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Cas (mathematics)
In mathematics, the Hartley transform (HT) is an integral transform closely related to the Fourier transform (FT), but which transforms real-valued functions to real-valued functions. It was proposed as an alternative to the Fourier transform by Ralph V. L. Hartley in 1942, and is one of many known Fourier-related transforms. Compared to the Fourier transform, the Hartley transform has the advantages of transforming real number, real functions to real functions (as opposed to requiring complex numbers) and of being its own inverse. The discrete version of the transform, the discrete Hartley transform (DHT), was introduced by Ronald N. Bracewell in 1983. The two-dimensional Hartley transform can be computed by an analog optical process similar to an optical Fourier transform (OFT), with the proposed advantage that only its amplitude and sign need to be determined rather than its complex phase. However, optical Hartley transforms do not seem to have seen widespread use. Definition T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symmetric Matrix
In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally, Because equal matrices have equal dimensions, only square matrices can be symmetric. The entries of a symmetric matrix are symmetric with respect to the main diagonal. So if a_ denotes the entry in the ith row and jth column then for all indices i and j. Every square diagonal matrix is symmetric, since all off-diagonal elements are zero. Similarly in characteristic different from 2, each diagonal element of a skew-symmetric matrix must be zero, since each is its own negative. In linear algebra, a real symmetric matrix represents a self-adjoint operator represented in an orthonormal basis over a real inner product space. The corresponding object for a complex inner product space is a Hermitian matrix with complex-valued entries, which is equal to its conjugate transpose. Therefore, in linear algebra over the complex numbers, it is often assumed that a symmetric ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Oxford University Press, Inc
Oxford () is a cathedral city and non-metropolitan district in Oxfordshire, England, of which it is the county town. The city is home to the University of Oxford, the oldest university in the English-speaking world; it has buildings in every style of English architecture since late Anglo-Saxon. Oxford's industries include motor manufacturing, education, publishing, science, and information technologies. Founded in the 8th century, it was granted city status in 1542. The city is located at the confluence of the rivers Thames (locally known as the Isis) and Cherwell. It had a population of in . It is north-west of London, south-east of Birmingham and north-east of Bristol. History The history of Oxford in England dates back to its original settlement in the Saxon period. The name “Oxford” comes from the Old English ''Oxenaforda'', meaning “ford of the oxen,” referring to a shallow crossing in the river where oxen could pass. The town was of strategic signific ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
McGraw-Hill
McGraw Hill is an American education science company that provides educational content, software, and services for students and educators across various levels—from K-12 to higher education and professional settings. They produce textbooks, digital learning tools, and adaptive technology to enhance learning experiences and outcomes. It is one of the "big three" educational publishers along with Houghton Mifflin Harcourt and Pearson Education. McGraw Hill also publishes reference and trade publications for the medical, business, and engineering professions. Formerly a division of The McGraw Hill Companies (later renamed McGraw Hill Financial, now S&P Global), McGraw Hill Education was divested and acquired by Apollo Global Management in March 2013 for $2.4 billion in cash. McGraw Hill was sold in 2021 to Platinum Equity for $4.5 billion. History McGraw Hill was founded in 1888, when James H. McGraw, co-founder of McGraw Hill, purchased the ''American Journal of Railway ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proceedings Of The IEEE
The ''Proceedings of the IEEE'' is a monthly peer-reviewed scientific journal published by the Institute of Electrical and Electronics Engineers (IEEE). The journal focuses on electrical engineering and computer science. According to the ''Journal Citation Reports'', the journal has a 2017 impact factor of 9.107, ranking it sixth in the category "Engineering, Electrical & Electronic." In 2018, it became fifth with an enhanced impact factor of 10.694. History of the ''Proceedings'' The journal was established in 1909, known as the ''Proceedings of the Wireless Institute''. Six issues were published under this banner by Greenleaf Pickard and Alfred Goldsmith. Then in 1911, a merger between the Wireless Institute (New York) and the Society of Wireless Telegraph Engineers (Boston) resulted in a society named the Institute of Radio Engineers (IRE). In January 1913 newly formed IRE published the first issue of the ''Proceedings of the IRE''. Later, a 1000-page special issue commemor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of The Optical Society Of America
The ''Journal of the Optical Society of America'' is a peer-reviewed scientific journal of optics, published by Optica. It was established in 1917 and in 1984 was split into two parts, A and B. ''Journal of the Optical Society of America A'' Part A covers various topics in optics, vision, and image science. The editor-in-chief is Olga Korotkova (University of Miami, USA). ''Journal of the Optical Society of America B'' Part B covers various topics in the field of optical physics, such as guided waves, laser spectroscopy, nonlinear optics, quantum optics, laser A laser is a device that emits light through a process of optical amplification based on the stimulated emission of electromagnetic radiation. The word ''laser'' originated as an acronym for light amplification by stimulated emission of radi ...s, organic and polymer materials for optics, and ultrafast phenomena. The editor-in-chief is Kurt Busch ( Humboldt University of Berlin, Germany). References ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proceedings Of The IRE
The ''Proceedings of the IEEE'' is a monthly peer-reviewed scientific journal published by the Institute of Electrical and Electronics Engineers (IEEE). The journal focuses on electrical engineering and computer science. According to the ''Journal Citation Reports'', the journal has a 2017 impact factor of 9.107, ranking it sixth in the category "Engineering, Electrical & Electronic." In 2018, it became fifth with an enhanced impact factor of 10.694. History of the ''Proceedings'' The journal was established in 1909, known as the ''Proceedings of the Wireless Institute''. Six issues were published under this banner by Greenleaf Pickard and Alfred Goldsmith. Then in 1911, a merger between the Wireless Institute (New York) and the Society of Wireless Telegraph Engineers (Boston) resulted in a society named the Institute of Radio Engineers (IRE). In January 1913 newly formed IRE published the first issue of the ''Proceedings of the IRE''. Later, a 1000-page special issue commemor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fractional Fourier Transform
In mathematics, in the area of harmonic analysis, the fractional Fourier transform (FRFT) is a family of linear transformations generalizing the Fourier transform. It can be thought of as the Fourier transform to the ''n''-th power, where ''n'' need not be an integer — thus, it can transform a function to any ''intermediate'' domain between time and frequency. Its applications range from filter design and signal analysis to phase retrieval and pattern recognition. The FRFT can be used to define fractional convolution, correlation, and other operations, and can also be further generalized into the linear canonical transformation (LCT). An early definition of the FRFT was introduced by Condon, by solving for the Green's function for phase-space rotations, and also by Namias, generalizing work of Wiener on Hermite polynomials. However, it was not widely recognized in signal processing until it was independently reintroduced around 1993 by several groups. Since th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cis (mathematics)
is a mathematical notation defined by , where is the cosine function, is the imaginary unit and is the sine function. is the argument of the complex number (angle between line to point and x-axis in polar form). The notation is less commonly used in mathematics than Euler's formula, which offers an even shorter notation for but cis(x) is widely used as a name for this function in software libraries. Overview The notation is a shorthand for the combination of functions on the right-hand side of Euler's formula: : e^ = \cos x + i\sin x, where . So, : \operatorname x = \cos x + i\sin x, i.e. "" is an acronym for "". It connects trigonometric functions with exponential functions in the complex plane via Euler's formula. While the domain of definition is usually x \in \mathbb, complex values z \in \mathbb are possible as well: : \operatorname z = \cos z + i\sin z, so the function can be used to extend Euler's formula to a more general complex version. The function ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trigonometry
Trigonometry () is a branch of mathematics concerned with relationships between angles and side lengths of triangles. In particular, the trigonometric functions relate the angles of a right triangle with ratios of its side lengths. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. The Greeks focused on the calculation of chords, while mathematicians in India created the earliest-known tables of values for trigonometric ratios (also called trigonometric functions) such as sine. Throughout history, trigonometry has been applied in areas such as geodesy, surveying, celestial mechanics, and navigation. Trigonometry is known for its many identities. These trigonometric identities are commonly used for rewriting trigonometrical expressions with the aim to simplify an expression, to find a more useful form of an expression, or to solve an equation. History Sumerian astronomers studied angle me ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convolution
In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions f and g that produces a third function f*g, as the integral of the product of the two functions after one is reflected about the y-axis and shifted. The term ''convolution'' refers to both the resulting function and to the process of computing it. The integral is evaluated for all values of shift, producing the convolution function. The choice of which function is reflected and shifted before the integral does not change the integral result (see #Properties, commutativity). Graphically, it expresses how the 'shape' of one function is modified by the other. Some features of convolution are similar to cross-correlation: for real-valued functions, of a continuous or discrete variable, convolution f*g differs from cross-correlation f \star g only in that either f(x) or g(x) is reflected about the y-axis in convolution; thus i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orthogonal Matrix
In linear algebra, an orthogonal matrix, or orthonormal matrix, is a real square matrix whose columns and rows are orthonormal vectors. One way to express this is Q^\mathrm Q = Q Q^\mathrm = I, where is the transpose of and is the identity matrix. This leads to the equivalent characterization: a matrix is orthogonal if its transpose is equal to its inverse: Q^\mathrm=Q^, where is the inverse of . An orthogonal matrix is necessarily invertible (with inverse ), unitary (), where is the Hermitian adjoint ( conjugate transpose) of , and therefore normal () over the real numbers. The determinant of any orthogonal matrix is either +1 or −1. As a linear transformation, an orthogonal matrix preserves the inner product of vectors, and therefore acts as an isometry of Euclidean space, such as a rotation, reflection or rotoreflection. In other words, it is a unitary transformation. The set of orthogonal matrices, under multiplication, forms the group , known as th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
