Statistical mechanical systems on complete graphs, infinite exchangeability, finite extensions and a discrete finite moment problem

We show that a large collection of statistical mechanical systems with quadratically represented Hamiltonians on the complete graph can be extended to infinite exchangeable processes. This extends a known result for the ferromagnetic Curie--Weiss Ising model and includes as well all ferromagnetic Curie--Weiss Potts and Curie--Weiss Heisenberg models. By de Finetti's theorem, this is equivalent to showing that these probability measures can be expressed as averages of product measures. We provide examples showing that ``ferromagnetism'' is not however in itself sufficient and also study in some detail the Curie--Weiss Ising model with an additional 3-body interaction. Finally, we study the question of how much the antiferromagnetic Curie--Weiss Ising model can be extended. In this direction, we obtain sharp asymptotic results via a solution to a new moment problem. We also obtain a ``formula'' for the extension which is valid in many cases.Comment: Published at http://dx.doi.org/10.1214/009117906000001033 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

On Fuzzy Sequences, Fixed Points and Periodicity in Iterated Fuzzy Maps

I exemplify various elementary cases of fuzzy sequences and results related to the iteration of fuzzy mappings and to fuzzy logic systems (FLS). Several types of fuzzy logic system iterations are exemplified in relationship with oscillations in FLS and with the problem of stability in fuzzy logic control. I establish several conditions for fixed points and periodicity of the iterations based on fuzzy systems

Transfunctions and Other Topics in Measure Theory

Measures are versatile objects which can represent how populations or supplies are distributed within a given space by assigning sizes to subregions (or subsets) of that space. To model how populations or supplies are shifted from one configuration to another, it is natural to use functions between measures, called transfunctions. Any measurable function can be identified with its push-forward transfunction. Other transfunctions exist such as convolution operators. In this manner, transfunctions are treated as generalized functions. This dissertation serves to build the theory of transfunctions and their connections to other mathematical fields. Transfunctions that identify with continuous or measurable push-forward operators are characterized, and transfunctions that map between measures concentrated in small balls -- called localized transfunctions -- can be spatially approximated with measurable functions or with continuous functions (depending on the setting). Some localized transfunctions have fat graphs in the product space and fat continuous graphs are necessarily formed by localized transfunctions. Any Markov transfunction -- a transfunction that is linear, variation-continuous, total-measure-preserving and positive -- corresponds to a family of Markov operators and a family of plans (indexed by their marginals) such that all objects have the same instructions of transportation between input and output marginals. An example of a Markov transfunction is a push-forward transfunction. In two settings (continuous and measurable), the definition and existence of adjoints of linear transfunctions are formed and simple transfunctions are implemented to approximate linear weakly-continuous transfunctions in the weak sense. Simple Markov transfunctions can be used both to approximate the optimal cost between two marginals with respect to a cost function and to approximate Markov transfunctions in the weak sense. These results suggest implementing future research to find more applications of transfunctions to optimal transport theory. Transfunction theory may have potential applications in mathematical biology. Several models are proposed for future research with an emphasis on local spatial factors that affect survivorship, reproducibility and other features. One model of tree population dynamics (without local factors) is presented with basic analysis. Some future directions include the use of multiple numerical implementations through software programs

Multiscale methods for traffic flow on networks

In this thesis we propose a model to describe traffic flows on network by the theory of measure-based equations. We first apply our approach to the initial/boundary-value problem for the measure-valued linear transport equation on a bounded interval, which is the prototype of an arc of the network. This simple case is the first step to build the solution of the respective linear problem on networks: we construct the global solution by gluing all the measure-valued solutions on the arcs by means of appropriate distribution rules at the vertices. The linear case is adopted to show the well-posedness for the transport equation on networks in case of nonlocal velocity fields, i.e. which depends not only on the state variable, but also on the solution itself. It is also studied a representation formula in terms of the push-forward of the initial and boundary data along the network along the admissible trajectories, weighted by a properly dened measure on curves space. Moreover, we discuss an example of nonlocal velocity eld tting our framework and show the related model features with numerical simulations. In the last part, we focus on a class of optimal control problems for measure-valued nonlinear transport equations describing traffic ow problems on networks. The objective is to optimize macroscopic quantities, such as traffic volume, average speed, pollution or average time in a fixed area, by controlling only few agents, for example smart traffic lights or automated cars. The measure-based approach allows to study in the same setting local and nonlocal drivers interactions and to consider the control variables as additional measures interacting with the drivers distribution. To complete our analysis, we propose a gradient descent adjoint-based optimization method and some numerical experiments in the case of smart traffic lights for a 2-1 junction

Holographic Aspects of Chaos and Integrability in String- and M-Theory

In this thesis we investigate classical integrability of the string worldsheet on different super-gravity backgrounds. We focus in particular on the class of half-supersymmetric AdS7 solutions of Massive Type IIA supergravity, that are thought to be the near-horizon limit of a D6-D8-NS5 Hanany-Witten brane set-up, and are dual to six-dimensional conformal field theories with N = (1, 0) supersymmetry. We use both analytical and numerical methods to show the (bosonic sector of the) string worldsheet is non-integrable on most of these backgrounds. The backgrounds on which the string is integrable are an infinite massless solution (corresponding to an infinite constant quiver), and a background corresponding to an infinite linear quiver theory.In addition we find that the (bosonic sector of the) string is integrable on a background that we call AdS7 × (S3)λ. For this background we show that it corresponds to a 6d SCFT with an infinitely long quiver with an infinite number of flavour groups, all proportional to the colour groups. We study this particular supergravity background in detail, and suggest it corresponds to the large-N limit of the dual SCFT in the limit where the Chern-Simons level k goes to infinity.This integrable AdS7 × (S3)λ background can be obtained as the λ-deformation of AdS7×S3. In this context we study integrable deformations of supergravity backgrounds in the last part of this thesis, in particular non-Abelian T-duality. We present another back-ground on which the string is integrable by performing two non-Abelian T-dualities on two three-spheres inside the AdS5×S5 solution and study the resulting background