Non-invertible symmetry refers to a type of symmetry in physical systems where certain transformations cannot be undone or reversed. In contrast to invertible symmetries, which have a clear operation that can be applied to return a system to its original state, non-invertible symmetries do not allow for such a straightforward correspondence. This concept often arises in the context of condensed matter physics and quantum field theory.

Non-topological solitons are a type of soliton that differ from their topological counterparts in the manner in which they maintain their shape and stability. Solitons are stable, localized wave packets that arise in various fields of physics, often characterized by their ability to propagate without changing shape due to a balance between nonlinearity and dispersion.

ShEx, or Shapes Expression, is a language used to describe the structure and constraints of RDF (Resource Description Framework) data. It provides a formal way to define what data should look like, including the properties and types of resources, to ensure that the data adheres to specific requirements or "shapes." The primary purpose of ShEx is to offer a mechanism for validating RDF datasets against defined schemas.

The on-shell renormalization scheme is a method used in quantum field theory to handle the divergences that arise in the calculation of physical quantities. In this approach, the parameters of a quantum field theory, such as mass and coupling constants, are renormalized in a way that relates the theoretical predictions directly to measurable physical quantities, specifically the observables associated with actual particles.

Pauli–Villars regularization is a method used in quantum field theory to manage divergences that arise in the calculation of loop integrals, particularly in the context of quantum electrodynamics (QED) and other quantum field theories. This technique introduces additional fields or particles with specific properties to modify the behavior of the underlying theory and render integrals convergent.

A credit default swap (CDS) is a financial derivative that allows an investor to "swap" or transfer the credit risk of a borrower to another party. Essentially, it is a contract between two parties where one party (the buyer of the CDS) pays a periodic fee to the other party (the seller of the CDS) in exchange for protection against the risk of default on a specified debt obligation, such as a bond or loan.

The R-matrix is an important concept in various fields of physics and mathematics, particularly within quantum mechanics and scattering theory. It serves as a mathematical framework for understanding interactions between particles. 1. **Quantum Mechanics and Scattering Theory**: In the context of quantum mechanics, the R-matrix can be used to analyze scattering processes. It relates to the wave functions of particles before and after a scattering event.

The total active reflection coefficient is a parameter used in the field of microwave engineering and antenna theory to describe how much of an incident wave is reflected back due to impedance mismatches at interfaces, such as at the feed point of an antenna. This coefficient can be particularly important when designing antennas and RF circuits, as it affects the efficiency and performance of the system.

A transformation matrix is a mathematical tool used to perform linear transformations on geometric objects, such as points, vectors, or shapes in space. In linear algebra, a transformation matrix represents a linear transformation, which is a function that maps vectors to other vectors while preserving the operations of addition and scalar multiplication. The properties of transformation matrices make them essential in various fields, including computer graphics, robotics, physics, and engineering.

The Pinsky phenomenon refers to a phenomenon in mathematics and physics involving the peculiar behavior of certain sequences or series, particularly those that exhibit rapid oscillations. One notable instance of the Pinsky phenomenon can be observed in the context of Fourier series or wave functions, where oscillations may become increasingly pronounced, leading to unexpected convergence properties or divergence in specific contexts.

"Triangles of numbers" can refer to several mathematical constructs that involve arranging numbers in a triangular formation. A common example is Pascal's Triangle, which is a triangular array of the binomial coefficients. Each number in Pascal's Triangle is the sum of the two numbers directly above it in the previous row. Here’s a brief overview of some well-known triangles of numbers: 1. **Pascal's Triangle**: Starts with a 1 at the top (the 0th row).

In linear algebra, commuting matrices are matrices that can be multiplied together in either order without affecting the result. That is, two matrices \( A \) and \( B \) are said to commute if: \[ AB = BA \] This property is significant in many areas of mathematics and physics, particularly in quantum mechanics and functional analysis, as it relates to the simultaneous diagonalization of matrices, the representation of observables in quantum systems, and other contexts where linear transformations play a crucial role.

Freivalds' algorithm is a randomized algorithm used to verify matrix products efficiently. It is particularly useful for checking whether the product of two matrices \( A \) and \( B \) equals a third matrix \( C \), i.e., whether \( A \times B = C \). The algorithm is notable for its efficiency and its ability to reduce the verification problem to a probabilistic one.

The Nullity Theorem, also known as the Nullity-Rank Theorem, is a fundamental result in linear algebra and relates to the structure of linear transformations and matrices.

A bilinear map is a mathematical function defined on two vector spaces (or modules) that is linear in each of its arguments when the other is held fixed.

The term "Essential extension" can refer to different concepts depending on the context, such as software development, web browsers, or various frameworks. Here are a few common interpretations: 1. **Web Browser Extensions**: In the context of web browsers, an "essential extension" typically refers to a browser add-on that significantly enhances usability, security, or productivity. Examples include ad blockers, password managers, and privacy-focused extensions.

The term "flat cover" can refer to a few different concepts depending on the context. Here are a couple of common meanings: 1. **Publishing and Graphic Design**: In the context of books, magazines, or other printed materials, a flat cover usually refers to a cover that is designed as a single flat piece, rather than having folds or layers. It can also mean that the cover does not have any additional features like embossing or die cuts and is printed uniformly on a single surface.

Module theory is a branch of abstract algebra that studies modules, which generalize vector spaces by allowing scalars to come from a ring instead of a field. Here's a glossary of key terms commonly used in module theory: 1. **Module**: A generalization of vector spaces where the scalars come from a ring instead of a field. A module over a ring \( R \) consists of an additive abelian group along with a scalar multiplication operation that respects the ring's structure.

In the context of algebra, particularly in the study of module theory over rings, a projective module is a type of module that generalizes the concept of free modules.

Monoidal categories are a fundamental concept in category theory, providing a framework that captures notions of multiplicative structures in a categorical setting. A monoidal category consists of a category equipped with a tensor product (which can be thought of as a kind of "multiplication" between objects), an identity object, and certain coherence conditions that ensure the structure behaves well.