In radio frequency (RF) applications, electric circuits produce signals exhibiting fast oscillations, whereas the amplitude and frequency may change slowly in time. Thus, solving a system of differential algebraic equations (DAEs), which describes the circuit's transient behaviour, becomes inefficient, since the fast rate restricts the step sizes in time. A multivariate model is able to decouple the widely separated time scales of RF signals and provides an alternative approach. Consequently, a system of DAEs changes into a system of multirate partial differential algebraic equations (MPDAEs). The determination of multivariate solutions allows for the exact reconstruction of corresponding time-dependent signals. Hence, an efficient numerical simulation is obtained by exploiting the periodicities in fast time scales. We outline the theory of this multivariate approach with respect to the simulation of amplitude as well as frequency modulated signals. Furthermore, a survey of numerical methods for solving the arising problems of MPDAEs is given.

Let X ⊂ ℙN be a geometrically integral cubic hypersurface defined over ℚ, with singular locus of dimension at most dim X − 4. The main result in this paper is a proof of the fact that X(ℚ) contains OɛX (BdimX + ɛ) points of height at most B.

Let K be a convex body of dimension at least 3, and let p0 be a point. If every section of K through p0 is centrally symmetric, then Rogers proved in [6] that K is centrally symmetric, although p0 may not be the centre of K. If this is the case, then Aitchison, Petty and Rogers [1] and Larman [2] proved that K must be an ellipsoid. Suppose now that, for every direction, we can choose continuously a section of K that is centrally symmetric; if K is strictly convex, then Montejano [3] proved that K must be centrally symmetric. Consider now the following example. Let D be a (euclidean) ball centred at the origin from which two symmetric caps are deleted. Then D is centrally symmetric with respect to the origin, and has a lot of circular sections whose centre is not the origin. In fact, we can choose continuously, for every direction, a section of D which is centrally symmetric, in such a way that not all these sections pass through the origin. Nevertheless, no matter how we choose these sections, there are always necessarily many of them that do pass through the origin. For those sections, of course, we have not imposed any condition, which explains the fact that D is not a quadric elsewhere.

A new lower bound is established for the distance between two roots of an integer polynomial, and a new upper bound for the distance between a given real number and the set of zeros of an integer polynomial. The latter result is applied to improve a metrical result in Diophantine approximation.

Let κ be an infinite cardinal. Okuyama showed that the product space X ×i Y of a paracompact weak P (ω)-space X and a K-analytic space Y is paracompact. In this paper, by using the notion of κ-K-analytic spaces which is basically defined by Hansell, Jayne and Rogers, the above result is extended and some other results are given related to normality, collectionwise normality and covering properties on products. An answer to a question of Okuyama and Watson is also given, as well as some applications to extensions of continuous functions on these products.

We propose a phase field model for stress and diffusion-induced interface motion. This model, in particular, can be used to describe diffusion-induced grain boundary motion and generalizes a model of Cahn, Fife and Penrose as it more accurately incorporates stress effects. In this paper we will demonstrate that the model can also be used to describe other stress-driven interface motion. As an example, interface motion resulting from interactions of interfaces with dislocations is studied.

Let E be a level 1, vector valued Eisenstein series of half-integral weight, normalized so that the coefficients are all in ℤ. It is shown that there is a level one vector valued cusp form f with the same weight as E and with coefficients in ℤ, which is congruent to E modulo the constant term of E.

The kth projection function of a convex body K ⊂ ℝn assigns to any k-dimensional linear subspace of ℝn the k-volume of the orthogonal projection of K to that subspace. Let K and K0 be convex bodies in ℝn, and let K0 be centrally symmetric and satisfy a weak regularity and curvature condition (which includes all K0 with ∂K0 of class C2 with positive radii of curvature). Assume that K and K0 have proportional 1st projection functions (i.e., width functions) and proportional kth projection functions. For 2 ≤ k < (n + 1)/2 and for k = 3, n = 5, it is shown that K and K0 are homothetic. In the special case where K0 is a Euclidean ball, characterizations of Euclidean balls as convex bodies of constant width and constant k-brightness are thus obtained.

Let υ be a Henselian valuation of arbitrary rank of a field K, and let ῡ be the (unique) extension of v to a fixed algebraic closure of K. For an element α ∈ \K, a chain α = α0, α1,…,αr of elements of , such that αi is of minimum degree over K with the property that ῡ(αi−1 − αi) = sup{ῡ(αi−1 − β) | [K (β) : K] < [K (αi−1) : K]} and that αr ∈ K, is called a saturated distinguished chain for α with respect to (K, υ). The notion of a saturated distinguished chain has been used to obtain results about the irreducible polynomials over any complete discrete rank one valued field K and to determine various arithmetic and metric invariants associated to elements of (cf. [J. Number Theory, 52 (1995), 98–118.] and [J. Algebra, 266 (2003), 14–26]). In this paper, a method is described of constructing a saturated distinguished chain for α, and also determining explicitly some invariants associated to α, when the degree of the extension K (α)/K is not divisible by the characteristic of the residue field of υ.

The flow of a thin layer of fluid down an inclined plane is modified by the presence of insoluble surfactant. For any finite surfactant mass, traveling waves are constructed for a system of lubrication equations describing the evolution of the free-surface fluid height and the surfactant concentration. The one-parameter family of solutions is investigated using perturbation theory with three small parameters: the coefficient of surface tension, the surfactant diffusivity, and the coefficient of the gravity-driven diffusive spreading of the fluid. When all three parameters are zero, the nonlinear PDE system is hyperbolic/degenerate-parabolic, and admits traveling wave solutions in which the free-surface height is piecewise constant, and the surfactant concentration is piecewise linear and continuous. The jumps and corners in the traveling waves are regularized when the small parameters are nonzero; their structure is revealed through a combination of analysis and numerical simulation.

We characterize a topological convex space $C$ in terms of the family $\mathcal{A}(C)$ of real continuous affine functions on $C$. Our main result states that two topological convex spaces $C_1$ and $C_2$ are affine-homeomorphic if and only if $\mathcal{A}(C_1)$ and $\mathcal{A}(C_2)$ are isomorphic as ordered unital vector spaces.

The stationary flow of a jet of a Newtonian fluid that is drawn by gravity onto a moving surface is analyzed. It is assumed that the jet has a convex shape and hits the moving surface tangentially. The flow is modelled by a third-order ODE on a domain of unknown length and with an additional integral condition. By solving part of the equation explicitly, the problem is reformulated as a first-order ODE with an integral constraint. The corresponding existence region in the three-dimensional parameter space is characterized in terms of an easily calculable quantity. In a qualitative sense, the results from the model are found to correspond with experimental observations.

We show the $L^1$ contraction and comparison principle for weak (and, more generally, renormalized) solutions of the elliptic–parabolic problem $j(v)_t-\text{div}(\nabla w+F(w))=f(t,x)$, $w=\varphi(v)$ in $(0,T)\times\varOmega\subset \mathbb{R}^+\times\mathbb{R}^N$ with inhomogeneous Dirichlet boundary datum $g\in L^2(0,T;W^{1,2}(\varOmega))$ for $w$ (the boundary datum is taken in the sense $w-g\in L^2(0,T;H^{1}_0(\varOmega))$) and initial datum $j_o\in L^1(\varOmega)$ for $j(v)$. Here $\varphi$ and $j$ are non-decreasing, and we assume that $F$ is just continuous.

Our proof consists in doubling of variables in the interior of $\varOmega$ as introduced by Carrillo in 1999, and in a careful treatment of the flux term near the boundary of $\varOmega$. For the latter argument, the result is restricted to the linear dependence on $\nabla w$ of the diffusion term. The proof allows for a wide class of domains $\varOmega$, including, for example, weakly Lipschitz domains with Lipschitz cracks.

We obtain the corresponding results for the associated stationary problem and discuss on generalization of our technique to the case of nonlinear diffusion operators.

In this paper a method is developed for the asymptotic expansion of some classes of integral as a parameter k → 0+. The procedure is analogous to the method of inner and outer sums for treating certain types of infinite series whose terms contain a small parameter, and can involve heavy algebra. However, this aspect of the process can be delegated to a symbolic manipulation package.

In a real Hilbert space $H$, we study the bifurcation points of equations of the form $F(\lambda,u)=0$, where $F:\mathbb{R}\times H\rightarrow H$ is a function with $F(\lambda,0)=0$ that is Hadamard differentiable, but not necessarily Fréchet differentiable, with respect to $u$ at $u=0$. In this context, there may be bifurcation at points $\lambda$ where $D_{u} F(\lambda,0):H\rightarrow H$ is an isomorphism. We formulate some additional conditions on $F$ that ensure that bifurcation does not occur at a point where $D_{u}F(\lambda,0):H\rightarrow H$ is an isomorphism. Then, in the case where $F(\lambda,\cdot)$ is a gradient, we give conditions that imply that bifurcation occurs at a point $\lambda$. These conditions may be satisfied at points where $D_{u}F(\lambda,0):H\rightarrow H$ is an isomorphism. We demonstrate the use of these abstract results in the context of nonlinear elliptic equations of the form

$$-\Delta u(x)+q(x)u(x)=\lambda\eta(x)^{-1}f(\eta(x)u(x))$$

and

$$-\Delta u(x)+q(x)u(x)-\eta(x)^{-1}f(\eta(x)u(x))=\lambda u(x),$$

where $u\in H^{2}(\mathbb{R}^{N})$.

Following pioneering work by Fan and Slemrod, who studied the effect of artificial viscosity terms, we consider the system of conservation laws arising in liquid–vapour phase dynamics with physical viscosity and capillarity effects taken into account. Following Dafermos, we consider self-similar solutions to the Riemann problem and establish uniform total variation bounds, allowing us to deduce new existence results. Our analysis covers both the hyperbolic and the hyperbolic–elliptic regimes and apply to arbitrarily large Riemann data.

The proofs rely on a new technique of reduction to two coupled scalar equations associated with the two wave fans of the system. Strong $L^1$ convergence to a weak solution of bounded variation is established in the hyperbolic regime, while in the hyperbolic–elliptic regime a stationary singularity near the axis separating the two wave fans, or more generally an almost-stationary oscillating wave pattern (of thickness depending upon the capillarity–viscosity ratio), is observed and the solution may not have globally bounded variation.

Algorithms are introduced that produce optimal Markovian couplings for large finite-state-space discrete-time Markov chains with sparse transition matrices; these algorithms are applied to some toy models motivated by fluid-dynamical mixing problems at high Peclét number. An alternative definition of the time-scale of a mixing process is suggested. Finally, these algorithms are applied to the problem of coupling diffusion processes in an acute-angled triangle, and some of the simplifications that occur in continuum coupling problems are discussed.

We study model-theoretic and stability-theoretic properties of the non-abelian free group in the light of Sela's recent result on stability and results announced by Bestvina and Feighn on ‘negligible subsets' of free groups. We point out analogies between the free group and so-called bad groups of finite Morley rank, and prove ‘non-CM-triviality' of the free group.

The Main Conjecture of Iwasawa theory for an elliptic curve $E$ over $\mathbb{Q}$ and the anticyclotomic $\mathbb{Z}_p$-extension of an imaginary quadratic field $K$ was studied in \cite{bertolini_darmon}, in the case where $p$ is a prime of ordinary reduction for $E$. Analogous results are formulated, and proved, in the case where $p$ is a prime of supersingular reduction. The foundational study of supersingular main conjectures carried out by Perrin-Riou, Pollack, Kurihara, Kobayashi and Iovita and Pollack are required to handle this case in which many of the simplifying features of the ordinary setting break down.

The Poisson and Martin boundaries for invariant random walks on the dual of the orthogonal quantum groups $A_{\mathrm{o}}(F)$ are identified with higher-dimensional Podleś spheres that we describe in terms of generators and relations. This provides the first such identification for random walks on non-amenable discrete quantum groups.