Killing wild ramification

We compute the inertia group of the compositum of wildly ramified Galois covers. It is used to show that even the p-part of the inertia group of a Galois cover of ?¹ branched only at infinity can be reduced if there is a jump in the lower ramification filtration at two and a certain linear disjointness statement holds.     

On strongly jump traceable reals

In this paper we show that there is no minimal bound for jump traceability. In particular, there is no single order function such that strong jump traceability is equivalent to jump traceability for that order. The uniformity of the proof method allows us to adapt the technique to showing that the index set of the c.e. strongly jump traceables is -complete.     

Jump conditions in stress relaxation and creep experiments of Burgers type fluids: a study in the application of Colombeau algebra of generalized functions

We discuss stress relaxation and creep experiments of fluids that are generalizations of the classical model due to Burgers by allowing the material moduli such as the viscosities and relaxation and retardation times to depend on the stress. The physical problem, which is cast within the context of one dimension, leads to an ordinary differential equation that involves nonlinear terms like product of a function with a jump discontinuity and the derivative of a function with a jump discontinuity. As the equations are nonlinear, standard techniques that are used to study problems concerning linear viscoelastic fluids such as Laplace transforms and the theory of distributions are not applicable. We find it necessary to seek the solution in a more general setting. We discuss the mathematical and physical issues concerning the jump discontinuities and nonlinearity of the governing equation, and we show that the solution to the governing equation can be found in the sense of the generalized functions introduced by Colombeau. In the framework of Colombeau algebra we, under certain assumptions, derive jump conditions that shall be used in stress relaxation and creep experiments of fluids of the Burgers type. We conclude the paper with a discussion of the physical relevance of these assumptions.     

Computability in structures representing a Scott set

Continuing work begun in [10], we utilize a notion of forcing for which the generic objects are structures and which allows us to determine whether these “generic” structures compute certain sets and enumerations. The forcing conditions are bounded complexity types which are consistent with a given theory and are elements of a given Scott set. These generic structures will “represent” this given Scott set, in the sense that the structure has a certain weak saturation property with respect to bounded complexity types in the Scott set. For example, if ? is a nonstandard model of PA, then ? represents the Scott set ? = n∈ω | ?⊧“the nth prime divides a” | a∈?. The notion of forcing yields two main results. The first characterizes the sets of natural numbers computable in all models of a given theory representing a given Scott set. We show that the characteristic function of such a set must be enumeration reducible to a complete existential type which is consistent with the given theory and is an element of the given Scott set. The second provides a sufficient condition for the existence of a structure ? such that ? represents a countable jump ideal and ? does not compute an enumeration of a given family of sets ?. This second result is of particular interest when the family of sets which cannot be enumerated is ? = Rep[Th(?)]. Under this additional assumption, the second result generalizes a result on TA [6] and on certain other completions of PA [10]. For example, we show that there also exist models of completions of ZF from which one cannot enumerate the family of sets represented by the theory. Received: 8 October 1997 / Published online: 25 January 2001     

Decomposition of default probability under a structural credit risk model with jumps?

In this paper, we consider the default probabilities caused by a jump or by oscillation under a structural credit risk model with jumps. We study the Laplace transforms of the times of default caused by a jump and by oscillation. We derive integro-differential equations and obtain some closed-form expressions for them. By inverting them, we numerically investigate the contributions of the jump component and the diffusion component to the default under a certain choice of the jump size distribution.     

Sto chastic Nonlinear Beam Equations with Levy Jump

In this paper, we study stochastic nonlinear beam equations with Levy jump, and use Lyapunov functions to prove existence of global mild solutions and asymptotic stability of the zero solution.     

The Jump of the Laplacian on a Submanifold

Assume that a submanifold M ? ?ⁿ of an arbitrary codimension k ? {1, …, n} is closed in some open set O→?ⁿ. With a given function u ? C²(O\M) we may associate its trivial extension u: O→? such that u|_O\M=u and u|m ≡ 0. The jump of the Laplacian of the function u on the submanifold M is defined by the distribution Δu — Δu. By applying some general version of the Fubini theorem to the nonlinear projection onto M we obtain the formula for the jump of the Laplacian (Theorem 2.2).     

Large Deviations for Hierarchical Systems of Interacting Jump Processes

We investigate the large deviations principle from the McKean–Vlasov limit for a collection of jump processes obeying a two-level hierarchy interaction. A large deviation upper bound is derived and it is shown that the associated rate function admits a Lagrangian representation as well as a nonvariational one. Moreover, it is proved that the admissible paths for the weak solution of the McKean–Vlasov equation enjoy certain strong differentiability properties.     

A Mixed Boundary Value Problem for Polyanalytic Function of Order n in the Sobolev Space Wn,p (D)

We discuss the construction of a polyanalytic function Φ of order n on a simple bounded domain D. The function satisfies n prescribed generalized Riemann-Hilbert boundary conditions on the boundary ?D and n generalized jump conditions on a simple closed smooth contour γ contained in D. The boundary conditions are transformed into n classical Riemann-Hilbert problems and the n jump conditions into n Riemann problems of conjugation for some 2n holomorphic functions. These transformed problems are solved using the standard methods from the literature.     

The Gibbs' phenomenon from a signal processing point of view

Recently, Fay and Kloppers gave two proofs to show that the well-known Gibbs' phenomenon for Fourier series at a jump discontinuity depends only on the size of the jump and is a multiple of the integral 1/π ∫₀ ^π (sin x / x) dx. We give another proof, based upon low-pass filtering of the Fourier transform, that uses the observation that a truncated Fourier series for a function ? (x) is ‘very nearly’ equal to the convolution integral 1/π ∫ _-∞ ^+∞ ? (x - t)(sin nt / t) dt.     

A thinning result for pure jump processes

A standard thinning procedure for point processes is extended to processes of pure jump type in which each jump is retained with probability p or deleted with probability 1 ? p, independently of everything else.Two theorems are proved, the first gives a sufficient condition for the existence of thinned pure jump processes, the second concerns the convergence of such processes to pure jump processes whose increments are generated by a Cox process. Some generalizations are discussed.     

Manipulating Gibbs'' Phenomenon for Fourier Interpolation

The Fourier interpolation polynomials of a periodic function with an isolated jump discontinuity at a node exhibit for growing order a Gibbs phenomenon. By a suitable definition of the function value at the jump the over- and undershoots on one side may be minimized.     

Controlled jump processes

Finite and infinite planning horizon Markov decision problems are formulated for a class of jump processes with general state and action spaces and controls which are measurable functions on the time axis taking values in an appropriate metrizable vector space. For the finite horizon problem, the maximum expected reward is the unique solution, which exists, of a certain differential equation and is a strongly continuous function in the space of upper semi-continuous functions. A necessary and sufficient condition is provided for an admissible control to be optimal, and a sufficient condition is provided for the existence of a measurable optimal policy. For the infinite horizon problem, the maximum expected total reward is the fixed point of a certain operator on the space of upper semi-continuous functions. A stationary policy is optimal over all measurable policies in the transient and discounted cases as well as, with certain added conditions, in the positive and negative cases.     

Necessary Maximum Principle of Stochastic Optimal Control with Delay and Jump Diffusion

In this paper, we have studied the necessary maximum principle of stochastic optimal control problem with delay and jump diffusion.     

On the construction of non-affine jump-diffusion models

We describe a method for construction of jump analogues of certain one-dimensional diffusion processes satisfying solvable stochastic differential equations. The method is based on the reduction of the original stochastic differential equations to the ones with linear diffusion coefficients, which are reducible to the associated ordinary differential equations, by using the appropriate integrating factor processes. The analogues are constructed by means of adding the jump components linearly into the reduced stochastic differential equations. We illustrate the method by constructing jump analogues of several diffusion processes and expand the notion of market price of risk to the resulting non-affine jump-diffusion models.     

On solutions of Kolmogorovʼs equations for nonhomogeneous jump Markov processes

This paper studies three ways to construct a nonhomogeneous jump Markov process: (i) via a compensator of the random measure of a multivariate point process, (ii) as a minimal solution of the backward Kolmogorov equation, and (iii) as a minimal solution of the forward Kolmogorov equation. The main conclusion of this paper is that, for a given measurable transition intensity, commonly called a Q-function, all these constructions define the same transition function. If this transition function is regular, that is, the probability of accumulation of jumps is zero, then this transition function is the unique solution of the backward and forward Kolmogorov equations. For continuous Q-functions, Kolmogorov equations were studied in Feller?s seminal paper. In particular, this paper extends Feller?s results for continuous Q-functions to measurable Q-functions and provides additional results.     

Arbitrarily large jumps of the Golovach function for trees

The ?-search problem on connected topological graphs is considered. The jumps of the Golovach function are studied for trees. As is known, the Golovach function for trees with at most 27 edges has only unit jumps. In the authors’ earlier papers, examples of trees on which the Golovach function has a jump of size 2 were constructed. In the present paper, it is shown that the jumps of the Golovach function for trees may be arbitrarily large. A sharp bound for the size of jumps is given for a sequence of trees constructed in the paper. A theorem about small perturbations of edge lengths for trees is proved, which asserts an arbitrarily small perturbation of the edge lengths of a given tree (whose Golovach function may be degenerate) may yield a new tree whose Golovach function has only unit jumps.     

An operator-based approach to the analysis of ruin-related quantities in jump diffusion risk models

Recent developments in ruin theory have seen the growing popularity of jump diffusion processes in modeling an insurer’s assets and liabilities. Despite the variations of technique, the analysis of ruin-related quantities mostly relies on solutions to certain differential equations. In this paper, we propose in the context of Lévy-type jump diffusion risk models a solution method to a general class of ruin-related quantities. Then we present a novel operator-based approach to solving a particular type of integro-differential equations. Explicit expressions for resolvent densities for jump diffusion processes killed on exit below zero are obtained as by-products of this work.     

Harnack Inequalities for Jump Processes

We consider a class of pure jump Markov processes in R ^d whose jump kernels are comparable to those of symmetric stable processes. We establish a Harnack inequality for nonnegative functions that are harmonic with respect to these processes. We also establish regularity for the solutions to certain integral equations.     

平面上的单跳双增函数的结构

本研究了平面上的单跳双增函数的结构，得到了平面上的单跳双增函数图像的明确，简洁，清楚的结构。这类函数可作为取非负整数值的两参数随机过程的样本轨道。