Necessary and sufficient conditions of some strong optimal solutions to the interval linear programming

This paper considers optimal solutions of general interval linear programming problems. Necessary and sufficient conditions of (A,b)$(\mathbf{A},\mathbf{b})$-strong and (A,b,c)$(\mathbf{A},\mathbf{b},\mathbf{c})$-strong optimal solutions to the interval linear programming with inequality constraints are proposed. The features of the proposed methods are illustrated by some examples.     

Generic global rigidity of body–bar frameworks

A basic geometric question is to determine when a given framework G(p)$G(\mathbf{p})$ is globally rigid in Euclidean space R^d${\mathbb{R}}^d$, where G is a finite graph and p is a configuration of points corresponding to the vertices of G  . G(p)$G(\mathbf{p})$ is globally rigid in  R^d${\mathbb{R}}^d$ if for any other configuration q for G   such that the edge lengths of G(q)$G(\mathbf{q})$ are the same as the corresponding edge lengths of G(p)$G(\mathbf{p})$, then p is congruent to q. A framework G(p)$G(\mathbf{p})$ is redundantly rigid, if it is rigid and it remains rigid after the removal of any edge of G.     

On the classification of projectively flat Finsler metrics with constant flag curvature

In this paper, we study locally projectively flat Finsler metrics with constant flag curvature K. We prove those are totally determined by their behaviors at the origin by solving some nonlinear PDEs. The classifications when K=0$\mathbf{K}=0$, K=−1$\mathbf{K}=-1$ and K=1$\mathbf{K}=1$ are given respectively in an algebraic way. Further, we construct a new projectively flat Finsler metric with flag curvature K=1$\mathbf{K}=1$ determined by a Minkowski norm with double square roots at the origin. As an application of our main theorems, we give the classification of locally projectively flat spherical symmetric Finsler metrics much easier than before.     

On multi-dimensional compressible flows of nematic liquid crystals with large initial energy in a bounded domain

We study the global existence of weak solutions to a multi-dimensional simplified Ericksen–Leslie system for compressible flows of nematic liquid crystals with large initial energy in a bounded domain Ω⊂R^N$\Omega \subset {\mathbb{R}}^N$, where N=2 or 3$N=2\text{or}3$. By exploiting a maximum principle, Nirenberg?s interpolation inequality and a smallness condition imposed on the N  -th component of initial direction field _d0${\mathbf{d}}_0$ to overcome the difficulties induced by the supercritical nonlinearity |∇d|²d${|\mathrm{\nabla }\mathbf{d}|}^2\mathbf{d}$ in the equations of angular momentum, and then adapting a modified three-dimensional approximation scheme and the weak convergence arguments for the compressible Navier–Stokes equations, we establish the global existence of weak solutions to the initial-boundary problem with large initial energy and without any smallness condition on the initial density and velocity.     

Homogenization of periodic semilinear parabolic degenerate PDEs

In this paper a second order semilinear parabolic PDE with rapidly oscillating coefficients is homogenized. The novelty of our result lies in the fact that we allow the second order part of the differential operator to be degenerate in some part of Rd${\mathbf{R}}^d$.     

A Batalin–Vilkovisky algebra structure on the Hochschild cohomology of truncated polynomials

We calculate the Batalin–Vilkovisky, and the induced Gerstenhaber structures of the Hochschild cohomology of the singular R-cochain complex of n-dimensional K-projective spaces, where K=C,H$\mathbf{K}=\mathbb{C},\mathbb{H}$ and R=Z$R=\mathbb{Z}$ and any field. In the special case that K=C$\mathbf{K}=\mathbb{C}$, n=1$n=1$ and R=Z$R=\mathbb{Z}$, we show that this structure cannot be identified with the BV-structure of the integral loop homology of the 2-dimensional sphere computed by L. Menichi, but their induced Gerstenhaber structures can still be identified. Combined with the work of Y. Félix and J. Thomas, the main result of the present paper calculates the BV-structure of the rational loop homology of projective spaces.     

Stable cohomology of spaces of non-singular hypersurfaces

We prove that the rational cohomology of the space _Xd,N$X_{d,N}$ of non-singular complex homogeneous polynomials of degree d in N   variables stabilizes to the cohomology of _GLN(C)${\mathrm{GL}}_N(\mathbf{C})$ for d sufficiently large.     

On Markov operators preserving polynomials

The paper is concerned with a special class of positive linear operators acting on the space C(K)$C(K)$ of all continuous functions defined on a convex compact subset K   of R^d${\mathbf{R}}^d$, d?1$d?1$, having non-empty interior. Actually, this class consists of all positive linear operators T   on C(K)$C(K)$ which leave invariant the polynomials of degree at most 1 and which, in addition, map polynomials into polynomials of the same degree. Among other things, we discuss the existence of such operators in the special case where K is strictly convex by also characterizing them within the class of positive projections. In particular we show that such operators exist if and only if ∂K   is an ellipsoid. Furthermore, a characterization of balls of R^d${\mathbf{R}}^d$ in terms of a special class of them is furnished. Additional results and illustrative examples are presented as well.     

Asymptotic profiles for wave equations with strong damping

We consider the Cauchy problem in Rⁿ${\mathbf{R}}^n$ for strongly damped wave equations. We derive asymptotic profiles of these solutions with weighted L^1,1(Rⁿ)$L^{1,1}({\mathbf{R}}^n)$ data by using a method introduced in [9] and/or [10].     

Recovery of sparsest signals via ℓq-minimization

In this paper, it is proved that every s-sparse vector $\mathbf{x}\in {\mathbb{R}}^n$ can be exactly recovered from the measurement vector $\mathbf{z}=\mathbf{Ax}\in {\mathbb{R}}^m$ via some ${?}^q$-minimization with $0<q?1$, as soon as each s-sparse vector $\mathbf{x}\in {\mathbb{R}}^n$ is uniquely determined by the measurement z. Moreover it is shown that the exponent q in the ${?}^q$-minimization can be so chosen to be about $0.6796\times (1?{\delta }_{2s}(\mathbf{A}))$, where ${\delta }_{2s}(\mathbf{A})$ is the restricted isometry constant of order 2s for the measurement matrix A.     

Generic residual intersections and intersection numbers of movable components

Given a sequence x of elements of a commutative equidimensional noetherian ring R  , cycles _zi(x,R)$z_i(\mathbf{x},R)$ (i∈N$i\in \mathbb{N}$) in the cycle group of polynomial rings over R are defined by generic residual intersections. The study of these cycles gives new insight into the theory for excess intersections in projective space developed by Stückrad and Vogel, in particular concerning the contribution to the intersection cycle of embedded components not defined over the base field.