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The Kuratowski closurecomplement theorem
 New Zealand Journal of Mathematics
"... The Kuratowski ClosureComplement Theorem 1.1. [29] If (X,T) is a topological space and A ⊆ X then at most 14 sets can be obtained from A by taking closures and complements. Furthermore there is a space in which this bound is attained. ..."
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The Kuratowski ClosureComplement Theorem 1.1. [29] If (X,T) is a topological space and A ⊆ X then at most 14 sets can be obtained from A by taking closures and complements. Furthermore there is a space in which this bound is attained.
PSEUDOCOMPLETENESS AND THE PRODUCT OF BAIRE SPACES
"... The class of pseudocomplete spaces defined by Oxtoby is one of the largest known classes ^ with the property that any member of & is a Baire space and ^ is closed under arbitrary products. Furthermore, all of the classical examples of Baire spaces belong to & * In this paper it is proved t ..."
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The class of pseudocomplete spaces defined by Oxtoby is one of the largest known classes ^ with the property that any member of & is a Baire space and ^ is closed under arbitrary products. Furthermore, all of the classical examples of Baire spaces belong to & * In this paper it is proved that if Xe & and if Y is any (quasiregular) Baire space, then J X 7 is a Baire space. The proof is based on the notion of Aembedding which makes it possible to recognize whether a dense subspace of a Baire space is a Baire space in its relative topology. Finally, examples are presented which relate pseudocompleteness to several other types of completeness. 1 * Introduction * A space X is a Baire space if every nonempty open subset is of second category [2] or, equivalently, if the intersection of countably many dense open subsets of X is dense in X. Locally compact Hausdorff spaces and completely metrizable spaces are the
On the Positivity Problem for simple linear recurrence sequences
 In Proceedings of ICALP’14, 2014. CoRR, abs/1309.1550
"... Abstract. Given a linear recurrence sequence (LRS) over the integers, the Positivity Problem asks whether all terms of the sequence are positive. We show that, for simple LRS (those whose characteristic polynomial has no repeated roots) of order 9 or less, Positivity is decidable, with complexity ..."
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Abstract. Given a linear recurrence sequence (LRS) over the integers, the Positivity Problem asks whether all terms of the sequence are positive. We show that, for simple LRS (those whose characteristic polynomial has no repeated roots) of order 9 or less, Positivity is decidable, with complexity in the Counting Hierarchy. 1
EXCISION IN BANACH SIMPLICIAL AND CYCLIC COHOMOLOGY
, 1996
"... Abstract. We prove that, for every extension of Banach algebras 0 → B → A → D → 0 such that B has a left or right bounded approximate identity, the existence of an associated long exact sequence of Banach simplicial or cyclic cohomology groups is equivalent to the existence of one for homology group ..."
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Abstract. We prove that, for every extension of Banach algebras 0 → B → A → D → 0 such that B has a left or right bounded approximate identity, the existence of an associated long exact sequence of Banach simplicial or cyclic cohomology groups is equivalent to the existence of one for homology groups. It follows from the continuous version of a result of Wodzicki that associated long exact sequences exist. In particular, they exist for every extension of C ∗algebras. 1.
Convergence of Automorphisms of Compact Projective Planes
"... Introduction Convergence and continuity properties of homomorphisms play an important role in the theory of topological projective planes. Grundhofer [8] showed that the set \Sigma of all automorphisms of a compact projective plane is a locally compact transformation group with respect to the topol ..."
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Introduction Convergence and continuity properties of homomorphisms play an important role in the theory of topological projective planes. Grundhofer [8] showed that the set \Sigma of all automorphisms of a compact projective plane is a locally compact transformation group with respect to the topology of uniform convergence; for the special case of compact connected projective planes see also Salzmann [21]. With regards to classification, compact connected projective planes have been successfully investigated by studying their automorphism group, see Salzmann, Betten, Grundhofer, Hahl, Lowen, and Stroppel [22] for a detailed exposition. Salzmann [20] proved that if \Pi is a 2dimensional compact projective plane, then on \Sigma the topology of pointwise convergence coincides with the topology of uniform convergence. He also showed ([19]) that any homomorphism between 2dimensional compact projective planes is in fact a homeomorphism. Grundhofer [9] characterized the continuity of non
A UNIFIED THEORY FOR WEAK SEPARATION PROPERTIES
, 2000
"... We devise a framework which leads to the formulation of a unified theory of normality (regularity), seminormality (semiregularity), snormality (sregularity), feeblynormality (feeblyregularity), prenormality (preregularity), and others. Certain aspects of theory are given by unified proof. ..."
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We devise a framework which leads to the formulation of a unified theory of normality (regularity), seminormality (semiregularity), snormality (sregularity), feeblynormality (feeblyregularity), prenormality (preregularity), and others. Certain aspects of theory are given by unified proof.
ON THE ACTION OF THE GROUP OF ISOMETRIES ON A LOCALLY COMPACT METRIC SPACE: CLOSEDOPEN PARTITIONS AND CLOSED ORBITS
, 902
"... Abstract. In the present work we study the dynamic behavior of the orbits of the natural action of the group G of isometries on a locally compact metric space X using suitable closedopen subsets of X. Precisely, we study the dynamic behavior of an orbit even in cases where G is not locally compact ..."
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Abstract. In the present work we study the dynamic behavior of the orbits of the natural action of the group G of isometries on a locally compact metric space X using suitable closedopen subsets of X. Precisely, we study the dynamic behavior of an orbit even in cases where G is not locally compact with respect to the compactopen topology. In case G is locally compact we decompose the space X into closedopen invariant disjoint sets that are related to various limit behaviors of the orbits. We also provide a simple example of a locally compact separable and complete metric space X with discrete group of isometries G such that the natural action of G on X has closed and nonclosed orbits. 1.
The Polyhedral Escape Problem is Decidable
"... Abstract. The Polyhedral Escape Problem for continuous linear dynamical systems consists of deciding, given an affine function f: Rd → Rd and a convex polyhedron P ⊂ Rd, whether, for some initial point x0 in P, the trajectory of the unique solution to the differential equation ẋ(t) = f(x(t)),x(0 ..."
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Abstract. The Polyhedral Escape Problem for continuous linear dynamical systems consists of deciding, given an affine function f: Rd → Rd and a convex polyhedron P ⊂ Rd, whether, for some initial point x0 in P, the trajectory of the unique solution to the differential equation ẋ(t) = f(x(t)),x(0) = x0, t ≥ 0, is entirely contained in P. We show that this problem is decidable, by reducing it in polynomial time to the decision version of linear programming with real algebraic coefficients, thus placing it in ∃R, which lies between NP and PSPACE. Our algorithm makes use of spectral techniques and relies among others on tools from Diophantine approximation. 1
INFINITEDIMENSIONAL DIAGONALIZATION AND SEMISIMPLICITY
"... Abstract. We characterize the diagonalizable subalgebras of End(V), the full ring of linear operators on a vector space V over a field, in a manner that directly generalizes the classical theory of diagonalizable algebras of operators on a finitedimensional vector space. Our characterizations are ..."
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Abstract. We characterize the diagonalizable subalgebras of End(V), the full ring of linear operators on a vector space V over a field, in a manner that directly generalizes the classical theory of diagonalizable algebras of operators on a finitedimensional vector space. Our characterizations are formulated in terms of a natural topology (the “finite topology”) on End(V), which reduces to the discrete topology in the case where V is finitedimensional. We further investigate when two subalgebras of operators can and cannot be simultaneously diagonalized, as well as the closure of the set of diagonalizable operators within End(V). Motivated by the classical link between diagonalizability and semisimplicity, we also give an infinitedimensional generalization of the WedderburnArtin theorem, providing a number of equivalent characterizations of left pseudocompact, Jacoboson semisimple rings that parallel various characterizations of artinian semisimple rings. This theorem unifies a number of related results in the literature, including the structure of linearly compact, Jacobson semsimple rings and of cosemisimple coalgebras over a field. 1.