pinecone/core-2020-05-10-deduplication huggingface dataset:
Lower CS-Closed Sets and Functions

AbstractIn this work, we introduce a class of convex sets, LCSCF(X), of a locally convex separated and not necessarily separable topological vector space X. They are called the lower CS-closed sets. This class contains the CS-closed sets, satisfies the property core(C)=int(C), ∀C∈LCSCF(X) when X is metrizable barrelled, and is stable under many operations. Among them, the projection and the denumerable intersection. We characterize the lower CS-closed functions (i.e., the functions who have a lower CS-closed epigraph) as marginal functions of CS-closed ones and show that they are very stable too. We establish an open mapping and a closed graph theorem for the lower CS-closed relations. Finally, we show that every real extended valued lower CS-closed function defined on a metrizable barrelled space is continuous on the interior of its domain. This result allows us to extend classical theorems of convex duality by replacing lower semicontinuous functions by lower CS-closed ones. More than that, it systematizes and extends some methods of convex analysis
General Weighted Opial Inequalities for Linear Differential Operators

AbstractA complete set of Lr (r≠0) form very general Opial type weighted inequalities is given for a general linear differential operator L. These involve its related initial value problem solution y, Ly, the associated Green's function H, and initial conditions point x0∈R. This work is inspired by work of R. P. Agarwal and P. Y. H. Pang (1995, “Opial Inequalities with Applications in Differential and Difference Equations,” Kluwer Academic, Dordrecht) and G. A. Anastassiou (1998, Math. Inequalities Appl.1, No. 2, 193–200) and generalizes their related results. An application to proving uniqueness in solutions of initial value problems is given at the end
The Bartle Bilinear Integration and Carleman Operators

AbstractSome characterizations of integrable functions in the bilinear sense of Bartle with respect to the injective tensor product are obtained. As a consequence it is shown that the kernels of Carleman compact operators coincide with these Bartle integrable functions. This result is applied to prove that every Carleman L-weak-compact operator is compact. An example showing the different behavior of the integrability with respect to the projective tensor product is given. A general Fubini theorem in this setting is shown
Hermite Polynomials in Asymptotic Representations of Generalized Bernoulli, Euler, Bessel, and Buchholz Polynomials

AbstractThis is the second paper on finite exact representations of certain polynomials in terms of Hermite polynomials. The representations have asymptotic properties and include new limits of the polynomials, again in terms of Hermite polynomials. This time we consider the generalized Bernoulli, Euler, Bessel, and Buchholz polynomials. The asymptotic approximations of these polynomials are valid for large values of a certain parameter. The representations and limits include information on the zero distribution of the polynomials. Graphs are given that indicate the accuracy of the first term approximations
Critical Blowup for Systems of Semilinear Wave Equations in Low Space Dimensions

AbstractWe consider the Cauchy problem for systems of semilinear wave equations in two and three space dimensions with small initial data. Del Santo et al. [“Geometric Optics and Related Topics” (F. Colombini and N. Lerner, Eds.), Progress in Nonlinear Differential Equations and Their Applications, Vol. 32, pp. 117–140, Birkhauser, Boston, 1997] have studied the existence and nonexistence of global classical solutions of the Cauchy problem except for the critical case. In this paper we study the critical case, and we show the nonexistence of global classical solutions and also give the upper bounds of the life span
On a Painlevé II Model in Steady Electrolysis: Application of a Bäcklund Transformation

AbstractA model equation of Painlevé II type was introduced by Bass in 1964 in connection with a boundary value problem which describes the electric field distribution in a region x>0 occupied by an electrolyte. This is possibly the earliest explicit physical application of a Painlevé equation to be found in the literature. Here we return to this problem informed by the subsequent discovery of a Bäcklund transformation for Painlevé II. This enables us to construct exact representations for the electric field and ion distributions for boundary value problems wherein the ratio of fluxes of the positive and negative ions at the boundary adopts one of an infinite sequence of values
Global Secondary Bifurcation in a Non-linear Boundary Value Problem

AbstractWe consider a steady-state non-linear boundary value problem which arises in modelling the formation of vascular networks in response to tumour growth. Global bifurcation from both trivial and non-trivial solution branches is considered, with emphasis on the latter. By investigating such secondary bifurcation, it is shown that positive, bounded solutions exist for all physically relevant values of a critical parameter. A certain class of these solutions is discussed with respect to the application to tumour growth
Discontinuity Waves, Shock Formation and Critical Temperature in Crystals

AbstractThe propagation of discontinuity waves in a rigid heat conductor at low temperatures is studied by using a generalized non-linear Maxwell–Cattaneo equation developed in the framework of extended thermodynamics. The critical time (i.e., the instant in which a shock wave formation occurs) is evaluated in both cases of infinite and finite heat conductivity. The critical temperature θ̃, pointed out in our previous papers concerning the propagation of shock and simple waves, once more plays an important role: in fact, now it determines two different regimes for the wave propagation and this phenomenon, from a mathematical point of view, is related to the loss of the genuine non-linearity when θ=θ̃. In the last sections some numerical results are given and a brief analysis about the evolution of a possible initial wave profile is performed
The Existence of Positive Solutions to Neutral Differential Equations

AbstractIn this paper, we shall consider a class of neutral differential equations of the form
				
					
						x
						
							t
							
						−
						r(t)
							r(t−τ)
							x
						
							t−τ
							
						
						
					′+Q
					
						t
						
					x
					
						t−σ
						
					=0,t≥t
					0
					,
					
						∗
						
					
				where τ∈(0,∞), σ∈[0,∞), Q(t)∈C([t0,∞),R+), r(t)∈C([t0,∞),(0,∞)) with r(t) nondecreasing on [t0−τ,∞). We shall show that all positive solutions of (∗) can be classified into four types, A, B, C, and D, and we shall obtain sufficient and necessary conditions for the existence of A-type, B-type, and D-type positive solutions of (∗), respectively. A sufficient condition for the existence of C-type positive solutions of (∗) is also given. Finally, we shall offer a sharp oscillation result for all solutions of (∗). Our results generalize and improve those established in B. Yang and B. G. Zhang (Funkcial. Ekvac.39 (1996), 347–362)
Barrelledness of Spaces with Toeplitz Decompositions

AbstractA Toeplitz decomposition of a locally convex space E into subspaces (Ek) with projections (Pk) is a decomposition of every x∈E as x=∑kPkx, where ordinary summability has been replaced by summability with respect to an infinite and lower triangular regular matrix. We extend to the setting of Toeplitz decompositions a couple of results about barrelledness of Schauder decompositions. The first result, given for Schauder decompositions by Noll and Stadler, links the barrelledness of a normed space E to the barrelledness of the pieces Ek via the fact that E′ is big enough so as to coincide with its summability dual. Our second theorem, given for Schauder decompositions by Dı́az and Miñarro, links the quasibarrelledness of an ℵ0-quasibarrelled (in particular, (DF)) space E to the quasibarrelledness of the pieces Ek via the fact that the decomposition is simple
The Stability of Subharmonic Solutions for Hamiltonian Systems

AbstractIn this paper, by using the dual Morse index theory, we study the stability of subharmonic solutions of the non-autonomous Hamiltonian systems. We obtain a (infinite) sequence of geometrically distinct periodic solutions such that every element has at most one direction of instability (i.e., it has at least 2n−2 Floquet multipliers lying on the unit circle in the complex plane if the periodic solution is non-degenerate) or it is elliptic (all its 2n Floquet multipliers are lying on the unit circle) if the periodic solution is degenerate
Positive Homoclinic Solutions for a Class of Second Order Differential Equations

AbstractWe study the existence of positive homoclinic solutions of the second order equationu″−αxu+βxu2+γxu3=0,x∈R, (I)where the coefficient functions α(x), β(x), and γ(x) are continuous and satisfy0<a≤αx,0≤b≤βx≤B,0<c≤γx≤C.Assuming that the coefficient functions are 2π-periodic, we prove the existence of a nontrivial positive homoclinic solution of Eq. (I) wheneverB2−b2<4ac.This homoclinic is derived as the limit of positive solutions of some approximating problems that are obtained by using the mountain pass theorem. Using the same method we also prove under adequate assumptions the existence of positive symmetric homoclinic solutions
A Necessary and Sufficient Condition for Existence of Large Solutions to Semilinear Elliptic Equations

AbstractWe consider the semilinear equation Δu=p(x)f(u) on a domain Ω⊆Rn, n≥3, where f is a nonnegative, nondecreasing continuous function which vanishes at the origin, and p is a nonnegative continuous function with the property that any zero of p is contained in a bounded domain in Ω such that p is positive on its boundary. For Ω bounded, we show that a nonnegative solution u satisfying u(x)→∞ as x→∂Ω exists if and only if the function ψ(s)≡∫s0f(t)dt satisfies ∫∞1(ψ(s))−1/2ds<∞. For Ω unbounded (including Ω=Rn), we show that a similar result holds where u(x)→∞ as |x|→∞ within Ω and u(x)→∞ as x→∂Ω if p(x) decays to zero rapidly as |x|→∞
Solvability of Semilinear Operator Equations with Growing Nonlinearity

AbstractLet U and V be Hilbert spaces, L: U→V be a bounded linear operator, and N: U→V be a nonlinear operator. Consider the semilinear operator equation Lu+Nu=v and the corresponding linear equation Lu=v where v∈V. In this paper we prove appropriate solvability of the semilinear system under the following assumptions: (i) the linear equation is approximately solvable; (ii) N is compact; (iii) ‖Nu‖≤A+B‖u‖ for A>0, B>0, and B is such that |B|<1c where c is a constant described in the text; (iv) range of N is contained in range of L
Self-Affine Fractal Functions and Wavelet Series

AbstractWe consider functions represented by series ∑g∈Gcgψ(g−1(x)) of wavelet-type, where G is a group generated by affine functions L1,…,Ln and ψ is piecewise affine. By means of those functions we characterize the class of self-affine fractal functions, previously studied by Barnsley et al. We compute their global and local Hölder exponents and investigate points of non-differentiability. Wavelet-representations for various continuous nowhere differentiable and singular functions are presented. Another application is the construction of functions with prescribed local Hölder exponents at each point
Stability of Non-Homentropic, Inviscid, Compressible Shear Flows

AbstractFor non-homentropic, inviscid, compressible shear flows, the equivalent of Squire's theorem is proved. It is shown that a shear free basic flow does not support subsonic modes. Further, it is shown that the instability region for subsonic disturbances is a semi-ellipse type region, which depends on the Mach number, wave number, and depth of the fluid layer. Under an approximation, two estimates for the growth rate of an unstable subsonic mode are obtained. For unbounded flows, a sufficient condition for stability to supersonic disturbances and an estimate for the growth rate of an unstable supersonic disturbance are given
Non-linear Elliptical Equations on the Sierpiński Gasket

AbstractThis paper investigates properties of certain nonlinear PDEs on fractal sets. With an appropriately defined Laplacian, we obtain a number of results on the existence of non-trivial solutions of the semilinear elliptic equation
				Δu+a
					
						x
						
					u=f
					
						x,u
						
					,
				with zero Dirichlet boundary conditions, where u is defined on the Sierpiński gasket. We use the mountain pass theorem and the saddle point theorem to study such equations for different classes of a and f. A strong Sobolev-type inequality leads to properties that contrast with those for classical domains
Numerical Methods for Systems of Nonlinear Parabolic Equations with Time Delays

AbstractThe purpose of this paper is to investigate some numerical aspects of a class of coupled nonlinear parabolic systems with time delays. The system of parabolic equations is discretized by the finite difference method which yields a coupled system of nonlinear algebraic equations. The mathematical analysis of the nonlinear system is by the method of upper and lower solutions and its associated monotone iterations. Three monotone iterative schemes are presented and it is shown that the sequence of iterations from each one of these iterative schemes converges monotonically to a unique solution of the finite difference system. A theoretical comparison result for the various monotone sequences and error estimates for the three monotone iterative schemes are obtained. It is also shown that the finite difference solution converges to the classical solution of the parabolic system as the mesh size decreases to zero
Pontryagin Duality for Spaces of Continuous Functions

AbstractA topological abelian group G is P-reflexive if the natural homomorphism of G to its Pontryagin bidual group is a topological isomorphism. Let Cp(X) be the space of continuous functions with the topology of pointwise convergence. We investigate for what spaces X the group Cp(X) is P-reflexive. We show that: (1) if Cp(X) is P-reflexive, then X is a P-space; (2) there exists a non-discrete space X such that Cp(X) is P-reflexive; (3) there exists a P-space X such that Cp(X) is not P-reflexive; (4) there exists a simple space X for which the question of whether Cp(X) is P-reflexive is undecidable in ZFC
Resonant Bifurcations

AbstractWe consider dynamical systems depending on one or more real parameters, and assuming that, for some “critical” value of the parameters, the eigenvalues of the linear part are resonant, we discuss the existence—under suitable hypotheses—of a general class of bifurcating solutions in correspondence with this resonance. These bifurcating solutions include, as particular cases, the usual stationary and Hopf bifurcations. The main idea is to transform the given dynamical system into normal form (in the sense of Poincaré and Dulac) and to impose that the normalizing transformation is convergent, using the convergence conditions in the form given by A. Bruno. Some specifically interesting situations, including the cases of multiple-periodic solutions and of degenerate eigenvalues in the presence of symmetry, are also discussed in some detail
C∞ Solvability of the ∂&#x0304; System on Wedges of CN

AbstractWe prove existence of C∞ solutions of the ∂&#x0304; system on open wedges of CN under a very general condition of weak q-pseudoconvexity. The method relies on the L2-estimates by Hörmander and in our generalization previously introduced in (G. Zampieri, 1998, J. Anal. Math.74, 99–111). Let us point out that in spite of the only slight improvements in the estimates, much stronger existence theorems are obtained. The ∂&#x0304; problem for wedges of CN is extensively treated in the literature (cf. G. M. Henkin and J. Leiterer, 1988, Andreotti–Grauert Theory by Integral Formulas, Progress in Math., Vol. 74, Birkhäuser and the references cited therein). However the usual approach employs integral representations rather than L2 estimates
Distributional Chaos on Compact Metric Spaces via Specification Properties

AbstractIn this paper we show that a continuous function on a compact metric space exhibits distributional chaos as introduced in [B. Schweizer and J. Smítal, Trans. Amer. Math. Soc.344 (1994), 737–754] and elucidated in [B. Schweizer, A. Sklar, and J. Smital, to appear] if the function has either a weaker form of the specification property (see [M. Denker, C. Grillenberger, and K. Sigmund, Springer Lecture Notes in Mathematics, Vol. 527, Springer-Verlag, New York/Heidelberg/Berlin, 1976]) or the generalized specification property introduced in [F. Balibrea, B. Schweizer, A. Sklar, and J. Smítal, to appear]. In particular, any Anosov diffeomorphism is distributionally chaotic, regardless of the fact that in this case the trajectories of a.e. pair of points exhibit regular, non-chaotic behavior
Some New Results of Nonautonomous Lotka–Volterra Competitive Systems with Delays

AbstractIn this paper, we improve and generalize the results of Montes de Oca and Zeeman [J. Math. Anal. Appl.192 (1995), 360–370; Proc. Amer. Math. Soc.124 (1996), 3677–3687] to general N-species nonautonomous Lotka–Volterra type competitive systems with finite or infinite delays. We establish a series of criteria under which a part of the n species of the systems is driven to extinction while the remaining part of the n species is persistent or uniformly persistent, or coexists globally asymptotically stable
Biweights on Partial *-Algebras

AbstractRepresentations of *-algebras are generally obtained by the familiar Gelfand–Naimark–Segal (GNS) construction, starting from a positive linear functional or even a (quasi-) weight, that is, a functional that may take infinite values. In previous works, we have extended this construction to the case of a partial *-algebra, using invariant positive sesquilinear forms, which play the same role as positive linear functionals for *-algebras. In this paper we extend the construction further, by introducing and studying systematically the notion of biweight on a partial *-algebra. In particular, we characterize, through the associated GNS representation, the so-called approximately admissible biweights (i.e., limits of biweights with a bounded GNS representation)
A Note on the Synthesis of Nonquadratic Optimal Control in a One-Dimensional Linear System

AbstractThe synthesis problem of the optimal control is investigated in a one-dimensional linear system with a cost function which is Hölderian with respect to the control and convex with respect to the final state. Under some conditions it is proved that the closed-loop synthesis is given by a nonlinear state feedback defined in terms of the solution of a Bernoulli type operator equation. It is also shown that the feedback operator can be explicitly expressed by solutions of associated algebraic equations
Generalized Stability of Isometries

AbstractLet X and Y be real Banach spaces and let ε,p≥0. A mapping f: X→Y is called an (ε,p)-isometry if |‖f(x)−f(y)‖−‖x−y‖|≤ε‖x−y‖p holds for all x,y∈X. A pair (X,Y) is p-stable with respect to isometries if there exists a function δ: [0,∞)→[0,∞) with limε→0δ(ε)=0 such that for every surjective (ε,p)-isometry f: X→Y there is a surjective isometry U: X→Y satisfying the estimate ‖f(x)−U(x)‖≤δ(ε)‖x‖p, x∈X. We show that every pair of Banach spaces (X,Y) is p-stable for 0≤p<1. The pair (R2,R2) is not 1-stable. When p>1 a superstability phenomenon occurs for finite-dimensional Banach spaces
The Asymptotic Behavior of a Transport Equation in Cell Population Dynamics with a Null Maturation Velocity

AbstractIn this paper we complete the study of Rotenberg's model (M. Rotenberg, 1983, J. Theoret. Biol.103, 181–199) describing the growth of a cell population and treated partially by M. Boulanouar and H. Emamirad (Differential Integral Equations, 13 (2000), 125–144). In contrast to our previously cited treatment, here we impose the condition that the maturation velocity for any cell can become null. This consideration implies that the cell population never completely leaves its initial distribution, because at every time we can find some cells of initial cell population that are not divided. In this case, the generated semigroup is not compact. To surmount this difficulty, after studying the irreducibility of the generated semigroup, we calculate explicitly its essential type and we show the asymptotic convergence of the generated semigroup to a projection of rank 1
Frictional Wear of a Thermoelastic Beam

AbstractWe present two versions of a mathematical model for the evolution of wear in a thermoelastic beam resulting from frictional contact with a rigid moving surface. One version is quasistatic and the other dynamic. We show that the quasistatic problem allows for the decoupling of the mechanical and thermal aspects of the process. The problem reduces to that of the heat equation with nonlinear and history-dependent boundary conditions. Then the displacements, shear stresses, and wear can be obtained by quadrature. We establish the existence of a local weak solution for the problem and a partial uniqueness result and obtain conditions for the solution's further regularity. The dynamic problem consists of the heat equation coupled with the equation of motion and the Archard wear equation. We prove the existence and uniqueness of the local weak solution of this problem too
Simple Food Chain in a Chemostat with Distinct Removal Rates

AbstractIn this paper, we consider a model describing predator–prey interactions in a chemostat that incorporates both general response functions and distinct removal rates. In this case, the conservation law fails. To overcome this difficulty, we make use of a novel way of constructing a Lyapunov function in the study of the global stability of a predator-free steady state. Local and global stability of other steady states, persistence analysis, as well as numerical simulations are also presented. Our findings are largely in line with those of an identical removal rate case
A Least-Squares Method for Optimal Control Problems for a Second-Order Elliptic System in Two Dimensions

AbstractA distributed optimal control problem for an elliptic problem is considered. A penalty/least-squares method for optimal control problems is used wherein the constraint equations are enforced via penalization. First, the existence of the optimal control problem is proved. The convergence, as the penalty parameter tends to zero, of the solution to the penalized optimal control problem to that unpenalized one is demonstrated as is the convergence of a gradient method for determining solutions of the penalized optimal control problem. Finally, finite element approximations of the penalized optimal control problem are studied and optimal error estimates are obtained
The Role of Critical Exponents in Blow-Up Theorems: The Sequel

AbstractIn [27] Fujita showed that for positive solutions, the initial value problem (in RN) for ut=Δu+up with p>1 exhibited the following behavior: If p<pc≡1+2/N, then the initial value problem does not have any nontrivial, non-negative solution existing on RN×[0,∞) (a global solution), whereas if p>pc, there exist global, small data, positive solutions as well as solutions which are non-global. We call such a result a blow-up theorem of Fujita type. In [50], Levine discussed the various theorems of this type that had appeared in the literature prior to 1990. In this paper we revisit the literature since 1990
Global and Blow-Up Solutions for Nonlinear Degenerate Parabolic Systems with Crosswise-Diffusion

AbstractIn this paper the nonlinear degenerate parabolic system ut=vα1(Uxx+au), vt=uα2(vxx+bv) with Dirichlet boundary condition is studied. The regularization method and upper–lower solutions technique are employed to show the local existence of a solution for the nonlinear degenerate parabolic system. The global existence of a solution is discussed. The finite time blow-up result together with an estimate of the blow-up time are found. The blow-up set with positive measure is analyzed in some detail