A sequent calculus for logic of knowledge and past time: Completeness and decidability

We consider logic of knowledge and past time. This logic involves the discrete-time linear temporal operators next, until, weak yesterday, and since. In addition, it contains an indexed set of unary modal operators agent i knows.We consider the semantic constraint of the unique initial states for this logic. For the logic, we present a sequent calculus with a restricted cut rule. We prove the soundness and completeness of the sequent calculus presented. We prove the decidability of provability in the considered calculus as well. So, this calculus can be used as a basis for automated theorem proving. The proof method for the completeness can be used to construct complete sequent calculi with a restricted cut rule for this logic with other semantical constraints as well. Published in Lietuvos Matematikos Rinkinys, Vol. 46, No. 3, pp. 427–437, July–September, 2006.     

Proof-theoretical investigation of temporal logic with time gaps

In the paper, the first-order intuitionistic temporal logic sequent calculus LBJ is considered. The invertibility of some of the LBJ rules, syntactic admissibility of the structural rules and the cut rule in LBJ, as well as Harrop and Craig's interpolation theorems for LBJ are proved. Gentzen's midsequent theorem is proved for the LBJ' calculus which is obtained from LBJ by removing the antecedent disjunction rule from it. Published in Lietuvos Matematikos Rinkinys, Vol. 40, No. 3, pp. 255–276, July–September, 2000.     

Sequent Calculi with Analytic Cut for Logics of Time and Knowledge with Perfect Recall

In this paper, we consider two logics of time and knowledge. These logics involve the discrete time linear temporal logic operators ``next' and ``until'. In addition, they contain an indexed set of unary epistemic modalities ``agent $i$ knows'. In these logics, the temporal and epistemic dimensions may interact. The particular interactions we consider capture perfect recall. We consider perfect recall in synchronously distributed systems and in systems without any assumptions. For these logics, we present sequent calculi with an analytic cut rule. Thus, we get proof systems where proof-search becomes decidable. The soundness and completeness of these calculi are proved.     

Analyzing completeness of axiomatic functional systems for temporal × modal logics

In previous works, we presented a modification of the usual possible world semantics by introducing an independent temporal structure in each world and using accessibility functions to represent the relation among them. Different properties ofthe accessibility functions (being injective, surjective, increasing, etc.) have been considered and axiomatic systems (called functional) which define these properties have been given. Only a few ofthese systems have been proved tobe complete. The aim ofthis paper is to make a progress in the study ofcompleteness for functional systems. For this end, we use indexes as names for temporal flows and give new proofs of completeness. Specifically, we focus our attention on the system which defines injectivity, because the system which defines this property without using indexes was proved to be incomplete in previous works. The only system considered which remains incomplete is the one which defines surjectivity, even ifwe consider a sequence ofnatural extensions ofthe previous one (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Extinction time and the total mass of the continuous-state branching processes with competition

Consider a general continuous-state branching process with additional interaction, which destroys the branching property. We give precise conditions on the interaction term, in order to decide whether the extinction time of the process remains or not bounded as the initial value tends to infinity, and similarly for the total mass of the process.     

A multisymplectic framework for classical field theory and the calculus of variations II: space + time decomposition

In a previous paper I laid the foundations of a covariant Hamiltonian framework for the calculus of variations in general. The purpose of the present work is to demonstrate, in the context of classical field theory, how this covariant Hamiltonian formalism may be space + time decomposed. It turns out that the resulting “instantaneous” Hamiltonian formalism is an infinite- dimensional version of Ostrogradski 's theory and leads to the standard symplectic formulation of the initial value problem. The salient features of the analysis are: (i) the instantaneous Hamiltonian formalism does not depend upon the choice of Lepagean equivalent; (ii) the space + time decomposition can be performed either before or after the covariant Legendre transformation has been carried out, with equivalent results; (iii) the instantaneous Hamiltonian can be recovered in natural way from the multisymplectic structure inherent in the theory; and (iv) the space + time split symplectic structure lives on the space of Cauchy data for the evolution equations, as opposed to the space of solutions thereof.     

Limit theorems for occupation time fluctuations of branching systems II: Critical and large dimensions

We give functional limit theorems for the fluctuations of the rescaled occupation time process of a critical branching particle system in R^d${\mathbb{R}}^d$ with symmetric α$\alpha$-stable motion in the cases of critical and large dimensions, d=2α$d=2\alpha$ and d>2α$d>2\alpha$. In a previous paper [T. Bojdecki, L.G. Gorostiza, A. Talarczyk, Limit theorems for occupation time fluctuations of branching systems I: long-range dependence, Stochastic Process. Appl., this issue.] we treated the case of intermediate dimensions, α<d<2α$\alpha <d<2\alpha$, which leads to a long-range dependence limit process. In contrast, in the present cases the limits are generalized Wiener processes. We use the same space–time random field method of the previous paper, the main difference being that now the tightness requires a new approach and the proofs are more difficult. We also give analogous results for the system without branching in the cases d=α$d=\alpha$ and d>α$d>\alpha$.     

Smoothing properties and approximation of time derivatives for parabolic equations: constant time steps

We study smoothing properties and approximation of time derivativesfor time discretization schemes with constant time steps fora homogeneous parabolic problem formulated as an abstract initial-valueproblem in a Banach space. The time stepping schemes are basedon using rational functions r(z) e^–z which are A()-stablefor suitable [0, /2] and satisfy |r()| < 1, and the approximationsof time derivatives are based on using difference quotientsin time. Both smooth and non-smooth data error estimates ofoptimal order for the approximation of time derivatives areproved. Further, we apply the results to obtain error estimatesof time derivatives in the supremum norm for fully discretemethods based on discretizing the spatial variable by a finite-elementmethod.     

On areas of attraction and repulsion in finite time dynamical systems and their numerical approximation

ABSTRACT

Stable and unstable fibre bundles with respect to a fixed point or a bounded trajectory are of great dynamical relevance in (non)autonomous dynamical systems. These sets are defined via an infinite limit process. However, the dynamics of several real world models are of interest on a short time interval only. This task requires finite time concepts of attraction and repulsion that have been recently developed in the literature. The main idea consists in replacing the infinite limit process by a monotonicity criterion and in demanding the end points to lie in a small neighbourhood of the reference trajectory. Finite time areas of attraction and repulsion defined in this way are fat sets and their dimension equals the dimension of the state space. We propose an algorithm for the numerical approximation of these sets and illustrate its application to several two- and three-dimensional dynamical systems in discrete and continuous time. Intersections of areas of attraction and repulsion are also calculated, resulting in finite time homoclinic orbits.     

Optimal quoting of delivery time by a third party logistics provider: The impact of shipment consolidation and temporal pricing schemes

Increased competition in business environments requires that firms provide not only quality but also timely service with minimal cost. Offering a delivery-time guarantee may increase the demand for a product or service, or allow the firm to charge a price premium. This paper investigates the effects of different pricing schemes for a Third Party Logistics (3PL) provider. The 3PL tenders a consolidated load to a carrier that line-hauls over a certain origin–destination lane. In a price- and time-sensitive logistics market, we derive the optimal quotations that should be made for price and delivery-time, with the objective of maximizing the profit rate of the 3PL provider. We propose four easy-to-use temporal pricing schemes, and derive the corresponding optimal length of shipment consolidation cycles and the prices. Depending on the logistics market parameters, we show that charging according to an order’s time of arrival is not necessarily the best pricing scheme. Various managerial insights and numerical examples with sensitivity analysis are provided.     

Joint distributions of first hitting time and first hitting location after explosion for birth and death processes

For a birth and death processX=|X(t),t <σ| with explosion and lifespanu distributions and joint distributions of first hitting time and first hitting location after explosion of setB _n = |0,1,...,n| ,n have been found.     

Two storage inventory problem with dynamic demand and interval valued lead-time over finite time horizon under inflation and time-value of money

A finite time horizon inventory problem for a deteriorating item having two separate warehouses, one is a own warehouse (OW) of finite dimension and other a rented warehouse (RW), is developed with interval-valued lead-time under inflation and time value of money. Due to different preserving facilities and storage environment, inventory holding cost is considered to be different in different warehouses. The demand rate of item is increasing with time at a decreasing rate. Shortages are allowed in each cycle and backlogged them partially. Shortages may or may not be allowed in the last cycle and under this circumstance, there may be three different types of model. Here it is assumed that the replenishment cycle lengths are of equal length and the stocks of RW are transported to OW in continuous release pattern. For each model, different scenarios are depicted depending upon the re-order point for the next lot. Representing the lead-time by an interval number and using the interval arithmetic, the single objective function for profit is changed to corresponding multi-objective functions. These functions are maximized and solved by Fast and Elitist Multi-objective Genetic Algorithm (FEMGA). The models are illustrated numerically and the results are presented in tabular form.     

Ergodic BSDE with unbounded and multiplicative underlying diffusion and application to large time behaviour of viscosity solution of HJB equation

We study ergodic backward stochastic differential equations (EBSDEs), for which the underlying diffusion is assumed to be multiplicative and of linear growth. The fact that the forward process has an unbounded diffusion is balanced with an assumption of weak dissipativity for its drift. Moreover, the forward equation is assumed to be non-degenerate. We study the existence and uniqueness of EBSDEs and we apply our results to an ergodic optimal control problem. In particular, we show the large time behaviour of viscosity solution of Hamilton–Jacobi–Bellman equation with an exponential rate of convergence when the underlying diffusion is multiplicative and unbounded.     

A multi‐regional epidemic model for controlling the spread of Ebola: awareness,treatment, and travel‐blocking optimal control approaches

Ebola virus disease (EVD) can rapidly cause death to animals and people, for less than 1month. In addition, EVD can emerge in one region and spread to its neighbors in unprecedented durations. Such cases were reported in Guinea, Sierra Leone, and Liberia. Thus, by blocking free travelers, traders, and transporters, EVD has had also impacts on economies of those countries. In order to find effective strategies that aim to increase public knowledge about EVD and access to possible treatment while restricting movements of people coming from regions at high risk of infection, we analyze three different optimal control approaches associated with awareness campaigns, treatment, and travel‐blocking operations that health policy‐makers could follow in the war on EVD. Our study is based on the application of Pontryagin's maximum principle, in a multi‐regional epidemic model we devise here for controlling the spread of EVD. The model is in the form of multi‐differential systems that describe dynamics of susceptible, infected, and removed populations belonging to p different geographical domains with three control functions incorporated. The forward–backward sweep method with integrated progressive‐regressive Runge–Kutta fourth‐order schemes is followed for resolving the multi‐points boundary value problems obtained. Copyright © 2016 John Wiley & Sons, Ltd.     

Stationarity and almost sure divergence of time averages in interval-valued probability

Our work is motivated by the study of empirical processes (such as flicker noise) that occur in stable systems yet give rise to observations with seemingly divergent time averages. Stationary models for such processes do not exist in the domain of numerical probability, as the ergodic theorems dictate the convergence of time averages of stationary and bounded processes. This has led us to investigate such models in the wider framework of interval-valued probability. In this paper we construct interval-valued probabilities on the space of infinite binary sequences that combine properties of (i) strict stationarity, (ii) unicity of extension from the algebra of cylinder sets to a wider collection containing salient asymptotic events, and (iii) almost sure support of divergence of time averages. These properties are not shared by conventional stochastic models.     

An exponential time differencing method of lines for the Burgers and the modified Burgers equations

A second–order exponential time differencing scheme using the method of lines is developed in this article for the numerical solution of the Burgers and the modified Burgers equations. For each case, the resulting nonlinear system is solved explicitly using a modified predictor‐corrector method. The efficiency of the method introduced is tested by comparing experimental results with others selected from the available literature.     

Optimal control of parabolic equation with an infinite number of variables for non-standard functional and time delay

A distributed control problem for the parabolic operator withan infinite number of variables and time delay is considered.The performance index has an integral form. Constraints on controlsare imposed. To obtain optimality conditions for the Neumannproblem, the generalization of the Dubovitskii–Milyutintheorem given by Walczak in WALCZAK, S. Folia Mathematics, 1,187–196 and WALCZAK, S. J. Optim. Theory Appl., 42, 561–582was applied.     

Existence and exponential stability of unique almost periodic solution for Lasota–Wazewska red blood cell model with perturbation on time scales

This paper deals with Lasota–Wazewska red blood cell model with perturbation on time scales. By applying the fixed point theorem of decreasing operator, we establish sufficient conditions for the existence of unique almost periodic positive solution. Particularly, we give iterative sequence which converges to the almost periodic positive solution. Moreover, we investigate exponential stability of the almost periodic positive solution by means of Gronwall inequality. Copyright © 2017 John Wiley & Sons, Ltd.     

Characterizing prospective secondary teachers’ foundation and contingency knowledge for definitions of transformations

One promising approach for connecting undergraduate content coursework to secondary teaching is using teacher-created representations of practice. Using these representations effectively requires seeing teachers' use of mathematical knowledge in the work of teaching. We argue that the dimensions of Rowland's (2013) Knowledge Quartet, especially Foundation and Contingency, form a fruitful framework for this purpose. We contribute an analytic framework to characterize the quality of mathematical knowledge observed in the Foundation and Contingency dimensions, developed using a purposive sampling from over 300 representations. These representations all featured geometry teaching. We showcase the framework with examples of "high" and "developing" Foundation and Contingency.Then, we compare our coding along these dimensions with performance on a measure of mathematical knowledge for teaching geometry. Finally, we describe the potential for generalizing this framework to other domains, such as algebra and mathematical modeling.