Stochastic Routh-Hurwitz problem

The stochastic Routh—Hurwitz problem is considered, i.e., the probability of stability is obtained for a polynomial xⁿ + a₁x^n–1 + + a_n with random coefficients.Translated from Kibernetika i Sistemnyi Analiz, No. 4, pp. 61–70, July–August, 1991.     

High Order Schemes for Resolving Waves: Number of Points per Wavelength

For energetic flows there are many advantages of high order schemes over low order schemes. Here we examine a previously unknown advantage. It is commonly thought that the number of points per wavelength in order to obtain a given error in a numerical approximation depends only on runtime and the order of the approximation. Using truncation error arguments and examples we will show that it is not a constant and depends also on the wavenumber. This dependence on the numerical order and wavenumber strongly favors high order schemes for use in flows which have significant energy in the high modes such at Rayleigh–Taylor and Richtmyer–Meshkov instabilities.     

Some Ergodic Results on Stochastic Iterative Discrete Events Systems

This paper deals with the asymptotic behavior of the stochastic dynamics of discrete event systems. In this paper we focus on a wide class of models arising in several fields and particularly in computer science. This class of models may be characterized by stochastic recurrence equations in ^K of the form T(n+1) = _n+1(T(n)) where _n is a random operator monotone and 1—linear. We establish that the behaviour of the extremas of the process T(n) are linear. The results are an application of the sub-additive ergodic theorem of Kingman. We also give some stability properties of such sequences and a simple method of estimating the limit points.     

A probabilistic scheme of independent random elements distributed over a finite lattice. II. The method of lattice moments

A new method is proposed for proving some theorems on the convergence of sequences of random quantities _n that assume values in a set {0,1,...,n} to discrete probability distributions. The method is based on the investigation of definite numerical characteristics (called lattice moments) of asymptotic behavior of distributions of _n and is illustrated by the examples of investigating the asymptotic behavior of the probability distribution of the solution space dimension of a system of independent random homogeneous linear equations over a finite field and that of the number of connected components of a random (unequiprobable) hypergraph with independent hyperedges.Translated from Kibernetika i Sistemnyi Analiz, No. 6, pp. 44– 65, November–December 2004.Part 1 was published in Cybernetics and Systems Analysis, No. 5, 2004.This revised version was published online in April 2005 with a corrected cover date.     

A relationship between difference hierarchies and relativized polynomial hierarchies

Chang and Kadin have shown that if the difference hierarchy over NP collapses to levelk, then the polynomial hierarchy (PH) is equal to thekth level of the difference hierarchy over ₂ ^p . We simplify their poof and obtain a slightly stronger conclusion: if the difference hierarchy over NP collapses to levelk, then PH collapses to (P _(k–1) ^NP )^NP, the class of sets recognized in polynomial time withk – 1 nonadaptive queries to a set in NP^NP and an unlimited number of queries to a set in NP. We also extend the result to classes other than NP: For any classC that has _m ^p -complete sets and is closed under _conj ^p -and _m ^NP -reductions (alternatively, closed under _disj ^p -and _m ^co-NP -reductions), if the difference hierarchy overC collapses to levelk, then PH ^C = (P _(k–1)–tt ^NP ) ^C . Then we show that the exact counting class C_P is closed under _disj ^p - and _m ^co-NP -reductions. Consequently, if the difference hierarchy over C_P collapses to levelk, then PH^PP(= PH^C_P) is equal to (P _(k–1)–tt ^NP )^PP. In contrast, the difference hierarchy over the closely related class PP is known to collapse.Finally we consider two ways of relativizing the bounded query class P _k–tt ^NP : the restricted relativization P _k–tt ^{NP C} and the full relativization (P _k–tt ^NP ) ^C . IfC is NP-hard, then we show that the two relativizations are different unless PH ^C collapses.Richard Beigel was supported in part by NSF Grants CCR-8808949 and CCR-8958528. Richard Chang was supported in part by NSF Research Grant CCR 88-23053. This work was done while Mitsunori Ogiwara was at the Department of Information Science, Tokyo Institute of Technology, Tokyo, Japan.     

Linear Transformation Groups and Shape Space

The problem of modelling variabilities of ensembles of objects leads to the study of the shape of point sets and of transformations between point sets. Linear models are only able to amply describe variabilities between shapes that are sufficiently close and require the computation of a mean configuration. Olsen and Nielsen (2000) introduced a Lie group model based on linear vector fields and showed that this model could describe a wider range of variabilities than linear models. The purpose of this paper is to investigate the mathematics of this Lie group model further and determine its expressibility. This is a necessary foundation for any future work on inference techniques in this model.Let ^k _m denote Kendall's shape space of sets of k points in m-dimensional Euclidean space (k > m): this consists of point sets up to equivalence under rotation, scaling and translation. Not all linear transformations on point sets give well-defined transformations of shapes. However, we show that a subgroup of transformations determined by invertible real matrices of size k – 1 does act on ^k _m. For m > 2, this group is maximal, whereas for m = 2, the maximal group consists of the invertible complex matrices. It is proved that these groups are able to transform any generic shape to any other. Moreover, we establish that for k > m + 1 this may be done via one-parameter subgroups. Each one-parameter subgroup is given by exponentiation of an arbitrary (k – 1) × (k – 1) matrix. Shape variabilities may thus be modelled by elements of a (k – 1)²-dimensional vector space.     

The mean-shift outlier model in general weighted regression and its applications

Consider the general weighted linear regression model y=Xβ+, where E()=0, Cov()=Vσ², σ² is an unknown positive scalar, and V is a symmetric positive-definite matrix not necessary diagonal. Two models, the mean-shift outlier model and the case-deletion model, can be employed to develop multiple case-deletion diagnostics for the linear model. The multiple case-deletion diagnostics are obtained via the mean-shift outlier model in this article and are shown to be equivalent to the deletion diagnostics via the case deletion model obtained by Preisser and Qaqish (1996, Biometrika, 83, 551–562). In addition, computing the multiple case-deletion diagnostics obtained via the mean-shift outlier model is faster than computing the one based on the more commonly used case-deletion model in some situations. Applications of the multiple deletion diagnostics developed from the mean-shift outlier model are also given for regression analysis with the likelihood function available and regression analysis based on generalized estimating equations. These applications include survival models and the generalized estimating equations of Liang and Zeger (1986, Biometrika, 73, 13–22). Several numerical experiments as well as a real example are given as illustrations.     

High Order Numerical Discretization for Hamilton–Jacobi Equations on Triangular Meshes

In this paper we construct several numerical approximations for first order Hamilton–Jacobi equations on triangular meshes. We show that, thanks to a filtering procedure, the high order versions are non-oscillatory in the sense of satisfying the maximum principle. The methods are based on the first order Lax–Friedrichs scheme [2] which is improved here adjusting the dissipation term. The resulting first order scheme is -monotonic (we explain the expression in the paper) and converges to the viscosity solution as for the L -norm. The first high order method is directly inspired by the ENO philosophy in the sense where we use the monotonic Lax–Friedrichs Hamiltonian to reconstruct our numerical solutions. The second high order method combines a spatial high order discretization with the classical high order Runge–Kutta algorithm for the time discretization. Numerical experiments are performed for general Hamiltonians and L ¹, L ² and L -errors with convergence rates calculated in one and two space dimensions show the k-th order rate when piecewise polynomial of degree k functions are used, measured in L ¹-norm.     

Design sensitivity analysis and optimization of nonlinear dynamic response for a motorcycle driving on a half-sine bump road

This paper analyses the design sensitivity of a suspension system with material and geometric nonlinearities for a motorcycle structure. The main procedures include nonlinear structural analysis, formulation of the problem with nonlinear dynamic response, design sensitivity analysis, and optimization. The incremental finite element method is used in structural analysis. The stiffness and damping parameters of the suspension system are considered as design variables. The maximum amplitude of nonlinear transient response at the seat is taken as the objective function during the optimization simulation. A more realistic finite element model for the motorcycle structure with elasto-damping elements of different material models is presented. A comparison is made of the optimum designs with and without geometric nonlinear response and is discussed.Nomenclature A amplitude of the excitation function - a ₀,a ₁ time integration constants for the Newmark method - ^t+t C _s secant viscous damping matrix at timet+t - ^t C _T tangent viscous damping matrix at timet - C linear part of ^t C _T - D _i ⁰ initial value of thei-th design variable - D _i instanenous value of thei-th design variables - ^t+t F^(t–1) total internal force vector at the end of iteration (i–1) and timet+t - ^t+t F _(NL) ^(i–1) nonlinear part of ^t+t F^(i–1) - f frequency of the excitation function - ^t+t K _s secant stiffness matrix at timet+t - ^t K _T tangent stiffness matrix at timet - K linear part of ^t K _T - effective stiffness matrix at timet - L distance between the wheel centres - M constant mass matrix - m _T number of solution time steps - NC number of constraint equations - Q nonlinear dynamic equilibrium equation of the structural system - ^t+t R external applied load vector at timet+t - t _e active time interval for the excitation function - ^t U displacement vector of the finite element assemblage at timet - velocity of the finite element assemblage at timet - ^t Ü acceleration vector of the finite element assemblage at timet - ^t+t U ⁽ⁱ⁾ displacement vector of the finite element assemblage at the end of iterationi and timet+t - velocity vector of the finite element assemblage at the end of iterationi and timet+t - ^t+t Ü⁽ⁱ⁾ acceleration vector of the finite element assemblage at the end of iterationi and timet+t - U ⁽ⁱ⁾ vector of displacement increments from the end of iteration (i–1) to the end of iterationi at timet+t - V driving speed of motorcycle - x vector of design variable - () quantities of variation - ₀ objective function - _i i-th constraint equation     

The BR1 Scheme is Stable for the Compressible Navier–Stokes Equations

In this work we prove that the original (Bassi and Rebay in J Comput Phys 131:267–279, 1997) scheme (BR1) for the discretization of second order viscous terms within the discontinuous Galerkin collocation spectral element method (DGSEM) with Gauss Lobatto nodes is stable. More precisely, we prove in the first part that the BR1 scheme preserves energy stability of the skew-symmetric advection term DGSEM discretization for the linearized compressible Navier–Stokes equations (NSE). In the second part, we prove that the BR1 scheme preserves the entropy stability of the recently developed entropy stable compressible Euler DGSEM discretization of Carpenter et al. (SIAM J Sci Comput 36:B835–B867, 2014) for the non-linear compressible NSE, provided that the auxiliary gradient equations use the entropy variables. Both parts are presented for fully three-dimensional, unstructured curvilinear hexahedral grids. Although the focus of this work is on the BR1 scheme, we show that the proof naturally includes the Local DG scheme of Cockburn and Shu.     

Primality testing with fewer random bits

In the usual formulations of the Miller-Rabin and Solovay-Strassen primality testing algorithms for a numbern, the algorithm chooses candidatesx ₁,x ₂, ...,x _k uniformly and independently at random from _n , and tests if any is a witness to the compositeness ofn. For either algorithm, the probabilty that it errs is at most 2^–k .In this paper, we study the error probabilities of these algorithms when the candidates are instead chosen asx, x+1, ..., x+k–1, wherex is chosen uniformly at random from _n . We prove that fork=[1/2log₂ n], the error probability of the Miller-Rabin test is no more thann ^–1/2+o(1), which improves on the boundn ^–1/4+o(1) previously obtained by Bach. We prove similar bounds for the Solovay-Strassen test, but they are not quite as strong; in particular, we only obtain a bound ofn ^–1/2+o(1) if the number of distinct prime factors ofn iso(logn/loglogn).     

Analytical Expansion and Numerical Approximation of the Fermi–Dirac Integrals F j(x) of Order j=−1/2 and j=1/2

This paper uses properties of the Weyl semiintegral and semiderivative, along with Oldham's representation of the Randles–Sevcik function from electrochemistry, to derive infinite series expansions for the Fermi–Dirac integrals _j (x), –j=–1/2, 1/2. The practical use of these expansions for the numerical approximation of _–1/2(x) and _1/2(x) over finite intervals is investigated and an extension of these results to the higher order cases j=3/2, 5/2, 7/2 is outlined.     

QFT template generation for time-delay plants based on zero-inclusion test

Plant template generation is the key step in applying quantitative feedback theory (QFT) to design robust control for uncertain systems. In this paper we propose a technique for generating plant templates for a class of linear systems with an uncertain time delay and affine parameter perturbations in coefficients. The main contribution lies in presenting a necessary and sufficient condition for the zero inclusion of the value set f(T,Q)={f(τ,q): τT[τ⁻,τ⁺], qQ∏_k=0^m−1[q_k⁻,q_k⁺]}, where f(τ,q)=g(q)+h(q)e^−jτω*, g(q) and h(q) are both complex-valued affine functions of the m-dimensional real vector q, and ω^* is a fixed frequency. Based on this condition, an efficient algorithm which involves, in the worst case, evaluation of m algebraic inequalities and solution of m2^m−1 one-variable quadratic equations, is developed for testing the zero inclusion of the value set f(T,Q). This zero-inclusion test algorithm allows one to utilize a pivoting procedure to generate the outer boundary of a plant template with a prescribed accuracy or resolution. The proposed template generation technique has a linear computational complexity in resolution and is, therefore, more efficient than the parameter gridding and interval methods. A numerical example illustrating the proposed technique and its computational superiority over the interval method is included.     

The most nonelementary theory

We give a direct proof by generic reduction that testing validity of formulas in a decidable rudimentary theory Ω of finite typed sets (Henkin, Fundamenta Mathematicæ 52 (1963) 323–344) requires space and time exceeding infinitely often(1)where n denotes the length of input. This gives the highest currently known lower bound for a decidable logical theory and affirmatively settles Problem 10.13 from (Compton and Henson, Ann. Pure Applied Logic 48 (1990) 1–79):

Is there a “natural” decidable theory with a lower bound of the form exp_∞(f(n)), where f is not linearly bounded?

The highest previously known lower (and upper) bounds for “natural” decidable theories, like WS1S, S2S, are of the form exp_∞(dn), with just linearly growing stacks of twos.Originally, the lower bound (1) for Ω was settled in (12th Annual IEEE Symposium on Logic in Computer Science (LICS’97), 1997, 294–305) using the powerful uniform lower bounds method due to Compton and Henson, and probably would never be discovered otherwise. Although very concise, the original proof has certain gaps, because the method was pushed out of the limits it was originally designed and intended for, and some hidden assumptions were violated. This results in slightly weaker bounds—the stack of twos in (1) grows subexponentially, but superpolynomially, namely, as for formulas with fixed quantifier prefix, or as 2^cn/log(n) for formulas with varying prefix. The independent direct proof presented in this paper closes the gaps and settles the originally claimed lower bound (1) for the minimally typed, succinct version of Ω.     

Spectral Versus Pseudospectral Solutions of the Wave Equation by Waveform Relaxation Methods

We examine spectral and pseudospectral methods as well as waveform relaxation methods for the wave equation in one space dimension. Our goal is to study block Gauss–Jacobi waveform relaxation schemes which can be efficiently implemented in a parallel computing environment. These schemes are applied to semidiscrete systems written in terms of sparse or dense matrices. It is demonstrated that the spectral formulations lead to the implicit system of ordinary differential equations Wã = Sã + g(t)w, with sparse matrices W and S which can be effectively solved by direct application of any Runge–Kutta method. We also examine waveform relaxation iterations based on splittings W = W ₁–W ₂ and S = S ₁ + S ₂ and demonstrate that these iterations are only linearly convergent on finite time windows. Waveform relaxation methods applied to the explicit system ã = W ^–1 Sã + g(t)W ^–1 w are somewhat faster but less convenient to implement since the matrix W ^–1 S is no longer sparse. The pseudospectral methods lead to the system = D + g(t)w with a differentiation matrix D of order one and the corresponding waveform relaxation iterations are much faster than the iterations corresponding to the spectral cases (both implicit and explicit).     

Turbulent boundary layer structure of flow over a smooth-curved ramp

The effects of a convex-curved wall followed by a recovery over a flat surface on a turbulent boundary layer structure are addressed via large-eddy simulation (LES). The curved wall constitutes a smooth ramp formed by a portion of circular arc. The statistically two-dimensional upstream boundary layer flow is realistically fed by an injected inflow boundary condition. The inflow is extracted from a simultaneously simulated flat-plate boundary layer which is computed based on a compressible rescaling method. After flowing over the curved surface the flow is allowed to recover its realistic condition by passing over a downstream flat surface. The Reynolds number introduced at the inlet section of the computational domain which starts 4 times the ramp length (L_r) upstream of the curved surface is Re_δo=U_∞δ_o/ν=9907. The Reynolds number is based on the inflow boundary layer thickness δ_o, the free-stream velocity U_∞ and the kinematic viscosity ν.Mean flow predictions obtained using the present LES with the rescaling–recycling inflow condition agree well with the available experimental data from literature. The Reynolds stress components match the experimental one. However, small deviation occurs due to the smaller-domain height used in the present simulation. The experiments showed that there is a generated pressure gradient on the upper wall and this in return affects the turbulence energy on the other wall. The numerical data as well as the experiments show an enhancement of the turbulent stresses in the adverse pressure gradient region. The increased level of turbulent stresses is accompanied with large peaks aligned with the inflection point of the velocity profiles. The high stress levels are nearly unchanged by reattachment process, decaying only after the mean velocity recovered and the high production of turbulence near the outer layer drops. The recovery of the outer layer is due to the turbulent eddies generated by the separation region. Numerical visualizations show strong elongation and lifting of eddies in the region of the adverse pressure gradient generated by the curved wall. Computations of two-point correlations are also performed to represent the formation and deformation of the turbulent eddies before, over and after the curved wall. Different effects on the eddy size and its structure angle are presented.     

Polynomial-time compression

This paper studies the class ofinfinite sets that have minimal perfect hash functions—one-to-one onto maps between the sets and ^*-computable in polynomial time. We will call such sets P-compressible. We show that all standard NP-complete sets are P-compressible, and give a structural condition,E = ₂ ^E , sufficient to ensure thatall infinite NP sets are P-compressible. On the other hand, we present evidence that some infinite NP sets, and indeed some infinite P sets, are not P-compressible: if an infinite NP setA is P-compressible, thenA has an infinite sparse NP subset, yet we construct a relativized world in which some infinite NP sets lack infinite sparse NP subsets. This world is built upon a result that is of interest in its own right; we determine optimally—with respect to any relativizable proof technique—the complexity of the easiest infinite sparse subsets that infinite P sets are guaranteed to have.     

Stabilizability of linear systems defined overC *-algebras

LetF andG be elements of aC ^*-algebraA. Assume that, for each irreducible^*-representation ofA on a Hilbert space210B; , there is a bounded linear operatorL B( ) such that the spectrum of(F) –(G)L is contained in the open left half plane. We prove that there is then an elementL A such that the spectrum ofF — GL is contained in the open left half plane. That is, if the system (F, G) is locally stabilizable, then it is stabilizable. We also consider the analogous problem with the open left half plane replaced by the open unit disk.This paper was supported in part by the National Science Foundation under Grant NSF-MCS-8002138.     

On control for linear systems with interval time-varying delay

This paper deals with the problem of delay-dependent robust H_∞ control for linear time-delay systems with norm-bounded, and possibly time-varying, uncertainty. The time-delay is assumed to be a time-varying continuous function belonging to a given interval, which means that the lower and upper bounds for the time-varying delay are available, and no restriction on the derivative of the time-varying delay is needed, which allows the time-delay to be a fast time-varying function. Based on an integral inequality, which is introduced in this paper, and Lyapunov–Krasovskii functional approach, a delay-dependent bounded real lemma (BRL) is first established without using model transformation and bounding techniques on the related cross product terms. Then employing the obtained BRL, a delay-dependent condition for the existence of a state feedback controller, which ensures asymptotic stability and a prescribed H_∞ performance level of the closed-loop systems for all admissible uncertainties, is proposed in terms of a linear matrix inequality (LMI). A numerical example is also given to illustrate the effectiveness of the proposed method.     

Functional completeness in P2×Pk

Two-sorted invariant relations are used to describe all maximum subalgebras of the iterative algebra P₂×P _k —the arity-calibrated product of the algebras of Boolean functions and of functions of k-valued logic with k3. Functional completeness criteria are obtained for P₂×P _k and also for the system P₂×...×P₂×P _k .Translated from Kibernetika, No. 1, pp. 1–8, 16, January–February, 1991.