The Wiener--Hopf Integral Equation in the Supercritical Case

We consider the scalar homogeneous equation $S(x) = \int_0^\infty {K(x - t)S(t)dt,{\text{ }}x \in \mathbb{R}^ + \equiv (0,\infty ),}$ with symmetric kernel $K:K( - x) = K(x),{\text{ }}x \in \mathbb{R}_1$ satisfying the conditions $0 \leqslant K \in L_1 (\mathbb{R}^ + ) \cap C^{\left( 2 \right)} (\mathbb{R}^ + )$ , $\int_0^\infty {K(t)dt > \frac{1}{2}} $ , $K' \leqslant 0{\text{ }}and 0 \leqslant K'' \downarrow {\text{ }}on \mathbb{R}^ + $ . We prove the existence of a real solution S of the equation given above with asymptotic behavior $S(x) = O(x){\text{ as }}x \to + \infty $ .     

The prime-counting function and its analytic approximations

The paper describes a systematic computational study of the prime counting function π(x) and three of its analytic approximations: the logarithmic integral \({\text{li}}{\left( x \right)}: = {\int_0^x {\frac{{dt}}{{\log \,t}}} }\), \({\text{li}}{\left( x \right)} - \frac{1}{2}{\text{li}}{\left( {{\sqrt x }} \right)}\), and \(R{\left( x \right)}: = {\sum\nolimits_{k = 1}^\infty {{\mu {\left( k \right)}{\text{li}}{\left( {x^{{1 \mathord{\left/ {\vphantom {1 k}} \right. \kern-\nulldelimiterspace} k}} } \right)}} \mathord{\left/ {\vphantom {{\mu {\left( k \right)}{\text{li}}{\left( {x^{{1 \mathord{\left/ {\vphantom {1 k}} \right. \kern-\nulldelimiterspace} k}} } \right)}} k}} \right. \kern-\nulldelimiterspace} k} }\), where μ is the Möbius function. The results show that π(x)x) for 2≤x≤10¹⁴, and also seem to support several conjectures on the maximal and average errors of the three approximations, most importantly \({\left| {\pi {\left( x \right)} - {\text{li}}{\left( x \right)}} \right|} < x^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}\) and \( - \frac{2}{5}x^{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2}} < {\int_2^x {{\left( {\pi {\left( u \right)} - {\text{li}}{\left( u \right)}} \right)}du < 0} }\) for all x>2. The paper concludes with a short discussion of prospects for further computational progress.     

Upper bounds for discrete moments of the derivatives of the riemann zeta-function on the critical line*

Assuming the Riemann hypothesis, we establish upper bounds for discrete moments of the Riemann zeta-function and its derivatives on the critical line. Moreover, we express continuous moments of the Riemann zeta-function and its derivatives in terms of these discrete moments. This allows us to give conditional upper bounds for $ {\int_0^T {\left| {{\zeta^{(l)}}\left( {{{1} \left/ {2} \right.} + {\text{i}}t} \right)} \right|}^{2k}}{\text{d}}t $ , where l and k are nonnegative integers.     

A general class of inequalities with mixed means

Suppose (T, Σ, μ) is a space with positive measure,f: ? → ? is a strictly monotone continuous function, and &;(T) is the set of real μ-measurable functions onT. Letx(·) ∈ &;(T) andf ○x)(·) ∈L ₁(T,μ). Comparison theorems are proved for the means $\mathfrak{M}_{(T,{\mathbf{ }}\mu ,{\mathbf{ }}f)} (x( \cdot ))$ and the mixed means $\mathfrak{M}_{(T_1 ,{\mathbf{ }}\mu _1 ,{\mathbf{ }}f_1 )} (\mathfrak{M}_{(T_2 ,{\mathbf{ }}\mu _2 ,{\mathbf{ }}f_2 )} (x( \cdot )))$ these inequalities imply analogs and generalizations of some classical inequalities, namely those of Hölder, Minkowski, Bellman, Pearson, Godunova and Levin, Steffensen, Marshall and Olkin, and others. These results are a continuation of the author's studies.     

A Problem with Nonlocal Boundary Conditions for a Quasilinear Parabolic Equation

The solvability of the nonlocal boundary value problem

An analog of the weyl decomposition of the spaceL q (ω, ℝ n ) for a first-order differential operatorfor a first-order differential operator

We prove the following theorem: Suppose the function f(x) belongs toL _q (ω, ? ⁿ ), ω ? ? ^m , q∈(1, ∞), and satisfies the inequality $$|\int\limits_\omega {(f(x),{\mathbf{ }}v(x)){\mathbf{ }}dx| \leqslant \mu ||} v||'_q ,{\mathbf{ }}\tfrac{1}{q} + \tfrac{1}{{q'}} = 1,$$ for all n-dimensional vector-valued functions in the kernel of a scalar-valued first-order differential operator £ for which the second-order operatorLL ^* is elliptic. Then there exists a function p(x)∈W _q ¹ (ω) such that $$||f(x) - \mathfrak{L}^* p(x)||q \leqslant C_q \mu .$$ Bibliography: 6 titles.     

ON THE ERROR FUNCTION OF THE SQUARE-FULL INTEGERS

Let L(x) denote the number of square-full integers not exceeding x. It is proved in [1] thatL(x)～(ζ(3/2)/ζ(3))x~(1/2) (ζ(2/3)/ζ(2))x~(1/3) as x→∞,where ζ(s) denotes the Riemann zeta function. Let △(x) denote the error function in the asymptotic formula for L(x). It was shown by D. Suryanaryana~([2]) on the Riemann hypothesis (RH) that1/x integral from n=1 to x |△(t)|dt=O(x~(1/10 s))for every ε>0. In this paper the author proves the following asymptotic formula for the mean-value of △(x) under the assumption of R. H.integral from n=1 to T (△~2(t/t~(6/5))) dt～c log T,where c>0 is a constant.     

Linear Volterra equations and integrated solution families

We consider the linear Volterra equation $${\text{(VE;}}A{\text{,}}a{\text{)}}u{\text{(}}t{\text{) = }}x {\text{ + }}\int_{\text{0}}^{\text{t}} { a{\text{(}}t{\text{ - }}s{\text{)}}Au{\text{(}}s{\text{)}}ds {\text{for }}t \geqslant {\text{0}}{\text{.}}} $$ HereA is an unbounded closed linear operator in a Banach spaceX anda is a scalar valued function. We study the theory of solution families which are not necessarily exponentially bounded and also, as their generalizations, consider the notion ofn-times integrated solution families for (VE;A, a). These families are characterized in terms of the associated Volterra integral equation $${\text{(VE;}}A{\text{,}}a{\text{)}}_n u{\text{(}}t{\text{) = }}\frac{{t^n }}{{n!}}x {\text{ + }}A{\text{ }}\int_{\text{0}}^{\text{t}} { a{\text{(}}t{\text{ - }}s{\text{)}}u{\text{(}}s{\text{)}}ds {\text{for }}t \geqslant {\text{0}}{\text{.}}} $$ The results are applied to additive and multiplicative perturbation theorems and adjoint problems.     

Presentations of the amalgamated free product of two infinite cycles

LetH=〈a,b;a ^k =b ^l 〉, wherek,l≧2 andk+l>4. McCool and Pietrowski have proved that any pair of generators forH is Nielsen equivalent to a pairx=a ^r andy=b ^s where $$(a){\text{ }}gcd(r, s) = gcd(r, k) = gcd(s, l) = 1,$$ $$(b){\text{ }}0< 2r \leqq ks{\text{ }}and{\text{ }}0< 2s \leqq lr.$$ In terms ofx andy,H can be presented as $$G = \left\langle {x,{\text{ }}y;{\text{ }}x^{ks} = y^{lr} ,\left[ {x,{\text{ }}y^l } \right] = \left[ {x^k ,{\text{ }}y} \right] = 1} \right\rangle$$ and Zieschang has shown that ifr=1 ors=1, thenH can be defined by a single relation inx andy. We establish the exact converse of Zieschang's result, namely thatH is not defined by a single relation inx andy unlessr=1 ors=1. The proof is based on an observation of Magnus which associates polynomials with relators and some elementary facts about cyclotomic polynomials.     

On the Univalence of an Integral on Subclasses of Meromorphic Functions

We study the integral operator $P_\lambda |f|(\zeta ) = \int {_{\zeta _0 }^\zeta } \left( {f\prime \left( t \right)} \right)^\lambda dt,{\text{ }}|\zeta |{\text{ }} > 1$ , acting on the class ∑ of functions meromorphic and univalent in the exterior of the unit disk. We refine the ranges of the parameter λ for which the operator preserves univalence either on ∑ or on its subclasses consisting of convex functions. As a consequence, a two-sided estimate is deduced for the separating constant in the sufficient condition for the univalent solvability of exterior inverse boundary-value problems.     

Evaluation of the limits of maximal means

It is proved that the limit $$\mathop {\lim }\limits_{\Delta \to \infty } \mathop {\sup }\limits_\gamma \tfrac{1}{\Delta }\int_0^\Delta {f(\gamma (t))dt} $$ , wheref: ? → ? is a locally integrable (in the sense of Lebesgue) function with zero mean and the supremum is taken over all solutions of the generalized differential equation γ ∈ [ω1, ω2], coincides with the limit $$\mathop {\lim }\limits_{T \to \infty } \mathop {\sup }\limits_{c \geqslant 0} \varphi _f (k,{\mathbf{ }}T,{\mathbf{ }}c)$$ , where $$\varphi _f = \frac{{(k - 1)\bar I_f (T,c)}}{{1 + (k - 1)\bar \lambda _f (T,c)}},k = \frac{{\omega _2 }}{{\omega _1 }}$$ . Here ¯λf = λf /T, ¯ I_f =If/T, and λf is the Lebesgue measure of the set $$\{ \gamma \in [\gamma _0 ,\gamma _0 + T]:f(\gamma ) \geqslant c\} = A_f ,I_f = \int_{A_f } {f(\gamma )d\gamma } $$ . It is established that this limit always exists for almost-periodic functionsf.     

On a result of Hardy and Littlewood concerning Fourier series

Let f ∈ L ¹( $ \mathbb{T} $ ) and assume that $$ f\left( t \right) \sim \frac{{a_0 }} {2} + \sum\limits_{k = 1}^\infty {\left( {a_k \cos kt + b_k \sin kt} \right)} $$ Hardy and Littlewood [1] proved that the series $ \sum\limits_{k = 1}^\infty {\frac{{a_k }} {k}} $ converges if and only if the improper Riemann integral $$ \mathop {\lim }\limits_{\delta \to 0^ + } \int_\delta ^\pi {\frac{1} {x}} \left\{ {\int_{ - x}^x {f(t)dt} } \right\}dx $$ exists. In this paper we prove a refinement of this result.     

Extremal functional interpolation in the mean with least value of thenth derivative for large averaging intervals

The smallest numberA<∞ is found such that for any sequenceY={y _k ,k ∈ ?} with ¦Δ ⁿ y _k ¦≤1 there exists au(t), ¦u(t)¦ ≤ A, for which the equationy ⁿ (t)=u(t) (?∞<t<∞) has a solution satisfying the conditions $$y_k = \frac{1}{h}\int_{ - h/2}^{h/2} {y(k + 1){\mathbf{ }}dt} ,{\mathbf{ }}where{\mathbf{ }}k{\mathbf{ }} \in {\mathbf{ }}\mathbb{Z},{\mathbf{ }}1{\mathbf{ }}< {\mathbf{ }}h{\mathbf{ }}< {\mathbf{ }}2.$$ , wherek ∈ ?, 1<h<2. A similar problem is treated inL _p (?∞, ∞). It is shown that forh=2m (m a natural number) no such finiteA exists.     

Characterization and recognition of d.c. functions

A function ${f : \Omega \to \mathbb{R}}$ , where Ω is a convex subset of the linear space X, is said to be d.c. (difference of convex) if f =  g ? h with ${g, h : \Omega \to \mathbb{R}}$ convex functions. While d.c. functions find various applications, especially in optimization, the problem to characterize them is not trivial. There exist a few known characterizations involving cyclically monotone set-valued functions. However, since it is not an easy task to check that a given set-valued function is cyclically monotone, simpler characterizations are desired. The guideline characterization in this paper is relatively simple (Theorem 2.1), but useful in various applications. For example, we use it to prove that piecewise affine functions in an arbitrary linear space are d.c. Additionally, we give new proofs to the known results that C ^1,1 functions and lower-C ² functions are d.c. The main goal remains to generalize to higher dimensions a known characterization of d.c. functions in one dimension: A function ${f : \Omega \to \mathbb{R}, \Omega \subset \mathbb{R}}$ open interval, is d.c. if and only if on each compact interval in Ω the function f is absolutely continuous and has a derivative of bounded variation. We obtain a new necessary condition in this direction (Theorem 3.8). We prove an analogous sufficient condition under stronger hypotheses (Theorem 3.11). The proof is based again on the guideline characterization. Finally, we obtain results concerning the characterization of convex and d.c. functions obeying some kind of symmetry.     

StrongL 1-norm consistency of data based histogram estimates of densities

In this paper we investigate the problem of estimating the d-dimensional probability densityf(x),x∈R~d from a sample of size n.The non-parametric estimator is the data based histogram f_n(x)as defined in (1).Under suitable conditions,we have proved the L_1-norm consistance of this estimate,that is(?)interal from R~α|f(x)-f_n(x)|dx=0, a.s.     

The pressure equation for fluid flow in a stochastic medium

An equation modelling the pressurep(x) =p(x, w) atx ∈D ⊂R ^d of an incompressible fluid in a heterogeneous, isotropic medium with a stochastic permeabilityk(x, w) ≥ 0 is the stochastic partial differential equation

The Laguerre inequalities with applications to a problem associated with the Riemann hypothesis

We investigate here a new numerical method, base on the Laguerre inequalities, for determining lower bounds for the de Bruijn-Newman constant ∧, which is related to the Riemann Hypothesis. (Specifically, the truth of the Riemann Hypothesis would imply that ∧≦0.) Unlike previous methods which involved either finding nonreal zeros of associated Jensen polynomials or finding nonreal zeros of a certain real entire function, this new method depends only on evaluating, in real arithmetic, the Laguerre difference $$L_1 (H_\lambda (x))\begin{array}{*{20}c} {\text{.}} \\ {\text{.}} \\ \end{array} = (H'_\lambda (x))^2 - H_\lambda (x) \cdot H''_{_\lambda } (x){\text{ (}}x,{\text{ }}\lambda \in \mathbb{R}{\text{)}}$$ where \((H_\lambda (z)\begin{array}{*{20}c} {\text{.}} \\ {\text{.}} \\ \end{array} = \int_0^\infty {e^{\lambda t^2 } \Phi (t)}\) cos(tz)dt is a real entire function. We apply this method to obtain the new lower bound for ∧, -0.0991 < ∧ which improves all previously published lower bounds for ∧.     

Solvability of convolution type integro-differential equations on ℝ

The paper looks for the solutions of integro-differential equations of the form

$ - \frac{{d\varphi }}{{dx}} + A\varphi (x) = g(x) + B\int_\mathbb{R} {k(x - t)\lambda (t)\varphi (t)dt, x \in \mathbb{R}} $

in the class of functions which are absolutely continuous and of slow growth on ?. It is assumed that A and B are nonnegative parameters, 0 ≤ g ∈ L ₁ (?), 0 ≤ k ∈ L ₁ (?), ∫_? k(x) dx = 1 and 0 ≤ λ(x) ≤ 1 is a measurable function in ?. The equation is solved by a special factorization of the corresponding integro-differential operator in combination with appropriately modified standard methods of the theory of convolution type integral equations.

     

Radial solutions of Neumann problems involving mean extrinsic curvature and periodic nonlinearities

We show that if ${{\mathcal A} \subset \mathbb{R}^N}$ is an annulus or a ball centered at zero, the homogeneous Neumann problem on ${{\mathcal A}}$ for the equation with continuous data $$\nabla \cdot \left(\frac{\nabla v}{\sqrt{1 - |\nabla v|^2}} \right) = g(|x|,v) + h(|x|)$$ has at least one radial solution when g(|x|,·) has a periodic indefinite integral and ${\int_{\mathcal A} h(|x|)\,{\rm{d}}x = 0.}$ The proof is based upon the direct method of the calculus of variations, variational inequalities and degree theory.