S. Albeverio

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Articles: 22

On the Hausdorff dimension faithfulness and the Cantor series expansion

S. Albeverio, Ganna Ivanenko, Mykola Lebid, Grygoriy Torbin

↓ Abstract   |   Article (.pdf)

MFAT 26 (2020), no. 4, 298-310


We study families $\Phi$ of coverings which are faithful for the Hausdorff dimension calculation on a given set $E$ (i. e., special relatively narrow families of coverings leading to the classical Hausdorff dimension of an arbitrary subset of $E$) and which are natural generalizations of comparable net-coverings. They are shown to be very useful for the determination or estimation of the Hausdorff dimension of sets and probability measures.

We give general necessary and sufficient conditions for a covering family to be faithful and new techniques for proving faithfulness/non-faithfulness for the family of cylinders generated by expansions of real numbers. Motivated by applications in the multifractal analysis of infinite Bernoulli convolutions, we study in details the Cantor series expansion and prove necessary and sufficient conditions for the corresponding net-coverings to be faithful. To the best of our knowledge this is the first known sharp condition of the faithfulness for a class of covering families containing both faithful and non-faithful ones.

Applying our results, we characterize fine fractal properties of probability measures with independent digits of the Cantor series expansion and show that a class of faithful net-coverings essentially wider that the class of comparable ones. We construct, in particular, rather simple examples of faithful families $\mathcal{A}$ of net-coverings which are "extremely non-comparable" to the Hausdorff measure.

Ми досліджуємо сім’ї $\Phi$ покриттів, які є довірчими для обчислення розмірності Хаусдорфа-Безиковича на певній множині $E$ (тобто, спеціальні відносно вузькі сім’ї покриттів, яких достатньо для коректного обчислення класичної розмірності Хаусдорфа-Безиковича довільної підмножини множини $E$) і які є природним узагальненням порівнянних мережевих покриттів. В роботі показано, що такі сім’ї є дуже корисними для обчислення чи оцінки розмірності Хаусдорфа-Бези\-ковича множин та ймовірнісних мір.

Нами отримано загальні необхідні та достатні умови довірчості для сімей покриттів та запропоновано нову техніку доведення довірчості/недовірчості для сімей циліндрів, породжених різними розкладами дійсних чисел. Маючи додатко\-ву мотивацію в мультифрактальному аналізі нескінченних згорток Бернуллі, ми детально дослідили розклади Кантора та довели необхідні та достатні умови довірчості відповідних сімей покриттів мережевими циліндрами. Наскільки нам відомо, ці результати є першими критеріями довірчості для класу сімей покриттів, що містить як довірчі, так і недовірчі сім’ї.

Застосовуючи отримані результати, ми дослідили тонкі фрактальні властивості ймовірнісних мір з незалежними символами розкладів Кантора і показали, що клас довірчих мережевих покриттів суттєво ширше за клас порівнянних. Ми побудували, зокрема, досить прості приклади довірчих сімей $\mathcal{A}$ мережевих покриттів, які є "екстремально непорівнянними" відносно міри Хаусдорфа.

Schrödinger operators with nonlocal potentials

Sergio Albeverio, Leonid Nizhnik

↓ Abstract   |   Article (.pdf)

MFAT 19 (2013), no. 3, 199-210


We describe selfadjoint nonlocal boundary-value conditions for new exact solvable models of Schrödinger operators with nonlocal potentials. We also solve the direct and the inverse spectral problems on a bounded line segment, as well as the scattering problem on the whole axis for first order operators with a nonlocal potential.

On fine structure of singularly continuous probability measures and random variables with independent $\widetilde{Q}$-symbols

S. Albeverio, V. Koshmanenko, M. Pratsiovytyi, G. Torbin

↓ Abstract   |   Article (.pdf)

MFAT 17 (2011), no. 2, 97-111


We introduce a new fine classification of singularly continuous probability measures on $R^1$ on the basis of spectral properties of such measures (topological and metric properties of the spectrum of the measure as well as local behavior of the measure on subsets of the spectrum). The theorem on the structural representation of any one-dimensional singularly continuous probability measure in the form of a convex combination of three singularly continuous probability measures of pure spectral type is proved.

We introduce into consideration and study a $\widetilde{Q}$-representation of real numbers and a family of probability measures with independent $\widetilde{Q}$-symbols. Topological, metric and fractal properties of the above mentioned probability distributions are studied in details. We also show how the methods of $\widetilde{P}-\widetilde{Q}$-measures can be effectively applied to study properties of generalized infinite Bernoulli convolutions.

Boundary problems for the wave equation with the Lévy Laplacian in Shilov's class

S. Albeverio, Ya. I. Belopolskaya, M. N. Feller

↓ Abstract   |   Article (.pdf)

MFAT 16 (2010), no. 3, 197-202


We present solutions to some boundary value and initial-boundary value problems for the "wave" equation with the infinite dimensional L\'evy Laplacian $\Delta _L$ $$\frac{\partial^2 U(t,x)}{\partial t^2}=\Delta_LU(t,x)$$ in the Shilov class of functions.

Algebras of unbounded operators over the ring of measurable functions and their derivations and automorphisms

S. Albeverio, Sh. A. Ayupov, A. A. Zaitov, J. E. Ruziev

↓ Abstract   |   Article (.pdf)

MFAT 15 (2009), no. 2, 177-187


In the present paper derivations and $*$-automorphisms of algebras of unbounded operators over the ring of measurable functions are investigated and it is shown that all $L^0$-linear derivations and $L^{0}$-linear $*$-automorphisms are inner. Moreover, it is proved that each $L^0$-linear automorphism of the algebra of all linear operators on a $bo$-dense submodule of a Kaplansky-Hilbert module over the ring of measurable functions is spatial.

Boundary problems for fully nonlinear parabolic equations with Lévy Laplacian

S. Albeverio, Ya. Belopolskaya, M. Feller

↓ Abstract   |   Article (.pdf)

MFAT 14 (2008), no. 1, 1-9


We suggest a method to solve boundary and initial-boundary value problems for a class of nonlinear parabolic equations with the infinite dimensional L'evy Laplacian $\Delta _L$ $$f\Bigl(U(t,x),\frac{\partial U(t,x)}{\partial t},\Delta_LU(t,x)\Bigl)=0$$ in fundamental domains of a Hilbert space.

Inverse spectral problems for coupled oscillating systems: reconstruction from three spectra

S. Albeverio, R. Hryniv, Ya. Mykytyuk

↓ Abstract   |   Article (.pdf)

MFAT 13 (2007), no. 2, 110-123


We study an inverse spectral problem for a compound oscillating system consisting of a singular string and $N$~masses joined by springs. The operator $A$ corresponding to this system acts in $L_2(0,1)\times C^N$ and is composed of a Sturm--Liouville operator in $L_2(0,1)$ with a distributional potential and a Jacobi matrix in~$C^N$ that are coupled in a special way. We solve the problem of reconstructing the system from three spectra---namely, from the spectrum of $A$ and the spectra of its decoupled parts. A complete description of possible spectra is given.

The Efimov effect for a model operator associated with the Hamiltonian of a non conserved number of particles

Sergio Albeverio, Saidakhmat N. Lakaev, Tulkin H. Rasulov

↓ Abstract   |   Article (.pdf)

MFAT 13 (2007), no. 1, 1-16


A model operator associated with the energy operator of a system of three non conserved number of particles is considered. The essential spectrum of the operator is described by the spectrum of a family of the generalized Friedrichs model. It is shown that there are infinitely many eigenvalues lying below the bottom of the essential spectrum, if a generalized Friedrichs model has a zero energy resonance.

Lévy-Dirichlet forms. II

S. Albeverio, Ya. Belopolskaya, M. Feller

↓ Abstract   |   Article (.pdf)

MFAT 12 (2006), no. 4, 302-314


A Dirichlet form associated with the infinite dimensional symmetrized Levy-Laplace operator is constructed. It is shown that there exists a natural connection between this form and a Markov process. This correspondence is similar to that studied in a previous paper by the same authors for the non-symmetric Levy Laplacian.

$*$-wildness of some classes of $C^*$-algebras

Sergio Albeverio, Kate Jushenko, Daniil Proskurin, Yurii Samoilenko

↓ Abstract   |   Article (.pdf)

MFAT 12 (2006), no. 4, 315-325


We consider the complexity of the representation theory of free products of $C^*$-algebras. Necessary and sufficient conditions for the free product of finite-dimensional $C^*$-algebras to be $*$-wild is presented. As a corollary we get criteria for $*$-wildness of free products of finite groups. It is proved that the free product of a non-commutative nuclear $C^*$-algebra and the algebra of continuous functions on the one-dimensional sphere is $*$-wild. This result is applied to estimate the complexity of the representation theory of certain $C^*$-algebras generated by isometries and partial isometries.

Dynamics of discrete conflict interactions between non-annihilating opponents

Sergio Albeverio, Maksym Bodnarchuk, Volodymyr Koshmanenko

MFAT 11 (2005), no. 4, 309-319


On real $AW^*$-algebras

A. H. Abduvaitov, S. Albeverio, Sh. A. Ayupov

MFAT 11 (2005), no. 2, 99-112


Dense subspaces in scales of Hilbert spaces

S. Albeverio, R. Bozhok, M. Dudkin, V. Koshmanenko

MFAT 11 (2005), no. 2, 156-169


Riquier problem for nonlinear elliptic equations with Lévy Laplacians

S. Albeverio, Ya. Belopolskaya, M. Feller

MFAT 11 (2005), no. 1, 1-9


Schrödinger operators with a number of negative eigenvalues equal to the number of point interactions

Sergio Albeverio, Leonid Nizhnik

MFAT 9 (2003), no. 4, 273-286


Fine structure of the singular continuous spectrum

Sergio Albeverio, Volodymyr Koshmanenko, Grygoriy Torbin

MFAT 9 (2003), no. 2, 101-119


Quasi-invariance and Gibbs structure of diffusion measures on infinite product groups

Sergio Albeverio, Alexei Daletskii

MFAT 6 (2000), no. 1, 28-42


Generalized eigenfunctions under singular perturbation

S. Albeverio, V. Koshmanenko, K. A. Makarov

MFAT 5 (1999), no. 1, 13-28


Some examples of Dirichlet operators associated with the actions of infinite dimensional Lie groups

S. Albeverio, A. Daletskii, Yu. Kondratiev

MFAT 4 (1998), no. 2, 1-15


Lippmann-Schwinger equation in the singular perturbation theory

S. Albeverio, J. F. Brasche, V. Koshmanenko

MFAT 3 (1997), no. 1, 1-27


Quantum hierarchical model

S. Albeverio, Yu. G. Kondratiev, Yu. V. Kozitsky

MFAT 2 (1996), no. 3, 1-35


Dirichlet operators semigroups in some Gibbs lattice spin systems

S. Albeverio, A. Val. Antoniouk, A. Vict. Antoniouk, Yu. G. Kondratiev

MFAT 1 (1995), no. 1, 3-27


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