Infinite-Dimensional Calculus Under Weak Spatial Regularity of the Processes

Two generalizations of Itô formula to infinite-dimensional spaces are given. The first one, in Hilbert spaces, extends the classical one by taking advantage of cancellations when they occur in examples and it is applied to the case of a group generator. The second one, based on the previous one and a limit procedure, is an Itô formula in a special class of Banach spaces having a product structure with the noise in a Hilbert component; again the key point is the extension due to a cancellation. This extension to Banach spaces and in particular the specific cancellation are motivated by path-dependent Itô calculus.     

Emergency facilities location

This paper considers the problem of locating two emergency stations in a small township so as to minimize the total response time to all emergencies. This problem was solved by a computer program which determines the response time for all possible solutions and finds the optimal one. The problem was first approached by assuming that emergencies occur only at the center of blocks and that stations are located only on street corners. The problem was then generalized to allow the emergencies and stations to be located anywhere along the streets. The paper concludes that the optimal solution for the general case is the same as that for the simpler case.For the particular case of the township of Rio Rancho, the optimal locations for the two stations were found to be at coordinates (3, 2) and (3, 6). For Rio Rancho the problem was complicated by the presence of obstacles; however, in this township they did not affect the optimal solution. Test cases confirmed the program's validity.     

Monitoring Phase II Comparative Clinical Trials with Two Endpoints and Penalty for Adverse Events

Adverse events in Phase II comparative clinical trials have received limited attention in the literature. Bersimis et al. (Stat Med 34:197–214, 2014) in proposed a class of comparative sequential designs with bivariate endpoints, where as a special case, the termination of the clinical trial due to the occurrence of a severe adverse event is treated. In this paper, using the Markov chain embedding technique, we extend this class of designs proposing two new designs, which treat cases where the development of an adverse event does not immediately stop the clinical trial, but penalizes appropriately the treatment that caused it. In both designs the penalty can be chosen either by assessing the severity of the adverse event or by optimizing the power. The numerical results show an excellent performance, achieving small expected sample sizes in conjunction with large values for power, satisfying in this way the ethical requirement for small sample sizes and fast decisions in clinical practice. The formulation of the procedure as a stochastic process is elegantly accomplished while it offers the necessary mathematical framework for further generalizing the designs covering more cases such as group sequential designs, etc.     

Estimation of second-order properties from jittered time series

This paper considers spectral and autocovariance estimation for a zero-mean, band-limited, stationary process that has been sampled at time points jittered from a regular, equi-interval, sampling scheme. The case of interest is where the sampling scheme is near regular so that the jitter standard deviation is small compared to the sampling interval. Such situations occur with many time series collected in the physical sciences including, in particular, oceanographic profiles.Spectral estimation procedures are developed for the case of independent jitter and autocovariance estimation procedures for both independent and dependent jitter. These are typically modifications of general estimation procedures proposed elsewhere, but tailored to the particular jittered sampling scheme considered. The theoretical properties of these estimators are developed and their relative efficiencies compared.The properties of the jittered sampling point process are also developed. These lead to a better understanding, in this situation, of more general techniques available for processes sampled by stationary point processes.     

Seeing chance: perceptual reasoning as an epistemic resource for grounding compound event spaces

The mathematics subject matter of probability is notoriously challenging, and in particular the content of random compound events. When students analyze experiments, they often omit to discern variations as distinct events, e.g., HT and TH in the case of flipping a pair of coins, and thus infer erroneous predictions. Educators have addressed this conceptual difficulty by engaging students in actual experiments whose outcomes contradict the erroneous predictions. Yet whereas empirical activities per se are crucial for any probability design, because they introduce the pivotal contents of randomness, variance, sample size, and relations among them, empirical activities may not be the unique or best means for students to accept the logic of combinatorial analysis. Instead, learners may avail of their own pre-analytic perceptual judgments of the random generator itself so as to arrive at predictions that agree rather than conflict with mathematical analysis. I support this view first by detailing its philosophical, theoretical, and didactical foundations and then by presenting empirical findings from a design-based research project. Twenty-eight students aged 9?C11 participated in tutorial, task-based clinical interviews that utilized an innovative random generator. Their predictions were mathematically correct even though initially they did not discern variations. Students were then led to recognize the formal event space as a semiotic means of objectifying these presymbolic notions. I elaborate on the thesis via micro-ethnographic analysis of key episodes from a paradigmatic case study.     

Problems Involving Infrequent but Significant Contingencies

Problematic situations often arise in which it is required to provide a solution which will tend to avoid events, which, if they occur, would be very costly, or, if not directly costable, they would be highly undesirable. Although direct approaches to this sort of problem exist, they can be unmanageable. If, however, we take as a posit, that the frequency with which the undesirable events arise, in the optimum solution, is small, considerable simplifications can be made. Naturally we need to check the posit once the solution has been found. This paper considers three applications of this principle, viz. determination of how many chargers are needed for steel furnaces, where the undesirable event is “a furnace waits for service”; determination of the number of emergency beds to set aside in a hospital unit, where the undesirable event is “an emergency case arrives and no bed is immediately available”; determination of an inventory reorder rule where the undesirable event is “stock run-out”. The general principle is formalized.     

基于故障树分析的堰塞湖溃坝概率估计方法

堰塞湖排险的一个关键问题是如何针对实施不同应对措施情况下的堰塞湖溃坝概率进行估计,这是一个值得关注的重要研究课题。本文提出了一种基于故障树分析(FTA)的堰塞湖溃坝概率估计方法。首先,通过堰塞湖排险问题的实际背景分析,基于FTA构建了堰塞湖溃坝故障树的基本架构;然后,通过相关领域知识、历史案例分析、专家主观判断和多位专家主观判断信息的融合,可以确定实施某一应对措施情形下故障树中各基本事件在不同时段内发生的概率;进一步地,依据构建的故障树和基本事件发生的概率,给出了在不同时段内堰塞湖溃坝事件发生的概率的估计方法。最后,通过一个实例分析说明了本文所提出方法的可行性与有效性。     

Computing isogenies between elliptic curves over using Couveignes's algorithm

The heart of the improvements by Elkies to Schoof's algorithm for computing the cardinality of elliptic curves over a finite field is the ability to compute isogenies between curves. Elkies' approach is well suited for the case where the characteristic of the field is large. Couveignes showed how to compute isogenies in small characteristic. The aim of this paper is to describe the first successful implementation of Couveignes's algorithm. In particular, we describe the use of fast algorithms for performing incremental operations on series. We also insist on the particular case of the characteristic 2.

     

THE SIZE OF FARSIGHTED STABLE COALITIONS IN A GAME OF POLLUTION ABATEMENT

Abstract The game theoretical analysis of international environmental problems has received increasing attention in recent years. Major issues are as follows. Under which conditions will an international environmental agreement (IEA) be signed? Will the agreement be stable? Game theory has given different answers to these questions; in particular, it is possible to show that if countries are myopic then only small stable coalitions occur, but if they are farsighted then both large and small stable coalitions exist. This paper studies the size of a farsighted stable IEA by considering a quadratic cost function in a pollution abatement model. Following Rubio and Ulph [2006] , we consider both the case of non‐negative emissions as well as the case of unrestricted ones.     

AN INVESTIGATION OF MATHEMATICS TEXTBOOKS IN ENGLAND,FRANCE AND GERMANY: SOME CHALLENGES FOR ENGLAND

This paper reports on an analysis of mathematics textbooks used in lower secondary classrooms in England, France and Germany. The examination was carried out using an analysis schedule based on existing literature on textbooks. In this paper we look first at the development of the analysis schedule and then at the how particular textbooks treat the topic of angle. From this we conclude that learners in the different countries are offered different mathematics. In particular, learners in England are offered few challenges in the mathematics they are offered compared with their French and German counterparts.     

Regularity of the Laws of Shot Noise Series and of Related Processes

We investigate the regularity of shot noise series and of Poisson integrals. We give conditions for the absolute continuity of their law with respect to the Lebesgue measure and for their continuity in total-variation norm. In particular, the case of truncated series in addressed. Our method relies on a disintegration of the probability space based on a mere conditioning by the first jumps of the underlying Poisson process.     

On stable composite capillary-gravitational waves of finite amplitude on the surface of a fluid of finite depth : PMM vol. 39, n 6, 1975, pp. 1023–1031

The problem of stable plane capillary-gravitational waves of finite amplitude on the surface of a perfect incompressible fluid stream of finite depth is considered. It is assumed that the waves are induced by pressure periodically distributed along the free surface, and that these, unlike induced waves, do not vanish when the pressure becomes constant, are transformed into free waves. Such waves are called composite; they exist similarly to free waves, for particular values of velocity of the stream.The problem, which is rigorously stated, reduces to solving a system of four nonlinear equations for two functions and two constants. One of the equations is integral and the remaining are transcendental. Pressure on the surface is defined by an infinite trigonometric series whose coefficients are proportional to integral powers of some dimensionless small parameter; these powers are by two units greater than the numbers of coefficients.The theorem of existence and uniqueness of solution is established, and the method of its proof is indicated. The derivation of solution in any approximation is presented in the form of series in powers of the indicated small parameter. Computation of the first three approximations is carried out to the end, and an approximate equation of the wave profile is presented.Composite capillary-gravitational waves in the case of fluid of infinite depth were considered by the author in [1].     

Some new properties of the Riesz probability distribution

The Riesz probability distribution on any symmetric cone and, in particular, on the cone of positive definite symmetric matrices represents an important generalization of the Wishart and of the matrix gamma distributions containing them as particular examples. The present paper is a continuation of the investigation of the properties of this probability distribution. We first establish a property of invariance of this probability distributions by a subgroup of the orthogonal group. We then show that the Pierce components of a Riesz random variable are independent, and we determine their probability distributions. Some moments and some useful expectations related to the Riesz probability distribution are also calculated. Copyright © 2017 John Wiley & Sons, Ltd.     

Nonparametric predictive pairwise comparison for real-valued data with terminated tails

This paper presents a statistical method for comparison of two groups of real-valued data, based on nonparametric predictive inference (NPI), with the tails of the data possibly terminated, leading to small values being left-censored and large values being right-censored. Such tails termination can occur due to several reasons, including limits of detection, consideration of outliers, and specific designs of experiments. NPI is a statistical approach based on few assumptions, with inferences strongly based on data and with uncertainty quantified via lower and upper probabilities. We present NPI lower and upper probabilities for the event that the value of a future observation from one group is less than the value of a future observation from the other group, and we discuss several special cases that relate to well-known statistical problems.     

On the curious series related to the elliptic integrals

By using the theory of the elliptic integrals, a new method of summation is proposed for a certain class of series and their derivatives involving hyperbolic functions. It is based on the termwise differentiation of the series with respect to the elliptic modulus and integral representations of several of the series in terms of the inverse Mellin transforms related to the Riemann zeta function. The relation with the corresponding case of the Voronoi summation formula is exhibited. The involved series are expressed in closed form in terms of complete elliptic integrals of the first and second kind, and some special cases are calculated in terms of particular values of the Euler gamma function.     

Renormalization group approach to function approximation and to improving subsequent approximations

We establish a relation between bijective functions and renormalization group transformations and find their renormalization group invariants. For these functions, taking into account that they are globally one-to-one, we propose several improved approximations (compared with the power series expansion) based on this relation. We propose using the obtained approximations to improve the subsequent approximations of physical quantities obtained, in particular, by one of the main calculation techniques in theoretical physics, i.e., by perturbation theory. We illustrate the effectiveness of the renormalization group approximation with several examples: renormalization group approximations of several analytic functions and calculation of the nonharmonic oscillator ground-state energy. We also generalize our approach to the case of set maps, both continuous and discrete.     

Construction of Periodic Prolate Spheroidal Wavelets Using Interpolation

Periodic prolate spheroidal wavelets (periodic PS wavelets), based on the periodizaton of the first prolate spheroidal wave function (PSWF), were recently introduced by the authors. Because of localization and other properties, these periodic PS wavelets could serve as an alternative to Fourier series for applications in modeling periodic signals. In this paper, we continue our work with periodic PS wavelets and direct our attention to their construction via interpolation. We show that they have a representation in terms of interpolation with the modified Dirichlet kernel. We then derive a group of formulas of interpolation type based on this representation. These formulas enable one to obtain a simple procedure for the calculation of the periodic PS wavelets and finding expansion coefficients. In particular, they are used to compute filter coefficients for the periodic PS wavelets. This is done for a number of concrete cases.     

A solution to the distribution problems arising in the studies of a two-sex population process

Given a population of two sexes, the birth rate of one sex of which depends upon the population size of the other, it is very difficult to find an explicit expression for the probability distribution of the former. In this paper we have explicitly found the probability generating function of the joint distribution from which individual probability distributions and, in particular, moments of all orders in each case can be obtained in principle. As an example, using this probability generating function we have worked out explicitly the first and second order moments of the male and female populations and the explicit expression for the distribution of the male population in a particular case. This method can be successfully applied for the same purpose in the studies of chemical and biological processes where the synthesis or production of one species depends upon the concentration of another species.     

On simulation of tempered stable random variates

Various simulation methods for tempered stable random variates with stability index greater than one are investigated with a view towards practical implementation, in particular cases of very small scale parameter, which correspond to increments of a tempered stable Lévy process with a very short stepsize. Methods under consideration are based on acceptance-rejection sampling, a Gaussian approximation of a small jump component, and infinite shot noise series representations. Numerical results are presented to discuss advantages, limitations and trade-off issues between approximation error and required computing effort. With a given computing budget, an approximative acceptance-rejection sampling technique Baeumer and Meerschaert (2009) [11] is both most efficient and handiest in the case of very small scale parameter and moreover, any desired level of accuracy may be attained with a small amount of additional computing effort.