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For any x 휖 (0, 1], let the series ∑n=1∞1/dnx be the Sylvester expansion of x, where {dj(x), j ≥ 1} is a sequence of positive integers satisfying d1(x) ≥ 2 and dj + 1(x) ≥ dj(x)(dj(x) − 1) + 1 for j ≥ 1. Suppose ϕ : ℕ → ℝ+ is a function satisfying ϕ(n+1) - ϕ (n) → ∞ as n → ∞. In this paper, we consider the setEϕ=x∈01:limn→∞logdnx−∑j=1n−1logdjxϕn=1and quantify the size of the set in the sense of Hausdorff dimension. As applications, for any β > 1 and γ > 0, we get the Hausdorff dimension of the set x∈01:limn→∞logdnx−∑j=1n−1logdjx/nβ=γ, and for any τ > 1 and η > 0, we get a lower bound of the Hausdorff dimension of the set x∈01:limn→∞logdnx−∑j=1n−1logdjx/τn=η. [ABSTRACT FROM AUTHOR]

The present paper continues a series of papers of the author (some of them are written in collaboration), in which the Yoneda algebras are calculated for several families of algebras of dihedral and semidihedral type (in K. Erdmann’s classification). In the paper, the Yoneda algebras are described (in terms of quivers with relations) for the algebras of semidihedral type that form the family SD(3 $$\mathcal{K}$$ ). Bibliography: 10 titles. [ABSTRACT FROM AUTHOR]

Let ${\mathfrak{g}}$ be a finite dimensional simple Lie algebra over an algebraically closed field $\mathbb{K}$ of characteristic 0. Let ${\mathfrak{g}}_{{\mathbb{Z}}}$ be a Chevalley ℤ-form of ${\mathfrak{g}}$ and ${\mathfrak{g}}_{\Bbbk}={\mathfrak{g}}_{{\mathbb{Z}}}\otimes _{{\mathbb{Z}}}\Bbbk$, where $\Bbbk$ is the algebraic closure of ${\mathbb{F}}_{p}$. Let $G_{\Bbbk}$ be a simple, simply connected algebraic $\Bbbk$-group with $\operatorname{Lie}(G_{\Bbbk})={\mathfrak{g}}_{\Bbbk}$. In this paper, we apply recent results of Rudolf Tange on the fraction field of the centre of the universal enveloping algebra $U({\mathfrak{g}}_{\Bbbk})$ to show that if the Gelfand–Kirillov conjecture (from 1966) holds for ${\mathfrak{g}}$, then for all p≫0 the field of rational functions $\Bbbk ({\mathfrak{g}}_{\Bbbk})$ is purely transcendental over its subfield $\Bbbk({\mathfrak{g}}_{\Bbbk})^{G_{\Bbbk}}$. Very recently, it was proved by Colliot-Thélène, Kunyavskiĭ, Popov, and Reichstein that the field of rational functions ${\mathbb{K}}({\mathfrak{g}})$ is not purely transcendental over its subfield ${\mathbb{K}}({\mathfrak{g}})^{\mathfrak{g}}$ if ${\mathfrak{g}}$ is of type B n, n≥3, D n, n≥4, E6, E7, E8 or F4. We prove a modular version of this result (valid for p≫0) and use it to show that, in characteristic 0, the Gelfand–Kirillov conjecture fails for the simple Lie algebras of the above types. In other words, if ${\mathfrak{g}}$ is of type B n, n≥3, D n, n≥4, E6, E7, E8 or F4, then the Lie field of ${\mathfrak{g}}$ is more complicated than expected. [ABSTRACT FROM AUTHOR]

The paper studies the region of values of the system { f( z1), f( z2), c2},where z j, j=1, 2, are arbitrary fixed points of the disk | z|<1; f ∈ T, and the class T consists of all functions f( z) = z + c2 z2 + ··· regular in the disk | z| < 1 and satisfying the condition Im f( z)·Im z>0 for Im z > 0 for Im z ≠ 0. The region of values of f( z1) in the subclass of functions f ( z) ∈ T with prescribed values c2 and f( z2) is determined. Bibliography: 8 titles. [ABSTRACT FROM AUTHOR]

The paper considers the capacity of a polycondenser and the module of a family of measures associated with the class of curves connecting the polycondenser’s plates. It is proved that these quantities are equal. Also a relation between the polycondenser’s capacity and the module of a family of vector measures associated with the class of functions admissible for the polycondenser’s capacity is established. The results obtained generalize the earlier results by M. Ohtsuka and H. Aikawa. [ABSTRACT FROM AUTHOR]

For a locally compact group G, the measure convolution algebra M( G) carries a natural coproduct. In previous work, we showed that the canonical predual C0( G) of M( G) is the unique predual which makes both the product and the coproduct on M( G) weak*-continuous. Given a discrete semigroup S, the convolution algebra ℓ1( S) also carries a coproduct. In this paper we examine preduals for ℓ1( S) making both the product and the coproduct weak*-continuous. Under certain conditions on S, we show that ℓ1( S) has a unique such predual. Such S include the free semigroup on finitely many generators. In general, however, this need not be the case even for quite simple semigroups and we construct uncountably many such preduals on ℓ1( S) when S is either ℤ+×ℤ or (ℕ,⋅). [ABSTRACT FROM AUTHOR]

This paper investigates the structure of the quasi-spectrum of divided powers and its behavior under the Taylor homomorphism. General as well as stronger particular results are obtained. In particular, the answer for the Hurwitz series is stated. [ABSTRACT FROM AUTHOR]

In this paper, we solve the problem proposed by Lan Wen for the case of dim M = 3. Roughly speaking, we prove that for fixed i, f has C1 persistently no small angles of index i if and only if f has a dominated splitting of index i on the C1 i-preperiodic set P( f). [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to establish the second-order nonsmooth sufficient unconstrained optimality condition for so called ℓ-stable at some point functions and in this way to generalize some previous results in this direction. We provide the comparisons with other results by examples. [ABSTRACT FROM AUTHOR]

The paper studies the structure of k-vertex cutsets in a k-connected graph. As an illustration of the method proposed, the well-known case of a biconnected graph is considered. Bibliography: 8 titles. [ABSTRACT FROM AUTHOR]

In this paper, it is proved that with at most $$ O{\left( {N^{{\frac{{65}} {{66}}}} } \right)} $$ exceptions, all even positive integers up to N are expressible in the form $$ p^{2}_{2} + p^{3}_{3} + p^{4}_{4} + p^{5}_{5} $$ . This improves a recent result $$ O{\left( {N^{{\frac{{19193}} {{19200}} + \varepsilon }} } \right)} $$ due to C. Bauer. [ABSTRACT FROM AUTHOR]

Let X be an irreducible algebraic variety over ℂ, endowed with an algebraic foliation $$ {\user1{\mathcal{F}}} $$ . In this paper, we introduce the notion of minimal invariant variety V( $$ {\user1{\mathcal{F}}} $$ , Y) with respect to ( $$ {\user1{\mathcal{F}}} $$ , Y), where Y is a subvariety of X. If Y = { x} is a smooth point where the foliation is regular, its minimal invariant variety is simply the Zariski closure of the leaf passing through x. First we prove that for very generic x, the varieties V( $$ {\user1{\mathcal{F}}} $$ , x) have the same dimension p. Second we generalize a result due to X. Gomez- Mont (see [G-M]). More precisely, we prove the existence of a dominant rational map F : X → Z, where Z has dimension ( n − p), such that for very generic x, the Zariski closure of F −1( F( x)) is one and only one minimal invariant variety of a point. We end up with an example illustrating both results. [ABSTRACT FROM AUTHOR]

This paper develops two necessary conditions for a local minimum of an arbitrary extended real-valued function (Theorem 2.2) and, quite separately, two conditions, each sufficient for such a function to have a strict local minimum (Theorem 2.3). [ABSTRACT FROM AUTHOR]

Łu logic plays a fundamental role among many-valued logics. However, the expressive power of this logic is restricted to piecewise linear functions. In this paper we enrich the language of Łu logic by adding a new connective which expresses multiplication. The resulting logic, PŁ, is defined, developed, and put into the context of other well-known many-valued logics. We also deal with several extensions of this propositional logic. A predicate version of PŁ logic is introduced and developed too. [ABSTRACT FROM AUTHOR]

In this paper we study the structure of positive homomorphisms on real function algebras. We prove that every positive homomorphism is completely characterized by a family of sets and when the algebra is invert-closed, by an ultrafilter of zero-sets of functions of the algebra. We show that the known sufficient conditions for every homomorphism of a real function algebra to be countably evaluating or a point evaluation are not necessary. Our results enable us to characterize the countably evaluating algebras as well as the Lindelöf spaces as the spaces in which for every algebra, each countably evaluating homomorphism is a point evaluation. [ABSTRACT FROM AUTHOR]

Pseudoeffect (PE) algebras have been introduced as a noncommutative generalization of effect algebras. We study in this paper PE algebras with the special property of having a nonempty state space. To this end, we consider PE algebras which are po-group intervals and which are, in a certain sense, noncommutative only in the small. Such a PE algebra is shown to possess a nontrivial commutative homomorphic image from which then follows that there exist states. A typical example is given by an interval of the lexicographical product of two po-groups the first of which is abelian. [ABSTRACT FROM AUTHOR]