pinecone/core-2020-05-10-deduplication huggingface dataset:
Squared Eigenfunction Symmetries for Soliton Equations: Part II

AbstractNonlinear integrable evolution equations in 1+1 dimensions arise from constraints of the 2+1-dimensional hierarchies associated with the Kadomtsev–Petviashvili (KP) equation, the modified KP equation and the Dym equation, respectively. Links of Bäcklund type and of reciprocal type are known to exist between the 2+1-dimensional systems. The corresponding links between the constrained flows are discussed in a general framework. In particular, squared eigenfunction symmetries generated by solutions of the associated linear scattering problems are considered. The links between the soliton hierarchies are extended to these symmetries
The Coexistence of a Community of Species with Limited Competition

AbstractThis paper is a study of a system modeling a biological community of species with limited competition. The community consists of two competing subcommunities, all species of which cooperate, and some species of one subcommunity can invade the steady state of another subcommunity, whereas others cannot. Sufficient conditions are given that all species can coexist. For Lotka–Volterra systems we can improve this result by showing that there is a unique, globally asymptotically stable steady state
Numerical Verification of Solutions for Nonlinear Elliptic Problems Using anL∞Residual Method

AbstractWe consider a numerical enclosure method with guaranteedL∞error bounds for the solution of nonlinear elliptic problems of second order. By using an a posteriori error estimate for the approximate solution of the problem with a higher orderC0-finite element, it is shown that we can obtain the guaranteedL∞error bounds with high accuracy. A particular emphasis is that our method needs no assumption of the existence of the solution of the original nonlinear equation, but it follows as the result of computation itself. A numerical example that confirms the effectiveness of the method is presented
Bounds for the Heat Diffusion through Windows of Given Area

AbstractWe consider the lowest eigenvalue for the Laplacian in a given Lipschitz domain under mixed boundary conditions: Dirichlet in a subset of the boundary (noninsulated window), Neumann otherwise. This eigenvalue can be interpreted as heat leakage rate due to diffusion. We give an explicit calculation for a model problem, a rigorous lower bound that depends only on the area, but not on the geometry of the window. This bound confirms the observations from the model problem. Finally, we show that no nontrivial upper bound is possible; i.e., any small area for the window being prescribed, its geometry can be made bad enough to cause heat leak rates arbitrarily close to the ones for no insulation anywhere. The most important techniques are the Aronszajn–Weinstein method of intermediate variational problems and the Gaussian upper bounds for the heat kernel by E. B. Davies
The Topology of AFS Structure and AFS Algebras

AbstractIn this paper, we will study the topological structure of the AFS structures published by the author; discuss the topological molecular lattice structures onEI, *EI,EII, *EIIalgebras published by the author; and give the relations of these topological structures. As applications, we study the topological produced by a family of fuzzy concepts in the AFS algebras and apply these to analyze relations among fuzzy concepts. Using these, we believe that we can study the law of human thinking. The most important fact is that all these can be operated by computers. We are also sure AFS theory will play an important role in information processing
Laminated Microstructure in a Variational Problem with a Non-Rank-One Connected Double Well Potential

AbstractA variational problem with a non-rank-one connected double well potential is considered. It is proved that the problem has a laminated microstructure solution which is uniquely determined by the two potential wells yet certainly not characterized by the fine oscillations between them because of the lack of rank-one connection. The laminated microstructure is explicitly worked out. It is also shown that an application of a nonconforming finite element method can cause overrelaxation and fail to approximate the right microstructure
Analytic and asymptotic properties of generalized Linnik probability densities

Cataloged from PDF version of article.This paper studies the properties of the probability density function pα,ν,n(x) of the n-variate generalized Linnik distribution whose characteristic function φα,ν,n(t) is given by where {norm of matrix}t{norm of matrix} is the Euclidean norm of t ∈ ℝn. Integral representations of pα,ν,n(x) are obtained and used to derive the asymptotic expansions of pα,ν,n(x) when {norm of matrix}x{norm of matrix}→0 and {norm of matrix}x{norm of matrix}→∞ respectively. It is shown that under certain conditions which are arithmetic in nature, pα,ν,n(x) can be represented in terms of entire functions. © 2009 Birkhäuser Boston
Differentiable Selections and Castaing Representations of Multifunctions

AbstractWe consider set-valued mappings defined on a linear normed space with convex closed images in Rn. Our aim is to construct selections that are (Hadamard) directionally differentiable using some approximation of the multifunction. The constructions assume the existence of a cone approximation given by a certain “derivative” of the mapping. The first one makes use of the properties of Steiner points. A notion of generalized Steiner points is introduced. The second construction defines a continuous selection that passes through given points of the graph of the multifunction and is Hadamard directionally differentiable at those points, with derivatives belonging to the corresponding “derivatives” of the multifunction. Both constructions lead to a directionally differentiable Castaing representation of a multifunction possessing appropriate differentiability properties. The results are applied to obtain statements about the asymptotic behavior of measurable selections of random sets via the delta method
Global Existence and Optimal Temporal Decay Estimates for Systems of Parabolic Conservation Laws. II. The Multidimensional Case

AbstractThis paper is a continuation of our previous paper. It is concerned with the global existence and the optimal temporal decay estimates for the Cauchy problem of the following multidimensional parabolic conservation laws[formula] Hereu(t,x)=(u1(t,x),…,un(t,x))tis the unknown vector,fj(u)=(fj1(u),…,fjn(u))t(j=1,2,…,N) are arbitraryn×1 smooth vector-valued flux functions defined inBr(u), a closed ball of radiusrcentered at some fixed vectoru∈Rn, andDis a constant, diagonalizable matrix with positive eigenvalues. Our results show that if the flux functionfj(u) satisfiesfj(u)/|u−u|s∈L∞(Br(u),Rn),j=1,2,…,Nfor somes>2+1/N,u∈Rn, then foru0(x)−u∈L∞∩L1(RN,Rn) with ‖u0(x)−u‖L1(RN,Rn)sufficiently small, the above Cauchy problem (*) admits a unique globally smooth solutionu(t,x) andu(t,x) satisfies the following temporal decay estimates. For eachk=0,1,2,… [formula]HereDk=∑|α|=k(∂|α|/∂xα11···∂xαNN). The above decay estimates are optimal in the sense that they coincide with the corresponding decay estimates for the solution to the linear part of the corresponding Cauchy problem
Relaxation Oscillators with Exact Limit Cycles

AbstractAs an inverse problem for relaxation oscillators modeled by the autonomous Liénard differential equation, we assume plausible exact expressions for the limit cycles in the phase plane (which correspond to periodic solutions) toward which all solutions converge, or from which they recede, as time tends to infinity. We illustrate this by two examples, one of which concerns an approximation to a Van der Pol oscillator. The results could be utilized for the synthesis problem, i.e., the construction of oscillator circuits yielding desired periodic behavior
A MaximumLpDistance Problem

AbstractThis paper deals with the problem of finding a point within a convex or nonconvex planar region, which is farthest from a given point. Equivalently, given a facility serving a geographical region, a point of this region is sought which receives the least amount of service, assuming that the amount of service received decreases with distance from the facility. Applications of this problem can be found in the military, transportation, telecommunications, and public safety services. In this study the generalizedLpnorm is considered as the distance metric. For this nonlinear and nonconvex problem, properties of the optimal solution are established. Based on these properties a solution algorithm is developed, consisting of (a) thesubdivision procedurethat decomposes the boundary into manageable nonlinear segments, and (b) a branch and bound basedenveloping procedurewhich determines the optimal solution on a nonlinear segment. Using Karush–Kuhn–Tucker conditions, it is proved that the optimal solution in the special cases of the rectilinear and Tchebycheff metrics is among a set of four candidate points. For the Tchebycheff metric, a converse problem, the farthest point Vornoi diagram is defined and constructed. Appropriate examples illustrate all cases
Linearizable Second Order Monge–Ampère Equations

AbstractIn this paper we consider the second order Monge–Ampère equations in (1+1), (2+1), and (3+1) dimensions. We face the problem of reducing to linear form the first order systems that are equivalent to the given Monge–Ampère equations. When the coefficients are assumed constant there are introduced invertible point transformations, suggested by the invariance with respect to one-parameter Lie groups of point symmetries, allowing us to get linear equations. Moreover, the case of nonconstant coefficients is discussed and some linearizable classes of Monge–Ampère equations with coefficients depending on first order derivatives are characterized
Existence of a Global Solution for a Viscoelastic System

AbstractIn this paper, a unique classical solution (v,u) of the Cauchy problem (1), (2) is obtained if |v0(x)|1+α≤M, |u0(x)|2+α≤M. The a priori estimates |v(x,t)|1+α, |u(x,t)|2+α≤M(T) are obtained by using the maximum principle without the restriction lim|x|→±∞(v0(x),u0(x))=(v±,u±), wherev±,u±are constants. This resolves a problem proposed by J. A. Smoller (“Shock Waves and Reaction-Diffusion Equations,” p. 444, Springer-Verlag, New York/Berlin, 1982) for the viscoelastic system. A parallel result (Theorem 8) for a model (42) of compressible adiabatic flow through porous media with a physical viscosity is also obtained
The Complete Solution for Constrained Delay Problems in the Calculus of Variations by Unconstrained Methods

AbstractThis paper has two main ideas. The first idea is that general constrained problems with delay in the calculus of variations can be associated with unconstrained calculus of variations problems by using multipliers. This allows us to obtain a true Lagrange multiplier rule where the original variables, the multipliers, and the slack variables for inequality constraints can be determined. The second idea is that critical point solutions to the delay problem, which include the determination of the multipliers, immediately follow from Euler–Lagrange equations for the unconstrained problem. This critical point solution is a necessary condition for the original problem. In later work we will show that these methods can be combined with previous methods by the first author to obtain efficient and accurate numerical solutions to the original problem where no such general numerical methods currently seem to exist. This seems to be an important development since the lack of such methods has hindered the usefulness of the theory
The Generalized Discrete Legendre Transformation

AbstractThe discrete transformation (LF)(x)=f(x)=∑∞n=0((2n+1)/2)Pn(x)F(n), where thePn(x)'s denote the well-known Legendre polynomials, is an isomorphism between the spaceL(N0) of the complex-valued sequences of rapid descent and the space L(−1,1) of all infinitely differentiable complex-valued functions defined on −1<x<1, of slow growth in the end points −1 and 1. This result is extended to the corresponding dual spaces. A space of multipliers for these spaces is considered and the translation operator and the convolution are investigated on them. The operational calculus generated involves certain finite-difference operators and is applied to find the solution of some ordinary and partial finite-difference equations
A Note on the Order of Contact between Sets in the Complex Plane

AbstractThis paper deals with the order of contact between arbitrary setsXandYlying in the complex plane C. Let ϕ be a rational function, and ξ0∈C. Suppose, at all pointsxi∈C∪{∞} with ϕ(xi)=ξ0, the setsX,Yhave contact of a given orderpi. In the stability analysis of numerical methods for solving differential equations, the problem arises to determine the order of contact, sayp, between ϕ(X) and ϕ(Y) at ξ0. In this paper a theorem is given according to whichp=maxipi/li, whereli≥1 denotes the order of a certain nonvanishing derivative related to ϕ andxi. The theorem is valid under the assumption that ϕ−1(ϕ(X))⊂cl(X) or ϕ−1(ϕ(Y))⊂cl(Y). This assumption is satisfied in the stability analysis just mentioned, so that the theorem settles the problem arising in that context. The paper also deals with various natural questions related to the specific concept of order (of contact) used in the above mentioned theorem. Some of these questions are important in view of applications
General Transmission Problems in the Theory of Elastic Oscillations of Anisotropic Bodies (Basic Interface Problems)

AbstractHere we discuss three-dimensional so-called basic and mixed boundary value problems (BVP) for steady state oscillations of piecewise homogeneous anisotropic bodies imbedded into an infinite elastic continuum. Uniqueness is shown with the help of generalized Sommerfeld–Kupradze radiation conditions, while existence follows for arbitrary values of the oscillation parameter by the reduction of the original interface transmission BVPs to equivalent uniquely solvable boundary integral or pseudodifferential equations on the interfaces. For the basic BVPs, we show classical regularity and, in addition for the mixed BVPs that the solutions are Hölder continuous with exponent α∈(0,1/2) in the neighbourhood of the curves of discontinuity of the boundary and transmission conditions
Extremity in Köthe–Bochner Function Spaces

AbstractLetEbe a Köthe function space over a complete measurable space andXa Banach space. Recall an elementhinEis said to beorder continuousif, for any decreasing sequence {gn} inSE, ⋀ngn=0 andgn≤|h| implies limn→∞gn=0. We show that every denting point of the unit ball ofEis order continuous. Using this result, we prove thatfis a denting point of the unit ball ofE(X) if and only if&#x02022;‖(·)‖is a denting point of the unit ball of.&#x02022;for almost all∈supp,()/‖()‖is a denting point of the unit ball of.Suppose thatEis order continuous. We also prove that for any unit vectorfinE(X), if ‖f(·)‖Xis a strongly exposed point of the unit ball ofEand for almost allt∈suppf,f(t)/‖f(t)‖Xis a strongly exposed point of the unit ball ofX, thenfis a strongly exposed point of the unit ball ofE(X)
Extensions of the Poisson Summation Formula

AbstractThe classical Poisson summation formula (1.1) and the corresponding distributional formula (1.2) have found extensive applications in various scientific fields. However, they are not universally valid. For instance, if φ(x) is a smooth function, the left-hand side of (1.1) is generally divergent. Even when both sides of (1.1) converge absolutely, they may do so to different numbers. Indeed, in Example 3 we are faced with the embarrassing situation where the series on the left-hand side of (1.1) converges for Res>1 while that on the right-hand side converges only for Res<0. Our aim is to extend formulas (1.1) and (1.2) with the help of some new results in distributional theory. For instance, the evaluation of the distribution with zero mean as given by (3.1) at a test function φ(x) yields the relation ∑∞−∞φ(k)−∫∞−∞φ(x)dx. Both the series and the integral in this expression are generally divergent. The concept of the Cesàro limit is then used to obtain the finite difference of these two terms. Thereafter we extend the analysis to higher dimensions. Various innovative examples are presented to illustrate these concepts
Discrete Dichotomies and Bifurcations from Critical Homoclinic Orbits

AbstractPerturbed discrete systems likexn+1=f(xn)+μg(xn,μ),xn∈RN,n∈Z, when the associated unperturbed map (μ=0) is not invertible and has acriticalorbit {γn} homoclinic to a hyperbolic fixed pointpare studied. Bycriticalwe mean that thef′(γn) are invertible for any integern≠0 butf′(γ0) is not invertible. The main goal is to give sufficient conditions for a bifurcationfrom zero to many homoclinicswhen the parameter crosses zero. We also give a Melnikov like result assuring thepersistence of homoclinicsin a complete neighborhood of μ=0. This result is similar to the ones obtained for diffeomorphisms and flows
Control of Dynamic Systems under the Influence of Singularly Perturbed Markov Chains

AbstractThis work is concerned with nearly optimal controls of nonlinear dynamic systems under the influence of singularly perturbed Markov chains. The underlying Markov chains have fast and slow components and their states can be divided into a number of groups. Within each group of states, the chain varies in a fast pace whereas the jumps from one group to another occur relative infrequently. To obtain the desired optimality, the states of the chain are naturally aggregated in accordance with the transition rates; i.e., replacing the states in a group by a single state to obtain an average system. Then the averaged system is used as a reference to develop the nearly optimal control for the actual system via comparison control methods. The technique used is the method of weak convergence together with the utilization of relaxed control representation
Stabilization of Uncertain Control Systems through Piecewise Constant Feedback

AbstractThis paper investigates the problem of stabilization of nonlinear systems submitted to a deterministic disturbance known by its bounds. The stabilization property is considered in terms of exponential decay of a Lyapunov-like function. As such systems cannot be stabilized by continuous state feedback laws, we consider an approximate stabilization problem with finite horizon. The main result of the paper is the existence of a—discontinuous—piecewise constant feedback control law, achieving the approximate stabilization property. Moreover, in deriving explicitly this feedback law, we show that the number of discontinuity jumps can be related to both precision of the approximate stabilization and time horizon of the control problem considered
A Functional Approach to Approximations for the Observed Open Times in Single Ion Channel Models

AbstractThe openings and closures of a single ion channel in a biological membrane are often modelled by a finite statespace continuous time Markov chain, whose states are aggregated into two classes, of open and closed states, respectively. Time interval omission occurs when short sojourns in either class of states are not observed. Many approximations for the distributions of the resulting apparent open and closed times have appeared in the literature. In this paper, a functional approach is used to derive new approximation formulae for these quantities, and to obtain first order correction terms for some existing approximations
A Convergence of ODE Method in Constrained Optimization

AbstractThis paper proposes a convergence theory of the ordinary differential equations (ODE) method for finding the local optima of general constrained optimization. We prove that solutions starting from the neighbourhood of a critical point of the differential equations in this paper about part variables always converge to the feasible point of the problem. We also study the initial relationship between neural computation in optimization (NCO) and ODE methods. Some detailed connections of the two methods in solving the linear and quadratic programming onRn+are shown in this paper
Mixed Markov Decision Processes in a Semi-Markov Environment with Discounted Criterion

AbstractThis paper presents a new model: the mixed Markov decision process (MDP) in a semi-Markov environment with discounted criterion. It describes a system which behaves like a MDP except that the system is influenced by its semi-Markov process environment. Following each state transition of the environment, the MDP model changes among discrete time MDP, continuous time MDP, and semi-MDP. After presenting the model, we show the validity of the optimality equation and the existence of ε-optimal policies. Finally, the mixed MDP in a Markov environment is transformed into a discrete time MDP
Further Properties of a Continuum of Model Equations with Globally Defined Flux

AbstractTo develop an understanding of singularity formation in vortex sheets, we consider model equations that exhibit shared characteristics with the vortex sheet equation but are slightly easier to analyze. A model equation is obtained by replacing the flux term in Burgers' equation by alternatives that contain contributions depending globally on the solution. We consider the continuum of partial differential equationsut=θ(H(u)u)x+(1−θ)(u)ux+νuxx, 0≤θ≤1, ν≥0, whereH(u) is the Hilbert transform ofu. We show that when θ=1/2, for ν>0, the solution of the equation exists for all time and is unique. We also show with a combination of analytical and numerical means that the solution when θ=1/2 and ν>0 is analytic. Using a pseudo-spectral method in space and the Adams–Moulton fourth-order predictor-corrector in time, we compute the numerical solution of the equation with θ=1/2 for various viscosities. The results confirm that for ν>0, the solution is well behaved and analytic. The numerical results also confirm that for ν>0 and θ=1/2, the solution becomes singular in finite time and finite viscosity prevents singularity formation. We also present, for a certain class of initial conditions, solutions of the equation, with 0<θ<1/3 and θ=1, that become infinite for ν≥0 in finite time
On the Ultimate Peano Derivative

AbstractA functionf: R→R is said to have annth Generalized Peano Derivative (GPD) atxiffis continuous in a neighborhood ofxand there exists an integerk≥0 such that thekth primitive offhas a (k+n)th Peano derivative atx. An example shows that sometimes no suchkexists. In this case, C.-M. Lee has proposed a further generalization when the sequence of derivates, indexed byk, converges to a common value. This value is termed thenth Ultimate Peano Derivative (UPD) atx. Here we show that these generalizations of the Peano derivative are related to a certain Laplace integral for which the Tauberian theorem shows that any finite UPD is in fact a GPD
On the Cauchy Problem of Some Dissipative Flows

AbstractThe Cauchy problem is studied for a system of nonlinear partial differential equations for some dissipative flows in Lagrangian formulation including heat conduction, damping relaxation, and coupling to electric field. The well-posedness of smooth solutions is investigated. It is proved that, for certain large initial data, the solution will develop singularities and shock waves in finite time, which indicates that the Cauchy problem does not have global smooth solutions even if the initial data are smooth, and one has to seek weak solutions