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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.

**Mathematics** includes the study of such topics as quantity, structure, space, and change.

**Thomas Ransford** Ph.D. Sc.D 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.

Let be a sequence of complex numbers, and let

A **complex number** is a number that can be expressed in the form *a* + *bi*, where *a* and *b* are real numbers, and *i* is a solution of the equation *x*^{2} = −1. Because no real number satisfies this equation, *i* is called an imaginary number. For the complex number *a* + *bi*, *a* is called the **real part**, and *b* is called the **imaginary part**. Despite the historical nomenclature "imaginary", complex numbers are regarded in the mathematical sciences as just as "real" as the real numbers, and are fundamental in many aspects of the scientific description of the natural world.

and

We remind that the binomial coefficients are defined by

Assume that, for some , we have and as . Then

- , as ,

where

Moreover, there is a universal constant such that

The precise value of is unknown. However, it is known that

In mathematics, the **limit inferior** and **limit superior** of a sequence can be thought of as limiting bounds on the sequence. They can be thought of in a similar fashion for a function. For a set, they are the infimum and supremum of the set's limit points, respectively. In general, when there are multiple objects around which a sequence, function, or set accumulates, the inferior and superior limits extract the smallest and largest of them; the type of object and the measure of size is context-dependent, but the notion of extreme limits is invariant. Limit inferior is also called **infimum limit**, **limit infimum**, **liminf**, **inferior limit**, **lower limit**, or **inner limit**; limit superior is also known as **supremum limit**, **limit supremum**, **limsup**, **superior limit**, **upper limit**, or **outer limit**.

In calculus, **Taylor's theorem** gives an approximation of a *k*-times differentiable function around a given point by a *k*-th order **Taylor polynomial**. For analytic functions the Taylor polynomials at a given point are finite-order truncations of its Taylor series, which completely determines the function in some neighborhood of the point. It can be thought of as the extension of linear approximation to higher order polynomials, and in the case of *k* equals 2 is often referred to as a *quadratic approximation*. The exact content of "Taylor's theorem" is not universally agreed upon. Indeed, there are several versions of it applicable in different situations, and some of them contain explicit estimates on the approximation error of the function by its Taylor polynomial.

In probability theory and statistics, the **beta distribution** is a family of continuous probability distributions defined on the interval [0, 1] parametrized by two positive shape parameters, denoted by *α* and *β*, that appear as exponents of the random variable and control the shape of the distribution. It is a special case of the Dirichlet distribution.

In mathematics, the **limit** of a sequence of sets *A*_{1}, *A*_{2}, ... is a set whose elements are determined by the sequence in either of two equivalent ways: **(1)** by upper and lower bounds on the sequence that converge monotonically to the same set and **(2)** by convergence of a sequence of indicator functions which are themselves real-valued. As is the case with sequences of other objects, convergence is not necessary or even usual.

In calculus and other branches of mathematical analysis, limits involving an algebraic combination of functions in an independent variable may often be evaluated by replacing these functions by their limits; if the expression obtained after this substitution does not give enough information to determine the original limit, it is said to take on an **indeterminate form**. The term was originally introduced by Cauchy's student Moigno in the middle of the 19th century.

In probability theory, the **law of the iterated logarithm** describes the magnitude of the fluctuations of a random walk. The original statement of the law of the iterated logarithm is due to A. Y. Khinchin (1924). Another statement was given by A. N. Kolmogorov in 1929.

**Multi-index notation** is a mathematical notation that simplifies formulas used in multivariable calculus, partial differential equations and the theory of distributions, by generalising the concept of an integer index to an ordered tuple of indices.

In mathematics, the **binomial series** is the Maclaurin series for the function given by , where is an arbitrary complex number. Explicitly,

In mathematics, a **matrix norm** is a vector norm in a vector space whose elements (vectors) are matrices.

In mathematics, **Grönwall's inequality** allows one to bound a function that is known to satisfy a certain differential or integral inequality by the solution of the corresponding differential or integral equation. There are two forms of the lemma, a differential form and an integral form. For the latter there are several variants.

In mathematics, a **compact quantum group** is an abstract structure on a unital separable C*-algebra axiomatized from those that exist on the commutative C*-algebra of "continuous complex-valued functions" on a compact quantum group.

In mathematics, a real or complex-valued function *f* on *d*-dimensional Euclidean space satisfies a **Hölder condition**, or is **Hölder continuous**, when there are nonnegative real constants *C*, α>0, such that

In mathematics, in the area of complex analysis, **Nachbin's theorem** is commonly used to establish a bound on the growth rates for an analytic function. This article will provide a brief review of growth rates, including the idea of a **function of exponential type**. Classification of growth rates based on type help provide a finer tool than big O or Landau notation, since a number of theorems about the analytic structure of the bounded function and its integral transforms can be stated. In particular, Nachbin's theorem may be used to give the domain of convergence of the **generalized Borel transform**, given below.

In mathematics, **convergence tests** are methods of testing for the convergence, conditional convergence, absolute convergence, interval of convergence or divergence of an infinite series .

In mathematics, the **hypergeometric function of a matrix argument** is a generalization of the classical hypergeometric series. It is a function defined by an infinite summation which can be used to evaluate certain multivariate integrals.

In mathematics, the **Jack function** is a generalization of the **Jack polynomial**, introduced by Henry Jack. The Jack polynomial is a homogeneous, symmetric polynomial which generalizes the Schur and zonal polynomials, and is in turn generalized by the Heckman–Opdam polynomials and Macdonald polynomials.

In mathematics, the **Cauchy–Hadamard theorem** is a result in complex analysis named after the French mathematicians Augustin Louis Cauchy and Jacques Hadamard, describing the radius of convergence of a power series. It was published in 1821 by Cauchy, but remained relatively unknown until Hadamard rediscovered it. Hadamard's first publication of this result was in 1888; he also included it as part of his 1892 Ph.D. thesis.

In the following we solve the second-order differential equation called the hypergeometric differential equation using Frobenius method, named after Ferdinand Georg Frobenius. This is a method that uses the series solution for a differential equation, where we assume the solution takes the form of a series. This is usually the method we use for complicated ordinary differential equations.

In queueing theory, a discipline within the mathematical theory of probability, a **heavy traffic approximation** is the matching of a queueing model with a diffusion process under some limiting conditions on the model's parameters. The first such result was published by John Kingman who showed that when the utilisation parameter of an M/M/1 queue is near 1 a scaled version of the queue length process can be accurately approximated by a reflected Brownian motion.

In mathematics, the **mean** (topological) **dimension** of a topological dynamical system is a non-negative extended real number that is a measure of the complexity of the system. Mean dimension was first introduced in 1999 by Gromov. Shortly after it was developed and studied systematically by Lindenstrauss and Weiss. In particular they proved the following key fact: a system with finite topological entropy has zero mean dimension. For various topological dynamical systems with infinite topological entropy, the mean dimension can be calculated or at least bounded from below and above. This allows mean dimension to be used to distinguish between systems with infinite topological entropy.

- Mashreghi, J.; Ransford, T. (2005). "Binomial sums and functions of exponential type".
*Bull. London Math. Soc*.**37**(01): 15–24..

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