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### Reciprocal Sum of Palindromes [article]

Phakhinkon Phunphayap, Prapanpong Pongsriiam
2018 arXiv   pre-print
Let s_b be the 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).  ...

### The Deny Proof of the Palindrome Number Conjecture

Maoze Wang, Zhenxiang He, Meiyi Wang
Theorem 2: The 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.  ...

### On (self-) reciprocal Appell polynomials: Symmetry and Faulhaber-type polynomials [article]

Bernd C. Kellner
2021 arXiv   pre-print
In the context 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.  ...

### On Polynomial Pairs of Integers [article]

Martianus Frederic Ezerman, Bertrand Meyer, Patrick Sole
2014 arXiv   pre-print
A pair (A,B) 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.  ...

### 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.  ...

### Generalized congruence properties of the restricted partition function p(n,m)

Brandt Kronholm
2012 The Ramanujan journal
The existence 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  ...

### Palindromic quadratization and structure-preserving algorithm for palindromic matrix polynomials of even degree

Tsung-Ming Huang, Wen-Wei Lin, Wei-Shuo Su
2011 Numerische Mathematik
In this paper, we propose a 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.  ...

### Implicit QR algorithms for palindromic and even eigenvalue problems

Daniel Kressner, Christian Schröder, David S. Watkins
2008 Numerical Algorithms
In the spirit 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  ...

### Stanley's Major Contributions to Ehrhart Theory [article]

Matthias Beck
2015 arXiv   pre-print
This expository paper features a few highlights 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.  ...

### Perturbation results related to palindromic eigenvalue problems

Eric King-wah Chu, Wen-Wei Lin, Chern-Shuh Wang
2009 ANZIAM Journal
From the transpose or Hermitian 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).  ...

### Reversing Palindromic Enumeration in Rank Two Free Groups [article]

Andrew E. Silverio
2016 arXiv   pre-print
The E-words, primitive words with 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.  ...

### Structured Polynomial Eigenvalue Problems: Good Vibrations from Good Linearizations

D. Steven Mackey, Niloufer Mackey, Christian Mehl, Volker Mehrmann
2006 SIAM Journal on Matrix Analysis and Applications
In this paper several useful classes 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  ...

### Palindromic Bernoulli distributions

Giovanni M. Marchetti, Nanny Wermuth
2016 Electronic Journal of Statistics
We introduce and study a subclass 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.  ...

### 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.)  ...

### Palindromic Bernoulli distributions [article]

Giovanni M. Marchetti, Nanny Wermuth
2016 arXiv   pre-print
We introduce and study a subclass 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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