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Displaying similar documents to “ L 2 well-posed Cauchy problems and symmetrizability of first order systems”

The parabolic Anderson model in a dynamic random environment: Basic properties of the quenched Lyapunov exponent

D. Erhard, F. den Hollander, G. Maillard (2014)

Annales de l'I.H.P. Probabilités et statistiques

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In this paper we study the parabolic Anderson equation u ( x , t ) / t = κ 𝛥 u ( x , t ) + ξ ( x , t ) u ( x , t ) , x d , t 0 , where the u -field and the ξ -field are -valued, κ [ 0 , ) is the diffusion constant, and 𝛥 is the discrete Laplacian. The ξ -field plays the role of athat drives the equation. The initial condition u ( x , 0 ) = u 0 ( x ) , x d , is taken to be non-negative and bounded. The solution of the parabolic Anderson equation describes the evolution of a field of particles performing independent simple random walks with binary branching: particles jump at rate 2 d κ ,...

Cutting the loss of derivatives for solvability under condition ( Ψ )

Nicolas Lerner (2006)

Bulletin de la Société Mathématique de France

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For a principal type pseudodifferential operator, we prove that condition  ( ψ ) implies local solvability with a loss of 3/2 derivatives. We use many elements of Dencker’s paper on the proof of the Nirenberg-Treves conjecture and we provide some improvements of the key energy estimates which allows us to cut the loss of derivatives from ϵ + 3 / 2 for any ϵ > 0 (Dencker’s most recent result) to 3/2 (the present paper). It is already known that condition  ( ψ ) doesimply local solvability with a loss of 1...

On the opial type criterion for the well-posedness of the Cauchy problem for linear systems of generalized ordinary differential equations

Malkhaz Ashordia (2016)

Mathematica Bohemica

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The Cauchy problem for the system of linear generalized ordinary differential equations in the J. Kurzweil sense d x ( t ) = d A 0 ( t ) · x ( t ) + d f 0 ( t ) , x ( t 0 ) = c 0 ( t I ) with a unique solution x 0 is considered. Necessary and sufficient conditions are obtained for a sequence of the Cauchy problems d x ( t ) = d A k ( t ) · x ( t ) + d f k ( t ) , x ( t k ) = c k ( k = 1 , 2 , ) to have a unique solution x k for any sufficiently large k such that x k ( t ) x 0 ( t ) uniformly on I . Presented results are analogous to the sufficient conditions due to Z. Opial for linear ordinary differential systems....

Some results on spaces with 1 -calibre

Wei-Feng Xuan, Wei-Xue Shi (2016)

Commentationes Mathematicae Universitatis Carolinae

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We prove that, assuming , if X is a space with 1 -calibre and a zeroset diagonal, then X is submetrizable. This gives a consistent positive answer to the question of Buzyakova in Observations on spaces with zeroset or regular G δ -diagonals, Comment. Math. Univ. Carolin. 46 (2005), no. 3, 469–473. We also make some observations on spaces with 1 -calibre.

Algebraic independence of the generating functions of Stern’s sequence and of its twist

Peter Bundschuh, Keijo Väänänen (2013)

Journal de Théorie des Nombres de Bordeaux

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Very recently, the generating function A ( z ) of the Stern sequence ( a n ) n 0 , defined by a 0 : = 0 , a 1 : = 1 , and a 2 n : = a n , a 2 n + 1 : = a n + a n + 1 for any integer n > 0 , has been considered from the arithmetical point of view. Coons [8] proved the transcendence of A ( α ) for every algebraic α with 0 < | α | < 1 , and this result was generalized in [6] to the effect that, for the same α ’s, all numbers A ( α ) , A ( α ) , A ( α ) , ... are algebraically independent. At about the same time, Bacher...

Coincidence for substitutions of Pisot type

Marcy Barge, Beverly Diamond (2002)

Bulletin de la Société Mathématique de France

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Let ϕ be a substitution of Pisot type on the alphabet 𝒜 = { 1 , 2 , ... , d } ; ϕ satisfies theif for every i , j 𝒜 , there are integers k , n such that ϕ n ( i ) and ϕ n ( j ) have the same k -th letter, and the prefixes of length k - 1 of ϕ n ( i ) and ϕ n ( j ) have the same image under the abelianization map. We prove that the strong coincidence condition is satisfied if d = 2 and provide a partial result for d 2 .

On a question of Schmidt and Summerer concerning 3 -systems

Johannes Schleischitz (2020)

Communications in Mathematics

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Following a suggestion of W.M. Schmidt and L. Summerer, we construct a proper 3 -system ( P 1 , P 2 , P 3 ) with the property ϕ ¯ 3 = 1 . In fact, our method generalizes to provide n -systems with ϕ ¯ n = 1 , for arbitrary n 3 . We visualize our constructions with graphics. We further present explicit examples of numbers ξ 1 , ... , ξ n - 1 that induce the n -systems in question.

Regularizing effect of the interplay between coefficients in some noncoercive integral functionals

Aiping Zhang, Zesheng Feng, Hongya Gao (2024)

Czechoslovak Mathematical Journal

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We are interested in regularizing effect of the interplay between the coefficient of zero order term and the datum in some noncoercive integral functionals of the type 𝒥 ( v ) = Ω j ( x , v , v ) d x + Ω a ( x ) | v | 2 d x - Ω f v d x , v W 0 1 , 2 ( Ω ) , where Ω N , j is a Carathéodory function such that ξ j ( x , s , ξ ) is convex, and there exist constants 0 τ < 1 and M > 0 such that | ξ | 2 ( 1 + | s | ) τ j ( x , s , ξ ) M | ξ | 2 for almost all x Ω , all s and all ξ N . We show that, even if 0 < a ( x ) and f ( x ) only belong to L 1 ( Ω ) , the interplay | f ( x ) | 2 Q a ( x ) implies the existence of a minimizer u W 0 1 , 2 ( Ω ) which belongs to L ( Ω ) .

On multiset colorings of generalized corona graphs

Yun Feng, Wensong Lin (2016)

Mathematica Bohemica

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A vertex k -coloring of a graph G is a if M ( u ) M ( v ) for every edge u v E ( G ) , where M ( u ) and M ( v ) denote the multisets of colors of the neighbors of u and v , respectively. The minimum k for which G has a multiset k -coloring is the χ m ( G ) of G . For an integer 0 , the - of a graph G , cor ( G ) , is the graph obtained from G by adding, for each vertex v in G , new neighbors which are end-vertices. In this paper, the multiset chromatic numbers are determined for - of all complete graphs, the regular complete...

On compactness and connectedness of the paratingent

Wojciech Zygmunt (2016)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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In this note we shall prove that for a continuous function ϕ : Δ n , where Δ ,  the paratingent of ϕ at a Δ is a non-empty and compact set in n if and only if ϕ satisfies Lipschitz condition in a neighbourhood of a . Moreover, in this case the paratingent is a connected set.

Local-global divisibility of rational points in some commutative algebraic groups

Roberto Dvornicich, Umberto Zannier (2001)

Bulletin de la Société Mathématique de France

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Let 𝒜 be a commutative algebraic group defined over a number field  k . We consider the following question:A complete answer for the case of the multiplicative group 𝔾 m is classical. We study other instances and in particular obtain an affirmative answer when r is a prime and  𝒜 is either an elliptic curve or a torus of small dimension with respect to r . Without restriction on the dimension of a torus, we produce an example showing that the answer can be negative even when r is a prime. ...