Showing papers on "Coprime integers published in 2009"

Fast Scaling in the Residue Number System

Abstract: A new scheme for precisely scaling numbers in the residue number system (RNS) is presented. The scale factor K can be any number coprime to the RNS moduli. Lookup table implementations are used as a basis for comparisons between the new scheme and scaling schemes from the literature. It is shown that new scheme decreases hardware complexity compared to previous schemes without affecting time complexity.

Optimal probabilistic ring exploration by semi-synchronous oblivious robots

Abstract: We consider a team of k identical, oblivious, semi-synchronous mobile robots that are able to sense (i.e., view) their environment, yet are unable to communicate, and evolve on a constrained path. Previous results in this weak scenario show that initial symmetry yields high lower bounds when problems are to be solved by deterministic robots. In this paper, we initiate research on probabilistic bounds and solutions in this context, and focus on the exploration problem of anonymous unoriented rings of any size. It is known that Θ(logn) robots are necessary and sufficient to solve the problem with k deterministic robots, provided that k and n are coprime. By contrast, we show that four identical probabilistic robots are necessary and sufficient to solve the same problem, also removing the coprime constraint. Our positive results are constructive.

On Coprime Modules and Comodules

Abstract: Many observations about coalgebras were inspired by comparable situations for algebras. Despite the prominent role of prime algebras, the theory of a corresponding notion for coalgebras was not well understood so far. Coalgebras C over fields may be called coprime provided the dual algebra C* is prime. This definition, however, is not intrinsic—it strongly depends on the base ring being a field. The purpose of the article is to provide a better understanding of related notions for coalgebras over commutative rings by employing traditional methods from (co)module theory, in particular (pre)torsion theory. Dualizing classical primeness condition, coprimeness can be defined for modules and algebras. These notions are developed for modules and then applied to comodules. We consider prime and coprime, fully prime and fully coprime, strongly prime and strongly coprime modules and comodules. In particular, we obtain various characterisations of prime and coprime coalgebras over rings and fields.

Numerical semigroups: Apéry sets and Hilbert series

Abstract: Let a 1,…,a n be relatively prime positive integers, and let S be the semigroup consisting of all non-negative integer linear combinations of a 1,…,a n . In this paper, we focus our attention on AA-semigroups, that is semigroups being generated by almost arithmetic progressions. After some general considerations, we give a characterization of the symmetric AA-semigroups. We also present an efficient method to determine an Apery set and the Hilbert series of an AA-semigroup.

Efficient isomorphism testing for a class of group extensions

Abstract: The group isomorphism problem asks whether two given groups are isomorphic or not. Whereas the case where both groups are abelian is well understood and can be solved efficiently, very little is known about the complexity of isomorphism testing for nonabelian groups. In this paper we study this problem for a class of groups corresponding to one of the simplest ways of constructing nonabelian groups from abelian groups: the groups that are extensions of an abelian group $A$ by a cyclic group $\mathbb{Z}_m$. We present an efficient algorithm solving the group isomorphism problem for all the groups of this class such that the order of $A$ is coprime with $m$. More precisely, our algorithm runs in time almost linear in the orders of the input groups and works in the general setting where the groups are given as black-boxes.

Further improvements of lower bounds for the least common multiples of arithmetic progressions

Abstract: For relatively prime positive integers uo and r, we consider the arithmetic progression {u k := u 0 + kr} n k=0 . Define L n := lcm {u 0 , u 1 , ..., u n } and let a ≥ 2 be any integer. In this paper, we show that for integers α,r ≥ a and n > 2αr, we have L n > u 0 r α+a-2 (r + 1) n . In particular, letting a = 2 yields an improvement to the best previous lower bound on L n (obtained by Hong and Yang) for all but three choices of α, r > 2.

On the Hasse principle for Shimura curves

Abstract: Let C be an algebraic curve defined over a number field K, of positive genus and without K-rational points. We conjecture that there exists some extension field L over which C has points everywhere locally but not globally. We show that our conjecture holds for all but finitely many Shimura curves of the form X0D (N)/ℚ or X1D (N)/ℚ, where D > 1 and N are coprime squarefree positive integers. The proof uses a variation on a theorem of Frey, a gonality bound of Abramovich, and an analysis of local points of small degree.

On the conjecture of jeśmanowicz concerning pythagorean triples

Abstract: Let a,b,c be relatively prime positive integers such that a2+b2=c2 with b even. In 1956 Jeśmanowicz conjectured that the equation ax+by=cz has no solution other than (x,y,z)=(2,2,2) in positive integers. Most of the known results of this conjecture were proved under the assumption that 4 exactly divides b. The main results of this paper include the case where 8 divides b. One of our results treats the case where a has no prime factor congruent to 1 modulo 4, which can be regarded as a relevant analogue of results due to Deng and Cohen concerning the prime factors of b. Furthermore, we examine parities of the three variables x,y,z, and give new triples a,b,c such that the conjecture holds for the case where b is divisible by 8. In particular, to prove our results, we shall show an important result which asserts that if x,y,z are all even, then x/2,y/2,z/2 are all odd. Our methods are based on elementary congruence and several strong results on generalized Fermat equations given by Darmon and Merel.

DEFINING POWER SUMS OF n AND φ(n) INTEGERS

Abstract: Let n be a positive integer and φ(n) denotes the Euler phi function. It is well known that the power sum of n can be evaluated in closed form in terms of n. Also, the sum of all those φ(n) positive integers that are coprime to n and not exceeding n, is expressible in terms of n and φ(n). Although such results already exist in literature, but here we have presented some new analytical results in these connections. Some functional and integral relations are derived for the general power sums.

Uniform distribution of fractional parts related to pseudoprimes

Abstract: We estimate exponential sums with the Fermat-like quotients where and are positive integers, is composite, and is the largest prime factor of . Clearly, both and are integers if is a Fermat pseudoprime to base , and if is a Carmichael number, this is true for all coprime to . Nevertheless, our bounds imply that the fractional parts and are uniformly distributed, on average over for , and individually for . We also obtain similar results with the functions and .

Binary partitions of integers and Stern-Brocot-like trees

Abstract: We study partitions of an integer n in which all parts are powers of 2, no power of 2 appearing more than twice. If b(n) is the number of these, and B(n) = P j n b(j), then we nd (a) that gcd (b(n);b(n + 1)) = 1, and that every coprime (i;j) occurs exactly once as a pair of consecutive values (b) the asymptotic behavior of B(t2 n ), for xed 1 < t < 2, as n ! 1 (c) descriptions of b(n) in terms of the structure of the blocks of 0’s and 1’s in the binary expansion of n, and many other properties of these functions. This partition function has previously been studied, under dierent guises, in other areas of discrete mathematics, such as in the theory of the Stern-Brocot tree.

Cubes in products of terms in arithmetic progression

Abstract: Euler proved that the product of four positive integers in arithmetic progression is not a square. Gy}ory, using a result of Darmon and Merel, showed that the product of three coprime positive integers in arithmetic progression cannot be an l-th power for l ¸ 3. There is an extensive literature on longer arithmetic progressions such that the product of the terms is an (almost) power. In this paper we extend the range of k's such that the product of k coprime integers in arithmetic progression cannot be a cube when 2 < k < 39. We prove a similar result for almost cubes.

On Relatively Prime Sets

Abstract: Abstract Functions counting the number of subsets of {1, 2, . . . , n} having particular properties are defined by Nathanson. Here, generalizations in two directions are given.

Conditions for triangular decoupling control

Abstract: The main purpose of this article is to explore the relationship of two existing conditions for the triangular decoupling problem. The first one is the triangular-diagonal-dominance condition proposed by Hung and Anderson. The second one is the stable coprime factorisation-described condition proposed by Gomez and Goodwin, which has been proven as a necessary and sufficient condition for the triangular decoupling problem. This article proves that the two conditions are actually equivalent. It also provides easy-to-use criteria for assessment of the solvability of the triangular decoupling problem.

On Newman's conjecture and prime trees.

Abstract: In 1980, Carl Pomerance and J. L. Selfridge proved D. J. Newman’s coprime mapping conjecture: If n is a positive integer and I is a set of n consecutive integers, then there is a bijection f :{1, 2, . . . , n}→ I such that gcd(i, f(i)) = 1 for 1 ≤ i ≤ n. The function f described in their theorem is called a coprime mapping. Around the same time, Roger Entringer conjectured that all trees are prime; that is, that if T is a tree with vertex set V , then there is a bijection L : V → {1, 2, . . . , |V |} such that gcd(L(x), L(y)) = 1 for all adjacent vertices x and y in V. So far, little progress towards a proof of this conjecture has been made. In this paper, we extend Pomerance and Selfridge’s theorem by replacing I with a set S of n integers in arithmetic progression and determining when there exist coprime mappings f : {1, 2, . . . , n} → S and g : {1, 3, . . . , 2n − 1} → S. We devote the rest of the paper to using coprime mappings to prove that various families of trees are prime, including palm trees, banana trees, binomial trees, and certain families of spider colonies.

On construction of one dimensional all order neighbor balanced designs by cyclic shifts

Abstract: Iqbal et al. (2006) presented a theorem to construct second order neighbor designs which is generalized here for all order neighbor balanced designs. By using method of cyclic shifts, two algorithms are developed to construct all order neighbor balanced designs for v odd prime when (i) k = v and (ii) k is relatively prime to v(v-1)/2.

Sum complexes - a new family of hypertrees

Abstract: A k-dimensional hypertree X is a k-dimensional complex on n vertices with a full (k-1)-dimensional skeleton and \binom{n-1}{k} facets such that H_k(X;Q)=0. Here we introduce the following family of simplicial complexes. Let n,k be integers with k+1 and n relatively prime, and let A be a (k+1)-element subset of the cyclic group Z_n. The sum complex X_A is the pure k-dimensional complex on the vertex set Z_n whose facets are subsets \sigma of Z_n such that |\sigma|=k+1 and \sum_{x \in \sigma}x \in A. It is shown that if n is prime then the complex X_A is a k-hypertree for every choice of A. On the other hand, for n prime X_A is k-collapsible iff A is an arithmetic progression in Z_n.

On the statistics of the minimal solution of a linear Diophantine equation and uniform distribution of the real part of orbits in hyperbolic spaces

Abstract: . We study a variant of a problem considered by Dinaburg and Sina˘ion the statistics of the minimal solution to a linear Diophantine equation.We show that the signed ratio between the Euclidean norms of the minimalsolution and the coefficient vector is uniformly distributed modulo one. Wereduce the problem to an equidistribution theorem of Anton Good concerningthe orbits of a point in the upper half-plane under the action of a Fuchsiangroup. 1. Statement of results1.1. For a pair of coprime integers (a,b), the linear Diophantine equationax − by = 1 is well known to have infinitely many integer solutions (x,y), anytwo differing by an integer multiple of (b,a). Dinaburg and Sina˘i [2] studied thestatistics of the “minimal” such solution v ′ = (x 0 ,y 0 ) when the coefficient vec-tor v = (a,b) varies over all primitive integer vectors lying in a large box withcommensurate sides. Their notion of “minimality” was in terms of the L ∞ -norm|v ′ | ∞ := max(|x 0 |,|y 0 |), and they studied the ratio |v

Optimal Probabilistic Ring Exploration by Asynchronous Oblivious Robots

Abstract: We consider a team of $k$ identical, oblivious, asynchronous mobile robots that are able to sense (\emph{i.e.}, view) their environment, yet are unable to communicate, and evolve on a constrained path. Previous results in this weak scenario show that initial symmetry yields high lower bounds when problems are to be solved by \emph{deterministic} robots. In this paper, we initiate research on probabilistic bounds and solutions in this context, and focus on the \emph{exploration} problem of anonymous unoriented rings of any size. It is known that $\Theta(\log n)$ robots are necessary and sufficient to solve the problem with $k$ deterministic robots, provided that $k$ and $n$ are coprime. By contrast, we show that \emph{four} identical probabilistic robots are necessary and sufficient to solve the same problem, also removing the coprime constraint. Our positive results are constructive.

Curves defined by Chebyshev polynomials

Abstract: Working over a field $\kk$ of characteristic zero, this paper studies line embeddings of the form $\phi = (T_i,T_j,T_k):\A^1\to\A^3$, where $T_n$ denotes the degree $n$ Chebyshev polynomial of the first kind. In {\it Section 4}, it is shown that (1) $\phi$ is an embedding if and only if the pairwise greatest common divisor of $i,j,k$ is 1, and (2) for a fixed pair $i,j$ of relatively prime positive integers, the embeddings of the form $(T_i,T_j,T_k)$ represent a finite number of algebraic equivalence classes. {\it Section 2} gives an algebraic definition of the Chebyshev polynomials, where their basic identities are established, and {\it Section 3} studies the plane curves $(T_i,T_j)$. {\it Section 5} establishes the Parity Property for Nodal Curves, and uses this to parametrize the family of alternating $(i,j)$-knots over the real numbers.