Precise rates in the law of the logarithm in the Hilbert space

Let be a sequence of i.i.d. random variables taking values in a real separable Hilbert space (H,‖⋅‖) with covariance operator Σ, and set Sn=X₁+?+Xn, n?1. Let . We prove that, for any 1<r<3/2 and a>−d/2,

     

On the exact constant in the quantitative Steinitz theorem in the plane

We determine the maximal value ofr with the following property. If the convex hull of a set inR ² contains a unit circleB, then a subset of at most four points can be selected so that the convex hull of this subset contains the circle of radiusr concentric withB. That the result is sharp is shown by the example when the original set is the set of vertices of a regular pentagon circumscribed aroundB. Imre Bárány was partially supported by Hungarian National Science Foundation Grant Nos. 1907 and 1909. Aladár Heppes was partially supported by Hungarian National Science Foundation Grant No. 2583.     

On the continuity of the sharp constant in the Jackson-Stechkin inequality in the space L 2

This paper deals with the continuity of the sharp constant K(T,X) with respect to the set T in the Jackson-Stechkin inequality $E(f,L) \leqslant K(T,X)\omega (f,T,X),$ , where E(f,L) is the best approximation of the function f ∈ X by elements of the subspace L ? X, and ω is a modulus of continuity, in the case where the space L ²( $\mathbb{T}^d $ , ?) is taken for X and the subspace of functions g ∈ L ²( $\mathbb{T}^d $ , ?), for L. In particular, it is proved that the sharp constant in the Jackson-Stechkin inequality is continuous in the case where L is the space of trigonometric polynomials of nth order and the modulus of continuity ω is the classical modulus of continuity of rth order.     

On the constants in the estimates of the rate of convergence in the Birkhoff Ergodic theorem

Mathematical Notes -     

Gaps in the poset of projections in the Calkin algebra

We study the gap structure of the partial order of projections of the Calkin algebra of a complex, separable, infinite-dimensional Hilbert space. We prove the existence of an analytic Hausdorff gap in this partial order. As a consequence we obtain that under Todorcevic’s Axiom and MA the gap spectrum of P(C(H)) is strictly bigger than the gap spectrum of P(ω)/Fin.     

On the maximality of the orthogonal groups in the symplectic groups in characteristic two

Sans résumé     

On the error term in the mean square formula for the Riemann zeta-function in the critical strip

For 1/2<<1 fixed, letE (T) denote the error term in the asymptotic formula for . We obtain some new bounds forE (T), and an _-result which is the analogue of the strongest _-result in the classical Dirichlet divisor problem.     

Dynamic decision methods in the presence of uncertainty in the state and the dynamics

The article considers the problem of dynamic decision under uncertainty in the state and in its dynamics. Game models and identification and decision methods are developed that ensure consistency and asymptotic stationarity of dynamic equilibria.This study was partially financed by the Russian Foundation of Basic Research (95-01-01137).Translated from Nelineinye Dinamicheskie Sistemy: Kachestvennyi Analiz i Upravlenie — Sbornik Trudov. No. 3. pp. 10–22, 1993.     

On the Gap in the Spectrum of the Translations

The spectrum of the translations in local quantum field theory will be analyzed in order to show that in a positive energy representation without vacuum vector and with lowest mass m₁ there is no gap in the spectrum which is larger than 2m₁. In particular in a zero mass representation there is no hole at all. These results are obtained with methods of analytic functions of several complex variables.     

Illumination in the presence of opaque line segments in the plane

What is the minimal number of light sources which is always sufficient to illuminate the plane in the presence of n disjoint opaque line segments? For n5, O'Rourke proved that 2n/3 light sources are always sufficient and sometimes necessary, if light sources can be placed on the line segments and thus they can illuminate both sides of a segment.

We prove that 2(n+1)/3 light sources are always sufficient and sometimes necessary, if light sources cannot be placed on the line segments. An O(nlogn) time algorithm is presented which allocates at most 2(n+1)/3 light sources collectively illuminating the plane.     

Facet Bending in the Driven Crystalline Curvature Flow in the Plane

We study simple cases of crystalline driven curvature flow with spatially nonuniform driving force term. We assume special monotonicity properties of the driving term, which are motivated by our previous work on Berg’s effect. We show that initially flat facets may begin to bend, provided, loosely speaking, they are too large.      

Exact constant in the Jackson inequality in the space L2

We investigate the lowest (exact) constants in Jackson-type inequalities with the second-order modulus of continuity for the case of approximation of differentiable periodic functions by trigonometric polynomials in the space L₂.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 9, pp. 1261–1265, September, 1994.     

Investment decisions in the rent-to-own industry in the absence of inventory

This study addresses the product investment decision faced by firms in the rent-to-own industry. In this setting, a customer arrives according to a random process and requests one unit of a product to rent (and eventually own should he/she choose to make all the required payments). At the time of request, if the product is available in inventory, the firm enters into a contractual agreement (by accepting the customer's offer) and rents the merchandise. More interesting and the case considered here, if the requested item is not in inventory, the firm must decide whether to purchase the item in order to rent it out or to simply reject the request. The customer's offer specifies the desired maximum contract length and the payment frequency—from which the firm determines the fixed periodic payment charged. The firm makes its investment decision based on the characteristics of the offer as well as those of the product (eg, initial and resale values, useful life and carrying costs) in essence performing a complicated cost benefit analysis. An extension is also considered whereby instead of simply rejecting the request the firm can adjust the required payment amount. Dynamic programming techniques are used to address the problem and to solve for the firm's optimal decision.     

Convergence in the mean of the Fourier series in orthogonal polynomials

For weights p(t) and q(t) with a finite number of power-law-type singularities we obtain necessary and sufficient conditions for the inequality $$\parallel s_n^{(p)} (f)q\parallel _{L^\eta - (1,1)} \leqslant C\parallel fq\parallel _{L^\eta ( - 1,1)'}$$ to hold, where s_n ^(p)(f) is a partial sum of the Fourier series of the function f in terms of polynomials orthogonal on [?1, 1] with weight p(t). This inequality is used to solve the problem concerning convergence in the mean and also convergence almost everywhere of the partial sums s_n ^(p)(f).     

Homogenization of boundary value problems in perforated domains with the third boundary condition and the resulting change in the character of the nonlinearity in the problem

We study the homogenization problem for the Poisson equation in a periodically perforated domain with a nonlinear boundary condition for the flux on the cavity boundaries. We show that, under certain relations on the problem scale, the homogenized equations may have different character of the nonlinearity. In each case considered, we obtain estimates for the convergence of solutions of the original problem to the solution of the homogenized problem in the corresponding Sobolev spaces.