Stefan Porubsky - Academia.edu (original) (raw)

Papers by Stefan Porubsky

Research paper thumbnail of Matyáš Lerch (1860-1922)

Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents ... more Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use.

Research paper thumbnail of Idempotents and Congruence <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow></mrow><annotation encoding="application/x-tex"></annotation></semantics></math></span><span class="katex-html" aria-hidden="true"></span></span>\boldsymbol{ax}\boldsymbol{ \equiv b\pmod n}$$

From Arithmetic to Zeta-Functions, 2016

Given a nonzero a 2 Z n , the ring of residues modulo n, such that gcd.a; n/ D djb, not only ther... more Given a nonzero a 2 Z n , the ring of residues modulo n, such that gcd.a; n/ D djb, not only there exists an element x 2 Z n such that x a Á b .mod n/, but that there even exists an invertible element x 2 Z n such that x a Á b .mod n/. Their sufficient and necessary condition for this says that gcd.b=d; n=d/ D 1 with d as above. A typical structure result on finite commutative semigroup says that the multiplicative semigroup of Z n decomposes into the so-called maximal subsemigroups belonging to the idempotents of Z n. Each such semigroup contains a maximal subgroup having for its identity the corresponding idempotent. In general this subgroup is a proper subset of the maximal subsemigroup containing it. However, the group of elements of Z n coprime to n is an example of the case when this maximal subsemigroup and the maximal subgroup coincide (both evidently belonging to the idempotent 1). In what follows we prove that if a congruence x a Á b .mod n/ is solvable there always exists a solution in the maximal semigroup belonging to the idempotent given by the divisor ı D gcd.b=d; n=d/ and if ı is a unitary divisor of n then there even exists a solution in the maximal subgroup belonging to the idempotent given by ı.

Research paper thumbnail of Covering systems, Kubert identities and difference equations

Mathematica Slovaca, 2000

A. S. Fraenkel proved tha t the following identities involving Bernoulli polynomials -»n(0) m • ч... more A. S. Fraenkel proved tha t the following identities involving Bernoulli polynomials -»n(0) m • ч for all n > 0 are t rue if and only if the system of ar i thmetic congruences {a^ (mod 6 )̂ : 1 < i < m} is an exact cover of Z . Generalizations of this result involving other functions and more general covering systems have been successively found by A. S. Fraenkel, J . Beebee, Z.-W. Sun and the author . Z.-W. Sun proved an alge­ braic characterization of functions capable of identities of this type, and indepen­ dently J. Beebee observed a connection of these results to the Raabe multiplica­ tion formula for Bernoulli polynomials and with the so-called Kubert identities. In this article, we shall analyze some analytic aspects of connections between finite systems of arithmetical progressions and the generalized Kubert identities «(»«)/(.-) = x ; г / f — ) • ІĚГo V m j 2000 M a t h e m a t i c s S u b j e c t C l a s s i f i c a t i o n : Pr imary 11B25, 11B68, 33E30; Seconda...

Research paper thumbnail of Semigroup Structure of Sets of Solutions to Equation X = X

We describe the semigroup and group structure of the set of solutions to equation X = X over the ... more We describe the semigroup and group structure of the set of solutions to equation X = X over the multiplicative semigroups of factor rings of residually finite commutative rings and of residually finite commutative PID’s. The analysis is done in terms of the structure of maximal unipotent subsemigroups and subgroups of semigroups of the corresponding rings. In case of residually finite PID’s we employ the available idempotents analysis of the Euler–Fermat Theorem in these rings used to determine minimal positive integers ν and μ such that for all elements x of these rings one has x = x. In particular, the case when this set of solutions is a union of groups is handled. As a simple application we show a not yet noticed group structure of the set of solutions to x = x (mod n) connected with the message space of RSA cryptosystems and Fermat pseudoprimes.

Research paper thumbnail of Uniform Distribution of the Weighted Sum-of-Digits Functions

Uniform distribution theory, 2021

The higher-dimensional generalization of the weighted q-adic sum-of-digits functions sq,γ (n), n ... more The higher-dimensional generalization of the weighted q-adic sum-of-digits functions sq,γ (n), n =0, 1, 2,..., covers several important cases of sequences investigated in the theory of uniformly distributed sequences, e.g., d-dimensional van der Corput-Halton or d-dimensional Kronecker sequences. We prove a necessary and sufficient condition for the higher-dimensional weighted q-adic sum-of-digits functions to be uniformly distributed modulo one in terms of a trigonometric product. As applications of our condition we prove some upper estimates of the extreme discrepancies of such sequences, and that the existence of distribution function g(x)= x implies the uniform distribution modulo one of the weighted q-adic sum-of-digits function sq,γ (n), n = 0, 1, 2,... We also prove the uniform distribution modulo one of related sequences h 1 sq, γ (n)+h 2 sq,γ (n +1), where h 1 and h 2 are integers such that h 1 + h 2 ≠ 0 and that the akin two-dimensional sequence sq,γ (n), sq,γ (n +1) canno...

Research paper thumbnail of On coverings of almost Dedekind domains

Czechoslovak Mathematical Journal, 1976

Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents ... more Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use.

Research paper thumbnail of On the density of certain sets in arithmetical semigroups

Czechoslovak Mathematical Journal, 1979

Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents ... more Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use.

Research paper thumbnail of Application and Misapplication of the Czechoslovak STP Cipher During WWII

Tatra Mountains Mathematical Publications, 2017

Lieutenant colonel Karol Cigáň (1921-2005), head of the cryptographic unit of the Czechoslovak Mi... more Lieutenant colonel Karol Cigáň (1921-2005), head of the cryptographic unit of the Czechoslovak Ministry of National Defence in the period 1949-1958 was after discharging from this position in Prague relocated to an insignificant and substandard command position at a district military administration in Slovakia. His cryptographic experience was of no use in his new position. To profit from his previous experience as a high qualified cryptographer he started to study the accessible literature and archive materials about the usage of the Czechoslovak cipher systems during the WWII. The result of this his activity were some manuscripts where he deciphered and analyzed some Czechoslovak military wireless telegrams. His critical analysis and his conclusions did not meet an understanding or a positive response of historians and were nor accepted for publication. He had no other chance as to send them to archives. Unfortunately only one (in two copies) and a collection of small notes surviv...

Research paper thumbnail of Remarks on systems of congruence classes

Research paper thumbnail of On theorems of Niven and Dressler

Mathematica Slovaca, 1978

Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digi... more Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use. This paper has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library http://project.dml.cz

Research paper thumbnail of Sperner Type Theorems for Generalized Divisors

sbc.org.pl

Abstract. The extensions of the well-known Sperner's result on antichains of sub sets of a g... more Abstract. The extensions of the well-known Sperner's result on antichains of sub sets of a given finite set for divisors of a positive integers are shown to hold also for sets of regular systems of divisors of elements of arithmetical semigroups.

Research paper thumbnail of On covering systems on rings

Mathematica Slovaca, 1978

Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digi... more Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use.

Research paper thumbnail of Identities involving covering systems. I

Mathematica Slovaca, 1994

Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digi... more Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use.

Research paper thumbnail of New solvability conditions for congruence ax≡b (mod n)

Tatra Mountains Mathematical Publications, 2015

K. Bibak et al. [arXiv:1503.01806v1 [math.NT],March 5 2015] proved that congruence ax ≡ b (mod n)... more K. Bibak et al. [arXiv:1503.01806v1 [math.NT],March 5 2015] proved that congruence ax ≡ b (mod n) has a solution x0 with t = gcd(x0, n) if and only if gcd thereby generalizing the result for t = 1 proved by B. Alomair et al. [J. Math. Cryptol. 4 (2010), 121-148] and O. Grošek et al. [ibid. 7 (2013), 217-224]. We show that this generalized result for arbitrary t follows from that for t = 1 proved in the later papers. Then we shall analyze this result from the point of view of a weaker condition that gcd . We prove that given integers a, b, n ≥ 1 and t ≥ 1, congruence ax ≡ b (mod n) has a solution x0 with t dividing gcd(x0, n) if and only if gcd divides gcd .

Research paper thumbnail of Prof. RNDr. Břetislav Novák, DrSc. (1938–2003) would be seventy

Research paper thumbnail of Transformations that preserve uniform distribution

Research paper thumbnail of On m times covering systems of congruences

Research paper thumbnail of Identities Involving Covering Systems II

Mathematica Slovaca, 1994

Research paper thumbnail of New necessary and sufficient conditions on (, ) in order that = (mod ) be a covering system

Electronic Notes in Discrete Mathematics, 2002

A covering system is a set of congruences x≡ ai (mod mi), i= 1,… k, such that every integer satis... more A covering system is a set of congruences x≡ ai (mod mi), i= 1,… k, such that every integer satisfies at least one of them. A new necessary and sufficient condition in order that a given set of congruences x≡ ai (mod mi) be a covering system is established. We show that (4) ...

Research paper thumbnail of Binary sequences generated by sequences n alpha , n=1,2, dots

Publicationes Mathematicae Debrecen, 2010

Research paper thumbnail of Matyáš Lerch (1860-1922)

Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents ... more Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use.

Research paper thumbnail of Idempotents and Congruence <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow></mrow><annotation encoding="application/x-tex"></annotation></semantics></math></span><span class="katex-html" aria-hidden="true"></span></span>\boldsymbol{ax}\boldsymbol{ \equiv b\pmod n}$$

From Arithmetic to Zeta-Functions, 2016

Given a nonzero a 2 Z n , the ring of residues modulo n, such that gcd.a; n/ D djb, not only ther... more Given a nonzero a 2 Z n , the ring of residues modulo n, such that gcd.a; n/ D djb, not only there exists an element x 2 Z n such that x a Á b .mod n/, but that there even exists an invertible element x 2 Z n such that x a Á b .mod n/. Their sufficient and necessary condition for this says that gcd.b=d; n=d/ D 1 with d as above. A typical structure result on finite commutative semigroup says that the multiplicative semigroup of Z n decomposes into the so-called maximal subsemigroups belonging to the idempotents of Z n. Each such semigroup contains a maximal subgroup having for its identity the corresponding idempotent. In general this subgroup is a proper subset of the maximal subsemigroup containing it. However, the group of elements of Z n coprime to n is an example of the case when this maximal subsemigroup and the maximal subgroup coincide (both evidently belonging to the idempotent 1). In what follows we prove that if a congruence x a Á b .mod n/ is solvable there always exists a solution in the maximal semigroup belonging to the idempotent given by the divisor ı D gcd.b=d; n=d/ and if ı is a unitary divisor of n then there even exists a solution in the maximal subgroup belonging to the idempotent given by ı.

Research paper thumbnail of Covering systems, Kubert identities and difference equations

Mathematica Slovaca, 2000

A. S. Fraenkel proved tha t the following identities involving Bernoulli polynomials -»n(0) m • ч... more A. S. Fraenkel proved tha t the following identities involving Bernoulli polynomials -»n(0) m • ч for all n > 0 are t rue if and only if the system of ar i thmetic congruences {a^ (mod 6 )̂ : 1 < i < m} is an exact cover of Z . Generalizations of this result involving other functions and more general covering systems have been successively found by A. S. Fraenkel, J . Beebee, Z.-W. Sun and the author . Z.-W. Sun proved an alge­ braic characterization of functions capable of identities of this type, and indepen­ dently J. Beebee observed a connection of these results to the Raabe multiplica­ tion formula for Bernoulli polynomials and with the so-called Kubert identities. In this article, we shall analyze some analytic aspects of connections between finite systems of arithmetical progressions and the generalized Kubert identities «(»«)/(.-) = x ; г / f — ) • ІĚГo V m j 2000 M a t h e m a t i c s S u b j e c t C l a s s i f i c a t i o n : Pr imary 11B25, 11B68, 33E30; Seconda...

Research paper thumbnail of Semigroup Structure of Sets of Solutions to Equation X = X

We describe the semigroup and group structure of the set of solutions to equation X = X over the ... more We describe the semigroup and group structure of the set of solutions to equation X = X over the multiplicative semigroups of factor rings of residually finite commutative rings and of residually finite commutative PID’s. The analysis is done in terms of the structure of maximal unipotent subsemigroups and subgroups of semigroups of the corresponding rings. In case of residually finite PID’s we employ the available idempotents analysis of the Euler–Fermat Theorem in these rings used to determine minimal positive integers ν and μ such that for all elements x of these rings one has x = x. In particular, the case when this set of solutions is a union of groups is handled. As a simple application we show a not yet noticed group structure of the set of solutions to x = x (mod n) connected with the message space of RSA cryptosystems and Fermat pseudoprimes.

Research paper thumbnail of Uniform Distribution of the Weighted Sum-of-Digits Functions

Uniform distribution theory, 2021

The higher-dimensional generalization of the weighted q-adic sum-of-digits functions sq,γ (n), n ... more The higher-dimensional generalization of the weighted q-adic sum-of-digits functions sq,γ (n), n =0, 1, 2,..., covers several important cases of sequences investigated in the theory of uniformly distributed sequences, e.g., d-dimensional van der Corput-Halton or d-dimensional Kronecker sequences. We prove a necessary and sufficient condition for the higher-dimensional weighted q-adic sum-of-digits functions to be uniformly distributed modulo one in terms of a trigonometric product. As applications of our condition we prove some upper estimates of the extreme discrepancies of such sequences, and that the existence of distribution function g(x)= x implies the uniform distribution modulo one of the weighted q-adic sum-of-digits function sq,γ (n), n = 0, 1, 2,... We also prove the uniform distribution modulo one of related sequences h 1 sq, γ (n)+h 2 sq,γ (n +1), where h 1 and h 2 are integers such that h 1 + h 2 ≠ 0 and that the akin two-dimensional sequence sq,γ (n), sq,γ (n +1) canno...

Research paper thumbnail of On coverings of almost Dedekind domains

Czechoslovak Mathematical Journal, 1976

Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents ... more Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use.

Research paper thumbnail of On the density of certain sets in arithmetical semigroups

Czechoslovak Mathematical Journal, 1979

Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents ... more Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use.

Research paper thumbnail of Application and Misapplication of the Czechoslovak STP Cipher During WWII

Tatra Mountains Mathematical Publications, 2017

Lieutenant colonel Karol Cigáň (1921-2005), head of the cryptographic unit of the Czechoslovak Mi... more Lieutenant colonel Karol Cigáň (1921-2005), head of the cryptographic unit of the Czechoslovak Ministry of National Defence in the period 1949-1958 was after discharging from this position in Prague relocated to an insignificant and substandard command position at a district military administration in Slovakia. His cryptographic experience was of no use in his new position. To profit from his previous experience as a high qualified cryptographer he started to study the accessible literature and archive materials about the usage of the Czechoslovak cipher systems during the WWII. The result of this his activity were some manuscripts where he deciphered and analyzed some Czechoslovak military wireless telegrams. His critical analysis and his conclusions did not meet an understanding or a positive response of historians and were nor accepted for publication. He had no other chance as to send them to archives. Unfortunately only one (in two copies) and a collection of small notes surviv...

Research paper thumbnail of Remarks on systems of congruence classes

Research paper thumbnail of On theorems of Niven and Dressler

Mathematica Slovaca, 1978

Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digi... more Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use. This paper has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library http://project.dml.cz

Research paper thumbnail of Sperner Type Theorems for Generalized Divisors

sbc.org.pl

Abstract. The extensions of the well-known Sperner's result on antichains of sub sets of a g... more Abstract. The extensions of the well-known Sperner's result on antichains of sub sets of a given finite set for divisors of a positive integers are shown to hold also for sets of regular systems of divisors of elements of arithmetical semigroups.

Research paper thumbnail of On covering systems on rings

Mathematica Slovaca, 1978

Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digi... more Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use.

Research paper thumbnail of Identities involving covering systems. I

Mathematica Slovaca, 1994

Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digi... more Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use.

Research paper thumbnail of New solvability conditions for congruence ax≡b (mod n)

Tatra Mountains Mathematical Publications, 2015

K. Bibak et al. [arXiv:1503.01806v1 [math.NT],March 5 2015] proved that congruence ax ≡ b (mod n)... more K. Bibak et al. [arXiv:1503.01806v1 [math.NT],March 5 2015] proved that congruence ax ≡ b (mod n) has a solution x0 with t = gcd(x0, n) if and only if gcd thereby generalizing the result for t = 1 proved by B. Alomair et al. [J. Math. Cryptol. 4 (2010), 121-148] and O. Grošek et al. [ibid. 7 (2013), 217-224]. We show that this generalized result for arbitrary t follows from that for t = 1 proved in the later papers. Then we shall analyze this result from the point of view of a weaker condition that gcd . We prove that given integers a, b, n ≥ 1 and t ≥ 1, congruence ax ≡ b (mod n) has a solution x0 with t dividing gcd(x0, n) if and only if gcd divides gcd .

Research paper thumbnail of Prof. RNDr. Břetislav Novák, DrSc. (1938–2003) would be seventy

Research paper thumbnail of Transformations that preserve uniform distribution

Research paper thumbnail of On m times covering systems of congruences

Research paper thumbnail of Identities Involving Covering Systems II

Mathematica Slovaca, 1994

Research paper thumbnail of New necessary and sufficient conditions on (, ) in order that = (mod ) be a covering system

Electronic Notes in Discrete Mathematics, 2002

A covering system is a set of congruences x≡ ai (mod mi), i= 1,… k, such that every integer satis... more A covering system is a set of congruences x≡ ai (mod mi), i= 1,… k, such that every integer satisfies at least one of them. A new necessary and sufficient condition in order that a given set of congruences x≡ ai (mod mi) be a covering system is established. We show that (4) ...

Research paper thumbnail of Binary sequences generated by sequences n alpha , n=1,2, dots

Publicationes Mathematicae Debrecen, 2010