An approach to numerical modeling of the dynamics of reacting gases in nozzles

An approach is proposed to computer simulation of gas-dynamic processes in chemically nonequilibrium flows in supersonic nozzles. An algorithm for the solution of the problem is developed. Convergence of iterative processes and stability of the linearized problem are investigated.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 58, pp. 89–95, 1986.     

Two storage inventory model with fuzzy deterioration over a random planning horizon

An inventory model for a deteriorating item with stock dependent demand is developed under two storage facilities over a random planning horizon, which is assumed to follow exponential distribution with known parameter. For crisp deterioration rate, the expected profit is derived and maximized via genetic algorithm (GA). On the other hand, when deterioration rate is imprecise then optimistic/pessimistic equivalent of fuzzy objective function is obtained using possibility/necessity measure of fuzzy event. Fuzzy simulation process is proposed to maximize the optimistic/pessimistic return and finally fuzzy simulation-based GA is developed to solve the model. The models are illustrated with some numerical data. Sensitivity analyses on expected profit function with respect to distribution parameter λ and confidence levels α₁ and α₂ are also presented.     

Rational approximation to formal power series

A general method for obtaining rational approximations to formal power series is defined and studied. This method is based on approximate quadrature formulas. Newton-Cotes and Gauss quadrature methods are used. It is shown that Padé approximants and the ε-algorithm are related to Gaussian formulas while linear summation processes are related to Newton-Cotes formulas. An example is exhibited which shows that Padé approximation is not always optimal. An application to e^−t is studied and a method for Laplace transform inversion is proposed.     

Notes on elation generalized quadrangles

Let be a finite generalized quadrangle of order (s,t),s,t>1. An “elation about a point p” of is an automorphism fixing p linewise and fixing no point which is not collinear with p. An elation that generates a cyclic group of elations is called a “standard elation”. One of the problems already considered in Payne and Thas (Finite Generalized Quadrangles (1984)) is to determine just when the set of elations about the point (∞) is a group. The purpose of this paper is to provide an example where this is not the case, and then to show that for a flock generalized quadrangle the usual group of elations about (∞) is the complete set of standard elations about (∞).     

Twice Q-polynomial distance-regular graphs are thin

Let Γ be a distance-regular graph of diameter d 3. For each vertex χ of Γ, let T(χ) denote the subconstituent algebra for Γ with respect to χ. An irreducible T(χ)-module W is said to be thin if dim E_i^*(χ) W 1 for 0 i d, where E_i^*(χ) is the projection onto the ith subconstituent for Γ with respect to χ. The graph Γ is said to be thin if, for each vertex χ of Γ, very irreducible T(χ)-module is thin. Our main result is the following Theorem: If Γ has two Q-polynomial structures, then Γ is thin.     

Checking the adequacy of the multivariate semiparametric location shift model

Let X,X₁,…,X_m,…, Y,Y₁,…,Y_n,… be independent d-dimensional random vectors, where the X_j are i.i.d. copies of X, and the Y_k are i.i.d. copies of Y. We study a class of consistent tests for the hypothesis that Y has the same distribution as X+μ for some unspecified . The test statistic L is a weighted integral of the squared modulus of the difference of the empirical characteristic functions of and Y₁,…,Y_n, where is an estimator of μ. An alternative representation of L is given in terms of an L²-distance between two nonparametric density estimators. The finite-sample and asymptotic null distribution of L is independent of μ. Carried out as a bootstrap or permutation procedure, the test is asymptotically of a given size, irrespective of the unknown underlying distribution. A large-scale simulation study shows that the permutation procedure performs better than the bootstrap.     

Algorithm for solving a new class of general mixed variational inequalities in Banach spaces

In this paper, a new concept of η-proximal mapping for a proper subdifferentiable functional (which may not be convex) on a Banach space is introduced. An existence and Lipschitz continuity of the η-proximal mapping are proved. By using properties of the η-proximal mapping, a new class of general mixed variational inequalities is introduced and studied in Banach spaces. An existence theorem of solutions is established and a new iterative algorithm for solving the general mixed variational inequality is suggested. A convergence criteria of the iterative sequence generated by the new algorithm is also given.     

Minimax estimation of means of multivariate normal mixtures

Assume X = (X₁, …, X_p)′ is a normal mixture distribution with density w.r.t. Lebesgue measure, , where Σ is a known positive definite matrix and F is any known c.d.f. on (0, ∞). Estimation of the mean vector under an arbitrary known quadratic loss function _Q(θ, a) = (a − θ)′ Q(a − θ), Q a positive definite matrix, is considered. An unbiased estimator of risk is obatined for an arbitrary estimator, and a sufficient condition for estimators to be minimax is then achieved. The result is applied to modifying all the Stein estimators for the means of independent normal random variables to be minimax estimators for the problem considered here. In particular the results apply to the Stein class of limited translation estimators.     

Convergence rates for exponential interpolation series

An analysis of the rate of convergence is made for the interpolation series based on the biorthogonal system (_n^Δ)ƒ(0) and e_n(z) = Δⁿx^z ¦_x−1, which was recently shown to be convergent for certain entire functions of exponential type. An error bound is obtained which is shown to vary as a negative power of the number of terms in the partial sum. Comparison is made with numerical calculations in a few simple cases and certain practical applications are mentioned.     

Operation properties and -equalities of complex fuzzy sets

A complex fuzzy set is a fuzzy set whose membership function takes values in the unit circle in the complex plane. This paper investigates various operation properties and proposes a distance measure for complex fuzzy sets. The distance of two complex fuzzy sets measures the difference between the grades of two complex fuzzy sets as well as that between the phases of the two complex fuzzy sets. This distance measure is then used to define δ-equalities of complex fuzzy sets which coincide with those of fuzzy sets already defined in the literature if complex fuzzy sets reduce to real-valued fuzzy sets. Two complex fuzzy sets are said to be δ-equal if the distance between them is less than 1-δ. This paper shows how various operations between complex fuzzy sets affect given δ-equalities of complex fuzzy sets. An example application of signal detection demonstrates the utility of the concept of δ-equalities of complex fuzzy sets in practice.     

An inverse polynomial method for the identification of the leading coefficient in the Sturm–Liouville operator from boundary measurements

An inverse polynomial method of determining the unknown leading coefficient k=k(x) of the linear Sturm–Liouville operator Au=−(k(x)u^′(x))^′+q(x)u(x), x(0,1), is presented. As an additional condition only two measured data at the boundary (x=0,x=1) are used. In absence of a singular point (u^′(x)≠0,u^″(x)≠0,x[0,1]) the inverse problem is classified as a well-conditioned . If there exists at least one singular point, then the inverse problem is classified as moderately ill-conditioned (u^′(x₀)=0,x₀(0,1);u^′(x)≠0,x≠x₀;u^″(x)≠0,x[0,1]) and severely ill-conditioned (u^′(x₀)=u^″(x₀)=0,x₀(0,1);u^′(x)≠0,u^″(x)≠0,x≠x₀). For each of the cases direct problem solution is approximated by corresponding polynomials and the inverse problem is reformulated as a Cauchy problem for to the first order differential equation with respect the unknown function k=k(x). An approximate analytical solution of the each Cauchy problems are derived in explicit form. Numerical simulations all the above cases are given for noise free and noisy data. An accuracy of the presented approach is demonstrated on numerical test solutions.     

Reverse Martingales and Approximation Operators

Let {ξ_n, _n, n ≥ m ≥ 1} be a reverse martingale such that the distribution of ξ_n depends on x I R =(− ∞, ∞)x. for each n ≥ m, and ξ_n[formula] For a continuous bounded function f on R let L_n(f, x) = Ef(ξ_n) be the associated positive linear operator. The properties of ξ_n are used to obtain the convergence properties of L_n(f, x), and some more details are given when ξ_n is a reverse martingale sequence of -statistics. Lipschitz properties for a subclass of these operators resulting from an exponential Family of distributions are also given. It is further shown that this class of operators of convex functions preserves convexity also. An example of a reverse supermartingale related to the Bleimann-Butzer-Hahn operator is also discussed.     

Scheduling jobs with time-resource tradeoff via nonlinear programming

We consider a scheduling problem where the processing time of any job is dependent on the usage of a discrete renewable resource, e.g. personnel. An amount of k units of that resource can be allocated to the jobs at any time, and the more of that resource is allocated to a job, the smaller its processing time. The objective is to find a resource allocation and a schedule that minimizes the makespan. We explicitly allow for succinctly encodable time-resource tradeoff functions, which calls for mathematical programming techniques other than those that have been used before. Utilizing a (nonlinear) integer mathematical program, we obtain the first polynomial time approximation algorithm for the scheduling problem, with performance bound (3+ε) for any ε>0. Our approach relies on a fully polynomial time approximation scheme to solve the nonlinear mathematical programming relaxation. We also derive lower bounds for the approximation.     

Algorithms and Methods for Solving Scheduling Problems and Other Extremum Problems on Large-Scale Graphs

We consider a large-scale directed graph G = (V, E) whose edges are endowed with a family of characteristics. A subset of vertices of the graph, V′ ⊂ V, is selected and some additional conditions are imposed on these vertices. An algorithm for reducing the optimization problem on the graph G to an optimization problem on the graph G′ = (V′, E′) of a lower dimension is developed. The main steps of the solution and some methods for constructing an approximate solution to the problem on the transformed graph G′ are presented.__________Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 9, No. 1, pp. 235–251, 2003.     

Adaptivity and computational complexity in the numerical solution of ODEs

In this paper we analyze the problem of adaptivity for one-step numerical methods for solving ODEs, both IVPs and BVPs, with a view to generating grids of minimal computational cost for which the local error is below a prescribed tolerance (optimal grids). The grids are generated by introducing an auxiliary independent variable τ and finding a grid deformation map, t=Θ(τ), that maps an equidistant grid {τ_j} to a non-equidistant grid in the original independent variable, {t_j}. An optimal deformation map Θ is determined by a variational approach. Finally, we investigate the cost of the solution procedure and compare it to the cost of using equidistant grids. We show that if the principal error function is non-constant, an adaptive method is always more efficient than a non-adaptive method.     

Numerical comparison of nonlinear subgridscale models via adaptive mesh refinement

In this paper, we provide a method of evaluating the efficacy of nonlinear subgridscale models for use in the large eddy simulation of incompressible viscous flow problems. We compare subgridscale “artificial” viscosity models using a posteriori error estimation and adaptive mesh refinement. Specifically, we compare α-Laplacian based subgridscale models and discuss the benefits and limitations of different values of α for some standard benchmark problems for the Navier–Stokes equations.     

Entropy numbers in sequence spaces with an application to weighted function spaces

We determine the exact asymptotic behaviour of entropy numbers of diagonal operators from ℓ_p to ℓ_q, 0<q<p∞, under mild regularity conditions on the generating diagonal sequence. On one hand, this is a quantitative version of Pitt's theorem for diagonal operators, and on the other hand it is a limiting case of results by Carl. An application to embeddings of weighted Besov and Triebel–Lizorkin spaces is also given.     

Labeling planar graphs with a condition at distance two

An L(2,1)-labeling of a graph is a mapping c:V(G)→{0,…,K} such that the labels assigned to neighboring vertices differ by at least 2 and the labels of vertices at distance two are different. The smallest K for which an L(2,1)-labeling of a graph G exists is denoted by λ_2,1(G). Griggs and Yeh [J.R. Griggs, R.K. Yeh, Labeling graphs with a condition at distance 2, SIAM J. Discrete Math. 5 (1992) 586–595] conjectured that λ_2,1(G)≤Δ² for every graph G with maximum degree Δ≥2. We prove the conjecture for planar graphs with maximum degree Δ≠3. All our results also generalize to the list-coloring setting.     

An adaptive algorithm for system level diagnosis

Consider a system of n units, at most t of which are faulty. An adaptive diagnosis algorithm is presented which uses a sequence of tests to identify a fault-free unit. The algorithm requires at most 2t − ν(t) tests, where ν(t) is the number of 1's in the binary representation of t. Moreover, many of the tests can be performed simultaneously. The previously best algorithms for the same purpose required 2t − 1 tests, none of which could be performed simultaneously.