Commutative Theory Research Articles

Integral Representation and the Computation of Multiple Combinatorial Sums from Hall’s Commutator Theory

In this paper we prove a series of combinatorial identities arising from computing the exponents of the commutators in P. Hall’s collection formula. We also compute a sum in closed form that arises from using the collection formula in Chevalley groups for solving B. A. F. Wehrfritz problem on the regularity of their Sylow subgroups

Bergman spaces over noncommutative domains and commutant lifting

The goal of the present paper is to provide analogues of Sarason interpolation theorem in the Hardy algebra H∞(D) and Sz.Nagy-Foiaş commutant lifting theorem for contractions on Hilbert spaces in the setting of noncommutative Hardy spaces associated with noncommutative regular domains and varieties. This is accompanied by the study of multi-analytic operators with respect to the universal models associated with the regular domains (resp. varieties) and the study of multipliers of noncommutative Bergman spaces.As applications, we obtain Toeplitz-corona theorems for multi-analytic operators, commutant lifting in several variables where the liftings are in certain Schur classes, factorization of multi-analytic operators, and Nevanlinna-Pick interpolation results for multipliers of Bergman spaces over Reinhardt domains in Cn. In addition, we obtain Andô type dilations and inequalities on noncommutative bi-domains and varieties.

Variadic equational matching in associative and commutative theories

In this paper we study matching in equational theories that specify counterparts of associativity and commutativity for variadic function symbols. We design a procedure to solve a system of matching equations and prove its termination, soundness, completeness, and minimality. The minimal complete set of matchers for such a system can be infinite, but our algorithm computes its finite representation in the form of solved set. From the practical side, we identify two finitary cases and impose restrictions on the procedure to get an incomplete algorithm, which, based on our experiments, describes the input-output behavior and properties of Mathematica's flat and orderless pattern matching.

The Commutator of Fuzzy Congruences in Universal Algebras

We develop the commutator theory for fuzzy congruence relations of general universal algebras. In particular, for algebras in modular varieties, we characterize the commutator of fuzzy congruences using the Day’s terms.

Commutator theory for racks and quandles

We adapt the commutator theory of universal algebra to the particular setting of racks and quandles, exploiting a Galois connection between congruences and certain normal subgroups of the displacement group. Congruence properties, such as abelianness and centrality, are reflected by the corresponding relative displacement groups, and the global properties, solvability and nilpotence, are reflected by the properties of the whole displacement group. To show the new tool in action, we present three applications: non-existence theorems for quandles (no connected involutory quandles of order $2^k$, no latin quandles of order $\equiv2\pmod4$), a non-colorability theorem (knots with trivial Alexander polynomial are not colorable by solvable quandles; in particular, by finite latin quandles), and a strengthening of Glauberman's results on Bruck loops of odd order.

Some Commutativity Theorems on Lie Ideals of Semiprime Rings

Some Commutativity Theorems on Lie Ideals of Semiprime Rings

Characterizations of dual truncated Toeplitz operators

We present an operator equation characterization of a dual truncated Toeplitz (DTT) operator using the compressed shift operator. This characterization is an analogue of the classical characterization of a Toeplitz operator [2] and the more recent characterization of a truncated Toeplitz operator [32]. As applications of this characterization, we give short proofs and refinements of some basic properties of DTT operators which were obtained in [9], [10] by more complicated methods. Furthermore, we determine when a DTT operator is normal, and when two DTT operators commute. We identify the commutant of the compressed shift which is a commutant lifting theorem in spirit.

Noncommutative field theory and composite Fermi liquids in some quantum Hall systems

Composite Fermi liquid metals arise at certain special filling fractions in the quantum Hall regime and play an important role as parent states of gapped states with quantized Hall response. They have been successfully described by the Halperin-Lee-Read (HLR) theory of a Fermi surface of composite fermions coupled to a $U(1)$ gauge field with a Chern-Simons term. However, the validity of the HLR description when the microscopic system is restricted to a single Landau has not been clear. Here for the specific case of bosons at filling $\nu = 1$, we build on earlier work from the 1990s to formulate a low energy description that takes the form of a {\em non-commutative} field theory. This theory has a Fermi surface of composite fermions coupled to a $U(1)$ gauge field with no Chern-Simons term but with the feature that all fields are defined in a non-commutative spacetime. An approximate mapping of the long wavelength, small amplitude gauge fluctuations yields a commutative effective field theory which, remarkably, takes the HLR form but with microscopic parameters correctly determined by the interaction strength. Extensions to some other composite fermi liquids, and to other related states of matter are discussed.

Medial and semimedial left quasigroups

In this paper, we investigate the class of semimedial left quasigroups, a class that properly contains racks and medial left quasigroups. We extend most of the results about commutator theory for racks collected in [M. Bonatto and D. Stanovský, Commutator theory for racks and quandles, preprint (2019), arXiv:1902.08980.] and we provide a structure theorem for finite medial connected racks.

The commutative justice and the doctrine of the restitution in the Summa Theologica of Saint Thomas

Este estudo pretende analisar a temática relativa ao conceito de Justiça e às suas espécies no pensamento de S. Tomás, apresentado particularmente em algumas questões da Suma Teológica, que é considerada sua obra principal. Por este motivo apresenta-se o esquema geral da Suma Teológica para poder entender melhor onde está colocada a temática sucessivamente estudada. Em seguida analisam-se as características fundamentais da Justiça e das suas espécies básicas, a partir da reflexão aristotélica sobre "justiça corretiva", retomada por Tomás na expressão "justiça comutativa", com destaque para a restituição, considerada por S. Tomás como um ato de justiça comutativa.

Isometric dilations and von Neumann inequality for finite rank commuting contractions

Motivated by Ball, Li, Timotin and Trent's Schur-Agler class version of commutant lifting theorem, we introduce a class, denoted by Pn(H), of n-tuples of commuting contractions on a Hilbert space H. We always assume that n≥3. The importance of this class of n-tuples stems from the fact that the von Neumann inequality or the existence of isometric dilation does not hold in general for n-tuples, n≥3, of commuting contractions on Hilbert spaces (even in the level of finite dimensional Hilbert spaces). Under some rank-finiteness assumptions, we prove that tuples in Pn(H) always admit explicit isometric dilations and satisfy a refined von Neumann inequality in terms of algebraic varieties in the closure of the unit polydisc in Cn.

Differential processes generated by two interpolators

We study couples of interpolators, the differentials they generate and their associated commutator theorems. An essential part of our analysis is the study of the intrinsic symmetries of the process. Since we work without any compatibility or categorical assumption, our results are flexible enough to generalize most known results for commutators or translation operators, in particular those of Cwikel, Kalton, Milman, Rochberg \cite{ckmr} for differential methods and those of Carro, Cerd\`a and Soria \cite{caceso} for compatible interpolators. We also generalize stability and singularity results in \cite{cfg,ccfg,correa} from the complex method to general differential methods and obtain new incomparability results.

Computing GIT-fans with symmetry and the Mori chamber decomposition of \overline{𝑀}_{0,6}

We propose an algorithm to compute the GIT-fan for torus actions on affine varieties with symmetries. The algorithm combines computational techniques from commutative algebra, convex geometry, and group theory. We have implemented our algorithm in the Singular library gitfan.lib. Using our implementation, we compute the Mori chamber decomposition of Mov ⁡ ( M ¯ 0 , 6 ) \operatorname {Mov}(\overline {M}_{0,6}) .

On a generalization of Ando’s dilation theorem

We introduce the notion of Q-commuting operators which includes commuting operators. We prove a generalized version of the commutant lifting theorem and Ando’s dilation theorem in the context of Q-commuting operators.

Commutant Lifting and Nevanlinna–Pick Interpolation in Several Variables

This paper concerns a commutant lifting theorem and a Nevanlinna–Pick type interpolation result in the setting of multipliers from vector-valued Drury–Arveson space to a large class of vector-valued reproducing kernel Hilbert spaces over the unit ball in \({\mathbb {C}}^n\). The special case of reproducing kernel Hilbert spaces includes all natural examples of Hilbert spaces like Hardy space, Bergman space and weighted Bergman spaces over the unit ball.

Some Commutativity Theorems for Near-Rings with Left Multipliers

Some Commutativity Theorems for Near-Rings with Left Multipliers

Constrained characteristic functions, multivariable interpolation, and invariant subspaces

In this paper, we present a functional model theorem for completely non-coisometric n-tuples of operators in the noncommutative variety mathcal{V}_{f,varphi,mathcal{I}}(mathcal{H}) in terms of constrained characteristic functions. As an application, we prove that the constrained characteristic function is a complete unitary invariant for this class of elements, which can be viewed as the noncommutative analogue of the classical Sz.-Nagy–Foiaş functional model for completely nonunitary contractions. On the other hand, we provide a Sarason-type commutant lifting theorem. Applying this result, we solve the Nevanlinna–Pick-type interpolation problem in our setting. Moreover, we also obtain a Beurling-type characterization of the joint invariant subspaces under the operators B_{1},ldots,B_{n}, where the n-tuple (B_{1},ldots,B_{n}) is the universal model associated with the abstract noncommutative variety mathcal{V}_{f,varphi,mathcal{I}}.

Multiplication operators on Hardy and weighted Bergman spaces over planar regions

This paper studies some aspects of commutant theory and functional calculus for analytic multiplication operators on Hardy and weighted Bergman spaces over bounded planar regions. Multiplication operators defined by univalent functions are shown to commute only with multiplication operators. This result is generalized to a tuple of operators, and a sufficient condition is given for irreducibility of that induced by finite Blaschke products. Operators defined by fairly general ancestral functions are shown to commute with no nonzero compact operators, and these include the ones by monomial functions over annuli. For such operators over annuli, we characterize a certain dense subalgebra of the commutant. Norm and sequential weak closures of the analytic functional calculus algebra generated by a multiplication operator are characterized and essential spectral mapping properties obtained. Generalizing the similarity for finite Blaschke products to a larger class of weighted Bergman spaces, the commutant classification of these operators is obtained and seen strictly finer than the similarity classification, among other related results.

Computing the size of zero divisor graphs

A recent subject of study linking commutative ring theory with graph theory has been the concept of the zero divisor graph of a commutative ring. The zero divisor graph of a commutative ring exhibits a remarkable amount of graphical structure. Let G(R) be the zero divisor graph introduced by Beck [9], whose vertices are the elements of a ring R such that two distinct vertices x, y are adjacent provided that xy = 0. Let Г(R) be the zero divisor graph introduced by Anderson, Livingston [5] whose vertices are the non-zero zero divisors of the ring R such that two distinct vertices x, y are adjacent provided that xy = 0. Here, the authors investigate the size of the graphs G(ℤ n ), Г(ℤ n ).

Toward an In-Depth Understanding of the Commutation Processes in a SiC MOSFET Switching Cell Including Parasitic Elements

This article provides in-depth insight into the commutation processes of high-speed SiC <sc xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">mosfet</small> s and investigates how they are influenced by various parasitic inductances. The switching dynamics of wide-bandgap power semiconductor devices are significantly larger compared to those of silicon devices. Therefore, the parasitic elements in the switching cell become increasingly important, as they limit the current and voltage slopes and cause oscillations. A thorough understanding of these effects is necessary for the design of highly efficient and integrated next-generation power electronic converters. However, the means of modeling hard-switched transitions, e.g., equivalent circuits, have not been adapted sufficiently compared to existing models on relatively slow-switching devices. This article presents the theory of commutation in high-speed semiconductor devices and outlines its key differences to traditional commutation models. These new insights are used to analyze the influence of parasitic inductances on the switching transitions. The theory is validated experimentally by double-pulse tests with variable parasitic inductances, performed on a test printed circuit board equipped with SiC <sc xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">mosfet</small> s. The results are in good agreement with the theoretical findings.