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.     

Area and the inradius of lemniscates

We study geometric properties of filled lemniscates of a complex polynomial p(z) of degree n. In particular, we answer a question raised by H.H. Cuenya and F.E. Levis (2007) by showing that there is a constant C(n) such that for every lemniscate E(p,c). Here μ(E(p,c)) and r(E(p,c)) denote the area and the inradius of E(p,c).     

On the prime power factorization of n!

In this paper, we prove two results. The first theorem uses a paper of Kim (J. Number Theory 74 (1999) 307) to show that for fixed primes p₁,…,p_k, and for fixed integers m₁,…,m_k, with , the numbers (e_p1(n),…,e_pk(n)) are uniformly distributed modulo (m₁,…,m_k), where e_p(n) is the order of the prime p in the factorization of n!. That implies one of Sander's conjectures from Sander (J. Number Theory 90 (2001) 316) for any set of odd primes. Berend (J. Number Theory 64 (1997) 13) asks to find the fastest growing function f(x) so that for large x and any given finite sequence , there exists n<x such that the congruences hold for all i?f(x). Here, p_i is the ith prime number. In our second result, we are able to show that f(x) can be taken to be at least , with some absolute constant c₁, provided that only the first odd prime numbers are involved.     

Expansions in non-integer bases: Lower, middle and top orders

Let q∈(1,2); it is known that each x∈[0,1/(q−1)] has an expansion of the form with an∈{0,1}. It was shown in [P. Erd?s, I. Joó, V. Komornik, Characterization of the unique expansions and related problems, Bull. Soc. Math. France 118 (1990) 377-390] that if , then each x∈(0,1/(q−1)) has a continuum of such expansions; however, if , then there exist infinitely many x having a unique expansion [P. Glendinning, N. Sidorov, Unique representations of real numbers in non-integer bases, Math. Res. Lett. 8 (2001) 535-543]. In the present paper we begin the study of parameters q for which there exists x having a fixed finite number m>1 of expansions in base q. In particular, we show that if q<q₂=1.71…, then each x has either 1 or infinitely many expansions, i.e., there are no such q in . On the other hand, for each m>1 there exists γm>0 such that for any q∈(2−γm,2), there exists x which has exactly m expansions in base q.     

On the residue classes of integer sequences satisfying a linear three term recurrence formula

In this paper we investigate linear three-term recurrence formulae with sequences of integers (T(n))_n?0 and (U(n))_n?0, which are ultimately periodic modulo m, e.g.

     

Feedback numbers of Kautz digraphs

A subset of vertices (resp. arcs) of a graph G is called a feedback vertex (resp. arc) set of G if its removal results in an acyclic subgraph. Let f(d,n) (f_a(d,n)) denote the minimum cardinality over all feedback vertex (resp. arc) sets of the Kautz digraph K(d,n). This paper proves that for any integers d?2 and n?1

     

Homogeneous polynomials as Lyapunov functions in the stability research of solutions of difference equations

The linear autonomous system of difference equations x(n+1)=Ax(n) is considered, where is a real nonsingular k×k matrix. In this paper it has been proved that if W(x) is any homogeneous polynomial of m-th degree in x, then there exists a unique homogeneous polynomial V(x) of m-th degree such that ΔV=V(Ax)-V(x)=W(x) if and only if where are the eigenvalues of the matrix A. The theorem on the instability has also been proved.     

On the parity of the partition function

Let N be the set of all positive integers and D a subset of N. Let p(D,n) be the number of partitions of n with parts in D and let |D(x)| denote the number of elements of D not exceeding x. It is proved that if D is an infinite subset of N such that p(D,n) is even for all n?n₀, then |D(x)|?logx/log2−logn₀/log2. Moreover, if D is an infinite subset of N such that p(D,n) is odd for all n?n₀ and , then |D(x)|?logx/log2−logn₀/log2. These lower bounds are essentially the best possible.     

Efficient Computation of Rectilinear Geodesic Voronoi Neighbor in the Presence of Obstacles

In this paper we present an algorithm to compute the rectilinear geodesic voronoi neighbor of an arbitrary query pointqamong a setSofmpoints in the presence of a set ofnvertical line segment obstacles inside a rectangular floor. The distance between a pair of points α and β is the shortest rectilinear distance avoiding the obstacles in and is denoted by δ(α, β). The rectilinear geodesic voronoi neighbor of an arbitrary query pointq,RGVN(q) is the pointp_i ∈ Ssuch that δ(q, p_i) is minimum. The algorithm suggests a preprocessing of the elements of the setsSand inO((m + n)log(m + n)) time such that for an arbitrary query pointq, theRGVNquery can be answered inO(log(m + n)) time. The space required for storing the preprocessed information isO(n + m log m). If the points inSare placed on the boundary of the rectangular floor, a different technique is adopted to decrease the space complexity toO(m + n). This technique works even if the obstacles are rectangles instead of line segments. Finally, the parallelization of the preprocessing steps for the latter algorithm is suggested, which takesO(log³(m + n)) time, usingO((m + n)^1.5/log²(m + n)) processors andO(log(m + n)) query time.     

Kummer subfields of tame division algebras over Henselian fields

By generalizing the method used by Tignol and Amitsur in [J.-P. Tignol, S.A. Amitsur, Kummer subfields of Malcev-Neumann division algebras, Israel Journal of Math. 50 (1985), 114-144], we determine necessary and sufficient conditions for an arbitrary central division algebra D over a Henselian valued field E to have Kummer subfields when the characteristic of the residue field of E does not divide the degree of D. We prove also that if D is a semiramified division algebra of degree n [resp., of prime power degree p^r] over E such that does not divide n and [resp., and p³ divides ], then D is non-cyclic [resp., D is not an elementary abelian crossed product].     

Coefficients of ternary cyclotomic polynomials

It is customary to define a cyclotomic polynomial Φn(x) to be ternary if n is the product of three distinct primes, p<q<r. Let A(n) be the largest absolute value of a coefficient of Φn(x) and M(p) be the maximum of A(pqr). In 1968, Sister Marion Beiter (1968, 1971) [3] and [4] conjectured that . In 2008, Yves Gallot and Pieter Moree (2009) [6] showed that the conjecture is false for every p≥11, and they proposed the Corrected Beiter conjecture: . Here we will give a sufficient condition for the Corrected Beiter conjecture and prove it when p=7.     

Determination of a differential pencil from interior spectral data

In this paper, inverse spectra problems for a differential pencil are studied. By using the approach similar to those in Hochstadt and Lieberman (1978) [14] and Ramm (2000) [26], we prove that (1) if p(x) (or q(x)) is full given on the interval [0,π], then a set of values of eigenfunctions at the mid-point of the interval [0,π] and one spectrum suffice to determine q(x) (or p(x)) on the interval [0,π] and all parameters in the boundary conditions; (2) if p(x) (or q(x)) is full given on the interval [0,π], then some information on eigenfunctions at some internal point and parts of two spectra suffice to determine q(x) (or p(x)) on the interval [0,π] and all parameters in the boundary conditions.     

Upper bounds on n-dimensional Kloosterman sums

Let p^m be any prime power and K_n(a,p^m) be the Kloosterman sum , where the x_i are restricted to values not divisible by p. Let m,n be positive integers with m?2 and suppose that p^γ||(n+1). We obtain the upper bound , for odd p. For p=2 we obtain the same bound, with an extra factor of 2 inserted.     

On the quadratic character of quadratic units

Let be a prime. Let a,b∈Z with p?a(a²+b²). In the paper we mainly determine by assuming p=c²+d² or p=Ax²+2Bxy+Cy² with AC−B²=a²+b². As an application we obtain simple criteria for εD to be a quadratic residue , where D>1 is a squarefree integer such that D is a quadratic residue of p, εD is the fundamental unit of the quadratic field with negative norm. We also establish the congruences for and obtain a general criterion for p|U(p−1)/4, where {Un} is the Lucas sequence defined by U₀=0, U₁=1 and Un+1=bUn+k²Un−1(n?1).     

Howe duality and combinatorial character formula for orthosymplectic Lie superalgebras

We study the Howe dualities involving the reductive dual pairs (O(d),spo(2m|2n)) and (Sp(d),osp(2m|2n)) on the (super)symmetric tensor of . We obtain complete decompositions of this space with respect to their respective joint actions. We also use these dualities to derive a character formula for these irreducible representations of spo(2m|2n) and osp(2m|2n) that appear in these decompositions.     

Note on the energy of regular graphs

For a simple graph G, the energy E(G) is defined as the sum of the absolute values of all the eigenvalues of its adjacency matrix A(G). Let n,m, respectively, be the number of vertices and edges of G. One well-known inequality is that , where λ₁ is the spectral radius. If G is k-regular, we have . Denote . Balakrishnan [R. Balakrishnan, The energy of a graph, Linear Algebra Appl. 387 (2004) 287-295] proved that for each ?>0, there exist infinitely many n for each of which there exists a k-regular graph G of order n with k<n-1 and , and proposed an open problem that, given a positive integer n?3, and ?>0, does there exist a k-regular graph G of order n such that . In this paper, we show that for each ?>0, there exist infinitely many such n that . Moreover, we construct another class of simpler graphs which also supports the first assertion that .     

Entire solutions to the Monge-Ampère equation

We consider the Monge-Ampère equation det(D²u)=Ψ(x,u,Du) in Rn, n?3, where Ψ is a positive function in C²(Rn×R×Rn). We prove the existence of convex solutions, provided there exist a subsolution of the form and a superharmonic bounded positive function φ satisfying: .