On the value distribution theory of elliptic functions

The Nevanlinna characteristic of a nonconstant elliptic function φ (z) satisfiesT(r, φ)=Kr ² (1+o(1)) asr→∞ whereK is a nonzero constant. In this paper, we completely answer the following question: For which polynomialsQ(z, u ₀,...,u _n ) inu ₀,...,u _n , having coefficientsa(z) satisfyingT(r, a)=o(r ²) asr→∞, will the meromorphic functionh _Q (z)=Q(z, ?(z),...,?⁽ⁿ⁾(z)) either be identically zero or satisfyN(r, 1/h _Q )=o(r ²) asr→∞? In fact, we answer this question for rational functionsQ(z, u ₀,...,u _n ) inu ₀,...,u _n , and also obtain analogous results for the Weierstrass functions ζ(z) and σ(z).     

Asymptotics of the complexity of systems of integer linear forms for additive computations

The computational complexity of integer linear forms is studied. By l ₂(A) we denote the minimal number of the additions and subtractions required for computing the system of p linear forms in q variables x ₁, x ₂, …, x _q that are defined by an integer matrix A of size p × q (repeated use of the results of intermediate computation is permitted). We show that l ₂(A) ? log D(A), where D(A) is the maximum of the absolute values of the minors of A over all minors from order 1 to order min (p, q) (Theorem 1). Moreover, for each sequence of matrices A(n) of size p(n) × q(n) satisfying the condition p + q = o ((log log D(A))^1/2) as n → ∞ the bound l ₂(A) ? log D(A) + o(log D(A)) is valid (Theorem 2). Hence, for all fixed (and even weakly increasing) sizes of matrices that determine a system of integer linear forms, the upper bound on the computational complexity of this system is asymptotically equal to the lower bound.     

Disjoint bases in a polymatroid

Let f : 2^N → ⁺ be a polymatroid (an integer‐valued non‐decreasing submodular set function with f(??) = 0). We call S ? N a base if f(S) = f(N). We consider the problem of finding a maximum number of disjoint bases; we denote by m* be this base packing number. A simple upper bound on m* is given by k* = max{k : Σ_iεNf_A(i) ≥ kf_A(N),?A ? N} where f_A(S) = f(A ∪ S) ‐ f(A). This upper bound is a natural generalization of the bound for matroids where it is known that m* = k*. For polymatroids, we prove that m* ≥ (1 ? o(1))k*/lnf(N) and give a randomized polynomial time algorithm to find (1 ? o(1))k*/lnf(N) disjoint bases, assuming an oracle for f. We also derandomize the algorithm using minwise independent permutations and give a deterministic algorithm that finds (1 ? ε)k*/lnf(N) disjoint bases. The bound we obtain is almost tight because it is known there are polymatroids for which m* < (1 + o(1))k*/lnf(N). Moreover it is known that unless NP ? DTIME(n^{log log n}), for any ε > 0, there is no polynomial time algorithm to obtain a (1 + ε)/lnf(N)‐approximation to m*. Our result generalizes and unifies two results in the literature. © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2009     

Mono-multi bipartite Ramsey numbers, designs, and matrices

Eroh and Oellermann defined BRR(G₁,G₂) as the smallest N such that any edge coloring of the complete bipartite graph K_N,N contains either a monochromatic G₁ or a multicolored G₂. We restate the problem of determining BRR(K_1,λ,K_r,s) in matrix form and prove estimates and exact values for several choices of the parameters. Our general bound uses Füredi's result on fractional matchings of uniform hypergraphs and we show that it is sharp if certain block designs exist. We obtain two sharp results for the case r=s=2: we prove BRR(K_1,λ,K_2,2)=3λ-2 and that the smallest n for which any edge coloring of K_λ,n contains either a monochromatic K_1,λ or a multicolored K_2,2 is λ².     

Hole probability for entire functions represented by Gaussian Taylor series

Consider the Gaussian entire functionf(z) = ?? _n=0 ^?? ?? _n a _n z ⁿ , where {?? _n } is a sequence of independent and identically distributed standard complex Gaussians and {a _n } is some sequence of non-negative coefficients, with a ₀ > 0. We study the asymptotics (for large values of r) of the hole probability for f (z), that is, the probability P _H (r) that f(z) has no zeros in the disk {|z| < r}. We prove that log P _H (r) = ?S(r) + o(S(r)), where S(r) = 2·?? _n??0log⁺(a _n r ⁿ ) as r tends to ?? outside a deterministic exceptional set of finite logarithmic measure.     

Partitions of natural numbers with the same representation functions

For a set A of nonnegative integers the representation functions R₂(A,n), R₃(A,n) are defined as the number of solutions of the equation n=a+a^′,a,a^′∈A with a<a^′, a?a^′, respectively. Let D(0)=0 and let D(a) denote the number of ones in the binary representation of a. Let A₀ be the set of all nonnegative integers a with even D(a) and A₁ be the set of all nonnegative integers a with odd D(a). In this paper we show that (a) if R₂(A,n)=R₂(N?A,n) for all n?2N−1, then R₂(A,n)=R₂(N?A,n)?1 for all n?12N²−10N−2 except for A=A₀ or A=A₁; (b) if R₃(A,n)=R₃(N?A,n) for all n?2N−1, then R₃(A,n)=R₃(N?A,n)?1 for all n?12N²+2N. Several problems are posed in this paper.     

Extension de quelques theoremes sur les densites de series d'elements de n a des series de sous-ensembles finis de n

For a sequence A = {A_k} of finite subsets of N we introduce: $\text{δ(A) =}\text{inf}{}_{\mathrm{m?n}}\text{A(m)}\text{2}{}^{\mathrm{n}}$, $\text{d(A) =}\text{lim inf}{}_{\mathrm{n\to \infty }}\text{A(n)}\text{2}{}^{\mathrm{n}}$, where A(m) is the number of subsets A_k ? {1, 2, …, m}.The collection of all subsets of {1, …, n} together with the operation $\text{a ∪ b, (a ∩ b), (a 1 b = a ∪ b ? a ∩ b)}$ constitutes a finite semi-group N^∪ (semi-group N^∩) (group $\text{N}{}^1$). For N^∪, N^∩ we prove analogues of the Erdös-Landau theorem: δ(A+B) ? δ(A)(1+(2λ)^?1(1?δ(A>))), where B is a base of N of the average order λ. We prove for $\text{N}{}^{\mathrm{\cup }}\text{, N}{}^{\mathrm{\cap }}\text{, N}{}^1$ analogues of Schnirelmann's theorem (that δ(A) + δ(B) > 1 implies δ(A + B) = 1) and the inequalities λ ? 2h, where h is the order of the base.We introduce the concept of divisibility of subsets: a|b if b is a continuation of a. We prove an analog of the Davenport-Erdös theorem: if d(A) > 0, then there exists an infinite sequence {A_kr}, where A_kr | A_kr+1 for r = 1, 2, …. In Section 6 we consider for $\text{N∪, N∩, N1}$ analogues of Rohrbach inequality: $\text{2n}\text{? g(n) ? 2}\text{n}$, where g(n) = min k over the subsets {a₁ < … < a_k} ? {0, 1, 2, …, n}, such that every m? {0, 1, 2, …, n} can be expressed as m = a_i + a_j.Pour une série A = {A_k} de sous-ensembles finis de N on introduit les densités: $\text{δ(A) =}\text{inf}{}_{\mathrm{m?n}}\text{A(m)}\text{2}{}^{\mathrm{m}}$, $\text{d(A) =}\text{lim inf}{}_{\mathrm{n\to \infty }}\text{A(n)}\text{2}{}^{\mathrm{n}}$ où A(m) est le nombre d'ensembles A_k ? {1, 2, …, m}. L'ensemble de toutes les parties de {1, 2, …, n} devient, pour les opérations $\text{a ∪ b, a ∩ b, a 1 b = a ∪ b ? a ∩ b}$, un semi-groupe fini N^∪, N^∩ ou un groupe N¹ respectivement. Pour N^∪, N^∩ on démontre l'analogue du théorème de Erdös-Landau: δ(A + B) ? δ(A)(1 + (2λ)^?1(1?δ(A))), où B est une base de N d'ordre moyen λ. On démontre pour $\text{N}{}^{\mathrm{\cup }}\text{, N}{}^{\mathrm{\cap }}\text{, N}{}^1$ l'analogue du théorème de Schnirelmann (si δ(A) + δ(B) > 1, alors δ(A + B) = 1) et les inégalités λ ? 2h, où h est l'ordre de base. On introduit le rapport de divisibilité des enembles: a|b, si b est une continuation de a. On démontre l'analogue du théorème de Davenport-Erdös: si d(A) > 0, alors il existe une sous-série infinie {A_kr}, où A_kr|A_kr+1, pour r = 1, 2, … . Dans le Paragraphe 6 on envisage pour N^∪, $\text{N}{}^{\mathrm{\cap }}\text{, N}{}^1$ les analogues de l'inégalité de Rohrbach: $\text{2n}\text{? g(n) ? 2}\text{n}$, où g(n) = min k pour les ensembles {a₁ < … < a_k} ? {0, 1, 2, …, n} tels que pour tout m? {0, 1, 2, …, n} on a m = a_i + a_j.     

Small subgraphs in random graphs and the power of multiple choices

The standard paradigm for online power of two choices problems in random graphs is the Achlioptas process. Here we consider the following natural generalization: Starting with G₀ as the empty graph on n vertices, in every step a set of r edges is drawn uniformly at random from all edges that have not been drawn in previous steps. From these, one edge has to be selected, and the remaining r−1 edges are discarded. Thus after N steps, we have seen rN edges, and selected exactly N out of these to create a graph GN.In a recent paper by Krivelevich, Loh, and Sudakov (2009) [11], the problem of avoiding a copy of some fixed graph F in GN for as long as possible is considered, and a threshold result is derived for some special cases. Moreover, the authors conjecture a general threshold formula for arbitrary graphs F. In this work we disprove this conjecture and give the complete solution of the problem by deriving explicit threshold functions N₀(F,r,n) for arbitrary graphs F and any fixed integer r. That is, we propose an edge selection strategy that a.a.s. (asymptotically almost surely, i.e. with probability 1−o(1) as n→∞) avoids creating a copy of F for as long as N=o(N₀), and prove that any online strategy will a.a.s. create such a copy once N=ω(N₀).     

A Second Main Theorem on ℙ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript> for difference operator

The main result of this article is an extension of the Second Main Theorem, of Halburd and Korhonen, for meromorphic functions of finite order. Their result replaces the counting function of the ramification divisor N _ramf(r) in the classical Second Main Theorem by the counting function of a finite difference divisor N _pair(r). In this article, the Second Main Theorem of Halburd and Korhonen is extended to the case of holomorphic maps into ℙ ⁿ of finite order.     

Incomplete Character Sums and Polynomial Interpolation of the Discrete Logarithm

In the first part of the paper, certain incomplete character sums over a finite field F_p^r are considered which in the case of finite prime fields F_p are of the form ∑^A+N−1_n=Aχ(g(n))ψ(f(n)), where A and N are integers with 1≤N<p, g and f are polynomials over F_p, and χ denotes a multiplicative and ψ an additive character of F_p. Excluding trivial cases, it is shown that the above sums are at most of the order of magnitude N^1/2p^r/4. Recently, Shparlinski showed that a polynomial f over the integers which coincides with the discrete logarithm of the finite prime field F_p for N consecutive elements of F_p must have a degree at least of the order of magnitude Np^−1/2. In this paper this result is extended to arbitrary F_p^r. The proof is based on the above new bound for incomplete hybrid character sums.     

On the Ramsey number of the triangle and the cube

The Ramsey number r(K ₃,Q _n ) is the smallest integer N such that every red-blue colouring of the edges of the complete graph K _N contains either a red n-dimensional hypercube, or a blue triangle. Almost thirty years ago, Burr and Erd?s conjectured that r(K ₃,Q _n )=2 ⁿ⁺¹?1 for every n∈?, but the first non-trivial upper bound was obtained only recently, by Conlon, Fox, Lee and Sudakov, who proved that r(K ₃,Q _n )?7000·2 ⁿ . Here we show that r(K ₃,Q _n )=(1+o(1))2 ⁿ⁺¹ as n→∞.     

Retraction spaces and the homotopy metric

Let X be a finite-dimensional compactum. Let $\text{R}$(X) and $\text{N}$(X) be the spaces of retractions and non-deformation retractions of X, respectively, with the compact-open (=sup-metric) topology. Let 2^X_h be the space of non-empty compact ANR subsets of X with topology induced by the homotopy metric. Let R^X_h be the subspace of 2^X_h consisting of the ANR's in X that are retracts of X.We show that $\text{N}$(S^m) is simply-connected for m > 1. We show that if X is an ANR and A₀?R^X_h, then lim_i→∞A_i=A₀ in 2^X_h if and only if for every retraction r₀ of X onto A₀ there are, for almost all i, retractions r_i of X onto A_i such that lim_i→∞r_i=r_o in $\text{R}$(X). We show that if X is an ANR, then the local connectedness of $\text{R}$(X) implies that of R^X_h. We prove that $\text{R}$(M) is locally connected if M is a closed surface. We give examples to show how some of our results weaken when X is not assumed to be an ANR.     

Superconvergence of Legendre projection methods for the eigenvalue problem of a compact integral operator

In this paper, we consider the Galerkin and collocation methods for the eigenvalue problem of a compact integral operator with a smooth kernel using the Legendre polynomials of degree ≤n. We prove that the error bounds for eigenvalues are of the order O(n^−2r) and the gap between the spectral subspaces are of the orders O(n^−r) in L₂-norm and O(n^1/2−r) in the infinity norm, where r denotes the smoothness of the kernel. By iterating the eigenvectors we show that the iterated eigenvectors converge with the orders of convergence O(n^−2r) in both L₂-norm and infinity norm. We illustrate our results with numerical examples.     

A weighted estimate for the square function on the unit ball in ℂ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

We show that the Luzin area integral or the square function on the unit ball of ℂ ⁿ , regarded as an operator in the weighted space L ²(w) has a linear bound in terms of the invariant A ₂ characteristic of the weight. We show a dimension-free estimate for the “area-integral” associated with the weighted L ²(w) norm of the square function. We prove the equivalence of the classical and the invariant A ₂ classes.     

Polynomial expansions and generating functions

We study expansions in polynomials {P_n(x)}^∞_o generated by ∑^∞_{n = o}P_n(x)tⁿ = A(t) φ(xt^kθ(t)), θ(0) ≠ 0, and ∑^∞_{n = 0}P_n(x)tⁿ = ∑^k_{j = 1}A_j(t) φ(xt?_j), ?₁,…,?_k being the k roots of unity. The case k = 1 is contained in a recent work by Fields and Ismail. We also prove a new generalization of Vandermond's inverse relations.     

Regions of meromorphy and value distribution of geometrically converging rational functions

Let D be a region, {rn}_n∈N a sequence of rational functions of degree at most n and let each rn have at most m poles in D, for m∈N fixed. We prove that if {rn}_n∈N converges geometrically to a function f on some continuum S⊂D and if the number of zeros of rn in any compact subset of D is of growth o(n) as n→∞, then the sequence {rn}_n∈N converges m₁-almost uniformly to a meromorphic function in D. This result about meromorphic continuation is used to obtain Picard-type theorems for the value distribution of m₁-maximally convergent rational functions, especially in Padé approximation and Chebyshev rational approximation.     

Hecke Eigenfunctions of Quantized Cat Maps Modulo Prime Powers

This paper continues the work done in Olofsson [Commun Math Phys 286(3):1051–1072, 2009] about the supremum norm of eigenfunctions of desymmetrized quantized cat maps. N will denote the inverse of Planck’s constant and we will see that the arithmetic properties of N play an important role. We prove the sharp estimate ||ψ||_∞ = O(N ^1/4) for all normalized eigenfunctions and all N outside of a small exceptional set. We are also able to calculate the value of the supremum norms for most of the so called newforms. For a given N = p ⁿ , with n > 2, the newforms can be divided in two parts (leaving out a small number of them in some cases), the first half all have supremum norm about ${2/\sqrt{1\pm 1/p}}This paper continues the work done in Olofsson [Commun Math Phys 286(3):1051–1072, 2009] about the supremum norm of eigenfunctions of desymmetrized quantized cat maps. N will denote the inverse of Planck’s constant and we will see that the arithmetic properties of N play an important role. We prove the sharp estimate ||ψ||_∞ = O(N ^1/4) for all normalized eigenfunctions and all N outside of a small exceptional set. We are also able to calculate the value of the supremum norms for most of the so called newforms. For a given N = p ⁿ , with n > 2, the newforms can be divided in two parts (leaving out a small number of them in some cases), the first half all have supremum norm about 2/?{1±1/p}{2/\sqrt{1\pm 1/p}} and the supremum norm of the newforms in the second half have at most three different values, all of the order N ^1/6. The only dependence of A is that the normalization factor is different if A has eigenvectors modulo p or not. We also calculate the joint value distribution of the absolute value of n different newforms.     

A specific family of cyclotomic polynomials of order three

Let A(n) be the largest absolute value of any coefficient of n-th cyclotomic polynomial Φn(x).We say Φn(x) is flat if A(n) = 1.In this paper,for odd primes p q r and 2r ≡ 1(mod pq),we prove that Φpqr(x) is flat if and only if p = 3 and q ≡ 1(mod 3).     

Bounds for the crossing number of the N-cube

Let Q_n denote the n-dimensional hypercube. In this paper we derive upper and lower bounds for the crossing number v(Q_n), i.e., the minimum number of edge-crossings in any planar drawing of Q_n. The upper bound is close to a result conjectured by Eggleton and Guy and the lower bound is a significant improvement over what was previously known. Let N = 2ⁿ be the number of vertices of Q_n. We show that v(Q_n) < 1/6N². For the lower bound we prove that v(Q_n) = Ω(N(lg N)^{c lg lg N}), where c > 0 is a constant and lg is the logarithm base 2. The best lower bound using standard arguments is v(Q_n) = Ω(N(lg N)²). The lower bound is obtained by constructing a large family of homeomorphs of a subcube with the property that no given pair of edges can appear in more than a constant number of the homeomorphs.