# Yaroslav Ivanovich GrushkaNational Academy of Sciences of Ukraine | ISP · Institute of Mathematics

Yaroslav Ivanovich Grushka

PhD

## About

64

Publications

6,537

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213

Citations

Introduction

Additional affiliations

January 2000 - present

Education

October 1993 - May 1997

**Graduate studies of Institute of Mathematics NAS of Ukraine**

Field of study

- Mathematical analysis

January 1989 - June 1992

## Publications

Publications (64)

In the present paper, based on the ideas of Algerian physicist M.E. Hassani, the generalizedHassani spatial-temporal transformations in real Hilbert space are introduced. The originaltransformations, introduced by M.E. Hassani, are the particular cases of the transformations,introduced in this paper. It is proven that the classes of generalized Has...

In the present paper, based on the ideas of Algerian physicist M.E. Hassani, the generalized Hassani spatio-temporal transformations in real Hilbert space are introduced. The original transformations, introduced by M.E. Hassani, are the particular cases of the transformations, introduced in this paper. It is proven that the classes of generalized H...

From an intuitive point of view universal kinematics are collections (sets) of changing objects, which evolve, being in a certain spatial-geometric environment, and evolution of whi- ch can be observed from many diﬀerent frames of reference. Moreover, the deﬁnition of uni- versal kinematics impose the existence of some (preassigned) universal coord...

This paper contains abstract of my talk in the conference. You can also download this document at:
https://www.winterschool.eu/files/1229-On_existence_of_one-point_time_on_an_oriented_set239675677.pdf

This document contains presentation of my talk in the conference.
You can also download this document at:
https://www.winterschool.eu/files/1229-On_existence_of_one-point_time_on_an_oriented_set1575031554.pdf

The subject of this article is closely related to the theory of changeable sets. The mathematically rigorous theory of changeable sets was constructed in 2012. From an intuitive point of view, the changeable sets are sets of objects which, unlike elements of ordinary (static) sets, can be in the process of continuous transformations, i.e., they can...

The notion of oriented set is the basic elementary concept of the theory of changeable sets. The main motivation for the introduction of changeable sets was the sixth Hilbert problem, that is, the problem of mathematically rigorous formulation of the fundamentals of theoretical physics. In the present paper the necessary and sufficient condition of...

Presentation for my talk "Necessary and sufficient condition for the existence of one-point time on an oriented set" at the XII International Algebraic Conference in Ukraine dedicated to the 215th anniversary of V. Bunyakovsky (Ukraine, Vinnytsia, July 02-06, 2019, in Ukrainian language).

Let ${\mathbb T}=({\bf T},\leq)$ and ${\mathbb T}_{1}=({\bf T}_{1},\leq_{1})$ be linearly ordered sets and $\mathscr{X}$ be a topological space. The main result of the paper is the following: If function $\boldsymbol{f}(t,x):{\bf T}\times\mathscr{X}\mapsto{\bf T}_{1}$ is continuous in each variable ("$t$"and "$x$") separately and function $\boldsym...

The notion of oriented set is the basic elementary concept of the theory of changeable sets. Oriented sets can be interpreted as the most primitive abstract models of sets of changing objects that evolve within a single (fixed) reference frame. Also oriented sets are mathematical objects, in the framework of which one can give the mathematically ri...

Presentation for my talk "On oriented sets and chronologization" on the conference "Modern problems of probability theory and mathematical analysis -- 2019" (Ukraine, Vorokhta, February 25 -- March 1, 2019, in Ukrainian language).

Universal kinematics as mathematical objects may be interesting for astrophysics, because there exists a hypothesis that, in the large scale of the Universe, physical laws (in particular, the laws of kinematics) may be different from the laws acting in a neighborhood of our solar System. The present paper is devoted to investigation of self-consist...

Let $\TT=\left(\T,\leq\right)$ and $\TT_{1}=\left(\T_{1},\leq_{1}\right)$
be linearly ordered sets and $\sX$ be a topological space. The main
result of the paper is the following: <\br>
If function $\bsf(t,x):\T\times\sX\mapsto\T_{1}$ is continuous in
each variable (``$t$'' and ``$x$'') separately and function $\bsf_{x}(t)=\bsf(t,x)$
is monotonous...

*** At the present time this paper is available only in UKRAINIAN language! *** Universal kinematics as mathematical objects may be interesting for astrophysics, because there exists the hypothesis, that in the large scale of the Universe, physical laws (in particular, the laws of kinematics) may be different from the laws, acting in the neighborho...

*** At the present time this paper is available only in UKRAINIAN language! *** Universal kinematics as mathematical objects may be interesting for astrophysics, because there exists the hypothesis, that in the large scale of the Universe, physical laws (in particular, the laws of kinematics) may be different from the laws, acting in the neighborho...

*** At the present time this paper is available only in UKRAINIAN language. ***
Universal kinematics as mathematical objects may be interesting for astrophysics, because there exists the hypothesis, that in the large scale of the Universe, physical laws (in particular, the laws of kinematics) may be different from the laws, acting in the neighborh...

Using the recently developed mathematical apparatus of the theory of universal kinematic sets, we prove that the hypothesis of the existence of material objects and inertial reference frames moving with superluminal velocities in the general case does not lead to the violation of the principle of causality, that is, to a possibility of the returnin...

This paper is devoted to study of coordinate transforms in abstract kinematic changeable sets. Investigations in this direction may be interesting for astrophysics, because there exists the hypothesis, that in large scale of the Universe, physical laws (in particular, the laws of kinematics) may be different from the laws, acting in the neighborhoo...

Document represents conspectus of my report at the XII Summer School "Algebra, Topology, Analysis" (Kolochava village, Zakarpatian district, Ukraine, July 10-23, 2017). Document is in Ukrainian language.

The present paper is devoted to the analysis of different versions of extended Lorentz transformations, proposed for reference frames moving with the velocity, greater then the velocity of light. In particular we point out some errors of individual authors in this field.

This work lays the foundations of the theory of kinematic changeable sets (" abstract kinematics "). Theory of kinematic changeable sets is based on the theory of changeable sets. From an intuitive point of view, changeable sets are sets of objects which, unlike elements of ordinary (static) sets, may be in the process of continuous transformations...

This work is devoted to a study of abstract coordinate transforms in kinematic changeable sets. Investigations in this direction may be interesting for astrophysics, because there exists a hypothesis that, in a large scale of the Universe, physical laws (in particular, the laws of kinematics) may be different from the laws acting in a neighbourhood...

We construct the logarithmic extension for the set of real numbers in which the numbers, less then −∞ exist. Using this logarithmic extension we give the single formula for hyperbolic representation of generalized tachyon Lorentz transforms.

*** This is old version of the preprint. More contemporary version you can find at http://dx.doi.org/10.13140/RG.2.2.28964.27521 as well as https://www.researchgate.net/publication/317368829 ***
This work lays the foundations of the theory of kinematic changeable sets (" abstract kinematics "). Theory of kinematic changeable sets is based on the t...

Short conspectus of the talk in International Conference dedicated to the 120-th anniversary of Kazimierz Kuratowski (Lviv, 27 September -- 1 October, 2016)

The paper is available in Ukrainian and English languages.
Using the recently developed mathematical apparatus of the theory of kinematic changeable sets, it is proved that the hypothesis of the existence of material objects and inertial reference systems moving with superluminal velocities does not lead to the violation of the principle of causal...

*** This paper is available only in UKRAINIAN language. Main results of the paper in English language were announced in the article "On time irreversibility of universal kinematics" (see at https://www.researchgate.net/publication/305417360_On_time_irreversibility_of_universal_kinematics).***
In the present paper we analyze the violation of the pr...

Analogues of the set-theoretic inclusion relation and the set-theoretic operation of union for universal kinematic sets are introduced. A theorem on evolutional extension for universal kinematic is proved. The obtained results can be used for formulation of mathematical foundations of special relativity in the framework of the theory of kinematic c...

Analogues of the set-theoretic inclusion relation and the set-theoretic operation of disjoint union for kinematic changeable sets and universal kinematic sets are introduced and their properties are investigated. The results may be used for formulation of mathematical foundations of special relativity in the framework of theory of kinematic changea...

This work is devoted to investigation of kinematic changeable sets (“abstract kinematics”), that are mathematical objects in which changeable sets are equipped by different geometrical or topological structures, namely metric, topological, linear, Banach, Hilbert and other spaces. Such mathematical objects allow simulating the evolution of physical...

In this comment some incorrect results published in the article "Extended Linear and Nonlinear Lorentz Transformations and Superluminality" are refuted. The article "Extended Linear and Nonlinear Lorentz Transformations and Superluminality" can be found in the journal "Advances in High Energy Physics", Volume 2013 (2013), article ID 760916.

One of the fundamental postulates of the special relativity theory is
existence of a single system of universal coordinate transforms for inertial
reference frames, that is coordinate transforms, which are uniquely determined
by space-time coordinates of a material point. In this paper the abstract
mathematical theory of coordinate transforms in ki...

This work is devoted to the study of kinematic changeable sets ("abstract kinematics"), that is the mathematical objects, in which changeable sets are equipped by different geometrical or topological structures, namely metric, topological, linear, Banach, Hilbert and other spaces. Investigations in this direction may be interesting for astrophysics...

This work is devoted to the investigation of kinematic changeable sets ("abstract kinematics"), that is the mathematical objects, in which changeable sets are equipped by different geometrical or topological structures, namely metric, topological, linear, Banach, Hilbert and other spaces. In the paper the definition of actual and universal coordina...

Analogues of set-theoretic inclusion relation and set-theoretic operation of union for basic (base) changeable sets are introduced, and their properties are studied.
Alternative title of the article is "Evolutional extensions and analogues of the operation of union for base changeable sets".
Document contains text of the original article in Ukrai...

We introduce the notion of base changeable sets and study the principal properties of these sets. Base changeable sets are required for the construction of the general theory of changeable sets. The problem studied in our paper is closely connected with the famous sixth Hilbert problem.
Article written in Ukrainian language in Ukrainian version of...

New results in the theory of changeable sets are obtained. Applications of the theory of changeable sets for construction of mathematically strict models of kinematics, based on generalized Lorentz transforms, is proposed. This kinematics includes the classical kinematics of special relativity theory but, also, permits the superluminal motion of th...

In the present paper the we investigate the algebraic properties of the generalized Lorentz transforms for superlight velocities in Minkowski space time over any real Hilbert space.
In particular, it is proved that, unlike the classical case, the set of generalized Lorentz transformations does not form a group.
Document contains text of the origin...

We investigate the generalized Lorents transforms in Minkowski space time $\mathcal{M}(\mathfrak{H})$ over any real Hilbert space $\mathfrak{H}$, which in the particular case $\mathfrak{H}=\mathbb{R}^{3}$ have been introduced in the papers of M. Hill, Barry J. Cox and E. Recami. The set of generalized Lorents transforms $\mathfrak{OT}(\mathfrak{H})...

In this paper we introduce and investigate gradations of visibility in the changeable sets with many areas of perception (reference frames).

The work lays the foundations of the theory of changeable sets. In author
opinion, this theory, in the process of it's development and improvement, can
become one of the tools of solving the sixth Hilbert problem at least for physics
of macrocosm.
From a formal point of view, changeable sets are sets of objects which,
unlike the elements of ordinar...

The work lays the foundations of the theory of changeable sets. Changeable sets can be considered as sets of objects which, unlike the elements of ordinary (static) sets may be in the process of continuous transformations, and which may change properties depending on the point of view on them (the area of observation or reference frame).
In author’...

In the present paper the concept of primitive changeable set is introduced and the properties of these sets are investigated.Primitive changeable set can be regarded as the mathematical abstraction of the most primitive models of physical processes in one (fixed) reference frame, and they are necessary to construct the more general theory of change...

A theory of Banach spaces of pseudo operators with bounded projection trace over a given Hilbert space is constructed. It is proved the possibility of building of certain groups of operators connected the evolution of infinite-particle quantum systems in these spaces.

We construct a theory of Banach spaces of “generalized” operators with bounded projection trace over a given Hilbert space.
This theory can be efficient in the investigation of evolution problems for quantum systems with infinitely many particles.

A theory of Banach spaces of “generalized” operators with bounded projection trace is constructed over a given Hilbert space. This theory may be useful in studying such problems of mathematical physics for many-particle systems, whose exploring is impossible in the space of operators with bounded trace.

A theory of Banach spaces of "generalized" operators with bounded projection trace over a Hilbert space is constructed. These spaces can be applied for the description of evolution problems of infnite-particle quantum systems.
Key Words: Hilbert space, translation-invariant operator, trace, difference variable trace, projection trace

An arbitrary operator A on a Banach space
. such that either A or iA generates the Co-group with certain growth condition at infinity is considered. The direct and inverse theorems on connection between the degree of smoothness of a vector
with respect to the operator A, the rate of convergence to zero of the best approximation of x by exponential...

Let $\left\{ T(t):t\geq0\right\} $ be a $C_{0}$-semigroup of bounded
linear operators in a complex Banach Space $(\X,\left\Vert \cdot\right\Vert )$
and $A$ be it's generator. The well known \textbf{uniform convergence
theorem} states that the generator $A$ is bounded for the semigroups
$\left\{ T(t)\right\} $, satisfying $\overline{\lim\limits _{t\...

For an arbitrary operator A on a Banach space X which is a generator of C_0-group with certain growth condition at the infinity, the direct theorems on connection between the smoothness degree of a vector $x\in X$ with respect to the operator A, the order of convergence to zero of the best approximation of x by exponential type entire vectors for t...

A sufficient condition for existence of an analytical solution of
the abstract Cauchy problem $\frac{dy}{dt}=A(t)y$, $y(0)=y_{0}$,
is obtained, where $A(t)$, $t\in U_{0}$, is an operator-valued function
defined in some neighborhood of zero $U_{0}\subseteq\mathbb{C}$ whose
values are linear (in general unbounded) operators in a Banach space
$\left(\...

For an arbitrary self-adjoint operator $B$ in a Hilbert space $\mathfrak{H}$,
we present direct and inverse theorems establishing the relationship
between the degree of smoothness of a vector $x\in\mathfrak{H}$ with
respect to the operator $B$, the rate of convergence to zero of its
best approximation by exponential-type entire vectors of the oper...

This is the other title and reference to the publication:
https://www.researchgate.net/publication/259531329

Let {T(t): t \geq 0} be a C_0-semigroup of bounded linear operators in a Banach space X and A be the generator of this semigroup. The well known uniform convergence theorem asserts that if for the semigroup T(t), $\overline{\lim\limits _{t\to+0}}\left\Vert T(t)-\mathbf{I}\right\Vert <1$, where $\mathbf{I}$ is the identity operator in X, then the ge...

Let $\left\{ T(t):t\geq0\right\} $ be a $C_{0}$-semigroup on a Hilbert
space $\mathcal{H}$ whose generator $A$ is normal. Then for any
element $f\in\mathcal{H}$, the norm $\left\Vert T(t)f-f\right\Vert \to0$
($t\to0+$). If $A$ is an unbounded operator then there is no function
$\gamma:[0,\infty)\to\mathbb{R}$, $\gamma(t)\to0$ ($t\to0+$) such
that...

Let $\left\{ T(t)\right\} t\geq0$ be a $C_{0}$ -semigroup of bounded
linear operators on a Hilbert space $H$ , and let $A$ be the generator
of this semigroup. In the paper, the author obtains an estimate near
zero of the magnitude $\delta_{f}(t)=\left\Vert T(t)f-f\right\Vert $
on the vectors $f\in D(\Phi(|A|))$ , where $\Phi:[0,\infty)\mapsto\mathb...

We describe classes of vectors $f$ from a Hilbert space $H$ for
which the quantity $\left\Vert T(t)f-f\right\Vert $, where $T(t)=e^{-tA}$,
$t \geq 0$, and $A$ is a self-adjoint non-negative operator in $H$,
has a certain order of convergence to zero as $t\to+0$.

We describe classes of vectors $f$ from a Hilbert space $H$ for
which the quantity $\left\Vert T(t)f-f\right\Vert $, where $T(t)=e^{-tA}$,
$t \geq 0$, and $A$ is a self-adjoint non-negative operator in $H$,
has a certain order of convergence to zero as $t\to+0$.

Let $A$ be a self-adjoint, non-negative operator in Hilbert space $H$ and $T(t)=e^{-tA}$ ($t \geq 0$) be the semigroup of operators, generated by the operator $(-A)$. We describe classes of vectors $f$ from a for which the quantity $\left\Vert T(t)f-f\right\Vert $, has a certain integral order of convergence to zero as $t\to+0$ in the terms of func...

## Questions

Questions (5)

Dear colleagues. The text of the question is available in the attached image or pdf-file.

Sincerely, Yaroslav Grushka.

The well-known Zermelio's theorem states that every set can be well-ordered. Since arbitrary well-ordering is a linear ordering, from this theorem it follows the following corollary:

(A) An arbitrary set can be linearly ordered.

It is well-known that Zermelio's theorem is equivalent to the axiom of choice.

Question: Can Corollary (A) be proven without axiom of choice?

The question details are contained in the attached pdf file.

Let $\alpha\in[0,\infty)$ be any nonnegative irrational number. Do for any $\varepsilon>0$ there exist $m,n\in\mathbb{N}$ such, that $|n-\alpha m|<\varepsilon$?