Displaying similar documents to “Two-level t-deformation”

The V a -deformation of the classical convolution

Anna Dorota Krystek (2007)

Banach Center Publications

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We study deformations of the classical convolution. For every invertible transformation T:μ ↦ Tμ, we are able to define a new associative convolution of measures by μ * T ν = T - 1 ( T μ * T ν ) . We deal with the V a -deformation of the classical convolution. We prove the analogue of the classical Lévy-Khintchine formula. We also show the central limit measure, which turns out to be the standard Gaussian measure. Moreover, we calculate the Poisson measure in the V a -deformed classical convolution and give the orthogonal...

The Lévy-Khintchine formula and Nica-Speicher property for deformations of the free convolution

Łukasz Jan Wojakowski (2007)

Banach Center Publications

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We study deformations of the free convolution arising via invertible transformations of probability measures on the real line T:μ ↦ Tμ. We define new associative convolutions of measures by μ T ν = T - 1 ( T μ T ν ) . We discuss infinite divisibility with respect to these convolutions, and we establish a Lévy-Khintchine formula. We conclude the paper by proving that for any such deformation of free probability all probability measures μ have the Nica-Speicher property, that is, one can find their convolution...

On the length of rational continued fractions over q ( X )

S. Driss (2015)

Discussiones Mathematicae - General Algebra and Applications

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Let q be a finite field and A ( Y ) q ( X , Y ) . The aim of this paper is to prove that the length of the continued fraction expansion of A ( P ) ; P q [ X ] , is bounded.

Simple fractions and linear decomposition of some convolutions of measures

Jolanta K. Misiewicz, Roger Cooke (2001)

Discussiones Mathematicae Probability and Statistics

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Every characteristic function φ can be written in the following way: φ(ξ) = 1/(h(ξ) + 1), where h(ξ) = ⎧ 1/φ(ξ) - 1 if φ(ξ) ≠ 0 ⎨ ⎩ ∞ if φ(ξ) = 0 This simple remark implies that every characteristic function can be treated as a simple fraction of the function h(ξ). In the paper, we consider a class C(φ) of all characteristic functions of the form φ a ( ξ ) = [ a / ( h ( ξ ) + a ) ] , where φ(ξ) is a fixed characteristic function. Using the well known theorem on simple fraction decomposition of rational functions we obtain...

Automatic continued fractions are transcendental or quadratic

Yann Bugeaud (2013)

Annales scientifiques de l'École Normale Supérieure

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We establish new combinatorial transcendence criteria for continued fraction expansions. Let  α = [ 0 ; a 1 , a 2 , ... ] be an algebraic number of degree at least three. One of our criteria implies that the sequence of partial quotients ( a ) 1 of  α is not ‘too simple’ (in a suitable sense) and cannot be generated by a finite automaton.

A convolution property of some measures with self-similar fractal support

Denise Szecsei (2007)

Colloquium Mathematicae

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We define a class of measures having the following properties: (1) the measures are supported on self-similar fractal subsets of the unit cube I M = [ 0 , 1 ) M , with 0 and 1 identified as necessary; (2) the measures are singular with respect to normalized Lebesgue measure m on I M ; (3) the measures have the convolution property that μ L p L p + ε for some ε = ε(p) > 0 and all p ∈ (1,∞). We will show that if (1/p,1/q) lies in the triangle with vertices (0,0), (1,1) and (1/2,1/3), then μ L p L q for any measure μ in our...

Generalised functions of bounded deformation

Gianni Dal Maso (2013)

Journal of the European Mathematical Society

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We introduce the space G B D of generalized functions of bounded deformation and study the structure properties of these functions: the rectiability and the slicing properties of their jump sets, and the existence of their approximate symmetric gradients. We conclude by proving a compactness results for G B D , which leads to a compactness result for the space G S B D of generalized special functions of bounded deformation. The latter is connected to the existence of solutions to a weak formulation...

A convolution property of the Cantor-Lebesgue measure, II

Daniel M. Oberlin (2003)

Colloquium Mathematicae

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For 1 ≤ p,q ≤ ∞, we prove that the convolution operator generated by the Cantor-Lebesgue measure on the circle is a contraction whenever it is bounded from L p ( ) to L q ( ) . We also give a condition on p which is necessary if this operator maps L p ( ) into L²().

Sets of β -expansions and the Hausdorff measure of slices through fractals

Tom Kempton (2016)

Journal of the European Mathematical Society

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We study natural measures on sets of β -expansions and on slices through self similar sets. In the setting of β -expansions, these allow us to better understand the measure of maximal entropy for the random β -transformation and to reinterpret a result of Lindenstrauss, Peres and Schlag in terms of equidistribution. Each of these applications is relevant to the study of Bernoulli convolutions. In the fractal setting this allows us to understand how to disintegrate Hausdorff measure by slicing,...

On the product formula on noncompact Grassmannians

Piotr Graczyk, Patrice Sawyer (2013)

Colloquium Mathematicae

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We study the absolute continuity of the convolution δ e X * δ e Y of two orbital measures on the symmetric space SO₀(p,q)/SO(p)×SO(q), q > p. We prove sharp conditions on X,Y ∈ for the existence of the density of the convolution measure. This measure intervenes in the product formula for the spherical functions. We show that the sharp criterion developed for SO₀(p,q)/SO(p)×SO(q) also serves for the spaces SU(p,q)/S(U(p)×U(q)) and Sp(p,q)/Sp(p)×Sp(q), q > p. We moreover apply our results to...

L p -improving properties of measures of positive energy dimension

Kathryn E. Hare, Maria Roginskaya (2005)

Colloquium Mathematicae

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A measure is called L p -improving if it acts by convolution as a bounded operator from L p to L q for some q > p. Positive measures which are L p -improving are known to have positive Hausdorff dimension. We extend this result to complex L p -improving measures and show that even their energy dimension is positive. Measures of positive energy dimension are seen to be the Lipschitz measures and are characterized in terms of their improving behaviour on a subset of L p -functions.

The type set for some measures on 2 n with n -dimensional support

E. Ferreyra, T. Godoy, Marta Urciuolo (2002)

Czechoslovak Mathematical Journal

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Let ϕ 1 , , ϕ n be real homogeneous functions in C ( n - { 0 } ) of degree k 2 , let ϕ ( x ) = ( ϕ 1 ( x ) , , ϕ n ( x ) ) and let μ be the Borel measure on 2 n given by μ ( E ) = n χ E ( x , ϕ ( x ) ) | x | γ - n d x where d x denotes the Lebesgue measure on n and γ > 0 . Let T μ be the convolution operator T μ f ( x ) = ( μ * f ) ( x ) and let E μ = { ( 1 / p , 1 / q ) T μ p , q < , 1 p , q } . Assume that, for x 0 , the following two conditions hold: det ( d 2 ϕ ( x ) h ) vanishes only at h = 0 and det ( d ϕ ( x ) ) 0 . In this paper we show that if γ > n ( k + 1 ) / 3 then E μ is the empty set and if γ n ( k + 1 ) / 3 then E μ is the closed segment with endpoints D = 1 - γ n ( k + 1 ) , 1 - 2 γ n ( k + 1 ) and D ' = 2 γ n ( 1 + k ) , γ n ( 1 + k ) . Also, we give some examples.

L p - L q estimates for some convolution operators with singular measures on the Heisenberg group

T. Godoy, P. Rocha (2013)

Colloquium Mathematicae

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We consider the Heisenberg group ℍⁿ = ℂⁿ × ℝ. Let ν be the Borel measure on ℍⁿ defined by ν ( E ) = χ E ( w , φ ( w ) ) η ( w ) d w , where φ ( w ) = j = 1 n a j | w j | ² , w = (w₁,...,wₙ) ∈ ℂⁿ, a j , and η(w) = η₀(|w|²) with η C c ( ) . We characterize the set of pairs (p,q) such that the convolution operator with ν is L p ( ) - L q ( ) bounded. We also obtain L p -improving properties of measures supported on the graph of the function φ ( w ) = | w | 2 m .

Lacunary formal power series and the Stern-Brocot sequence

Jean-Paul Allouche, Michel Mendès France (2013)

Acta Arithmetica

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Let F ( X ) = n 0 ( - 1 ) ε X - λ be a real lacunary formal power series, where εₙ = 0,1 and λ n + 1 / λ > 2 . It is known that the denominators Qₙ(X) of the convergents of its continued fraction expansion are polynomials with coefficients 0, ±1, and that the number of nonzero terms in Qₙ(X) is the nth term of the Stern-Brocot sequence. We show that replacing the index n by any 2-adic integer ω makes sense. We prove that Q ω ( X ) is a polynomial if and only if ω ∈ ℤ. In all the other cases Q ω ( X ) is an infinite formal power series; we discuss...

The type set for homogeneous singular measures on ℝ ³ of polynomial type

E. Ferreyra, T. Godoy (2006)

Colloquium Mathematicae

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Let φ:ℝ ² → ℝ be a homogeneous polynomial function of degree m ≥ 2, let μ be the Borel measure on ℝ ³ defined by μ ( E ) = D χ E ( x , φ ( x ) ) d x with D = x ∈ ℝ ²:|x| ≤ 1 and let T μ be the convolution operator with the measure μ. Let φ = φ e φ e be the decomposition of φ into irreducible factors. We show that if e i m / 2 for each φ i of degree 1, then the type set E μ : = ( 1 / p , 1 / q ) [ 0 , 1 ] × [ 0 , 1 ] : | | T μ | | p , q < can be explicitly described as a closed polygonal region.

Multidimensional Gauss reduction theory for conjugacy classes of SL ( n , )

Oleg Karpenkov (2013)

Journal de Théorie des Nombres de Bordeaux

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In this paper we describe the set of conjugacy classes in the group SL ( n , ) . We expand geometric Gauss Reduction Theory that solves the problem for SL ( 2 , ) to the multidimensional case, where ς -reduced Hessenberg matrices play the role of reduced matrices. Further we find complete invariants of conjugacy classes in GL ( n , ) in terms of multidimensional Klein-Voronoi continued fractions.

On Ordinary and Standard Lebesgue Measures on

Gogi Pantsulaia (2009)

Bulletin of the Polish Academy of Sciences. Mathematics

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New concepts of Lebesgue measure on are proposed and some of their realizations in the ZFC theory are given. Also, it is shown that Baker’s both measures [1], [2], Mankiewicz and Preiss-Tišer generators [6] and the measure of [4] are not α-standard Lebesgue measures on for α = (1,1,...).

Osgood type conditions for an m th-order differential equation

Stanisaw Szufla (1998)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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We present a new theorem on the differential inequality u ( m ) w ( u ) . Next, we apply this result to obtain existence theorems for the equation x ( m ) = f ( t , x ) .

Self-affine measures that are L p -improving

Kathryn E. Hare (2015)

Colloquium Mathematicae

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A measure is called L p -improving if it acts by convolution as a bounded operator from L q to L² for some q < 2. Interesting examples include Riesz product measures, Cantor measures and certain measures on curves. We show that equicontractive, self-similar measures are L p -improving if and only if they satisfy a suitable linear independence property. Certain self-affine measures are also seen to be L p -improving.

Real deformations and invariants of map-germs

J. H. Rieger, M. A. S. Ruas, R. Wik Atique (2008)

Banach Center Publications

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A stable deformation f t of a real map-germ f : , 0 p , 0 is said to be an M-deformation if all isolated stable (local and multi-local) singularities of its complexification f t are real. A related notion is that of a good real perturbation f t of f (studied e.g. by Mond and his coworkers) for which the homology of the image (for n < p) or discriminant (for n ≥ p) of f t coincides with that of f C t . The class of map germs having an M-deformation is, in some sense, much larger than the one having a good...

On L p - L q boundedness for convolutions with kernels having singularities on a sphere

Alexey N. Karapetyants (2001)

Studia Mathematica

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For the convolution operators A a α with symbols a ( | ξ | ) | ξ | - α e x p i | ξ | , 0 ≤ Re α < n, a ( | ξ | ) L , we construct integral representations and give the exact description of the set of pairs (1/p,1/q) for which the operators are bounded from L p to L q .

Estimates of L p norms for sums of positive functions

Ilgiz Kayumov (2013)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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We present new inequalities of L p norms for sums of positive functions. These inequalities are useful for investigation of convergence of simple partial fractions in L p ( ) .

A note on certain partial sum operators

Marek Bożejko, Gero Fendler (2006)

Banach Center Publications

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We show that for the t-deformed semicircle measure, where 1/2 < t ≤ 1, the expansions of L p functions with respect to the associated orthonormal polynomials converge in norm when 3/2 < p < 3 and do not converge when 1 ≤ p < 3/2 or 3 < p. From this we conclude that natural expansions in the non-commutative L p spaces of free group factors and of free commutation relations do not converge for 1 ≤ p < 3/2 or 3 < p.

On the isotropic constant of marginals

Grigoris Paouris (2012)

Studia Mathematica

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We show that if μ₁, ..., μₘ are log-concave subgaussian or supergaussian probability measures in n i , i ≤ m, then for every F in the Grassmannian G N , n , where N = n₁ + ⋯ + nₘ and n< N, the isotropic constant of the marginal of the product of these measures, π F ( μ μ ) , is bounded. This extends known results on bounds of the isotropic constant to a larger class of measures.

Perfect unary forms over real quadratic fields

Dan Yasaki (2013)

Journal de Théorie des Nombres de Bordeaux

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Let F = ( d ) be a real quadratic field with ring of integers 𝒪 . In this paper we analyze the number h d of GL 1 ( 𝒪 ) -orbits of homothety classes of perfect unary forms over F as a function of d . We compute h d exactly for square-free d 200000 . By relating perfect forms to continued fractions, we give bounds on h d and address some questions raised by Watanabe, Yano, and Hayashi.