A note on meromorphic convexity

We give examples of Stein domains D in C ² such that C ² ? D is either completely pluripolar or union of germs of (principal) hypersurfaces not intersecting D such that D fails to be meromorphically convex in C ². © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

Stack structures on GIT quotients parametrizing hypersurfaces

We suggest to endow Mumford's GIT quotient scheme with a stack structure, by replacing Proj(?) of the invariant ring with its stack theoretic analogue. We analyse the stacks resulting in this way from classically studied invariant rings, and in particular for binary forms of low degree. Our viewpoint is that the stack structure carries interesting geometric information that is intrinsically present in the invariant ring, but lost when passing to its Proj(?). © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

On the number of groups in clustering

Clustering is the problem of partitioning data into a finite number k of homogeneous and separate groups, called clusters. A good choice of k is essential for building meaningful clusters. In this paper, this task is addressed from the point of view of model selection via penalization. We design an appropriate penalty shape and derive an associated oracle-type inequality. The method is illustrated on both simulated and real-life data sets.     

Analytic and numerical analysis of some statistical features of fragmentation processes

In this work, we investigate the following inhomogeneous model of fragmentation with an exponential splitting probability: suppose that an initial particle having a mass x₀>0 (large enough) is broken, with a size-dependent probability , into b>1 pieces. With probability 1−p(x₀), this fragment is left unchanged for ever. The process of fragmentation is repeated recursively for all pieces.Some statistical features are studied, chiefly the fragment size distribution, number of fragments and partition function of the random fragmentation tree. Also, we give details of the numerical analysis of these features.     

CERES in higher-order logic

We define a generalization of the first-order cut-elimination method CERES to higher-order logic. At the core of lies the computation of an (unsatisfiable) set of sequents (the characteristic sequent set) from a proof π of a sequent S. A refutation of in a higher-order resolution calculus can be used to transform cut-free parts of π (the proof projections) into a cut-free proof of S. An example illustrates the method and shows that can produce meaningful cut-free proofs in mathematics that traditional cut-elimination methods cannot reach.     

Logic of secrets in collaboration networks

The article proposes Logic of Secrets in Collaboration Networks, a formal logical system for reasoning about a set of secrets established over a fixed configuration of communication channels. The system’s key feature, a multi-channel relation called independence, is a generalization of a two-channel relation known in the literature as nondeducibility. The main result is the completeness of the proposed system with respect to a semantics of secrets.     

On open problems concerning distributional chaos for triangular maps

We show that in the class T of the triangular maps (x,y)?(f(x),g_x(y)) of the square there is a map of type 2^∞ with non-minimal recurrent points which is not DC3. We also show that every DC1 continuous map of a compact metric space has a trajectory which cannot be (weakly) approximated by trajectories of compact periodic sets. These two results make possible to answer some open questions concerning classification of maps in T with zero topological entropy, and contribute to an old problem formulated by A.N. Sharkovsky.     

Positive solutions of systems of singular Hammerstein integral equations with applications to semilinear elliptic equations in annuli

The existence of multiple positive solutions of systems of singular Hammerstein integral equations is studied, where the nonlinearities involved are allowed to have singularities in their second variables and satisfy weaker conditions involving the first eigenvalues of the corresponding linear Hammerstein integral operators. Such systems contain some mathematical models arising in science and engineering. Applications are given to the existence of multiple positive radial solutions of systems of semilinear singular elliptic equations in annuli on which, to the best of our knowledge, there has been little study.     

A Reilly inequality for the first Steklov eigenvalue

Let M be a compact submanifold with boundary of a Euclidean space or a Sphere. In this paper, we derive an upper bound for the first non-zero eigenvalue p₁ of Steklov problem on M in terms of the r-th mean curvatures of its boundary ∂M. The upper bound obtained is sharp.     

On definability of types of finite Cantor‐Bendixson rank

We prove that every type of finite Cantor‐Bendixson rank over a model of a first‐order theory without the strict order property is definable and has a unique nonforking extension to a global type. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim