pinecone/core-2020-05-10-deduplication huggingface dataset:
The Nonclassical Group Analysis of the Heat Equation

AbstractThe nonclassical method of reduction was devised originally by Bluman and Cole in 1969, to find new exact solutions of the heat equation. Much success has been had by many authors using the method to find new exact solutions of nonlinear equations of mathematical and physical significance. However, the defining equations for the nonclassical reductions of the heat equation itself have remained unsolved, although particular solutions have been given. Recently, Arrigo, Goard, and Broadbridge showed that there are no nonclassical reduction solutions of constant coefficient linear equations that are not already classical Lie symmetry reduction solutions. Their arguments leave open the problem of what is the general nonclassical group action, and its effect on the relevant solution of the heat equation. In this article, both these problems are solved. In the final section we use the methods developed to solve the remaining outstanding case of nonclassical reductions of Burgers' equation
Stability of Inequalities in the Dual Brunn-Minkowski Theory

AbstractStability versions are given of several inequalities from E. Lutwak's dual Brunn-Minkowski theory. These include the dual Aleksandrov-Fenchel inequality, the dual Brunn-Minkowski inequality, and the dual isoperimetric inequality. Two methods are used. One involves the application of strong forms of Clarkson's inequality forLpnorms that hold for nonnegative functions, and the other utilizes a refinement of the arithmetic-geometric mean inequality. A new and more informative proof of the equivalence of the dual Brunn-Minkowski inequality and the dual Minkowski inequality is given. The main results are shown to be the best possible up to constant factors
Asymptotic Behavior in Linear Thermoelasticity

AbstractThis paper studies a linear thermoelastic material which exhibits a constitutive equation for heat flux with memory. An approximated theory of thermodynamics is developed for this model and amaximal pseudo free energyis explicitly constructed in the frequency domain. This thermodynamic potential is used to prove stability and domain of dependence results.Moreover, by the thermodynamic restrictions on the heat flux kernel, existence, uniqueness, and asymptotic behavior theorems are proved for the unidimensional model
Extremal Faces of the Range of a Vector Measure and a Theorem of Lyapunov

AbstractA theorem of Lyapunov states that the range R(μ) of a nonatomic vector measure μ is compact and convex. In this paper we give a condition to detect the dimension of the extremal faces of R(μ) in terms of the Radon–Nikodym derivative of μ with respect to its total variation |μ|: namely, R(μ) has an extremal face of dimension less than or equal tokif and only if the set (x1,…,xk+1) such thatf(x1),…,f(xk+1) are linear dependent has positive |μ|⊗(k+1)-measure. Decomposing the setXin a suitable way, we obtain R(μ) as a vector sum of sets which are strictly convex. This result allows us to study the problem of the description of the range of μ if μ has atoms, achieving an extension of Lyapunov's theorem
Criteria for Closedness of Spectral Measures and Completeness of Boolean Algebras of Projections

AbstractThe intimate connection between spectral measures and σ-complete Boolean algebras of projections in Banach spaces was intensively investigated by W. Bade in the 1950s, [“Linear Operators III: Spectral Operators,” Wiley-Interscience, New York, 1971; Chap. XVII]. It is well known that the most satisfactory situation occurs when the Boolean algebra is actually complete rather than just σ-complete. This makes it desirable to have available criteria (not just in Banach spaces but also in the nonnormable setting) which can be used to determine the completeness of Boolean algebras of projections. Such criteria, which should be effective and applicable in practice, as general as possible and apply to extensive classes of spaces, are presented in this article. A series of examples shows that these criteria are close to optimal
Reliable Solution for a 1D Quasilinear Elliptic Equation with Uncertain Coefficients

AbstractThe solution of a quasilinear elliptic state equation depends on the coefficient function belonging to an admissible set. The solution is evaluated by a cost functional the value of which is to be maximized over the admissible set, i.e., the reliable (safe) solution is searched for. Due to the nature of the equation, the Kirchhoff transformation can be applied to obtain both the existence of the true state solution and a cost sensitivity formula. In many cases, the latter makes it possible to determine the reliable solution immediately. The problem is approximated by means of the finite element method, and some convergence results are proven. Numerical examples illustrate the theory which can be directly generalized to spatial problems
Singularly Perturbed Boundary Value Problems in Case of Exchange of Stabilities

AbstractWe consider a mixed boundary value problem for a system of two second-order nonlinear differential equations where one equation is singularly perturbed. We assume that the associated equation has two intersecting families of equilibria. This property excludes the application of standard results. By means of the method of upper and lower solutions, we prove the existence of a solution of the boundary value problem and determine its asymptotic behavior with respect to the small parameter. The results can be used to study differential systems modeling bimolecular reactions with fast reaction rates
Nondegenerate Necessary Conditions for Nonconvex Optimal Control Problems with State Constraints

AbstractStandard versions of the maximum principle for optimal control problems with pathwise state inequality constraints are satisfied by a trivial set of multipliers in the case when the left endpoint is fixed and lies in the boundary of the state constraint set, and so give no useful information about optimal controls. Recent papers have addressed the problem of overcoming this degenerate feature of the necessary conditions. In these papers it is typically shown that, if a constraint qualification is imposed, requiring existence of inward pointing velocities, then sets of multipliers exist in addition to the trivial ones. A simple, new approach for deriving nondegenerate necessary conditions is presented, which permits relaxation of hypotheses previously imposed, concerning data regularity and convexity of the velocity set
A Splitting Theorem for Transitive Maps

AbstractIn this paper we prove a splitting theorem for continuous transitive maps on locally connected compact metric spaces which generalizes the results of Barge and Martin [4] for transitive maps on the interval and of Blokh [7] for transitive graph maps. As a consequence, lower bounds for the topological entropy of transitive graph maps in terms of the cardinality of their splittings are obtained. Also we show that for every connected graph which is not a tree, the infimum of the topological entropy of the transitive maps having a periodic point is zero. Since the topological entropy of transitive maps without periodic points is zero [7] and positive lower bounds of the entropy were given for the transitive maps on trees [2], this paper completes the problem of obtaining lower bounds of the topological entropy for transitive maps of any connected graph
Operator-Valued Typically Real Functions Induced by a Contraction

AbstractLet H be a complex Hilbert space and let L(H) denote the algebra of all bounded linear operators on H. LetF(z)=∑∞n=0znAnbe an operator-valued analytic function whose coefficients are bounded operators in L(H) forzin the open unit disc D and the series is convergent in the strong operator topology. In this paper, we discuss operator-valued typically real functionsF(z)=∑∞n=0znAnwhich generalize complex-valued typically real functions. We characterize operator-valued typically real functions and study such functionsF(z) induced by a contraction on H. In addition, we considerm×n-tuple operator-valued typically real functions and positively real functions. Finally, we charaterize operator-valued typically real functions in the finite-dimensional cases
Generalized Convexity in Multiobjective Programming

AbstractFor the scalar programming problem, some characterizations for optimal solutions are known. In these characterizations convexity properties play a very important role. In this work, we study characterizations for multiobjective programming problem solutions when functions belonging to the problem are differentiable. These characterizations need some conditions of convexity. In differentiable scalar programming problems the concept of invexity is very important. We prove that it is also necessary for the multiobjective programming problem and give some characterizations of multiobjective programming problem solutions under weaker conditions. We define analogous concepts to those of stationary points and to the conditions of Kuhn–Tucker and Fritz–John for the multiobjective programming problem
Optimality Conditions for Vector Mathematical Programming via a Theorem of the Alternative

AbstractThis paper is concerned with the optimality conditions for nonsmooth and nonconvex vector mathematical programming but with the existence of the Hadamard directional derivatives of objective and constraint functions. By using a generalization of the Gordan alternative theorem proved by Jeyakumar [8], we establish the optimality conditions in terms of Lagrange–Kuhn–Tucker multipliers. Also the sufficient optimality conditions are given whenever the data are convex. In the last, we give an application to vector fractional mathematical programming
Dilation Equations with Exponential Decay Coefficients

AbstractDilation equations with finitely many nonzero coefficients have been discussed by many people, since those equations are related to compactly supported wavelets, which constitute an important family of wavelets. However, some properties required in applications are not compatible with the compactness of the support; e.g., it is impossible to find an orthonormal and compactly supported scaling function with the sampling property except for a trivial case. Based on this fact, Xia and Zhang constructed a cardinal and orthonormal scaling function with exponential decay coefficients. It seems such a class of wavelets is interesting. We shall investigate that in this paper and in particular study the relationship between the exponential decay property of a scaling function and that of the corresponding coefficients
On the Lowest Eigenvalue of General Operators for the Intersection of Two Domains

AbstractE. Lieb (Invent. Math.74(1983), 441–448) has proved that ifAandBare two bounded domains in RN, then there exists a translation τysuch that λ1(A∩τyB)<λ1(A)+λ1(B), where λ1is the first eigenvalue of the laplacian with Dirichlet boundary conditions. Here we extend this result to elliptic operators in divergence fromL≔−divQ(x)∇+c(x) with mixed Dirichlet–Neumann boundary conditions onA. If μL(A) is the corresponding eigenvalue, we show that there exists a translation τysuch that μL(A∩τyB)<μL(A)+βλ1(B), iftξQ(x)ξ≤β|ξ|2for allx∈A, ξ∈RN. One can further improve the estimate for non-isotropic operators (where β can be large) by taking into account rotations ofB. In that case, a similar inequality holds iftξQ̄(x)ξ≤β|ξ|2, whereQ̄is a “mean value ofQin different directions.
Extendible Polynomials on Banach Spaces

AbstractWe are concerned with the following question: when can a polynomialP:E→X(EandXare Banach spaces) be extended to a Banach space containingE? We prove that the polynomials that are extendible to any larger space are precisely those which can be extended toC(BE′), ifXis complemented in its bidual, andl∞(BE′) in general. We also show that the extendibility is a property that is preserved by Aron–Berner extensions and composition with linear operators. We construct a predual of the space of extendible polynomials for the case thatXis a dual space
First- and Second-Order Epi-Differentiability in Eigenvalue Optimization

AbstractThe subject of this work is the first- and second-order sensitivity analysis of some spectral functions which are essential in eigenvalue optimization by the way of epi-differentiability. We show that the sum of the m largest eigenvalues of a real symmetric matrix is twice epi-differentiable and we derive an explicit expression of its second-order epi-derivative. We also prove that the mth largest eigenvalue function is twice epi-differentiable if and only if it ranks first in a group of equal eigenvalues. Finally, we derive chain rules and then we obtain optimality conditions for an important class of eigenvalue optimization problems
All Solutions of Standard Symmetric Linear Partial Differential Equations Have Classical Lie Symmetry

AbstractIt is proven that every solution of any linear partial differential equation with an independent-variable-deforming classical Lie point symmetry is invariant under someclassicalLie point symmetry. This is true for any number of independent variables and for equations of any order higher than one. Although this result makes use of the infinite-dimensional component of the Lie symmetry algebra due to linear superposition, it is shown that new similarity solutions, previously thought not to be classical, can be recovered prospectively by allowing symmetries to include superposition of similarity solutions already known from the finite part of the symmetry algebra. This result applies to all constant-coefficient equations and to many variable-coefficient equations such as Fokker–Planck equations
On Diffusive Population Models with Toxicants and Time Delays

AbstractIn this article we study the global stability in reaction-diffusion models for single-species population growth under environmental toxicants with or without time delays. The existence and uniqueness of a positive steady-state solution are established in those models. It is shown that as long as the magnitude of the instantaneous self-limitation and toxicant effects is larger than that of the time-delay effects in the model with delays, the solution of both reaction-diffusion systems has the same asymptotic behavior (extinction or converging to the positive steady-state solution, depending on the growth rate of the species). Numerical simulations for both cases (with or without time delays) are demonstrated for the purpose of comparison
Polynomial Interpolation and Marcinkiewicz-Zygmund Inequalities on the Unit Circle

AbstractThe objective of this paper is to derive an intimate relationship among three important mathematical tools, namely: polynomial interpolation, Marcinkiewicz-Zygmund inequalities, andAp-weights. In particular, it is shown that minimum separation of sample points on the unit circle together with certain uniformAp-weights generated by these sample points constitute a necessary and sufficient condition for the validity of the Marcinkiewicz-Zygmund inequality evaluated at these points, which in turn, is equivalent to the Jackson-type estimate, using the Popov-Andreev module of continuity, of polynomial interpolation, again at these sample points
Generalized Symmetric Duality for Multiobjective Variational Problems with Invexity

AbstractWe formulate a pair of multiobjective nonlinear generalized symmetric dual variational problems involving vector-valued functions which unify the Wolfe and Mond–Weir models. For a special case, our problems become the symmetric dual pair of D. S. Kim and W. J. Lee (1998,J. Math. Anal. Appl.218, 34–48). Under invexity assumptions, we establish weak, strong, and converse duality theorems for our variational problems by using the concept of efficiency. Also, we give the static case of our multiobjective generalized symmetric duality results. Several known results are obtained as special cases
Inverse/Observability Estimates for Second-Order Hyperbolic Equations with Variable Coefficients

AbstractWe consider a general second-order hyperbolic equation defined on an open bounded domain Ω⊂Rn with variable coefficients in both the elliptic principal part and in the first-order terms as well. At first, no boundary conditions (B.C.) are imposed. Our main result (Theorem 3.5) is a reconstruction, or inverse, estimate for solutions w: under checkable conditions on the coefficients of the principal part, the H1(Ω)×L2(Ω)-energy at time t=T, or at time t=0, is dominated by the L2(Σ)-norms of the boundary traces ∂w/∂νA and wt, modulo an interior lower-order term. Once homogeneous B.C. are imposed, our results yield—under a uniqueness theorem, needed to absorb the lower-order term—continuous observability estimates for both the Dirichlet and Neumann case, with an explicit, sharp observability time; hence, by duality, exact controllability results. Moreover, no artificial geometrical conditions are imposed on the controlled part of the boundary in the Neumann case. In contrast with existing literature, the first step of our method employs a Riemann geometry approach to reduce the original variable coefficient principal part problem in Ω⊂Rn to a problem on an appropriate Riemann manifold (determined by the coefficients of the principal part), where the principal part is the Laplacian. In our second step, we employ explicit Carleman estimates at the differential level to take care of the variable first-order (energy level) terms. In our third step, we employ micro-local analysis yielding a sharp trace estimate, to remove artificial geometrical conditions on the controlled part of the boundary, in the Neumann case
Asymptotic Behavior of an Age-Structured Population Model and Optimal Maturation Age

AbstractA continuous age-structured population model with two population groups, juveniles and adults, and with a dynamics for the resource, is considered. The only density dependence is through a uniform increase of the death rates. A complete description of the asymptotic behavior of the population dynamics system is obtained showing under natural hypotheses the existence of a nontrivial (“ecological”) equilibrium which attracts any initial nonzero population with nonzero initial amount of resources. Assuming that the offspring production rate is an increasing function of the maturation age, the evolutionarily stable value of this variable is briefly discussed
Weakly Divisible MV-Algebras and Product

AbstractWe introduce weakly divisible MV-algebras and we show that every weakly divisible σ-complete MV-algebra is isomorphic to the system of all continuous fuzzy functions defined on some compact Hausdorff space which generalizes a result of Di Nola and Sesse (in“Non Classical Logics and Their Application to Fuzzy Subsets,” Kluwer Academic, Dordrecht, to appear). This enables us to define an associative and commutative product on weakly divisible σ-complete MV-algebras with a multiplicative unity 1. As an application, we show that any system of observables on this MV-algebra admits a joint observable with respect to this product
Energy Decay of Solutions to the Boussinesq, Primitive, and Planetary Geostrophic Equations

AbstractIn this paper we study the long time behavior of the energy of solutions to the Boussinesq, planetary geostrophic, and primitive equations. The equations are considered in the whole space R3. The asymptotic behavior will depend on the type of data and how many damping constants are nonzero in the equations. In several cases we are able to establish an algebraic rate of decay of the same order as the solutions of the underlying linear equations. In the case with less damping our results establish that either the energy of the solutions decays with no rate to an equilibrium or it will be oscillating
Stochastic Comparability and Dual Q-Functions

AbstractIn this paper we discuss the relationship and characterization of stochastic comparability, duality, and Feller–Reuter–Riley transition functions which are closely linked with each other for continuous time Markov chains. A necessary and sufficient condition for two Feller minimal transition functions to be stochastically comparable is given in terms of their density q-matrices only. Moreover, a necessary and sufficient condition under which a transition function is a dual for some stochastically monotone q-function is given in terms of, again, its density q-matrix. Finally, for a class of q-matrices, the necessary and sufficient condition for a transition function to be a Feller–Reuter–Riley transition function is also given
On Infeasibility of Systems of Convex Analytic Inequalities

AbstractIn this paper we address the problem of the infeasibility of systems defined by convex analytic inequality constraints. In particular, we investigate properties of irreducible infeasible sets and provide an algorithm that identifies a set of all constraints (K) that may affect the feasibility status of the system after some perturbation of the right-hand sides. We analyze properties of the irreducible sets, as well as infeasibility sets in connection with the setK, showing in particular that every infeasible system contains an inconsistent subsystem of cardinality not greater than the number of variables plus one.The results presented in this paper are generalizations of a theory developed for the systems of quadratic and linear inequality constraints
General Transmission Problems in the Theory of Elastic Oscillations of Anisotropic Bodies (Mixed Interface Problems)

AbstractHere we extend our previous paper [L. Jentsch, D. Natroshvili, and W. Wendland, J. Math. Anal. Appl.220 (1998), 397–433], where the mixed interface problems were formulated. With the aid of the explicit solution for the rigid contact problem, here the mixed interface problems are reduced to pseudodifferential equations on manifolds with boundary. We obtain uniqueness and existence in Bessel-potential and Besov spaces. Embedding theorems yield Hölder continuity with any exponent α∈(0,1/2)
Symmetries of First Integrals and Their Associated Differential Equations

AbstractThe relationship between the reduction of order through point symmetries and integration is explored with particular emphasis on the loss and gain of point (contact for third order) symmetries to and from nonlocal symmetries. It is seen that reduction of order can even lead to the loss of all point symmetries at the third order level and their replacement at the second order level from nonlocal symmetries. It is evident that nonlocal symmetries should be given more attention in applications
Lp(μ)-Estimation of Tangent Maps

AbstractWe analyze the conditions under which the best Lp(μ)-linear fittings of the action of a mapping f on small balls give reliable estimates of the tangent map Df. We show that there is an inverse relationship between the conditions on the regularity, in terms of local densities, of the measure μ and the smoothness of the mapping f which are required to ensure the goodness of the estimates. The above results can be applied to the estimation of tangent maps in two empirical settings: from finite samples of a given probability distribution on Rn and from finite orbits of smooth dynamical systems. As an application of the results of this paper we obtain sufficient conditions on the measure μ to ensure the convergence of the Eckmann–Ruelle algorithm for computing the Liapunov exponents of smooth dynamical systems
Upper and Lower Solutions and Topological Degree

AbstractThis paper deals with second order nonlinear boundary value problems. We suppose the existence of upper and lower solutions of the problems which are well ordered, i.e., the lower solution is less than the upper one, and we also consider the case of upper and lower solutions having the opposite ordering. We prove the relation between the topological degree and strict upper and lower solutions in both cases and using this we get the existence and multiplicity results for the boundary value problems under consideration