Tensor products of power Köthe spaces of different types

The multirectangular characteristics µ _m ^(λ,c) are applied to the isomorphic classification of tensor products of the form $ E_0 (a)\widehat \otimes E_\infty (b) $ . We single out a subclass of tensor products such that the two-rectangular characteristic µ ₂ ^(λ,c) is a complete invariant on this class.     

On the cosets of the 2q-power group in the unit group modulop

Letp be a prime number ≡ 3 mod 4,G ^p the unit group of ?/p?, andg a generator ofG _p. Letq be an odd divisor ofp - 1 andG _p ^2q = {t ^2q;t ∈G _pthe subgroup of index2q inG _p. The groupG _p ² / _p ^2q consists of the classes \(\bar g^{2j} \) ,j = 0,...,q – 1. In this paper we study the ’excesses’ of the classes \(\bar g^{2j} \) in {l,...,(_p–l)/2}, i.e., the numbers \(\Phi _j = \left| {\left\{ {k;1 \leqslant k \leqslant \left( {p - 1} \right)/2,\bar k \in \bar g^{2j} } \right\}} \right| - \left| {\left\{ {k;\left( {p - 1} \right)/2 \leqslant k \leqslant p - 1,\bar k \in \bar g^{2j} } \right\}} \right|\) ,j = 0.....q — 1. First we express therelative class number h _2q of the subfieldK _2q? ?(e^2#x03C0;i/p ) of degree [K _2q: ?] =2q in terms of these excesses. We use this formula to establish certaincongruences for the Ф^j. E.g., ifq ∈ {3,5,11}, each number Фj is congruent modulo 4 to each other iff 2 dividesh _2q ^- . Finally we study thevariance of the excesses, i.e., the number \(\sigma ^2 = ((\Phi _0 - \hat \Phi )^2 + \ldots + (\Phi _{q - 1} - \hat \Phi )^2 )/(q - 1)\) , where \(\hat \Phi \) is the mean value of the numbers Ф_j. We obtain an explicit lower bound for σ² in terms ofh _2q ^- /h ₂ ^- . Moreover, we show that log σ² is asymptotically equal to 21og(h _2q ^- h ₂ ^- )/(q - 1) forp→∞. Three tables illustrate the results.     

A construction for infinite families of semisymmetric graphs revealing their full automorphism group

We give a general construction leading to different non-isomorphic families $\varGamma_{n,q}(\mathcal{K})$ of connected q-regular semisymmetric graphs of order 2q ⁿ⁺¹ embedded in $\operatorname{PG}(n+1,q)$ , for a prime power q=p ^h , using the linear representation of a particular point set $\mathcal{K}$ of size q contained in a hyperplane of $\operatorname{PG}(n+1,q)$ . We show that, when $\mathcal{K}$ is a normal rational curve with one point removed, the graphs $\varGamma_{n,q}(\mathcal{K})$ are isomorphic to the graphs constructed for q=p ^h in Lazebnik and Viglione (J. Graph Theory 41, 249–258, 2002) and to the graphs constructed for q prime in Du et al. (Eur. J. Comb. 24, 897–902, 2003). These graphs were known to be semisymmetric but their full automorphism group was up to now unknown. For q≥n+3 or q=p=n+2, n≥2, we obtain their full automorphism group from our construction by showing that, for an arc $\mathcal{K}$ , every automorphism of $\varGamma_{n,q}(\mathcal{K})$ is induced by a collineation of the ambient space $\operatorname{PG}(n+1,q)$ . We also give some other examples of semisymmetric graphs $\varGamma _{n,q}(\mathcal{K})$ for which not every automorphism is induced by a collineation of their ambient space.     

Non-classical polar unitals in finite Dickson semifield planes

We give three proofs, two intrinsic and one extrinsic, that every Dickson–Ganley unital ${\mathcal{U}(\sigma)}$ , parametrized by a field automorphism σ, is non-classical if σ is not the identity, extending a result of Ganley’s (Math Z 128:34–42, 1972); we prove that ${\mathcal{U}(\sigma_1)}$ is isomorphic to ${\mathcal{U}(\sigma_2)}$ if and only if σ ₁ = σ ₂ or σ ₁ = σ ₂ ^?1 ; and we determine the (design) automorphism group of ${\mathcal{U}(\sigma)}$ as the collineation subgroup of the ambient Dickson semifield plane stabilizing the unital. This contains as a special case the corresponding result of O’Nan’s (J Algebra 20:495–511, 1965) on the classical unital.     

Метод многопараметрической интерполяции и теоремы вложения пространств БесоваB_{\vec p}^{\vec \alpha } \left[ {0,2\pi } \right)

In this paper we determine the method of multi-parameter interpolation and the scales of Lebesgue spaces $B_{\vec p} \left[ {0,2\pi } \right)$ and Besov spaces $B_{\vec p}^{\vec \alpha } \left[ {0,2\pi } \right)$ , which are generalizations of the Lorentz spacesL _pq [0, 2π) and Besov spacesB _pq ^α [0, 2π). We also prove imbedding theorems.     

On the eigenvalues of the Sturm-Liouville operator with potentials from Sobolev spaces

We study the asymptotic behavior of the eigenvalues the Sturm-Liouville operator Ly = ?y″ + q(x)y with potentials from the Sobolev space W ₂ ^θ?1 , θ ≥ 0, including the nonclassical case θ ∈ [0, 1) in which the potential is a distribution. The results are obtained in new terms. Let s _2k (q) = λ _k ^1/2 (q) ? k, s _2k?1(q) = μ _k ^1/2 (q) ? k ? 1/2, where {λ _k } ₁ ^∞ and {μ _k } ₁ ^∞ are the sequences of eigenvalues of the operator L generated by the Dirichlet and Dirichlet-Neumann boundary conditions, respectively,. We construct special Hilbert spaces t ₂ ^θ such that the mapping F:W ₂ ^θ?1 →t ₂ ^θ defined by the equality F(q) = {s _n } ₁ ^∞ is well defined for all θ ≥ 0. The main result is as follows: for θ > 0, the mapping F is weakly nonlinear, i.e., can be expressed as F(q) = Uq + Φ(q), where U is the isomorphism of the spaces W ₂ ^θ?1 and t ₂ ^θ , and Φ(q) is a compact mapping. Moreover, we prove the estimate ∥Ф(q)∥_τ ≤ C∥q∥_θ?1, where the exact value of τ = τ(θ) > θ ? 1 is given and the constant C depends only on the radius of the ball ∥q∥_θ? ≤ R, but is independent of the function q varying in this ball.     

Multivariate model with a Kronecker product covariance structure: S. N. Roy method of estimation

In this paper we consider a (p × q)-matrix X = (X ₁, ..., X _q ), where a pq-vector vec (X) = (X ₁ ^T , ...,X _q ^T ) ^T is assumed to be distributed normally with mean vector vec (M) = (M ₁ ^T , ...,M _q ^T ) ^T and a positive definite covariance matrix Λ. Suppose that Λ follows a Kronecker product covariance structure, that is Λ = Φ?Σ, where Φ = (? _ij ) is a (q × q)-matrix and Σ = (σ _ij ) is a (p × p)-matrix and the matrices Φ, Σ are positive definite. Such a model is considered in [4], where the maximum likelihood estimates of the parameters M, Φ, Σ are obtained. Using S. N. Roy’s technique (see, e.g., [3]) of the multivariate statistical analysis, we obtain consistent and unbiased estimates of M, Φ, Σ as in [4], but with less calculations.     

Some class of analytic functions related to conic domains

For q ∈ (0, 1) let the q-difference operator be defined as follows $$\partial _q f(z) = \frac{{f(qz) - f(z)}} {{z(q - 1)}} (z \in \mathbb{U}),$$ where \(\mathbb{U}\) denotes the open unit disk in a complex plane. Making use of the above operator the extended Ruscheweyh differential operator R _q ^λ f is defined. Applying R _q ^λ f a subfamily of analytic functions is defined. Several interesting properties of a defined family of functions are investigated.     

An example of asymmetry of convolution operators

Esistono un gruppo compatto non commutativoG ^～ ed un operatore di convoluzioneT tale che: perp∈[2,4] e perq∈[1,2),T∈L _p ^p (G ^～) eT?L _q ^q (G ^～).     

On Isomorphisms of Marušič–Scapellato Graphs

Maru?i?–Scapellato graphs are vertex-transitive graphs of order \(m(2^k + 1)\), where m divides \(2^k - 1\), whose automorphism group contains an imprimitive subgroup that is a quasiprimitive representation of \(\mathrm{SL}(2,2^k)\) of degree \(m(2^k + 1)\). We show that any two Maru?i?–Scapellato graphs of order pq, where p is a Fermat prime, and q is a prime divisor of \(p - 2\), are isomorphic if and only if they are isomorphic by a natural isomorphism derived from an automorphism of \(\mathrm{SL}(2,2^k)\). This work is a contribution towards the full characterization of vertex-transitive graphs of order a product of two distinct primes.     

An embedding theorem for a weighted space of Sobolev type and correct solvability of the Sturm-Liouville equation

We consider the weighted space W ₁ ⁽²⁾ (?,q) of Sobolev type $$W_1^{(2)} (\mathbb{R},q) = \left\{ {y \in A_{loc}^{(1)} (\mathbb{R}):\left\| {y''} \right\|_{L_1 (\mathbb{R})} + \left\| {qy} \right\|_{L_1 (\mathbb{R})} < \infty } \right\} $$ and the equation $$ - y''(x) + q(x)y(x) = f(x),x \in \mathbb{R} $$ Here f ε L ₁(?) and 0 ? q ∈ L ₁ ^loc (?). We prove the following:

1. The problems of embedding W ₁ ⁽²⁾ (?q) ? L ₁(?) and of correct solvability of (1) in L ₁(?) are equivalent
2. an embedding W ₁ ⁽²⁾ (?,q) ? L ₁(?) exists if and only if $$\exists a > 0:\mathop {\inf }\limits_{x \in R} \int_{x - a}^{x + a} {q(t)dt > 0} $$

     

A property of algebraic univoque numbers

Consider the set $ {\mathcal{U}} $ of real numbers q ≧ 1 for which only one sequence (c _i ) of integers 0 ≦ c _i ≦ q satisfies the equality Σ _i=1 ^∞ c _i q ^?i = 1. We show that the set of algebraic numbers in $ {\mathcal{U}} $ is dense in the closure $ \overline {\mathcal{U}} $ of $ {\mathcal{U}} $ .     

General function spaces. III(SpacesB p,q g(x) andF p,q g(x) , 1<p<∞: basic properties)

qNАстОьЩАь РАБОтА (А тАк жЕ ЕЕ пРОДОлжЕНИЕ — РА БОтА «ОБЩИЕ ФУНкцИОНАльН ыЕ пРО-стРАНстВА, IV») пОсВьЩЕНА ИсслЕДОВ АНИУ БАНАхОВых пРОст РАНстВB _{p, q} ^g(x) ИF _p,q ^g(x) РАспРЕДЕлЕНИИ (ОБОБЩЕННых) ВR _n . В спЕц ИАльНых слУЧАьх ЁтИ пРОстРАНстВА РОДстВ ЕННы ИжВЕстНыМ клАсс АМ сОБОлЕВА—лЕБЕгА—БЕ сОВА, ИжОтРОпНыМ И АНИ жОтРОпНыМ. жДЕсь РАссМАтРИВАУт сь слЕДУУ-ЩИЕ сВОИстВА: плОтНОсть г лАДкИх ФУНкцИИ, ЁкВИВ АлЕНтНыЕ НОРМы, ИНтЕРпОльцИь, В клУЧЕНИь И сРАВНЕНИь. ОБЩИЕ ФУНкцИОНАльНы Е пРОстРАНстВА. III (пРОстРАНстВАB _p,q ^g(x) ИF _p,q ^g(x) , 1     

On the multipliers of Fourier series with respect to the Haar system

qV РАБОтЕ РАссМАтРИВАУ тсь МУльтИплИкАтИВН ыЕ пРЕОБРАжОВАНИь РьДО В ФУРьЕ-хААРА. ИжУЧЕНы слЕДУУЩИЕ кл Ассы МУльтИплИкАтОР ОВ: (H _p ^α ,H _q ^β ), (H _p ^α ,L ^q ) И (с, с).     

The quantum toroidal algebra \widehat {\widehat {\mathfrak{g}{\mathfrak{l}_1}}}: Calculation of characters of some representations as generating functions of plane partitions

The generating function of plane partitions {a _i,j } subject to the constraint a _m,n = 0 is expressed and calculated as the character of an irreducible representation of the quantum toroidal algebra $\widehat {\widehat {\mathfrak{g}{\mathfrak{l}_1}}}$ in the case K = q ₁ ^m q ₂ ⁿ .     

Interpolation of Besov B pτ σq and Lizorkin-Triebel F pτ σq spaces

An interpolation method is introduced for anisotropic spaces which generalizes the method by D. L. Fernandez [4]. By means of this method, interpolation properties of Besov B _pτ ^σq and Lizorkin-Triebel F _pτ ^σq spaces are investigated. Among others, the completeness of the scale of these spaces is proved with respect to the considered interpolation method.     

Between\widehat{gl}(\infty ) and\widehat{sl}_N affine algebras. I. Geometrical actions

We consider the central extended $\widehat{gl}(\infty )$ Lie algebra and a set of its subalgebras parametrized by |q|=1, which coincides with the embedding of the quantum tori Lie algebras (QTLA) in $\widehat{gl}(\infty )$ . Forq ^N=1 there exists an ideal, and a factor over this ideal is isomorphic to an $\widehat{sl}_{N(z)} $ affine algebra. For a generic valueq the corresponding subalgebras are dense in $\widehat{gl}(\infty )$ . Thus, they interpolate between $\widehat{gl}(\infty )$ and $\widehat{sl}_{N(z)} $ . All these subalgebras are fixed points of automorphism of $\widehat{gl}(\infty )$ . Using the automorphisms, we construct geometrical actions for the subalgebras, starting from the Kirillov-Kostant form and the corresponding geometrical action for $\widehat{gl}(\infty )$ .     

Matrix Maps of Quasi-Convex Null Sequences

In [7] Stieglitz and Tietz identify the space q ^α of all quasi-convex convergent sequences as a BK-space. They characterize all infinite matrices which map q ^α into an arbitrary FK-space. In [6] they do so for matrices which map a particular class of sequence spaces into q ^α . In [10] Zygmund introduces q ² in connexion with convergence factors of Fourier series. Dawson considers in [3] and [4] matrix maps of the space q ₀ ^α of all quasi-convex null sequences. In Section 2 we characterize all matrices which map q ₀ ^α into an arbitrary FK-space. Prior to that, a particular matrix map on q ₀ ^α gives us the BK-topology on q ₀ ^α . As an application we characterize in Section 3 the matrices which map q ₀ ^α into the FK-spaces considered by Stieglitz and Tietz in [8]. Based on [6], we determine the matrices which map these spaces into q ₀ ^α . Using methods similar to those in [7] our results in Section 2 depend on Theorems 2.1 and 4.1 in [5] due to Jakimovski and Livne. Theorem 2.1 gives for suitable pairs of sequence spaces necessary and sufficient conditions for an infinite matrix to map one space into the second one. In Theorem 4.1 a special sequence which is useful in applications of quasi-convexity is constructed. We close our paper with two remarks concerning three results in [8].     

Wavelet estimations for density derivatives

Donoho et al. in 1996 have made almost perfect achievements in wavelet estimation for a density function f in Besov spaces Bsr,q(R). Motivated by their work, we define new linear and nonlinear wavelet estimators flin,nm, fnonn,m for density derivatives f(m). It turns out that the linear estimation E(‖flinn,m-f(m)‖p) for f(m) ∈ Bsr,q(R) attains the optimal when r≥ p, and the nonlinear one E(‖fnonn,m-f(m)‖p) does the same if r≤p/2(s+m)+1 . In addition, our method is applied to Sobolev spaces with non-negative integer exponents as well.     

To the Sobolev embedding theorem for the limiting exponent

We establish embeddings of the Sobolev space W _p ^s and the space B _pq ^s (with the limiting exponent) in certain spaces of locally integrable functions of zero smoothness. This refines the embedding of the Sobolev space in the Lorentz and Lorentz-Zygmund spaces. Similar problems are considered for the case of irregular domains and for the potential space.