A global minimization algorithm for Lipschitz functions

The global optimization problem with and f(x) satisfying the Lipschitz condition , is considered. To solve it a region-search algorithm is introduced. This combines a local minimum algorithm with a procedure that at the ith iteration finds a region S _i where the global minimum has to be searched for. Specifically, by making use of the Lipschitz condition, S _i , which is a sequence of intervals, is constructed by leaving out from S _i-1 an interval where the global minimum cannot be located. A convergence property of the algorithm is given. Further, the ratio between the measure of the initial feasible region and that of the unexplored region may be used as stop rule. Numerical experiments are carried out; these show that the algorithm works well in finding and reducing the measure of the unexplored region.     

On formulas for the index of the circular distributions

A circular distribution is a Galois equivariant map ψ from the roots of unity μ _∞ to an algebraic closure of such that ψ satisfies product conditions, for ϵ ∈ μ _∞ and , and congruence conditions for each prime number l and with (l, s) = 1, modulo primes over l for all , where μ _l and μ _s denote respectively the sets of lth and sth roots of unity. For such ψ, let be the group generated over by and let be , where U _s denotes the global units of . We give formulas for the indices and of and inside the circular numbers P _s and units C _s of Sinnott over . This work was supported by the SRC Program of Korea Science and Engineering Foundation (KOSEF) grant funded by the Korea government (MOST) (No. R11-2007-035-01001-0). This work was supported by the Korea Research Foundation Grant funded by the Korean Government (MOEHRD, Basic Research Promotion Fund) (KRF-2006-312-C00455).     

A priori bounds and a Liouville theorem on a half-space for higher-order elliptic Dirichlet problems

We consider the 2m-th order elliptic boundary value problem Lu = f (x, u) on a bounded smooth domain with Dirichlet boundary conditions on ∂Ω. The operator L is a uniformly elliptic operator of order 2m given by . For the nonlinearity we assume that , where are positive functions and q > 1 if N ≤ 2m, if N > 2m. We prove a priori bounds, i.e, we show that for every solution u, where C > 0 is a constant. The solutions are allowed to be sign-changing. The proof is done by a blow-up argument which relies on the following new Liouville-type theorem on a half-space: if u is a classical, bounded, non-negative solution of ( − Δ) ^m u  =  u ^q in with Dirichlet boundary conditions on and q > 1 if N ≤ 2m, if N > 2m then .      

Solutions of elliptic problems with nonlinearities of linear growth

In this paper, we study existence of nontrivial solutions to the elliptic equation

A representation of bivariate extreme value distributions via norms on \mathbb{R}^{2}

It is known that a bivariate extreme value distribution (EVD) with reverse exponential margins can be represented as , , where is a suitable norm on . We prove in this paper the converse implication, i.e., given an arbitrary norm on , , , defines an EVD with reverse exponential margins, if and only if the norm satisfies for the condition . This result is extended to bivariate EVDs with arbitrary margins as well as to extreme value copulas. By identifying an EVD , , with the unit ball corresponding to the generating norm , we obtain a characterization of the class of EVDs in terms of compact and convex subsets of .     

Strong majorization in a free ✱-algebra

We study, in the spirit of modern real algebra, the interplay between left ideals of the free ∗-algebra with n generators, and their suitably defined zero sets; and similarly between quadratic submodules of and their positivity sets. Partially supported by grants from the National Science Foundation and the Ford Motor Co.     

Quasi-locally finite polynomial endomorphisms

If F is a polynomial endomorphism of , let denote the field of rational functions such that . We will say that F is quasi-locally finite if there exists a nonzero such that p(F) = 0. This terminology comes out from the fact that this definition is less restrictive than the one of locally finite endomorphisms made in Furter, Maubach (J Pure Appl Algebra 211(2):445–458, 2007). Indeed, F is called locally finite if there exists a nonzero such that p(F) = 0. In the present paper, we show that F is quasi-locally finite if and only if for each the sequence is a linear recurrent sequence. Therefore, this notion is in some sense natural. We also give a few basic results on such endomorphisms. For example: they satisfy the Jacobian conjecture.     

Bounds for a constrained optimal stopping problem

In this short letter, we present an explicit upper bound for the optimal value of a bidimensional optimal stopping problem over stopping times τ subject to a constraint , where x(.) is a geometric Brownian motion coupled with an arbitrary diffusion process y(.), θ(., .) and c(.) are given positive, continuous functions and β > 0 is a fixed constant. The present result is derived from a corresponding Lagrangian dual problem, and using a recent result of Makasu (Seq Anal 27:435–440, 2008). Examples are given to illustrate our main result. Partial results of this note were obtained when the author was holding a postdoc grant PRO12/1003 at the Mathematics Institute, University of Oslo, Norway.     

Multi-bump Bound States of Schrödinger Equations with a Critical Frequency

We consider existence and qualitative properties of standing wave solutions $\Psi(x,t) = e^{-iEt/h}u(x)We consider existence and qualitative properties of standing wave solutions to the nonlinear Schr?dinger equation with E being a critical frequency in the sense that inf . We verify that if the zero set of W − E has several isolated points x _i () near which W − E is almost exponentially flat with approximately the same behavior, then for h > 0 small enough, there exists, for any integer k, , a standing wave solution which concentrates simultaneously on , where is any given subset of . This generalizes the result of Byeon and Wang in 3 (Arch Rat Mech Anal 165: 295–316, 2002).Supported by the Alexander von Humboldt foundation and NSFC(No:10571069).     

Bubbling solutions for an anisotropic Emden–Fowler equation

We consider the following anisotropic Emden–Fowler equation where is a bounded smooth domain and a(x) is a positive smooth function. We investigate the effect of anisotropic coefficient a(x) on the existence of bubbling solutions. We show that at given local maximum points of a(x), there exists arbitrarily many bubbles. As a consequence, the quantity can approach to as . These results show a striking difference with the isotropic case [ Constant].     

Necessary and Sufficient Conditions for the Solvability of the L p Dirichlet Problem on Lipschitz Domains

We study the homogeneous elliptic systems of order $2\ellWe study the homogeneous elliptic systems of order with real constant coefficients on Lipschitz domains in, . For any fixed p > 2, we show that a reverse H?lder condition with exponent p is necessary and sufficient for the solvability of the Dirichlet problem with boundary data in L ^p . We also obtain a simple sufficient condition. As a consequence, we establish the solvability of the L ^p Dirichlet problem for and . The range of p is known to be sharp if and . For the polyharmonic equation, the sharp range of p is also found in the case n = 6, 7 if , and if .Research supported in part by the NSF.     

Rigidity and holomorphic Segre transversality for holomorphic Segre maps

Let and denote the complexifications of Heisenberg hypersurfaces in and , respectively. We show that non-degenerate holomorphic Segre mappings from into with possess a partial rigidity property. As an application, we prove that the holomorphic Segre non-transversality for a holomorphic Segre map from into with propagates along Segre varieties. We also give an example showing that this propagation property of holomorphic Segre transversality fails when N > 2n − 2.     

Hausdorff Dimension of Ruptures for Solutions of a Semilinear Elliptic Equation with Singular Nonlinearity

We consider the following semilinear elliptic equation with singular nonlinearity:

Brake orbits type solutions to some class of semilinear elliptic equations

We consider a class of semilinear elliptic equations of the form

Finite groups with minimal 1-PIM

Let be a field of characteristic and let G be a finite group. It is well-known that the dimension of the minimal projective cover (the so-called 1-PIM) of the trivial left -module is a multiple of the -part of the order of G. In this note we study finite groups G satisfying . In particular, we classify the non-abelian finite simple groups G and primes satisfying this identity (Theorem A). As a consequence we show that finite soluble groups are precisely those finite groups which satisfy this identity for all prime numbers (Corollary B). Another consequence is the fact that the validity of this identity for a finite group G and for a small prime number implies the existence of an -Hall subgroup for G (Theorem C). An important tool in our proofs is the super-multiplicativity of the dimension of the 1-PIM over short exact sequences (Proposition 2.2).     

An extension of Takahashi theorem for the linearized operators of the higher order mean curvatures

We study hypersurfaces in Euclidean space whose position vector x satisfies the condition L _k x = Ax + b, where L _k is the linearized operator of the (k + 1)th mean curvature of the hypersurface for a fixed , is a constant matrix and is a constant vector. For every k, we prove that the only hypersurfaces satisfying that condition are hypersurfaces with zero (k + 1)th mean curvature and open pieces of round hyperspheres and generalized right spherical cylinders of the form , with . This extends a previous classification for hypersurfaces in satisfying , where is the Laplacian operator of the hypersurface, given independently by Hasanis and Vlachos [J. Austral. Math. Soc. Ser. A 53, 377–384 (1991) and Chen and Petrovic [Bull. Austral. Math. Soc. 44, 117–129 (1991)].      

Reconstruction and Subgaussian Operators in Asymptotic Geometric Analysis

We present a randomized method to approximate any vector from a set . The data one is given is the set T, vectors of and k scalar products , where are i.i.d. isotropic subgaussian random vectors in , and . We show that with high probability, any for which is close to the data vector will be a good approximation of , and that the degree of approximation is determined by a natural geometric parameter associated with the set T. We also investigate a random method to identify exactly any vector which has a relatively short support using linear subgaussian measurements as above. It turns out that our analysis, when applied to {−1, 1}-valued vectors with i.i.d. symmetric entries, yields new information on the geometry of faces of a random {−1, 1}-polytope; we show that a k- dimensional random {−1, 1}-polytope with n vertices is m-neighborly for The proofs are based on new estimates on the behavior of the empirical process when F is a subset of the L ₂ sphere. The estimates are given in terms of the γ ₂ functional with respect to the ψ ₂ metric on F, and hold both in exponential probability and in expectation. Received: November 2005, Revision: May 2006, Accepted: June 2006     

Characterizations of simultaneous farthest point in normed linear spaces with applications

In this paper, we consider a problem of best approximation (simultaneous farthest point) for bounded sets in a real normed linear space X. We study simultaneous farthest point in X by elements of bounded sets, and present various characterizations of simultaneous farthest point of elements by bounded sets in terms of the extremal points of the closed unit ball of X ^*, where X ^* is the dual space of X. We establish the characterizations of simultaneous farthest points for bounded sets in , the space of all real-valued continuous functions on a compact topological space Q endowed with the usual operations and with the norm . It is important to state clearly that the contribution of this paper in relation with the previous works (see, for example, [9, Theorem 1.13]) is a technical method to represent the distance from a bounded set to a compact convex set in X which specifically concentrates on the Hahn-Banach Theorem in X.     

The integral monodromy of hyperelliptic and trielliptic curves

We compute the and monodromy of every irreducible component of the moduli spaces of hyperelliptic and trielliptic curves. In particular, we provide a proof that the monodromy of the moduli space of hyperelliptic curves of genus g is the symplectic group . We prove that the monodromy of the moduli space of trielliptic curves with signature (r,s) is the special unitary group . Rachel Pries was partially supported by NSF grant DMS-04-00461.     

Weak convergence of diffusion processes on Wiener space

Let γ be a Gaussian measure on a Suslin space X, H be the corresponding Cameron–Martin space and {e _i } ⊂ H be an orthonormal basis of H. Suppose that μ _n = ρ _n · γ is a sequence of probability measures which converges weakly to a probability measure μ = ρ · γ Consider a sequence of Dirichlet forms , where and . We prove some sufficient conditions for Mosco convergence where . In particular, if X is a Hilbert space, and can be uniformly approximated by finite dimensional conditional expectations for every fixed e _i , then under broad assumptions Mosco and the distributions of the associated stochastic processes converge weakly.