"The Hot Troll Deviation" is the title of an episode from the popular TV show *The Big Bang Theory*, specifically season 4, episode 14. In this episode, the characters navigate various personal relationships and social dynamics. The storyline revolves around Raj's interest in a woman he meets online after he gets drunk and posts a risqué photo of himself, which leads to humorous situations. The episode explores themes of attraction and identity through its comedic lens, typical of the show's style.

Input: a sequence of $N$ complex numbers $x_k$.

Output: another sequence of $N$ complex numbers $X_k$ such that:Intuitively, this means that we are braking up the complex signal into $N$ sinusoidal frequencies:and is the amplitude of each sine.

We use Zero-based numbering in our definitions because it just makes every formula simpler.

Motivation: similar to the Fourier transform:In particular, the discrete Fourier transform is used in signal processing after a analog-to-digital converter. Digital signal processing historically likely grew more and more over analog processing as digital processors got faster and faster as it gives more flexibility in algorithm design.

Sample software implementations:

Previously, updates were being done with more focus to sponsors in the format of the child sections to this section. That format is now retired in favor of the more direct Section "Updates" format.

Basically, a "representation" means associating each group element as an invertible matrices, i.e. a matrix in (possibly some subset of) $GL(n)$, that has the same properties as the group.

Or in other words, associating to the more abstract notion of a group more concrete objects with which we are familiar (e.g. a matrix).

Each such matrix then represents one specific element of the group.

This is basically what everyone does (or should do!) when starting to study Lie groups: we start looking at matrix Lie groups, which are very concrete.

Or more precisely, mapping each group element to a linear map over some vector field $V$ (which can be represented by a matrix infinite dimension), in a way that respects the group operations:

As shown at Physics from Symmetry by Jakob Schwichtenberg (2015)

Motivation:

Bibliography:

In lattice theory, which is a branch of abstract algebra, a lattice is a partially ordered set (poset) in which any two elements have a unique supremum (least upper bound) and an infimum (greatest lower bound). Theorems in lattice theory often deal with the properties and relationships of these structures.

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.

Whitehead's Lemma is a result in the field of algebraic topology, particularly in the study of homotopy theory and the properties of topological spaces. It deals with the question of when a certain kind of map induces an isomorphism on homotopy groups.

The Wallace–Bolyai–Gerwien theorem is a result in geometry related to the transformation of polygons. Specifically, it states that any two simple polygons of equal area can be dissected into a finite number of polygonal pieces that can be rearranged to form one another. The theorem has important implications in the study of geometric dissections, a topic that has intrigued mathematicians for centuries.

The Circle Packing Theorem is a result in mathematics that concerns arrangements of circles in a plane. Specifically, the theorem states that given any simple closed curve (a curve that does not intersect itself), it is possible to pack a finite number of circles within that curve such that all the circles are tangent to each other and to the curve.

MacMahon's Master Theorem is a mathematical tool used in the analysis of combinatorial structures, particularly in the enumeration of various combinatorial objects. While it's not as widely known as some other results in combinatorics, it provides a framework for counting partitions, arrangements, and related structures using generating functions. The theorem is named after the British mathematician Percy MacMahon, who made significant contributions to the theory of partitions and generating functions.

Robert May, Baron May of Oxford, is an eminent British scientist known for his significant contributions to the fields of ecology and theoretical biology. Born on April 8, 1936, he is particularly recognized for his work in mathematical ecology, biodiversity, and the dynamics of ecosystems. He served as the Chief Scientific Adviser to the UK government and held the position of President of the Royal Society from 2000 to 2005.

Frank A. Weinhold may refer to a specific individual known for contributions in a certain field, likely in academia or research, given that I don't have specific information on him. As of my last update in October 2021, he was recognized in the domain of chemistry, particularly in the context of chemical education and organometallic chemistry.

Stephen Altschul is a prominent figure in the field of computational biology and bioinformatics. He is known for his work on the development of algorithms and methodologies for analyzing biological data, particularly in the context of sequence alignment and phylogenetic analysis. One of his significant contributions is the development of the BLAST (Basic Local Alignment Search Tool) algorithm, which is widely used for comparing sequences of DNA, RNA, and proteins.

The Journal of Chemical Theory and Computation (JCTC) is a peer-reviewed scientific journal that focuses on the application of computational methods to the field of chemistry.

Gernot Frenking is a prominent chemist known for his work in theoretical and computational chemistry. His research often focuses on the study of electronic structures, reaction mechanisms, and molecular properties using quantum chemical methods. Throughout his career, Frenking has contributed to the understanding of chemical bonding and molecular interactions, and he has published numerous papers in scientific journals. His contributions also extend to the development and application of computational tools in chemistry.