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Unbounded Fredholm Operators and Spectral Flow
, 2001
"... Abstract. We study the gap ( = “projection norm ” = “graph norm”) topology of the space of (not necessarily bounded) self–adjoint Fredholm operators in a separable Hilbert space by the Cayley transform and direct methods. In particular, we show that the space is connected contrary to the bounded ca ..."
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Abstract. We study the gap ( = “projection norm ” = “graph norm”) topology of the space of (not necessarily bounded) self–adjoint Fredholm operators in a separable Hilbert space by the Cayley transform and direct methods. In particular, we show that the space is connected contrary to the bounded case. Moreover, we present a rigorous definition of spectral flow of a path of such operators (actually alternative but mutually equivalent definitions) and prove the homotopy invariance. As an example, we discuss operator curves on manifolds with boundary.
Constructive Closed Range and Open Mapping Theorems
 Indag. Math. N.S
, 1998
"... We prove a version of the closed range theorem within Bishop's constructive mathematics. This is applied to show that if an operator T on a Hilbert space has an adjoint and a complete range, then both T and T are sequentially open. 1 Introduction In this paper we continue our constructive exp ..."
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Cited by 3 (2 self)
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We prove a version of the closed range theorem within Bishop's constructive mathematics. This is applied to show that if an operator T on a Hilbert space has an adjoint and a complete range, then both T and T are sequentially open. 1 Introduction In this paper we continue our constructive exploration of the theory of operators, in particular operators on a Hilbert space ([4], [5], [6]). We work entirely within Bishop's constructive mathematics, which we regard as mathematics with intuitionistic logic. For discussions of the merits of this approach to mathematics in particular, the multiplicity of its modelssee [3] and [10]. The technical background needed in our paper is found in [1] and [9]. Our main aim is to prove the following result, the constructive Closed Range Theorem for operators on a Hilbert space (cf. [16], pages 99103): Theorem 1 Let H be a Hilbert space, and T a linear operator on H such that T exists and ran(T ) is closed. Then ran(T ) and ker(T ) are bo...
A theoretical analysis of hierarchical proofs
 In Asperti et al
, 2003
"... www.uclic.ucl.ac.uk/imp Abstract. Hierarchical proof presentations are ubiquitous within logic and computer science, but have made little impact on mathematics in general. The reasons for this are not currently known, and need to be understood if mathematical knowledge management systems are to gain ..."
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Cited by 3 (2 self)
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www.uclic.ucl.ac.uk/imp Abstract. Hierarchical proof presentations are ubiquitous within logic and computer science, but have made little impact on mathematics in general. The reasons for this are not currently known, and need to be understood if mathematical knowledge management systems are to gain acceptance in the mathematical community. We report on some initial experiments with three users of a set of webbased hierarchical proofs, which suggest that usability problems could be a factor. In order to better understand these problems we present a theoretical analysis of hierarchical proofs using Cognitive Dimensions [6]. The analysis allows us to formulate some concrete hypotheses about the usability of hierarchical proof presentations. 1
KTheory for Algebras of Operators on Banach Spaces
 Jour. London Math. Soc
, 1997
"... We prove that, for each pair (m; n) of nonnegative integers, there is a Banach space X for which K 0 (B(X)) ¸ = Z m and K 1 (B(X)) ¸ = Z n : Along the way we compute the Kgroups of all closed ideals of operators contained in the ideal of strictly singular operators, and we derive some resu ..."
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Cited by 2 (0 self)
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We prove that, for each pair (m; n) of nonnegative integers, there is a Banach space X for which K 0 (B(X)) ¸ = Z m and K 1 (B(X)) ¸ = Z n : Along the way we compute the Kgroups of all closed ideals of operators contained in the ideal of strictly singular operators, and we derive some results about the existence of splittings of certain short exact sequences. 1 Introduction and Notation Throughout, all vector spaces and algebras are tacitly assumed to be complex. In addition, all Banach spaces are assumed to be infinitedimensional. Banach spaces are denoted by X; Y;Z; and W. The term `operator' means a continuous linear map. The Banach space of all operators from X to Y is denoted by B(X;Y) (or B(X) in the case where X = Y). By an ideal of operators we understand an assignment I which associates to each pair (X; Y) of Banach spaces a subspace I (X; Y) of B(X;Y) satisfying TSR 2 I (W;Z) whenever R 2 B(W;X); S 2 I (X; Y); and T 2 B(Y;Z): In the case where X = Y, we write ...
On the complexity of the equivalence problem for probabilistic automata
 In Proc. of FoSSaCS’12, volume 7213 of LNCS
, 2012
"... Abstract. Deciding equivalence of probabilistic automata is a key problem for establishing various behavioural and anonymity properties of probabilistic systems. In recent experiments a randomised equivalence test based on polynomial identity testing outperformed deterministic algorithms. In this pa ..."
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Abstract. Deciding equivalence of probabilistic automata is a key problem for establishing various behavioural and anonymity properties of probabilistic systems. In recent experiments a randomised equivalence test based on polynomial identity testing outperformed deterministic algorithms. In this paper we show that polynomial identity testing yields efficient algorithms for various generalisations of the equivalence problem. First, we provide a randomized NC procedure that also outputs a counterexample trace in case of inequivalence. Second, we consider equivalence of probabilistic cost automata. In these automata transitions are labelled with integer costs and each word is associated with a distribution on costs, corresponding to the cumulative costs of the accepting runs on that word. Two automata are equivalent if they induce the same cost distributions on each input word. We show that equivalence can be checked in randomised polynomial time. Finally we show that the equivalence problem for probabilistic visibly pushdown automata is logspace equivalent to the problem of whether a polynomial represented by an arithmetic circuit is identically zero. 1
Operator ordering and Classical soliton path in Twodimensional N = 2 supersymmetry
, 2005
"... We investigate a twodimensional N = 2 supersymmetric model which consists of n chiral superfields with Kähler potential. When we define quantum observables, we are always plagued by operator ordering problem. Among various ways to fix the operator order, we rely upon the supersymmetry. We demonstra ..."
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We investigate a twodimensional N = 2 supersymmetric model which consists of n chiral superfields with Kähler potential. When we define quantum observables, we are always plagued by operator ordering problem. Among various ways to fix the operator order, we rely upon the supersymmetry. We demonstrate that the correct operator order is given by requiring the superPoincaré algebra by carrying out the canonical Dirac bracket quantization. This is shown to be also true when the supersymmetry algebra has a central extension by the presence of topological soliton. It is also shown that the path of soliton is a straight line in the complex plane of superpotential W and triangular mass inequality holds. One half of supersymmetry is broken by the presence of soliton. Keywords: Supersymmetry; operator ordering; soliton; Bogomol’nyi bound.