Powers and Logarithms

There are applied power mappings in algebras with logarithms induced by a given linear operator D in order to study particular properties of powers of logarithms. Main results of this paper will be concerned with the case when an algebra under consideration is commutative and has a unit and the operator D satisfies the Leibniz condition, i.e. D(xy) = xDy + yDx for x, y ∈ dom D. Note that in the Number Theory there are well-known several formulae expressed by means of some combinations of powers of logarithmic and antilogarithmic mappings or powers of logarithms and antilogarithms (cf. for instance, the survey of Schinzel S[1].

Smale's Conjecture on Mean Values of Polynomials and Electrostatics

2000 Mathematics Subject Classification: Primary 30C10, 30C15, 31B35.

Interaction of Kelvin waves and nonlocality of energy transfer in superfluids

We argue that the physics of interacting Kelvin Waves (KWs) is highly nontrivial and cannot be understood on the basis of pure dimensional reasoning. A consistent theory of KW turbulence in superfluids should be based upon explicit knowledge of their interactions. To achieve this, we present a detailed calculation and comprehensive analysis of the interaction coefficients for KW turbuelence, thereby, resolving previous mistakes stemming from unaccounted contributions. As a first application of this analysis, we derive a local nonlinear (partial differential) equation. This equation is much simpler for analysis and numerical simulations of KWs than the Biot-Savart equation, and in contrast to the completely integrable local induction approximation (in which the energy exchange between KWs is absent), describes the nonlinear dynamics of KWs. Second, we show that the previously suggested Kozik-Svistunov energy spectrum for KWs, which has often been used in the analysis of experimental and numerical data in superfluid turbulence, is irrelevant, because it is based upon an erroneous assumption of the locality of the energy transfer through scales. Moreover, we demonstrate the weak nonlocality of the inverse cascade spectrum with a constant particle-number flux and find resulting logarithmic corrections to this spectrum.

On optimal completions of incomplete pairwise comparison matrices

An important variant of a key problem for multi-attribute decision making is considered. We study the extension of the pairwise comparison matrix to the case when only partial information is available: for some pairs no comparison is given. It is natural to define the inconsistency of a partially filled matrix as the inconsistency of its best, completely filled completion. We study here the uniqueness problem of the best completion for two weighting methods, the Eigen-vector Method and the Logarithmic Least Squares Method. In both settings we obtain the same simple graph theoretic characterization of the uniqueness. The optimal completion will be unique if and only if the graph associated with the partially defined matrix is connected. Some numerical experiences are discussed at the end of the paper.

A method for solving LSM problems of small size in the AHP

The Analytic Hierarchy Process (AHP) is one of the most popular methods used in Multi-Attribute Decision Making. It provides with ratio-scale measurements of the prioirities of elements on the various leveles of a hierarchy. These priorities are obtained through the pairwise comparisons of elements on one level with reference to each element on the immediate higher level. The Eigenvector Method (EM) and some distance minimizing methods such as the Least Squares Method (LSM), Logarithmic Least Squares Method (LLSM), Weighted Least Squares Method (WLSM) and Chi Squares Method (X2M) are of the tools for computing the priorities of the alternatives. This paper studies a method for generating all the solutions of the LSM problems for 3 × 3 matrices. We observe non-uniqueness and rank reversals by presenting numerical results.

Influence of parameters of drying on laser induced diffuse reflectance of banana discs

Banana discs of 1 cm thickness were immersed into different antioxidant solutions to slow down potentially disturbing discoloration during drying. Samples were randomly split into 8 groups according to the 2^p experimental design. Two antioxidant solutions with 1.66% and 4.59% ascorbic acid, two levels of drying temperature with 50°C and 80°C, two levels of drying time with 6h and 8h were used or adjusted. Laser diodes of seven wavelengths (532, 635, 650, 780, 808, 850 and 1064 nm) were selected to illuminate the surface and light penetration pattern was evaluated on the basis of radial profiles. Profiles acquired at three wavelengths (532, 635 and 650 nm) were found to respond sensitively to adjusted parameters. As a result of drying, intensity decay was observed to move closer to incident point. Significant effect (p<0.01) of temperature, drying time and their interaction was found on extracted descriptive attributes of intensity profiles: full width at half maximum (FWHM), distance of inflection point (DIP) and slope of logarithmic decay (SLD). Beyond their presence, antioxidant concentration was neutral factor without significant contribution to the model. Results were in agreement with reference spectroscopic measurements, especially with NDVI index. Promising results suggest that evaluated method might be suitable for monitoring purposes during drying of fruits.

Incomplete pairwise comparison matrices and weighting methods

A special class of preferences, given by a directed acyclic graph, is considered. They are represented by incomplete pairwise comparison matrices as only partial information is available: for some pairs no comparison is given in the graph. A weighting method satisfies the property linear order preservation if it always results in a ranking such that an alternative directly preferred to another does not have a lower rank. We study whether two procedures, the Eigenvector Method and the Logarithmic Least Squares Method meet this axiom. Both weighting methods break linear order preservation, moreover, the ranking according to the Eigenvector Method depends on the incomplete pairwise comparison representation chosen.

Contributions of humic substances to the dissolved organic carbon pool in wetlands from different climates

Wetlands are an important source of DOM. However, the quantity and quality of wetlands’ DOM from various climatic regions have not been studied comprehensively. The relationship between the concentrations of DOM (DOC), humic substances (HS) and non-humic substances (NHS) in wetland associated sloughs, streams and rivers, in cool temperate (Hokkaido, Japan), sub-tropical (Florida, USA), and tropical (Sarawak, Malaysia) regions was investigated. The DOC ranged from 1.0 to 15.6 mg C L−1 in Hokkaido, 6.0–24.4 mg C L−1 in Florida, and 18.9–75.3 mg C L−1 in Sarawak, respectively. The relationship between DOC and HS concentrations for the whole sample set was regressed to a primary function with y-intercept of zero (P < 0.005) and a slope value of 0.841. A similar correlation was observed between DOC and NHS concentrations, with a smaller slope value of 0.159. However, the correlation coefficient of the latter was much larger when the data was regressed to a logarithmic curve. These observations suggest the presence of a general tendency that the increased DOC in the river waters was mainly due to the increased supply of HS from wetland soils, whereas the rate of the increase in the NHS supply has an upper limit which may be controlled by primary productivity.

The Relationship of Heteroptera Species Richness, Abundance, and Morphology to Elevation Gradients and Land-use Regimes on Mt. Kilimanjaro

Insect biodiversity is unevenly distributed on local, regional, and global scales. Elevation is a key factor in the uneven distribution of insect diversity, serving as a proxy for a host of environmental variables. My study examines the relationship of Heteroptera (true bugs) species diversity, abundance, and morphology to elevational gradients and land-use regimes on Mt. Kilimanjaro, Tanzania, East Africa. Heteroptera specimens were collected from 60 research sites covering an elevational range of 3684m (866-4550m above sea level). Thirty of the sites were classified as natural, while the remaining 30 were classified as disturbed (e.g., agricultural use or converted to grasslands). I measured aspects of the body size of adult specimens and recorded their location of origin. I used regression models to analyze the relationships of Heteroptera species richness, abundance, and body measurements to elevation and land-use regime. Richness and abundance declined with greater elevation, controlling for land use. The declines were linear or logarithmic in form, depending on the model. Richness and abundance were greater in natural than disturbed sites, controlling for elevation. According to an interaction, richness decreased more in natural than disturbed sites with rising elevation. Body length increased as a quadratic function of elevation, adjusting for land use. Body width X length decreased as a logarithmic function of elevation, while leg length/body length decreased as a quadratic function. Leg length/body length was greater in disturbed than natural sites. Interactions indicated that body length and body width X length were greater in natural than disturbed sites as elevation rose, although the general trend was downward. Future research should examine the relative importance of land area, temperature, and resource constraints for Heteroptera diversity and morphology on Mt. Kilimanjaro.

Estimation of sea surface temperature from sediment core PS1768-8

The quantitative diatom analysis of 218 surface sediment samples recovered in the Atlantic and western Indian sector of the Southern Ocean is used to define a base of reference data for paleotemperature estimations from diatom assemblages using the Imbrie and Kipp transfer function method. The criteria which justify the exclusion of samples and species out of the raw data set in order to define a reference database are outlined and discussed. Sensitivity tests with eight data sets were achieved evaluating the effects of overall dominance of single species, different methods of species abundance ranking, and no-analog conditions (e.g., Eucampia Antarctica) on the estimated paleotemperatures. The defined transfer functions were applied on a sediment core from the northern Antarctic zone. Overall dominance of Fragilariopsis kerguelensis in the diatom assemblages resulted in a close affinity between paleotemperature curve and relative abundance pattern of this species downcore. Logarithmic conversion of counting data applied with other ranking methods in order to compensate the dominance of F. kerguelensis revealed the best statistical results. A reliable diatom transfer function for future paleotemperature estimations is presented.

(Table T2) Rock magnetic parameters of ODP Leg 180 sites

Rock magnetic measurements were performed on sediments above 20 meters below seafloor (mbsf) (general) and above 2.5 mbsf (detailed) at Sites 1109, 1110, and 1115 (Ocean Drilling Program Leg 180) in the western Woodlark Basin. Rock magnetic parameters imply the presence of magnetite as a principal magnetic mineral in the sediments. The hysteresis ratios lay in the pseudo-single domain field and generally showed the trend close to that for the mixture of single domain and multidomain magnetite The sediments in the oxidized zones at the top at Sites 1109 and 1115 provided a different trend in the logarithmic plot of the hysteresis ratios, and the oxidized samples were characterized by higher coercivity.

Mineralogical, geochemical and magnetic signatures of surface sediments from the Canadian Beaufort Shelf and Amundsen Gulf (Canadian Arctic)

Mineralogical, geochemical, magnetic, and siliciclastic grain-size signatures of 34 surface sediment samples from the Mackenzie-Beaufort Sea Slope and Amundsen Gulf were studied in order to better constrain the redox status, detrital particle provenance, and sediment dynamics in the western Canadian Arctic. Redox-sensitive elements (Mn, Fe, V, Cr, Zn) indicate that modern sedimentary deposition within the Mackenzie-Beaufort Sea Slope and Amundsen Gulf took place under oxic bottom-water conditions, with more turbulent mixing conditions and thus a well-oxygenated water column prevailing within the Amundsen Gulf. The analytical data obtained, combined with multivariate statistical (notably, principal component and fuzzy c-means clustering analyses) and spatial analyses, allowed the division of the study area into four provinces with distinct sedimentary compositions: (1) the Mackenzie Trough-Canadian Beaufort Shelf with high phyllosilicate-Fe oxide-magnetite and Al-K-Ti-Fe-Cr-V-Zn-P contents; (2) Southwestern Banks Island, characterized by high dolomite-K-feldspar and Ca-Mg-LOI contents; (3) the Central Amundsen Gulf, a transitional zone typified by intermediate phyllosilicate-magnetite-K-feldspar-dolomite and Al-K-Ti-Fe-Mn-V-Zn-Sr-Ca-Mg-LOI contents; and (4) mud volcanoes on the Canadian Beaufort Shelf distinguished by poorly sorted coarse-silt with high quartz-plagioclase-authigenic carbonate and Si-Zr contents, as well as high magnetic susceptibility. Our results also confirm that the present-day sedimentary dynamics on the Canadian Beaufort Shelf is mainly controlled by sediment supply from the Mackenzie River. Overall, these insights provide a basis for future studies using mineralogical, geochemical, and magnetic signatures of Canadian Arctic sediments in order to reconstruct past variations in sediment inputs and transport pathways related to late Quaternary climate and oceanographic changes.

Matemática e música

Esta dissertação propõe atividades a fim de estabelecer uma aproximação entre a matemática e a música, aproveitando a estreita relação existente entre esses dois assuntos. Aplica-se a história da matemática como ferramenta para o entendimento de conceitos matemáticos e musicais, bem como a evolução da música. Além disso, utiliza-se a resolução de problemas para estabelecer a relação entre a matemática e a música através das funções exponencial e logarítmica e a sequência numérica chamada progressão geométrica (P.G.) Ações efetivas, como questionários relativos ao assunto a ser estudado, debates com o intuito de perceber o nível de conhecimento dos alunos sobre a música e uma atividade prática, a construção de um xilofone de garrafas, envolvendo os conhecimentos adquiridos, auxiliaram para a obtenção do resultado final deste trabalho. Apresenta-se também a análise das atividades que embasaram a proposta desse trabalho quando aplicadas em uma turma de segundo ano do Colégio Sinodal Alfredo Simon, localizada na cidade de Pelotas no estado do Rio Grande do Sul.

THE PHYSICS OF IDEAS: INFERRING THE MECHANICS OF OPINION FORMATION FROM MACROSCOPIC STATISTICAL PATTERNS

In a microscopic setting, humans behave in rich and unexpected ways. In a macroscopic setting, however, distinctive patterns of group behavior emerge, leading statistical physicists to search for an underlying mechanism. The aim of this dissertation is to analyze the macroscopic patterns of competing ideas in order to discern the mechanics of how group opinions form at the microscopic level. First, we explore the competition of answers in online Q&A (question and answer) boards. We find that a simple individual-level model can capture important features of user behavior, especially as the number of answers to a question grows. Our model further suggests that the wisdom of crowds may be constrained by information overload, in which users are unable to thoroughly evaluate each answer and therefore tend to use heuristics to pick what they believe is the best answer. Next, we explore models of opinion spread among voters to explain observed universal statistical patterns such as rescaled vote distributions and logarithmic vote correlations. We introduce a simple model that can explain both properties, as well as why it takes so long for large groups to reach consensus. An important feature of the model that facilitates agreement with data is that individuals become more stubborn (unwilling to change their opinion) over time. Finally, we explore potential underlying mechanisms for opinion formation in juries, by comparing data to various types of models. We find that different null hypotheses in which jurors do not interact when reaching a decision are in strong disagreement with data compared to a simple interaction model. These findings provide conceptual and mechanistic support for previous work that has found mutual influence can play a large role in group decisions. In addition, by matching our models to data, we are able to infer the time scales over which individuals change their opinions for different jury contexts. We find that these values increase as a function of the trial time, suggesting that jurors and judicial panels exhibit a kind of stubbornness similar to what we include in our model of voting behavior.

Secagem da microalga Spirulina em camada delgada utilizando secador com bomba de calor

A Spirulina apresenta propriedades antioxidantes o que favorece seu uso como alimento funcional, fato que tem motivado a sua comercialização para a formulação de alimentos diversos e com finalidades terapêuticas. A secagem ganha importância durante produção de Spirulina, uma vez que a umidade necessária, para garantir que não ocorra degradação da biomassa desidratada durante o armazenamento, é alcançada através do conhecimento dos parâmetros que caracterizam a operação. Neste estudo foi utilizada a secagem com bomba de calor, um método alternativo, pois viabiliza a operação com temperaturas inferiores as tradicionalmente utilizadas, além de seu funcionamento ser independente das condições meteorológicas do ambiente. O trabalho experimental da secagem de Spirulina sp. foi iniciado com um estudo comparativo entre a secagem com bomba de calor (SBC) e a secagem tradicional (ST). O efeito dos diferentes métodos utilizados sob a amostra foi comparado em relação à cinética da operação e as características da microalga desidratada (cor, ficocianina, compostos fenólicos totais e atividade antioxidante total). As temperaturas do ar foram de 50 e 60ºC e a umidade absoluta da SBC foi dez vezes inferior a utilizada durante a ST. Os parâmetros que caracterizam a secagem foram influenciados pela temperatura do ar, bem como, pela baixa umidade absoluta na SBC. Os valores do tempo total da SBC foram 40% inferiores aos encontrados para a secagem ST, em ambas as temperaturas do ar. A maior preservação das características da Spirulina foi obtida na SBC e temperatura do ar de 50°C, e nesta condição os valores foram 14% (ficocianina), 60% (compostos fenólicos) e 10% (atividade antioxidante) superiores aos encontrados na mesma condição para a ST. Isto evidencia que o método de secagem é determinante na qualidade do produto desidratado. Posteriormente, foi realizado o estudo da cinética da SBC, bem como a otimização da operação de secagem e a reidratação das amostras desidratadas nas diferentes condições de secagem. O estudo foi realizado através de um planejamento fatorial 32, tendo como fatores de estudo a temperatura do ar (30, 40 e 50ºC) e a espessura da bandeja (1, 3 e 5 mm). As respostas utilizadas foram ficocianina, compostos fenólicos, atividade antioxidante total e cor da microalga desidratadas. Também foram realizadas a microscopia eletrônica de varredura (MEV) e as curvas termogravimétricas (DSC) das amostras desidratadas. A secagem apresentou um curto período de taxa constante, delimitado pela umidade crítica, sendo que, seus valores foram influenciados apenas pela temperatura do ar de secagem. O modelo Logarítmico forneceu elevados valores de R2ajust e os menores valores de soma dos erros quadráticos (SSE) e de critério informativo de Akaike (AIC). Os valores das energias de ativação para as espessuras de 1, 3 e 5 mm, foram na faixa de 20-23 kJ mol-1. A condição de operação mais adequada, para a secagem de Spirulina sp. com bomba de calor, foi obtida na temperatura do ar de 50°C e espessura da bandeja de 5 mm, com valores de ficocianina, compostos fenólicos, atividade antioxidante total e diferença de cor de 19,60 mg g-1, 1508 µgEAG g-1, 52,6% e 5,71, respectivamente. Os termogramas (DSC) evidenciaram que em 50 ºC e espessura de 5 mm, o produto apresentou maior estabilidade térmica. As amostras de Spirulina sp. desidratadas apresentaram estrutura morfológica (MEV), aparentemente, rígida e heterogênea, e os seus percentuais de reidratação corresponderam a 85-91% da umidade da microalga in natura.