A linearized polynomial is a polynomial that has been transformed into a linear form, often for the purpose of simplification or analysis.

An **integer-valued polynomial** is a polynomial function that takes integer values for all integer inputs.

A P-recursive equation (also known as a polynomially recursive equation) is a type of recurrence relation that can be defined by polynomial expressions.

Shapiro polynomials, also known as Shapiro's polynomials or Shapiro's equations, are a specific sequence of polynomials that arise in the study of certain mathematical problems, particularly in the context of probability and combinatorics. These polynomials are associated with various mathematical constructs, such as generating functions and interpolation. The Shapiro polynomials are defined recursively, and they exhibit properties related to roots and symmetry, making them useful in various theoretical frameworks.

A **polynomial ring** is a mathematical structure formed from polynomials over a given coefficient ring or field. Formally, if \( R \) is a ring (or a field), then the polynomial ring \( R[x] \) consists of all polynomials in the variable \( x \) with coefficients in \( R \).

**Polynomial solutions of P-recursive equations** refer to solutions of certain types of recurrence relations, specifically ones that can be characterized as polynomial equations. Let's break down the concepts involved: 1. **P-recursive Equations (or P-recursions)**: These are recurrence relations defined by polynomial expressions.

The Tutte polynomial is a two-variable polynomial associated with a graph, which encodes various combinatorial properties of the graph. It is named after the mathematician W. T. Tutte, who introduced it in the 1950s.

The Laguerre–Forsyth invariant is a concept in the field of differential geometry and the theory of differential equations. It arises in the context of studying the properties of certain mathematical objects under transformations, particularly in the context of higher-order differential equations. The Laguerre–Forsyth invariant specifically relates to the form of a class of differential equations known as ordinary differential equations (ODEs), particularly those of the type that can be transformed into a canonical form by appropriate changes of variables.

In the context of formal logic, mathematics, and computer science, the concepts of **free variables** and **bound variables** are important in understanding the structure of expressions, particularly in terms of quantification and function definitions. ### Free Variables A **free variable** is a variable that is not bound by a quantifier or by the scope of a function. In simpler terms, free variables are those that are not limited to a specific context or definition, meaning they can represent any value.

Intensional logic is a type of logic that focuses on the meaning and intention behind statements, as opposed to just their truth values or reference. Unlike extensional logic, which primarily deals with truth conditions and the relationships between objects and their properties, intensional logic takes into account the context, use, and meaning of the terms involved. Key features of intensional logic include: 1. **Intensions vs.

Monadic predicate calculus is a type of logical system that focuses on predicates involving only one variable (hence "monadic"). In mathematical logic, predicate calculus (or predicate logic) is an extension of propositional logic that allows for the use of quantifiers and predicates. In monadic predicate calculus, predicates are unary, meaning they take a single argument. For example, if \( P(x) \) is a predicate, it can express properties of individual elements in a domain.

In logic, a second-order predicate is an extension of first-order logic that allows quantification not only over individual variables but also over predicates or sets of individuals. In first-order logic, you can have statements that quantify over objects in a domain (like "for every \(x\), \(P(x)\)").

In probability theory, Bernstein inequalities are a set of concentration inequalities that provide bounds on the probability that the sum of independent random variables deviates from its expected value. They are particularly useful in the context of random variables that exhibit bounded variance.

The Borell–TIS (Truncation and Integration for Sums) inequality is a result in probability theory and the theory of Gaussian measures. It provides bounds on the tail probabilities of sums of independent random variables that have a certain structure, particularly in relation to Gaussian distributions. In simple terms, the Borell–TIS inequality helps to quantify how much the sum of independent random variables deviates from its expected value.

Multidimensional Chebyshev's inequality is an extension of the classical Chebyshev's inequality to the context of multivariate distributions. The classical Chebyshev's inequality provides a probabilistic bound on how far a random variable can deviate from its mean.

Ville's inequality is a result in probability theory that provides an upper bound on the probability of a certain event involving a martingale. Specifically, it deals with the behavior of a non-negative submartingale and relates to stopping times.

In the context of mathematical optimization and differential geometry, the term "Hessian pair" generally refers to a specific combination of the Hessian matrix and a function that is being analyzed. The Hessian matrix, which represents the second-order partial derivatives of a scalar function, provides important information about the curvature of the function, and thus about the nature of its critical points (e.g., whether they are minima, maxima, or saddle points).

The term "Helium planet" is not a commonly used designation in planetary science, but it can refer to certain types of exoplanets that are characterized by a significant presence of helium in their atmospheric composition. One specific type of exoplanet that could be described as a "Helium planet" is a "hot Jupiter," which is a class of exoplanets that are gas giants orbiting very close to their host stars.

Circular points at infinity are a concept from projective geometry, particularly relating to the projective plane and the study of lines and conics. In the context of projective geometry, the idea is to extend the usual Euclidean plane by adding "points at infinity," which allows us to treat parallel lines as if they meet at a point. In the case of conics, specifically circles, there are two points at infinity that are referred to as the "circular points at infinity.