"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 Jordan Curve Theorem is a fundamental result in topology, a branch of mathematics that studies properties of spaces that are preserved under continuous deformations. The theorem states that any simple closed curve in a plane (a curve that does not intersect itself and forms a complete loop) divides the plane into two distinct regions: an "inside" and an "outside.

A dipole magnet is a type of magnet that has two poles: a north pole and a south pole. These magnets produce a magnetic field that is characterized by a distinct orientation. In a basic sense, dipole magnets can be thought of as having a magnetic moment that points from the south pole to the north pole.

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).

In ring theory, a branch of abstract algebra, theorems describe properties and structures of rings, which are algebraic objects consisting of a set equipped with two binary operations: addition and multiplication. Here are some fundamental theorems and results related to ring theory: 1. **Ring Homomorphisms**: A function between two rings that preserves the ring operations.

"The Big Bang Theory" Season 9 is the ninth installment of the popular American sitcom created by Chuck Lorre and Bill Prady. This season premiered on September 21, 2015, and concluded on May 16, 2016.

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.

Descartes' Rule of Signs is a mathematical theorem that provides a way to determine the number of positive and negative real roots of a polynomial function based on the signs of its coefficients. Here’s a concise breakdown of the rule: 1. **Positive Roots**: To find the number of positive real roots of a polynomial \(P(x)\), count the number of sign changes in the sequence of the coefficients of \(P(x)\).

Gauss's lemma in the context of polynomials states that if \( f(x) \) is a polynomial with integer coefficients, and if it can be factored into the product of two non-constant polynomials over the integers, then it can also be factored into polynomials of degree less than or equal to \( \deg(f) \) over the integers.

The Gauss–Lucas theorem is a result in complex analysis and polynomial theory concerning the roots of a polynomial. Specifically, it provides insight into the relationship between the roots of a polynomial and the roots of its derivative.

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.

Fejér's theorem is a result in the theory of Fourier series, specifically concerning the convergence of the Fourier series of a periodic function. It states that if \( f \) is a piecewise continuous function on the interval \([-L, L]\), then the sequence of partial sums of its Fourier series converges uniformly to the average of the left-hand and right-hand limits of \( f \) at each point.

Chebotarev's theorem is a result in number theory that deals with the distribution of roots of unity in relation to polynomial equations over finite fields. Specifically, it is often associated with the density of certain classes of primes in number fields, but it can be stated in a context relevant to roots of unity.

The Routh–Hurwitz theorem is a mathematical criterion used in control theory and stability analysis of linear time-invariant (LTI) systems. It provides a systematic way to determine whether all roots of a given polynomial have negative real parts, which indicates that the system is stable.

Wirtinger's representation theorem and projection theorem are fundamental results in mathematical analysis, particularly in the fields of functional analysis and the theory of Sobolev spaces. They are often applied in the study of harmonic functions, the solution of partial differential equations, and variational problems. ### Wirtinger's Representation Theorem: The Wirtinger representation theorem provides a way to connect the Dirichlet energy of functions to their boundary conditions.

The AF + BG theorem is a concept in the field of mathematics, specifically in the area of set theory and topology. However, the notation AF + BG does not correspond to a widely recognized theorem or principle within standard mathematical literature or education. It's possible that this notation is specific to a certain context, course, or area of research that is not broadly covered.

Blum's speedup theorem is a result in the field of computational complexity theory, specifically dealing with the relationship between the time complexity of algorithms and the computation of functions. Formulated by Manuel Blum in the 1960s, the theorem essentially asserts that if a certain function can be computed by a deterministic Turing machine within a certain time bound, then there exists an alternative algorithm (or Turing machine) that computes the same function more quickly.

Algebraic number theory is a branch of mathematics that studies the properties of numbers and the relationships between them, particularly through the lens of algebraic structures such as rings, fields, and ideals. Within this field, theorems often address the properties of algebraic integers, the structure of algebraic number fields, and the behavior of various arithmetic objects.

The Friedlander–Iwaniec theorem is a result in number theory, specifically in the area of additive number theory concerning the distribution of prime numbers. It was established by the mathematicians J. Friedlander and H. Iwaniec in the early 1990s.