The dynamic behaviour of magneto-mechanical sensors and actuators can be completely described by Maxwell's and Navier-Lamé's partial differential equations (PDEs) with appropriate coupling terms reflecting the interactions of these fields and with the corresponding initial, boundary and interface conditions. Neglecting the displacement currents, which can be done for the classes of problems considered in this paper, and introducing the vector potential for the magnetic field, we arrive at a system of degenerate parabolic PDEs for the vector potential coupled with the hyperbolic PDEs for the displacements.Usually the computational domain, the finite element discretization, the time integration, and the solver are different for the magnetic and mechanical parts. For instance, the vector potential is approximated by edge elements whereas the finite element discretization of the displacements is based on nodal elements on different meshes. The most time consuming modules in the solution procedure are the solvers for both, the magnetical and the mechanical finite element equations arising at each step of the time integration procedure. We use geometrical multigrid solvers which are different for both parts. These multigrid solvers enable us to solve quite efficiently not only academic test problems, but also transient 3D technical magneto-mechanical systems of high complexity such as solenoid valves and electro-magnetic-acoustic transducers. The results of the computer simulation are in very good agreement with the experimental data.

We consider the following system of equations: where the spatial average ⟨ B ⟩ = 0 and μ > σ > 0. This system plays an important role as a Ginzburg-Landau equation with a mean field in several areas of the applied sciences and the steady-states of this system extend to periodic steady-states with period L on the real line which are observed in experiments. Our approach is by combining methods of nonlinear functional analysis such as nonlocal eigenvalue problems and the variational characterization of eigenvalues with Jacobi elliptic integrals. This enables us to give a complete classification of all stable steady-states for any positive L.

Let , N ≥2 be a bounded smooth domain and α > 1. We are interested in the singular elliptic equationwith Neumann boundary conditions. In this paper, a complete description of all continuous radially symmetric solutions is given. In particular, we construct nontrivial smooth solutions as well as rupture solutions. Here a continuous solution is said to be a rupture solution if its zero set is nonempty. When N = 2 and α = 3, the equation is used to model steady states of van der Waals force driven thin films of viscous fluids. We also consider the physical problem when total volume of the fluid is prescribed.

The intrusion of a constant density fluid at the interface of a two-layer fluid is considered. Numerical solutions are computed for a model of a steady intrusion resulting from flow down a bank and across a broad lake or reservoir. The incoming fluid is homogeneous and spreads across the lake at its level of neutral buoyancy. Solutions are obtained for a range of different inflow angles, flow rate and density differences. Except in extreme cases, the nature of the solution is predicted quite well by linear theory, with the wavelength at any Froude number given by a dispersion relation and wave steepness determined largely by entry angle. However, some extreme solutions with rounded meandering flows and non-unique solutions in the parameter space are also obtained.

We explain three methods for showing that the $p$-adic monodromy of a modular family of abelian varieties is ‘as large as possible', and illustrate them in the case of the ordinary locus of the moduli space of $g$-dimensional principally polarized abelian varieties over a field of characteristic $p$. The first method originated from Ribet's proof of the irreducibility of the Igusa tower for Hilbert modular varieties. The second and third methods both exploit Hecke correspondences near a hypersymmetric point, but in slightly different ways. The third method was inspired by work of Hida, plus a group theoretic argument for the maximality of $\ell$-adic monodromy with $\ell\neq p$.

Nous démontrons des théorèmes d'immersion holomorphe dans un espace projectif complexe pour des variétés kählériennes non compactes et des laminations par variétés complexes qui admettent un fibré en droites holomorphe strictement positif. En particulier, nous montrons que sur une lamination compacte par surfaces de Riemann, les fonctions méromorphes séparent les points si et seulement si aucun cycle feuilleté n'est homologue à $0$.

We prove holomorphic immersion theorems in a finite dimensional complex projective space for kählerian non-compact manifolds and for laminations by complex manifolds that carry a line bundle of positive curvature. In particular, we prove that on a Riemann surfaces lamination of a compact space, the space of meromorphic functions separates points if and only if every foliation cycle is non homologous to $0$.

Consider two graphs G1 and G2 on the same vertex set V and suppose that Gi has mi edges. Then there is a bipartition of V into two classes A and B so that, for both i = 1, 2, we have . This gives an approximate answer to a question of Bollobás and Scott. We also prove results about partitions into more than two vertex classes. Our proofs yield polynomial algorithms.

We introduce the minor-closed, dual-closed class of multi-path matroids. We give a polynomial-time algorithm for computing the Tutte polynomial of a multi-path matroid, we describe their basis activities, and we prove some basic structural properties. Key elements of this work are two complementary perspectives we develop for these matroids: on the one hand, multi-path matroids are transversal matroids that have special types of presentations; on the other hand, the bases of multi-path matroids can be viewed as sets of lattice paths in certain planar diagrams.

The partition functions of the Ising and Potts models in statistical mechanics are well known to be partial evaluations of the Tutte–Whitney polynomial of the appropriate graph. The Ashkin–Teller model generalizes the Ising model and the four-state Potts model, and has been extensively studied since its introduction in 1943. However, its partition function (even in the symmetric case) is not a partial evaluation of the Tutte–Whitney polynomial. In this paper, we show that the symmetric Ashkin–Teller partition function can be obtained from a generalized Tutte–Whitney function which is intermediate in a precise sense between the usual Tutte–Whitney polynomialof the graph and that of its dual.

Consider the following communication problem, which leads to a new notion of edge colouring. The communication network is represented by a bipartite multigraph, where the nodes on one side are the transmitters and the nodes on the other side are the receivers. The edges correspond to messages, and every edge e is associated with an integer c(e), corresponding to the time it takes the message to reach its destination. A proper k-edge-colouring with delays is a function f from the edges to {0, 1, . . ., k − 1}, such that, for every two edges e1 and e2 with the same transmitter, f(e1) ≠ f(e2), and for every two edges e1 and e2 with the same receiver, f(e1) + c(e1) ≢ f(e2) + c(e2) (mod k). Ross, Bambos, Kumaran, Saniee and Widjaja [18] conjectured that there always exists a proper edge colouring with delays using k = Δ + o(Δ) colours, where Δ is the maximum degree of the graph. Haxell, Wilfong and Winkler [11] conjectured that a stronger result holds: k = Δ + 1 colours always suffice. We prove the weaker conjecture for simple bipartite graphs, using a probabilistic approach, and further show that the stronger conjectureholds for some multigraphs, applying algebraic tools.

We show that a random 4-regular graph asymptotically almost surely (a.a.s.) has chromatic number 3. The proof uses an efficient algorithm which a.a.s. 3-colours a random 4-regular graph. The analysis includes use of the differential equation method, and exponential bounds on the tail of random variables associated with branching processes. A substantial part of the analysis applies to random d-regular graphs in general.

Let A be a set of N matrices. Let g(A) ≔ |A + A| + |A · A|, where A + A = {a1 + a2 ∣ ai ∈ A} and A · A = {a1a2 ∣ ai ∈ A} are the sum set and product set. We prove that if the determinant of the difference of any two distinct matrices in A is nonzero, then g(A) cannot be bounded below by cN for any constant c. We also prove that if A is a set of d × d symmetric matrices, then there exists ϵ = ϵ(d)>0 such that g(A)>N1+ϵ. For the first result, we use the bound on the number of factorizations in a generalized progression. For the symmetric case, we use a technical proposition which provides an affine space V containing a large subset E of A, with the property that if an algebraic property holds for a large subset of E, then it holds for V. Then we show that the system a2 : a ∈ V is commutative, allowing us to decompose as eigenspaces simultaneously, so we can finish the proof with induction and a variant of the Erdős–Szemerédi argument.

We describe how non-crossing partitions arise in substitution method calculations. By using efficient algorithms for computing non-crossing partitions we are able to substantially reduce the computational effort, which enables us to compute improved bounds on the percolation thresholds for three percolation models. For the Kagomé bond model we improve bounds from 0.5182 ≤ pc ≤ 0.5335 to 0.522197 ≤ pc ≤ 0.526873, improving the range from 0.0153 to 0.004676. For the (3, 122) bond model we improve bounds from 0.7393 ≤ pc ≤ 0.7418 to 0.739773 ≤ pc ≤ 0.741125, improving the range from 0.0025 to 0.001352. We also improve the upper bound for the hexagonal site model, from 0.794717 to 0.743359.

We show that every regular tournament on n vertices has at least n!/(2 + o(1))n Hamiltonian cycles, thus answering a question of Thomassen [17] and providing a partial answer to a question of Friedgut and Kahn [7]. This compares to an upper bound of about O(n0.25n!/2n) for arbitrary tournaments due to Friedgut and Kahn (somewhat improving Alon's bound of O(n0.5n!/2n)). A key ingredient of the proof is a martingale analysis of self-avoiding walks on a regular tournament.

The strong isometric dimension of a graph G is the least number k such that G isometrically embeds into the strong product of k paths. Using Sperner's theorem, the strong isometric dimension of the Hamming graphs K2 □ Kn is determined.

We study a dynamically evolving random graph which adds vertices and edges using preferential attachment and is ‘attacked by an adversary’. At time t, we add a new vertex xt and m random edges incident with xt, where m is constant. The neighbours of xt are chosen with probability proportional to degree. After adding the edges, the adversary is allowed to delete vertices. The only constraint on the adversarial deletions is that the total number of vertices deleted by time n must be no larger than δn, where δ is a constant. We show that if δ is sufficiently small and m is sufficiently large then with high probability at time n the generated graph has a component of size at least n/30.

One of the main obstacles in applying converse theorems to prove new cases of functoriality is that of stability of $\gamma$-factors for a certain class of $L$-functions obtained from the ‘Langlands–Shahidi’ method, where the $\gamma$-factors are defined inductively by means of ‘local coefficients’. The problem then becomes that of stability of local coefficients upon twisting the representation by a highly ramified character. In this paper we first establish that the inverses of certain local coefficients are, up to an abelian $\gamma$-factor, genuine Mellin transforms of partial Bessel functions of the type we analysed in our previous paper. The second main result is then the resulting stability of the local coefficients in this situation, which include all the cases of interest for functoriality. Hopefully, the analysis given here will open the door to a proof of the general stability and the equality of $\gamma$-factors obtained from different methods through integration over certain quotient spaces whose generic fibres are closed. They do not seem to have been studied before in any generality.

We consider the focusing nonlinear Schrödinger equation on the quarter plane. Initial data vanish at infinity while boundary data are time-periodic. The main goal of this paper is to introduce a Riemann–Hilbert problem whose solution gives the solution of our initial–boundary-value problem. This is a preliminary step to obtain uniform long-time asymptotics for the solution of this equation using both the stationary phase and the Deift–Zhou methods.

The pseudovariety $\mathbf{DG}$ of all finite monoids all of whose regular $\mathcal{D}$-classes are subgroups is shown to be local, that is, it is verified that the pseudovariety $g\mathbf{DG}$ of finite categories generated by $\mathbf{DG}$ coincides with the pseudovariety $\ell\mathbf{DG}$ of all finite categories whose local monoids all belong to $\mathbf{DG}$. Yet more general statements of this kind are deduced, yielding results such as that, for every prime number $p$, the pseudovariety $\mathbf{DG}_p$ of all finite monoids all of whose regular $\mathcal{D}$-classes are $p$-groups is local, or that the pseudovarieties $\mathbf{DG}_{\mathrm{sol}}$ and $\mathbf{DG}_{\mathrm{nil}}$ of all finite monoids all of whose regular \mbox{$\mathcal{D}$-classes} are, respectively, solvable groups and nilpotent groups are local.

Nous proposons une façon simple de compter le nombre de $G$-chtoucas de Drinfeld avec modifications arbitraires, inspirée par les travaux de Kottwitz sur les variétés de Shimura.

We propose a simple method to count the number of Drinfeld $G$-shtukas with arbitrary modifications which is inspired by works by Kottwitz on Shimura varieties.