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Publications
Publications (70)
It is, by now, classical that lattices in higher rank semisimple groups have various rigidity properties. In this work, we add another such rigidity property to the list, namely uniformly stability with respect to the family of unitary operators on finite-dimensional Hilbert spaces equipped with submultiplicative norms. Towards this goal, we first...
In this paper, we show that residually-finite-by-weakly-sofic extensions are weakly sofic. More precisely, we show that if in an exact sequence of groups 1\to N\hookrightarrow K\twoheadrightarrow G\to 1 the group G is residually finite and N is weakly sofic, then K is weakly sofic.
Let $\Gamma$ be a group and $\mathscr{C}$ a class of groups endowed with bi-invariant metrics. We say that $\Gamma$ is $\mathscr{C}$-stable if every $\varepsilon$-homomorphism $\Gamma \rightarrow G$, $(G,d) \in \mathscr{C}$, is $\delta_\varepsilon$-close to a homomorphism, $\delta_\varepsilon\to 0$ when $\varepsilon\to 0$. If $\delta_\varepsilon <...
Ratio-based encoding has recently been proposed for single-level resistive memory cells, in which the resistance ratio of a pair of resistance-switching devices, rather than the resistance of a single device (i.e. resistance-based encoding), is used for encoding single-bit information, which significantly reduces the bit error probability. Generali...
Let C be a class of groups. (For example, C is a class of all finite groups, or C is a class of all finite symmetric groups.) I give a definition of approximations of a group G by groups from C. For example, the groups approximable by symmetric groups are, by definition, sofic groups.
For some classes C the following result holds:
G is approximab...
We show that the defect diminishing is equivalent to the stability with linera speed.
We show that residually finite by residually finite extensions are weakly sofic.
Let σ be a permutation on n letters. We say that a permutation τ is an even (resp. odd) kth root of σ if τ k = σ and τ is an even (resp. odd) permutation. In this article we obtain generating functions for the number of even and odd kth roots of permutations. Our result implies know generating functions of Moser and Wyman and also some generating f...
In this paper we deal with a finite abelian group $G$ and the abstract Fourier transform ${\mathcal F}:{\mathbb C}^G\to {\mathbb C}^\hat{G}$. Then, we consider $\tilde{j}\circ {\mathcal F}:{\mathbb C}^G\to {\mathbb C}^\hat{G}$ where $\tilde j:{\mathbb C}^\hat{G}\to {\mathbb C}^G$ is defined by the composition with a bijection $j:G\to \hat{G}$. ($\t...
We show that the unrestricted wreath product of a sofic group by an amenable group is sofic. We use this result to present an alternative proof of the known fact that any group extension with sofic kernel and amenable quotient is again a sofic group. Our approach exploits the famous Kaloujnine-Krasner theorem and extends, with an additional argumen...
We show that the unrestricted wreath product of a sofic group by an amenable group is sofic. We use this result to present an alternative proof of the known fact that any group extension with sofic kernel and amenable quotient is again a sofic group. Our approach exploits the famous Kaloujnine-Krasner theorem and extends, with an additional argumen...
Several well-known open questions (such as: are all groups sofic/hyperlinear?) have a common form: can all groups be approximated by asymptotic homomorphisms into the symmetric groups $\mathrm{Sym}(n)$ (in the sofic case) or the finite dimensional unitary groups ${\rm U}(n)$ (in the hyperlinear case)? In the case of ${\rm U}(n)$, the question can b...
Several well-known open questions (such as: are all groups sofic/hyperlinear?) have a common form: can all groups be approximated by asymptotic homomorphisms into the symmetric groups $\mathrm{Sym}(n)$ (in the sofic case) or the finite dimensional unitary groups ${\rm U}(n)$ (in the hyperlinear case)? In the case of ${\rm U}(n)$, the question can b...
Let H be a subgroup of F and 〈 〈 H 〉 〉 F {\langle\kern-1.422638pt\langle H\rangle\kern-1.422638pt\rangle_{F}} the normal closure of H in F. We say that H has the Almost Congruence Extension Property (ACEP) in F if there is a finite set of nontrivial elements ϝ ⊂ H {\digamma\subset H} such that for any normal subgroup N of H one has H ∩ 〈 〈 N 〉 〉 F...
In this paper we provide a sufficient condition for a subgroup of a free group not to have the almost congruence extension property (ACEP). We also show that any finitely generated subgroup of a free group satisfies some a generalization of ACEP.
These notes are based on the mini-course "On the Graham Higman group", given at the Erwin Schr\"odinger Institute in Vienna, January 20, 22, 27 and 29, 2016, as a part of the Measured Group Theory program. The main purpose is to describe p-quotients of the Higman group $H(k)$ for $p|(k-1)$. (One may check that the condition $p|(k-1)$ is necessary f...
We give a definition of approximations of groups and new characterizations
of sofic and weakly sofic groups. Keywords: sofic groups,approximations,
equation over groups.
We give a definition of approximations of groups and new characterizations of sofic and weakly sofic groups.
We give the following characterization of sofic (weakly sofic) groups: a
group $G$ is sofic (weakly sofic) if and only if any system of equations
solvable in any alternating group (any finite group) is solvable over $G$.
As an application we prove that hyperlinear groups are weakly sofic.
We consider a dynamical system generated by exponentiation modulo r, that is, by the map u ↦ fq(u), where fq(u) ≡ qu (mod r) and 0 ≤ fg(u) ≤ r − 1. The number of cycles is estimated from above in the case when r = pⁿ with a prime integer p and gcd(q, p) = 1. Also a more general class of functions is considered.
Este es una introduccion a los polinomios en varias variables. Explicamos porque los polinomios en varias varibles son esencialmente diferentes de los polinomios en una solo variable. Damos una introduccion a las bases de Groebner. Pensamos que puede servir a los que quieren saber que es una base de Groebner sin entrar mucho en el algebra abstracta...
We introduce and study the notion of a directional complexity and entropy for
maps of degree 1 on the circle. For piecewise affine Markov maps we use
symbolic dynamics to relate this complexity to the symbolic complexity. We
apply a combinatorial machinery to obtain exact formulas for the directional
entropy, to find the maximal directional entropy...
The aim of this note is to present an easy proof of Hilbert's Nullstellensatz
using Groebner basis. I believe, that the proof has some methodical advantage
in a course on Groebner bases.
Key words: Hilbert's Nullstellensatz, Groebner bases.
The trivial proof of the ergodic theorem for a finite set $Y$ and a
permutation $T:Y\to Y$ shows that for an arbitrary function $f:Y\to{\mathbb R}$
the sequence of ergodic means $A_n(f,T)$ stabilizes for $n \gg |T|$. We show
that if $|Y|$ is very large and $|f(y)| \ll |Y|$ for almost all $y$, then
$A_n(f,T)$ stabilizes for significantly long segmen...
We study the number of $0,1$-words where the fraction of 0 is "almost" fixed
for any initial subword. It turns out that this study use and reveal the
structure of the Galois group (the monodromy group) of the polynomials
$(x+1)^n-\lambda x^p$. ($p$ is not necessary a prime here.)
Although the G.Birkhoff Ergodic Theorem (BET) is trivial for finite spaces,
this does not help in proving it for hyperfinite Loeb spaces. The proof of the
BET for this case, suggested by T. Kamae, works, actually, for arbitrary
probability spaces, as it was shown by Y. Katznelson and B. Weiss.
In this paper we discuss the reason why the usual appro...
Metric complexity functions measure an amount of instability of trajectories in dynamical systems acting on metric spaces. They reflect an ability of trajectories to diverge by the distance of ε during the time interval n. This ability depends on the position of initial points in the phase space, so, there are some distributions of initial points w...
Given a number $r$, we consider the dynamical system generated by repeated exponentiations modulo $r$, that is, by the map $u \mapsto f_g(u)$, where $f_g(u) \equiv g^u \pmod r$ and $0 \le f_g(u) \le r-1$. The number of cycles of the defined above dynamical system is considered for $r=p^n$. Comment: 4 pages
Almost-commuting matrices with respect to the normalized Hilbert-Schmidt norm are considered. Normal almost commuting matrices are proved to be near commuting. Comment: 11 pages
Given a prime $p$, we consider the dynamical system generated by repeated exponentiations modulo $p$, that is, by the map $u \mapsto f_g(u)$, where $f_g(u) \equiv g^u \pmod p$ and $0 \le f_g(u) \le p-1$. This map is in particular used in a number of constructions of cryptographically secure pseudorandom generators. We obtain nontrivial upper bounds...
In this article, several theorems on perturbations ofa complex matrix by a matrix ofa given rank are presented. These theorems may be divided into two groups. The first group is about spectral properties ofa matrix under such perturbations; the second is about almost-near relations with respect to the rank distance.
The Schelling segregation models are ''agent based'' population models, where individual members of the population (agents) interact directly with other agents and move in space and time. In this note we study one-dimensional Schelling pop-ulation models as finite dynamical systems. We define a natural notion of entropy which measures the complexit...
We give a definition of weakly sofic groups (w-sofic groups). Our definition is a rather natural extension of the definition of sofic groups where instead of the Hamming metric on symmetric groups we use general bi-invariant metrics on finite groups. The existence of non-w-sofic groups is equivalent to some conjecture about profinite topology on fr...
The article is devoted to different aspects of the question "What can be done with a matrix by low rank perturbation?" It is proved that one can change a geometrically simple spectrum drastically by a rank 1 permutation, but the situation is quite different if one restricts oneself to normal matrices. Also, the Jordan normal form of a perturbed mat...
We will say that the permutations f1,..., fn is an ǫ-solution of an equation if the normalized Hamming distance between its l.h.p. and r.h.p. is ≤ ǫ. We give a sufficient conditions when near to an ǫ-solution exists an exact solution and some examples when there does not exist such a solution. Key words Permutations, equations, sofic groups. Mathem...
The (e,n)-complexity functions describe total instability of trajectories in dynamical systems. They reflect an ability of trajectories going through a Borel set to diverge on the distance $\epsilon$ during the time interval n. Behavior of the (e, n)-complexity functions as n goes to infinity is reflected in the properties of special measures. Thes...
We give an amenability criterion for groups in terms of their approximation by finite quasigroups. Bibliography: 20 titles.
We introduce and discuss a concept of approximation of a
topological algebraic system A by finite algebraic systems from
a given class 𝔎. If A is discrete, this concept agrees with
the familiar notion of a local embedding of A in a class
𝔎 of algebraic systems. One characterization of this concept
states that A is locally embedded in 𝔎 iff it is a...
A (partial) Latin square is a table of multiplication of a (partial) quasigroup. Multiplication of a (partial) quasigroup may be considered as a set of triples. We give a necessary and sufficient condition for a set of triples to be a quotient of a (partial) Latin square.
We study some measures which are related to the notion of the ε-complexity. We prove that measure of ε-complexity defined on the base of the notion of ε-separability is equivalent to the dual measure that is defined through ε-nets.
We give a simple proof of the fact that the following property: any injective transition function of a cellular automaton on a group G is surjective, holds for any group G approximable by amenable groups. For finitely generated groups, this was proved by G. Gromov [J. Eur. Math. Soc. (JEMS) 1, 109–197 (1999; Zbl 0998.14001)].
We study some measures which are related to the notion of the epsilon-complexity. We prove that measure of epsilon-complexity defined on the base of the notion of epsilon-separability is equivalent to the dual measure that is defined through epsilon-nets.
Recall that a locally compact group $G$ is called
unimodular if the left Haar measure on $G$ is equal to the
right one. It is shown that $G$ is unimodular iff it is
approximable by finite quasigroups (Latin squares).
It is known that any locally compact group that is approximable by
finite groups must be unimodular. However, this condition is not
sufficient. For example, simple Lie groups are not approximable by
finite ones. In this paper we consider the approximation of
locally compact groups by more general finite algebraic systems.
We prove that a locally co...
Instability of orbits in dynamical systems is the reason for their complex behavior. Main characteristics of this complexity are ε-complexity, topological entropy and fractal dimension. In this two lectures we give a short introduction to ideas, results and machinery of this part of modern nonlinear dynamics
It has been long conjectured that the crossing number of Cm × Cn is (m−2)n, for all m, n such that n ≥ m ≥ 3. In this paper, it is shown that if n ≥ m(m + 1) and m ≥ 3, then this conjecture holds. That is, the crossing number of Cm × Cn is as conjectured for all but finitely many n, for each m. The proof is largely based on techniques from the...
We discuss the approximability of locally compact groups by finite semigroups and finite quasigroups (latin squares). We show that if a locally compact group G is approximable by finite semigroups, then it is approximable by finite groups, and thus many important groups are not approximable by finite semigroups. This result implies, in particular,...
We study some measures which are related to the notion of the $\e$-complexity. We prove that measure of $\e$-complexity defined on the base of the notion of $\e$-separability is equivalent to the dual measure that is defined through $\e$-nets. Keywords: Complexity, Separability, Bernoulli measure
Recall that a locally compact group G is called unimodular if the left Haar measure on G is equal to the right one. It is proved in this paper that G is unimodular iff it is approximable by finite quasigroups (Latin squares).
A (partial) Latin square is a table of multiplication of a (partial) quasigroup. Multiplication of a (partial) quasigroup may be considered as a set of triples. We give a necessary and sufficient condition when a set of triples is a quotient of a (partial) Latin square.
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It is known that locally compact groups approximable by finite ones are unimodular, but this condition is not sufficient, for example, the simple Lie groups are not approximable by finite ones as topological groups. In this paper the approximations of locally compact groups by more general finite algebraic systems are investigated. It is proved tha...
We give new examples and criteria in the theory of approximation of groups by finite groups. Bibliography: 17 titles.
It has been long congectured that the crossing number of $C_m\times C_n$ is $(m-2)n$ for $2<m<=n$. In this paper we proved that conjecture is true for all but finitely many $n$ for each $m$. More specifically we proved conjecture for $n>=(m/2)((m+3)^2/2+1)$.The proof is largely based on the theory of arrangements introduced by Adamsson and further...
We study the spectral properties of the stationary solutions u(t, x) = u(x) that arise near the threshold of the time instability of the stationary homogeneous solution u0(t, x) ≡ 0 in families of partial differential equations with two independent variables (time and space). We consider bifurcations of reversible ordinary differential equations for...
We study stability properties of small stationary localized and periodic solutions for a class of PDEs reversible with respect to a spatial variable x in the case where the related stationary equation undergoes the reversible Hopf bifurcation. Under general assumptions we prove instability of localized and stability of periodic solutions. Proofs us...
Theorems are proved establishing a relationship between the spectra of the linear operators of the formA+Ωg
iBigi
−1 andA+B
i, whereg
i∈G, andG is a group acting by linear isometric operators. It is assumed that the closed operatorsA andB
i possess the following property: ‖B
iA−1gBjA−1‖→0 asd(e,g)→∞. Hered is a left-invariant metric onG ande is the...
A number of stationary localized solutions to the well-known pattern-forming gradient system mentioned in the title have been found. Their search is based on the theory of homoclinic orbits to a saddle-focus equilibrium and some results of linear symmetric differential operators with decaying coefficients along with computer simulations.
A three-degrees-of-freedom Hamiltonian systems with a 1:-1 resonance periodic orbit lying in a one-parameter family of periodic orbits is studied. Along such the family, when a family parameter h goes through zero, the type of the orbit is changed from elliptic to hyperbolic of the saddle-focus type. The normal form method gives an asymptotic integ...
We prove the existence of small localized stationary solutions for the generalized Swift-Hohenberg equation and find under some assumption a part of a boundary of their existence in the parameter plane. The related stationary equation creates a reversible Hamiltonian system with two degrees of freedom that undergoes the Hamiltonian-Hopf bifurcation...
It has been long conjectured that the crossing number of Cm ×Cn is (m − 2)n, for all m,n such that n ≥ m ≥ 3. In this paper it is proved that this conjecture holds for all but finitely many n, for each m. More specifically, it is shown that if n ≥ (m/2)((m + 3)2/2 + 1) and m ≥ 3, then the crossing number of Cm ×Cn is exactly (m − 2)n, as conjecture...
Questions
Question (1)
I know that there are experiments with distant entangled photons. The photons may be sent more then 100 km from each other (http://optics.org/article/30121). What is about entangled fermions? Could them be far from each other? It seams to be clear that the time of entanglement is short and speed of fermions is low, but may be one can prepare distant fermions?
(To be honest all experiments with entangled fermions looks more subtle then experiments with photons. Why?)