The Khovanov complex for virtual links

One of the most outstanding achievements of modern knot theory is Khovanov’s categorification of Jones polynomials. In the present paper, we construct the homology theory for virtual knots. An important obstruction to this theory (unlike the case of classical knots) is the nonorientability of “atoms”; an atom is a two-dimensional combinatorial object closely related with virtual link diagrams. The problem is solved directly for the field ℤ₂ and also by using some geometrical constructions applied to atoms. We discuss a generalization proposed by Khovanov; he modifies the initial homology theory by using the Frobenius extension. We construct analogs of these theories for virtual knots, both algebraically and geometrically (by using atoms). __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 4, pp. 127–152, 2005.     

On hyperbolic virtual polytopes and hyperbolic fans

Hyperbolic virtual polytopes arose originally as polytopal versions of counterexamples to the following A.D.Alexandrov’s uniqueness conjecture: Let K ⊂ ℝ³ be a smooth convex body. If for a constant C, at every point of ∂K, we have R ₁ ≤ C ≤ R ₂ then K is a ball. (R ₁ and R ₂ stand for the principal curvature radii of ∂K.) This paper gives a new (in comparison with the previous construction by Y.Martinez-Maure and by G.Panina) series of counterexamples to the conjecture. In particular, a hyperbolic virtual polytope (and therefore, a hyperbolic hérisson) with odd an number of horns is constructed. Moreover, various properties of hyperbolic virtual polytopes and their fans are discussed.     

Virtual and effective control for distributed systems and decomposition of everything

For the numerical approximation of the solution of boundary value problems (BVP), decomposition techniques are very important, in particular in view of parallel computations. The same is true, in principle, for optimal control of distributed systems, i.e., systems governed (modelled) by partial differential equations (PDE). Very many techniques have been studied for the approximation of BVP, such as DDM (domain decomposition method), decomposition of operators (splitting up, for instance). In contrast, not so many techniques of decomposition have been used in control problems for distributed systems, as pointed out in the contributions of Benamou [1], Benamou and Després [2], and Lagnese and Leugering [11]. However, it has been observed by Pironneau and Lions [24], [26] that by using so-calledvirtual controls, systematic DDM can be obtained, and that problems of optimal control and analysis of BVP can be considered in the same framework. We then deal with virtual control problems for BVP, virtual and effective control problems for the control of PDE (cf. Pironneau and Lions [25]). Using the idea of virtual control in other guises, Glowinski, Lions and Pironneau [9] have shown how to obtain new decomposition methods for the energy spaces (cf. Section 3), and Pironneau and Lions [27] have shown how to obtain systematically operator decomposition in BVP. In the present paper, we show (without assuming prior knowledge) how to apply the virtual control ideas in several different guises to the “decomposition of everything” for PDE of evolution and for their control. In this way, one can decompose the geometrical domain, the energy space and the operator. This is briefly presented in Sections 2, 3 and 4. We show in Section 5 how one can simultaneously apply two of the decomposition techniques and also indicate briefly how virtual control ideas can be used in case of bilinear control. The content of this paper is presented here for the first time. It is part of a systematic program which is in progress, developed with several colleagues. I wish to thank particularly F. Hecht, R. Glowinski, J. Periaux, O. Pironneau, H. Q. Chen and T. W. Pan. Of course we do not claim by any means that the methods based on “virtual control” are “better” than the many decomposition techniques already available (no attempt has been made to compile a Bibliography on these topics). Numerical works in progress show that the methods are “not bad”, but no serious benchmarking has been made yet. Possible interest lies in the fact thatone technique, with some variants, seems to lead to all possible Decompositions of Everything.     

Additive functionals with application to sojourn times in infinite-server and processor sharing systems

We deal with additive functionals of stationary processes. It is shown that under some assumptions a stationary model of the time-changed process exists. Further, bounds for the expectation of functions of additive functionals are derived. As an application we analyze virtual sojourn times in an infinite-server system where the service speed is governed by a stationary process. It turns out that the sojourn time of some kind of virtual requests equals in distribution an additive functional of a stationary time-changed process, which provides bounds for the expectation of functions of virtual sojourn times, in particular bounds for fractional moments and the distribution function. Interpreting the GI(n)/GI(n)/∞ system or equivalently the GI(n)/GI system under state-dependent processor sharing as an infinite-server system where the service speed is governed by the number n of requests in the system provides results for sojourn times of virtual requests. In the case of M(n)/GI(n)/∞, the sojourn times of arriving and added requests equal in distribution sojourn times of virtual requests in modified systems, which yields several results for the sojourn times of arriving and added requests. In case of positive integer moments, the bounds generalize earlier results for M/GI(n)/∞. In particular, the mean sojourn times of arriving and added requests in M(n)/GI(n)/∞ are proportional to the required service time, generalizing Cohen’s famous result for M/GI(n)/∞.     

On the guaranteed throughput and efficiency of closed re-entrant lines

A closed network is said to be “guaranteed efficient” if the throughput converges under all non-idling policies to the capacity of the bottlenecks in the network, as the number of trapped customers increases to infinity. We obtain a necessary condition for guaranteed efficiency of closed re-entrant lines. For balanced two-station systems, this necessary condition is almost sufficient, differing from it only by the strictness of an inequality. This near characterization is obtained by studying a special type of virtual station called “alternating visit virtual station”. These special virtual stations allow us to relate the necessary condition to certain indices arising in heavy traffic studies using a Brownian network approximation, as well as to certain policies proposed as being extremal with respect to the asymptotic loss in the throughput. Using the near characterization of guaranteed efficiency we also answer the often pondered question of whether an open network or its closed counterpart has greater throughput - the answer is that neither can assure a greater guaranteed throughput. This revised version was published online in June 2006 with corrections to the Cover Date.     

Lie groupoids as generalized atlases

Starting with some motivating examples (classical atlases for a manifold, space of leaves of a foliation, group orbits), we propose to view a Lie groupoid as a generalized atlas for the “virtual structure” of its orbit space, the equivalence between atlases being here the smooth Morita equivalence. This “structure” keeps memory of the isotropy groups and of the smoothness as well. To take the smoothness into account, we claim that we can go very far by retaining just a few formal properties of embeddings and surmersions, yielding a very polymorphous unifying theory. We suggest further developments.     

Virtual knots and links

This paper is an introduction to the subject of virtual knot theory and presents a discussion of some new specific theorems about virtual knots. The new results are as follows: Using a 3-dimensional topology approach, we prove that if a connected sum of two virtual knots K ₁ and K ₂ is trivial, then so are both K ₁ and K ₂. We establish an algorithm for recognizing virtual links that is based on the Haken-Matveev technique. Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2006, Vol. 252, pp. 114–133.     

Heterogeneous coupling by virtual control methods

The virtual control method, recently introduced to approximate elliptic and parabolic problems by overlapping domain decompositions (see [7–9]), is proposed here for heterogeneous problems. Precisely, we address the coupling of an advection equation with a diffusion-advection equation, with the aim of modelling boundary layers. We investigate both overlapping and non-overlapping (disjoint) subdomain decompositions. In the latter case, several cost functions are considered and a numerical assessment of our theoretical conclusions is carried out.     

Integrated fuzzy AHP and fuzzy VIKOR model for supplier selection in an agile and modular virtual enterprise

In this ever-changing world, organizations need to outsource parts of their processes for having agile response to market’s needs and varying demands. Because of temporal nature of virtual enterprises (VE’s), the situation of outsourcing process in this kind of organizations is a vital situation. The main idea of this paper aims to present a decision-making framework for specific area that is appropriated for complex states. Its contribution is developing a fuzzy VlseKriterijumska Optimizacija I Kompromisno Resenje (VIKOR) method and combining it with fuzzy analytic hierarchy process (AHP). This extension suitable for decision-making situations which is faced with mixture appraisement that simultaneously regarded to both “group utility” or majority and “individual regret” of the opponent. The Integrated and developed model suits to inconsistent conditions that we face to collection of criteria and subcriteria that should satisfy some of them collectively and simultaneously and in other attainment of some individual criteria is desirable. This framework then extended to a case study with varied criteria for outsourcing process.     

Transient solution of a non-empty chemical queueing system

In this paper, we illustrate that a power series technique can be used to derive explicit expressions for the transient state distribution of a queueing problem having “chemical” rules with an arbitrary number of customers present initially in the system. Based on generating function and Laplace techniques Conolly et al. (in Math Sci 22:83–91, 1997) have obtained the distributions for a non-empty chemical queue. Their solution enables us only to recover the idle probability of the system in explicit form. Here, we extend not only the model of Conolly et al. but also get a new and simple solution for this model. The derived formula for the transient state is free of Bessel function or any integral forms. The transient solution of the standard M/M/1/∞ queue with λ = μ is a special case of our result. Furthermore, the probability density function of the virtual waiting time in a chemical queue is studied. Finally, the theory is underpinned by numerical results.      

Gauged Floer Theory Of Toric Moment Fibers

We investigate the small area limit of the gauged Lagrangian Floer cohomology of Frauenfelder [Fr1]. The resulting cohomology theory, which we call quasimap Floer cohomology, is an obstruction to displaceability of Lagrangians in the symplectic quotient. We use the theory to reproduce the results of Fukaya–Oh–Ohta–Ono [FuOOO3,1] and Cho–Oh [CO] on non-displaceability of moment fibers of not-necessarily-Fano toric varieties and extend their results to toric orbifolds, without using virtual fundamental chains. Finally, we describe a conjectural relationship with Floer cohomology in the quotient.     

Busy period,virtual waiting time and number of customers in G<Superscript><Emphasis Type="Italic">δ</Emphasis></Superscript>|M<Superscript><Emphasis Type="Italic">ϰ</Emphasis></Superscript>|1|B system

In this article, we determine the integral transforms of several two-boundary functionals for a difference of a compound Poisson process and a compound renewal process. Another part of the article is devoted to studying the above-mentioned process reflected at its infimum. We use the results obtained to study a G ^δ |M ^ϰ |1|B system with batch arrivals and finite buffer in the case when δ∼ge(λ). We derive the distributions of the main characteristics of the queuing system, such as the busy period, the time of the first loss of a customer, the number of customers in the system, the virtual waiting time in transient and stationary regimes. The advantage is that these results are given in a closed form, namely, in terms of the resolvent sequences of the process.     

A BMAP/SM/1 queue with service times depending on the arrival process

We study a BMAP/>SM/1 queue with batch Markov arrival process input and semi‐Markov service. Service times may depend on arrival phase states, that is, there are many types of arrivals which have different service time distributions. The service process is a heterogeneous Markov renewal process, and so our model necessarily includes known models. At first, we consider the first passage time from level {κ+1} (the set of the states that the number of customers in the system is κ+1) to level {κ} when a batch arrival occurs at time 0 and then a customer service included in that batch simultaneously starts. The service descipline is considered as a LIFO (Last‐In First‐Out) with preemption. This discipline has the fundamental role for the analysis of the first passage time. Using this first passage time distribution, the busy period length distribution can be obtained. The busy period remains unaltered in any service disciplines if they are work‐conserving. Next, we analyze the stationary workload distribution (the stationary virtual waiting time distribution). The workload as well as the busy period remain unaltered in any service disciplines if they are work‐conserving. Based on this fact, we derive the Laplace–Stieltjes transform for the stationary distribution of the actual waiting time under a FIFO discipline. In addition, we refer to the Laplace–Stieltjes transforms for the distributions of the actual waiting times of the individual types of customers. Using the relationship between the stationary waiting time distribution and the stationary distribution of the number of customers in the system at departure epochs, we derive the generating function for the stationary joint distribution of the numbers of different types of customers at departures. This revised version was published online in June 2006 with corrections to the Cover Date.     

Separoids,Their Categories and a Hadwiger-Type Theorem for Transversals

In this paper we study the topology of transversals to a family of convex sets as a subset of a Grassmanian manifold. This topology seems to be ruled by a combinatorial structure which we call a separoid. With these combinatorial objects and the topological notion of virtual transversal we prove a Borsuk—Ulam-type theorem which has as a corollary a generalization of Hadwiger's theorem. Received October 25, 2000, and in revised form September 27, 2001, October 6, 2001, and October 11, 2001. Online publication March 1, 2002.     

Stick breaking process generated by virtual permutations with Ewens distribution

Given a sequence x of points in the unit interval, we associate with it a virtual permutation w=w(x) (that is, a sequence w of permutationsw _n such that for all n=1,2,..., w_n−1=w′_n is obtained from w_n by removing the last element n from its cycle). We introduce a detailed version of the well-known stick breaking process generating a random sequence x. It is proved that the associated random virtual permutation w(x) has a Ewens distribution. Up to subsets of zero measure, the space of virtual permutations is identified with the cube [0, 1]^∞. Bibliography: 8 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 223, 1995, pp. 162–180.     

Relative Gromov-Witten invariants and the mirror formula

Let X be a smooth complex projective variety, and let be a smooth very ample hypersurface such that is nef. Using the technique of relative Gromov-Witten invariants, we give a new short and geometric proof of (a version of) the “mirror formula”, i.e. we show that the generating function of the genus zero 1-point Gromov-Witten invariants of Y can be obtained from that of X by a certain change of variables (the so-called “mirror transformation”). Moreover, we use the same techniques to give a similar expression for the (virtual) numbers of degree-d plane rational curves meeting a smooth cubic at one point with multiplicity 3d, which play a role in local mirror symmetry. Received: 11 July 2001 / Published online: 4 February 2003 Funded by the DFG scholarships Ga 636/1–1 and Ga 636/1–2.     

Generalization of Braids and Homologies

In this paper, we give a survey of recent results devoted to the homology of generalizations of braids: the homological properties of virtual braids and the generalized homology of Artin groups studied by C. Broto and the author. Virtual braid groups VB _n correspond to virtual knots in the same way that classical braids correspond to usual knots. Virtual knots arise in the study of Gauss diagrams and Vassiliev invariants of usual knots. The Burau representation to GL _n ℤ[t, t ⁻¹] is extended from classical braids to virtual ones. Its homological properties are also studied. The following splitting of infinite loop spaces for the plus-construction of the classifying space of the virtual braid group on an infinite number of strings exists:

Galois module structure of unramified covers

We use the theory of n-cubic structures to study the Galois module structure of the coherent cohomology groups of unramified Galois covers of varieties over the integers. Assuming that all the Sylow subgroups of the covering group are abelian, we show that the invariant that measures the obstruction to the existence of a “virtual normal integral basis” is annihilated by a product of certain Bernoulli numbers with orders of even K-groups of Z. We also show that the existence of such a basis is closely connected to the truth of the Kummer-Vandiver conjecture for the prime divisors of the degree of the cover. Partially supported by NSF grants # DMS05-01049 and # DMS01-11298 (via the Institute for Advanced Study).