Diagonal form refers to a way of representing matrices or linear transformations that simplifies the analysis and computation of systems of equations. Specifically, a matrix is said to be in diagonal form when all of its non-zero elements are located along its main diagonal, and all other elements are zero.

SOS-convexity, or Sum of Squares convexity, is a concept in optimization and mathematical programming that relates to certain types of convex functions. A function is said to be SOS-convex if it allows for a polynomial representation that can be described using sums of squares.

The Eilenberg–Steenrod axioms are a set of axioms in algebraic topology that characterize (reduced) singular homology and cohomology theories. Formulated by Samuel Eilenberg and Norman Steenrod in the mid-20th century, these axioms provide a rigorous framework for what constitutes a generalized homology or cohomology theory. They serve as a foundation for the study of topological spaces through algebraic means.

The Kirby–Siebenmann class is a concept in the field of algebraic topology, particularly within the study of manifolds and their embeddings. It is named after mathematicians Robion Kirby and Louis Siebenmann, who introduced it in their work on the topology of high-dimensional manifolds. In particular, the Kirby–Siebenmann class arises in the context of the study of manifold structures and their differentiability.

Singular homology is an important concept in algebraic topology, which provides a way to associate a sequence of abelian groups or vector spaces (called homology groups) to a topological space. These groups encapsulate information about the space's structure, such as its number of holes in various dimensions. ### Key Concepts: 1. **Simplices**: The building blocks of singular homology are simplices, which are generalizations of triangles.

Equivariant stable homotopy theory is a branch of algebraic topology that studies the stable homotopy categories of topological spaces or spectra with a group action, particularly focusing on the actions of a compact Lie group or discrete group. The theory extends classical stable homotopy theory, which examines stable phenomena in topology, into the context where symmetry plays an important role.

Cotriple homology is a concept that arises in the context of homological algebra and category theory. It is associated with the study of coalgebras and cohomological methods, akin to how traditional homology theories apply to algebraic structures like groups, rings, and spaces.

The list of the tallest wrestlers in WWE history includes several notable figures, many of whom are known for their imposing stature and larger-than-life personas. Here are some of the tallest wrestlers: 1. **Giant González** (Jorge González) - 7'6" (229 cm) 2. **The Big Show** (Paul Wight) - 7'0" (213 cm) 3.

The Generalized Whitehead product is a concept in algebraic topology, specifically within the context of homotopy theory. It generalizes the classical Whitehead product, which arises in the study of higher homotopy groups and the structure of loop spaces. ### Background In algebraic topology, the Whitehead product is a way of constructing a new homotopy class of maps from two existing homotopy classes.

The Nilpotence Theorem, often referred to in the context of algebra, pertains primarily to the properties of nilpotent elements or nilpotent operators in various algebraic structures, such as rings and linear operators. In a general sense, an element \( a \) of a ring \( R \) is said to be **nilpotent** if there exists a positive integer \( n \) such that \( a^n = 0 \).

A "phantom map" typically refers to a theoretical or conceptual representation in various contexts, including geography, fantasy mapping, or even in virtual reality and gaming. However, the term can also have specific meanings in different fields: 1. **Theoretical Geography or Cartography**: A phantom map might refer to a map that represents an area that doesn't exist in reality, such as a fictional world in a novel or game. It often serves as a tool for storytelling and world-building.

In algebraic topology, a homotopy group is an important algebraic invariant that captures the topological structure of a space. The most common homotopy groups are the fundamental group and higher homotopy groups. 1. **Fundamental Group (\(\pi_1\))**: The fundamental group is the first homotopy group and provides a measure of the "loop structure" of a space.

Anna Krylov is a prominent theoretical chemist known for her contributions to the fields of electronic structure theory, quantum chemistry, and computational chemistry. She has worked on a variety of topics, including the development of new methods for understanding molecular electronic properties, as well as applications of quantum chemistry to complex systems. Krylov is a professor at the University of Southern California, where she conducts research and teaches. Her work often focuses on the development and application of computational tools to study molecular behavior and reactions.

In category theory, a **Quillen adjunction** is a specific type of adjunction between two categories that arises within the context of homotopy theory, particularly when dealing with model categories.

The Seifert–Van Kampen theorem is a fundamental result in algebraic topology that provides a method for computing the fundamental group of a topological space that can be decomposed into simpler pieces. Specifically, it relates the fundamental group of a space to the fundamental groups of its subspaces when certain conditions are satisfied.

Shape theory is a branch of mathematics that studies the properties and classifications of shapes in a more abstract sense. It primarily deals with the concept of "shape" in topological spaces and focuses on understanding how shapes can be analyzed and compared based on their intrinsic properties, rather than their exact geometrical measurements. One of the key aspects of shape theory is the idea that two shapes can be considered equivalent if they can be continuously transformed into one another without cutting or gluing.

Simple homotopy equivalence is a concept in algebraic topology that provides a way to compare topological spaces in terms of their deformation properties. More specifically, it focuses on the notion of homotopy equivalence under certain simplifications. Two spaces \( X \) and \( Y \) are said to be *simple homotopy equivalent* if there exists a sequence of simple homotopy equivalences between them.

The Body Adiposity Index (BAI) is a method used to estimate body fat percentage based on a person's hip circumference and height. Unlike the Body Mass Index (BMI), which uses weight as a factor, BAI specifically focuses on providing an estimate of body fat percentage by considering abdominal fat distribution.

Spanier–Whitehead duality is a concept in algebraic topology, named after the mathematicians Edwin Spanier and Frank W. Whitehead. It provides a duality between certain types of topological spaces regarding their homotopy and homology theories. More specifically, it relates the category of pointed spaces to the category of pointed spectra, allowing one to translate problems in unstable homotopy theory into stable homotopy theory, and vice versa.