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.

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)

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)

Two solutions are offered to the problem of distinguishing "historical" from "functional" associations in cross-cultural surveys. The underlying logic of the mathematical model is discussed and three kinds of association distinguished: hyperdiffusional or purely "historical" association, undiffusional or purely "functional" association, and semidiffusional or mixed "historical-functional" association. Two overland diffusion arcs constitute the test sample; the relationship of social stratification to political complexity constitutes the test problem. A sifting test establishes a bimodal distribution of interval lengths between like types (...) and sifts out repetitions with a lesser interval length than the second mode. A cluster test shows that for the test problem, the "hits" cluster more than the "misses". (shrink)

From the data of the 1989 Bangladesh Fertility Survey, aggregate deaths reported at ages 0-12 and 13-60 months are used to estimate infant and child mortality. Multivariate analysis shows that preceding birth interval length, followed by survival status of the immediately preceding child, are the most important factors associated with differential infant and child mortality risks; sex of the index child and mother's and father's education are also significant. Demographic factors are influential during infancy as well as childhood, (...) but social factors, particularly mother's and father's education, now emerge as significant predictors of infant mortality risks. This indicates a change in the role of socioeconomic factors, since the earlier Bangladesh Fertility Survey in 1975. (shrink)

The current contents of visual working memory can be disrupted by previously formed memories. This phenomenon is known as proactive interference, and it can be used to index the availability of old memories. However, there is uncertainty about the robustness and lifetime of proactive interference, which raises important questions about the role of temporal factors in forgetting. The present study assessed different factors that were expected to influence the persistence of proactive interference over an inter-trial interval in the visual (...) recent probes task. In three experiments, participants encoded arrays of targets and then determined whether a single probe matched one of those targets. On some trials, the probe matched an item from the previous trial, whereas on other trials the probe matched a more distant item. Prior studies have found that recent negative probes can increase errors and slow response times in comparison to non-recent negative probes, and this offered a behavioral measure of proactive interference. In Experiment 1, factors of array size and inter-trial interval were manipulated in the recent probes task. There was a reduction in proactive interference when a longer delay separated trials on one measure, but only when participants encoded two targets. When working memory capacity was strained by increasing the array size to four targets, proactive interference became stronger after the long delay. In Experiment 2, the inter-trial interval length was again manipulated, along with stimulus novelty. Proactive interference was modestly stronger when a smaller number of stimuli were used throughout the experiment, but proactive interference was minimally affected by the inter-trial interval. These findings are problematic for temporal models of forgetting, but Experiment 3 showed that proactive interference also resisted disruption produced by a secondary task presented within the inter-trial interval. Proactive interference was constantly present and generally resilient to the different manipulations. The combined data suggest a relatively durable, passive representation that can disrupt current working memory under a variety of different circumstances. (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.

In this paper, a repetitive control scheme of a 2-DOF robotic manipulator based on the improved cubic B-spline curve is proposed. Firstly, a repetitive controller for robotic manipulator is designed, which is composed of an iterative controller and disturbance observer. Then, an improved B-spline optimization scheme is introduced to divide the task of the robotic manipulator into three intervals. A correction function is added to each interval of cubic spline interpolation. Finally, a variety of cases are designed and simulated (...) by MATLAB. The experimental results show that, compared with the conventional B-spline, the improved B-spline has better performance in tracking accuracy and smoothness of motion trajectory. By changing the mechanism of the manipulator, the cases of different weights and lengths are designed. The experimental results in these cases show that the proposed scheme can be applied to most of the 2-DOF robotic manipulator control systems. (shrink)

There is no uniquely standard concept of an effectively decidable set of real numbers or real n-tuples. Here we consider three notions: decidability up to measure zero [M.W. Parker, Undecidability in Rn: Riddled basins, the KAM tori, and the stability of the solar system, Phil. Sci. 70(2) (2003) 359–382], which we abbreviate d.m.z.; recursive approximability [or r.a.; K.-I. Ko, Complexity Theory of Real Functions, Birkhäuser, Boston, 1991]; and decidability ignoring boundaries [d.i.b.; W.C. Myrvold, The decision problem for entanglement, in: R.S. (...) Cohen et al. (Eds.), Potentiality, Entanglement, and Passion-at-a-Distance: Quantum Mechanical Studies fo Abner Shimony, Vol. 2, Kluwer Academic Publishers, Great Britain, 1997, pp. 177–190]. Unlike some others in the literature, these notions apply not only to certain nice sets, but to general sets in Rn and other appropriate spaces. We consider some motivations for these concepts and the logical relations between them. It has been argued that d.m.z. is especially appropriate for physical applications, and on Rn with the standard measure, it is strictly stronger than r.a. [M.W. Parker, Undecidability in Rn: Riddled basins, the KAM tori, and the stability of the solar system, Phil. Sci. 70(2) (2003) 359–382]. Here we show that this is the only implication that holds among our three decidabilities in that setting. Under arbitrary measures, even this implication fails. Yet for intervals of non-zero length, and more generally, convex sets of non-zero measure, the three concepts are equivalent. (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 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)

In The Explanatory Dispensability of Idealizations, Sam Baron suggests a possible strategy enabling the indispensability argument to break the symmetry between mathematical claims and idealization assumptions in scientific models. Baron’s distinction between mathematical and non-mathematical idealization, I claim, is in need of a more compelling criterion, because in scientific models idealization assumptions are expressed through mathematical claims. In this paper I argue that this mutual dependence of idealization and mathematics cannot be read in terms of symmetry and that Baron’s non-causal (...) notion of mathematical difference-making is not effective in justifying any symmetry-breaking between mathematics and idealization. The function of making a difference that Baron attributes to mathematics cannot be referred to physical facts, but to the features of quantities, such as step lengths or time intervals taken into account in the models. It appears, indeed, that it does not follow from Baron’s argument that idealizations do not help to carry the explanatory load at least for two reasons: mathematics is not independent of idealizations in modelling and idealizations help mathematics to carry the explanatory load of a model in different degrees. (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)

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)

Hempel's proposal of covering laws which explain historical events has a certain plausibility, but can never be actually realized due to the chaotic nature of history. The natural laws that would govern both individual lives and greater history would be nonlinear; consequently, in the terminology of chaos theory, the final states of both are extremely sensitive to initial conditions. Initial conditions would need to be exactly known in order to account correctly for historic phenomena, especially for causes and effects which (...) span long, historically interesting, lengths of time. Covering-law history therefore fails because the details of initial conditions are generally unknowable. Since this constraint diminishes as the time over which covering laws operate is divided into smaller consecutive intervals of scenes, covering-law explanations resolve into those having a narrative temporal structure. (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)

We report two issues concerning diverging sets of Bayesian (conditional) probabilities-divergence of "posteriors"-that can result with increasing evidence. Consider a set P of probabilities typically, but not always, based on a set of Bayesian "priors." Fix E, an event of interest, and X, a random variable to be observed. With respect to P, when the set of conditional probabilities for E, given X, strictly contains the set of unconditional probabilities for E, for each possible outcome X = x, call this (...) phenomenon dilation of the set of probabilities (Seidenfeld and Wasserman 1993). Thus, dilation contrasts with the asymptotic merging of posterior probabilities reported by Savage (1954) and by Blackwell and Dubins (1962). (1) In a wide variety of models for Robust Bayesian inference the extent to which X dilates E is related to a model specific index of how far key elements of P are from a distribution that makes X and E independent. (2) At a fixed confidence level, (1-α), Classical interval estimates A n for, e.g., a Normal mean θ have length O(n -1/2 ) (for sample size n). Of course, the confidence level correctly reports the (prior) probability that θ ∈ A n ,P(A n )=1-α , independent of the prior for θ . However, as shown by Pericchi and Walley (1991), if an ε -contamination class is used for the prior on the parameter θ , there is asymptotic (posterior) dilation for the A n , given the data. If, however, the intervals A ′ n are chosen with length $O(\sqrt{\log (\text{n})/\text{n})}$ , then there is no asymptotic dilation. We discuss the asymptotic rates of dilation for ClassClassical and Bayesian interval estimates and relate these to Bayesian hypothesis testing. (shrink)

In The Explanatory Dispensability of Idealizations, Sam Baron suggests a possible strategy enabling the indispensability argument to break the symmetry between mathematical claims and idealization assumptions in scientific models. Baron’s distinction between mathematical and non-mathematical idealization, I claim, is in need of a more compelling criterion, because in scientific models idealization assumptions are expressed through mathematical claims. In this paper I argue that this mutual dependence of idealization and mathematics cannot be read in terms of symmetry and that Baron’s non-causal (...) notion of mathematical difference-making is not effective in justifying any symmetry-breaking between mathematics and idealization. The function of making a difference that Baron attributes to mathematics cannot be referred to physical facts, but to the features of quantities, such as step lengths or time intervals taken into account in the models. It appears, indeed, that it does not follow from Baron’s argument that idealizations do not help to carry the explanatory load at least for two reasons: mathematics is not independent of idealizations in modelling and idealizations help mathematics to carry the explanatory load of a model in different degrees. (shrink)

The first attempt to represent the Periodic system graphically was the Telluric Helix presented in 1862 by Alexandre-Emile Béguyer de Chancourtois, in which the sequence of elements was wound round a cylinder. This has hardly been attempted since, because the intervals between periodic returns vary in length from 2 to 32 elements, but Charles Janet presented a model wound round four nested cylinders. The rows in Janet’s table are defined by a constant sum of the first two quantum numbers, (...) n and l, so that they end with the s-block, headed by hydrogen and helium. By combining Janet’s table, Edward Mazurs’ version, in which each row represents an electron shell and Valery Tsimmerman’s use of a half square for each element, I have produced a representation that can be printed out and wound round to make a cylinder with manageable dimensions. In the unwound version, I have placed the s-block in the middle, to emphasise its pivotal nature, since it both ends each row and contributes electrons to the valence of elements in the next row; it thus does not necessarily belong either on the left or the right side of a table. The downward arrows that link subshells within each series graphically illustrate the Janet Effect. To acknowledge my debt to Chancourtois, Janet, Mazurs and Tsimmerman, I call my design the ‘Telluric Remix’. (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)

An intuitive idea concerning degrees of precision is widely accepted, and it is that we increase precision of theories by paying attention to ever decreasing orders of magnitude of measurements which we incorporate in these theories. We increase precision of measuring or of predicting measurement of length, for instance, if we pay attention not only to centimeters but also to millimeters, microns, angstroms, and so on. And our theories are precise to centimeters, then to millimeters, and so on respectively. (...) The idea is that increased precision is the process of capturing a point within nested intervals, and that this is reflected both in experimental and in theoretical progress. An intuitive analogy may be drawn with series of photos made by a camera which approaches the moon's surface. A corollary from this intuitive idea which has been stated by a number of writers is that observation reports and theories are not overthrown even when superseded by more accurate observation because within the limits of their own precision-range they are still valid. When a law in physics is stated, it is stated usually without such qualifications; but this is a mere matter of convenience, we are told. The idea is that a tacit proviso is understood and the law should be understood to be not the statement “such and such is the case” but the statement “within this and that limit of accuracy such and such is the case”; in such a formulation, the law, once verified, remains unshaken regardless of further progress in the field. (shrink)

Adequate temporal abilities are required for most daily activities. Traumatic brain injury (TBI) patients often present with cognitive dysfunctions, but few studies have investigated temporal impairments associated with TBI. The aim of the present work is to review the existing literature on temporal abilities in TBI patients. Particular attention is given to the involvement of higher cognitive processes in temporal processing in order to determine if any temporal dysfunction observed in TBI patients is due to the disruption of an internal (...) clock or to the dysfunction of general cognitive processes. The results showed that temporal dysfunctions in TBI patients are related to the deficits in cognitive functions involved in temporal processing rather than to a specific impairment of the internal clock. In fact, temporal dysfunctions are observed when the length of temporal intervals exceeds the working memory span or when the temporal tasks require high cognitive functions to be performed. The consistent higher temporal variability observed in TBI patients is a sign of impaired frontally mediated cognitive functions involved in time perception. (shrink)