328 research outputs found
Hilbert evolution algebras, weighted digraphs, and nilpotency
Hilbert evolution algebras generalize evolution algebras through a framework
of Hilbert spaces. In this work we focus on infinite-dimensional Hilbert
evolution algebras and their representation through a suitably defined weighted
digraph. By means of studying such a digraph we obtain new properties for these
structures extending well-known results related to the nilpotency of finite
dimensional evolution algebras. We show that differently from what happens for
the finite dimensional evolution algebras, the notions of nil and nilpotency
are not equivalent for Hilbert evolution algebras. Furthermore, we exhibit
necessary and sufficient conditions under which a given Hilbert evolution
algebra is nil or nilpotent. Our approach includes illustrative examples
Isotopisms of nilpotent Leibniz algebras and Lie racks
In this paper we study the isotopism classes of two-step nilpotent Leibniz
algebras. We show that every nilpotent Leibniz algebra with one-dimensional
commutator ideal is isotopic to the Heisenberg algebra or
to the Heisenberg Leibniz algebra , where is
the Jordan block of eigenvalue . We also prove that two such
algebras are isotopic if and only if the Lie racks integrating them are
isotopic. This gives the classification, up to isotopism, of Lie racks whose
tangent space at the unit element is a nilpotent Leibniz algebra with
commutator ideal of dimension one. Eventually we introduce new isotopism
invariants for Leibniz algebras and Lie racks.Comment: arXiv admin note: text overlap with arXiv:2106.1290
Selected Topics in Gravity, Field Theory and Quantum Mechanics
Quantum field theory has achieved some extraordinary successes over the past sixty years; however, it retains a set of challenging problems. It is not yet able to describe gravity in a mathematically consistent manner. CP violation remains unexplained. Grand unified theories have been eliminated by experiment, and a viable unification model has yet to replace them. Even the highly successful quantum chromodynamics, despite significant computational achievements, struggles to provide theoretical insight into the low-energy regime of quark physics, where the nature and structure of hadrons are determined. The only proposal for resolving the fine-tuning problem, low-energy supersymmetry, has been eliminated by results from the LHC. Since mathematics is the true and proper language for quantitative physical models, we expect new mathematical constructions to provide insight into physical phenomena and fresh approaches for building physical theories
A random Hall-Paige conjecture
A complete mapping of a group is a bijection such
that is also bijective. Hall and Paige conjectured in 1955
that a finite group has a complete mapping whenever is
the identity in the abelianization of . This was confirmed in 2009 by
Wilcox, Evans, and Bray with a proof using the classification of finite simple
groups. In this paper, we give a combinatorial proof of a far-reaching
generalisation of the Hall-Paige conjecture for large groups. We show that for
random-like and equal-sized subsets of a group , there exists a
bijection such that is a bijection from
to whenever in the
abelianization of . Using this result, we settle the following conjectures
for sufficiently large groups. (1) We confirm in a strong form a conjecture of
Snevily by characterising large subsquares of multiplication tables of finite
groups that admit transversals. Previously, this characterisation was known
only for abelian groups of odd order. (2) We characterise the abelian groups
that can be partitioned into zero-sum sets of specific sizes, solving a problem
of Tannenbaum, and confirming a conjecture of Cichacz. (3) We characterise
harmonious groups, that is, groups with an ordering in which the product of
each consecutive pair of elements is distinct, solving a problem of Evans. (4)
We characterise the groups with which any path can be assigned a cordial
labelling. In the case of abelian groups, this confirms a conjecture of Patrias
and Pechenik
Topics in Moufang Loops
We will begin by discussing power graphs of Moufang loops. We are able to show that as in groups the directed power graph of a Moufang loop is uniquely determined by the undirected power graph. In the process of proving this result we define the generalized octonion loops, a variety of Moufang loops which behave analogously to the generalized quaternion groups. We proceed to investigate para-F quasigroups, a variety of quasigroups which we show are antilinear over Moufang loops. We briefly depart from the context of Moufang loops to discuss solvability in general loops. We then prove some results on the cosets of subloops of Moufang loops. Finally, we investigate generalizations of the variety of Moufang loops, the varieties of universally and semi-universally flexible loops
Row-Hamiltonian Latin squares and Falconer varieties
A \emph{Latin square} is a matrix of symbols such that each symbol occurs
exactly once in each row and column. A Latin square is
\emph{row-Hamiltonian} if the permutation induced by each pair of distinct rows
of is a full cycle permutation. Row-Hamiltonian Latin squares are
equivalent to perfect -factorisations of complete bipartite graphs. For the
first time, we exhibit a family of Latin squares that are row-Hamiltonian and
also achieve precisely one of the related properties of being
column-Hamiltonian or symbol-Hamiltonian. This family allows us to construct
non-trivial, anti-associative, isotopically -closed loop varieties, solving
an open problem posed by Falconer in 1970
A Geometric Approach to the Projective Tensor Norm
The main focus of this thesis is on the projective norm on finite-dimensional real or complex tensor products. There are various mathematical subjects with relations to the projective norm. For instance, it appears in the context of operator algebras or in quantum physics.
The projective norm on multipartite tensor products is considered to be less accessible. So we use a method from convex algebraic geometry to approximate the projective unit ball by convex supersets, so-called theta bodies. For real multipartite tensor products we obtain theta bodies which are close to the projective unit ball, leading to a generalisation of the Schmidt decomposition. In a second step the method is applied to complex tensor products, in a third step to separable states.
In a more general context, the projective norm can be related to binomial ideals, especially to so-called Hibi relations. In this respect, we also focus on a generalisation of the projective unit ball, here called Hibi body, and its theta bodies. It turns out that many statements also hold in this general context
Collected Papers (on various scientific topics), Volume XIII
This thirteenth volume of Collected Papers is an eclectic tome of 88 papers in various fields of sciences, such as astronomy, biology, calculus, economics, education and administration, game theory, geometry, graph theory, information fusion, decision making, instantaneous physics, quantum physics, neutrosophic logic and set, non-Euclidean geometry, number theory, paradoxes, philosophy of science, scientific research methods, statistics, and others, structured in 17 chapters (Neutrosophic Theory and Applications; Neutrosophic Algebra; Fuzzy Soft Sets; Neutrosophic Sets; Hypersoft Sets; Neutrosophic Semigroups; Neutrosophic Graphs; Superhypergraphs; Plithogeny; Information Fusion; Statistics; Decision Making; Extenics; Instantaneous Physics; Paradoxism; Mathematica; Miscellanea), comprising 965 pages, published between 2005-2022 in different scientific journals, by the author alone or in collaboration with the following 110 co-authors (alphabetically ordered) from 26 countries: Abduallah Gamal, Sania Afzal, Firoz Ahmad, Muhammad Akram, Sheriful Alam, Ali Hamza, Ali H. M. Al-Obaidi, Madeleine Al-Tahan, Assia Bakali, Atiqe Ur Rahman, Sukanto Bhattacharya, Bilal Hadjadji, Robert N. Boyd, Willem K.M. Brauers, Umit Cali, Youcef Chibani, Victor Christianto, Chunxin Bo, Shyamal Dalapati, Mario Dalcín, Arup Kumar Das, Elham Davneshvar, Bijan Davvaz, Irfan Deli, Muhammet Deveci, Mamouni Dhar, R. Dhavaseelan, Balasubramanian Elavarasan, Sara Farooq, Haipeng Wang, Ugur Halden, Le Hoang Son, Hongnian Yu, Qays Hatem Imran, Mayas Ismail, Saeid Jafari, Jun Ye, Ilanthenral Kandasamy, W.B. Vasantha Kandasamy, Darjan Karabašević, Abdullah Kargın, Vasilios N. Katsikis, Nour Eldeen M. Khalifa, Madad Khan, M. Khoshnevisan, Tapan Kumar Roy, Pinaki Majumdar, Sreepurna Malakar, Masoud Ghods, Minghao Hu, Mingming Chen, Mohamed Abdel-Basset, Mohamed Talea, Mohammad Hamidi, Mohamed Loey, Mihnea Alexandru Moisescu, Muhammad Ihsan, Muhammad Saeed, Muhammad Shabir, Mumtaz Ali, Muzzamal Sitara, Nassim Abbas, Munazza Naz, Giorgio Nordo, Mani Parimala, Ion Pătrașcu, Gabrijela Popović, K. Porselvi, Surapati Pramanik, D. Preethi, Qiang Guo, Riad K. Al-Hamido, Zahra Rostami, Said Broumi, Saima Anis, Muzafer Saračević, Ganeshsree Selvachandran, Selvaraj Ganesan, Shammya Shananda Saha, Marayanagaraj Shanmugapriya, Songtao Shao, Sori Tjandrah Simbolon, Florentin Smarandache, Predrag S. Stanimirović, Dragiša Stanujkić, Raman Sundareswaran, Mehmet Șahin, Ovidiu-Ilie Șandru, Abdulkadir Șengür, Mohamed Talea, Ferhat Taș, Selçuk Topal, Alptekin Ulutaș, Ramalingam Udhayakumar, Yunita Umniyati, J. Vimala, Luige Vlădăreanu, Ştefan Vlăduţescu, Yaman Akbulut, Yanhui Guo, Yong Deng, You He, Young Bae Jun, Wangtao Yuan, Rong Xia, Xiaohong Zhang, Edmundas Kazimieras Zavadskas, Zayen Azzouz Omar, Xiaohong Zhang, Zhirou Ma.
New Development of Neutrosophic Probability, Neutrosophic Statistics, Neutrosophic Algebraic Structures, and Neutrosophic Plithogenic Optimizations
This volume presents state-of-the-art papers on new topics related to neutrosophic theories, such as neutrosophic algebraic structures, neutrosophic triplet algebraic structures, neutrosophic extended triplet algebraic structures, neutrosophic algebraic hyperstructures, neutrosophic triplet algebraic hyperstructures, neutrosophic n-ary algebraic structures, neutrosophic n-ary algebraic hyperstructures, refined neutrosophic algebraic structures, refined neutrosophic algebraic hyperstructures, quadruple neutrosophic algebraic structures, refined quadruple neutrosophic algebraic structures, neutrosophic image processing, neutrosophic image classification, neutrosophic computer vision, neutrosophic machine learning, neutrosophic artificial intelligence, neutrosophic data analytics, neutrosophic deep learning, and neutrosophic symmetry, as well as their applications in the real world
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