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Serguei Iakovlev - One of the best experts on this subject based on the ideXlab platform.

  • On the Singular Behavior of the Inverse Laplace Transforms of the Functions In(s) sI ′ n(s)
    2014
    Co-Authors: Serguei Iakovlev
    Abstract:

    Abstract. Exact analytical expressions for the inverse Laplace transforms of the functions In(s) sI ′ are n (s) obtained in the form of trigonometric series. The convergence of the series is analyzed theoretically, and it is proven that those diverge on an infinite Denumerable Set of points. Therefore it is shown that the inverse transforms have an infinite number of singular points. This result, to the best of the author’s knowledge, is new, as the inverse transforms of In(s) sI ′ have previously been considered to be n (s) piecewise smooth and continuous. It is also found that the inverse transforms have an infinite number of points of finite discontinuity with different left- and right-side limits. The points of singularity and points of finite discontinuity alternate, and the sign of the infinity at the singular points also alternates depending on the order n. The behavior of the inverse transforms in the proximity of the singular points and the points of finite discontinuity is addressed as well

  • on the singular behavior of the inverse laplace transforms of the functions
    Canadian Mathematical Bulletin, 2002
    Co-Authors: Serguei Iakovlev
    Abstract:

    Exact analytical expressions for the inverse Laplace transforms of the functions are obtained in the form of trigonometric series. The convergence of the series is analyzed theoretically, and it is proven that those diverge on an infinite Denumerable Set of points. Therefore it is shown that the inverse transforms have an infinite number of singular points. This result, to the best of the author’s knowledge, is new, as the inverse transforms of have previously been considered to be piecewise smooth and continuous. It is also found that the inverse transforms have an infinite number of points of finite discontinuity with different left- and right-side limits. The points of singularity and points of finite discontinuity alternate, and the sign of the infinity at the singular points also alternates depending on the order . The behavior of the inverse transforms in the proximity of the singular points and the points of finite discontinuity is addressed as well.

Peter Yuditskii - One of the best experts on this subject based on the ideXlab platform.

  • asymptotic behavior of polynomials orthonormal on a homogeneous Set
    Journal D Analyse Mathematique, 2003
    Co-Authors: Franz Peherstorfer, Peter Yuditskii
    Abstract:

    LetE be a homogeneous compact Set, for instance a Cantor Set of positive length. Further, let σ be a positive measure with supp(σ)=E. Under the condition that the absolutely continuous part of σ satisfies a Szego-type condition, we give an asymptotic representation, on and off the support, for the polynomials orthonomal with respect to σ. For the special case thatE consists of a finite number of intervals and that σ has no singular component, this is a well-known result of Widom. IfE=[a,b], it becomes a classical result due to Szego; and in case that there appears in addition a singular component, it is due to Kolmogorov-krein. In fact, the results are presented for the more general case that the orthogonality measure may have a Denumerable Set of mass-points outside ofE which are supposed to accumulate only onE and to satisfy (together with the zeros of the associated Stieltjes function) the free-interpolation Carleson-type condition. Up to the case of a finite number of mass points, this is even new for the single interval case. Furthermore, as a byproduct of our representations, we obtain that the recurrence coefficients of the orthonormal polynomials behave asymptotically almost periodic. In other words, the Jacobi matrices associated with the above discussed orthonomal polynomials are compact perturbations of a onesided restriction of almost periodic Jacobi matrices with homogeneous spectrum. Our main tool is a theory of Hardy spaces of character-automorphic functions and forms on Riemann surfaces of Widom type; we use also some ideas of scattering theory for one-dimensional Schrodinger equations.

  • asymptotic behavior of polynomials orthonormal on a homogeneous Set
    arXiv: Functional Analysis, 2002
    Co-Authors: Franz Peherstorfer, Peter Yuditskii
    Abstract:

    Let $E$ be a homogeneous compact Set, for instance a Cantor Set of positive length. Further let $\sigma$ be a positive measure with $\text{supp}(\sigma)=E$. Under the condition that the absolutely continuous part of $\sigma$ satisfies a Szeg\"o--type condition we give an asymptotic representation, on and off the support, for the polynomials orthonormal with respect to $\sigma$. For the special case that $E$ consists of a finite number of intervals and that $\sigma$ has no singular component this is a nowaday well known result of Widom. If $E=[a,b]$ it becomes a classical result due to Szeg\"o and in case that there appears in addition a singular component, it is due to Kolmogorov--Krein. In fact the results are presented for the more general case that the orthogonality measure may have a Denumerable Set of mass--points outside of $E$ which are supposed to accumulate on $E$ only and to satisfy (together with the zeros of the associated Stieltjes function) the free--interpolation Carleson--type condition. Up to the case of a finite number of mass points this is even new for the single interval case. Furthermore, as a byproduct of our representations, we obtain that the recurrence coefficients of the orthonormal polynomials behave asymptotically almost periodic. Or in other words the Jacobi matrices associated with the above discussed orthonormal polynomials are compact perturbations of a one--sided restriction of almost periodic Jacobi matrices with homogeneous spectrum. Our main tool is a theory of Hardy spaces of character--automorphic functions and forms on Riemann surfaces of Widom type, we use also some ideas of scattering theory for one--dimensional Schr\"odinger equations.

  • Asymptotics of orthonormal polynomials in the presence of a Denumerable Set of mass points
    Proceedings of the American Mathematical Society, 2001
    Co-Authors: Franz Peherstorfer, Peter Yuditskii
    Abstract:

    Let a be a positive measure whose support is an interval E plus a Denumerable Set of mass points which accumulate at the boundary points of E only. Under the assumptions that the mass points satisfy Blaschke's condition and that the absolutely continuous part of a satisfies Szego's condition, asymptotics for the orthonormal polynomials on and off the support are given. So far asymptotics were only available if the Set of mass points is finite.

Thiéry, Nicolas M. - One of the best experts on this subject based on the ideXlab platform.

  • Classification of P-oligomorphic groups, conjectures of Cameron and Macpherson
    2020
    Co-Authors: Falque Justine, Thiéry, Nicolas M.
    Abstract:

    Let G be a group of permutations of a Denumerable Set E. The profile of G is the function phi which counts, for each n, the number phi(n) of orbits of G acting on the n-subSets of E. Counting functions arising this way, and their associated generating series, form a rich yet apparently strongly constrained class. In particular, Cameron conjectured in the late seventies that, whenever the profile phi(n) is bounded by a polynomial -- we say that G is P-oligomorphic --, it is asymptotically equivalent to a polynomial. In 1985, Macpherson further asked whether the orbit algebra of G -- a graded commutative algebra invented by Cameron and whose Hilbert function is phi -- is finitely generated. In this paper we establish a classification of (closed) P-oligomorphic permutation groups in terms of finite permutation groups with decorated blocks. It follows from the classification that the orbit algebra of any P-oligomorphic group is isomorphic to (a straightforward quotient of) the invariant ring of some finite permutation group. This answers positively both Cameron's conjecture and Macpherson's question. The orbit algebra is in fact Cohen-Macaulay; therefore the generating series of phi is a rational fraction whose numerator has positive coefficients, while the denominator admits a combinatorial description. In addition, the classification provides a finite data structure for encoding closed P-oligomorphic groups. This paves the way for computing with them and enumerating them as well as for proofs by structural induction. Finally, the relative simplicity of the classification gives hopes to extend the study to, e.g., the class of (closed) permutations groups with sub-exponential profile. The proof exploits classical notions from group theory -- notably block systems and their lattice properties --, commutative algebra, and invariant theory.Comment: 39 pages v2: fixed some typo

  • The orbit algebra of a permutation group with polynomial profile is Cohen-Macaulay
    2018
    Co-Authors: Falque Justine, Thiéry, Nicolas M.
    Abstract:

    Let $G$ be a group of permutations of a Denumerable Set $E$. The profile of $G$ is the function $\phi_G$ which counts, for each $n$, the (possibly infinite) number $\phi_G(n)$ of orbits of $G$ acting on the $n$-subSets of $E$. Counting functions arising this way, and their associated generating series, form a rich yet apparently strongly constrained class. In particular, Cameron conjectured in the late seventies that, whenever $\phi_G(n)$ is bounded by a polynomial, it is asymptotically equivalent to a polynomial. In 1985, Macpherson further asked if the orbit algebra of $G$ - a graded commutative algebra invented by Cameron and whose Hilbert function is $\phi_G$ - is finitely generated. In this paper, we announce a proof of a stronger statement: the orbit algebra is Cohen-Macaulay. The generating series of the profile is a rational fraction whose numerator has positive coefficients and denominator admits a combinatorial description. The proof uses classical techniques from group actions, commutative algebra, and invariant theory; it steps towards a classification of ages of permutation groups with profile bounded by a polynomial.Comment: 12 pages. To be presented at FPSAC 2018 Hanover, July 2018. This version includes some minor improvements. Full proofs and additional examples and figures will be published in a long version of this extended abstrac

  • The orbit algebra of a permutation group with polynomial profile is Cohen-Macaulay
    HAL CCSD, 2018
    Co-Authors: Falque Justine, Thiéry, Nicolas M.
    Abstract:

    12 pages. To be presented at FPSAC 2018 Hanover, July 2018. This version includes some minor improvements. Full proofs and additional examples and figures will be published in a long version of this extended abstractInternational audienceLet $G$ be a group of permutations of a Denumerable Set $E$. The profile of $G$ is the function $\phi_G$ which counts, for each $n$, the (possibly infinite) number $\phi_G(n)$ of orbits of $G$ acting on the $n$-subSets of $E$. Counting functions arising this way, and their associated generating series, form a rich yet apparently strongly constrained class. In particular, Cameron conjectured in the late seventies that, whenever $\phi_G(n)$ is bounded by a polynomial, it is asymptotically equivalent to a polynomial. In 1985, Macpherson further asked if the orbit algebra of $G$ - a graded commutative algebra invented by Cameron and whose Hilbert function is $\phi_G$ - is finitely generated. In this paper, we announce a proof of a stronger statement: the orbit algebra is Cohen-Macaulay. The generating series of the profile is a rational fraction whose numerator has positive coefficients and denominator admits a combinatorial description. The proof uses classical techniques from group actions, commutative algebra, and invariant theory; it steps towards a classification of ages of permutation groups with profile bounded by a polynomial

Franz Peherstorfer - One of the best experts on this subject based on the ideXlab platform.

  • asymptotic behavior of polynomials orthonormal on a homogeneous Set
    Journal D Analyse Mathematique, 2003
    Co-Authors: Franz Peherstorfer, Peter Yuditskii
    Abstract:

    LetE be a homogeneous compact Set, for instance a Cantor Set of positive length. Further, let σ be a positive measure with supp(σ)=E. Under the condition that the absolutely continuous part of σ satisfies a Szego-type condition, we give an asymptotic representation, on and off the support, for the polynomials orthonomal with respect to σ. For the special case thatE consists of a finite number of intervals and that σ has no singular component, this is a well-known result of Widom. IfE=[a,b], it becomes a classical result due to Szego; and in case that there appears in addition a singular component, it is due to Kolmogorov-krein. In fact, the results are presented for the more general case that the orthogonality measure may have a Denumerable Set of mass-points outside ofE which are supposed to accumulate only onE and to satisfy (together with the zeros of the associated Stieltjes function) the free-interpolation Carleson-type condition. Up to the case of a finite number of mass points, this is even new for the single interval case. Furthermore, as a byproduct of our representations, we obtain that the recurrence coefficients of the orthonormal polynomials behave asymptotically almost periodic. In other words, the Jacobi matrices associated with the above discussed orthonomal polynomials are compact perturbations of a onesided restriction of almost periodic Jacobi matrices with homogeneous spectrum. Our main tool is a theory of Hardy spaces of character-automorphic functions and forms on Riemann surfaces of Widom type; we use also some ideas of scattering theory for one-dimensional Schrodinger equations.

  • asymptotic behavior of polynomials orthonormal on a homogeneous Set
    arXiv: Functional Analysis, 2002
    Co-Authors: Franz Peherstorfer, Peter Yuditskii
    Abstract:

    Let $E$ be a homogeneous compact Set, for instance a Cantor Set of positive length. Further let $\sigma$ be a positive measure with $\text{supp}(\sigma)=E$. Under the condition that the absolutely continuous part of $\sigma$ satisfies a Szeg\"o--type condition we give an asymptotic representation, on and off the support, for the polynomials orthonormal with respect to $\sigma$. For the special case that $E$ consists of a finite number of intervals and that $\sigma$ has no singular component this is a nowaday well known result of Widom. If $E=[a,b]$ it becomes a classical result due to Szeg\"o and in case that there appears in addition a singular component, it is due to Kolmogorov--Krein. In fact the results are presented for the more general case that the orthogonality measure may have a Denumerable Set of mass--points outside of $E$ which are supposed to accumulate on $E$ only and to satisfy (together with the zeros of the associated Stieltjes function) the free--interpolation Carleson--type condition. Up to the case of a finite number of mass points this is even new for the single interval case. Furthermore, as a byproduct of our representations, we obtain that the recurrence coefficients of the orthonormal polynomials behave asymptotically almost periodic. Or in other words the Jacobi matrices associated with the above discussed orthonormal polynomials are compact perturbations of a one--sided restriction of almost periodic Jacobi matrices with homogeneous spectrum. Our main tool is a theory of Hardy spaces of character--automorphic functions and forms on Riemann surfaces of Widom type, we use also some ideas of scattering theory for one--dimensional Schr\"odinger equations.

  • Asymptotics of orthonormal polynomials in the presence of a Denumerable Set of mass points
    Proceedings of the American Mathematical Society, 2001
    Co-Authors: Franz Peherstorfer, Peter Yuditskii
    Abstract:

    Let a be a positive measure whose support is an interval E plus a Denumerable Set of mass points which accumulate at the boundary points of E only. Under the assumptions that the mass points satisfy Blaschke's condition and that the absolutely continuous part of a satisfies Szego's condition, asymptotics for the orthonormal polynomials on and off the support are given. So far asymptotics were only available if the Set of mass points is finite.

Falque Justine - One of the best experts on this subject based on the ideXlab platform.

  • Classification of P-oligomorphic groups, conjectures of Cameron and Macpherson
    2020
    Co-Authors: Falque Justine, Thiéry, Nicolas M.
    Abstract:

    Let G be a group of permutations of a Denumerable Set E. The profile of G is the function phi which counts, for each n, the number phi(n) of orbits of G acting on the n-subSets of E. Counting functions arising this way, and their associated generating series, form a rich yet apparently strongly constrained class. In particular, Cameron conjectured in the late seventies that, whenever the profile phi(n) is bounded by a polynomial -- we say that G is P-oligomorphic --, it is asymptotically equivalent to a polynomial. In 1985, Macpherson further asked whether the orbit algebra of G -- a graded commutative algebra invented by Cameron and whose Hilbert function is phi -- is finitely generated. In this paper we establish a classification of (closed) P-oligomorphic permutation groups in terms of finite permutation groups with decorated blocks. It follows from the classification that the orbit algebra of any P-oligomorphic group is isomorphic to (a straightforward quotient of) the invariant ring of some finite permutation group. This answers positively both Cameron's conjecture and Macpherson's question. The orbit algebra is in fact Cohen-Macaulay; therefore the generating series of phi is a rational fraction whose numerator has positive coefficients, while the denominator admits a combinatorial description. In addition, the classification provides a finite data structure for encoding closed P-oligomorphic groups. This paves the way for computing with them and enumerating them as well as for proofs by structural induction. Finally, the relative simplicity of the classification gives hopes to extend the study to, e.g., the class of (closed) permutations groups with sub-exponential profile. The proof exploits classical notions from group theory -- notably block systems and their lattice properties --, commutative algebra, and invariant theory.Comment: 39 pages v2: fixed some typo

  • The orbit algebra of a permutation group with polynomial profile is Cohen-Macaulay
    2018
    Co-Authors: Falque Justine, Thiéry, Nicolas M.
    Abstract:

    Let $G$ be a group of permutations of a Denumerable Set $E$. The profile of $G$ is the function $\phi_G$ which counts, for each $n$, the (possibly infinite) number $\phi_G(n)$ of orbits of $G$ acting on the $n$-subSets of $E$. Counting functions arising this way, and their associated generating series, form a rich yet apparently strongly constrained class. In particular, Cameron conjectured in the late seventies that, whenever $\phi_G(n)$ is bounded by a polynomial, it is asymptotically equivalent to a polynomial. In 1985, Macpherson further asked if the orbit algebra of $G$ - a graded commutative algebra invented by Cameron and whose Hilbert function is $\phi_G$ - is finitely generated. In this paper, we announce a proof of a stronger statement: the orbit algebra is Cohen-Macaulay. The generating series of the profile is a rational fraction whose numerator has positive coefficients and denominator admits a combinatorial description. The proof uses classical techniques from group actions, commutative algebra, and invariant theory; it steps towards a classification of ages of permutation groups with profile bounded by a polynomial.Comment: 12 pages. To be presented at FPSAC 2018 Hanover, July 2018. This version includes some minor improvements. Full proofs and additional examples and figures will be published in a long version of this extended abstrac

  • The orbit algebra of a permutation group with polynomial profile is Cohen-Macaulay
    HAL CCSD, 2018
    Co-Authors: Falque Justine, Thiéry, Nicolas M.
    Abstract:

    12 pages. To be presented at FPSAC 2018 Hanover, July 2018. This version includes some minor improvements. Full proofs and additional examples and figures will be published in a long version of this extended abstractInternational audienceLet $G$ be a group of permutations of a Denumerable Set $E$. The profile of $G$ is the function $\phi_G$ which counts, for each $n$, the (possibly infinite) number $\phi_G(n)$ of orbits of $G$ acting on the $n$-subSets of $E$. Counting functions arising this way, and their associated generating series, form a rich yet apparently strongly constrained class. In particular, Cameron conjectured in the late seventies that, whenever $\phi_G(n)$ is bounded by a polynomial, it is asymptotically equivalent to a polynomial. In 1985, Macpherson further asked if the orbit algebra of $G$ - a graded commutative algebra invented by Cameron and whose Hilbert function is $\phi_G$ - is finitely generated. In this paper, we announce a proof of a stronger statement: the orbit algebra is Cohen-Macaulay. The generating series of the profile is a rational fraction whose numerator has positive coefficients and denominator admits a combinatorial description. The proof uses classical techniques from group actions, commutative algebra, and invariant theory; it steps towards a classification of ages of permutation groups with profile bounded by a polynomial