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Convergence tests are mathematical techniques used to determine whether a series or sequence converges (approaches a finite limit) or diverges (grows indefinitely or does not settle at any finite value). These tests are particularly important in the study of infinite series in calculus and analysis, as they help evaluate the behavior of sums of infinitely many terms.

Fixed-point theorems are fundamental results in mathematics that establish conditions under which a function will have a point that maps to itself. In simpler terms, if you have a function \( f \) defined on a certain space, a fixed point \( x \) satisfies the equation \( f(x) = x \). Fixed-point theorems are widely applicable in various areas such as analysis, topology, and applied mathematics.

In mathematical analysis and other fields of mathematics, a "lemma" is a preliminary proposition or statement that is proven to aid in the proof of a larger theorem. The term "lemma" comes from the Greek word "lemma," which means "that which is received" or "that which is taken." In effect, results that are designated as lemmas are often foundational results that help establish more complex results.

Analytic number theory is a branch of number theory that uses techniques from mathematical analysis to solve problems about integers and prime numbers. Several important theorems form the foundation of this field. Here are some of the prominent theorems and concepts within analytic number theory: 1. **Prime Number Theorem**: This fundamental theorem describes the asymptotic distribution of prime numbers.

In approximation theory, several theorems provide fundamental insights into how functions can be approximated by simpler functions, such as polynomials, trigonometric series, or other basis functions. Here are some key theorems and concepts in approximation theory: 1. **Weierstrass Approximation Theorem**: This theorem states that any continuous function defined on a closed interval can be uniformly approximated as closely as desired by a polynomial function.

The ATS theorem, also known as the Aharonov–Bohm theorem, is a fundamental result in quantum mechanics that illustrates the importance of electromagnetic potentials in the behavior of charged particles, even in regions where the electric and magnetic fields are zero.

Ahlfors' finiteness theorem is a result in complex analysis and several complex variables, particularly in the study of Riemann surfaces and compact complex manifolds. The theorem is named after Lars Ahlfors, a prominent mathematician known for his contributions to complex analysis. The theorem essentially states that for a compact Riemann surface (or a compact complex manifold), the number of non-constant meromorphic functions (or rational functions) that can be defined on it is finite.

The Ahlfors measure conjecture is a conjecture in the field of complex analysis and geometric function theory, specifically relating to quasiconformal mappings and the properties of certain topological spaces. Named after the mathematician Lars Ahlfors, this conjecture deals with the existence of a specific type of measure associated with quasiconformal mappings.

The Atkinson-Mingarelli theorem is a result in the field of differential equations, particularly in the context of boundary value problems for second-order ordinary differential equations (ODEs). It deals with the existence of multiple solutions to certain types of boundary value problems. The theorem essentially states conditions under which a second-order linear differential equation can have multiple solutions based on its boundary conditions and the nature of the functions involved.

The Babuška–Lax–Milgram theorem is a result in functional analysis and the theory of partial differential equations (PDEs), particularly concerning the solvability of boundary value problems. It is named after mathematicians Ivo Babuška, Gilbert Lax, and Alexander Milgram, who contributed to its development. The theorem provides conditions under which a linear operator associated with a boundary value problem possesses a unique solution and characterizes this solution in terms of bounded linear functionals.

The Besicovitch covering theorem is a result in measure theory and geometric measure theory that deals with the covering of sets in Euclidean space by balls. It is particularly important in the context of studying properties of sets of points in \(\mathbb{R}^n\) and has applications in various areas such as size theory, geometric measure theory, and analysis.

The Brezis–Gallouët inequality is an important result in functional analysis and partial differential equations, particularly in the context of Sobolev spaces. It provides a bound for a certain type of functional involving the fractional Sobolev norms. Specifically, the inequality can be stated as follows: Let \( n \geq 1 \) and \( p \in (1, n) \).

Carathéodory's existence theorem is a fundamental result in the theory of ordinary differential equations (ODEs). It provides conditions under which a first-order ordinary differential equation has at least one solution. The theorem is particularly important for equations that may not have Lipschitz continuity, allowing for broader applications.

The Cartan–Kuranishi prolongation theorem is a result in the field of differential geometry and the theory of differential equations, particularly in relation to the existence of local solutions to differential equations and the structures of their solutions. The theorem is attributed to the work of Henri Cartan and Masao Kuranishi, who contributed fundamentally to the understanding of deformation theory and the theory of analytic structures on manifolds.

The Cartan–Kähler theorem is a fundamental result in the field of differential geometry and partial differential equations, dealing with the integration of partial differential equations. It establishes conditions under which solutions exist for a certain class of systems of partial differential equations. Specifically, the theorem provides criteria for the existence of "integral submanifolds" of a given system of differential equations.

The Cauchy formula for repeated integration is a result in calculus that provides a way to express the \( n \)-th repeated integral of a function in terms of its derivatives. Specifically, it relates the \( n \)-fold integral of a function to its \( n \)-th derivative.

The Cauchy–Kovalevskaya theorem is a fundamental result in the theory of partial differential equations (PDEs) that provides conditions under which a certain class of initial value problems has solutions. Named after Augustin-Louis Cauchy and Sofia Kovalevskaya, the theorem essentially states that if the initial conditions of a certain type of PDE are satisfied, then there exists a unique analytic solution in a neighborhood of the initial value.

The Chebyshev–Markov–Stieltjes inequalities refer to a set of results in probability theory and analysis that provide estimates for the probabilities of deviations of random variables from their expected values. These inequalities are generalizations of the well-known Chebyshev inequality and are closely related to concepts from measure theory and Stieltjes integrals.

Danskin's theorem is a result in the field of optimization and convex analysis. It provides a result on the sensitivity of the optimal solution of a parametric optimization problem.

The Denjoy–Koksma inequality is a key result in the field of numerical integration and approximation theory, particularly in the context of uniform distribution theory. It provides a bound on the discrepancy of a sequence of points used in numerical integration and describes how well a given numerical method approximates the integral of a function.