Sard's theorem is a result in differential topology that pertains to the behavior of smooth functions between manifolds. Specifically, it addresses the notion of the image of a smooth function and the measure of its critical values.

The Remez inequality is a result in approximation theory that provides a bound on the deviation of a continuous function from its best approximation by a polynomial. Specifically, it relates the norm of a polynomial approximation to the maximum deviation of the approximated function over a given interval.

The Rellich–Kondrachov theorem is a significant result in functional analysis and the theory of differential equations, particularly in the context of Sobolev spaces. It essentially states conditions under which the embedding of Sobolev spaces into Lp spaces is compact.

The Rademacher–Menchov theorem is a result in the field of measure theory and functional analysis. It is particularly significant in the study of series of functions, specifically in the context of rearrangement of series in Banach spaces.

The Poincaré inequality is a fundamental result in mathematical analysis and partial differential equations. It provides a bound on the integral of a function in terms of the integral of its derivative.

The Picard–Lindelöf theorem, also known as the Picard existence theorem or the Picard-Lindelöf 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 a unique solution in a specified interval.

The Peano existence theorem, often referred to in the context of ordinary differential equations (ODEs), is a fundamental result that provides conditions under which solutions to certain initial value problems exist.

The Narasimhan-Seshadri theorem is a fundamental result in the theory of vector bundles over complex curves (or Riemann surfaces). It establishes a deep connection between the geometry of vector bundles and the representation theory of groups, particularly in the context of holomorphic vector bundles on Riemann surfaces and unitary representations of the fundamental group.

Markov's inequality is a result in probability theory that provides an upper bound on the probability that a non-negative random variable is greater than or equal to a positive constant. The inequality is named after the Russian mathematician Andrey Markov. The statement of Markov's inequality is as follows: Let \(X\) be a non-negative random variable (i.e., \(X \geq 0\)), and let \(a > 0\) be a positive constant.

Malmquist's theorem, also known as the Malmquist interpolation theorem, is a result in the field of complex analysis and functional analysis that pertains to the behavior of holomorphic functions. Specifically, it addresses the existence of holomorphic functions defined on a certain domain that agree with prescribed values on a collection of points.

The Malgrange–Ehrenpreis theorem is a result in the theory of partial differential equations (PDEs). It pertains to the existence of solutions to systems of linear partial differential equations, particularly in the context of several variables. More specifically, it addresses the question of whether one can find solutions to a given system of linear PDEs with specified boundary or initial conditions.

The Malgrange preparation theorem is a result in complex analysis and algebraic geometry that is concerned with the behavior of analytic functions and their singularities. It provides a way to analyze and decompose certain classes of analytic functions near isolated singular points.

The Lagrange reversion theorem is a result in mathematical analysis and combinatorics that relates to the coefficients of a power series. More specifically, it provides a method to express the coefficients of the inverse of a power series in terms of the coefficients of the original series.

Krein's condition refers to a specific criterion used in the mathematical field of functional analysis, particularly in the study of operators on Hilbert spaces. It is particularly associated with the stability of operators and the spectral properties of certain classes of linear operators, especially in the context of self-adjoint operators. In its most well-known form, Krein's condition provides a way to characterize the stability of a linear operator with respect to perturbations.

Komlós' theorem, also known as Komlós' conjecture, is a result in combinatorial mathematics, specifically in the field of graph theory. The theorem deals with the concept of almost perfect matchings in large graphs.

Kneser's theorem is a result in the theory of differential equations, particularly in the context of linear differential equations with variable coefficients. It addresses the behavior of solutions for higher-order linear ordinary differential equations.

The Khintchine inequality is a result in mathematical analysis, particularly in the study of probability theory and functional analysis. It pertains to the properties of sums of independent random variables, specifically regarding their expected values and moments.

The Kantorovich inequality is a result in the realm of functional analysis, specifically associated with the theory of measures and integrable functions. It provides a crucial estimate related to the norms of integral operators defined on vector spaces of measurable functions. In one of its common forms, the Kantorovich inequality relates to the notion of integrable functions and their norms.

Jensen's inequality is a fundamental result in convex analysis and probability theory that relates to convex functions.

Integration using Euler's formula involves the application of Euler's formula to express complex exponentials in terms of sine and cosine functions, which can simplify the integration of certain functions, especially those involving trigonometric terms.