Mukhtar Ahmad

This book presents a collection of recent advances in neutrosophic mathematics, highlighting the growing role of neutrosophic structures in addressing uncertainty, indeterminacy, and inconsistency within modern mathematical systems. The volume integrates developments from neutrosophic graph theory, algebra, metric spaces, and functional analysis, demonstrating both theoretical depth and practical applicability. The first part of the book investigates novel classes of neutrosophic graphs, including Min-Max neutrosophic graphs and neutrosophic bidirected graphs. New combinatorial constructions, structural properties, graph operations, and network modeling applications are developed, providing mathematical tools for the analysis of uncertain and complex networks. The second part focuses on neutrosophic algebraic structures through the study of neutrosophic Lie subalgebras and ideals. Fundamental results concerning homomorphisms, images, inverse images, quotient structures, and preservation properties are established, extending classical Lie algebra theory to neutrosophic environments. The third part explores analytical aspects of neutrosophic mathematics. New common fixed-point results in neutrosophic metric spaces are developed under integral-type contractive conditions and applied to dynamic programming problems. Furthermore, the theory of rough I-statistical convergence of order α′ in neutrosophic normed spaces is introduced and characterized, with applications to dynamical systems, iterative processes, and sensor-network data fusion. The book concludes with a comprehensive discussion of emerging research directions, including neutrosophic artificial intelligence, machine learning, quantum neutrosophic structures, network science, algebraic topology under uncertainty, and sustainability applications. By bringing together diverse mathematical perspectives, this volume provides a unified overview of contemporary developments in neutrosophic mathematics and serves as a valuable resource for researchers, graduate students, and professionals working in uncertainty modeling and advanced mathematical sciences.

This book develops several directions in generalized functional analysis within fuzzy, intuitionistic fuzzy, Fermatean intuitionistic fuzzy, and neutrosophic normed spaces. It investigates fundamental analytical concepts including convergence, Cauchy sequences, completeness, compactness, total boundedness, finite-dimensionality, equivalence of generalized norms, and fixed point principles. The first part studies compactness and finite-dimensional structures in intuitionistic algebra fuzzy normed spaces, establishing generalized versions of classical results such as completeness of finite-dimensional subspaces, equivalence of norms, and the Heine–Borel theorem. The second part introduces Fermatean intuitionistic fuzzy (n)-normed linear spaces and develops their convergence, completeness, and fixed point theory. The third part examines Tribonacci ideal and ideal-star convergence in non-Archimedean neutrosophic normed spaces, combining matrix transformations, ideal convergence, and neutrosophic analytical methods. Together, these contributions provide a unified framework for extending classical functional analysis to mathematical environments characterized by uncertainty, hesitation, and indeterminacy, with potential applications in applied mathematics, artificial intelligence, optimization, decision-making, and information sciences.