Regular variation of a random length sequence of random variables and application to risk assessment

When assessing risks on a finite-time horizon, the problem can often be reduced to the study of a random sequence C(N) = (C ₁,…,C _N ) of random length N, where C(N) comes from the product of a matrix A(N) of random size N × N and a random sequence X(N) of random length N. Our aim is to build a regular variation framework for such random sequences of random length, to study their spectral properties and, subsequently, to develop risk measures. In several applications, many risk indicators can be expressed from the extremal behavior of ∥C(N)∥, for some norm ∥?∥. We propose a generalization of Breiman’s Lemma that gives way to a tail estimate of ∥C(N)∥ and provides risk indicators such as the ruin probability and the tail index for Shot Noise Processes on a finite-time horizon. Lastly, we apply our main result to a model used in dietary risk assessment and in non-life insurance mathematics to illustrate the applicability of our method.     

A Polynomial Time Algorithm for the Resource Allocation Problem with a Convex Objective Function

Consider the resource allocation problem:minimize ∑ⁿ_i=1 f_i(x_i) subject to ∑ⁿ_i=1 x_i = N and x_i's being nonnegative integers, where each f_i is a convex function. The well-known algorithm based on the incremental method requires O(N log n + n) time to solve this problem. We propose here a new algorithm based on the Lagrange multiplier method, requiring O[n²(log N)²] time. The latter is faster if N is much larger than n. Such a situation occurs, for example, when the optimal sample size problem related to monitoring the urban air pollution is treated.     

Costing a Finite Minimal Repair Replacement Policy

In this paper an integral equation technique is used to evaluate the expected cost for the period (0, t] of a policy involving minimal repair at failure with replacement after N failures. This cost function provides an appropriate criterion to determine the optimal replacement number N* for a system required for use over a finite time horizon. In an example, it is shown that significant cost savings can be achieved using N* from the new finite time horizon model rather than the value predicted by the usual asymptotic model.     

Approximation of Gaussian Random Fields: General Results and Optimal Wavelet Representation of the Lévy Fractional Motion

We investigate the approximation rate for certain centered Gaussian fields by a general approach. Upper estimates are proved in the context of so–called Hölder operators and lower estimates follow from the eigenvalue behavior of some related self–adjoint integral operator in a suitable L ₂(μ)–space. In particular, we determine the approximation rate for the Lévy fractional Brownian motion X _H with Hurst parameter H∈(0,1), indexed by a self–similar set T?? ^N of Hausdorff dimension D. This rate turns out to be of order n ^?H/D (log?n)^1/2. In the case T=[0,1] ^N we present a concrete wavelet representation of X _H leading to an approximation of X _H with the optimal rate n ^?H/N (log?n)^1/2.     

A new algorithm for minimizing makespan, <Emphasis Type="Italic">C</Emphasis><Subscript>max</Subscript>, in blocking flow-shop problem through slowing down the operations

In this paper, a new algorithm with complexity O(nm²) is presented, which finds the optimal makespan, C_max, for a blocking flow-shop problem by slowing down the operations of a no-wait flow-shop problem, F _m ∣no-wait∣C_max, for a given sequence where restriction on the slowing down is committed. However, the problem with performance measure makespan, C_max, in a non-cyclic environment, is a special case of cyclic problem with cycle time, C _t , as its performance measure. This new algorithm is much faster than the previously developed algorithms for cyclical scheduling problems.     

Difference scheme for an initial–boundary value problem for a singularly perturbed transport equation

An initial–boundary value problem for a singularly perturbed transport equation with a perturbation parameter ε multiplying the spatial derivative is considered on the set ? = G ∪ S, where ? = D? × [0 ≤ t ≤ T], D? = {0 ≤ x ≤ d}, S = S ^l ∪ S, and S ^l and S₀ are the lateral and lower boundaries. The parameter ε takes arbitrary values from the half-open interval (0,1]. In contrast to the well-known problem for the regular transport equation, for small values of ε, this problem involves a boundary layer of width O(ε) appearing in the neighborhood of S ^l ; in the layer, the solution of the problem varies by a finite value. For this singularly perturbed problem, the solution of a standard difference scheme on a uniform grid does not converge ε-uniformly in the maximum norm. Convergence occurs only if h=dN^-1 ? ε and N₀^-1 ? 1, where N and N₀ are the numbers of grid intervals in x and t, respectively, and h is the mesh size in x. The solution of the considered problem is decomposed into the sum of regular and singular components. With the behavior of the singular component taken into account, a special difference scheme is constructed on a Shishkin mesh, i.e., on a mesh that is piecewise uniform in x and uniform in t. On such a grid, a monotone difference scheme for the initial–boundary value problem for the singularly perturbed transport equation converges ε-uniformly in the maximum norm at an ?(N^?1 + N₀^?1) rate.     

Improved difference scheme of the solution decomposition method for a singularly perturbed reaction-diffusion equation

A Dirichlet problem is considered for a singularly perturbed ordinary differential reaction-diffusion equation. For this problem, a new approach is developed in order to construct difference schemes that converge uniformly with respect to the perturbation parameter ?, ? ∈ (0, 1]. The approach is based on the decomposition of a discrete solution into regular and singular components, which are solutions of discrete subproblems on uniform grids. Using the asymptotic construction technique, a difference scheme of the solution decomposition method is constructed that converges ?-uniformly in the maximum norm at the rate O (N ^?2 ln² N), where N + 1 is the number of nodes in the grid used; for fixed values of the parameter ?, the scheme converges at the rate O(N ^?2). Using the Richardson technique, an improved scheme of the solution decomposition method is constructed, which converges ?-uniformly in the maximum norm at the rate O(N ^?4 ln⁴ N).     

Stability of nonlocal difference schemes in a subspace

We consider a family of two-layer difference schemes for the heat equation with nonlocal boundary conditions containing the parameter γ. In some interval γ ∈ (1, γ ₊), the spectrum of the main difference operator contains a unique eigenvalue λ ₀ in the left complex half-plane, while the remaining eigenvalues λ ₁, λ ₂, …, λ _N?1 lie in the right half-plane. The corresponding grid space H _N is represented as the direct sum H _N = H ₀⊕H _N?1 of a one-dimensional subspace and the subspace H _N?1 that is the linear span of eigenvectors µ⁽¹⁾, µ⁽²⁾, …, µ^(N?1). We introduce the notion of stability in the subspace H _N?1 and derive a stability criterion.     

Asymptotic formula for the convolution of a generalized divisor function

An asymptotic formula is obtained for the sum of terms σ _it (n)σ_-it (N - n) (t is real) over 0 < n < N with a remainder estimated by O _ε((1+|t|)^1+ε N ^3/4+ε) for any ε > 0. As a consequence, Porter’s result on a power scale for the average number of steps in the Euclidean algorithm is improved.     

Positivity for polyharmonic problems on domains close to a disk

We study the problem of positivity preserving of the Green operator for the polyharmonic operator (?Δ) ^m under homogeneous Dirichlet boundary conditions on domains Ω of ?R ². Here we will treat only Ω, which are ε-close to a disk B in C ^m,γ-sense, meaning, there exists a C ^m,γ-mapping g : \( \bar{B}\longrightarrow \bar{\Omega}\) such that g?(B) = ?Ω and \(||g -- Id||_{C^{m,\gamma}}(\bar{B})\!\leq\!\varepsilon\). We show that ε-closeness in C ^{m, γ}-sense is enough in order to ensure positivity preserving. For the clamped plate equation (i.e. m = 2), this means that it is a Hölder norm of the curvature of ? Ω, which governs the positivity behavior. This improves the previous work by Grunau and Sweers, where closeness to the disk in C ^2m -sensewas required (in C ⁴-sense for thethe clamped plate).     

Equiconvergence of expansions in multiple Fourier series and in fourier integrals with “lacunary sequences of partial sums”

We investigate the equiconvergence on T^N = [?π, π)^N of expansions in multiple trigonometric Fourier series and in the Fourier integrals of functions f ∈ L_p(T^N) and g ∈ L_p(R^N), p > 1, N ≥ 3, g(x) = f(x) on T^N, in the case where the “partial sums” of these expansions, i.e., S_n(x; f) and J_α(x; g), respectively, have “numbers” n ∈ Z^N and α ∈ R^N (n_j = [α_j], j = 1,..., N, [t] is the integral part of t ∈ R¹) containing N ? 1 components which are elements of “lacunary sequences.”     

Lower Bounds for the Degree of a Branched Covering of a Manifold

The problem of finding new lower bounds for the degree of a branched covering of a manifold in terms of the cohomology rings of this manifold is considered. This problem is close to M. Gromov’s problem on the domination of manifolds, but it is more delicate. Any branched (finite-sheeted) covering of manifolds is a domination, but not vice versa (even up to homotopy). The theory and applications of the classical notion of the group transfer and of the notion of transfer for branched coverings are developed on the basis of the theory of n-homomorphisms of graded algebras.The main result is a lemma imposing conditions on a relationship between the multiplicative cohomology structures of the total space and the base of n-sheeted branched coverings of manifolds and, more generally, of Smith–Dold n-fold branched coverings. As a corollary, it is shown that the least degree n of a branched covering of the N-torus T ^N over the product of k 2-spheres and one (N ? 2k)-sphere for N ≥ 4k + 2 satisfies the inequality n ≥ N ? 2k, while the Berstein–Edmonds well-known 1978 estimate gives only n ≥ N/(k + 1).     

Approximation of a system of singularly perturbed reaction-diffusion parabolic equations in a rectangle

The Dirichlet problem for a system of singularly perturbed reaction-diffusion parabolic equations in a rectangle is considered. The higher order derivatives of the equations are multiplied by a perturbation parameter ?², where ? takes arbitrary values in the interval (0, 1]. When ? vanishes, the system of parabolic equations degenerates into a system of ordinary differential equations with respect to t. When ? tends to zero, a parabolic boundary layer with a characteristic width ? appears in a neighborhood of the boundary. Using the condensing grid technique and the classical finite difference approximations of the boundary value problem, a special difference scheme is constructed that converges ?-uniformly at a rate of O(N ^?2ln² N + N ₀ ^?1 , where \(N = \mathop {\min }\limits_s N_s \), N ^s + 1 and N ₀ + 1 are the numbers of mesh points on the axes x _s and t, respectively.     

A Note on the Joint Replenishment Problem under Constant Demand

This note considers the joint replenishment inventory problem for N items under constant demand. The frequently-used cyclic strategy (T; k₁, …, k _N ) is investigated: a family replenishment is made every T time units and item i is included in each k _i th replenishment. Goyal proposed a solution to find the global optimum within the class of cyclic strategies. However, we will show that the algorithm of Goyal does not always lead to the optimal cyclic strategy. A simple correction is suggested.     

The <Emphasis Type="Italic">p</Emphasis>-Centre Problem—Heuristic and Optimal Algorithms

The p-centre problem, or minimax location-allocation problem in location theory terminology, is the following: given n demand points on the plane and a weight associated with each demand point, find p new facilities on the plane that minimize the maximum weighted Euclidean distance between each demand point and its closest new facility. We present two heuristics and an optimal algorithm that solves the problem for a given p in time polynomial in n. Computational results are presented.     

Bose–Einstein distribution as a problem of analytic number theory: The case of less than two degrees of freedom

The problem of finding the number and the most likely shape of solutions of the system \(\sum\nolimits_{j = 0}^\infty {{\lambda _j}{n_j}} \leqslant M,\;\sum\nolimits_{j = 1}^\infty {{n_j}} = N\), where λ_j,M,N > 0 and N is an integer, as M,N →∞, can naturally be interpreted as a problem of analytic number theory. We solve this problem for the case in which the counting function of λ_j is of the order of λ^d/2, where d, the number of degrees of freedom, is less than two.     

Optimal Ordering Policies for Inventory Systems with Emergency Ordering

This paper considers dynamic single- and multi-product inventory problems in which the demands in each period are independent and identically distributed random variables. The problems considered have the following common characteristics. At the beginning of each period two order quantities are determined for each product. A “normal order” quantity with a constant positive lead time of λ _n periods and an “emergency order” quantity with a lead time of λ _e periods, where λ _e = λ _n - 1. The ordering decisions are based on linear procurement costs for both methods of ordering and convex holding and penalty costs. The emergency ordering costs are assumed to be higher than the normal ordering costs. In addition, future costs are discounted.For the single-product problem the optimal ordering policy is shown to be the same for all periods with the exception of the last period in the N-period problem. For the multi-product problem the one- and N-period optimal ordering policy is characterized where it is assumed that there are resource constraints on the total amount that can be ordered or produced in each period.     

Multi-period empty container repositioning with stochastic demand and lost sales

This paper considers repositioning empty containers between multi-ports over multi-periods with stochastic demand and lost sales. The objective is to minimize the total operating cost including container-holding cost, stockout cost, importing cost and exporting cost. First, we formulate the single-port case as an inventory problem over a finite horizon with stochastic import and export of empty containers. The optimal policy for period n is characterized by a pair of critical points (A _n , S _n ), that is, importing empty containers up to A _n when the number of empty containers in the port is fewer than A _n ; exporting empty containers down to S _n when the number of empty containers in the port is more than S _n ; and doing nothing, otherwise. A polynomial-time algorithm is developed to determine the two thresholds, that is, A _n and S _n , for each period. Next, we formulate the multi-port problem and determine a tight lower bound on the cost function. On the basis of the two-threshold optimal policy for a single port, a polynomial-time algorithm is developed to find an approximate repositioning policy for multi-ports. Simulation results show that the proposed approximate repositioning algorithm performs very effectively and efficiently.     

On the failure of concentration for the ℓ∞-ball

Let (X, d) be a compact metric space and µ a Borel probability on X. For each N ≥ 1 let d_∞^N be the ?_∞-product on X^N of copies of d, and consider 1-Lipschitz functions X^N → ? for d_∞^N.     

Construction of optimal quadrature formulas for Fourier coefficients in Sobolev space $L_{2}^{(m)}(0,1)$

This paper studies the problem of construction of optimal quadrature formulas in the sense of Sard in the \(L_{2}^{(m)}(0,1)\) space for numerical calculation of Fourier coefficients. Using the S.L.Sobolev’s method, we obtain new optimal quadrature formulas of such type for N+1≥m, where N+1 is the number of nodes. Moreover, explicit formulas for the optimal coefficients are obtained. We study the order of convergence of the optimal formula for the case m=1. The obtained optimal quadrature formulas in the \(L_{2}^{(m)}(0,1)\) space are exact for P _m?1(x), where P _m?1(x) is a polynomial of degree m?1. Furthermore, we present some numerical results, which confirm the obtained theoretical results.