Some properties of bivariate Gumbel Type A distributions with proportional hazard rates

We call a set of univariate distributions with the same mathematical form but different parameter values a family $\text{J}$. Consider a bivariate Gumbel Type A survival distribution, S₁₂(x₁, x₂), defined in (2.1), for which both marginal distributions, S₁(x₁), S₂(x₂), belong to the same family, $\text{J}$ of distributions. It is proved in this paper that subject to weak conditions, the crude hazard rates, h₁(t) and h₂(t), are proportional if and only if the marginal hazard rates, λ₁(t) and λ₂(t), are proportional (Theorem 1). It is also shown that the survival functions of W = min(X₁, X₂), and of the identified minimum, W_i = X_i, for X_i < X_j, j ≠ i, belong to the same family $\text{J}$ as do S₁(x₁), S₂(x₂) (Corollary 1). Counter-examples of distributions other than Gumbel Type A, for which these properties do not hold, are given. Some applications to the analysis of competing risks, using a family of Gompertz distributions, are discussed.     

Target set reachability criteria for dynamical systems described by inaccurate models

S is taken to be a dynamical system (described by Banach space operators) whose outputy we wish to regulate. The structural complexity ofS (nonlinearities, distributed parameters, etc.) forces us to design a controller forS using an approximate modelM ofS. A convex error bound ? describes the accuracy of the approximation ofS byM. For a prescribed target setY _t , we considered the problem of driving the output ofS toY _t subject to worst possible error excursions betweenM andS. The notion of areconstructed support function is instrumental to the derivation of the main result, Theorem 6.1, which we can paraphrase as follows. IfM is linear (S need not be), then we can describe a finite-dimensional convex programming Problem (P), whose solution tells us whether or notY _t is reachable. Theorem 6.1 is then specialized to differential systems approximated in the norm. The computation of numerical solutions is also discussed.     

High-order curvilinear mesh generation technique based on an improved radius basic function approach

A high-order curvilinear hybrid mesh generation technique is developed for high-order numerical method (eg, discontinuous Galerkin method) applications to improve the accuracy for problems with curve boundary. The grid generation technique is based on an improved radius basic function (RBF) approach by which the straight-edge mesh is converted into high-order curve mesh. Firstly, an initial straight-edge mesh is prepared by traditional grid generation software. Then, high-order interpolation points are inserted into the mesh entities such as edges, faces, and cells according to the final demand of mesh order. To preserve the original geometry, the inserted points on solid wall are then projected onto the CAD model using an open source tool “Open Cascade.” Finally, other inserted points in the field near the solid wall are moved to appropriate positions by the improved RBF approach to avoid tangled cells. If we use the original RBF approach, then the inserted points on the edge and face entities normal to the solid boundary in the region of boundary layer will move to improper positions. To overcome this problem, a weighting based on the local grid aspect ratio between normal direction and tangential direction is introduced into the baseline RBF approach. Three typical configurations are tested to validate the mesh generator. Meanwhile, a third-order solution of subsonic flow over an analytical 3D body of revolution in the second International Workshop on High-Order CFD Methods is supplied by a discontinuous Galerkin solver. These numerical tests demonstrate the potential capability of present technique for high-order simulations of complex geometries.     

A class of loss functions of catenary form

The catenary form of loss function is considered in the framework of Bayesian decision theory. The mathematical tractability of this form seems to be unrecognized; it contains quadratic loss as a limiting case. For various probability distributions expressions are given for posterior analysis, and limiting properties are investigated.     

Multi-funnel optimization using Gaussian underestimation

In several applications, underestimation of functions has proven to be a helpful tool for global optimization. In protein–ligand docking problems as well as in protein structure prediction, single convex quadratic underestimators have been used to approximate the location of the global minimum point. While this approach has been successful for basin-shaped functions, it is not suitable for energy functions with more than one distinct local minimum with a large magnitude. Such functions may contain several basin-shaped components and, thus, cannot be underfitted by a single convex underestimator. In this paper, we propose using an underestimator composed of several negative Gaussian functions. Such an underestimator can be computed by solving a nonlinear programming problem, which minimizes the error between the data points and the underestimator in the L ¹ norm. Numerical results for simulated and actual docking energy functions are presented.     

Algorithms for Computing U(N) Clebsch Gordan Coefficients

If V ^(m) is an irreducible representation space for the unitary group U(N), then the r-fold tensor product space, is in general reducible. Such a reducible representation can be reduced to a direct sum of irreducible representation spaces, albeit with multiplicity. Clebsch–Gordan coefficients are the overlap coefficients between an orthonormal basis in the tensor product space and an orthonormal basis in the direct sum space. Since such coefficients are basis dependent, bases in the U(N) irrep spaces must be chosen. In this paper we use the Gelfand–Zetlin basis, built out of the chain of subgroups U(N)⊃...⊃U(1). Our first result is to derive algorithms for generating Gelfand–Zetlin bases from Gelfand–Zetlin tableaux. Given these concrete basis realizations we develop algorithms for computing the Clebsch–Gordan coefficients themselves.      

Fast evaluation of polyharmonic splines in three dimensions

M. J. D. Powell ^{This paper concerns the fast evaluation of radial basis functions.It describes the mathematics of hierarchical and fast multipolemethods for fast evaluation of splines of the form where is a positive integer andp is a low-degree polynomial. Splines s of this form are polyharmonicsplines in 3 and have been found to be very useful for providingsolutions to scattered data interpolation problems in 3. Asit is now well known, hierarchical methods reduce the incrementalcost of a single extra evaluation from O(N) to O(log N) operationsand reduce the cost of a matrix–vector product (evaluationof s at all the centres) from O(N2) to O(N log N) operations.We give appropriate far- and near-field expansions, togetherwith error estimates, uniqueness theorems and translation formulae.A hierarchical code based on these formulae is detailed andsome numerical results are given.}     

Short interval asymptotics for a class of arithmetic functions

Summary We provide a general asymptotic formula which permits applications to sums like <InlineEquation ID=IE"1"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"2"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"3"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"4"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"5"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"6"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"7"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"8"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"9"><EquationSource Format="TEX"><![CDATA[$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation> \sum_{x< n\le x+y} \big(d(n)\big)^2, \quad \sum_{x< n\le x+y} d(n^3),\quad \sum_{x< n\le x+y}\big(r(n)\big)^2, \quad \sum_{x< n\le x+y}r(n^3), $$ where $d(n)$ and $r(n)$ are the usual arithmetic functions (number of divisors, sums of two squares), and $y$ is small compared to~$x$.     

Two-dimensional problem of two Coulomb centers at small intercenter distances

We use analytic methods to analyze the discrete spectrum for the problem (Z₁eZ₂)₂ in the united-atom limit (R ≪ 1) and obtain asymptotic expansions for the quantum defect and energy terms of the system (Z₁eZ₂)₂ at small intercenter distances R up to terms of the order O(R⁶). We investigate the effect of the dimensionality factor on the energy spectrum of the hydrogen molecular ion H ₂ ⁺ . __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 148, No. 2, pp. 269–287, August, 2006.     

Positivity and positive definiteness in generalized function algebras

The definitions of positivity and positive definiteness are extended to generalized function algebras in coherence with the corresponding notions for distributions. Versions of Bochner's theorem for a positive definite Colombeau generalized function are given.