Commutative Structures Research Articles

Dispersion and Limit Theorems for Random Walks Associated with Hypergeometric Functions of Type BC

The spherical functions of the non-compact Grassmann manifolds $$G_{p,q}({\mathbb {F}})=G/K$$ over the (skew-)fields $${\mathbb {F}}={\mathbb {R}}, {\mathbb {C}}, {\mathbb {H}}$$ with rank $$q\ge 1$$ and dimension parameter $$p>q$$ can be described as Heckman–Opdam hypergeometric functions of type BC, where the double coset space G / / K is identified with the Weyl chamber $$ C_q^B\subset {\mathbb {R}}^q$$ of type B. The corresponding product formulas and Harish-Chandra integral representations were recently written down by M. Rosler and the author in an explicit way such that both formulas can be extended analytically to all real parameters $$p\in [2q-1,\infty [$$ , and that associated commutative convolution structures $$*_p$$ on $$C_q^B$$ exist. In this paper, we study the associated moment functions and the dispersion of probability measures on $$C_q^B$$ with the aid of this generalized integral representation. This leads to strong laws of large numbers and central limit theorems for associated time-homogeneous random walks on $$(C_q^B, *_p)$$ where the moment functions and the dispersion appear in order to determine drift vectors and covariance matrices of these limit laws explicitly. For integers p, all results have interpretations for G-invariant random walks on the Grassmannians G / K. Besides the BC-cases, we also study the spaces $$GL(q,{\mathbb {F}})/U(q,{\mathbb {F}})$$ , which are related to Weyl chambers of type A, and for which corresponding results hold. For the rank-one-case $$q=1$$ , the results of this paper are well known in the context of Jacobi-type hypergroups on $$[0,\infty [$$ .

On the geometry and algebra of networks with state

In [1] an algebra of automata with interfaces, Span(Graph), was introduced with main operation being communicating-parallel composition – a system is represented by an expression in this algebra. A system so described has two aspects: an informal network geometry arising from the algebraic expression, and a space of states and transition given by its evaluation in Span(Graph). Note that Span(Graph) yields purely compositional descriptions of systems.In this paper we give an alternative globally (non-compositional) view of the same systems. In order to do this we make mathematically precise the network geometry in terms of monoidal graphs, and assignment of state in terms of morphisms of monoidal graphs. We call such globally described systems networks with state. To relate networks with state and Span(Graph) systems we use the algebra of cospans of monoidal graphs.As an illustration we give a new representation of a (non-compositionally described) class of Petri nets in the compositional setting of Span(Graph). We also present a tool for calculating with networks with state.Both algebras, of spans and of cospans, are symmetric monoidal categories with commutative separable algebra structures on the objects.We include a short section, following [2], in which we show that a simple modification of the algebra allows the description of networks in which the tangling of connectors may be represented, yielding a connection with the theory of knots.

A multivariate version of the disk convolution

We present an explicit product formula for the spherical functions of the compact Gelfand pairs (G,K1)=(SU(p+q),SU(p)×SU(q)) with p≥2q, which can be considered as the elementary spherical functions of one-dimensional K-type for the Hermitian symmetric spaces G/K with K=S(U(p)×U(q)). Due to results of Heckman, they can be expressed in terms of Heckman–Opdam Jacobi polynomials of type BCq with specific half-integer multiplicities. By analytic continuation with respect to the multiplicity parameters we obtain positive product formulas for the extensions of these spherical functions as well as associated compact and commutative hypergroup structures parametrized by real p∈]2q−1,∞[. We also obtain explicit product formulas for the involved continuous two-parameter family of Heckman–Opdam Jacobi polynomials with regular, but not necessarily positive multiplicities. The results of this paper extend well known results for the disk convolutions for q=1 to higher rank.

Operadic multiplications in equivariant spectra, norms, and transfers

We study homotopy-coherent commutative multiplicative structures on equivariant spaces and spectra. We define N∞ operads, equivariant generalizations of E∞ operads. Algebras in equivariant spectra over an N∞ operad model homotopically commutative equivariant ring spectra that only admit certain collections of Hill–Hopkins–Ravenel norms, determined by the operad. Analogously, algebras in equivariant spaces over an N∞ operad provide explicit constructions of certain transfers. This characterization yields a conceptual explanation of the structure of equivariant infinite loop spaces.To explain the relationship between norms, transfers, and N∞ operads, we discuss the general features of these operads, linking their properties to families of finite sets with group actions and analyzing their behavior under norms and geometric fixed points. A surprising consequence of our study is that in stark contract to the classical setting, equivariantly the little disks and linear isometries operads for a general incomplete universe U need not determine the same algebras.Our work is motivated by the need to provide a framework to describe the flavors of commutativity seen in recent work of the second author and Hopkins on localization of equivariant commutative ring spectra.

Tree- versus graph-level quasilocal Poincaré duality on $$S^{1}$$ S 1

Among its many corollaries, Poincare duality implies that the de Rham cohomology of a compact oriented manifold is a shifted commutative Frobenius algebra --- a commutative Frobenius algebra in which the comultiplication has cohomological degree equal to the dimension of the manifold. We study the question of whether this structure lifts to a "homotopy" shifted commutative Frobenius algebra structure at the cochain level. To make this question nontrivial, we impose a mild locality-type condition that we call "quasilocality": strict locality at the cochain level is unreasonable, but it is reasonable to ask for homotopically-constant families of operations that become local "in the limit." To make the question concrete, we take the manifold to be the one-dimensional circle. The answer to whether a quasilocal homotopy-Frobenius algebra structure exists turns out to depend on the choice of context in which to do homotopy algebra. There are two reasonable worlds in which to study structures (like Frobenius algebras) that involve many-to-many operations: one can work at "tree level," corresponding roughly to the world of operadic homotopy algebras and their homotopy modules; or one can work at "graph level," corresponding to the world of PROPs. For the tree-level version of our question, the answer is the unsurprising "Yes, such a structure exists" --- indeed, it is unique up to a contractible space of choices. But for the graph-level version, the answer is the surprising "No, such a structure does not exist." Most of the paper consists of computing explicitly this nonexistence, which is controlled by the numerical value of a certain obstruction, and we compute this value explicitly via a sequence of integrals.

Quantum stabilizer codes from Abelian and non-Abelian groups association schemes

A new method for the construction of the binary quantum stabilizer codes is provided, where the construction is based on Abelian and non-Abelian groups association schemes. The association schemes based on non-Abelian groups are constructed by bases for the regular representation from U6n, T4n, V8n and dihedral D2n groups. By using Abelian group association schemes followed by cyclic groups and non-Abelian group association schemes a list of binary stabilizer codes up to 40 qubits is given in tables 4, 5 and 10. Moreover, several binary stabilizer codes of minimum distances 5, 7 and 8 with good quantum parameters is presented. The preference of this method specially for Abelian group association schemes is that one can construct any binary quantum stabilizer code with any distance by using the commutative structure of association schemes.

Estimation and Orthogonal Block Structure

Estimators with good behaviors for estimable vectors and variance components are obtained for a class of models that contains the well known models with orthogonal block structure, OBS, see [15], [16] and [1], [2]. The study observations of these estimators uses commutative Jordan Algebras, CJA, and extends the one given for a more restricted class of models, the models with commutative orthogonal block structure, COBS, in which the orthogonal projection matrix on the space spanned by the means vector commute with all variance-covariance matrices, see [7].

Tensors, !-graphs, and non-commutative quantum structures

Categorical quantum mechanics (CQM) and the theory of quantum groups rely heavily on the use of structures that have both an algebraic and co-algebraic component, making them well-suited for manipulation using diagrammatic techniques. Diagrams allow us to easily form complex compositions of (co)algebraic structures, and prove their equality via graph rewriting. One of the biggest challenges in going beyond simple rewriting-based proofs is designing a graphical language that is expressive enough to prove interesting properties (e.g. normal form results) about not just single diagrams, but entire families of diagrams. One candidate is the language of !-graphs, which consist of graphs with certain subgraphs marked with boxes (called !-boxes) that can be repeated any number of times. New !-graph equations can then be proved using a powerful technique called !-box induction. However, previously this technique only applied to commutative (or cocommutative) algebraic structures, severely limiting its applications in some parts of CQM and (especially) quantum groups. In this paper, we fix this shortcoming by offering a new semantics for non-commutative !-graphs using an enriched version of Penrose's abstract tensor notation.

Frobenius map for the centers of Hecke algebras

We introduce a commutative associative graded algebra structure on the direct sum Z of the centers of the Hecke algebras associated to the symmetric groups in n letters for all n. As a natural deformation of the classical construction of Frobenius, we establish an algebra isomorphism from the algebra Z to the ring of symmetric functions. This isomorphism provides an identification between several distinguished bases for the centers (introduced by Geck-Rouquier, Jones, Lascoux) and explicit bases of symmetric functions.

Inference for types and structured families of commutative orthogonal block structures

Models with commutative orthogonal block structure, COBS, have orthogonal block structure, OBS, and their least square estimators for estimable vectors are, as it will be shown, best linear unbiased estimator, BLUE. Commutative Jordan algebras will be used to study the algebraic structure of the models and to define special types of models for which explicit expressions for the estimation of variance components are obtained. Once normality is assumed, inference using pivot variables is quite straightforward. To illustrate this class of models we will present unbalanced examples before considering families of models. When the models in a family correspond to the treatments of a base design, the family is structured. It will be shown how, under quite general conditions, the action of the factors in the base design on estimable vectors, can be studied.

Method of reconstructing the structure of the group telemetric signals based on the graph model

Evaluation of a complex technical object during the test is based on analysis of recorded sensor parameters. Parameters are passed from the object to a data center in the form of group telemetric signal (GTS). When transferring GTS in some cases, their accompanying descriptions of the structure are missing or contain inaccuracies. To solve the problem of reconstructing or validating GTS structure authors describe a new approach to GTS structure analysis, based on application of graph theory and methods of correlation analysis. The article considers the graph model describing GTS, which allows submitting a GTS as a union of sensors commutation structure and characteristics of parameters, and method of GTS construction.

A generalized geometrical piecewise‐affine model of DC‐DC power electronic converters

AbstractA generalized model of the dynamics (GMD) of DC‐DC power electronic converters (PECs) is discussed in this paper. It is a geometrical piecewise‐affine continuous‐time model. The general idea of the GMD is to determine the local dynamic behavior of trajectories on the faces of the PEC commutation structure, which is a geometrical model of its commutation. This allows us to establish the direction of PEC dynamics on these faces. It can be either ‘entering’ into specific regions in state space or ‘exiting’ from them. Therefore, the local PEC dynamics can be treated as logical (two‐state). In practice, the GMD can be used for the analysis of PEC practical stability, which is a completely different concept from the concept of PEC stability in the classical Lyapunov sense. An outline of the design‐oriented approach to PEC practical stability analysis, which is based on the GMD, has also been presented.As illustrative examples, the GMD of a boost converter under peak current‐mode control and its application are presented. These examples show that the Lyapunov stability of a given PEC does not imply its practical stability, and that the results of PEC Lyapunov stability analysis and practical stability analysis are complementary to each other. Copyright © 2013 John Wiley & Sons, Ltd.

Local realism does not impose the commutative algebraic structure into all quantum observables

Malley and Fine (Phys. Rev. A 2006 73 066102) discussed that all quantum observables would commute simultaneously if we accept a realistic theory of the Bell type (a local realistic theory) for quantum events, provided that all quantum events, including every quantum state and every observable (including every projector), are reproduced by a realistic theory of the Bell type for quantum events. We point out that their claim leads us to a contradiction. We study the relation between a local realistic theory and commutativity of quantum observables in a finite-dimensional space. We show that a realistic theory of the Bell type for quantum events does not impose the commutative algebraic structure into the set of all quantum observables if all quantum events are reproduced by a local realistic theory. We discuss that a violation of Bell locality is derived within a realistic theory of the Kochen–Specker type within quantum events.

The piecewise‐affine model of buck converter suitable for practical stability analysis

AbstractA generalized geometrical piecewise‐affine continuous‐time model (GMD) of buck converter under pulse‐width modulated (PWM) voltage‐mode control is presented in this paper. In general, such a model can be applied to any DC‐DC power electronic converter (PEC) in which the valves are modelled as ideal switches. The GMD is suitable and convenient to analyse PEC practical stability which is a completely different concept in relation to the notion of its stability in the classical Lyapunov sense.The PEC GMD is based on its commutation structure which is a general geometrical model of its commutation. The general idea of this model consists in determining the local dynamic behaviour of PEC trajectories on the faces of its commutation structure and/or their sections. These faces and sections are treated as geometrical objects with generalized local dynamics.The analysis of buck converter practical stability is carried out using a new method based directly on the definition of this term but not Lyapunov‐like functions as in the direct method. It has been shown that PEC Lyapunov stability does not imply its practical stability. These two concepts are complementary to each other. Copyright © 2013 John Wiley & Sons, Ltd.

On fully regular $$\mathcal{AG }$$ -groupoids

One of the best approaches to study one type of algebraic structure is to connect it with other type of algebraic structure which is better explored. In this paper we have accomplished this aim by connecting \({\mathcal{AG }}\)-groupoids with some useful associative and commutative algebraic structures. We have also introduced a fully regular class of an \( {\mathcal{AG }}\)-groupoid and shown that an \({\mathcal{AG }}\)-groupoid \(\mathcal S \) with left identity is fully regular if and only if \(\mathcal{L=L }^{i+1}\), for any left ideal \(\mathcal L \) of \(\mathcal S \), where \(i=1,\ldots ,n\).

On Holomorphic Hypercomplex Connections

The main purpose of this paper is to study some properties of integrable commutative hypercomplex structures endowed with a holomorphic torsion-free affine connection whose curvature tensor satisfies the purity condition with respect to the covariantly constant structure affinors.

Pauli graphs, Riemann hypothesis, and Goldbach pairs

Let consider the Pauli group $\mathcal{P}_q=<X,Z>$ with unitary quantum generators $X$ (shift) and $Z$ (clock) acting on the vectors of the $q$-dimensional Hilbert space via $X|s> =|s+1>$ and $Z|s> =\omega^s |s>$, with $\omega=\exp(2i\pi/q)$. It has been found that the number of maximal mutually commuting sets within $\mathcal{P}_q$ is controlled by the Dedekind psi function $\psi(q)=q \prod_{p|q}(1+\frac{1}{p})$ (with $p$ a prime) \cite{Planat2011} and that there exists a specific inequality $\frac{\psi (q)}{q}>e^{\gamma}\log \log q$, involving the Euler constant $\gamma \sim 0.577$, that is only satisfied at specific low dimensions $q \in \mathcal {A}=\{2,3,4,5,6,8,10,12,18,30\}$. The set $\mathcal{A}$ is closely related to the set $\mathcal{A} \cup \{1,24\}$ of integers that are totally Goldbach, i.e. that consist of all primes $p2$) is equivalent to Riemann hypothesis. Introducing the Hardy-Littlewood function $R(q)=2 C_2 \prod_{p|n}\frac{p-1}{p-2}$ (with $C_2 \sim 0.660$ the twin prime constant), that is used for estimating the number $g(q) \sim R(q) \frac{q}{\ln^2 q}$ of Goldbach pairs, one shows that the new inequality $\frac{R(N_r)}{\log \log N_r} \gtrapprox e^{\gamma}$ is also equivalent to Riemann hypothesis. In this paper, these number theoretical properties are discusssed in the context of the qudit commutation structure.

Commutative orthogonal block structure and error orthogonal models

A model has orthogonal block structure, OBS, if it has variance-covariance matrix&nbsp;that is a linear combination of known pairwise orthogonal orthogonal projection matrices that sum&nbsp;to the identity matrix. These models were introduced by Nelder is 1965, and continue to play an&nbsp;important part in randomized block designs. Two important types of OBS are related, and necessary and sufficient conditions for model of&nbsp;one type belonging to the other are determined. The first type, is that of models with commutative orthogonal block structure in which T,&nbsp;the orthogonal projection matrix on the space spanned by the mean vector, commutes with the&nbsp;orthogonal projection matrices in the expression of the variance-covariance matrix. The second type, is that of error orthogonal models. These results open the possibility of deepening the study of the important class of models with&nbsp;OBS.

Weak attractors for the dissipative fractional Heisenberg equation

In this short paper, we consider the long time behaviors of the fractional Heisenberg equation and the existence of a global weak attractor is proved for the shift dynamics in the path space. The key ingredient is some new type of commutator structure introduced in this paper, which seems indispensable in proving the compactness of the dynamics. The technique introduced in this paper may also be useful to other fractional order partial differential equations.

On a VOA Associated with the Simple Jordan Algebra of Type D

If a vertex operator algebra satisfies dim V 0 = 1 and V 1 = {0}, then V 2 has a commutative (not necessarily associative) algebra structure, called the Griess algebra. By using a vertex operator algebra associated with the simple Jordan algebra of type D, this article gives a counterexample to the following assertion: If R is a subalgebra of the Griess algebra, then the weight two space of the vertex operator subalgebra VOA(R) generated by R coincides with R.