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In algebraic geometry, "theorems" typically refer to significant results and findings that pertain to the study of geometric objects defined by polynomial equations. This field, which bridges algebra, geometry, and number theory, has many important theorems that provide insights into the properties of algebraic varieties, their structures, and relationships.

In lattice theory, which is a branch of abstract algebra, a lattice is a partially ordered set (poset) in which any two elements have a unique supremum (least upper bound) and an infimum (greatest lower bound). Theorems in lattice theory often deal with the properties and relationships of these structures.

In representation theory, theorems often refer to fundamental results that describe the structure and behavior of representations of groups, algebras, or other algebraic structures. Representation theory is a branch of mathematics that studies how algebraic structures can be represented through linear transformations of vector spaces.

"The Big Bang Theory" Season 5 is the fifth installment of the popular American television sitcom created by Chuck Lorre and Bill Prady. The season originally aired from September 22, 2011, to May 10, 2012, and consists of 24 episodes. In this season, viewers continue to follow the lives of physicists Sheldon Cooper and Leonard Hofstadter, along with their friends Penny, Howard Wolowitz, and Rajesh Koothrappali.

"The Big Bang Theory" Season 7 is the seventh installment of the popular American sitcom created by Chuck Lorre and Bill Prady. The season originally aired from September 26, 2013, to May 15, 2014, consisting of 24 episodes.

"The Big Bang Theory" Season 8 is the eighth installment of the popular American sitcom created by Chuck Lorre and Bill Prady. It originally aired on CBS from September 22, 2014, to May 14, 2015. The season consists of 24 episodes and continues to follow the lives of physicists Leonard Hofstadter and Sheldon Cooper, along with their friends and fellow scientists, Penny, Howard Wolowitz, and Bernadette Rostenkowski-Wolowitz.

Fenchel's theorem, often referred to in the context of convex analysis, deals with the correspondence between the convex functions and their subgradients. Specifically, it provides a characterization of convex functions through their conjugate functions.

Newton's theorem, often referred to as the "Newton's theorem on ovals," relates to the properties of an oval, particularly in the context of projective geometry and combinatorial geometry. The theorem essentially states that given a set of points in the plane, if these points are located on a smooth convex curve (an oval), then there exists a certain relationship concerning the tangents, secants, and other lines drawn from these points.

The Pestov–Ionin theorem is a result in the field of mathematical logic that deals with the preservation of certain properties in structures, particularly in the context of countable models. Although it is a specialized topic, the theorem itself is typically discussed within the framework of model theory, which studies the relationships between formal languages and their interpretations (or models).

The Abel–Ruffini theorem is a result in algebra that states there is no general solution in radicals to polynomial equations of degree five or higher. In other words, it is impossible to express the roots of a general polynomial of degree five or greater using only radicals (i.e., through a finite sequence of operations involving addition, subtraction, multiplication, division, and taking roots).

Bernstein's theorem in the context of polynomials refers to results concerning the approximation of continuous functions by polynomials, particularly in relation to the uniform convergence of polynomial sequences. One of the key results of Bernstein's theorem states that if \( f \) is a continuous function defined on a closed interval \([a, b]\), then \( f \) can be approximated arbitrarily closely by polynomials in the uniform norm.

The Binomial Theorem is a fundamental result in algebra that provides a formula for expanding expressions of the form \((a + b)^n\), where \(n\) is a non-negative integer. The theorem states that: \[ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \] In this formula: - \(\sum\) denotes summation.

Cohn's theorem is a result in the field of algebra, particularly concerning the representation of semigroups and rings. The theorem primarily addresses the structure of commutative semigroups and explores conditions under which a commutative semigroup can be embedded into a given algebraic structure. In more specific terms, Cohn's theorem states that every commutative semigroup can be represented as a certain kind of matrix semigroup over a certain commutative ring.

The Complex Conjugate Root Theorem states that if a polynomial has real coefficients and a complex number \( a + bi \) (where \( a \) and \( b \) are real numbers and \( i \) is the imaginary unit) as a root, then its complex conjugate \( a - bi \) must also be a root of the polynomial.

The Factor Theorem is a fundamental principle in algebra that relates to polynomials. It provides a way to determine whether a given polynomial has a particular linear factor. Specifically, the theorem states: If \( f(x) \) is a polynomial and \( c \) is a constant, then \( (x - c) \) is a factor of \( f(x) \) if and only if \( f(c) = 0 \).

The Grace–Walsh–Szegő theorem is a significant result in complex analysis and polynomial theory, particularly concerning the behavior of polynomials and their roots. The theorem deals with the location of the roots of a polynomial \( P(z) \) in relation to the roots of another polynomial \( Q(z) \). Specifically, it provides conditions under which all roots of \( P(z) \) lie within the convex hull of the roots of \( Q(z) \).

Hilbert's irreducibility theorem is a result in algebraic number theory, specifically related to the behavior of certain types of polynomial equations. Formulated by David Hilbert in the early 20th century, the theorem provides a significant insight into the irreducibility of polynomials over number fields.

Kharitonov's theorem is a result in control theory, particularly in the study of linear time-invariant (LTI) systems and the stability of polynomial systems. It is often used in the analysis of systems with polynomials that have parameters, allowing for the examination of how variations in those parameters affect stability. The theorem provides a method to determine the stability of a family of linear systems defined by a parameterized characteristic polynomial.

Marden's theorem is a result in complex analysis that deals with the roots of a polynomial and their geometric properties, particularly concerning the locations of the roots in the complex plane.

Mason–Stothers theorem is a result in complex analysis and the theory of meromorphic functions, specifically concerning the growth and distribution of the zeros of these functions. It is a generalization of the classical results about the growth of entire functions and provides a way to relate the growth of a meromorphic function to the distribution of its zeros and poles.