An Integral Representation Formula for Systems of Convolution Equations

AMS(MOS): 46F10

We give an integral representation formula for entire holomorphic solutions of exponential type to systems of convolution equations.     

一元向量有理插值的误差公式

Ｇｒａｖｅｓ－Ｍｏｒｒｉｓ于１９８３年利用向量的Ｓａｍｅｌｓｏｎ逆变换建立了一种实用的向量有理插值方法。本文得到了该向量有理插值的一个精确的误差公式。     

二元Thile型向量有理插值的误差公式

借助于Ｓｏｍｅｌｓｏｎ广义逆，文［１］首次讨论了多元向量有理插值问题．本文得到了二元Ｔｈｉｅｌｅ型向量有理插值的一个精确的误差公式．     

Lax Representation for a Triplet of Scalar Fields

We construct a 3×3 matrix zero-curvature representation for the system of three two-dimensional relativistically invariant scalar fields. This system belongs to the class described by the Lagrangian L = [g _ij(u)u _x ⁱ u _t ^j]/2+f(u), where g _ij is the metric tensor of a three-dimensional reducible Riemannian space. We previously found all systems of this class that have higher polynomial symmetries of the orders 2, 3, 4, or 5. In this paper, we find a zero-curvature representation for one of these systems. The calculation is based on the analysis of an evolutionary system u _t = S(u), where S is one of the higher symmetries. This approach can also be applied to other hyperbolic systems. We also find recursion relations for a sequence of conserved currents of the triplet of scalar fields under consideration.     

On Certain Commutator Estimates for Vector Fields

A unifying approach for proving certain commutator estimates involving smooth, not-necessarily divergence-free vector fields is introduced and implemented in the scales of weighted Triebel–Lizorkin and Besov spaces and certain variable exponent Triebel–Lizorkin and Besov spaces. Such commutator estimates are motivated by the study of well-posedness results for some models in incompressible fluid mechanics.     

Improving the Reconstruction of Vector Fields Using Mixed Finite Element Methods and Optimal Preconditioning

In this article, we study numerically a diagnostic model, based on mass conservation, to recover solenoidal vector fields from experimental data. Based on a reformulation of the mathematical model as a saddle‐point problem, we introduce an iterative preconditioned conjugate gradient algorithm, applied to an associated operator equation of elliptic type, to solve the problem. To obtain a stable algorithm, we use a second‐order mixed finite element approximation for discretization. We show, using synthetic vector fields, that this new approach, yields very accurate solutions at a low computational cost compared to traditional methods with the same order of approximation. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1137–1154, 2016     

The Plancherel Formula for an Inhomogeneous Vector Group

Journal of Fourier Analysis and Applications - We give a concrete realization of the Plancherel measure for a semi-direct product $$N rtimes H$$ where N and H are vector groups for which the...     

Sparse Data Representation of Random Fields

Mathematical models with uncertainties are often described by stochastic partial differential equations (SPDEs) with multiplicative noise. The coefficients, the right-hand side, the boundary conditions are modelled by random fields. As a result the solution is also a random field. We offer to use the Karhunen-Loève expansion (KLE) to compute a sparse data format for the fast generation and representation of these random fields. The KLE of a random field requires the solution of a large eigenvalue problem. Usually it is solved by a Krylov subspace method with a sparse matrix approximation. We demonstrate the use of both, the sparse hierarchical matrix format as well as the low-rank Kronecker tensor format. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Poisson Summation Formula for Extensions of Number Fields

A formula of Poisson summation type is established for sumsinvolving relative norms in an extension of number fields.     

Further Reduction of Normal Forms for Vector Fields

We study Lie's method in the way of Ushiki for further reduction of normal forms for vector fields with singularity at the origin. We give further reduction of normal forms in two typical cases for vector fields in dimension 2: one with a rotation as its linear part and the other with a nilpotent linear part.     

On Sums of Vector Fields

We discuss one case where the integration of a sum of vector fields is reducible to the integration of the summands. Applications include the construction of a class of additive group actions on affine space and a proof that these are stably tame, and also the explicit solution of a class of differential equations from mathematical biology.     

有限域上一类方程组解数的直接公式

本文给出有限域F=F_q(q=p~f,f≥1,p是一个奇素数)上一类方程组∑_(i=s_(r-1)+1~(s_r)∑_(j=1)~(m_i-m_(i-1))a_(m_(i-1)+j)x_1~(d_m(i-1)+j,1)…x_(n_i)~d_(m_(i-1)+j,n_i)=b_r,r=1,…,k当指数满足一定条件时,在F~(n_s_k)上解数的一个直接公式,这里d_(ij)＞0,a_i∈F~*,b_i∈F,0= s_0＜s_1＜…＜s_k,0=m_0＜m_1＜…＜m_(s_k),0=n_0＜n_1＜…＜n_(s_k), m_1≤n_1,…,m_(s_k)≤n_(s_k)．     

New Representation Formula for Convex Functions on Separable Normed Spaces

We derive a new representation formula for lower-semicontinuous convex functions on separable normed spaces. As a consequence of this formula, we obtain a C -approximation method for convex functions which are not necessarily differentiable.     

关于算子半群的一个一般概率表示公式

In this paper we present a general probabilistic representation theorem for semigroups of operators which essentially comprises almost all known representation theorems of this kind. Especially, so-called first and second main theorems turn out to be special cases of the same general formula. The general theorem also permits the calculation of rates of convergence involving the second modulus of continuity.     

Harmonic and Minimal Radial Vector Fields

We present new examples of harmonic and minimal unit vector fields. These are radial vector fields on tubular neighbourhoods about points and submanifolds in two-point homogeneous spaces and harmonic manifolds, and about characteristic curves in Sasakian space forms. This revised version was published online in August 2006 with corrections to the Cover Date.     

Functions Holomorphic along Holomorphic Vector Fields

The main result of the paper is the following generalization of Forelli’s theorem (Math. Scand. 41:358–364, 1977): Suppose F is a holomorphic vector field with singular point at p, such that F is linearizable at p and the matrix is diagonalizable with eigenvalues whose ratios are positive reals. Then any function φ that has an asymptotic Taylor expansion at p and is holomorphic along the complex integral curves of F is holomorphic in a neighborhood of p. We also present an example to show that the requirement for ratios of the eigenvalues to be positive reals is necessary. K.T. Kim and G. Schmalz were supported by the Scientific visits to Korea program of the AAS and KOSEF. E. Poletsky was supported by NSF Grant DMS-0500880. G. Schmalz gratefully acknowledges support and hospitality of the Max-Planck-Institut für Mathematik Bonn.     

Stratified Transversality via Time-Dependent Vector Fields

For a stratification with suitable regularity, in particularfor any Whitney stratification and, via regular embedding, forany abstract stratified set, time-dependent vector fields areused to prove an extension theorem for diffeomorphisms nearthe identity defined on strata of a given dimension. Then itis shown that after isotopy a stratified map h : Z can bemade transverse to a fixed stratified map g : y .     

Eikonal型方程粘性解的表达式

对于圆锥型和棱锥型Hamiltonian的Eikonal型方程,本文给出了一种几何方法,得出其初值问题解的表达式并且说明由此式给出的解为原初值问题的粘性解．首先用一个凸函数序列逼近Eikonal型方程中的Hamiltonian,再由Hopf-Lax公式给出方程序列的粘性解,最后证明了该粘性解序列会收敛到Eikonal方程的粘性解．