pinecone/core-2020-05-10-deduplication huggingface dataset:
Absolute Norms on Cn

AbstractIn this paper, we study the absolute normalized norms on Cn. We show that the set of all absolute normalized norms on Cn is in one-to-one correspondence with the set Ψn of all continuous convex functions on Δn with some suitable conditions, where Δn={(s1,…,sn−1)∈Rn−1; s1+s2+···+sn−1≤1, si≥0 (1≤i≤n−1)}. As some applications, we show that an absolute norm on Cn is strictly convex if and only if the corresponding function ψ on Δn is strictly convex. Further, we consider the von Neumann–Jordan constant of Cn with an absolute normalized norm
The Determination of Nonlocal Symmetries by the Technique of Reduction of Order

AbstractAutonomous systems of ordinary differential equations can be rewritten as systems of first order ordinary differential equations and one of the dependent variables chosen as a new independent variable. Some of the variables are eliminated to give a mixed system of first and second order equations for which the determination of point symmetries can be automated without having to make an Ansatz on the detailed structure of the symmetry. Because the coefficient function for the original independent variable appears only as its derivative in the reduced system, symmetries which are nonlocal in this variable become local symmetries of the reduced system and can be computed algorithmically
Diagonalization of Non-selfadjoint Analytic Semigroups and Application to the Shape Memory Alloys Operator

AbstractTo a densely defined, but not necessarily selfadjoint, operator A on a Hilbert space H we consider on R+×H the following abstract “elliptic” problem of Dirichlet type:[formula] Then, in this paper, we establish that for every t>0, the solution [formula] can be expanded into a series of generalized eigenvectors of the operator A provided that its resolvent belongs to Carleman class Cp for some p∈]0,12[. A similar result holds for t large enough if the inverse A−1 belongs to Carleman class Cp for every p>12. (See Theorem 3.1 and Theorem 3.2.) Furthermore, we apply these obtained results to the shape memory alloys non-selfadjoint operator [formula] and Dn=∂n/∂xn when acting on an appropriate Hilbert space E of functions on the interval [0,1], by establishing that the inverse C−1 belongs to the Carleman class Cp for every p>12, so that we get in this case more regularity in the sense that the operatorial series ∑∞k=1etCPk converges strongly in E to the analytic semigroup etC for every t>0 (the Pk are the projectors into the root subspaces of C). A similar result holds for [formula] provided that t is large enough. (See Theorem 4.1 and Theorem 4.2 in Section 4 for the precise result.
Oscillation Criteria for Second Order Nonlinear Differential Equations of Euler Type

AbstractThe purpose of this paper is to solve the oscillation problem for the nonlinear Euler differential equation t2x″+g(x)=0 and the extended equation x″+a(t)g(x)=0. Here g(x) satisfies the sign condition xg(x)>0 if x≠=0, but is not assumed to be monotone. We give necessary and sufficient conditions for all nontrivial solutions to be oscillatory. To this end, we use phase plane analysis of the Liénard system and the oscillation result on the Riemann–Weber version of the linear Euler differential equation t2y″+(1/4+δ/(logt)2)y=0. Our results are a negative answer to a conjecture which was given by Wong. Finally, we illustrate our results by two examples
Natural Dualities for Varieties of MV-Algebras, I

AbstractMV-algebras are the Lindenbaum algebras for Łukasiewicz's infinite-valued logic, just as Boolean algebras correspond to the classical propositional calculus. The finitely generated subvarieties of the variety M of all MV-algebras are generated by finite chains. We develop a natural duality, in the sense of Davey and Werner, for each subvariety generated by a finite chain, and use it to describe the free and the injective members of these classes. Finally, we point out the relations between the dualities and some categorical equivalences discovered by A. Di Nola and A. Lettieri
A Note on Riesz Bases of Eigenvectors of Certain Holomorphic Operator-Functions

AbstractOperator-valued functions of the form A(λ)≔A−λ+Q(λ) with λ↦Q(λ)(A−μ)−1 compact-valued and holomorphic on certain domains Ω⊂C are considered in separable Hilbert space. Assuming that the resolvent of A is compact, its eigenvalues are simple and the corresponding eigenvectors form a Riesz basis for H of finite defect, it is shown that under certain growth conditions on ‖Q(λ)(A−λ)−1‖ the eigenvectors of A corresponding to a part of its spectrum also form a Riesz basis of finite defect. Applications are given to operator-valued functions of the form A(λ)=A−λ+B(λ−D)−1C and to spectral problems in L2(0,1) of the form −f″(x)+p(x,λ)f′(x)+q(x,λ)f(x)=λf(x) with, for example, Dirichlet boundary conditions
Inhomogeneous Quadratic Functionals on Time Scales

AbstractIn a previous paper by the author (Math. Comput. Modelling32 (2000), 507–527, Linear Hamiltonian systems on time scales. Positivity of quadratic functionals) a unified theory for continuous (T=R) and discrete (T=Z) linear Hamiltonian systems on an arbitrary time scale T is developed. In particular, it is stated that disconjugacy of the Hamiltonian system is a sufficient condition for positivity of the corresponding (homogeneous) quadratic functional. In this paper we present a similar condition for quadratic functionals on time scales with general (inhomogeneous) boundary conditions
Generalized Coherent Pairs

AbstractA pair of quasi-definite moment functionals {u0,u1} is a generalized coherent pair if monic orthogonal polynomials {Pn(x)}∞n=0 and {Rn(x)}∞n=0 relative to u0 and u1, respectively, satisfy a relationRnx=1n+1P′n+1x−σnnP′nx−τn−1n−1P′n−1x,n≥2,where σn and τn are arbitrary constants, which may be zero. If {u0,u1} is a generalized coherent pair, then u0 and u1 must be semiclassical. We find conditions under which either u0 or u1 is classical. In such a case, we also determine the types of the “companion” moment functionals. Also some illustrating examples and two ways of generating generalized coherent pairs are given. We also discuss the corresponding Sobolev orthogonal polynomials
Orthogonal Polynomials in Stein's Method

AbstractStein's method provides a way of finding approximations to the distribution, ρ say, of a random variable, which at the same time gives estimates of the approximation error involved. In essence the method is based on a defining equation, or equivalently an operator, of the distribution ρ and a related Stein equation. Up to now it was not clear which equation to take. One could think of a lot of equations. We show how for a broad class of distributions, there is one convenient equation. We give a systematic treatment, including the Stein equation and its solution. A key tool in Stein's theory is the generator method developed by Barbour and we suggest employing the generator of a Markov process for the operator of the Stein equation. For a given distribution there may be various Markov processes which fit in Barbour's method and, up to now, it was still not clear which Markov process to take to obtain good results. We show how for a broad class of distributions there is a special Markov process, a birth and death process, or a diffusion, which takes a leading role in the analysis. Furthermore, a key role is played by the classical orthogonal polynomials. It turns out that the defining operator is based on a hypergeometric difference or differential equation, which lies at the heart of the classical orthogonal polynomials. Furthermore, the spectral representation of the transition probabilities of the Markov process involved are in terms of orthogonal polynomials closely related to the distribution to be approximated. This systematic treatment together with the introduction of orthogonal polynomials in the analysis seems to be new. Furthermore some earlier uncovered examples like the beta, the Student's t, and the hypergeometric distribution are now worked out
Yosida–Hewitt and Lebesgue Decompositions of States on Orthomodular Posets

AbstractOrthomodular posets are usually used as event structures of quantum mechanical systems. The states of the systems are described by probability measures (also called states) on it. It is well known that the family of all states on an orthomodular poset is a convex set, compact with respect to the product topology. This suggests using geometrical results to study its structure. In this line, we deal with the problem of the decomposition of states on orthomodular posets with respect to a given face of the state space. For particular choices of this face, we obtain, e.g., Lebesgue-type and Yosida–Hewitt decompositions as special cases. Considering, in particular, the problem of existence and uniqueness of such decompositions, we generalize to this setting numerous results obtained earlier only for orthomodular lattices and orthocomplete orthomodular posets
Algebraic Univalence Theorems for Nonsmooth Functions

AbstractA well known univalence result due to D. Gale and H. Nikaido (1965, Math. Ann.159, 81–93) asserts that if the Jacobian matrix of a differentiable function from a closed rectangle K in Rn into Rn is a P-matrix at each point of K, then f is one-to-one on K. In this paper, by introducing the concepts of H-differentiability and H-differential of a function (as a set of matrices), we generalize the Gale–Nikaido result to nonsmooth functions. Our results further extend those of other authors valid for compact rectangles. We show that our results are applicable when the H-differential is any one of the following: the Jacobian matrix of a differentiable function, the generalized Jacobian of a locally Lipschitzian function, the Bouligand subdifferential of a semismooth function, and the C-differential of L. Qi (1993, Math. Oper. Res.18, 227–244)
Convergence of Solutions for an Equation with State-Dependent Delay

AbstractA result of Smith and Thieme shows that if a semiflow is strongly order preserving, then a typical orbit converges to the set of equilibria. For the equation with state-dependent delay ẋ(t)=−μx(t)+f(x(t−r(x(t)))), where μ>0 and f and r are smooth real functions with f(0)=0 and f′>0, we construct a semiflow which is monotone but not strongly order preserving. We prove a convergence result under a monotonicity condition different from the strong order preserving property, and apply it to the above equation to obtain generic convergence
Improvability of Feedback Systems

AbstractThis paper introduces the notion of improvability and bottlenecks of feedback systems in the context of instrumentation cost. Specifically, a feedback system is improvable if its performance can be enhanced by re-allocating sensor and actuator costs under budget constraints. We derive a criterion which determines when the system is improvable, using the LQG performance index. In addition, we introduce and analyze the notion of instrumentation bottleneck and provide a criterion for bottleneck identification. An important feature of the results derived is that both improvability and BN indicators can be evaluated using on-line measurements in the feedback loop, without requiring precise knowledge of the plant data. Examples illustrating results are provided
Stability of a Lyapunov Controller for a Semi-active Structural Control System with Nonlinear Actuator Dynamics

AbstractWe investigate semi-active control for a wide class of systems with scalar nonlinear semi-active actuator dynamics and consider the problem of designing control laws that guarantee stability and provide sufficient performance. Requiring the semi-active actuator to satisfy two general conditions, we present a method for designing quickest descent controllers generated from quadratic Lyapunov functions that guarantee asymptotic stability within the operating range of the semiactive device for the zero disturbance case. For the external excitation case, bounded-input, bounded-output stability is achieved and a stable attractor (ball of ultimate boundedness) of the system is computed based on the upper bound of the disturbances. We show that our wide class of systems covers, in particular, two nonlinear actuator models from the literature. Tuning the performance of the simple Lyapunov controllers is straightforward using either modal or state penalties. Simulation results are presented which indicate that the Lyapunov control laws can be selected to provide similar decay rates as a “time-optimal” controller for a semi-actively controlled single degree of freedom structure with no external excitation
A Simple Proof of an Aomoto-Type Extension of Askey's Last Conjectured Selberg q-Integral

AbstractWe establish an Aomoto-type extension of Askey's last conjectured Selberg q-integral, which was recently proved by Evans. We follow the lines of our proofs of Aomoto-type extensions of the Morris constant term q-identity and Gustafson's Askey–Wilson Selberg q-integral. We require integral forms of the q-transportation theory and its alternative for the root system An−1, which are related to the simple reflections and minuscule weight of An−1. We use an elementary symmetry which is related to the minuscule weight of An−1 to lift a proof of the one dimensional q-integral to the multivariable setting
Stability and Bifurcation for a Delayed Predator–Prey Model and the Effect of Diffusion

AbstractWe consider a predator–prey system with one or two delays and a unique positive equilibrium E∗. Its dynamics are studied in terms of the local stability of E∗ and of the description of the Hopf bifurcation that is proven to exist as one of the delays (taken as a parameter) crosses some critical values. We also consider a reaction–diffusion system with Neumann conditions, resulting from adding one spatial variable and diffusion terms in the previous model. The spectral and bifurcation analysis in the neighborhood of E∗, now as a stationary point of this latter system, is addressed and the results obtained for the case without diffusion are applied
Analysis of Fractional Differential Equations

AbstractWe discuss existence, uniqueness, and structural stability of solutions of nonlinear differential equations of fractional order. The differential operators are taken in the Riemann–Liouville sense and the initial conditions are specified according to Caputo's suggestion, thus allowing for interpretation in a physically meaningful way. We investigate in particular the dependence of the solution on the order of the differential equation and on the initial condition, and we relate our results to the selection of appropriate numerical schemes for the solution of fractional differential equations
A Comparison Inequality for Sums of Independent Random Variables

AbstractWe give a comparison inequality that allows one to estimate the tail probabilities of sums of independent Banach space valued random variables in terms of those of independent identically distributed random variables. More precisely, let X1,…,Xn be independent Banach-valued random variables. Let I be a random variable independent of X1,…,Xn and uniformly distributed over {1,…,n}. Put X̃1=XI, and let X̃2,…,X̃n be independent identically distributed copies of X̃1. Then, P(‖X1+···+Xn‖≥λ)≤cP(‖X̃1+···+X̃n‖≥λ/c) for all λ≥0, where c is an absolute constant
Singularities of the Hamiltonian Vectorfield in Optimal Control Problems

AbstractVariational problems with n degrees of freedom give rise (by the Pontriaguine maximum principle) to a hamiltonian vectorfield in T*Rn, that presents singularities (non-smoothness points) when the lagrangean is not convex. For the problems of the calculus of variations, the singularities that occur are points where the hamiltonian vectorfield is not C0. For optimal control problems, we show that besides these singularities there appear other ones: points where the hamiltonian vectorfield is C0 but not C1, and we classify them
Characterizations of QT Spaces

AbstractWe introduce a new space, QT space, of analytic functions on the unit disk in terms of a nondecreasing function T. The relation between QT and Qp spaces, which have attracted considerable attention, is given by studying the growth order of T. The counterpart Q#T of QT for the meromorphic case is also considered and investigated. We note that some characterizations of Q#T and QT are different. Moreover, our results answer a question raised by P. Wu (1998, Complex Variables35, 157–170) in the negative
Ultradistributions of Roumieu Type and Projective Descriptions

AbstractWe show that the topology of the weighted inductive limit of Fréchet spaces of entire functions which is obtained as the Fourier Laplace transform of the space of ultradistributions with compact support of Roumieu type cannot be described by means of canonical weighted sup-seminorms. This is a natural weighted LF-space of entire functions for which the problem of projective description of Bierstedt, Meise, and Summers has a negative answer. However, the projective hull of the LF-space coincides algebraically with the inductive limit. The behaviour of the corresponding inductive limits of spaces of continuous functions is completely opposite: the topology can be described by weighted sup-seminorms, but the LF-space is a proper subspace of its projective hull
Starplus-Compactness and Starplus-Compact Open Fuzzy Topologies on Function Spaces

AbstractExtending Lowen's notion of strong fuzzy compactness to an arbitrary fuzzy set the notion of a starplus-compact fuzzy set is introduced. It is shown that the category of starplus-compact fuzzy topological spaces is productive, and that starplus-compactness is a good extension of the notion of compactness. It is shown that the class of starplus-compact fuzzy sets is pseudo closed hereditary and invariant under fuzzy continuous maps. Moreover, the notion of starplus-compact open fuzzy topology on a function space is introduced and its interrelations with the fuzzy topology of pointwise convergence and the fuzzy topology of joint fuzzy continuity are studied. It is shown that a fuzzy topology on a function space which is jointly fuzzy continuous on starplus-compacta is finer than the starplus-compact open fuzzy topology. Sufficient conditions on function spaces are obtained for the starplus-compact open fuzzy topology and the fuzzy topology of joint fuzzy continuity on starplus-compacta to coincide
On the Equation ∇u=uV

AbstractThe PDE ∇u=uV can be treated as its ODE equivalent u′=au if the vector field V satisfies the integrability condition curl V=0. In this case there exist solutions v of the equation ∇v=V and solutions of the original equation have the form u=cev (c∈C). However, in devising meaningful solution concepts, the equation displays some surprising features related to local and global integrability of its solutions. In the course of the investigation technical tools like a lifting theorem with respect to the exponential map and Poincaré type theorems are proved for distributions and functions of Beppo Levi type
Inequalities for Trigonometric Polynomials and Some Integral Means

AbstractSome inequalities associated with the Laplacian for trigonometric polynomials are given, which will be applied to investigate the behavior in approximation by trigonometric polynomials in higher dimensions and the best lower and upper estimates for some linear operators. In particular, we obtain a complete characterization for the approximation behavior of the classical Jackson operator. Thus the approximation error of a function by Jackson polynomials is equivalent to a K-functional defined by the Laplacian
Stabilization of Vibrating Beam with a Tip Mass Controlled by Combined Feedback Forces

AbstractA flexible structure consisting of a Euler–Bernoulli beam with a tip mass is considered. To stabilize this system we use a boundary control laws: −uxxx(1,t)+mutt(1,t)=−αut(1,t)+βuxxxt(1,t) and uxx(1,t)=−γuxt(1,t). A sensitivity asymptotic analysis of the system's eigenvalues and eigenfunctions is set up. We prove that all of the generalized eigenfunctions of (2.9) form a Riesz basis of H. By a new method, we prove that the operator A generates a C0 contraction semigroup T(t), t≥0. Furthermore T(t), t≥0, is uniformly exponentially stable and the optimal exponential decay rate can be obtained from the spectrum of the system
Two Radius Support Theorem for the Sphere Transform

AbstractWe prove support theorems and injectivity for the sphere transform on restricted sets of spheres. We consider spheres with restricted sets of centers and two fixed radii. The proof of these theorems is based on E. T. Quinto's (1993, Israel J. Math., 353–363) microlocal regularity theorem and a theorem of L. Hörmander (1983, “The Analysis of Linear Partial Differential Operators I,” Springer-Verlag, Berlin/New York) and M. Sato et al. (1973, “Hyperfunctions and Pseudodifferential Equations,” Lecture Notes in Math., Vol. 287, pp. 265–529, Springer-Verlag, Berlin/New York)
Nonglobal Existence of Solutions for a Generalized Ginzburg–Landau Equation Coupled with a Poisson Equation

AbstractIn this article, we consider a system of a Ginzburg–Landau equation in u coupled with a Poisson equation in φ,∂u∂t=1+iαΔu+|u|2u+k∂φ∂x1u+γu,t∈0,T, x∈Rn,−Δφ=∂∂x1|u|2,x∈Rn, t∈0,T,u0,x=u0x,x=x1,x2,…,xn∈Rn,where T>0, α, γ, and k are real parameters, u and φ are, respectively, a complex and real valued function. For α∈(−1,1) and k<1, we prove the existence of initial data u0∈H1(Rn), n=1 or 2, for which the solution u is nonglobal. Our method uses energy arguments. We establish differential inequalities having only nonglobal solutions
Structure and Entropy for Positive-Definite Toeplitz Kernels on Free Semigroups

AbstractIn this article, we study the structure of positive-definite Toeplitz kernels on free semigroups (called also multi-Toeplitz) and its implications in noncommutative dilation theory, harmonic analysis on Fock spaces, prediction and interpolation theory for stationary stochastic processes. A parametrization of positive-definite multi-Toeplitz kernels in terms of generalized Schur sequences of contractions is obtained. This leads to explicit minimal Naimark dilations, Cholesky factorizations, Szegö type limit theorems, and entropy for positive-definite multi-Toeplitz kernels. Maximal outer factors are associated with positive-definite multi-Toeplitz kernels and are used to obtain inner–outer factorization for operators on Fock spaces with coefficients in Hilbert spaces. The Kolmogorov–Wiener prediction problem for stochastic processes having as covariance kernels positive-definite multi-Toeplitz kernels is considered. The predication-error operator is calculated in terms of Schur parameters (resp., maximal outer factor) associated with the covariance kernel of the process, and a connection with a Szegö type infimum problem is established. We solve the Carathéodory interpolation problem for positive-definite multi-Toeplitz kernels, we obtain a parametrization of all solutions in terms of Schur sequences, and we find the maximal entropy solution. The results of this article can be used to develop a theory of stochastic n-linear systems
Application of Graph Theory: Relationship of Eccentric Connectivity Index and Wiener's Index with Anti-inflammatory Activity

AbstractGraph theory was successfully applied in developing a relationship between chemical structure and biological activity. The relationship of two graph invariants—the eccentric connectivity index and the Wiener's index was investigated with regard to anti-inflammatory activity, for a data set consisting of 76 pyrazole carboxylic acid hydrazide analogues. The values of the eccentric connectivity index and the Wiener's index of each analogue in the data set were computed and active ranges were identified. Subsequently, each analogue was assigned a biological activity that was compared with the anti-inflammatory activity reported as percent reduction in paw swelling. Prediction with an accuracy of ∼90% was obtained using the eccentric connectivity index as compared to 84% in the case of Wiener's index