The importance of the prime factorization problem is very well known (e.g., many security protocols are based on the impossibility of a fast factorization of integers on traditional computers). It is necessary from a number k to establish two primes a and b giving k = a · b. Usually, k is written in a positional numeral system. However, there exists a variety of numeral systems that can be used to represent numbers. Is it true that the (...) prime factorization is difficult in any numeral system? In this paper, a numeral system with partial carrying is described. It is shown that this system contains numerals allowing one to reduce the problem of prime factorization to solving [K/2] − 1 systems of equations, where K is the number of digits in k (the concept of digit in this system is more complex than the traditional one) and [u] is the integer part of u. Thus, it is shown that the difficulty of prime factorization is not in the problem itself but in the fact that the positional numeral system is used traditionally to represent numbers participating in the prime factorization. Obviously, this does not mean that P=NP since it is not known whether it is possible to re-write a number given in the traditional positional numeral system to the new one in a polynomial time. (shrink)

In this survey, a recent computational methodology paying a special attention to the separation of mathematical objects from numeral systems involved in their representation is described. It has been introduced with the intention to allow one to work with infinities and infinitesimals numerically in a unique computational framework in all the situations requiring these notions. The methodology does not contradict Cantor’s and non-standard analysis views and is based on the Euclid’s Common Notion no. 5 “The whole is greater (...) than the part” applied to all quantities (finite, infinite, and infinitesimal) and to all sets and processes (finite and infinite). The methodology uses a computational device called the Infinity Computer (patented in USA and EU) working numerically (recall that traditional theories work with infinities and infinitesimals only symbolically) with infinite and infinitesimal numbers that can be written in a positional numeral system with an infinite radix. It is argued that numeral systems involved in computations limit our capabilities to compute and lead to ambiguities in theoretical assertions, as well. The introduced methodology gives the possibility to use the same numeral system for measuring infinite sets, working with divergent series, probability, fractals, optimization problems, numerical differentiation, ODEs, etc. (recall that traditionally different numerals lemniscate; Aleph zero, etc. are used in different situations related to infinity). Numerous numerical examples and theoretical illustrations are given. The accuracy of the achieved results is continuously compared with those obtained by traditional tools used to work with infinities and infinitesimals. In particular, it is shown that the new approach allows one to observe mathematical objects involved in the Hypotheses of Continuum and the Riemann zeta function with a higher accuracy than it is done by traditional tools. It is stressed that the hardness of both problems is not related to their nature but is a consequence of the weakness of traditional numeral systems used to study them. It is shown that the introduced methodology and numeral system change our perception of the mathematical objects studied in the two problems. (shrink)

The idea that there is a “Number Sense” (Dehaene, 1997) or “Core Knowledge” of number ensconced in a modular processing system (Carey, 2009) has gained popularity as the study of numerical cognition has matured. However, these claims are generally made with little, if any, detailed examination of which modular properties are instantiated in numerical processing. In this article, I aim to rectify this situation by detailing the modular properties on display in numerical cognitive processing. In the process, I review literature (...) from across the cognitive sciences and describe how the evidence reported in these works supports the hypothesis that numerical cognitive processing is modular. I outline the properties that would suffice for deeming a certain processing system a modular processing system. Subsequently, I use behavioral, neuropsychological, philosophical, and anthropological evidence to show that the number module is domain specific, informationally encapsulated, neurally localizable, subject to specific pathological breakdowns, mandatory, fast, and inaccessible at the person level; in other words, I use the evidence to demonstrate that some of our numerical capacity is housed in modular casing. (shrink)

In a recent paper in Astrophysics and Space Science Vol. 364 no. 11 (2019), S. Ershkov & D. Leschenko presented a new solving procedure for Euler-Poisson equations for solving momentum equations of the CR3BP near libration points for uniformly rotating planets having inclined orbits in the solar system with respect to the orbit of the Earth. The system of equations of the CR3BP has been explored with regard to the existence of an analytic way of presentation of the approximated solution (...) in the vicinity of libration points. A new and elegant ansatz has been suggested in their publication whereby, in solving, the momentum equation is reduced to a system two coupled Riccati ODEs. In this paper, we presented a numerical solution of such coupled Riccati ODEs using Mathematica software package. (shrink)

Four perspectives on numerical origins are examined. The nativist model sees numbers as an aspect of numerosity, the biologically endowed ability to appreciate quantity that humans share with other species. The linguistic model sees numbers as a function of language. The embodied model sees numbers as conceptual metaphors informed by physical experience and expressed in language. Finally, the extended model sees numbers as conceptual outcomes of a cognitive system that includes material forms as constitutive components. If numerical origins are to (...) be found, each perspective must address one or more critical questions that will require working across discipline boundaries. (shrink)

Number systems differ cross-culturally in characteristics like how high counting extends and which number is used as a productive base. Some of this variability can be linked to the way the hand is used in counting. The linkage shows that devices like the hand used as external representations of number have the potential to influence numerical structure and organization, as well as aspects of numerical language. These matters suggest that cross-cultural variability may be, at least in part, a matter (...) of whether devices are used in counting, which ones are used, and how they are used. (shrink)

Four perspectives on numerical origins are examined. The nativist model sees numbers as an aspect of numerosity, the biologically endowed ability to appreciate quantity that humans share with other species. The linguistic model sees numbers as a function of language. The embodied model sees numbers as conceptual metaphors informed by physical experience and expressed in language. Finally, the extended model sees numbers as conceptual outcomes of a cognitive system that includes material forms as constitutive components. If numerical origins are to (...) be found, each perspective must address one or more critical questions that will require working across discipline boundaries. (shrink)

A recently developed computational methodology for executing numerical calculations with infinities and infinitesimals is described in this paper. The approach developed has a pronounced applied character and is based on the principle “The part is less than the whole” introduced by the ancient Greeks. This principle is applied to all numbers (finite, infinite, and infinitesimal) and to all sets and processes (finite and infinite). The point of view on infinities and infinitesimals (and in general, on Mathematics) presented in this paper (...) uses strongly physical ideas emphasizing interrelations that hold between a mathematical object under observation and the tools used for this observation. It is shown how a new numeral system allowing one to express different infinite and infinitesimal quantities in a unique framework can be used for theoretical and computational purposes. Numerous examples dealing with infinite sets, divergent series, limits, and probability theory are given. (shrink)

Plant diseases are numerous in the world of agriculture. These diseases cause a lot of trouble to most farmers. Among these common diseases, we single out the diseases that affect the Passion fruit, which is affected by about seven diseases, with different symptoms for each disease. Today, technology is facilitating human life in all areas of life, and among these facilities are expert system, a computer program that uses artificial-intelligence methods to solve problems within a specialized domain that ordinarily requires (...) human expertise. The first expert system was developed in 1965 by Edward Feigen Baum and Joshua Lederberg of Stanford University in California, U.S. Dendral, as their expert system was later known, was designed to analyses chemical compounds. Expert systems now have commercial applications in fields as diverse as medical diagnosis, petroleum engineering, and financial investing and other areas, and with reference to expert systems and their importance to humans, an integrated expert system has been created in the agricultural field that diagnoses Passion diseases using CLIPS Expert System language. The system was used to design and implement the proposed expert system. The system facilitates the diagnosis of Passion -related diseases. There is no doubt that this expert system will help farmers and those involved in the agricultural field to diagnose Passion -related diseases. Objectives: is to help farmers diagnose Passion diseases in the correct way and how to treat these diseases. Method: The system contains a program project that diagnoses 7 diseases that affect Passion and the seven diseases are: Brown spot, Septoria spot, Root and crown rot, Fusarium wilt, Anthracnose, Woodiness virus, Scab. Results: The expert system was evaluated by farmers and praised for helping them with it. Conclusion: The expert system for diagnosing Passion diseases is effective and usable. (shrink)

Developing earlier studies of the system of numbers in Mundurucu, this paper argues that the Mundurucu numeral system is far more complex than usually assumed. The Mundurucu numeral system provides indirect but insightful arguments for a modular approach to numbers and numerals. It is argued that distinct components must be distinguished, such as a system of representation of numbers in the format of internal magnitudes, a system of representation for individuals and sets, and one-to-one correspondences between the numerosity (...) expressed by the number and its metrics. It is shown that while many-number systems involve a compositionality of units, sets and sets composed of units, few-number languages, such as Mundurucu, do not have access to sets composed of units in the usual way. The nonconfigurational character of the Mundurucu language, which is related to a property for which we coin the term 'low compositionality power', accounts for this and explains the curious fact that Mundurucus make use of marked one-to-one correspondence strategies in order to overcome the limitations of the core system at the perceptual/motor interface of the language faculty. We develop an analysis of a particular construction, parallel numbers, which has not been studied before, elucidating the whole system. This analysis, we argue, sheds new light on classical philosophical, psychological and linguistic debates about numbers and numerals and their relation to language, and more particularly, sheds light on few-number languages. (shrink)

PARC is an "appended numeral" system of natural deduction that I learned as an undergraduate and have taught for many years. Despite its considerable pedagogical strengths, PARC appears to have never been published. The system features explicit "tracking" of premises and assumptions throughout a derivation, the collapsing of indirect proofs into conditional proofs, and a very simple set of quantificational rules without the long list of exceptions that bedevil students learning existential instantiation and universal generalization. The system can be (...) used with any Copi-style set of inference rules, so it is quite adaptable to many mainstream symbolic logic textbooks. Consequently, PARC may be especially attractive to logic teachers who find Jaskowski/Gentzen-style introduction/elimination rules to be far less "natural" than Copi-style rules. The PARC system is also keyboard-friendly in comparison to the widely adopted Jaskowski-style graphical subproof system of natural deduction, viz., Fitch diagrams and Copi "bent arrow" diagrams. (shrink)

The First Hilbert problem is studied in this paper by applying two instruments: a new methodology distinguishing between mathematical objects and mathematical languages used to describe these objects; and a new numeral system allowing one to express different infinite numbers and to use these numbers for measuring infinite sets. Several counting systems are taken into consideration. It is emphasized in the paper that different mathematical languages can describe mathematical objects (in particular, sets and the number of their elements) (...) with different accuracies. The traditional and the new approaches are compared and discussed. (shrink)

This chapter examines the pioneering work of Gottfried Wilhelm Leibniz (1646-1716) on various number systems, in particular binary, which he independently invented in the mid-to-late 1670s, and hexadecimal, which he invented in 1679. The chapter begins with the oft-debated question of who may have influenced Leibniz’s invention of binary, though as none of the proposed candidates is plausible I suggest a different hypothesis, that Leibniz initially developed binary notation as a tool to assist his investigations in mathematical problems that (...) were exercising him at the time, namely those concerning the divisibility of composite numbers, primality, and perfect numbers. The chapter then explores Leibniz’s development of binary, his little-known work on binary fractions and expansions, his use of binary as a symbol for creation in his philosophical-theology, and his response to the suggestion that there was a correlation between binary numeration and the hexagrams of the ancient Chinese divinatory text, the Yijing. The chapter then focuses on Leibniz’s work on other number systems, in particular his invention and exploration of hexadecimal as well as his work on duodecimal. The chapter concludes by revealing a hitherto unknown practical application of binary that Leibniz devised in the last year of his life. (shrink)

Several ways used to rank countries with respect to medals won during Olympic Games are discussed. In particular, it is shown that the unofficial rank used by the Olympic Committee is the only rank that does not allow one to use a numerical counter for ranking – this rank uses the lexicographic ordering to rank countries: one gold medal is more precious than any number of silver medals and one silver medal is more precious than any number of bronze medals. (...) How can we quantify what do these words, more precious, mean? Can we introduce a counter that for any possible number of medals would allow us to compute a numerical rank of a country using the number of gold, silver, and bronze medals in such a way that the higher resulting number would put the country in the higher position in the rank? Here we show that it is impossible to solve this problem using the positional numeral system with any finite base. Then we demonstrate that this problem can be easily solved by applying numerical computations with recently developed actual infinite numbers. These computations can be done on a new kind of a computer – the recently patented Infinity Computer. Its working software prototype is described briefly and examples of computations are given. It is shown that the new way of counting can be used in all situations where the lexicographic ordering is required. (shrink)

The boundless nature of the natural numbers imposes paradoxically a high formal bound to the use of standard artificial computer programs for solving conceptually challenged problems in number theory. In the context of the new cognitive foundations for mathematics' and physics' program immersed in the setting of artificial mathematical intelligence, we proposed a refined numerical system, called the physical numbers, preserving most of the essential intuitions of the natural numbers. Even more, this new numerical structure additionally possesses the property of (...) being a bounded object allowing us to work with quite similar axioms like the classic Peano axioms, but in a finite environment and with an enriched physical dimension. Finally, we present several enlightening examples and we conclude that the physical numbers provide a natural formal setting for approaching classic problems in number theory from a more hybrid perspective, i.e. with a potential participation in the generation of solutions, not only of working mathematicians but also of more sophisticated artificial interactive (mathematical) intelligent agents (within the context of cognitive-computational metamathematics or artificial mathematical intelligence) and more controlled by physically-inspired principles. Finally, this paper addresses a highly new, bottom-up and paradigm-shifting approach to the foundations of physics: One should refine and improve the conceptual formal setting of the language that we use for understanding physical phenomena (e.g. the mathematical grounding concepts and theories) for being able to understand better more subtle and complex physical phenomena. (shrink)

The idea the New Zealand Māori once counted by elevens has been viewed as a cultural misunderstanding originating with a mid-nineteenth-century dictionary of their language. Yet this “remarkable singularity” had an earlier, Continental origin, the details of which have been lost over a century of transmission in the literature. The affair is traced to a pair of scientific explorers, René-Primevère Lesson and Jules Poret de Blosseville, as reconstructed through their publications on the 1822–1825 circumnavigational voyage of the Coquille, a French (...) corvette. Possible explanations for the affair are briefly examined, including whether it might have been a prank by the Polynesians or a misunderstanding or hoax on the part of the Europeans. Reasons why the idea of counting by elevens remains topical are discussed. First, its very oddity has obscured the counting method actually used—setting aside every tenth item as a tally. This “ephemeral abacus” is examined for its physical and mental efficiencies and its potential to explain aspects of numerical structure and vocabulary (e.g., Mangarevan binary counting; the Hawaiian number word for twenty, iwakalua), matters suggesting material forms have a critical if underappreciated role in realising concepts like exponential value. Second, it provides insight into why it can be difficult to appreciate highly elaborated but unwritten numbers like those found throughout Polynesia. Finally, the affair illuminates the difficulty of categorising number systems that use multiple units as the basis of enumeration, like Polynesian pair-counting; potential solutions are offered. (shrink)

I argue that the human mind includes an innate domain-specific system for representing precise small numerical quantities. This theory contrasts with object-tracking theories and with domain-general theories that only make use of mental models. I argue that there is a good amount of evidence for innate representations of small numerical quantities and that such a domain-specific system has explanatory advantages when infants’ poor working memory is taken into account. I also show that the mental models approach requires previously unnoticed domain-specific (...) structure and consequently that there is no domain-general alternative to an innate domain-specific small number system. (shrink)

Emotion recognition systems (ERS) offer the ability to identify the emotions of others, based on an analysis of their facial expressions and regardless of culture, ethnicity, context, gender or social class. By claiming universalism in the expression as well as in the recognition of emotions, we believe that ERS present significant risks of causing great harm to some individuals, in addition to targeting, in some contexts, specific social groups. Drawing on a wide range of multidisciplinary knowledge - including philosophy, (...) psychology, computer science and anthropology - this research project aims to identify the current limitations of ERS and the main risks that their use brings, with the goal of providing a clear and robust analysis of the use of ERS and their contribution to greater social justice. Pointing to technical limitations, we refute, on the one hand, the idea that ERS are able to prove the causal link between specific emotions and specific facial expressions. We support our argument with evidence of the inability of ERS to distinguish facial expressions of emotions from facial expressions as communication signals. On the other hand, due to the contextual and cultural limitations of current ERS, we refute the idea that ERS are able to recognise, with equal performance, the emotions of individuals, regardless of their culture, ethnicity, gender and social class. Our ethical analysis shows that the risks are considerably more numerous and important than the benefits that could be derived from using ERS. However, we have separated out a specific type of ERS, whose use is limited to the field of care, and which shows a remarkable potential to actively participate in social justice, not only by complying with the minimum requirements, but also by meeting the criterion of beneficence. While ERS currently pose significant risks, it is possible to consider the potential for specific types to participate in social justice and provide emotional and psychological support and assistance to certain members of society. (shrink)

In this study, the archaic counting systems of Mesopotamia as understood through the Neolithic tokens, numerical impressions, and proto-cuneiform notations were compared to the traditional number-words and counting methods of Polynesia as understood through contemporary and historical descriptions of vocabulary and behaviors. The comparison and associated analyses capitalized on the ability to understand well-known characteristics of Uruk-period numbers like object-specific counting, polyvalence, and context-dependence through historical observations of Polynesian counting methods and numerical language, evidence unavailable for ancient numbers. Similarities (...) between the two number systems were then used to argue that archaic Mesopotamian numbers, like those of Polynesia, were highly elaborated and would have served as cognitively efficient tools for mental calculation. Their differences also show the importance of material technologies like tokens, impressions, and notations to developing mathematics. This project has received funding from the European Union’s Horizon 2020 research and innovation program under grant agreement No. 785793. (shrink)

The approximate number system (ANS) enables organisms to represent the approximate number of items in an observed collection, quickly and independently of natural language. Recently, it has been proposed that the ANS goes beyond representing natural numbers by extracting and representing rational numbers (Clarke & Beck, 2021a). Prior work demonstrates that adults and children discriminate ratios in an approximate and ratio-dependent manner, consistent with the hallmarks of the ANS. Here, we use a well-known “connectedness illusion” to provide evidence that these (...) ratio-dependent ratio discriminations are (a) based on the perceived number of items in seen displays (and not just non-numerical confounds), (b) are not dependent on verbal working memory, or explicit counting routines, and (c) involve representations with a part-whole (or subset-superset) format, like a fraction, rather than a part-part format, like a ratio. These results vindicate key predictions of the hypothesis that the ANS represents rational numbers. (shrink)

As the digitization of information accelerates, the push to encode our surrounding numerically instead of linguistically increases. The role that language has traditionally played in the nomenclature of an integrative taxonomy is being replaced by the numeric identification of one or few quantitative characteristics. Nineteenth-century scientific systems of color identification divided, grouped, and named colors according to multiple characteristics. Now color identification relies on numeric values applied to spectrographic readings. This means of identification of color lacks the taxonomic rigor (...) of nineteenth century systems. Identifying color by numeric value instead of by grouping and naming them, strips color taxonomy of all but one quantitative aspect of a color. I use the case of color taxonomy to argue against a similar trend of numeric identification in the biological sciences. Unlike historically more integrative approaches to taxonomy in biology, genomic sequencing identifies one or few quantitative characteristics to encode an organism. If genomic sequencing becomes the primary means of identification in the biological sciences, just as in numeric systems of color identification, scientific taxonomy would suffer. Basing my analysis on theories of perception of division and on theories of language, I use the cases of color and species to argue for the advantages of an integrative taxonomic system of naming and categorizing over a method of identification, which encodes limited characteristics numerically. I hold that language is the most sophisticated tool for systematic taxonomy and that taxonomic nomenclature should be retained. (shrink)

All humans share a universal, evolutionarily ancient approximate number system (ANS) that estimates and combines the numbers of objects in sets with ratio-limited precision. Interindividual variability in the acuity of the ANS correlates with mathematical achievement, but the causes of this correlation have never been established. We acquired psychophysical measures of ANS acuity in child and adult members of an indigene group in the Amazon, the Mundurucú, who have a very restricted numerical lexicon and highly variable access to mathematics education. (...) By comparing Mundurucú subjects with and without access to schooling, we found that education significantly enhances the acuity with which sets of concrete objects are estimated. These results indicate that culture and education have an important effect on basic number perception. We hypothesize that symbolic and nonsymbolic numerical thinking mutually enhance one another over the course of mathematics instruction. (shrink)

This study examines problems related to the representation of music. It constructs the sender/message/perceiver/result model, a prototype broad enough to incorporate a large variety of music and other notation systems, including those having to do with computers. The work defines music notation itself, describes various models for studying the subject--including the binary types prescriptive/descriptive, and symbolic/iconic--and assesses music notation as a contemporary practice. It encompasses a review of the actions and intentions of composers, performers, and audiences, and a consideration (...) of the message and how it is affected by media , reference systems , and meanings. Finally, through an examination of the possibilities of the self acting as a notation system within models posited by feminist and queer theory, I attempt to demonstrate the power of notation to affect other systems and worlds, including the worlds of music, art, and politics. ;The consideration of these topics draws on theories from linguistics, semiotics, structuralism, post-structuralism, cybernetics, and music, as well as the philosophies of John Cage, Jean-Jacques Nattiez, Nelson Goodman, and others. The work examines various languages and applications used to program computers for musical purposes, though this examination does not incorporate programs dedicated exclusively to printing music. Nor does the study focus on non-Western notations, suggestions for reforms of notation, or encyclopedic lists of notation devices. ;The expository writing is augmented by excerpts from material I gathered in numerous interviews. There are separate sections containing interviews of Leonard Meyer, Allan Kaprow, John Cage, Earle Brown, Roger Reynolds, and Heidi von Gunden. There is also a large, partially-annotated bibliography. ;My hope is that an inclusive, interdisciplinary exploration of music notation will be useful in broadening the theoretical and creative boundaries of the field, as well as bridging gaps among musicians using various music technologies, and artists and theorists from diverse fields and perspectives. (shrink)

Autonomy is often considered a core value of Western society that is deeply entrenched in moral, legal, and political practices. The development and deployment of artificial intelligence (AI) systems to perform a wide variety of tasks has raised new questions about how AI may affect human autonomy. Numerous guidelines on the responsible development of AI now emphasise the need for human autonomy to be protected. In some cases, this need is linked to the emergence of increasingly ‘autonomous’ AI (...) class='Hi'>systems that can perform tasks without human control or supervision. Do such ‘autonomous’ systems pose a risk to our own human autonomy? In this article, I address the question of a trade-off between human autonomy and system ‘autonomy’. (shrink)

Abstract: The aims of this study are to introduce acceleration methods that are called triangular acceleration methods, which come within the series of several acceleration methods that generally known as Al-Tememe's acceleration methods of the second kind which are discovered by (Ali Hassan Mohammed). These methods are useful in improving the results of determining numerical integrals of continuous integrands where the main error is of the forth order with respect to accuracy, partial intervals and the fasting of calculating the results (...) specifically to accelerate results come out by Simpson's method. Also, it is possible to make use of it to improve the results of solving differential equations numerically of the main error of the forth order. (shrink)

Abstract: The aims of this study are to introduce acceleration methods that called relative and algorithmic acceleration methods, which we generally call Al-Tememe's acceleration methods of the second kind discovered by (Ali Hassan Mohammed). It is very useful to improve the numerical results of continuous integrands in which the main error is of the 4th order, and related to accuracy, the number of used partial intervals and how fast to get results especially to accelerate the results got by Simpson's method. (...) Also, it is possible to utilize it in improving the results of differential equations numerically of the main error of the forth order. (shrink)

Emotion recognition systems (ERS) offer the ability to identify the emotions of others, based on an analysis of their facial expressions and regardless of culture, ethnicity, context, gender or social class. By claiming universalism in the expression as well as in the recognition of emotions, we believe that ERS present significant risks of causing great harm to some individuals, in addition to targeting, in some contexts, specific social groups. Drawing on a wide range of multidisciplinary knowledge - including philosophy, (...) psychology, computer science and anthropology - this research project aims to identify the current limitations of ERS and the main risks that their use brings, with the goal of providing a clear and robust analysis of the use of ERS and their contribution to greater social justice. Pointing to technical limitations, we refute, on the one hand, the idea that ERS are able to prove the causal link between specific emotions and specific facial expressions. We support our argument with evidence of the inability of ERS to distinguish facial expressions of emotions from facial expressions as communication signals. On the other hand, due to the contextual and cultural limitations of current ERS, we refute the idea that ERS are able to recognise, with equal performance, the emotions of individuals, regardless of their culture, ethnicity, gender and social class. Our ethical analysis shows that the risks are considerably more numerous and important than the benefits that could be derived from using ERS. However, we have separated out a specific type of ERS, whose use is limited to the field of care, and which shows a remarkable potential to actively participate in social justice, not only by complying with the minimum requirements, but also by meeting the criterion of beneficence. While ERS currently pose significant risks, it is possible to consider the potential for specific types to participate in social justice and provide emotional and psychological support and assistance to certain members of society. (shrink)

Abstract: The main aim of this work is to introduce acceleration methods called a Trigonometric acceleration methods which are of series of numerated methods. In general, these methods named as AL-Tememe’s acceleration methods of first kind to his discoverer ''Ali Hassan Mohammed''. These are very beneficial to acceleration the numerical results for definite integrations with continuous integrands which are of 2nd order main error, with respect to the accuracy and the number of the used subintervals and the fasting obtaining results. (...) Especially, for acceleration the results which are obviously obtained by trapezoidal and midpoint methods. Moreover, these methods could be enhancing the results of the ordinary differential equations numerically which are of 2nd order main error. (shrink)

Abstract: The main aim of this work is to introduce acceleration methods called a hyperbolic acceleration methods which are of series of numerated methods. In general, these methods named as AL-Tememe’s acceleration methods of first kind discovered by (Ali Hassan Mohammed). These are very beneficial to acceleration the numerical results for definite integrations with continuous integrands which are of 2nd order main error, with respect to the accuracy and the number of the used subintervals and the fasting obtaining results. Especially, (...) for acceleration the results which are obviously obtained by trapezoidal and midpoint methods. Moreover, these methods could be enhancing the results of the ordinary differential equations numerically which are of 2nd order main error. (shrink)

Xinxue （The philosophy of mind） founded by ancient Chinese philosopher Wang Yangming of the Ming Dynasty for over 700 years. Its ideas have deeply influenced East Asian countries such as China, Japan, and Korea in the field of social philosophy, and even indirectly promoted Japan's Meiji Restoration movement. At the same time, scholars from all over the world have conducted numerous studies and explorations on it, but overall, there is a lack of systematic exploration and research on it. This article (...) starts with the concept of a system, combines the content of Xinxue and integrates it into a complete, logical, and comprehensive system of Xinxue. (shrink)

Xinxue （The philosophy of mind） founded by ancient Chinese philosopher Wang Yangming of the Ming Dynasty for over 700 years. Its ideas have deeply influenced East Asian countries such as China, Japan, and Korea in the field of social philosophy, and even indirectly promoted Japan's Meiji Restoration movement. At the same time, scholars from all over the world have conducted numerous studies and explorations on it, but overall, there is a lack of systematic exploration and research on it. This article (...) starts with the concept of a system, combines the content of Xinxue and integrates it into a complete, logical, and comprehensive system of Xinxue. (shrink)

Better journey comfort and controllability of automobile are pursued via car industries with the aid of considering using suspension system which plays a very crucial function in handling and ride comfort characteristics. This paper presents the design of an active suspension of quarter automobile system using robust H2 optimal controller and robust μ - synthesis controller with a second order hydraulic actuator. Parametric uncertainties have been additionally considered to model within the system. Numerical simulation become completed to the designed controllers. (...) Results display that during spite of introducing uncertainties, the designed μ - synthesis controller improves ride consolation and road protecting of the automobile while as compared to the H2 optimal controller. (shrink)

This work is a complete mathematical guide to lottery games, covering all of the problems related to probability, combinatorics, and all parameters describing the lottery matrices, as well as the various playing systems. The mathematics sections describe the mathematical model of the lottery, which is in fact the essence of the lotto game. The applications of this model provide players with all the mathematical data regarding the parameters attached to the gaming events and personal playing systems. By applying (...) these data, one can find all the winning probabilities for the play with one line, and how these probabilities change with playing the various types of systems containing several lines, depending on their structure. Also, each playing system has a formula attached that provides the number of possible multiple prizes in various circumstances. Other mathematical parameters of the playing systems and the correlations between them are also presented. The generality of the mathematical model and of the obtained formulas allows their application for any existent lottery and any playing system. Each formula is followed by numerical results covering the most frequent lottery matrices worldwide and by multiple examples predominantly belonging to the 6/49 lottery. The listing of the numerical results in dozens of well-organized tables, along with instructions and examples of using them, makes possible the direct usage of this guide by players without a mathematical background. The author also discusses from a mathematical point of view the strategies of choosing involved in the lotto game. The book does not offer so-called winning strategies, but helps players to better organize their own playing systems and to confront their own convictions with the incontestable reality offered by the direct applications of the mathematical model of the lotto game. As a must-have handbook for any lottery player, this book offers essential information about the game itself and can provide the basis for gaming decisions of any kind. (shrink)

Risk assessment is one of the prerequisites for understanding its causes and possible consequences. We base our risk assessment on the principles described in the European standard EN 31000 - Risk Management Process. This standard comprehensively describes the continuous activities that are necessary in managing risks and minimizing their possible adverse effects on the operation of the system under investigation. In this activity, it is necessary to first identify the existing risks, then analyze and evaluate the identified risks. In the (...) analysis of existing risks, it is possible to use both qualitative and quantitative analytical methods, or combine them. We use qualitative methods in cases where we do not have a sufficient amount of input information, these are more subjective. Quantitative methods are more accurate, but also more demanding on input information and time. The choice of a suitable analytical method is a basic prerequisite for knowledge of risks and their evaluation. The values of individual risks obtained in this way are the basis for determining the measures that are ecessary to minimize them, i.e., to adjust them to an acceptable level. The draft measures are always based on the value of the individual components used to calculate the risk number, as well as on the value of the asset , which needs to be protected. Appropriately chosen analytical methods are one of the basic prerequisites for the consistent application of the principles of risk management, as a continuous process aimed at increasing the overall security of the system under study. In the article, the author describes the procedures used in risk assessment, as well as specific analytical methods that can be used in working with risks. The aim of identifying risk factors is to create a list of events that could cause undesirable disruption to ongoing processes. At this stage, we define all the risks that will be subsequently analyzed and evaluated. When identifying, we can use methods such as, e.g. SWOT, PHA (Preliminary Hazard Analysis) or CA (Checklist Analysis). Methods suitable for determining the causes and creating scenarios for the course of a risk event are ETA (Event Tree Analysis) or FTA (Fault Tree Analysis). The basic analysis of the system can be performed using the FMEA method (Failure Mode and Effect Analysis), which provides a numerical risk assessment. By comparison with the numerical value of the risk that we are willing to accept, we obtain 2 groups of risks. Acceptable, which will be given regular attention and unacceptable, which we will focus on in risk management and we will try to minimize its negative affect on the functioning of the system under study. (shrink)

Many biological processes and objects can be described by fractals. The paper uses a new type of objects – blinking fractals – that are not covered by traditional theories considering dynamics of self-similarity processes. It is shown that both traditional and blinking fractals can be successfully studied by a recent approach allowing one to work numerically with infinite and infinitesimal numbers. It is shown that blinking fractals can be applied for modeling complex processes of growth of biological systems including (...) their season changes. The new approach allows one to give various quantitative characteristics of the obtained blinking fractals models of biological systems. (shrink)

In his [1937, 1938], Paul Dirac proposed his “Large Number Hypothesis” (LNH), as a speculative law, based upon what we will call the “Large Number Coincidences” (LNC’s), which are essentially “coincidences” in the ratios of about six large dimensionless numbers in physics. Dirac’s LNH postulates that these numerical coincidences reflect a deeper set of law-like relations, pointing to a revolutionary theory of cosmology. This led to substantial work, including the development of Dirac’s later [1969/74] cosmology, and other alternative cosmologies, such (...) as the Brans-Dicke modification of GTR, and to extensive empirical tests. We may refer to the generic hypothesis of “Large Number Relations” (LNR’s), as the proposal that there are lawlike relations of some kind between the dimensionless numbers, not necessarily those proposed in Dirac’s early LNH. Such relations would have a profound effect on our concepts of physics, but they remain shrouded in mystery. Although Dirac’s specific proposals for LNR theories have been largely rejected, the subject retains interest, especially among cosmologists seeking to test possible variations in fundamental constants, and to explain dark energy or the cosmological constant. In the first two sections here we briefly summarize the basic concepts of LNR’s. We then introduce an alternative LNR theory, using a systematic formalism to express variable transformations between conventional measurement variables and the true variables of the theory. We demonstrate some consistency results and review the evidence for changes in the gravitational constant G. The theory adopted in the strongest tests of Ġ/G, by the Lunar Laser Ranging (LLR) experiments, assumes: Ġ/G = 3(dr/dt)/r – 2(dP/dt)/P – (dm/dt)/m, as a fundamental relationship. Experimental measurements show the RHS to be close to zero, so it is inferred that significant changes in G are ruled out. However when the relation is derived in our alternative theory it gives: Ġ/G = 3(dr/dt)/r – 2(dP/dt)/P – (dm/dt)/m – (dR/dt)/R. The extra final term (which is the Hubble constant) is not taken into account in conventional derivations. This means the LLR experiments are consistent with our LNR theory (and others), and they do not really test for a changing value of G at all. This failure to transform predictions of LNR theories correctly is a serious conceptual flaw in current experiment and theory. (shrink)

The objective of this article is to shed light on some challenging questions regarding public health and medical ethics that the Turkish healthcare system has recently been forced to confront. In recent years, terrorists in eastern Turkey have launched increasingly destructive attacks, including numerous attempts to undermine the social order by targeting not only government agencies but also the healthcare system. In this study, 54 terrorist incidents specifically targeting the Turkish healthcare system and healthcare professionals were analyzed and divided into (...) 6 categories according to the type of attack. Each category was evaluated in terms of the relevant ethical issues, with regard to both public health and medical ethics. This study shows that terrorist activity may lead to numerous breaches of public health and clinical ethics rules. Therefore, healthcare policy must involve special precautions to minimize breaches of public health and medical ethics in the face of terrorist attacks. (shrink)

The objective of this article is to shed light on some challenging questions regarding public health and medical ethics that the Turkish healthcare system has recently been forced to confront. In recent years, terrorists in eastern Turkey have launched increasingly destructive attacks, including numerous attempts to undermine the social order by targeting not only government agencies but also the healthcare system. In this study, 54 terrorist incidents specifically targeting the Turkish healthcare system and healthcare professionals were analyzed and divided into (...) 6 categories according to the type of attack. Each category was evaluated in terms of the relevant ethical issues, with regard to both public health and medical ethics. This study shows that terrorist activity may lead to numerous breaches of public health and clinical ethics rules. Therefore, healthcare policy must involve special precautions to minimize breaches of public health and medical ethics in the face of terrorist attacks. (shrink)

Strong nativist views about numerical concepts claim that human beings have at least some innate precise numerical representations. Weak nativist views claim only that humans, like other animals, possess an innate system for representing approximate numerical quantity. We present a new strong nativist model of the origins of numerical concepts and defend the strong nativist approach against recent cross-cultural studies that have been interpreted to show that precise numerical concepts are dependent on language and that they are restricted to speakers (...) of languages with the right kind of structure. (shrink)

Human biological memory systems have adapted to use technological artifacts to overcome some of the limitations of these systems. For example, when performing a difficult calculation, we use pen and paper to create and store external number symbols; when remembering our appointments, we use a calendar; when remembering what to buy, we use a shopping list. This chapter looks at the history of memory artifacts, describing the evolution from cave paintings to virtual reality. It first characterizes memory artifacts, (...) memory systems, and the two main functions such artifacts have, which are to aid individual users in completing memory tasks and as a cultural inheritance channel (section 2). It then outlines some of our first symbolic practices such as making cave paintings and figurines, and then moves on to outline several key developments in external representational systems and the artifacts that support these such as written language, numeral systems and counting devices, diagrams and maps, measuring devices, libraries and archives, photographs, analogue and digital computational artifacts, the World Wide Web, virtual reality, and smartphones (section 3). After that, it makes some brief points about the cumulative nature of the cultural evolution of memory artifacts and speculates about the possible future of memory artifacts, arguing that it is very difficult to look beyond an epistemological horizon of more than five years (section 4). (shrink)

The paper considers a new type of objects – blinking fractals – that are not covered by traditional theories studying dynamics of self-similarity processes. It is shown that the new approach allows one to give various quantitative characteristics of the newly introduced and traditional fractals using infinite and infinitesimal numbers proposed recently. In this connection, the problem of the mathematical modelling of continuity is discussed in detail. A strong advantage of the introduced computational paradigm consists of its well-marked numerical character (...) and its own instrument – Infinity Computer – able to execute operations with infinite and infinitesimal numbers. (shrink)

Numbers are concepts whose content, structure, and organization are influenced by the material forms used to represent and manipulate them. Indeed, as argued here, it is the inclusion of multiple forms (distributed objects, fingers, single- and two-dimensional forms like pebbles and abaci, and written notations) that is the mechanism of numerical elaboration. Further, variety in employed forms explains at least part of the synchronic and diachronic variability that exists between and within cultural number systems. Material forms also impart characteristics (...) like linearity that may persist in the form of knowledge and behaviors, ultimately yielding numerical concepts that are irreducible to and functionally independent of any particular form. Material devices used to represent and manipulate numbers also interact with language in ways that reinforce or contrast different aspects of numerical cognition. Not only does this interaction potentially explain some of the unique aspects of numerical language, it suggests that the two are complementary but ultimately distinct means of accessing numerical intuitions and insights. The potential inclusion of materiality in contemporary research in numerical cognition is advocated, both for its explanatory power, as well as its influence on psychological, behavioral, and linguistic aspects of numerical cognition. (shrink)

Context: Consistency of mathematical constructions in numerical analysis and the application of computerized proofs in the light of the occurrence of numerical chaos in simple systems. Purpose: To show that a computer in general and a numerical analysis in particular can add its own peculiarities to the subject under study. Hence the need of thorough theoretical studies on chaos in numerical simulation. Hence, a questioning of what e.g. a numerical disproof of a theorem in physics or a prediction in (...) numerical economics could mean. Method: An algebraic simple model system is subjected to a deeper structure of underlying variables. With an algorithm simulating the steps in taking a limit of second order difference quotients the error terms are studied at the background of their algebraic expression. Results: With the algorithm that was applied to a simple quadratic polynomial system we found unstably amplified round-off errors. The possibility of numerical chaos is already known but not in such a simple system as used in our paper. The amplification of the errors implies that it is not possible with computer means to constructively show that the algebra and numerical analysis will ‘on the long run’ converge to each other and the error term will vanish. The algebraic vanishing of the error term cannot be demonstrated with the use of the computer because the round-off errors are amplified. In philosophical terms, the amplification of the round-off error is equivalent to the continuum hypothesis. This means that the requirement of (numerical) construction of mathematical objects is no safeguard against inference-only conclusions of qualities of (numerical) mathematical objects. Unstably amplified round-off errors are a same type of problem as the ordering in size of transfinite cardinal numbers. The difference is that the former problem is created within the requirements of constructive mathematics. This can be seen as the reward for working numerically constructive. (shrink)

What are numbers, and where do they come from? A novel answer to these timeless questions is proposed by cognitive archaeologist Karenleigh A. Overmann, based on her groundbreaking study of material devices used for counting in the Ancient Near East—fingers, tallies, tokens, and numerical notations—as interpreted through the latest neuropsychological insights into human numeracy and literacy. The result, a unique synthesis of interdisciplinary data, outlines how number concepts would have been realized in a pristine original condition to develop into one (...) of the ancient world’s greatest mathematical traditions, a foundation for mathematical thinking today. In this view, numbers are abstract from their inception and materially bound at their most elaborated. The research updates historical work on Neolithic tokens and interpretations of Mesopotamian numbers, challenging several longstanding assumptions about numbers in the process. The insights generated are also applied to the role of materiality in human cognition more generally, including how concepts become distributed across and independent of the material forms used for their representation and manipulation; how societies comprised of average individuals use material structures to create elaborated systems of numeracy and literacy; and the differences between thinking through and thinking about materiality. (shrink)

Numerous high-profile ethics scandals, rising inequality, and the detrimental effects of climate change dramatically underscore the need for business schools to instill a commitment to social purpose in their students. At the same time, the rising financial burden of education, increasing competition in the education space, and overreliance on graduates’ financial success as the accepted metric of quality have reinforced an instrumentalist climate. These conflicting aims between social and financial purpose have created an existential crisis for business education. To resolve (...) this impasse, we draw on the concept of moral self-awareness to offer a system-theoretical strategy for crowding-in a culture of ethics within business schools. We argue that to do so, business schools will need to (1) reframe the purpose of business, (2) reframe the meaning of professional success, and (3) reframe the ethos of business education itself. (shrink)

A new computational methodology for executing calculations with infinite and infinitesimal quantities is described in this paper. It is based on the principle ‘The part is less than the whole’ introduced by Ancient Greeks and applied to all numbers (finite, infinite, and infinitesimal) and to all sets and processes (finite and infinite). It is shown that it becomes possible to write down finite, infinite, and infinitesimal numbers by a finite number of symbols as particular cases of a unique framework. The (...) new methodology has allowed us to introduce the Infinity Computer working with such numbers (its simulator has already been realized). Examples dealing with divergent series, infinite sets, and limits are given. (shrink)

On a now orthodox view, humans and many other animals possess a “number sense,” or approximate number system, that represents number. Recently, this orthodox view has been subject to numerous critiques that question whether the ANS genuinely represents number. We distinguish three lines of critique – the arguments from congruency, confounds, and imprecision – and show that none succeed. We then provide positive reasons to think that the ANS genuinely represents numbers, and not just non-numerical confounds or exotic substitutes for (...) number, such as “numerosities” or “quanticals,” as critics propose. In so doing, we raise a neglected question: numbers of what kind? Proponents of the orthodox view have been remarkably coy on this issue. But this is unsatisfactory since the predictions of the orthodox view, including the situations in which the ANS is expected to succeed or fail, turn on the kind of number being represented. In response, we propose that the ANS represents not only natural numbers, but also non-natural rational numbers. It does not represent irrational numbers, however, and thereby fails to represent the real numbers more generally. This distances our proposal from existing conjectures, refines our understanding of the ANS, and paves the way for future research. (shrink)

Numerous studies indicate that racial minorities are both more likely to be executed for murder and that those who murder them are less likely to be executed than if they murder whites. Death penalty opponents have long attempted to use these studies to argue for a moratorium on capital punishment. Whatever the merits of such arguments, they overlook the fact that such discrimination alters the costs of murder; racial discrimination imposes higher costs on minorities for murdering through tougher sentences, and (...) it imposes lower costs on whites for murdering minorities by dispensing weaker sentences. These cost differentials constitute an injustice not simply to actual minority defendants in capital cases, nor simply to the actual minority victims of murder, but to all members of minority communities. I here offer two arguments for a moratorium on capital punishment: The first draws upon evidence of racial discrimination against minority defendants in capital cases, and claims that such discrimination modifies the costs of murder in such a way that minority individuals do not enjoy equal status under the law. The second draws upon the evidence regarding racial discrimination in relation to the race of victims, and claims that such discrimination modifies the costs of murder in such a way that minority individuals do not enjoy the equal protection of the law. Thus, by not assigning equal costs to murder, the American criminal justice system fails to provide racial minorities the equality under the law and discounts the value of their lives and liberties. A moratorium is the least unjust response to such a social injustice. I also reply to the criticism that a moratorium prevents us from executing deserving murderers. (shrink)

This chapter presents a re-understanding of the contents of our analog magnitude representations (e.g., approximate duration, distance, number). The approximate number system (ANS) is considered, which supports numerical representations that are widely described as fuzzy, noisy, and limited in their representational power. The contention is made that these characterizations are largely based on misunderstandings—that what has been called “noise” and “fuzziness” is actually an important epistemic signal of confidence in one’s estimate of the value. Rather than the ANS having noisy (...) or fuzzy numerical content, it is suggested that the ANS has exquisitely precise numerical content that is subject to epistemic limitations. Similar considerations will arise for other analog representations. The chapter discusses how this new understanding of ANS representations recasts the learnability problem for number and the conceptual changes that children must accomplish in the number domain. (shrink)

Previous discussions of the origins of writing in the Ancient Near East have not incorporated the neuroscience of literacy, which suggests that when southern Mesopotamians wrote marks on clay in the late-fourth millennium, they inadvertently reorganized their neural activity, a factor in manipulating the writing system to reflect language, yielding literacy through a combination of neurofunctional change and increased script fidelity to language. Such a development appears to take place only with a sufficient demand for writing and reading, such as (...) that posed by a state-level bureaucracy; the use of a material with suitable characteristics; and the production of marks that are conventionalized, handwritten, simple, and non-numerical. From the perspective of Material Engagement Theory, writing and reading represent the interactivity of bodies, materiality, and brains: movements of hands, arms, and eyes; clay and the implements used to mark it and form characters; and vision, motor planning, object recognition, and language. Literacy is a cognitive change that emerges from and depends upon the nexus of interactivity of the components. (shrink)