The Polynomial Remainder Theorem is a fundamental result in algebra that relates to the division of polynomials. It states that if a polynomial \( f(x) \) is divided by a linear polynomial of the form \( (x - c) \), the remainder of this division is equal to the value of the polynomial evaluated at \( c \).
The Rational Root Theorem is a useful tool in algebra for finding the possible rational roots of a polynomial equation. It states that if a polynomial \( P(x) \) with integer coefficients has a rational root \( \frac{p}{q} \) (in lowest terms), where \( p \) and \( q \) are integers, then: - \( p \) (the numerator) must be a divisor of the constant term of the polynomial.
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.
The Gap Theorem is a concept in the field of mathematics, particularly in the study of algebraic geometry and topology, though there are applications and related ideas in other areas of mathematics as well. In one of its forms, the Gap Theorem refers to a result concerning the existence of "gaps" in the spectrum of certain types of operators, particularly in the context of spectral theory.
The Geiringer–Laman theorem is a result in the field of graph theory and combinatorial geometry, specifically concerning the rigidity of frameworks. The theorem provides a criterion for determining when a certain kind of graph, known as a "framework", can be considered rigid, meaning that its vertices cannot be moved without distorting the distances between them.
Abhyankar's lemma is a result in the area of algebraic geometry, specifically dealing with the properties of algebraic varieties and their points over fields. Named after the mathematician Shivaramakrishna Abhyankar, the lemma provides a criterion for the existence of certain types of points in the context of algebraic varieties defined over a field.
The Kronecker limit formula is an important result in the theory of modular forms and number theory. It relates the behavior of certain L-functions to the special values of those functions at integers. Specifically, it provides a way to compute the special value of an L-function associated with a point on a certain modular curve. The formula can be stated in the context of the Dedekind eta function and the Eisenstein series.
The Landau prime ideal theorem is a result in the field of algebra, specifically in commutative algebra and the theory of rings. It concerns the structure of prime ideals in a non-zero commutative ring.
The Landsberg–Schaar relation is a concept in the field of thermodynamics, particularly in relation to the thermoelectric properties of materials. It establishes a relationship between the electrical conductivity, the Seebeck coefficient, and the thermal conductivity of a material. This relation is significant because it helps to optimize materials for thermoelectric applications, such as in power generation or cooling devices.
Maier's theorem is a result in number theory related to the distribution of prime numbers. Specifically, it deals with the existence of certain arithmetic progressions among prime numbers. The theorem is typically discussed in the context of additive number theory and is named after the mathematician Helmut Maier, who contributed to the understanding of the distribution of primes.
The Petersson trace formula is an important result in the theory of modular forms and number theory. It provides a relationship between the eigenvalues of Hecke operators on modular forms and the values of L-functions at certain critical points. The formula is named after the mathematician Heinrich Petersson, who was instrumental in its development. In its most common form, the Petersson trace formula connects the spectral theory of automorphic forms with the arithmetic of numbers through the Fourier coefficients of modular forms.
Vinogradov's mean-value theorem is a result in additive number theory that concerns the distribution of the values of additive functions. It has significant implications for the study of Diophantine equations and is particularly important in the field of analytic number theory. The theorem essentially states that for a certain class of additive functions (typically of the type that can be exhibited as sums of integers), the average number of representations of a number as a sum of other integers can be understood in a mean-value sense.
The Erdős–Turán inequality is a result in combinatorial number theory that deals with the distribution of sums in sequences of integers.
Cramer's Rule is a mathematical theorem used to solve systems of linear equations with as many equations as unknowns, provided that the system has a unique solution. It is applicable when the coefficient matrix is non-singular (i.e., its determinant is non-zero).
The Goddard–Thorn theorem is a result in the field of theoretical physics, particularly in string theory. It addresses the conditions under which certain types of models, specifically those involving extended objects or strings, can achieve a consistent description of physical phenomena. The theorem is named after physicists Peter Goddard and David Thorn, who developed it in the context of string theory in the early 1980s.
The Hawkins–Simon condition is a criterion used in economics, particularly in input-output analysis, to determine the feasibility of a production system. It is named after the economists R. J. Hawkins and R. L. Simon, who introduced this condition in the context of linear production models. In simple terms, the Hawkins–Simon condition states that a certain system of production can be sustained in equilibrium if the total inputs required for production do not exceed the total outputs available.
The Erdős–Pósa theorem is a result in graph theory that deals with the relationship between the presence of certain subgraphs and the presence of certain structures in a graph. Specifically, it provides a relationship between the existence of a set of vertex-disjoint cycles in a graph and the existence of a set of vertices that intersects all these cycles. To state the theorem more formally, it addresses the case of cycles in graphs.
The Erdős–Stone theorem is a fundamental result in extremal graph theory, which deals with understanding the maximum number of edges in a graph that does not contain a particular subgraph. Specifically, the theorem provides a way to determine the asymptotic behavior of the maximum number of edges in a graph on \( n \) vertices that does not contain a complete subgraph \( K_r \) (the complete graph on \( r \) vertices) as a subgraph.
Fleischner's theorem is a result in graph theory that relates to the properties of cycles in Eulerian graphs. Specifically, it states that every 2-edge-connected graph (a graph where there are at least two vertex-disjoint paths between any two vertices) contains a cycle that includes every edge of the graph. This is closely associated with the concept of an Eulerian circuit, which is a cycle that visits every edge of a graph exactly once.