On the index of composition of integers from various sets

Given an integer n ≥ 2, let λ(n) := (log n)/(log γ(n)), where γ(n) = Π _p|n p, stand for the index of composition of n, with λ(1) = 1. We study the distribution function of (λ(n) – 1) log n as n runs through particular sets of integers, such as the shifted primes, the values of a given irreducible cubic polynomial and the shifted powerful numbers. Research supported in part by a grant from NSERC. Research supported by the Applied Number Theory Research Group of the Hungarian Academy of Science and by a grant from OTKA. Professor M.V. Subbarao passed away on February 15, 2006. Received: 3 March 2006 Revised: 28 October 2006     

Binomial Thue equations,ternary equations and power values of polynomials

We explicitly solve the equation Ax ⁿ   − By ⁿ  = ±1 and, along the way, we obtain new results for a collection of equations Ax ⁿ   − By ⁿ  = z ^m with m ∈ {3, n}, where x, y, z, A, B, and n are unknown nonzero integers such that n ≥ 3, AB = p ^α q ^β with nonnegative integers α and β and with primes 2 ≤ p < q < 30. The proofs depend on a combination of several powerful methods, including the modular approach, recent lower bounds for linear forms in logarithms, somewhat involved local considerations, and computational techniques for solving Thue equations of low degree.     

À propos de l’indice de composition des nombres

 We define the index of composition λ(n) of an integer n ⩾ 2 as λ(n) = log n/log γ(n), where γ(n) stands for the product of the primes dividing n, and first establish that λ and 1/λ both have asymptotic mean value 1. We then establish that, given any ɛ > 0 and any integer k ⩾ 2, there exist infinitely many positive integers n such that . Considering the distribution function F(z,x) := #{n < x : λ(n) > z}, we prove that, given 1 < z < 2 and ɛ > 0, then, if x is sufficiently large,

Constrained energy problems with applications to orthogonal polynomials of a discrete variable

Given a positive measure Σ with gs > 1, we write Με ℳ^Σ if Μ is a probability measure and Σ—Μ is a positive measure. Under some general assumptions on the constraining measure Σ and a weight functionw, we prove existence and uniqueness of a measure λ^Σ _w that minimizes the weighted logarithmic energy over the class ℳ^Σ. We also obtain a characterization theorem, a saturation result and a balayage representation for the measure λ^Σ _w As applications of our results, we determine the (normalized) limiting zero distribution for ray sequences of a class of orthogonal polynomials of a discrete variable. Explicit results are given for the class of Krawtchouk polynomials. The research done by this author is in partial fulfillment of the Ph.D. requirements at the University of South Florida. The research done by this author was supported, in part, by U.S. National Science Foundation under grant DMS-9501130.     

Optimization of convex functions on w*-compact sets

We give a direct, self-contained, and iterative proof that for any convex, Lipschitz andw ^*-lower semicontinuous function ϕ defined on aw ^*-compact convex setC in a dual Banach spaceX ^* and for any ε>0 there is anx∈X, with ‖x‖≤ε, such that ϕ+x attains its supremum at an extreme point ofC. This result is implicitly contained in the work of Lindenstrauss [9] and the work of Ghoussoub and Maurey on strongw ^*−H _σ sets [8]. In addition, we discuss the applications of this result to the geometry of convex sets. Research supported in part by the NSERC of Canada under grant OGP41983 for the first author and grant OGP7926 for the second author.     

Reverse hypercontractivity over manifolds

Suppose thatX is a vector field on a manifoldM whose flow, exptX, exists for all time. If μ is a measure onM for which the induced measuresμ _t ≡(exptX)_* μ are absolutely continuous with respect to μ, it is of interest to establish bounds on theL ^p (μ) norm of the Radon-Nikodym derivativedμ _t /dμ. We establish such bounds in terms of the divergence of the vector fieldX. We then specilizeM to be a complex manifold and derive reverse hypercontractivity bounds and reverse logarithmic Sololev inequalities in some holomorphic function spaces. We give examples onC ^m and on the Riemann surface forz ^1/n . Research supported in part by CONACyT, Mexico, grant 32725-E. Research supported in part by CONACyT, Mexico, grant 32146-E.     

Carmichael numbers with a totient of the form a2?+?nb2

Let φ be the Euler function. Fix , and let be an arbitrary set of primes of positive lower natural density. Using a variant of the Alford–Granville–Pomerance construction, we show that there are infinitely many Carmichael numbers N with a totient of the form , where and is a nonempty product of primes from the set . In particular, for any fixed natural number n, there are infinitely many Carmichael numbers N such that φ(N) = a ² + nb ² for some positive integers a and b.     

On the Scope of Averaging for Frankl’s Conjecture

Let be a union-closed family of subsets of an m-element set A. Let . For b ∈ A let w(b) denote the number of sets in containing b minus the number of sets in not containing b. Frankl’s conjecture from 1979, also known as the union-closed sets conjecture, states that there exists an element b ∈ A with w(b) ≥ 0. The present paper deals with the average of the w(b), computed over all b ∈ A. is said to satisfy the averaged Frankl’s property if this average is non-negative. Although this much stronger property does not hold for all union-closed families, the first author (Czédli, J Comb Theory, Ser A, 2008) verified the averaged Frankl’s property whenever n ≥ 2 ^m  − 2 ^m/2 and m ≥ 3. The main result of this paper shows that (1) we cannot replace 2 ^m/2 with the upper integer part of 2 ^m /3, and (2) if Frankl’s conjecture is true (at least for m-element base sets) and then the averaged Frankl’s property holds (i.e., 2 ^m/2 can be replaced with the lower integer part of 2 ^m /3). The proof combines elementary facts from combinatorics and lattice theory. The paper is self-contained, and the reader is assumed to be familiar neither with lattices nor with combinatorics. This research was partially supported by the NFSR of Hungary (OTKA), grant no. T 049433, T 48809 and K 60148.     

Classification of manifolds with weakly 1/4-pinched curvatures

We show that a compact Riemannian manifold with weakly pointwise 1/4-pinched sectional curvatures is either locally symmetric or diffeomorphic to a space form. More generally, we classify all compact, locally irreducible Riemannian manifolds M with the property that M × R ² has non-negative isotropic curvature. The first author was partially supported by a Sloan Foundation Fellowship and by NSF grant DMS-0605223. The second author was partially supported by NSF grant DMS-0604960.     

Finite Element Method and Discontinuous Galerkin Method for Stochastic Scattering Problem of Helmholtz Type in ℝ<Superscript><Emphasis Type="Italic">d</Emphasis></Superscript> (<Emphasis Type="Italic">d</Emphasis> = 2, 3)

In this paper, we address the finite element method and discontinuous Galerkin method for the stochastic scattering problem of Helmholtz type in ℝ ^d (d = 2, 3). Convergence analysis and error estimates are presented for the numerical solutions. The effects of the noises on the accuracy of the approximations are illustrated. Results of the numerical experiments are provided to examine our theoretical results. The first author is supported by NSF under grand number 0609918 and AFOSR under grant numbers FA9550-06-1-0234 and FA9550-07-1-0154, the second author is supported by NSFC(10671082, 10626026, 10471054), and the third author is supported by NNSF (No. 10701039 of China) and 985 program of Jilin University.     

Neumann Bessel Heat Kernel Monotonicity

We prove that the diagonal of the transition probabilities for the d-dimensional Bessel processes on (0, 1], reflected at 1, which we denote by , is an increasing function of r for d > 2 and that this is false for d = 2. The first and third authors were supported in part by NSF grant # 0603701-DMS. The second author was supported in part by KBN Grant 1 P03A 020 28.     

A decomposition of the beta distribution,related order and asymptotic behavior

Let β _{v, w} be any beta variate with p.d.f.(Γ(v+w)/Γ(v)Γ(w))x ^v−1(1−x) ^w−1 and letU _{v, w} =-log β _{v, w} . ThenU _v,w =U ^CM +U ^PF , whereU ^CM andU ^PF are independent with completely monotone andPF _∞ densities, respectively. It is shown thatU _{v, w} is infinitely divisible and β _{v, w} correspondingly infinitely factorable. The asymptotic behavior ofU _{v, w} and β _{v, w} for largev, w is described. For different modes of increase ofv andw,U _{v, w} is asymptotically normal, gamma or extreme value distributed. The decomposition is employed to provide an algorithm for generating random β _{v, w} distributed numbers. Many of the results are based on insights provided by the classical theory of the Gamma function in the complex plane. This work was supported in part by the United States Air Force, Office of Scientific Research, under grant No. AFOSR-79-0043.     

Irreducible algebraic integers in short intervals

We study distribution of irreducible algebraic integers in short intervals and prove that if the class number of an algebraic number field K exceeds 2, every interval of the form (x, x + x ^θ) with a fixed θ > 1/2 contains absolute value of the norm of an irreducible algebraic integer from K provided x ≥ x ₀. The constant x ₀ effectively depends on K and θ. The author was supported in part by the grant no. N N201 1482 33 from the Polish Ministry of Science and Higher Education.     

Locally Closed Semirings

 We call a semiring S locally closed if for all a ∈ S there is some integer k such that 1 + a + ⋯ + a ^k  =1 + a + ⋯ + a ^k + 1 . In any locally closed semiring we may define a star operation a ↦ a ^*, where a ^* is the above finite sum. We prove that when S is locally closed and commutative, then S is an iteration semiring. Partially supported by grant no. T30511 from the National Foundation of Hungary for Scientific Research and the Austrian–Hungarian Bilateral Research and Development Fund, no. A-4/1999, and by the Austrian–Hungarian Action Foundation. Partially supported by the Austrian–Hungarian Bilateral Research and Development Fund, no. A-4/1999, and by the Austrian–Hungarian Action Foundation. Received March 16, 2001     

The Lambert W-functions and some of their integrals: a case study of high-precision computation

The real-valued Lambert W-functions considered here are w ₀(y) and w _− 1(y), solutions of we ^w  = y, − 1/e < y < 0, with values respectively in ( − 1,0) and ( − ∞ , − 1). A study is made of the numerical evaluation to high precision of these functions and of the integrals ò₁^￥ [-w₀(-xe^-x)]^a x^-bx\int_1^\infty [-w_0(-xe^{-x})]^\alpha x^{-\beta}\d x, α > 0, β ∈ ℝ, and ò₀¹ [-w_-1(-x e^-x)]^a x^-bx\int_0^1 [-w_{-1}(-x e^{-x})]^\alpha x^{-\beta}\d x, α > − 1, β < 1. For the latter we use known integral representations and their evaluation by nonstandard Gaussian quadrature, if α ≠ β, and explicit formulae involving the trigamma function, if α = β.     

The logic of pseudo-S-integers

Let {n _i } be a sequence of natural numbers and let {p _i } be a listing of rational primes. Then an abelian groupG={x ∈ √| ord _pi x ≥ −n _i } is called a group of pseudo-integers. We investigate the logical properties of such groups of pseudo-integers and the counterparts of such groups in global fields in the case the number of primes allowed to appear in the denominator is infinite. We show that, while the addition problem of any recursive group of pseudo-integers is decidable, the Diophantine problem for some recursive groups of pseudo-integers with infinite number of primes allowed in the denominator, is not decidable. More precisely, there exist recursive groups of pseudo-integers, where infinite number of primes are allowed to appear in the denominator, such that there is no uniform algorithm to decide whether a polynomial equation over ℤ in several variables has solutions in the group. This result is obtained by giving a Diophantine definition of ℤ over these groups. The proof is based on the strong Hasse norm principal. The research for this paper has been partially supported by NSA grant MDA904-96-1-0019.     

Bounds on Maximal Families of Sets Not Containing Three Sets with A ∩ B ⊂ C, A ⊄ B

Lower and upper estimates are given on the size of a family of subsets of an n-element set containing no three distinct sets satisfying A ∩ B ⊂ C, A ⊄ B. This is a sharpening of an earlier result where the same question was solved under the condition that there are no three distinct sets such that A ∩ B ⊂ C. The second author was supported by the Hungarian National Foundation for Scientific Research grant numbers NK062321, AT048826, the Bulgarian National Science Fund under Grant IO-03/2005 and the projects of the European Community: INTAS 04-77-7171, FIST–MTKD-CT-2004-003006.     

Transcendence of Certain Reciprocal Sums of Linear Recurrences

 Suppose that {R _n } _n ⩾ 0 is a sequence of integers satisfying a binary linear recurrence relation with suitable conditions. We prove the transcendency of the numbers

Measure-valued Markov branching processes conditioned on non-extinction

We consider a particular class of measure-valued Markov branching processes that are constructed as “superprocesses” over some underlying Markov process. Such a processX dies out almost surely, so we introduce various conditioning schemes which keepX alive at large times. Under suitable hypotheses, which include the convergence of the semigroup for the underlying process to some limiting probability measureν, we show that the conditional distribution oft ⁻¹ X _t converges to that ofZν ast → ∞, whereZ is some strictly positive, real random variable. Research supported in part by NSF grant DMS 8701212. Research supported in part by an NSERC operating grant.