Maxima and Minima of Finite Sums

1936 ◽  
Vol 43 (3) ◽  
pp. 164
Author(s):  
Tomlinson Fort
Keyword(s):  
Finite Sums

On finite sums of periodic functions

2020 ◽  
Vol 27 (2) ◽  
pp. 265-269
Author(s):  
Alexander Kharazishvili
Keyword(s):  
Analytic Function ◽  
Real Line ◽  
Real Analytic Function ◽  
Periodic Functions ◽  
Uniform Limit ◽  
The Real ◽  
Pointwise Limit ◽  
Finite Sums ◽  
Real Analytic

AbstractIt is shown that any function acting from the real line {\mathbb{R}} into itself can be expressed as a pointwise limit of finite sums of periodic functions. At the same time, the real analytic function {x\rightarrow\exp(x^{2})} cannot be represented as a uniform limit of finite sums of periodic functions and, simultaneously, this function is a locally uniform limit of finite sums of periodic functions. The latter fact needs the techniques of Hamel bases.

scholarly journals Complete and incomplete Bell polynomials associated with Lah–Bell numbers and polynomials

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Taekyun Kim ◽  
Dae San Kim ◽  
Lee-Chae Jang ◽  
Hyunseok Lee ◽  
Han-Young Kim
Keyword(s):  
Bell Polynomials ◽  
Bell Numbers ◽  
Bell Number ◽  
Finite Sums

AbstractThe nth r-extended Lah–Bell number is defined as the number of ways a set with $n+r$ n + r elements can be partitioned into ordered blocks such that r distinguished elements have to be in distinct ordered blocks. The aim of this paper is to introduce incomplete r-extended Lah–Bell polynomials and complete r-extended Lah–Bell polynomials respectively as multivariate versions of r-Lah numbers and the r-extended Lah–Bell numbers and to investigate some properties and identities for these polynomials. From these investigations we obtain some expressions for the r-Lah numbers and the r-extended Lah–Bell numbers as finite sums.

scholarly journals Elements of finite order in Stone-Čech compactifications

1993 ◽  
Vol 36 (1) ◽  
pp. 49-54 ◽  
Author(s):  
John Baker ◽  
Neil Hindman ◽  
John Pym
Keyword(s):  
Compact Group ◽  
Finite Order ◽  
Free Semigroup ◽  
Continuous Homomorphism ◽  
Discrete Topology ◽  
Natural Extensions ◽  
Finite Sums

Let S be a free semigroup (on any set of generators). When S is given the discrete topology, its Stone-Čech compactification has a natural semigroup structure. We give two results about elements p of finite order in βS. The first is that any continuous homomorphism of βS into any compact group must send p to the identity. The second shows that natural extensions, to elements of finite order, of relationships between idempotents and sequences with distinct finite sums, do not hold.

scholarly journals A Three-Dimensional Magnetic Force SolutionBetween Axially-Polarized Permanent-Magnet Cylinders for Different Magnetic Arrangements

2020 ◽  
Vol 7 ◽  
Keyword(s):  
Experimental Data ◽  
Rare Earth ◽  
Field Strength ◽  
Permanent Magnet ◽  
Magnetic Force ◽  
Three Dimensional ◽  
Method Of Images ◽  
Field Equations ◽  
Rare Earth Magnet ◽  
Finite Sums

A three-dimensional field solution is presented foraxially polarized permanent magnet cylinders. The fieldcomponents are expressed in terms of finite sums of elementaryfunctions and are easily programmable. They can be used todetermine the operating point of rare-earth magnet cylinders.They are also useful for performing rapid parametriccalculations of field strength as a function of materialproperties and dimensions. The field components aredeveloped for different magnet arrangements by taking intoaccount the back iron. Also the method of images is used. Usingthe field equations, three-dimensional analytical expressionsare derived for computing the magnetic force between axiallypolarized permanent-magnet cylinders for different magneticarrangements. The field calculated results are in goodagreement with the experimental data.

CLASS NUMBER FORMULAS FOR CERTAIN BICYCLIC BIQUADRATIC NUMBER FIELDS AS FINITE SUMS INVOLVING JACOBI ELLIPTIC FUNCTIONS

2012 ◽  
Vol 08 (05) ◽  
pp. 1257-1270
Author(s):  
M. A. GÓMEZ-MOLLEDA ◽  
JOAN-C. LARIO
Keyword(s):  
Class Number ◽  
Quadratic Field ◽  
Elliptic Functions ◽  
Number Fields ◽  
Jacobi Elliptic Functions ◽  
Classical Class ◽  
Class Number Formula ◽  
Number Formula ◽  
Finite Sums ◽  
Finite Sum

We give formulas for the class numbers of bicyclic biquadratic number fields containing an imaginary quadratic field of class number one. The class number is expressed as a finite sum in terms of the basic Jacobi elliptic functions, playing a similar role as the trigonometric sine in Dirichlet classical class number formula.

scholarly journals Independent finite sums in graphs defined on the natural numbers

1998 ◽  
Vol 181 (1-3) ◽  
pp. 289-294 ◽  
Author(s):  
Tomasz Luczak ◽  
Vojtěch Rödl ◽  
Tomasz Schoen
Keyword(s):  
Natural Numbers ◽  
Finite Sums

Finite Heisenberg Quantum Spin Chain

2019 ◽  
pp. 667-686
Author(s):  
Hans-Peter Eckle
Keyword(s):  
Bethe Ansatz ◽  
Conformal Symmetry ◽  
Finite Size ◽  
Quantum Spin ◽  
Ansatz Method ◽  
Finite Sums ◽  
Mathematical Techniques ◽  
Higher Order Corrections ◽  
Finite Size Corrections ◽  
Maclaurin Formula

The Bethe ansatz genuinely considers a finite system. The extraction of finite-size results from the Bethe ansatz equations is of genuine interest, especially against the background of the results of finite-size scaling and conformal symmetry in finite geometries. The mathematical techniques introduced in chapter 19 permit a systematic treatment in this chapter of finite-size corrections as corrections to the thermodynamic limit of the system. The application of the Euler-Maclaurin formula transforming finite sums into integrals and finite-size corrections transforms the Bethe ansatz equations into Wiener–Hopf integral equations with inhomogeneities representing the finite-size corrections solvable using the Wiener–Hopf technique. The results can be compared to results for finite systems obtained from other approaches that are independent of the Bethe ansatz method. It briefly discusses higher-order corrections and offers a general assessment of the finite-size method.

Hyperplans poissoniens et compacts de Steiner

1974 ◽  
Vol 6 (03) ◽  
pp. 563-579 ◽  
Author(s):  
G. Matheron
Keyword(s):  
Stationary Process ◽  
Convex Set ◽  
Compact Convex ◽  
Minkowski Sum ◽  
Isotropic Case ◽  
Geometrical Properties ◽  
Steiner Formula ◽  
Compact Convex Set ◽  
Finite Sums ◽  
Linear Variety

A compact convex set in RN is Steiner if it is a finite Minkowski sum of line segments, or a limit of such finite sums, and then satisfies an extension of the Steiner formula. With each Poisson hyperplane stationary process A is uniquely associated a Steiner set M, and for any linear variety V, the Steiner set associated with is the projection of M on V. The density of the order k network Ak (i.e., the set of the intersections of k hyperplanes belonging to A) is linked with simple geometrical properties of M. In the isotropic case, the expression of the covariance measures associated with Ak is derived and compared with the analogous results obtained for (N — k)-dimensional Poisson flats.

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