Authors:M. Khrypchenko Pages: 48 - 55 Abstract: Let P be a partially ordered set, R a commutative ring with identity and FI(P,R) the finitary incidence algebra of P over R. We prove that each R-linear local derivation of FI(P,R) is a derivation, which partially generalizes Theorem 3 of [21]. PubDate: 2018-02-01 DOI: 10.1007/s10474-017-0758-7 Issue No:Vol. 154, No. 1 (2018)

Authors:S. De Marchi; A. Kroó Pages: 69 - 89 Abstract: We investigate Marcinkiewicz–Zygmund type inequalities for multivariate polynomials on various compact domains in \({\mathbb{R}^d}\) . These inequalities provide a basic tool for the discretization of the L p norm and are widely used in the study of the convergence properties of Fourier series, interpolation processes and orthogonal expansions. Recently Marcinkiewicz–Zygmund type inequalities were verified for univariate polynomials for the general class of doubling weights, and for multivariate polynomials on the ball and sphere with doubling weights. The main goal of the present paper is to extend these considerations to more general multidimensional domains, which in particular include polytopes, cones, spherical sectors, toruses, etc. Our approach will rely on application of various polynomial inequalities, such as Bernstein–Markov, Schur and Videnskii type estimates, and also using symmetry and rotation in order to generate results on new domains. PubDate: 2018-02-01 DOI: 10.1007/s10474-017-0769-4 Issue No:Vol. 154, No. 1 (2018)

Authors:C. L. Samuels Pages: 105 - 123 Abstract: For an algebraic number α, the metric Mahler measure \({m_1(\alpha)}\) was first studied by Dubickas and Smyth [4] and was later generalized to the t-metric Mahler measure \({m_t(\alpha)}\) by the author [16]. The definition of \({m_t(\alpha)}\) involves taking an infimum over a certain collection N-tuples of points in \(\overline{\mathbb{Q}}\) , and from previous work of Jankauskas and the author, the infimum in the definition of \({m_t(\alpha)}\) is attained by rational points when \({\alpha\in \mathbb{Q}}\) . As a consequence of our main theorem in this article, we obtain an analog of this result when \({\mathbb{Q}}\) is replaced with any imaginary quadratic number field of class number equal to 1. Further, we study examples of other number fields to which our methods may be applied, and we establish various partial results in those cases. PubDate: 2018-02-01 DOI: 10.1007/s10474-017-0770-y Issue No:Vol. 154, No. 1 (2018)

Authors:E. Scheidecker; A. Stanley Pages: 124 - 133 Abstract: A space is left-separated if it has a well ordering for which initial segments are closed. We explore when the union of two left-separated spaces must be left-separated. We prove that if X and Y are left-separated and \({X \cup Y}\) is locally countable, then whenever \({{\rm ord}_\ell(Y ) \leq \omega_{1}, X \cup Y}\) is left-separated. In 1986, Fleissner [2] proved that if a space has a point-countable base, then it is left-separated if and only if it is \({\sigma}\) -weakly separated. We provide a new proof of this result using elementary submodels and add an additional characterization. PubDate: 2018-02-01 DOI: 10.1007/s10474-017-0771-x Issue No:Vol. 154, No. 1 (2018)

Authors:S. Hu; M.-S. Kim Pages: 134 - 146 Abstract: We provide several properties of hypergeometric Bernoulli numbers and polynomials, including sums of products identity, differential equations and recurrence formulas. PubDate: 2018-02-01 DOI: 10.1007/s10474-017-0767-6 Issue No:Vol. 154, No. 1 (2018)

Authors:P. Komjáth Pages: 215 - 222 Abstract: Let \({\mu \geq \omega}\) be regular, assume the Generalized Continuum Hypothesis and the principle \({\square_\lambda}\) holds for every singular \({\lambda}\) with \({{\rm cf}(\lambda) \leq \mu}\) . Let X be a graph with chromatic number greater than \({\mu^+}\) . Then X contains a \({\mu}\) -connected subgraph Y of X whose chromatic number is greater than \({\mu^+}\) . PubDate: 2018-02-01 DOI: 10.1007/s10474-017-0752-0 Issue No:Vol. 154, No. 1 (2018)

Authors:Z. Wang Pages: 223 - 230 Abstract: We generalize [4, Theorem 4.3] to the case of Hopf–Galois extension, by introducing the cotensor product of a comodule algebra and its opposite algebra, and then give some applications. PubDate: 2018-02-01 DOI: 10.1007/s10474-017-0765-8 Issue No:Vol. 154, No. 1 (2018)

Authors:A. Karagila Pages: 231 - 242 Abstract: We show that it is equiconsistent with \({\mathsf{ZF}}\) that Fodor’s lemma fails everywhere, and furthermore that the club filter on every regular cardinal is not even \({\sigma}\) -complete. Moreover, these failures can be controlled in a very precise manner. PubDate: 2018-02-01 DOI: 10.1007/s10474-017-0768-5 Issue No:Vol. 154, No. 1 (2018)

Authors:F. Matúš Abstract: Given finitely many events in a probability space, conditional independences among the indicators of events are considered simultaneously with the signs of covariances. Resulting discrete structures are studied restricting attention mostly to all couples and triples of events. Necessary and sufficient conditions for such structures to be represented by events are found. Consequences of the results for the patterns of conjunctive forks are discussed. PubDate: 2018-02-22 DOI: 10.1007/s10474-018-0799-6

Authors:S. N. Begum; C. Nag; M. R. Talukder Abstract: We discuss congruences of p-algebras. We characterize kernel ideals of a p-algebra. Indeed, we show that an ideal of a p-algebra is a p-ideal if and only if it is a kernel ideal. We study cokernel filters of a p-algebra. We construct a class of p-algebras in which every cokernel filter is a p-filter. We also give some characterizations of Boolean congruences of a p-algebra. PubDate: 2018-02-22 DOI: 10.1007/s10474-018-0793-z

Authors:V. Bernik; N. Budarina; H. O’Donnell Abstract: An upper bound for the number of cubic polynomials which have small discriminant in terms of the Euclidean and p-adic metrics simultaneously is obtained. PubDate: 2018-02-22 DOI: 10.1007/s10474-018-0794-y

Authors:G. Hegedűs Abstract: We combine here Tao’s slice-rank bounding method and Gröbner basis techniques and apply it to the Erdős–Rado Sunflower Conjecture. Let \({0\leq k\leq n}\) be integers. We prove that if \({\mathcal{F}}\) is a k-uniform family of subsets of [n] without a sunflower with 3 petals, then $$ \mathcal{F} \leq3 \left(\begin{array}{c} {n }\\ \lfloor n/3\rfloor \end{array}\right).$$ This result allows us to improve slightly a recent upper bound of Naslund and Sawin for the size of a sunflower-free family in 2[n]. PubDate: 2018-02-22 DOI: 10.1007/s10474-018-0798-7

Authors:A. V. Osipov Abstract: For a Tychonoff space X, we denote by C p (X) the space of all real-valued continuous functions on X with the topology of pointwise convergence. In this paper we prove that: If every finite power of X is Lindelöf then C p (X) is strongly sequentially separable iff X is \({\gamma}\) -set. \({B_{\alpha}(X)}\) (= functions of Baire class \({\alpha}\) ( \({1 < \alpha \leq \omega_1}\) ) on a Tychonoff space X with the pointwise topology) is sequentially separable iff there exists a Baire isomorphism class \({\alpha}\) from a space X onto a \({\sigma}\) -set. \({B_{\alpha}(X)}\) is strongly sequentially separable iff \({iw(X)=\aleph_0}\) and X is a \({Z^{\alpha}}\) -cover \({\gamma}\) -set for \({0 < \alpha \leq \omega_1}\) . There is a consistent example of a set of reals X such that C p (X) is strongly sequentially separable but B1(X) is not strongly sequentially separable. B(X) is sequentially separable but is not strongly sequentially separable for a \({\mathfrak{b}}\) -Sierpiński set X. PubDate: 2018-02-22 DOI: 10.1007/s10474-018-0800-4

Authors:J. Cossey; Y.-M. Li Abstract: Let a finite group \({G = AB}\) be the product of the mutually permutable subgroups A and B. We investigate the structure of G given by conditions on conjugacy class sizes of elements in \({A \cup B}\) . Some recent results are extended. PubDate: 2018-02-22 DOI: 10.1007/s10474-018-0796-9

Authors:Z. Boros; E. Garda-Mátyás Abstract: We consider quadratic functions f that satisfy the additional equation y2 f(x) = x2 f(y) for the pairs \({ (x,y) \in \mathbb{R}^2}\) that fulfill the condition P(x, y) = 0 for some fixed polynomial P of two variables. If P(x, y) = ax + by + c with \({ a , b , c \in \mathbb{R}}\) and \({(a^2 + b^2)c \neq 0}\) or P(x,y) = x n − y with a natural number \({n \geq 2}\) , we prove that f(x) = f(1) x2 for all \({x \in \mathbb{R}}\) . Some related problems, admitting quadratic functions generated by derivations, are considered as well. PubDate: 2018-02-22 DOI: 10.1007/s10474-018-0795-x

Authors:A. Fošner; X.-F. Liang; F. Wei Abstract: Let \({\mathcal{T}}\) be a triangular algebra over a commutative ring \({\mathcal{R}}\) , \({\xi}\) be an automorphism of \({\mathcal{T}}\) and \({\mathcal{Z}_{\xi}(\mathcal{T})}\) be the \({\xi}\) -center of \({\mathcal{T}}\) . Suppose that \({\mathfrak{q} \colon \mathcal{T} \times \mathcal{T} \longrightarrow \mathcal{T}}\) is an \({\mathcal{R}}\) -bilinear mapping and that \({\mathfrak{T}_{\mathfrak{q}} \colon \mathcal{T} \longrightarrow \mathcal{T}}\) is a trace of \({\mathfrak{q}}\) . The aim of this article is to describe the form of \({\mathfrak{T}_{\mathfrak{q}}}\) satisfying the commuting condition \({[\mathfrak{T}_{\mathfrak{q}}(x), x]_{\xi}=0}\) (resp. the centralizing condition \({[\mathfrak{T}_{\mathfrak{q}}(x), x]_{\xi} \in \mathcal{Z}_\xi(\mathcal{T})}\) ) for all \({x\in \mathcal{T}}\) . More precisely, we will consider the question of when \({\mathfrak{T}_{\mathfrak{q}}}\) satisfying the previous condition has the so-called proper form. PubDate: 2018-02-22 DOI: 10.1007/s10474-018-0797-8

Authors:M. Almahalebi Abstract: Using the fixed point method, we investigate the stability of a generalization of Jensen functional equation $$ \sum_{k=0}^{n-1} f(x+ b_{k}y)=nf(x),$$ where \({n \in \mathbb{N}_{2}}\) , \({b_{k}=\exp(\frac{2i\pi k}{n})}\) for \({0\leq k \leq n-1}\) , in Banach spaces. Also, we prove the hyperstability results of this equation by the fixed point method. PubDate: 2017-12-15 DOI: 10.1007/s10474-017-0781-8

Authors:Y. Y. Zheng Abstract: A hierarchy of topological Ramsey spaces \({\mathcal{R}_\alpha}\) ( \({\alpha < \omega_1}\) ), generalizing the Ellentuck space, were built by Dobrinen and Todorcevic in order to completely classify certain equivalent classes of ultrafilters Tukey (resp. Rudin–Keisler) below \({\mathcal{U}_\alpha}\) \({(\alpha < \omega_1)}\) , where \({\mathcal{U}_\alpha}\) are ultrafilters constructed by Laflamme satisfying certain partition properties and have complete combinatorics over the Solovay model. We show that Nash–Williams, or Ramsey ultrafilters in these spaces are preserved under countable-support side-by-side Sacks forcing. This is achieved by proving a parametrized theorem for these spaces, and showing that Nash–Williams ultrafilters localizes the theorem. We also show that every Nash–Williams ultrafilter in \({\mathcal{R}_\alpha}\) is selective. PubDate: 2017-12-14 DOI: 10.1007/s10474-017-0780-9

Authors:C.-Y. Cao; Y.-H. Chen Abstract: For any real number β > 1, let S n (β) be the partial sum of the first n items of the β-expansion of 1. It was known that the approximation order of 1 by S n (β) is β −n for Lebesgue almost all β > 1. We consider the size of the set of β > 1 for which 1 can be approximated with the other orders \({\beta^{-\varphi(n)}}\) , where \({\varphi}\) is a positive function defined on \({\mathbb N}\) . More precisely, the size of the sets $$\left\{\beta\in \mathfrak{B}:\limsup_{n\rightarrow\infty}\frac{\log_{\beta}(1-S_n(\beta))}{\varphi(n)}=-1\right\}$$ and $$\left\{\beta\in \mathfrak{B}:\liminf_{n\rightarrow\infty}\frac{\log_{\beta}(1-S_n(\beta))}{\varphi(n)}=-1\right\}$$ are determined, where \({\mathfrak{B}=\{ \beta>1:\beta \text{ is not a simple Parry number}\}}\) . PubDate: 2017-12-14 DOI: 10.1007/s10474-017-0776-5

Authors:T. Akhobadze; Sh. Zviadadze Abstract: The assertion that the Salem test [5] for the uniform convergence of a trigonometric Fourier series is improvable, is proved. In particular, an example of a continuous function, which does not fulfill the condition of the Salem test but satisfies the condition of the generalized Salem test [10], is constructed. Besides, the theorem which improves Golubov’s [3,4] result for continuous functions of two variables, is given. PubDate: 2017-12-14 DOI: 10.1007/s10474-017-0774-7