pinecone/core-2020-05-10-deduplication huggingface dataset:
The Cauchy Problem for an Axially Symmetric Equation and the Schwarz Potential Conjecture for the Torus

AbstractWe present our results in this paper in two parts. In the first part, we consider the Cauchy problem for the axially symmetric equation ∂2xu+∂2yu+kx∂xu=0 with entire Cauchy data given on an initial plane (see Eq. (2.1)). We solve the Cauchy problem and obtain its solutions in two cases, depending on whether k is a positive even integer or k is a positive odd integer. For k odd, we demonstrate that the solution has more singularities due to the propagation of the singularities of the coefficients. In the second part, the Cauchy problem for the same equation is considered, but instead, its entire Cauchy data are given on an initial sphere (see Eq. (3.1)). Whenever k is a positive even integer, we obtain the global existence of the solution and determine all possible singularities. Whenever k is a positive odd integer, we discuss both local and global solutions. As a consequence of our results in this paper, we show that the Schwarz Potential Conjecture (see the Introduction) for the even dimensional torus is true
Some Convergence Results for a Class of Nonlinear Phase-Field Evolution Equations

AbstractTwo heat diffusion problems in the framework of the parabolic phase-field model are presented. The first problem is related to a single isotropic fluid and the other describes the heat transmission between two different substances in contact. Some known existence and uniqueness results are briefly recalled. Then, an asymptotic analysis of both situations is carried out as the kinetic equation collapses to a temperature-phase relation of Stefan type, in the first case in the whole material, and in the second in only one of the substances. In both cases, a convergence result for the solutions is proved. The second problem shows some more mathematical difficulties that are due to the presence of nontrivial terms on the common boundary. In order to control the latter, some tools are used from the Γ-convergence theory for convex functionals
Phase-Field Models with Hysteresis

AbstractPhase-field systems as mathematical models for phase transitions have drawn increasing attention in recent years. However, while capable of capturing many of the experimentally observed phenomena, they are only of restricted value in modelling hysteresis effects occurring during phase transition processes. To overcome this shortcoming of existing phase-field theories, the authors have recently proposed a new approach to phase-field models which is based on the mathematical theory of hysteresis operators developed in the past 15 years. In particular, they have proved well-posedness and thermodynamic consistency for hysteretic phase field models which are related to the Caginalp and Penrose–Fife models. In this paper, these results are extended into different directions: we admit temperature-dependent relaxation coefficients and relax the growth conditions for the hysteresis operators considerably; also, a unified approach is used for a general class of systems that includes both the Caginalp and Penrose–Fife analogues
Oscillation Results for a Second Order Damped Differential Equation with Nonmonotonous Nonlinearity

AbstractWe present new oscillation criteria for a nonlinear second order differential equation with a damping term. An essential feature of our results is that we do not require the nonlinearity to be nondecreasing. Furthermore, as opposed to the recent results by S. R. Grace (1989, Math. Nachr.141, 117–127; 1992, J. Math. Anal. Appl.171, 220–241) and S. R. Grace and B. S. Lalli (1990, J. Math. Anal. Appl.149, 277–311), we do not impose any additional condition on the damping term p(t). Consequently, our theorems apply to new classes of equations which are not covered by the known criteria. The relevance of the results obtained is illustrated with a number of carefully selected examples
Asymptotic Analysis and Stability of Inviscid Liquid Sheets

AbstractAsymptotic methods are employed to determine the leading-order equations that govern the fluid dynamics of slender, and thin and slender, inviscid, irrotational, planar liquid sheets subject to pressure differences and gravity. Two flow regimes have been identified depending on the Weber number, and analytical solutions to the steady state equations are provided. Linear stability studies indicate that the sinuous mode corresponds to Weber numbers on the order of unity, while the varicose mode is associated with small Weber numbers. For small Weber numbers, the nonlinear stability of liquid sheets is determined analytically in terms of elliptic integrals of the first and second kinds. It is also shown that the sinuous mode of thin and slender liquid sheets is identical to the same mode for slender sheets
Existence and Stability of Periodic Quasisolutions in Nonlinear Parabolic Systems with Discrete Delays

AbstractThe existence of periodic quasisolutions and the dynamics for a couple system of semilinear parabolic equations with discrete time delays and periodic coefficients are investigated using the method of upper and lower solutions. It is shown that if the reaction function in the system possesses a mixed quasimonotone property and the periodic boundary value system has a pair of coupled T-upper and lower solutions then the periodic boundary value problem has a pair of periodic quasisolutions and the sector between the quasisolutions is an attractor of the delayed periodic parabolic system. Under some additional conditions the periodic quasisolutions are exactly true periodic solutions of the periodic boundary value system. Finally, three models arising from ecology are given to illustrate the obtained results
Auxiliary Principle and Iterative Algorithms for Generalized Set-Valued Strongly Nonlinear Mixed Variational-like Inequalities

AbstractIn this paper, we extend the auxiliary principle technique to study a class of generalized set-valued strongly nonlinear mixed variational-like inequalities. We prove the existence of a solution of the auxiliary problem for the generalized set-valued strongly nonlinear mixed variational-like inequalities, construct the iterative algorithm for the generalized set-valued strongly nonlinear mixed variational-like inequalities, and show the existence of a solution of the generalized set-valued strongly nonlinear mixed variational-like inequality by using the auxiliary principle technique. We also prove the convergence of iterative sequences generated by the algorithm
Duality for a Class of Nondifferentiable Multiobjective Programming Problems

AbstractIn this paper, both the Wolfe type and Mond–Weir type dual problems for a class of nondifferentiable multiobjective programs in which every component of the objective function contains a term involving the support function of a compact convex set are formulated. Weak duality theorems are established under suitable conditions. A special case which appears repeatedly in the literature is that the support function is the square root of a positive semidefinite quadratic form. This and other special cases can be readily generated from our results
Hamiltonian Systems on Time Scales

AbstractLinear and nonlinear Hamiltonian systems are studied on time scales T. We unify symplectic flow properties of discrete and continuous Hamiltonian systems. A chain rule which unifies discrete and continuous settings is presented for our so-called alpha derivatives on generalized time scales. This chain rule allows transformation of linear Hamiltonian systems on time scales under simultaneous change of independent and dependent variables, thus extending the change of dependent variables recently obtained by Došlý and Hilscher. We also give the Legendre transformation for nonlinear Euler–Lagrange equations on time scales to Hamiltonian systems on time scales
Nonlinear Control of Incompressible Fluid Flow: Application to Burgers' Equation and 2D Channel Flow

AbstractThis paper proposes a methodology for the synthesis of nonlinear finite-dimensional feedback controllers for incompressible Newtonian fluid flows described by two-dimensional Navier–Stokes equations. Combination of Galerkin's method with approximate inertial manifolds is employed for the derivation of low-order ordinary differential equation (ODE) systems that accurately describe the dominant dynamics of the flow. These ODE systems are subsequently used as the basis for the synthesis of nonlinear output feedback controllers that guarantee stability and enforce the output of the closed-loop system to follow the reference input asymptotically. The method is successfully used to synthesize nonlinear finite-dimensional output feedback controllers for the Burgers' equation and the two-dimensional channel flow that enhance the convergence rate to the spatially uniform steady-state and the parabolic velocity profile, respectively. The performance of the proposed controllers is successfully tested through simulations and is shown to be superior to the one of linear controllers
On Discrete-Time Linear Systems with Almost-Periodic Kernels

AbstractThe paper considers several different stability notions for discrete-time linear time-varying systems specified by almost-periodic kernels. Various input-output stability notions are analysed, including those in which the signal spaces consist of bounded power and asymptotically almost-periodic functions. Further, it is proved that every stable time-varying discrete time system commuting with a polynomial in the shift operator is periodic. An example shows that this result does not hold for unstable time-varying systems
Mixed Type Duality for Multiobjective Variational Problems

AbstractThe concept of mixed-type duality has been extended to the class of multiobjective variational problems. A number of duality relations are proved to relate the efficient solutions of the primal and its mixed-type dual problems. The results are obtained for ρ-convex (generalized ρ-convex) functions. These studies have been generalized to the case of ρ-invex (generalized ρ-invex) functions. Our results apparently generalize a fairly large number of duality results previously obtained for finite-dimensional nonlinear programming problems under various convexity assumptions
Julia Sets of Certain Exponential Functions

AbstractWe characterize the Julia sets of certain exponential functions. We show that the Julia sets J(Fλn) of Fλn(z)=λnezn where λn>0 is the whole plane C, provided that limk→∞Fkλn(0)=∞. In particular, this is true when λn are real numbers such that λn>1ne1/n. On the other hand, if 0<λn<1ne1/n, then J(Fλn) is nowhere dense in C and is the complement of the basin of attraction of the unique real attractive fixed point of Fλn. We then prove similar results for the functions[formula] where λi∈C−{0}, 1≤i≤n+1, aj>1, 1≤j≤n, and m, n≥1
Spectrum of a Dynamical System and Applied Symbolic Dynamics

AbstractThe paper introduces a constructive method for localization of the Morse spectrum of a dynamical system on a vector bundle. The Morse spectrum is a limit set of Lyapunov exponents of periodic pseudo-trajectories. The proposed method does not demand any preliminary information on a system. An induced dynamical system on the projective bundle is associated with a directed graph called the symbolic image. The symbolic image can be considered as a finite discrete approximation of a dynamical system. Valuable information about the system may come from the analysis of a symbolic image. In particular, a neighborhood of the Morse spectrum can be found. A special sequence of symbolic images is considered to obtain a sequence of embedded neighborhoods which converges to the Morse spectrum. The main results of this article were announced in a previous paper (“Proceedings of the Fifteenth IMACS World Congress,” 1997, Vol. 1, pp. 15–30)
Zero Relaxation Limit for Piecewise Smooth Solutions to a Rate-Type Viscoelastic System in the Presence of Shocks

AbstractWe study a rate-type viscoelastic system proposed in I. Suliciu (Int. J. Engng. Sci.28 (1990), 827–841), which is a 3×3 hyperbolic system with relaxation. As the relaxation time tends to zero, this system converges to the well-known p-system formally. In the case that the solutions of the p-system are piecewise smooth, including finitely many noninteracting shock waves, we show that there exist smooth solutions for Suliciu's model which converge to those of the p-system strongly as the relaxation time goes to zero. The method used here is the so-called matched asymptotic analysis suggested in J. Goodman and Z. P. Xin (Arch. Ration. Mech. Anal.121 (1992), 235–265), which includes two parts: the matched asymptotic expansion and stability analysis
Asymptotic Oscillations of Continua of Positive Solutions of a Semilinear Sturm–Liouville Problem

AbstractWe consider the set of positive solutions (λ,u) of the semilinear Sturm–Liouville boundary value problem−u″=λu+fuin 0,π,u0=uπ=0,where f:[0,∞)→R is Lipschitz continuous and λ is a real parameter. We suppose that f(s) oscillates, as s→∞, in such a manner that the problem is not linearizable at u=∞ but does, nevertheless, have a continuum C of positive solutions bifurcating from infinity. We investigate the relationship between the oscillations of f and those of C in the λ–|u|0 plane at large |u|0. In particular, we discuss whether C oscillates infinitely often over a single point λ, or over an interval I (of positive length) of λ values. An immediate consequence of such oscillations over I is the existence of infinitely many solutions, of arbitrarily large norm |u|0, of the problem for all values of λ∈I
Conditional Convexity

AbstractIt is well known that a function f:D→R Fréchet differentiable on an open convex subset D of a real normed linear space is convex, i.e.,fλx+1−λy≤λfx+1−λfyx,y∈D,λ∈0,11holds if and only iff′xy−x≤fy−fxx,y∈I2is valid [see, e.g., Roberts and Varberg (“Convex Functions,” Academic Press, New York and London, 1973)].It is shown that (1) with a fixedy=w (or with fixed λx+(1−λ)y=w) is equivalent to the inequality (2) with fixedy=w (or with fixedx=w, respectively).Then these results are applied to study some conditional inequalities for deviation means
Direct Adaptive Rejection of Persistent Disturbances

AbstractGiven any finite set of persistent disturbances that can be represented by a finite sequence in the Banach space of bounded continuous functions, it is shown that stable disturbance rejection is possible for command generator tracker (CGT) model reference adaptive control if certain conditions are satisfied. If the system in question is disturbance-free, the addition of disturbance accommodating adaptive gains into the control law will not impede the asymptotic convergence of the tracking error. Almost periodic disturbances are explored and an illustrative example of the method is given
Some Nonlinear Evolution Equations in Water Waves

AbstractThis paper deals with the two- and three-dimensional nonlinear water waves generated by a steady or oscillating pressure distribution acting at the free surface of a running stream on water of finite depth h. The solutions of the corresponding linearized problems become singular depending on the relative values of the water depth, frequency, and the speed of the applied pressure leading to a resonant phenomenon. When the applied oscillatory pressure distribution moves at a resonant speed, the finite amplitude response is governed by the forced nonlinear Schrödinger (fNLS) equation. Under certain circumstances, the generated wave disturbance may not reach a steady state; in particular, for deep water, a steady state is never attained. For the case of a two-dimensional wave generated by a localized steady pressure moving at a resonant speed, the generated waves are actually of bounded amplitude and are governed by a forced Korteweg–de Vries (fKdV) equation subject to appropriate asymptotic initial conditions. A computational study of the forced KdV equation reveals that a series of solitons is generated in front of the pressure distribution. On the other hand, for the case of the three-dimensional waves induced by a localized steady pressure traveling at a resonant speed, the nonlinear response is governed by a forced Kadomtsev–Petviashvili (fKP) equation. In order to extend the range of applicability of the Boussinesq-type equations in the theory of water waves, the modified Boussinesq equations are derived in terms of the velocity potential on an arbitrary elevation and the free surface displacement
Asymptotic Analysis of Perturbed Mathematical Programs

AbstractWe consider a perturbed mathematical programming problem where both the objective and the constraint functions are analytical in both the underlying decision variables and in the perturbation variable/parameter that is denoted by ϵ. The following question arises: what is the description of the solutions of such a perturbed problem when ϵ→0? We demonstrate that, under weak conditions, the solutions of the perturbed problems are obtained as Puiseux series expansions in ϵ. The results are obtained by application of the Remmert–Stein representation theorem for complex analytic varieties
On a Theorem of Halphen and its Application to Integrable Systems

AbstractWe extend Halphen's theorem which characterizes solutions of certain nth-order differential equations with rational coefficients and meromorphic fundamental systems to a first-order n×n system of differential equations. As an application of this circle of ideas we consider stationary rational algebro-geometric solutions of the KdV hierarchy and illustrate some of the connections with completely integrable models of the Calogero–Moser type. In particular, our treatment recovers the complete characterization of the isospectral class of such rational KdV solutions in terms of a precise description of the Airault–McKean–Moser locus of their poles
Nonexpansive Retractions in Hyperconvex Spaces

AbstractThis paper is primarily concerned with the study of conditions on a hyperconvex subset D of a hyperconvex metric space M which assure that there exists a nonexpansive retraction R of M\D onto D which has the property that R(M\D)⊂∂D. A related question we take up is, when is such a retraction R proximinal, that is, when does R have the propertydx,Rx=distx,Dfor each x∈M? Among other things, we show that if a subset D of a hyperconvex metric space M has nonempty interior and is externally hyperconvex relative to M in a very weak sense, then there always exists a nonexpansive retraction of M onto D which maps M\D onto ∂D. We also show that any compact weakly externally hyperconvex subset of a hyperconvex M is a proximinal nonexpansive retract of M
Orders of Absolute Measurability

AbstractA subset A of the torus [0,1)k is called absolute measurable if the value of μ(A) is the same for every finitely-additive translation-invariant probability measure μ defined on all subsets of [0,1)k. We define four set functions (called orders) that measure how “strongly” a set A is absolute measurable. The order o(A) equals k−dimB(∂ A) and is connected to the Jordan measurability of A. The order δ(A) measures how small the oscillation of the average of n translates of χA can be. The order τ(A) is related to the absolute inner and outer measures defined by Tarski; finally, σ(A) is defined by the oscillation of those functions that are “scissor-congruent” to χA. We prove that o⪡δ⪡τ⪡σ, that is, each of the orders o, δ, τ, σ is “finer” than the previous one. We investigate the connection between the orders and questions of equidecomposability. We show that, under certain conditions, a set of large order is equidecomposable to a cube and present some results in the other direction as well
Symmetry, Singularities, and Integrability in Complex Dynamics II. Rescaling and Time-Translation in Two-Dimensional Systems

AbstractThe explicit integrability of second-order ordinary differential equations invariant under time-translation and rescaling is investigated. Quadratic systems generated from the linearisable version of this class of equations are analysed to determine the relationship between the Painlevé and singularity properties of the different systems. The transformation contains a parameter and for critical values, intimately related to the possession of the Painlevé property in the parent second-order equation, one finds a difference from the generic behaviour. This study is a prelude to a full discussion of the class of transformations which preserve the Painlevé property in the construction of quadratic systems from scalar nth-order ordinary differential equations invariant under time invariant under time translation and rescaling
Critical Exponents of Fujita Type for Inhomogeneous Parabolic Equations and Systems

AbstractWe consider the large-time behavior of sign-changing solutions of inhomogeneous parabolic equations and systems. For example, for ut=Δu+|u|p+w(x) in Rn×[0,T), we show the following: If n≥3 and ∫Rnw(x)dx>0 and 1<p≤n/(n−2), then all solutions blow up in finite time, while if p>n/(n−2) there are both global and nonglobal solutions. We show by example that global solutions exist for all p>1 and w satisfying ∫Rnw(x)dx<0. When n=1,2 and ∫Rnw(x)dx>0, no solution can exist for all time. Extensions of the above result to various geometries and some other problems are indicated
Some Categorical Aspects of Fuzzy Topology

AbstractResearch in recent years has revealed that the construct of fuzzy topological spaces behaves quite differently from that of topological spaces with respect to certain categorical properties. In this paper we discuss some of these aspects. Since the topological construct L-FTS contains nontrivial both initially and finally closed full subconstructs, and each such construct gives rise to a natural autonomous theory of fuzzy topology, it can be said to some extent that fuzzy topology should consist of a system of closely related topology theories, including the classical topology theory as a special case, with each applying to one such subconstruct. Therefore in the first part of this paper the theory of sobriety is established for each finally and initially closed full subconstruct of L-FTS to illustrate this idea. The second topic of this paper is the relationship between the construct of stratified L-fuzzy topological spaces and several other familiar constructs in fuzzy topology, for example, the constructs of Šostak fuzzy topological spaces and L-fuzzifying topological spaces
The Guaranteed Cost Control Problem of Uncertain Singularly Perturbed Systems

AbstractIn this paper we study the algebraic Riccati equation corresponding to the guaranteed cost control theory for an uncertain singularly perturbed system. The construction of the controller involves solving the full-order algebraic Riccati equation with small parameter ε. Under control-oriented assumptions, we first provide the sufficient conditions such that the full-order algebraic Riccati equation has a positive semi-definite stabilizing solution. Next we propose an iterative algorithm based on the Kleinman algorithm to solve the algebraic Riccati equation which depends on the parameter ε. Our new idea is to use the solutions of the reduced-order algebraic Riccati equations for the initial condition. By using the iterative algorithm, we can easily obtain a required solution of the algebraic Riccati equation. Moreover, using the initial conditions without ε, we show that there exists an ε̃ such that the proposed algorithm has quadratic convergence. Finally, in order to show the effectiveness of the proposed algorithm, numerical examples are included