On a robust version of the integral representation formula of nonlinear filtering

The paper is concerned with completing “unfinished business” on a robust representation formula for the conditional expectation operator of nonlinear filtering. Such a formula, robust in the sense that its dependence on the process of observations is continuous, was stated in [2] without proof. The main purpose of this paper is to repair this deficiency.The formula is “almost obvious” as it can be derived at a formal level by a process of integration-by-parts applied to the stochastic integrals that appear in the integral representation formula. However, the rigorous justification of the formula is quite subtle, as it hinges on a measurability argument the necessity of which is easy to miss at first glance. The continuity of the representation (but not its validity) was proved by Kushner [9] for a class of diffusions.Here we follow the definition given in [11].     

Formal and precise derivation of the Green functions for a simple potential

In formal scattering theory, the Green functions are obtained as solutions of a distributional equation. In this paper, we use the Sturm–Liouville theory to compute the Green functions within a rigorous mathematical theory. We shall show that both the Sturm–Liouville theory and the formal treatment yield the same Green functions. We shall also show how the analyticity of the Green functions as functions of the energy keeps track of the so-called “incoming” and “outgoing” boundary conditions.     

模糊联盟合作对策的收益分配研究

针对现实环境中联盟组成的不确定性, 本文研究了具有模糊联盟的合作对策求解问题。提出了模糊联盟合作对策的一种新的分配方式,即平均分摊解,并给出了这种解与模糊联盟合作对策Shapley值一致的充分条件。同时,还提出了模糊联盟合作对策的Shapley值的一个重要性质。最后,结合算例进行了分析论证。     

AIC, Overfitting Principles, and the Boundedness of Moments of Inverse Matrices for Vector Autotregressions and Related Models

In his somewhat informal derivation, Akaike (in “Proceedings of the 2nd International Symposium Information Theory” (C. B. Petrov and F. Csaki, Eds.), pp. 610–624, Academici Kiado, Budapest, 1973) obtained AIC's parameter-count adjustment to the log-likelihood as a bias correction: it yields an asymptotically unbiased estimate of the quantity that measures the average fit of the estimated model to an independent replicate of the data used for estimation. We present the first mathematically complete derivation of an analogous property of AIC for comparing vector autoregressions fit to weakly stationary series. As a preparatory result, we derive a very general “overfitting principle,” first formulated in a more limited context in Findley (Ann. Inst. Statist. Math.43, 509–514, 1991), asserting that a natural measure of an estimated model's overfit due to parameter estimation is equal, asymptotically, to a measure of its accuracy loss with independent replicates. A formal principle of parsimony for fitted models is obtained from this, which for nested models, covers the situation in which all models considered are misspecified. To prove these results, we establish a set of general conditions under which, for each τ1, the absolute τth moments of the entries of the inverse matrices associated with least squares estimation are bounded for sufficiently large sample sizes.     

Multiway Dependence in Exponential Family Conditional Distributions

Conditionally specified statistical models are frequently constructed from one-parameter exponential family conditional distributions. One way to formulate such a model is to specify the dependence structure among random variables through the use of a Markov random field (MRF). A common assumption on the Gibbsian form of the MRF model is that dependence is expressed only through pairs of random variables, which we refer to as the “pairwise-only dependence” assumption. Based on this assumption, J. Besag (1974, J. Roy. Statist. Soc. Ser. B36, 192–225) formulated exponential family “auto-models” and showed the form that one-parameter exponential family conditional densities must take in such models. We extend these results by relaxing the pairwise-only dependence assumption, and we give a necessary form that one-parameter exponential family conditional densities must take under more general conditions of multiway dependence. Data on the spatial distribution of the European corn borer larvae are fitted using a model with Bernoulli conditional distributions and several dependence structures, including pairwise-only, three-way, and four-way dependencies.     

Fashion cycle dynamics induced by agents' heterogeneity for generic bell-shaped attractiveness functions

ABSTRACT

We extend the exchange economy evolutionary discrete-time model with heterogeneous agents introduced in ‘Fashion cycle dynamics in a model with endogenous discrete evolution of heterogeneous preferences’, by A. Naimzada and M. Pireddu, appeared in Chaos: An Interdisciplinary Journal of Nonlinear Science 28, 055907 (2018), by considering generic bell-shaped attractiveness functions for the two groups of agents, increasing for low visibility levels, but decreasing when the visibility of the group exceeds a given threshold value, due to a congestion effect. The reproduction level of a group is related to its attractiveness degree, which depends on its social visibility level, determined by the consumption choices of the agents in that group. We investigate the asymptotic heterogeneity among agents, finding that the global dynamics may differ according to the chosen functional form for the attractiveness. Nonetheless, the model is still able to reproduce the recurrent dynamic behaviour typical of the fashion cycle.     

The perception of randomness

Psychologists have studied people's intuitive notions of randomness by two kinds of tasks: judgment tasks (e.g., “is this series like a coin?” or “which of these series is most like a coin?”), and production tasks (e.g., “produce a series like a coin”). People's notion of randomness is biased in that they see clumps or streaks in truly random series and expect more alternation, or shorter runs, than are there. Similarly, they produce series with higher than expected alternation rates. Production tasks are subject to other biases as well, resulting from various functional limitations. The subjectively ideal random sequence obeys “local representativeness”; namely, in short segments of it, it represents both the relative frequencies (e.g., for a coin, 50%–50%) and the irregularity (avoidance of runs and other patterns). The extent to which this bias is a handicap in the real world is addressed.     

Random creation and dispersion of mass

Consider evolution of density of a mass or a population, geographically situated in a compact region of space, assuming random creation-annihilation and migration, or dispersion of mass, so the evolution is a random measure. When the creation-annihilation and dispersion are diffusions the situation is described formally by a stochastic partial differential equation; ignoring dispersion make approximations to the initial density by atomic measures and if the corresponding discrete random measures converge “in law” to a unique random measure call it a solution. To account for dispersion Trotter's product formula is applied to semiflows corresponding to dispersion and creation-annihilation. Existence of solutions has been a conjecture for several years despite a claim in ([2], J. Multivariate Anal. 5, 1–52). We show that solutions exist and that non-deterministic solutions are “smeared” continuous-state branching diffusions.     

A market approach to decentralized control of a manufacturing cell

Based on a fictitious market model, a decentralized approach is presented for the workstation scheduling in a CNC workshop. A multi-agent framework is proposed, where job agents and resource agents act as buyers and sellers of resource in the virtual market. With cost and benefit calculation of these agent activities, which reflects the state of the production environment, various, and often conflicting goals and interests influencing the scheduling process in practice can be balanced through a unified instrument offered by the markets. The paper first introduces a heuristic procedure that makes scheduling reservations in a periodic manner. A multi-agent framework is then introduced, in which job agents and resource agents seek appropriate job–workstation matches through bidding in the construction of the above periodic “micro-schedules”. A pricing policy is proposed for the price-directed coordination of agent activities in this. Simulation results demonstrate the feasibility of the proposed approach and give some insights on the effects of some decision making parameters. Future work will be focused on the designing of some more sophisticated coordination mechanism and its deployment.     

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.     

Self-enforcing coalitions with power accumulation

Agents endowed with power compete for a divisible resource by forming coalitions with other agents. The coalition with the greatest power wins the resource and divides it among its members. The agents’ power increases according to their share of the resource.We study two models of coalition formation where winning agents accumulate power and losing agents may participate in further coalition formation processes. An axiomatic approach is provided by focusing on variations of two main axioms: self-enforcement, which requires that no further deviation happens after a coalition has formed, and rationality, which requires that agents pick the coalition that gives them their highest payoff. For these alternative models, we determine the existence of stable coalitions that are self-enforcing and rational for two traditional sharing rules. The models presented in this paper illustrate how power accumulation, the sharing rule, and whether losing agents participate in future coalition formation processes, shape the way coalitions will be stable throughout time.     

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.     

Multilinear forms and measures of dependence between random variables

Based on an idea of Rosenblatt, the methods of interpolation theory are used to establish moment inequalities and equivalence relations for measures of dependence between two or more families of random variables. A couple of “interpolation” theorems proved here appear to be new.     

Gauss map and Lyapunov exponents of interacting particles in a billiard

We show that the Lyapunov exponent (LE) of periodic orbits with Lebesgue measure zero from the Gauss map can be used to determine the main qualitative behavior of the LE of a Hamiltonian system. The Hamiltonian system is a one-dimensional box with two particles interacting via a Yukawa potential and does not possess Kolmogorov–Arnold–Moser (KAM) curves. In our case the Gauss map is applied to the mass ratio (γ = m₂/m₁) between particles. Besides the main qualitative behavior, some unexpected peaks in the γ dependence of the mean LE and the appearance of ‘stickness’ in phase space can also be understand via LE from the Gauss map. This shows a nice example of the relation between the “instability” of the continued fraction representation of a number with the stability of non-periodic curves (no KAM curves) from the physical model. Our results also confirm the intuition that pseudo-integrable systems with more complicated invariant surfaces of the flow (higher genus) should be more unstable under perturbation.     

The rate of convergence for the cyclic projections algorithm III: Regularity of convex sets

The cyclic projections algorithm is an important method for determining a point in the intersection of a finite number of closed convex sets in a Hilbert space. That is, for determining a solution to the “convex feasibility” problem. This is the third paper in a series on a study of the rate of convergence for the cyclic projections algorithm. In the first of these papers, we showed that the rate could be described in terms of the “angles” between the convex sets involved. In the second, we showed that these angles often had a more tractable formulation in terms of the “norm” of the product of the (nonlinear) metric projections onto related convex sets.In this paper, we show that the rate of convergence of the cyclic projections algorithm is also intimately related to the “linear regularity property” of Bauschke and Borwein, the “normal property” of Jameson (as well as Bakan, Deutsch, and Li’s generalization of Jameson’s normal property), the “strong conical hull intersection property” of Deutsch, Li, and Ward, and the rate of convergence of iterated parallel projections. Such properties have already been shown to be important in various other contexts as well.     

Norm property of Hilbert pairing

The natural analog of the property (x, 1 – x)=1, x 0,1 of the multiplicative Hilbert symbol is considered for Hilbert pairing in Lubin-Tate formal groups; in this paper the property is called the fundamental norm property. Criteria are given for satisfaction of the fundamental norm property in Lubin-Tate formal groups, and -symbols generalizing this property are described. A new approach to constructing formal Lubin-Tate groups is presented.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 175, pp. 30–45, 1989.     

On single-machine scheduling without intermediate delays

This paper introduces the non-idling machine constraint where no intermediate idle time between the operations processed by a machine is allowed. In its first part, the paper considers the non-idling single-machine scheduling problem. Complexity aspects are first discussed. The “Earliest Non-Idling” property is then introduced as a sufficient condition so that an algorithm solving the original problem also solves its non-idling variant. Moreover it is shown that preemptive problems do have that property. The critical times of an instance are then introduced and it is shown that when their number is polynomial, as for equal-length jobs, a polynomial algorithm solving the original problem has a polynomial variant solving its non-idling version.