scholarly journals Decompositions of Complete Multigraphs into Cyclic Designs

2020 ◽  
Vol 4 (2) ◽  
pp. 80
Author(s):  
Mowafaq Alqadri ◽  
Haslinda Ibrahim ◽  
Sharmila Karim
Keyword(s):  
Positive Integer ◽  
Difference Set ◽  
Cycle System ◽  
The Difference ◽  
Complete Multigraphs

Let  and  be positive integer,  denote a complete multigraph. A decomposition of a graph  is a set of subgraphs of  whose edge sets partition the edge set of . The aim of this paper, is to decompose a complete multigraph  into cyclic -cycle system according to specified conditions. As the main consequence, construction of decomposition of  into cyclic Hamiltonian wheel system, where , is also given. The difference set method is used to construct the desired designs.

A method of version merging for computer-aided design files based on feature extraction

2010 ◽  
Vol 225 (2) ◽  
pp. 463-471 ◽  
Author(s):  
C Sun ◽  
D Guo ◽  
H Gao ◽  
L Zou ◽  
H Wang
Keyword(s):  
Feature Extraction ◽  
Computer Aided Design ◽  
Application Programming Interface ◽  
Difference Set ◽  
Computer Aided ◽  
Application Programming ◽  
Feature Difference ◽  
The Difference ◽  
Aided Design ◽  
The Given

In order to manage the version files and maintain the latest version of the computer-aided design (CAD) files in asynchronous collaborative systems, one method of version merging for CAD files is proposed to resolve the problem based on feature extraction. First of all, the feature information is extracted based on the feature attribute of CAD files and stored in a XML feature file. Then, analyse the feature file, and the feature difference set is obtained by the given algorithm. Finally, the merging result of the difference set and the master files with application programming interface (API) interface functions is achieved, and then the version merging of CAD files is also realized. The application in Catia validated that the proposed method is feasible and valuable in engineering.

scholarly journals On the 2-adic order of Stirling numbers of the second kind and their differences

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Tamás Lengyel
Keyword(s):  
Positive Integer ◽  
Stirling Numbers ◽  
Bell Polynomials ◽  
Binary Representation ◽  
Exact Order ◽  
Special Cases ◽  
International Audience ◽  
The Difference ◽  
Divisibility Properties ◽  
Positive Integers

International audience Let $n$ and $k$ be positive integers, $d(k)$ and $\nu_2(k)$ denote the number of ones in the binary representation of $k$ and the highest power of two dividing $k$, respectively. De Wannemacker recently proved for the Stirling numbers of the second kind that $\nu_2(S(2^n,k))=d(k)-1, 1\leq k \leq 2^n$. Here we prove that $\nu_2(S(c2^n,k))=d(k)-1, 1\leq k \leq 2^n$, for any positive integer $c$. We improve and extend this statement in some special cases. For the difference, we obtain lower bounds on $\nu_2(S(c2^{n+1}+u,k)-S(c2^n+u,k))$ for any nonnegative integer $u$, make a conjecture on the exact order and, for $u=0$, prove part of it when $k \leq 6$, or $k \geq 5$ and $d(k) \leq 2$. The proofs rely on congruential identities for power series and polynomials related to the Stirling numbers and Bell polynomials, and some divisibility properties.

scholarly journals Statistics on Lattice Walks and q-Lassalle Numbers

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Lenny Tevlin
Keyword(s):  
Positive Integer ◽  
Generating Function ◽  
Catalan Numbers ◽  
Binomial Coefficients ◽  
Integer Coefficients ◽  
Major Index ◽  
Lattice Walks ◽  
International Audience ◽  
The Difference ◽  
Positive Integers

International audience This paper contains two results. First, I propose a $q$-generalization of a certain sequence of positive integers, related to Catalan numbers, introduced by Zeilberger, see Lassalle (2010). These $q$-integers are palindromic polynomials in $q$ with positive integer coefficients. The positivity depends on the positivity of a certain difference of products of $q$-binomial coefficients.To this end, I introduce a new inversion/major statistics on lattice walks. The difference in $q$-binomial coefficients is then seen as a generating function of weighted walks that remain in the upper half-plan. Cet document contient deux résultats. Tout d’abord, je vous propose un $q$-generalization d’une certaine séquence de nombres entiers positifs, liés à nombres de Catalan, introduites par Zeilberger (Lassalle, 2010). Ces $q$-integers sont des polynômes palindromiques à $q$ à coefficients entiers positifs. La positivité dépend de la positivité d’une certaine différence de produits de $q$-coefficients binomial.Pour ce faire, je vous présente une nouvelle inversion/major index sur les chemins du réseau. La différence de $q$-binomial coefficients est alors considérée comme une fonction de génération de trajets pondérés qui restent dans le demi-plan supérieur.

scholarly journals On the Number of Witnesses in the Miller–Rabin Primality Test

Symmetry ◽  
2020 ◽  
Vol 12 (6) ◽  
pp. 890
Author(s):  
Shamil Talgatovich Ishmukhametov ◽  
Bulat Gazinurovich Mubarakov ◽  
Ramilya Gakilevna Rubtsova
Keyword(s):  
Positive Integer ◽  
Euler’S Totient Function ◽  
Number Of Factors ◽  
Primality Test ◽  
Average Probability ◽  
Test Errors ◽  
The Difference

In this paper, we investigate the popular Miller–Rabin primality test and study its effectiveness. The ability of the test to determine prime integers is based on the difference of the number of primality witnesses for composite and prime integers. Let W ( n ) denote the set of all primality witnesses for odd n. By Rabin’s theorem, if n is prime, then each positive integer a < n is a primality witness for n. For composite n, the power of W ( n ) is less than or equal to φ ( n ) / 4 where φ ( n ) is Euler’s Totient function. We derive new exact formulas for the power of W ( n ) depending on the number of factors of tested integers. In addition, we study the average probability of errors in the Miller–Rabin test and show that it decreases when the length of tested integers increases. This allows us to reduce estimations for the probability of the Miller–Rabin test errors and increase its efficiency.

scholarly journals A lower bound for a remainder term associated with the sum of digits function

1991 ◽  
Vol 34 (1) ◽  
pp. 121-142 ◽  
Author(s):  
D. M. E. Foster
Keyword(s):  
Lower Bound ◽  
Positive Integer ◽  
Upper Bound ◽  
Remainder Term ◽  
Fixed Integer ◽  
Sum Of Digits ◽  
Sum Of Digits Function ◽  
The Difference

For a fixed integer q≧2, every positive integer k = Σr≧0ar(q, k)qr where each ar(q, k)∈{0,1,2,…, q−1}. The sum of digits function α(q, k) Σr≧0ar(q, k) behaves rather erratically but on averaging has a uniform behaviour. In particular if , where n>1, then it is well known that A(q, n)∼½((q − 1)/log q)n logn as n → ∞. For odd values of q, a lower bound is now obtained for the difference 2S(q, n) = A(q, n)−½(q − 1))[log n/log q, where [log n/log q] denotes the greatest integer ≦log n /log q. This complements an upper bound already found.

scholarly journals Continuous functions of bounded nth variation

1969 ◽  
Vol 16 (3) ◽  
pp. 205-214
Author(s):  
Gavin Brown
Keyword(s):  
Continuous Function ◽  
Positive Integer ◽  
Decomposition Theorem ◽  
Continuous Functions ◽  
Unit Interval ◽  
Jordan Decomposition ◽  
The Real ◽  
Elementary Construction ◽  
The Difference

Let n be a positive integer. We give an elementary construction for the nth variation, Vn(f), of a real valued continuous function f and prove an analogue of the classical Jordan decomposition theorem. In fact, let C[0, 1] denote the real valued continuous functions on the closed unit interval, let An denote the semi-algebra of non-negative functions in C[0, 1] whose first n differences are non-negative, and let Sn denote the difference algebra An - An. We show that Sn is precisely that subset of C[0, 1] on which Vn(f)<∞. (Theorem 1).

scholarly journals Evidence for the s count macrogene

2015 ◽  
Author(s):  
Russell Schexnayder
Keyword(s):  
Cerebral Cortex ◽  
Positive Integer ◽  
Brain Size ◽  
Inbred Mice ◽  
Mouse Strains ◽  
Major Genes ◽  
Quantum Variable ◽  
Second Phases ◽  
The Difference ◽  
Repeat Count

Background: A macrogene is defined here as a gene on which successive mutations incrementing a repeat count produces successive punctuated evolutionary events in species that are homogeneous for it. The set of repeat count on the asp (abnormal spindle) family of gene is thought to affect brain size in mammals. Corticogenesis requires two integer valued (quantum) variables, the f and s counts, to determine the number of division cycles during the first and second phases, respectively, of neuron production in the cerebral cortex. Quantum ‘extra’ neuron theory hypothesizes that increments in a quantum variable, the n count, cause punctuated encephalization events in species that are homogenous for it. There is evidence in six pairs of inbred mice strains for one or more major genes affecting brain size. Results: The s count is probably equal to the n count plus a positive integer. The calculated n counts are different in three of the four pairs of strains studied where encephalization data has been previously published. Five different n counts have been found in eleven mouse strains. The difference between the n counts of humans and mice is about 25. Conclusions: Encephalization in mammals may be caused by a macrogene that determines the s count. This theory can be tested by determining the s counts of the various mice strains. However, the asp family of gene is probably not the s count macrogene because the difference in the asp counts of humans and mice of 13 (= 74 – 61) is much smaller than the difference in their s counts of around 25.

scholarly journals Sets with Few Differences in Abelian Groups

10.37236/5502 ◽  
2018 ◽  
Vol 25 (3) ◽  
Author(s):  
Mitchell Lee
Keyword(s):  
Abelian Group ◽  
Explicit Formula ◽  
Abelian Groups ◽  
Difference Set ◽  
The Difference ◽  
Fixed Cardinality

Let $(G, +)$ be an abelian group. In 2004, Eliahou and Kervaire found an explicit formula for the smallest possible cardinality of the sumset $A+A$, where $A \subseteq G$ has fixed cardinality $r$. We consider instead the smallest possible cardinality of the difference set $A-A$, which is always greater than or equal to the smallest possible cardinality of $A+A$ and can be strictly greater. We conjecture a formula for this quantity and prove the conjecture in the case that $G$ is an elementary abelian $p$-group. This resolves a conjecture of Bajnok and Matzke on signed sumsets.

scholarly journals About an even as the Sum or the Difference of Two Primes

2013 ◽  
Vol 5 ◽  
pp. 35-43
Author(s):  
Jamel Ghanouchi
Keyword(s):  
Positive Integer ◽  
Finite Number ◽  
Prime Numbers ◽  
The Difference ◽  
Definition Of

The present algebraic development begins by an exposition of the data of the problem. The definition of the primal radius r>0 is : For all positive integer x≥3 exists a finite number of integers called the primal radius r>0, for which x+r and x-r are prime numbers. The corollary is that 2x=(x+r)+(x-r) is always the sum of a finite number of primes. Also, for all positive integer x≥0, exists an infinity of integers r>0, for which x+r and r-x are prime numbers. The conclusion is that 2x=(x+r)-(r-x) is always an infinity of differences of primes.

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