W1,p-gradient Young measures supported in the set Q2(K) of two-dimensional K-quasiconformal matrices are studied. We prove that these Young measures can be generated by gradients of K-quasiregular mappings. This leads, for example, to the 0-1 law for quasiregular W1,p-gradient Young measures and other quasiregular properties such as higher integrability.

In this paper, we study n-dimensional complete submanifolds with constant scalar curvature in the Euclidean space En+p and n-dimensional compact submanifolds with constant scalar curvature in the unit sphere Sn+p(1). We prove that the totally umbilical sphere Sn(r), totally geodesic Euclidean space En and generalized cylinder Sn−1(c) × E1 are the only n-dimensional (n > 2) complete submanifolds Mn with constant scalar curvature n(n − 1)r in the Euclidean space En+p, which satisfy the following condition: where S denotes the squared norm of the second fundamental form of Mn. For compact submanifolds with constant scalar curvature in the unit sphere Sn+p(1), we also obtain a corresponding result (see theorem 1.3).

The notion of gluing of abelian categories was introduced in a paper by Kazhdan and Laumon in 1988 and studied furtherby Polishchuk. We observe that this notion is a particular case of a general categorical construction.

We then apply this general notion to the study of the ring of global differential operators $\mathcal{D}$ on the basic affinespace $G/U$ (here $G$ is a semi-simple simply connected algebraic group over $\mathbb{C}$ and $U\subset G$ is a maximalunipotent subgroup).

We show that the category of $\mathcal{D}$-modules is glued from $|W|$ copies of the category of $D$-modules on $G/U$ where$W$ is the Weyl group, and the Fourier transform is used to define the gluing data. As an application we prove that thealgebra $\mathcal{D}$ is Noetherian, and get some information on its homological properties.

AMS 2000 Mathematics subject classification: Primary 13N10; 16S32; 17B10; 18C20

The connection between the discrete and the continuous coagulation–fragmentation models is investigated. A weak stability principle relying on a priori estimates and weak compactness in L1 is developed for the continuous model. We approximate the continuous model by a sequence of discrete models and, writing the discrete models as modified continuous ones, we prove the convergence of the latter towards the former with the help of the above-mentioned stability principle. Another application of this stability principle is the convergence of an explicit time and size discretization of the continuous coagulation-fragmentation model.

We use singularity theory to classify forced symmetry-breaking bifurcation problems where f1 is 𝕆(2)-equivariant and f2 is 𝔻n-equivariant with the orthogonal group actions on z ∈ ℝ2. Forced symmetry breaking occurs when the symmetry of the equation changes when parameters are varied. We explicitly apply our results to the branching of subharmonic solutions in a model periodic perturbation of an autonomous equation and sketch further applications.

Convergence theorems for the practical eigenvector free methods of Gay and Goerisch are obtained under a variety of hypotheses, so that our theorems apply to both traditional boundary-value problems and atomic problems. In addition, we prove convergence of the T*T method of Bazley and Fox without an alignment of projections hypothesis required in previous literature.

In this paper, we study n-dimensional complete submanifolds with constant scalar curvature in the Euclidean space En+p and n-dimensional compact submanifolds with constant scalar curvature in the unit sphere Sn+p(1). We prove that the totally umbilical sphere Sn(r), totally geodesic Euclidean space En and generalized cylinder Sn−1(c) × E1 are the only n-dimensional (n > 2) complete submanifolds Mn with constant scalar curvature n(n − 1)r in the Euclidean space En+p, which satisfy the following condition:where S denotes the squared norm of the second fundamental form of Mn. For compact submanifolds with constant scalar curvature in the unit sphere Sn+p(1), we also obtain a corresponding result (see theorem 1.3).

We consider the inverse connection problem consisting of determining a gauge field on $\mathbb{R}^d$ from its non-abelian Radontransform along oriented straight lines. The determination is considered modulo gauge transformations. Our resultsinclude: global uniqueness theorems for $d\geq3$, new local uniqueness theorems for $d=2$, constructive proofs(i.e. proofs containing reconstruction procedures), counterexamples to the global uniqueness for $d=2$, a reduction tothe attenuated X-ray transform.

AMS 2000 Mathematics subject classification: Primary 53C65; 81U40

We introduce the totally multiplicatively prime algebras as those normed algebras for which there exists a positive number K such that K‖F‖‖a‖ ≤ ‖WF,a‖ for all F in M(A) (the multiplication algebra of A) and a in A, where WF,a denotes the operator from M(A) into A defined by WF,a(T) = FT(a) for all T in M(A). These algebras are totally prime and their multiplication algebra is ultraprime. We get the stability of the class of totally multiplicatively prime algebras by taking central closure. We prove that prime H*-algebras are totally multiplicatively prime and that the ℓ1-norm is the only classical norm on the free non-associative algebras for which these are totally multiplicatively prime.

Surfaces arising in amorphous thin-film-growth are often described by certain classes of stochastic PDEs. In this paper we address the question of existence of a solution and statistical quantities (e.g. mean interface width or correlation functions). Moreover, we discuss the approximations of such statistical quantities by the spectral Galerkin method. This is an important question, as the numerical computation of statistical quantities plays a key role in the verification of the models.

We consider a one-dimensional stochastic model of sediment deposition in which the complete time history of sedimentation is the sum of a linear trend and a fractional Brownian motion wH(t) with self-similarity parameter H ∈ (0, 1). The thickness of the sedimentary layer as a function of time, d(t), looks like the Cantor staircase. The Hausdorff dimension of the points of growth of d(t) is found. We obtain the statistical distribution of gaps in the sedimentary record, periods of time during which the sediments have been eroded. These gaps define sedimentary unconformities. In the case H = 1/2 we obtain the statistical distribution of layer thicknesses between unconformities and investigate the multifractality of d(t). We show that the multifractal structures of d(t) and the local time function of Brownian motion are identical; hence d(t) is not a standard multifractal object. It follows that natural statistics based on local estimates of the sedimentation rate produce contradictory estimates of the range of local dimension for d(t). The physical object d(t) is interesting in that it involves the above anomalies, and also in its mechanism of fractality generation, which is different from the traditional multiplicative process.

We consider a mathematical model for the quasi-static deformation of a thinning sheet. The model couples a first-order equation for the thickness of the sheet to a prescribed curvature equation for the displacement of the sheet. We prove a local in time existence and uniqueness theorem for this system when the sheet can be written as a graph. A contact problem is formulated for a sheet constrained to be above a mould. Finally we present some computational results.

The existence of classical solutions with a free boundary changing its topology in time is proved for the multi-dimensional Hele-Shaw problem. We prove that, if some symmetry and monotonicity conditions on the problem data hold, then a cusp, arising at the contact between two free boundary components, disappears instantly and the free boundary becomes smooth and singly-connected.

Image inpainting is an image restoration problem, in which image models play a critical role, as demonstrated by Chan, Kang & Shen's [12] recent inpainting schemes based on the bounded variation and the elastica [11] image models. In this paper, we propose two novel inpainting models based on the Mumford–Shah image model [41], and its high order correction – the Mumford–Shah–Euler image model. We also present their efficient numerical realization based on the Γ-convergence approximations of Ambrosio & Tortorelli [2, 3] and De Giorgi [21].

Let {Av}v∈V be a finite collection of events and G = (V, E) be a chordal graph. Our main result – the chordal graph sieve – is a Bonferroni-type inequality where the selection of intersections in the estimates is determined by a chordal graph G. It interpolates between Boole's inequality (G empty) and the sieve formula (G complete). By varying G, several inequalities both well-known and new are obtained in a concise and unified way.

We show that the limiting distribution of the number of comparisons used by Hoare's quickselect algorithm when given a random permutation of n elements for finding the mth-smallest element, where m = o(n), is the Dickman function. The limiting distribution of the number of exchanges is also derived.

We show that, if G is a graph of order n with maximal degree Δ(G) and minimal degree δ(G) whose complement contains no K2,s, s [ges ] 2, then G contains every tree T of order n−s+1 whose maximal degree is at most Δ(G) and whose vertex of second-largest degree is at most δ(G). We then show that this result implies that special cases of two conjectures are true. We verify that the Erdös–Sós conjecture, which states that a graph whose average degree is larger than k−1 contains every tree of order k+1, is true for graphs whose complement does not contain a K2,4, and the Komlós–Sós conjecture, which states that every graph of median degree at least k contains every tree of order k+1, is true for graphs whose complement does not contain a K2,3.

Consider a finite alphabet Ω and patterns which consist of characters from Ω. For a given pattern w, let cor(w) denote its autocorrelation, which can be seen as a measure of the amount of overlap in w. Letting aw(n) denote the number of strings over Ω of length n which do not contain w as a substring, the main result of this paper is: If cor(w) > cor(w′) then aw(n)−aw′(n) > (|Ω|−1)(aw(n−1)−aw′(n−1)) for n [ges ] N, and the value of N is given. This result confirms a conjecture by Eriksson [2], which was previously proved to be true by Cakir, Chryssaphinou and Månsson [1] when |Ω| [ges ] 3.

In this note it is shown that resistance and resistance dimension are rough isometry invariant.

Consider the class of graphs on n vertices which have maximum degree at most 1/2n−1+τ, where τ [ges ] −n1/2+ε for sufficiently small ε > 0. We find an asymptotic formula for the number of such graphs and show that their number of edges has a normal distribution whose parameters we determine. We also show that expectations of random variables on the degree sequences of such graphs can often be estimated using a model based on truncated binomial distributions.