In abstract algebra, an additive monoid is said to be zerosumfree, conical, centerless or positive if nonzero elements do not sum to zero. Formally:
This means that the only way zero can be expressed as a sum is as .
In abstract algebra, an additive monoid is said to be zerosumfree, conical, centerless or positive if nonzero elements do not sum to zero. Formally:
This means that the only way zero can be expressed as a sum is as .
In mathematics, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural setting for studying divisibility. In an integral domain, every nonzero element a has the cancellation property, that is, if a ≠ 0, an equality ab = ac implies b = c.
In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0.
In the mathematical subfield of numerical analysis, a B-spline or basis spline is a spline function that has minimal support with respect to a given degree, smoothness, and domain partition. Any spline function of given degree can be expressed as a linear combination of B-splines of that degree. Cardinal B-splines have knots that are equidistant from each other. B-splines can be used for curve-fitting and numerical differentiation of experimental data.
The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter ζ (zeta), is a mathematical function of a complex variable defined as
In the branch of mathematics known as real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval. It was presented to the faculty at the University of Göttingen in 1854, but not published in a journal until 1868. For many functions and practical applications, the Riemann integral can be evaluated by the fundamental theorem of calculus or approximated by numerical integration.
In linear algebra, the trace of a square matrix A, denoted tr(A), is defined to be the sum of elements on the main diagonal of A. The trace is only defined for a square matrix.
In complex analysis, a pole is a certain type of singularity of a complex-valued function of a complex variable. In some sense, it is the simplest type of singularity. Technically, a point z0 is a pole of a function f if it is a zero of the function 1/f and 1/f is holomorphic in some neighbourhood of z0.
In mathematics, the dot product or scalar product is an algebraic operation that takes two equal-length sequences of numbers, and returns a single number. In Euclidean geometry, the dot product of the Cartesian coordinates of two vectors is widely used. It is often called the inner product of Euclidean space, even though it is not the only inner product that can be defined on Euclidean space.
In mathematics, an empty sum, or nullary sum, is a summation where the number of terms is zero. The natural way to extend non-empty sums is to let the empty sum be the additive identity.
In mathematics, the prime-counting function is the function counting the number of prime numbers less than or equal to some real number x. It is denoted by π(x).
In transcendental number theory, the Lindemann–Weierstrass theorem is a result that is very useful in establishing the transcendence of numbers. It states the following:
In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring formed from the set of polynomials in one or more indeterminates with coefficients in another ring, often a field.
In number theory, the Mertens function is defined for all positive integers n as
In mathematics, a harshad number in a given number base is an integer that is divisible by the sum of its digits when written in that base. Harshad numbers in base n are also known as n-harshad numbers. Harshad numbers were defined by D. R. Kaprekar, a mathematician from India. The word "harshad" comes from the Sanskrit harṣa (joy) + da (give), meaning joy-giver. The term "Niven number" arose from a paper delivered by Ivan M. Niven at a conference on number theory in 1977.
In number theory, Kaprekar's routine is an iterative algorithm that, with each iteration, takes a natural number in a given number base, creates two new numbers by sorting the digits of its number by descending and ascending order, and subtracts the second from the first to yield the natural number for the next iteration. It is named after its inventor, the Indian mathematician D. R. Kaprekar.
The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a more elementary kind of structure, the abelian group. The direct sum of two abelian groups and is another abelian group consisting of the ordered pairs where and . To add ordered pairs, we define the sum to be ; in other words addition is defined coordinate-wise. For example, the direct sum , where is real coordinate space, is the Cartesian plane, . A similar process can be used to form the direct sum of two vector spaces or two modules.
In number theory, the Padovan sequence is the sequence of integers P(n) defined by the initial values
In mathematics, especially in probability and combinatorics, a doubly stochastic matrix (also called bistochastic matrix) is a square matrix of nonnegative real numbers, each of whose rows and columns sums to 1, i.e.,
In mathematics, Dirichlet's test is a method of testing for the convergence of a series. It is named after its author Peter Gustav Lejeune Dirichlet, and was published posthumously in the Journal de Mathématiques Pures et Appliquées in 1862.
In mathematics, convergence tests are methods of testing for the convergence, conditional convergence, absolute convergence, interval of convergence or divergence of an infinite series .
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