pinecone/core-2020-05-10-deduplication huggingface dataset:
A Bliss-Type Multiplier Rule for Constrained Variational Problems with Time Delay

AbstractThis paper gives a Bliss-type multiplier rule for general constrained variational problems with time delay. Our formulation allows the time delay terms and their derivatives to be present in both the objective functional and the equality and inequality constraint equations. Our major results include a Lagrange Multiplier Rule involving Euler–Lagrange equations for delay differential equations and transversality conditions. Of special note is that these results will be used in later work by the authors to obtain new analytical techniques to solve general constrained problems involving delay differential equations for the Calculus of Variations and for Optimal Control Theory. They will also lead to new general, accurate, and efficient numerical methods to solve these important problems where no such methods exist at this time. Thus, for the first time, we will be able to solve this important class of problems obtaining analytic solutions for simpler problems and accurate numerical results for more complex problems when such a solution exists
Complex Roots of a Random Algebraic Polynomial

AbstractThis paper, for any constantK, provides an exact formula for the average density of the distribution of the complex roots of equation η0+η1z+η2z2+···+ηn−1zn−1=Kwhere ηj=aj+ibjand {aj}n−1j=0and {bj}n−1j=0are sequences of independent identically and normally distributed random variables andKis a complex number withKas its real and imaginary parts. The case of real roots of the above equation with real coefficients andK,z∈Ris well known. Further we obtain the limiting behaviour of this distribution function asntends to infinity
Iteration Functions in Some Nonsmooth Optimization Algorithms

AbstractIn this paper, it is proved that a locally Lipschitzian function has a PHR iteration function or a QS iteration function if and only if it is pseudo-regular, and a locally Lipschitzian function has a positive homogeneous PHR iteration function or a positive homogeneous QS iteration function if and only if it is continuously differentiable. In addition, we also give the computable PHR iteration functions or QS iteration functions for a piece-wise smooth function and a subsmooth function, respectively
Self-Similar Subsolutions and Blowup for Nonlinear Parabolic Equations

AbstractFor a wide class of nonlinear parabolic equations of the formut−Δu=F(u,∇u), we prove the nonexistence of global solutions for large initial data. We also estimate the maximal existence time. To do so we use a method of comparison with suitable blowing up self-similar subsolutions. As a consequence, we improve several known results onut−Δu=up, on generalized Burgers' equations, and on other semilinear equations. This method can also apply to degenerate equations of porous medium type and provides a unified treatment for a large class of problems, both semilinear and quasilinear
An Analytical Solution to the MinimumLp-Norm of a Hyperplane

AbstractWe consider in this paper the problem of determining the minimumLp-norm of a hyperplane inn-dimensional space. A subset of the hyperplane is identified first that contains the optimal solution. On this reduced feasible space, the sets of optimal solutions for all values ofp, 1≤p≤∞, are analytically derived. Several interesting mathematical properties of the optimal solution are presented. Forp, 1<p<∞, it is proved that a unique solution exists, while for the limiting valuesp=1,∞, conditions on the equation coefficients of the hyperplane are found for which an infinite number of optimal solutions exist. The minimumLp-distance of a point from a hyperplane is also analytically derived
On the Minimal Extension ofC0-Semigroups for Second-Order Damped Equations

AbstractThe present paper deals with a minimal extension of the classical semigroup theory for second-order damped differential equations in Banach spaces with closed, densely defined linear operators as coefficients. We do not ask anymore from our operators than in the case of first-order equations, i.e., semigroups. We present here generalizations of the Miyadera–Phillips–Feller theorem, the Hille type theorem, and the Trotter–Kato type theorem. The method is quite general and could be used for equations of any order. We focus our attention on a particular dynamical operator solution or main propagator and we assume some properties about it. From this we can obtain some information about the complementary basis operator solutions or secondary propagators
The Multifractal Spectrum of Quasi Self-Similar Measures

AbstractWe define the notion of quasi self-similar measures and show that for such measures their generalised Hausdorff and packing measures are positive and finite at the critical exponent. In practice this allows easy calculation of their dimension functions. We then show that a coarse form of the multifractal formalism automatically holds for quasi self-similar measures. Examples of quasi self-similar measures include many of the standard constructions of multifractal measures satisfying a strong separation condition, in particular, self-similar measures
Exact Internal Controllability for the Semilinear Heat Equation

AbstractUsing the Browder–Minty surjective theorem from the theory of monotone operators, we consider the exact internal controllability for the semilinear heat equation. We show that the system is exactly controllable inL2(Ω) if the nonlinearities are globally Lipschitz continuous. Furthermore, we prove that the controls depend Lipschitz continuously on the terminal states, and discuss the behaviour of the controls as the nonlinear terms tend to zero in some sense. A variant of the Hilbert Uniqueness Method is presented to cope with the nonlinear nature of the problem
On an Unbounded Linear Operator Arising in the Theory of Growing Cell Population

AbstractThis paper deals with the spectral analysis of a class of unbounded linear operators corresponding to a partial differential equation originally proposed by15to model an age structured proliferating cell population. Individual cells are distinguished by age and cell cycle length. The cell cycle length is considered as an inherited property determined at birth. After a detailed spectral analysis (for general boundary conditions which model the process of cell division of mother cell and the inherence of cycle length by daughter cells) it is shown that the associated Cauchy problem is governed by aC0-semigroup. A spectral decomposition of the solutions into an asymptotic term and a transient one which will be estimated for smooth initial data is given
Radii of Convexity and Strong Starlikeness for Some Classes of Analytic Functions

AbstractLet φ(z) be an analytic function with positive real part on Δ={z;|z|<1} with φ(0)=1, φ′(0)>0 which maps the unit disk Δ onto a region starlike with respect to 1 and symmetric with respect to the real axis. LetST(φ) denote the class of analytic functionsf(z) withf(0)=0=f′(0)−1 for whichzf′(z)/f(z) ≺φ(z),z∈Δ. The radius of convexity of order β of uniformly convex functions and the radius of starlikeness of order β of the classST(φ) are computed. The radii of strong starlikeness of certain classes of analytic functions are computed
Transmutation Method for Solving Erdélyi–Kober Fractional Differintegral Equations

AbstractBy means of fractional calculus techniques we find explicit solutions of Volterra integral equations of the second kind and fractional differential and differintegral equations, involving Erdélyi–Kober fractional integrals and derivatives. Also, some hypergeometric integral equations have been considered and solved as double Erdélyi–Kober equations of the second kind. We use the transmutation method to reduce the solutions of all these equations to known solutions of simpler (Riemann–Liouville) equations of the same type. Some examples are given
Multiple Homoclinics for a Class of Singular Hamiltonian Systems

AbstractWe consider a Hamiltonian system ü+∇V(u)=0 where the potentialV:RN\S→Rhas a unique strict global maximum at a pointp∈RNand a singular setS⊂RN\{p} such thatRN\Sis open, path-connected and the fundamental groupG=π1(RN\S) is nontrivial. Under some compactness conditions onVat infinity and around the singular setSwe study the existence of homoclinic orbits topwhich link withS. WhenVandGsatisfy suitable geometrical conditions, we can prove the existence of multiple homoclinics, each one belonging to a different homotopy class ofG
Evolutions of Tubes under Operability Constraints

AbstractIn this paper, we study the evolutions of tubes that can be regarded as set-valued maps with compact images from R+to a finite dimensional vector spaceXunder some constraints that we call operability constraints. We first recall some properties of the mutational calculus in the metric space (K(X),D) where K(X) is the set of nonempty compact subsets ofX, and D is the Hausdorff distance. The main result is the geometric characterization of the dynamics of the tubes that satisfy the operability constraint
Integral Separation, Lyapunov Transformations, and the Small Change in the Direction of the Solutions of Periodic Linear Systems of Ordinary Differential Equations

AbstractFirst, we discuss the relation between the integral separation of solutions of a linear periodic systemx′=A(t)x(1)and the structure of its characteristic multipliers. Second, we investigate the reducibility of (1) by a complex or a real Lyapunov transformation. Third, we prove the property of the small change in the direction of solutions of ω-periodic system (1) as the result of a small perturbation of (1) under the following assumptions:(a)When the characteristic multipliers of (1) are distinct and the perturbation of (1) is admissible. The perturbationC(t)=B(t)−A(t) is calledadmissibleto the system (1), if the characteristic multipliers ρkand ρkof the systems (1) andy′=B(t)y,(2)respectively, have the same argument, i.e., argρk=argρk,k=1,…,n. In this case, we also say that the perturbed ω-periodic system (2) is admissible to the system (1).(b)When the characteristic multipliers of (1) are real and distinct and the perturbation is arbitrary
Two Dimensional Zonoids and Chebyshev Measures

AbstractWe give an alternative proof to the well known fact that each convex compact centrally symmetric subset of R2containing the origin is a zonoid, i.e., the range of a two dimensional vector measure, and we prove that a two dimensional zonoid whose boundary contains the origin is strictly convex if and only if it is the range of a Chebyshev measure. We give a condition under which a two dimensional vector measure admits a decomposition as the difference of two Chebyshev measures, a necessary condition on the density function for the strict convexity of the range of a measure and a characterization of two dimensional Chebyshev measures
Integrated Semigroups and Related Partial Differential Equations

AbstractWe investigate the relationship between abstract linear evolution equations of heat, wave, and Schrödinger types in terms of well-posedness in Banach spaces. More precisely, we study our operators as generators of integrated semigroups and integrated cosine functions. As applications, we consider in a Banach space context an abstract Laplacian which generalizes the ordinary Laplacian Δ in theLp(RN)-spaces. We obtain optimal results when compared to the classical situation where the generation results were obtained by17using completely different methods. Our main tools are the boundary value theorem for holomorphic semigroups and the abstract Weierstrass formula. Related partial differential equations have been considered previously by Bragg and Dettman
Linear-Nonquadratic Optimal Control Problems with Terminal Inequality Constraints

AbstractThis paper considers a linear-nonquadratic optimal control problem subject to nonlinear terminal inequality constraints. We approximate it by a series of approximate problems via the penalty method. It is shown that the optimal control functions of the approximate problems uniformly converge to the optimal control function of the original terminal constrained problem. Furthermore, the optimal values of the approximate problems also converge to the optimal value of the original problem under some mild conditions
The Continuity Equation with Random Velocity Field and the Expectation of Its Solutions

AbstractThe continuity equation with the random velocity field in thed-dimensional space is studied and the existence of the corresponding family of random evolution operators is proved. In case of the random velocities defined by means of the Gaussian random fields the expectations of the evolution operators are derived and represented in terms of the diffusion operator. The obtained result is applied to the environmental transport problem on a bounded domain. The obtained transport models generalize the conventional transport model defined by means of the diffusion advection equation
A Note about Differentiability of Maps Defined on Convex Subsets of Banach Spaces That May Be Nowhere Dense

AbstractIn connection with continuum mechanics there are physically meaningful choices of infinite-dimensional Banach spaces such that the domain of constitutive maps is nowhere dense in them, as4pointed out. Thus the usual differential calculus on open sets cannot be applied there. Here we give a differentiability notion for mapsfdefined on any convex subset of a Banach space that may be nowhere dense. When the domain offis open, this notion coincides with the usual one. We give the definitions and prove the theorems related to first and higher order derivatives off
Identifiability of the Landau–Ginzburg Potential in a Mathematical Model of Shape Memory Alloys

AbstractThe nonlinear partial differential equations considered here arise from the conservation laws of linear momentum and energy, and describe structural phase transitions in one-dimensional shape memory solids with non-convex Landau–Ginzburg free energy potentials. In this article the theories of analytic semigroups and real interpolation spaces for maximal accretive operators are used to show that the solutions of the model depend continuously on the admissible parameters. Also, we show that the non-physical parameters that define the free energy are identifiable from the model
Necessary Conditions for Optimal Control Problems Involving Nonlinear Differential Algebraic Equations

AbstractDynamic models which take the form of a coupled set of differential and algebraic equations (DAEs) are widely used in process systems engineering. Necessary conditions of optimality for optimal control problems involving such models are derived. A strong Maximum Principle is obtained under a convexity hypothesis on the velocity set. An example illustrates that the strong Maximal Principle may be violated when this hypothesis is dropped. For problems involving nonconvex velocity sets, however, a weak Maximum Principle is valid
Complex Extremal Structure in Spaces of Continuous Functions

AbstractThis paper considers the spaceY=C(T,X) of all continuous and bounded functions from a topological spaceTto a complex normed spaceXwith the sup norm. The extremal structure of the closed unit ballB(Y) inYhas been intensively studied whenXis strictly convex, that is, in terms of its unitary functions (mappings fromTinto the unit sphere ofX). We prove that ifTis completely regular andXhas finite dimension, then every function inB(Y) is expressible as a convex combination of three unitary functions if and only if the condition dimT<dimXRis satisfied (where dimTis the covering dimension ofTandXRdenotesXconsidered as a real normed space). IfXis infinite-dimensional the above decomposition is always possible without restrictions aboutT. These results are interesting whenXis complex strictly convex. As a consequence we state a surprising fact: The identity function on the unit ball of an infinite-dimensional complex normed space can be expressed as the average of three retractions of the unit ball onto the unit sphere. Really, such a representation is the best possible
Almost Sure Exponential Stability of Neutral Stochastic Differential Difference Equations

AbstractA neutral stochastic differential difference equationd[x(t)−G(x(t−τ))]=f(t,x(t),x(t−τ))dt+σ(t,x(t),x(t−τ))dw(t)was introduced by V. B. Kolmanovskii and Y. R. Nosov (“Stability and Periodic Modes of Control Systems with Aftereffect,” Nauka, Moscow, 1981) several years ago. However, so far little is known about the almost sure exponential stability for such equations and the aim of this paper is to close this gap. The convergence of nonnegative special semimartingales established by R. Sh. Lipster and A. N. Shiryayev (“Theory of Martingales,” Kluwer Academic, Dordrecht, 1989) and the Itô formula will play a key role in this paper
A New Product Integral Representation for Differential Equations in Separable Banach Spaces

AbstractLetXbe a separable Banach space. Consider the problem[formula]where the continuous functionf: [0,∞)×X→Xis locally Lipschitz continuous inx, uniformly inton bounded intervals, and continuous intuniformly w.r.t.x. The product integral formulax(T)=limn→∞∏i=0n[I+Tnf(iTn)]x0,0≤T<tmaxfor the solutionx(t) of (D) has been shown to converge. We also show that iff(t,.) is Lipschitz continuous onXwith constantL, then the mappingx0→x(T) is Lipschitz continuous with constanteLTfor anyT>0. This formula has been recently developed for differential inclusions inRnby Wolenski, but the infinite dimensional case is considerably more involved
On Some Degenerate Differential Operators on Weighted Function Spaces

AbstractWe deal with the degenerate differential operatorAu(x)≔α(x)u″(x) (x≥0) defined for everyu∈W20∩C2(]0,+∞[) satisfying limx→0+α(x)u″(x)=limx→+∞(α(x)/(1+x2))u″(x)=0. HereW20denotes the Banach space of all continuous functionsf:[0,+∞[→R such thatf(x)/(1+x2) vanishes at infinity, endowed with the weighted norm ‖f‖2≔supx≥0(|f(x)|/(1+x2)) (f∈W20). Moreover, we assume that the function α is continuous and positive on [0,+∞[, it is differentiable at 0, and satisfies the inequalities 0<α0≤α(x)/x≤α1(x>0) for suitable constants α0and α1. We show that the operatorAgenerates aC0-semigroup (T(t))t≥0of positive operators onW20. Moreover, we prove that everyT(t) can be represented as a limit of powers of suitable discrete-type positive linear operators that are constructed by means of the coefficient α
Asymptotic Behavior of the Eigenfrequency of a One-Dimensional Linear Thermoelastic System

AbstractIn this paper, we are concerned with the asymptotic behavior of the eigenvalues arising from a one-dimensional linear thermoelastic system with the Dirichlet–Dirichlet boundary condition. It is shown that the eigenfrequency asymptotically falls on two branches: one branch is along the negative horizontal axis in the complex plane and the other branch is asymptotic to the vertical line Reλ=−γ2/2k. These results lead to the exponential stability of the system and also provide a proof for the numerical simulation results by1
Nonexistence of Global Solutions of Some Quasilinear Hyperbolic Equations

AbstractIn this work, the nonexistence of the global solutions to initial boundary value problems with dissipative terms in the boundary conditions is considered for a class of quasilinear hyperbolic equations. The nonexistence proof is achieved by the usage of the so-called concavity method. In this method one writes down a functional which reflects the properties of dissipative boundary conditions and represents the norm of the solution in some sense. Then it is proved that this functional satisfies the hypotheses of the concavity lemma. Hence from the conclusion of the lemma one concludes that this functional and hence the norm of the solution blows up in a finite time
Non-reflecting Inflow and Outflow in a Wind Tunnel for Transonic Time-Accurate Simulation

AbstractWe consider a time-dependent flow problem in an infinitely long wind tunnel of circular cross-section. It is assumed that beginning with a distance up- and downstream from the stream-lined body, the flow is approximately described by the time-dependent Euler equations linearized about a uniform free-stream flow. By using this linear model, we transfer the boundary conditions from infinity to the front and back boundaries of a finite computational domain. The obtained boundary conditions called transparent ones (TBCs) are represented by explicit formulae in the physical space. The TBCs contain operators of Fourier expansion with respect to eigenfunctions of the cross-section and convolutions with respect to time, i.e., they are non-local in both space and time. One possibility of their numerical implementation is described; the main attention is paid to the evaluation of the convolution operator. Due to the representation of the convolution kernel by an exponential polynomial, the evaluation formulae become even local in time
Robust Stability and Performance Analysis for the System with Linear Dependent Perturbations

AbstractIn this paper, the sufficient conditions of the robust stability and robust performance for both continuous-time and discrete-time systems with structured perturbations are obtained using the matrix measure based approach. Using the derived results along with the Lyapunov based approach, several results for determining the robust criterion for the control system with time-varying perturbations are also obtained. Numerical examples have been given to illustrate the effectiveness of the proposed results. It shows that the proposed approach is less conservative than those proposed in the previous literature
On the Solvability of a Class of Second Kind Integral Equations on Unbounded Domains

AbstractWe consider integral equations of the form ψ(x)=φ(x)+∫Ωk(x,y)z(y)ψ(y)dy(in operator form ψ=φ+Kzψ), where Ω is some subset ofRn(n≥1). The functionsk,z, and φ are assumed known, withz∈L∞(Ω) and φ∈Y, the space of bounded continuous functions on Ω. The function ψ∈Yis to be determined. The class of domains Ω and kernelskconsidered includes the case Ω=Rnandk(x,y)=κ(x−y) with κ∈L1(Rn), in which case, ifzis the characteristic function of some setG, the integral equation is one of Wiener–Hopf type. The main theorems, proved using arguments derived from collectively compact operator theory, are conditions on a setW⊂L∞(Ω) which ensure that ifI−Kzis injective for allz∈WthenI−Kzis also surjective and, moreover, the inverse operators (I−Kz)−1onYare bounded uniformly inz. These general theorems are used to recover classical results on Wiener–Hopf integral operators of21and19, and generalisations of these results, and are applied to analyse the Lippmann–Schwinger integral equation
On the Analytic Continuation of the Minakshisundaram–Pleijel Zeta Function for Compact Symmetric Spaces of Rank One

AbstractWe give two equivalent analytic continuations of the Minakshisundaram–Pleijel zeta function ζU/K(z) for a Riemannian symmetric space of the compact type of rank oneU/K. First we prove that ζU/Kcan be written asζU/K(z)=eiπ(z−N/2)VU/KζG/K(z)+F(z),whereN=dimU/K,VU/Kis the volume ofU/K, ζG/K(z) is the local zeta function forG/K(the noncompact symmetric space dual toU/K), andF(z) is an analytic function which is given explicitly as a contour integral (cf. Eq. (4.11)). To prove the above formula we use a relation (first noticed by Vretare) between the scalar degeneracies of the Laplacian onU/Kand the Plancherel measure onG/K. The second expression we obtain for ζU/K(z) is in terms of a series of (generalized) Riemann zeta functions ζR(z,q) (cf. Eq. (5.9)). The doubly connected case of real projective spaces is also discussed