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Reciprocal Sum of Palindromes
[article]

2018
*
arXiv
*
pre-print

Let s_b be the

arXiv:1803.00161v2
fatcat:xjteqxgfhvcpxbofys6emnoyg4
*reciprocal**sum**of*all b-adic*palindromes*. It is not difficult to show that s_b converges. ... A positive integer n is said to be a*palindrome*in base b (or b-adic*palindrome*) if the representation*of*n = (a_k a_k-1... a_0)_b in base b with a_k ≠ 0 has the symmetric property a_k-i = a_i for every ... Acknowledgments Phakhinkon Phunphayap receives a scholarship from Science Achievement Scholarship*of*Thailand(SAST). ...##
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The Deny Proof of the Palindrome Number Conjecture

2022
*
Advances in Pure Mathematics
*

Theorem 2: The

doi:10.4236/apm.2022.124026
fatcat:ejwkb54qzra57g7wpvmnx6ngf4
*sum**of*any*palindrome*number H and any non-*palindrome*j number must be a non-*palindrome*number. ...*Palindrome*number conjecture: Take any non-*palindromic*natural number with two or more digits, add its inverse ordinal number, continue to use the inverted number*of**sum*plus*sum*, repeat this process continuously ... Conflicts*of*Interest The authors declare no conflicts*of*interest regarding the publication*of*this paper. ...##
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On (self-) reciprocal Appell polynomials: Symmetry and Faulhaber-type polynomials
[article]

2021
*
arXiv
*
pre-print

In the context

arXiv:2105.15169v2
fatcat:ovacqdjz5vg5dc7ms5fvep7bo4
*of*power*sums*the results transfer to the classical Faulhaber polynomials. ... Furthermore, the coefficients*of*the latter polynomials are given by values*of*derivatives*of*generalized*reciprocal*Appell polynomials. ... If Pn (x) = Pn⋆ (x), then Pn is called self-*reciprocal*or*palindromic*, since the coefficients*of*Pn form a*palindromic*sequence such that cν = cn−ν for 0 ≤ ν ≤ n. ...##
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On Polynomial Pairs of Integers
[article]

2014
*
arXiv
*
pre-print

A pair (A,B)

arXiv:1210.7593v2
fatcat:sadq6mja5fajjbr6rcn7um3fje
*of*positive integers is said to be*palindromic*if the reversal*of*the product A × B is equal to the product*of*the reversals*of*A and*of*B. ... In this paper, we use polynomial pairs in constructing and in studying the properties*of**palindromic*pairs. It is shown that polynomial pairs are always*palindromic*. ... Thus, by Proposition 9, (A × A * ) ∞ = d i=0 a 2 i , (3) which is the*sum**of*the squares*of*the digits*of*A. ...##
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Page 1302 of Mathematical Reviews Vol. 40, Issue 6
[page]

1970
*
Mathematical Reviews
*

The law

*of*quadratic*reciprocity*has been intentionally omitted from the book. ... Let 7(n)=>$in.a¢n=the*sum**of*all proper divisors*of*n and then iterate this function: r™(n) = 7(r*~(n)), r(n) =n. ...##
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Generalized congruence properties of the restricted partition function p(n,m)

2012
*
The Ramanujan journal
*

The existence

doi:10.1007/s11139-012-9382-x
fatcat:5t6ky426qrgwjh3vfnvqzyxqca
*of*Ramanujan-type congruences are virtually unknown for p(n, m), the closely related restricted partition function that enumerates the number*of*partitions*of*n into exactly m parts. ... In this paper we establish explicit Ramanujan-type congruences for p(n, ) modulo any power*of*that prime α . In addition, we establish general congruence relations for p(n, ) modulo α for any n. ... When n is a multiple*of*the prime , the congruence relation resides in the difference*of*a single case*of**palindromic*partners and not the*sum**of*the differences*of*two instances*of**palindromic*partners ...##
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Palindromic quadratization and structure-preserving algorithm for palindromic matrix polynomials of even degree

2011
*
Numerische Mathematik
*

In this paper, we propose a

doi:10.1007/s00211-011-0370-7
fatcat:74qlivtuhbgrpiskbjffsi5nmq
*palindromic*quadratization approach, transforming a*palindromic*matrix polynomial*of*even degree to a*palindromic*quadratic pencil. ... Numerical experiments show that the relative residuals for eigenpairs*of**palindromic*polynomial eigenvalue problems computed by*palindromic*quadratized eigenvalue problems are better than those via*palindromic*... The jth column*of*an identity matrix is denoted by e j . The direct*sum**of*two matrices is denoted by "⊕". This paper is organized as follows. ...##
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Implicit QR algorithms for palindromic and even eigenvalue problems

2008
*
Numerical Algorithms
*

In the spirit

doi:10.1007/s11075-008-9226-3
fatcat:xawj2ofxirhbrbhrgg2at4yzuq
*of*the Hamiltonian QR algorithm and other bidirectional chasing algorithms, a structure-preserving variant*of*the implicit QR algorithm for*palindromic*eigenvalue problems is proposed. ... By an extension*of*the implicit Q theorem, the*palindromic*QR algorithm is shown to be equivalent to a previously developed explicit version. ... In order to assess the solvability*of*(39), (40), we note that their*sum*(M 31 + N 31 )X + X (M 31 − N 31 ) = rhs (41) (with rhs the*sum**of*the right hand sides*of*(39), (40)) is*of*the form (20) and that ...##
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Stanley's Major Contributions to Ehrhart Theory
[article]

2015
*
arXiv
*
pre-print

This expository paper features a few highlights

arXiv:1407.0255v2
fatcat:cyk4hszsdrgp7m7zrttnadc6za
*of*Richard Stanley's extensive work in Ehrhart theory, the study*of*integer-point enumeration in rational polyhedra. ...*RECIPROCITY*Theorem 2 is an example*of*a combinatorial*reciprocity*theorem: we get interesting information out*of*a counting function when we evaluate it at a negative integer (and so, a priori the counting ... Furthermore, since all ingredients for the*sum*in Theorem 12 are nonnegative, this gives another (and the first combinatorial) proof*of*Theorem 6. ...##
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Perturbation results related to palindromic eigenvalue problems

2009
*
ANZIAM Journal
*

From the transpose or Hermitian

doi:10.21914/anziamj.v50i0.388
fatcat:44c4w5fsojdd7akbda5mexq74e
*of*(1), a*palindromic*eigenvalue problem possesses a spectrum σ(P ) containing both λ and its "*reciprocal*" 1/λ (with 0 and ∞ considered to be*reciprocal*to each other). ... Notice that only simple eigenvalues (or*sum*and averages*of*multiple eigenvalues) are ... Note also that the perturbation in δA 0 may be nonsymmetric, pushing a pair*of**reciprocal**palindromic*eigenvalues to one which are not*reciprocal*(or approximately so when δA 0 is small). ...##
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Reversing Palindromic Enumeration in Rank Two Free Groups
[article]

2016
*
arXiv
*
pre-print

The E-words, primitive words with

arXiv:1311.6358v4
fatcat:llkepmeeqbbldc2diwxdvri6si
*palindromic*product forms, are also indexed by rational numbers and infinity. ... The Gilman-Maskit algorithm for determining the discreteness or non-discreteness*of*a two-generator subgroup*of*PSL_2R terminates with a pair*of*generators that are Farey words. ... It is well-known that the Farey*sum**of*the parents*of*p/q is equal to p/q. ...##
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Structured Polynomial Eigenvalue Problems: Good Vibrations from Good Linearizations

2006
*
SIAM Journal on Matrix Analysis and Applications
*

In this paper several useful classes

doi:10.1137/050628362
fatcat:ni5syrieujh6zlmyo2vsadldh4
*of*structured polynomials (e.g.,*palindromic*, even, odd) are identified and the relationships between them explored. ... A special class*of*linearizations which reflect the structure*of*these polynomials, and therefore preserve symmetries in their spectra, is introduced and investigated. ... We thank the mathematics departments*of*the University*of*Manchester, Technische Universität Berlin, and Western Michigan University, and the Banff International Research Station for giving us the opportunity ...##
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Palindromic Bernoulli distributions

2016
*
Electronic Journal of Statistics
*

We introduce and study a subclass

doi:10.1214/16-ejs1175
fatcat:vo75uv2errdoviwsckcer3o34u
*of*joint Bernoulli distributions which has the*palindromic*property. ... For such distributions the vector*of*joint probabilities is unchanged when the order*of*the elements is reversed. ... Bulgarelli, Univ.*of*Florence, for the data; G. Ottaviani and the referees for their helpful comments. We used Matlab R , 2015, and for exact median-dichotomizing, 'jitter.m' by R. ...##
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Page 3697 of Mathematical Reviews Vol. , Issue 82i
[page]

1982
*
Mathematical Reviews
*

Using the

*reciprocity*theorem for Dedekind*sums*, the author shows that m,(h)+m,(k)+(h— 1(k— 1)/4=0 (mod 4). ... A congruence (mod 2) is used in a familiar proof*of*Gauss*of*the law*of*quadratic*reciprocity*. Bruce C. Berndt (Urbana, II1.) ...##
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Palindromic Bernoulli distributions
[article]

2016
*
arXiv
*
pre-print

We introduce and study a subclass

arXiv:1510.09072v2
fatcat:gznrja5udzahdnc25bdadw6jze
*of*joint Bernoulli distributions which has the*palindromic*property. ... For such distributions the vector*of*joint probabilities is unchanged when the order*of*the elements is reversed. ... Bulgarelli, Univ.*of*Florence, for the data; G. Ottaviani and the referees for their helpful comments. We used Matlab R , 2015, and for exact median-dichotomizing, 'jitter.m' by R. ...
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