Davenport constant for finite abelian groups

For a finite abelian group G, we investigate the length of a sequence of elements of G that is guaranteed to have a subsequence with product identity of G. In particular, we obtain a bound on the length which takes into account the repetitions of elements of the sequence, the rank and the invariant factors of G. Consequently, we see that there are plenty of such sequences whose length could be much shorter than the best known upper bound for the Davenport constant of G, which is the least integer s such that any sequence of length s in G necessarily contains a subsequence with product identity. We also show that the Davenport constant for the multiplicative group of reduced residue classes modulo n is comparatively large with respect to the order of the group, which is φ(n),when n is in certain thin subsets of positive integers. This is done by studying the Carmichael’s lambda function, defined as the maximal multiplicative order of any reduced residue modulo n, along these subsets.     

An explicit result of the sum of seven cubes

We prove that every integer ≥ exp(524) is a sum of seven non negative cubes.     

Davenport constant for semigroups

Let G be a finite commutative semigroup. The Davenport constant of G is the smallest integer d such that, every sequence S of d elements in G contains a subsequence T (≠S) with the same product of S. Let . Among other results, we determine D(R ^×)−D(U(R)), where R ^× is the multiplicative semigroup of R and U(R) is the group of units of R.     

A new partial regularity result for non-autonomous convex integrals with non-standard growth conditions

We establish C1,γ-partial regularity of minimizers of non-autono-mous convex integral functionals of the type: , with non-standard growth conditions into the gradient variable

     

On Romanoff's constant

In 1934, Romanoff proved that there are a positive proportion natural numbers which can be expressed as the sum of a prime and a power of 2. In this paper, a quantitative version of this theorem is given. We show that the proportion is larger than 0.0868 and for a positive proportion of odd integers the number of such representations is between 1 and 16.     

LDU decomposition of an extension matrix of the Pascal matrix

Let Ω_m,n(α,β,γ) denote a set of all elements of weighted lattice paths with weight (α,β,γ) in the xy-plane from (0,0) to (m,n) such that a vertical step V=(0,1), a horizontal step H=(1,0), and a diagonal step D=(1,1) are endowed with weights α,β, and γ respectively and let ω(Ω_m,n(α,β,γ)) denote the weight of Ω_m,n(α,β,γ) defined by

     

The critical number of finite abelian groups

Let G be an additive, finite abelian group. The critical number cr(G) of G is the smallest positive integer ? such that for every subset S⊂G?{0} with |S|?? the following holds: Every element of G can be written as a nonempty sum of distinct elements from S. The critical number was first studied by P. Erd?s and H. Heilbronn in 1964, and due to the contributions of many authors the value of cr(G) is known for all finite abelian groups G except for G≅Z/pqZ where p,q are primes such that . We determine that cr(G)=p+q−2 for such groups.     

Remarks on a generalization of the Davenport constant

A generalization of the Davenport constant is investigated. For a finite abelian group G and a positive integer k, let denote the smallest ? such that each sequence over G of length at least ? has k disjoint non-empty zero-sum subsequences. For general G, expanding on known results, upper and lower bounds on these invariants are investigated and it is proved that the sequence is eventually an arithmetic progression with difference exp(G), and several questions arising from this fact are investigated. For elementary 2-groups, is investigated in detail; in particular, the exact values are determined for groups of rank four and five (for rank at most three they were already known).     

Correction to: Davenport constant for semigroups

Semigroup Forum - Given a finite commutative semigroup (written additively), denote by     

On the parity of exponents in the standard factorization of n!

Let p₁,p₂,… be the sequence of all primes in ascending order. The following result is proved: for any given positive integer k and any given , there exist infinitely many positive integers n with

     

On the number of zero-sum subsequences

For a sequence S of elements from an additive abelian group G, let f(S) denote the number of subsequences of S the sum of whose terms is zero. In this paper we characterize all sequences S in G with f(S)>2^|S|-2, where |S| denotes the number of terms of S.     

On the prime power factorization of n!, II

Let p, q be primes and m be a positive integer. For a positive integer n, let ep(n) be the nonnegative integer with pep(n)|n and pep(n)+1?n. The following results are proved: (1) For any positive integer m, any prime p and any ε∈Zm, there are infinitely many positive integers n such that ; (2) For any positive integer m, there exists a constant D(m) such that if ε,δ∈Zm and p, q are two distinct primes with max{p,q}?D(m), then there exist infinitely many positive integers n such that , . Finally we pose four open problems.     

Generalized Fermat, double Fermat and Newton sequences

In this paper, we discuss the relationship among the generalized Fermat, double Fermat, and Newton sequences. In particular, we show that every double Fermat sequence is a generalized Fermat sequence, and the set of generalized Fermat sequences, as well as the set of double Fermat sequences, is closed under term-by-term multiplication. We also prove that every Newton sequence is a generalized Fermat sequence and vice versa. Finally, we show that double Fermat sequences are Newton sequences generated by certain sequences of integers. An approach of symbolic dynamical systems is used to obtain congruence identities.     

On the Fürstenberg closure of a class of binary recurrences

In this paper, we determine the closure in the full topology over Z of the set , where (un)_n?0 is a nondegenerate binary recurrent sequence with integer coefficients whose characteristic roots are quadratic units. This generalizes the result for the case when un=Fn was the nth Fibonacci number.     

A new formula for the volume of lattice polyhedra

LetL _n be the lattice consisting of all pointsx inR ^N such thatnx belongs to the fundamental latticeL ₁ of points with integer coordinates. When the vertices of a polyhedronP inR ^N are restricted to lie inL ₁ there is a formula which relates the volume ofP to the numbers of points ofL ₁,...,L _N in the interior and on the boundary ofP. The aim of this note is to show that the volume ofP can be determined only by means of the numbers of points ofL ₁,...,L _N lying in the interior ofP and cannot be expressed by the numbers of points ofL ₁,...,L _N lying on the boundary ofP. The latter numbers in turn can be used to compute to comopute the Euler characteristic of the boundary ofP.     

On the value set of the Ramanujan function

We give lower bounds on the number of distinct values of the Ramanujan function τ(n), n ≦ x, and on the number of distinct residues of τ(n), n ≦ x, modulo a prime ℓ. We also show that for any prime ℓ the values τ(n), n ≦ ℓ⁴, form a finite additive basis modulo ℓ. Received: 6 October 2004     

A uniqueness result on the solution for a Landau–Lifshitz system with penalization

The author proves the uniqueness of the solution to an evolution Landau–Lifshitz type problem, when the parameter tends to zero. In addition, the unique solution is just a heat flow of a harmonic map. This uniqueness result is derived by establishing a uniform estimate for the solution.     

Minimal zero sum sequences of length four over finite cyclic groups

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Let G be a finite cyclic group. Every sequence S over G can be written in the form S=(n₁g)⋅…⋅(nlg) where g∈G and n₁,…,nl∈[1,ord(g)], and the index ind(S) of S is defined to be the minimum of (n₁+?+nl)/ord(g) over all possible g∈G such that 〈g〉=〈supp(S)〉. The problem regarding the index of sequences has been studied in a series of papers, and a main focus is to determine sequences of index 1. In the present paper, we show that if G is a cyclic of prime power order such that gcd(|G|,6)=1, then every minimal zero-sum sequence of length 4 has index 1.

Video

For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=BC7josX_xVs.     

A well-posedness result for hyperbolic operators with Zygmund coefficients

In this paper we prove an energy estimate with no loss of derivatives for a strictly hyperbolic operator with Zygmund continuous second order coefficients both in time and in space. In particular, this estimate implies the well-posedness for the related Cauchy problem. On the one hand, this result is quite surprising, because it allows to consider coefficients which are not Lipschitz continuous in time. On the other hand, it holds true only in the very special case of initial data in H^1/2×H^−1/2$H^{1/2}\times H^{-1/2}$. Paradifferential calculus with parameters is the main ingredient to the proof.