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We establish a kind of subadjunction formula for quasi-log canonical pairs. As an application, we prove that a connected projective quasi-log canonical pair whose quasi-log canonical class is anti-ample is simply connected and rationally chain connected. We also supplement the cone theorem for quasi-log canonical pairs. More precisely, we prove that every negative extremal ray is spanned by a rational curve. Finally, we treat the notion of Mori hyperbolicity for quasi-log canonical pairs. [ABSTRACT FROM AUTHOR]

This paper proposes full convexity as an alternative definition of digital convexity, which is valid in arbitrary dimension. It solves many problems related to its usual definitions, like possible non-connectedness or non-simple connectedness, while encompassing its desirable features. Fully convex sets are digitally convex, but are also connected and simply connected. They have a morphological characterization, which induces a simple convexity test algorithm. Arithmetic planes are fully convex too. Full convexity implies local full convexity, hence it enables local shape analysis, with an unambiguous definition of convex, concave and planar points. As a kind of relative full convexity, we propose a natural definition of tangent subsets to a digital set. It gives rise to the tangential cover in 2D, and to consistent extensions in arbitrary dimension. Finally, we present two applications of tangency: the first one is a simple algorithm for building a polygonal mesh from a set of digital points, with reversibility property, the second one is the definition and computation of shortest paths within digital sets. [ABSTRACT FROM AUTHOR]

In this paper, we prove that the topological spaces of nonzero weighted composition operators acting on some Hilbert spaces of analytic functions on the unit open ball are simply connected.

We show that any union of slc strata of a Fano log pair with semi-log canonical singularities is simply connected. In particular, Fano log pairs with semi-log canonical singularities are simply connected, which confirms a conjecture of the first author. [ABSTRACT FROM AUTHOR]

The present paper aims at the study of a topology on Lorentzian manifolds, defined by Göbel [ 4 ] using the ideas of Zeeman [ 16 ]. Observing that on the Minkowski space it is the same as Zeeman's time topology, it has been found that a Lorentzian manifold with this topology is path connected, nonfirst countable and nonsimply connected while the Minkowski space with time topology is, in addition nonregular and separable. Furthermore, using the notion of Zeno sequences it is obtained that a compact set does not contain a nonempty open set and that a set is compact if and only if each of its infinite subsets has a limit point if and only if each of its sequences has a convergent subsequence. [ABSTRACT FROM AUTHOR]

Abstract: For a class of parapolar spaces that includes the geometries , , and with lines of size at least three, the metasymplectic spaces with lines of size at least four, and the polar line Grassmannians with lines of size at least four except , we show that the subgraph of the point-collinearity graph induced on the complement of a hyperplane is simply connected. We also show that these parapolar spaces have Veldkamp lines. [ABSTRACT FROM AUTHOR]

Abstract: Suppose is a dual polar space of rank and is a hyperplane of . Cardinali, De Bruyn and Pasini have already shown that if and the line size is greater than or equal to 4 then the hyperplane complement is simply connected. This paper is a follow-up, where we investigate the remaining cases. We prove that the hyperplane complements are simply connected in all cases except for three specific types of hyperplane occurring in the smallest case, when the rank and the line size are both 3. [Copyright &y& Elsevier]

We investigate the Galois coverings of weakly shod algebras. For a weakly shod algebra not quasi-tilted of canonical type, we establish a correspondence between its Galois coverings and the Galois coverings of its connecting component. As a consequence we show that a weakly shod algebra which is not quasi-tilted of canonical type is simply connected if and only if its first Hochschild cohomology group vanishes. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we study the covering theory of laura algebras. We prove that if a connected laura algebra is standard (that is, has a standard connecting component), then it has Galois coverings associated to the coverings of the connecting component. As a consequence, the first Hochschild cohomology group of a standard laura algebra vanishes if and only if it has no proper Galois coverings. [Copyright &y& Elsevier]

Abstract: Let be a dual polar space of rank , be a hyperplane of and be the complement of in . We shall prove that, if all lines of have more than 3 points, then is simply connected. Then we show how this theorem can be exploited to prove that certain families of hyperplanes of dual polar spaces, or all hyperplanes of certain dual polar spaces, arise from embeddings. [Copyright &y& Elsevier]

Abstract: Let A be a basic connected finite dimensional algebra over an algebraically closed field k. Let T be a basic tilting A-module with arbitrary finite projective dimension. For a fixed group G we compare the set of isoclasses of connected Galois coverings of A with group G and the set of isoclasses of connected Galois coverings of with group G. Using the Hasse diagram (see [D. Happel, L. Unger, On a partial order of tilting modules, Algebr. Represent. Theory 8 (2) (2005) 147–156] and [C. Riedtmann, A. Schofield, On a simplicial complex associated with tilting modules, Comment. Math. Helv. 66 (1) (1991) 70–78]) of basic tilting A-modules, we give sufficient conditions on T under which there is a bijection between these two sets (these conditions are always verified when A is of finite representation type). Then we apply these results to study when the simple connectedness of A implies the one of (see [I. Assem, E.N. Marcos, J.A. de la Peña, The simple connectedness of a tame tilted algebra, J. Algebra 237 (2) (2001) 647–656]). Finally, using an argument due to W. Crawley-Boevey, we prove that the type of any simply connected tilted algebra is a tree and that its first Hochschild cohomology group vanishes. [Copyright &y& Elsevier]

Abstract: Let Δ be a dual polar space of rank , H be a hyperplane of Δ and be the complement of H in Δ. We shall prove that, if all lines of Δ have more than 3 points, then Γ is simply connected. Then we show how this theorem can be exploited to prove that certain families of hyperplanes of dual polar spaces, or all hyperplanes of certain dual polar spaces, arise from embeddings. [Copyright &y& Elsevier]

Abstract: We consider the problem of achieving a desired steady-state effect through periodic behavior for a class of control systems with and without drift. The problem of using periodic behavior to achieve set-point regulation for the control systems with drift is directly related to that of achieving unbounded effect for the corresponding driftless control systems. We prove that in both cases, the ability to use periodic behavior, and more generally, bounded behavior, to achieve the desired goal implies, under a certain topological condition, the non-holonomicity of the control systems. We also prove that under a regularity condition, the resulting system trajectories must be area-generating in a precise sense. [Copyright &y& Elsevier]