Curvature Dimension Inequalities and Subelliptic Heat Kernel Gradient Bounds on Contact Manifolds

We study curvature dimension inequalities for the sub-Laplacian on contact Riemannian manifolds. This new curvature dimension condition is then used to obtain:

• Geometric conditions ensuring the compactness of the underlying manifold (Bonnet–Myers type results);
• Volume estimates of metric balls;
• Gradient bounds and stochastic completeness for the heat semigroup generated by the sub-Laplacian;
• Spectral gap estimates.

     

On the geometry of the characteristic vector of an lcQS-manifold

We study conditions under which the characteristic vector of a normal lcQS-manifold is a torsion-forming or even a concircular vector field. We prove that the following assertions are equivalent:

• An lcQS-structure is normal, and its characteristic vector is a torsion-forming vector field.
• An lcQS-structure is normal, and its characteristic vector is a concircular vector field.
• An lcQS-structure is locally conformally cosymplectic and has a closed contact form.

     

The Critical Group of a Line Graph

The critical group of a graph is a finite abelian group whose order is the number of spanning forests of the graph. This paper provides three basic structural results on the critical group of a line graph.

• The first deals with connected graphs containing no cut-edge. Here the number of independent cycles in the graph, which is known to bound the number of generators for the critical group of the graph, is shown also to bound the number of generators for the critical group of its line graph.

• The second gives, for each prime p, a constraint on the p-primary structure of the critical group, based on the largest power of p dividing all sums of degrees of two adjacent vertices.

• The third deals with connected graphs whose line graph is regular. Here known results relating the number of spanning trees of the graph and of its line graph are sharpened to exact sequences which relate their critical groups.

The first two results interact extremely well with the third. For example, they imply that in a regular nonbipartite graph, the critical group of the graph and that of its line graph determine each other uniquely in a simple fashion.     

Probabilistic inequalities and remarks

The authors use their recently proved integral inequality to obtain bounds for the covariance of two random variables

• 1.in a general setup and
• 2.for a class of special joint distributions.

The same inequality is also used to estimate the difference of the expectations of two random variables. Finally, the authors study the attainability of a related inequality.     

Maximal independent, analytic sets in Abelian polish groups

The paper deals with the questions:

1. whether a topological module admits maximal linearly independent subsets that are analytic
2. whether an Abelian topological group admits maximal independent subsets that are analytic
3. whether a topological field extension admits transcendence bases that are analytic.

     

A generalization of Norton's theorem for queueing networks

A general framework for aggregation and decomposition of product form queueing networks with state dependent routing and servicing is presented. By analogy with electrical circuit theory, the stations are grouped into clusters of subnetworks such that the process decomposes into a global process and a local process. Moreover, the local process factorizes into the subnetworks. The global process and the local processes can be analyzed separately as if they were independent. The global process describes the behaviour of the queuing network in which each cluster is aggregated into a single station, whereas the local process describes the behaviour of the subnetworks as if they are not part of the queueing network. The decomposition and aggregation method formalized in this paper allows us to first analyze the global behaviour of the queueing network and subsequently analyze the local behaviour of the subnetworks of interest or to aggregate clusters into single stations without affecting the behaviour of the rest of the queueing network. Conditions are provided such that:

- the global equilibrium distribution for aggregated clusters has a product form;

- this form can be obtained by merely monitoring the global behaviour;

- the computation of a detailed distribution, including its normalizing constant, can be decomposed into the computation of a global and a local distribution;

- the marginal distribution for the number of jobs at the stations of a cluster can be obtained by merely solving local behaviour.

As a special application, Norton's theorem for queueing networks is extended to queueing networks with state dependent routing such as due to capacity constraints at stations or at clusters of stations and state dependent servicing such as due to service delays for clusters of stations.     

Harmonic Analysis of Neural Networks

It is known that superpositions of ridge functions (single hidden-layer feedforward neural networks) may give good approximations to certain kinds of multivariate functions. It remains unclear, however, how to effectively obtain such approximations. In this paper, we use ideas from harmonic analysis to attack this question. We introduce a special admissibility condition for neural activation functions. The new condition is not satisfied by the sigmoid activation in current use by the neural networks community; instead, our condition requires that the neural activation function be oscillatory. Using an admissible neuron we construct linear transforms which represent quite general functionsfas a superposition of ridge functions. We develop

• • • a continuous transform which satisfies a Parseval-like relation;
• • • a discrete transform which satisfies frame bounds.

Both transforms representfin a stable and effective way. The discrete transform is more challenging to construct and involves an interesting new discretization of time–frequency–direction space in order to obtain frame bounds for functions inL²(A) whereAis a compact set of Rⁿ. Ideas underlying these representations are related to Littlewood–Paley theory, wavelet analysis, and group representation theory.     

Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below

This paper is devoted to a deeper understanding of the heat flow and to the refinement of calculus tools on metric measure spaces $(X,\mathsf {d},\mathfrak {m})$ . Our main results are:

• A general study of the relations between the Hopf–Lax semigroup and Hamilton–Jacobi equation in metric spaces (X,d).
• The equivalence of the heat flow in $L^{2}(X,\mathfrak {m})$ generated by a suitable Dirichlet energy and the Wasserstein gradient flow of the relative entropy functional $\mathrm {Ent}_{\mathfrak {m}}$ in the space of probability measures .
• The proof of density in energy of Lipschitz functions in the Sobolev space $W^{1,2}(X,\mathsf {d},\mathfrak {m})$ .
• A fine and very general analysis of the differentiability properties of a large class of Kantorovich potentials, in connection with the optimal transport problem, is the fourth achievement of the paper.

Our results apply in particular to spaces satisfying Ricci curvature bounds in the sense of Lott and Villani (Ann. Math. 169:903–991, 2009) and Sturm (Acta Math. 196: 65–131, 2006, and Acta Math. 196:133–177, 2006) and require neither the doubling property nor the validity of the local Poincaré inequality.     

Order stars and stability theorems

This paper clears up to the following three conjectures:

1. The conjecture of Ehle [1] on theA-acceptability of Padé approximations toe ^z , which is true;
2. The conjecture of Nørsett [5] on the zeros of the “E-polynomial”, which is false;
3. The conjecture of Daniel and Moore [2] on the highest attainable order of certainA-stable multistep methods, which is true, generalizing the well-known Theorem of Dahlquist.

We further give necessary as well as sufficient conditions forA-stable (acceptable) rational approximations, bounds for the highest order of “restricted” Padé approximations and prove the non-existence ofA-acceptable restricted Padé approximations of order greater than 6. The method of proof, just looking at “order stars” and counting their “fingers”, is very natural and geometric and never uses very complicated formulas.     

Coding and control for communication networks

The purpose of this paper is to survey techniques for constructing effective policies for controlling complex networks, and to extend these techniques to capture special features of wireless communication networks under different networking scenarios. Among the key questions addressed are:

1. The relationship between static network equilibria, and dynamic network control.
2. The effect of coding on control and delay through rate regions.
3. Routing, scheduling, and admission control.

Through several examples, ranging from multiple-access systems to network coded multicast, we demonstrate that the rate region for a coded communication network may be approximated by a simple polyhedral subset of a Euclidean space. The polyhedral structure of the rate region, determined by the coding, enables a powerful workload relaxation method that is used for addressing complexity—the relaxation technique provides approximations of a highly complex network by a far simpler one. These approximations are the basis of a specific formulation of an h-MaxWeight policy for network routing. Simulations show a 50% improvement in average delay performance as compared to methods used in current practice.     

Compactness and the axiom of choice

In the absence of the axiom of choice four versions of compactness (A-, B-, C-, and D-compactness) are investigated. Typical results:

1. C-compact spaces form the epireflective hull in Haus of A-compact completely regular spaces.
2. Equivalent are:
3. the axiom of choice,
4. A-compactness = D-compactness,
5. B-compactness = D-compactness,
6. C-compactness = D-compactness and complete regularity,
7. products of spaces with finite topologies are A-compact,
8. products of A-compact spaces are A-compact,
9. products of D-compact spaces are D-compact,
10. powers X ^k of 2-point discrete spaces are D-compact,
11. finite products of D-compact spaces are D-compact,
12. finite coproducts of D-compact spaces are D-compact,
13. D-compact Hausdorff spaces form an epireflective subcategory of Haus,
14. spaces with finite topologies are D-compact.

1. Equivalent are:
2. the Boolean prime ideal theorem,
3. A-compactness = B-compactness,
4. A-compactness and complete regularity = C-compactness,
5. products of spaces with finite underlying sets are A-compact,
6. products of A-compact Hausdorff spaces are A-compact,
7. powers X ^k of 2-point discrete spaces are A-compact,
8. A-compact Hausdorff spaces form an epireflective subcategory of Haus.

1. Equivalent are:
2. either the axiom of choice holds or every ultrafilter is fixed,
3. products of B-compact spaces are B-compact.

1. Equivalent are:
2. Dedekind-finite sets are finite,
3. every set carries some D-compact Hausdorff topology,
4. every T ₁-space has a T ₁-D-compactification,
5. Alexandroff-compactifications of discrete spaces and D-compact.

     

Symplectic geometry,minors and graph Laplacians

In my talk, I will present some works done in the nineties on Laplacians on graphs: from eigenvalue problems to inverse problem for resistor networks. I will focus on the motivations and the main results as well as on the main ideas:

• •A differential topology point of view on the minor relation: a nice stratification associated to a finite graph Γ whose strata are associated to the minors of Γ
• •“Discrete” (graphs) versus “continuous” (Riemannian manifolds)
• •Stability of spectra with respect to singular limits: a finite dimensional theory of operators with domains (Von Neumann theory).

The link with topology will appear in some results about my graph parameter μ, in particular the planarity and the linkless embedding properties.     

Symmetric differentials, Kähler groups and ball quotients

While Margulis’ superrigidity theorem completely describes the finite dimensional linear representations of lattices of higher rank simple real Lie groups, almost nothing is known concerning the representation theory of complex hyperbolic lattices. The main result of this paper (Theorem 1.3) is a strong rigidity theorem for a certain class of cocompact arithmetic complex hyperbolic lattices. It relies on the following two ingredients:

• Theorem 1.6 showing that the representations of the topological fundamental group of a compact Kähler manifold X are controlled by the global symmetric differentials on X.
• An arithmetic vanishing theorem for global symmetric differentials on certain compact ball quotients using automorphic forms, in particular deep results of Clozel on base change (Theorem 1.11).

     

Modelling the motion of a three-link manipulator based on a plane with a time-dependent inclination

This work deals with the modelling of a three-link manipulator mounted on a plane with a time-dependent inclination. Two cases are considered.

• (i)The plane is part of a rigid body.
• (ii)The plane is in a moored ship.

     

System dynamics and feedback control problem formulations for real time dynamic traffic routing

This paper formulates the Dynamic Traffic Routing (DTR) problem as a real-time feedback control problem. Three different forms of the formulation are presented:

• 1.(1) distributed parameter system form derived from the conservation law;
• 2.(2) space discretized continuous lumped parameter form;
• 3.(3) space and time discretized lumped parameter form.

These formulations can be considered as the starting points for development of feedback control laws for the different control problems stated in this paper. This paper presents the feedback control problems, and does not discuss in detail the methodology of solution techniques which could be used to solve these problems. However, for the sake of completeness a brief treatment of the three forms are included in this paper to show possible ways to design the controllers.     

Free energy and solutions of the Vlasov-Poisson-Fokker-Planck system: external potential and confinement (Large time behavior and steady states)

This paper is devoted to the characterization of external electrostatic potentials for which the Vlasov-Poisson-Fokker-Planck system satisfies one of the following properties:

• (i) the system admits stationary solutions,
• (ii) any solution to the evolution problem converges to a stationary solution, or, equivalently, no mass vanishes for large times,
• (iii) the free energy is bounded from below, We give conditions under which these different notions of confinement are equivalent.

     

Algebraic Theory of the Dependability of Binary Sensors

Binary sensor systems are analog sensors of various types (optical, microelectromechanical (MEMS) systems, X-ray, gamma-ray, acoustic, electronic, etc.) based on the binary decision process. Typical examples of such “binary sensors” are X-ray luggage inspection systems, product quality control systems, automatic target recognition systems, numerous medical diagnostic systems, and many others. In all these systems, the binary decision process provides only two mutually exclusive responses. There are also two types of key parameters that characterize either a system or external conditions in relation to the system that are determined by their prior probabilities. In this paper, by using a formal neuron model, we analyze the problem of threshold redundancy of binary sensors of a critical state. The following three major tasks are solved:

1. implementation of the algorithm of calculation of error probabilities for threshold redundancy of a group of sensors;
2. computation of the minimal upper bound for the probability in a closed analytical form and determination of its link with Claude Shannons theorem;
3. derivation of the expression (estimate) for sensor “weights” when the probability of the binary system error does not exceed the specified minimal upper bound.

     

An optimal time to sleep for an auto-sleep system considering multi-usage states

An auto-sleep system is defined by the following two properties:

• 1.(i) a call for the system occurs randomly and intermittently
• 2.(ii) the system automatically goes to sleep if there occurs no call during a prespecified time T.

It considers four states:

• 1.(a) sleep
• 2.(b) warm-up
• 3.(c) nonusage
• 4.(d) usage.

For such a system, the time to sleep has been discussed based on suitable criteria. This study extends the model for an auto-sleep system so that the model can deal with multi-usage states. With a view to determining an optimal time to sleep under the extended model, the expected energy consumed per unit time is formulated as a criterion to be minimized. The existence of an optimal time to sleep is examined under a general call distribution. Numerical examples are also provided for a Weibull as well as a log-normal call distribution.     

Chirurgies d'indice un et isotopies de sphères dans les variétés de contact tendues

We prove the following theorems:

• 1)Any surgery of index one on u tight contact manifold (of dimension three) gives rise to a manifold which carries a natural tight contact structure.
• 2)In a tight contact manifold, any two isotopic spheres which carry the same characteristic foliation are isotopic through a contact isolopy.
• 3)In a tight contact manifold, any two isotopic spheres have isomorphic complements.

     

Convex rearrangements of stable processes

The asymptotic behavior of convex rearrangements for smooth approximations of random processes is considered. The main results are.

- the relations between the convergence of convex rearrangements of absolutely continuous on [0, 1] functions and the weak convergence of its derivatives considered as random variables on the probability space {[0, 1], ß_{[0, 1]}, λ} are established:

- a strong law of large numbers for convex rearrangements of polygonal approximations of stable processes with the exponent α, 1<α≦2, is proved:

- the relations with the results by M. Wshebor (see references) on oscillations of the Wiener process and with the results by Yu. Davydov and A. M. Vershik (see references) on convex rearrangements of random walks are discussed.