pinecone/core-2020-05-10-deduplication huggingface dataset:
Extension of the Unit Disk Gyrogroup into the Unit Ball of Any Real Inner Product Space

AbstractThe group of all holomorphic automorphisms of the complex unit disk consists of Möbius transformations involving translation-like holomorphic automorphisms and rotations. The former are calledgyrotranslations. As opposed to translations of the complex plane, which are associative-commutative operations (i.e., their composition law is associative and commutative) forming a group, gyrotranslations of the complex unit disk fail to form a group. Rather, left gyrotranslations are gyroassociative-gyrocommutative operations (i.e., their composition law is gyroassociative and gyrocommutative) forming agyrogroup. The complex unit disk gyrogroup has previously been studied by the author (Aequationes Math.47, 1994, 240–254). Employing analogies shared by complex numbers and linear transformations of vector spaces, we extend in this article the complex disk gyrogroup and its Möbius transformations into the ball of any real inner product space and its generalized Möbius transformations. A gyrogroup is a mathematical object which first arose in the study of relativistic velocities which, under velocity addition, form a nongroup gyrogroup, as opposed to prerelativistic velocities, which form a group under velocity addition. It has been discovered that the mathematical regularity, seemingly lost in the transition from prerelativistic to relativistic velocities, is concealed in a relativistic effect known as Thomas precession. In its abstract context, Thomas precession is called Thomas gyration, giving rise to our “gyroterminology.” Our gyroterminology, developed by the author (Amer. J. Phys.59, 1991, 824–834), involves terms like gyrogroups, gyroassociative-gyrocommutative laws, and gyroautomorphisms, in which we extensively use the prefix “gyro.
Steady States of the Vlasov–Poisson–Fokker–Planck System

AbstractThe form of steady state solutions to the Vlasov–Poisson–Fokker–Planck system is known from the works of Dressler and others. In these papers an external potential is present which tends to infinity as |x|→∞. It is shown here that this assumption is needed to obtain nontrivial steady states. This is achieved by showing that for a given nonnegative background density satisfying certain integrability conditions, only the trivial solution is possible. This result is sharp and exactly matches the known existence criteria of F. Bouchut and J. Dolbeault (Differential Integral Equations8, 1995, 487–514) and others. These steady states are solutions to a nonlinear elliptic equation with an exponential nonlinearity. For a given background density which is asymptotically constant, it is directly shown by elementary means that this nonlinear elliptic equation possesses a smooth and uniquely determined global solution
Diffraction at a Curved Grating: Approximation by an Infinite Plane Grating

AbstractWe study the influence of the curvature of a circular cylindrical grating on the diffraction of a TE or TM polarized electromagnetic plane wave. For every ratio of the wave length λ to the periodd, we prove the local approximation in the planar field by a plane infinite grating, when the radiusRof the circular grating goes to infinity. The error estimates are optimal and of orderO(d/R) outside some transition regions corresponding to a tangent Bragg's mode or to transition between the region of direct illumination and the region of shadow, where the error is likeO(d/(λ2/3R1/3)). In each region, we calculate the first corrector, solution of an infinite plane grating diffraction problem
Maximal Elements and Equilibria of Generalized Games for Condensing Correspondences

AbstractIn this paper, we first obtain some fixed point theorems in locally convex topological vector spaces which in turn imply some existence theorems of maximal elements for Ψ-condensing correspondences. Then by employing approximation methods, we prove some existence theorems of equilibria for generalized games in which the constraint correspondences are Ψ-condensing and lower or upper semicontinuous instead of having open lower sections or open graphs in infinite dimensional locally convex topological vector spaces. Finally, an existence theorem for theN-person game is also given. In particular, the corresponding results in the literature have been improved
Collision–Ejection Manifold and Collective Analytic Continuation of Simultaneous Binary Collisions in the PlanarN-Body Problem

AbstractWe study simultaneous binary collision (SBC) singularities ofLbinaries in the planarN-body problem, 2L≤N. We introduce the generalized Levi-Civita transformation and follow it by a new transformation which we call the projective transformation near a SBC singularity. We use this transformation to show the following near a SBC singularity:(1) In the generalized Levi-Civita variables, near a SBC singularity, the collection of collision and ejection orbits together with the singularity form a real analytic submanifold, which we call the collision–ejection (CE) manifold.(2) LetRc(Re) be the collection of SBC (SBE) orbits in phase space, i.e., in the original variables. Then, bothRcandReare real analytic.(3) Each collision orbit corresponds to a unique ejection orbit. Together, they form a real analytic orbit in the generalized Levi-Civita variables, which we call a collision–ejection orbit.(4) LetC⊂Rc(E⊂Re) be a submanifold of initial conditions that end (start) in a simultaneous binary collision (ejection) singularity. We show thatC(E) can be chosen to be a real analytic submanifold of codimension 1 inRc(Re), and that the correspondence in item (3) above defines a real analytic section mapping fromCtoE.(5) Collision and ejection orbits can be collectively analytically continued, i.e., each collision–ejection orbit can be written as a convergent power series int1/3, with coefficients that depend real analytically on the initial conditions inC.(6) SBC orbits ofJ<Lbinaries do not accumulate on any SBC orbit ofLbinaries.(7) A single binary collision singularity in theN-body problem is real analytic block-regularizable.We also give the asymptotic behaviour of collision and ejection orbits
Constructing Singular Functions via Farey Fractions

AbstractTo illustrate some points about continued fractions, H. Minkowski in 1904 introduced the so-called ?-function. This function and some generalizations of it are known to be singular, i.e., strictly monotone with derivative 0 almost everywhere. They can be characterized by systems of functional equations, such as f\biggl({x\over x+1}\biggr)=tf(x),\qquad f\biggl({1\over x+1}\biggr)=1-(1-t)f(x)\qquad\hbox{for all $x\in[0,1]$},\eqno{\hbox{(F)}} wheref:[0,1]→Ris the unknown, and r\biggl({x\over x+1}\biggr)=tr(x),\qquad r\biggl({1\over 2-x}\biggr)=t+(1-t)r(x)\qquad\hbox{for all $x\in[0,1]$.}\eqno{\hbox{(R)}} wherer:[0,1]→Ris the unknown. In both cases,t∈(0,1) is a given parameter. In the present note we give a general construction of singular functions, based on the Farey fractions and including, as a special case, the Minkowski function and its generalizations. In contrast to earlier proofs, the methods presented here do not make explicit use of the theory of continued fractions
Fuzzy Measures Defined by Fuzzy Integral and their Absolute Continuity

AbstractGiven a measurable space (X,F), a fuzzy measure μ on (X,F), and a nonnegative functionfonXthat is measurable with respect toF, we can define a new set function ν on (X,F) by the fuzzy integral. It is known that ν is a lower semicontinuous fuzzy measure on (X,F) and, moreover, if μ is finite, then ν is a finite fuzzy measure as well. In this paper, we generalize in several different ways the concept of absolute continuity of set functions, as defined in classical measure theory. In addition, we investigate the relationship among these generalizations by using the structural characteristics of set functions such as null-additivity and autocontinuity, and determine which types of absolute continuity of fuzzy measures are possessed by the fuzzy measure (or the lower semicontinuous fuzzy measure) obtained by the fuzzy integral
Existence of Degenerate Zero-Energy Ground States of the Dirac Hamiltonian for a Spin 1/2 Neutral Particle with an Anomalous Magnetic Moment in an Arbitrary Dimensional Electrostatic Field

AbstractIt is proven that the Dirac HamiltonianHfor a spin 1/2 neutral particle with an anomalous magnetic moment in an arbitrary dimensional electrostatic field has supersymmetric quantum mechanical structure. By estimating the number of the zero-energy ground states ofHfrom below, it is proven that the ground states are degenerate. The estimate is stated in terms of the order of the scalar potential at infinity. In particular, under suitable conditions, existence of infinitely degenerate zero-energy ground states ofHis proven
Box Splines with Rational Directions and Linear Diophantine Equations

AbstractA necessary and sufficient condition for the linear independence of integer translates of Box splines with rational directions is presented in terms of intrinsic properties of the defining matrices. We also give a necessary and sufficient condition for the space of linear dependence relations to be finite dimensional. A method to compute the approximation order of these Box spline spaces is obtained. All these conditions can be tested by finite steps of computations based on elementary properties of the matrices. The method of proofs is from linear diophantine equations
Contractive Completions of Hankel Partial Contractions

AbstractA Hankel partial contraction is a Hankel matrix such that not all of its entries are determined, but in which every well-defined submatrix is a contraction. We address the problem of whether a Hankel partial contraction in which the upper left triangle is known can be completed to a contraction. It is known that the 2×2 and 3×3 cases can be solved, and that 4×4 Hankel partial contractions cannot always be completed. We introduce a technique that allows us to exhibit concrete examples of such 4×4 matrices, and to analyze in detail the dependence of the solution set on the given data. At the same time, we obtain necessary and sufficient conditions on the given cross-diagonals in order for the matrix to be completed. We also study the problem of extending a contractive Hankel block of sizento one of sizen+1
A Constructive Approach to Generalized Integral Transforms

AbstractResults of recent work on distributional extensions of normal operators are used to show that distributional versions of a large number of classical integral transforms can be developed in a systematic way by noting that they can be regarded as functions of a particular normal operator. As examples, we consider logarithmic fractional integrals, Erdélyi–Kober operators, radially symmetric Riesz potentials, the semi-infinite Hilbert transform, and the Hankel transform. We show how particular spaces of test and generalized functions which have been used for developing distributional versions of various integral transforms arise in a natural way
Blocking Probabilities for a Single Link with Trunk Reservation

AbstractA single link in a circuit-switched network is considered. The link hasCcircuits,Rof which are reserved for the primary (directly offered) traffic. Offered calls arrive in independent Poisson streams with mean rates λ and ν for the primary and secondary (rerouted) traffic, respectively, and corresponding independent and exponentially distributed holding times with means 1 and 1/κ. A primary call requires just 1 circuit, whereas a secondary call requirestcircuits, wheretis a positive integer. A primary call is blocked on arrival if allCcircuits are busy, whereas a secondary call is blocked if more thanC−R−tcircuits are busy. Blocked calls are lost to the link. The critically loaded case in which λ⪢1,C−λ=O( $Rλ$t ),R=O( $Rλ$T ), and ν=γ $Rλ$T , where γ=O(1), is investigated. Asymptotic approximations toB1andB2, the blocking probabilities for the primary and secondary traffic, are derived. The results are explicit if κ=1, but involve expansions in powers of γ for κ≠1, in which it is shown how to determine the coefficients recursively. The first two terms in powers of γ are given explicitly. An alternate approach, which involves truncation rather than power series expansions, is presented. The caseR=O(1) is also considered, and explicit results are obtained. Finally, an approximation proposed by J. W. Roberts is investigated. The approximation is shown to be asymptotically correct if κ=1, or ifR=O(1), but not if κ≠1 andR=O( $Rλ$T ). Interestingly, Roberts' approximation corresponds to truncation with just 2 coefficients. Truncation with more coefficients leads to refinements of his approximation
A General Framework for Regularized Evaluation of Unstable Operators

AbstractA general approach is developed for stable evaluation of unbounded operators when the data are corrupted by error. The essence of the approach is to recast the direct problem of evaluating the unstable operator as an inverse problem involving an associated bounded operator. Classical general regularization methods for inverse problems are then elucidated within the new setting, and new order optimal parameter choice strategies for iterative and noniterative regularization methods are developed. Finally, a regularizing stopping rule for a conjugate gradient acceleration of Lardy's method is provided
Real-Time Target Tracking: Optimal Determination of the Characteristics, Using Martingale Tool

AbstractWe study the problem of real-time target tracking: How do we use all the previous observations in order to get, at a given time, the best knowledge of the position of the target? A realistic approach is to take into account the processing time for each observation, and if the precision on the position is low, the processing will take longer. As was shown by C. Olivier (Real-time observability of targets with constrained processing power, to appear), the tracking is efficient when some affine random walk (characterizing the precision of the tracking) is almost surely bounded. Using martingale techniques, we show that this is the case under proper (and realistic) conditions on the parameters
H∞-Optimal Observer-Based Controller Design for Nonlinear Multivariable Systems

AbstractWe propose a novel algorithm to synthesize anH∞-optimal observer-based controller for a nonlinear multivariable system. Based on the parametrization of the observer-based controller, a necessary and sufficient condition to achieve robust stability is developed. Moreover, the Nevanlinna–Pick interpolation theory is employed to test the solvability condition of the robust stabilization problem. Optimal robustness can be achieved via the technique ofH∞-optimization. The exact solution of the model-matching problem induced by the stability criterion is solved by the method of noniterative computation of theH∞-norm
Random Fixed Point Theorems for Various Classes of 1-Set-Contractive Maps in Banach Spaces

AbstractVarious random fixed point theorems for different classes of 1-set-contractive random operator are proved. The class of 1-set-contractive random operators includes condensing and nonexpansive random operators. It also includes semicontractive type random operators and locally almost nonexpansive random operators. Thus results due to S. Itoh [J. Math. Anal. Appl.67(1979), 261–273], T. C. Lin [Proc. Amer. Math. Soc.103(1988), 1129–1135;Proc. Amer. Math. Soc.123(1995), 1167–1176], and H. K. Xu [Proc. Amer. Math. Soc.110(1990), 495–500] are either extended or improved
Generalized Spaces of Difference Sequences

AbstractLetl∞,candc0be the Banach spaces of bounded convergent and null sequencesx=(xk)∞1, respectively. Write Δx=(xk−xk+1)∞1and Δ2x=(Δxk−Δxk+1)1∞. In [Canad. Math. Bull.24(1981), 169–176] H. Kizmaz has introduced and studied the sequence spaces,E(Δ)={x:Δx∈E}, whereE∈{c0,c,l∞}. Recently, [Doga Mat.17(1993), 18–24] Mikail Et defined the setsE(Δ2)={x:Δ2x∈E}. He obtained α-duals of these sets and characterized the matrix class (E,F(Δ2)), whereE,F∈{c0,c,l∞}. In this paper, we generalize these sets and defineE(u;Δ2)={x:u·Δ2x∈E}, whereu=(uk) is another sequence such thatuk≠0 (k=1,2,…). We obtain α- and β-duals of these sets and further we characterize the matrix classes (E(u;Δ2),F) and (E,F(u;Δ2))
Extinction and Persistence of Species in Discrete Competitive Systems with a Safe Refuge

AbstractWe analyze a two species discrete competition model in which one species diffuses between two patches, A and B. In this model, two species, species 1 and 2, compete in patch A with species 1 being the sedentary species. Thus, patch B is a safe refuge for species 2. We obtain sufficient conditions for the extinction of species 1. Species 2 is the superior competitor whenever a linear combination of its growth rates always exceeds the growth rate of the sedentary species 1. By using a specific example, we demonstrate that providing a safe refuge does not always make a species a superior competitor. In fact, without diffusion, species 2 drives species 1 to extinction. However, with the addition of diffusion, there is stable coexistence of the two species. If the safe refuge is not suitable for its growth and reproduction, species 2 may go extinct. We obtain sufficient conditions for the extinction of species 2. We also show that a species persists whenever all of its carrying capacities are sufficiently large. This result rules out the possibility of a population becoming arbitrarily close to zero and therefore risking extinction
A Class of Degenerate Totally Nonlinear Parabolic Equations

AbstractOf concern is the following totally nonlinear parabolic equation, as well as its higher space dimensional analogue \eqalign{u_t(x,t)& =\beta(\phi(x,u_x)u_{xx}+f(x,u,u_x)),\qquad (x,t)\in(0,1)\times(0,\infty)\cr u_x(j,t)&\in(-1)^j\beta_j(u(j,t)),\qquad j=0,1\cr u(x,0)&=u_0(x).\cr} Here β0and β1are maximal monotone graphs inR×R, and β(t) or β′(t) might equal zero for somet, at which the equation is not uniformly parabolic. It is shown by the method of lines and nonlinear operator semigroup theory that the equation has a unique global solution
Unbounded Continuum of Periodic Solutions for Autonomous Delay Equations

AbstractThis work is devoted to the study of the existence of an unbounded continuum of periodic solutions that appear by Hopf bifurcation in non-linear delay differential equations. Our main objective is to give a theorem that guarantees the appearance of an unbounded continuum of periodic solutions. Furthermore, the verification of the hypotheses requires much less effort than the known results in the literature. In the last section some applications to examples already studied show the power of the theorem
Angular Derivatives of Holomorphic Maps in Infinite Dimensions

AbstractLetHbe a complex Hilbert space and letL(H,H) be the complex Banach space of all bounded linear operators fromHtoHwith the operator norm. In the generalized right half-planes and the unit balls contained inHandL(H,H), infinite-dimensional angular sets are defined, various new generalizations of the classical Pick–Julia theorem to infinite dimensions are proved, and conditions of Carathéodory–Fan type, guaranteeing the existence of angular limits and angular derivatives of holomorphic maps of these generalized right half-planes and unit balls, are established
Norm Continuity (fort>0) of Propagators of Arbitrary Order Abstract Differential Equations in Hilbert Spaces

AbstractLet (ACPn),u(n)(t)+∑n−1k=0Aku(k)(t)=0 (t≥0),u(k)(0)=uk(0≤k≤n−1) be strongly wellposed where theAk(0≤k≤n−1) are densely defined closed linear operators in a Hilbert space. LetSk(t) (0≤k≤n−1) be the propagators of (ACPn) andPλ=λnI+∑n−1k=0λkAk. It is shown thatS(k)k(t) (∀ 0≤k≤n−1) is norm continuous fort>0 if and only if \lim_{|\omega|\to\infty}\left\(\tau_0+i\omega)^{n-1}P_{\tau_0+i\omega}^{-1}\right\=0. \lim_{|\omega|\to\infty}\left\(\tau_0+i\omega)^{k-1}\overline{P_{\tau_0+i\omega}^{-1}A_k}\right\=0,\qquad 1\le k,n\le n-1, for a suitable real number τ0
The Effect of a Nonlinear Viscous Regularization on the Solution of a Cauchy–Riemann Equation

AbstractHere we consider a variant of the Cauchy–Riemann equation, in which the Cauchy–Riemann equation has been regularized with a nonlinear second-order viscous term [(ϵ+|ux|2)ux]x. The equation is degenerate of parabolic type when ϵ=0 and has a weak solution for all time. We use an embedding process to analyze the properties of the solution. We also show that the smallest scale of the solution is ϵ, the coefficient of the regularizing second-order term. We numerically confirm the result. We show that from the sequence of regularized solutions, we can extract a converging subsequence. A limit of the subsequenceubelongs toH1([0,1]×[0,T]). For each fixedt, the first-order spatial derivative ofualso belongs toL∞([0,1]). Thereforeubelongs toC0([0,1]×[0,Trsqb;) and, for fixedt, is Hölder continuous with exponent α, α<1. We numerically see that the solution has some of the features of the solution of the porous medium equation even though the equation is complex. We also numerically see the influence of the regularizing second-order diffusion term on the solution
Blow-Up of Solutions with Sign Changes for a Semilinear Diffusion Equation

AbstractThis paper is concerned with the initial-boundary value problem[formula]with the Dirichlet, Neumann, or periodic boundary condition. Here λ>0 is a parameter, andfis an odd function ofusatisfyingf′(0)>0 and some convexity condition. Letz(U) be the number of times of sign changes forU∈C[0,1]. It is shown that there exists an increasing sequence of positive numbers {λk}k=0,1,2,…such that any solution withz(u0)=kblows up in finite time if λ≥λk, and there exists a global solution withz(u0)=kif λ<λk
A Nonlinear Nonlocal Multi-dimensional Conservation Law

AbstractWe study a continuous model which describes the mass transport in droplet dispersions. The model is formulated as a conservation law with several space-variables, {\partial n\over \partial t}+{\partial(Gn)\over\partial r}+\sum_{j=1}^{m}{\partial(H_jn)\over\partial x_j}=0, fort>0,r>0,xj>0, and ∑mj=1xj=1, whereG,Hj(j=1,…,m) are nonlocal functions of the unknownn(t,r,x1,…,xm); moreover, they have exponential singularities atr=0. We prove global existence and uniqueness and establish some numerical properties of the solution
Completely Monotonic and Related Functions

AbstractWe study completely monotonic and related functions whose firstNderivatives are definite on an interval. LetLNdenote the class of functions defined by f\in{\cal L}^N \Leftrightarrow (-1)^k f^{(k)}(t)\ge0,\qquad \forall t>0, \forall k,0\le k\le N. ForN→∞ we writef∈L; such functions are called completely monotonic on (0,∞). We prove in particular that the implication f\in{\cal L}^N \Rightarrow \bigl[\forall\alpha>1\colon f^{\alpha}\in{\cal L}^N\bigr], is true for 0≤N≤5, but false forN≥6. In contrast, L. Lorch and D. J. Newman (J. London Math. Soc.(2),28, 1983, 31–45, Theorem 8) claimed this implication to be false forN=5. Further we prove that the implication f\in{\cal L}\Rightarrow\bigl[\forall\alpha>1\colon f^{\alpha}\in{\cal L}^N\bigr], is true for 0≤N≤6, but false forN≥20
Cesàro Summability of Multi-dimensional Trigonometric-Fourier Series

AbstractThed-dimensional classical Hardy spacesHp(Td) are introduced and it is shown that the maximal operator of the Cesàro means of a distribution is bounded fromHp(Td) toLp(Td) ((2d+1)/(2d+2)<p≤∞) and is of weak type (1,1) provided that the supremum in the maximal operator is taken over a positive cone. As a consequence we obtain the summability result due to Marcinkievicz and Zygmund, more exactly, the Cesàro means of a functionf∈L1(Td) converge a.e. to the function in question, provided again that the limit is taken over a positive cone. Similar results for the (C,β) summability are also formulated