
Vassily A LyubetskyRussian Academy of Sciences | RAS · Institute for information transmission problems
Vassily A Lyubetsky
Doctor of Sciences, PhD, professor
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179
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Introduction
Skills and Expertise
Publications
Publications (179)
The following two consequences of the axiom of constructibility V = L are established for every n ≥ 3: 1. Every linear $\bf\Sigma^1_n$ set is the projection of a uniform planar $\bf\Pi^1_{n-1}$ set. 2. There is a planar $\bf\Pi^1_{n-1}$ set with countable cross-sections, not covered by a union of countably many uniform $\bf\Sigma^1_n$ sets. If n =...
This article proposes a methodology for establishing a relationship between the change rate of a given gene (relative to a given taxon) together with the amino acid composition of the proteins encoded by this gene and the traits of the species containing this gene. The methodology is illustrated based on the mammalian genes responsible for regulati...
We make use of generalized iterations of a version of the Jensen forcing to define a cardinal-preserving generic model of ZF for any $n\ge 1$ and each of the following four Choice hypotheses: (1) $\text{DC}(\mathbf\Pi^1_n)\land\neg\text{AC}_\omega(\varPi^1_{n+1})\,;$ (2) $\text{AC}_\omega(\text{OD})\land\text{DC}(\varPi^1_{n+1})\land \neg\text{AC}_...
The parameter-free part \(\textbf{PA}_2^*\) of \(\textbf{PA}_2\), second order Peano arithmetic, is considered. We make use of a product/iterated Sacks forcing to define an \(\omega \)-model of \(\textbf{PA}_2^*+ \textbf{CA}(\Sigma ^1_2)\), in which an example of the full Comprehension schema \(\textbf{CA}\) fails. Using Cohen’s forcing, we also de...
The mathematical side of applied problems in multiple subject areas (biology, pattern recognition, etc.) is reduced to the problem of discrete optimization in the following mathematical method. We were provided a network and graphs in its leaves, for which we needed to find a rearrangement of graphs by non-leaf nodes, in which the given functional...
It was established by Jensen in 1970 that there is a generic extension L[a] of the constructible universe L by a non-constructible real a ∉ L, minimal over L, such that a is Δ31 in L[a]. Our first main theorem generalizes Jensen’s result by constructing, for each n≥2, a generic extension L[a] by a non-constructible real a ∉ L, still minimal over L,...
Background
It is generally accepted that most evolutionary transformations at the phenotype level are associated either with rearrangements of genomic regulatory elements, which control the activity of gene networks, or with changes in the amino acid contents of proteins. Recently, evidence has accumulated that significant evolutionary transformati...
A model of set theory ZFC is defined in our recent research, in which, for a given n≥3, (An) there exists a good lightface Δn1 well-ordering of the reals, but (Bn) no well-orderings of the reals (not necessarily good) exist in the previous class Δn−11. Therefore, the conjunction (An)∧(Bn) is consistent, modulo the consistency of ZFC itself. In this...
We consider the problem of the existence of well-orderings of the reals, definable at a certain level of the projective hierarchy. This research is motivated by the modern development of descriptive set theory. Given n≥3, a finite support product of forcing notions similar to Jensen’s minimal-13-real forcing is applied to define a model of set theo...
The paper solves the problem of constructing an evolutionary tree and the evolution of structures along it. This problem has long been posed and extensively researched; it is formulated and discussed below. As a result, we construct an exact cubic-time algorithm which outputs a tree with the minimum cost of embedding into it and of embedding it int...
We make use of generalized iterations of the Sacks forcing to define cardinal-preserving generic extensions of the constructible universe L in which the axioms of ZF hold and in addition either (1) the parameter-free countable axiom of choice ACω* fails, or (2) ACω* holds but the full countable axiom of choice ACω fails in the domain of reals. In a...
We make use of generalized iterations of the Sacks forcing to define cardinal-preserving generic extensions of the constructible universe L in which the axioms of ZF hold and in addition either 1) the parameter-free countable axiom of choice AC* fails, or 2) AC* holds but the full countable axiom of choice AC fails in the domain of reals. In anothe...
We make use of generalized iterations of the Sacks forcing to define cardinal-preserving generic extensions of the constructible universe L in which the axioms of ZF hold and in addition either 1) the parameter-free countable axiom of choice AC* fails, or 2) AC* holds but the full countable axiom of choice AC fails in the domain of reals. In anothe...
Multiple primary lung cancer (MPLC) is an increasingly prevalent subtype of lung cancer. According to recent genomic studies, the different lesions of a single MPLC patient exhibit functional similarities that may reflect evolutionary convergence. We performed whole-exome sequencing for a unique cohort of MPLC patients with multiple samples from ea...
Alfred Tarski [J. Symbolic Logic 13 (1948), pp. 107–111] defined D p m \mathbf {D}_{pm} to be the set of all sets of type p p , type-theoretically definable by parameterfree formulas of type ≤ m {\le m} , and asked whether it is true that D 1 m ∈ D 2 m \mathbf {D}_{1m}\in \mathbf {D}_{2m} for m ≥ 1 m\ge 1 . Tarski noted that the negative solution i...
Parasitic life-strategies in the phylum Nematoda (roundworms) are remarkably diverse and intricate in terms of evolution and taxonomy. By analysing novel rDNA data obtained on rare host-associated groups with unusual biology, we reveal paraphyly of the last major taxon with uncertain higher-rank classification that united solely parasitic nematodes...
The parameter-free part $\text{PA}_2^\ast$ of $\text{PA}_2$, the 2nd order Peano arithmetic, is considered. We make use of a product/iterated Sacks forcing to define an $\omega$-model of $\text{PA}_2^\ast + \text{CA}(\Sigma^1_2)$, in which an example of the full Comprehension schema $\text{CA}$ fails. Using Cohen's forcing, we also define an $\omeg...
The problem of the existence of analytically definable well-orderings at a given level of the projective hierarchy is considered. This problem is important as a part of the general problem of the study of the projective hierarchy in the ongoing development of descriptive set theory. We make use of a finite support product of the Jensen-type forcing...
In this paper, we prove the following: If $n\ge3$, there is a generic extension of $L$ -- the constructible universe -- in which it is true that the Separation principle holds for both effective (lightface) classes $\varSigma^1_n$ and $\varPi^1_n$ for sets of integers. The result was announced long ago by Leo Harrington with a sketch of the proof f...
In this paper, we prove the following: If n≥3, there is a generic extension of L—the constructible universe—in which it is true that the Separation principle holds for both effective (lightface) classes 𝛴n1 and Πn1 of sets of integers. The result was announced long ago by Leo Harrington with a sketch of the proof for n=3; its full proof has never b...
[Preprint published: https://doi.org/10.1093/zoolinnean/zlac070]
Nematodes (roundworms) are ubiquitous animals commonly dominating in ecological communities and networks, with many parasites and pathogen vectors of great economic and medical significance. Nematode parasites are remarkably diverse in life strategies and adaptations at a great range...
The notion of ordinal definability and the related notions of ordinal definable sets (class OD) and hereditarily ordinal definable sets (class HOD) belong to the key concepts of modern set theory. Recent studies have discovered more general types of sets, still based on the notion of ordinal definability, but in a more blurry way. In particular, Tz...
A set is nontypical in the Russell sense, if it belongs to a countable ordinal definable set. The class HNT of all hereditarily nontypical sets satisfies all axioms of ZF and the double inclusion HOD $\subseteq$ HNT $\subseteq$ V holds. Solving a problem recently proposed by Tzouvaras, a generic extension L$[a,x]$ of L, by two reals $a,x$, is prese...
By Tzouvaras, a set is nontypical in the Russell sense, if it belongs to a countable ordinal definable set. The class HNT of all hereditarily nontypical sets satisfies all axioms of ZF and the double inclusion HOD$\subseteq$HNT$\subseteq$V holds. Several questions about the nature of such sets, recently proposed by Tzouvaras, are solved in this pap...
For any weighted directed path-cycle graphs, a and b (referred to as structures), and any equal costs of operations (intermergings and duplication), we obtain an algorithm which, by successively applying these operations to a, outputs b if the first structure contains no paralogs (i.e., edges with a repeated name) and the second has no more than tw...
Examples of effectively indiscernible projective sets of real numbers in various models of set theory are presented. We prove that it is true, in Miller and Laver generic extensions of the constructible universe, that there exists a lightface Π21 equivalence relation on the set of all nonconstructible reals, having exactly two equivalence classes,...
If a real a is random over a model M and x∈M[a] is another real then either (1) x∈M, or (2) M[x]=M[a], or (3) M[x] is a random extension of M and M[a] is a random extension of M[x]. This result may belong to the old set theoretic folklore. It appeared as Exapmle 1.17 in Jech’s book “Multiple forcing” without the claim that M[x] is a random extensio...
In this paper we prove that for any m≥1 there exists a generic extension of L, the constructible universe, in which it is true that the set of all constructible reals (here subsets of ω) is equal to the set D1m of all reals definable by a parameter free type-theoretic formula with types bounded by m, and hence the Tarski ‘definability of definable’...
We propose a novel linear time algorithm which, given any directed weighted graphs a and b with vertex degrees 1 or 2, constructs a sequence of operations transforming a into b. The total cost of operations in this sequence is minimal among all possible ones or differs from the minimum by an additive constant that depends only on operation costs bu...
An original bioinformatics technique is developed to identify the protein-coding genes in rodents, lagomorphs and nonhuman primates that are pseudogenized in humans. The method is based on per-gene verification of local synteny, similarity of exon-intronic structures and orthology in a set of genomes. It is applicable to any genome set, even with t...
In this paper, we prove the following. If n≥3, then there is a generic extension of L, the constructible universe, in which it is true that the set P(ω)∩L of all constructible reals (here—subsets of ω) is equal to the set P(ω)∩Δn1 of all (lightface) Δn1 reals. The result was announced long ago by Leo Harrington, but its proof has never been publish...
Models of set theory are defined, in which nonconstructible reals first appear on a given level of the projective hierarchy. Our main results are as follows. Suppose that n ≥ 2 . Then: 1. If it holds in the constructible universe L that a ⊆ ω and a ∉ Σ n 1 ∪ Π n 1 , then there is a generic extension of L in which a ∈ Δ n + 1 1 but still a ∉ Σ n 1 ∪...
A "genome structure" is a labeled directed graph with vertices of degree 1 or 2. A set of operations over such graphs is fixed, and each of the operations has a certain cost, a strictly positive number. The transformation problem consists in the following: for given structures a and b and given costs, find a minimum total cost sequence of operation...
The lengths of intergenic regions between neighboring genes that are convergent, divergent, or unidirectional were calculated for plastids of the rhodophytic branch and complete archaeal and bacterial genomes. Statistically significant linear relationships between any pair of the medians of these three length types have been revealed in each genomi...
Using an invariant modification of Jensen's "minimal $\varPi^1_2$ singleton" forcing, we define a model of ZFC, in which, for a given $n\ge2$, there exists a lightface $\varPi^1_n$ unordered pair of non-OD (hence, OD-indiscernible) countable sets of reals, but there is no $\varSigma^1_n$ unordered pairs of this kind.
Age-related dysfunctions are accompanied by impairments in the mitochondrial morphology, activity of signaling pathway, and protein interactions. Cardiolipin is one of the most important phospholipids that maintains the curvature of the cristae and facilitates assembly and interaction of complexes and supercomplexes of the mitochondrial respiratory...
Background:
Gerontogenes include those that modulate life expectancy in various species and may be the actual longevity genes. We believe that a long (relative to body weight) lifespan in individual rodent and primate species can be due, among other things, to the loss of particular genes that are present in short-lived species of the same orders....
We prove that the Solovay set Σ is absolutely definable in a sufficiently wide sense; in particular, Σ does not depend on the choice of the ground model.
Cardiolipin interacts with many proteins of the mitochondrial inner membrane and, together with cytochrome C and creatine kinase, activates them. It can be considered as an integrating factor for components of the mitochondrial respiratory chain, which provides for an efficient transfer of electrons and protons. The major, if not the only, factor o...
The molecular basis of higher regenerative capacity
of cold-blooded animals comparing to warmblooded ones is poorly understood. Although this difference in regenerative capacities is commonly
thought to be a result of restructuring of the same regulatory gene network, we hypothesized that it may be
due to loss of some genes essential for regenerati...
Two enigmatic groups of morphologically simple parasites of invertebrates, the Dicyemida (syn. Rhombozoa) and the Orthonectida, since the 19th century have been usually considered as two classes of the phylum Mesozoa. Early molecular evidence suggested their relationship within the Spiralia (=Lophotrochozoa), however, high rates of dicyemid and ort...
We make use of a finite support product of the Jensen minimal forcing to define a model of set theory in which the separation theorem fails for projective classes $\mathbf\Sigma^1_n$ and $\mathbf\Pi^1_n$, for a given $n\ge3$.
We prove that it is true in Sacks, Cohen, and Solovay generic extensions that any ordinal definable Borel set of reals necessarily contains an ordinal definable element. This result has previously been known only for countable sets.
The genetic basis of higher regenerative capacity of fishes, amphibians and reptiles compared to birds and mammals is still poorly understood. Though it is thought to be a result of restructuring in the regulatory network of a static set of genes, we argued that it could be due to the loss of genes essential for regeneration. In the present work, w...
We propose an algorithm, linear in both running time and memory, that constructs a sequence of operations that transform any given directed graph with degree of any vertex at most two to any other given graph of the same type with minimal total cost. This sequence is called the shortest one. We allow four standard operations of re-gluing graphs wit...
Social insects with identical genotype that form castes with radically different lifespans are a promising model system for studying the mechanisms underlying longevity. The main direction of progressive evolution of social insects, in particular, ants, is the development of the social way of life inextricably linked with the increase in the colony...
The paper is the second of our series of notes aimed to bring back in circulation some bright ideas of early modern set theory, mainly due to Harrington and Sami, which have never been adequately presented in set theoretic publications. We prove that if a real $a$ is random over a model $M$ and $x\in M[a]$ is another real then either (1) $x\in M$,...
It is true in the Cohen, Solovay-random, and Sacks generic extensions that every ordinal-definable Borel set of reals has a Borel code in the ground model, and hence if non-empty, then has an element in the ground model.
We propose a linear time and linear space algorithm that constructs a minimal (in the total cost) sequence of operations required to transform a directed graph consisting of disjoint cycles into any graph of the same type. The following operations are allowed: double-cut-and-join of vertices and insertion or deletion of a connected fragment of edge...
In a handwtitten note of 1975, Leo Harrington sketched a construction of a model of ZFC (no large cardinals or anything beyond ZFC!) in which $\mathbf\Pi^1_3$-Separation holds but $\mathbf\Sigma^1_3$-Reduction fails. The result has never appeared in a journal or book publication except for a few of old references.
It is true in the Cohen, Solovay-random, dominaning, and Sacks generic extension, that every countable ordinal-definable set of reals belongs to the ground universe. It is true in the Solovay collapse model that every non-empty OD countable set of sets of reals consists of OD\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \u...
A canonization scheme for Borel reduction between smooth equivalence relations on $\mathbb R^\omega$ modulo restriction to infinite perfect products is proposed. It shows that given a pair of Borel smooth equivalence relations $\mathsf E,\mathsf F$ on $\mathbb R^\omega$, there is an infinite perfect product $P\subseteq\mathbb R^\omega$ such that ei...
Background:
Chromosome structure is a very limited model of the genome including the information about its chromosomes such as their linear or circular organization, the order of genes on them, and the DNA strand encoding a gene. Gene lengths, nucleotide composition, and intergenic regions are ignored. Although highly incomplete, such structure ca...
A generic extension of $L$, the constructible universe, is defined, in which it is true for a given $n\ge2$ that there exists a non-ROD-uniformizable lightface $\varPi^1_n$ set in $\mathbb R\times\mathbb R$ with all vertical cross-sections being Vitali classes (hence, countable sets of reals), and in the same time every boldface $\bf\Sigma^1_n$ set...
We prove that the Solovay set Σ is generic over the ground model in the sense of a forcing whose order relation extends the order relation of the given forcing.
A complete proof that algorithms proposed by the authors solve the problem of minimum-cost transformation of a graph into another graph is given. The problem is solved both by a direct algorithm of linear complexity and by a reduction to quadratic integer linear programming.
Using a non-Laver modification of Uri Abraham's minimal $\varDelta^1_3$ collapse function, we define a generic extension $L[a]$ by a real $a$, in which, for a given $n\ge3$, $\{a\}$ is a lightface $\varPi^1_n$ singleton, $a$ effectively codes a cofinal map $\omega\to\omega_1^L$ minimal over $L$, while every $\varSigma^1_n$ set $X\subseteq\omega$ is...
We modify the definable ultrapower construction of Kanovei and Shelah (2004) to develop a ZF-definable extension of the continuum with transfer provable using countable choice only, with an additional mild hypothesis on well-ordering implying properness. Under the same assumptions, we also prove the existence of a definable, proper elementary exten...
A novel time- and memory-efficient algorithm for solving the problem of finding the most economical (i.e., having the lowest overall cost) transformation of an arbitrary oriented graph representing a disjoint union of chains and cycles into another graph of the same type is proposed. The correctness of this algorithm (i.e., the fact that it always...
Using a modification of the invariant Jensen forcing, we define a model of ZFC, in which, for a given $n\ge3$, there exists a lightface $\varPi^1_n$ set of reals, which is a ${\mathsf E}_0$ equivalence class, hence a countable set, and which does not contain any OD element, while every non-empty countable $\varSigma^1_n$ set of reals is necessarily...
Using a modification of the invariant Jensen forcing, we define a model of ZFC, in which, for a given $n\ge3$, there exists a lightface $\varPi^1_n$ set of reals, which is a ${\mathsf E}_0$ equivalence class, hence a countable set, and which does not contain any OD element, while every non-empty countable $\varSigma^1_n$ set of reals is necessarily...
We propose a linear time and linear space algorithm which constructs a minimal sequence of operations rearranging one structure (directed graph of cycles and paths) into another. Structures in such a sequence may have a varying number of edges; a list of operations is fixed and includes deletion and insertion of a fragment of a structure. We give a...
The authors study orgraphs with any number of chains and cycles. Edges of orgraphs have unique names - natural numbers. There is a fixed list of operations that transform one graph into another. A cost is assigned to each operation. The task is to find the path of transformations with minimal total cost. This problem has a wide range of practical a...
Recent phylogenetic analyses are incorporating ultraconserved elements (UCEs) and highly conserved elements (HCEs). Models of evolution of the genome structure and HCEs initially faced considerable algorithmic challenges, which gave rise to (often unnatural) constraints on these models, even for conceptually simple tasks such as the calculation of...
We make use of a finite support product of $\omega_1$ clones of the Jensen minimal $\varPi^1_2$ singleton forcing to obtain a model of ZFC in which every non-empty lightface analytically definable set of reals contains a lightface analytically definable real (the full basis theorem), but there is no analytically definable wellordering of the contin...
We make use of a finite support product of $\omega_1$ clones of the Jensen minimal $\varPi^1_2$ singleton forcing to obtain a model of ZFC in which every non-empty lightface analytically definable set of reals contains a lightface analytically definable real (the full basis theorem), but there is no analytically definable wellordering of the contin...
Short leader genes usually do not encode stable proteins, although their importance in expression control of bacterial genomes is widely accepted. Such genes are often involved in the control of attenuation regulation. However, the abundance of leader genes suggests that their role in bacteria is not limited to regulation. Specifically, we hypothes...
We prove that Solovay's set $\Sigma$ is generic over the ground model via a forcing notion whose order relation $\subseteq$-extends the given order relation.
Background:
Perfectly or highly conserved DNA elements were found in vertebrates, invertebrates, and plants by various methods. However, little is known about such elements in protists. The evolutionary distance between apicomplexans can be very high, in particular, due to the positive selection pressure on them. This complicates the identificatio...
A novel efficient algorithm for solution of the problem of equal partitioning of a set with predefined weights of elements is proposed. The algorithm is based on calculation of a linear group preserving an invariant: the set of zeros of a cubic form. Algorithms for solution of related problems, including the problem of the search for the second sol...
A novel algorithm and original software were used to cluster all proteins encoded in plastids of 72 species of the rhodophytic branch. The results are publicly available at http://lab6.iitp.ru/ppc/redline72/ in a database that allows fast identification of clusters (protein families) both by a fragment of an amino acid sequence and by a phylogeneti...
Background
One of the main aims of phylogenomics is the reconstruction of objects defined in the leaves along the whole phylogenetic tree to minimize the specified functional, which may also include the phylogenetic tree generation. Such objects can include nucleotide and amino acid sequences, chromosomal structures, etc. The structures can have an...
We make use of a finite support product of the Jensen minimal singleton forcing to define a model in which uniformization fails for a set with countable cross-sections. We also define appropriate submodels of the same model in which separation fails for .
The well-known Σ-construction in forcing by Solovay is generalized to the case of intermediate sets that are not subsets of the initial model. Our method gives a more transparent construction of a forcing over an intermediate model than that in the classical paper [1] by Grigorieff on intermediate models.
A generic extension $L[x,y]$ of $L$ by reals $x,y$ is defined, in which the
union of $\mathsf E_0$-classes of $x$ and $y$ is a $\Pi^1_2$ set, but neither
of these two $\mathsf E_0$-classes is separately ordinal-definable.
Two dichotomy theorems on the effective σ-boundedness and effective σ-compactness of ordinal definable point sets in Solovay’s model are proved.
A generic extension L[x] by a real x is defined, in which the E0-class of x is a lightface Π12 (hence, ordinal-definable) set containing no ordinal-definable reals.
We report the database of plastid protein families from red algae, secondary and tertiary rhodophyte-derived plastids, and Apicomplexa constructed with the novel method to infer orthology. The families contain proteins with maximal sequence similarity and minimal paralogous content. The database contains 6509 protein entries, 513 families and 278 n...
The chromosome structure is defined as a set of chromosomes that consist of genes assigned to one of the DNA strands and represented in a circular or linear arrangement. A widely investigated problem is to define the shortest algorithmic path of chromosome rearrangements that transforms one chromosome structure into another. When equal rearrangemen...
The aim of this paper is to demonstrate that several non-rigorous methods of mathematical reasoning in the field of divergent series, mostly related to the Euler and Hutton transforms, may be developed in a correct and consistent way by methods of the grossone analysis.
We prove that in some cases definable thin sets (including chains) of Borel
partial orderings are necessarily countably cofinal. This includes the
following cases: analytic thin sets, ROD thin sets in the Solovay model, and
$\Sigma^1_2$ thin sets in the assumption that $\omega_1^{L[x]}<\omega_1$ for
all reals $x$. We also prove that definable thin...
We make use of a finite support product of the Jensen minimal $\varPi^1_2$
singleton forcing to define a model in which $\varPi^1_2$ Uniformization fails
for a set with countable cross-sections. We also define appropriate submodels
of the same model in which Separation fails for $\varPi^1_3$.
A generic extension $L[x]$ of $L$ by a real $x$ is defined, in which the
$\mathsf E_0$-class of $x$ is a lightface $\Pi^1_2$ set containing no
ordinal-definable reals.
We make use of a finite support product of Jensen forcing to define a model
in which there is a countable non-empty lightface $\Pi^1_2$ set of reals
containing no ordinal-definable real.
We modify arguments in Vladimir Kanovei, Linearization of definable order
relations, APAL, 102(1-2):69--100, 2000, to reprove a linearization theorem on
real-ordinal definable partial quasi-orderings in the Solovay model.