Displaying similar documents to “Mean field limit for the one dimensional Vlasov-Poisson equation”

On the global regularity of subcritical Euler–Poisson equations with pressure

Eitan Tadmor, Dongming Wei (2008)

Journal of the European Mathematical Society

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We prove that the one-dimensional Euler–Poisson system driven by the Poisson forcing together with the usual γ -law pressure, γ 1 , admits global solutions for a large class of initial data. Thus, the Poisson forcing regularizes the generic finite-time breakdown in the 2 × 2 p -system. Global regularity is shown to depend on whether or not the initial configuration of the Riemann invariants and density crosses an intrinsic critical threshold.

Poisson's equation and characterizations of reflexivity of Banach spaces

Vladimir P. Fonf, Michael Lin, Przemysław Wojtaszczyk (2011)

Colloquium Mathematicae

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Let X be a Banach space with a basis. We prove that X is reflexive if and only if every power-bounded linear operator T satisfies Browder’s equality x X : s u p n | | k = 1 n T k x | | < = (I-T)X . We then deduce that X (with a basis) is reflexive if and only if every strongly continuous bounded semigroup T t : t 0 with generator A satisfies A X = x X : s u p s > 0 | | 0 s T t x d t | | < . The range (I-T)X (respectively, AX for continuous time) is the space of x ∈ X for which Poisson’s equation (I-T)y = x (Ay = x in continuous time) has a solution y ∈ X; the above equalities...

Canonical Poisson-Nijenhuis structures on higher order tangent bundles

P. M. Kouotchop Wamba (2014)

Annales Polonici Mathematici

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Let M be a smooth manifold of dimension m>0, and denote by S c a n the canonical Nijenhuis tensor on TM. Let Π be a Poisson bivector on M and Π T the complete lift of Π on TM. In a previous paper, we have shown that ( T M , Π T , S c a n ) is a Poisson-Nijenhuis manifold. Recently, the higher order tangent lifts of Poisson manifolds from M to T r M have been studied and some properties were given. Furthermore, the canonical Nijenhuis tensors on T A M are described by A. Cabras and I. Kolář [Arch. Math. (Brno) 38 (2002),...

Prolongation of Poisson 2 -form on Weil bundles

Norbert Mahoungou Moukala, Basile Guy Richard Bossoto (2016)

Archivum Mathematicum

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In this paper, M denotes a smooth manifold of dimension n , A a Weil algebra and M A the associated Weil bundle. When ( M , ω M ) is a Poisson manifold with 2 -form ω M , we construct the 2 -Poisson form ω M A A , prolongation on M A of the 2 -Poisson form ω M . We give a necessary and sufficient condition for that M A be an A -Poisson manifold.

The Dixmier-Moeglin equivalence and a Gel’fand-Kirillov problem for Poisson polynomial algebras

K. R. Goodearl, S. Launois (2011)

Bulletin de la Société Mathématique de France

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The structure of Poisson polynomial algebras of the type obtained as semiclassical limits of quantized coordinate rings is investigated. Sufficient conditions for a rational Poisson action of a torus on such an algebra to leave only finitely many Poisson prime ideals invariant are obtained. Combined with previous work of the first-named author, this establishes the Poisson Dixmier-Moeglin equivalence for large classes of Poisson polynomial rings, including semiclassical limits of quantum...

Involutivity of truncated microsupports

Masaki Kashiwara, Térésa Monteiro Fernandes, Pierre Schapira (2003)

Bulletin de la Société Mathématique de France

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Using a result of J.-M. Bony, we prove the weak involutivity of truncated microsupports. More precisely, given a sheaf F on a real manifold and k , if two functions vanish on SS k ( F ) , then so does their Poisson bracket.

Kontsevich Deformation Quantization on Lie Algebras

Nabiha Ben Amar, Mouna Chaabouni, Mabrouka Hfaiedh (2007)

Bollettino dell'Unione Matematica Italiana

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We consider Kontsevich star product on the dual 𝔤 * of a general Lie algebra g equipped with the linear Poisson bracket. We show that this star product provides a deformation quantization by partial embeddings in the direction of the Poisson bracket.

Cauchy-Poisson transform and polynomial inequalities

Mirosław Baran (2009)

Annales Polonici Mathematici

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We apply the Cauchy-Poisson transform to prove some multivariate polynomial inequalities. In particular, we show that if the pluricomplex Green function of a fat compact set E in N is Hölder continuous then E admits a Szegö type inequality with weight function d i s t ( x , E ) - ( 1 - κ ) with a positive κ. This can be viewed as a (nontrivial) generalization of the classical result for the interval E = [-1,1] ⊂ ℝ.

About a Variant of the 1 d Vlasov equation, dubbed “Vlasov-Dirac-Benney Equation"

Claude Bardos (2012-2013)

Séminaire Laurent Schwartz — EDP et applications

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This is a report on project initiated with Anne Nouri [3], presently in progress, with the collaboration of Nicolas Besse [2] ([2] is mainly the material of this report) . It concerns a version of the Vlasov equation where the self interacting potential is replaced by a Dirac mass. Emphasis is put on the relations between the linearized version, the full non linear problem and also on natural connections with several other equations of mathematical physic.

GCD sums from Poisson integrals and systems of dilated functions

Christoph Aistleitner, István Berkes, Kristian Seip (2015)

Journal of the European Mathematical Society

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Upper bounds for GCD sums of the form k , = 1 N ( gcd ( n k , n ) ) 2 α ( n k n ) α are established, where ( n k ) 1 k N is any sequence of distinct positive integers and 0 < α 1 ; the estimate for α = 1 / 2 solves in particular a problem of Dyer and Harman from 1986, and the estimates are optimal except possibly for α = 1 / 2 . The method of proof is based on identifying the sum as a certain Poisson integral on a polydisc; as a byproduct, estimates for the largest eigenvalues of the associated GCD matrices are also found. The bounds for such GCD sums are used to...

One-parameter contractions of Lie-Poisson brackets

Oksana Yakimova (2014)

Journal of the European Mathematical Society

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We consider contractions of Lie and Poisson algebras and the behaviour of their centres under contractions. A polynomial Poisson algebra 𝒜 = 𝕂 [ 𝔸 n ] is said to be of Kostant type, if its centre Z ( 𝒜 ) is freely generated by homogeneous polynomials F 1 , ... , F r such that they give Kostant’s regularity criterion on 𝔸 n ( d x F i are linear independent if and only if the Poisson tensor has the maximal rank at x ). If the initial Poisson algebra is of Kostant type and F i satisfy a certain degree-equality, then the contraction...

A Hardy space related to the square root of the Poisson kernel

Jonatan Vasilis (2010)

Studia Mathematica

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A real-valued Hardy space H ¹ ( ) L ¹ ( ) related to the square root of the Poisson kernel in the unit disc is defined. The space is shown to be strictly larger than its classical counterpart H¹(). A decreasing function is in H ¹ ( ) if and only if the function is in the Orlicz space LloglogL(). In contrast to the case of H¹(), there is no such characterization for general positive functions: every Orlicz space strictly larger than L log L() contains positive functions which do not belong to H ¹ ( ) , and no Orlicz...

Noncommutative del Pezzo surfaces and Calabi-Yau algebras

Pavel Etingof, Victor Ginzburg (2010)

Journal of the European Mathematical Society

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The hypersurface in 3 with an isolated quasi-homogeneous elliptic singularity of type E ˜ r , r = 6 , 7 , 8 , has a natural Poisson structure. We show that the family of del Pezzo surfaces of the corresponding type E r provides a semiuniversal Poisson deformation of that Poisson structure. We also construct a deformation-quantization of the coordinate ring of such a del Pezzo surface. To this end, we first deform the polynomial algebra [ x 1 , x 2 , x 3 ] to a noncommutative algebra with generators x 1 , x 2 , x 3 and the following 3 relations...

The evolution and Poisson kernels on nilpotent meta-abelian groups

Richard Penney, Roman Urban (2013)

Studia Mathematica

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Let S be a semidirect product S = N⋊ A where N is a connected and simply connected, non-abelian, nilpotent meta-abelian Lie group and A is isomorphic to k , k>1. We consider a class of second order left-invariant differential operators on S of the form α = L a + Δ α , where α k , and for each a k , L a is left-invariant second order differential operator on N and Δ α = Δ - α , , where Δ is the usual Laplacian on k . Using some probabilistic techniques (e.g., skew-product formulas for diffusions on S and N respectively)...

On a modification of the Poisson integral operator

Dariusz Partyka (2011)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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Given a quasisymmetric automorphism γ of the unit circle 𝕋 we define and study a modification P γ of the classical Poisson integral operator in the case of the unit disk 𝔻 . The modification is done by means of the generalized Fourier coefficients of γ . For a Lebesgue’s integrable complexvalued function f on 𝕋 , P γ [ f ] is a complex-valued harmonic function in 𝔻 and it coincides with the classical Poisson integral of f provided γ is the identity mapping on 𝕋 . Our considerations are motivated by...

Quantization of Drinfeld Zastava in type A

Michael Finkelberg, Leonid Rybnikov (2014)

Journal of the European Mathematical Society

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Drinfeld Zastava is a certain closure of the moduli space of maps from the projective line to the Kashiwara flag scheme of the affine Lie algebra 𝔰𝔩 ^ n . We introduce an affine, reduced, irreducible, normal quiver variety Z which maps to the Zastava space bijectively at the level of complex points. The natural Poisson structure on the Zastava space can be described on Z in terms of Hamiltonian reduction of a certain Poisson subvariety of the dual space of a (nonsemisimple) Lie algebra. The...

Existence of solutions to the Poisson equation in L p -weighted spaces

Joanna Rencławowicz, Wojciech M. Zajączkowski (2010)

Applicationes Mathematicae

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We examine the Poisson equation with boundary conditions on a cylinder in a weighted space of L p , p≥ 3, type. The weight is a positive power of the distance from a distinguished plane. To prove the existence of solutions we use our result on existence in a weighted L₂ space.

Poisson geometry and deformation quantization near a strictly pseudoconvex boundary

Eric Leichtnam, Xiang Tang, Alan Weinstein (2007)

Journal of the European Mathematical Society

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Let X be a complex manifold with strongly pseudoconvex boundary M . If ψ is a defining function for M , then log ψ is plurisubharmonic on a neighborhood of M in X , and the (real) 2-form σ = i ¯ ( log ψ ) is a symplectic structure on the complement of M in a neighborhood of M in X ; it blows up along M . The Poisson structure obtained by inverting σ extends smoothly across M and determines a contact structure on M which is the same as the one induced by the complex structure. When M is compact, the Poisson structure...

Universal lifting theorem and quasi-Poisson groupoids

David Inglesias-Ponte, Camille Laurent-Gengoux, Ping Xu (2012)

Journal of the European Mathematical Society

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We prove the universal lifting theorem: for an α -simply connected and α -connected Lie groupoid Γ with Lie algebroid A , the graded Lie algebra of multi-differentials on A is isomorphic to that of multiplicative multi-vector fields on Γ . As a consequence, we obtain the integration theorem for a quasi-Lie bialgebroid, which generalizes various integration theorems in the literature in special cases. The second goal of the paper is the study of basic properties of quasi-Poisson groupoids....

The structure of split regular Hom-Poisson algebras

María J. Aragón Periñán, Antonio J. Calderón Martín (2016)

Colloquium Mathematicae

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We introduce the class of split regular Hom-Poisson algebras formed by those Hom-Poisson algebras whose underlying Hom-Lie algebras are split and regular. This class is the natural extension of the ones of split Hom-Lie algebras and of split Poisson algebras. We show that the structure theorems for split Poisson algebras can be extended to the more general setting of split regular Hom-Poisson algebras. That is, we prove that an arbitrary split regular Hom-Poisson algebra is of the form...

Stability and semiclassics in self-generated fields

László Erdős, Soren Fournais, Jan Philip Solovej (2013)

Journal of the European Mathematical Society

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We consider non-interacting particles subject to a fixed external potential V and a self-generated magnetic field B . The total energy includes the field energy β B 2 and we minimize over all particle states and magnetic fields. In the case of spin-1/2 particles this minimization leads to the coupled Maxwell-Pauli system. The parameter β tunes the coupling strength between the field and the particles and it effectively determines the strength of the field. We investigate the stability and...

Cluster ensembles, quantization and the dilogarithm

Vladimir V. Fock, Alexander B. Goncharov (2009)

Annales scientifiques de l'École Normale Supérieure

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A cluster ensemble is a pair ( 𝒳 , 𝒜 ) of positive spaces (i.e. varieties equipped with positive atlases), coming with an action of a symmetry group Γ . The space 𝒜 is closely related to the spectrum of a cluster algebra [12]. The two spaces are related by a morphism p : 𝒜 𝒳 . The space 𝒜 is equipped with a closed 2 -form, possibly degenerate, and the space 𝒳 has a Poisson structure. The map p is compatible with these structures. The dilogarithm together with its motivic and quantum avatars plays a central...

Regular behavior at infinity of stationary measures of stochastic recursion on NA groups

Dariusz Buraczewski, Ewa Damek (2010)

Colloquium Mathematicae

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Let N be a simply connected nilpotent Lie group and let S = N ( ) d be a semidirect product, ( ) d acting on N by diagonal automorphisms. Let (Qₙ,Mₙ) be a sequence of i.i.d. random variables with values in S. Under natural conditions, including contractivity in the mean, there is a unique stationary measure ν on N for the Markov process Xₙ = MₙXn-1 + Qₙ. We prove that for an appropriate homogeneous norm on N there is χ₀ such that l i m t t χ ν x : | x | > t = C > 0 . In particular, this applies to classical Poisson kernels on symmetric...

On a theorem of Lindelof

Vladimir Ya. Gutlyanskii, Olli Martio, Vladimir Ryazanov (2011)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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We give a quasiconformal version of the proof for the classical Lindelof theorem: Let f map the unit disk 𝔻 conformally onto the inner domain of a Jordan curve 𝒞 : Then 𝒞 is smooth if and only if arg f ' ( z ) has a continuous extension to 𝔻 ¯ . Our proof does not use the Poisson integral representation of harmonic functions in the unit disk.

Eigenfunctions of the Laplace Operators for a Building of Type G ~ 2

A. M. Mantero, A. Zappa (2009)

Bollettino dell'Unione Matematica Italiana

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Let Δ be thick affine building of type G ~ 2 the Laplace operators of Δ ; associated with a pair ( y 1 , y 2 ) ; is the Poisson transform of a suitable finitely additive measure on the maximal boundary Ω of Δ ; by using only the combinatorial structure of Δ .

Asymptotic behavior of a stochastic combustion growth process

Alejandro Ramírez, Vladas Sidoravicius (2004)

Journal of the European Mathematical Society

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We study a continuous time growth process on the d -dimensional hypercubic lattice 𝒵 d , which admits a phenomenological interpretation as the combustion reaction A + B 2 A , where A represents heat particles and B inert particles. This process can be described as an interacting particle system in the following way: at time 0 a simple symmetric continuous time random walk of total jump rate one begins to move from the origin of the hypercubic lattice; then, as soon as any random walk visits a site...