An Existence Theorem for Room Squares*

1969 ◽  
Vol 12 (4) ◽  
pp. 493-497 ◽  
Author(s):  
R. C. Mullin ◽  
E. Nemeth
Keyword(s):  
Positive Integer ◽  
Existence Theorem ◽  
Prime Power ◽  
Unordered Pair ◽  
Room Squares ◽  
Square Array

It is shown that if v is an odd prime power, other than a prime of the form 22n + 1, then there exists a Room square of order v + 1.A room square of order 2n, where n is a positive integer, is an arrangement of 2n objects in a square array of 2 side 2n - 1, such that each of the (2n - 1)2 cells of the array is either-empty or contains exactly two distinct objects; each of the 2n objects appears exactly once in each row and column; and each (unordered) pair of objects occurs in exactly one cell.

A Room Design of Order 14

1968 ◽  
Vol 11 (2) ◽  
pp. 191-194 ◽  
Author(s):  
C D. O'Shaughnessy
Keyword(s):  
Positive Integer ◽  
Unordered Pair ◽  
Room Design ◽  
The Right ◽  
Square Array ◽  
Being Moved

A Room design of order 2n, where n is a positive integer, is an arrangement of 2n objects in a square array of side 2n - 1, such that each of the (2n - 1)2 cells of the array is either empty or contains exactly two distinct objects; each of the 2n objects appears exactly once in each row and column; and each (unordered) pair of objects occurs in exactly one cell. A Room design of order 2n is said to be cyclic if the entries in the (i + l) th row are obtained by moving the entries in the i th row one column to the right (with entries in the (2n - l)th column being moved to the first column), and increasing the entries in each occupied cell by l(mod 2n - 1), except that the digit 0 remains unchanged.

scholarly journals Duplication of Room squares

1972 ◽  
Vol 14 (1) ◽  
pp. 75-81 ◽  
Author(s):  
W. D. Wallis
Keyword(s):  
Unordered Pair ◽  
Room Squares ◽  
Square Array

ARoom squareRof order 2nis a way of arranging 2nobjects (usually 1,2,…, 2n) in a square arrayRof side 2n– 1 so that:(i) every cell of the array is empty or contains two objects;(ii) each unordered pair of objects occurs once inR(iii) every row and column ofRcontains one copy of each object.

Sign changes of Fourier coefficients of cusp forms supported on prime power indices

2014 ◽  
Vol 10 (08) ◽  
pp. 1921-1927 ◽  
Author(s):  
Winfried Kohnen ◽  
Yves Martin
Keyword(s):  
Positive Integer ◽  
Prime Power ◽  
Fourier Coefficients ◽  
Cusp Forms ◽  
Power Indices ◽  
Hecke Operator ◽  
Sign Changes ◽  
Integral Weight ◽  
Hecke Eigenform ◽  
Almost All

Let f be an even integral weight, normalized, cuspidal Hecke eigenform over SL2(ℤ) with Fourier coefficients a(n). Let j be a positive integer. We prove that for almost all primes p the sequence (a(pjn))n≥0 has infinitely many sign changes. We also obtain a similar result for any cusp form with real Fourier coefficients that provide the characteristic polynomial of some generalized Hecke operator is irreducible over ℚ.

A Room Design of Order 10

1964 ◽  
Vol 7 (3) ◽  
pp. 377-378 ◽  
Author(s):  
Louis Weisner
Keyword(s):  
Positive Integer ◽  
Unordered Pair ◽  
Room Design

A Room design of order 2n, where n is a positive integer, is an arrangement of 2n objects in a square of side 2n - 1, so that each of the (2n - 1)2 cells of the array is either empty or contains just two distinct objects; each of the 2n objects occurs just once in each row and in each column; and each (unordered) pair of objects occurs in just one cell.

scholarly journals Applying Balanced Generalized Weighing Matrices to Construct Block Designs

10.37236/1556 ◽  
2001 ◽  
Vol 8 (1) ◽  
Author(s):  
Yury J. Ionin
Keyword(s):  
Positive Integer ◽  
Prime Power ◽  
Block Designs ◽  
Weighing Matrices ◽  
Symmetric Designs

Balanced generalized weighing matrices are applied for constructing a family of symmetric designs with parameters $(1+qr(r^{m+1}-1)/(r-1),r^{m},r^{m-1}(r-1)/q)$, where $m$ is any positive integer and $q$ and $r=(q^{d}-1)/(q-1)$ are prime powers, and a family of non-embeddable quasi-residual $2-((r+1)(r^{m+1}-1)/(r-1),r^{m}(r+1)/2,r^{m}(r-1)/2)$ designs, where $m$ is any positive integer and $r=2^{d}-1$, $3\cdot 2^{d}-1$ or $5\cdot 2^{d}-1$ is a prime power, $r\geq 11$.

scholarly journals A congruence involving harmonic sums modulo pαqβ

2017 ◽  
Vol 13 (05) ◽  
pp. 1083-1094 ◽  
Author(s):  
Tianxin Cai ◽  
Zhongyan Shen ◽  
Lirui Jia
Keyword(s):  
Positive Integer ◽  
Prime Power ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Prime Factors ◽  
Necessary And Sufficient ◽  
Positive Integers

In 2014, Wang and Cai established the following harmonic congruence for any odd prime [Formula: see text] and positive integer [Formula: see text], [Formula: see text] where [Formula: see text] and [Formula: see text] denote the set of positive integers which are prime to [Formula: see text]. In this paper, we obtain an unexpected congruence for distinct odd primes [Formula: see text], [Formula: see text] and positive integers [Formula: see text], [Formula: see text] and the necessary and sufficient condition for [Formula: see text] Finally, we raise a conjecture that for [Formula: see text] and odd prime power [Formula: see text], [Formula: see text], [Formula: see text] However, we fail to prove it even for [Formula: see text] with three distinct prime factors.

ON THE CLASSIFICATION OF GROUPS OF PRIME-POWER ORDER BY COCLASS: THE 3-GROUPS OF COCLASS 2

2013 ◽  
Vol 23 (05) ◽  
pp. 1243-1288 ◽  
Author(s):  
BETTINA EICK ◽  
C. R. LEEDHAM-GREEN ◽  
M. F. NEWMAN ◽  
E. A. O'BRIEN
Keyword(s):  
Positive Integer ◽  
Prime Power ◽  
Prime Power Order ◽  
Computational Tools ◽  
Significant Step ◽  
New Phenomena ◽  
Full Classification

In this paper we take a significant step forward in the classification of 3-groups of coclass 2. Several new phenomena arise. Theoretical and computational tools have been developed to deal with them. We identify and are able to classify an important subset of the 3-groups of coclass 2. With this classification and further extensive computations, it is possible to predict the full classification. On the basis of the work here and earlier work on the p-groups of coclass 1, we formulate another general coclass conjecture. It implies that, given a prime p and a positive integer r, a finite computation suffices to determine the p-groups of coclass r.

scholarly journals On the Twin Designs with the Ionin–type Parameters

10.37236/1479 ◽  
1999 ◽  
Vol 7 (1) ◽  
Author(s):  
H. Kharaghani
Keyword(s):  
Positive Integer ◽  
Prime Power ◽  
Hadamard Matrix ◽  
Hadamard Matrices ◽  
Symmetric Designs ◽  
New Class

Let $4n^2$ be the order of a Bush-type Hadamard matrix with $q=(2n-1)^2$ a prime power. It is shown that there is a weighing matrix $$ W(4(q^m+q^{m-1}+\cdots+q+1)n^2,4q^mn^2) $$ which includes two symmetric designs with the Ionin–type parameters $$ \nu=4(q^m+q^{m-1}+\cdots+q+1)n^2,\;\;\; \kappa=q^m(2n^2-n), \;\;\; \lambda=q^m(n^2-n) $$ for every positive integer $m$. Noting that Bush–type Hadamard matrices of order $16n^2$ exist for all $n$ for which an Hadamard matrix of order $4n$ exist, this provides a new class of symmetric designs.

scholarly journals MÖBIUS–FROBENIUS MAPS ON IRREDUCIBLE POLYNOMIALS

2020 ◽  
pp. 1-12
Author(s):  
F. E. BROCHERO MARTÍNEZ ◽  
DANIELA OLIVEIRA ◽  
LUCAS REIS
Keyword(s):  
Positive Integer ◽  
Finite Field ◽  
Fixed Points ◽  
Prime Power ◽  
Irreducible Polynomials ◽  
Natural Action

Abstract Let n be a positive integer and let $\mathbb{F} _{q^n}$ be the finite field with $q^n$ elements, where q is a prime power. We introduce a natural action of the projective semilinear group on the set of monic irreducible polynomials over the finite field $\mathbb{F} _{q^n}$ . Our main results provide information on the characterisation and number of fixed points.

Symmetric graphs of order four times a prime power and valency seven

2018 ◽  
Vol 17 (05) ◽  
pp. 1850093 ◽  
Author(s):  
J. M. Pan ◽  
F. G. Yin
Keyword(s):  
Positive Integer ◽  
Prime Power ◽  
Arbitrary Positive Integer ◽  
Symmetric Graphs ◽  
Order Four

In this paper, we study symmetric graphs of valency seven and order [Formula: see text] with [Formula: see text] a prime and [Formula: see text] an arbitrary positive integer. It is proved that no such graph exists for any prime [Formula: see text], thus reducing the study to the case [Formula: see text]. The result is then used to determine all such graphs of order [Formula: see text], and such graphs of order [Formula: see text] with ‘[Formula: see text]’ can be inductively investigated similarly.

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