Florentin Smarandache

his book introduces Type-k Neutrosophic Sets as a recursive extension of classical neutrosophic logic. While standard neutrosophic sets represent each proposition through three independent components—truth, indeterminacy, and falsity—Type-k Neutrosophic Sets allow each of these components to be recursively characterized by further neutrosophic triplets. This creates a hierarchical epistemic structure capable of representing nested uncertainty, meta-uncertainty, contradiction, ignorance, and paraconsistent states that cannot be adequately captured by classical probability, fuzzy logic, or intuitionistic fuzzy models. The book develops the formal foundations of this recursive framework, proves its expressive hierarchy, and extends it through single-valued neutrosophic tensors and plithogenic tensors for multi-criteria decision-making. It also explores connections with neutrosophic paraconsistent logic and applies the framework to the epistemic auditing of Large Language Models. Particular attention is given to hallucination detection, the limitations of softmax-based confidence, and the need for triadic and recursive models of truth, falsity, and indeterminacy in AI evaluation. By combining non-classical logic, tensorial structures, decision theory, and artificial intelligence auditing, the work proposes a new formal and computational framework for understanding uncertainty in complex epistemic systems. It is intended for researchers in neutrosophic logic, philosophy of logic, formal epistemology, artificial intelligence, paraconsistent reasoning, and decision sciences.

When a newspaper systematically omits a relevant causal explanation, it does not merely fail to inform — it performs an illocutionary act. Drawing on Austin's speech act theory and Grice's cooperative principle, this paper argues that the systematic press silence on structural drivers of urban violence in Guayaquil, Ecuador (2025) constitutes an act of structural erasure: a communicative action that produces a distorted public understanding of violence causation with real perlocutionary effects on policy, investment, and citizen perception. We provide the first empirical, quantitative evidence for this claim at scale. Using four Large Language Models (LLMs) to score 31 press sources (16 local Spanish-language, 15 international English-language) across eight structural causal conditions, and applying Neutrosophic fsQCA (N-fsQCA) to compute illocutionary consistency and indeterminacy, we find that five of eight structural drivers are systematically under-represented in the press relative to a validated structural baseline: weapons trafficking (gap=-0.444), extortion (gap=-0.345), prison linkages (gap=-0.340), drug routes (gap=-0.290), and economic pressure (gap=-0.214). Crucially, this silence is not random: it is patterned, stable across 31 sources and 4 LLMs, and exhibits a local/international asymmetry that reveals two distinct illocutionary communities enacting different acts of erasure about the same referential reality. We term this empirical pragmatics of violent silence: the systematic, measurable breach of Gricean maxims in press accounts of urban violence.

This book provides a comprehensive and systematic survey of modern set-theoretic frameworks developed to model uncertainty, vagueness, indeterminacy, and partial truth in real-world phenomena. It synthesizes a wide spectrum of classical and advanced theories, including fuzzy sets, intuitionistic fuzzy sets, neutrosophic sets, plithogenic sets, and numerous related extensions. The work organizes these concepts across multiple structural and conceptual dimensions—such as interval, granular, probabilistic, and complex extensions—highlighting their mathematical foundations, interrelations, and distinguishing characteristics. By unifying diverse approaches under a coherent analytical perspective, the book clarifies how different uncertainty models complement or generalize one another. It also explores emerging directions in set theory, including hybrid and dynamic frameworks, offering insights into their applicability in fields such as decision-making, control systems, artificial intelligence, and data analysis. Serving both as a reference and a conceptual guide, the book contributes to advancing the theoretical understanding and practical deployment of uncertainty-oriented mathematical models.

This book presents a comprehensive neutrosophic evaluation of the Mid-Day Meal Scheme (MDMS) in India, a large-scale public welfare initiative aimed at improving child nutrition, educational participation, and social equity. Recognizing the inherent complexity, uncertainty, and contradictions in policy implementation, the authors propose a neutrosophic framework that extends beyond classical and fuzzy evaluation models by incorporating degrees of truth, indeterminacy, and falsity. Using a micro-level case study approach, the research systematically analyzes key dimensions of the scheme, including nutritional adequacy, enrolment and attendance, food quality and hygiene, administrative efficiency, and social inclusion. The methodology integrates primary and secondary data, expert judgment, and neutrosophic aggregation techniques to construct a multidimensional performance assessment. Special emphasis is placed on identifying zones of indeterminacy and contradictory evidence, which are often overlooked in traditional evaluations. The study further develops neutrosophic indices, applies sensitivity and robustness analyses, and explores the implications of uncertainty-aware decision-making in public policy. The findings demonstrate that neutrosophic logic provides a more nuanced and realistic representation of policy outcomes, offering valuable insights for policymakers, researchers, and stakeholders. Ultimately, the book contributes both methodologically and practically to the application of neutrosophic theory in socio-economic policy analysis.

Real-world systems frequently involve vagueness, partial truth, incomplete information, and various forms of indeterminacy. To address these challenges, numerous generalized set-theoretic frameworks have been developed, including fuzzy sets, intuitionistic fuzzy sets, neutrosophic sets, plithogenic sets, and related extensions. These approaches provide powerful tools for modeling uncertainty in a mathematically rigorous and flexible manner. This book presents a comprehensive and systematic survey of fuzzy and uncertainty-oriented concepts in applied mathematics. It brings together a wide range of theoretical frameworks and applications, spanning discrete structures, algebraic systems, analytical models, and quantitative methods. By organizing these diverse topics within a unified perspective, the work clarifies their underlying mathematical structures, highlights their interrelationships, and compares their expressive capabilities. The book aims to serve both as a reference and as a conceptual guide, offering readers a coherent overview of the landscape of uncertainty modeling. It emphasizes the role of advanced frameworks—particularly neutrosophic and plithogenic systems—in extending classical fuzzy theory and enabling more refined representations of complex, inconsistent, and multi-attribute information in applied mathematics.

This book introduces and develops neutrosophy as a unifying philosophical and mathematical framework for the study of neutralities and their interactions with truth and falsity. Extending beyond classical, fuzzy, and intuitionistic logics, the work formalizes neutrosophic logic, in which every proposition is characterized by three independent components: truth (T), indeterminacy (I), and falsity (F). These components are defined over a non-standard unit interval, allowing values beyond conventional probabilistic constraints and enabling the modeling of incomplete, inconsistent, and paradoxical information. The volume systematically generalizes several foundational domains, including logic, set theory, probability, and statistics, by introducing corresponding neutrosophic structures such as neutrosophic sets, neutrosophic probability, and neutrosophic statistics. The framework accommodates contradictions, uncertainty, and vagueness in a unified mathematical structure, overcoming limitations of existing theories that cannot adequately represent indeterminacy or simultaneous truth and falsity. The book also establishes formal definitions based on non-standard analysis and hyper-real numbers, providing a rigorous mathematical foundation for neutrosophic components and operations. By integrating philosophical insight with mathematical formalism, the work opens new directions for research across disciplines, including artificial intelligence, decision theory, physics, and social sciences, where complex and uncertain phenomena require more expressive representational models.

Filosofia greacă antică reinterpretată: în dialog cu neutrosofia și plitogenia propune o reinterpretare sistematică a gândirii filozofice grecești timpurii prin prisma unor cadre logice contemporane – neutrosofia și plitogenia. Cartea analizează principalele concepte dezvoltate de filozofii presocratici, sofiști și reprezentanți ai Academiei platonice, evidențiind tensiunile logice dintre concepte precum ființă și devenire, unitate și pluralitate, adevăr și relativism. Autorul argumentează că logica clasică, bazată pe binaritate, este insuficientă pentru a surprinde complexitatea acestor idei și introduce neutrosofia ca un sistem care integrează simultan grade de adevăr, falsitate și indeterminare. Plitogenia extinde acest cadru prin modelarea realității ca sistem multi-atribut, caracterizat de grade de apartenență și contradicție. Printr-o metodă comparativă, fiecare filozof este reinterpretat în acest nou cadru conceptual, fără a-i atribui anacronic idei moderne, ci evidențiind convergențele posibile. Lucrarea contribuie astfel la dezvoltarea unei filozofii a cunoașterii extinse, în care indeterminarea, contradicția și complexitatea devin elemente fundamentale ale analizei filozofice.

Many real-world systems involve uncertainty not only in individual attributes but also in the relationships between entities. To address this, a variety of graph-theoretic frameworks have been developed that incorporate uncertainty directly into vertices, edges, and higher-level structural properties. Among these, fuzzy graphs, intuitionistic fuzzy graphs, neutrosophic graphs, and plithogenic graphs represent key approaches to modeling uncertainty in network structures. This book presents a comprehensive and systematic survey of graph theory under uncertainty, with particular emphasis on the unifying role of the uncertain graph framework. It reviews fundamental concepts, structural properties, graph classes, and graph parameters within fuzzy, neutrosophic, and related models, while also introducing a wide range of extensions such as uncertain digraphs, hypergraphs, superhypergraphs, and dynamic graphs. In addition to theoretical developments, the book explores practical applications, including uncertain molecular graphs, decision-making systems, graph neural networks, knowledge graphs, and cognitive maps. By organizing diverse uncertainty-aware graph models within a common perspective, this work provides a coherent framework for understanding their relationships, capabilities, and applications in complex systems.

Mategrams represents an original artistic expression situated at the intersection of literature, mathematics, and visual art. The book introduces the concept of “mategrams”—condensed thoughts and reflections that naturally organize themselves into geometric and algebraic structures, becoming mathematical-poetic figures akin to anti-calligrams. Rather than following a linear textual format, the content is embedded within visual compositions that exploit space, form, and symbolic arrangement, creating a nontraditional reading experience. Each mategram functions simultaneously as idea, image, and structure, inviting reflection on human condition, society, and the relationship between logic and creativity. By merging poetic language with graphical representations and mathematical concepts, the volume proposes an innovative interdisciplinary aesthetic in which meaning emerges from the interaction between text and form.

Paradoxica propune o reconstrucție radicală a limbajului prin integrarea paradoxului ca principiu fundamental de generare a sensului. Pornind de la critica logicii clasice, care privilegiază stabilitatea, identitatea și coerența, autorul dezvoltă un sistem alternativ în care contradicția nu este exclusă, ci devine condiție esențială a semnificației. Cartea introduce „Paradoxica” ca limbă formală și poetică în care fiecare enunț conține o tensiune semantică internă, bazată pe opoziție, inversiune sau autoreferențialitate. Sensul nu mai este un rezultat fix, ci un proces emergent generat din conflictul dintre concepte și anticoncepte. Lucrarea dezvoltă, de asemenea, un sistem formal de reguli, structuri și axiome, precum și o reprezentare vizuală (Scriptul Paradoxica), unde spațiul, simbolurile și relațiile dintre ele devin purtători de sens. În plan poetic, sunt prezentate multiple forme paradoxiste (distihuri duale, oxymorfoze, anacoluturi, structuri recursive), fiecare ilustrând mecanisme distincte de producere a sensului prin instabilitate și tensiune. În ansamblu, Paradoxica nu este doar o colecție de texte, ci o propunere epistemologică și lingvistică, în care adevărul nu este afirmat, ci generat prin contradicție, iar limbajul devine un sistem dinamic, deschis și autoreflexiv.

This book develops a comprehensive comparative analysis of plithogeny as a meta-philosophical framework capable of interpreting the relationships among diverse philosophical traditions. Plithogeny, originally formulated within the domain of mathematical logic, describes the formation and evolution of entities through the interaction of multiple elements that may be compatible, contradictory, or neutral. Extending this formal concept into philosophy, the work examines how philosophical systems can be understood not as isolated or mutually exclusive doctrines, but as interacting conceptual structures within a broader field of intellectual plurality. The study systematically explores the connections between plithogeny and major philosophical movements, including empiricism, rationalism, idealism, realism, phenomenology, and epiphenomenalism, as well as ethical theories such as deontology, utilitarianism, and hedonism. It further engages with postmodernism, pragmatism, and contemporary domains such as digital ethics, bioethics, environmental philosophy, and justice theory. A distinctive contribution of the expanded edition lies in its integration of Latin American and Indigenous philosophical traditions, including concepts such as yanantin, ch’ixi, pachakuti, ayni, Sumak Kawsay (Buen Vivir), and teotl, highlighting their structural affinities with plithogenic reasoning. By employing a consistent comparative methodology—focusing on similarities, distinctions, and conceptual connections—the book demonstrates that philosophical plurality can be modeled as a dynamic system of interacting perspectives. Plithogeny is thus presented not as a replacement for existing philosophies, but as a unifying analytical lens that enables dialogue across traditions while preserving their differences. The work contributes to contemporary philosophy by offering a framework suited to complexity, interdisciplinarity, and the coexistence of diverse epistemologies in both theoretical and applied contexts.

This third edition of Representing Higher-Order Networks: A Survey of Graph-Based Frameworks presents a substantially expanded and conceptually enriched treatment of higher-order network models. While retaining the foundational survey character of earlier editions, the present version introduces not only editorial refinements but also significant conceptual and structural developments that deepen and broaden the scope of the work. Beyond corrections and improved exposition, this edition incorporates a range of newly developed frameworks, including advanced tensor-based representations, recursive and iterated graph constructions, and extended superhypergraph formalisms. These additions reflect a shift from a primarily descriptive catalog of models toward a more systematic and integrative perspective, emphasizing relationships, hierarchies, and interoperability among diverse higher-order structures. The book further strengthens its unifying framework by organizing higher-order graph models into coherent families—combinatorial, topological, tensorial, and semantic—while highlighting their underlying principles and modeling capabilities. In doing so, it advances a more comprehensive theoretical landscape that supports both comparative analysis and practical application across disciplines such as network science, data modeling, and knowledge representation. Overall, this edition represents a qualitative evolution of the work, transitioning from a survey of existing approaches to a more cohesive and forward-looking synthesis of higher-order network theory.

SuperHyperScapes: An Ontological Atlas is a visual–conceptual exploration situated within the theoretical framework of oUTER-aRT, extending its principles into a structured yet open-ended system of image production. Composed of 77 generative visual fields, the album organizes reality into phased domains—ranging from perception and cognition to matter, society, and abstraction—each articulated through a distinct visual grammar. Grounded in neutrosophic logic, the project rejects binary representation and instead constructs images as spaces of simultaneous affirmation, negation, and indeterminacy. These “scapes” are not representations of reality, but ontological propositions: dynamic environments where contradictions coexist, structures destabilize, and meaning emerges through tension rather than resolution. By integrating superhyperstructural thinking with recursive and self-referential visual systems, SuperHyperScapes proposes an expanded ontology of art—one that transcends medium, dissolves disciplinary boundaries, and situates artistic practice within a continuum between system and anomaly, order and breakdown. As both atlas and anti-atlas, the work maps not what reality is, but how it can be simultaneously constructed, deconstructed, and reimagined.

This book develops a comprehensive and unified theory of SuperHyperStructures and their uncertain extensions, introducing a new paradigm for modeling multi-level, hierarchical, and non-deterministic systems. Extending classical structures and hyperstructures, the framework generalizes operations from single-valued mappings to higher-order mappings of the form f:X^n→P^k (X), enabling the representation of sets of sets and recursively organized relational systems. SuperHyperStructures were introduced by Florentin Smarandache. At its core, the theory addresses fundamental limitations of traditional mathematical models, which are often inadequate for describing systems characterized by hierarchy, interdependence, and uncertainty. By integrating concepts from set theory, hyperstructures, n-ary operations, and neutrosophic logic, the book establishes a rigorous foundation for Uncertain SuperHyperStructures, incorporating truth, indeterminacy, and falsity within a unified formalism. The work systematically develops the theory across multiple domains, including SuperHyperAlgebra, SuperHyperGeometry, SuperHyperLogic, SuperHyperGraphs, and SuperHyperCalculus, and introduces corresponding uncertain and neutrosophic extensions. A comparative framework is provided to position SuperHyperStructures relative to classical structures and hyperstructures, highlighting their increased expressive power and structural depth. Beyond theoretical development, the book demonstrates the applicability of the framework to a wide range of disciplines, including artificial intelligence, complex systems, data science, and physics. In these contexts, SuperHyperStructures enable the modeling of systems-of-systems, multi-agent interactions, hierarchical networks, and multi-scale phenomena under uncertainty. The final chapters explore advanced directions such as SuperHyperCategory Theory and outline open problems, emphasizing the foundational nature of the theory and its potential to support a broad research program. Overall, this work proposes a conceptual and mathematical shift—from flat, deterministic representations to hierarchical, multi-level, and uncertainty-aware structures—offering a unified approach to complexity across scientific domains.

This book presents a comprehensive and systematic survey of fuzzy and uncertain logical operators developed to model reasoning under imprecision, incompleteness, and partial reliability. Building upon foundational frameworks such as fuzzy sets, intuitionistic fuzzy sets, neutrosophic sets, vague sets, and hesitant fuzzy sets, the work focuses on the extension of classical logical connectives into uncertainty-aware environments. It provides a structured overview of a wide range of operators, including basic set-theoretic operations, conjunctive–disjunctive families (t-norms, t-conorms, uninorms), relational and inferential operators, aggregation mechanisms, and integral-based models such as Choquet and Sugeno integrals. The book organizes these operators within a unified uncertain-set framework, clarifying their mathematical properties, interrelationships, and functional roles in reasoning processes. By developing a coherent taxonomy and highlighting connections across different uncertainty paradigms, it establishes a solid theoretical foundation for future research. The survey also emphasizes applications in approximate reasoning, decision-making systems, control theory, and artificial intelligence, making it a valuable reference for both theoretical investigation and practical implementation of uncertainty-based logic.

Florentin Smarandache întreprinde o (nouă) incursiune radicală în logica paradoxului în volumul “Țara oamenilor care merg în cap”, un “experiment superparadoxist” care sfidează toate convențiile. Lucrarea nu se rezumă la o simplă poveste, ci se constituie ca un dispozitiv complex de deconstrucție a realității, alcătuit dintr-un “roman întors pe dos”, un “raport strict secret” de teren și un “studiu științific incorect”. Această succesiune nu este o simplă aglomerare de texte, ci o progresie deliberată a absurdului, de la experiența subiectivă a eroului, la instituționalizarea acestuia prin birocrație și, în final, la teoretizarea sa ca o realitate ontologică validă.

Uncertainty, indeterminacy, and inconsistency are intrinsic features of real-world systems, requiring mathematical and conceptual frameworks that extend beyond classical binary logic. Building upon earlier developments in fuzzy sets and intuitionistic fuzzy sets, neutrosophy introduces a triadic structure in which each element is characterized by independent degrees of truth, indeterminacy, and falsity. This book represents a revised and expanded synthesis of prior work, particularly the volume Neutrosophy Revisited, while incorporating further developments inspired by the Nidus idearum series and related research. It systematically re-formalizes and consolidates a wide spectrum of concepts in neutrosophic logic, plithogenic set theory, and associated mathematical structures. The work emphasizes the transition from exploratory and conceptual formulations toward coherent, rigorous, and unified theoretical frameworks. It presents refined definitions, extended structures (including n-refined and multi-valued neutrosophic systems), and connections to diverse domains such as discrete mathematics, algebra, applied sciences, and emerging neutrosophic technologies. By organizing and clarifying previously scattered or partially developed ideas, this volume aims to provide a solid foundation for further theoretical advancement and to stimulate interdisciplinary applications in mathematics, physics, computer science, and complex systems analysis.

This book addresses a fundamental epistemological limitation of contemporary artificial intelligence systems: their inability to represent and communicate uncertainty. Current AI architectures, particularly large language models, generate outputs based on probabilistic pattern matching without an intrinsic mechanism to distinguish between verified knowledge, partial knowledge, and fabricated information. This results in what the authors define as “confident machines,” capable of producing fluent yet potentially unreliable responses without signaling epistemic boundaries. To address this structural deficiency, the book introduces a practical and conceptual framework grounded in neutrosophic logic, a triadic system developed by Florentin Smarandache, in which every proposition is characterized by degrees of truth (T), indeterminacy (I), and falsity (F). The authors extend this formalism beyond mathematics, integrating it with philosophical traditions from Latin America and other non-Western epistemologies that embrace ambiguity, complementarity, and productive doubt. The work develops an operational methodology—referred to as “The Third Answer”—which enables users to critically evaluate AI-generated outputs through structured questioning, classification of uncertainty, and decision protocols. By translating abstract logical principles into accessible tools (e.g., decision frameworks, uncertainty mapping, and prompt strategies), the book provides a practical guide for professionals interacting with AI systems across domains. Ultimately, the book argues that improving AI reliability is not merely a matter of increasing data or model size, but requires a paradigm shift toward uncertainty-aware reasoning. It positions neutrosophic logic as a bridge between ancient philosophical insights and modern computational systems, offering a pathway toward more transparent, trustworthy, and epistemically responsible artificial intelligence.

In this paper, for the first time, we extend the Over/Under/Off Set/Logic/Probability used in uncertain theories (such as: fuzzy, neutrosophic and extensions) to the Over/Under/Off Mass that could be used in information fusion. The approach is exemplified in three scenarios: (1) wildfire evacuation and resource allocation with satellite, IoT, and social media; (2) coverage gaps where indeterminacy must be managed; and (3) security monitoring where contradictory or erroneous reports are discounted.

This book provides a comprehensive and systematic survey of modern set-theoretic frameworks developed to model uncertainty, vagueness, indeterminacy, and partial truth in real-world phenomena. It synthesizes a wide spectrum of classical and advanced theories, including fuzzy sets, intuitionistic fuzzy sets, neutrosophic sets, plithogenic sets, and numerous related extensions. The work organizes these concepts across multiple structural and conceptual dimensions-such as interval, granular, probiliistic, and complex extensions-highlighting their mathematical foundations, interrelations, and distinguishing characteristics. By unifying diverse approaches under a coherent analytical perspective, the book clarifies how different uncertainty models A complement or generalize one another. It also explores emerging directions in set theory, including hybrid and dynamic frameworks, offering insights into their applicability in fields such as decisionmaking, control systems, artificial intelligence, and data analysis.

Rough set theory models uncertainty by approximating target concepts via lower and upper sets induced by indiscernibility (or more general granulation) relations in data tables. This perspective captures vagueness caused by limited observational resolution and supports settheoretic reasoning about what can be determined with certainty versus what remains only possible. The present book is written as a model map: rather than developing a single algorithmic pipeline in depth, we provide a systematic survey of the main rough-set paradigms and their extension routes. Concretely, we organize representative variants according to (i) the underlying granulation mechanism (e.g., equivalence-, tolerance-, covering-, neighborhood-, and probabilistic-based approximations) and (ii) the uncertainty semantics attached to data and relations (e.g., crisp, fuzzy, intuitionistic fuzzy, neutrosophic, and plithogenic settings), and we explain how each choice changes the form of approximations and the interpretation of boundary regions. Throughout, small illustrative examples are used to clarify modeling intent and typical use-cases in classification and decision support. Finally, we note an important scope clarification: if the purpose of this book is limited to a model map, then the Abstract/Introduction should not lead readers to expect that feature reduction and rule induction are primary goals. Those topics are central in the rough-set literature, but here they are discussed mainly as motivating applications and as pointers to the broader ecosystem; the main objective is to align the book’s stated aim with its actual focus on surveying and positioning rough-set models and extensions.

Decision-making in real applications is often affected by vagueness, incomplete information, heterogeneous data, and conflicting expert opinions. This survey reviews uncertainty-aware multi-criteria decision-making (MCDM) and organizes the field into a concise, task-oriented taxonomy. We summarize problem-level settings (discrete, group/consensus, dynamic, multi-stage, multi-level, multi agent, and multi-scenario), weight elicitation (subjective and objective schemes under fuzzy/linguistic in puts), and inter-criteria structure and causality modelling. For solution procedures, we contrast compen satory scoring methods, distance-to-reference and compromise approaches, and non-compensatory outrank ing frameworks for ranking or sorting. We also outline rule/evidence-based and sequential decision models that produce interpretable rules or policies. The survey highlights typical inputs, core computational steps, and primary outputs, and provides guidance on choosing methods according to robustness, interpretability, and data availability. It concludes with open directions on explainable uncertainty integration, stability, and scalability in large-scale and dynamic decision environments.

This book presents a systematic survey of the concepts of Z-Numbers, D-Numbers, and ZD-Numbers, which represent advanced mathematical frameworks for modeling uncertainty, imprecision, and incomplete information. Building upon classical uncertainty theories such as fuzLy sets, intuitionistic fuzzy sets, and neutrosophic sets, the work examines how Z-Numbers and D-Numbers extend traditional approaches by incorporating reliability assessments and flexible belief structures. The book reviews a wide range of extensions and variants, including linguistic, probabilistic, multidimensional, hyper, and plithogenic ZNumber models, as well as numerous D-Number generalizations designed to address non-exclusive hypotheses and incomplete knowledge. In addition to surveying existing developments, the book introduces the concept of the ZD-Number, which integrates the perspectives of Z-Numbers and D-Numbers into a unified representation capable of capturing both value-reliability relations and flexible evidence structures. Through definitions, comparisons, and illustrative examples, the book clarifies the relationships among these models and highlights their relevance in areas such as decision-making, information fusion, artificial intelligence, and complex systems analysis. The resulting survey provides both a reference for researchers and a foundation for further theoretical and applied developments in uncertainty modeling.

Many real-world phenomena are naturally modeled by graphs and networks. However, classical graph models are often limited to pairwise interactions and may not adequately capture the richer structures that arise in practice. Higher-order graph formalisms extend this framework by incorporating multiway, hierarchical, temporal, multilayer, recursive, and tensor-based interactions, thereby providing more expressive representations of complex systems. This book presents a comprehensive overview of mathematical notions that can be used to model higher-order networks. It surveys foundational concepts, extensional frameworks, and newly introduced formalisms, with an emphasis on their structural principles, relationships, and modeling roles. The aim is to provide a unified perspective that helps readers compare diverse higher-order network models and identify appropriate tools for theoretical study and practical applications. This book is Edition 2.0. It mainly includes the addition of several concepts, as well as corrections and improvements of typographical errors and explanations. Keywords: Hypergraph, Superhypergraph, Higher-Order Graphs MSC2010 (Mathematics Subject Classification 2010): 05C65 - Hypergraphs, 05C82 - Graph theory with applications

Waves are among the most fundamental phenomena in nature. From sound in air and light in vacuum to seismic tremors and quantum matter waves, their mathematical descriptions are traditionally expressed through continuum field equations. Yet every measurement of a wave ultimately yields discrete outcomes. Infinitesimally Punctured Waves proposes a unifying framework in which this apparent tension between continuity and discreteness is resolved at a structural level. The Infinitesimally Punctured Wave (IPW) approach models a wavefield as the macroscopic limit of an ultra-dense lattice of interacting “punctures”—elementary carriers of physical quantities coupled through local dynamical rules. In the limit of vanishing lattice spacing, the familiar partial differential equations of acoustics, elasticity, electromagnetism, gravity, and quantum mechanics emerge. For finite spacing, the same formalism predicts systematic corrections, controlled dispersion, and experimentally testable deviations. Developed rigorously and applied across multiple physical domains, the IPW framework connects: • acoustic and elastic wave propagation, • electromagnetic and optical phenomena, • linearised gravitational dynamics, • quantum wave evolution and measurement theory, and • nonlinear effects including shocks and solitons. Beyond conceptual unification, the lattice structure provides a transferable computational methodology applicable to multiphysics metamaterials, analogue gravity systems, and high-frequency wave engineering. The book integrates theoretical development with proposed experimental tests, offering concrete pathways to probe finite-scale effects. Written for researchers and advanced students in physics, applied mathematics, and engineering, Infinitesimally Punctured Waves presents a coherent, cross-disciplinary language for wave phenomena and outlines a research programme extending from practical device design to foundational questions in spacetime physics.

infinitesimally punctured geometry formulated within Extended Nonstandard Analysis (ENSA). It introduces the MoBiNad manifold as a hyperreal extension of classical differential geometry, where monads, binads, and infinitesimal punctures generate structured micro-deviations in metric, connection, and curvature operators. Through neutrosophic logic, each geometric and physical quantity acquires a triplet valuation (truth, indeterminacy, falsity), allowing curvature, topology, and field dynamics to encode graded logical structure at the infinitesimal level. Building on this foundation, the book develops MoBiNad differential operators, supersymmetric extensions, Smarandache curves and surfaces, and a neutrosophic Gauss–Bonnet theorem, demonstrating how mixed geometric regimes emerge as standard-part projections of a rigorously defined hyperreal substratum. Gauge–gravitational coupling, curvature oscillations, and topological invariants are reformulated within this punctured continuum, providing a deterministic infinitesimal basis for phenomena traditionally attributed to quantum fluctuation. The result is a coherent bridge between differential geometry, topology, logic, and high-energy physics: classical geometry appears as the macroscopic limit of a structured infinitesimal network, while Smarandache geometries arise naturally from local curvature sign alternations induced by monadic asymmetries. This work lays the analytical groundwork for a neutrosophic Chern–Weil theory and higher-dimensional topological invariants in hyperreal spacetime.

For more than a century, singularities and ultraviolet divergences have stood at the frontiers of modern theoretical physics, marking points where our most successful theories cease to be mathematically well defined. Infinitesimal Punctures proposes a structural shift in perspective: instead of inserting point-like sources into smooth manifolds, matter and physical attributes are interpreted as intrinsic geometric defects—measure-zero punctures—within spacetime itself. In this framework, curvature, charge, and quantum behavior arise not as external additions but as distributionally supported features of geometry. The series develops this idea systematically, moving from foundational geometry, through dynamical physical laws, to a unified S-MultiSpace and S-MultiStructure structural formalism. This volume elevates the framework to a unified structural and meta-geometric level. Drawing on Smarandache multi-space concepts and Neutrosophic logic, it develops a rigorous theory of S-MultiSpaces and S-MultiStructures in which multiple geometric regimes coexist within a single manifold. Neutrosophic differential geometry, hybrid connections, and generalized curvature tensors are constructed, together with modified Gauss–Codazzi relations and Smarandache variational principles. This volume situates infinitesimally punctured geometry within a broader mathematical landscape that includes topology, holonomy, generalized algebras, and spectral reconstruction of geometry. It provides the formal completion of the trilogy, demonstrating how the structural triad underlying punctured manifolds leads to a coherent, logically consistent framework for matter, fields, and spacetime.

For more than a century, singularities and ultraviolet divergences have stood at the frontiers of modern theoretical physics, marking points where our most successful theories cease to be mathematically well defined. Infinitesimal Punctures proposes a structural shift in perspective: instead of inserting point-like sources into smooth manifolds, matter and physical attributes are interpreted as intrinsic geometric defects—measure-zero punctures—within spacetime itself. In this framework, curvature, charge, and quantum behavior arise not as external additions but as distributionally supported features of geometry. The series develops this idea systematically, moving from foundational geometry, through dynamical physical laws, to a unified S-MultiSpace and S-MultiStructure structural formalism. Building on the geometric framework of Infinitesimally Punctured Geometry, this volume develops the dynamical laws governing infinitesimally punctured spacetime. A hybrid variational principle is introduced in which smooth curvature, defect-supported curvature, and an indeterminate transition sector enter on equal footing. Variation of this action yields a master field equation that generalizes the Einstein equations and naturally incorporates defect density as a geometric source rather than an externally prescribed stress-energy tensor. Within this setting, an approach termed Infinitesimally Punctured Quantum Physics is proposed, in which a quantum object is visualized as an aggregation of infinitely many infinitesimally spaced constituents. When these constituents are densely packed, the ensemble manifests as a continuous wave, surface, space, or manifold; when a measurement effectively isolates a single constituent, particle-like behavior emerges. This picture is linked to Neutrosophic Quantum Theory, which supplies a logical framework for handling indeterminacy in the transition between continuous and discrete regimes. Applications include regularised black hole geometries, punctured wave and Schrödinger equations, modified cosmological evolution, and geometric interpretations of dark matter and dark energy. This volume centralizes the predictive content of the theory and develops its phenomenological and observational consequences.