Given i, j, k ∈ {1,2,3,4}, the notion of -length neutrosophic subalgebras in BCK=BCI-algebras is introduced, and their properties are investigated. Characterizations of length neutrosophic subalgebras are discussed by using level sets of interval neutrosophic sets. Conditions for level sets of interval neutrosophic sets to be subalgebras are provided.

This study examines Babylonian records of the new moon interval NA (sunset to moonset on the day of first lunar visibility) and the connection of this interval to the length of the moon. I show that the NA intervals in the Normal Star Almanacs were computed using the goal-year method and were then used in turn to predict the lengths of each month of the year. I further argue that these predicted month lengths, adjusted occasionally on the (...) basis of observation in cases where the moon’s visibility was considered marginal, formed the basis of the Late Babylonian calendar. (shrink)

Let κ be a regular cardinal, and let B be a subalgebra of an interval algebra of size κ. The existence of a chain or an antichain of size κ in B is due to M. Rubin (see [7]). We show that if the density of B is countable, then the same conclusion holds without this assumption on κ. Next we also show that this is the best possible result by showing that it is consistent with 2 ℵ 0 (...) = ℵ ω 1 that there is a boolean algebra B of size ℵ ω 1 such that length(B) = ℵ ω 1 is not attained and the incomparability of B is less than ℵ ω 1 . Notice that B is a subalgebra of an interval algebra. For more on chains and antichains in boolean algebras see, e.g. [1] and [2]. (shrink)

Chords are not pure sets of tones or notes. They are mainly characterized by their matrices. A chord matrix is the pattern of all the lengths of intervals given without further context. Chords are well-structured invariants. They show their inner logical form. This opens up the possibility to develop a molecular logic of chords. Chords are our primitive, but, nevertheless, already interrelated expressions. The logical space of internal harmony is our well-known chromatic scale represented by an infinite line of integers. (...) Internal harmony is nothing more than the pure interrelatedness of two or more chords. We consider three cases: chords inferentially related to subchords, pairs of chords in the space of major–minor tonality and arbitrary chords as arguments of unary chord operators in relation to their outputs. One interesting result is that chord negation transforms any pure major chord into its pure minor chord and vice versa. Another one is the fact that the negation of chords with symmetric matrices does not change anything. A molecular logic of chords is mainly characterized by combining general rules for chord operators with the inner logical form of their arguments. (shrink)

This dissertation offers new proposals, based on a philosophical appraisal of scientific findings, to address old philosophical problems regarding our immediate acquaintance with time. It focuses on two topics: our capacity to determine the length of intervals and our acquaintance with the present moment. A review of the relevant scientific findings concerning these topics grounds the main contributions of this dissertation. Thus, this study introduces to the philosophical literature an empirically adequate way to talk about how the mind represents (...) time at the most fundamental level. In order to account for our immediate acquaintance with the duration of intervals, a theoretical framework for classifying clocks is used to explain why the representational outputs of the circadian clock and the stopwatch have metric structure. A philosophical analysis of these outputs is proposed to classify them and explain their properties. With respect to our immediate acquaintance with the present, a novel two-phase model of the present is proposed. This model shows that there are two forms of acquaintance with the present, which has consequences for contemporary debates in the philosophy of time. (shrink)

Indeed, this is the first serious book-length study of the subject by a philosopher of science.

We present an elementary system of axioms for the geometry of Minkowski spacetime. It strikes a balance between a simple and streamlined set of axioms and the attempt to give a direct formalization in first-order logic of the standard account of Minkowski spacetime in [Maudlin 2012] and [Malament, unpublished]. It is intended for future use in the formalization of physical theories in Minkowski spacetime. The choice of primitives is in the spirit of [Tarski 1959]: a predicate of betwenness and a (...) four place predicate to compare the square of the relativistic intervals. Minkowski spacetime is described as a four dimensional ‘vector space’ that can be decomposed everywhere into a spacelike hyperplane - which obeys the Euclidean axioms in [Tarski and Givant, 1999] - and an orthogonal timelike line. The length of other ‘vectors’ are calculated according to Pythagora’s theorem. We conclude with a Representation Theorem relating models of our system that satisfy second order continuity to the mathematical structure called ‘Minkowski spacetime’ in physics textbooks. (shrink)

We present an elementary system of axioms for the geometry of Minkowski spacetime. It strikes a balance between a simple and streamlined set of axioms and the attempt to give a direct formalization in first-order logic of the standard account of Minkowski spacetime in Maudlin and Malament. It is intended for future use in the formalization of physical theories in Minkowski spacetime. The choice of primitives is in the spirit of Tarski : a predicate of betwenness and a four place (...) predicate to compare the square of the relativistic intervals. Minkowski spacetime is described as a four dimensional ‘vector space’ that can be decomposed everywhere into a spacelike hyperplane—which obeys the Euclidean axioms in Tarski and Givant, 175–214 1999)—and an orthogonal timelike line. The length of other ‘vectors’ are calculated according to Pythagoras’ theorem. We conclude with a Representation Theorem relating models \ of our system \ that satisfy second order continuity to the mathematical structure \, called ‘Minkowski spacetime’ in physics textbooks. (shrink)

Parameters of a Bertalanffy type of temperature dependent growth model are fitted using data from a population of stone loach ( Barbatula barbatula ). Over two periods respectively in 1990 and 2010 length data of this population has been collected at a lowland stream in the central part of the Netherlands. The estimation of the maximum length of a fully grown individual is given special attention because it is in fact found as the result of an extrapolation over (...) a large interval of the entire lifetime. It is concluded that this parameter should not at forehand be set at one fixed value for the population at that location due to varying conditions over the years. (shrink)

Despite numerous studies on the treatment of developmental language disorder, the intervention effect has long been debated. Systematic reviews of the effect of language therapy alone are rare. This evidence-based study investigated the effect of language therapy alone for different expressive and receptive language levels in children with DLD. Publications in databases including PubMed, the Cochrane Library, the Wanfang Database and the China National Knowledge Infrastructure were searched. Randomized controlled trials were selected. The methodological quality of the included trials was (...) assessed using the modified Jadad method. RevMan 5.3 software was used for the data analysis. Fifteen trials were included in this study. Compared with the control group, the intervention group showed significant differences in overall expressive language development [standard mean differences, 0.46; 95% confidence interval, 0.12–0.80], mean length of utterances in a language sample, number of utterances in a language sample, parent reports of expressive phrase complexity, overall expressive vocabulary development and different words used in a language sample. However, language therapy did not show satisfactory long-term effects on DLD. Although language therapy is helpful in improving the performance of children with DLD, its long-term effect is unsatisfactory. (shrink)

The variance effective population size (NeV\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N_{eV}$$\end{document}) is frequently used to quantify the expected rate at which a population’s allele frequencies change over time. The purpose of this paper is to find expressions for the global NeV\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N_{eV}$$\end{document} of a spatially structured population that are of interest for conservation of species. Since NeV\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N_{eV}$$\end{document} depends on allele (...) frequency change, we start by dividing the cause of allele frequency change into genetic drift within subpopulations (I) and a second component mainly due to migration between subpopulations (II). We investigate in detail how these two components depend on the way in which subpopulations are weighted as well as their dependence on parameters of the model such a migration rates, and local effective and census sizes. It is shown that under certain conditions the impact of II is eliminated, and NeV\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N_{eV}$$\end{document} of the metapopulation is maximized, when subpopulations are weighted proportionally to their long term reproductive contributions. This maximal NeV\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N_{eV}$$\end{document} is the sought for global effective size, since it approximates the gene diversity effective size NeGD\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N_{eGD}$$\end{document}, a quantifier of the rate of loss of genetic diversity that is relevant for conservation of species and populations. We also propose two novel versions of NeV\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N_{eV}$$\end{document}, one of which (the backward version of NeV\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N_{eV}$$\end{document}) is most stable, exists for most populations, and is closer to NeGD\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N_{eGD}$$\end{document} than the classical notion of NeV\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N_{eV}$$\end{document}. Expressions for the optimal length of the time interval for measuring genetic change are developed, that make it possible to estimate any version of NeV\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N_{eV}$$\end{document} with maximal accuracy. (shrink)

You can search this site: Note that this analysis of a beginning of time concerns intervals ’of the same length' ; if this qualifying phrase is not added, then the analysis would be invalid for a dense time. If time is dense and began, then for each interval of time there is another interval of a shorter length that is a part of that interval and which completely elapses before the interval of which it (...) is a part completely elapses. Before the first hour completely elapses, the first minute does so, and before the first minute, the first second, and so on ad infinitum. This entails that there is no 'first moment' of time in the sense of an interval that precedes every other interval, but there is a 'first moment' in the sense that there is a first interval of each length of time : there is a first hour, a first minute, etc. (shrink)

In this series of papers I attempt to provide an answer to the question how the Babylonian scholars arrived at their mathematical theory of planetary motion. Papers I and II were devoted to system A theory of the outer planets and of the planet Venus. In this third and last paper I will study system A theory of the planet Mercury. Our knowledge of the Babylonian theory of Mercury is at present based on twelveEphemeridesand sevenProcedure Texts. Three computational systems of (...) Mercury are known, all of system A. System A1is represented by nineEphemeridescovering the years 190 BC to 100 BC and system A2by twoEphemeridescovering the years 310 to 290 BC. System A3is known from aProcedure Textand from Text M, anEphemerisof the last evening visibility of Mercury for the years 424 to 403 BC. From an analysis of the Babylonian observations of Mercury preserved in theAstronomical DiariesandPlanetary Textswe find: (1) that dates on which Mercury reaches its stationary points are not recorded, (2) that Normal Star observations on or near dates of first and last appearance of Mercury are rare (about once every twenty observations), and (3) that about one out of every seven pairs of first and last appearances is recorded as “omitted” when Mercury remains invisible due to a combination of the low inclination of its orbit to the horizon and the attenuation by atmospheric extinction. To be able to study the way in which the Babylonian scholars constructed their system A models of Mercury from the available observational material I have created a database of synthetic observations by computing the dates and zodiacal longitudes of all first and last appearances and of all stationary points of Mercury in Babylon between 450 and 50 BC. Of the data required for the construction of an ephemeris synodic time intervals Δt can be directly derived from observed dates but zodiacal longitudes and synodic arcs Δλ must be determined in some other way. Because for Mercury positions with respect to Normal Stars can only rarely be determined at its first or last appearance I propose that the Babylonian scholars used the relation Δλ = Δt −3;39,40, which follows from the period relations, to compute synodic arcs of Mercury from the observed synodic time intervals. An additional difficulty in the construction of System A step functions is that most amplitudes are larger than the associated zone lengths so that in the computation of the longitudes of the synodic phases of Mercury quite often two zone boundaries are crossed. This complication makes it difficult to understand how the Babylonian scholars managed to construct System A models for Mercury that fitted the observations so well because it requires an excessive amount of computational effort to find the best possible step function in a complicated trial and error fitting process with four or five free parameters. To circumvent this difficulty I propose that the Babylonian scholars used an alternative more direct method to fit System A-type models to the observational data of Mercury. This alternative method is based on the fact that after three synodic intervals Mercury returns to a position in the sky which is on average only 17.4° less in longitude. Using reduced amplitudes of about 14°–25° but keeping the same zone boundaries, the computation of what I will call 3-synarc system A models of Mercury is significantly simplified. A full ephemeris of a synodic phase of Mercury can then be composed by combining three columns of longitudes computed with 3-synarc step functions, each column starting with a longitude of Mercury one synodic event apart. Confirmation that this method was indeed used by the Babylonian astronomers comes from Text M (BM 36551+), a very early ephemeris of the last appearances in the evening of Mercury from 424 to 403 BC, computed in three columns according to System A3. Based on an analysis of Text M I suggest that around 400 BC the initial approach in system A modelling of Mercury may have been directed towards choosing “nice” sexagesimal numbers for the amplitudes of the system A step functions while in the later final models, dating from around 300 BC onwards, more emphasis was put on selecting numerical values for the amplitudes such that they were related by simple ratios. The fact that different ephemeris periods were used for each of the four synodic phases of Mercury in the later models may be related to the selection of a best fitting set of System A step function amplitudes for each synodic phase. (shrink)

A poset is well‐partially ordered (WPO) if all its linear extensions are well orders; the supremum of ordered types of these linear extensions is the length, of p. We prove that if the vertex set X is infinite, of cardinality κ, and the ordering ⩽ is the intersection of finitely many well partial orderings of X,, then, letting, with, denote the euclidian division by κ (seen as an initial ordinal) of the length of each corresponding poset: where denotes (...) the least initial ordinal greater than the ordinal. This inequality is optimal. This result answers questions of Forster. (shrink)

Relatives play an important role in human reproduction according to evolutionary theories of reproductive behavior, but previous empirical studies show large differences in the effects of kin on fertility outcomes. In our paper we examine the effect of co-resident kin and non-kin on the length of birth intervals over the reproductive life course of Dutch women born between 1842 and 1920. We estimate Cox proportional hazard models for parity progression based on the presence of kin and non-kin in the (...) household while controlling for a large number of individual and community-level characteristics. We find that couples living with their brothers experienced shorter birth intervals whereas couples residing with a widowed father had relatively longer birth intervals. The effects of these types of kin on reproduction were most pronounced up to the birth of the fifth child, but not thereafter. We found no effect for mothers or other types of kin. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:Commentary on Augustine City of God, Books 1–5 by Gillian ClarkJames J. O'DonnellCommentary on Augustine City of God, Books 1–5. By Gillian Clark. Oxford: Oxford University Press, 2021. Pp. xii + 281. ISBN: 978-0-19-887007-4.Pierre Bayard's masterful How to Talk about Books You Haven't Read offers soothing balm for readers in the daunting presence of Augustine's City of God. Weighing in at a third of a million words, Augustine's (...) magnum opus et arduum,1 offers a Moby-Dick and a half of verbiage spread out over twenty-two books. I make the claim, nervously and looking about for intelligent contradiction, that it is the longest work of sustained and coherent argument to survive from Greco-Roman antiquity. Other behemoths are compilations in the main of one kind or another, but this one has a structure and a linear direction and was tightly enough written that a theme opened in the second book with a promise of returning with later treatment is in fact picked up in the nineteenth book written perhaps a dozen years later.Those who really will read, or at least read in, Civ. benefit from all the help they can get. There are good translations (English alone in my lifetime by McCracken et al. in Loeb, Bettenson, Dyson, and Babcock) but none provides more than a minimalist annotation. Francophone readers have been better served since 1960 by the five volume edition with more ample annotation in the Bibliothèque Augustinienne of G. Bardy et al., and it is the standard point of entrance to serious study for scholars. That work has many strengths but the limitations of arising from a devout tradition of scholarship and of coming before the efflorescence in late antique and Augustinian studies through which we have lived.More help is now at hand in the form of the volume under review and its promise of more to come. Gillian Clark is known and admired among Augustinian scholars for many works, not least the earlier exercise in commentary of her Cambridge "green and yellow"2 edition of Confessions 1–4 (1995). The present work began as an ambitious multi-author collaborative project that proved unwieldy (see the present volume, pp. 31–32) and results in this first installment of [End Page 179] a series now in Clark's hands exclusively, drawing on her strengths as a classical historian specializing in late antiquity and patristics. Four additional volumes are anticipated, ideally on two-year intervals. The task requires multiple competences: at least, the history of the fourth and fifth centuries, the history and presence of ancient philosophy, the history of Christian reception and interpretation of scripture, and the reception of classical literature in late antiquity. It is a work that soars with no middle flight.As commentary on a behemoth, the commentary begins with the virtue of concision. At a rough calculation, commentary here runs at about half again the length of the text under review, producing volumes that can be usefully read alongside Augustine without swamping either Augustine or the reader in an ocean of erudition and detail. After an introduction that is as much to the whole work as to books 1–5, the book is organized strictly by lemmata to passages commented. The lemmata are all in Latin to facilitate use by readers of various editions and translations, with references given by book/chapter and page/line to the standard CCSL 1955 reprinting of the edition of Dombart-Kalb (itself originally a Teubner of 1928).3The focus of the annotation is on the work as historical artifact, well situated in both the moment and controversies of writing and in the apparatus of learning that Augustine brought to his work. Inevitably this includes welcome discussion of argument, rhetoric, and strategy (the very short discussion of how Augustine wrote on pp. 9–12 is remarkably insightful), but readers will welcome the lucid guidance offered to how the text made sense in Aug.'s own time, to what he was evidently reading as he wrote, and to the range of historical realia that make up his argument. Commentary on later... (shrink)

To know the scale of an aulos it is necessary to have a complete instrument. None of the surviving auloi are complete. It is the purpose of this article to attempt to reconstruct the missing parts of several auloi and thus to determine their scales. All musical sounds consist of regular vibrations. The interval between two notes may be expressed as the rate of vibration of the higher note divided by the rate of vibration of the lower note. This (...) ratio is very roughly the same as the length of the air column for the lower note divided by the length of the air column for the higher note. This gross over-simplification has been used for all calculations in this article. The width of the air column, the size of the holes, and the reed itself affect the pitch considerably. The player also has a large control over the precise note played. And so the notes obtained by the ancients might have been very different from the notes obtained by ignoring these factors; although the player's control over pitch might have been used to lessen the effects of the other factors. (shrink)

It is a commonplace in the analytic geometry of physical space-time that an extended straight line segment, having positive length, is treated as “consisting of” unextended points, each of which has zero length. Analogously, time intervals of positive duration are resolved into instants, each of which has zero duration.

If is often taken for granted that on a rotating disk it is possible to operate a global 3+1 splitting of spacetime such that both lengths and time intervals are uniquely defined in terms of measurements performed by real rods and real clocks at rest on the platform. This paper shows that this assumption, although widespread and apparently trivial, leads to an anisotropy of the velocity of two light beams traveling in opposite directions along the rim of the disk, which (...) in turn implies some recently pointed out paradoxical consequences undermining the self-consistency of the special theory of relativity (SRT). A correct application of the SRT solves the problem and recovers complete internal consistency for the theory. As an immediate consequence, it is shown that the Sagnac effect only depends on the nonhomogeneity of time of the platform and has nothing to do with any anisotropy of the speed of light along the rim of the disk, contrary to an incorrect but widely held idea. (shrink)

In the Christian imaginary, the ternary representation of the universe is reiterated by appealing either to the Platonic texts, or to the Stoic ones. The triadic scheme of the worlds certifies an ambiguous status of man, of an individual placed neither here nor there, by the force of some circumstances which he cannot resist. Situated at equal distance from sidereal heights - credited as having the monopoly on perfection - and from the terrifying shadows, managed in a totalitarian manner by (...) the instances of evil, in “the world between the worlds”, he thinks of the interval as of a space of communication, filled with signs, shapes and characters, by means of which distances can be “neared, compressed and “humanised”. Each step, stage, climb or descent is perceived as a “rupture of level”, as overcoming of the human condition by assuming a trans-mundane axiological repertoire. (shrink)

In recent years, several remarkable results have shown that certain theorems of finite combinatorics are unprovable in certain logical systems. These developments have been instrumental in stimulating research in both areas, with the interface between logic and combinatorics being especially important because of its relation to crucial issues in the foundations of mathematics which were raised by the work of Kurt Godel. Because of the diversity of the lines of research that have begun to shed light on these issues, there (...) was a need for a comprehensive overview which would tie the lines together. This volume fills that need by presenting a balanced mixture of high quality expository and research articles that were presented at the August 1985 AMS-IMS-SIAM Joint Summer Research Conference, held at Humboldt State University in Arcata, California.With an introductory survey to put the works into an appropriate context, the collection consists of papers dealing with various aspects of 'unprovable theorems and fast-growing functions'. Among the topics addressed are: ordinal notations, the dynamical systems approach to Ramsey theory, Hindman's finite sums theorem and related ultrafilters, well quasiordering theory, uncountable combinatorics, nonstandard models of set theory, and a length-of-proof analysis of Godel's incompleteness theorem. Many of the articles bring the reader to the frontiers of research in this area, and most assume familiarity with combinatorics and/or mathematical logic only at the senior undergraduate or first-year graduate level. (shrink)

Goat is a small ruminant with hollow horn that is found throughout the world. A survey was carried out to evaluate the performance of locally adopted goats under farmers’ management practices in Sundarbazar municipality of Lamjung district between February and August, 2017. The study focused on two breeds, Khari and Jamunaparias well as their crossbred. Individual goats’ primary data, collected using a convenient sampling technique and pretested questionnaires with open and close ended questions, were analyzed using SPSS 20 and MS (...) Excel 2010. Results revealed significant effects of a Doe’s breed on age at first estrus, age at first kidding, gestation length, postpartum estrus, and kidding interval. Age at first estrus and age at first kidding were lowest for Khari. Highly significant lowest kidding interval and significantly shortest gestation length were found for Khari. No significant effect of parity was observed on age at first estrus, age at first kidding, kidding interval, and postpartum estrus, but a significant effect was observed on gestation length with lowest values in 4th to 6th parity. Doe’s breed showed a significant effect on litter size at birth. However, no significant effect of the breed was found on litter birth weight and litter size at weaning. The results demonstrated Khari as the desirable breed for the locality. (shrink)