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1
Content available remote The largest higher commutator sequence
EN
Given the congruence lattice L of a finite algebra A that generates a congruence permutable variety, we look for those sequences of operations on L that have the properties of higher commutator operations of expansions of A. If we introduce the order of such sequences in the natural way the question is whether exists or not the largest one. The answer is positive. We provide a description of the largest element and as a consequence we obtain that the sequences form a complete lattice.
PL
W pracy zaproponowano konstelację trójwymiarową dla techniki 3D-OFDM (ang. Three Dimensional Orthogonal Frequency Division Multiplexing) wykorzystującej dwuwymiarową odwrotnaą transformacją Fouriera – 2D-IDFT (ang. Two Dimensional Inverse Discrete Fourier Transform). Konstelację znaleziono przeszukując bezpośrednio przestrzeń trójwymiarową. Zysk względem konstelacji znanych z literatury wynosi ok. 0.45 [dB] przy BER = 10−5.
EN
This paper proposes an alternative threedimensional signal constellation for 3D-OFDM based on 2D-IDFT. Unlike similar solutions that can be found in literature our three dimensional modulation scheme is designed directly in a three dimensional space. We achieve approximately 0.45 [dB] gain at the BER = 10−5 when compared to other 3D constellations.
PL
W pracy zaproponowano algorytm demodulacji dla konstelacji wielowymiarowych odmienny od powszechnie używanego dekodera sferycznego. Wykorzystując względną złożoność´ obliczeniową pokazujemy, że zaproponowane rozwiązanie może być nie więcej niż dziesięciokrotnie bardziej wydajne od rozwiązania znanego z literatury. Rozważania teoretyczne zweryfikowane zostały za pomocą symulacji komputerowej.
EN
This paper proposes an algorithm for symbol detection in multidimensional constellations as an alternative to widely used Sphere Decoder. We use a relative complexity of algorithms in order to measure the performance gain of the one solution over the other. Measurements and comparisons of relative execution time of computer simulations verify the theory.
EN
Signed partitions are used in order to describe a new discrete dynamical model whose configurations have fixed sum and whose evolution rules act in balancing from left and right on the configurations of the system. The resulting model can be considered as an extension to the case of signed partitions of the discrete dynamical system introduced by Brylawski in his classical paper concerning the dominance order of integer partitions. We provide a possible interpretation of our model as a simplified description of p − n junction between two semiconductors. We also show as our model can be embedded in a specific Brylawski dynamical system by means of the introduction of a new evolution rule.
5
Content available remote Distributivity versus associativity in the homology theory of algebraic structures
EN
While homology theory of associative structures, such as groups and rings, has been extensively studied in the past beginning with the work of Hopf, Eilenberg, and Hochschild, homology of non-associative distributive structures, such as quandles, were neglected until recently. Distributive structures have been studied for a long time. In 1880, C.S. Peirce emphasized the importance of (right) self-distributivity in algebraic structures. However, homology for these universal algebras was introduced only sixteen years ago by Fenn, Rourke, and Sanderson. We develop this theory in the historical context and propose a general framework to study homology of distributive structures. We illustrate the theory by computing some examples of 1-term and 2-term homology, and then discussing 4-term homology for Boolean algebras. We outline potential relations to Khovanov homology, via the Yang-Baxter operator.
6
Content available remote Equational bases for k-normal identities
EN
The depth of a term may be used as a measurement of complexity of identities. For any natural number [...] have depth at least k. For any variety V, the k-normalization of V is the variety Nk(V) defined by all k-normal identities of V. We describe a process to produce from a basis for V a basis for Nk(V), for any variety V which has an idempotent term; when the type of V is finite and V is finitely based, this results in a finite basis for Nk(V) as well. This process encompasses several known examples, for varieties of bands and lattices, and allows us to give a new basis for the normalization of the variety PL of pseudo-complemented lattices.
7
Content available remote Notes on BCK-algebraa with condition (s)
EN
Some further properties related to BCK-algebras with the condition (S) are obtained. The main results are as follows: (i) If a commutative BCK-algebra X is a lattice with respect to the BCK-ordering <, then X need not be with the condition (S); (ii) A positive implicative BCK-algebra X with the condition (S) may not be a lattice with respect to <, moreover, if (X; <) is a lattice, it must be a distributive lattice; (iii) Each involutory BCK-algebra is with the condition (S).
8
Content available remote Archimedean and block-finite lattice efect algebras
EN
We show that every complete effect algebra is Archimedean. Moreover, a block-finite lattice effect algebra has the MacNeille completion which is a complete effect algebra iff it is Archimedean. We apply our results to orthomodular lattices.
9
Content available remote A few notes on subalgebra lattices, part 1
EN
First, we apply results proved in [Pió1] and some results of graph theory to formulate and prove a necessary condition for partial (and thus also total) unary algebras to have isomorphic (strong) subalgebra lattices. Although this condition is not sufficient for arbitrary partial unary algebras, we can form, having this fact, a lot of new partial unary algebras with the same subalgebra lattices. Moreover, we use this result to characterize arbitrary two partial (thus in particular also total) monounary algebras with isomorphic (strong) subalgebra lattices. Having this result we can also describe all pairs (A, L), where A is a partial monounary algebra and L a lattice, such that the subalgebra lattice of A is isomorphic to L. In the next part [Pió2] we apply the results of this paper to characterize connections between weak and strong subalgebra lattices of partial (thus also total) monounary algebras.
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