IEEE/ACM Transactions on Computational Biology and Bioinformatics, 2010
We introduce a simple algorithm for reconstructing phylogenies from multiple gene trees in the pr... more We introduce a simple algorithm for reconstructing phylogenies from multiple gene trees in the presence of incomplete lineage sorting, that is, when the topology of the gene trees may differ from that of the species tree. We show that our technique is statistically consistent under standard stochastic assumptions, that is, it returns the correct tree given sufficiently many unlinked loci. We also show that it can tolerate moderate estimation errors.
The main aim of the paper is Fredholm properties of a class of bounded linear operators acting on... more The main aim of the paper is Fredholm properties of a class of bounded linear operators acting on weighted Lebesgue spaces on an infinite metric graph Γ which is periodic with respect to the action of the group Z n. The operators under consideration are distinguished by their local behavior: they act as (Fourier) pseudodifferential operators in the class OP S 0 on every open edge of the graph, and they can be represented as a matrix Mellin pseudodifferential operator on a neighborhood of every vertex of Γ. We apply these results to study the Fredholm property of a class of singular integral operators and of certain locally compact operators on graphs.
Let Γ be a finitely generated discrete exact group. We consider operators on l 2 (Γ) which are co... more Let Γ be a finitely generated discrete exact group. We consider operators on l 2 (Γ) which are composed by operators of multiplication by a function in l ∞ (Γ) and by the operators of left-shift by elements of Γ. These operators generate a C *-subalgebra of L(l 2 (Γ)) the elements of which we call band-dominated operators on Γ. We study the stability of the finite sections method for band-dominated operators with respect to a given generating system of Γ. Our approach is based on the equivalence of the stability of a sequence and the Fredholmness of an associated operator, and on Roe's criterion for the Fredholmness of a band-dominated operator on a exact discrete group, which we formulate in terms of limit operators. Special emphasis is paid to the quasicommutator ideal of the algebra generated by the finite sections sequences and to the stability of sequences in that algebra. For both problems, the sequence of the discrete boundaries plays an essential role.
Approximation methods for singular integral operators with continuous coefficients and conjugatio... more Approximation methods for singular integral operators with continuous coefficients and conjugation on curves with corners are investigated with respect to their stability. Particular emphasis is devoted to index constraints for the local stability conditions. It turns out that, if an associated local operator is Fredholm, then the absolute value of its Fredholm index is necessarily bounded by 2, and this maximal value is attained in some instances (whereas it is known that in the case of pure singular integral equations Fredholmness of the associated local operators already implies vanishing index).
The goal of this chapter is to provide a possible tool to study the invertibility of the local co... more The goal of this chapter is to provide a possible tool to study the invertibility of the local cosets which arise after localization of convolution type operators. The basic observation is that the local algebras in some cases are generated by a finite number of elements p which are idempotent in the sense that p 2=p. Under some additional conditions, it turns out that such algebras possess matrix-valued symbols. Thus, one can associate with every element of the algebra a matrix-valued function such that the element is invertible if and only if the associated function is invertible at every point. In this way, one gets an effective criterion for the invertibility of elements in local algebras.
Zeitschrift für Analysis und ihre Anwendungen, 1997
This paper is concerned with the applicability of the finite section method to operators belongin... more This paper is concerned with the applicability of the finite section method to operators belonging to the closed subalgebra of L(L 2 (R)) generated by operators of multiplication by piecewise continuous functions inṘ, convolution operators-also with piecewise continuous generating functions-and the flip operator (Ju)(x) = u(−x). For this, a larger algebra of sequences is introduced, which contains the special sequences we are interested in. There is a direct relationship between the applicability of the finite section method for a given operator and the invertibility of the corresponding sequence in this algebra. Exploring this relationship, the methods of essentialization, localization and identification of the local algebras through construction of locally equivalent representations are used and so useful invertibility criteria are derived. Finally, examples are presented, including explicit conditions for the applicability of the finite section method to a Wiener-Hopf plus Hankel operator with piecewise continuous symbols, and some relations between the approximation operators and the limit operator are discussed.
Approximation methods for singular integral operators with continuous coefficients and conjugatio... more Approximation methods for singular integral operators with continuous coefficients and conjugation as well as for the double layer potential operator on curves with corners are investigated with respect to their stability. The methods under consideration include several kinds of quadrature rules, collocation and qualocation methods. The approach is based on the thorough use of Banach algebra techniques and local principles. Complete necessary and sufficient stability conditions are derived.
Journal of Mathematical Analysis and Applications, 2012
ABSTRACT We study several algebras generated by convolution, multiplication and flip operators on... more ABSTRACT We study several algebras generated by convolution, multiplication and flip operators on Lp(R)Lp(R), their Calkin counterparts and derive new isomorphism relations. We introduce a new class of homogenization strong limits, compatible with the flip operator and which explore the properties of the Fourier transform in Lp(R)Lp(R), when p≠2p≠2.
The topics of this paper are Fredholm properties and the applicability of the finite section meth... more The topics of this paper are Fredholm properties and the applicability of the finite section method for band operators onlp-spaces as well as for their norm limits which we call band-dominated operators. The derived criteria will be established in terms of the limit operators of the given band-dominated operator. After presenting the general theory, we present its specifications to concrete
The central theme of the present paper are band and band-dominated operators, i.e. norm limits of... more The central theme of the present paper are band and band-dominated operators, i.e. norm limits of band operators. In the first part, we generalize the results from [24] and [25] concerning the Fredholm properties of band-dominated operators and the applicability of the finite section method to the case of operators with operator-valued coefficients. We characterize these properties in terms of the limit operators of the given band-dominated operator. The main objective of the second part is to apply these results to pseudodifferential operators on cones in IR n which is possible after a suitable discretization.
The image in the Oalkin algebra of theBanach algebra generated by all singular integral operators... more The image in the Oalkin algebra of theBanach algebra generated by all singular integral operators with piecewise continuous coefficients on some composed centou~ is described up te isomorphy and isometry for weighted L~-spaces,
IEEE/ACM Transactions on Computational Biology and Bioinformatics, 2010
We introduce a simple algorithm for reconstructing phylogenies from multiple gene trees in the pr... more We introduce a simple algorithm for reconstructing phylogenies from multiple gene trees in the presence of incomplete lineage sorting, that is, when the topology of the gene trees may differ from that of the species tree. We show that our technique is statistically consistent under standard stochastic assumptions, that is, it returns the correct tree given sufficiently many unlinked loci. We also show that it can tolerate moderate estimation errors.
The main aim of the paper is Fredholm properties of a class of bounded linear operators acting on... more The main aim of the paper is Fredholm properties of a class of bounded linear operators acting on weighted Lebesgue spaces on an infinite metric graph Γ which is periodic with respect to the action of the group Z n. The operators under consideration are distinguished by their local behavior: they act as (Fourier) pseudodifferential operators in the class OP S 0 on every open edge of the graph, and they can be represented as a matrix Mellin pseudodifferential operator on a neighborhood of every vertex of Γ. We apply these results to study the Fredholm property of a class of singular integral operators and of certain locally compact operators on graphs.
Let Γ be a finitely generated discrete exact group. We consider operators on l 2 (Γ) which are co... more Let Γ be a finitely generated discrete exact group. We consider operators on l 2 (Γ) which are composed by operators of multiplication by a function in l ∞ (Γ) and by the operators of left-shift by elements of Γ. These operators generate a C *-subalgebra of L(l 2 (Γ)) the elements of which we call band-dominated operators on Γ. We study the stability of the finite sections method for band-dominated operators with respect to a given generating system of Γ. Our approach is based on the equivalence of the stability of a sequence and the Fredholmness of an associated operator, and on Roe's criterion for the Fredholmness of a band-dominated operator on a exact discrete group, which we formulate in terms of limit operators. Special emphasis is paid to the quasicommutator ideal of the algebra generated by the finite sections sequences and to the stability of sequences in that algebra. For both problems, the sequence of the discrete boundaries plays an essential role.
Approximation methods for singular integral operators with continuous coefficients and conjugatio... more Approximation methods for singular integral operators with continuous coefficients and conjugation on curves with corners are investigated with respect to their stability. Particular emphasis is devoted to index constraints for the local stability conditions. It turns out that, if an associated local operator is Fredholm, then the absolute value of its Fredholm index is necessarily bounded by 2, and this maximal value is attained in some instances (whereas it is known that in the case of pure singular integral equations Fredholmness of the associated local operators already implies vanishing index).
The goal of this chapter is to provide a possible tool to study the invertibility of the local co... more The goal of this chapter is to provide a possible tool to study the invertibility of the local cosets which arise after localization of convolution type operators. The basic observation is that the local algebras in some cases are generated by a finite number of elements p which are idempotent in the sense that p 2=p. Under some additional conditions, it turns out that such algebras possess matrix-valued symbols. Thus, one can associate with every element of the algebra a matrix-valued function such that the element is invertible if and only if the associated function is invertible at every point. In this way, one gets an effective criterion for the invertibility of elements in local algebras.
Zeitschrift für Analysis und ihre Anwendungen, 1997
This paper is concerned with the applicability of the finite section method to operators belongin... more This paper is concerned with the applicability of the finite section method to operators belonging to the closed subalgebra of L(L 2 (R)) generated by operators of multiplication by piecewise continuous functions inṘ, convolution operators-also with piecewise continuous generating functions-and the flip operator (Ju)(x) = u(−x). For this, a larger algebra of sequences is introduced, which contains the special sequences we are interested in. There is a direct relationship between the applicability of the finite section method for a given operator and the invertibility of the corresponding sequence in this algebra. Exploring this relationship, the methods of essentialization, localization and identification of the local algebras through construction of locally equivalent representations are used and so useful invertibility criteria are derived. Finally, examples are presented, including explicit conditions for the applicability of the finite section method to a Wiener-Hopf plus Hankel operator with piecewise continuous symbols, and some relations between the approximation operators and the limit operator are discussed.
Approximation methods for singular integral operators with continuous coefficients and conjugatio... more Approximation methods for singular integral operators with continuous coefficients and conjugation as well as for the double layer potential operator on curves with corners are investigated with respect to their stability. The methods under consideration include several kinds of quadrature rules, collocation and qualocation methods. The approach is based on the thorough use of Banach algebra techniques and local principles. Complete necessary and sufficient stability conditions are derived.
Journal of Mathematical Analysis and Applications, 2012
ABSTRACT We study several algebras generated by convolution, multiplication and flip operators on... more ABSTRACT We study several algebras generated by convolution, multiplication and flip operators on Lp(R)Lp(R), their Calkin counterparts and derive new isomorphism relations. We introduce a new class of homogenization strong limits, compatible with the flip operator and which explore the properties of the Fourier transform in Lp(R)Lp(R), when p≠2p≠2.
The topics of this paper are Fredholm properties and the applicability of the finite section meth... more The topics of this paper are Fredholm properties and the applicability of the finite section method for band operators onlp-spaces as well as for their norm limits which we call band-dominated operators. The derived criteria will be established in terms of the limit operators of the given band-dominated operator. After presenting the general theory, we present its specifications to concrete
The central theme of the present paper are band and band-dominated operators, i.e. norm limits of... more The central theme of the present paper are band and band-dominated operators, i.e. norm limits of band operators. In the first part, we generalize the results from [24] and [25] concerning the Fredholm properties of band-dominated operators and the applicability of the finite section method to the case of operators with operator-valued coefficients. We characterize these properties in terms of the limit operators of the given band-dominated operator. The main objective of the second part is to apply these results to pseudodifferential operators on cones in IR n which is possible after a suitable discretization.
The image in the Oalkin algebra of theBanach algebra generated by all singular integral operators... more The image in the Oalkin algebra of theBanach algebra generated by all singular integral operators with piecewise continuous coefficients on some composed centou~ is described up te isomorphy and isometry for weighted L~-spaces,
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Papers by S. Roch