Multilinear Control Systems Theory and its Applications

In biological and engineering systems, structure, function, and dynamics are highly coupled. Such multiway interactions can be naturally and compactly captured via tensor-based representations. Exploiting recent advances in tensor algebraic methods, we develop novel theoretical and computational approaches for data-driven model learning, analysis, and control of such tensor-based representations. In one line of work, we extend classical linear time-invariant (LTI) system notions including stability, reachability, and observability to multilinear time-invariant (MLTI) systems, in which the state, inputs, and outputs are represented as tensors, and express these notions in terms of more standard concepts of tensor ranks/decompositions. We also introduce a tensor decomposition-based model reduction framework which can significantly reduce the number of MLTI system parameters. In another line of work, we develop the notion of tensor entropy for uniform hypergraphs, which can capture higher order interactions between entities than classical graphs. We show that this tensor entropy is an extension of von Neumann entropy for graphs and can be used as a measure of regularity for uniform hypergraphs. Moreover, we employ uniform hypergraphs for studying controllability of high-dimensional networked systems. We propose another tensor-based multilinear system representation for characterizing the multidimensional state dynamics of uniform hypergraphs, and derive a Kalman-rank-like condition to identify the minimum number of control nodes (MCN) needed to achieve full control of the whole hypergraph. We demonstrate these new tensor-based theoretical and computational developments in a variety of biological and engineering examples.PHDApplied and Interdisciplinary MathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/169968/1/canc_1.pd

Fuzzy n-ary polygroups related to fuzzy points

AbstractRecently, fuzzy n-ary sub-polygroups were introduced and studied by Davvaz, Corsini and Leoreanu-Fotea [B. Davvaz, P. Corsini, V. Leoreanu-Fotea, Fuzzy n-ary sub-polygroups, Comput. Math. Appl. 57 (2008) 141–152]. Now, in this paper, the concept of (∈,∈∨q)-fuzzy n-ary sub-polygroups, (∈¯,∈¯∨q¯)-fuzzy n-ary sub-polygroups and fuzzy n-ary sub-polygroup with thresholds of an n-ary polygroup are introduced and some characterizations are described. Also, we give the definition of implication-based fuzzy n-ary sub-polygroups in an n-ary polygroup, in particular, the implication operators in Łukasiewicz system of continuous-valued logic are discussed

Coding Theory

This book explores the latest developments, methods, approaches, and applications of coding theory in a wide variety of fields and endeavors. It consists of seven chapters that address such topics as applications of coding theory in networking and cryptography, wireless sensor nodes in wireless body area networks, the construction of linear codes, and more

Discrete Mathematics and Symmetry

Some of the most beautiful studies in Mathematics are related to Symmetry and Geometry. For this reason, we select here some contributions about such aspects and Discrete Geometry. As we know, Symmetry in a system means invariance of its elements under conditions of transformations. When we consider network structures, symmetry means invariance of adjacency of nodes under the permutations of node set. The graph isomorphism is an equivalence relation on the set of graphs. Therefore, it partitions the class of all graphs into equivalence classes. The underlying idea of isomorphism is that some objects have the same structure if we omit the individual character of their components. A set of graphs isomorphic to each other is denominated as an isomorphism class of graphs. The automorphism of a graph will be an isomorphism from G onto itself. The family of all automorphisms of a graph G is a permutation group

Neutrosophic Sets and Systems, Vol. 10, 2015

This volume is a collection of thirteen papers, written by different authors and co-authors (listed in the order of the papers): J. J. Peng and J. Q. Wang, E. Marei, S. Kar, K. Basu, S. Mukherjee, I. M. Hezam, M. Abdel-Baset and F. Smarandache, K. Mondal, S. Pramanik, A. Ionescu, M. R. Parveen and P. Sekar, B. Teodorescu, D. Kour and K. Basu, P. P. Dey and B. C. Giri, A. A. A. Agboola. In first paper, the authors studied Multi-valued Neutrosophic Sets and its Application in Multi-criteria Decision-Making Problems. More on neutrosophic soft rough sets and its modification is discussed in the second paper. Solution of Multi-Criteria Assignment Problem using Neutrosophic Set Theory are studied in third paper. In fourth paper, Taylor Series Approximation to Solve Neutrosophic Multiobjective Programming Problem. Similarly in fifth paper, Decision Making Based on Some similarity Measures under Interval Rough Neutrosophic Environment is discussed. In paper six, Neutralité neutrosophique et expressivité dans le style journalistique is studied by the author. Neutrosophic Semilattices and Their Properties given in seventh paper. Liminality and Neutrosophy is proposed in the next paper. Application of Extended Fuzzy Program-ming Technique to a real life Transportation Problem in Neutrosophic environment in the next paper. Further, TOPSIS for Single Valued Neutrosophic Soft Expert Set Based Multi-attribute Decision Making Problems is discussed by the authors in the tenth paper. In eleventh paper, Neutrosophic Quadruple Numbers, Refined Neutrosophic Quadruple Numbers, Absorbance Law, and the Multiplication of Neutrosophic Quadruple Numbers have been studied by the author. In the next paper, On Refined Neutrosophic Algebraic Structures. At the end, Neutrosophic Actions, Prevalence Order, Refinement of Neutrosophic Entities, and Neutrosophic Literal Logical Operators are introduced by the author

Hyperstructures and Idempotent Semistructures

Much of this thesis concerns hypergroups, multirings, and hyperfields. These are analogous to abelian groups, rings, and fields, but have a multivalued addition operation. M. Krasner introduced the notion of a valued hyperfield; The prototypical example is $K/(1+\mathfrak{m}_K^n)$ where $K$ is a local field. P. Deligne introduced a category of triples whose objects have the form $\mathrm{Tr}_n(K)=(\mathcal{O}_K/\mathfrak{m}_K^n,\mathfrak{m}_K/\mathfrak{m}_K^{n+1},\epsilon)$ where $\epsilon:\mathfrak{m}_K/\mathfrak{m}_K^{n+1}\rightarrow \mathcal{O}_K/\mathfrak{m}_K^n$ is the obvious map. In this thesis I relate the category of discretely valued hyperfields to Deligne's category of triples. An extension of a local field is arithmetically profinite if the upper ramification subgroups are open. Given such an extension $L/K$, J.P. Wintenberger defined the norm field $X_K(L)$ as the inverse limit of the finite subextensions of $L/K$ along the norm maps. Wintenberger has defined an addition operation on $X_K(L)$, and shown that $X_K(L)$ is a local field of finite characteristic. Using Deligne's triples, I have given a new proof of Wintenberger's characterization of its Galois group. The semifield $\mathbb{Z}_{\max}$ is defined as $\{0\}\cup\{u^k\mid k\in\mathbb{Z}\}$ with addition given by $u^m+u^n=u^{\max(m,n)}$. An extension of $\mathbb{Z}_{\max}$ is a semifield containing $\mathbb{Z}_{\max}$. The extension is finite if $S$ is finitely generated as a $\mathbb{Z}_{\max}$-semimodule. In this thesis I classify the finite extensions of $\mathbb{Z}_\mathrm{max}$. There are two previously known methods for constructing a hypergroup from a totally ordered set. In this thesis I generalize these to a family of constructions parametrized by hypergroups $H$ satisfying $x-x=H$ for all $x\in H$. We say a hyperfield $K$ is selective if $1+1-1-1=1-1$ and for all $x,y\in K$ one has either $x\in x+y$ or $y=x+y$. In this thesis, I show that a selective hyperfield is characterized by a totally ordered group $\Gamma$, a hyperfield $H$ satisfying $1-1=H$, and an extension $\phi\in\mathrm{Ext}^1(\Gamma,H^\times)$. We say a triple of elements $(x,y,z)$ of an idempotent semiring is a corner triple if $x+y=y+z=x+z$. We say an idempotent semiring is regular if whenever $(x,y,a)$ and $(z,w,a)$ are corner triples, there exists $b$ such that $(x,z,b)$ and $(y,w,b)$ are also corner triples. I prove in this thesis that the category of regular idempotent semirings is a reflective subcategory of the category of multirings

The Encyclopedia of Neutrosophic Researchers - vol. 1

This is the first volume of the Encyclopedia of Neutrosophic Researchers, edited from materials offered by the authors who responded to the editor’s invitation. The authors are listed alphabetically. The introduction contains a short history of neutrosophics, together with links to the main papers and books. Neutrosophic set, neutrosophic logic, neutrosophic probability, neutrosophic statistics, neutrosophic measure, neutrosophic precalculus, neutrosophic calculus and so on are gaining significant attention in solving many real life problems that involve uncertainty, impreciseness, vagueness, incompleteness, inconsistent, and indeterminacy. In the past years the fields of neutrosophics have been extended and applied in various fields, such as: artificial intelligence, data mining, soft computing, decision making in incomplete / indeterminate / inconsistent information systems, image processing, computational modelling, robotics, medical diagnosis, biomedical engineering, investment problems, economic forecasting, social science, humanistic and practical achievements

Collected Papers (on Neutrosophic Theory and Its Applications in Algebra), Volume IX

This ninth volume of Collected Papers includes 87 papers comprising 982 pages on Neutrosophic Theory and its applications in Algebra, written between 2014-2022 by the author alone or in collaboration with the following 81 co-authors (alphabetically ordered) from 19 countries: E.O. Adeleke, A.A.A. Agboola, Ahmed B. Al-Nafee, Ahmed Mostafa Khalil, Akbar Rezaei, S.A. Akinleye, Ali Hassan, Mumtaz Ali, Rajab Ali Borzooei , Assia Bakali, Cenap Özel, Victor Christianto, Chunxin Bo, Rakhal Das, Bijan Davvaz, R. Dhavaseelan, B. Elavarasan, Fahad Alsharari, T. Gharibah, Hina Gulzar, Hashem Bordbar, Le Hoang Son, Emmanuel Ilojide, Tèmítópé Gbóláhàn Jaíyéolá, M. Karthika, Ilanthenral Kandasamy, W.B. Vasantha Kandasamy, Huma Khan, Madad Khan, Mohsin Khan, Hee Sik Kim, Seon Jeong Kim, Valeri Kromov, R. M. Latif, Madeleine Al-Tahan, Mehmat Ali Ozturk, Minghao Hu, S. Mirvakili, Mohammad Abobala, Mohammad Hamidi, Mohammed Abdel-Sattar, Mohammed A. Al Shumrani, Mohamed Talea, Muhammad Akram, Muhammad Aslam, Muhammad Aslam Malik, Muhammad Gulistan, Muhammad Shabir, G. Muhiuddin, Memudu Olaposi Olatinwo, Osman Anis, Choonkil Park, M. Parimala, Ping Li, K. Porselvi, D. Preethi, S. Rajareega, N. Rajesh, Udhayakumar Ramalingam, Riad K. Al-Hamido, Yaser Saber, Arsham Borumand Saeid, Saeid Jafari, Said Broumi, A.A. Salama, Ganeshsree Selvachandran, Songtao Shao, Seok-Zun Song, Tahsin Oner, M. Mohseni Takallo, Binod Chandra Tripathy, Tugce Katican, J. Vimala, Xiaohong Zhang, Xiaoyan Mao, Xiaoying Wu, Xingliang Liang, Xin Zhou, Yingcang Ma, Young Bae Jun, Juanjuan Zhang