Algebraic operation

In mathematics, a basic algebraic operation is any one of the common operations of arithmetic, which include addition, subtraction, multiplication, division, raising to a whole number power, and taking roots (fractional power). These operations may be performed on numbers, in which case they are often called arithmetic operations. They may also be performed, in a similar way, on variables, algebraic expressions, and more generally, on elements of algebraic structures, such as groups and fields. An algebraic operation may also be defined simply as a function from a Cartesian power of a set to the same set. The term algebraic operation may also be used for operations that may be defined by compounding basic algebraic operations, such as the dot product. In calculus and mathematical analysis, algebraic operation is also used for the operations that may be defined by purely algebraic methods. For example, exponentiation with an integer or rational exponent is an algebraic operation, but not the general exponentiation with a real or complex exponent. Also, the derivative is an operation that is not algebraic. Multiplication symbols are usually omitted, and implied, when there is no operator between two variables or terms, or when a coefficient is used. For example, 3 × x2 is written as 3x2, and 2 × x × y is written as 2xy. Sometimes, multiplication symbols are replaced with either a dot or center-dot, so that x × y is written as either x . y or x · y. Plain text, programming languages, and calculators also use a single asterisk to represent the multiplication symbol, and it must be explicitly used; for example, 3x is written as 3 * x. Rather than using the ambiguous division sign (÷), division is usually represented with a vinculum, a horizontal line, as in 3/x + 1. In plain text and programming languages, a slash (also called a solidus) is used, e.g. 3 / (x + 1). Exponents are usually formatted using superscripts, as in x2. In plain text, the TeX mark-up language, and some programming languages such as MATLAB and Julia, the caret symbol, ^, represents exponents, so x2 is written as x ^ 2.

CHIMERA: Combining Ring-LWE-based Fully Homomorphic Encryption Schemes

Dimitar Petkov Jetchev

This paper proposes a practical hybrid solution for combining and switching between three popular Ring-LWE-based FHE schemes: TFHE, B/FV and HEAAN. This is achieved by first mapping the different plaintext spaces to a common algebraic structure and then by ...

WALTER DE GRUYTER GMBH2020

The Tonal Diffusion Model

Martin Alois Rohrmeier, Fabian Claude Moss, Robert Lieck

Pitch-class distributions are of central relevance in music information retrieval, computational musicology and various other fields, such as music perception and cognition. However, despite their structure being closely related to the cognitively and musi ...

2020

Geodesic Distance on Optimally Regularized Functional Connectomes Uncovers Individual Fingerprints

CHIMERA: Combining Ring-LWE-based Fully Homomorphic Encryption Schemes

The Tonal Diffusion Model

Geodesic Distance on Optimally Regularized Functional Connectomes Uncovers Individual Fingerprints

CHIMERA: Combining Ring-LWE-based Fully Homomorphic Encryption Schemes

The Tonal Diffusion Model