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1

Geometric properties of the lattice of polynomials with integer coefficients

Lipnicki Artur, Śmietański Marek J.

Opuscula Mathematica

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2024

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Vol. 44, no. 4

565--585

EN

This paper is related to the classic but still being examined issue of approximation of functions by polynomials with integer coefficients. Let r, n be positive integers with n ≥ 6r. Let Pn ∩Mr be the space of polynomials of degree at most n on [0, 1] with integer coefficients such that P(k)(0)/k! and P(k)(1)/k! are integers for k = 0, . . . , r − 1 and let PZn ∩Mr be the additive group of polynomials with integer coefficients. We explore the problem of estimating the minimal distance of elements of PZn ∩Mr from Pn ∩Mr in L2(0, 1). We give rather precise quantitative estimations for successive minima of PZn in certain specific cases. At the end, we study properties of the shortest polynomials in some hyperplane in Pn ∩Mr.

2

Point process of clusters for a stationary Gaussian random field on a lattice

Lu Yingyin, Guo Jinhui

Probability and Mathematical Statistics

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2023

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Vol. 43, Fasc. 2

247--262

EN

It is well established that the normalized exceedances resulting from a standard stationary Gaussian triangular array at high levels follow a Poisson process under the Berman condition. To model frequent cluster phenomena, we consider the asymptotic distribution of the point process of clusters for a Gaussian random field on a lattice. Our analysis demonstrates that the point process of clusters also converges to a Poisson process in distribution, provided that the correlations of the Gaussian random field meet certain conditions. Additionally, we provide a numerical example to illustrate our theoretical results.

3

Algebras of Definable Sets vs. Concept Lattices

Wasilewski Piotr

Fundamenta Informaticae

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2019

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Vol. 167, nr 3

235--256

EN

The paper is aimed at comparing Rough Set Theory (RST) and Formal Concept Analysis (FCA) with respect to algebraic structures of concepts appearing in both theories, namely algebras of definable sets and concept lattices. The paper presents also basic ideas and concepts of RST and FCA together with some set theoretical concepts connected with set spaces which can serve as a convenient platform for a comparison of RST and FCA. In the last section there are shown necessary and sufficient conditions for the fact, that families of definable sets and concept extents determined by the same formal contexts are equal. This in finite cases is equivalent to an isomorphism of respective structures and generally reflects a very specific situation when both theories give the same conceptual hierarchies.

4

Uniform approximation by polynomials with integer coefficients

Lipnicki A.

Opuscula Mathematica

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2016

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Vol. 36, no. 4

489--498

EN

Let r, n be positive integers with n ≥ 6r. Let P be a polynomial of degree at most n on [0,1] with real coefficients, such that [formula] are integers for k = 0,…, r — 1. It is proved that there is a polynomial Q of degree at most n with integer coefficients such that [formula] for x ∈ [0,1], where C1, C2 are some numerical constants. The result is the best possible up to the constants.

5

An Energy Efficient QAM Modulation with Multidimensional Signal Constellation

Markiewicz T. G.

International Journal of Electronics and Telecommunications

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2016

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Vol. 62, No. 2

159--165

EN

Packing constellations points in higher dimensions, the concept of multidimensional modulation exploits the idea drawn from geometry for searching dense sphere packings in a given dimension, utilising it to minimise the average energy of the underlying constellations. The following work analyses the impact of spherical shaping of the constellations bound instead of the traditional, hyper-cubical bound. Balanced constellation schemes are obtained with the N-dimensional simplex merging algorithm. The performance of constellations of dimensions 2, 4 and 6 is compared to the performance of QAM modulations of equivalent throughputs in the sense of bits transmitted per complex (twodimensional) symbols. The considered constellations give an approximately 0.7 dB to 1 dB gain in terms of BER over a standard QAM modulation.

6

Identity-based Signatures from Lattices : Simpler, Faster, Shorter

Tian M., Huang L.

Fundamenta Informaticae

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2016

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Vol. 145, nr 2

171--187

EN

Identity-based signature is an important technique for light-weight authentication. Recently, many efforts have been made to construct identity-based signatures over lattice assumptions since they would remain secure in future quantum age. In this paper we present a new identitybased signature scheme from lattice problems. This scheme is more efficient than other lattice-based identity-based signature schemes in terms of both computation and communication complexities. We prove its security in the random oracle model under short integer solution assumption that is as hard as approximating several worst-case lattice problems. We also extend the scheme to an identity-based message recovery signature scheme that has better performance.

7

Lattices of annihilators in commutative algebras over fields

Jastrzębska M., Krempa J.

Demonstratio Mathematica

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2015

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Vol. 48, nr 4

546--553

EN

Let K be any field and L be any lattice. In this note we show that L is a sublattice of annihilators in an associative and commutative K-algebra. If L is finite, then our algebra will be finite dimensional over K.

8

Involutory BE-algebras

Borzooei R. A., Borumand Saeid A., Ameri R., Rezaei A.

Journal of Mathematics and Applications

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2014

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Vol. 37

13--26

EN

This paper is devoted to the study of some structural properties of bounded and involutory BE-algebras and investigate the relationship between them. We construct a commutative monoid by definition of proper operation in an involutory BE-algebra. Some rules of calculus for BE-algebras with a semi-lattice structure are provided. Many results related to the natural order of a BE-algebras were found. Finally, we show that an involutory bounded BE-algebra X is semi-simple.

9

Ocenianie projektów technologii informacyjnej

Pomykała Janusz A.

Studia Informatica

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2013

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Vol. 34, nr 4

63--76

PL

Artykuł prezentuje możliwości zastosowania sieci telemetrycznych oraz opisuje problematykę bezprzewodowej transmisji danych pomiędzy mikrokomputerami jednoukładowymi o niewielkiej mocy obliczeniowej. Przedstawiony został cykl pracy sensora i modułu radiowego. Zaprezentowano porównanie układu teoretycznego z rzeczywistym w badaniu czasu życia układu przy zmiennych parametrach wejściowych dla protokołu Aloha.

EN

The paper presents possible applications od telemetry networks and aims to describe the problem of wireless data transmission between single-chip microcomputers with low processing power. The work describes the duty cycles of sensors and wireless module. It also delivers a comparison of the theoretical and real-life approaches in a study of their lifespan with variable input parameters for the Aloha protocol.

10

On duality between order and algebraic structures in Boolean systems

Dadej A., Halik K.

Journal of Mathematics and Applications

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2013

|

Vol. 36

35--62

EN

We present an extension of the known one-to-one correspondence between Boolean algebras and Boolean rings with unit being two types of Boolean systems endowed with order and algebraic structures, respectively. Two equivalent generalizations of Boolean algebras are discussed. We show that there is a one-to-one correspondence between any of the two mentioned generalized Boolean algebras and Boolean rings without unit.

11

On Characterizing Hierarchies of Granulation Structures via Distances

Yang X, Qian Y., Yang J.

Fundamenta Informaticae

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2013

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Vol. 123, nr 3

365--380

EN

Hierarchy plays a crucial role in the development of the granular computing. In this paper, three different hierarchies are considered for judging whether a granulation structure is finer or coarser than another one. The first hierarchy is based on the set containment of information granulations, the second hierarchy is based on the cardinal numbers of information granulations while the third hierarchy is based on the sum of cardinal numbers of information granulations. Through introducing set distance and knowledge distance, we investigate the algebraic lattices, in which the derived partial orders are corresponding to the three different hierarchies, respectively. From the viewpoint of distance, these results look forward to provide a more comprehensible perspective for the study of hierarchies on granulation structures.

12

Insertion and extension theorems for completely distributive lattice valued functions

García J. G., Kubiak T.

Commentationes Mathematicae

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2013

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Vol. 53, No. 2

313--319

EN

Insertion and extension theorems are presented for lattice-valued functions where the lattice is an appropriately based completely distributive lattice endowed with its interval topology.

13

Design of non-uniform truss structures for improved part properties

Boruciński M., Królikowski M.

Advances in Manufacturing Science and Technology

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2013

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Vol. 37, nr 4

77--85

EN

This paper discusses generation of lattice structures with non-uniform shape and their optimization in a computer aided design (CAD) model of an exemplary part. Feasibility of encapsulating the whole process within one off-the-shelf CAD suite is analyzed. A method of design of lattice structures of variable density is proposed. A series of finite element method (FEM) analyses leading to achieving improved structures is presented.

PL

W artykule przedstawiono tworzenie nieregularnych struktur kratownicowych. Prowadzono proces optymalizacji z użyciem modelu CAD na przykładzie przyjętego elementu. Ustalono możliwość realizacji zadania za pomocą jednego niezmodyfikowanego narzędzia projektowego. Opracowano metodę tworzenia nieregularnych struktur kratownicowych. Wykonano analizę z zastosowaniem metody elementów skończonych, umożliwiających opracowanie modeli stanowiących podstawy do poprawy właściwości elementów struktury kratownicy.

14

A Strategy for Automatic Elimination of Mechanical Instability in Structural Analysis of Spatial Truss Tower Model

Pelc J.

Technical Sciences / University of Warmia and Mazury in Olsztyn

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2012

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nr 15(2)

319-330

EN

A method for structural analysis of lattice transmission towers considered to be spatial trusses is proposed. It consists in automatic elimination of mechanical instability from the structure. The instability result from the existence of the so-called out-of-plane nodes in a structure model. The method is based on blocking the possible displacements of out-of-plane nodes and the information concerning the need for blocking the specific displacement is obtained from analysis of the system stiffness matrix. Through application of the spatial rod finite elements lattice transmission towers may be analysed with the use of spatial truss models. The short characteristic of the in-house computer program applied for identification of truss members not satisfying the load capacity criterion, comparison of the results obtained by use of the method and ones obtained by applying the plane truss model and the frame model with the reduced member flexural stiffnesses are also presented.

15

Kształtowanie wieży telekomunikacyjnej o nietypowym kształcie

Lenar J., Ślęczka L.

Zeszyty Naukowe Politechniki Rzeszowskiej. Budownictwo i Inżynieria Środowiska

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2011

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z. 58, nr 1

227-236

EN

First part of this paper presents requirements for designing of steel lattice telecommunication towers and presents typical shapes of such towers used in Poland. Second part is devoted to forming new aesthetic type of tower, which can become an architectural fit into the landscape. Various draft proposals were made, and after that two towers were analyzed in details.

16

The uncountable cofinality of the automorphism group of the countable universal distributive lattice

Droste M., Truss J. K.

Demonstratio Mathematica

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2011

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Vol. 44, nr 3

473-479

EN

We show that the automorphism group of the countable universal distributive lattice has strong uncountable cofinality, and we adapt the method to deduce the strong uncountable cofinality of the automorphism group of the countable universal generalized boolean algebra.

17

Some characterizations of pseudo-BL-chains

Walendziak A., Wojciechowska-Rysiawa M.

Commentationes Mathematicae

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2011

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Vol. 51, [Z] 2

121-124

EN

Pseudo-BL-chains are linearly ordered pseudo-BL-algebras. Characterizations of them in terms of concepts of lattice theory are given.

18

Quantifiers on lattices with an antitone involution

Chajda I., Langer H.

Demonstratio Mathematica

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2009

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Vol. 42, nr 2

241-246

EN

Quantifiers on lattices with an antitone involution are considered and it is proved that the poset of existential quantifiers is antiisomorphic to the poset of relatively complete sublattices.

19

On a complete lattice of retracts of a free monoid generated by three elements

Foryś W.

Opuscula Mathematica

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2008

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Vol. 28, no. 2

123-127

EN

We prove that the family of retracts of a free monoid generated by three elements, partially ordered with respect to the inclusion, is a complete lattice.

20

Fuzzification of Rational and Recognizable Sets

Konstantinidis S., Nicolae S., Yu S

Fundamenta Informaticae

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2007

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Vol. 76, nr 4

413-447

EN

In this paper we present a different framework for the study of fuzzy finite machines and their fuzzy languages. Unlike the previous work on fuzzy languages, characterized by fuzzification at the level of their acceptors/generators, here we follow a top-down approach by starting our fuzzification with more abstract entities: monoids and particular families in monoids. Moreover, we replace the unit interval (in fact, a finite subset of the unit interval) as support for fuzzy values with the more general structure of a lattice. We have found that completely distributive complete lattices allow the fuzzification at the highest level, that of recognizable and rational sets. Quite surprisingly, the fuzzification process has not followed thoroughly the classical (crisp) theory. Unlike the case of rational sets, the fuzzification of recognizable sets has revealed a few remarkable exceptions from the crisp theory: for example the difficulty of proving closure properties with respect to complement, meet and inverse morphisms. Nevertheless, we succeeded to prove the McKnight and Kleene theorems for fuzzy sets by making the link between fuzzy rational/recognizable sets on the one hand and fuzzy regular languages and FT-NFA languages (languages defined by NFA with fuzzy transitions) on the other. Finally, we have drawn the attention to fuzzy rational transductions, which have not been studied extensively and which bring in a strong note of applicability.