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1
Content available remote Jordan centralizer maps on trivial extension algebras
EN
The structure of Jordan centralizer maps is investigated on trivial extension algebras. One may obtain some conditions under which a Jordan centralizer map on a trivial extension algebra is a centralizer map. As an application, we characterize the Jordan centralizer map on a triangular algebra.
2
Content available Automatic search of rational self-equivalences
EN
Two Witt rings that are not strongly isomorphic (i.e., two Witt rings over two fields that are not Witt equivalent) have different groups of strong automorphisms. Therefore, the description of a group of strong automorphisms is different for almost every Witt ring, which requires the use various tools in proofs. It is natural idea to use computers to generate strong automorphisms of the Witt rings, which is especially effective in the case of the finitely generated Witt rings, where a complete list of strong automorphisms can be created. In this paper we present the algorithm that was used to generate strong automorphisms from the infinite group of strong automorphisms of the Witt ring of rational numbers W(Q).
EN
In this paper we introduce a new ranking algorithm, called Collaborative Judgement (CJ), that takes into account peer opinions of agents and/or humans on objects (e.g. products, exams, papers) as well as peer judgements over those opinions. The combination of these two types of information has not been studied in previous work in order to produce object rankings. Here we apply Collaborative Judgement to the use case of scientific paper assessment and we validate it over simulated data. The results show that the rankings produced by our algorithm improve current scientific paper ranking practice, which is based on averages of opinions weighted by their reviewers’ self-assessments.
EN
We give a new simplified proof for undecidability of the Bi-Infinite Post Correspondence Problem (ℤPCP). We reduce the special case of the word problem of semi-Thue systems to ℤPCP.
PL
Praca dotyczy praktycznego wykorzystania elementów teorii Galois w zastosowaniach kryptograficznych w aspekcie bezpieczeństwa narodowego. W związku z powyższym zaprezentowano historię narodzin matematyki, w tym genezę powstania algebry oraz znaczenie tego terminu. Następnie przedstawiono krótką charakterystykę rozwoju algebry w kierunku abstrakcyjnym oraz omówiono niektóre elementy klasycznej teorii Galois, które mogą być wykorzystywane w implementacjach kryptograficznych na potrzeby bezpieczeństwa i obronności kraju. Szczególną uwagę zwrócono na ciała skończone oraz ich rozszerzenia, wykorzystywane do budowy algorytmów szyfrujących, a także omówiono kilka WAT-owskich wynalazków bazujących na rozwiązaniach tego typu.
EN
The following paper focuses on the practical use of some elements of the Galois’ Theory in the field of cryptography. Therefore, the paper briefly presents the origin of the algebra and the meaning of this term, as well as short characteristic of its development in the abstract aspects and its applicability to security protection. Subsequently, crucial points of the classic Galois’ Theory which can be used in cryptographic implementation for the national defence needs, are discussed here. Special attention is paid to the finite fields and their developments which can constitute the basis of the construction of the cryptographic algorithms. Additionally, some WAT inventions based on the solution of this type are introduced.
6
Content available remote Derivable maps and generalized derivations on nest and standard algebras
EN
For an algebra A, an A-bimodule M, and m (…) M, define a relation on A by RA(m, 0) = {(a, b) (…) A x A : amb = 0}. We show that generalized derivations on unital standard algebras on Banach spaces can be characterized precisely as derivable maps on these relations. More precisely, if A is a unital standard algebra on a Banach space X then (…)L(A,B(X)) is a generalized derivation if and only if (..) is derivable on RA(M,0), for some M (…) B(X). We give an example to show this is not the case in general for nest algebras. On the other hand, for an idempotent P in a nest algebra A = algN on a Hilbert space H such that P is either left-faithful to N or right-faithful to N﬩, if δ (…)L(A, B)H)) is derivable on RA(P, 0) then δ is a generalized derivation.
7
Content available remote Soft set theory applied to general algebras
EN
The notions of a soft general algebra and a soft subalgebra are introduced and studied. The operations on them such as a restricted intersection, an extended intersection, a restricted union, a ^-intersection, a ˅-union and a cartesian product are established.
8
Content available remote On the directly and subdirectly irreducible many-sorted algebras
EN
A theorem of single-sorted universal algebra asserts that every finite algebra can be represented as a product of a finite family of finite directly irreducible algebras. In this article, we show that the many-sorted counterpart of the above theorem is also true, but under the condition of requiring, in the definition of directly reducible many-sorted algebra, that the supports of the factors should be included in the support of the many-sorted algebra. Moreover, we show that the theorem of Birkhoff, according to which every single-sorted algebra is isomorphic to a subdirect product of subdirectly irreducible algebras, is also true in the field of many-sorted algebras.
9
Content available remote Free algebras over a poset in varieties of Łukasiewicz-Moisil algebras
EN
A general construction of the free algebra over a poset in varieties finitely generated is given in [8]. In this paper, we apply this to the varieties of Łukasiewicz–Moisil algebras, giving a detailed description of the free algebra over a finite poset (X, ≤), Free n ((X, ≤)). As a consequence of this description, the cardinality of Free n ((X, ≤)) is computed for special posets.
EN
The theory of rough sets provides a widely used modern tool, and in particular, rough sets induced by quasiorders are in the focus of the current interest, because they are strongly interrelated with the applications of preference relations and intuitionistic logic. In this paper, a structural characterisation of rough sets induced by quasiorders is given. These rough sets form Nelson algebras defined on algebraic lattices. We prove that any Nelson algebra can be represented as a subalgebra of an algebra defined on rough sets induced by a suitable quasiorder. We also show that Monteiro spaces, rough sets induced by quasiorders and Nelson algebras defined on T0-spaces that are Alexandrov topologies can be considered as equivalent structures, because they determine each other up to isomorphism.
11
Content available remote On the canonical connection for smooth envelopes
EN
A notion known as smooth envelope, or superposition closure, appears naturally in several approaches to generalized smooth manifolds, which were proposed in the last decades. Such an operation is indispensable in order to perform differential calculus. A derivation of the enveloping algebra can be restricted to the original one, but it is a delicate question if the the vice–versa can be done as well. In a physical language, this would correspond to the existence of a canonical connection. In this paper, we show an example of an algebra which always possesses such a connection.
12
Content available remote Endomorphisms of implication algebras
EN
In this note we prove that if two implication algebras have isomorphic monoids of endomorphisms then they are isomorphic.
13
Content available Application of geogebra for teaching mathematics
EN
This paper shows how GeoGebra can be helpful in teaching mathematics. GeoGebra is an interactive geometry, algebra, statistics and calculus application, intended for learning and teaching mathematics and science from primary school to university level. It can be used for active and problem oriented teaching and fosters mathematical experiments and discoveries both in classroom and at home. In this work we show the sketch of using the above-mentioned software to build, solve and illustrate mathematical problems.
EN
By applying computer algebra tools (mainly, Maple and C++), given the Dynkin diagram ∆ = An, with n > 2 vertices and the Euler quadratic form q∆ : Zn → Z, we study the problem of classifying mesh root systems and the mesh geometries of roots of A (see Section 1 for details). The problem reduces to the computation of the Weyl orbits in the set Mor∆ ⊆ Mn(Z) of all matrix modifications A of q∆, i.e., the non-singular matrices A ∈ Mn(Z) such that (i) q∆ (v) = v A vtr, for all v ∈ Zn, and (ii) the Coxeter matrix CoxA := —A A-tr lies in Gl(n, Z). The Weyl group W∆ ⊆ Gl(n, Z) acts on MorA and the determinant det A ∈ Z, the order cA > 2 of CoxA (i.e. the Coxeter number), and the Coxeter polynomial coxA(t) := det(t E — CoxA) ∈ Z[t] are W∆ -invariant. The problem of determining the W∆ -orbits Orb(A) of MorA and the Coxeter polynomials coxA(t), with A e MorA, is studied in the paper and we get its solution for n < 8, and A = [aij] ∈ MorAn, with [aij] | < 1. In this case, we prove that the number of the W∆ - orbits Orb(A) and the number of the Coxeter polynomials coxA(t) equals two or three, and the following three conditions are equivalent: (i) Orb(A) = Orb(A'), (ii) coxA(t) = coxA> (t), (iii) cA det A = cA det A!. We also construct: (a) three pairwise different W∆ -orbits in Mor∆, with pairwise different Coxeter polynomials, if ∆ = A2m-i and m > 3; and (b) two pairwise different W∆ -orbits in Mor∆, with pairwise different Coxeter polynomials, if ∆ = A2m and m > 1.
PL
W artykule przedstawiono rozwój metod minimalizacji opisów funkcji w algebrze bramek prądowych jakie opracowano w przeciągu ostatnich dwudziestu lat. Technologia bramek prądowych związana jest z koncepcją bramki prądowej zaproponowanej przez prof. Andrzeja Guzińskiego w latach 90 ubiegłego wieku. Wskazane zostaną również kierunki dalszego rozwoju badań nad optymalizacją układów prądowych.
EN
In this paper we present the key points of the public address of F. Sapalski at the Cracow Scientific Society on 16 November 1817. F. Sapalski presenting the essence of the new science, descriptive geometry, refers in his speech to the practical benefits of its application in art, science, construction, architecture etc. An important element is also the assessment of the impact of descriptive geometry on the socio - economic development of the country. The author thus becomes the proponent of the idea of shaping the social being through the development of modern technical education.
PL
W artykule przedstawiono zasadnicze tezy, pierwszego publicznego, wystąpienia F. Sapalskiego w Krakowskim Towarzystwie Naukowym w dniu 16 listopada 1817 roku. F. Sapalski prezentując istotę nowej nauki, geometrii wykreślnej, odnosi się w swoim wystąpieniu do praktycznych korzyści wynikających z jej stosowania w sztuce, nauce, budownictwie, architekturze itp. Ważnym elementem jest także ocena wpływu geometrii wykreślnej na rozwój społeczno-ekonomiczny kraju. Autor przez to staje się także propagatorem idei kształtowania bytu społecznego poprzez rozwijanie nowoczesnej edukacji technicznej.
EN
Let F be a field of characteristic p >0,S=F[[X]],S∗ the unit group of S, and W a subgroup of S ∗. We characterize finite groups depending on a projective (S,W)-representation type. We also give necessary and sufficient conditions for a finite group and its Sylow p-subgroups to be of the same projective (S,W) -representation type.
18
Content available remote Relations between various continuity concepts in effect algebras
EN
The aim of the present paper is to establish relations between continuity concepts for a nonnegative extended real-valued function [...] defined on an effect algebra. Examples and counterexamples are given to illustrate various situations arising in this study.
19
Content available remote Stability of the equation of ring homomorphisms
EN
Let R be a unitary ring and (A║-║) stand for a Banach algebra with a unit. In connection with some stability results of R. Badora [1] and D.G. Bourgin [2] concerning the system of two Cauchy functional equations [formula] ( f(x + y) = f(x) + f(y) f(xy) = f(x)f(y)(*) for mappings f : R→ A, we deal with Hyers-Ulam stability problem for a single equation [formula] f(x + y) + f(xy) = f(x) + f(y) + f(x)f(y) (**) The basic question whether or not equation (**) is equivalent to the system (*) has widely been examined by J. Dhombres [3] and the present author in [4] and [5].
EN
Construction of retracts of a general algebra is described by using retracts of mono-unary algebras. Inspirational influence of the theory of Pawlak machines is mentioned.
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