An arithmetical theorem for partially ordered sets

The simplest multiplicative systems in which arithmetical ideas can be defined are semigroups. For such systems irreducible (prime) elements can be introduced and conditions under which the fundamental theorem of arithmetic holds have been investigated (Clifford (3)). After identifying associates, the elements of the semigroup form a partially ordered set with respect to the ordinary division relation. This suggests the possibility of an analogous arithmetical result for abstract partially ordered sets. Although nothing corresponding to product exists in a partially ordered set, there is a notion similar to g.c.d. This is the meet operation, defined as greatest lower bound. Thus irreducible elements, namely those elements not expressible as meets of proper divisors can be introduced. The assumption of the ascending chain condition then implies that each element is representable as a reduced meet of irreducibles. The central problem of this thesis is to determine conditions on the structure of the partially ordered set in order that each element have a unique such representation.

Part I contains preliminary results and introduces the principal tools of the investigation. In the second part, basic properties of the lattice of ideals and the connection between its structure and the irreducible decompositions of elements are developed. The proofs of these results are identical with the corresponding ones for the lattice case (Dilworth (2)). The last part contains those results whose proofs are peculiar to partially ordered sets and also contains the proof of the main theorem.

FUZZY SOLID SETS FOR MATHEMATICAL MORPHOLOGY AND OPTICAL IMPLEMENTATION

Fuzzy sets in the subject space are transformed to fuzzy solid sets in an increased object space on the basis of the development of the local umbra concept. Further, a counting transform is defined for reconstructing the fuzzy sets from the fuzzy solid sets, and the dilation and erosion operators in mathematical morphology are redefined in the fuzzy solid-set space. The algebraic structures of fuzzy solid sets can lead not only to fuzzy logic but also to arithmetic operations. Thus a fuzzy solid-set image algebra of two image transforms and five set operators is defined that can formulate binary and gray-scale morphological image-processing functions consisting of dilation, erosion, intersection, union, complement, addition, subtraction, and reflection in a unified form. A cellular set-logic array architecture is suggested for executing this image algebra. The optical implementation of the architecture, based on area coding of gray-scale values, is demonstrated. (C) 1995 Optical Society of America

A generalized Hausdorff dimension for functions and sets

Let E be a compact subset of the n-dimensional unit cube, 1_n, and let C be a collection of convex bodies, all of positive n-dimensional Lebesgue measure, such that C contains bodies with arbitrarily small measure. The dimension of E with respect to the covering class C is defined to be the number

d_C(E) = sup(β:H_{β, C}(E) > 0),

where H_{β, C} is the outer measure

inf(Ʃm(C_i)^β:UC_i Ↄ E, C_i ϵ C) .

Only the one and two-dimensional cases are studied. Moreover, the covering classes considered are those consisting of intervals and rectangles, parallel to the coordinate axes, and those closed under translations. A covering class is identified with a set of points in the left-open portion, 1’_n, of 1_n, whose closure intersects 1_n - 1’_n. For n = 2, the outer measure H_{β, C} is adopted in place of the usual:

Inf(Ʃ(diam. (C_i))^β: UC_i Ↄ E, C_i ϵ C),

for the purpose of studying the influence of the shape of the covering sets on the dimension d_C(E).

If E is a closed set in 1₁, let M(E) be the class of all non-decreasing functions μ(x), supported on E with μ(x) = 0, x ≤ 0 and μ(x) = 1, x ≥ 1. Define for each μ ϵ M(E),

d_C(μ) = lim/c → inf/0 log ∆μ(c)/log c , (c ϵ C)

where ∆μ(c) = v/x (μ(x+c) – μ(x)). It is shown that

d_C(E) = sup (d_C(μ):μ ϵ M(E)).

This notion of dimension is extended to a certain class Ӻ of sub-additive functions, and the problem of studying the behavior of d_C(E) as a function of the covering class C is reduced to the study of d_C(f) where f ϵ Ӻ. Specifically, the set of points in 1₁,

(*) {d_B(F), d_C(f)): f ϵ Ӻ}

is characterized by a comparison of the relative positions of the points of B and C. A region of the form (*) is always closed and doubly-starred with respect to the points (0, 0) and (1, 1). Conversely, given any closed region in 1₂, doubly-starred with respect to (0, 0) and (1, 1), there are covering classes B and C such that (*) is exactly that region. All of the results are shown to apply to the dimension of closed sets E. Similar results can be obtained when a finite number of covering classes are considered.

In two dimensions, the notion of dimension is extended to the class M, of functions f(x, y), non-decreasing in x and y, supported on 1₂ with f(x, y) = 0 for x · y = 0 and f(1, 1) = 1, by the formula

d_C(f) = lim/s · t → inf/0 log ∆f(s, t)/log s · t , (s, t) ϵ C

where

∆f(s, t) = V/x, y (f(x+s, y+t) – f(x+s, y) – f(x, y+t) + f(x, t)).

A characterization of the equivalence d_C1(f) = d_C2(f) for all f ϵ M, is given by comparison of the gaps in the sets of products s · t and quotients s/t, (s, t) ϵ C_i (I = 1, 2).

Should EFL Teachers Present Vocabulary in Semantically Related Sets?

[EN] Teaching vocabulary in semantically related sets is common practice among EFL teachers. The present study tests the effectiveness of this method by comparing it to the alternative technique: presenting vocabulary in an unrelated way. In the study two intact classes of Spanish learners of English in high-school were presented with a set of unrelated and related words and were then asked to complete a post-test to measure the impact of both techniques on learning. The results indicate that, while both techniques successfully help the learners to acquire new words, presenting words in unrelated sets seems to be more effective.

Results on proximal and generalized weak proximal contractions including the case of iteration-dependent range sets

This paper presents some further results on proximal and asymptotic proximal contractions and on a class of generalized weak proximal contractions in metric spaces. The generalizations are stated for non-self-mappings of the forms for and , or , subject to and , such that converges uniformly to T, and the distances are iteration-dependent, where , , and are non-empty subsets of X, for , where is a metric space, provided that the set-theoretic limit of the sequences of closed sets and exist as and that the countable infinite unions of the closed sets are closed. The convergence of the sequences in the domain and the image sets of the non-self-mapping, as well as the existence and uniqueness of the best proximity points, are also investigated if the metric space is complete. Two application examples are also given, being concerned, respectively, with the solutions through pseudo-inverses of both compatible and incompatible linear algebraic systems and with the parametrical

Duration of unassisted swimming activity for spotted dolphin (Stenella attenuata) calves: implications for mother-calf separation during tuna purse-seine sets

Size-related differences in power production and swim speed duration may contribute to the observed deficit of nursing calves in relation to lactating females killed in sets by tuna purse-seiners in the eastern tropical Pacific Ocean (ETP). Power production and swim-speed duration were estimated for northeastern spotted dolphins (Stenella attenuata), the species (neonate through adult) most often captured by the fishery. Power required by neonates to swim unassisted was 3.6 times that required of an adult to swim the same speed. Estimated unassisted burst speed for neonates is only about 3 m/s compared to about 6 m/s for adults. Estimated long-term sustainable speed is about 1 m/s for neonates compared to about 2.5 m/s for adults. Weight-specific power requirements decrease as dolphin calves increase in size, but power estimates for 2-year-old spotted dolphin calves are still about 40% higher than power estimates for adults, to maintain the same speed. These estimated differences between calves and adults are conservative because the calculations do not include accommodation for reduced aerobic capacity in dolphin calves compared to adults. Discrepancies in power production are probably ameliorated under normal circumstances by calves drafting next to their mothers, and by employing burst-coast or leap-burst-coast swimming, but the relatively high speeds associated with evasion behaviors during and after tuna sets likely diminish use of these energy-saving strategies by calves.

Length-weight relationships of fishes from tributaries of the Volta River, Ghana Part 1: Analysis of pooled data sets

The parameters a and b of length-weight relationships of the form W = a L super(b) were estimated for 45 fish species sampled in the Oti, Pru and Black Volta rivers, Ghana. Also, the slope and intercepts of regressional enabling standard to total length conversions were estimated for each of these same species. The estimates of b, which ranged from 2.35 to 3.27 have a mean of 2.98, with a s.e. of 0.036. These results are complemented with a brief discussion of the need for data summaries such as presented in this article.

Differences in Dolphin Mortality Rates in Night and Day Sets for the U.S. Eastern Tropical Pacific Tuna Purse Seine Fishery

Because dolphins sometimes travel with yellowfin tuna, Thunnus albacares, in the eastern tropical Pacific (ETP), purse seiners use the dolphins to locate and capture tuna schools. During the process of setting the purse seine nets, dolphins often become entangled and drown before they can be released. Data for the U.S. purse seine fleet in the ETP during 1979-88 show that dolphin mortality rates in sets made during the night are higher than mortality rates in sets made during the day. Even with efforts to reduce nightset mortality rates through the use of high intensity floodlights, night set mortality rates remain higher. The data are also used to simulate a regulation on the fishery aimed at eliminating night sets and show that dolphin mortality rates would decrease.

Consumo de água e tarifa social em áreas de baixa renda: estudo de caso das comunidades de Santa Marta, Complexo do Borel/Casa Branca e Complexo da Mangueira.

O município do Rio de Janeiro, desde sua fundação luta com o grave problema de abastecimento de água. Com o passar do tempo todas as pequenas captações no entorno da cidade foram exauridas tanto em termos de quantidade como em qualidade, afetadas pela poluição decorrente do lançamento de efluentes in natura. Solução foi a busca por novas fontes de abastecimento em outros municípios. Atualmente 80% (oitenta por cento) do abastecimento da região metropolitana do Rio de Janeiro, RMRJ, é proveniente de uma única fonte: o rio Guandu. Isto foi possível devido à transposição da bacia do rio Paraíba do Sul para o rio Guandu ocorrida na década de 50. Esta fonte essencial de abastecimento, onde foi construída a maior Estação de Tratamento de Água do mundo - ETA Guandu - está à beira da exaustão. Somente uma pequena parcela de água bruta foi captada e tratada nesta última década. Visando minimizar o grave problema que já se apresenta e enquanto investimentos em novas alternativas não forem alcançados, o consumo com responsabilidade e sustentabilidade passa a ser a tônica da discussão. Neste contexto, as áreas de baixa renda do município com suas 801 favelas e mais de 1.500 loteamentos irregulares representando em 2010 aproximadamente 1/3 (um terço) da população total, consumindo de 10% a 15% de toda a produção de água tratada da região metropolitana deve ser permanentemente estimulada a contribuir com esta redução de consumo. Estudos apresentados nesta dissertação em três metodologias distintas apontam para um consumo acima da média nacional deste mesmo perfil de população. Os resultados obtidos indicam a necessidade real da redução de consumo observando, entretanto que este é um trabalho extremamente árduo e difícil uma vez que exige mudança de hábitos e a envolvimento de todos, desde a população até a Companhia de Saneamento local.