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Register a new userThe Kirch space is topologically rigid
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The $Golomb$ $space$ (resp. the $Kirch$ $space$) is the set $mathbb N$ of positive integers endowed with the topology generated by the base consisting of arithmetic progressions $a+bmathbb N_0={a+bn:nge 0}$ where $ainmathbb N$ and $b$ is a (square-free) number, coprime with $a$. It is known that the Golomb space (resp. the Kirch space) is connected (and locally connected). By a recent result of Banakh, Spirito and Turek, the Golomb space has trivial homeomorphism group and hence is topologically rigid. In this paper we prove the topological rigidity of the Kirch space.
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Given a class $mathcal P$ of Banach spaces, a locally convex space (LCS) $E$ is called {em multi-$mathcal P$} if $E$ can be isomorphically embedded into a product of spaces that belong to $mathcal P$. We investigate the question whether the free locally convex space $L(X)$ is strongly nuclear, nuclear, Schwartz, multi-Hilbert or multi-reflexive. If $X$ is a Tychonoff space containing an infinite compact subset then, as it follows from the results of cite{Aus}, $L(X)$ is not nuclear. We prove that for such $X$ the free LCS $L(X)$ has the stronger property of not being multi-Hilbert. We deduce that if $X$ is a $k$-space, then the following properties are equivalent: (1) $L(X)$ is strongly nuclear; (2) $L(X)$ is nuclear; (3) $L(X)$ is multi-Hilbert; (4) $X$ is countable and discrete. On the other hand, we show that $L(X)$ is strongly nuclear for every projectively countable $P$-space (in particular, for every Lindelof $P$-space) $X$. We observe that every Schwartz LCS is multi-reflexive. It is known that if $X$ is a $k_omega$-space, then $L(X)$ is a Schwartz LCS cite{Chasco}, hence $L(X)$ is multi-reflexive. We show that for any first-countable paracompact (in particular, metrizable) space $X$ the converse is true, so $L(X)$ is multi-reflexive if and only if $X$ is a $k_omega$-space, equivalently, if $X$ is a locally compact and $sigma$-compact space. Similarly, we show that for any first-countable paracompact space $X$ the free abelian topological group $A(X)$ is a Schwartz group if and only if $X$ is a locally compact space such that the set $X^{(1)}$ of all non-isolated points of $X$ is $sigma$-compact.
In this paper, we show that deciding rigid foldability of a given crease pattern using all creases is weakly NP-hard by a reduction from Partition, and that deciding rigid foldability with optional creases is strongly NP-hard by a reduction from 1-in-3 SAT. Unlike flat foldability of origami or flexibility of other kinematic linkages, whose complexity originates in the complexity of the layer ordering and possible self-intersection of the material, rigid foldability from a planar state is hard even though there is no potential self-intersection. In fact, the complexity comes from the combinatorial behavior of the different possible rigid folding configurations at each vertex. The results underpin the fact that it is harder to fold from an unfolded sheet of paper than to unfold a folded state back to a plane, frequently encountered problem when realizing folding-based systems such as self-folding matter and reconfigurable robots.
We propose a new method to construct rigid $G$-automorphic representations and rigid $widehat{G}$-local systems for reductive groups $G$. The construction involves the notion of euphotic representations, and the proof for rigidity involves the geometry of certain Hessenberg varieties.
On any smooth algebraic variety over a $p$-adic local field, we construct a tensor functor from the category of de Rham $p$-adic etale local systems to the category of filtered algebraic vector bundles with integrable connections satisfying the Griffiths transversality, which we view as a $p$-adic analogue of Delignes classical Riemann--Hilbert correspondence. A crucial step is to construct canonical extensions of the desired connections to suitable compactifications of the algebraic variety with logarithmic poles along the boundary, in a precise sense characterized by the eigenvalues of residues; hence the title of the paper. As an application, we show that this $p$-adic Riemann--Hilbert functor is compatible with the classical one over all Shimura varieties, for local systems attached to representations of the associated reductive algebraic groups.
We construct examples of smooth proper rigid-analytic varieties admitting formal model with projective special fiber and violating Hodge symmetry for cohomology in degrees $geq 3$. This answers negatively a question raised by Hansen and Li.
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