scholarly journals A Generalization of Partition Identities for First Differences of Partitions of $n$ Into at Most $m$ Parts

10.37236/8199 ◽  
2021 ◽  
Vol 28 (3) ◽  
Author(s):  
Acadia Larsen
Keyword(s):  
Finite Number ◽  
Linear Combination ◽  
Prime Power ◽  
Power Number ◽  
Partition Identities ◽  
Partition Identity ◽  
Combinatorial Interpretations ◽  
Finite Set

We show for a prime power number of parts $m$ that the first differences of partitions into at most $m$ parts can be expressed as a non-negative linear combination of partitions into at most $m-1$ parts. To show this relationship, we combine a quasipolynomial construction of $p(n,m)$ with a new partition identity for a finite number of parts. We prove these results by providing combinatorial interpretations of the quasipolynomial of $p(n,m)$ and the new partition identity.  We extend these results by establishing conditions for when partitions of $n$ with parts coming from a finite set $A$ can be expressed as a non-negative linear combination of partitions with parts coming from a finite set $B$.

The inverse problem of recovering the coefficients of a differential equations on a graph

2020 ◽  
Vol 28 (5) ◽  
pp. 727-738
Author(s):  
Victor Sadovnichii ◽  
Yaudat Talgatovich Sultanaev ◽  
Azamat Akhtyamov
Keyword(s):  
Inverse Problem ◽  
Inverse Problems ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Finite Number ◽  
Characteristic Equation ◽  
Arbitrary Number ◽  
New Class ◽  
Finite Set

AbstractWe consider a new class of inverse problems on the recovery of the coefficients of differential equations from a finite set of eigenvalues of a boundary value problem with unseparated boundary conditions. A finite number of eigenvalues is possible only for problems in which the roots of the characteristic equation are multiple. The article describes solutions to such a problem for equations of the second, third, and fourth orders on a graph with three, four, and five edges. The inverse problem with an arbitrary number of edges is solved similarly.

scholarly journals Proofs of some partition identities conjectured by Kanade and Russell

2021 ◽  
Author(s):  
Hjalmar Rosengren
Keyword(s):  
Lie Algebra ◽  
Partition Identities ◽  
Partition Identity ◽  
Affine Lie Algebra ◽  
Level 2

AbstractKanade and Russell conjectured several Rogers–Ramanujan-type partition identities, some of which are related to level 2 characters of the affine Lie algebra $$A_9^{(2)}$$ A 9 ( 2 ) . Many of these conjectures have been proved by Bringmann, Jennings-Shaffer and Mahlburg. We give new proofs of five conjectures first proved by those authors, as well as four others that have been open until now. Our proofs for the new cases use quadratic transformations for Askey–Wilson and Rogers polynomials. We also obtain some related results, including a partition identity conjectured by Capparelli and first proved by Andrews.

scholarly journals On linear independence for integer translates of a finite number of functions

1993 ◽  
Vol 36 (1) ◽  
pp. 69-85 ◽  
Author(s):  
Rong-Qing Jia ◽  
Charles A. Micchelli
Keyword(s):  
Finite Number ◽  
Linear Combination ◽  
Linear Independence ◽  
Sufficient Conditions ◽  
Basis Functions ◽  
Partial Difference ◽  
Integer Translates ◽  
First Case ◽  
Compactly Supported Functions ◽  
Necessary And Sufficient

We investigate linear independence of integer translates of a finite number of compactly supported functions in two cases. In the first case there are no restrictions on the coefficients that may occur in dependence relations. In the second case the coefficient sequences are restricted to be in some lp space (1 ≦ p ≦ ∞) and we are interested in bounding their lp-norms in terms of the Lp-norm of the linear combination of integer translates of the basis functions which uses these coefficients. In both cases we give necessary and sufficient conditions for linear independence of integer translates of the basis functions. Our characterization is based on a study of certain systems of linear partial difference and differential equations, which are of independent interest.

Permutation endomorphisms and refinement of a theorem of Birkhoff

1960 ◽  
Vol 56 (4) ◽  
pp. 322-328 ◽  
Author(s):  
H. K. Farahat ◽  
L. Mirsky
Keyword(s):  
Abelian Group ◽  
Finite Number ◽  
Linear Combination ◽  
Finite Linear Combination ◽  
Usual Manner ◽  
Restricted Permutation ◽  
Image Position ◽  
Ring Of Endomorphisms ◽  
Finite Linear

Let be a free additive abelian group, and let be a basis of , so that every element of can be expressed in a unique way as a (finite) linear combination with integral coefficients of elements of . We shall be concerned with the ring of endomorphisms of , the sum and product of the endomorphisms φ, χ being defined, in the usual manner, by the equationsA permutation of a set will be called restricted if it moves only a finite number of elements. We call an endomorphism of a permutation endomorphism if it induces a restricted permutation of the basis .

A Polynomial Model for Logics with a Prime Power Number of Truth Values

2010 ◽  
Vol 46 (2) ◽  
pp. 205-221 ◽  
Author(s):  
Antonio Hernando ◽  
Eugenio Roanes-Lozano ◽  
Luis M. Laita
Keyword(s):  
Prime Power ◽  
Polynomial Model ◽  
Power Number ◽  
Truth Values

On Carleman Integral Operators

1970 ◽  
Vol 13 (3) ◽  
pp. 351-357
Author(s):  
Charles G. Costley
Keyword(s):  
Finite Number ◽  
Integral Operators ◽  
Finite Set ◽  
Limit Points ◽  
Image Position ◽  
Set Of Points

L2(a, b)1with the property2were originally defined by T. Carleman [4]. Here he imposed on the kernel the conditions of measurability and hermiticity,3for all x with the exception of a countable set with a finite number of limit points and4where Jδ denotes the interval [a, b] with the exception of subintervals |x - ξv| < δ; here ξv represents a finite set of points for which (3) fails to hold.

scholarly journals SHIFTED AND SHIFTLESS PARTITION IDENTITIES II

2007 ◽  
Vol 03 (01) ◽  
pp. 43-84 ◽  
Author(s):  
FRANK G. GARVAN ◽  
HAMZA YESILYURT
Keyword(s):  
Positive Integer ◽  
Theta Function ◽  
Computer Search ◽  
Modular Functions ◽  
Partition Identities ◽  
Partition Identity ◽  
Function Identity ◽  
Theta Function Identity ◽  
Fixed Positive Integer ◽  
Positive Integers

Let S and T be sets of positive integers and let a be a fixed positive integer. An a-shifted partition identity has the form [Formula: see text] Here p(S,n) is the number partitions of n whose parts are elements of S. For all known nontrivial shifted partition identities, the sets S and T are unions of arithmetic progressions modulo M for some M. In 1987, Andrews found two 1-shifted examples (M = 32, 40) and asked whether there were any more. In 1989, Kalvade responded with a further six. In 2000, the first author found 59 new 1-shifted identities using a computer search and showed how these could be proved using the theory of modular functions. Modular transformation of certain shifted identities leads to shiftless partition identities. Again let a be a fixed positive integer, and S, T be distinct sets of positive integers. A shiftless partition identity has the form [Formula: see text] In this paper, we show, except in one case, how all known 1-shifted and shiftless identities follow from a four-parameter theta-function identity due to Jacobi. New shifted and shiftless partition identities are proved.

An Algorithmic Solution for a Word Problem in Group Theory

1964 ◽  
Vol 16 ◽  
pp. 509-516 ◽  
Author(s):  
N. S. Mendelsohn
Keyword(s):  
Group Theory ◽  
Finite Number ◽  
Word Problem ◽  
Finite Index ◽  
Unit Element ◽  
Systematic Procedure ◽  
Algorithmic Solution ◽  
Finite Set ◽  
The Right

This paper describes a systematic procedure which yields in a finite number of steps a solution to the following problem. Let G be a group generated by a finite set of generators g1, g2, g3, . . . , gr and defined by a finite set of relations R1 = R2 = . . . = Rk = I, where I is the unit element of G and R1R2, . . . , Rk are words in the gi and gi-1. Let H be a subgroup of G, known to be of finite index, and generated by a finite set of words, W1, W2, . . . , Wt. Let W be any word in G. Our problem is the following. Can we find a new set of generators for H, together with a set of representatives h1 = 1, h2, . . . , hu of the right cosets of H (i.e. G = H1 + Hh2 + . . . + Hhu) such that W can be expressed in the form W = Uhp, where U is a word in .

Hole dissections for planar figures

2021 ◽  
Vol 105 (563) ◽  
pp. 237-244
Author(s):  
Greg N. Frederickson
Keyword(s):  
Finite Number ◽  
Geometric Figure ◽  
Finite Set

A geometric dissection is a cutting of a geometric figure (or a finite set of figures) into pieces that we can rearrange to form another geometric figure (or finite set of figures). If our figures are required to be polygons, then there is always a dissection that has just a finite number of pieces. This was established by John Lowry [1], William Wallace [2], Farkas Bolyai [3], and Karl Gerwien [4]. The American Sam Loyd [5] and the Englishman Henry Ernest Dudeney [6, 7] emphasised the goal of minimising the number of pieces that resulted from such a standard dissection.

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