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Mashreghi–Ransford Inequality
In Mathematics, the Mashreghi–Ransford inequality is a bound on the growth rate of certain sequences. It is named after J. Mashreghi and T. Ransford. Let (a_n)_ be a sequence of complex numbers, and let : b_n = \sum_^n a_k, \qquad (n \geq 0), and : c_n = \sum_^n (-1)^ a_k, \qquad (n \geq 0). Here the binomial coefficients are defined by : = \frac. Assume that, for some \beta>1, we have b_n = O(\beta^n) and c_n = O(\beta^n) as n \to \infty. Then Mashreghi-Ransford showed that : a_n = O(\alpha^n), as n \to \infty, where \alpha=\sqrt. Moreover, there is a universal constant A physical constant, sometimes fundamental physical constant or universal constant, is a physical quantity that cannot be explained by a theory and therefore must be measured experimentally. It is distinct from a mathematical constant, which has a ... \kappa such that : \left( \limsup_ \frac \right) \leq \kappa \, \left( \limsup_ \frac \right)^ \left( \limsup_ \frac \right)^. The pre ... [...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] |
Sequences
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called the ''length'' of the sequence. Unlike a set, the same elements can appear multiple times at different positions in a sequence, and unlike a set, the order does matter. Formally, a sequence can be defined as a function from natural numbers (the positions of elements in the sequence) to the elements at each position. The notion of a sequence can be generalized to an indexed family, defined as a function from an ''arbitrary'' index set. For example, (M, A, R, Y) is a sequence of letters with the letter "M" first and "Y" last. This sequence differs from (A, R, M, Y). Also, the sequence (1, 1, 2, 3, 5, 8), which contains the number 1 at two different positions, is a valid sequence. Sequences can be '' finite'', as in these examples, or '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Javad Mashreghi
Javad Mashreghi is a mathematician and author working in function space theory, functional analysis and complex analysis. He is a professeur titulaire at Université Laval and was the 35th President of the Canadian Mathematical Society (2020–2022). Early life and education Mashreghi was born in Kashan, Iran, in 1968. He studied electrical engineering, electronics, (B.Sc., 1991), and pure mathematics (M.Sc., 1993) at the University of Tehran. He moved to Canada in 1996 and earned his Ph.D. from the McGill University in 2001. Since then he lives in the Province of Quebec. Service to Canadian mathematical community Mashreghi is immensely involved in various aspects of North America's mathematical community, having served on numerous editorial, administrative and selection committees all across Canada and the U.S. (CMS, AMS, Fields Institute, CRM, AARMS, NSERC, FQRNT, NSF). He was the 35th President of the Canadian Mathematical Society (2020–2022), and is currently the editor-in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Thomas Ransford
Thomas Ransford (born 1958) is a British-born Canadian mathematician, known for his research in spectral theory and complex analysis. He holds a Canada Research Chair in mathematics at Université Laval. Ransford earned his PhD from the University of Cambridge in 1984. Career He was a fellow of Trinity College, University of Cambridge, from 1983 to 1987. In addition to over 90 research papers on mathematics, he has written a research monograph "Potential Theory in the Complex Plane" in 1995, and the graduate book "A Primer on the Dirichlet Space" with Omar El-Fallah, Karim Kellay and Javad Mashreghi in 201 He has proved results on potential theory, functional analysis, the theory of capacity, and probability. For example, with Javad Mashreghi he proved the Mashreghi–Ransford inequality. He also derived a short elementary proof of Stone–Weierstrass theorem In mathematical analysis, the Weierstrass approximation theorem states that every continuous function defined on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a + bi, where and are real numbers. Because no real number satisfies the above equation, was called an imaginary number by René Descartes. For the complex number is called the , and is called the . The set of complex numbers is denoted by either of the symbols \mathbb C or . Despite the historical nomenclature, "imaginary" complex numbers have a mathematical existence as firm as that of the real numbers, and they are fundamental tools in the scientific description of the natural world. Complex numbers allow solutions to all polynomial equations, even those that have no solutions in real numbers. More precisely, the fundamental theorem of algebra asserts that every non-constant polynomial equation with real or complex coefficie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binomial Coefficients
In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the term in the polynomial expansion of the binomial power ; this coefficient can be computed by the multiplicative formula : \binom nk = \frac, which using factorial notation can be compactly expressed as : \binom = \frac. For example, the fourth power of is : \begin (1 + x)^4 &= \tbinom x^0 + \tbinom x^1 + \tbinom x^2 + \tbinom x^3 + \tbinom x^4 \\ &= 1 + 4x + 6 x^2 + 4x^3 + x^4, \end and the binomial coefficient \tbinom =\tfrac = \tfrac = 6 is the coefficient of the term. Arranging the numbers \tbinom, \tbinom, \ldots, \tbinom in successive rows for gives a triangular array called Pascal's triangle, satisfying the recurrence relation : \binom = \binom + \binom . The binomial coefficients occur in many areas of mathematics, and especia ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Universal Constant
A physical constant, sometimes fundamental physical constant or universal constant, is a physical quantity that cannot be explained by a theory and therefore must be measured experimentally. It is distinct from a mathematical constant, which has a fixed numerical value, but does not directly involve any physical measurement. There are many physical constants in science, some of the most widely recognized being the speed of light in vacuum ''c'', the gravitational constant ''G'', the Planck constant ''h'', the electric constant ''ε''0, and the elementary charge ''e''. Physical constants can take many dimensional forms: the speed of light signifies a maximum speed for any object and its dimension is length divided by time; while the proton-to-electron mass ratio is dimensionless. The term "fundamental physical constant" is sometimes used to refer to universal-but-dimensioned physical constants such as those mentioned above. Increasingly, however, physicists reserve the expression ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
