pinecone/core-2020-05-10-deduplication huggingface dataset:
Instant Chaos Is Chaos in Slow Motion

AbstractInstant chaos is the onset of chaotic behaviour as a local bifurcation directly from a trivial steady state. We describe a systematic method for constructing examples of instant chaos, by scaling spatial variables and time. In this way we generalize properties of examples previously studied by other authors. We show that whenever a chaotic attractor of limited amplitude is obtained using a scaling property then it appears in slow motion—for any setStransverse to the vector field, the return time toStends to infinity as we approach the bifurcation point. When instant chaos appears for a family of vector fields with a nontrivial scaling property, if it is not in slow motion then the amplitude of the chaotic attractor becomes arbitrarily large around the bifurcation point. We use this method to obtain the Lorenz attractor in a bifurcation directly from an asymptotically stable equilibrium
Koebe's and Bieberbach's Inequalities in the Banach Algebra of Continuous Functions

AbstractThroughout the paper,C(T) denotes the Banach algebra, with sup norm, of continuous complex-valued functions defined on a compact metric spaceT. ByB(f;r) we denote an open ball inC(T) centered atf∈C(T) of radiusr>0, and byU(a;r) the open ball in the complex planeCata∈Cof radiusr>0 (a neighborhood ofa). In compliance with usual notations concerningunivalent functions, bySwe denote the class of functionsf(z)=z+∑∞p=1apzpunivalentin the unit diskE=U(0;1). By means offractional derivative, interesting generalizations of certain inequalities are obtained. The object of the present article is to generalize the classSin the algebraC(T) and to prove a generalization ofKoebe'sandBieberbach's inequalities:Iff∈Sandz∈E,then[formula
Oscillation and Block-Diagonalization

AbstractAsymptotic methods are used for the two-term differential equationy(n)=p(x)y, withp(x) a real analytic function at infinity. The differential equation is written as a systemY′=A(x)Y, and block diagonalization is utilized to study the asymptotic solutions in the case of coalescing eigenvalues. It is shown that the system may be reduced to one standard system of dimension 2×2. We provide a necessary and sufficient condition for various types of oscillation of the two-term differential equation near infinity. This complements Hille–Kneser theory which gives a sufficient condition and, separately, a necessary condition for oscillation, while it leaves some cases unresolved
Boundary Value Problems for Functional Differential Equations

AbstractThis paper is concerned with the existence of solutions of a boundary value problem for second-order functional differential equations. Specifically the following problem is considered: x''(t) = f(t,x,x'(t)), t \in [0,T], \alpha_0 x_0 - \alpha_1 x'(0) = \varphi \in C_r \beta_0 x(T) + \beta_1 x'(T) = A \in {\Bbb R}^n. The results are based on a nonlinear alternative of Granas and the use of a priori bounds on solutions. Some examples are also discussed to illustrate how these results may be used to yield the existence of solutions of specific boundary value problems
Interior Regularity of the Dirichlet Problem for the Tricomi Equation

AbstractFor the Tricomi equation with Dirichlet boundary conditions, we study the relationship between singularites at the boundary and singularities in the interior of a bounded planar region with smooth non-characteristic boundary. Necessary and sufficient conditions for interior smoothness are stated in terms of microlocal regularity at the boundary and are proven via known microlocal propagation of singularities results along the generalized bicharacteristic flow. In particular, a trapped gliding ray phenomenon at parabolic boundary points is demonstrated under a sharp geometric hypothesis, which provides a microlocal explanation for the possibility of having only isolated singularities at the boundary, which is a question left open in the work of Morawetz
Fuzzy Criterion Set and Fuzzy Criterion Dynamic Programming

AbstractThis paper presents a framework for fuzzy criterion set and fuzzy criterion dynamic programming which generalizes the tools applicable to an array of fuzzy decision models, particularly those represented by the generalized reservoir operation and the stochastic inventory models. The objective is to maximize the expected fuzzy criterion function of the product of fuzzy criterion sets. We prove certain properties of the resultant models including existence, uniqueness, and stability theorems of solutions to this model and show that under the regular hypothesis assumption, the optimal control is a bounded critical number policy
Dulac Criteria for Autonomous Systems Having an Invariant Affine Manifold

AbstractFor a class of higher dimensional autonomous systems that have an invariant affine manifold, conditions are derived to preclude the existence of periodic solutions on the invariant manifold. It is established that these conditions are robust under certain types of local perturbations of the vector field. As a consequence, each bounded semitrajectory on the invariant manifold is shown to converge to a single equilibrium using aC1closing lemma for this class. Applications to autonomous systems that are homogeneous of degree 1 are also considered
Local Reconstructions in Exponential Tomography

AbstractIn this paper, pseudolocal and local approaches to the tomographic reconstruction of discontinuities of an unknown functionffrom its exponential Radon transform datag(θ,p) are developed. A functionfis supposed to be piecewise-continuous and compactly supported. A pseudolocal tomography functionfd(x) is introduced and it is proved that the differencef−fdis continuous. Therefore, locations and values of jumps offcan be recovered fromfd, the computation of which is pseudolocal: for the reconstruction offdat a pointxone needs to knowg(θ,p) only for (θ,p) satisfying |Θ·x−p|≤d. Investigation of the properties offdasd→0 is given. Also a local exponential tomography functionfΛ(μ)is proposed and it is proved thatfΛ(μ)is the result of action onfof an elliptic pseudo-differential operator with the principal symbol |ξ|. Thus singsupp(fΛ(μ))=singsupp(f) and, moreover, the asymptotics offΛ(μ)(x) asx→S, the discontinuity curve off, are established. These asymptotics allow one to find values of jumps offacrossSusing local exponential tomography
An Infinite Dimensional 0-epi Mapping with Degree Zero

AbstractLetXbe a real Banach space and letD⊂Xbe an open and bounded set. A mappingT:D→Xwith 0∉T(∂D) is called 0-epi mapping onDif the equationTx=g(x) is solvable inDfor every compact continuous mappingg:D→Xwhich vanishes identically on ∂D. We show that there exists a mappingT:l2⊃D→l2such that the mappingI−Tis 0-epi, but the degree deg(I−T−g,D,0) is well defined and equals zero, for any such functiong. This says that the degree theory cannot be applied directly toI−T−gin order to guarantee the solvability of (I−T)x=g(x) inDfor any mappinggas above. This fact provides a good justification for the study of 0-epi mappings by M. Furi, M. Martelli, and A. Vignoli (Ann. Mat. Pura Appl.124, 1980, 321–343) in infinite dimensional spaces
Asymptotic Periodicity, Monotonicity, and Oscillation of Solutions of Scalar Neutral Functional Differential Equations

AbstractWe consider the periodic scalar neutral functional differential equation (d/dt)[x(t)−c(t)x(t−τ)]=−h(t,x(t))+h(t−σ,x(t−σ)), wherecis continuously differentiable,his increasing in its second argument, and bothcandhare 1-periodic in thet-variable. The two time-lags τ and σ are not required to be the same. It is shown that, under certain conditions, (i) the set of 1-periodic solutions is an ordered arc and each solution is convergent to a periodic solution; (ii) the asymptotic and oscillatory behaviors of each solution are completely classified in terms of the value of the first integral at the initial condition
First Return Approachability

AbstractWe consider real-valued functions defined on [0,1]. Both the class of Baire one functions and the class of Baire-one, Darboux functions have been characterized using first return limit notions. The former class consists of the first return recoverable functions and the latter consists of the first return continuous functions. Here we introduce a natural intermediate type of first return limiting process, first return approachability, and show that the first return approachable functions are precisely those Baire class one functions whose graphs are dense in themselves. Also, the set of points at which a function is first return approachable, but not first return continuous, is shown to be σ-porous
On Spătaru's Extension of the Hsu-Robbins-Erdo&#x030B;s Law of Large Numbers

AbstractThe Hsu-Robbins-Erdo&#x030B;s law of large numbers (1947, 1949) states that ifX1,X2,… are independent identically distributed random variables andSn=X1+···+Xn, then[formula]for every ε>0 if and only ifE[X21]<∞ andE[X1]=0. Under some auxiliary conditions, Spătaru (1994) extended this to the case where theXnare no longer identically distributed, but rather their distributions come from a finite set of distributions. We improve Spătaru's conditions, and present a counterexample to a conjecture of his
A Pseudospectral Method for Hammerstein Equations

AbstractThis paper considers Hammerstein equations[formula]and presents a pseudospectral method for solving them numerically. The method is applied to approximate the solution not to the equation in its original form, but rather to an equivalent equationz(t)=g(t,y(t)). The functionzis approximated by a polynomialzNof degreeN, with coefficients determined by collocation at the Legendre–Gauss–Lobatto nodes. We define the desire approximation toyby[formula]and establish that, under suitable conditions, limN→∞yN(t)=y(t), uniformly int. Advantages of the proposed method, compared with existing methods in the literature, are discussed. An illustrative example is included to show the accuracy of the proposed method
Quasi-homogeneous Critical Swirling Flows in Expanding Pipes, Part II

AbstractTime-independent swirling flows in rotationally symmetric pipes of constant and varying diameter are constructed using variational techniques. In Part I, by E. van Groesen, B. W. van de Fliert, and E. Fledderus (J. Math. Anal. Appl.,192(1995), 764–788.) critical flows in pipes of uniform cross-section were found by extremizing the cross-sectional energy at constrained value of the cross-sectional helicity and the axial flow rate. In this paper we add the cross-sectional angular momentum to the constraints and find an unfolding of the family of flows obtained in Part I. The flows are steady Beltrami-type flows, independent of the axial coordinate, and show the phenomenon of flow-reversal at the central axis of the pipe. Closely related to the unfolding is the introduction of a complexity parameter; instead of taking only discrete values for the flows studied in Part I, now the complexity can change in a continuous way. An approximation for swirling flows in a pipe with varying circular cross-section is constructed in a quasi-homogeneous way as a succession in the axial direction of critical flows in uniform pipes. The evolution of the constraint values in the axial direction is determined by the conservation laws for the energy, angular momentum, and the axial flux. A clear presentation of the mechanism behind the evolution of parameters in the base-flow is so obtained. The consistency conditions that follow from the conservation laws are shown to be related to the solvability conditions of the linearized equation
Asymptotic Behavior of Solutions of Functional Differential Equations with Finite Delays

AbstractIn this paper we consider a sufficient condition forW(t,x(t)) to approach zero ast→∞, wherex(t) is a solution of a non-autonomous functional differential equation with finite delays andW(t,x) is a so-called Lyapunov function. We shall show that in the applications this provides useful information for asymptotic behavior of the solutionx(t). For example, we generalize examples given by J. R. Haddock and J. Terjéki (J. Differential Equations48, 1983, 95–122) to the case of non-autonomous systems
Linear Hamiltonian Difference Systems: Disconjugacy and Jacobi-Type Conditions

AbstractWe consider a linear Hamiltonian Difference System for the so-called singular case so that discrete Sturm–Liouville Equations of higher order are included in our theory. We introduce the concepts of focal points for matrix-valued and generalized zeros for vector-valued solutions of the system and define disconjugacy for linear Hamiltonian Difference Systems. We prove a Reid Roundabout Theorem which gives conditions equivalent to positive definiteness of a certain discrete quadratic functional, among them the strengthened Jacobi's Condition and a condition on a certain Riccati Difference Equation. The key to this theorem is a discrete version of Picone's Identity. Furthermore, for the sake of generalization of our theorem, we introduce controllability for linear Hamiltonian Difference Systems and prove a Reid Roundabout Theorem for a more general functional and more general boundary conditions
A Mollification Method for a Noncharacteristic Cauchy Problem for a Parabolic Equation

AbstractIn this paper the noncharacteristic Cauchy problem (NCP)[formula]is considered. With ϕ∈Lp(R),p∈[1,∞], it is proved that a solution of NCP exists if and only if ϕ is infinitely differentiable and ‖ϕ(n)‖Lp(R)≤c(2n)!s2n, ∀n∈N, for certain constantscands. NCP is well known to be severely ill-posed: a small perturbation in the Cauchy data may cause a dramatically large error in the solution. The following mollification method is suggested for solving NCP in a stable way: If ϕ∈Lp(R) is given inexactly by ϕϵ∈Lp(R) then we mollify ϕϵby convolutions with the Dirichlet kernel and the de la Vallée Poussin kernel. The exact solution of NCP is approximated by the solution of the mollified problem with a reasonable choice of mollification parameters which yields error estimates of the Hölder type. By the method we can work with the data inLp(R),p∈[1,∞] and obtain several sharp stability estimates inLp- andL∞-norms of the Hölder type for the solution of the problem. The method can easily be implemented numerically using the fast Fourier transform. A stable marching difference scheme based on this method is suggested. Several numerical examples are given, which show that the method is effective
Nonhomogeneous Boundary Value Problems

AbstractThe diffusion equation is solved under stochastic nonhomogeneity using eigen function expansion and the Georges method. The statistical moments of the solution process are computed through the two previously mentioned techniques and proved to be the same. A general solution is obtained under general initial and boundary conditions. A random source composed of deterministic and stochastic parts is taken into consideration. The stochastic part is then restricted to a generalized Gaussian field, mainly modulated white noise. A special case is considered under constant noise level and constant average noise. A numerical case study concerning pollution in a stream is solved and a parametric study is achieved through various figures
Means, Generalized Divided Differences, and Intersections of Osculating Hyperplanes

AbstractWe define means innvariables by taking the intersection point inRnofnosculating hyperplanes to a given curve inRn. These planes are the natural extensions of the osculating plane inR3. More precisely, letCbe a curve inRn, and let 0<a1<···<an<∞. LetOkbe the osculating hyperplane toCatak. Under certain assumptions on the component functionsxk(t) ofC, theOkwill have a unique point of intersectionP=(i1,…,in) inRn. Furthermore,a1<xk−1(ik)<anfork=1,2,…,n. This definesnsymmetric meansMk(a1,…,an)=xk−1(ik). Ifxk(t)=tpk,k=1,…,n, then the meansMkare homogeneous. In particular, ifxk(t)=tk,k=1,…,n, thenM1(a1,…,an)=arithmetic mean=(∑i=1nai)/nandMn(a1,…,an)=geometric mean=(a1···an)1/n. Also, ifxk(t)=t−k,k=1,…,n, thenM1equals the harmonic meanH=n/(1/a1+···+1/an). Finally, ifxk(t)=tk,k=1,…,n−2,xn−1(t)=log(t), andxn(t)=1/t, thenMn(a1,…,an)=L(a1,…,an). HereLis the logarithmic mean innvariables defined by A. O. Pittenger (Amer. Math. Monthly92, 1995, 99–104), given by[formula]wherep(1,i,n)=∏nj=1,≠i(ai−aj). The meansMkare generalizations of meansm(a,b) defined by taking intersections of tangent lines to curvesCin the plane, discussed in an earlier paper by the author (J. Math. Anal. Appl.149, 1990, 220–235)
Conformal Mapping of Nonsmooth Domains via the Kerzman–Stein Integral Equation

AbstractThe connection between the Riemann map and the Szego&#x030B; kernel is classical. But the fact that there is an efficient numerical procedure, based on the Kerzman–Stein integral equation, for computing the Szego&#x030B; kernel of a smoothly bounded domain in the plane is more recent. In this paper it is shown how to extend the results of Kerzman and Stein to certain nonsmooth domains. This provides a new method for numerically computing the Szego&#x030B; kernel, and hence the Riemann map, of these domains
A Spectral Approach to Distributional Extensions of Normal Operators

AbstractA constructive method for producing a test function space and hence a generalized function space suitable for a given normal operator is developed. Spectral theory and a functional calculus are shown to be valid in these spaces. Application of the theory is illustrated by considering a normal realisation of the operatorx(d/dx) and demonstrating that the test and generalized function spaces for this operator coincide with certain spaces originally introduced by McBride in the theory of fractional calculus. It is shown that the so-called Hadamard operators arise as functions ofx(d/dx) and so a distributional theory of these operators can be obtained using the functional calculus in the generalized function space
Positive Homogeneous Functionals Related toLp-Norms

AbstractOne of the results reads as follows. Let (ΩΣ,μ) be a measure space with at least two disjoint sets of finite and positive measure. Suppose thatm,φ,ψ:(0,∞)→(0,∞) are functions such that φ and ψ are bijective, and φ(1)=1=ψ(1). Then[formula]for all nonnegative simple functionsx:Ω→R,x≢0 μ-a.e., and allt<0, where Ω(x)≔{ω∈Ω:x(ω)≠0}, if, and only if,m,φ,ψ are multiplicative and ψ=m∘φ−1. If, moreover, arbitrary two functions chosen from the set {m,φ,ψ} satisfy some modest regularity assumptions then the homogeneity relation (∗) holds true if, and only if,m, φ, and ψ are the power functions
Iterative Solution of Nonlinear Equations of the Φ-Strongly Accretive Type

AbstractLetq>1, and letEbe a realq-uniformly smooth Banach space. LetT:E→Ebe a Lipschitz φ-strongly accretive operator and suppose the equationTx=f,f∈E, has a solution. It is proved that under suitable conditions on the real sequences {αn}n=0∞and {βn}n=0∞, the iteration process[formula]converges strongly to the unique solution of the equationTx=f. A related result deals with the approximation of fixed points of φ-hemicontractive operators—a class of operators which is much more general than the important class of strongly pseudocontractive operators
An Optimal Control Formulation of the Blaschke–Lebesgue Theorem

AbstractThe Blaschke–Lebesgue theorem states that of all plane sets of given constant width the Reuleaux triangle has least area. The area to be minimized is a functional involving the support function and the radius of curvature of the set. The support function satisfies a second order ordinary differential equation where the radius of curvature is the control parameter. The radius of curvature of a plane set of constant width is non-negative and bounded above. Thus we can formulate and analyze the Blaschke–Lebesgue theorem as an optimal control problem
LpEstimates for Some Ultraparabolic Operators with Discontinuous Coefficients

AbstractWe consider a class of ultraparabolic operators of the kind[formula](z=(x,t)∈RN+1), where the principal part is uniformly elliptic on Rq,q≤N, and the constant matrixBis upper triangular and such that the operator obtained by freezing the coefficientsaijat any pointz0∈RN+1is hypoelliptic. We prove local Lp-estimates for the derivatives ∂xixju(i,j=1,…,q) of a solution to the equationLu=f, under the assumption that the coefficientsaijbelong to the space VMO (“vanishing mean oscillation”) with respect to a suitable metric related toB
A Sonine-Gegenbauer Integral of the Neumann Function

AbstractMotivated by a desire to express in closed form a certain potential occurring in the study of diffraction of a plane electromagnetic wave by a wedge, Miller and Exton developed computable expressions for a large class of Sonine-Gegenbauer type integrals. In the present paper one of these integrals, given previously in terms of generalized hypergeometric functions in three variables, is now obtained in terms of hypergeometric functions in two variables, thereby much reducing the complexity of the representation. In the course of this investigation certain identities involving Srivastava'sF(3)-function, Kampé de Fériet functions, and Wright functions are deduced. In addition, evaluations of integrals of Bessel functions related to the above analysis are provided
Lipschitz–Nikolskii Constants for the Trotter–Feller Operator

AbstractIn this paper we determine the Lipschitz–Nikolskii constants for the Trotter–Feller operator which contains various well-known operators. As corollaries to our general settings we obtain the Lipschitz–Nikolskii constants for a series of concrete operators including the Bernstein, Szász, Gamma, Weierstrass, and Baskakov operators, their generalizations such as the Cheney–Sharma and Bleimann–Butzer–Hahn operators, and many others. Our results also improve Rathore's on the Meyer-König and Zeller operator and the Gamma operator of Müller as well as Rathore and Singh's on the Post–Widder operator. Throughout the paper the probabilistic method is used intensively while a result in probability theory on normal approximation plays a key role
Bifurcation Theory for a Rod with Small Bending Stiffness

AbstractIn this paper we study the equilibrium states of a conducting rod with small bending stiffness in a magnetic field. The magnetic field is produced by current flowing in a pair of infinitely long parallel wires. The line between the supports of the rod is in the plane of the wires and equidistant from them. The rod is clamped at both ends. We consider planar deformations of the rod. We prove a bifurcation theorem describing the set of equilibrium states. Our analysis of this problem brings together two important theories in modern applied mathematics; bifurcation theory and the theory of singular perturbations for systems of nonlinear ordinary differential equations
Convergence and Error Bounds for Passive Stochastic Algorithms Using Vanishing Step Size

AbstractThis work is concerned with passive stochastic approximation (PSA) algorithms having vanishing or decreasing step size and window width. Unlike the traditional stochastic approximation methods, the passive stochastic approximation algorithms utilize passive strategies. Under the framework of PSA, not only the measurement noise is unobservable, but also the “state” {xn} is a randomly generated sequence. In our formulation, both the observation noise and the randomly generated {xn} are correlated random processes. Under rather general conditions, w.p.1. convergence of the algorithms is established. Then upper bounds on estimation errors are obtained. It is shown that the bounds depend on the smoothness of the function under consideration in an essential way, which reveals another distinct feature of the passive algorithms