We deal with the initial-value problem for parabolic equations with discontinuous nonlinearities and establish the existence of its weak solution. Next, we show that for a suitable class of initial data, the weak solution is locally or globally unique in time. Lastly, we prove that there exist at least two different weak solutions in general if initial data do not belong to this class.

In this note, we first consider the monotonicity of the Maslov-type index theory. More precisely, for any two 1-periodic symmetric continuous matrix functions B0(t) and B1(t) with B0(t) < B1(t), we consider the relations between the Maslov-type indices (i(B0), ν (B0)) and (i(B1), ν (B1)). We then apply this theory to study the existence and multiplicity of some kinds of asymptotically linear Hamiltonian systems

This is a study of relations between pure cubic fields and their normal closures. Explicit formula shows how the discriminant, regulator and class number of the normal closure can be expressed in terms of the cubic field.

In this paper the absolute value or distance from the origin analogue of the classical Khintchine-Groshev theorem [5] is established for a single linear form with a “slowly decreasing” error function. To explain this in more detail, some notation is introduced. Throughout this paper, m, n are positive integers; i.e., m, n ∈ ℕ; x = (x1,…, xn) will denote a point or vector in ℝn, q = (q1,…, qn) will denote a non-zero vector in ℤn and

|x| := max{|x1|,…,|xn|} = ‖X‖∞

will denote the height of the vector x. Let Ψ : ℕ → (0, ∞) be a (non-zero) function which converges to 0 at ∞. The notion of a slowly decreasing functionΨ is defined in [3] as a function for which, given c ∈ (0, 1), there exists a K = K(c) > 1 such that Ψ(ck) ≤ KΨ(k). Of course, since Ψ is decreasing, Ψ(k) ≤ Ψ(ck). For any set X, |X| will denote the Lebesgue measure of X (there should be no confusion with the height of a vector).

We study nonlinear Landau–Ginzburg-type equations on the half-line in the critical casewhere β ∈ C, ρ > 2. The linear operator K is a pseudodifferential operator defined by the inverse Laplace transform with dissipative symbol K(p) = αpρ, M = [1/2ρ]. The aim of this paper is to prove the global existence of solutions to the initial–boundary-value problem and to find the main term of the asymptotic representation of solutions in the critical case, when the time decay of the nonlinearity has the same rate as that of the linear part of the equation.

The Ehrhart polynomials for the class of 0-symmetric convex lattice polytopes in Euclidean n-space ℝn are investigated. It turns out that the roots of the Ehrhart polynomial and Minkowski's successive minima of such polytopes are closely related by their geometric and arithmetic means. It is also shown that the roots of the Ehrhart polynomials of lattice n-polytopes with or without interior lattice points differ essentially. Furthermore, the structure of the roots in the planar case is studied. Here it turns out that their distribution reflects basic properties of lattice polygons.

In this paper is considered the average size of the 2-Selmer groups of a class of quadratic twists of each elliptic curve over ℚ with ℚ-torsion group ℤ2 × ℤ2. The existence is shown of a positive proportion of quadratic twists of such a curve, each of which has rank 0 Mordell-Weil group.

Let us consider the set SA(Rn) of rapidly decreasing functions G: Rn → A, where A is a separable C*-algebra. We prove a version of the Calderón–Vaillancourt theorem for pseudodifferential operators acting on SA(Rn) whose symbol is A-valued. Given a skew-symmetric matrix, J, we prove that a pseudodifferential operator that commutes with G(x + JD), G ∈ SA(Rn), is of the form F(x − JD), for F a C∞-function with bounded derivatives of all orders.

In this paper, an improvement of a large sieve type inequality in high dimensions is presented, and its implications on a related problem are discussed.

The following conjecture generalizing the Contraction Mapping Theorem was made by Stein.

Let (X, ρ) be a complete metric space and let ℱ = {T1,…, Tn} be a finite family of self-maps of X. Suppose that there is a constant γ ∈ (0, 1) such that, for any x, y ∈ X, there exists T ∈ ℱ with ρ(T(x), T(y)) ≤ γρ(x, y). Then some composition of members of ℱ has a fixed point.

In this paper this conjecture is disproved, We also show that it does hold for a (continuous) commuting ℱ in the case n = 2. It is conjectured that it holds for commuting ℱ for any n.

We characterize those holomorphic symbols ϕ: D → D for which the induced composition operator Cϕ: Bω → Bμ (respectively, Bω,0 → Bμ,0) is bounded or compact, where D is the unit disc in the complex plane C, ω is a normal function on [0, 1) and μ is a non-negative function on [0, 1) with μ(tn) > 0 for some sequence satisfying limn→∞ tn= 1.

Equifacetal simplices, all of whose codimension one faces are congruent to one another, are studied. It is shown that the isometry group of such a simplex acts transitively on its set of vertices and, as an application, equifacetal simplices are shown to have unique centres. It is conjectured that a simplex with a unique centre must be equifacetal. The notion of the combinatorial type of an equifacetal simplex is introduced and analysed, and all possible combinatorial types of equifacetal simplices are constructed in even dimensions.

Linear stability of an incompressible triple-deck flow over a wall roughness is considered for disturbances of high frequency. The wall roughness consists of two relatively short obstacles placed far apart on an otherwise flat surface. It is shown that the flow is unstable to feedback or global mode disturbances. The feedback loop is formed by algebraically decaying disturbances propagating upstream and weakly growing Tollmien-Schlichting waves travelling downstream and as such represents an interaction between modes from continuous and discrete spectra of the corresponding parallel-flow problem. An example of growth rate calculation for a specific roughness is considered.

Conical shock waves are generated as sharp conical solid projectiles fly supersonically in the air. We study such conical shock waves in steady supersonic flow using an isentropic Euler system. The stability of such attached conical shock waves for non-symmetrical conical projectiles and non-uniform incoming supersonic flow is established. Meanwhile, the existence of the solution to the Euler system with such attached conical shock as a free boundary is also proved for solid projectiles close to a regular solid cone.

This paper treats finite lattice packings Cn + K of n copies of some centrally symmetric convex body K in Ed for large n. Assume that Cn is a subset of a lattice Λ, and ϱ is at least the covering radius; namely, Λ + ϱK covers the space. The parametric density δ(Cn, ϱ) is defined by δ(Cn, ϱ) = n · V(K)/V(convCn + ϱK). It is shown that, if δ(Cn, ϱ) is minimal for n large, then the shape of conv Cn is approximately given by Wulff's condition, well-known from crystallography. Thus maximizing parametric density is equivalent to optimizing a certain Gibbs–Curie energy. It is also proved that, in case of lattice packings of K (allowing any packing lattice), for large n the optimal shape with respect to the parametric density is approximately a Wulff-shape associated to some densest packing lattice of K.

We use Arkhipov's twisting functors to show that the universal enveloping algebra of a semi-simple complex finite-dimensional Lie algebra surjects onto the space of ad-finite endomorphisms of the simple highest weight module $L(\lambda)$, whose highest weight is associated (in the natural way) with a subset of simple roots and a simple root in this subset. This is a new step towards a complete answer to a classical question of Kostant. We also show how one can use the twisting functors to reprove the classical results related to this question.

In this paper we establish a uniform distribution result for the time spent by a billiard particle in the unit square having vertices (0,0), (1,0), (1,1), (0,1), with small triangular pockets of size $\epsilon$ removed from its corners.

Let X and Y be Banach spaces such that each one is isomorphic to a complemented subspace of the other. In 1996, W. T. Gowers solved the Schroeder-Bernstein Problem for Banach spaces by showing that X is not necessarily isomorphic to Y. In this paper, we give suitable conditions on X, Y, the supplemented subspaces of Y in X and of X in Y to yield that X is isomorphic to Y. In other words, we obtain generalizations of Pełczyński's decomposition method via supplemented subspaces. In order to determine all the possible generalizations, we introduce the notion of Mixed Schroeder-Bernstein Quadruples for Banach spaces. Then, we use some Banach spaces constructed by W. T. Gowers and B. Maurey in 1997 to characterize them.

The discrete universality of the derivative and logarithmic derivative of zeta-functions of normalized eigenforms is obtained. This is used to estimate the number of zeros of the derivatives in the critical strip. For the proof the method of functional limit theorems in the sense of weak convergence of probability measures is applied.

Let $\varphi(\cdot)$ denote the Euler function, and let $a>1$ be a fixed integer. We study several divisibility conditions which exhibit typographical similarity with the standard formulation of the Euler theorem, such as $a^n \equiv 1\!\!\!\!\pmod{\varphi(n)}$, and we estimate the number of positive integers $n\le x$ satisfying these conditions.