This paper is an addendum to the paper, published in Bulletin of Taras Shevchenko National University of Kyiv (Series: Physics & Mathematics), 2022, No. 3, pp. 42-44. In the original version does not ...given a clear definition of the corepresentation of an algebra with identity what led to some formal inaccuracies (regarding notations, writing of defining relations, etc.). All the results of the original paper are essentially correct, but an error was made in the last part of the proof of Theorem 1, which does not affect the correctness of the idea of proof, but requires extended considerations. In this addendum the author provides detailed information on everything related to statements about the generating elements and defining relations of the Munn algebras over algebras with identity.
In this paper, the realisation problem of linear multi-input multi-output, time-varying systems is studied. The approach, based on the theory of non-commutative polynomial rings, yields explicit and ...simple formulas for computation of the state coordinates as well as for state equations in observable canonical form. The formulas are based on (left) Euclidean polynomial division.
Authentication is a term very important for data communication security. We see many frauds due to authentication failure. The problem manifolds when communication is over insecure channel. Entity ...authentication is a term which involves proof of sender’s identity and very useful in various applications like in banking sector and various other client server mechanisms. Availability of quantum computers increases the vulnerability of breaking old protocols. Researchers are finding new platforms to overcome this problem and one such example is non commutative polynomial rings NCPR. In 2012, M.R.Vallauri MRV, in his paper suggested an authentication protocol using NCPR. He has proved security analysis under the assumption that polynomial symmetrical decomposition problem (PSDP) is hard. In this paper we show that the protocol suggested by him is breakable without solving PSDP. We also provide corrected protocol to overcome this problem.
Recently, inspired by the Connes-Kreimer Hopf algebra of rooted trees, the second named author introduced rooted tree maps as a family of linear maps on the non-commutative polynomial algebra in two ...letters. These give a class of relations among multiple zeta values,which are known to be a subclass of the so-called linear part of the Kawashima relations. In this paper we show the opposite implication, that is, the linear part of the Kawashima relations is implied by the relations coming from rooted tree maps.
The (classical) truncated moment problem, extensively studied by Curto and Fialkow, asks to characterize when a finite sequence of real numbers indexes by words in commuting variables can be ...represented with moments of a positive Borel measure
μ
on
R
n
. Burgdorf and Klep (J Oper Theory 68:141–163,
2012
) introduced its tracial analog, the truncated tracial moment problem, which replaces commuting variables with non-commuting ones and moments of
μ
with tracial moments of matrices. In the bivariate quartic case, where indices run over words in two variables of degree at most four, every sequence with a positive definite
7
×
7
moment matrix
M
2
can be represented with tracial moments (Burgdorf and Klep in C R Math Acad Sci Paris 348:721–726,
2010
,
2012
). In this article the case of singular
M
2
is studied. For
M
2
of rank at most 5 the problem is solved completely; namely, concrete measures are obtained whenever they exist and the uniqueness question of the minimal measures is answered. For
M
2
of rank 6 the problem splits into four cases, in two of which it is equivalent to the feasibility problem of certain linear matrix inequalities. Finally, the question of a flat extension of the moment matrix
M
2
is addressed. While this is the most powerful tool for solving the classical case, it is shown here by examples that, while sufficient, flat extensions are mostly not a necessary condition for the existence of a measure in the tracial case.
The paper gives an overview of an algebraic approach based on differential 1-forms, developed for the study of nonlinear control systems. The purpose of the paper is to describe the approach, comment ...on the necessary assumptions made, and demonstrate the effectiveness and limitations of the approach. Two very important aspects of the approach are as follows: (1) one works with differentials and not with functions, meaning that computations are, up to integration similar to the linear case and (2) the approach is used to study generic properties of control systems that hold for almost every point of a suitable domain. The first point means that solutions to various problems are found in terms of 1-forms and the integrability properties allow transformation of the solution back to the level of functions. The study of generic properties simplifies the presentation of the solutions, since there is no need to specify the working point and its neighbourhood. Finally, the paper includes an extensive list of publications, where the approach of 1-forms is studied or applied to solve different control problems.
Celotno besedilo
Dostopno za:
DOBA, IZUM, KILJ, NUK, ODKLJ, PILJ, PNG, SAZU, UILJ, UKNU, UL, UM, UPUK
Canonical bases, also called SAGBI bases, for subalgebras of the non-commutative polynomial ring are investigated. The process of subalgebra reduction is defined. Methods, including generalizations ...of the standard Gröbner bases techniques, are developed for the test whether bases are canonical, and for the completion procedure of constructing canonical bases. The special case of homogeneous subalgebras is discussed.
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