pinecone/core-2020-05-10-deduplication huggingface dataset:
On Strongly Degenerate Convection–Diffusion Problems Modeling Sedimentation–Consolidation Processes

AbstractWe investigate initial-boundary value problems for a quasilinear strongly degenerate convection–diffusion equation with a discontinuous diffusion coefficient. These problems come from the mathematical modeling of certain sedimentation–consolidation processes. The existence of entropy solutions belonging to BV is shown by the vanishing viscosity method. The existence proof for one of the models includes a new regularity result for the integrated diffusion coefficient. New uniqueness proofs for entropy solutions are also presented. These proofs rely on a recent extension to second-order equations of Kružkov's method of “doubling the variables.” The application to a sedimentation–consolidation model is illustrated by two numerical examples
Periodic Near Optimal Control

AbstractWe consider the infinite time optimal control problem of minimizing an average cost functional for nonlinear singularly perturbed systems. In particular, we are interested in the existence of near optimal trajectories which are periodic. For this purpose, we investigate nonlinear control systems with three time scales and construct a limiting system for the slowest motion by an iterated averaging procedure. We furthermore show that global tools, invariant control sets, and local tools, Lie algebra ranks of vector fields, are preserved under singular perturbations. The existence of periodic near optimal trajectories is verified under the condition that the fast subsystem and an unperturbed slow subsystem are controllable and that a standard Lie algebra rank condition for local accessibility holds
Substitution Hamiltonians with Bounded Trace Map Orbits

AbstractWe investigate discrete one-dimensional Schrödinger operators with aperiodic potentials generated by primitive invertible substitutions on a two-letter alphabet. We prove that the spectrum coincides with the set of energies having a bounded trace map orbit and show that it is a Cantor set of zero Lebesgue measure. This result confirms a suggestion arising from a study of Roberts and complements results obtained by Bovier and Ghez. As an application we present a class of models exhibiting a purely singular continuous spectrum with probability one
An Application of Krasnoselskii Fixed Point Theorem to the Asymptotic Behavior of Solutions of Difference Equations in Banach Spaces

AbstractIn this paper we consider the first order difference equationΔxn=∑i=0∞ainfxn+i+∑i=0∞bingxn+i+ynand the second order difference equationΔqnΔxn+rnfxn+sngxn+zn=0,where f is a Lipschitz mapping and g is a compact operator, both defined on a Banach space X. We give sufficient conditions so that there exist solutions which are asymptotically constant. These results generalize those given by A. Drozdowicz and J. Popenda (1987, Proc. Amer. Math. Soc.99, 135–140), J. Popenda and E. Schmeidel (1994, Publ. Mat.38, 3–9; 1997, Indian J. Pure Appl. Math.28, 319–327), and E. Schmeidel (1997, Demonstratio Math.30, 193–197; 1997; Comm. Appl. Nonlinear Anal.4, 87–92)
Instability of Equilibria in Some Delay Reaction-Diffusion Systems

AbstractA new result is derived which extends a known instability result for a class of reaction-diffusion equations to a corresponding system incorporating time delay effects. For a significant class of nonlinear equations it is shown that an unstable equilibrium solution of the reaction-diffusion system cannot be stabilised by the introduction of delay. The result is applied to problems posed on convex domains with homogeneous Neumann boundary conditions. Finally, global methods in bifurcation theory are applied to a delay reaction-diffusion system of Lotka–Volterra type representing the interaction of two mobile species. The existence of a countable set of continua consisting of unstable equilibrium solutions for this system is proved
Non-monotonic Random Schrödinger Operators: The Anderson Model

AbstractWe prove exponential localization at all energies for one-dimensional continuous Anderson-type models with single site potentials of changing sign. A periodic background potential is allowed. The main problem arises from non-monotonicity; i.e., the operator does not depend monotonically in the form sense on the random parameters. We show that the method of “two-parameter spectral averaging,” recently devised by Buschmann and Stolz to prove localization for Poisson and random displacement models, can be modified to work for the type of Anderson model considered here
Energy Estimates in Hierarchical Plate Theories

AbstractThe starting assumption of hierarchical plate theories is a representation of the displacement field in the form u(n)(x,ζ)=∑nk=0φk(ζ)b(n)k(x), where x stands for the in-plane coordinates and ζ for the transverse coordinate, where the linearly independent functions φk are the first (n+1) in a complete system, and where the functions b(n)k are determined by solving a minimum problem in an approximation space V(n). For V the function space where the exact solution u is sought to the three-dimensional problem that a given hierarchical plate theory is meant to approximate, we show that the sequence {u(n)} converges in energy to a limit element u(∞)∈V, whatever the functions φk; and that, if φk(ζ)=ζk, then u and u(∞) coincide pointwise, provided their difference is smooth
On the Exponential Stability and Periodic Solutions of Delayed Cellular Neural Networks

AbstractA set of criteria is presented for the global exponential stability and the existence of periodic solutions of delayed cellular neural networks (DCNNs) by constructing suitable Lyapunov functionals, introducing many parameters and combining with the elementary inequality technique. These criteria have important leading significance in the design and applications of globally stable DCNNs and periodic oscillatory DCNNs. In addition, earlier results are extended and improved; other results are contained. Two examples are given to illustrate the theory
Precise Asymptotics in the Baum–Katz and Davis Laws of Large Numbers

AbstractLet X,X1,X2,… be a sequence of i.i.d. random variables such that EX=0, let Z be a random variable possessing a stable distribution G with exponent α, 1<α≤2, assume the distribution of X is attracted to G, and set Sn=X1+···+Xn. We prove that∑n≥1nr/p−2P|Sn|≥εn1/p∼ε−(αp/(α−p))(r/p−1)pr−pE|Z|(αp/(α−p))(r/p−1)as ε↘0,for 1≤p<r<α, under the additional assumption that there is normal attraction to G, and that∑n≥11nP|Sn|≥εn1/p∼αpα−p−logεas ε↘0,for 1≤p<α.We close with a related result for the limiting case α=r=p=2
Global Existence for Coupled Reaction–Diffusion Systems

AbstractWe prove here global existence in time of classical solutions for reaction–diffusion systems with strong coupling in the diffusion and with natural structure conditions on the nonlinear reactive terms. This extends some similar results in the case of a diagonal diffusion-operator associated with nonlinearities preserving the positivity and the total mass of the solutions or for which the total mass is a priori bounded. Here, however, no positivity assumption is made since nondiagonal systems do not preserve it
Periodic Solutions of Infinite Delay Evolution Equations

AbstractFor A(t) and f(t,x,y) T-periodic in t, we consider the differential equation with infinite delay in a general Banach space X,u′t+Atut=ft,ut,ut,t>0,us=φs,s≤0,0.1where the resolvent of the unbounded operator A(t) is compact and f is continuous in its variables, and ut(s)=u(t+s), s≤0. We first show that the Poincaré operator given by P(φ)=uT(φ) (i.e., T units along the unique solution u(φ) determined by the initial function φ) is a condensing operator with respect to Kuratowski's measure of non-compactness in a phase space Cg, and then derive periodic solutions from bounded solutions by using Sadovskii's fixed point theorem. This extends the study of deriving periodic solutions from bounded solutions to infinite delay differential equations in general Banach spaces
On the Range of Certain Pendulum-Type Equations

AbstractLet us consider the BVPmx″t+g1x′t=ft,t∈0,Tx0=xT,x′0=x′T,where g1 is a continuous function. The range R1 of the operator related to this problem is very well known. In this paper we treat the perturbed problemmx″t+g1x′t+g0xt=ft,t∈0,Tx0=xT,x′0=x′T,where g0 is of pendulum type, showing that, in general, the range of the perturbed operator is not contained in R1. This points out an important qualitative difference with respect to the case where g0 is of the Landesmann–Lazer type. On the other hand we prove that if f is small then the mentioned inclusion is true in general
Effects of Varying Nonlinearity and Their Singular Perturbation Flavour

AbstractAutonomous differential equations y″+f(y,p)=0 whose nonlinearity varies with a parameter p are studied. As a prototype, one may think of y″+f0(y)+|y|p−1g(y)=0. We discuss periodic solutions with initial values taken from various domains and their different types of convergence as p→∞. Equations y″−f(y,p)=0, yf(y,p)>0 are similarly discussed, with the “period” of a solution replaced by its “maximal interval of existence.” The study shows a natural link to singularly perturbed problems. It turns out that the family of ODEs under consideration are essentially a family of singularly perturbed problems. Solutions may develop “kinks” and higher order derivatives of solutions possess “boundary layers,” namely sets of non-uniform convergence. Similarities and differences between this family and the more common singularly perturbed problems which abound in the literature emerge
Roundness and Metric Type

AbstractWe prove that if X is a Banach space containing lnp uniformly in n, and if Y is a metric space with metric type q>p, then the inverse of any uniform homeomorphism T from X onto Y cannot satisfy a Lipschitz condition for large distances of order α<q/p. It follows that if Y is a midpoint-convex subset of a Banach space Z with type q larger than the type supremum of a Banach space X, then X and Y cannot be uniformly homeomorphic. In particular, we prove the non-existence of uniform homeomorphisms between certain non-commutative Lp-spaces and midpoint-convex subsets of another such space. We also prove that if a Banach space X has cotype infimum q larger than two, then it has maximal generalized roundness zero and maximal roundness at most q′. As a consequence, infinite-dimensional C*-algebras are seen to have maximal generalized roundness zero and maximal roundness one
Cesàro Summability of Hardy Spaces on the Ring of Integers in a Local Field

AbstractLet O be the ring of integers in a local field K. We solve an open problem due to M. H. Taibleson (1975, “Math. Notes,” Vol. 15, Princeton Univ. Press, Princeton, NJ): Suppose f∈L1(O). Does the Cesàro means of f converge to f almost everywhere if K has characteristic zero? To this end we study the (Hp,Lp) boundedness of the associated maximal operator σ* to get the corresponding interpolation result on Hardy–Lorentz spaces; in particular we obtain that σ* is of weak type (1,1). The proof mainly depends on certain estimates for the oscillatory Dirichlet kernels, which are refinements of those obtained earlier by the author (1997, J. Math. Anal. Appl.208, 528–552)
On a Functional Equation in Actuarial Mathematics

AbstractIn 1839, De Morgan gave a mathematical justification of Gompertz's law of mortality through a composite functional equation, f(x+y)+f(x+z)=f(x+h(y,z)). A slightly more general version of this equation was studied in 1905 by M. Chini. Both solved their equations in the class of differentiable functions on the real line. Here we solve the equation f(x)+f(x+y)=cf(x+g(y)), which is a generalization of Chini's equation, on intervals in the class of locally bounded functions and in the class of continuous functions
A Resource-Constrained Optimal Control Model for Crackdown on Illicit Drug Markets

AbstractIn this paper we present a budget-constrained optimal control model aimed at finding the optimal enforcement profile for a street-level, illicit drug crackdown operation. The objective is defined as minimizing the number of dealers dealing at the end of the crackdown operation, using this as a surrogate measure of residual criminal activity. Analytical results show that optimal enforcement policy will invariably use the budget resources completely. Numerical analysis using realistic estimates of parameters shows that crackdowns normally lead to significant results within a matter of a week, and if they do not, it is likely that they will be offering very limited success even if pursued for a much longer duration. We also show that a ramp-up enforcement policy will be most effective in collapsing a drug market if the drug dealers are risk-seeking, and the policy of using maximum enforcement as early as possible is usually optimal in the case when the dealers are risk averse or risk neutral. The work then goes on to argue that the underlying model has some general characteristics that are both reasonable and intuitive, allowing possible applications in focused, local enforcement operations on other similar illegal activities
Nearest Route Problem

AbstractIn this paper we propose a new class of optimization problems on oriented graphs with weight, which are called nearest route problems. A nearest problem has a target value, which generates an error function in a sense. Which route is nearest to the designated value? The nearest route problem includes both the shortest route problem and longest one. We give an invariant imbedding solution through both forward-backward and backward-forward approaches. A typical nearest problem is solved by both approaches
Jump-Diffusions with Controlled Jumps: Existence and Numerical Methods

AbstractA comprehensive development of effective numerical methods for stochastic control problems in continuous time, for reflected jump-diffusion models, is given in earlier work by the author. While these methods cover the bulk of models which have been of interest to date, they do not explicitly deal with the case where the jump itself is controlled in the sense that the value of the control just before the jump affects the distribution of the jump. We do not deal explicitly with the numerical algorithms but develop some of the concepts which are needed to provide the background which is necessary to extend the proofs of earlier work to this case. A critical issue is that of closure, i.e., defining the model such that any sequence of (systems, controls) has a convergent subsequence of the same type. One needs to introduce an extension of the Poisson measure (which serves a purpose analogous to that served by relaxed controls), which we call the relaxed Poisson measure, analogously to the use of the martingale measure concept given earlier to deal with controlled variance. The existence of an optimal control is a consequence of the development
Neuro-Fuzzy Estimation in Spatial Statistics

AbstractAlthough spatial statistics was developed based on probability and classic statistics, the data usually handled by them are frequently very approximate and linguistic and certainly are not suited for the probability concept. Furthermore, the traditional spatial statistics is developed principally for mining situations. When the approach is applied to problems under other situations such as air and water pollution, certain basic assumptions need to be modified. In an earlier paper, fuzzy spatial statistics was proposed. In this paper, neural learning combined with fuzzy representation is suggested for handling the variogram, which is essentially a covariance correlation, and the kriging, which is an unbiased method to estimate the missing data. Based on the fuzzy adaptive network, various computational methods are proposed to solve the resulting spatially distributed problem
The Optimal Age-Dependent Checkpoint Strategy for a Stochastic System Subject to General Failure Mode

AbstractIn this paper, we consider an age-dependent stochastic system with checkpointing and rollback recovery, subject to general failure mode. In the case where the queueing effect for transaction processing is not remarkable, we investigate the stochastic behavior of the system and derive analytically the optimal checkpointing strategy which maximizes the system availability. When the queueing effect due to the idle period in the system is remarkable, the optimal checkpoint strategy is characterized approximately
N Degrees of Separation: Influences of Dynamic Programming on Computer Science

AbstractBellman's dynamic programming methodology can be applied to a wide range of computer science optimization problems. Some of these applications are briefly reviewed here. This work has led to advances in numerous other areas of computer science, including programming languages, computer simulation, and computer architecture. In turn, advances in these other areas suggest other applications of and to dynamic programming. In summary, we survey interrelationships of disparate computer science areas, and how progress in these areas evolved from Bellman's work
Optimization of Disturbance Rejection in Systems with Saturating Actuators

AbstractAn extension of the LQR/LQG methodology to systems with saturating actuators, referred to as SLQR/SLQG, is obtained. The development is based on the method of stochastic linearization. Using this method and the Lagrange multiplier technique, a solution to the SLQR/SLQG problem is derived. This solution is given by the standard Riccati equations coupled with two transcendental equations, which define the variance of the signal at the input of the saturation and the Lagrange multiplier associated with the quadratic performance index. It is shown that, under the standard stabilizability and detectability conditions, these equations have a unique solution, which can be found by a simple bisection algorithm. When the saturation is removed, these equations reduce to the standard LQR/LQG solution
The Parametric Solutions of Eigenstructure Assignment for Controllable and Uncontrollable Singular Systems

AbstractThe previous results about the parametric solutions of eigenstructure assignment for singular systems applied by pure proportional or proportional-plus-derivative state feedback can only apply to controllable systems. It is because the parametric solutions of the right generalized eigenvectors corresponding to the infinite and uncontrollable finite eigenvalues have still not been found. In this paper, the parametric solutions of the right generalized eigenvectors for finite controllable (or uncontrollable) eigenvalues and infinite controllable (or uncontrollable) eigenvalues, when the system is applied by pure proportional or proportional-plus-derivative state feedback, are given. Hence, the parametric solution of eigenstructure assignment can be used to design the state feedback of both the controllable and uncontrollable systems. The condition for detecting the regularity of the resulting system is also given
Existence and Stability of Almost Periodic Solutions for Quasilinear Delay Systems and the Halanay Inequality

AbstractWe present some easily verifiable conditions for the existence and global asymptotical stability of almost periodic solutions to systems of delay differential equations in the form u′(t)=−Au(t)+Wg(t,ut)+f(t), where A,W∈L(Rn) and f(t),g(t,ϕ) are almost periodic in t uniformly on ϕ from bounded subsets of C([−h,0],Rn). With this purpose, we use the exponential dichotomy theory together with usual positivity arguments. In particular, the semigroup version of the so-called Perron–Frobenius theorem is applied to study the generalized Halanay inequality. Finally, systems with maxima are studied in detail
Asymptotic Behavior of Trigonometric Series with O-Regularly Varying Quasimonotone Coefficients

AbstractIn this paper, the asymptotic behavior of a trigonometric series with O-regularly varying quasimonotone coefficients is investigated. Our results generalize the work done by S. Aljančić, R. Bojanić, and M. Tomić (1956, Publ. Inst. Math. (Beograd) (N.S.) 10, 101–120), R. Bojanić and J. Karamata (1963, Tech Rep 432, Math Research Center, Madison, WI), G. H. Hardy (1928, J. London Math Soc.3, 12–13; 1931, Proc. London Math. Soc. (2)32, 441–448), G. H. Hardy and W. W. Rogosinski (1945, Quart. J. Math. Oxford16, 49–58), J. R. Nurcombe (1993, J. Math. Anal. Appl.178, 63–69), C. H. Yong (1969, Publ. Inst. Math. (Beograd) (N.S.) 9, 29–35), and A. Zygmund (1968, “Trigonometric Series,” Vol. 1, 2nd ed., Cambridge Univ. Press, Cambridge, UK)
H∞ Control of Linear Singularly Perturbed Systems with Small State Delay

AbstractAn infinite horizon H∞ state-feedback control problem for singularly perturbed linear systems with a small state delay is considered. An asymptotic solution of the hybrid system of Riccati-type algebraic, ordinary differential, and partial differential equations with deviating arguments, associated with this problem, is constructed. Based on this asymptotic solution, conditions for the existence of a solution of the original H∞ problem, independent of the singular perturbation parameter, are derived. A simplified controller with parameter-independent gain matrices, solving the original problem for all sufficiently small values of this parameter, is obtained. An illustrative example is presented
Global Solutions for Dissipative Kirchhoff Strings with m(r)=rp (p<1)

AbstractWe investigate the evolution problemu″+δu′+m|A1/2u|2HAu=0,u0=u0,u′0=u1,where H is a Hilbert space, A is a self-adjoint non-negative operator on H with domain D(A), δ>0 is a parameter, and m(r)=rp with p<1. We prove that this problem has a unique global solution for positive times, provided that the initial data (u0,u1)∈D(Aαi/2)×D(A(αi−1)/2) satisfy a suitable smallness assumption and the non-degeneracy condition m(|A1/2u0|2H)>0 (where p≥2−i and αi=2i+1). Moreover, we prove for this solution decay with a polynomial rate as t→+∞. These results apply to degenerate hyperbolic PDEs with non-local non-linearities
Triangles Inscribed in a Semicircle, in Minkowski Planes, and in Normed Spaces

AbstractIn this paper we mainly consider triangles inscribed in a semicircle of a normed space; in two-dimensional spaces, their perimeter has connections with the perimeter of the sphere. Moreover, by using the largest values the perimeter of such triangles can have, we define two new, simple parameters in real normed spaces: one of these parameters is strictly connected with the modulus of convexity of the space, while the study of the other one seems to be more complicated. We calculate the value of our two parameters and we bring out a few connections among their values and the geometry of real normed spaces
The Structure of the Efficient Frontier of Finite-Dimensional Completely-Shaded Sets

AbstractThe aim of this paper is to study the contractibility of the efficient frontier of closed free-disposal sets with respect to general ordering cones in the Euclidean space. In fact, it is shown that, under certain suitable continuity assumptions, the efficient frontier of a completely-shaded set is a strong deformation retract of its weakly-efficient frontier, the last one being homeomorphic to a hyperplane. In particular, for the standard positive cone, we can apply our result to multicriteria optimization to get the contractibility of the efficient outcome set