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In One Line and Close. Permutations as Linear Orders.
Published in Miklós Bóna, Combinatorics of Permutations, 2022
The number of alternating n-permutations is called an Euler number (not to be confused with the Eulerian numbers A(n,k)) and is denoted by En. The reader is invited to verify that E2=1, E3=2, E4=5, and E5=16. The Euler numbers have a very interesting exponential generating function. This is the content of the next theorem.
E
Published in Philip A. Laplante, Comprehensive Dictionary of Electrical Engineering, 2018
Euclidean distance is the special case of Minkowski distance when = 2. See also Minkowski distance. Euler number a topological invariant of an object having an orientable surface. Assuming that the surface is endowed with the structure of a graph with vertices, edges, and faces (where two neighboring faces have in common either a vertex or an edge with its two end-vertices, their interiors being disjoint): the Euler number is V - E + F, where V , E, and F are respectively the number of vertices, edges and faces; this number V - E + F does not depend on the choice of the subdivision into vertices, edges, and faces. For a bounded 2-D object in a Euclidean or digital plane, the Euler number is equal to the number of connected components of that object, minus the number of holes in it. For 2-D binary digital figures on a bounded grid, the Euler number can easily be computed by counting the number of occurrences of some local configurations of on and off pixels. Also called genus. eureka in a multiprocessor system, a coordination (synchronization) operation generating a completion signal that is logically ORed among all processors participating in an asynchronously parallel action. The interpretation and name come
E
Published in Phillip A. Laplante, Dictionary of Computer Science, Engineering, and Technology, 2017
Elder number a topological invariant of an object having an orientable surface. Assuming that the surface is endowed with the structure of a graph with vertices, edges, and faces (where two neighboring faces have in common either a vertex or an edge with its two end-vertices, their interiors being disjoint): the Euler number is V − E + F, where V, E, and F are, respectively, the number of vertices, edges, and faces; this number V−E+F does not depend on the choice of the subdivision into vertices, edges, and faces. For a bounded 2D object in a Euclidean or digital plane, the Euler number is equal to the number of connected components of that object, minus the number of holes in it. For 2D binary digital figures on a bounded grid, the Euler number can easily be computed by counting the number of occurrences of some local configurations of on and off pixels. Also called genus.
Reciprocity of degenerate poly-Dedekind-type DC sums
Published in Applied Mathematics in Science and Engineering, 2023
Lingling Luo, Yuankui Ma, Wenpeng Zhang, Taekyun Kim
In [1–9,16–30], the Euler polynomials are usually defined by From [19], Carlitz introduced the ordinary degenerate Euler polynomials The poly-Euler polynomials are defined by the generating function [29] when x = 0, are called the poly-Euler numbers. In particular, are called Euler polynomials.
On generalized degenerate Euler–Genocchi polynomials
Published in Applied Mathematics in Science and Engineering, 2023
Taekyun Kim, Dae San Kim, Hye Kyung Kim
For any nonzero , the degenerate exponentials are defined by where the generalized falling factorials are given by In [1,2], Carlitz introduced the degenerate Euler polynomials which are given by When x = 0, are called the degenerate Euler numbers.
A new family of Apostol–Genocchi polynomials associated with their certain identities
Published in Applied Mathematics in Science and Engineering, 2023
Nabiullah Khan, Saddam Husain, Talha Usman, Serkan Araci
The following generating function (see [4]) defines the Bell polynomials of two variables (i.e. bivariate Bell polynomials): When u = 0, are known as classical Bell polynomials (or exponential polynomials) and are described by the following generating function (see [11–14]): If v = 1 in (2) i.e. are known as Bell numbers described by the following generating function (see [11]): The following generating function defines Euler polynomials and Euler numbers (see [15,16]): If u = 0 then the Euler numbers are described by the following generating function: Dattoli et al. [17] introduced the Bernoulli polynomials and Bernoulli numbers, which are defined by the following generating function If u = 0 then the Bernoulli numbers are described by the following generating function: The following generating function defines Genocchi polynomials and Genocchi numbers (see [18,19]): If u = 0 then the Genocchi numbers are described by the following generating function: The Euler polynomials, Bernoulli polynomials and Genocchi polynomials of order (see [20–22]) are defined by the following generating function as follows: If we take u = 0 in (10), (11) and (12) i.e. , and are called Euler numbers, Bernoulli numbers and Genocchi numbers of order η are defined as follows: The Apostol–Bernoulli polynomials of order η (see [10,23]) are defined by the generating function as: with and where are known as Apostol–Bernoulli numbers of order η.