Moment inequalities and the Riemann hypothesis

It is known that the Riemann hypothesis is equivalent to the statement that all zeros of the Riemann ξ-function are real. On writingξ(x/2)=8 ∫ ₀ ^∞ Φ(t) cos(xt)dt, it is known that a necessary condition that the Riemann hypothesis be valid is that the moments \(\hat b_m (\lambda ): = \int_0^\infty {t^{2m} e^{\lambda t^2 } \Phi (t)dt}\) satisfy the Turán inequalities (*) $$(\hat b_m (\lambda ))^2 > \left( {\frac{{2m - 1}}{{2m + 1}}} \right)\hat b_{m - 1} (\lambda )\hat b_{m + 1} (\lambda )(m \geqslant 1,\lambda \geqslant 0).$$ We give here a constructive proof that log \(\Phi (\sqrt t )\) is strictly concave for 0 <t < ∞, and with this we deduce in Theorem 2.4 a general class of moment inequalities which, as a special case, establishes that the inequalities (*) are in fact valid for all real λ. As the case λ=0 of (*) corresponds to the Pólya conjecture of 1927, this gives a new proof of the Pólya conjecture.     

A method of multipliers for mathematical programming problems with equality and inequality constraints

This paper deals with the numerical solution of the general mathematical programming problem of minimizing a scalar functionf(x) subject to the vector constraints φ(x)=0 and ψ(x)≥0. The approach used is an extension of the Hestenes method of multipliers, which deals with the equality constraints only. The above problem is replaced by a sequence of problems of minimizing the augmented penalty function Ω(x, λ, μ,k)=f(x)+λ ^T φ(x)+kφ ^T (x)φ(x) ?μ ^T \(\tilde \psi \) (x)+k \(\tilde \psi \) ^T (x) \(\tilde \psi \) (x). The vectors λ and μ, μ ≥ 0, are respectively the Lagrange multipliers for φ(x) and \(\tilde \psi \) (x), and the elements of \(\tilde \psi \) (x) are defined by \(\tilde \psi \) _(j)(x)=min[ψ_(j)(x), (1/2k) μ_(j)]. The scalark>0 is the penalty constant, held fixed throughout the algorithm. Rules are given for updating the multipliers for each minimization cycle. Justification is given for trusting that the sequence of minimizing points will converge to the solution point of the original problem.     

Die metrisierbaren linearen Teilräume des Raumes\mathfrak{O}_M von L. Schwartz

In Schwartz' terminology, a real or complex valued functionf, defined and infinitely differentiable on ? _n , belongs to \(\mathfrak{O}_M \) iff, as well as any of its derivatives, is at most of polynomial growth. The topology of \(\mathfrak{O}_M \) is defined by the seminorms sup{∣?(x)D ^p f(x)∣;x∈? ⁿ }, where ? belongs to \(\mathfrak{S}\) andD ^p is any derivative. It is well-known that \(\mathfrak{O}_M \) is non-metrisable. For any μ: ? ⁿ →?, let \(\mathfrak{B}_\mu \) be the space of all infinitely differentiable functionsf satisfying, for eachp, sup{∣(1+∣x∣²)^?μ(p) D ^p f(x)∣;x∈? ⁿ }<∞, with the obvious topology. These spaces, which are of very little use elsewhere in the theory of distributions, can be conveniently applied to characterise the metrisable linear subspaces of \(\mathfrak{O}_M \) : A linear subspace of \(\mathfrak{O}_M \) is metrisable if and only if it is, algebraically and topologically, a subspace of some \(\mathfrak{B}_\mu \) .     

Grid approximation of singularly perturbed parabolic convection-diffusion equations subject to a piecewise smooth initial condition

A boundary value problem for a singularly perturbed parabolic convection-diffusion equation on an interval is considered. The higher order derivative in the equation is multiplied by a parameter ? that can take arbitrary values in the half-open interval (0, 1]. The first derivative of the initial function has a discontinuity of the first kind at the point x ₀. For small values of ?, a boundary layer with the typical width of ? appears in a neighborhood of the part of the boundary through which the convective flow leaves the domain; in a neighborhood of the characteristic of the reduced equation outgoing from the point (x ₀, 0), a transient (moving in time) layer with the typical width of ?^1/2 appears. Using the method of special grids that condense in a neighborhood of the boundary layer and the method of additive separation of the singularity of the transient layer, special difference schemes are designed that make it possible to approximate the solution of the boundary value problem ?-uniformly on the entire set $\bar G$ , approximate the diffusion flow (i.e., the product ?(?/?x)u(x, t)) on the set $\bar G^ * = \bar G\backslash \{ (x_0 ,0)\} $ , and approximate the derivative (?/?x)u(x, t) on the same set outside the m-neighborhood of the boundary layer. The approximation of the derivatives ?²(?²/?x ²)u(x, t) and (?/?t)u(x, t) on the set $\bar G^ * $ is also examined.     

On the regularity of solutions of model nonlinear elliptic systems with the oblique derivative type boundary condition

Properties of generalized solutions of model nonlinear elliptic systems of second order are studied in the semiball $B_1^ + = B_1 (0) \cap \{ x_n > 0\} \subset $ ? ⁿ , with the oblique derivative type boundary condition on $\Gamma _1 = B_1 (0) \cap \{ x_n = 0\} $ . For solutionsu∈H ¹(B ₁ ⁺ ) of systems of the form $\frac{d}{{dx_\alpha }}a_\alpha ^k (u_x ) = 0, k \leqslant {\rm N}$ , it is proved that the derivatives u_x are Hölder in $B_1^ + \cup \Gamma _1 )\backslash \Sigma $ , where H_n?p(σ)=0,p>2. It is shown for continuous solutions u from H¹(B₁/⁺) of systems $\frac{d}{{dx_\alpha }}a_\alpha ^k (u,u_x ) = 0$ that the derivatives u_x are Hölder on the set $(B_1^ + \cup \Gamma _1 )\backslash \Sigma , dim_\kappa \Sigma \leqslant n - 2$ . Bibliography: 13 titles.     

Existence and asymptotic behaviour of solutions of the very fast diffusion equation

Let n ≥ 3, 0 < m ≤ (n ? 2)/n, p > max(1, (1 ? m)n/2), and ${0 \le u_0 \in L_{loc}^p(\mathbb{R}^n)}$ satisfy ${{\rm lim \, inf}_{R\to\infty}R^{-n+\frac{2}{1-m}} \int_{|x|\le R}u_0\,dx = \infty}$ . We prove the existence of unique global classical solution of u _t = Δu ^m , u > 0, in ${\mathbb{R}^n \times (0, \infty), u(x, 0) = u_0(x)}$ in ${\mathbb{R}^n}$ . If in addition 0 < m < (n ? 2)/n and u ₀(x) ≈ A|x|^?q as |x| → ∞ for some constants A > 0, q < n/p, we prove that there exist constants α, β, such that the function v(x, t) = t ^α u(t ^β x, t) converges uniformly on every compact subset of ${\mathbb{R}^n}$ to the self-similar solution ψ(x, 1) of the equation with ψ(x, 0) = A|x|^?q as t → ∞. Note that when m = (n ? 2)/(n + 2), n ≥ 3, if ${g_{ij} = u^{\frac{4}{n+2}}\delta_{ij}}$ is a metric on ${\mathbb{R}^n}$ that evolves by the Yamabe flow ?g _ij /?t = ?Rg _ij with u(x, 0) = u ₀(x) in ${\mathbb{R}^n}$ where R is the scalar curvature, then u(x, t) is a global solution of the above fast diffusion equation.     

Strong capacity-estimates for “fractional” norms

It is proved that for all fractionall the integral \(\int\limits_0^\infty {(p,\ell ) - cap(M_t )} dt^p\) is majorized by the P-th power norm of the functionu in the space ? _p ^l (Rⁿ) (here M_t={x∶¦u(x)¦?t} and (p,l)-cap(e) is the (p,l)-capacity of the compactum e?Rⁿ). Similar results are obtained for the spaces W _p ^l (Rⁿ) and the spaces of M. Riesz and Bessel potentials. One considers consequences regarding imbedding theorems of “fractional” spaces in ?_q(dμ), whereμ is a nonnegative measure in Rⁿ. One considers specially the case p=1.     

Attractors of dissipative hyperbolic equations with singularly oscillating external forces

We study a uniform attractor $\mathcal{A}^\varepsilon $ for a dissipative wave equation in a bounded domain Ω ? ?ⁿ under the assumption that the external force singularly oscillates in time; more precisely, it is of the form g ₀(x, t)+ ε^?α g ₁ (x, t/ε), x ∈ Ω, t ∈ ?, where α > 0, 0 < ε ≤ 1. In E = H ₀ ¹ × L ₂, this equation has an absorbing set B ^ε estimated as ‖B ^ε‖ _E ≤C ₁+C ₂ε^?α and, therefore, can increase without bound in the norm of E as ε → 0+. Under certain additional constraints on the function g ₁(x, z), x ∈ Ω, z ∈ ?, we prove that, for 0 < α ≤ α ₀, the global attractors $\mathcal{A}^\varepsilon $ of such an equation are bounded in E, i.e., $\parallel \mathcal{A}^\varepsilon \parallel _E \leqslant C_3 $ , 0 < ε ≤ 1. Along with the original equation, we consider a “limiting” wave equation with external force g ₀(x, t) that also has a global attractor $\mathcal{A}^0 $ . For the case in which g ₀(x, t) = g ₀(x) and the global attractor $\mathcal{A}^0 $ of the limiting equation is exponential, it is established that, for 0 < α ≤ α ₀, the Hausdorff distance satisfies the estimate $dist_E (\mathcal{A}^\varepsilon ,\mathcal{A}^0 ) \leqslant C\varepsilon ^{\eta (\alpha )} $ , where η(α) > 0. For η(α) and α ₀, explicit formulas are given. We also study the nonautonomous case in which g ₀ = g ₀(x, t). It is assumed that sufficient conditions are satisfied for which the “limiting” nonautonomous equation has an exponential global attractor. In this case, we obtain upper bounds for the Hausdorff distance of the attractors $\mathcal{A}^\varepsilon $ from $\mathcal{A}^0 $ , similar to those given above.     

Optimal discounted linear control of the wiener process

The following stochastic control problem is considered: to minimize the discounted expected total cost $$J(x;u) = E\int_0^\infty {\exp ( - at)[\phi } (x_l ) + |u_l (x)|]dt,$$ subject todx _t =u _t (x)dt+dw _t ,x ₀=x, |u _t |≤1, (w _t ) a Wiener process, α>0. All bounded by unity, measurable, and nonanticipative functionalsu _t (x) of the state processx _t are admissible as controls. It is proved that the optimal law is of the form $$\begin{gathered} u_t^* (x) = - 1,x_t > b, \hfill \\ u_t^* (x) = 0,|x_t | \leqslant b, \hfill \\ u_t^* (x) = 1,x_t< - b, \hfill \\ \end{gathered}$$ for some switching pointb > 0, characterized in terms of the function ø(·) through a transcendental equation.     

Large deviations of the first passage time for a random walk with semiexponentially distributed jumps

Suppose that ξ, ξ(1), ξ(2), ... are independent identically distributed random variables such that ?ξ is semiexponential; i.e., $P( - \xi \geqslant t) = e^{ - t^\beta L(t)} $ is a slowly varying function as t → ∞ possessing some smoothness properties. Let E ξ = 0, D ξ = 1, and S(k) = ξ(1) + ? + ξ(k). Given d > 0, define the first upcrossing time η ₊(u) = inf{k ≥ 1: S(k) + kd > u} at nonnegative level u ≥ 0 of the walk S(k) + kd with positive drift d > 0. We prove that, under general conditions, the following relation is valid for $u = (n) \in \left[ {0, dn - N_n \sqrt n } \right]$ : 0.1 $P(\eta + (u) > n) \sim \frac{{E\eta + (u)}}{n}P(S(n) \leqslant x) as n \to \infty $ , where x = u ? nd < 0 and an arbitrary fixed sequence N _n not exceeding $d\sqrt n $ tends to ∞. The conditions under which we prove (0.1) coincide exactly with the conditions under which the asymptotic behavior of the probability P(S(n) ≤ x) for $x \leqslant - \sqrt n $ was found in [1] (for $x \in \left[ { - \sqrt n ,0} \right]$ it follows from the central limit theorem).     

Descriptions of exceptional sets in the circles for functions from the Bergman space

Let D be a domain in $\mathbb{C}^2 $ . For w ∈ $\mathbb{C}$ , let D_w=\{z \in $\mathbb{C}$ \, \vert \, (z,w)\in D\}. If f is a holomorphic and square-integrable function in D, then the set E(D, f) of all w such that f(., w) is not square-integrable in D _w is of measure zero. We call this set the exceptional set for f. In this note we prove that for every 0 < r < 1, and every G _δ-subset E of the circle C(0,r)=\{z \in $\mathbb{C}$ \, \vert \, \vert z \vert = r \},there exists a holomorphic square-integrable function f in the unit ball B in $\mathbb{C}$ ² such that E(B, f) = E.     

A markov type inequality for higher derivatives of polynomials

Let ‖·‖ be the weightedL ²-norm with weightw(t). LetP _n be the set of all complex polynomials whose degree does not exceedn and let \(\gamma _n^{(r)} : = \sup _{f \in P_n } \) (‖f ^(r)‖/‖f‖). In this paper we given upper and lower bound for γ _n ^(r) in the case of the Laguerre weight functionw(t)=exp (?t) and investigate its behaviour asn→∞. Moreover, we derive some identities concerningorthogonal polynomials.     

Solution of the Szili type conjugate problem on the roots of the laguerre polynomials

Let ?1<α≤0 and let $$L_n^{(\alpha )} (x) = \frac{1}{{n!}}x^{ - \alpha } e^x \frac{{d^n }}{{dx^n }}(x^{\alpha + n} e^{ - x} )$$ be the generalizednth Laguerre polynomial,n=1,2,… Letx ₁,x ₂,…,x _n andx*₁,x*₂,…,x* _n?1 denote the roots ofL _n ^(α) (x) andL _n ^(α)′ (x) respectively and putx*₀=0. In this paper we prove the following theorem: Ify ₀,y ₁,…,y _n ?1 andy ₁ ^′ ,…,y _n ^′ are two systems of arbitrary real numbers, then there exists a unique polynomialP(x) of degree 2n?1 satisfying the conditions $$\begin{gathered} P\left( {x_k^* } \right) = y_k (k = 0,...,n - 1) \hfill \\ P'\left( {x_k } \right) = y_k^\prime (k = 1,...,n). \hfill \\ \end{gathered} $$ .     

Multiobjective programming and penalty functions

This paper generalizes the penalty function method of Zang-will for scalar problems to vector problems. The vector penalty function takes the form $$g(x,\lambda ) = f(x) + \lambda ^{ - 1} P(x)e,$$ wheree ?R ^m, with each component equal to unity;f:R ⁿ →R ^m, represents them objective functions {f ⁱ} defined onX \( \subseteq \) R ⁿ; λ ∈R ¹, λ>0;P:R ⁿ →R ¹ X \( \subseteq \) Z \( \subseteq \) R ⁿ,P(x)≦0, ∨x ∈R ⁿ,P(x) = 0 ?x ∈X. The paper studies properties of {E (Z, λ _r )} for a sequence of positive {λ _r } converging to 0 in relationship toE(X), whereE(Z, λ _r ) is the efficient set ofZ with respect tog(·, λ_r) andE(X) is the efficient set ofX with respect tof. It is seen that some of Zangwill's results do not hold for the vector problem. In addition, some new results are given.     

The extreme bodies in the set of plane convex bodies with a given width function

Let \(\bar K\) (w) denote the class of plane convex bodies having a width functionw. Examining the length measure of the boundary of a convex body in \(\bar K\) (w), a characterization is given for the extreme (indecomposable) bodies in \(\bar K\) (w). This is a generalization of the solution previously given by the author in Israel J. Math. (1974) for the case wherew′ is absolutely continuous.     

Zeros of analytic functions and norms of inverse matrices

Letk _n be the smallest constant such that for anyn-dimensional normed spaceX and any invertible linear operatorT ∈L(X) we have $|\det (T)| \cdot ||T^{ - 1} || \le k_n |||T|^{n - 1} $ . LetA ₊ be the Banach space of all analytic functionsf(z)=Σ _k≥0 a _kz^k on the unit diskD with absolutely convergent Taylor series, and let ‖f‖A ₊=σ _k≥0 |ακ|; define ? _n on $\overline D ^n $ by $ \begin{array}{l} \varphi _n \left( {\lambda _1 ,...,\lambda _n } \right) \\ = inf\left\{ {\left\| f \right\|_{A + } - \left| {f\left( 0 \right)} \right|; f\left( z \right) = g\left( z \right)\prod\limits_{i = 1}^n {\left( {\lambda _1 - z} \right), } g \in A_ + , g\left( 0 \right) = 1 } \right\} \\ \end{array} $ . We show thatk _n=sup {? _n(λ₁,…, λ _n ); (λ₁,…, λ _n )∈ $\overline D ^n $ }. Moreover, ifS is the left shift operator on the space ?∞:S(x ₀,x ₁, …,x _p, …)=(x ₁,…,x _p,…) and if J_n(S) denotes the set of allS-invariantn-dimensional subspaces of ?∞ on whichS is invertible, we have $k_n = \sup \{ |\det (S|_E )|||(S|_E )^{ - 1} ||E \in J_n (S)\} $ . J. J. Schäffer (1970) proved thatk _n≤√en and conjectured thatk _n=2, forn≥2. In factk ₃>2 and using the preceding results, we show that, up to a logarithmic factor,k _n is of the order of √n whenn→+∞.     

О приближении функци й средними Чезаро вто рого

Let σ _n ² (f, x) be the Cesàro means of second order of the Fourier expansion of the function f. Upper bounds of the deviationf(x)-σ _n ² (f, x) are studied in the metricC, while f runs over the class \(\bar W^1 C\) , i. e., of the deviation $$F_n^2 (\bar W^1 ,C) = \mathop {\sup }\limits_{f \in \bar W^1 C} \left\| {f(x) - \sigma _n^2 (f,x)} \right\|_c$$ . It is proved that the function $$g^* (x) = \frac{4}{\pi }\mathop \sum \limits_{v = 0}^\infty ( - 1)^v \frac{{\cos (2v + 1)x}}{{(2v + 1)^2 }}$$ , for whichg ^*′(x)=sign cosx, satisfies the following asymptotic relation: $$F_n^2 (\bar W^1 ,C) = g^* (0) - \sigma _n^2 (g^* ,0) + O\left( {\frac{1}{{n^4 }}} \right)$$ , i.e.g ^* is close to the extremal function. This makes it possible to find some of the first terms in the asymptotic formula for \(F_n^2 (\bar W^1 ,C)\) asn → ∞. The corresponding problem for approximation in the metricL is also considered.     

Estimates of maximal functions measuring local smoothness

Letη be a nondecreasing function on (0, 1] such thatη(t)/t decreases andη(+0)=0. Letf ∈L(I ⁿ ) (I≡[0,1]. Set $${\mathcal{N}}_\eta f(x) = \sup \frac{1}{{\left| Q \right|\eta (\left| Q \right|^{1/n} )}} \smallint _Q \left| {f(t) - f(x)} \right|dt,$$ , where the supremum is taken over all cubes containing the pointx. Forη=t ^α (0<α≤1) this definition was given by A.Calderón. In the paper we prove estimates of the maximal functions ${\mathcal{N}}_\eta f$ , along with some embedding theorems. In particular, we prove the following Sobolev type inequality: if $$1 \leqslant p< q< \infty , \theta \equiv n(1/p - 1/q)< 1, and \eta (t) \leqslant t^\theta \sigma (t),$$ , then $$\parallel {\mathcal{N}}_\sigma {f} {\parallel_{q,p}} \leqslant c \parallel {\mathcal{N}}_\eta {f} {\parallel_p} .$$ . Furthermore, we obtain estimates of ${\mathcal{N}}_\eta f$ in terms of theL ^p -modulus of continuity off. We find sharp conditions for ${\mathcal{N}}_\eta f$ to belong toL ^p (I ⁿ ) and the Orlicz class?(L), too.     

An initial value problem for a singular parabolic equation of even order

At first Cauchy-problem for the equation: \(L[u(X,t)] \equiv \sum\limits_{i = 1}^n {\frac{{\partial ^2 u}}{{\partial x_1^2 }} + \frac{{2v}}{{\left| X \right|^2 }}} \sum\limits_{i = 1}^n {x_i \frac{{\partial u}}{{\partial x_i }} - \frac{{\partial u}}{{\partial t}} = 0} \) wheren≥1,v—an arbitrary constant,t>0,X=(x ₁, …, x_n)∈E ⁿ/{0}, |X|= =(x ₁ ² +…+x _n ² )^1/2, with 0 being a centre of coordinate system, is studied. Basing on the above, the solution of Cauchy-Nicolescu problem is given which consist in finding a solution of the equationL ^p [u (X, t)]=0, withp∈N subject the initial conditions \(\mathop {\lim }\limits_{t \to \infty } L^k [u(X,t)] = \varphi _k (X)\) ,k=0, 1,…,p?1 and ?_k(X) are given functions.     

An inverse problem of the three-dimensional wave equation for a general annular vibrating membrane with piecewise smooth boundary conditions

This paper deals with the very interesting problem about the influence of piecewise smooth boundary conditions on the distribution of the eigenvalues of the negative Laplacian inR ³. The asymptotic expansion of the trace of the wave operator $\widehat\mu (t) = \sum\limits_{\upsilon = 1}^\infty {\exp \left( { - it\mu _\upsilon ^{1/2} } \right)} $ for small ?t? and $i = \sqrt { - 1} $ , where $\{ \mu _\nu \} _{\nu = 1}^\infty $ are the eigenvalues of the negative Laplacian $ - \nabla ^2 = - \sum\limits_{k = 1}^3 {\left( {\frac{\partial }{{\partial x^k }}} \right)} ^2 $ in the (x ¹,x ²,x ³), is studied for an annular vibrating membrane Ω inR ³ together with its smooth inner boundary surfaceS ₁ and its smooth outer boundary surfaceS ₂. In the present paper, a finite number of Dirichlet, Neumann and Robin boundary conditions on the piecewise smooth componentsS ^* _i(i=1, …,m) ofS ₁ and on the piecewise smooth componentsS ^* _i(i=m+1, …,n) ofS ₂ such that $S_1 = \bigcup\limits_{i = 1}^m {S_i^* } $ and $S_2 = \bigcup\limits_{i = m + 1}^n {S_i^* } $ are considered. The basic problem is to extract information on the geometry of the annular vibrating membrane ω from complete knowledge of its eigenvalues by analyzing the asymptotic expansions of the spectral function $\widehat\mu (t)$ for small ?t?.