Quaternions are a number system that has become increasingly useful for representing the rotations of objects in three-dimensional space and has important applications in theoretical and applied ...mathematics, physics, computer science, and engineering. This is the first book to provide a systematic, accessible, and self-contained exposition of quaternion linear algebra. It features previously unpublished research results with complete proofs and many open problems at various levels, as well as more than 200 exercises to facilitate use by students and instructors. Applications presented in the book include numerical ranges, invariant semidefinite subspaces, differential equations with symmetries, and matrix equations.
Designed for researchers and students across a variety of disciplines, the book can be read by anyone with a background in linear algebra, rudimentary complex analysis, and some multivariable calculus. Instructors will find it useful as a complementary text for undergraduate linear algebra courses or as a basis for a graduate course in linear algebra. The open problems can serve as research projects for undergraduates, topics for graduate students, or problems to be tackled by professional research mathematicians. The book is also an invaluable reference tool for researchers in fields where techniques based on quaternion analysis are used.
For the vanishing deformation parameter Formula omitted, the full structure of the (anti)commutator relations in the Formula omitted supersymmetric linear Formula omitted algebra is obtained for ...arbitrary weights Formula omitted and Formula omitted of the currents appearing on the left hand sides in these (anti)commutators. The Formula omitted algebra can be seen from this by taking the vanishing limit of other deformation parameter q with the proper contractions of the currents. For the nonzero Formula omitted, the complete structure of the Formula omitted supersymmetric linear Formula omitted algebra is determined for the arbitrary weight Formula omitted together with the constraint Formula omitted. The additional structures on the right hand sides in the (anti)commutators, compared to the above Formula omitted case, arise for the arbitrary weights Formula omitted and Formula omitted where the weight Formula omitted is outside of above region.
In this paper, we study the Aggarwal, Ciambelli, Detournay, and Somerhausen (ACDS) boundary conditions (Aggarwal et al. in JHEP 22:013, 2020) for Warped AdS Formula omitted (WAdS Formula omitted) in ...the framework of General Massive Gravity (GMG) in the quadratic ensemble. We construct the phase space, the asymptotic structure, and the asymptotic symmetry algebra. We show that the global surface charges are finite, but not integrable, and also we find the conditions to make them integrable. In addition, to confirm that the phase space has the same symmetries as that of a Warped Conformal Field Theory (WCFT), we compare the bulk entropy of Warped BTZ (WBTZ) black holes with the number of states belonging to a WCFT.
By using the K-free complex bosons and the K-free complex fermions, we construct the Formula omitted supersymmetric Formula omitted algebra which is the matrix generalization of previous Formula ...omitted supersymmetric Formula omitted algebra. By twisting this Formula omitted supersymmetric Formula omitted algebra, we obtain the Formula omitted supersymmetric Formula omitted algebra which is the matrix generalization of known Formula omitted supersymmetric topological Formula omitted algebra. From this two-dimensional symmetry algebra, we propose the operator product expansion (OPE) between the soft graviton and gravitino (as a first example), at nonzero deformation parameter, in the supersymmetric Einstein-Yang-Mills theory explicitly. Other six OPEs between the graviton, gravitino, gluon and gluino can be determined completely. At vanishing deformation parameter, we reproduce the known result of Fotopoulos, Stieberger, Taylor and Zhu on the above OPEs via celestial holography.
We present the first example of Formula omitted formulation for the extended higher-spin Formula omitted supergravity with the most general boundary conditions as an extension of the Formula omitted ...work, discovered recently by us (Özer and Filiz in Eur Phys J C 80(11):1072, 2020). Using the method proposed by Grumiller and Riegler, we restrict a consistent class of the most general boundary conditions to extend it. An important consequence of our method is that, for the loosest set of boundary conditions it ensures that their asymptotic symmetry algebras consist of two copies of the Formula omitted. Moreover, we impose some restrictions on the gauge fields for the most general boundary conditions, leading to the supersymmetric extensions of the Brown and Henneaux boundary conditions. Based on these results, we finally find out that the asymptotic symmetry algebras are two copies of the super Formula omitted algebra for Formula omitted extended higher-spin supergravity theory in Formula omitted.
We obtain the Courant bracket twisted simultaneously by a 2-form B and a bi-vector Formula omitted by calculating the Poisson bracket algebra of the symmetry generator in the basis obtained acting ...with the relevant twisting matrix. It is the extension of the Courant bracket that contains well known Schouten-Nijenhuis and Koszul bracket, as well as some new star brackets. We give interpretation to the star brackets as projections on isotropic subspaces.
This book is the natural continuation of Computational Commutative Algebra 1 with some twists.
The main part of this book is a breathtaking passeggiata through the computational domains of graded ...rings and modules and their Hilbert functions. Besides Gröbner bases, we encounter Hilbert bases, border bases, SAGBI bases, and even SuperG bases.
The tutorials traverse areas ranging from algebraic geometry and combinatorics to photogrammetry, magic squares, coding theory, statistics, and automatic theorem proving. Whereas in the first volume gardening and chess playing were not treated, in this volume they are.
This is a book for learning, teaching, reading, and most of all, enjoying the topic at hand. The theories it describes can be applied to anything from children's toys to oil production. If you buy it, probably one spot on your desk will be lost forever!
This book demonstrates current trends in research on combinatorial and computational commutative algebra with a primary emphasis on topics related to monomial ideals. In the text, theory is ...complemented by a number of examples and exercises.
Let Formula omitted be a locally compact quantum group with dual Formula omitted. Suppose that the left Haar weight Formula omitted and the dual left Haar weight Formula omitted are tracial, e.g. ...Formula omitted is a unimodular Kac algebra. We prove that for Formula omitted, the Fourier multiplier Formula omitted is bounded from Formula omitted to Formula omitted whenever the symbol x lies in Formula omitted, where Formula omitted. Moreover, we have ||mx:Lp(G^,phi^)right arrowLq(G^,phi^)||less than or equal tocp,q||x||Lr,infinity(G,phi),where Formula omitted is a constant depending only on p and q. This was first proved by Hörmander (Acta Math 104:93-140, 1960) for Formula omitted, and was recently extended to more general groups and quantum groups. Our work covers all these results and the proof is simpler. In particular, this also yields a family of Formula omitted-Fourier multipliers over discrete group von Neumann algebras. A similar result for Formula omitted- Formula omitted Schur multipliers is also proved.
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