Inverse problems in magneto-electroscanning (in encephalography, for magnetic microscopes, etc.)

Contrary to the prevailing opinion about the incorrectness of the inverse MEEG-problem, we prove its unique solvability in the framework of the system of Maxwell''s equations [3]. The solution of this problem is the distribution of ${\bf y} \mapsto {\bf q}({\bf y})$ current dipoles of brain neurons that occupies the region $Y \subset \mathbb{R}^3 $. It is uniquely determined by the non-invasive measurements of the electric and magnetic fields induced by the current dipoles of neurons on the patient"s head. The solution can be represented in the form ${\bf q}={\bf q}_0+{\bf p}_0\delta\Big|_{\partial Y}$, where ${\bf q}_0$ is the usual function defined in $Y,$ and ${\bf p}_0\delta\Big|_{\partial Y} $ is a $\delta$-function on the boundary of the domain $Y$ with a certain density ${\bf p}_0$. It is essential that ${\bf p}_0$ and ${\bf q}_0$ are interrelated. This ensures the correctness of the inverse MEEG-problem. However, the components of the required 3-dimensional distribution $ {\bf q} $ must turn out to be linearly dependent if only the magnetic field ${\bf B}$ is taken into account. This question is considered in detail in a flat model of the situation.     

Wavelet Transforms Associated to Square Integrable Group Representations on L 2 ( C,H ; dz )

Let SL (2, C ) be the special linear group of 2 ‐ 2 complex matrices with determinant 1 and SU (2) its maximal compact subgroup. Then SL (2, C )/ SU (2) can be realized as the quaternionic upper half-plane $ {\cal H}^c $ . Let SL (2, C ) = NASU (2) be the Iwasawa decomposition and M the centerlizer of A in SU (2). Then P = NA and P a = NAM are the automorphism groups of $ {\cal H}^c $ . In this article, we define the unitary representations of P and P a on L 2 ( C , H ; dz ). From the viewpoint of square integrable group representations we discuss the wavelet transforms, and obtain the orthogonal direct sum decompositions for the function spaces $ L^2({\cal H}^c, \fraca {(dz\, d\rho)}{\rho ^3}) $ and $ L^2({\bf R}^2\times {\bf R}^2, \fraca {dx\, dy\, dx^{\prime }dy^{\prime }}{{({x^{\prime }}^2 + {y^{\prime }}^2)^{\fraca {3}{2}})}} $ .     

Invariant Theory for Non-Associative Real Two-Dimensional Algebras and Its Applications

The set ${\mathcal A}$ of all non-associative algebra structures on a fixed 2-dimensional real vector space $A$ is naturally a ${\mbox{\rm GL}}(2,{\mbox{\bf R}})$-module. We compute the ring of ${\mbox{\rm SL}}(2,{\mbox{\bf R}})$-invariants in the ring of polynomial functions, ${\mathcal P}$, on ${\mathcal A}$. We use invariant theory to compute the exact number of nonzero idempotents of an arbitrary 2-dimensional real division algebra. We show that the absolute invariants (i.e.,the ${\mbox{\rm GL}}(2, {\mbox{\bf R}})$-invariants in the field of fractions of ${\mathcal P}$) distinguish the isomorphism classes of 2-dimensional non-associative real division algebras. We show that the (open) set $\Omega^+\subset{\mathcal A}$ of all division algebra structures on $A$ has four connected components. A similar result is proved for another class of regular 2-dimensional real algebras (the principal isotopes of the algebra ${\mbox{\bf R}}\oplus{\mbox{\bf R}}$).     

Hardy Spaces on the Plane and Double Fourier Transforms

We provide a direct computational proof of the known inclusion where is the product Hardy space defined for example by R. Fefferman and is the classical Hardy space used, for example, by E.M. Stein. We introduce a third space of Hardy type and analyze the interrelations among these spaces. We give simple sufficient conditions for a given function of two variables to be the double Fourier transform of a function in and respectively. In particular, we obtain a broad class of multipliers on and respectively. We also present analogous sufficient conditions in the case of double trigonometric series and, as a by-product, obtain new multipliers on and respectively.     

Optimal recovery of anisotropic Besov--Wiener classes

This paper deals with the problem of optimal recovery of some anisotropic Besov--Wiener classes $S^{\bf r}_{pq\theta} B({\bf R}^d)$ in $L_q({\bf R}^d)$ and the dual case $S^{\bf r}_{p\theta} B({\bf R}^d)$ in $L_{qp} ({\bf R}^d)$ $(1相似文献   

Function Spaces in Lipschitz Domains and Optimal Rates of Convergence for Sampling

Assume that we want to recover $f : \Omega \to {\bf C}$ in the $L_r$-quasi-norm ($0 < r \le \infty$) by a linear sampling method $$ S_n f = \sum_{j=1}^n f(x^j) h_j , $$ where $h_j \in L_r(\Omega )$ and $x^j \in \Omega$ and $\Omega \subset {\bf R}^d$ is an arbitrary bounded Lipschitz domain. We assume that $f$ is from the unit ball of a Besov space $B^s_{pq} (\Omega)$ or of a Triebel--Lizorkin space $F^s_{pq} (\Omega)$ with parameters such that the space is compactly embedded into $C(\overline{\Omega})$. We prove that the optimal rate of convergence of linear sampling methods is $$ n^{ -{s}/{d} + ({1}/{p}-{1}/{r})_+} , $$ nonlinear methods do not yield a better rate. To prove this we use a result from Wendland (2001) as well as results concerning the spaces $B^s_{pq} (\Omega) $ and $F^s_{pq}(\Omega)$. Actually, it is another aim of this paper to complement the existing literature about the function spaces $B^s_{pq} (\Omega)$ and $F^s_{pq} (\Omega)$ for bounded Lipschitz domains $\Omega \subset {\bf R}^d$. In this sense, the paper is also a continuation of a paper by Triebel (2002).     

线性约束下Hermite-广义反Hamilton矩阵的最佳逼近问题

本文利用对称向量与反对称向量的特征性质,给出了约束矩阵集合非空的充分必要条件及矩阵的一般表达式.运用空间分解理论和闭凸集上的逼近理论,得到了任一n阶复矩阵在约束矩阵集合中的惟一最佳逼近解.     

Generalized Shift-Invariant Systems

A countable collection $X$ of functions in $L_2(\mbox{\footnotesize\bf R})$ is said to be a Bessel system if the associated analysis operator $$ \txs{X}:L_2(\mbox{\smallbf R}^d)\to \ell_2(X) : f\mapsto (\inpro{f,x})_{x\in X} $$ is well-defined and bounded. A Bessel system is a fundamental frame if $\txs{X}$ is injective and its range is closed. This paper considers the above two properties for a generalized shift-invariant system $X$. By definition, such a system has the form $$ X=\bigcup_{j\in J} Y_j, $$ where each $Y_j$ is a shift-invariant system (i.e., is comprised of lattice translates of some function(s)) and $J$ is a countable (or finite) index set. The definition is general enough to include wavelet systems, shift-invariant systems, Gabor systems, and many variations of wavelet systems such as quasi-affine ones and nonstationary ones. The main theme of this paper is the fiberization of $\txs{X}$, which allows one to study the frame and Bessel properties of $X$ via the spectral properties of a collection of finite-order Hermitian nonnegative matrices.     

Refined asymptotic profiles for a semilinear heat equation

We study the large time behavior of the solutions of the Cauchy problem for a semilinear heat equation,

eO-代数簇公理

扩张Ockham代数簇$e{\bf O}$是由所有$(L;\wedge,\vee, f, k,0,1)$所组成的代数类,其中$(L;\wedge,\vee,0,1)$是有界分配格, $f$是$L$上的偶同态, $k$是$L$ 是$L$上的同态且满足条件: $fk=kf$. 在本文中,我们把Urquhart定理推广到$e{\bf O}$-代数类,并特别考虑$e{\bf O}$-代数的子代数类 $e_2{\bf M}$.在子代数类$e_2{\bf M}$中, $f$和$k$满足条件: $f^{2}=id_L$及$k^{2}=id_L$. 我们证明: 在子代数类$e_2{\bf M}$中,有19个非等价公理.同时我们给出其蕴含关系的表达图式.     

A Characterization of Best φ-Approximants with Applications to Multidimensional Isotonic Approximation

Some properties of best monotone approximants in several variables are obtained. We prove the following abstract characterization theorem. Let $(\om, {\cal A},\mu)$ be a measurable space and let ${\cal L}\subset{\cal A}$ be a $\sigma$-lattice. If $f$ belongs to a Musielak–Orlicz space $L_{\varphi}(\Omega, {\cal A},\mu),$ then there exists a $\sigma$-algebra ${\cal A}_f\subset{\cal A}$ such that $g$ is a best $\varphi$-approximant to $f$ from $L_{\varphi}({\cal L})$ iff $g$ is a best $\varphi$-approximant to $f$ from $L_{\p}({\cal A}_f)$. The $\sigma$-algebra ${\cal A}_f$ depends only on $f$. When $\Omega\subset\mbox{{\bf R}}^n$ and $L_{\varphip}({\cal L})$ is the set of monotone functions in several variables, we give sufficient conditions on the geometry of $\Omega$ to obtain a uniqueness theorem. This result extends and unifies previous ones. Finally, we prove a coincidence relation between a function and its best $\varphi$-approximant. Our main results are new, even in the classical Lebesgue spaces $L_p$.     

Higher Order Optimality Conditions in Nonsmooth Optimization

For an arbitrary function $\font\Opr=msbm10 at 8pt \def\Op#1{\hbox {\Opr{#1}}}f\!\!: {\bf E} \rightarrow {\bar {\Op R}}$     

The Partial-Fractions Method for Counting Solutions to Integral Linear Systems

We present a new tool to compute the number $\phi_{\bf A} (b)$ of integer solutions to the linear system $$ x \geq 0, A x = b, $$ where the coefficients of $A$ and $b$ are integral. $\phi_{\bf A} (b)$ is often described as a vector partition function. Our methods use partial fraction expansions of Eulers generating function for $\phi_{\bf A} (\b)$. A special class of vector partition functions are Ehrhart (quasi-)polynomials counting integer points in dilated polytopes.     

Functional equations for vector products and quaternions

In our paper we find all functions ${f : \mathbb {R} \times \mathbb {R}^{3} \rightarrow \mathbb {H}}$ and ${g : \mathbb {R}^{3} \rightarrow \mathbb {H}}$ satisfying ${f (r, {\bf v}) f (s, {\bf w}) = -\langle{\bf v},{\bf w}\rangle + f (rs, s{\bf v} + r{\bf w} + {\bf v} \times {\bf w})}$ ${(r, s \in \mathbb {R}, {\bf v},{\bf w} \in \mathbb {R}^{3})}$ , and ${g({\bf v})g({\bf w}) = -\langle{\bf v}, {\bf w}\rangle + g({\bf v} \times {\bf w})}$ $({{\bf v},{\bf w} \in \mathbb {R}^{3}})$ . These functional equations were motivated by the well-known identities for vector products and quaternions, which can be obtained from the solutions f (r, (v ₁, v ₂, v ₃)) = r + v ₁ i + v ₂ j + v ₃ k and g((v ₁ ,v ₂, v ₃)) = v ₁ i + v ₂ j + v ₃ k.     

Entropy Numbers of Embeddings of Weighted Besov Spaces

We investigate the asymptotic behavior of the entropy numbers of the compact embedding $$ B^{s_1}_{p_1,q_1} \!\!(\mbox{\footnotesize\bf R}^d, \alpha) \hookrightarrow B^{s_2}_{p_2,q_2} \!\!({\xxR}). $$ Here $B^s_{p,q} \!({\mbox{\footnotesize\bf R}^d}, \alpha)$ denotes a weighted Besov space, where the weight is given by $w_\alpha (x) = (1+| x |^2)^{\alpha/2}$, and $B^{s_2}_{p_2,q_2} \!({\mbox{\footnotesize\bf R}^d})$ denotes the unweighted Besov space, respectively. We shall concentrate on the so-called limiting situation given by the following constellation of parameters: $s_2 < s_1$, $0 < p_1,p_2 \le \infty$, and $$ \alpha = s_1 - \frac{d}{p_1} - s_2 + \frac{d}{p_2} > d \, \max \Big(0, \frac{1}{p_2}-\frac{1}{p_1}\Big). $$ In almost all cases we give a sharp two-sided estimate.     

Problems of Lifts in Symplectic Geometry

Let(M,ω)be a symplectic manifold.In this paper,the authors consider the notions of musical(bemolle and diesis)isomorphisms ω~b:T M→T~*M and ω~?:T~*M→TM between tangent and cotangent bundles.The authors prove that the complete lifts of symplectic vector field to tangent and cotangent bundles is ω~b-related.As consequence of analyze of connections between the complete lift ~cω_(T M )of symplectic 2-form ω to tangent bundle and the natural symplectic 2-form dp on cotangent bundle,the authors proved that dp is a pullback o f~cω_(TM)by ω~?.Also,the authors investigate the complete lift ~cφ_T~*_M )of almost complex structure φ to cotangent bundle and prove that it is a transform by ω~?of complete lift~cφ_(T M )to tangent bundle if the triple(M,ω,φ)is an almost holomorphic A-manifold.The transform of complete lifts of vector-valued 2-form is also studied.     

Direct Sum Decomposition of L2(Rn 1) into Subspaces Invariant under Fourier Transformation

Denote by the real-linear span of , where Under the concept of left-monogeneity defined through the generalized Cauchy-Riemann operator we obtain the direct sum decomposition of