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Ilias Zadik - One of the best experts on this subject based on the ideXlab platform.
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pade approximants density of rational functions in a ω and smoothness of the integration operator
Journal of Mathematical Analysis and Applications, 2015Co-Authors: Vassili Nestoridis, Ilias ZadikAbstract:First we establish some generic universalities for Pade approximants in the closure X∞(Ω) in the space A∞(Ω) of all rational functions with poles off Ω¯. The closure Ω¯ of the domain Ω⊂C is taken with respect to the finite plane C. Next we give sufficient conditions on Ω so that X∞(Ω)=A∞(Ω). Some of these conditions imply that, even if the boundary ∂Ω of a Jordan domain Ω has infinite length, the integration operator on Ω preserves H∞(Ω) and A(Ω) as well. We also give an example of a Jordan domain Ω and a function f∈A(Ω), such that its antiderivative is not bounded on Ω. Finally we restate these results for Volterra operators on the open unit disc D and we complete them by some generic results.
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pad e approximants density of rational functions in bbb a infty oo and smoothness of the integration operator
arXiv: Complex Variables, 2012Co-Authors: Vassili Nestoridis, Ilias ZadikAbstract:First we establish some generic universalities for Pad\'{e} approximants in the closure $X^\infty(\OO)$ in $A^\infty(\OO)$ of all rational functions with poles off $\oO$, the closure taken in $\C$ of the domain $\OO\subset\C$.\ Next we give sufficient conditions on $\OO$ so that $X^\infty(\OO)=A^\infty(\OO)$.\ Some of these conditions imply that, even if the boundary $\partial\OO$ of a Jordan domain $\OO$ has infinite length, the integration operator on $\OO$ preserves $H^\infty(\OO)$ and $A(\OO)$ as well.\ We also give an example of a Jordan domain $\OO$ and a function $f\in A(\OO)$, such that its antiderivative is not bounded on $\OO$.\ Finally we restate these results for Volterra operators on the open unit disc $D$ and we complete them by some generic results.
Vassili Nestoridis - One of the best experts on this subject based on the ideXlab platform.
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pade approximants density of rational functions in a ω and smoothness of the integration operator
Journal of Mathematical Analysis and Applications, 2015Co-Authors: Vassili Nestoridis, Ilias ZadikAbstract:First we establish some generic universalities for Pade approximants in the closure X∞(Ω) in the space A∞(Ω) of all rational functions with poles off Ω¯. The closure Ω¯ of the domain Ω⊂C is taken with respect to the finite plane C. Next we give sufficient conditions on Ω so that X∞(Ω)=A∞(Ω). Some of these conditions imply that, even if the boundary ∂Ω of a Jordan domain Ω has infinite length, the integration operator on Ω preserves H∞(Ω) and A(Ω) as well. We also give an example of a Jordan domain Ω and a function f∈A(Ω), such that its antiderivative is not bounded on Ω. Finally we restate these results for Volterra operators on the open unit disc D and we complete them by some generic results.
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pad e approximants density of rational functions in bbb a infty oo and smoothness of the integration operator
arXiv: Complex Variables, 2012Co-Authors: Vassili Nestoridis, Ilias ZadikAbstract:First we establish some generic universalities for Pad\'{e} approximants in the closure $X^\infty(\OO)$ in $A^\infty(\OO)$ of all rational functions with poles off $\oO$, the closure taken in $\C$ of the domain $\OO\subset\C$.\ Next we give sufficient conditions on $\OO$ so that $X^\infty(\OO)=A^\infty(\OO)$.\ Some of these conditions imply that, even if the boundary $\partial\OO$ of a Jordan domain $\OO$ has infinite length, the integration operator on $\OO$ preserves $H^\infty(\OO)$ and $A(\OO)$ as well.\ We also give an example of a Jordan domain $\OO$ and a function $f\in A(\OO)$, such that its antiderivative is not bounded on $\OO$.\ Finally we restate these results for Volterra operators on the open unit disc $D$ and we complete them by some generic results.
Victor G Kac - One of the best experts on this subject based on the ideXlab platform.
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poisson vertex algebras in the theory of hamiltonian equations
Japanese Journal of Mathematics, 2009Co-Authors: Aliaa Barakat, Alberto De Sole, Victor G KacAbstract:We lay down the foundations of the theory of Poisson vertex algebras aimed at its applications to integrability of Hamiltonian partial differential equations. Such an equation is called integrable if it can be included in an infinite hierarchy of compatible Hamiltonian equations, which admit an infinite sequence of linearly independent integrals of motion in involution. The construction of a hierarchy and its integrals of motion is achieved by making use of the so called Lenard scheme. We find simple conditions which guarantee that the scheme produces an infinite sequence of closed 1-forms \(\omega_j, j \in {\mathbb {Z}}_+\), of the variational complex Ω. If these forms are exact, i.e., ωj are variational derivatives of some local functionals ∫ hj, then the latter are integrals of motion in involution of the hierarchy formed by the corresponding Hamiltonian vector fields. We show that the complex Ω is exact, provided that the algebra of functions is \(\fancyscript {V}\) is “normal”; in particular, for arbitrary \(\fancyscript {V}\), any closed form in Ω becomes exact if we add to \(\fancyscript {V}\) a finite number of Antiderivatives. We demonstrate on the examples of the KdV, HD and CNW hierarchies how the Lenard scheme works. We also discover a new integrable hierarchy, which we call the CNW hierarchy of HD type. Developing the ideas of Dorfman, we extend the Lenard scheme to arbitrary Dirac structures, and demonstrate its applicability on the examples of the NLS, pKdV and KN hierarchies.
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poisson vertex algebras in the theory of hamiltonian equations
arXiv: Mathematical Physics, 2009Co-Authors: Aliaa Barakat, Alberto De Sole, Victor G KacAbstract:We lay down the foundations of the theory of Poisson vertex algebras aimed at its applications to integrability of Hamiltonian partial differential equations. Such an equation is called integrable if it can be included in an infinite hierarchy of compatible Hamiltonian equations, which admit an infinite sequence of linearly independent integrals of motion in involution. The construction of a hierarchy and its integrals of motion is achieved by making use of the so called Lenard scheme. We find simple conditions which guarantee that the scheme produces an infinite sequence of closed 1-forms \omega_j, j in Z_+, of the variational complex \Omega. If these forms are exact, i.e. \omega_j are variational derivatives of some local functionals \int h_j, then the latter are integrals of motion in involution of the hierarchy formed by the corresponding Hamiltonian vector fields. We show that the complex \Omega is exact, provided that the algebra of functions V is "normal"; in particular, for arbitrary V, any closed form in \Omega becomes exact if we add to V a finite number of Antiderivatives. We demonstrate on the examples of KdV, HD and CNW hierarchies how the Lenard scheme works. We also discover a new integrable hierarchy, which we call the CNW hierarchy of HD type. Developing the ideas of Dorfman, we extend the Lenard scheme to arbitrary Dirac structures, and demonstrate its applicability on the examples of the NLS, pKdV and KN hierarchies.
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Poisson vertex algebras in the theory of Hamiltonian equations
'Springer Science and Business Media LLC', 2009Co-Authors: Aliaa Barakat, Alberto De Sole, Victor G KacAbstract:We lay down the foundations of the theory of Poisson vertex algebras aimed at its applications to integrability of Hamiltonian partial differential equations. Such an equation is called integrable if it can be included in an infinite hierarchy of compatible Hamiltonian equations, which admit an infinite sequence of linearly independent integrals of motion in involution. The construction of a hierarchy and its integrals of motion is achieved by making use of the so called Lenard scheme. We find simple conditions which guarantee that the scheme produces an infinite sequence of closed 1-forms omega(j), j is an element of Z(+), of the variational complex Omega. If these forms are exact, i.e., omega(j) are variational derivatives of some local functionals integral h(j), then the latter are integrals of motion in involution of the hierarchy formed by the corresponding Hamiltonian vector fields. We show that the complex Omega is exact, provided that the algebra of functions V is "normal"; in particular, for arbitrary V, any closed form in Omega becomes exact if we add to V a finite number of Antiderivatives. We demonstrate on the examples of the KdV, HD and CNW hierarchies how the Lenard scheme works. We also discover a new integrable hierarchy, which we call the CNW hierarchy of HD type. Developing the ideas of Dorfman, we extend the Lenard scheme to arbitrary Dirac structures, and demonstrate its applicability on the examples of the NLS, pKdV and KN hierarchies
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euler maclaurin formula
2002Co-Authors: Victor G Kac, Pokman CheungAbstract:In q-calculus, the Jackson formula (19.2) provides a way to compute explicitly a q-antiderivative of any function. Recall that the Jackson formula was deduced formally using operators. We will do a similar thing for the h-antiderivative in this chapter.
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Poisson vertex algebras in the theory of Hamiltonian equations
2026Co-Authors: Aliaa Barakat, Alberto De Sole, Victor G KacAbstract:We lay down the foundations of the theory of Poisson vertex algebras aimed at its applications to integrability of Hamiltonian partial differential equations. Such an equation is called integrable if it can be included in an infinite hierarchy of compatible Hamiltonian equations, which admit an infinite sequence of linearly independent integrals of motion in involution. The construction of a hierarchy and its integrals of motion is achieved by making use of the so called Lenard scheme. We find simple conditions which guarantee that the scheme produces an infinite sequence of closed 1-forms ωj, j ∈ Z+, of the variational complex Ω. If these forms are exact, i.e. ωj are variational derivatives of some local functionals hj, then the latter are integrals of motion in involution of the hierarchy formed by the corresponding Hamiltonian vector fields. We show that the complex Ω is exact, provided that the algebra of functions V is “normal”; in particular, for arbitrary V, any closed form in Ω becomes exact if we add to V a finite number of Antiderivatives. We demonstrate on the examples of KdV, HD and CNW hierarchies how the Lenard scheme works. We also discover a new integrable hierarchy, which we call the CNW hierarchy of HD type. Developing the ideas of Dorfman, we extend the Lenard scheme to arbitrary Dirac structures, and demonstrate its applicability on the examples of the NLS, pKdV and KN hierarchies
Fedor Pakovich - One of the best experts on this subject based on the ideXlab platform.
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solution of the parametric center problem for the abel differential equation
Journal of the European Mathematical Society, 2017Co-Authors: Fedor PakovichAbstract:The Abel differential equation y′ = p(x)y + q(x)y with p, q ∈ R[x] is said to have a center on a segment [a, b] if all its solutions with the initial value y(a) small enough satisfy the condition y(b) = y(a). The problem of description of conditions implying that the Abel equation has a center may be interpreted as a simplified version of the classical Center-Focus problem of Poincare. The Abel equation is said to have a “parametric center” if for each e ∈ R the equation y′ = p(x)y + eq(x)y has a center. In this paper we show that the Abel equation has a parametric center if and only if the Antiderivatives P = ∫ p(x)dx, Q = ∫ q(x)dx satisfy the equalities P = P ◦ W, Q = Q ◦ W for some polynomials P , Q, and W such that W (a) = W (b). We also show that the last condition is necessary and sufficient for the “generalized moments” ∫ b a P dQ and ∫ b a QdP to vanish for all i ≥ 0.
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solution of the parametric center problem for the abel differential equation
arXiv: Classical Analysis and ODEs, 2014Co-Authors: Fedor PakovichAbstract:The Abel differential equation $y'=p(x)y^2+q(x)y^3$ with $p,q\in \mathbb R[x]$ is said to have a center on a segment $[a,b]$ if all its solutions, with the initial value $y(a)$ small enough, satisfy the condition $y(b)=y(a)$. The problem of description of conditions implying that the Abel equation has a center may be interpreted as a simplified version of the classical Center-Focus problem of Poincar\'e. The Abel equation is said to have a "parametric center" if for each $\varepsilon \in \mathbb R$ the equation $y'=p(x)y^2+\varepsilon q(x)y^3$ has a center. In this paper we show that the Abel equation has a parametric center if and only if the Antiderivatives $P=\int p(x) dx,$ $Q=\int q(x) dx$ satisfy the equalities $P=\widetilde P \circ W,\ $ $Q=\widetilde Q\circ W$ for some polynomials $\widetilde P,$ $\widetilde Q,$ and $W$ such that $W(a)=W(b)$. We also show that the last condition is necessary and sufficient for the "generalized moments" $\int_a^b P^id Q$ and $\int_a^b Q^id P$ to vanish for all $i\geq 0.$
Gary Mccartor - One of the best experts on this subject based on the ideXlab platform.
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The Indispensability of Ghost Fields in the Light-Cone Gauge Quantization of Gauge Fields
1999Co-Authors: Yuji Nakawaki, Gary MccartorAbstract:We continue McCartor and Robertson’s recent demonstration of the indispensability of ghost fields in the light-cone gauge quantization of gauge fields. It is shown that the ghost fields are indispensable in deriving well-defined Antiderivatives and in regularizing the most singular component of the gauge field propagator. To this end it is sufficient to confine ourselves to noninteracting abelian fields. Furthermore, to circumvent dealing with constrained systems, we construct the temporal gauge canonical formulation of the free electromagnetic field in auxiliary coordinates xµ = (x−, x+, x1, x2), where x − = x0cosθ − x3sinθ, x+ = x0sinθ+x3cosθ and x − plays the role of time. In so doing we can quantize the fields canonically without any constraints, unambiguously introduce “static ghost fields ” as residual gauge degrees of freedom and construct the light-cone gauge solution in the light-cone representation by simply taking the light-cone limit (θ → π 4). As a by product we find that, with a suitable choice of vacuum, the Mandelstam-Leibbrandt form of the propagator can be derived in the θ = 0 case (the temporal gauge formulation in the equal-time representation)
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SMUHEP/99-02 The Indispensability of Ghost Fields in the Light-Cone Gauge Quantization of Gauge Fields
1999Co-Authors: Yuji Nakawaki, Gary MccartorAbstract:We continue McCartor and Robertson’s recent demonstration of the indispensability of ghost fields in the light-cone gauge quantization of gauge fields. It is shown that the ghost fields are indispensable in deriving well-defined Antiderivatives and in regularizing the most singular component of gauge field propagator. To this end it is sufficient to confine ourselves to noninteracting abelian fields. Furthermore to circumvent dealing with constrained systems, we construct the temporal gauge canonical formulation of the free electromagnetic field in auxiliary coordinates x µ = (x − , x +, x 1, x 2) where x − = x 0 cosθ − x 3 sinθ, x + = x 0 sinθ + x 3 cosθ and x − plays the role of time. In so doing we can quantize the fields canonically without any constraints, unambiguously introduce ”static ghost fields ” as residual gauge degrees of freedom and construct the light-cone gauge solution in the light-cone representation by simply taking the lightcone limit (θ → π). As a by product we find that, with a suitable choice of vacuum 4 the Mandelstam-Leibbrandt form of the propagator can be derived in the θ = 0 cas