On Almost Compactness of Some Partially Integral Operators in $ L_{p} $

We establish sufficient conditions for the almost compactness of partially integral operators in \( L_{p} \).

     

The Maximal Cardinality of the Base in $ P_{2}times P_{2} $

Siberian Mathematical Journal - Some structural properties are discussed of the functions in $ P_{2}times P_{2} $ . We describe the properties of functions not belonging to the maximal...     

Interpolation spaces between(L_1^{W_0 } ,L_1^{W_1 } ) and (L1, L∞)

Let A₀, A₁ be a pair of normed spaces, having the property that the difference K(x, t; A₀, A₁) ?K(x, s; A₀, A₁) regarded as a function of xε A₀+ A₁ is a seminorm for t>s (here K is the Oklander-Peetre functional). All the pairs A, L of normed spaces, such that, if a linear operator is bounded from A₀ into L₁ and from A₁ into L_∞, then it is bounded from A into L, are characterized in the following article.     

Atomic subspaces of {L_1}-martingale spaces

In Part I of the present paper the following problem was investigated. Let G be a finite simple graph, and S be a finite set of primes. We say that G is representable with S if it is possible to attach rational numbers to the vertices of G such that the vertices v₁, v₂ are connected by an edge if and only if the difference of the attached values is an S-unit. In Part I we gave several results concerning the representability of graphs in the above sense.     

On the {H^1}-stability of the {L_2}-projection onto finite element spaces

We study the stability in the $H^1$ -seminorm of the $L_2$ -projection onto finite element spaces in the case of nonuniform but shape regular meshes in two and three dimensions and prove, in particular, stability for conforming triangular elements up to order twelve and conforming tetrahedral elements up to order seven.     

Estimates of Mixed Moduli of Smoothness in Metrics of <Emphasis Type="Italic">L</Emphasis><Subscript>q</Subscript> Through Mixed Moduli of Smoothness in Metrics of <Emphasis Type="Italic">L</Emphasis><Subscript>1</Subscript>

Interrelations between mixed fractional moduli of smoothness considered in metrics of L₁ and L_q are studied in the paper.     

Moduli of Continuity of Harmonic Quasiregular Mappings in $\mathbb{B}^n$

We prove that ω _u (δ)?≤?Cω _f (δ), where \(u : \overline{\mathbb{B}^n} \rightarrow \mathbb{R}^n\) is the harmonic extension of a continuous map \(f:\mathbb{S}^{n-1}\rightarrow\mathbb{R}^n\), if u is a K-quasiregular map. Here C is a constant depending only on n, ω _f and K and ω _h denotes the modulus of continuity of h.     

Geodesics on the Moduli Space of Oriented Circles in \mathbb{S}^{3}

Let ${\mathbb{Q}^3}$ be the moduli space of oriented circles in the three dimensional unit sphere ${\mathbb{S}^3}$ . Given a natural complex structure such space becomes a three dimensional complex manifold, with a M?bius invariant Hermitian metric h of type (2, 1). Up to M?bius transformations, all geodesics with respect to the Lorentz metric g = Re(h) on ${\mathbb{Q}^3}$ are determined to form a one-parameter family of circles on a helicoid in a space form ${\mathbb{R}^3, \mathbb{H}^3}$ or ${\mathbb{S}^{3}}$ , resp. We show also that any two oriented circles in ${\mathbb{S}^3}$ are connected by countably infinitely many geodesics in ${\mathbb{Q}^3}$ .     

Signature Characters of Highest-Weight Representations of $U_{q}(\mathfrak {gl}_{n})$

We consider \(U_{q}(\mathfrak {gl}_{n})\), the quantum group of type A for |q| = 1, q generic. We provide formulas for signature characters of irreducible finite-dimensional highest weight modules and Verma modules. In both cases, the technique involves combinatorics of the Gelfand-Tsetlin bases. As an application, we obtain information about unitarity of finite-dimensional irreducible representations for arbitrary q: we classify the continuous spectrum of the unitarity locus. We also recover some known results in the classical limit \(q \rightarrow 1\) that were obtained by different means. Finally, we provide several explicit examples of signature characters.     

Characterizations of the Vertex Operator Algebras ${V_{L}^{T}}$ and VLO${V_{L}^{O}}$

In this paper, we give characterizations of the rational vertex operator algebras \({V_{L}^{T}}\) and \({V_{L}^{O}}\), where L is the root lattice of type A ₁, T is the tetrahedral group, and O is the octahedral group. By these two characterizations, the classification of rational VOAs of central charge 1 is reduced to the characterization of \({V_{L}^{I}}\) where I is the icosahedral group.     

Harmonic Analysis Associated with the Heckman-Opdam-Jacobi Operator on $\mathbb{R}^{d+1}$

In this paper we consider the Heckman-Opdam-Jacobi operator $∆_{HJ}$ on $\mathbb{R}^{d+1}.$ We define the Heckman-Opdam-Jacobi intertwining operator $V_{HJ},$ which turns out to be the transmutation operator between $∆_{HJ}$ and the Laplacian $∆_{d+1}.$ Next we construct $^tV_{HJ}$ the dual of this intertwining operator. We exploit these operators to develop a new harmonic analysis corresponding to $∆_{HJ}.$     

On the Coincidence of the Classes of Finite Groups $ E_{\pi_{x}} $ and $ D_{\pi_{x}} $

Let \( \pi_{x} \) be the set of primes greater than \( x \). We prove that for all \( x\in{??} \) the classes of finite groups \( D_{\pi_{x}} \) and \( E_{\pi_{x}} \) coincide; i.e., a finite group \( G \) possesses a \( \pi_{x} \)-Hall subgroup if and only if \( G \) satisfies the complete analog of the Sylow Theorems for a \( \pi_{x} \)-subgroup.

     

The Cauchy Harish-Chandra Integral, for the pair \mathfrak{u}_{p,q} ,\mathfrak{u}_1

For the dual pair considered, the Cauchy Harish-Chandra Integral, as a distribution on the Lie algebra, is the limit of the holomorphic extension of the reciprocal of the determinant. We compute that limit explicitly in terms of the Harish-Chandra orbital integrals.      

Lipschitzian Properties of Integral Functionals on Lebesgue Spaces $\bf{L_p, 1 \leqslant {p} < \infty}$

We give complete characterizations of integral functionals which are Lipschitzian on a Lebesgue space L _p with p ≠ ∞. When the measure is atomless, we characterize the integral functionals which are locally Lipschitzian on such Lebesgue spaces. In every cases, the Lipchitzian properties of the integral functional can be described by growth conditions on the subdifferentials of the integrand which are equivalent to Lipschitzian properties of the integrand.     

On the Number of Solutions of the Equation a_{1}x_{1}^{m_{1}}+\cdots +a_{n}x_{n}^{m_{n}}=bx_{1}\cdots x_{n} in a Finite Field

We obtain an explicit formula for the number of solutions of a special equation in a finite field under a certain restriction on the exponents. Mathematics Subject Classifications (2000) primary: 11G25, secondary: 11T24.     

Recognizing $ A_{7} $ by Its Set of Element Orders

Siberian Mathematical Journal - Let $ G $ be a periodic group, and let $ \omega(G)\subseteq{??} $ be the spectrum of&nbsp; $ G $ that is the set of orders of elements in&nbsp;...