Evgeny V. Shchepin

Russian Academy of Sciences | RAS · Steklov Mathematical Institute

We discuss the question of extending homeomorphism between closed subsets of the Cantor discontinuum $D^\tau$. For every set $P\subset D^\tau$ let $\mathfrak{L}_P$ be the set of cardinality $\lambda$ such that the $\lambda$-interior of $P$ is not empty. It is established that any homeomorphism $f$ between two proper closed subsets $P$ and $K$ of $D...

We discuss the question of extending homeomorphisms between closed subsets of the Cantor cube $D^{\tau }$ . It is established that any homeomorphism between two closed negligible subsets of $D^{\tau }$ can be extended to an autohomeomorphism of $D^{\tau }$ .

We introduce an algorithm for a search of extremal fractal curves in large curve classes. It heavily uses SAT-solvers—heuristic algorithms that find models for CNF boolean formulas. Our algorithm was implemented and applied to the search of fractal surjective curves \(\gamma :[0,1]\rightarrow [0,1]^d\) with minimal dilation $$\begin{aligned} \sup _...

It is shown that any homeomorphism between two compact subsets of $\mathbb N^\tau$ can be extended to an autohomeomorphism of $\mathbb N^\tau$.

We discuss the question of extending homeomorphisms between closed subsets of the Cantor cube D τ. It is established that any homeomorphism between two closed negligible subsets of D τ can be extended to an autohomeomorphism of D τ .

One studies spaces in which disjoint closures of open sets have disjoint neighborhoods.

We discuss the question of extending homeomorphism between closed subsets of the Cantor discontinuum D τ. It is established that any homeomorphism f between two closed subsets of D τ can be extended to an autohomeomorphism of D τ provided f preserves the λ-interiors of the sets for every cardinal λ.

We discuss the question of extending homeomorphism between closed subsets of the Cantor cube D τ. It is established that any homeomorphism between two closed negligible subset of D τ can be extended to an autohomeomorphism of D τ .

Доказано, что для любого отображения единичного отрезка на единичный квадрат найдется пара точек отрезка, для которой квадрат евклидова расстояния между их образами превосходит расстояние между ними на отрезке не менее чем в $3.625$ раз. Дополнительное условие на принадлежность образов начала и конца отрезка противоположным сторонам квадрата повыша...

We will call an operator array a family of bounded linear operators {U s : H 1 → H 2 } s∈S between complex or real Banach spaces (S be an arbitrary set). Given an operator array {U s } s∈S and a positive number ε, we refer to the array {U s | U s > ε} as the ε-cut of the array. If all ε-cuts of operator array are finite and its sums strongly (i.e....

The paper describes a structurizer based on superfast algorithms of hierarchical cluster analysis for optical recognition of compound-organized texts.

We introduce an algorithm for a search of extremal fractal curves in large curve classes. It heavily uses SAT-solvers -- heuristic algorithms that find models for CNF boolean formulas. Our algorithm was implemented and applied to the search of fractal surjective curves $\gamma\colon[0,1]\to[0,1]^d$ with minimal dilation $$\sup_{t_1<t_2}\frac{\|\gam...

It is proved that for any mapping of a unit segment to a unit square, there is a pair of points of the segment for which the square of the Euclidean distance between their images exceeds the distance between them on the segment by at least $3\frac58$ times. And the additional condition that the images of the beginning and end of the segment belong...

It is proved that the algebraic sum of curves lying in the Hilbert space in the General position has a topological dimension of at least the number of curves

The Serpinsky-Knopp curve is characterized as the only curve (up to isometry) that maps a unit segment onto a triangle of a unit area, so for any pair of points in the segment, the square of the distance between their images does not exceed four times the distance between them. keywords plane Peano curves, square-to-linear ratio, locality Sierpinsk...

The paper presents a plane regular fractal Peano curve with a Euclidean square-to-line ratio (L 2-locality) of , which is minimal among all known curves of this class. The presented curve has a fractal genus of 25. Performed calculations allow us to state that all the other regular curves with a fractal genus not exceeding 36 have a strictly greate...

Two models of integral theory based on the concept of a differential as a certain infinitesimal quantity are considered. One theory treats an infinitesimal quantity as a zero-tending sequence. The second is as an infinitesimal Hyper-real.

The designation of the integral of a function by a measure usually contains a differential symbol, which is not given an independent meaning. In this article, we will show how to naturally define the concept of a measure differential in both standard and non-standard analysis. In the standard analysis, the differential-based scheme for determining...

Professor Mikhail Ivanovich Shtogrin (born September 25, 1938) is widely known due to his contributions to discrete geometry (including regular tilings and Dirichlet-Voronoi partitions) and geometrical crystallography (including cubical complexes). The paper contains a short description of his life, scientific activities, and a photo.

Professor Mikhail Ivanovich Shtogrin (born September 25, 1938) is widely known due to his contributions to discrete geometry (including regular tilings and Dirichlet-Voronoi partitions) and geometrical crystallography (including cubical complexes). The paper contains a short description of his life, scientific activities, and a photo.

A direction d is called a tangent direction to the unit sphere S of a normed linear space s S and lin(s + d) is a tangent line to the sphere S at s imply that lin(s + d) is a one-sided tangent to the sphere S, i. e., it is the limit of secant lines at s. A set M is called convex with respect to a direction d if [x, y] M whenever x, y in M, (y - x)...

A direction d is called a tangent direction to the unit sphere S if the conditions s ∈ S and aff(s + d) is a tangent line to the sphere S at s imply that aff(s + d) is a one-sided tangent to the sphere S, i.e., it is the limit of secant lines at the point s. A set M is called convex with respect to a direction d if [x, y] ⊂ M whenever x, y ∈ M , (y...

В статье разработана теория и основанные на ней алгоритмы для точного вычисления кубо-линейного отношения (локальности) трехмерных кривых Пеано в чебышевской метрике.

We implement Leibniz’s idea about the differential as the length of an infinitesimally small elementary interval (a monad) in a form satisfying modern standards of rigor. The concept of sequential differential introduced in this paper is shown to be in good alignment with the standard convention of the integral calculus. As an application of this c...

A theory and corresponding algorithms are developed for fast and exact calculation of the L ∞-locality (i.e., the greatest cube-to-linear ratio in the maximum metric) for polyfractal three-dimensional Peano curves.

В данной статье излагается подход к суммированию неупорядоченных числовых и матричных массивов на основе их упорядочения по абсолютной величине (жадное суммирование). Доказаны теоремы о произведении жадных сумм. Вскрыта связь теории жадного суммирования с теорией обобщенных рядов Дирихле. Рассмотрено понятие асимптотического ряда Дирихле.

An approach to the summation of unordered number and matrix arrays based on ordering them by absolute value (greedy summation) is proposed. Theorems on products of greedy sums are proved. A relationship between the theory of greedy summation and the theory of generalized Dirichlet series is revealed. The notion of asymptotic Dirichlet series is con...

We study the periodic homeomorphisms of the Sphere

The ranking approach using the Bayesian classifier for a variable number of features with the factual theory, which allows us to add more information to the classifier - characteristics of selfsimilarity. For this the Naive Bayes classifier is modified and defins Hurst data that is associated with traditional fractal dimension.

This article provides an approach to summation of unordered numeric and matrix arrays based on their ordering by absolute value (greedy summation). A theorem on the product of greedy sums is proved. Reveals the connection of the theory of greedy summation with the theory of generalized Dirichlet series. Examines the concept of asymptotic series Dir...

Chain distance between points in a metric space is defined as the infimum of epsilon such that there is an epsilon-chain connecting these points. We call a mapping of a metric compact into the real line a chain development if it preserves chain distances. We give a criterium of existence of the chain development for metric compacts. We prove the di...

Chain distance between points in a metric space is defined as the infimum of epsilon such that there is an epsilon-chain connecting these points. We call a mapping of a metric compact into the real line a chain development if it preserves chain distances. We give a criterium of existence of the chain development for metric compacts. We prove the di...

The class of q-adic Peano curves is defined. This class is large enough to include all Peano curves considered by Haverkort, for which the maximum cube-to-linear ratio not only is bounded but also attains its maximum value, which can be found by a finite exhaustive search implementable on modern computers.

The class of so-called q-adic Peano curves is defined, which is large enough to include the polyfractal curves. The cube-to-linear ratio for this class attains its maximum value, which can be effectively determined by an exhaustive search implementable on modern computers.

We present a full and correct proof of the fact that the problem of constructing an optimal schedule for the open shop problem with at most m − 3 preemptions for an m-processor system is NP-hard. We also show that the proof of this result given by E. Shchepin and N. Vakhania in Ann. Oper. Res. 159, 183–213 (2008) is incorrect.

In a finite set X with distance, we introduce a so-called chain distance. This distance generates a partition of X into clusters such that any point inside each cluster can be connected with any other point of the same cluster by a chain whose every link does not exceed a given threshold value. We construct a chain development, by which we mean a m...

The mystery of infinitesimally small values was unveiled by the theory of limits of Cauchy-Weierstrass. The "epsilon-delta" language establish a new standard of the rigor in analysis. Unfortunately the new rigor language occurs inaccessible for the majority of students. From pedagogical point of view many achievement turns into it opposite.

A regular fractal Peano curve $p\: [0,1]\to[0,1]^3$ has the following self-similarity property: there are decompositions $[0,1]=\bigcup\limits_{0\le k<n} [\frac k{n},\frac {k+1}{n}]$ (for arbitrary large $n$) such that the images of all intervals $[\frac k{n},\frac {k+1}{n}]$ are cubes of volumes equal to $\frac1n$. The main result of the article d...

Proposed algorithm in a linear time produces a fuzzy clustering of a nieghbohr of a weighted graph. The algorithm has many parameters to be adopted for solution of different problems of clusterization and relevant relations. The action of the algorithm is demonstrated for the cathegorization of retrieval requests.

It is proposed a new code for contours of plane images. This code was applied for optical character recognition of printed and handwritten characters. One can apply it to recognition of any visual images.

One introduces the concept of greedy k-summability in such a way, that the direct product of one greedy k-summable numeric array onto another greedy n-summable numeric array to be greedy (n+k+1)-summable.

We give a complement note on the proof of the NP-hardness of preemptive acyclic open-shop problem presented earlier by the authors in Ann. Oper. Res. 159:183–213, 2008.

We prove that the greedy sum of a direct product of two numeric arrays of complex numbers is equal to the product of the greedy sums of the factors provided that all the mentioned sums exist.

A map of metric spaces f: X → Y satisfying the inequality $$ \left| {f(x) - f(y)} \right| \leqslant C\left| {x - y} \right|^\alpha $$ for some C and α and all x, y ∈ X is called a Hölder map with exponent α. V. I. Arnold posed the following problem: Does there exist a Höldermap from the square onto the cube with exponent 2/3? The firstmain theo...

It was proved by H. Whitney in 1933 that a Serre fibration of compact metric spaces admits a global section provided every fiber is homeomorphic to the unit interval [0,1]. An extension of the Whitney's theorem to the case when all fibers are homeomorphic to some fixed compact two-dimensional manifold was proved by the authors \cite{BCS}. The main...

A Peano curve p(x) with maximum square-to-linear ratio |p(x)−p(y)|2/|x−y| equal to 5 2/3 is constructed; this ratio is smaller than that of the classical Peano-Hilbert curve, whose maximum square-to-linear ratio is 6. The curve constructed is of fractal genus 9 (i.e., it is decomposed into nine fragments that are similar to the whole curve) and of...

It was proved by H. Whitney in 1933 that a Serre fibration of compact metric spaces admits a global section provided every fiber is homeomorphic to the unit interval [0,1]. Results of this paper extend Whitney's theorem to the case when all fibers are homeomorphic to a given compact two-dimensional manifold.

In this paper we study multiprocessor and open shop scheduling problems from several points of view. We explore a tight dependence of the polynomial solvability/intractability on the number of allowed preemptions. For an exhaustive interrelation, we address the geometry of problems by means of a novel graphical representation. We use the so-called...

This paper is concerned with maps without antipodal coincidence from spheres into compacta and polyhedra of a smaller dimension and to obstructions for embeddings of polyhedra and compacta in Euclidean spaces. Estimates of the dimension of the antipodal coincidence set are given for maps of spheres into compacta. The theory of the Yang homology ind...

In this paper the author studies spaces in which one can define a "distance" from points to canonically closed sets (the -metric). It is proved that products of metric spaces and locally compact groups are examples of such spaces, and in these cases the -metric can be constructed so that an analogue of the triangle axiom is satisfied. The topologic...

We show that an n-dimensional compactum X embeds in R^m, where m>3(n+1)/2, if and only if X x X - \Delta admits an equivariant map to S^{m-1}. In particular, X embeds in R^{2n}, n>3, iff the top power of the (twisted) Euler class of the factor-exchanging involution on X x X - \Delta is trivial. Assuming that X quasi-embeds in R^{2n} (i.e. is an inv...

One of the main restrictions in scheduling problems are the machine (resource) restrictions: each machine can perform at most one job at a time. We explore machine dependencies is shop scheduling problems representing them as graphs. As it is turning out, the structure of machine dependency graphs is important in the complexity analysis of shop sch...

Non-preemptive scheduling of n independent jobs on m unrelated machines so as to minimize the maximal job completion time is considered. A polynomial algorithm with the worst-case absolute error of min{(1 − 1/m)pmax, p} is presented, where pmax is the largest job processing time and p is the mth element from the non-increasing list of job processin...

One of the main restrictions in scheduling problems are the machine (resource) re-strictions: each machine can perform at most one job at a time. We explore machine dependencies is shop scheduling problems representing them as graphs. Our study shows that the structure of machine dependency graphs is important in the complex-ity analysis of shop sc...

A polynomial-time algorithm is suggested for non-preemptive scheduling of n-independent jobs on m-unrelated machines to minimize the makespan. The algorithm has a worst-case performance ratio of 2−1/m. This is better than the earlier known best performance bound 2. Our approach gives the best possible approximation ratio that can be achieved using...

We show that preemptive versions of some NP-hard scheduling problems re-main NP-hard, but only for a restricted number of preemptions. If we allow a "sufficient" number of preemptions, then these problems become polynomi-ally solvable. We find, as we call, the critical number of preemptions for our problems, i.e., the minimal number of preemptions...

It is proved that for every fractal continuous mapping F: I\to I^2 of the unit interval onto the unit square there is a pair of points x,y\in I, such that |F(x)-F(y)|^2\ge 5|x-y|.

It was proved by H. Whitney in 1933 that it is possible to mark a point in all curves in a continuous way. The main result of this paper extends the Whitney theorem to dimensions 2 and 3. Namely, we prove that it is possible to choose a point continuously in all two-dimensional surfaces sufficiently close to a given surface, and in all 3-manifolds...

We consider the multiprocessor job shop scheduling problem (JSP) with unrelated processors, an extension of the classical JSP. The precedence relations between the operations are given by an acyclic directed weighted graph, in which nodes represent operations and arcs represent precedence relations. The whole set of operations is partitioned into m...

The main theorem of this paper generalizes a classic Aumann's characterization of compact convex sets, via the acyclicity of their hypersections, to arbitrary weakly closed subsets of a locally convex linear space.

We consider the preemptive scheduling of n independent jobs on m unrelated machines to minimize the makespan. Preemptive schedules with at most 2m−3 preemptions are built, which are optimal when the maximal job processing time is no more than the optimal schedule makespan. We further restrict the maximal job processing time and obtain optimal sched...

The main theorem of this paper generalizes a classic Aumann's characterization of compact convex sets, via the acyclicity of their hypersec-tions, to arbitrary weakly closed subsets of a locally convex linear space.

The Borsuk-Sieklucki theorem says that for every uncountable family {Xα}α∈A of n-dimensional closed subsets of an n-dimensional ANR-compactum, there exist α ≠ β such that dim(Xα ∩ Xβ) = n. In this paper we show a cohomological version of that theorem: THEOREM. Suppose a compactum X is clcℤn+1, where n ≥ 1, and G is an Abelian group. Let {Xα}α∈J be...

An algorithm for solving the linear programming problem known as the multipro� cessor distribution �or scheduling� problem is suggested� The problem is to distribute a given set of tasks among given processors so as to minimize the load time of the most loaded processor� Dividing the tasks into parts and distributing the parts among di�erent proces...

The purpose of this paper is to study how small orbits of peri-odic homemorphisms of spheres can be.

The paper is devoted to studying the linking of cycles with compacta in LCn-spaces and in particular homology Z-sets. The main two consequences of our considerations are the following:(1) It is proved that a k-dimensional polyhedron cannot link a (n−k−1)-dimensional cycle in an n-dimensional Menger manifold.(2) It is proved that a compact set in an...

We consider the problem of scheduling of n independent jobs on m unrelated machines to minimize the max(t 1 , t 2 ,..., t m ), t i being the completion time of machine i. In [1] was suggested a polynomial 2-approximation algorithm for this problem. It was also proved that there can exist no polynomial 1.5-approximation algorithm unless P = NP. Here...

A selection problem for convex-valued mappings is studied. Two general results, so called "sandwich" theorems, are proved.

A proof of the Hilbert-Smith conjecture for a free Lipschitz action is given. The proof is elementary in the sense that it does not rely on Yang's theorem about the cohomology dimension of the orbit space of the p-adic action. The result turns out to be true for the class of spaces of finite Hausdorff volume, which is considerably wider than Rieman...

A proof of the Hilbert-Smith conjecture for a free Lipschitz action is given. The proof is elementary in the sense that it does not rely on Yang’s theorem about the cohomology dimension of the orbit space of thep-acid action. The result turns out to be true for the class of spaces of finite Hausdorff volume, which is considerably wider than Riemann...

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