Displaying similar documents to “Resonant delocalization for random Schrödinger operators on tree graphs”

On the convergence and character spectra of compact spaces

István Juhász, William A. R. Weiss (2010)

Fundamenta Mathematicae

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An infinite set A in a space X converges to a point p (denoted by A → p) if for every neighbourhood U of p we have |A∖U| < |A|. We call cS(p,X) = |A|: A ⊂ X and A → p the convergence spectrum of p in X and cS(X) = ⋃cS(x,X): x ∈ X the convergence spectrum of X. The character spectrum of a point p ∈ X is χS(p,X) = χ(p,Y): p is non-isolated in Y ⊂ X, and χS(X) = ⋃χS(x,X): x ∈ X is the character spectrum of X. If κ ∈ χS(p,X) for a compactum X then κ,cf(κ) ⊂ cS(p,X). A selection of our...

Simultaneous solutions of operator Sylvester equations

Sang-Gu Lee, Quoc-Phong Vu (2014)

Studia Mathematica

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We consider simultaneous solutions of operator Sylvester equations A i X - X B i = C i (1 ≤ i ≤ k), where ( A , . . . , A k ) and ( B , . . . , B k ) are commuting k-tuples of bounded linear operators on Banach spaces and ℱ, respectively, and ( C , . . . , C k ) is a (compatible) k-tuple of bounded linear operators from ℱ to , and prove that if the joint Taylor spectra of ( A , . . . , A k ) and ( B , . . . , B k ) do not intersect, then this system of Sylvester equations has a unique simultaneous solution.

Borel parts of the spectrum of an operator and of the operator algebra of a separable Hilbert space

Piotr Niemiec (2012)

Studia Mathematica

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For a linear operator T in a Banach space let σ p ( T ) denote the point spectrum of T, let σ p , n ( T ) for finite n > 0 be the set of all λ σ p ( T ) such that dim ker(T - λ) = n and let σ p , ( T ) be the set of all λ σ p ( T ) for which ker(T - λ) is infinite-dimensional. It is shown that σ p ( T ) is σ , σ p , ( T ) is σ δ and for each finite n the set σ p , n ( T ) is the intersection of an σ set and a δ set provided T is closable and the domain of T is separable and weakly σ-compact. For closed densely defined operators in a separable Hilbert space a more...

Horocyclic products of trees

Laurent Bartholdi, Markus Neuhauser, Wolfgang Woess (2008)

Journal of the European Mathematical Society

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Let T 1 , , T d be homogeneous trees with degrees q 1 + 1 , , q d + 1 3 , respectively. For each tree, let 𝔥 : T j be the Busemann function with respect to a fixed boundary point (end). Its level sets are the horocycles. The horocyclic product of T 1 , , T d is the graph 𝖣𝖫 ( q 1 , , q d ) consisting of all d -tuples x 1 x d T 1 × × T d with 𝔥 ( x 1 ) + + 𝔥 ( x d ) = 0 , equipped with a natural neighbourhood relation. In the present paper, we explore the geometric, algebraic, analytic and probabilistic properties of these graphs and their isometry groups. If d = 2 and q 1 = q 2 = q then we obtain a Cayley graph...

On a characterization of k -trees

De-Yan Zeng, Jian Hua Yin (2015)

Czechoslovak Mathematical Journal

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A graph G is a k -tree if either G is the complete graph on k + 1 vertices, or G has a vertex v whose neighborhood is a clique of order k and the graph obtained by removing v from G is also a k -tree. Clearly, a k -tree has at least k + 1 vertices, and G is a 1-tree (usual tree) if and only if it is a 1 -connected graph and has no K 3 -minor. In this paper, motivated by some properties of 2-trees, we obtain a characterization of k -trees as follows: if G is a graph with at least k + 1 vertices, then G is...

Weighted Frobenius-Perron operators and their spectra

Mohammad Reza Jabbarzadeh, Rana Hajipouri (2017)

Mathematica Bohemica

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First, some classic properties of a weighted Frobenius-Perron operator 𝒫 ϕ u on L 1 ( Σ ) as a predual of weighted Koopman operator W = u U ϕ on L ( Σ ) will be investigated using the language of the conditional expectation operator. Also, we determine the spectrum of 𝒫 ϕ u under certain conditions.

Size of the giant component in a random geometric graph

Ghurumuruhan Ganesan (2013)

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

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In this paper, we study the size of the giant component C G in the random geometric graph G = G ( n , r n , f ) of n nodes independently distributed each according to a certain density f ( · ) in [ 0 , 1 ] 2 satisfying inf x [ 0 , 1 ] 2 f ( x ) g t ; 0 . If c 1 n r n 2 c 2 log n n for some positive constants c 1 , c 2 and n r n 2 as n , we show that the giant component of G contains at least n - o ( n ) nodes with probability at least 1 - e - β n r n 2 for all n and for some positive constant β . We also obtain estimates on the diameter and number of the non-giant components of G .

A-Browder-type theorems for direct sums of operators

Mohammed Berkani, Mustapha Sarih, Hassan Zariouh (2016)

Mathematica Bohemica

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We study the stability of a-Browder-type theorems for orthogonal direct sums of operators. We give counterexamples which show that in general the properties ( SBaw ) , ( SBab ) , ( SBw ) and ( SBb ) are not preserved under direct sums of operators. However, we prove that if S and T are bounded linear operators acting on Banach spaces and having the property ( SBab ) , then S T has the property ( SBab ) if and only if σ SBF + - ( S T ) = σ SBF + - ( S ) σ SBF + - ( T ) , where σ SBF + - ( T ) is the upper semi-B-Weyl spectrum of T . We obtain analogous preservation results for the properties ( SBaw ) ,...

On γ-labelings of trees

Gary Chartrand, David Erwin, Donald W. VanderJagt, Ping Zhang (2005)

Discussiones Mathematicae Graph Theory

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Let G be a graph of order n and size m. A γ-labeling of G is a one-to-one function f:V(G) → 0,1,2,...,m that induces a labeling f’: E(G) → 1,2,...,m of the edges of G defined by f’(e) = |f(u)-f(v)| for each edge e = uv of G. The value of a γ-labeling f is v a l ( f ) = Σ e E ( G ) f ' K ( e ) . The maximum value of a γ-labeling of G is defined as v a l m a x ( G ) = m a x v a l ( f ) : f i s a γ - l a b e l i n g o f G ; while the minimum value of a γ-labeling of G is v a l m i n ( G ) = m i n v a l ( f ) : f i s a γ - l a b e l i n g o f G ; The values v a l m a x ( S p , q ) and v a l m i n ( S p , q ) are determined for double stars S p , q . We present characterizations of connected graphs G of order n for which...

Uniform mixing time for random walk on lamplighter graphs

Júlia Komjáthy, Jason Miller, Yuval Peres (2014)

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

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Suppose that 𝒢 is a finite, connected graph and X is a lazy random walk on 𝒢 . The lamplighter chain X associated with X is the random walk on the wreath product 𝒢 = 𝐙 2 𝒢 , the graph whose vertices consist of pairs ( f ̲ , x ) where f is a labeling of the vertices of 𝒢 by elements of 𝐙 2 = { 0 , 1 } and x is a vertex in 𝒢 . There is an edge between ( f ̲ , x ) and ( g ̲ , y ) in 𝒢 if and only if x is adjacent to y in 𝒢 and f z = g z for all z x , y . In each step, X moves from a configuration ( f ̲ , x ) by updating x to y using the transition rule of X and then...

Positivity of integrated random walks

Vladislav Vysotsky (2014)

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

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Take a centered random walk S n and consider the sequence of its partial sums A n : = i = 1 n S i . Suppose S 1 is in the domain of normal attraction of an α -stable law with 1 l t ; α 2 . Assuming that S 1 is either right-exponential (i.e. ( S 1 g t ; x | S 1 g t ; 0 ) = e - a x for some a g t ; 0 and all x g t ; 0 ) or right-continuous (skip free), we prove that { A 1 g t ; 0 , , A N g t ; 0 } C α N 1 / ( 2 α ) - 1 / 2 as N , where C α g t ; 0 depends on the distribution of the walk. We also consider a conditional version of this problem and study positivity of integrated discrete bridges.

Symmetries of the nonlinear Schrödinger equation

Benoît Grébert, Thomas Kappeler (2002)

Bulletin de la Société Mathématique de France

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Symmetries of the defocusing nonlinear Schrödinger equation are expressed in action-angle coordinates and characterized in terms of the periodic and Dirichlet spectrum of the associated Zakharov-Shabat system. Application: proof of the conjecture that the periodic spectrum &lt; λ k - λ k + &lt; λ k + 1 - of a Zakharov-Shabat operator is symmetric,. λ k ± = - λ - k for all k , if and only if the sequence ( γ k ) k of gap lengths, γ k : = λ k + - λ k - , is symmetric with respect to k = 0 .

On graceful colorings of trees

Sean English, Ping Zhang (2017)

Mathematica Bohemica

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A proper coloring c : V ( G ) { 1 , 2 , ... , k } , k 2 of a graph G is called a graceful k -coloring if the induced edge coloring c ' : E ( G ) { 1 , 2 , ... , k - 1 } defined by c ' ( u v ) = | c ( u ) - c ( v ) | for each edge u v of G is also proper. The minimum integer k for which G has a graceful k -coloring is the graceful chromatic number χ g ( G ) . It is known that if T is a tree with maximum degree Δ , then χ g ( T ) 5 3 Δ and this bound is best possible. It is shown for each integer Δ 2 that there is an infinite class of trees T with maximum degree Δ such that χ g ( T ) = 5 3 Δ . In particular, we investigate for each...

Generalized 3-edge-connectivity of Cartesian product graphs

Yuefang Sun (2015)

Czechoslovak Mathematical Journal

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The generalized k -connectivity κ k ( G ) of a graph G was introduced by Chartrand et al. in 1984. As a natural counterpart of this concept, Li et al. in 2011 introduced the concept of generalized k -edge-connectivity which is defined as λ k ( G ) = min { λ ( S ) : S V ( G ) and | S | = k } , where λ ( S ) denotes the maximum number of pairwise edge-disjoint trees T 1 , T 2 , ... , T in G such that S V ( T i ) for 1 i . In this paper we prove that for any two connected graphs G and H we have λ 3 ( G H ) λ 3 ( G ) + λ 3 ( H ) , where G H is the Cartesian product of G and H . Moreover, the bound is sharp. We also...

H p spaces associated with Schrödinger operators with potentials from reverse Hölder classes

Jacek Dziubański, Jacek Zienkiewicz (2003)

Colloquium Mathematicae

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Let A = -Δ + V be a Schrödinger operator on d , d ≥ 3, where V is a nonnegative potential satisfying the reverse Hölder inequality with an exponent q > d/2. We say that f is an element of H A p if the maximal function s u p t > 0 | T t f ( x ) | belongs to L p ( d ) , where T t t > 0 is the semigroup generated by -A. It is proved that for d/(d+1) < p ≤ 1 the space H A p admits a special atomic decomposition.

Properties of Wiener-Wintner dynamical systems

I. Assani, K. Nicolaou (2001)

Bulletin de la Société Mathématique de France

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In this paper we prove the following results. First, we show the existence of Wiener-Wintner dynamical system with continuous singular spectrum in the orthocomplement of their respective Kronecker factors. The second result states that if f L p , p large enough, is a Wiener-Wintner function then, for all γ ( 1 + 1 2 p - β 2 , 1 ] , there exists a set X f of full measure for which the series n = 1 f ( T n x ) e 2 π i n ϵ n γ converges uniformly with respect to ϵ .

On the perturbation functions and similarity orbits

Haïkel Skhiri (2008)

Studia Mathematica

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We show that the essential spectral radius ϱ e ( T ) of T ∈ B(H) can be calculated by the formula ϱ e ( T ) = inf · ( X T X - 1 ) : X an invertible operator, where · ( T ) is a Φ₁-perturbation function introduced by Mbekhta [J. Operator Theory 51 (2004)]. Also, we show that if · ( T ) is a Φ₂-perturbation function [loc. cit.] and if T is a Fredholm operator, then d i s t ( 0 , σ e ( T ) ) = sup · ( X T X - 1 ) : X an invertible operator.

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 .