We introduce the concept of a relative Tutte polynomial of coloured graphs. We show that this relative Tutte polynomial can be computed in a way similar to the classical spanning tree expansion used by Tutte in his original paper on this subject. We then apply the relative Tutte polynomial to virtual knot theory. More specifically, we show that the Kauffman bracket polynomial (and hence the Jones polynomial) of a virtual knot can be computed from the relative Tutte polynomial of its face (Tait) graph with some suitable variable substitutions. Our method offers an alternative to the ribbon graph approach, using the face graph obtained from the virtual link diagram directly.

Let H be some fixed graph. We call a graph Gvicarious for H if G is maximal H-free and, for every edge e of G, there is an edge f not in G such that G − e + f is also H-free. We demonstrate various properties of vicarious graphs and several examples are given. It is conjectured that a graph of order n which is vicarious for K3 has size at most (1/4 + o(1))().

We show that the number of independent sets in an N-vertex, d-regular graph is at most (2d+1 − 1)N/2d, where the bound is sharp for a disjoint union of complete d-regular bipartite graphs. This settles a conjecture of Alon in 1991 and Kahn in 2001. Kahn proved the bound when the graph is assumed to be bipartite. We give a short proof that reduces the general case to the bipartite case. Our method also works for a weighted generalization, i.e., an upper bound for the independence polynomial of a regular graph.

In this paper we study the use of spectral techniques for graph partitioning. Let G = (V, E) be a graph whose vertex set has a ‘latent’ partition V1,. . ., Vk. Moreover, consider a ‘density matrix’ Ɛ = (Ɛvw)v, sw∈V such that, for v ∈ Vi and w ∈ Vj, the entry Ɛvw is the fraction of all possible Vi−Vj-edges that are actually present in G. We show that on input (G, k) the partition V1,. . ., Vk can (very nearly) be recovered in polynomial time via spectral methods, provided that the following holds: Ɛ approximates the adjacency matrix of G in the operator norm, for vertices v ∈ Vi, w ∈ Vj ≠ Vi the corresponding column vectors Ɛv, Ɛw are separated, and G is sufficiently ‘regular’ with respect to the matrix Ɛ. This result in particular applies to sparse graphs with bounded average degree as n = #V → ∞, and it has various consequences on partitioning random graphs.

The following first problem is posed:is a correct ‘entropy solution’ of the Cauchy problem for the fifth-order degenerate non-linear dispersion equations (NDEs), same as for the classic Euler one ut + uux = 0,These two quasi-linear degenerate partial differential equations (PDEs) are chosen as typical representatives; so other (2m + 1)th-order NDEs of non-divergent form admit such shocks waves. As a related second problem, the opposite initial shock S+(x) = −S−(x) = sign x is shown to be a non-entropy solution creating a rarefaction wave, which becomes C∞ for any t > 0. Formation of shocks leads to non-uniqueness of any ‘entropy solutions’. Similar phenomena are studied for a fifth-order in time NDE uttttt = (uux)xxxx in normal form.

On the other hand, related NDEs, such asare shown to admit smooth compactons, as oscillatory travelling wave solutions with compact support. The well-known non-negative compactons, which appeared in various applications (first examples by Dey, 1998, Phys. Rev. E, vol. 57, pp. 4733–4738, and Rosenau and Levy, 1999, Phys. Lett. A, vol. 252, pp. 297–306), are non-existent in general and are not robust relative to small perturbations of parameters of the PDE.

This paper deals with a new method to determine the dependence of the electrical conductivity of metals or semiconductors on temperature. It is based on the fact that the current–voltage relationship is easily measurable. This inverse problem is solved by the classical Abel integral equation.

A two component system driven by both interface area and interface curvature is studied with a new phase field model. We show that if the curvature impact in the system is strong enough, there exist bubble profiles. A bubble profile describes a pattern of an inner core of one component surround by an outer membrane of the other component. It is a radial solution to a fourth-order nonlinear partial differential equation. We show the existence of such profiles in all dimensions, although the profile is unstable if the dimension is greater than 2.

We study the Benney equation and show that the associated initial-value problem is locally well-posed in Sobolev spaces Hs(ℝ2) for s > −2. Furthermore, we use a priori estimates to establish the global well-posedness for s ≥ 0. We also prove that these results are in some sense sharp. In addition, we obtain some exact travelling-wave solutions of the equation.

We study the multiplicity of positive solutions for the following semilinear elliptic equation:

where 1 < q < 2 < p < 2* (2* = 2N/(N − 2) if N ≥ 3, 2* = ∞ if N = 2), the parameters λ, μ ≥ 0, is an infinite strip in ℝN and Θ is a bounded domain in ℝN−1 We assume that fλ(x) = λf+(x) + f−(x) and gμ(x) = a(x) + μb(x), where the functions f±, a and b satisfy suitable conditions.

Given a bounded domain G ⊂ ℝd, d ≥ 3, we study smooth solutions of a linear parabolic equation with non-constant coefficients in G, which at the boundary have to C1-match with some harmonic function in ℝd \ Ḡ vanishing at spatial infinity.

This problem arises in the framework of magnetohydrodynamics if certain dynamo-generated magnetic fields are considered: for example, in the case of axisymmetry, or for non-radial flow fields, the poloidal scalar of the magnetic field solves the above problem.

We first investigate the Poisson problem in G with the boundary condition described above as well as the associated eigenvalue problem and prove the existence of smooth solutions. As a by-product we obtain the completeness of the well-known poloidal ‘free decay modes’ in ℝ3 if G is a ball. Smooth solutions of the evolution problem are then obtained by Galerkin approximation based on these eigenfunctions.

We prove sharp upper bounds for invariant measures of Markov processes in ℝN associated with second-order elliptic differential operators with unbounded coefficients.

Building on a recent work, we consider a two-dimensional viscous fluid in the exterior of a thin obstacle shrinking to a curve, proving convergence to a solution of the Navier–Stokes equations in the exterior of a curve. The uniqueness of the limit solution is also shown.>

Assuming that (Ω, Σ, μ) is a complete probability space and that X is a Banach space, we evaluate both the semivariation and the variation norm of a wide class of Pettis integrable functions f : Ω → X.

As stated in the preceding paper, the method of expressing, at sight, shillings, pence, and farthings as a decimal of a pound to 3 places has long been known. It is sometimes referred to as the actuaries' rule. According to De Morgan, it occurs for the first time in Kersey's edition of Wingate's Arithmetic, 1673 (p. 191). It is also to be found in Cocker's Decimal Arithmetic, 1685 (although in a form which is not quite accurate). In some of the earlier books the method of conversion at sight from the decimal form is given, but not vice versâ. It is now found in most modern text-books in one form or another.

We discuss travelling wavefronts of a degenerate and singular parabolic equation in non-divergent form with changing sign sources. Necessary and sufficient conditions will be given for the existence of smooth or non-smooth and non-decreasing or non-increasing solutions. We also study the regularity of such solutions.

We consider a semilinear elliptic system with both concave—convex nonlinearities and critical growth terms in bounded domains. The existence and multiplicity results of positive solutions are obtained by variational methods.

We describe decompositions of the group of units of a ring and of its subgroups, induced by idempotents with certain properties. The results apply to several classes of rings, most notably to semi-perfect rings.