On (k,\varepsilon)-saddle submanifolds of Riemannian manifolds

The paper continues our (in collaboration with A. Borisenko [J. Differential Geom. Appl. 20 p., to appear]) discovery of the new classes of $(k,\varepsilon)The paper continues our (in collaboration with A. Borisenko [J. Differential Geom. Appl. 20 p., to appear]) discovery of the new classes of -saddle, -parabolic, and -convex submanifolds ( ). These are defined in terms of the eigenvalues of the 2nd fundamental forms of each unit normal of the submanifold, extending the notion of k-saddle, k-parabolic, k-convex submanifolds ( ). It follows that the definition of -saddle submanifolds is equivalent to the existence of -asymptotic subspaces in the tangent space. We prove non-embedding theorems of -saddle submanifolds, theorems about 1-connectedness and homology groups of these submanifolds in Riemannian spaces of positive (sectional or qth Ricci) curvature, in particular, spherical and projective spaces. We apply these results to submanifolds with ‘small’ normal curvature, , and for submanifolds with extrinsic curvature (resp., non-positive) and small codimension, and include some illustrative examples. The results of the paper generalize theorems about totally geodesic, minimal and k-saddle submanifolds by Frankel; Borisenko, Rabelo and Tenenblat; Kenmotsu and Xia; Mendon?a and Zhou.      

Convex sets with semidefinite representation

We provide a sufficient condition on a class of compact basic semialgebraic sets for their convex hull co(K) to have a semidefinite representation (SDr). This SDr is explicitly expressed in terms of the polynomials g _j that define K. Examples are provided. We also provide an approximate SDr; that is, for every fixed , there is a convex set such that (where B is the unit ball of ), and has an explicit SDr in terms of the g _j ’s. For convex and compact basic semi-algebraic sets K defined by concave polynomials, we provide a simpler explicit SDr when the nonnegative Lagrangian L _f associated with K and any linear is a sum of squares. We also provide an approximate SDr specific to the convex case.      

Projective re-normalization for improving the behavior of a homogeneous conic linear system

In this paper we study the homogeneous conic system . We choose a point that serves as a normalizer and consider computational properties of the normalized system . We show that the computational complexity of solving F via an interior-point method depends only on the complexity value of the barrier for C and on the symmetry of the origin in the image set , where the symmetry of 0 in is

On Approximate Efficiency in Multiobjective Programming

This paper is focused on approximate ( -efficient) solutions of multiobjective mathematical programs. We introduce a new -efficiency concept which extends and unifies different notions of approximate solution defined in the literature. We characterize these -efficient solutions in convex multiobjective programs through approximate solutions of linear scalarizations, which allow us to obtain parametric representations of different -efficiency sets. Several classical -efficiency notions are considered in order to show the concepts introduced and the results obtained.This research was partially supported by Ministerio de Ciencia y Tecnología (Spain), project BFM2003-02194.     

Laguerre geometry of hypersurfaces in $$\mathbb{R}^{n}$$

Laguerre geometry of surfaces in is given in the book of Blaschke [Vorlesungen über Differentialgeometrie, Springer, Berlin Heidelberg New York (1929)], and has been studied by Musso and Nicolodi [Trans. Am. Math. soc. 348, 4321–4337 (1996); Abh. Math. Sem. Univ. Hamburg 69, 123–138 (1999); Int. J. Math. 11(7), 911–924 (2000)], Palmer [Remarks on a variation problem in Laguerre geometry. Rendiconti di Mathematica, Serie VII, Roma, vol. 19, pp. 281–293 (1999)] and other authors. In this paper we study Laguerre differential geometry of hypersurfaces in . For any umbilical free hypersurface with non-zero principal curvatures we define a Laguerre invariant metric g on M and a Laguerre invariant self-adjoint operator : TM → TM, and show that is a complete Laguerre invariant system for hypersurfaces in with n≥ 4. We calculate the Euler–Lagrange equation for the Laguerre volume functional of Laguerre metric by using Laguerre invariants. Using the Euclidean space , the semi-Euclidean space and the degenerate space we define three Laguerre space forms , and and define the Laguerre embeddings and , analogously to what happens in the Moebius geometry where we have Moebius space forms S ⁿ , and (spaces of constant curvature) and conformal embeddings and [cf. Liu et al. in Tohoku Math. J. 53, 553–569 (2001) and Wang in Manuscr. Math. 96, 517–534 (1998)]. Using these Laguerre embeddings we can unify the Laguerre geometry of hypersurfaces in , and . As an example we show that minimal surfaces in or are Laguerre minimal in .C. Wang Partially supported by RFDP and Chuang-Xin-Qun-Ti of NSFC.     

Gröbner bases in Asymptotic Analysis of Perturbed Polynomial Programs

We consider a perturbed mathematical programming problem where both the objective and the constraint functions are polynomial in all underlying decision variables and in the perturbation parameter $\varepsilon.We consider a perturbed mathematical programming problem where both the objective and the constraint functions are polynomial in all underlying decision variables and in the perturbation parameter We study the behaviour of the solutions of such a perturbed problem as Though the solutions of the programming problems are real, we consider the Karush–Kuhn–Tucker optimality system as a one-dimensional complex algebraic variety in a multi-dimensional complex space. We use the Buchberger’s elimination algorithm of the Gr?bner bases theory to replace the defining equations of the variety by its Gr?bner basis, that has the property that one of its elements is bivariate, that is, a polynomial in and one of the decision variables only. Changing the elimination order in the Buchberger’s algorithm, we obtain such a bivariate polynomial for each of the decision variables. Thus, the solutions of the original system reduces to a number of algebraic functions in that can be represented as a Puiseux series in a neighbourhood of . A detailed analysis of the branching order and the order of the pole is also provided. The latter is estimated via characteristics of these bivariate polynomials of Gr?bner bases.This research was supported by a grant from the Australian Research Council no. DP0343028. We are indebted to K. Avrachenkov, P. Howlett, and V. Gaitsgory for many helpful discussions.     

Corrector–predictor methods for monotone linear complementarity problems in a wide neighborhood of the central path

Two corrector–predictor interior point algorithms are proposed for solving monotone linear complementarity problems. The algorithms produce a sequence of iterates in the neighborhood of the central path. The first algorithm uses line search schemes requiring the solution of higher order polynomial equations in one variable, while the line search procedures of the second algorithm can be implemented in arithmetic operations, where n is the dimension of the problems, is a constant, and m is the maximum order of the predictor and the corrector. If then both algorithms have iteration complexity. They are superlinearly convergent even for degenerate problems.      

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.     

An inexact primal–dual path following algorithm for convex quadratic SDP

We propose primal–dual path-following Mehrotra-type predictor–corrector methods for solving convex quadratic semidefinite programming (QSDP) problems of the form: , where is a self-adjoint positive semidefinite linear operator on , b∈R ^m , and is a linear map from to R ^m . At each interior-point iteration, the search direction is computed from a dense symmetric indefinite linear system (called the augmented equation) of dimension m + n(n + 1)/2. Such linear systems are typically very large and can only be solved by iterative methods. We propose three classes of preconditioners for the augmented equation, and show that the corresponding preconditioned matrices have favorable asymptotic eigenvalue distributions for fast convergence under suitable nondegeneracy assumptions. Numerical experiments on a variety of QSDPs with n up to 1600 are performed and the computational results show that our methods are efficient and robust. Research supported in part by Academic Research Grant R146-000-076-112.     

Extremal stochastic integrals: a parallel between max-stable processes and α-stable processes

We construct extremal stochastic integrals of a deterministic function with respect to a random Fréchet () sup-measure. The measure is sup-additive rather than additive and is defined over a general measure space , where is a deterministic control measure. The extremal integral is constructed in a way similar to the usual stable integral, but with the maxima replacing the operation of summation. It is well-defined for arbitrary , and the metric metrizes the convergence in probability of the resulting integrals.This approach complements the well-known de Haan's spectral representation of max-stable processes with Fréchet marginals. De Haan's representation can be viewed as the max-stable analog of the LePage series representation of stable processes, whereas the extremal integrals correspond to the usual stable stochastic integrals. We prove that essentially any strictly stable process belongs to the domain of max-stable attraction of an Fréchet, max-stable process. Moreover, we express the corresponding Fréchet processes in terms of extremal stochastic integrals, involving the kernel function of the stable process. The close correspondence between the max-stable and stable frameworks yields new examples of max-stable processes with non-trivial dependence structures.This research was partially supported by a fellowship of the Horace H. Rackham School of Graduate Studies at the University of Michigan and the NSF Grant DMS-0505747 at Boston University.     

On the complexity of general matrix scaling and entropy minimization via the RAS algorithm

Given an n × m nonnegative matrix A = (a _ij ) and positive integral vectors and having a common one-norm h, the (r,c)-scaling problem is to obtain positive diagonal matrices X and Y, if they exist, such that XAY has row and column sums equal to r and c, respectively. The entropy minimization problem corresponding to A is to find an n × m matrix z = (z _ij ) having the same zero pattern as A, the sum of whose entries is a given number h, its row and column sums are within given integral vectors of lower and upper bounds, and such that the entropy function consisting of the sum of the terms z _ij ln (z _ij /a _ij ) is minimized. When the lower and upper bounds coincide, matrix scaling and entropy minimization are closely related. In this paper we present several complexity bounds for the -approximate (r,c)-scaling problem, polynomial in n,m,h, , and ln , where V and v are the largest and the smallest positive entries of A, respectively. These bounds, although not polynomial in , not only provide alternative complexities for the polynomial time algorithms, but could result in better overall complexities. In particular, our theoretical analysis supports the practicality of the well-known RAS algorithm. In our analysis we obtain bounds on the norm of scaling vectors which will be used in deriving not only some of the above complexities, but also a complexity for square nonnegative matrices having positive permanent. In particular, our results extend, nontrivially, many bounds for the doubly stochastic scaling of square nonnegative matrices previously given by Kalantari and Khachiyan to the case of general (r,c)-scaling. Finally, we study a more general entropy minimization problem where row and column sums are constrained to lie in prescribed intervals, and the sum of all entries is also prescribed. Balinski and Demange described an RAS type algorithm for its continuous version, but did not analyze its complexity. We show that this algorithm produces an -approximate solution within complexity polynomial in n, m, h, and .     

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.     

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].     

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 .     

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)].      

Dy namic of Abelia n Subgroups of GL( n, $$\mathbb{C}$$): A Structure Theorem

In this paper, we characterize the dynamic of every Abelian subgroups of , or . We show that there exists a -invariant, dense open set U in saturated by minimal orbits with a union of at most n -invariant vector subspaces of of dimension n−1 or n−2 over . As a consequence, has height at most n and in particular it admits a minimal set in . This work is supported by the research unit: systèmes dynamiques et combinatoire: 99UR15-15     

A note on simultaneous Diophantine approximation on planar curves

Let denote the set of simultaneously - approximable points in and denote the set of multiplicatively ψ-approximable points in . Let be a manifold in . The aim is to develop a metric theory for the sets and analogous to the classical theory in which is simply . In this note, we mainly restrict our attention to the case that is a planar curve . A complete Hausdorff dimension theory is established for the sets and . A divergent Khintchine type result is obtained for ; i.e. if a certain sum diverges then the one-dimensional Lebesgue measure on of is full. Furthermore, in the case that is a rational quadric the convergent Khintchine type result is obtained for both types of approximation. Our results for naturally generalize the dimension and Lebesgue measure statements of Beresnevich et al. (Mem AMS, 179 (846), 1–91 (2006)). Moreover, within the multiplicative framework, our results for constitute the first of their type. The research of Victor V. Beresnevich was supported by an EPSRC Grant R90727/01. Sanju L. Velani is a Royal Society University Research Fellow. For Iona and Ayesha on No. 3.     

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).     

Deep Matrix Algebras of Finite Type

Deep matrix algebras based on a set over a ring are introduced and studied by McCrimmon when has infinite cardinality. Here, we construct a new -module that is faithful for of any cardinality. For a field of arbitrary characteristic and , is studied in depth. The algebra is shown to possess a unique proper non-zero ideal, isomorphic to . This leads to a new and interesting simple algebra, , the quotient of by its unique ideal. Both and the quotient algebra are shown to have centers isomorphic to . Via the new faithful representation, all automorphisms of are shown to be inner in the sense of Definition 18.Presented by D. Passman.     

On the minimality of the p-harmonic ma p $$x/\|x\|$$ for weighted energy

In this paper, we study the minimality of the map for the weighted energy functional , where is a continuous function. We prove that for any integer and any non-negative, non-decreasing continuous function f, the map minimizes E _f,p among the maps in which coincide with on . The case p = 1 has been already studied in [Bourgoin J.-C. Calc. Var. (to appear)]. Then, we extend results of Hong (see Ann. Inst. Poincaré Anal. Non-linéaire 17: 35–46 (2000)). Indeed, under the same assumptions for the function f, we prove that in dimension n ≥  7 for any real with , the map minimizes E _f,p among the maps in which coincide with on .