The Summation Formulae of Euler�CMaclaurin, Abel�CPlana, Poisson, and their Interconnections with the Approximate Sampling Formula of Signal Analysis

This paper is concerned with the two summation formulae of Euler?CMaclaurin (EMSF) and Abel?CPlana (APSF) of numerical analysis, that of Poisson (PSF) of Fourier analysis, and the approximate sampling formula (ASF) of signal analysis. It is shown that these four fundamental propositions are all equivalent, in the sense that each is a corollary of any of the others. For this purpose ten of the twelve possible implications are established. Four of these, namely the implications of the grouping ${\text{APSF}\Leftarrow\text{ASF}\Rightarrow\text{EMSF}\Leftrightarrow\text{PSF}}$ are shown here for the first time. The proofs of the others, which are already known and were established by three of the above authors, have been adapted to the present setting. In this unified exposition the use of powerful methods of proof has been avoided as far as possible, in order that the implications may stand in a clear light and not be overwhelmed by external factors. Finally, the four propositions of this paper are brought into connection with four propositions of mathematical analysis for bandlimited functions, including the Whittaker?CKotel??nikov?CShannon sampling theorem. In conclusion, all eight propositions are equivalent to another. Finally, the first three summation formulae are interpreted as quadrature formulae.     

On the Estimation of Parameters of Poisson Processes

Let(N(t),t≥0)be a poisson process on probability spaces,i.e.(N(t),t≥0)is stochastic process with independent increments under P_θ and satisfies (a)N(0)=0 a.s.P_θ, (b)For all.     

On the global well-posedness of a class of Boussinesq�CNavier�CStokes systems

In this paper we consider the following 2D Boussinesq–Navier–Stokes systems

On Lipschitz continuity of solutions of hyperbolic Poisson’s equation

In this paper, we investigate solutions of the hyperbolic Poisson equation \(\Delta _{h}u(x)=\psi (x)\), where \(\psi \in L^{\infty }(\mathbb {B}^{n}, {\mathbb R}^n)\) and

$$\begin{aligned} \Delta _{h}u(x)= (1-|x|^2)^2\Delta u(x)+2(n-2)\left( 1-|x|^2\right) \sum _{i=1}^{n} x_{i} \frac{\partial u}{\partial x_{i}}(x) \end{aligned}$$

is the hyperbolic Laplace operator in the n-dimensional space \(\mathbb {R}^n\) for \(n\ge 2\). We show that if \(n\ge 3\) and \(u\in C^{2}(\mathbb {B}^{n},{\mathbb R}^n) \cap C(\overline{\mathbb {B}^{n}},{\mathbb R}^n )\) is a solution to the hyperbolic Poisson equation, then it has the representation \(u=P_{h}[\phi ]-G_{ h}[\psi ]\) provided that \(u\mid _{\mathbb {S}^{n-1}}=\phi \) and \(\int _{\mathbb {B}^{n}}(1-|x|^{2})^{n-1} |\psi (x)|\,d\tau (x)<\infty \). Here \(P_{h}\) and \(G_{h}\) denote Poisson and Green integrals with respect to \(\Delta _{h}\), respectively. Furthermore, we prove that functions of the form \(u=P_{h}[\phi ]-G_{h}[\psi ]\) are Lipschitz continuous.

     

Regularity of solutions of Poisson’s equation in multiplier spaces

The most important result stated in this paper is to show that the solutions of the Poisson equation −Δu = f, where f ∈ (Ḣ¹(ℝ ^d ) → (Ḣ⁻¹(ℝ ^d )) is a complex-valued distribution on ℝ ^d , satisfy the regularity property D ^k u ∈ (Ḣ¹ → Ḣ⁻¹) for all k, |k| = 2. The regularity of this equation is well studied by Maz’ya and Verbitsky [12] in the case where f belongs to the class of positive Borel measures.      

Existence of Dyons in the Coupled Georgi�CGlashow�CSkyrme Model

We prove the existence of a continuous family of finite-energy particle-like solutions in the coupled Georgi–Glashow–Skyrme model carrying both electric and magnetic charges, known as dyons. Due to the presence of electricity and the Minkowski spacetime signature, we need to solve a variational problem with an indefinite action functional. Our results show that, while the magnetic charge is uniquely determined by the topological monopole number, the electric charge of a solution can be arbitrarily prescribed in an open interval.     

Averaging of stochastic systems of integral-differential equations with ≪Poisson noise≫

An analog of N. N. Bogolyubov's theorem is proved concerning averaging, on a finite time interval, of a system of integral-differential equations with a Poisson noise.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 2, pp. 273–277, February, 1991.     

On the Existence of Solutions of Poisson Equation and Poincaré–Lelong Equation

One of the main goals of this paper is to solve the Poincaré–Lelong equation on a class of Kähler manifolds with nonnegative holomorphic bisectional curvature, $\mathrm{Ric}(x)\geq \left(a\ln\ln\left(10+r(x)\right)\right)\Big/\big.\left(\left(1+r^2(x)\right)\ln(10+r(x))\right)One of the main goals of this paper is to solve the Poincaré–Lelong equation on a class of K?hler manifolds with nonnegative holomorphic bisectional curvature, for some a > 67(n + 4)². We will also study the Poisson equation on complete noncompact manifolds which satisfy volume doubling and Poincaré inequality.     

On Approximation of Functions by Generalized Abel–Poisson Operators

We find the exact constant in the principal term of the deviation of a function in the Zygmund class from the generalized Abel-Poisson operators determined by the Fourier series over the trigonometric system with particular summation factors.     

Calabi–Yau Algebras Viewed as Deformations of Poisson Algebras

From any algebra A defined by a single non-degenerate homogeneous quadratic relation f, we prove that the quadratic algebra B defined by the potential w?=?fz is 3-Calabi–Yau. The algebra B can be viewed as a 3-Calabi–Yau completion of Keller. The algebras A and B are both Koszul. The classification of the algebras B in three generators, i.e., when A has two generators, leads to three types of algebras. The second type (the most interesting one) is viewed as a deformation of a Poisson algebra S whose Poisson bracket is non-diagonalizable quadratic. Although the potential of S has non-isolated singularities, the homology of S is computed. Next the Hochschild homology of B is obtained.     

On the Poisson geometry of the Adler-Gel’fand-Dikii brackets

In this paper, we consider a symplectic leaf that goes through a singular point of the Adler-Gel’fand-Dikii Poisson bracket associated to SL(n,R). We find a finite-dimensional transverse section2 at the singular point and we prove that one can induce a Poisson structure on2 (the transverse structure) that is linearizable and equivalent to the Lie-Poisson structure on sl(n,R)*. This problem is closely related to finding normal forms for nth order scalar differential operators with periodic coefficient. We partially generalize a well-known result for Hill’s operators to the higher order case.     

On Characterization of Poisson Integrals of Schrödinger Operators with Morrey Traces

Let L be a Schr?dinger operator of the form L =-Δ + V acting on L~2(R~n) where the nonnegative potential V belongs to the reverse H?lder class B_q for some q ≥ n. In this article we will show that a function f ∈ L~(2,λ)(R~n), 0 λ n, is the trace of the solution of L_u =-u_(tt) + L_u =0, u(x, 0) = f(x), where u satisfies a Carleson type condition sup x_B,r_Br_B~(-λ)∫_0~(rB)∫_(B(x_B,r_B))t|u(x,t)|~2dxdt≤C∞.Its proof heavily relies on investigate the intrinsic relationship between the classical Morrey spaces and the new Campanato spaces L_L~(2,λ)(R~n) associated to the operator L, i.e.L_L~(2,λ)(R~n)=L~(2,λ)(R~n).Conversely, this Carleson type condition characterizes all the L-harmonic functions whose traces belong to the space L~(2,λ)(R~n) for all 0 λ n.     

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本文考虑了常利力下带干扰的双复合Poisson风险过程, 借助微分和伊藤公式, 分别获得了无限时和有限时生存概率的积分微分方程. 当保费服从指数分布时, 得到了无限时生存概率的微分方程.     

Decay of the two-species Vlasov–Poisson–Boltzmann system

We establish the time decay rates of the solution to the Cauchy problem for the two-species Vlasov–Poisson–Boltzmann system near Maxwellians via a refined pure energy method. The total density of two species of particles decays at the optimal algebraic rate as the Boltzmann equation, but the disparity between two species and the electric field decay at an exponential rate. This phenomenon reveals the essential difference when compared to the one-species Vlasov–Poisson–Boltzmann system or the Navier–Stokes–Poisson equations in which the electric field decays at the optimal algebraic rate, and compared to the Vlasov–Boltzmann system in which the disparity between two species decays at the optimal algebraic rate. Our achievement heavily relies on a reformulation of the problem which well displays the cancelation property of the two-species system, and our proof is based on a family of scaled energy estimates with minimum derivative counts and interpolations among them without linear decay analysis.     

A Generalization of the Heine�CStieltjes Theorem

The Heine?CStieltjes theorem describes the polynomial solutions, (v,f) such that T(f)=vf, to specific second-order differential operators, T, with polynomial coefficients. We extend the theorem to concern all (nondegenerate) differential operators preserving the property of having only real zeros, thus solving a conjecture of B. Shapiro. The new methods developed are used to describe intricate interlacing relations between the zeros of different pairs of solutions. This extends recent results of Bourget, McMillen and Vargas for the Heun equation and answers their question of how to generalize their results to higher degrees. Many of the results are new even for the classical case.     

Stochastic Description of a Bose�CEinstein Condensate

In this work we give a positive answer to the following question: does Stochastic Mechanics uniquely define a three-dimensional stochastic process which describes the motion of a particle in a Bose?CEinstein condensate? To this extent we study a system of N trapped bosons with pair interaction at zero temperature under the Gross?CPitaevskii scaling, which allows to give a theoretical proof of Bose?CEinstein condensation for interacting trapped gases in the limit of N going to infinity. We show that under the assumption of strictly positivity and continuous differentiability of the many-body ground state wave function it is possible to rigorously define a one-particle stochastic process, unique in law, which describes the motion of a single particle in the gas and we show that, in the scaling limit, the one-particle process continuously remains outside a time dependent random ??interaction-set?? with probability one. Moreover, we prove that its stopped version converges, in a relative entropy sense, toward a Markov diffusion whose drift is uniquely determined by the order parameter, that is the wave function of the condensate.     

Robust estimation of efficient mean�Cvariance frontiers

Standard methods for optimal allocation of shares in a financial portfolio are determined by second-order conditions which are very sensitive to outliers. The well-known Markowitz approach, which is based on the input of a mean vector and a covariance matrix, seems to provide questionable results in financial management, since small changes of inputs might lead to irrelevant portfolio allocations. However, existing robust estimators often suffer from masking of multiple influential observations, so we propose a new robust estimator which suitably weights data using a forward search approach. A Monte Carlo simulation study and an application to real data show some advantages of the proposed approach.