An optimal stopping problem for a geometric Brownian motion with poissonian jumps

This paper examines an optimal stopping problem for a geometric Brownian motion with random jumps. It is assumed that jumps occur according to a time-homogeneous Poisson process and the proportions of these sizes are independent and identically distributed nonpositive random variables. The objective is to find an optimal stopping time of maximizing the expected discounted terminal reward which is defined as a nondecreasing power function of the stopped state. By applying the “smooth pasting technique” [1,2], we derive almost explicitly an optimal stopping rule of a threshold type and the optimal value function of the initial state. That is, we express the critical state of the optimal stopping region and the optimal value function by formulae which include only given problem parameters except an unknown to be uniquely determined by a nonlinear equation.     

Self-intersection local time for Gaussian ′( )-processes: Existence, path continuity and examples

In this paper we develop a criterion for existence or non-existence of self-intersection local time (SILT) for a wide class of Gaussian ′( ^d)-valued processes, we show that quite generally the SILT process has continuous paths, and we give several examples which illustrate existence of SILT for different ranges of dimensions (e.g., d ≤ 3, d ≤ 7 and 5 ≤ d ≤ 11 in the Brownian case). Some of the examples involve branching and exhibit “dimension gaps”. Our results generalize the work of Adler and coauthors, who studied the special case of “density processes” and proved that SILT paths are cadlag in the Brownian case making use of a “particle picture” approximation (this technique is not available for our general formulation).     

Linear multi-secret sharing schemes based on multi-party computation

In multi-secret sharing schemes, publishing shares during the process of reconstructing partial secrets may leak some information of the secrets unrecovered yet. By using a multi-party computation (MPC) protocol, we solve this problem for any linear multi-secret sharing scheme (MSSS). We also show that LMSSS usually involve more complicated reconstruction algorithms than “direct sum” schemes, but from the point of reducing share expansion, the former is preferred.     

On stopping points in the plane that lie on a unique optional increasing path

Given a stopping point in the plane, is there an optional increasing path that passes through the stopping point with probability one? The commutation Hypothesis F4 of Cairoli and Walsh was known to be a sufficient condition in continuous time, and we prove that the conditional qualitative independence Hypothesis CQI of Krengel and Sucheston is in fact necessary and sufficient. Rather surprisingly, certain stopping points may lie on a unique optional increasing path, even when the two-parameter nitration is the natural filtration ot a Rrownian sheet. In the construction of these examples, we prove a result of independent interest, namely that the solutions to certain random differentia! equations are optional increasing paths     

Exploiting structure in quantified formulas

We study the computational problem “find the value of the quantified formula obtained by quantifying the variables in a sum of terms.” The “sum” can be based on any commutative monoid, the “quantifiers” need only satisfy two simple conditions, and the variables can have any finite domain. This problem is a generalization of the problem “given a sum-of-products of terms, find the value of the sum” studied in [R.E. Stearns and H.B. Hunt III, SIAM J. Comput. 25 (1996) 448–476]. A data structure called a “structure tree” is defined which displays information about “subproblems” that can be solved independently during the process of evaluating the formula. Some formulas have “good” structure trees which enable certain generic algorithms to evaluate the formulas in significantly less time than by brute force evaluation. By “generic algorithm,” we mean an algorithm constructed from uninterpreted function symbols, quantifier symbols, and monoid operations. The algebraic nature of the model facilitates a formal treatment of “local reductions” based on the “local replacement” of terms. Such local reductions “preserve formula structure” in the sense that structure trees with nice properties transform into structure trees with similar properties. These local reductions can also be used to transform hierarchical specified problems with useful structure into hierarchically specified problems having similar structure.     

Permuting in place: analysis of two stopping rules

Permuting in place has been first analyzed by Knuth. It uses the cycle structure of the permutation. The elements of an array to be permuted are only moved when one sees a cycle leader (smallest element in its cycle). So the essential part of such an algorithm is to test an element i about whether it is a cycle leader.Recently, Keller [Inform. Process. Lett. 81 (2002) 119–125] introduced two stopping rules: “If the last cycle leader has been detected, all elements have been moved, and no further tests are necessary” (heuristic 1), respectively “If only r elements have not been moved, then proceeding along a cycle is only useful for r steps” (heuristic 2).We analyze the average costs of these modifications applied to the standard algorithm of Knuth; they are (n+2)H_n−5n/2−1/2nlogn and respectively ((2n+1)/4)H_(n+1)/2+(1/2)H_2(n+1)/2−(1/2)((n+1)/2−n/2)−(n+1)/2(n/2)logn, as opposed to (n+1)H_n−2nnlogn in the classical case.     

Existence of solutions for elliptic equations with discontinuous nonlinearities strong resonance at infinity

In the present paper, our main purposes are to study nonlinear elliptic equations with strong resonance at infinity. Some existence theorems for nontrivial solutions are obtained by using some nonsmooth critical point theorems in [N. C. Kourogenis, N. S. Papageorgiou, Nonsmooth critical point theory and Nonlinear elliptic equations at resonance, J. Austral. Math Soc. (Ser. A) 69 (2000) 245–271]. The two of our theorems generalize Theorems 0.1 and 5.2 in [P. Bartolo, V. Benci, D. Fortunato, Abstract critical point theorems and applications to some nonlinear problems with “strong” resonance at infinity, Nonlinear Anal. TMA 7 (1983) 981–1012] to nonsmooth cases. Another theorem is new even if for the smooth case.     

Māl, enunciations, and the prehistory of Arabic algebra

Medieval Arabic algebra books intended for practical training generally have in common a first “book” which is divided into two sections: one on the methods of solving simplified equations and manipulating expressions, followed by one consisting of worked-out problems. By paying close attention to the wording of the problems in the books of al-Khwārizmī, Abū Kāmil, and Ibn Badr, we reveal the different ways the word māl was used. In the enunciation of a problem it is a common noun meaning “quantity,” while in the solution it is the proper noun naming the square of “thing” (shay '). We then look into the differences between the wording of enunciations and equations, which clarify certain problems solved without “thing,” and help explain the development of algebra before the time of al-Khwārizmī.     

Genetic algorithms with noisy fitness

The convergence properties of genetic algorithms with noisy fitness information are studied here. In the proposed scheme, hypothesis testing methods are used to compare sample fitness values. The “best” individual of each generation is kept and a greater-than-zero mutation rate is used so that every individual will be generated with positive probability in each generation. The convergence criterion is different from the frequently-used uniform population criterion; instead, the sequence of the “best” individual in each generation is considered, and the algorithm is regarded as convergent if the sequence of the “best” individuals converges with probability one to a point with optimal average fitness.     

A stopping game in a stochastic and fuzzy environment

A zero-sum stopping game for a sequence of fuzzy-valued random variables is discussed. The fuzzy random variables are estimated by probabilistic expectation and fuzzy expectation. A saddle point is given for the stopping game.     

Several sufficient conditions of solvability for a nonlinear higher-order three-point boundary value problem with all derivatives

In this paper, firstly, some errors in the proof of our paper “Several sufficient conditions of solvability for a nonlinear higher-order three-point boundary value problem on time scales, Appl. Math. Comput. 190 (2007) 566–575” are pointed, and we make the corresponding correction when T=R. Then, the more general problem with all derivatives is considered. Under certain growth conditions on the nonlinearity, several sufficient conditions for the existence and uniqueness of nontrivial solution are obtained by using Leray–Schauder nonlinear alternative and Banach fixed point theorem.     

各种停点及其关系

给出 了一个 尽可能 完善 的各种 停点的 关系． 仿照 Ｂａｋｒｙ 引 入 ｉ强停点, 对亚强 停点 本文给出了它的准 确位置,它处于 停点和强停点之 间,且可以像停 点和强停点一样 用停时点来刻画     

L-fuzzy quantifiers of type determined by fuzzy measures

The aim of this paper is, first, to introduce two new types of fuzzy integrals, namely, -fuzzy integral and →-fuzzy integral. The first integral is based on a fuzzy measure of L-fuzzy sets and the second one on a complementary fuzzy measure of L-fuzzy sets, where L is a complete residuated lattice. Some of their properties and a relation to the fuzzy (Sugeno) integral are investigated. Second, using these integrals, two classes of monadic L-fuzzy quantifiers of type 1 are defined. These L-fuzzy quantifiers can be used for modeling the semantics of natural language quantifiers like “all”, “some”, “many”, “none”, “at most half”, etc. Several semantic properties of these L-fuzzy quantifiers are studied.     

Influence diagnostics and outlier tests for varying coefficient mixed models

In this paper, we consider subset deletion diagnostics for fixed effects (coefficient functions), random effects and one variance component in varying coefficient mixed models (VCMMs). Some simple updated formulas are obtained, and based on which, Cook’s distance, joint influence and conditional influence are also investigated. Besides, since mean shift outlier models (MSOMs) are also efficient to detect outliers, we establish an equivalence between deletion models and MSOMs, which is not only suitable for fixed effects but also for random effects, and test statistics for outliers are then constructed. As a byproduct, we obtain the nonparametric “delete = replace” identity. Our influence diagnostics methods are illustrated through a simulated example and a real data set.     

CDS simulation and pattern formation of phase separation

Several properties of the generation and evolution of phase separating patterns for binary material studied by CDS model are proposed. The main conclusions are (1) for alloys spinodal decomposition, the conceptions of “macro-pattern” and “micropattern” are posed by “black-and- white graph” and “gray-scale graph” respectively. We find that though the four forms of map f that represent the self-evolution of order parameter in a cell (lattice) are similar to each other in “macro-pattern”, there are evident differences in their micro-pattern, e.g., some different fine netted structures in the black domain and the white domain are found by the micro-pattern, so that distinct mechanical and physical behaviors shall be obtained. (2) If the two constitutions of block copolymers are not symmetric (i.e. r ≠ 0.5), a pattern called “grain-strip cross pattern” is discovered, in the 0.43 <r <0.45.     

The Injective Spectrum of a Noncommutative Space

For a noncommutative space X, we study Inj(X), the set of isomorphism classes of indecomposable injective X-modules. In particular, we look at how this set, suitably topologized, can be viewed as an underlying “spectrum” for X. As applications we discuss noncommutative notions of irreducibility and integrality, and a way of associating an integral subspace of X to each element of Inj(X) which behaves like a “weak point.”     

Newton's solution to the equiangular spiral problem and a new solution using only the equiangular property

This paper is a study of Proposition IX of Book I of Newton's Principia, the problem of determining the centripetal force for an equiangular spiral. In Newton's main proof of this proposition there is an error concerning his reason for the figure SPRQT being “given in kind,” and a very interesting technique of varying things in the neighborhood of a limit. This main proof utilized Newton's formula for the limit of SP²QT²/QR given in Corollary I to Proposition VI of the Principia. Newton also gave an alternate proof which utilized his formula for SY²PV given in Corollary III to Proposition VI. The “given” of Proposition IX was “a spiral PQS, cutting all the radii SP, SQ, &c., in a given angle.” Both the main proof and the alternate proof implicitly depend on the property of the equiangular spiral that the radius of curvature at any point is proportional to the pole distance SP. We here offer a new proof of Newton's proposition which does not depend on this implicit assumption.     

Maximum thick paths in static and dynamic environments

We consider the problem of finding a large number of disjoint paths for unit disks moving amidst static or dynamic obstacles. The problem is motivated by the capacity estimation problem in air traffic management, in which one must determine how many aircraft can safely move through a domain while avoiding each other and avoiding “no-fly zones” and predicted weather hazards. For the static case we give efficient exact algorithms, based on adapting the “continuous uppermost path” paradigm. As a by-product, we establish a continuous analogue of Menger's Theorem.Next we study the dynamic problem in which the obstacles may move, appear and disappear, and otherwise change with time in a known manner; in addition, the disks are required to enter/exit the domain during prescribed time intervals. Deciding the existence of just one path, even for a 0-radius disk, moving with bounded speed is NP-hard, as shown by Canny and Reif [J. Canny, J.H. Reif, New lower bound techniques for robot motion planning problems, in: Proc. 28th Annu. IEEE Sympos. Found. Comput. Sci., 1987, pp. 49–60]. Moreover, we observe that determining the existence of a given number of paths is hard even if the obstacles are static, and only the entry/exit time intervals are specified for the disks. This motivates studying “dual” approximations, compromising on the radius of the disks and on the maximum speed of motion.Our main result is a pseudopolynomial-time dual-approximation algorithm. If K unit disks, each moving with speed at most 1, can be routed through an environment, our algorithm finds (at least) K paths for disks of radius somewhat smaller than 1 moving with speed somewhat larger than 1.     

Point Evaluation and Hardy Space on a Homogeneous Tree

We consider stationary multiscale systems as defined by Basseville, Benveniste, Nikoukhah and Willsky. We show that there are deep analogies with the discrete time non stationary setting as developed by the first author, Dewilde and Dym. Following these analogies we define a point evaluation with values in a C*–algebra and the corresponding “Hardy space” in which Cauchy’s formula holds. This point evaluation is used to define in this context the counterpart of classical notions such as Blaschke factors.