In recent times, a previously unchallenged and longstanding communis opinio concerning the extant manuscript tradition of Valerius Flaccus' Argonautica has been shattered by Prof. W.-W. Ehlers in his revelation that the fifteenth-century Laurentianus plut. 39.38, L, written by the Florentine scholar, Niccolò Niccoli, is independent of the much exalted oldest witness, Vaticanus Latinus 3277, V, copied in Fulda in the second quarter of the ninth century. With equally silent subservience to the hazardous and now discredited principle, vetustissimus et optimus, (...) second place in the tradition had commonly been given to a manuscript of similar age to that of V, namely S, discovered in 1416 at the monastery of St. Gall by three humanist scholars, Poggio Bracciolini, Cencio Rustici and Bartolomeo da Montepulciano, and subsequently lost, except through reconstruction from six extant fifteenth-century apographa, each instantly recognizable from the fact that they contain only 1.1–4.317. Between V and S, however, as Ehlers and others before him have argued, a common exemplar, a, must be deduced to exist. (shrink)

One of the most primary, as well as intriguing, questions in the field of St. Bonaventure studies is the authorship of several disputed questions. After the publication of the monumental Quaracchi edition some scholars incorporated into the Corpus Bonaventurianum questions De superfluo, De caritate, De novissimis, and De theologia. The ascription of the above questions to Bonaventure met with a severe criticism which disproved his authorship of all of them. Of the spurious questions a series conserved in the Florentine (...) manuscript (Biblioteca Nazionale Centrale, Conv. Soppr. D.IV.27) has remained until now unedited and its authenticity unsettled. This volume presents a critical edition of all the Florentine questions -- De productione rerum, De imagine, De anima. It aims to throw some light, on the one hand, on the problem of their authenticity. It formulates arguments for Bonaventure's authorship which are based mostly on the doctrinal affinity of the questions with non-spurious Bonaventure's works. On the other hand, it discloses the merely hypothetical character of these arguments."--Back cover. (shrink)

Published in 1880–1, this three-volume edition of the extant works of the Greek mathematician Archimedes of Syracuse was edited by the Danish philologist and historian Johan Ludvig Heiberg, whose Quaestiones Archimedeae is also reissued in this series. He later discovered a medieval palimpsest containing lost works by Archimedes, which significantly expanded the canon, but the present collection was produced long before this and therefore contains the works known at the time of publication. Heiberg consulted a Florentine codex, which (...) he painstakingly compared with other sources to produce his edition. This second volume contains On Spirals, On the Equilibrium of Planes, The Sand Reckoner, The Quadrature of the Parabola, On Floating Bodies, the Liber Assumptorum, the cattle problem and fragments. The texts are given in the original Greek with parallel Latin translation, notes and introductory material. (shrink)

Published in 1880–1, this three-volume edition of the extant works of the Greek mathematician Archimedes of Syracuse was edited by the Danish philologist and historian Johan Ludvig Heiberg, whose Quaestiones Archimedeae is also reissued in this series. He later discovered a medieval palimpsest containing lost works by Archimedes, which significantly expanded the canon, but the present collection was produced long before this and therefore contains the works known at the time of publication. Heiberg consulted a Florentine codex, which (...) he painstakingly compared with other sources to produce his edition. This first volume contains On the Sphere and the Cylinder, On the Measurement of a Circle and On Conoids and Spheroids. The texts are given in the original Greek with parallel Latin translation, notes and introductory material. (shrink)

Published in 1880–1, this three-volume edition of the extant works of the Greek mathematician Archimedes of Syracuse was edited by the Danish philologist and historian Johan Ludvig Heiberg, whose Quaestiones Archimedeae is also reissued in this series. He later discovered a medieval palimpsest containing lost works by Archimedes, which significantly expanded the canon, but the present collection was produced long before this and therefore contains the works known at the time of publication. Heiberg consulted a Florentine codex, which (...) he painstakingly compared with other sources to produce his edition. This third volume contains the editor's Latin prolegomena - his own extended essay on the works of Archimedes - followed by the commentaries on Archimedes by Eutocius of Ascalon and indexes. The texts are given in the original Greek with parallel Latin translation, notes and introductory material. (shrink)

Published in 1880–1, this three-volume edition of the extant works of the Greek mathematician Archimedes of Syracuse was edited by the Danish philologist and historian Johan Ludvig Heiberg, whose Quaestiones Archimedeae is also reissued in this series. He compiled this edition from a Florentine codex, which he compared with other extant sources. Volume 1 contains On the Sphere and the Cylinder, On the Measurement of a Circle and On Conoids and Spheroids. Volume 2 contains On Spirals, On the (...) Equilibrium of Planes, The Sand Reckoner, The Quadrature of the Parabola, On Floating Bodies, the Liber Assumptorum, the cattle problem and fragments. Volume 3 contains the editor's own Latin prolegomena, the commentaries on Archimedes by Eutocius of Ascalon and indexes. The texts are given in Greek with parallel Latin translation, notes and introductory material. (shrink)

Recent work in the history of science has questioned the Eurocentric nature of the field and sought to include a more global approach that would serve to displace center–periphery models in favor of approaches that take seriously local knowledge production. Historians of Iberian colonial science have taken up this approach, which involves reliance on indigenous knowledge traditions of the Americas. These traditions present a number of challenges to modern researchers, including availability and reliability of source material, issues of translation and (...) identification, and lack of systematization. This essay explores the challenges that emerged in the author’s attempt to compile a pre-contact Nahua pharmacopeia, the reasons for these challenges, and the ways they may – or may not – be overcome. (shrink)

The mathematical scenario in Italy during the Late Middle Ages and the Renaissance is mainly dominated by the treatises on the abacus, which developed together with the abacus schools. In that context, between approximately the last thirty years of the fourteenth century and the first twenty years of the sixteenth century, the manuscript and printed tradition tell us of queries and challenges, barely known or totally unknown, in which the protagonists were abacus masters. We report in this work on the (...) most significant examples and draw out interesting cues for thoughts and remarks of a scientific, historical and biographical nature. Five treatises, written in the fifteenth and sixteenth century, have been the main source of inspiration for this article: the Trattato di praticha d’arismetricha and the Tractato di praticha di geometria included in the codices Palat. 573 and Palat. 577 (c. 1460) kept in the Biblioteca Nazionale of Florence and written by an anonymous Florentine disciple of the abacist Domenico di Agostino Vaiaio; another Trattato di praticha d’arismetrica written by Benedetto di Antonio da Firenze in 1463 and included in the codex L.IV.21 kept in the Biblioteca Comunale of Siena; the Tractatus mathematicus ad discipulos perusinos written by Luca Pacioli between 1477 and 1480, manuscript Vat. Lat. 3129 of the Biblioteca Apostolica Vaticana; and Francesco Galigai’s Summa de arithmetica, published in Florence in 1521. (shrink)

The author makes an introduction to non-standard analysis, then extends the dialectics to “neutrosophy” – which became a new branch of philosophy. This new concept helps in generalizing the intuitionistic, paraconsistent, dialetheism, fuzzy logic to “neutrosophic logic” – which is the first logic that comprises paradoxes and distinguishes between relative and absolute truth. Similarly, the fuzzy set is generalized to “neutrosophic set”. Also, the classical and imprecise probabilities are generalized to “neutrosophic probability”.

A simple, direct, fast point of view regarding my perception of Administration Philosophy, Teaching Philosophy, Research Philosophy, and What I Can Bring to This Institution.

We have parameterized Einstein’s thought experiment with atomic clocks, supposing that we knew neither if the space and time are relative or absolute, nor if the speed of light was ultimate speed or not. We have obtained a Parameterized Special Theory of Relativity (PSTR) (1982). Our PSTR generalized not only Einstein’s Special Theory of Relativity, but also our Absolute Theory of Relativity, and introduced three more possible Relativities to be studied in the future. After the 2011 CERN’s superluminal neutrino experiments, (...) we recall our ideas and invite researchers to deepen the study of PSTR, ATR, and check the three new mathematically emerged Relativities 4.3, 4.4, and 4.5. (shrink)

In this paper we present the development from paradoxism to neutrosophy, which gave birth to neutrosophic set and logic and especially to NeutroAlgebraic Structures (or NeutroAlgebras) and AntiAlgebraic Structures (or AntiAlgebras) that are generalizations and alternatives of the classical algebraic structures.

El presente fotorreportaje relata, en texto e imágenes, encuentros científicos y personales del autor con motivo de un congreso sobre neutrosofía y plitogenia que tuvo lugar los días 8 y 9 de julio de 2024 en la Universidad César Vallejo, en Lima (Perú), y contó con la asistencia de más de un centenar de participantes, estudiantes y profesores de Perú, Ecuador, Cuba, México, República Dominicana, Estados Unidos. Las actas de la conferencia se pueden descargar del Repertorio Digital de la Universidad (...) de Nuevo. Méxic o, EE.UU. (shrink)

In this short note many conjectures on partitions of integers as summations of prime numbers are presented, which are extension of Goldbach conjecture.

In this short paper we extend the dialectical triad thesis-antithesis-synthesis to the neutrosophic tetrad thesis-antithesis-neutrothesis-neutrosynthesis. We do this for better reflecting our world, since the neutralities between opposites play an important role. The neutrosophic synthesis (neutrosynthesis) is more refined that the dialectical synthesis. It carries on the unification and synthesis regarding the opposites and their neutrals too.

Neutrosophy considers a proposition, theory, event, concept, or entity, "A" in relation to its opposite, "Anti A" and that which is not A, "Non-A", and that which is neither "A" nor "Anti-A", denoted by "Neut-A". Neutrosophy is the basis of neutrosophic logic, neutrosophic probability, neutrosophic set, and neutrosophic statistics.

ArgumentThis paper argues that the economics discipline is highly concentrated, which may inhibit scientific innovation and change in the future. The argument is based on an empirical investigation of six dimensions of concentration in economics between 1956 and 2016 using a large-scale data set. The results show that North America accounts for nearly half of all articles and three quarters of all citations. Twenty institutions reap a share of 42 percent of citations, five journals a share of 28.5 percent, and (...) 100 authors a share of 15.5 percent. A total of 2.8 percent of citations may be attributed to heterodox schools of thought. Also top articles are concentrated along these dimensions. Overall, concentration has strongly increased over the last six decades. (shrink)

In this paper we define the Soft Set Product as a product of many soft sets and afterwards we extend it to the HyperSoft Set. Similarly, the IndetermSoft Product is extended to the IndetermHyperSoft Set. We also present several applications of the Soft Set Product to Fuzzy (and fuzzy-extensions) Soft Set Product and to IndetermSoft Set and IndetermHyperSoft Set.

Although we should respect the autonomy of our patients and engage in shared decision-making with them, the debate over Physician Assisted Suicide (PAS) has not given sufficient attention to other principles that need to be addressed in these discussions. In trying to offer a sound ethical analysis of this movement, we must weigh consequences that go beyond the individual patient and the physician. Concerns about the unintended consequences of PAS to the social dimension of human life are explored in this (...) paper. (shrink)

This book contains concrete examples from history, economy, biology, digital world, nuclear physics, agriculture and so on about breaking a neutrosophic dynamic system (i.e. a dynamic system that has indeterminacy) from inside. We define a neutrosophic mathematical model using a system of ordinary differential equations and the neutrosophic probability in order to approximate the process of breaking from inside a neutrosophic complex dynamic system. It shows that for breaking from inside it is needed a smaller force than for breaking from (...) outside the neutrosophic complex dynamic system. Methods that have been used in the past for breaking from inside are listed. Simulation and animation of this neutrosophic dynamical system are needed for the future since, by changing certain parameters, various types of breaking from inside may be simulated. Abstract: Această carte conține exemple concrete din istorie, economie, biologie, spațiu digital, fizică nucleară, agricultură și altele, de distrugere unui sistem neutrosofic dinamic (adică a unui sistem care are indeterminări), acționând din interiorul acestuia. Descriem un model neutrosofic matematic folosind un sistem de ecuații diferențiale ordinare și probabilitatea neutrosofică, cu scopul de a aproxima procesul de distrugere din interior a unui sistem neutrosofic dinamic complex. Se demonstrează că, pentru distrugerea din interior a unui astfel de sistem, este necesară o forță mai mică decât pentru distrugerea din exterior. Sunt enumerate metode folosite în trecut pentru distrugerea din interior. Simularea și animația acestui sistem neutrosofic dinamic sunt necesare pentru viitor, deoarece, schimbând diferiți parametri, se pot simula diferite tipuri de distrugeri din interior. (shrink)

Classes of linguistic paradoxes are introduced with examples and explanations. They are part of the author's work on the Paradoxist Philosophy based on mathematical logic.

La pensée eschatologique d'lsraël surgit et s'élabore au temps et sur les lieux de l'Exil, dans une population de personnes déplacées qui ont perdu leurs repères à leur passé et à leur terre. Elle inspire de rédiger un Livre pour redonner à ce peuple un lieu habitable, de lui restituer une histoire en lui transmettant des récits et des textes du passé réinterprétés par une vision d'avenir. Paradoxalement, l'espérance eschatologique se fait remémoration, historiographie. Faisant le deuil du passé en le (...) relisant, elle rend à nouveau possible l'histoire du peuple avec son Dieu, qui paraissait absent alors qu'il n'était que souffrant. Le « dernier jour » se profile dès le début de cette histoire, accomplissement inaugural, promesse d'un renouveau toujours possible. The eschatological thinking of Israel arose and was developed at the time and in the lands of the Exile. This took place in a population of displaced persons who had lost their connection with the past as well as their land. This thinking inspired them to compose a Book that would return a hospitable place and restore a history to them. This was done by the transmission of stories and texts reinterpreting the past, with a vision of the future. Paradoxically, eschatological hope became remembering, historiography. Burying the past through a rereading, hope again makes possible the history of a people with its God, who seemed absent while He was only suffering. The « Last Day » develops its profile from the start of this history, as an inaugural fulfilments, a renewal always possible. (shrink)

Essays in Romanian, English, French, and Portuguese, some of which are translations.

In this article, the concepts of Nth Power Set of a Set, Super-Hyper-Oper-Operation, Super-Hyper-Axiom, SuperHyper-Algebra, and their corresponding Neutrosophic Super-Hyper-Oper-Operation, Neutrosophic Super-Hyper-Axiom and Neutrosophic Super-Hyper-Algebra are reviewed. In general, in any field of knowledge, really what are found are Super-HyperStructures (or more specifically Super-Hyper-Structures (m, n)).

In this paper we recall, improve, and extend several definitions, properties and applications of our previous 2019 research referred to NeutroAlgebras and AntiAlgebras (also called NeutroAlgebraic Structures and respectively AntiAlgebraic Structures). Let <A> be an item (concept, attribute, idea, proposition, theory, etc.). Through the process of neutrosphication, we split the nonempty space we work on into three regions {two opposite ones corresponding to <A> and <antiA>, and one corresponding to neutral (indeterminate) <neutA> (also denoted <neutroA>) between the opposites}, which may (...) or may not be disjoint – depending on the application, but they are exhaustive (their union equals the whole space). A NeutroAlgebra is an algebra which has at least one NeutroOperation or one NeutroAxiom (axiom that is true for some elements, indeterminate for other elements, and false for the other elements). A Partial Algebra is an algebra that has at least one Partial Operation, and all its Axioms are classical (i.e. axioms true for all elements). Through a theorem we prove that NeutroAlgebra is a generalization of Partial Algebra, and we give examples of NeutroAlgebras that are not Partial Algebras. We also introduce the NeutroFunction (and NeutroOperation). (shrink)

Neutrosophic theory and its applications have been expanding in all directions at an astonishing rate especially after of the introduction the journal entitled “Neutrosophic Sets and Systems”. New theories, techniques, algorithms have been rapidly developed. One of the most striking trends in the neutrosophic theory is the hybridization of neutrosophic set with other potential sets such as rough set, bipolar set, soft set, hesitant fuzzy set, etc. The different hybrid structures such as rough neutrosophic set, single valued neutrosophic rough set, (...) bipolar neutrosophic set, single valued neutrosophic hesitant fuzzy set, etc. are proposed in the literature in a short period of time. Neutrosophic set has been an important tool in the application of various areas such as data mining, decision making, e-learning, engineering, law, medicine, social science, and some more. -/- This book explores the emerging field of Neutrosophic Algebraic Structures, focusing on both their theoretical foundations and practical applications. We apply innovative algorithmic methods to investigate the complex interactions of neutrosophic elements, such as neutrosophic numbers, sets, and functions, within algebraic systems. Our goal is to show how neutrosophic structures challenge and expand traditional algebraic approaches, offering solutions to problems across diverse fields like computer science, engineering, artificial intelligence, and decision-making. (shrink)

Fotojurnalul de față relatează, în text și imagini, întâlniri științifice și personale ale autorului ocazionate de o conferință de neutrosofie și plitogenie desfășurată în 8 și 9 iulie 2024, organizată la Universidad César Vallejo, în Lima (Peru), la care au luat parte peste o sută de participanți, mulți studenți și profesori din Peru, Ecuador, Cuba, Mexic, Republica Dominicană, Statele Unite ale Americii. Lucrările conferinței se pot descărca din repetoriul digital al Universității New Mexico.

This article explores the formation of a tranimal, hippopotamus alter-ego. Confronting transgender with transpecies, the author claims that his hippopotamus “identity” allowed him to escape, all at once, several sets of categorization that govern human bodies. He starts with an account of how his metaphorical hippo-self is collectively produced and performed, distinguishing the subjective, the intersubjective and the social. The article then investigates the politics of equating transgender and transpecies, critically examining the question of the inclusion of “xenogenders” in the (...) trans political movement. Finally, the author returns to the magical power of metaphors, arguing that metaphors do materialize insofar as the flesh does not remain unchanged by them. Analogizing his hippo-self to a “cut” as theorized by Eva Hayward – a regeneration of the boundaries of the self – he offers a final crossing to the world of fiction by showing how the His Dark Materials trilogy outlines an aesthetics of porosity, which suggests that the self is, as much as a novel, a work of fiction. (shrink)

In this paper, we generalize the soft set to the hypersoft set by transforming the function F into a multi-attribute function. Then we introduce the hybrids of Crisp, Fuzzy, Intuitionistic Fuzzy, Neutrosophic, and Plithogenic Hypersoft Set.

Graph theory is a specific concept that has numerous applications throughout many industries. Despite the advancement of this technique, graph theory can still yield ambiguous and imprecise results. In order to cut down on these indeterminate factors, neutrosophic logic has emerged as an applicable solution that is gaining significant attention in solving many real-life decision-making problems that involve uncertainty, impreciseness, vagueness, incompleteness, inconsistency, and indeterminacy. However, empirical research on this specific graph set is lacking. Neutrosophic Graph Theory and Algorithms is (...) a collection of innovative research on the methods and applications of neutrosophic sets and logic within various fields including systems analysis, economics, and transportation. While highlighting topics including linear programming, decision-making methods, and homomorphism, this book is ideally designed for programmers, researchers, data scientists, mathematicians, designers, educators, researchers, academicians, and students seeking current research on the various methods and applications of graph theory. (shrink)

Drawing on the theory of engagement, the present study aims to examine the outcomes of the co-creation experience in a realistic co-creation setting, a hotpot restaurant. To this end, the current research links the relationship marketing literature to hospitality and tourism research and formulates a novel framework by incorporating tourists’ co-creation experience, brand evangelism, brand trust, and brand passion in an integrated conceptual model. Using a quantitative research design, a total of 453 international tourists were surveyed in China. The findings (...) revealed that co-creation experience dimensions positively impact brand evangelism, trust, and passion. Additionally, we found that brand trust and brand passion positively affect brand evangelism. We also confirmed the mediating effect of brand trust and brand passion in bridging the co-creation experience and brand evangelism. This study offers valuable insights for restaurant brand managers regarding attracting and engaging foreign travelers with their service businesses. (shrink)

We recall and improve our 2019 and 2020 concepts of n-SuperHyperGraph, Plithogenic nSuperHyperGraph, n-Power Set of a Set, and we present some application from the real world. The nSuperHyperGraph is the most general form of graph today and it is able to describe the complex reality we live in, by using n-SuperVertices (groups of groups of groups etc.) and nSuperHyperEdges (edges connecting groups of groups of groups etc.).

Issues related to research integrity receive increasing attention in policy discourse and beyond with most universities having introduced by now courses addressing issues of good scientific practice. While communicating expectations and regulations related to good scientific practice is essential, criticism has been raised that integrity courses do not sufficiently address discipline and career-stage specific dimensions, and often do not open up spaces for in-depth engagement. In this article, we present the card-based engagement method RESPONSE_ABILITY, which aims at supporting researchers in (...) developing their ability to respond to challenges of good scientific practice. The method acknowledges that what counts and what does not count as acceptable practice may not be as clear-cut as imagined and that research environments matter when it comes to integrity issues. Using four sets of cards as stimulus material, participants are invited to reflect individually and collectively about questions of research integrity from different perspectives. This approach is meant to train them to negotiate in which contexts certain practices can still be regarded as acceptable and where possible transgressions might begin. RESPONSE_ABILITY can be seen as fostering the creation of an integrity culture as it invites a more reflexive engagement with ideals and realities of good practice and opens a space to address underlying value conflicts researchers may be confronted with. Concluding the article, we call for caution that addressing issues of integrity meaningfully requires striking a delicate balance between raising researchers’ awareness of individual responsibilities and creating institutional environments that allow them to be response-able. (shrink)

In this paper, we prove that Neutrosophic Statistics is more general than Interval Statistics, since it may deal with all types of indeterminacies (with respect to the data, inferential procedures, probability distributions, graphical representations, etc.), it allows the reduction of indeterminacy, and it uses the neutrosophic probability that is more general than imprecise and classical probabilities and has more detailed corresponding probability density functions. While Interval Statistics only deals with indeterminacy that can be represented by intervals. And we respond to (...) the arguments by Woodall et al. [1]. We show that not all indeterminacies (uncertainties) may be represented by intervals. Also, in some cases, we should better use hesitant sets (that have less indeterminacy) instead of intervals. We redirect the authors to the Plithogenic Probability and Plithogenic Statistics which are the most general forms of MultiVariate Probability and Multivariate Statistics respectively (including, of course, the Imprecise Probability and Interval Statistics as subclasses). (shrink)

This short article pairs the realms of “Mathematics”, “Philosophy”, and “Poetry”, presenting some corners of intersection of this type of scientocreativity. Poetry have long been following mathematical patterns expressed by stern formal restrictions, as the strong metrical structure of ancient Greek heroic epic, or the consistent meter with standardized rhyme scheme and a “volta” of Italian sonnets. Poetry was always connected to Philosophy, and further on, notable mathematicians, like the inventor of quaternions, William Rowan Hamilton, or Ion Barbu, the creator (...) of the Barbilian spaces, have written appreciated poems. We will focus here on an avant-garde movement in literature, art, philosophy, and science, called Paradoxism, founded in Romania in 1980 by a mathematician, philosopher and poet, and on the laboured writing exercises of the Oulipo group, founded in Paris in 1960 by mathematicians and poets, both of them still in act. (shrink)

In the fifth version of our response-paper [26] to Imamura’s criticism, we recall that NonStandard Neutrosophic Logic was never used by neutrosophic community in no application, that the quarter of century old neutrosophic operators (1995-1998) criticized by Imamura were never utilized since they were improved shortly after but he omits to tell their development, and that in real world applications we need to convert/approximate the NonStandard Analysis hyperreals, monads and binads to tiny intervals with the desired accuracy – otherwise they (...) would be inapplicable. We point out several errors and false statements by Imamura [21] with respect to the inf/sup of nonstandard subsets, also Imamura’s “rigorous definition of neutrosophic logic” is wrong and the same for his definition of nonstandard unit interval, and we prove that there is not a total order on the set of hyperreals (because of the newly introduced Neutrosophic Hyperreals that are indeterminate), whence the Transfer Principle from R to R* is questionable. After his criticism, several response publications on theoretical nonstandard neutrosophics followed in the period 2018-2022. As such, I extended the NonStandard Analysis by adding the left monad closed to the right, right monad closed to the left, pierced binad (we introduced in 1998), and unpierced binad - all these in order to close the newly extended nonstandard space (R*) under nonstandard addition, nonstandard subtraction, nonstandard multiplication, nonstandard division, and nonstandard power operations [23, 24]. Improved definitions of NonStandard Unit Interval and NonStandard Neutrosophic Logic, together with NonStandard Neutrosophic Operators are presented. (shrink)

Since the world is full of indeterminacy, the Neutrosophics found their place into contemporary research. We now introduce for the first time the notions of Neutrosophic Crisp Sets and Neutrosophic Topology on Crisp Sets. We develop the 2012 notion of Neutrosophic Topological Spaces and give many practical examples. Neutrosophic Science means development and applications of Neutrosophic Logic, Set, Measure, Integral, Probability etc., and their applications in any field. It is possible to define the neutrosophic measure and consequently the neutrosophic integral (...) and neutrosophic probability in many ways, because there are various types of indeterminacies, depending on the problem we need to solve. Indeterminacy is different from randomness. Indeterminacy can be caused by physical space, materials and type of construction, by items involved in the space, or by other factors. In 1965 [51], Zadeh generalized the concept of crisp set by introducing the concept of fuzzy set, corresponding to the situation in which there is no precisely defined set;there are increasing applications in various fields, including probability, artificial intelligence, control systems, biology and economics. Thus, developments in abstract mathematics using the idea of fuzzy sets possess sound footing. In accordance, fuzzy topological spaces were introduced by Chang [12] and Lowen [33]. After the development of fuzzy sets, much attention has been paid to the generalization of basic concepts of classical topology to fuzzy sets and accordingly developing a theory of fuzzy topology [1-58]. In 1983, the intuitionistic fuzzy set was introduced by K. Atanassov [55, 56, 57] as a generalization of the fuzzy set, beyond the degree of membership and the degree of non-membership of each element. In 1999 and 2002, Smarandache [71, 72, 73, 74] defined the notion of Neutrosophic Sets, which is a generalization of Zadeh’s fuzzy set and Atanassov's intuitionistic fuzzy set. Some neutrosophic concepts have been investigated by Salama et al. [61-70]. Forwarding the study of neutrosophic sets, this book consists of seven chapters, targeting to:  generalize the previous studies in [1-59], and[91-94] so to define the neutrosopic crisp set and neutrosophic set concepts;  discuss their main properties; A. A. Salama & Florentin Smarandache  introduce and study some concepts of neutrosophic crisp and neutrosophic topological spaces and deduce their properties;  deduce many types of functions and give the relationships between different neutrosophic topological spaces, which helps to build new properties of neutrosophic topological spaces;  stress once more the importance of Neutrosophic Ideal as a nontrivial extension of neutrosophic set and neutrosophic logic [71, 72, 73, 74];  propose applications on computer sciences by using neutrosophic sets. (shrink)

In this paper one introduces for the first time the IndetermSoft Set, as extension of the classical (determinate) Soft Set, that deals with indeterminate data, and similarly the HyperSoft Set extended to IndetermHyperSoft Set, where ‘Indeterm’ stands for ‘Indeterminate’ (uncertain, conflicting, not unique outcome). They are built on an IndetermSoft Algebra that is an algebra dealing with IndetermSoft Operators resulted from our real world. Afterwards, the corresponding Fuzzy / Intuitionistic Fuzzy / Neutrosophic / and other fuzzy-extension IndetermSoft Set & IndetermHyperSoft (...) Set are presented together with their applications. (shrink)

This paper presents for the first time the IndetermSoft Set, as an extension of the classical (determinate) Soft Set, which operates on indeterminate data, and similarly the HyperSoft Set extended to the IndetermHyperSoft Set, where 'Indeterm' means 'Indeterminate' (uncertain, conflicting, non-unique result). They are built on an IndetermSoft Algebra which is an algebra dealing with IndetermSoft Operators resulting from our real world. Subsequently, the IndetermSoft and IndetermHyperSoft Sets and their Fuzzy/Fuzzy Intuitionistic/Neutrosophic and other fuzzy extensions and their applications are presented.