Holmgren's uniqueness theorem is a result in the theory of partial differential equations (PDEs), particularly concerning elliptic equations. It addresses the uniqueness of solutions to certain boundary value problems.

Helly's selection theorem is a result in combinatorial geometry and convex analysis, named after the mathematician Eduard Helly. The theorem asserts conditions under which a family of convex sets possesses a point in common, based on the intersections of smaller subfamilies of those sets. The precise statement of Helly's selection theorem typically involves a finite collection of convex sets in \(\mathbb{R}^d\).

The Goldbach–Euler theorem is a result in number theory that relates to the representation of even integers as sums of prime numbers. More specifically, it builds on the ideas of the original Goldbach conjecture. While the conjecture itself states that every even integer greater than 2 can be expressed as the sum of two prime numbers, the Goldbach–Euler theorem provides a more generalized framework.

Godunov's theorem is a result in the field of numerical analysis, specifically related to the numerical solution of hyperbolic partial differential equations (PDEs). It is named after the Russian mathematician S. K. Godunov, who contributed significantly to the development of finite volume methods for solving these types of equations.

Glaeser's continuity theorem is a result in the field of real analysis, specifically concerning the continuity properties of certain functions. While I cannot provide the specific wording of the theorem, I can summarize its significance and implications. The theorem is often related to the concepts of continuity in functions defined on certain spaces. It typically deals with the conditions under which a function can be approximated continuously by other functions, or under which certain limits exist as parameters change.

The Gaussian integral refers to the integral of the function \( e^{-x^2} \) over the entire real line.

Fuchs' theorem is a result in the field of complex analysis, particularly in the study of ordinary differential equations with singularities. The theorem provides conditions under which a linear ordinary differential equation with an irregular singular point can be solved using power series methods. Specifically, Fuchs' theorem states that if a linear differential equation has only regular singular points, then around each regular singular point, there exist solutions that can be expressed as a Frobenius series.

The Fraňková–Helly selection theorem is a result in the field of functional analysis and topology, specifically concerning the selection of points from family of sets. It builds upon the classical Helly's theorem, which deals with finite intersections of convex sets in Euclidean spaces. The Fraňková–Helly selection theorem provides conditions under which one can extract a sequence from a family of sets that converges in a certain sense.

The Fenchel–Moreau theorem is a fundamental result in convex analysis that relates the concepts of convex conjugates and duality. It characterizes the relationship between a convex function and its conjugate. Let \( f : \mathbb{R}^n \to \mathbb{R} \) be a proper, convex, and lower semicontinuous function.

Fenchel's duality theorem is a fundamental result in convex analysis and optimization, which establishes a relationship between a convex optimization problem and its dual problem. Specifically, it provides conditions under which the solution of a primal convex optimization problem can be found by solving its dual.

The Dirichlet–Jordan test is a criterion used in analysis, particularly in the study of the convergence of series of functions, such as Fourier series. The test is useful for determining the pointwise convergence of a series of functions, especially when dealing with orthogonal functions or trigonometric series.

The Denjoy–Young–Saks theorem is a result in measure theory concerning the decomposition of the Lebesgue measurable sets. It is named after mathematicians Arne Magnus Denjoy, John Willard Young, and Aleksandr Yakovlevich Saks, who contributed to the development of this area of mathematics.

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.

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 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.

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 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 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 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.

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.