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1782. A note on the three-segment problem
- Creator:
- Doležal, Martin
- Format:
- bez média and svazek
- Type:
- model:article and TEXT
- Subject:
- three-segment problem and cluster sets
- Language:
- English
- Description:
- We improve a theorem of C. L. Belna (1972) which concerns boundary behaviour of complex-valued functions in the open upper half-plane and gives a partial answer to the (still open) three-segment problem.
- Rights:
- http://creativecommons.org/publicdomain/mark/1.0/ and policy:public
1783. A note on the transcendence of infinite products
- Creator:
- Hančl, Jaroslav, Kolouch, Ondřej, Pulcerová, Simona, and Štěpnička, Jan
- Format:
- bez média and svazek
- Type:
- model:article and TEXT
- Subject:
- transcendence and infinite product
- Language:
- English
- Description:
- The paper deals with several criteria for the transcendence of infinite products of the form $\prod _{n=1}^\infty {[b_n\alpha ^{a_n}]}/{b_n\alpha ^{a_n}}$ where $\alpha >1$ is a positive algebraic number having a conjugate $\alpha ^*$ such that $\alpha \not =|\alpha ^*|>1$, $\{a_n\}_{n=1}^\infty $ and $\{b_n\}_{n=1}^\infty $ are two sequences of positive integers with some specific conditions. \endgraf The proofs are based on the recent theorem of Corvaja and Zannier which relies on the Subspace Theorem ({P. Corvaja, U. Zannier}: On the rational approximation to the powers of an algebraic number: solution of two problems of Mahler and Mendès France, Acta Math. 193, (2004), 175–191).
- Rights:
- http://creativecommons.org/publicdomain/mark/1.0/ and policy:public
1784. A note on topological groups and their remainders
- Creator:
- Peng, Liang-Xue and He, Yu-Feng
- Format:
- bez média and svazek
- Type:
- model:article and TEXT
- Subject:
- topological group, remainder, compactification, metrizable space, and weak base
- Language:
- English
- Description:
- In this note we first give a summary that on property of a remainder of a non-locally compact topological group $G$ in a compactification $bG$ makes the remainder and the topological group $G$ all separable and metrizable. If a non-locally compact topological group $G$ has a compactification $bG$ such that the remainder $bG\setminus G$ of $G$ belongs to $\mathcal {P}$, then $G$ and $bG\setminus G$ are separable and metrizable, where $\mathcal {P}$ is a class of spaces which satisfies the following conditions: (1) if $X\in \mathcal {P}$, then every compact subset of the space $X$ is a $G_\delta $-set of $X$; (2) if $X\in \mathcal {P}$ and $X$ is not locally compact, then $X$ is not locally countably compact; (3) if $X\in \mathcal {P}$ and $X$ is a Lindelöf $p$-space, then $X$ is metrizable. Some known conclusions on topological groups and their remainders can be obtained from this conclusion. As a corollary, we have that if a non-locally compact topological group $G$ has a compactification $bG$ such that compact subsets of $bG\setminus G$ are $G_{\delta }$-sets in a uniform way (i.e., $bG\setminus G$ is CSS), then $G$ and $bG\setminus G$ are separable and metrizable spaces. In the last part of this note, we prove that if a non-locally compact topological group $G$ has a compactification $bG$ such that the remainder $bG\setminus G$ has a point-countable weak base and has a dense subset $D$ such that every point of the set $D$ has countable pseudo-character in the remainder $bG\setminus G$ (or the subspace $D$ has countable $\pi $-character), then $G$ and $bG\setminus G$ are both separable and metrizable.
- Rights:
- http://creativecommons.org/publicdomain/mark/1.0/ and policy:public
1785. A note on transitively $D$-spaces
- Creator:
- Peng, Liang-Xue
- Format:
- bez média and svazek
- Type:
- model:article and TEXT
- Subject:
- transitively $D$, sequential, discretely Lindelöf, and $wcs^*$-network
- Language:
- English
- Description:
- In this note, we show that if for any transitive neighborhood assignment $\phi $ for $X$ there is a point-countable refinement ${\mathcal F}$ such that for any non-closed subset $A$ of $X$ there is some $V\in {\mathcal F}$ such that $|V\cap A|\geq \omega $, then $X$ is transitively $D$. As a corollary, if $X$ is a sequential space and has a point-countable $wcs^*$-network then $X$ is transitively $D$, and hence if $X$ is a Hausdorff $k$-space and has a point-countable $k$-network, then $X$ is transitively $D$. We prove that if $X$ is a countably compact sequential space and has a point-countable $wcs^*$-network, then $X$ is compact. We point out that every discretely Lindelöf space is transitively $D$. Let $(X, \tau )$ be a space and let $(X, {\mathcal T})$ be a butterfly space over $(X, \tau )$. If $(X, \tau )$ is Fréchet and has a point-countable $wcs^*$-network (or is a hereditarily meta-Lindelöf space), then $(X, {\mathcal T})$ is a transitively $D$-space.
- Rights:
- http://creativecommons.org/publicdomain/mark/1.0/ and policy:public
1786. A note on triangular schemes for weak congruences
- Creator:
- Chajda, Ivan, Šešelja, Branimir, and Tepavčević, Andreja
- Format:
- bez média and svazek
- Type:
- model:article and TEXT
- Subject:
- triangular scheme, triangular principle, weak congruence, weak congruence modularity, and weak congruence distributivity
- Language:
- English
- Description:
- Some geometrical methods, the so called Triangular Schemes and Principles, are introduced and investigated for weak congruences of algebras. They are analogues of the corresponding notions for congruences. Particular versions of Triangular Schemes are equivalent to weak congruence modularity and to weak congruence distributivity. For algebras in congruence permutable varieties, stronger properties—Triangular Principles—are equivalent to weak congruence modularity and distributivity.
- Rights:
- http://creativecommons.org/publicdomain/mark/1.0/ and policy:public
1787. A note on ultrametric matrices
- Creator:
- Zhang, Xiao-Dong
- Format:
- bez média and svazek
- Type:
- model:article and TEXT
- Subject:
- generalized ultrametric matrix, $ \mathcal U$ matrix, weighted graph, and inverse $M$-matrix
- Language:
- English
- Description:
- It is proved in this paper that special generalized ultrametric and special $\mathcal U$ matrices are, in a sense, extremal matrices in the boundary of the set of generalized ultrametric and $ \mathcal U$ matrices, respectively. Moreover, we present a new class of inverse $M$-matrices which generalizes the class of $\mathcal U$ matrices.
- Rights:
- http://creativecommons.org/publicdomain/mark/1.0/ and policy:public
1788. A note on weak solutions to stochastic differential equations
- Creator:
- Ondreját, Martin and Seidler, Jan
- Format:
- bez média and svazek
- Type:
- model:article and TEXT
- Subject:
- stochastic differential equations, continuous coefficients, and weak solutions
- Language:
- English
- Description:
- We revisit the proof of existence of weak solutions of stochastic differential equations with continuous coeficients. In standard proofs, the coefficients are approximated by more regular ones and it is necessary to prove that: i) the laws of solutions of approximating equations form a tight set of measures on the paths space, ii) its cluster points are laws of solutions of the limit equation. We aim at showing that both steps may be done in a particularly simple and elementary manner.
- Rights:
- http://creativecommons.org/publicdomain/mark/1.0/ and policy:public
1789. A note on weakly (µ, λ)-closed functions
- Creator:
- Roy, Bishwambhar
- Format:
- bez média and svazek
- Type:
- model:article and TEXT
- Subject:
- µ-open set, weakly (µ, λ)-closed function, contra (µ, λ)-open function, and strongly (µ, λ)-continuous function
- Language:
- English
- Description:
- In this paper we introduce a new class of functions called weakly (µ, λ)-closed functions with the help of generalized topology which was introduced by Á. Császár. Several characterizations and some basic properties of such functions are obtained. The connections between these functions and some other similar types of functions are given. Finally some comparisons between different weakly closed functions are discussed. This weakly (µ, λ)- closed functions enable us to facilitate the formulation of certain unified theories for different weaker forms of closed functions.
- Rights:
- http://creativecommons.org/publicdomain/mark/1.0/ and policy:public
1790. A note on weakly Lindelöf determined Banach spaces
- Creator:
- Gonzáles, A. and Montesinos, Vicente
- Format:
- bez média and svazek
- Type:
- model:article and TEXT
- Subject:
- projectional generator, projectional resolution of the identity, weakly Lindelöf determined Banach space, Markushevich base, and Corson compacta
- Language:
- English
- Description:
- We prove that weakly Lindelöf determined Banach spaces are characterized by the existence of a ``full'' projectional generator. Some other results pertaining to this class of Banach spaces are given.
- Rights:
- http://creativecommons.org/publicdomain/mark/1.0/ and policy:public