 Contoh Penggunaan Kata nya dalam Kalimat" [Twenty Examples of the Use of 'nya' in Sentences]. kumparan.com (in Indonesian). Retrieved 2024-05-26. External links edit Look up article, definite article, or indefinite article in Wiktionary, the free dictionary. Wikisource has the text of the 1921 Collier's Encyclopedia article Article. "The Definite Article, 'The': The Most Frequently Used Word in World's Englishes" Ing, John (September 17, 2019). "How to Use Articles Correctly in English (A, An & The)". Retrieved May 3, 2020. Last edited 20 days ago by Kaloan-koko Related articles French articles and determiners Definiteness Semantic feature of noun phrases in linguistics English determiners Determiners in the English language Wikipedia Wikimedia Foundation Powered by MediaWiki Content is available under CC BY-SA 4.0 unless otherwise noted. Privacy policy Contact Wikipedia Code of Conduct Developers Statistics Cookie statement Terms of Use DesktopMaths Euclidean Alogrithms From geeksforgeeks. The Eucliean alogrithm is a way of find the greatest common divisor of two positive integers. GCD of two numbers is the largest number that divides both of them. A simple way to find GCD is to factorize both numbers and multiply common prime factors. 36 = 2 * 2 * 3 * 3 60 = 2 * 2 * 3 * 5 GCD = Multiplication of common factors = 2 * 2 * 3 = 12 Basic Euclidean Algorithm for GCD: The alogirithm is based on the below facts: - If we subtract a smaller number from a larger one ( we reduce a large number ), GCD does not change. So, if we keep subtracting repeatedly the larger of two, we end up with GCD. - Now instead of subtraction, if we divide the larger number, the algorithm stops when we find the reminder 0. # frozen_string_literal: true def gcd(first, second) if first.zero? return second end gcd(second % first, first) end first = 36 second = 60 puts "GCD of #{first} and #{second} is #{gcd(first, second)}" CommentsBlog Open user menuuser photo Featured Ruby Maths Fourier theorem A Fourier series is an expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series. By expressing a function as a sum of sines and cosines, may problems involving the function become easier to analyze because trigonometric functions are well understood. For example, Fourier series were first used by Josheph Fourier(1786-1830) to find solutions to the heat equation. This application is possible because the derivatives of trigonometric functions fall into simple patterns. Fourier series cannot be used to approximate arbitrary functions, because most functions fall into simple patterns. Fourier series cannot be used to approximate arbitrary functions, because most functions have infinitely many terms in their Fourier series, and the series do not always converge. Well-behaved functions, for example smooth functions, have Fourier series that converge to the original function. The coefficients of the Fourier series are determined by integrals of the function multiplied by trigonometric functions, described in Fourier series Definition. The study of the convergence of Fourier series focus on the behaviors of the partial sums, which means studying the behavior of the sum as more and more terms from the series are summed. Fourier series are closely related to the Fourier transform. a more general tool that can even find the frequency information for functions that are not periodic. Periodic functions are identified with functions on a circle; for this reason Fourier series are the subject of Fourier analysis on the circle group, denoted by T or S1. The Fourier transform is also part of Fourier analysis, but is defined for functions on Rn. Since Fourier's time, many different approaches to defining and understanding the concept of Fourier series have been discovered, all of which are consistent with one another, but each of which emphasizes different aspects of the topic. Some of the more powerful and elegant approaches are based on the mathematical ideas and tools that were not available in Fourier's time. Fourier originally defined the Fourier series for real-valued functions of real arguments, and used the sine and cosine functions in the decomposition. May other Fourier-related transforms have since been defined, extending his initial idea to many applications and birthing an area of mathematics called Fourier analysis. Fourier transform The Fourier Transform is an integral transform that makes a function as input and outputs another function that describes the extent to which various frequencies are present in the original function. The output of the transform is a complex-valued function of frequency. The output of the transform is a complex valued function of frequency. The term Fourier transform refers to both this complex-valued function and the mathematical operation. When a distinction needs to be made, the output of the operation is sometimes called the frequency domain representation of the original function. The Fourier transform is analogous to decomposing the sound of a musical chord into the intensities of its constituent pitches. Functions that are localized in the time domain have Fourier transforms that are spread out across the frequency domain and vice versa,a phenomenon known as the uncertainty principle. The critical case for this principle is the Gaussian function, of substantial importance in probability theory and statistics as well as in the study of physical phenomena exhibiting normal distribution(example: diffusion). The Fourier transform of a Gaussian function is another Gaussian function. Joseph Fourier introduced the sine and cosine transforms ( which correspond to the imaginary and real components of the modern Fourier transform) in his study of heat transfer, where Gaussian functions appear as solutions of the heat equation. CommentsMaths Gaussian function Gaussian is a function of the base form with f(x) = exp(-x^2) and with parametric extension f(x) = aexp(-(x-b)^2)/2c^2) for arbitrary real constants a, b and non-zero c. It is named after the mathematician Carl Friedrich Gauss. The graph of a Gaussian is a characteristic symmetric "bell curve" shape. The parameter is the height of the curve's peak, b is the position of the center of the peak, and c (the standard deviation, sometimes called the Gaussian RMS width) controls the width of the "bell". Gaussian functions are widely used in statistics to describe the normal distributions, in signal processing to define the Gaussian filters, in image processing where two-dimensional Gaussians are used for Gaussian blurs, and in mathematics to solve heat equations and diffusion equations and to define the Weierstrass transform. They are also abundantly used in quantum chemistry to form basic sets. The most common method for estimating the Gaussian parameters is to take the logarithm of data and fit a parabola to the resulting data set. While this provides a simple curve fitting procedure, the resulting algorithm may be biased by excessively weighting small data values, which can produce large errors in the profile estimate. One can partially compensate for this problem through weighted least squares estimation, reducing the weight of small data values, but this too can be biased by allowing the tail of the Gaussian to dominate the fit. In order to remove the bias, one can instead use an iteratively reweighted least squares procedure, in which the weights are updated at each iteration. It is also possible to perform non-linear regression directly on the data without involving the logarithmic data transformation. Parameter Precision Once one has an algorithm for estimating the Gaussian parameters, it is also important to know how precise those estimates are. Any least squares estimation algorithm can provide numerical estimates for the variance of each parameter ( i.e. the variance of estimated height, position, and width of the function). One can also use Cramer-Rao bound theory to obtain an analytical expression for the lower bound on the parameter variances, given certain assumptions about the data. 1. The noise in the measured profile is either independent and identically distributed Gaussian, or the noise is Poissin-distributed. 2. The spacing between each sampling ( i.e the distance between pixels measuring the data) is uniform. 3. The peak is "well-sampled", so that less than 10% of the area or volume under the peak(area if a 1D Gaussian, volume if a 2D Gaussian) lies outside the measurement region. 4. The width of the peak is much larger than the distance between sample locations( i.e. the detector pixels must be al least 5 time smaller than the Gaussian FWHM). Applications Gaussian functions appear in many context in natural sciences, the social sciences, mathematics and engineering. Some examples include: 1. In statistics and probability theory, Gaussian functions appear as the density function of the normal distribution, which is a limiting probability distribution of complicated sums, according to central limit theorem. 2. A Gaussian function is the wave function of the ground state of the quantum harmonic oscillator. 3. The molecular orbitals used in computational chemistry can be linear combinations of Gaussian functions called Gaussian orbitals. 3. Gaussian functions are also associated with the vacuum state in quantum field theory. 4. Gaussian beams are used in optical systems, microwave systems and lasers. 5. In scale space representation, Gaussian functions are used as smoothing kernels for generating multi-scale representations in computer vision and image processing. Specifically, derivatives of Gaussians ( Hermite functions ) are used as a basis for defining a large number of types of visual operations. 6. Gaussian functions are used to define some types of artificial neural networks. 7. In fluorescence microscopy a 2D Gaussian function is used to approximate the Airy disk, describing the intensity distribution produced by a point source. 8. In signal processing they serve to define Gaussian filter, such as in image processing where 2D Gaussian kernel, which may be approximated by the Binomial coefficient or sampling a Gaussina. 9. In geostatistics, they have been used for understanding the variability between the patterns of a complex training image. They are used with kernel methods to cluster the patterns in the feature space. [email protected]# frozen_string_literal: true def gcd(first, second) if first.zero? return second end gcd(second % first, first) end first = 36 second = 60 puts "GCD of #{first} and #{second} is #{gcd(first, second)}"Ruby Maths Fourier theorem A Fourier series is an expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series. By expressing a function as a sum of sines and cosines, may problems involving the function become easier to analyze because trigonometric functions are well understood. For example, Fourier series were first used by Josheph Fourier(1786-1830) to find solutions to the heat equation. This application is possible because the derivatives of trigonometric functions fall into simple patterns. Fourier series cannot be used to approximate arbitrary functions, because most functions fall into simple patterns. Fourier series cannot be used to approximate arbitrary functions, because most functions have infinitely many terms in their Fourier series, and the series do not always converge. Well-behaved functions, for example smooth functions, have Fourier series that converge to the original function. The coefficients of the Fourier series are determined by integrals of the function multiplied by trigonometric functions, described in Fourier series Definition. The study of the convergence of Fourier series focus on the behaviors of the partial sums, which means studying the behavior of the sum as more and more terms from the series are summed. Fourier series are closely related to the Fourier transform. a more general tool that can even find the frequency information for functions that are not periodic. Periodic functions are identified with functions on a circle; for this reason Fourier series are the subject of Fourier analysis on the circle group, denoted by T or S1. The Fourier transform is also part of Fourier analysis, but is defined for functions on Rn. Since Fourier's time, many different approaches to defining and understanding the concept of Fourier series have been discovered, all of which are consistent with one another, but each of which emphasizes different aspects of the topic. Some of the more powerful and elegant approaches are based on the mathematical ideas and tools that were not available in Fourier's time. Fourier originally defined the Fourier series for real-valued functions of real arguments, and used the sine and cosine functions in the decomposition. May other Fourier-related transforms have since been defined, extending his initial idea to many applications and birthing an area of mathematics called Fourier analysis. Fourier transform The Fourier Transform is an integral transform that makes a function as input and outputs another function that describes the extent to which various frequencies are present in the original function. The output of the transform is a complex-valued function of frequency. The output of the transform is a complex valued function of frequency. The term Fourier transform refers to both this complex-valued function and the mathematical operation. When a distinction needs to be made, the output of the operation is sometimes called the frequency domain representation of the original function. The Fourier transform is analogous to decomposing the sound of a musical chord into the intensities of its constituent pitches. Functions that are localized in the time domain have Fourier transforms that are spread out across the frequency domain and vice versa,a phenomenon known as the uncertainty principle. The critical case for this principle is the Gaussian function, of substantial importance in probability theory and statistics as well as in the study of physical phenomena exhibiting normal distribution(example: diffusion). The Fourier transform of a Gaussian function is another Gaussian function. Joseph Fourier introduced the sine and cosine transforms ( which correspond to the imaginary and real components of the modern Fourier transform) in his study of heat transfer, where Gaussian functions appear as solutions of the heat equation. CommentsFourier theorem A Fourier series is an expansion of a periodic function into a sum of trigonometric functions. Read more Gaussian function Gaussian is a function of the base form with f(x) = exp(-x^2) and with parametric extension f(x) = aexp(-(x-b)^2)/2c^2) for arbitrary real constants a, b and non-zero c. Read more © 2025 bikashpandey.com. All Rights Reserved. Privacy Policy [email protected]://youtube.com/watch?v=GUS_MnJQ5Uo&lc=UgxkvBdpKjTYrynRE2V4AaABAg&si=m2nw-jfZ1pQMq0EgPls admin monkey bou 🐒 👉 bkp by Om Luitel 10 minutes ago by Janak Shrestha 11 days ago