On chordal and bilateral SLE in multiply connected domains

We discuss the possible candidates for conformally invariant random non-self-crossing curves which begin and end on the boundary of a multiply connected planar domain, and which satisfy a Markovian-type property. We consider both, the case when the curve connects a boundary component to itself (chordal), and the case when the curve connects two different boundary components (bilateral). We establish appropriate extensions of Loewner’s equation to multiply connected domains for the two cases. We show that a curve in the domain induces a motion on the boundary and that this motion is enough to first recover the motion of the moduli of the domain and then, second, the curve in the interior. For random curves in the interior we show that the induced random motion on the boundary is not Markov if the domain is multiply connected, but that the random motion on the boundary together with the random motion of the moduli forms a Markov process. In the chordal case, we show that this Markov process satisfies Brownian scaling and discuss how this limits the possible conformally invariant random non-self-crossing curves. We show that the possible candidates are labeled by two functions, one homogeneous of degree zero, the other homogeneous of degree minus one, which describes the interaction of the random curve with the boundary. We show that the random curve has the locality property for appropriate choices of the interaction term. The research of the first author was supported by NSA grant H98230-04-1-0039. The research of the second author was supported by a grant from the Max-Planck-Gesellschaft.     

On radial stochastic Loewner evolution in multiply connected domains

We discuss the extension of radial SLE to multiply connected planar domains. First, we extend Loewner's theory of slit mappings to multiply connected domains by establishing the radial Komatu-Loewner equation, and show that a simple curve from the boundary to the bulk is encoded by a motion on moduli space and a motion on the boundary of the domain. Then, we show that the vector-field describing the motion of the moduli is Lipschitz. We explain why this implies that “consistent,” conformally invariant random simple curves are described by multidimensional diffusions, where one component is a motion on the boundary, and the other component is a motion on moduli space. We argue what the exact form of this diffusion is (up to a single real parameter κ) in order to model boundaries of percolation clusters. Finally, we show that this moduli diffusion leads to random non-self-crossing curves satisfying the locality property if and only if κ=6.     

A limit theorem for randomly stopped independent increment processes on separable metrizable groups

In the spirit of the classical random central limit theorem a general limit theorem for random stopping in the scheme of infinitesimal triangular arrays on a separable metrizable group is presented. The approach incorporates and generalizes earlier results for normalized sequences of independent random variables on both separable Banach spaces and simply connected nilpotent Lie groups originated by Siegel and Hazod, respectively. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Quasi-symmetric functions and the KP hierarchy

Quasi-symmetric functions arise in an approach to solve the Kadomtsev-Petviashvili (KP) hierarchy. This moreover features a new nonassociative product of quasi-symmetric functions that satisfies simple relations with the ordinary product and the outer coproduct. In particular, supplied with this new product and the outer coproduct, the algebra of quasi-symmetric functions becomes an infinitesimal bialgebra. Using these results we derive a sequence of identities in the algebra of quasi-symmetric functions that are in formal correspondence with the equations of the KP hierarchy.     

Optimality in Infinite-Horizon Variational Problems under Sign Conditions

We consider infinite-horizon variational problems on several spaces of curves. We establish relations between these problems and the properties of their solutions. Notably, we exhibit situations where optimality in a given space of curves implies optimality in a bigger space of curves. We work with a domain of definition of the Lagrangian which has a very general form and we provide assumptions to ensure a satisfactory theory of the necessary conditions of optimality. We apply these results to actualized Lagrangians.     

A note on Torelli‐type theorems for Gorenstein curves

Using the notion of generalized divisors introduced by Hartshorne (see 16 ), we adapt the theory of adjoint forms to the case of Gorenstein curves. We show an infinitesimal Torelli‐type theorem for line bundles on Gorenstein curves. We also construct explicit counterexamples to the infinitesimal Torelli claim in the case of a reduced reducible Gorenstein curve.     

Hitting time of large subsets of the hypercube

We study the simple random walk on the n‐dimensional hypercube, in particular its hitting times of large (possibly random) sets. We give simple conditions on these sets ensuring that the properly rescaled hitting time is asymptotically exponentially distributed, uniformly in the starting position of the walk. These conditions are then verified for percolation clouds with densities that are much smaller than (n log n)^‐1. A main motivation behind this article is the study of the so‐called aging phenomenon in the Random Energy Model, the simplest model of a mean‐field spin glass. Our results allow us to prove aging in the REM for all temperatures, thereby extending earlier results to their optimal temperature domain. © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2008     

Traveling in randomly embedded random graphs

We consider the problem of traveling among random points in Euclidean space, when only a random fraction of the pairs are joined by traversable connections. In particular, we show a threshold for a pair of points to be connected by a geodesic of length arbitrarily close to their Euclidean distance, and analyze the minimum length Traveling Salesperson Tour, extending the Beardwood‐Halton‐Hammersley theorem to this setting.     

On families of distributions characterized by certain properties of ordered random variables

In this work, we obtain new characterizations of certain probability distributions by relations with different ordered random variables. Such variables include order statistics, sequential maxima, and records. We consider relations that include not only upper, but also lower record values. The presented ordered objects are based on sequences of independent random variables with a common continuous distribution function. We also investigate equalities in the distribution of sequential maxima exposed by various random shifts. These shifts (one-sided or two-sided) have exponential distributions. Certain theorems and their corollaries present corresponding characterizations of distributions by relations of such a type. In addition, we consider exponentially shifted order statistics such that simple relations among them also characterize certain probability distributions. All of the presented results yield a set of characterizations of various distributions. For particular cases, we present the relations that characterize families of classical exponential and logistic distributions.     

Cover time and mixing time of random walks on dynamic graphs

The application of simple random walks on graphs is a powerful tool that is useful in many algorithmic settings such as network exploration, sampling, information spreading, and distributed computing. This is due to the reliance of a simple random walk on only local data, its negligible memory requirements, and its distributed nature. It is well known that for static graphs the cover time, that is, the expected time to visit every node of the graph, and the mixing time, that is, the time to sample a node according to the stationary distribution, are at most polynomial relative to the size of the graph. Motivated by real world networks, such as peer‐to‐peer and wireless networks, the conference version of this paper was the first to study random walks on arbitrary dynamic networks. We study the most general model in which an oblivious adversary is permitted to change the graph after every step of the random walk. In contrast to static graphs, and somewhat counter‐intuitively, we show that there are adversary strategies that force the expected cover time and the mixing time of the simple random walk on dynamic graphs to be exponentially long, even when at each time step the network is well connected and rapidly mixing. To resolve this, we propose a simple strategy, the lazy random walk, which guarantees, under minor conditions, polynomial cover time and polynomial mixing time regardless of the changes made by the adversary.     

On a topological description of solutions of complex differential equations

In this article we give a topological approach to the global behavior of arbitrary single-valued solutions in a simple connected domain of some general classes of complex differential equations with multi-valued coefficients. In particular, this permits us to describe certain globally multi-valued solutions as well as algebraic and algebroid solutions.     

Interaction of acoustic waves and piezoelectric structures

In the paper, we investigate the basic transmission problems arising in the model of fluid‐solid acoustic interaction when a piezo‐ceramic elastic body ( Ω ⁺ ) is embedded in an unbounded fluid domain ( Ω ^? ). The corresponding physical process is described by boundary‐transmission problems for second order partial differential equations. In particular, in the bounded domain Ω ⁺ , we have 4 × 4 dimensional matrix strongly elliptic second order partial differential equation, while in the unbounded complement domain Ω ^? , we have a scalar Helmholtz equation describing acoustic wave propagation. The physical kinematic and dynamic relations mathematically are described by appropriate boundary and transmission conditions. With the help of the potential method and theory of pseudodifferential equations, the uniqueness and existence theorems are proved in Sobolev–Slobodetskii spaces. Copyright © 2014 John Wiley & Sons, Ltd.     

Generation of Analytic Semigroups by Elliptic Operators with Dirichlet Boundary Conditions in a Cylindrical Domain

We consider an elliptic operator of second order with Dirichlet boundary conditions in a cylindrical domain. We show that a suitable interpretation of this operator in a certain space of continuous functions vanishing on the boundary is the infinitesimal generator of an analytic semigroup in the space of continuous functions. We prove several inclusions of the domain of the infinitesimal generator and of the real interpolation spaces between this domain and the basic space of continuous functions.     

Slit Holomorphic Stochastic Flows and Gaussian Free Field

It was realized recently that the chordal, radial and dipolar Schramm–Löwner evolution (SLEs) are special cases of a general slit holomorphic stochastic flow. We characterize those slit holomorphic stochastic flows which generate level lines of the Gaussian free field. In particular, we describe the modifications of the Gaussian free field (GFF) corresponding to the chordal and dipolar SLE with drifts. Finally, we develop a version of conformal field theory based on the background charge and Dirichlet boundary condition modifications of GFF and present martingale-observables for these types of SLEs.     

Geometry of the vacant set left by random walk on random graphs,Wright's constants,and critical random graphs with prescribed degrees

We provide an explicit algorithm for sampling a uniform simple connected random graph with a given degree sequence. By products of this central result include: (1) continuum scaling limits of uniform simple connected graphs with given degree sequence and asymptotics for the number of simple connected graphs with given degree sequence under some regularity conditions, and (2) scaling limits for the metric space structure of the maximal components in the critical regime of both the configuration model and the uniform simple random graph model with prescribed degree sequence under finite third moment assumption on the degree sequence. As a substantive application we answer a question raised by ?erný and Teixeira study by obtaining the metric space scaling limit of maximal components in the vacant set left by random walks on random regular graphs.     

The development of children's understanding of mathematical patterns through mathematical activities

In this paper we report how children (aged 8) developed their mathematical understanding through number tasks based on the Fibonacci sequence (Bamboo numbers) used in the context of a Substantial Learning Environment (SLE), which is designed to be mathematically rich, have a clear purpose and give opportunities to utilise mathematical thinking. The flexible nature of the SLEs makes it possible for teachers and children to explore various mathematical patterns. To capture children's activities when working within SLEs, we make particular reference to Pegg and Tall's work in 2005, and consider a theoretical framework based on the SOLO taxonomy (Biggs and Collis 1982) and the developmental process of understanding mathematical concepts. It was found that the key progression to be made through learning using our Bamboo number-based SLEs is from Multi-structural to Relational levels. It was also suggested that it is difficult for many children to understand the structural aspects of number patterns.     

The Mixed Problem for the Laplace Equation in an Exterior Domain with an Arbitrary Partition of the Boundary

In this paper we propose a method for solving the mixed boundary-value problem for the Laplace equation in a connected exterior domain with an arbitrary partition of the boundary. All simple closed curves making up the boundary are divided into three sets. On the elements of the first set the Dirichlet condition is given, on the elements of the second set the third boundary condition is prescribed, and the third set, in turn, is divided into two subsets of simple closed arcs, with the Dirichlet condition prescribed on the elements of one of these subsets and the third boundary condition on the elements of the other subset. The problem is reduced to a uniquely solvable Fredholm equation of the second kind in a Banach space. The third boundary-value problem and the mixed Dirichlet--Neumann problem are particular cases of the problem under study.     

Connectivity of a general class of inhomogeneous random digraphs

We study a family of directed random graphs whose arcs are sampled independently of each other, and are present in the graph with a probability that depends on the attributes of the vertices involved. In particular, this family of models includes as special cases the directed versions of the Erd?s‐Rényi model, graphs with given expected degrees, the generalized random graph, and the Poissonian random graph. We establish a phase transition for the existence of a giant strongly connected component and provide some other basic properties, including the limiting joint distribution of the degrees and the mean number of arcs. In particular, we show that by choosing the joint distribution of the vertex attributes according to a multivariate regularly varying distribution, one can obtain scale‐free graphs with arbitrary in‐degree/out‐degree dependence.     

The modulus problem for families of classes of curves in a circular annulus

Earlier, the author has considered the extremal metric problem for the families of several homotopic classes of curves in the closed plane ¯C or in a simply connected domain in ¯C. In this paper, these results are extended to the case of the modulus problem for families of classes of curves in a doubly connected domain.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 144, pp. 115–128, 1985.     

Courbes rigidifiant les surfaces

The infinitesimal rigidity of surfaces is addressed in this Note. Let us consider a bidimensional metric manifold imbedded in R³: we seek the infinitesimal changes of imbedding conserving its metric to first order. A criterion for the existence of such deformations is presented in the particular case of surfaces of revolution. It reveals rigidifying curves on the surface, called “a-curves”. A geometric interpretation of the rigidity brought about by a-curves leads to the identification of rigidifying curves on arbitrary surfaces.