pinecone/core-2020-05-10-deduplication huggingface dataset:
Weighted Inequalities on John Domains

AbstractWe study two-weighted inequalities on John domains. We first introduce domains that generalize the C(λ,M) domains defined by HajLasz and Koskela (J. London Math. Soc., to appear) and then show that our domains are actually just John domains. We then extend an interesting and nice result of HajLasz and Koskela by modifying their proof and then obtain some interesting consequences that include weighted Sobolev interpolation inequalities and an almost necessary and sufficient condition for two-weighted Poincaré inequality on cubes
Boundary Effect on a Stationary Viscous Shock Wave for Scalar Viscous Conservation Laws

AbstractThe initial-boundary value problem on the negative half-line R−[formula]is considered, subsequently to T.-P. Liu and K. Nishihara (1997, J. Differential Equations133, 296–320). Here, the flux f is a smooth function satisfying f(u±)=0 and the Oleinik shock condition f(φ)<0 for u+<φ<u− if u+<u− or f(φ)>0 for u+>φ>u− if u+<u−. In this situation the corresponding Cauchy problem on the whole line R=(−∞,∞) to (∗) has a stationary viscous shock wave φ(x+x0) for any fixed x0. Our aim in this paper is to show that the solution u(x,t) to (∗) behaves as φ(x+d(t)) with d(t)=O(lnt) as t→∞ under the suitable smallness conditions. When f=u2/2, the fact was shown by T.-P. Liu and S.-H. Yu (1997, Arch. Rational Mech. Anal.139, 57–82), based on the Hopf–Cole transformation. Our proof is based on the weighted energy method
Flow Invariance Preserving Feedback Controllers for the Navier–Stokes Equation

AbstractIn this paper we develop a concrete procedure for designing feedback controllers to ensure that the resultant dynamics of turbulence will preserve certain prescribed physical constraints. Examples of such constraints include, in particular, the level sets of well known invariants of the inviscid flow such as helicity. We also bring to light a certain m-accretivity property of suitable quantization of the nonlinearity in the Navier–Stokes equation and utilize the theory of nonlinear semigroups to resolve the controlled Navier–Stokes inclusion with a multi-valued feedback term
Weak Almost-Convergence Theorem without Opial's Condition

AbstractLet E be Banach space with property (U,m,m+1,λ),λ∈R,m∈N, and a uniformly Gateaux differentiable norm; J: E→E* a duality mapping; D a nonempty closed convex bounded subset of E; and T: D→D a uniformly L-Lipschitzian asymptotically hemicontractive mapping with L<N(E)1/2 where N(E) is the normal structure coefficient of E satisfying the condition ‖x−Tny‖2≤〈x−Tny,J(x−y)〉 for all x,y∈D, n∈N∪{0}. Under the above conditions, the convergence of {J(xn−v)} for the sequence {xn} of the modified Ishikawa iteration process is established and then it is used to prove weak convergence of the process. The modified Ishikawa iteration process is defined as follows: For D a convex subset of a Banach space X and T a mapping D into itself, the sequence {xn}∞n=0 in D is defined by x0∈D,xn+1=(1−αn)xn+αnT((1−βn)xn+βnTxn),n≥0, where {αn} and {βn} satisfy 0≤αn,βn≤1 for all n and ∑∞n=0αn=∞
A Remark on Goldschmidt's Theorem on Formal Integrability

AbstractIn this paper we apply Weil bundle techniques to the study of formal integrability for systems of nonlinear partial differential equations. We clarify the role of the curvature and show that both the statement and proof of Goldschmidt's criterion on formal integrability are of algebraic nature. This fact allows us to obtain a version of this theorem which holds for smooth, algebraic, or analytic manifolds. Finally, we give a definition for the characteristic co-vectors of a system of partial differential equations and how their relationship with the Cauchy–Kowalevski normal form
The Hyperbolic Derivative in the Poincaré Ball Model of Hyperbolic Geometry

AbstractThe generic Möbius transformation of the complex open unit disc induces a binary operation in the disc, called the Möbius addition. Following its introduction, the extension of the Möbius addition to the ball of any real inner product space and the scalar multiplication that it admits are presented, as well as the resulting geodesics of the Poincaré ball model of hyperbolic geometry. The Möbius gyrovector spaces that emerge provide the setting for the Poincaré ball model of hyperbolic geometry in the same way that vector spaces provide the setting for Euclidean geometry. Our summary of the presentation of the Möbius ball gyrovector spaces sets the stage for the goal of this article, which is the introduction of the hyperbolic derivative. Subsequently, the hyperbolic derivative and its application to geodesics uncover novel analogies that hyperbolic geometry shares with Euclidean geometry
Well-posedness Results for Models of Elastomers

AbstractExistence and uniqueness of weak solutions are shown for different models of the dynamic behavior of elastomers. The models are based on a nonlinear stress-strain relationship (satisfying a locally Lipschitz and affine domination property) and incorporate hysteretic effects as well. The results provide alternatives to previous theories that required monotonicity assumptions on the nonlinearities. Results with a nonlinear constitutive law and nonlinear internal dynamics are presented for the first time
Duality for a Class of Multiobjective Control Problems with Generalized Invexity

AbstractB. Mond and I. Smart (J. Math. Anal. Appl.136 (1988), 325–333) defined a kind of invexity and discussed the duality and sufficiency in scalar control problems with such invexity. D. Bhatia and P. Kumar (J. Math. Anal. Appl.189 (1995), 676–692) defined another kind of invexity, corresponding generalized invexity, and discussed the duality for multiobjective control problems with such generalized invexity. In this paper, the duality results for multiobjective control problems with Mond and Smart's generalized invexity are discussed
On the Unique Solvability of a Nonlinear Reaction-Diffusion Model with Convection

AbstractNonnegative solutions of a general reaction-diffusion model with convection are known to be unique if the reaction, convection, and diffusion terms are all Lipschitz continuous with respect to their dependence on the solution variable. However, it is also known that such a Lipschitz condition is not necessary for the unique solvability of the model if either convection or reaction is not present. We introduce monotonicity conditions which, when imposed on the reaction and convection, are sufficient for the uniqueness of all nonnegative solutions of the general model. Consideration of the model where reaction, diffusion, and convection are governed by power laws also reveals the extent to which these conditions are necessary
Multipoint rational approximants with preassigned poles

20 pages, no figures.-- MSC1991 codes: 41A21, 42C05, 30E10.MR#: MR1820073 (2002i:41021)Zbl#: Zbl 1160.41305Let $\mu$ be a finite positive Borel measure whose support $S(\mu)$ is a compact regular set contained in $\Bbb R$. For a function of Markov type $\hat\mu(z)=\int_{S(\mu)}d\mu(x)/(z-x)$, $z\in\Bbb C\sbs S(\mu)$, we consider multipoint Padé-type approximants (MPTAs), where some poles are preassigned and interpolation is carried out along a table of points contained in $\overline{\Bbb C}\sbs {\rm Co}(S(\mu))$ which is symmetrical with respect to the real line. The main purpose of this paper is the study of the `exact rate of convergence' of the MPTAs to the function $\hat\mu$.Research by first author (F.C.) partially carried out at the Mathematics Department of Umeå University under Guest Scholarship from the Swedish Institute. Research by second author (G.L.L.) partially supported by Dirección General de Enseñanza Superior under grant PB 96-0120-CO3-01 and by INTAS under grant 93-0219 EXT.Publicad
On the Connection and Linearization Problem for Discrete Hypergeometric q-Polynomials

AbstractIn the present paper, starting from the second-order difference hypergeometric equation on the non-uniform lattice x(s) satisfied by the set of discrete hypergeometric orthogonal q-polynomials {pn}, we find analytical expressions of the expansion coefficients of any q-polynomial rm(x(s)) on x(s) and of the product rm(x(s))qj(x(s)) in series of the set {pn}. These coefficients are given in terms of the polynomial coefficients of the second-order difference equations satisfied by the involved discrete hypergeometric q-polynomials
Analysis of a Mathematical Model of the Growth of Necrotic Tumors

AbstractIn this paper we study a model of necrotic tumor growth. The tumor comprises necrotic cells which occupy a radially symmetric core and life proliferating cells which occupy a radially symmetric shell adjacent to the core. The proliferating cells receive nutrients through diffusion from the outer boundary as well as by means of blood flow through a network of capillary vessels. The mathematical model describes the evolution of the nutrient concentration σ between the boundary of the necrotic core r=ρ(t) and the outer boundary of the tumor r=R(t); within the core itself the concentration is a constant σ=σnec, a level under which life cells cannot be sustained. Both of the surfaces r=ρ(t) and r=R(t) are free boundaries, which are unknown in advance. Under some assumptions on the parameters, we prove that (i) there exists a stationary solution with radii r=ρs, r=Rs; (ii) for any initial data near the stationary solution, the time dependent model has a unique solution σ(r,t) with free boundaries r=ρ(t), r=R(t); and (iii) ρ(t)→ρs and R(t)→Rs as t→∞
Existence for a Diffusing Population with Discrete Returns to the Interior

AbstractWe examine a one-dimensional steady-state diffusion model for a biological population in which individuals collected either throughout the domain or from one endpoint of the domain are returned to the population at a discrete set of locations. Specifically, the modeling equations are u″(x)+f(x,u(x))+(∑ni=1κiδβi(x))g(∫10α(s)u(s)ds)=0 in the former case and u″(x)+f(x,u(x))+(∑ni=1κiδβi(x))g(u′(0)) in the latter case, where the function g specifies the rate of return and f is the usual net population growth term. With Dirichlet boundary conditions imposed at both ends of the domain, the topological transversality theorem is used to show that a positive solution exists provided g does not increase too rapidly
On Conditions for the Stability of a Two Component Mixed Quasimonotone Reaction Diffusion Equation

AbstractA spatially heterogeneous two component mixed quasimonotone system of reaction diffusion equations is considered. The kinetic functions exhibit mixed quasimonotonicity and are, in general, non-autonomous, while the boundary conditions are given by one of three possibilities: homogeneous Dirichlet, homogeneous Neumann, or Neumann with a constant inhomogeneity. Some conditions which yield the stability of such equations are deduced via comparison theorems and subsequently used to investigate the stability of a simple example system
Local Stability and Lyapunov Functionals for n-Dimensional Quasipolynomial Conservative Systems

AbstractWe present a method for determining the local stability of equilibrium points of conservative generalizations of the Lotka–Volterra equations. These generalizations incorporate both an arbitrary number of species—including odd-dimensional systems—and nonlinearities of arbitrarily high order in the interspecific interaction terms. The method combines a reformulation of the equations in terms of a Poisson structure and the construction of their Lyapunov functionals via the energy-Casimir method. These Lyapunov functionals are a generalization of those traditionally known for Lotka–Volterra systems. Examples are given
Conservation Laws and Global Solutions of Linear First Order PDEs with Distributional Coefficients

AbstractWe treat linear partial differential equations of first order with distributional coefficients naturally related to physical conservation laws in the spirit of our preceding papers (which concern ordinary differential equations): the solutions are consistent with the classical ones. Under compatibility conditions we prove uniqueness and existence results. As an example we consider the problem ut+δtux=0,u(x,−1)=h(x)(h∈C2(R) is given); our theory grants that the unique solution in C2(R2)⊕D′ℓ(R2) is u(x,t)=h(x)−h′(0)δ(x,t) and this has a physical meaning (D′ℓ(R2) is the space of distributions with discrete support and δ is the Dirac measure at (0,0))
Spectral Properties at a Threshold for Two-Channel Hamiltonians II. Applications to Scattering Theory

AbstractSpectral properties and scattering theory in the low-energy limit are investigated for two-channel Hamiltonians with Schrödinger operators as component Hamiltonians. In various, mostly fairly “singular” settings asymptotic expansions of the resolvent are deduced as the spectral parameter tends to the threshold zero. Furthermore scattering theory for pairs of two-channel Hamiltonians is established. As an application of the expansions of the resolvent, asymptotic expansions of the scattering matrix are derived as the energy parameter tends to the threshold zero
One-Sided Resonance Problems for Quasilinear Elliptic Operators

AbstractWe present some new existence results for a quasilinear elliptic problem with an unbounded driving force. The quasilinear elliptic operator is assumed to be variational and is such that 0 acts like an isolated eigenvalue with a corresponding eigenfunction which does not change sign. The driving force is further assumed to be in one-sided resonance around the eigenvalue 0, and a solvability condition of potential type is imposed. Variational methods are used to obtain existence. Our results significantly improve earlier results of the authors
Bounded Composition Operators on Weighted Bergman Spaces

AbstractWe answer two questions asked by T. L. Kriete (1998, in Contemp. Math., Vol. 213, pp. 73–91, Amer. Math. Soc., Providence) concerning bounded composition operators on weighted Bergman spaces of the unit disk. The main result is the following: if Gi=e−hi, for i=1,2, are weight functions in a certain range for which h′1(r)/h′2(r)→∞ as r→1 then there is a self-map of the unit disk such that the induced composition operator Cϕ maps A2G2 boundedly into itself but does not map A2G1 into itself
A Phase-Field Model with Temperature Dependent Constraint

AbstractThis paper is concerned with a phase-field model with temperature dependent constraint for the order parameter. In the kinetic equation of the order parameter, we take account of diffusion as well as the undercooled/superheated state which is modelled by means of a hysteresis input–output relation; the input is the temperature field and the output is the order parameter. In this paper we shall show that there exists a solution “globally” in time in a physically reasonable sense, when the coefficient of interfacial energy is sufficiently small
Convergence of Cascade Algorithms in Sobolev Spaces Associated with Multivariate Refinement Equations

AbstractThis paper is concerned with multivariate inhomogeneous refinement equations written in the form ϕ(x)=∑α∈Zsa(α)ϕ (Mx−α)+g(x),x∈Rs, where ϕ is the unknown function defined on the s-dimentional Euclidean space Rs,g is a given compactly supported function on Rs,a is a finitely supported sequence on Zs, and M is an s×s dilation matrix with m=|detM|. Let ϕ0 be an initial function in the Sobolev space Wk2(Rs). For n=1,2,…, define ϕn(x)=∑α∈Zsa(α)ϕn−1(Mx−α)+g(x),x∈Rs. In this paper, we give a characterization for the strong convergence in the Sobolev space Wk2(Rs)(k∈N) of the cascade sequence (ϕn)n∈N for the case in which M is isotropic
On a Question of Pietsch about Hilbert–Schmidt Multilinear Mappings

AbstractIn 1983, Pietsch asked if, for n≥3 and all Hilbert spaces E1,…,En, the vector space of the scalar valued absolutely (r;r1,…,rn)-summing multilinear mappings on E1×···×En coincides with the vector space of the n-linear Hilbert–Schmidt functionals on E1×···×En, for some choice of r,r1,…,rn∈]0,+∞], satisfying 1/r≤1/r1+···+1/rn. We show that the answer to this question is no. Moreover, we show that the same question, for n≥2 and mappings with values in infinite dimensional Hilbert spaces, has the answer no
Global Iterative Schemes for Accretive Operators

AbstractLet E be a real q-uniformly smooth Banach space and A:E→2E be an m-accretive operator which satisfies a linear growth condition of the form ‖Ax‖≤c(1+‖x‖) for some constant c>0 and for all x∈E. It is proved that if two real sequences {λn} and {θn} satisfy appropriate conditions, the sequence {xn} generated from arbitrary x0∈E by xn+1=xn−λn(un+θn(xn−z)); un∈Axnn≥0, converges strongly to some x*∈A−1(0). Furthermore, if E is a reflexive Banach space with a uniformly Gâteaux differentiable norm, and if every weakly compact convex subset of E has the fixed-point property for nonexpansive mappings and A:D(A)≔E→2E is m-accretive, then for arbitrary, z,x0∈E the sequence {xn} defined by xn+1+λn(un+1+θn(xn+1−z))=xn+en, for un∈Axn, where en∈E is such that ∑‖en‖<∞∀n≥0, converges strongly to some x*∈A−1(0)
Generalization of a Theorem of Bohr for Bases in Spaces of Holomorphic Functions of Several Complex Variables

AbstractIn the first part, we generalize the classical result of Bohr by proving that an analogous phenomenon occurs whenever D is an open domain in Cm (or, more generally, a complex manifold) and (ϕn)∞n=0 is a basis in the space of holomorphic functions H(D) such that ϕ0=1 and ϕn(z0)=0, n≥1, for some z0∈D. Namely, then there exists a neighborhood U of the point z0 such that, whenever a holomorphic function on D has modulus less than 1, the sum of the suprema in U of the moduli of the terms of its expansion is less than 1 too. In the second part we consider some natural Hilbert spaces of analytic functions and derive necessary and sufficient conditions for the occurrence of Bohr's phenomenon in this setting
On the Regularity of an Obstacle Control Problem

AbstractAn optimal control problem of the obstacle for an elliptic variational inequality is considered, in which the obstacle is regarded as the control. To get the regularity of the optimal pair, a new related control problem is introduced. By proving the existence of an optimal pair to such a new control problem, the regularity of the optimal pair to the original problem is obtained. It turns out that the regularity obtained is sharp in general. Some other interesting properties of the optimal pair are also established
Subsonic Solutions to a One-Dimensional Non-isentropic Hydrodynamic Model for Semiconductors

AbstractThe one-dimensional stationary full hydrodynamic model for semiconductor devices with non-isentropic pressure is studied. This model consists of the equations for the electron density, electron temperature, and electric field in a bounded domain supplemented with boundary conditions. The existence of a classical subsonic solution with positive particle density and positive temperature is shown in two situations: non-constant and constant heat conductivities. Moreover, we prove uniqueness of a classical solution in the latter case. The existence proofs are based on elliptic estimates, Stampacchia truncation methods, and fixed-point arguments
A Subdifferential Condition for Calmness of Multifunctions

AbstractA condition ensuring calmness of a class of multifunctions between finite-dimensional spaces is derived in terms of subdifferential concepts developed by Mordukhovich. The considered class comprises general constraint set mappings as they occur in optimization or mappings associated with a certain type of variational system. The condition ensuring calmness is obtained as an appropriate reduction of Mordukhovich's well-known characterization of the stronger Aubin property. (Roughly spoken, one may pass to the boundaries of normal cones or subdifferentials when aiming at calmness.) It allows one to derive dual constraint qualifications in nonlinear optimization that are weaker than conventional ones (e.g., Mangasarian–Fromovitz) but still sufficient for the existence of Lagrange multipliers
An Inverse Function Theorem in Sobolev Spaces and Applications to Quasi-Linear Schrödinger Equations

AbstractA Nash–Moser type inverse function theorem in Banach spaces with loss of derivatives is proved, and applications are given to singular quasi-linear Schrödinger equations like the superfluid film equation in plasma physics. Based on an implicit function theorem in Sobolev spaces, a linearization method is introduced for the local and global well-posedness of the Cauchy problem for nonlinear evolution equations. The technique compensates for a loss of derivatives in the linearized problem. This is illustrated by an application to strongly singular quasi-linear Schrödinger equations where the nonlinearities include derivatives of second order. The local well-posedness of the Cauchy problem is proved in Sobolev spaces for arbitrary space dimension without assuming smallness assumptions on the initial value. Here the linearized problem is solved using hyperbolic semigroup theory, including evolution systems
Asymmetric Sufficient Condition for Stability Robustness of Singular Systems With Structured Parameter Uncertainties

AbstractIn this paper, under assumptions that the linear nominal continuous-time (or discrete-time) singular system is regular, impulse-free, and asymptotically stable, an asymmetric sufficient condition is proposed to preserve the assumed properties when the structured (elemental) parameter uncertainties are added into the nominal singular system. The corresponding stable region in the parameter space that is obtained by using the proposed asymmetric sufficient condition is not necessarily symmetric with respect to the origin of the parameter space. An illustrated example, in which the signs and ranges of the parameters are known although their values are uncertain, is given to show that the proposed sufficient condition may yield less conservative results than the existing ones reported recently
A New Method of Solving Noisy Abel-Type Equations

AbstractA new approach to solving noisy integral equations of the first kind is applied to the family of Abel equations. Such equations play a role in stereology (Wicksell's unfolding problem), medicine, engineering, and astronomy. The method is based on an expansion in an arbitrary orthonormal basis, coupled with exact inversion of the integral operator. The inverse appears in the Fourier coefficients of the expansion, where it can be carried over to the usually well-behaved basis elements in the form of the adjoint. This method is an alternative to Tikhonov regularization, regularization of the inverse of the operator itself, or a wavelet-vaguelette/singular-value decomposition. The method is particularly interesting in irregularity of the kernel, the input, or both. Because knowledge of the spectral properties of the operator is not required, the method is also of interest in regular cases where these spectral properties are not sufficiently known or are hard to deal with. For smooth input functions, the simple basis of trigonometric functions yields input estimators whose mean integrated squared error converges at the optimal rate for the entire family of Abel operators. This can be shown when smooth wavelets are used for Abel operators with index smaller than 1/2, and when the Haar wavelet is used for operators with index larger than 1/2
Instability in Diffusive Ecological Models with Nonlocal Delay Effects

AbstractWe propose a reaction–diffusion system with nonlocal delays to model the growth of plankton communities feeding on a limiting nutrient supplied at a constant rate. Two delays are incorporated into the model: one describes the delayed nutrient recycling process and the other models the delayed growth response of the plankton. It is assumed that both delays are nonlocal in the sense that there are delayed not only in time but also in space. Local and bifurcation analyses are carried out. It has been shown that Turing-type spatial patterns occur when the diffusion coefficients vary and temporal or spatial-temporal patterns occur when the delay involved in the growth response changes