Khodakhast Bibak

On a restricted linear congruence

Let b,n ∈ ℤ, n ≥ 1, and 𝒟1,…,𝒟τ(n) be all positive divisors of n. For 1 ≤ l ≤ τ(n), define 𝒞l := {1 ≤ x ≤ n:(x,n) = 𝒟l}. In this paper, by combining ideas from the finite Fourier transform of arithmetic functions and Ramanujan sums, we give a short proof for the following result: the number of solutions of the linear congruence x1 + ⋯ + xk ≡ b(modn), with κl = |{x1,…,xk}∩𝒞l|, 1 ≤ l ≤ τ(n), is 1 n∑d|ncd(b)∏l=1τ(n)(c n 𝒟l(d))κl, where cd(b) is a Ramanujan sum. Some special cases and other forms of this problem have been already studied by several authors. The problem has recently found very interesting applications in number theory, combinatorics, computer science, and cryptography. The above explicit formula generalizes the main results of several papers, for example, the main result of the paper by Sander and Sander [J. Number Theory 133 (2013) 705–718], one of the main results of the paper by Sander [J. Number Theory 129 (2009) 2260–2266], and also gives an equivalent formula for the main result of the paper by Sun and Yang [Int. J. Number Theory 10 (2014) 1355–1363].

Additive combinatorics: with a view towards Computer Science and Cryptography - An exposition

Recently, additive combinatorics has blossomed into a vibrant area in mathematical sciences. But it seems to be a difficult area to define—perhaps because of a blend of ideas and techniques from several seemingly unrelated contexts which are used there. One might say that additive combinatorics is a branch of mathematics concerning the study of combinatorial properties of algebraic objects, for instance, Abelian groups, rings, or fields. This emerging field has seen tremendous advances over the last few years and has recently become a focus of attention among both mathematicians and computer scientists. This fascinating area has been enriched by its formidable links to combinatorics, number theory, harmonic analysis, ergodic theory, and some other branches; all deeply cross-fertilize each other, holding great promise for all of them! In this exposition, we attempt to provide an overview of some breakthroughs in this field, together with a number of seminal applications to sundry parts of mathematics and some other disciplines, with emphasis on computer science and cryptography.

Counting surface-kernel epimorphisms from a co-compact Fuchsian group to a cyclic group with motivations from string theory and QFT

Abstract Graphs embedded into surfaces have many important applications, in particular, in combinatorics, geometry, and physics. For example, ribbon graphs and their counting is of great interest in string theory and quantum field theory (QFT). Recently, Koch et al. (2013) [12] gave a refined formula for counting ribbon graphs and discussed its applications to several physics problems. An important factor in this formula is the number of surface-kernel epimorphisms from a co-compact Fuchsian group to a cyclic group. The aim of this paper is to give an explicit and practical formula for the number of such epimorphisms. As a consequence, we obtain an ‘equivalent’ form of Harveys famous theorem on the cyclic groups of automorphisms of compact Riemann surfaces. Our main tool is an explicit formula for the number of solutions of restricted linear congruence recently proved by Bibak et al. using properties of Ramanujan sums and of the finite Fourier transform of arithmetic functions.

On the determinant of bipartite graphs

Abstract The nullity of a graph G , denoted by η ( G ) , is the multiplicity of 0 in the spectrum of G . Nullity of a (molecular) graph (e.g., a bipartite graph corresponding to an alternant hydrocarbon) has important applications in quantum chemistry and Huckel molecular orbital (HMO) theory. A famous problem, posed by Collatz and Sinogowitz in 1957, asks to characterize all graphs with positive nullity. Clearly, det A ( G ) = 0 if and only if η ( G ) > 0 . So, examining the determinant of a graph is a way to attack this problem. For a graph G , we define the matching core of G to be the graph obtained from G by successively deleting each pendant vertex along with its neighbour. In this paper, we show that the determinant of a graph G with all cycle lengths divisible by four (e.g., the 1-subdivision of a bipartite graph), is 0 or ( − 1 ) | V ( G ) | / 2 . Furthermore, the determinant is 0 if and only if the matching core of G is nonempty.

MMH* with arbitrary modulus is always almost-universal

Universal hash functions, discovered by Carter and Wegman in 1979, are of great importance in computer science with many applications. MMH* is a well-known ?-universal hash function family, based on the evaluation of a dot product modulo a prime. In this paper, we introduce a generalization of MMH*, that we call GMMH*, using the same construction as MMH* but with an arbitrary integer modulus n 1 , and show that GMMH* is 1 p -almost-?-universal, where p is the smallest prime divisor of n. This bound is tight. Universal hashing is connected to the number of solutions of linear congruences.The connection is made via a classical result of D.N. Lehmer.A generalization of MMH*, that we call GMMH*, is introduced.GMMH* uses the same construction as MMH* but with an arbitrary modulus n 1 .GMMH* is 1 p -almost-?-universal, where p is the smallest prime divisor of n.

The Cayley Graphs Associated With Some Quasi-Perfect Lee Codes Are Ramanujan Graphs

Let Z_n[i] be the ring of Gaussian integers modulo a positive integer n. Very recently, Camarero and Martinez et al. showed that for every prime number p > 5 such that p ≡ ±5 (mod 12), the Cayley graph ς_p = Cay(Z_p[i], S₂), where S₂ is the set of units of Z_p[i], induces a two-quasi-perfect Lee code over Z_p^m, where m = 2[p/4]. They also conjectured that ς_p is a Ramanujan graph for every prime p, such that p ≡ 3 (mod 4). In this paper, we solve this conjecture. Our main tools are Delignes bound from 1977 for estimating a particular kind of trigonometric sum and a result of Lovász from 1975 (or of Babai from 1979) which gives the eigenvalues of Cayley graphs of finite Abelian groups. Our proof techniques may motivate more work in the interactions between spectral graph theory, character theory, and coding theory, and may provide new ideas toward the famous Golomb-Welch conjecture on the existence of perfect Lee codes.

Determinants of grids, tori, cylinders and Möbius ladders

Recently, Bien A.?Bien, The problem of singularity for planar grids, Discrete Math. 311 (2011) 921-931] obtained a recursive formula for the determinant of a grid. Also, recently, Pragel D.?Pragel, Determinants of box products of paths, Discrete Math. 312 (2012) 1844-1847], independently, obtained an explicit formula for this determinant. In this paper, we give a short proof for this problem. Furthermore, applying the same technique, we get explicit formulas for the determinant of a torus, that of a cylinder, and that of a Mobius ladder.

Unweighted linear congruences with distinct coordinates and the Varshamov–Tenengolts codes

In this paper, we first give explicit formulas for the number of solutions of unweighted linear congruences with distinct coordinates. Our main tools are properties of Ramanujan sums and of the discrete Fourier transform of arithmetic functions. Then, as an application, we derive an explicit formula for the number of codewords in the Varshamov–Tenengolts code

Restricted linear congruences

VT_b(n)