pinecone/core-2020-05-10-deduplication huggingface dataset:
Solutions to Three Functional Equations Arising from Different Ways of Measuring Utility

AbstractUtility of gains (losses) can be measured in four distinct ways: riskless vs risky choices and gains (losses) alone vs the gain-loss trade-off. Conditions forcing these measures all to be the same lead to functional equations, three of which are displaylines{F^{-1}[F(x)+F(-Y)]Z=F^{-1}[F(XZ)+F(-YZ)]\cr \hfill (F\colon \left]-k,k'\right[\to \left]-K.K'\right[;\ k,k',K.K'>0)\quad\hbox{(i)}\cr F(X-R)[1-F(Y)]+F(Y)=F[F^{-1}(F(X)[1-F(Y)]+F(Y))-S]\cr \hfill(F\colon \left[0,1\right[\to\left[0,\right[)\quad\hbox{(ii)}\cr \qquad F^{-1}[F(X)+F(Y)-F(X)F(Y)]Z\hfill\cr {}=F^{-1}[F(XZ)+F[YP(X,Z)]-F(XY)F[YP(X,Z)]]\cr \hfill(F\colon\left[0,1\right[\to\left[0,1\right[,\ P\colon\left[0,1\right[\times[0,1]\to[0,1]).\quad\hbox{(iii)}\cr} We determine all strictly increasing, surjective (and thus continuous) solutions of (i) and (ii) and all strictly increasing, surjective solutions of (iii) that are differentiable on [0,1[ as are their inverses (thus,F′≠0 on ]0,1[)
Complex Strongly Extreme Points in Quasi-Normed Spaces

AbstractWe study the complex strongly extreme points of (bounded) subsets of continuously quasi-normed vector spacesXoverC. WhenXis a complex normed linear space, these points are the complex analogues of the familiar (real) strongly extreme points. We show that ifXis a complex Banach space then the complex strongly extreme points ofBXadmit several equivalent formulations some of which are in terms of “pointwise” versions of well known moduli of complex convexity. We use this result to obtain a characterization of the complex extreme points ofBlp(Xj)j∈IandBLp(μ,X)where 0<p<∞,Xand eachXj,j∈I, are complex Banach spaces
Structural Operators and Eigenmanifold Decomposition for Functional Differential Equations in Hilbert Spaces

AbstractThis paper studies spectral properties of linear retarded functional differential equations in Hilbert spaces with the emphasis on their relations to structural operators. The equations involve unbounded operators acting on the discrete and distributed delayed terms, and the operators acting on the instantaneous term are defined through sesquilinear forms. The main concern of this paper is studying the spectral properties of the infinitesimal generators associated with the solution semigroups by means of structural operators. The characterizations of eigenmanifolds are derived and the relations between the manifolds and structural operators are shown by using the properties of structural operators
Perturbedp-Laplacian inRN: Bifurcation from the Principal Eigenvalue

AbstractIn this paper we consider the perturbed bifurcation problem inRN, -\mathop{\rm div}\bigl(a(x,u)|\ abla u|^{p-2}\ abla u\bigr)=\lambda g(x)|u|^{p-2}u+f(\lambda,x,u),\eqno(1) and the associated problem with homogeneous principal part, -\mathop{\rm div}\bigl(a_0(x)|\ abla u|^{p-2}\ abla u\bigr)=\lambda g(x)|u|^{p-2}u+f(\lambda,x,u),\eqno(2) where 1<p<N. We prove that the principal eigenvalue λ1of the homogeneous eigenvalue problem -\mathop{\rm div}\bigl(a_0(x)|\ abla u|^{p-2}\ abla u\bigr)=\lambda g(x)|u|^{p-2}u is a bifurcation point for (1) and (2). We emphasize thataanda0are both positive inRNand may be singular or degenerate at infinity, no growth restriction ona(x,·) is postulated, and bothfandgmay change sign
Generalizations of Dobrushin's Inequalities and Applications

AbstractLetf:Rn→Rbe a seminorm and let (ei)1≤i≤nbe the canonical base ofRn. DenoteM=12 maxr,sf(er−es),K=maxrf(er). We prove the inequality f(x)\le M\biggl(\sum_{i=1}^n|x_i|\biggr)+(K-M)\biggl|\sum_{i=1}^n x_i\biggr|,\qquad x=(x_1,x_2,\ldots,x_n)\in{\Bbb R}^n. We use the above inequality to prove some generalizations of Dobrushin's inequalities and a generalization of an inequality due to J. E. Cohenet al.(Linear Algebra Appl.179, 1993, 211–235). Hilbert space generalizations of the above inequalities are proved using Levi's reduction theorem. As special cases of our results we obtain several inequalities given previously by Adamovici, Djokovic, Hlawka, and Hornich
Bifurcation Analysis of a Chemostat Model with a Distributed Delay

AbstractA chemostat model of a single species feeding on a limiting nutrient supplied at a constant rate is proposed. The model incorporates a general nutrient uptake function and a distributed delay. The delay indicates that the growth of the species depends on the past concentration of nutrient. Using the average time delay as a bifurcation parameter, it is proven that the model undergoes a sequence of Hopf bifurcations. Stability criteria for the bifurcating periodic solutions are derived. It is also found that the periodic solutions become unstable if the dilution rate is increased. Computer simulations illustrate the results
Another Approach to the Existence of Value Functions of Stochastic Differential Games

AbstractThe existence of value functions for general two-player, zero-sum stochastic differential games has been obtained by Fleming and Souganidis. In this paper we present a new approach to this problem. We prove optimality inequalities of dynamic programming for viscosity sub- and supersolutions of the associated Bellman–Isaacs equations. These inequalities are well known for deterministic differential games but are new for stochastic differential games. It then easily follows that value functions are the unique viscosity solutions of the Bellman–Isaacs equations and satisfy the principle of dynamic programming. The results presented here are not the same as those of Fleming and Souganidis because we work with different reference spaces and the independence of value functions of the choice of reference spaces is not clear to us
Travelling Wave-Like Solutions of the Navier–Stokes and the Related Equations

AbstractWe present a new family of travelling wave-like solutions to the Navier–Stokes equations of incompressible fluid flows, and other regularized equations of the Euler equations, obtain their trend to the solutions of the Euler equations as the viscosity tends to zero, and estimate the rate of convergence. We also find a “singularizing effect” of the viscosity term in the Navier–Stokes equations, i.e., we have a local moving blow-up of unbounded solutions with the blow-up's speed depending on viscosity. We demonstrate that if the initial function is the Beltrami flow then the solution of the Navier–Stokes equations conserves the Beltrami flow property for all time
On the Stability of Some Exponential Polynomials

AbstractThis paper studies elementary transcendental equations of the type (z2+pz+q)eτz+rzn=0, wherep,q,r∈ R , τ,p>0,q≥0,r≠0,n=0,1,2. We are mainly interested in the casen=0 for which a characterization of stability is accomplished; that is, we state a necessary and sufficient condition for all the roots to lie to the left of the imaginary axis. Also a characterization of stability independent of delays is given. Sufficient conditions for instability are stated for the casesn=1,2. The proofs are carried by elementary arguments independent of usual tools
Integration over Unbounded Multidimensional Intervals

AbstractIn this paper we define an integral of Kurzweil–Henstock type over multidimensional unbounded intervals having the property that a multiple series converges in the sense of Hardy–Móricz if and only if the associated step function is integrable in this sense. We show that this integral has good properties and in particular Hake's property characterizing the integrability in terms of some convergence property of the associated indefinite integral, and a divergence theorem on unbounded intervals
On the Two-Dimensional Navier–Stokes Equations in Stream Function Form

AbstractIn this paper the initial-boundary value problem of the Navier–Stokes equations in stream function form is considered. A trilinear form is introduced to deal with the nonlinear term. A weak formulation of this problem is provided. The existence of a weak solution is proved by an auxiliary semi-discrete Faedo–Galerkin scheme and a compactness argument. The uniqueness and regularity of the solution are discussed. Finally the convergence of the numerical solution and the converge rate with a certain choice of basis in the Faedo–Galerkin method are given
Systems of Parabolic Equations with Continuous and Discrete Delays

AbstractIn this paper we investigate the global existence and the dynamics of a coupled system of nonlinear parabolic equations where the nonlinear “reaction function” may depend on both continuous (infinite or finite) and discrete delays. It is shown that if the reaction function is locally Lipschitz continuous and the system possesses a pair of coupled upper and lower solutions then there exists a unique global solution to the system without any quasimonotone condition on the reaction function. For systems with mixed quasimonotone reaction functions we use the monotone method to establish more dynamic property of the parabolic system in terms of the quasisolutions of the corresponding elliptic system. This approach yields a (global) attractor of the parabolic system, and under some additional conditions this attractor leads to the existence and asymptotic stability of a solution of the elliptic system. Application to three model problems in population dynamics and chemical reactions is given
On Explicitly Solvable Gradient Systems of Moser–Karmarkar Type

AbstractA generalization and a classification of explicitly solvable gradient systems which appear in linear programming and neurodynamics are studied, It is shown that nonlinear dynamical systems of Moser–Karmarkar type generically take both a Lax representation and a gradient form. A related optimization problem on a simplex is discussed. An integration procedure and a complete description of the invariant manifolds and the phase portrait of the dynamical systems of Moser–Karmarkar type are also investigated
Hamiltonian Dynamics of Polygons and Integrability

AbstractWe introduce a class of Hamiltonian systems generating motion of polygons in plane and define “geometrical” Hamilton functions depending arbitrarily on distances between the vertices and the areas of the inscribed triangles. The motion preserves the centroid and the moment of inertia if unit masses are assigned to the vertices. Therefore the 3-vertex systems corresponding to the dynamics of triangles are completely integrable. In the case of Hamiltonians depending only on the areas there is an additional integral leading to integrable dynamics of the quadrilaterals. For a particular subclass of such Hamiltonians the total area is also preserved, which results in integrable motion of pentagons. We present several numerical illustrations of the polygon dynamics
Inequalities for Indefinite Forms

AbstractLetn≥2 be a fixed integer andΦp(x)≔(x1p−x2p−···−xnp)1/p(x∈Rp),whereRpdenotes the set of all vectorsx=(x1,…,xn) for whichxi≥0 (i=1,…,n),x1p≥x2p+···+xnpifp>0 andxi>0 (i=1,…,n),x1p>x2p++···+xnpifp<0.Three inequalities are presented for Φp. The first is a comparison theorem. The second is a “Hölder-like” generalization of Aczél's inequality (an extension of Popoviciu's inequality), while the third is a generalization of Bellman's inequality to all possible values ofp. The proofs show that the above inequalities are consequences of some well-known inequalities for power means and power sums
On Nilpotent Semigroups and Solutions with Finite Stopping Time

AbstractWe consider here the evolution equationu′(t)=Bu(t), whereBis some unbounded closed operator with dense domain in some separable Hilbert space. We consider the non-trivial classical solutionu(t) of the last equation such thatu(t)=0 fort>T. We are interested in finding conditions on operatorBfor this to occur. There are two cases: in the first case operatorBgenerates a nice semigroup and the inverse to it is an abstract Volterra operator without point spectra, the Cauchy problem is well-posed in this case, and every solution will be zero in finite time; in the second case every point of the complex plane is in the spectral of operatorBand so it cannot generate any semigroup and the Cauchy problem in this case is not well-posed. More precisely, there is no uniqueness for solution of the Cauchy problem in the last case. It is interesting to note that such a solution can occur only in two extreme situations: when the spectra of operatorBare trivial, or when every point of the complex plane is in it
A Two-Line Algorithm for Provingq-Hypergeometric Identities

AbstractWe show thatq-hypergeometric identities ∑kF(n,k)=1 can be proved by checking that they are correct for only finitely many,Nsay, values ofn. We give a specifica prioriformula forN, as a polynomial of degree 24 in the parameters ofF(n,k). We see this because of the presence of “q”, the estimates ofNcan be made smaller than the general estimates that were found in the author's thesis8. As an example of the method we show that theq-Vandermonde identity can beprovedby “only” checking that its first 2358 cases (i.e., values ofn) are correct, by direct computation
The Classification of Linearly Polarized Transverse Electric Waves

AbstractRecently the author has undertaken the classification of continuous solutions to some common vector PDEs. These include the simplest of Beltrami solutions to hydrodynamic flows and electromagnetic wave equations (which are, in fact, closely related). In this paper, we consider linearly polarized transverse electric wave solutions to the electromagnetic wave equation:∇×E=iωH;∇×H=−iωE. Using Clebsch functions and differential geometric techniques the author is able to give a discussion of the most general possible forms of propagation of linearly polarized waves. A zero curvature condition (which asserts that the Clebsch functions necessary to represent the solutions are expressible in terms of Cartesian variables) reduces the study of vector PDEs to nonlinear PDEs, which may be solved in special cases. The benefit is that the Clebsch representation is the one that is the most useful for analyzing the structure of the flow. This method of analysis is now a feasible solution method due to the tremendous advances in mathematical software allowing one to compute curvatures, given metric functions. For the special case of linearly polarized transverse electric waves the analysis may be applied to a great extent
Generalized Solutions of Boundary Problems for Layered Composites with Notches or Cracks

AbstractA method is presented for solutions of a class of boundary value problems corresponding to problems of a layered composite with a notch, or in particular, a crack. In this paper the method is applied to problems reducible to Poisson's partial differential equation, namely heat conduction, mass diffusion in solid bodies, consolidation, and antiplane fracture mechanics. The examples which we discuss in this paper refer to problems of heat conduction in solids. Such problems have a direct physical explanation. It is a matter of replacing the coefficients of thermal conductivity λiby the shear moduli μito obtain antiplane problems of fracture mechanics. We apply the Fourier and Mellin transforms technique for generalized functions and reduce the problem to solving a singular integral equation with fixed singularities on the semi-axis. The method is a generalization of the classical approach on the cases when we deal with distributions
Products of Vector Measures

AbstractWe consider the problem of the existence of the product of two vector measures with respect to a continuous bilinear map. We study conditions on the linear map induced by the bilinear form in order to assure the existence of the product of any two vector measures with respect to it. We show that every measure with values inlpadmits product with any measure with values inlqwith respect to any continuous bilinear map defined onlp×lqif and only ifp,q∈[1,2] and min{p,q}=1. We also show that there exist measures that admit product with any vector measure with respect to a fixed bilinear map although their semivariations in the sense of Bartle do not have a control measure
A Semantics for Possibility Theory Based on Likelihoods

AbstractIn this paper, a semantic basis for Possibility Theory based on likelihood functions is presented. In some cases, possibilities have been considered as approximations of Shafer plausibility measures. This approximation exchanges exactness of plausibility values for the simplicity of use of possibility values. In this paper, a different direction is followed. Possibility measures are considered as the supremum of a family of likelihood functions. This is an exact interpretation, not an approximation. The minimum rule to combine possibility distributions is justified in this framework under general conditions. Conditions under which other rules can be applied are also studied
A Class of Orthogonal Polynomials on the Radial Rays in the Complex Plane

AbstractWe introduce a class of polynomials orthogonal on some radial rays in the complex plane and investigate their existence and uniqueness. A recurrence relation for these polynomials, a representation, and the connection with standard polynomials orthogonal on (0,1) are derived. It is shown that their zeros are simple and distributed symmetrically on the radial rays, with the possible exception of a multiple zero at the origin. An analogue of the Jacobi polynomials and the corresponding problem with the generalized Laguerre polynomials are also treated
On 2-Homogeneous Polynomials on Some Non-stable Banach and Fréchet Spaces

AbstractLetFbe a Banach or a nuclear Fréchet space isomorphic to its square. ThenP(2F), the space of 2-homogeneous polynomials onF, is isomorphic to the space of continuous linear operatorsL(F,F′), both of them endowed with the topology of uniform convergence on bounded sets. In this note we prove that the isomorphism can fail ifFis not stable by studying two kind of examples: First, for Banach spaces, we consider James spacesJpconstructed with thelp-norm, withp>2; second, we treat nuclear power spaces of finite or infinite type
Oscillation Results for Emden–Fowler Type Differential Equations

AbstractThe third order nonlinear differential equationx‴+a(t)x′+b(t)f(x)=0, (∗)is considered. We present oscillation and nonoscillation criteria which extend and improve previous results existing in the literature, in particular some results recently stated by M. Greguš and M. Greguš, Jr., (J. Math. Anal. Appl.181, 1994, 575–585). In addition, contributions to the classification of solutions are given. The techniques used are based on a transformation which reduces (∗) to a suitable disconjugate form. To this aim auxiliary results on the asymptotic behavior of solutions of a second order linear differential equation associated to (∗) are stated. They are presented in an independent form because they may be applied also to simplify and improve other qualitative problems concerning differential equations with quasiderivatives
Convergence Theorems and Asymptotic Integration for Functional Differential Equations

AbstractThis paper is devoted to the investigation on convergence of solutions and asymptotic integration of some functional differential equations (FDEs). In the first part of the paper, some convergence criteria are obtained, and an invariance principle is established for the convergence of solutions of linear FDEs under perturbations and disturbances withL1-integrable coefficients. These theorems are applicable to some nonlinear or nonhomogeneous equations. In the second part, the convergence theorems are applied to investigate the asymptotic integration of some linear FDEs. This research is made mainly to discuss a situation that was avoided by some previous literature. Some examples are given to illustrate the theorems
The Characterization of Third Order Ordinary Differential Equations Admitting a Transitive Fiber-Preserving Point Symmetry Group

AbstractThird order ordinary differential equations admitting a transitive symmetry group of fiber-preserving point transformations are completely characterized. We give the general expressions of these equations, as well as the canonical equations representing the equivalence classes. The symmetry groups admitted by these equations are also characterized. In some cases we give the transformation taking the canonical equation into the general one. The linearization problem for third order equations is solved
The Solution of the Riemann Problem for a Hyperbolic System Modeling Polymer Flooding with Hysteresis

AbstractIt is well known that multiphase flow in porous media exhibits hysteresis. This is typically modeled by modifying the saturation dependence of the relative permeabilities. In this paper, a model for hysteretic relative permeabilities is built into the polymer flooding model and the analytical solution to the corresponding Riemann problem is constructed. This produces a nonstrictly hyperbolic system of conservation laws with a history-dependent flux function. Because the polymer model without hysteresis possesses Riemann problem solutions that are not monotonic, the introduction of hysteresis necessarily produces structurally different solutions. We show that hysteresis produces more complicated solutions with more fronts and expansions; and removes some nonuniqueness of solutions
Smoothing Effect and Propagations of Singularities for Viscoelastic Plates

AbstractWe study dissipative models for plates and we show that the solutions have a smoothing effect on the initial data for viscous plates, while for materials with memory the solution propagates singularities, that is, the solution of the plate equation of memory type is as regular as the initial data. Moreover, we show that when both dissipations are present, the memory type prevails in the sense that the solution propagates singularities. Finally, we prove the existence of global solutions for non-linear dissipative equations, with small data, which decay exponentially as time goes to infinity
A Nonlocal Parabolic Equation Arising in a Turbulence Model

AbstractWe consider a class of parabolic partial differential equations with nonlocal diffusion coefficient. The study is motivated by a model of turbulent viscosity for a developed flow in a duct between parallel walls. Under rather more general conditions, we use a rescaling intto prove existence and well-posedness for weak solutions. Sharp comparison estimates enable us to bound from below the spatial derivative at the wall. The steady state problem is also nonstandard and we show local stability by a spectral continuation argument. Finally numerical results for the steady state solution are compared with available experimental data
Bounds for Certain Harmonic Sums

AbstractThe monotonicity properties of the functionΦ(n)=(pn+r+1)−1+(pn+r+2)−1+···+(qn+s)−1are determined, wherep,q,r, andsare fixed integers such that 0<p<qand 0≤p+r<q+s. The results extend earlier results of Adamović and Tasković (1969) and Simić (1979) for the casesr=s=0 andr=0,s=1. We settle negatively a conjecture of Simić that Φ(n) is always monotonic when 0≤r≤s. The results enable us to obtain sharp bounds for the function Φ(n), a problem initially raised, in the special caser=0,s=1, by Mitrinović. The analysis uses properties of the psi function ψ(x)=Γ′(x)/Γ(x). However, an elementary proof is also given for the main result of the above-mentioned authors (r=0,s=1)
Nonoscillation of First Order Impulse Differential Equations with Delay

AbstractOscillation properties of impulse functional-differential equations are studied for equations of the typex(t)=∑i=1mpi(t)x(t−τi(t))=0,t∈[a,b],x(ξ)=0,ξ∉[a,b],x(tj)=βjx(tj−0),j=1,…,k,a<t1<t2<···<tk<b.$PThe proven test for oscillation generalizes the known ones and allows consideration of the solvability of boundary value problems for the corresponding nonhomogeneous impulse equations. In particular, for the scalar impulse equationx(t)+p(t)x(t−τ(t))=0,t∈[0,∞),x(ξ)=0for ξ<0,x(tj)=βjx(tj−0),βj>0,j=1,2,…,denoteB(t)=∏j∈Dtβj, whereDt={i:ti∈[t−τ(t),t]},p+(t)=max{p(t),0}.Proposition.Let1+lnB(t)e≥∫r(t)tp+(s)dswherer(t)=max{t−τ(t),0},t>0.Then the nontrivial solution of this equation has no zeros on[0,∞)