A LIMIT THEOREM FOR INTERACTING MEASURE-VALUED BRANCHING PROCESSES

1IntroductionItiswellknownthatthe(classical)DW-superprocessiscloselyassociatedwiththe(c1assical)FV-superprocess.Tl1isconnectionisfiIStobservedbyKonnoandShiga(1988),rigorouslyprovedbyEtheridgeandMarch(1991),andgeneralizedbyPerkins(l992).Infact,intermofapproximationofparticlesystemsthisrelationshipisratherintuitive,buttoproveitisnoteasy.I11thispaper,weshallconsiderthesanequestionforthemeasure-valuedprocesseswithinteractions.Forthesakeofbrevity,weomittheintroductionoftl1iskindofmeasure-value…     

Fuzzy logic in measurements

Our main interest in this paper is to translate from “natural language” into “system theoretical language”. This is of course important since a statement in system theory can be analyzed mathematically or computationally. We assume that, in order to obtain a good translation, “system theoretical language” should have great power of expression. Thus we first propose a new frame of system theory, which includes the concepts of “measurement” as well as “state equation”. And we show that a certain statement in usual conversation, i.e., fuzzy modus ponens with the word “very”, can be translated into a statement in the new frame of system theory. Though our result is merely one example of the translation from “natural language” into “system theoretical language”, we believe that our method is fairly general.     

Absolutely continuous measure for a jump-type Fleming-Viot process

In this paper, we prove that the random measure of the one-dimensional jump-type Fleming-Viot process is absolutely continuous with respect to the Lebesgue measure in R, provided the mutation operator satisfies certain regularity conditions. This result is an important step towards the representation of the Fleming-Viot process with jumps in terms of the solution of a stochastic partial differential equation.     

A quantum mechanical approach to a fuzzy theory

Recently, we proposed a general measurement theory for classical and quantum systems (i.e., “objective fuzzy measurement theory”). In this paper, we propose “subjective fuzzy measurement theory”, which is characterized as the statistical method of the objective fuzzy measurement theory. Our proposal of course has a lot of advantages. For example, we can directly see “membership functions” (= “fuzzy sets”) in this theory. Therefore, we can propose the objective and the subjective methods of membership functions. As one of the consequences, we assert the objective (i.e., individualistic) aspect of Zadeh's theory. Also, as a quantum application, we clarify Heisenberg's uncertainty relation.     

Mathematical modelling of the development of mechanical instability in fracturing systems

The dynamics of processes accompanying a loss of stability in a mechanical system are investigated. The mechanical system is in the form of an elastic rod, stretched by an axial load, with one of its lateral surfaces “glued” to a rigid wall. The “glue” is a low-strength elastic material which is subject to brittle fracture at a certain value of the load acting on it. In a fractured segment, the rod surface slides over the wall surface under the action of a dry friction force which is less than the breaking stress. The high sensitivity of the process of the development of instability to small perturbations which initiate the development of instability is established. The system considered is the simplest model of the zone of contact between lithospheric plates which generate earthquakes.     

A theory of tensor products for module categories for a vertex operator algebra, III

This is the third part in a series of papers developing a tensor product theory for modules for a vertex operator algebra. The goal of this theory is to construct a “vertex tensor category” structure on the category of modules for a suitable vertex operator algebra. The notion of vertex tensor category is essentially a “complex analogue” of the notion of symmetric tensor category, and in fact a vertex tensor category produces a braided tensor category in a natural way. In this paper, we focus on a particular element P(z) of a certain moduli space of three-punctured Riemann spheres; in general, every element of this moduli space will give rise to a notion of tensor product, and one must consider all these notions in order to construct a vertex tensor category. Here we present the fundamental properties of the P(z)-tensor product of two modules for a vertex operator algebra. We give two constructions of a P(z)-tensor product, using the results, established in Parts I and II of this series, for a certain other element of the moduli space. The definitions and results in Parts I and II are recalled.     

Comment on “The Steiner number of a graph” by G. Chartrand and P. Zhang: [Discrete Mathematics 242 (2002) 41–54]

We show by counterexample that one of the main results in the paper “The Steiner number of a graph” by Chartrand and Zhang (Disc. Math. 242 (2002) 41–54) does not hold. To be more precise, we prove both that not every Steiner set is a geodetic set and that there are connected graphs whose Steiner number is strictly lower than its geodetic number.     

Interleaving two-phased jobs on a single machine

In this paper, we consider a single machine that processes a set of jobs having two (ordered) phases. After processing the first phase of a job, this job must be removed from the machine for some exact amount of time, after which the machine must immediately begin processing its second phase. During this “dead time” between job phases, the machine may be used to process other similar jobs. We first prove that the problem of interleaving these jobs in order to minimize the makespan (or to process as many jobs as possible by a given deadline) is strongly NP-hard. Next, we compare the effectiveness of a mixed-integer programming formulation based on a continuous time domain to that of a discrete-time integer programming model for solving problems having different data characteristics. These comparisons are performed on a set of realistic synthetic problems based on different scenarios arising in radar pulsing applications.     

The basin of attraction of the Chen attractor

By geometric analysis, we discuss the riddled property of the basin of attraction of the Chen attractor based on Milnor’s definition, and prove that any neighborhood of the Chen attractor contains repelled sets with positive Lebesque measures. Our analytic and numerical results show that the Chen attractor indeed has some unusual properties leading to a “strange attractor” in the sense of Milnor.     

Drawing series parallel digraphs symmetrically

In this paper we present algorithms for drawing series parallel digraphs with as much symmetry as possible. The first step is to compute a certain kind of automorphism, called an “upward planar automorphism” for an input series parallel digraph. The next step uses these automorphisms to construct a symmetric drawing of the graph. We present several variations of the second step, with visibility drawings, “bus-orthogonal” drawings, and polyline drawings. All algorithms run in linear time.     

Classes of pretopological spaces closed under the formation of final structures

We investigate coreflective subconstructs of the construct Prtop of pretopological spaces and continuous maps and in particular the inclusion “order” between these subconstructs. We describe the smallest, second and third coreflective class and then all minimal elements that are strictly larger. Using these minimal elements we obtain a “partition” of the whole conglomerate of coreflective subconstructs of Prtop. The results dealing with classes in one member of this partition have an immediate interpretation in the framework of reflexive relations.     

On an observer-related unequivalence between spatial dimensions of a generic Cremonian universe

Given a generic Cremonian space-time, its three spatial dimensions are shown to exhibit an intriguing, “two-plus-one” partition with respect to standard observers. Such observers are found to form three distinct, disjoint groups based on which one out of the three dimensions stands away from the other two. These two subject-related properties have, to our knowledge, no analogue in any of the existing physical theories of space-time; yet, in one of them, the so-called Cantorian model, a closer inspection may reveal some traits of such a “space split-up.”     

Statistical learning control of uncertain systems: theory and algorithms

It has recently become clear that many control problems are too difficult to admit analytic solutions. New results have also emerged to show that the computational complexity of some “solved” control problems is prohibitive. Many of these control problems can be reduced to decidability problems or to optimization questions. Even though such questions may be too difficult to answer analytically, or may not be answered exactly given a reasonable amount of computational resources, researchers have shown that we can “approximately” answer these questions “most of the time”, and have “high confidence” in the correctness of the answers.     

New results on the conjecture of Rhodes and on the topological conjecture

The Conjecture of Rhodes, originally called the “type II conjecture” by Rhodes, gives an algorithm to compute the kernel of a finite semigroup. This conjecture has numerous important consequences and is one of the most attractive problems on finite semigroups. It was known that the conjecture of Rhodes is a consequence of another conjecture on the finite group topology for the free monoid. In this paper, we show that the topological conjecture and the conjecture of Rhodes are both equivalent to a third conjecture and we prove this third conjecture in a number of significant particular cases.     

Universal deformation formulas and breaking symmetry

We show that an algebra with a non-nilpotent Lie group of automorphisms or “symmetries” (e.g., smooth functions on a manifold with such a group of diffeomorphisms) may generally be deformed (in the function case, “quantized”) in such a way that only a proper subgroup of the original group acts. This symmetry breaking is a consequence of the existence of certain “universal deformation formulas” which are elements, independent of the original algebra, in the tensor algebra of the enveloping algebra of the Lie algebra of the group.     

A covariance inequality under a two-part dependence assumption

A covariance inequality is proved under a certain “two-part” dependence assumption. It generalizes and sharpens, with a simpler and more transparent proof, two earlier covariance inequalities used in central limit theory under certain “two-part” strong mixing assumptions.     

Continuous iterations of dynamical maps: An axiomatic approach

In this paper, an axiomatic definition of continuous iterations of a dynamical map is provided. From the axioms that define common properties of all continuous iterations, it will be demonstrated that continuous iterations that are also derivable must satisfy a certain nonlinear differential equation, herein referred as the “Equation of Derivable Continuous Iterations”. A general solution of this equation will be obtained by means of the Laplace transform and it will be shown that derivable continuous iterations of a map must have a certain functional form. A formula for analytically calculating derivable continuous iterations of maps with at least a fixed point is provided.     

Dynamic spanning without probabilities

The paper presents a non-probabilistic approach to continuous-time trading where, in analogy to the binomial option-pricing model, terminal payoffs resulting from a given trading strategy are meaningful ‘state-by-state’, i.e., path-by-path. In particular, we obtain results of the form: “If a certain trading strategy is applied and if the realized price trajectory satisfies a certain analytical property, then the terminal payoff is.…” This way, derivation of the Black and Scholes formula and its extension become an exercise in the analysis of a certain class of real functions. While results of the above forms are of great interest if the analytical property in question is believed to be satisfied for almost all realized price trajectories (for example, if the price is believed to follow a certain stochastic process which has this property with probability 1), they are valid regardless of the stochastic process which presumably generates the possible price trajectories or the probability assigned to the set of all paths having this analytical property.