On a semilattice of numberings

We study some properties of a $ \mathfrak{c} $ \mathfrak{c} -universal semilattice $ \mathfrak{A} $ \mathfrak{A} with the cardinality of the continuum, i.e., of an upper semilattice of m-degrees. In particular, it is shown that the quotient semilattice of such a semilattice modulo any countable ideal will be also $ \mathfrak{c} $ \mathfrak{c} -universal. In addition, there exists an isomorphism $ \mathfrak{A} $ \mathfrak{A} such that $ {\mathfrak{A} \mathord{\left/ {\vphantom {\mathfrak{A} {\iota \left( \mathfrak{A} \right)}}} \right. \kern-\nulldelimiterspace} {\iota \left( \mathfrak{A} \right)}} $ {\mathfrak{A} \mathord{\left/ {\vphantom {\mathfrak{A} {\iota \left( \mathfrak{A} \right)}}} \right. \kern-\nulldelimiterspace} {\iota \left( \mathfrak{A} \right)}} will be also $ \mathfrak{c} $ \mathfrak{c} -universal. Furthermore, a property of the group of its automorphisms is obtained. To study properties of this semilattice, the technique and methods of admissible sets are used. More exactly, it is shown that the semilattice of mΣ-degrees $ L_{m\Sigma }^{\mathbb{H}\mathbb{F}\left( S \right)} $ L_{m\Sigma }^{\mathbb{H}\mathbb{F}\left( S \right)} on the hereditarily finite superstructure $ \mathbb{H}\mathbb{F} $ \mathbb{H}\mathbb{F} (S) over a countable set S will be a $ \mathfrak{c} $ \mathfrak{c} -universal semilattice with the cardinality of the continuum.     

On holomorphic solutions of some inhomogeneous linear differential equations in a banach space over a non-archimedean field

Let A be a closed linear operator on a Banach space $ \mathfrak{B} $ \mathfrak{B} over the field Ω of complex p-adic numbers having an inverse operator defined on the whole $ \mathfrak{B} $ \mathfrak{B} , and f be a locally holomorphic at 0 $ \mathfrak{B} $ \mathfrak{B} -valued vector function. The problem of existence and uniqueness of a locally holomorphic at 0 solution of the differential equation y ^(m) − Ay = f is considered in this paper. In particular, it is shown that this problem is solvable under the condition $ \mathop {\lim }\limits_{n \to \infty } \sqrt[n]{{\left\| {A^{ - n} } \right\|}} $ \mathop {\lim }\limits_{n \to \infty } \sqrt[n]{{\left\| {A^{ - n} } \right\|}} = 0. It is proved also that if the vector-function f is entire, then there exists a unique entire solution of this equation. Moreover, the necessary and sufficient conditions for the Cauchy problem for such an equation to be correctly posed in the class of locally holomorphic functions are presented.     

Strict topology over the space of measures with continuous translations on a locally compact foundation semigroup

Let $ \mathfrak{S} $ \mathfrak{S} be a locally compact semigroup, ω be a weight function on $ \mathfrak{S} $ \mathfrak{S} , and M _a ($ \mathfrak{S} $ \mathfrak{S} , ω) be the weighted semigroup algebra of $ \mathfrak{S} $ \mathfrak{S} . Let L ₀^∞ ($ \mathfrak{S} $ \mathfrak{S} ; M _a ($ \mathfrak{S} $ \mathfrak{S} , ω)) be the C*-algebra of all M _a ($ \mathfrak{S} $ \mathfrak{S} , ω)-measurable functions g on $ \mathfrak{S} $ \mathfrak{S} such that g/ω vanishes at infinity. We introduce and study a strict topology β ¹($ \mathfrak{S} $ \mathfrak{S} , ω) on M _a ($ \mathfrak{S} $ \mathfrak{S} , ω) and show that the Banach space L ₀^∞ ($ \mathfrak{S} $ \mathfrak{S} ; M _a ($ \mathfrak{S} $ \mathfrak{S} , ω)) can be identified with the dual of M _a ($ \mathfrak{S} $ \mathfrak{S} , ω) endowed with β ¹($ \mathfrak{S} $ \mathfrak{S} , ω). We finally investigate some properties of the locally convex topology β ¹($ \mathfrak{S} $ \mathfrak{S} , ω) on M _a ($ \mathfrak{S} $ \mathfrak{S} , ω).     

Weight modules over some Block algebras

In this paper, the Harish-Chandra modules and Verma modules over Block algebra $ \mathfrak{L} $ \mathfrak{L} [G] are investigated. More precisely, the irreducibility of the Verma modules over $ \mathfrak{L} $ \mathfrak{L} [G] is completely determined, and the Harish-Chandra modules over $ \mathfrak{L} $ \mathfrak{L} [ℤ] are classified.     

Cardinalities of definable sets in superstructures over models

We introduce the notion of a superstructure over a model. This is a generalization of the notion of the hereditarily finite superstructure ℍ$ \mathbb{F}\mathfrak{M} $ \mathbb{F}\mathfrak{M} over a model $ \mathfrak{M} $ \mathfrak{M} . We consider the question on cardinalities of definable (interpretable) sets in superstructures over λ-homogeneous and λ-saturated models.     

Double sequence spaces defined by a modulus

This paper begins with new definitions for double sequence spaces. These new definitions are constructed, in general, by combining modulus function and nonnegative four-dimensional matrix. We use these definitions to establish inclusion theorems between various sequence spaces such as: If A = (a _m,n,k,l ) be a nonnegative four-dimensional matrix such that

Integral-type operators from a mixed norm space to a bloch-type space on the unit ball

Let $ \mathbb{B} $ \mathbb{B} be the unit ball in ℂ ⁿ and let H($ \mathbb{B} $ \mathbb{B} ) be the space of all holomorphic functions on $ \mathbb{B} $ \mathbb{B} . We introduce the following integral-type operator on H($ \mathbb{B} $ \mathbb{B} ):

Fourier-Feynman transforms of unbounded functionals on abstract Wiener space

Huffman, Park and Skoug established several results involving Fourier-Feynman transform and convolution for functionals in a Banach algebra S on the classical Wiener space. Chang, Kim and Yoo extended these results to abstract Wiener space for a more generalized Fresnel class $ \mathcal{F}_{\mathcal{A}_1 ,\mathcal{A}_2 } $ \mathcal{F}_{\mathcal{A}_1 ,\mathcal{A}_2 } _A1,A2 than the Fresnel class $ \mathcal{F} $ \mathcal{F} (B)which corresponds to the Banach algebra S. In this paper we study Fourier-Feynman transform, convolution and first variation of unbounded functionals on abstract Wiener space having the form

A note on nil power serieswise Armendariz rings

A ring R is called nil power serieswise Armendariz if $ \forall f = \sum\limits_{i = 0}^\infty {a_i X^i } $ \forall f = \sum\limits_{i = 0}^\infty {a_i X^i } and $ g = \sum\limits_{i = 0}^\infty {b_i X^i } $ g = \sum\limits_{i = 0}^\infty {b_i X^i } in R[[X]] such that f g ∈ Nil(R)[[X]], then a _i b _j ∈ Nil(R) for all i and j. In this note we characterize completely nil power serieswise Armendariz rings with their nilradical Nil(R) (where the nilradical is the set of nilpotent elements). We prove that a ring is nil power serieswise Armendariz if and only if Nil(R) is an ideal of R. We prove that each power serieswise Armendariz ring is nil power serieswise Armendariz and we give examples of nil power serieswise Armendariz rings.     

Hereditarily indecomposable Banach algebras of diagonal operators

We provide a characterization of the Banach spaces X with a Schauder basis (e _n ) _n∈ℕ which have the property that the dual space X* is naturally isomorphic to the space L _diag(X) of diagonal operators with respect to (e _n ) _n∈ℕ. We also construct a Hereditarily Indecomposable Banach space $ \mathfrak{X} $ \mathfrak{X} _D with a Schauder basis (e _n ) _n∈ℕ such that $ \mathfrak{X} $ \mathfrak{X} *_D is isometric to L _diag($ \mathfrak{X} $ \mathfrak{X} _D) with these Banach algebras being Hereditarily Indecomposable. Finally, we show that every T ∈ L _diag($ \mathfrak{X} $ \mathfrak{X} _D) is of the form T = λI + K, where K is a compact operator.     

The best extension of algebraic polynomials from the unit circle

We study the class $ \mathfrak{P}_n $ \mathfrak{P}_n of algebraic polynomials P _n (x, y) in two variables of total degree n whose uniform norm on the unit circle Γ₁ centered at the origin is at most 1: $ \left\| {P_n } \right\|_{C(\Gamma _1 )} $ \left\| {P_n } \right\|_{C(\Gamma _1 )} ≤ 1. The extension of polynomials from the class $ \mathfrak{P}_n $ \mathfrak{P}_n to the plane with the least uniform norm on the concentric circle Γ _r of radius r is investigated. It is proved that the values θ _n (r) of the best extension of the class $ \mathfrak{P}_n $ \mathfrak{P}_n satisfy the equalities θ _n (r) = r ⁿ for r > 1 and θ _n (r) = r ⁿ⁻¹ for 0 < r < 1.     

Imaginary powers of a Laguerre differential operator

Imaginary powers associated to the Laguerre differential operator $ L_\alpha = - \Delta + |x|^2 + \sum _{i = 1}^d \frac{1} {{x_i^2 }}(\alpha _i^2 - 1/4) $ L_\alpha = - \Delta + |x|^2 + \sum _{i = 1}^d \frac{1} {{x_i^2 }}(\alpha _i^2 - 1/4) are investigated. It is proved that for every multi-index α = (α₁,...α _d ) such that α _i ≧ −1/2, α _i ∉ (−1/2, 1/2), the imaginary powers $ \mathcal{L}_\alpha ^{ - i\gamma } ,\gamma \in \mathbb{R} $ \mathcal{L}_\alpha ^{ - i\gamma } ,\gamma \in \mathbb{R} , of a self-adjoint extension of L _α, are Calderón-Zygmund operators. Consequently, mapping properties of $ \mathcal{L}_\alpha ^{ - i\gamma } $ \mathcal{L}_\alpha ^{ - i\gamma } follow by the general theory.     

Multidimensional analog of the Hardy condition for power series

In the middle of the 20th century Hardy obtained a condition which must be imposed on a formal power series f(x) with positive coefficients in order that the series f ⁻¹(x) = $ \sum\limits_{n = 0}^\infty {b_n x^n } $ \sum\limits_{n = 0}^\infty {b_n x^n } b _n x ⁿ be such that b ₀ > 0 and b _n ≤ 0, n ≥ 1. In this paper we find conditions which must be imposed on a multidimensional series f(x ₁, x ₂, …, x _m ) with positive coefficients in order that the series f ⁻¹(x ₁, x ₂, …, x _m ) = $ \sum i_1 ,i_2 , \ldots ,i_m \geqslant 0^b i_1 ,i_2 , \ldots ,i_m ^{x_1^{i_1 } x_2^{i_2 } \ldots x_m^{i_m } } $ \sum i_1 ,i_2 , \ldots ,i_m \geqslant 0^b i_1 ,i_2 , \ldots ,i_m ^{x_1^{i_1 } x_2^{i_2 } \ldots x_m^{i_m } } satisfies the property b _{0, …, 0} > 0, $ bi_1 ,i_2 , \ldots ,i_m $ bi_1 ,i_2 , \ldots ,i_m ≤ 0, i ₁² + i ₂² + … + i _m ² > 0, which is similar to the one-dimensional case.     

Local limit theorem for the first crossing time of a fixed level by a random walk

Let X,X(1),X(2),... be independent identically distributed random variables with mean zero and a finite variance. Put S(n) = X(1) + ... + X(n), n = 1, 2,..., and define the Markov stopping time η _y = inf {n ≥ 1: S(n) ≥ y} of the first crossing a level y ≥ 0 by the random walk S(n), n = 1, 2,.... In the case $ \mathbb{E} $ \mathbb{E} |X|³ < ∞, the following relation was obtained in [8]: $ \mathbb{P}\left( {\eta _0 = n} \right) = \frac{1} {{n\sqrt n }}\left( {R + \nu _n + o\left( 1 \right)} \right) $ \mathbb{P}\left( {\eta _0 = n} \right) = \frac{1} {{n\sqrt n }}\left( {R + \nu _n + o\left( 1 \right)} \right) as n → ∞, where the constant R and the bounded sequence ν _n were calculated in an explicit form. Moreover, there were obtained necessary and sufficient conditions for the limit existence $ H\left( y \right): = \mathop {\lim }\limits_{n \to \infty } n^{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2}} \mathbb{P}\left( {\eta _y = n} \right) $ H\left( y \right): = \mathop {\lim }\limits_{n \to \infty } n^{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2}} \mathbb{P}\left( {\eta _y = n} \right) for every fixed y ≥ 0, and there was found a representation for H(y). The present paper was motivated by the following reason. In [8], the authors unfortunately did not cite papers [1, 5] where the above-mentioned relations were obtained under weaker restrictions. Namely, it was proved in [5] the existence of the limit $ \mathop {\lim }\limits_{n \to \infty } n^{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2}} \mathbb{P}\left( {\eta _y = n} \right) $ \mathop {\lim }\limits_{n \to \infty } n^{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2}} \mathbb{P}\left( {\eta _y = n} \right) for every fixed y ≥ 0 under the condition $ \mathbb{E} $ \mathbb{E} X ² < ∞ only; In [1], an explicit form of the limit $ \mathop {\lim }\limits_{n \to \infty } n^{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2}} \mathbb{P}\left( {\eta _0 = n} \right) $ \mathop {\lim }\limits_{n \to \infty } n^{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2}} \mathbb{P}\left( {\eta _0 = n} \right) was found under the same condition $ \mathbb{E} $ \mathbb{E} X ² < ∞ in the case when the summand X has an arithmetic distribution. In the present paper, we prove that the main assertion in [5] fails and we correct the original proof. It worth noting that this corrected version was formulated in [8] as a conjecture.     

Bicompletion of metrically generated constructs

We develop a general technique to study completeness in a metrically generated construct X, the objects of which can be isomorphically described as sets endowed with a gauge of quasi metrics, saturated in a certain way. Our approach depends on a given functor F on X, describing the transition to some known topological construct A in which there already exists a completion theory described by some reflector R and for which also the associated firm class of morphisms $ \mathfrak{L} $ \mathfrak{L} (R) is known. This functor will permit to “lift” the notion of completeness from A to X. We describe sufficient conditions on R and F, ensuring that the lifting produces a completion theory R^F for the construct X₀ of T ₀ objects and that the associated firm class of morphisms can be derived from $ \mathfrak{L} $ \mathfrak{L} (R). Application of this technique to concrete examples results in many interesting completion theories and provides us with a far better understanding of several existing ones.     

Higher derivations of ore extensions

Let R be a prime ring and δ a derivation of R. Divided powers $ D_n ^{\underline{\underline {def.}} } \tfrac{1} {{n!}}\tfrac{{d^n }} {{dx^n }} $ D_n ^{\underline{\underline {def.}} } \tfrac{1} {{n!}}\tfrac{{d^n }} {{dx^n }} of ordinary differentiation d/dx form Hasse-Schmidt higher derivations of the Ore extension (skew polynomial ring) R[x; δ]. They have been used crucially but implicitly in the investigation of R[x; δ]. Our aim is to explore this notion. The following is proved among others: Let Q be the left Martindale quotient ring of R. It is shown that $ S^{\underline{\underline {def.}} } Q[x;\delta ] $ S^{\underline{\underline {def.}} } Q[x;\delta ] is a quasi-injective (R, R)-module and that any (R,R)-bimodule endomorphism of S can be uniquely expressed in the form

On the shift semigroup on the Hardy space of Dirichlet series

We develop a Wold decomposition for the shift semigroup on the Hardy space $ \mathcal{H}^2 $ \mathcal{H}^2 of square summable Dirichlet series convergent in the half-plane $ \Re (s) > 1/2 $ \Re (s) > 1/2 . As an application we have that a shift invariant subspace of $ \mathcal{H}^2 $ \mathcal{H}^2 is unitarily equivalent to $ \mathcal{H}^2 $ \mathcal{H}^2 if and only if it has the form $ \phi \mathcal{H}^2 $ \phi \mathcal{H}^2 for some $ \mathcal{H}^2 $ \mathcal{H}^2 -inner function φ.     

Convergence rates of the LIL for random fields in Hilbert spaces

Considering the positive d-dimensional lattice point Z ₊ ^d (d ≥ 2) with partial ordering ≤, let {X _k: k ∈ Z ₊ ^d } be i.i.d. random variables taking values in a real separable Hilbert space (H, ‖ · ‖) with mean zero and covariance operator Σ, and set $ S_n = \sum\limits_{k \leqslant n} {X_k } $ S_n = \sum\limits_{k \leqslant n} {X_k } , n ∈ Z ₊ ^d . Let σ _i ², i ≥ 1, be the eigenvalues of Σ arranged in the non-increasing order and taking into account the multiplicities. Let l be the dimension of the corresponding eigenspace, and denote the largest eigenvalue of Σ by σ ². Let logx = ln(x ∨ e), x ≥ 0. This paper studies the convergence rates for $ \sum\limits_n {\frac{{\left( {\log \log \left| n \right|} \right)^b }} {{\left| n \right|\log \left| n \right|}}} P\left( {\left\| {S_n } \right\| \geqslant \sigma \varepsilon \sqrt {2\left| n \right|\log \log \left| n \right|} } \right) $ \sum\limits_n {\frac{{\left( {\log \log \left| n \right|} \right)^b }} {{\left| n \right|\log \left| n \right|}}} P\left( {\left\| {S_n } \right\| \geqslant \sigma \varepsilon \sqrt {2\left| n \right|\log \log \left| n \right|} } \right) . We show that when l ≥ 2 and b > −l/2, E[‖X‖²(log ‖X‖) ^d−2(log log ‖X‖) ^b+4] < ∞ implies $ \begin{gathered} \mathop {\lim }\limits_{\varepsilon \searrow \sqrt {d - 1} } (\varepsilon ^2 - d + 1)^{b + l/2} \sum\limits_n {\frac{{\left( {\log \log \left| n \right|} \right)^b }} {{\left| n \right|\log \left| n \right|}}P\left( {\left\| {S_n } \right\| \geqslant \sigma \varepsilon \sqrt 2 \left| n \right|\log \log \left| n \right|} \right)} \hfill \\ = \frac{{K(\Sigma )(d - 1)^{\frac{{l - 2}} {2}} \Gamma (b + l/2)}} {{\Gamma (l/2)(d - 1)!}} \hfill \\ \end{gathered} $ \begin{gathered} \mathop {\lim }\limits_{\varepsilon \searrow \sqrt {d - 1} } (\varepsilon ^2 - d + 1)^{b + l/2} \sum\limits_n {\frac{{\left( {\log \log \left| n \right|} \right)^b }} {{\left| n \right|\log \left| n \right|}}P\left( {\left\| {S_n } \right\| \geqslant \sigma \varepsilon \sqrt 2 \left| n \right|\log \log \left| n \right|} \right)} \hfill \\ = \frac{{K(\Sigma )(d - 1)^{\frac{{l - 2}} {2}} \Gamma (b + l/2)}} {{\Gamma (l/2)(d - 1)!}} \hfill \\ \end{gathered} , where Γ(·) is the Gamma function and $ \prod\limits_{i = l + 1}^\infty {((\sigma ^2 - \sigma _i^2 )/\sigma ^2 )^{ - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} } $ \prod\limits_{i = l + 1}^\infty {((\sigma ^2 - \sigma _i^2 )/\sigma ^2 )^{ - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} } .     

On $\frak {F}$-normal Fitting classes of finite soluble groups

Let \frak X, \frak F,\frak X\subseteqq \frak F\frak {X}, \frak {F},\frak {X}\subseteqq \frak {F}, be non-trivial Fitting classes of finite soluble groups such that G_{\frak X}G_{\frak {X}} is an \frak X\frak {X}-injector of G for all G ? \frak FG\in \frak {F}. Then \frak X\frak {X} is called \frak F\frak {F}-normal. If \frak F=\frak S_p\frak {F}=\frak {S}_{\pi }, it is known that (1) \frak X\frak {X} is \frak F\frak {F}-normal precisely when \frak X^*=\frak F^*\frak {X}^{\ast }=\frak {F}^{\ast }, and consequently (2) \frak F í \frak X\frak N\frak {F}\subseteq \frak {X}\frak {N} implies \frak X^*=\frak F^*\frak {X}^{\ast }=\frak {F}^{\ast }, and (3) there is a unique smallest \frak F\frak {F}-normal Fitting class. These assertions are not true in general. We show that there are Fitting classes \frak F\not = \frak S_p\frak {F}\not =\frak {S}_{\pi } filling property (1), whence the classes \frak S_p\frak {S}_{\pi } are not characterized by satisfying (1). Furthermore we prove that (2) holds true for all Fitting classes \frak F\frak {F} satisfying a certain extension property with respect to wreath products although there could be an \frak F\frak {F}-normal Fitting class outside the Lockett section of \frak F\frak {F}. Lastly, we show that for the important cases \frak F=\frak Nⁿ, n\geqq 2\frak {F}=\frak {N}^{n},\ n\geqq 2, and \frak F=\frak S_p1?\frak S_pr, p_i \frak {F}=\frak {S}_{p_{1}}\cdots \frak {S}_{p_{r}},\ p_{i} primes, there is a unique smallest \frak F\frak {F}-normal Fitting class, which we describe explicitly.     

A note on the powers of Cesàro bounded operators

In this note we give a negative answer to Zemánek’s question (1994) of whether it always holds that a Cesàro bounded operator T on a Hilbert space with a single spectrum satisfies $ \mathop {\lim }\limits_{n \to \infty } $ \mathop {\lim }\limits_{n \to \infty } ∥T ⁿ⁺¹ − T ⁿ ∥ = 0.