pinecone/core-2020-05-10-deduplication huggingface dataset:
An Algorithm for the Multiple Lot Sizing Problem with Rigid Demand and Interrupted Geometric Yield

AbstractIn this paper, we consider the multiple lot sizing problem with interrupted geometric yield distribution. In the literature, this problem has been modeled in a dynamic programming form, and a closed-form solution for an optimal lot size has not yet been discovered. Our aim here is to develop an algorithm to search for an optimal lot size for this problem. Numerical results show that our algorithm performs superior to Zhang and Guu's algorithm. A closed-form solution of an optimal lot size has been proposed as well but for small demands and for special cases only
On the Convergence of Waveform Relaxation Methods for Differential-Functional Systems of Equations

AbstractIn this paper the convergence of a waveform relaxation method applied to an initial value problem for the Volterra functional-differential system is discussed. It is shown that the method is convergent under the assumption that the splitting function satisfies only the one side Lipschitz condition with respect to some arguments and the Lipschitz condition with respect to the others. The conditions given in the paper also guarantee the existence and uniqueness of the solution to the initial problem discussed in the paper. The convergence of the perturbed continuous time waveform relaxation method is also discussed
Global Asymptotic Stability in Some Discrete Dynamical Systems

AbstractFor a discrete dynamical system xn+1=Txn on M⊂Rr some general conditions will be specified under which the unique equilibrium is globally asymptotically stable. As a special result we obtain the strong negative feedback property established in A. M. Amleh, N. Kruse, and G. Ladas (J. Differ. Equations Appl. to appear). Finally we apply our result to show that the equilibrium x*=1 of the Putnam difference equation,xn+1=xn+xn−1+xn−2xn−3xnxn−1+xn−2+xn−3,with positive initial conditions x0,…,x−3 is globally asymptotically stable
Ergodic Control of Semilinear Stochastic Equations and the Hamilton–Jacobi Equation

AbstractIn this paper we consider optimal control of stochastic semilinear equations with Lipschitz continuous drift and cylindrical noise. We show existence and uniqueness (up to an additive constant) of solutions to the stationary Hamilton–Jacobi equation associated with the cost functional given by the asymptotic average per unit time cost. As a consequence we find the optimizing controls given in the feedback form. To obtain these results we prove also some new results on the transition semigroups of semilinear diffusion acting in the spaces of continuous function with the weighted sup norms and on the optimal control of semilinear diffusions for the discounted cost functional
A Class of Degenerate Multivalued Second-Order Boundary Value Problems

AbstractThe main purpose of this paper is to investigate existence and uniqueness for 1D, double nonlinear and multivalued, degenerate, second-order boundary value problems of the form0∈−prGu′r′+qrHur,0<r<1,1.10∈prGu′r|r=0+,C∈p1Gu′1.1.2Our assumptions are sufficiently general to cover a wide class of applications. We show by some appropriate examples that the assumptions are quite natural and essential for the validity of our main result stated in Theorem 2.1. A variational interpretation of the above equations is also given
Consistency of Monomial and Difference Representations of Functions Arising from Empirical Phenomena

AbstractChoice probabilities in the behavioral sciences are often analyzed from the standpoint of a differencerepresentation such as P(x,x,y)=F[u(x,x)−g(y)]. Here, x and y are real, positive vector variables, x is a positive real variable, P(x,x,y) is the probability of choosing alternative (x,x) over alternative y, and u, g and F are real valued, continuous functions, strictly increasing in all arguments. In some situations (e.g. in psychophysics), the researchers are more interested in the functions u and g than in the function F. In such cases, they investigate the choice phenomenon by estimating empirically the value x such that P(x,x,y)=ρ, for some values of ρ, and for many values of the variables involved in x and y. In other words, they study the function ξ satisfying ξ(x,y;ρ)=x⇔P(x,x,y)=ρ. A reasonable model to consider for the function ξ is offered by the monomialrepresentationξx,y;ρ=∏n−1i=1x−ηi(ρ)i∏mj=1yζj(ρ)jCρ,in which the ηi's, the ζj's and C are functions of ρ. In this paper we investigate the consistency of these difference and monomial representations. The main result is that, under some background conditions, if both the difference and the monomial representations are assumed, then: (i) all functions ηi (1≤i≤n−1) must be constant; (ii) if one of the functions ζj is nonconstant, then all of them must be of the form ζj(ρ)=θjexp[δF−1(ρ)], for some constants θj>0 (1≤j≤m) and δ≠0, where F−1 is the inverse of the function F of the difference representation. Surprisingly, F can be chosen rather arbitrarily
Lifetime and Illness in Almost Deterministic Maps

AbstractThis paper presents a completely new approach to analysing the effect of stochasticity upon the stability of sets in deterministic maps. Such stability is quantified via the definition of an expected lifetime. More relevant to dynamical systems modelling is the case where the stochasticity is small; the almost deterministic case. Formal arguments are employed to quantify the illness, through which the effect on the lifetime expectancy due to small stochasticity, is obtained. In the spirit of perturbative analysis, such equations are derived in terms of the deterministic dynamics, and the leading order stochasticity alone; the stochastic trajectories need not be computed. Of help is the definition of the ageing (for mortal trajectories) and the ageing exponent (for the immortal case). Special cases (attracting sets, Hamiltonian maps, etc.) are analysed in some detail, leading to a better understanding of the equations derived
Hyperbolic Systems with Relaxation: Characterization of Stiff Well-Posedness and Asymptotic Expansions

AbstractThe Cauchy problem for linear constant-coefficient hyperbolic systems ut+∑jA(j)uxj=(1/δ)Bu+Cu in d space dimensions is analyzed. Here (1/δ)Bu is a large relaxation term, and we are mostly interested in the critical case where B has a non-trivial null-space. A concept of stiff well-posedness is introduced that ensures solution estimates independent of 0<δ⪡1. Stiff well-posedness is characterized algebraically and—under mild assumptions on B—is shown to be equivalent to the existence of a limit of the L2-solution as δ→0. The evolution of the limit is governed by a reduced hyperbolic system, the so-called equilibrium system, which is related to the original system by a phase speed condition. We also show that stiff well-posedness—which is a weaker requirement than the existence of an entropy—leads to the validity of an asymptotic expansion. As an application, we consider a linearized version of a generic model of two-phase flow in a porous medium and show stiff well-posedness using a general result on strictly hyperbolic systems. To confirm the theory, the leading terms of the asymptotic expansion are computed and compared with a numerical solution of the full problem
The Characterization of Generalized Metric KKM Mappings with Open Values in Hyperconvex Metric Spaces and Some Applications

AbstractIn this note, the characterization for a set-valued mapping with finitely metrically open values being a generalized metric KKM mapping in hyperconvex metric spaces is established. This result could be regarded as a dual form of corresponding results for the Fan–KKM principle in hyperconvex metric spaces obtained recently by M. A. Khamsi [J. Math. Anal. Appl.204 (1996) 298–306] and W. A. Kirk, B. Sims, and X. Z. Yuan [Nonlinear Analysis (1999) (in press)]. Then we show that the finite intersection property of generalized metric KKM mappings with finitely metrically open values indeed is equivalent to the finite intersection property of generalized metric KKM mappings with finitely metrically closed values in hyperconvex spaces. As applications, we first obtain Ky Fan type matching theorems for both closed and open covers in hyperconvex spaces, which, in turn, are used to establish fixed point theorems for set-valued mappings in hyperconvex metric spaces
Runge–Walsh-Wavelet Approximation for the Helmholtz Equation

AbstractMetaharmonic wavelets are introduced for constructing the solution of the Helmholtz equation (reduced wave equation) corresponding to Dirichlet's or Neumann's boundary values on a closed surface Σ in three-dimensional Euclidean space R3. A consistent scale continuous and scale discrete wavelet approach leading to exact reconstruction formulas is considered in more detail. A scale discrete version of multiresolution is described for potential functions metaharmonic outside the closed surface and satisfying the radiation condition at infinity. Moreover, we discuss fully discrete wavelet representations of band-limited metaharmonic potentials. Finally, a decomposition and reconstruction (pyramid) scheme for economical numerical implementation is presented for Runge-wavelet approximation
Convex Constrained Programmes with Unattained Infima

AbstractWe consider a problem of minimization of convex function f(x) over the convex region R where the objective function and the feasible region have a common direction of recession. In cases when one of these directions is not in the constancy space of the objective function, then the minimal solution is not achieved even if the function f(x) is bounded below over the region R. Many algorithms, if applied to this class of programmes, do not guarantee convergence to the global infimum. Our approach to this problem leads to derivation of the equation of the feasible parametrized curve C(t), such that the infimum of the logarithmic penalty function along this curve is equal to the global infimum of the objective function over the region R. We show that if all functions defining the program are analytic, then C(t) is also an analytic function. The equation of the curve can be successfully used to determine the global infimum (in particular, unboundedness) of the convex constrained programmes in cases when the application of classical methods, such as the steepest descent method, fails to converge to the global infimum
Periodic Solutions of Parabolic Systems with Nonlinear Boundary Conditions

AbstractThis paper is concerned with the existence and stability of periodic solutions for a coupled system of nonlinear parabolic equations under nonlinear boundary conditions. The approach to the problem is by the method of upper and lower solutions and its associated monotone iterations. This method leads to the existence of maximal and minimal periodic solutions which can be computed from a linear iteration process in the same fashion as for parabolic initial-boundary value problems. A sufficient condition for the stability of a periodic solution is also given. These results are applied to three model problems arising from chemical kinetics, ecology, and population biology
Stability and Bifurcation in Delay–Differential Equations with Two Delays

AbstractThe purpose of this paper is to study a class of differential–difference equations with two delays. First, we investigate the local stability of the zero solution of the equation by analyzing the corresponding characteristic equation of the linearized equation. General stability criteria involving the delays and the parameters are obtained. Second, by choosing one of the delays as a bifurcation parameter, we show that the equation exhibits the Hopf bifurcation. The stability of the bifurcating periodic solutions are determined by using the center manifold theorem and the normal form theory. Finally, as an example, we analyze a simple motor control equation with two delays. Our results improve some of the existing results on this equation
On the Derivatives of Radial Positive Definite Functions

AbstractA radial positive definite function ϕ(‖·‖) on Rd which is 2k times differentiable at zero has 2k+[(d−1)/2] continuous derivatives away from zero, and the estimate is sharp. Furthermore, ϕ(iv)(0)≥3d|ϕ″(0)|2/(d+2) if the derivatives exist; this proves a conjecture of Paul Switzer and has implications on the smoothness of isotropic random fields and their local averages. The techniques used have evolved in geostatistics, and side results refer to Matheron's turning bands operator and the montée
On the Subdifferentials of Quasiconvex and Pseudoconvex Functions and Cyclic Monotonicity

AbstractThe notions of cyclic quasimonotonicity and cyclic pseudomonotonicity are introduced. A classical result of convex analysis concerning the cyclic monotonicity of the (Fenchel–Moreau) subdifferential of a convex function is extended to corresponding results for the Clarke–Rockafellar subdifferential of quasiconvex and pseudoconvex functions.The notion of proper quasimonotonicity is also introduced. It is shown that this new notion retains the characteristic property of quasimonotonicity (i.e., a lower semicontinuous function is quasiconvex if and only if its Clarke–Rockafellar subdifferential is properly quasimonotone), while it is also related to the KKM property of multivalued maps; this makes it more useful in applications to variational inequalities
Mean and Almost Everywhere Convergence of Fourier–Neumann Series

AbstractLet Jμ denote the Bessel function of order μ. The functions x−α/2−β/2−1/2Jα+β+2n+1(x1/2), n=0,1,2,…, form an orthogonal system in L2((0,∞),xα+βdx) when α+β>−1. In this paper we analyze the range of p, α, and β for which the Fourier series with respect to this system converges in the Lp((0,∞),xαdx)-norm. Also, we describe the space in which the span of the system is dense and we show some of its properties. Finally, we study the almost everywhere convergence of the Fourier series for functions in such spaces
Analysis of a Mathematical Model of Protocell

AbstractIn this paper we study a mathematical model of growth of protocell proposed by Tarumi and Schwegler. The model comprises three unknown functions: the concentration u(r,t) of nutrient, the density v(r,t) of building material, and the radius R(t) of the organism which is assumed to be spherically symmetric. The functions u(r,t),v(r,t) satisfy a system of reaction–diffusion equations in the region 0≤r<R(t), t>0, and v satisfies a Stefan condition on the free-boundary r=R(t). We give precise conditions for existence of one stationary solution, two stationary solutions, or none. We then prove that (a) in the first case the stationary solution is unstable so that the transient protocell either disappears in finite time or expands unboundedly; (b) in the second case the stationary solution with the larger radius is stable whereas the one with the smaller radius is unstable, so that the transient protocell generally either disappears in finite time or converges to the stationary configuration with the larger radius; and (c) in the last case the transient protocell disappears in finite time
Quasi-Steady-State Solutions of Some Population Models

AbstractWe consider a class of first-order differential equations generalizing the logistic equation of population growth, together with a two-point boundary condition of the form y(0)=η(y(1)) (where y(t) is the size of the population at time t). Thus the population, defined for t∈[0,1], resets itself at the end of the unit time interval to its initial value. If y satisfies the boundary condition and we define Y(t+n)=y(t) for t∈[0,1) and n=0,1,…, then Y is a 1-periodic solution of the differential equation (extended to t∈[0,∞ by periodicity) for t≠1,2,… and Y has a jump of magnitude η(y(1))−y(0) at t=1,2,…. This quasi-steady-state solution corresponds to a population growing or declining on n−1<t<n (n=1,2,…) and decreasing or increasing impulsively at t=1,2,…. Y plays a role for the jump condition y(n+)=η(y(n−)) analogous to that played by constant solutions to the differential equation with zero jump condition (i.e., y(n+)=y(n−)). We show, under hypotheses motivated by biological considerations, that a strictly positive solution exists, is unique, and is monotone and continuous in its dependence on η
Adaptive Pole Positioning in MIMO Linear Systems by Periodic Multirate-Input Controllers

AbstractIn this paper, the certainty equivalence principle is used to combine the identification method with a control structure derived from the pole placement problem, which rely on periodic multirate-input controllers. The proposed adaptive pole placers, contain a sampling mechanism with different sampling period to each system input and rely on a periodically varying controller which suitably modulates the sampled outputs and reference signals of the plant under control. Such a control strategy allows us to arbitrarily assign the poles of the sampled closed-loop system in desired locations and does not make assumptions on the plant other than controllability and observability of the continuous and the sampled system, and the knowledge of a set of structural indices, namely the locally minimum controllability indices of the continuous-time plant. An indirect adaptive control scheme is derived, which estimates the unknown plant parameters (and consequently the controller parameters) on-line, from sequential data of the input and outputs of the plant, which are recursively updated within the time limit imposed by a fundamental sampling period T0. Using the proposed algorithm, the controller determination is based on the transformation of the discrete analogous of the system under control to a phase-variable canonical form, prior to the application of the control design procedure. The solution of the problem can, then, be obtained by a quite simple utilization of the concept of state similarity transformation. Known indirect adaptive pole placement schemes usually resort to the computation of dynamic controllers through the solution of a polynomial Diophantine equation, thus introducing high order exogenous dynamics in the control loop. Moreover, in many cases, the solution of the Diophantine equation for a desired set of closed-loop eigenvalues might yield an unstable controller, and the overall adaptive pole placement scheme is unstable with unstable compensators because their outputs are unbounded. The proposed control strategy avoids these problems, since here gain controllers are essentially needed to be designed. Moreover, persistency of excitation and, therefore, parameter convergence, of the continuous-time plant is provided without making any assumption either the richness of the reference signals or on the existence of specific convex sets in which the estimated parameters belong or, finally, on the coprimeness of the polynomials describing the ARMA model, as in known adaptive pole placement schemes
Automated Target Tracking in Real-Time: Optimal Adjustment of the Parameters

AbstractWe study a problem in real-time target tracking by means of a one-dimensional “random” Kalman–Bucy filter. The tracking device has constrained processing power and hence has to choose (1) a subregion to be observed out of a larger region in which the target motion takes place and (2) a finite resolution scale. As a consequence the variance of the best prediction of the location of the target becomes a random process. The tracking is satisfactory if the expected time it takes the variance process to enter a finite strip is finite. We determine the optimal choice of subregion and resolution and give conditions on the model that insure a satisfactory tracking procedure for this choice. In addition, using large deviation estimates for Bernoulli random walks, we obtain exponential bounds for the tail probabilities of the expected entrance time of the variance process
Elliptic Norms and the Convergence of Iterated Function Systems in the Plane

AbstractThe convergence of Iterated Function Systems (IFSs) is guaranteed by Banach's fixed point theorem, which requires that all functions in the IFS are contractions for the same distance function. Here we consider IFSs composed of affine maps in the plane, and distance functions induced by elliptic norms (the unit ball is an ellipse). Every affine map of spectral radius less than 1 is contractive for some elliptic norm, but there exists no norm for which all such maps are contractive. Here we seek the set of all elliptic norms for which a given affine map is contractive (the compatibility domain), and we show that the geometry of the compatibility domain depends on the nature of the eigenvalues: real and distinct, double or complex. An IFS will converge if and only if the compatibility domains have a nonempty intersection
Time Decay of Solutions for Generalized Boussinesq Equations in Two Space Dimensions

AbstractIn this article, we study the Cauchy problem of generalized Boussinesq equations. We prove the local existence in time in Sobolev and weighted Sobolev space through Fourier transforms. Then our main result is to prove that the supremum norm of the solution (n,v) with sufficiently small and regular initial data decays to zero like t−1/3. The proof of this result is based on the analysis of the linear part of these Boussinesq equations. After diagonalization of the symbol of the matrix operator associated with the linearized equations, it appears that the components of the eigenvectors associated with the eigenvalues of this matrix valued symbol play a significant role in the difficulties we encountered in our study
Complex-Analytic Theory of the μ-Function

AbstractIn this paper, we consider the determinant of the multivariable return difference Nyquist map, crucial in defining the complex μ-function, as a holomorphic function defined on a polydisk of uncertainty. The key property of holomorphic functions of several complex variables that is crucial in our argument is that it is an open mapping. From this single result only, we show that, in the diagonal perturbation case, all preimage points of the boundary of the Horowitz template are included in the distinguished boundary of the polydisk. In the block-diagonal perturbation case, where each block is norm-bounded by one, a preimage of the boundary is shown to be a unitary matrix in each block. Finally, some algebraic geometry, together with the Weierstrass preparation theorem, allows us to show that the deformation of the crossover under (holomorphic) variations of “certain” parameters is continuous
Real versus Complex Robustness Margin Continuity as a Smooth versus Holomorphic Singularity Problem

AbstractIt is well known that the real robustness margin can be discontinuous, while the complex robustness margin is always continuous relative to problem data. Using some concepts from set-valued analysis, continuity of μ can be viewed as structural stability of the neutral stability region. From this point of view, the crucial issue is whether 0+j0 is a critical value of the return difference map. This paper shows that the discrepancy between real and complex cases is due to the additional holomorphic property of the Nyquist return difference mapping of the complex μ-function. The critical points of the Nyquist map in the complex case are at most finite in number; in contrast, the critical points of the Nyquist map of the real smooth case form, generically, a curve. Furthermore and more importantly, in the complex case, even when 0+j0 is critical, the stability crossover is continuously deformed under the variation of “certain” parameters, while in the real case, the crossover could sustain a catastrophic change