Identification of a time-dependent UV-photon source in an interstellar cloud

Abstract Consider an interstellar cloud that occupies the region , bounded by the known surface , and assume that the scattering cross section σ_s and the total cross section σ are also known. Then, we prove that it is possible to identify the source q=q(x,t) that produces UV-photons inside the cloud, provided that the UV-photon distribution function arriving at a location , far from the cloud, is measured at times , , ..., . Keywords: Photon transport, Semigroups and linear evolution equations, Inverse problems Mathematics Subject Classification (2000): 82A25, 82C70, 34K29, 65M32     

A new bound for an extension of Mason’s theorem for functions of several variables

In this paper, using the generalized Wronskian, we obtain a new sharp bound for the generalized Masons theorem [1] for functions of several variables. We also show that the Diophantine equation (The generalized Fermat-Catalan equation) where , such that k out of the n-polynomials are constant, and under certain conditions for has no non-constant solution. Received: 20 March 2003     

On the Homological Dimension of Lattices

Let L be a finite lattice and let . It is shown that if the order complex satisfies then |L| ≥ 2 ^k . Equality |L| = 2 ^k holds iff L is isomorphic to the Boolean lattice {0,1} ^k . Research supported by the Israel Science Foundation.     

Fundamental groups of moduli and the Grothendieck-Teichmüller group

Let denote the moduli space of Riemann spheres with ordered marked points. In this article we define the group of quasi-special symmetric outer automorphisms of the algebraic fundamental group for all to be the group of outer automorphisms respecting the conjugacy classes of the inertia subgroups of and commuting with the group of outer automorphisms of obtained by permuting the marked points. Our main result states that is isomorphic to the Grothendieck-Teichmüller group for all .

     

Fundamental Solution of the Anisotropic Porous Medium Equation

We establish the existence of fundamental solutions for the anisotropic porous medium equation, ut = ∑n i=1（u^mi）xixi in R^n × （O,∞）, where m1,m2,..., and mn, are positive constants satisfying min1≤i≤n{mi}≤ 1, ∑i^n=1 mi 〉 n - 2, and max1≤i≤n{mi} ≤1/n（2 ＋ ∑i^n=1 mi）.     

Time-Frequency Mean and Variance Sequences of Orthonormal Bases

We show that there exists an orthonormal basis for such that and are bounded sequences. We also show that there does not exist any orthonormal basis for , and being bounded sequences. This is motivated by a question posed by H.S. Shapiro on the mean and variance sequences associated to orthonormal bases.     

Bases for some reciprocity algebras I

For a complex vector space , let be the algebra of polynomial functions on . In this paper, we construct bases for the algebra of all highest weight vectors in , where and for all , and the algebra of highest weight vectors in .

     

Quasi-finite modules for Lie superalgebras of infinite rank

We classify the quasi-finite irreducible highest weight modules over the infinite rank Lie superalgebras , and , and determine the necessary and sufficient conditions for such modules to be unitarizable. The unitarizable irreducible modules are constructed in terms of Fock spaces of free quantum fields, and explicit formulae for their formal characters are also obtained by investigating Howe dualities between the infinite rank Lie superalgebras and classical Lie groups.

     

Equivalence Bimodule between Spherical Noncommutative Tori

Let n _i ,m _j > 1. In [5], the sperical noncommutative torus was defined by twisting in by a totally skew multiplier p on for T ^pd a pd-homogeneous C*-algebra over . It is shown that is strongly Morita equivalent to . This work is supported by Grant No. 1999-2-102-001-3 from the interdisciplinary research program year of the KOSEF     

On the dependency for asymptotically independent estimates

Suppose ${\widehat{\theta}_1}$ and ${\widehat{\theta}_2}$ are asymptotically independent non-lattice with a joint second order Edgeworth expansion in n ^?1/2. Then the ?? dependency coefficient is $$\alpha \left(\widehat{\theta}_1, \widehat{\theta}_2 \right) = n^{-1/2} C + O \left(n^{-1} \right),$$ where ${C = (4 \pi)^{-1}\exp (-1/2) (\tau^2_1 + \tau^2_2) ^{1/2}}$ for ${\tau_1, \tau_2}$ their joint skewness coefficients.     

Some exact sequences for Toeplitz algebras of spherical isometries

A family of commuting bounded operators on a Hilbert space is said to be a spherical isometry if in the weak operator topology. We show that every commuting family of spherical isometries is jointly subnormal, which means that it has a commuting normal extension on some Hilbert space Suppose now that the normal extension is minimal. Then we show that every bounded operator in the commutant of has a unique norm preserving extension to an operator in the commutant of Moreover, if is the commutator ideal in then is *-isomorphic to We also show that the commutant of the minimal normal extension is completely isometric, via the compression mapping, to the space of Toeplitz-type operators associated to We apply these results to construct exact sequences for Toeplitz algebras on generalized Hardy spaces associated to strictly pseudoconvex domains.

     

Triebel-Lizorkin Spaces of Para-Accretive Type and a Tb Theorem

In this article, we use a discrete Calderón-type reproducing formula and Plancherel-Pôlya-type inequality associated to a para-accretive function to characterize the Triebel-Lizorkin spaces of para-accretive type $\dot{F}^{\alpha,q}_{b,p}In this article, we use a discrete Calderón-type reproducing formula and Plancherel-P?lya-type inequality associated to a para-accretive function to characterize the Triebel-Lizorkin spaces of para-accretive type , which reduces to the classical Triebel-Lizorkin spaces when the para-accretive function is constant. Moreover, we give a necessary and sufficient condition for the boundedness of paraproduct operators. From this, we show that a generalized singular integral operator T with M _b TM _b ∈WBP is bounded from to if and only if and T ^* b=0 for , where ε is the regularity exponent of the kernel of T. Chin-Cheng Lin supported by National Science Council, Republic of China under Grant #NSC 97-2115-M-008-021-MY3. Kunchuan Wang supported by National Science Council, Republic of China under Grant #NSC 97-2115-M-259-009 and NCU Center for Mathematics and Theoretic Physics.     

Projektive Skalen,Tschebyscheff-Polynome und Fibonacci-Zahlen

Summary. Let $\widehat{\widehat T}_n$ and $\overline U_n$ denote the modified Chebyshev polynomials defined by $\widehat{\widehat T}_n (x) = {T_{2n + 1} \left(\sqrt{x + 3 \over 4} \right) \over \sqrt{x + 3 \over 4}}, \quad \overline U_{n}(x) = U_{n} \left({x + 1 \over 2}\right) \qquad (n \in \mathbb{N}_{0},\ x \in \mathbb{R}).$ For all $n \in \mathbb{N}_{0}$ define $\widehat{\widehat T}_{-(n + 1)} = \widehat{\widehat T}_n$ and $\overline U_{-(n + 2)} = - \overline U_n$, furthermore $\overline U_{-1} = 0$. In this paper, summation formulae for sums of type $\sum\limits^{+\infty}_{k = -\infty} \mathbf a_{\mathbf k}(\nu; x)$ are given, where $\bigl(\mathbf a_{\mathbf k}(\nu; x)\bigr)^{-1} = (-1)^k \cdot \Bigl( x \cdot \widehat{\widehat T}_{\left[k + 1 \over 2\right] - 1} (\nu) +\widehat{\widehat T}_{\left[k + 1 \over 2\right]}(\nu)\Bigr) \cdot \Bigl(x \cdot \overline U_{\left[k \over 2\right] - 1} (\nu) + \overline U_{\left[k \over 2\right]} (\nu)\Bigr)$ with real constants $ x, \nu $. The above sums will turn out to be telescope sums. They appear in connection with projective geometry. The directed euclidean measures of the line segments of a projective scale form a sequence of type $(\mathbf a_{\mathbf k} (\nu;x))_{k \in \mathbb{Z}}$ where $ \nu $ is the cross-ratio of the scale, and x is the ratio of two consecutive line segments once chosen. In case of hyperbolic $(\nu \in \mathbb{R} \setminus] - 3,1[)$ and parabolic $\nu = -3$ scales, the formula $\sum\limits^{+\infty}_{k = -\infty} \mathbf a_{\mathbf k} (\nu; x) = {\frac{1}{x - q_{{+}\atop(-)}}} - {\frac{1}{x - q_{{-}\atop(+)}}} \eqno (1)$ holds for $\nu > 1$ (resp. $\nu \leq - 3$), unless the scale is geometric, that is unless $x = q_+$ or $x = q_-$. By $q_{\pm} = {-(\nu + 1) \pm \sqrt{(\nu - 1)(\nu + 3)} \over 2}$ we denote the quotient of the associated geometric sequence.

     

Graph von Neumann Algebras

In this paper, we will define a graph von Neumann algebra over a fixed von Neumann algebra M, where G is a countable directed graph, by a crossed product algebra = M × _α , where is the graph groupoid of G and α is the graph-representation. After defining a certain conditional expectation from onto its M-diagonal subalgebra we can see that this crossed product algebra is *-isomorphic to an amalgamated free product where = vN(M × _α where is the subset of consisting of all reduced words in {e, e ^–1} and M × _α is a W ^*-subalgebra of as a new graph von Neumann algebra induced by a graph G _e . Also, we will show that, as a Banach space, a graph von Neumann algebra is isomorphic to a Banach space ⊕ where is a certain subset of the set E(G)* of all words in the edge set E(G) of G. The author really appreciates to Prof F. Radulescu and Prof P. Jorgensen for the valuable discussion and kind advice. Also, he appreciates all supports from St. Ambrose Univ.. In particular, he thanks to Prof T. Anderson and Prof V. Vega for the useful conversations and suggestions.     

Distributional Point Values and Convergence of Fourier Series and Integrals

In this article we show that the distributional point values of a tempered distribution are characterized by their Fourier transforms in the following way: If and , and is locally integrable, then distributionally if and only if there exists k such that , for each a > 0, and similarly in the case when is a general distribution. Here means in the Cesaro sense. This result generalizes the characterization of Fourier series of distributions with a distributional point value given in [5] by . We also show that under some extra conditions, as if the sequence belongs to the space for some and the tails satisfy the estimate ,\ as , the asymmetric partial sums\ converge to . We give convergence results in other cases and we also consider the convergence of the asymmetric partial integrals. We apply these results to lacunary Fourier series of distributions.     

On cyclotomic schemes over finite near-fields

We introduce a concept of cyclotomic association scheme over a finite near-field . It is proved that any isomorphism of two such nontrivial schemes is induced by a suitable element of the group AGL(V), where V is the linear space associated with . A sufficient condition on a cyclotomic scheme that guarantee the inclusion where is a finite field with elements, is given. I. Ponomarenko partially supported by RFFI, grants 03-01-00349, NSH-2251.2003.1.     

The universal central extension of the three-point loop algebra

We consider the three-point loop algebra,

where denotes a field of characteristic 0 and is an indeterminate. The universal central extension of was determined by Bremner. In this note, we give a presentation for via generators and relations, which highlights a certain symmetry over the alternating group . To obtain our presentation of , we use the realization of as the tetrahedron Lie algebra.

     

Relations between different types of spectra and spectral characterizations

In the study of the asymptotic behaviour of solutions of differential-difference equations the -spectrum has been useful, where and implies Fourier transform , with given , φ∈L ^∞(ℝ,X), X a Banach space, (half)line. Here we study and related concepts, give relations between them, especially weak Laplace half-line spectrum of φ, and thus ⊂ classical Beurling spectrum = Carleman spectrum =  ; also  = Beurling spectrum of “φ modulo ” (Chill-Fasangova). If satisfies a Loomis type condition (L _U ), then countable and uniformly continuous ∈U are shown to imply ; here (L _U ) usually means , indefinite integral Pf of f in U imply Pf in (the Bohl-Bohr theorem for = almost periodic functions, U=bounded functions). This spectral characterization and other results are extended to unbounded functions via mean classes , ℳ ^m U ((2.1) below) and even to distributions, generalizing various recent results for uniformly continuous bounded φ. Furthermore for solutions of convolution systems S*φ=b with in some we show . With these above results, one gets generalizations of earlier results on the asymptotic behaviour of solutions of neutral integro-differential-difference systems. Also many examples and special cases are discussed.     

Inner Product Space and Concept Classes Induced by Bayesian Networks

Bayesian networks have become one of the major models used for statistical inference. In this paper we discuss the properties of the inner product spaces and concept class induced by some special Bayesian networks and the problem whether there exists a Bayesian network such that lower bound on dimensional inner product space just is some positive integer. We focus on two-label classification tasks over the Boolean domain. As main results we show that lower bound on the dimension of the inner product space induced by a class of Bayesian networks without v-structures is where m _i denotes the number of parents for ith variable. As the variable’s number of Bayesian network is n≤5, we also show that for each integer m∈[n+1,2 ⁿ −1] there is a Bayesian network such that VC dimension of concept class and lower bound on dimensional inner product space induced by all are m. This work was supported by National Natural Science Foundation of China (60574075).