A generalization of Cooke's integral inversion formula with application to remote-sensing theory

The remote sensing of environmental particulate pollutants, particularly their size distribution, frequently leads to the solution of first-kind Fredholm integral equations. The corresponding physical kernel tends to smooth the behavior of the required function for all values of the dependent variable. Thus, the problem is ill posed and needs regularization by the introduction of constraints on the solution (closure condition). However, under physically realistic conditions, the original problem can be transformed so that it presents a unique and stable solution. One such condition is the so-called anomalous-diffraction approximation, for which we provide two alternate inversion formulae. We derive a new inversion formula (see our theorem) which generalizes that of Cooke and which also provides, as a special case, one of Titchmarsh's formulae. We propose a unifying viewpoint for a number of known integral inversion formulae, including those of Fox (his first theorem), Hardy, Hankel, Titchmarsh, Cooke, and our own, along with the mutual interrelationships that exist between them (Fig. 1 and Table 1). One solution to the particulate sounding problem is then obtained from a direct application of our formula [Eq. (25)]. An alternate solution is likewise obtained by applying Titchmarsh's formula (II) [Eq. (27)]. Both solutions can be independently recovered from Fox's first theorem, although under somewhat more restrictive conditions. They are shown to be identical, and to provide the unique solution to the remote sensing problem considered.     

Clifford分析中双正则函数的Taylor展式及其性质

首先借助实Clifford分析中双正则函数的累次积分的换序公式,给出了双正则函数的Cauchy积分公式,然后由特征边界的Cauchy积分公式,得到了双正则函数的Taylor展式,并由此给出了双正则函数的唯一性定理,柯西不等式和Weierstrass定理.     

Itô’s calculus under sublinear expectations via regularity of PDEs and rough paths

In this paper, we first study the martingale problem in a sublinear expectation space. The critical tool is the Evans–Krylov theorem on regularity properties for solutions of fully nonlinear PDEs. Based on the analysis for the martingale problem and inspired by the rough path theory, we then develop stochastic calculus with respect to a general stochastic process, and derive an Itô type formula and the integration-by-parts formula. Our framework is analytic in that it does not rely on the probabilistic concept of “independence” as in the $G$-expectation theory.     

The propagation problem and far-field patterns in a stratified finite-depth ocean

In this paper we investigate the direct problem associated with the scattering of ‘plane waves’ from an object submerged in an ocean of finite depth. An integral representation for the Dirichlet problem is found, from which a formula for the far-field pattern evolves. A density theorem is established concerning the set of all far-field patterns. This theorem is essential for the reconstruction of the submerged object, the ‘inverse’ problem [2], [4], [5].     

Inhabitation of polymorphic and existential types

This paper shows that the inhabitation problem in the lambda calculus with negation, product, polymorphic, and existential types is decidable, where the inhabitation problem asks whether there exists some term that belongs to a given type. In order to do that, this paper proves the decidability of the provability in the logical system defined from the second-order natural deduction by removing implication and disjunction. This is proved by showing the quantifier elimination theorem and reducing the problem to the provability in propositional logic. The magic formulas are used for quantifier elimination such that they replace quantifiers. As a byproduct, this paper also shows the second-order witness theorem which states that a quantifier followed by negation can be replaced by a witness obtained only from the formula. As a corollary of the main results, this paper also shows Glivenko’s theorem, Double Negation Shift, and conservativity for antecedent-empty sequents between the logical system and its classical version.     

B.A. Sevastyanov’s famous theorem

In 1956, at the Third All-Union Mathematical Congress, Boris Aleksandrovich Sevastyanov gave a talk on the ergodic theorem proved by him for Markov processes and on its application to queueing systems. In 1957, this result was published in the journal Teoriya Veroyatnostei i Ee Primeneniya (Theory of Probability and Its Applications). An important corollary to the ergodic theorem is a generalization of Erlang’s well-known formula to a queueing system with a Poisson input flow and an arbitrary distribution of the service time. This result of Sevastyanov has served as a starting point for numerous studies on the problem, which was later called the insensitivity (invariance) problem for queueing systems with losses. There are hundreds of references to this result of Sevastyanov.     

On a theorem of Hermite and Hurwitz

The Hermite-Hurwitz theorem computes the degree, over $\text{R}$, of a real rational function ? in terms of the signature of an associated quadratic form—known today as the Hankel matrix of ?. This formula, which Hermite was led to by his work on the problem of representing integers as sums of squares, gave rise to striking applications in the theory of equations and in the stability theory of ordinary differential equations. In this paper, this theorem and various generalizations to the matrix-valued case are discussed and described in terms of signature formulae. These include its relation to stability theory and the matrix Hermite-Hurwitz theorem of Bitmead-Anderson as applied to questions of circuit synthesis. This also includes a global form of Hörmander's signature formula for the Maslov index of a rational loop in a Lagrangian Grassmannian, due to Byrnes and Duncan, and applications to the topology of spaces of rational matrix-valued functions, following the work of Brockett, Byrnes, and Duncan. This includes, in particular, a topological proof of the matrix Hermite-Hurwitz theorem.     

On potential theory for two-dimensional quasistatic problems of uncoupled thermoelasticity

The construction of potential theory for two-dimensional quasistatic problems of uncoupled thermoelasticity is carried out by considering the full system of differential equations of the problem as a nonselfadjoint differential operator. Green's second formula for this operator is interpreted as a duality theorem that differs from Mizel's duality theorem. In the case of a homogeneous isotropic medium we construct new integral equations for the basic initial-boundary value problems.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 33, 1991, pp. 48–52.     

极限的一个定理及其应用

给出了一个极限定理,能较好地解决一类特殊“和式”的极限问题.同时,利用对数函数的性质,又能够用来解决一些“积式”的极限.     

Semiclassical limit of the lowest eigenvalue of a Schrödinger operator on a Wiener space

We study a semiclassical limit of the lowest eigenvalue of a Schrödinger operator on a Wiener space. Key results are semiboundedness theorem of the Schrödinger operator, Laplace-type asymptotic formula and IMS localization formula. We also make a remark on the semiclassical problem of a Schrödinger operator on a path space over a Riemannian manifold.     

广义双正则函数向量的带位移带共轭的边值问题

本讨论实Clifford分析中广义双正则函数向量的带位移带共轭的边值问题,首先得到其Plemelj公式,然后用积分方程的方法和压缩映射原理讨论了这个边值问题的存在唯一性。     

Half Space and Quarter Space Dirichlet Problems for the Partial Differential Equation

A customary, heuristic, method, by which the Poisson integral formula for the Dirichlet problem, for the half space, for Laplace's equation is obtained, involves Green's function, and Kelvin's method of images. Although this heuristic method leads one to guess the correct result, this Poisson formula still has to be verified directly, independently of the method by which it was arrived at, in order to be absolutely certain that a solution of the Dirichlet problem for the half space, for Laplace's equation, has been actually obtained. A similar heuristic method, as seems to be generally known, could be followed in solving the Dirichlet problem, for the half space, for the equation where is a real constant. However, in Part 1, a different, labor-saving, method is used to study Dirichlet problems for the equation. This method is essentially based on what Hadamard called the method of descent. Indeed, it is shown that he who has solved the half space Dirichlet problem for Laplace's equation has already solved the half space Dirichlet problem for the equation In Part 2, the solution formula for the quarter space Dirichlet problem for Laplace's equation is obtained from the Poisson integral formula for the half space Dirichlet problem for Laplace's equation. A representation theorem for harmonic functions in the quarter space is deduced. The method of descent is used, in Part 3, to obtain the solution formula for the quarter space Dirichlet problem for the equation by means of the solution formula for the quarter space Dirichlet problem for Laplace's equation. So that, indeed, it is also shown that he who has solved the quarter space Dirichlet problem for Laplace's equation has already solved the quarter space Dirichlet problem for the " equation" For the sake of completeness and clarity, and for the convenience of the reader, the appendix, at the end of Part 3, contains a detailed proof that the Poisson integral formula solves the half space Dirichlet problem for Laplace's equation. The Bibliography for Parts 1,2, 3 is to be found at the end of Part 1.     

Surjections‐‐a classroom discussion

Learning pure mathematics through problem solving, group work and classroom discussion can be very motivating for students provided that they are given suitable problems, and appropriate guidance and instruction. Problems should be simple to state and they should yield some results easily but have very much more challenging components so that, with some early success, students will have the confidence and determination to learn new mathematics if necessary in order to reach a final solution. Students can be encouraged to work in groups and then to compare the advantages and disadvantages of different approaches. The problem outlined here is one suitable for undergraduates, and for very able school students, involving the definition of a function and simple combinatorics. It can be solved by a variety of methods involving algebraic expansions using the multinomial theorem, or solution of sets of linear equations using matrices and inverse matrices, or the inclusion exclusion formula as applied to the number of elements in the union of sets.     

Clifford 分析中超正则函数向量的一个边值问题

By the Plemelj formula and the compressed fixed point theorem,this paper discusses a kind of boundary value problem for hypermonogenic function vectors in Clifford analysis.And the paper proves the existence and uniqueness of the solution to the boundary value problem for hypermonogenic function vectors in Clifford analysis.     

Diffraction of a plane electromagnetic wave on a wedge-shaped inclusion

A technique is developed for reducing the problem of diffraction by a wedge-shaped inclusion to functional equations in the complex plane. An original formulation of the radiation conditions is proposed that enables one to formulate the diffraction problem in this case. An existence and uniqueness theorem is proven. For the case of a sufficiently small opening span of the angle a formula is derived which is a good approximation to the solution to the problem.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Stekova AN SSSR, Vol. 195, pp. 29–39, 1991.     

广义双正则函数带共轭值带位移的边值问题

本文讨论实Clifford分析中广义双正则函数的带共轭值和带位移的非线性边值问 题,首先得到其Plemelj公式,然后用积分方程的方法和Schauder 不动点定理讨论了这个边值 问题的可解性,     

Moving-centre monotonicity formulae for minimal submanifolds and related equations

Monotonicity formulae play a crucial role for many geometric PDEs, especially for their regularity theories. For minimal submanifolds in a Euclidean ball, the classical monotonicity formula implies that if such a submanifold passes through the centre of the ball, then its area is at least that of the equatorial disk. Recently Brendle and Hung proved a sharp area bound for minimal submanifolds when the prescribed point is not the centre of the ball, which resolved a conjecture of Alexander, Hoffman and Osserman. Their proof involves asymptotic analysis of an ingeniously chosen vector field, and the divergence theorem.In this article we prove a sharp ‘moving-centre’ monotonicity formula for minimal submanifolds, which implies the aforementioned area bound. We also describe similar moving-centre monotonicity formulae for stationary p-harmonic maps, mean curvature flow and the harmonic map heat flow.     

A BLACK-SCHOLES FORMULA FOR OPTION PRICING WITH DIVIDENDS

Abstract. We obtain a Black-Scholes formula for the arbitrage-free pricing of Eu-ropean Call options with constant coefficients when the underlylng stock generatesdividends. To hedge the Call option, we will always borrow money from bank. We seethe influence of the dividend term on the option pricing via the comparison theoremof BSDE(backward stochastic di~erential equation [5], [7]). We also consider the option pricing problem in terms of the borrowing rate R whichis not equal to the interest rate r. The corresponding Black-Sdxoles formula is given.We notice that it is in fact the borrowing rate that plays the role in the pricing formula.     

Generalized reflected BSDEs driven by a Lévy process and an obstacle problem for PDIEs with a nonlinear Neumann boundary condition

In this paper, we derive the existence and uniqueness of the solution for a class of generalized reflected backward stochastic differential equations (GRBSDEs in short) driven by a Lévy process, which involve the integral with respect to a continuous process by means of the Snell envelope, the penalization method and the fixed point theorem. In addition, we obtain the comparison theorem for the solutions of the GRBSDEs. As an application, we give a probabilistic formula for the viscosity solution of an obstacle problem for a class of partial differential-integral equations (PDIEs in short) with a nonlinear Neumann boundary condition.     

A free boundary problem in the theory of electrically actuated microdevices

Electrostatically deflected elastic systems are increasingly used in technical applications. We study a typical situation as a free boundary problem and prove, under quite general hypotheses, the existence of a local branch of solutions. Hadamard’s variational formula is crucial in the proof, which is based on the implicit function theorem in Banach spaces.