Quadratic Extensions of Special Hyperfields and the general Arason-Pfister Hauptsatz

Multialgebraic structures -- an "algebraic like" structure but endowed with multiple valued operations -- are considered and studied since 1930s. The use of hyperfields/hyperrings/multirings in connection with Real Algebraic Geometry started 15 years ago \cite{marshall2006real}. In the present work, we develop some fragments of the theory of algebraic extension for hyperfields, through the development of results concerning the superrings of polynomials. The significance of these multialgebraic methods to (univalent) Commutative Algebra is indicated by applying these results to algebraic theory of quadratic forms: we have obtained extensions of the validity of the Arason-Pfister Hauptsatz (\cite{arason1971hauptsatz}) - a positive answer to a question posed by J. Milnor in a classical paper of 1970 (\cite{milnor1970algebraick}).Comment: arXiv admin note: substantial text overlap with arXiv:2208.0853

Entropy algebras and Birkhoff factorization

We develop notions of Rota-Baxter structures and associated Birkhoff factorizations, in the context of min-plus semirings and their thermodynamic deformations, including deformations arising from quantum information measures such as the von Neumann entropy. We consider examples related to Manin's renormalization and computation program, to Markov random fields and to counting functions and zeta functions of algebraic varieties.Comment: 28 pages, LaTe

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

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

Field extensions, Derivations, and Matroids over Skew Hyperfields

We show that a field extension $K\subseteq L$ in positive characteristic $p$ and elements $x_e\in L$ for $e\in E$ gives rise to a matroid $M^\sigma$ on ground set $E$ with coefficients in a certain skew hyperfield $L^\sigma$. This skew hyperfield $L^\sigma$ is defined in terms of $L$ and its Frobenius action $\sigma:x\mapsto x^p$. The matroid underlying $M^\sigma$ describes the algebraic dependencies over $K$ among the $x_e\in L$ , and $M^\sigma$ itself comprises, for each $m\in \mathbb{Z}^E$, the space of $K$-derivations of $K\left(x_e^{p^{m_e}}: e\in E\right)$. The theory of matroid representation over hyperfields was developed by Baker and Bowler for commutative hyperfields. We partially extend their theory to skew hyperfields. To prove the duality theorems we need, we use a new axiom scheme in terms of quasi-Pl\"ucker coordinates.Comment: Changed the signing convention for coordinates to better conform to existing concepts in the literature (Tutte group, quasi-determinants

Normalizer Circuits and Quantum Computation

(Abridged abstract.) In this thesis we introduce new models of quantum computation to study the emergence of quantum speed-up in quantum computer algorithms. Our first contribution is a formalism of restricted quantum operations, named normalizer circuit formalism, based on algebraic extensions of the qubit Clifford gates (CNOT, Hadamard and $\pi/4$-phase gates): a normalizer circuit consists of quantum Fourier transforms (QFTs), automorphism gates and quadratic phase gates associated to a set $G$, which is either an abelian group or abelian hypergroup. Though Clifford circuits are efficiently classically simulable, we show that normalizer circuit models encompass Shor's celebrated factoring algorithm and the quantum algorithms for abelian Hidden Subgroup Problems. We develop classical-simulation techniques to characterize under which scenarios normalizer circuits provide quantum speed-ups. Finally, we devise new quantum algorithms for finding hidden hyperstructures. The results offer new insights into the source of quantum speed-ups for several algebraic problems. Our second contribution is an algebraic (group- and hypergroup-theoretic) framework for describing quantum many-body states and classically simulating quantum circuits. Our framework extends Gottesman's Pauli Stabilizer Formalism (PSF), wherein quantum states are written as joint eigenspaces of stabilizer groups of commuting Pauli operators: while the PSF is valid for qubit/qudit systems, our formalism can be applied to discrete- and continuous-variable systems, hybrid settings, and anyonic systems. These results enlarge the known families of quantum processes that can be efficiently classically simulated. This thesis also establishes a precise connection between Shor's quantum algorithm and the stabilizer formalism, revealing a common mathematical structure in several quantum speed-ups and error-correcting codes.Comment: PhD thesis, Technical University of Munich (2016). Please cite original papers if possible. Appendix E contains unpublished work on Gaussian unitaries. If you spot typos/omissions please email me at JLastNames at posteo dot net. Source: http://bit.ly/2gMdHn3. Related video talk: https://www.perimeterinstitute.ca/videos/toy-theory-quantum-speed-ups-based-stabilizer-formalism Posted on my birthda