Deadlock properties of queueing networks with finite capacities and multiple routing chains

Blocking in queueing network models with finite capacities can lead to deadlock situations. In this paper, deadlock properties are investigated in queueing networks with multiple routing chains. The necessary and sufficient conditions for deadlockfree queueing networks with blocking are provided. An optimization algorithm is presented for finding deadlock-free capacity assignments with the least total capacity. The optimization algorithm maps the queueing network into a directed graph and obtains the deadlock freedom conditions from a specified subset of cycles in the directed graph. In certain network topologies, the number of deadlock freedom conditions can be large, thus, making any optimization computationally expensive. For a special class of topologies, so-calledtandem networks, it is shown that a minimal capacity assignment can be directly obtained without running an optimization algorithm. Here, the solution to the minimal capacity assignment takes advantage of the regular topology of tandem networks.This work was supported by the National Science Foundation under Grant No. CCR-90-11981.     

A study concerning splitting and jointly continuous topologies on C(Y,Z)

Let Y and Z be two fixed topological spaces and C(Y, Z) the set of all continuous maps from Y into Z. We construct and study topologies on C(Y, Z) that w^{e call F}n(τn)-family-open topologies. Furthermore, we find necessary and su?cient conditions such that these topologies to be splitting and jointly continuous. Finally, we present questions concerning a further study on this area.     

Strong approximations for Markovian service networks

Inspired by service systems such as telephone call centers, we develop limit theorems for a large class of stochastic service network models. They are a special family of nonstationary Markov processes where parameters like arrival and service rates, routing topologies for the network, and the number of servers at a given node are all functions of time as well as the current state of the system. Included in our modeling framework are networks of M _t /M _t /n _t queues with abandonment and retrials. The asymptotic limiting regime that we explore for these networks has a natural interpretation of scaling up the number of servers in response to a similar scaling up of the arrival rate for the customers. The individual service rates, however, are not scaled. We employ the theory of strong approximations to obtain functional strong laws of large numbers and functional central limit theorems for these networks. This gives us a tractable set of network fluid and diffusion approximations. A common theme for service network models with features like many servers, priorities, or abandonment is “non-smooth” state dependence that has not been covered systematically by previous work. We prove our central limit theorems in the presence of this non-smoothness by using a new notion of derivative. This revised version was published online in June 2006 with corrections to the Cover Date.     

Representation and approximation of convex dynamic risk measures with respect to strong–weak topologies

We provide a representation for strong-weak continuous dynamic risk measures from L^p into L^p_t spaces where these spaces are equipped respectively with strong and weak topologies and p is a finite number strictly larger than one. Conversely, we show that any such representation that admits a compact (with respect to the product of weak topologies) sub-differential generates a dynamic risk measure that is strong--weak continuous. Furthermore, we investigate sufficient conditions on the sub-differential for which the essential supremum of the representation is attained. Finally, the main purpose is to show that any convex dynamic risk measure that is strong-weak continuous can be approximated by a sequence of convex dynamic risk measures which are strong--weak continuous and admit compact sub-differentials with respect to the product of weak topologies. Throughout the arguments, no conditional translation invariance or monotonicity assumptions are applied.     

Lattice-based threshold-changeability for standard CRT secret-sharing schemes

We consider the problem of increasing the threshold parameter of a secret-sharing scheme after the setup (share distribution) phase, without further communication between the dealer and the shareholders. Previous solutions to this problem require one to start off with a non-standard scheme designed specifically for this purpose, or to have secure channels between shareholders. In contrast, we show how to increase the threshold parameter of the standard CRT secret-sharing scheme without secure channels between the shareholders. Our method can thus be applied to existing CRT schemes even if they were set up without consideration to future threshold increases.Our method is a positive cryptographic application for lattice reduction algorithms, and we also use techniques from lattice theory (geometry of numbers) to prove statements about the correctness and information-theoretic security of our constructions.     

Finite Intervals in the Lattice of Topologies

We discuss the question whether every finite interval in the lattice of all topologies on some set is isomorphic to an interval in the lattice of all topologies on a finite set – or, equivalently, whether the finite intervals in lattices of topologies are, up to isomorphism, exactly the duals of finite intervals in lattices of quasiorders. The answer to this question is in the affirmative at least for finite atomistic lattices. Applying recent results about intervals in lattices of quasiorders, we see that, for example, the five-element modular but non-distributive lattice cannot be an interval in the lattice of topologies. We show that a finite lattice whose greatest element is the join of two atoms is an interval of T ₀-topologies iff it is the four-element Boolean lattice or the five-element non-modular lattice. But only the first of these two selfdual lattices is an interval of orders because order intervals are known to be dually locally distributive.     

MacNeille completions of lattice expansions

There are two natural ways to extend an arbitrary map between (the carriers of) two lattices, to a map between their MacNeille completions. In this paper we investigate which properties of lattice maps are preserved under these constructions, and for which kind of maps the two extensions coincide. Our perspective involves a number of topologies on lattice completions, including the Scott topologies and topologies that are induced by the original lattice. We provide a characterization of the MacNeille completion in terms of these induced topologies. We then turn to expansions of lattices with additional operations, and address the question of which equational properties of such lattice expansions are preserved under various types of MacNeille completions that can be defined for these algebras. For a number of cases, including modal algebras and residuated (ortho)lattice expansions, we provide reasonably sharp sufficient conditions on the syntactic shape of equations that guarantee preservation. Generally, our results show that the more residuation properties the primitive operations satisfy, the more equations are preserved. Received August 21, 2005; accepted in final form October 17, 2006.     

Abel maps and presentation schemes

We sharpen the two main tools used to treat the compactified Jacobian of a singular curve: Abel maps and presentation schemes. First we prove a smoothness theorem for bigraded Abel maps. Second we study the two complementary filtrations provided by the images of certain Abel maps and certain presentation schemes. Third we study a lifting of the Abel map of bidegree (m, 1) to the corresponding presentation scheme. Fourth we prove that, if a curve is blown up at a double point, then the corresponding presentation scheme is a ¹-bundle. Finally, using Abel maps of bidegree (m, 1), we characterize the curves having double points at worst.     

F-actions and parallel-product decomposition of reflexible maps

The parallel product of two rooted maps was introduced by S.E. Wilson in 1994. The main question of this paper is whether for a given reflexible map M one can decompose the map into a parallel product of two reflexible maps. This can be achieved if and only if the monodromy (or the automorphism) group of the map has at least two minimal normal subgroups. All reflexible maps up to 100 edges, which are not parallel-product decomposable, are calculated and presented. For this purpose, all degenerate and slightly-degenerate reflexible maps are classified. In this paper the theory of F-actions is developed including a classification of quotients and parallel-product decomposition. Projections and lifts of automorphisms for quotients and for parallel products are studied. The theory can be immediately applied on rooted maps and rooted hypermaps as they are special cases of F-actions.     

Remarks on products of generalized topologies

We study the relationship between the product and other basic operations (namely σ, π, α and β) of generalized topologies. Also we discuss the connectedness, generalized connectedness and compactness of products of generalized topologies. It is proved that the connectedness and compactness are preserved under the product of generalized topologies, which shows that the definition of product of generalized topologies is quite reasonable.     

Bitopologies on products and ratios

The purpose of the paper is to extend the notions of splitting and jointly continuous topologies on function spaces to products of spaces, to present a new theory of duality between topologies on function spaces and topologies on products of spaces, and to generalize these theories to bitopological spaces. The author starts from the definitions well known in the literature. Bibliography: 1 title. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 242, 1997, pp. 217–229. Translated by N. Yu. Netsvetaev.     

Nonsmooth one-dimensional maps: some basic concepts and definitions

The main purpose of the present survey is to contribute to the theory of dynamical systems defined by one-dimensional piecewise monotone maps. We recall some definitions known from the theory of smooth maps, which are applicable to piecewise smooth ones, and discuss the notions specific for the considered class of maps. To keep the presentation clear for the researchers working in other fields, especially in applications, many examples are provided. We focus mainly on the notions and concepts which are used for the investigation of various kinds of attractors of a map and related bifurcation structures observed in its parameter space.     

Overdetermined partial boundary value problems on finite networks

In this study, we define a class of non-self-adjoint boundary value problems on finite networks associated with Schrödinger operators. The novel feature of this study is that no data are prescribed on part of the boundary, whereas both the values of the function and of its normal derivative are given on another part of the boundary. We show that overdetermined partial boundary value problems are crucial for solving inverse boundary value problems on finite networks since they provide the theoretical foundations for the recovery algorithm. We analyze the uniqueness and the existence of solution for overdetermined partial boundary value problems based on the nonsingularity of partial Dirichlet-to-Neumann maps. These maps allow us to determine the value of the solution in the part of the boundary where no data was prescribed. We also execute full conductance recovery for spider networks.     

Proximal Set-Open Topologies on Partial Maps

Let X, Y be T ₁ topological spaces. A partial map from X to Y is a continuous function f whose domain is a subspace D of X and whose codomain is Y. Let P(X, Y) be the set of partial maps with domains in a fixed class D. In analogy with the global case, we introduce on P(X, Y), whatever be the nature of the domain class D, new function space topologies, the proximal set-open topologies, briefly PSOTs, deriving from general networks on X and proximity on Y by replacing inclusion with strong inclusion. The PSOTs include the already known generalized compact-open topology on partial maps with closed domains. When domains are supposed closed, the network α closed and hereditarily closed and the proximity δ on Y Efremovic, then the PSOT attached to α and δ is uniformizable iff α is a Urysohn family in X. This revised version was published online in June 2006 with corrections to the Cover Date.     

Topologies on function spaces

In the present paper we introduce notions of A-splitting and A-jointly continuous topology on the set C(Y,Z) of all continuous maps of a topological space Y into a topological space Z, where A is any family of spaces. These notions satisfy the basic properties of splitting and jointly continuous topologies on C(Y,Z). In particular, for every A, the greatest A-splitting topology on C(Y,Z) (denoted by τ(A) always exists. We indicate some families A of spaces for which the topology τ(A) coincides with the greatest splitting topology on C(X,Y). We give a notion of equivalent families of spaces and try to find a “simple” family which is equivalent to a given family. In particular, we prove that every family is equivalent to a family consisting of one space, and the family of all spaces is equivalent to a family of all T₁-spaces containing at most one nonisolated point. We compare the topologies τ({X}) for distinct compact metrizable spaces X and give some examples. Bibliography: 13 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 208, 1993, pp. 82–97. Translated by A. A. Ivanov.     

α-Irresoluteness and α-compactness based on continuous valued logic

This paper considers fuzzifying topologies, a special case of I-fuzzy topologies (bifuzzy topologies), introduced by Ying [1]. It investigates topological notions defined by means of α-open sets when these are planted into the framework of Ying’s fuzzifying topological spaces (by ?ukasiewicz logic in [0, 1]) . The concept of α-irresolute functions and α-compactness in the framework of fuzzifying topology are introduced and some of their properties are obtained. We use the finite intersection property to give a characterization of fuzzifying α-compact spaces. Furthermore, we study the image of fuzzifying α-compact spaces under fuzzifying α-continuity and fuzzifying α-irresolute maps.     

A note on representation of pseudovariant maps

We define and study the notions of pseudovariant maps and pseudogeny maps on G-spaces. We prove that the set of all pseudogenies of a locally comapct G-space is a group. We further obtain representation of pseudovariant maps in terms of pseudogenies. Finally, we obtain Tietze type extension result for pseudovariant homotopies defined on locally compact second countable G-spaces.     

An application of Newton–Puiseux charts to the Jacobian problem

We study 2-dimensional Jacobian maps using so-called Newton–Puiseux charts. These are multi-valued coordinates near divisors of resolutions of indeterminacies at infinity of the Jacobian map in the source space as well as in the target space. The map expressed in these charts takes a very simple form, which allows us to detect a series of new analytical and topological properties. We prove that the Jacobian Conjecture holds true for maps (f,g) whose topological degree is ≤5, for maps with gcd(degf,degg)≤16 and for maps with. gcd(degf,degg) equal to 2 times a prime.     

Variational results on flag manifolds: Harmonic maps, geodesics and Einstein metrics

In this paper, we study variational aspects for harmonic maps from M to several types of flag manifolds and the relationship with the rich Hermitian geometry of these manifolds. We consider maps that are harmonic with respect to any invariant metric on each flag manifold. They are called equiharmonic maps. We survey some recent results for the case where M is a Riemann surface or is one dimensional; i.e., we study equigeodesics on several types of flag manifolds. We also discuss some results concerning Einstein metrics on such manifolds.