A note on variance bounding for continuous time Markov Chains

In this paper the notion of variance bounding introduced by Roberts and Rosenthal (2008) is extended to continuous time Markov Chains. Moreover, it is proven that, as in the discrete time case, the notion of variance bounding for reversible Markov Chains is equivalent to the existence of a central limit theorem. A connection with the continuous time Peskun ordering, introduced by Leisen and Mira (2008), concludes the paper.     

A Cut-Invariant Law of Large Numbers for Random Heaps

We consider the framework of Bernoulli measures for heap monoids. We introduce in this framework the notion of asynchronous stopping time, which generalizes the notion of stopping time for classical probabilistic processes. A strong Bernoulli property is proved. A notion of cut-invariance is formulated for convergent ergodic means. Then, a version of the strong law of large numbers is proved for heap monoids with Bernoulli measures. We study a sub-additive version of the law of large numbers in this framework.     

Polynomial contractions and Lyapunov regularity

The purpose of this note is twofold: to introduce the notion of polynomial contraction for a linear nonautonomous dynamics with discrete time, and to show that it persists under sufficiently small linear and nonlinear perturbations. The notion of polynomial contraction mimics the notion of exponential contraction, but with the exponential decay replaced by a polynomial decay. We show that this behavior is exhibited by a large class of dynamics, by giving necessary conditions in terms of “polynomial” Lyapunov exponents. Finally, we establish the persistence of the asymptotic stability of a polynomial contraction under sufficiently small linear and nonlinear perturbations. We also consider the case of nonuniform polynomial contractions, for which the Lyapunov stability is not uniform.     

KKT Optimality Conditions and Nonsmooth Continuous Time Optimization Problems

We introduce a concept of generalized invexity for the nonsmooth continuous time optimization problems, namely, the concept of Karush-Kuhn-Tucker (KKT) invexity. Then, we prove that this notion is necessary and sufficient for global optimality of a KKT point. We also extend the notion of weak-invexity for nonsmooth continuous time optimization problems. Further, we show that weak-invexity is a necessary and sufficient condition for weak duality.     

Transportation distances and noise sensitivity of multiplicative Lévy SDE with applications

This article assesses the distance between the laws of stochastic differential equations with multiplicative Lévy noise on path space in terms of their characteristics. The notion of transportation distance on the set of Lévy kernels introduced by Kosenkova and Kulik yields a natural and statistically tractable upper bound on the noise sensitivity. This extends recent results for the additive case in terms of coupling distances to the multiplicative case. The strength of this notion is shown in a statistical implementation for simulations and the example of a benchmark time series in paleoclimate.     

Almost periodic solutions to dynamic equations on time scales

In this paper, we first present a notion of almost periodic functions on time scales and study their basic properties. Then by means of Liapunov functionals, we study the existence of almost periodic solutions for an almost periodic dynamic equation on time scales.     

Jacobi's condition for singular extremals: An extended notion of conjugate points

For the simple fixed endpoint problem in the calculus of variations, Jacobi's condition (“there are no conjugate points in the interior of the underlying time interval”) is necessary for optimality if the trajectory under consideration is nonsingular. In this paper, we extend the notion of conjugate points so that the above condition (in terms of this new notion) is necessary also for singular extremals. This is achieved by showing that, without any additional assumption on the trajectory, the nonnegativity of the second variation on the space of admissible variations is equivalent to the nonexistence of these “extended conjugate points”.     

On the complexity of recognizing perfectly orderable graphs

The question whether a polynomial time recognition algorithm for the class of perfectly orderable graphs exists was posed by Chvátal in 1981 when he introduced the notion of perfect orders. Since then several classes of perfectly orderable graphs have been identified. In this note we prove that recognizing perfectly orderable graphs is NP-complete.     

DEGREE OF ECONOMIC UNSUSTAINABILITY IN THE MANAGEMENT OF RENEWABLE RESOURCES

A criterion commonly used to determine economic sustainability posits that the discounted welfare should not decrease over time. Resource management problems involving fluctuating stocks may be exacerbated by a lack of sustainable policies unless some violations of this condition are accepted. Moreover, sustainable policies may lead to the minimal welfare levels. To alleviate these problems, a new notion of sustainability is suggested. In this new form of sustainability, violations of nondecreasing welfare are accepted to a certain degree. The policies that maximize the discounted welfare under the constraint for the largest accepted violation for sustainability satisfy a dynamic programming type of fixed‐point condition that can be effectively utilized for finding optimal policies. The new notion of sustainability can also be utilized in defining an index for measuring the degree of overall unsustainability of resource management problems and different harvesting policies.     

Dynamic consistency for stochastic optimal control problems

For a sequence of dynamic optimization problems, we aim at discussing a notion of consistency over time. This notion can be informally introduced as follows. At the very first time step?t ₀, the decision maker formulates an optimization problem that yields optimal decision rules for all the forthcoming time steps?t ₀,t ₁,??,T; at the next time step?t ₁, he is able to formulate a new optimization problem starting at time?t ₁ that yields a new sequence of optimal decision rules. This process can be continued until the final time?T is reached. A?family of optimization problems formulated in this way is said to be dynamically consistent if the optimal strategies obtained when solving the original problem remain optimal for all subsequent problems. The notion of dynamic consistency, well-known in the field of economics, has been recently introduced in the context of risk measures, notably by Artzner et al. (Ann. Oper. Res. 152(1):5?C22, 2007) and studied in the stochastic programming framework by Shapiro (Oper. Res. Lett. 37(3):143?C147, 2009) and for Markov Decision Processes (MDP) by Ruszczynski (Math. Program. 125(2):235?C261, 2010). We here link this notion with the concept of ??state variable?? in MDP, and show that a significant class of dynamic optimization problems are dynamically consistent, provided that an adequate state variable is chosen.     

The definition and measurement of electromagnetic chirality

The notion of electromagnetic chirality, recently introduced in the Physics literature, is investigated in the framework of scattering of time‐harmonic electromagnetic waves by bounded scatterers. This type of chirality is defined as a property of the farfield operator. The relation of this novel notion of chirality to that of geometric chirality of the scatterer is explored. It is shown for several examples of scattering problems that geometric achirality implies electromagnetic achirality. On the other hand, a chiral material law, as for example given by the Drude‐Born‐Fedorov model, yields an electromagnetically chiral scatterer. Electromagnetic chirality also allows the definition of a measure. Scatterers invisible to fields of one helicity turn out to be maximally chiral with respect to this measure. For a certain class of electromagnetically chiral scatterers, we provide numerical calculations of the measure of chirality through solutions of scattering problems computed by a boundary element method.     

Well-setness and scalarization in set optimization

In this paper, we consider a notion of global well-setness for a set optimization problem using the excess function. We give sufficient conditions for well-setness and establish Dontchev–Zolezzi type measure for well-setness. Moreover, we relate this notion with well-setness notion for a scalar problem obtained by using generalized Gerstewitz’s function.     

New perspectives in algebraic logic,from neat embeddings to Erdos graphs

The notion of neat reducts is an old venerable notion in cylindric algebra theory invented by Henkin. This notion is regaining momentum. In this paper we explain why. This notion is discussed in connection to the algebraic notions of representability and complete representability, and the corresponding metalogical ones of completeness and omitting types, particularly for finite variable fragments. Also it is shown how such a notion has found intersection with non-trivial topics in model theory (like finite forcing) and set theory (forcing).     

Colorings and interpretations: a connection between graphs and grammar forms

A mechanism for constructing families of “similar” graphs is introduced. This mechanism is essentially identical to the one considered in the theory of grammar forms and, at the same time, generalizes the notion of coloring of graphs. The mechanism applies to both directed and undirected graphs. The paper investigate hierarchies of families of graphs obtained by this mechanism, both in the directed and undirected case.     

Graphical modelling of multivariate time series

We introduce graphical time series models for the analysis of dynamic relationships among variables in multivariate time series. The modelling approach is based on the notion of strong Granger causality and can be applied to time series with non-linear dependences. The models are derived from ordinary time series models by imposing constraints that are encoded by mixed graphs. In these graphs each component series is represented by a single vertex and directed edges indicate possible Granger-causal relationships between variables while undirected edges are used to map the contemporaneous dependence structure. We introduce various notions of Granger-causal Markov properties and discuss the relationships among them and to other Markov properties that can be applied in this context. Examples for graphical time series models include nonlinear autoregressive models and multivariate ARCH models.     

Proper relaxation of optimal control problems

In a recent paper, we proposed different relaxation procedures for optimal control problems involving time delays in the control functions. We introduced a notion of proper relaxation, applicable to problems without side constraints, where we required the minimum cost of the relaxed problem to coincide with the infimum cost of the original one. In this paper, a new and appropriate notion of relaxation for problems with side constraints is introduced. As examples of proper extensions, in the new sense, we describe in detail the standard procedure for delay-free problems and the procedure for one-delay systems which we recently proposed.     

Conditioned stochastic differential equations: theory, examples and application to finance

We generalize the notion of Brownian bridge. More precisely, we study a standard Brownian motion for which a certain functional is conditioned to follow a given law. Such processes appear as weak solutions of stochastic differential equations that we call conditioned stochastic differential equations. The link with the theory of initial enlargement of filtration is made and after a general presentation several examples are studied: the conditioning of a standard Brownian motion (and more generally of a Markov diffusion) by its value at a given date, the conditioning of a geometric Brownian motion with negative drift by its quadratic variation and finally the conditioning of a standard Brownian motion by its first hitting time of a given level. As an application, we introduce the notion of weak information on a complete market, and we give a “quantitative” value to this weak information.     

α-FUZZY PAIRWISE RETRACT OF L-VALUED PAIRWISE STRATIFICATION SPACES

The notion of a fuzzy retract was introduced by Rodabaugh (1981). The notion of a fuzzy pairwise retract was introduced in 2001. Some weak forms and some strong forms of α-continuous mappings were introduced in 1988 and 1997. The authors extend some of these forms to the L-fuzzy bitopological setting and construct various α-fuzzy pairwise retracts. The concept of weakly induced spaces in the case L = [0,1] was introduced by Martin (1980). Liu and Luo (1987) generalized this notion to the case that L is an arbitrary F-lattice and introduced the notion of induced L-fts. Several results are obtained, especially, for L-valued pairwise stratification spaces.     

α-FUZZY PAIRWISE RETRACT OF L-VALUED PAIRWISE STRATIFICATION SPACES

The notion of a fuzzy retract was introduced by Rodabaugh (1981). The notion of a fuzzy pairwise retract was introduced in 2001. Some weak forms and some strong forms of α-continuous mappings were introduced in 1988 and 1997. The authors extend some of these forms to the L-fuzzy bitopological setting and construct various α-fuzzy pairwise retracts. The concept of weakly induced spaces in the case L = [0, 1] was introduced by Martin (1980). Lin and Luo (1987) generalized this notion to the case that L is an arbitrary F-lattice and introduced the notion of induced L-fts. Several results are obtained, especially, for L-valued pairwise stratification spaces.     

Introducing Young Children to the Role of Assumptions in Proving

The notion of assumptions permeates school mathematics, but instruction tends to highlight this notion only in the advanced grades. In this article, I argue that it is important for even young children to develop a sense of the role of assumptions in proving, and I investigate what it might mean and look like for instruction to promote this goal. Toward this end, I study an episode from third grade that describes the first time that the students in the class were introduced in a deliberate and explicit way to the role of assumptions in proving. The central role of the mathematical task in the episode is identified, and features of mathematical tasks that can generate rich mathematical activity in the intersection of assumptions and proving are discussed. In addition, issues of the role of teachers in fostering productive interactions between students and mathematical tasks that have those features are considered.