Dimension polynomials of difference local algebras

We extend the concept of the difference dimension polynomial of a difference field extension to difference local algebras. The main theorem of the paper establishes the existence and form of the dimension polynomial associated with the localization of a finitely generated well-mixed difference algebra at a prime reflexive difference ideal.     

Improved inhomogeneous Strichartz estimates for the Schrödinger equation

We study inhomogeneous Strichartz estimates for the Schrödinger equation for dimension n?3. Using a frequency localization, we obtain some improved range of Strichartz estimates for the solution of inhomogeneous Schrödinger equation except dimension n=3.     

Trees and circle orders

This paper continues a recent resurgence of interest in combinatorial properties of a poset that are associated with graph properties of its cover graph and order diagram. The following two theorems appearing in a 1977 paper of Trotter and Moore have played important roles in motivating this more modern research: (1) The dimension of a poset is at most 3 when its cover graph is at tree; (2) The dimension of a poset is at most 3 when the poset has a zero and its order diagram is planar. Although the underlying ideas lay dormant for more than 30 years, the first of these two results has become the base case for recent results bounding the dimension of a poset in terms of (a) the tree-width of its cover graph, and (b) the maximum dimension of its blocks. The second result is the base case for bounding the dimension of a planar poset in terms of the number of minimal elements. Continuing with this line of research, we show that every poset whose cover graph is a tree is a circle order, i.e., it has a representation as a family of circular disks in the Euclidean plane partially ordered by inclusion.     

On the Dimension of Posets with Cover Graphs of Treewidth 2

In 1977, Trotter and Moore proved that a poset has dimension at most 3 whenever its cover graph is a forest, or equivalently, has treewidth at most 1. On the other hand, a well-known construction of Kelly shows that there are posets of arbitrarily large dimension whose cover graphs have treewidth 3. In this paper we focus on the boundary case of treewidth 2. It was recently shown that the dimension is bounded if the cover graph is outerplanar (Felsner, Trotter, and Wiechert) or if it has pathwidth 2 (Biró, Keller, and Young). This can be interpreted as evidence that the dimension should be bounded more generally when the cover graph has treewidth 2. We show that it is indeed the case: Every such poset has dimension at most 1276.     

关于测度的Hausdorff维数的局部化

主要研究测度的豪斯道夫维数的局部化.通过定义一个测度μx,ε,从而给出dim·Hμ在点x的局部化维数dim·Hμ(x).进而得到局部化维数dim·Hμ(x)与dim·Hμ之间的关系,并得到了一个等式关系.     

NONLINEAR STOCHASTIC HEAT EQUATION DRIVEN BY SPATIALLY COLORED NOISE:MOMENTS AND INTERMITTENCY

In this article, we study the nonlinear stochastic heat equation in the spatial domain R~d subject to a Gaussian noise which is white in time and colored in space. The spatial correlation can be any symmetric, nonnegative and nonnegative-definite function that satisfies Dalang's condition. We establish the existence and uniqueness of a random field solution starting from measure-valued initial data. We find the upper and lower bounds for the second moment. With these moment bounds, we first establish some necessary and sufficient conditions for the phase transition of the moment Lyapunov exponents, which extends the classical results from the stochastic heat equation on Z~d to that on R~d. Then,we prove a localization result for the intermittency fronts, which extends results by Conus and Khoshnevisan [9] from one space dimension to higher space dimension. The linear case has been recently proved by Huang et al [17] using different techniques.     

Asymptotic spectral analysis in semiconductor nanowire heterostructures

Mathematical settings in which heterogeneous structures affect electron transport through a tube-shaped quantum waveguide are studied, highlighting the interaction between material composition and geometric parameters like curvature and torsion. First, the macroscopic behaviour of a nanowire made of composite fibres with microscopic periodic texture is analysed, which amounts to determining the asymptotic behaviour of the spectrum of an elliptic Dirichlet eigenvalue problem with finely oscillating coefficients in a tube with shrinking cross-section. A suitable formal expansion suggests that the effective one-dimensional limit problem is of Sturm–Liouville type and yields the explicit formula for the underlying potential. In the torsion-free case, these findings are made rigorous by performing homogenization and 3d–1d dimension reduction for the two-scale problem in a variational framework by means of Γ-convergence. Second, waveguides with non-oscillating inhomogeneities in the cross-section are investigated. This leads to explicit criteria for propagation and localization of eigenmodes.     

On the dimension of trivariate spline spaces with the highest order smoothness on 3D T-meshes

T-meshes are a type of rectangular partitions of planar domains which allow hanging vertices. Because of the special structure of T-meshes, adaptive local refinement is possible for splines defined on this type of meshes, which provides a solution for the defect of NURBS. In this paper, we generalize the definitions to the three-dimensional (3D) case and discuss a fundamental problem – the dimension of trivariate spline spaces on 3D T-meshes. We focus on a special case where splines are C ^d?1 continuous for degree d. The smoothing cofactor method for trivariate splines is explored for this situation. We obtain a general dimension formula and present lower and upper bounds for the dimension. At last, we introduce a type of 3D T-meshes, where we can give an explicit dimension formula.     

B-splines on 3-D tetrahedron partition in four-directional mesh

It is more difficult to construct 3-D splines than in 2-D case. Some results in the three directional meshes of bivariate case have been extended to 3-D case and corresponding tetrahedron partition has been constructed. The support of related B-splines and their recurrent formulas on integration and differentiation-difference are obtained. The results of this paper can be extended into higher dimension spaces, and can be also used in wavelet analysis, because of the relationship between spline and wavelets.     

Beurling Dimension of Gabor Pseudoframes for Affine Subspaces

Pseudoframes for subspaces have been recently introduced by Li and Ogawa as a tool to analyze lower dimensional data with arbitrary flexibility of both the analyzing and the dual sequence. In this paper we study Gabor pseudoframes for affine subspaces by focusing on geometrical properties of their associated sets of parameters. We first introduce a new notion of Beurling dimension for discrete subsets of ℝ ^d by employing a certain generalized Beurling density. We present several properties of Beurling dimension including a comparison with other notions of dimension showing, for instance, that our notion includes the mass dimension as a special case. Then we prove that Gabor pseudoframes for affine subspaces satisfy a certain Homogeneous Approximation Property, which implies invariance under time–frequency shifts of an approximation by elements from the pseudoframe. The main result of this paper is a classification of Gabor pseudoframes for affine subspaces by means of the Beurling dimension of their sets of parameters. This provides us, in particular, with a Nyquist dimension which separates sets of parameters of pseudoframes from those of non-pseudoframes and which links a fixed value to sets of parameters of pseudo-Riesz sequences. These results are even new for the special case of Gabor frames for an affine subspace.      

Commutative Power-Associative Nilalgebras of Nilindex 5

In this paper we study the structure of commutative power-associative nilalgebras of dimension 8 and nilindex ≤ 5 over a field of characteristic different from 2, 3 and 5. We prove that every algebra in this class verifies the identities x⁴y = 0 and x(x(x(x(xy)))) = 0. In particular, we finish the proof of the Albert’s problem [0] in the following case: every commutative power-associative nilalgebra of dimension ≤ 8 over a field of characteristic ≠ 2, 3 and 5 is solvable. The solvability of these algebras for dimension 4, 5 and 6 were proved in [0], [0] and [0] respectively.     

Metric sparsification and operator norm localization

We study an operator norm localization property and its applications to the coarse Novikov conjecture in operator K-theory. In particular, we introduce a sufficient geometric condition (called metric sparsification) for the operator norm localization property. This is used to give many examples of finitely generated groups with infinite asymptotic dimension and the operator norm localization property. We also show that a sequence of expanding graphs does not possess the operator norm localization property.     

On the existence and uniqueness of smooth solution for a generalized Zakharov equation

The paper deals with the existence and uniqueness of smooth solution for a generalized Zakharov equation. We establish local in time existence and uniqueness in the case of dimension d=2,3. Moreover, by using the conservation laws and Brezis-Gallouet inequality, the solution can be extended globally in time in two dimensional case for small initial data. Besides, we also prove global existence of smooth solution in one spatial dimension without any small assumption for initial data.     

Characterization of Piecewise-Smooth Surfaces Using the 3D Continuous Shearlet Transform

One of the most striking features of the Continuous Shearlet Transform is its ability to precisely characterize the set of singularities of multivariable functions through its decay at fine scales. In dimension n=2, it was previously shown that the continuous shearlet transform provides a precise geometrical characterization for the boundary curves of very general planar regions, and this property sets the groundwork for several successful image processing applications. The generalization of this result to dimension n=3 is highly nontrivial, and so far it was known only for the special case of 3D bounded regions where the boundary set is a smooth 2-dimensional manifold with everywhere positive Gaussian curvature. In this paper, we extend this result to the general case of 3D bounded regions with piecewise-smooth boundaries, and show that also in this general situation the continuous shearlet transform precisely characterizes the geometry of the boundary set.     

Partial regularity for the 3D magneto-hydrodynamics system with hyper-dissipation

We prove that for the 3D MHD equations with hyper-dissipations(-△)~α(1α5/4)the Hausdorff dimension of singular set at the first blowing up time is at most 5-4α,by means of physical and frequency localization,Bony's paraproduct and Littlewood-Paley theory.     

Dimension of the global attractor for the discretized damped sine-Gordon equation

This paper presents a more precise estimate on the dimension of the global attractor for spatially discretized damped sine-Gordon equation with periodic boundary condition. The gained Hausdorff dimension is consistent with the continuous case and conforms to physics.     

On Oka’s extra-zero problem and examples

After the solution of Cousin II problem by Oka III in 1939, he thought an extra-zero problem in 1945 (his posthumous paper) asking if it is possible to solve an arbitrarily given Cousin II problem adding some extra-zeros whose support is disjoint from the given one. By the secondly named author, some special case was affirmatively confirmed in dimension two and a counter-example in dimension three or more was given. The purpose of the present paper is to give a complete solution of this problem with examples and some new questions.     

c-Graded filiform Lie algebras

In this paper we generalize naturally graded filiform Lie algebras as well as filiform Lie algebras admitting a connected gradation of maximal length, by introducing the concept of c-graded complex filiform Lie algebras. We deal with the particular case of 3-graded filiform Lie algebras and we obtain their classification in arbitrary dimension. We finally show a link among derived algebras, graded filiform and rigid solvable Lie algebras.     

Z-eigenvalue methods for a global polynomial optimization problem

As a global polynomial optimization problem, the best rank-one approximation to higher order tensors has extensive engineering and statistical applications. Different from traditional optimization solution methods, in this paper, we propose some Z-eigenvalue methods for solving this problem. We first propose a direct Z-eigenvalue method for this problem when the dimension is two. In multidimensional case, by a conventional descent optimization method, we may find a local minimizer of this problem. Then, by using orthogonal transformations, we convert the underlying supersymmetric tensor to a pseudo-canonical form, which has the same E-eigenvalues and some zero entries. Based upon these, we propose a direct orthogonal transformation Z-eigenvalue method for this problem in the case of order three and dimension three. In the case of order three and higher dimension, we propose a heuristic orthogonal transformation Z-eigenvalue method by improving the local minimum with the lower-dimensional Z-eigenvalue methods, and a heuristic cross-hill Z-eigenvalue method by using the two-dimensional Z-eigenvalue method to find more local minimizers. Numerical experiments show that our methods are efficient and promising. This work is supported by the Research Grant Council of Hong Kong and the Natural Science Foundation of China (Grant No. 10771120).