We pose a new and intriguing question motivated by distributed computing regarding random walks on graphs: How long does it take for several independent random walks, starting from the same vertex, to cover an entire graph? We study the cover time – the expected time required to visit every node in a graph at least once – and we show that for a large collection of interesting graphs, running many random walks in parallel yields a speed-up in the cover time that is linear in the number of parallel walks. We demonstrate that an exponential speed-up is sometimes possible, but that some natural graphs allow only a logarithmic speed-up. A problem related to ours (in which the walks start from some probabilistic distribution on vertices) was previously studied in the context of space efficient algorithms for undirected s–t connectivity and our results yield, in certain cases, an improvement upon some of the earlier bounds.

Homological properties of several Banach left L1(G)-modules have been studied by Dales and Polyakov and recently by Ramsden. In this paper, we characterize some homological properties of and as Banach left L1(G)-modules, such as flatness, injectivity and projectivity.

We discuss some basic properties of the graded centre of a triangulated category and compute examples arising in representation theory of finite-dimensional algebras.

We study the boundedness and compactness of Toeplitz operators Ta on Bergman spaces , 1 < p < ∞. The novelty is that we allow distributional symbols. It turns out that the belonging of the symbol to a weighted Sobolev space of negative order is sufficient for the boundedness of Ta. We show the natural relation of the hyperbolic geometry of the disc and the order of the distribution. A corresponding sufficient condition for the compactness is also derived.

Poincaré's Polyhedron Theorem is a widely known valuable tool in constructing manifolds endowed with a prescribed geometric structure. It is one of the few criteria providing discreteness of groups of isometries. This work contains a version of Poincaré's Polyhedron Theorem that is applicable to constructing fibre bundles over surfaces and also suits geometries of non-constant curvature. Most conditions of the theorem, being as local as possible, are easy to verify in practice.

We go back to the results of Poincaré [Poincare, H (1891) Sur lintegration des equations differentielles du premier ordre et du premier degre I and II, Rendiconti del circolo matematico di Palermo5, 161–191] on the multipliers of a periodic orbit for proving the C1 non-integrability of differential systems. We apply these results to Lorenz, Rossler and Michelson systems, among others.

We first describe a reduction from the problem of lower-bounding the number of distinct distances determined by a set S of s points in the plane to an incidence problem between points and a certain class of helices (or parabolas) in three dimensions. We offer conjectures involving the new set-up, but are still unable to fully resolve them.

Instead, we adapt the recent new algebraic analysis technique of Guth and Katz [9], as further developed by Elekes, Kaplan and Sharir [6], to obtain sharp bounds on the number of incidences between these helices or parabolas and points in ℝ3. Applying these bounds, we obtain, among several other results, the upper bound O(s3) on the number of rotations (rigid motions) which map (at least) three points of S to three other points of S. In fact, we show that the number of such rotations which map at least k ≥ 3 points of S to k other points of S is close to O(s3/k12/7).

One of our unresolved conjectures is that this number is O(s3/k2), for k ≥ 2. If true, it would imply the lower bound Ω(s/logs) on the number of distinct distances in the plane.

We investigate the functors from modules to modules that occur as the summands of tensor powers and the functors from modules to Hopf algebras that occur as natural coalgebra summands of tensor algebras. The main results provide some explicit natural coalgebra summands of tensor algebras. As a consequence, we obtain some decompositions of Lie powers over the general linear groups.

We review recent nonlinear partial differential equation techniques developed to address questions concerning solitons for the quartic generalized Korteweg—de Vries equation (gKdV) and other generalizations of the KdV equation. We draw a comparison between results obtained in this way and some elements of the classical integrability theory for the original KdV equation, which serve as a reference for soliton and multi-soliton problems. First, known results on stability and asymptotic stability of solitons for gKdV equations are reviewed from several different sources. Second, we consider the problem of the interaction of two solitons for the quartic gKdV equation. We focus on recent results and techniques from a previous paper by the present authors concerning the interaction of two almost-equal solitons.

This paper is concerned with the limit of the vanishing ratio of the electron mass to the ion mass in the hydrodynamic models for plasmas in critical Besov spaces. We give a new construction of approximation solutions and show that periodic initial-value problems of certain scaled hydrodynamic models have smooth solutions in a (finite) time interval where the Euler solution is known to exist. Furthermore, it is justified that, as the electron mass tends to zero, the smooth solutions converge rigorously to solutions of the incompressible Euler equations, and the definite convergence orders are also obtained.

We are interested in the positive doubly periodic solutions, which are even in each variable, of a stationary nonlinear Schrödinger equation in ℝ2, with a small parameter. For any pair of periods (2a, 2b), we construct a branch of solutions that concentrate uniformly to the ground-state solution of the equation.

We review some recent results concerning tug-of-war games and their relation to some well-known partial differential equations (PDEs). In particular, we will show that solutions to certain PDEs can be obtained as limits of values of tug-of-war games when the parameter that controls the length of the possible movements goes to zero. Since the equations being studied are nonlinear and are not in divergence form, we will make extensive use of the concept of viscosity solutions.

In 1987, Weidmann proved that, for a symmetric differential operator τ and a real λ, if there exist fewer square-integrable solutions of (τ−λ)y = 0 than needed and if there is a self-adjoint extension of τ such that λ is not its eigenvalue, then λ belongs to the essential spectrum of τ. However, he posed an open problem of whether the second condition is necessary and it has been conjectured that the second condition can be removed. In this paper, we first set up a formula of the dimensions of null spaces for a closed symmetric operator and its closed symmetric extension at a point outside the essential spectrum. We then establish a formula of the numbers of linearly independent square-integrable solutions on the left and the right subintervals, and on the entire interval for nth-order differential operators. The latter formula ascertains the above conjecture. These results are crucial in criteria of essential spectra in terms of the numbers of square-integrable solutions for real values of the spectral parameter.

If Ω is an unbounded domain in ℝN and p > N, the Sobolev space W1,p(Ω) is not compactly embedded into L∈(Ω). Nevertheless, we prove that if Ω is a strip-like domain, then the subspace of W1,p(Ω) consisting of the cylindrically symmetric functions is compactly embedded into L∈(Ω). As an application, we study a Neumann problem involving the p-Laplacian operator and an oscillating nonlinearity, proving the existence of infinitely many weak solutions. Analogous results are obtained for the case of partial symmetry.

After a brief introduction and physical motivation, we show how the nonlinear Schrödinger (NLS) equation can be derived from a general class of nonlinear hyperbolic systems. Its purpose is to describe the behaviour of high-frequency oscillating wave packets over a large time-scale that requires us to take into account diffractive effects. We then show that the NLS approximation fails for short pulses and propose some alternative models, including a modified Schrödinger equation with improved frequency dispersion. It turns out that these models have better properties and are quite accurate for short pulses. For ultrashort pulses, however, they must also be abandoned for more complex approaches. We give the main steps for such an analysis and explain one striking fact about ultrashort pulses: their dynamics in dispersive media is linear.

We examine the regularity of the solution of a quasilinear system involving the curl of vector fields. This system arises in the mathematical theory of superconductivity. The C2+α regularity was obtained by Bates and Pan under the condition that Ω is simply connected and has no holes, and that the normal component of the curl of the boundary data vanishes. The aim of this paper is to remove these technical restrictions on the topology of the domain and on the boundary data. By carefully studying the related quasilinear Neumann problem, we obtain the C2+α regularity without assuming these technical conditions.

The self-adjointness of Δ2 + κ|x|−4 (κ>κ0 = κ0(N)) in L2(ℝN) is established as an application of the perturbation theorem in terms of Re(Au, Bεu), u ∈ D(A), for two non-negative self-adjoint operators A, B in a Hilbert space, where the family {Bε}ε>0 is the Yosida approximation of B. A key to the proof lies in a new inequality for the functions ν ∈ L2(ℝN) with |x|2Δν ∈ L2(ℝN) derived by using two real parameters.