Displaying similar documents to “Automata, algebraicity and distribution of sequences of powers”

Oscillation conditions for difference equations with several variable arguments

George E. Chatzarakis, Takaŝi Kusano, Ioannis P. Stavroulakis (2015)

Mathematica Bohemica

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Consider the difference equation Δ x ( n ) + i = 1 m p i ( n ) x ( τ i ( n ) ) = 0 , n 0 x ( n ) - i = 1 m p i ( n ) x ( σ i ( n ) ) = 0 , n 1 , where ( p i ( n ) ) , 1 i m are sequences of nonnegative real numbers, τ i ( n ) [ σ i ( n ) ], 1 i m are general retarded (advanced) arguments and Δ [ ] denotes the forward (backward) difference operator Δ x ( n ) = x ( n + 1 ) - x ( n ) [ x ( n ) = x ( n ) - x ( n - 1 ) ]. New oscillation criteria are established when the well-known oscillation conditions lim sup n i = 1 m j = τ ( n ) n p i ( j ) > 1 lim sup n i = 1 m j = n σ ( n ) p i ( j ) > 1 and lim inf n i = 1 m j = τ i ( n ) n - 1 p i ( j ) > 1 e lim inf n i = 1 m j = n + 1 σ i ( n ) p i ( j ) > 1 e are not satisfied. Here τ ( n ) = max 1 i m τ i ( n ) [ σ ( n ) = min 1 i m σ i ( n ) ] . The results obtained essentially improve known results in the literature. Examples illustrating the results are also given.

Global approximations for the γ-order Lognormal distribution

Thomas L. Toulias (2013)

Discussiones Mathematicae Probability and Statistics

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A generalized form of the usual Lognormal distribution, denoted with γ , is introduced through the γ-order Normal distribution γ , with its p.d.f. defined into (0,+∞). The study of the c.d.f. of γ is focused on a heuristic method that provides global approximations with two anchor points, at zero and at infinity. Also evaluations are provided while certain bounds are obtained.

The right tail exponent of the Tracy–Widom β distribution

Laure Dumaz, Bálint Virág (2013)

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

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The Tracy–Widom β distribution is the large dimensional limit of the top eigenvalue of β random matrix ensembles. We use the stochastic Airy operator representation to show that as a the tail of the Tracy–Widom distribution satisfies P ( 𝑇𝑊 β g t ; a ) = a - ( 3 / 4 ) β + o ( 1 ) exp - 2 3 β a 3 / 2 .

Summation equations with sign changing kernels and applications to discrete fractional boundary value problems

Christopher S. Goodrich (2016)

Commentationes Mathematicae Universitatis Carolinae

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We consider the summation equation, for t [ μ - 2 , μ + b ] μ - 2 , y ( t ) = γ 1 ( t ) H 1 i = 1 n a i y ξ i + γ 2 ( t ) H 2 i = 1 m b i y ζ i + λ s = 0 b G ( t , s ) f ( s + μ - 1 , y ( s + μ - 1 ) ) in the case where the map ( t , s ) G ( t , s ) may change sign; here μ ( 1 , 2 ] is a parameter, which may be understood as the order of an associated discrete fractional boundary value problem. In spite of the fact that G is allowed to change sign, by introducing a new cone we are able to establish the existence of at least one positive solution to this problem by imposing some growth conditions on the functions H 1 and H 2 . Finally, as an application of the abstract existence...

Construction techniques for some thin sets in duals of compact abelian groups

D. J. Hajela (1986)

Annales de l'institut Fourier

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Various techniques are presented for constructing Λ (p) sets which are not Λ ( p + ϵ ) for all ϵ > 0 . The main result is that there is a Λ (4) set in the dual of any compact abelian group which is not Λ ( 4 + ϵ ) for all ϵ > 0 . Along the way to proving this, new constructions are given in dual groups in which constructions were already known of Λ (p) not Λ ( p + ϵ ) sets, for certain values of p . The main new constructions in specific dual groups are: – there is a Λ (2k) set which is not Λ ( 2 k + ϵ ) in Z ( 2 ) Z ( 2 ) for all 2 k , k N and...

Explicit construction of normal lattice configurations

Mordechay B. Levin, Meir Smorodinsky (2005)

Colloquium Mathematicae

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We extend Champernowne’s construction of normal numbers to base b to the d case and obtain an explicit construction of a generic point of the d shift transformation of the set 0 , 1 , . . . , b - 1 d .

Scale-free percolation

Maria Deijfen, Remco van der Hofstad, Gerard Hooghiemstra (2013)

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

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We formulate and study a model for inhomogeneous long-range percolation on d . Each vertex x d is assigned a non-negative weight W x , where ( W x ) x d are i.i.d. random variables. Conditionally on the weights, and given two parameters α , λ g t ; 0 , the edges are independent and the probability that there is an edge between x and y is given by p x y = 1 - exp { - λ W x W y / | x - y | α } . The parameter λ is the percolation parameter, while α describes the long-range nature of the model. We focus on the degree distribution in the resulting graph, on whether...

About a Pólya-Schiffer inequality

Bodo Dittmar, Maren Hantke (2011)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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For simply connected planar domains with the maximal conformal radius 1 it was proven in 1954 by G. Pólya and M. Schiffer that for the eigenvalues λ of the fixed membrane for any n the following inequality holds k = 1 n 1 λ k k = 1 n 1 λ k ( σ ) , where λ k ( σ ) are the eigenvalues of the unit disk. The aim of the paper is to give a sharper version of this inequality and for the sum of all reciprocals to derive formulas which allow in some cases to calculate exactly this sum.

Complete monotonicity of the remainder in an asymptotic series related to the psi function

Zhen-Hang Yang, Jing-Feng Tian (2024)

Czechoslovak Mathematical Journal

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Let p , q with p - q 0 , σ = 1 2 ( p + q - 1 ) and s = 1 2 ( 1 - p + q ) , and let 𝒟 m ( x ; p , q ) = 𝒟 0 ( x ; p , q ) + k = 1 m B 2 k ( s ) 2 k ( x + σ ) 2 k , where 𝒟 0 ( x ; p , q ) = ψ ( x + p ) + ψ ( x + q ) 2 - ln ( x + σ ) . We establish the asymptotic expansion 𝒟 0 ( x ; p , q ) - n = 1 B 2 n ( s ) 2 n ( x + σ ) 2 n as x , where B 2 n ( s ) stands for the Bernoulli polynomials. Further, we prove that the functions ( - 1 ) m 𝒟 m ( x ; p , q ) and ( - 1 ) m + 1 𝒟 m ( x ; p , q ) are completely monotonic in x on ( - σ , ) for every m 0 if and only if p - q [ 0 , 1 2 ] and p - q = 1 , respectively. This not only unifies the two known results but also yields some new results.

On the topology of polynomials with bounded integer coefficients

De-Jun Feng (2016)

Journal of the European Mathematical Society

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For a real number q > 1 and a positive integer m , let Y m ( q ) : = i = 0 n ϵ i q i : ϵ i 0 , ± 1 , ... , ± m , n = 0 , 1 , ... . In this paper, we show that Y m ( q ) is dense in if and only if q < m + 1 and q is not a Pisot number. This completes several previous results and answers an open question raised by Erdös, Joó and Komornik [8].