On the number of semismooth integers

Let [Formula: see text] be a fixed positive number. Define [Formula: see text] as the number of positive integers [Formula: see text] having no prime factors [Formula: see text], and define [Formula: see text] as the number of positive integers [Formula: see text] having [Formula: see text] prime factors [Formula: see text], with all the other prime factors [Formula: see text]. In this paper, we give an asymptotic estimate for the ratio [Formula: see text], provided that [Formula: see text], [Formula: see text], and [Formula: see text] as [Formula: see text]. Also, combining this estimate with conventional ones for [Formula: see text], we provide sharp estimates for [Formula: see text].

A Note on the Primitive Roots and the Golomb Conjecture

In this paper, we use the elementary methods and the estimates for character sums to prove the following conclusion. Let p be a prime large enough. Then, for any positive integer n with p 1 / 2 + ɛ ≤ n < p , there must exist two primitive roots α and β modulo p with 1 < α , β ≤ n − 1 such that the equation n = α + β holds, where 0 < ɛ < 1 / 2 is a fixed positive number. In other words, n can be expressed as the exact sum of two primitive roots modulo p .

Lattice Simplices with a Fixed Positive Number of Interior Lattice Points: A Nearly Optimal Volume Bound

We give an explicit upper bound on the volume of lattice simplices with fixed positive number of interior lattice points. The bound differs from the conjectural sharp upper bound only by a linear factor in the dimension. This improves significantly upon the previously best results by Pikhurko from 2001.

ON DISCRETE UNIVERSALITY OF COMPOSITE FUNCTIONS

In 1975, S.M. Voronin proved that the Riemann zeta-function ζ (s) is universal in the sense that its shifts approximate uniformly on some sets any analytic function. Let h be a fixed positive number such that exp is irrational for all . In the paper, the classes of functions F such that the shifts F (ζ (s + imh)), , approximate any analytic function are presented. For the proof of theorems, some elements of the space of analytic functions are applied.

Inverse Pressure Estimates and the Independence of Stable Dimension for Non-Invertible Maps

We study the case of an Axiom A holomorphic non-degenerate (hence non-invertible) mapf: ℙ2ℂ → ℙ2ℂ, where ℙ2ℂ stands for the complex projective space of dimension 2. Letδs(x)denote a basic set for f of unstable index 1, and x an arbitrary point of Λ; we denote byδs(x)the Hausdorff dimension of∩ Λ, whereris some fixed positive number andis the local stable manifold atxof sizer;δs(x)is calledthe stable dimension at x. Mihailescu and Urba ńnski introduced a notion of inverse topological pressure, denoted by P−, which takes into consideration preimages of points. Manning and McCluskey studied the case of hyperbolic diffeomorphisms on real surfaces and give formulas for Hausdorff dimension. Our non-invertible situation is different here since the local unstable manifolds are not uniquely determined by their base point, instead they depend in general on whole prehistories of the base points. Hence our methods are different and are based on using a sequence of inverse pressures for the iterates off, in order to give upper and lower estimates of the stable dimension. We obtain an estimate of the oscillation of the stable dimension on Λ. When each pointxfrom Λ has the same numberd′of preimages in Λ, then we show thatδs(x)is independent of x; in factδs(x)is shown to be equal in this case with the unique zero of the mapt → P(tϕs−log d′). We also prove the Lipschitz continuity of the stable vector spaces over Λ; this proof is again different than the one for diffeomorphisms (however, the unstable distribution is not always Lipschitz for conformal non-invertible maps). In the end we include the corresponding results for a real conformal setting.

On the fractional parts of lacunary sequences

In this paper, we prove that if $t_0, t_1, t_2, \dots$ is a lacunary sequence, namely, $t_{n+1}/t_n\geq 1+r^{-1}$ for each $n\geq 0$, where $r$ is a fixed positive number, then there are two positive constants $c(r)=\max(1-r, 2(3r+6)^{-2})$ and $\xi=\xi(t_0, t_1,\dots)$ such that the fractional parts $\{\xi t_n\}$, $n=0,1,2,\dots$, all belong to a subinterval of $[0,1)$ of length $1-c(r)$. Some applications of this theorem to the chromatic numbers of certain graphs and to some fast growing sequences are discussed. We prove, for instance, that the number $\sqrt{10}$ can be written as a quotient of two positive numbers whose decimal expansions contain the digits $0$, $1$, $2$, $3$ and $4$ only.

Floquet solutions of non-linear ordinary differential equations

SynopsisIn this paper we study the solutions of the boundary value problemwhere t ∊ℝ, x ∊ ℝN, f is a continuous function of (t,x)and locally Lipschitz in x and ω is a fixed positive number and λ ∊ ℝ. By using degree theory we prove results on the existence of solutions of (*) and the dependence of such solutions on λ. We shall prove that (*) does not have an isolated solution, and study the topological properties of the components of solutions of (*).

Optimization and α-Disfocality for Ordinary Differential Equations

For f ∊ L−1(0, T), we define the distribution functionwhere T is a fixed positive number and |·| denotes Lebesgue measure. Let Φ:[0, T] → [0, m] be a nonincreasing, right continuous function. In an earlier paper [3], we discussed the equation(0.1)when the coefficient q was allowed to vary in the classWe were in particular interested in finding the supremum and infimum of y(T) when q was in or in the convex hull Ω(Φ) of (see below).

On the generalized heat-equation

The general heat equation is defined aswhere v is a fixed positive number and α is a fixed number. If v = α = 0, then (1.1) reduces to the ordinary heat equationwhere u(x,t) is regarded as the temperature at a point x at time t, in an infinite insulated rod extended along the x-axis in the xt-plane. If we set , then (1.1) becomes

Any 2-Sphere in E3 with Uniform Interior Tangent Balls is Flat

This paper addresses some flatness properties of an (n – 1)-sphere Σ in Euclidean n-space En resulting from the presence of round balls in En tangent to Σ. The notion of tangency used here is geometric rather than differentiable, for a round n-cell Bp (that is, the set of points whose distance, in the standard metric, from some center point is less than or equal to a fixed positive number) is said to be tangent to the (n – 1)-sphere Σ in En at a point p ∈ Σ if p ∈ Bp and Int Bp ⊂ Σ = ∅. The ball Bp is called an interior tangent ball at p if Int Bp ⊂ Int Σ; otherwise, it is called an exterior tangent ball at p.