Meixner class of non-commutative generalized stochastic processes with freely independent values II. The generating function

Let $T$ be an underlying space with a non-atomic measure $\sigma$ on it. In [{\it Comm.\ Math.\ Phys.}\ {\bf 292} (2009), 99--129] the Meixner class of non-commutative generalized stochastic processes with freely independent values, $\omega=(\omega(t))_{t\in T}$, was characterized through the continuity of the corresponding orthogonal polynomials. In this paper, we derive a generating function for these orthogonal polynomials. The first question we have to answer is: What should serve as a generating function for a system of polynomials of infinitely many non-commuting variables? We construct a class of operator-valued functions $Z=(Z(t))_{t\in T}$ such that $Z(t)$ commutes with $\omega(s)$ for any $s,t\in T$. Then a generating function can be understood as $G(Z,\omega)=\sum_{n=0}^\infty \int_{T^n}P^{(n)}(\omega(t_1),...,\omega(t_n))Z(t_1)...Z(t_n)\sigma(dt_1)...\sigma(dt_n)$, where $P^{(n)}(\omega(t_1),...,\omega(t_n))$ is (the kernel of the) $n$-th orthogonal polynomial. We derive an explicit form of $G(Z,\omega)$, which has a resolvent form and resembles the generating function in the classical case, albeit it involves integrals of non-commuting operators. We finally discuss a related problem of the action of the annihilation operators $\partial_t$, $t\in T$. In contrast to the classical case, we prove that the operators \di_t related to the free Gaussian and Poisson processes have a property of globality. This result is genuinely infinite-dimensional, since in one dimension one loses the notion of globality

Interacting Fock spaces and Gaussianization of probability measures

We prove that any probability measure on $\mathbb R$, with moments of all orders, is the vacuum distribution, in an appropriate interacting Fock space, of the field operator plus (in the non symmetric case) a function of the number operator. A corollary of this is that all the momenta of such a measure are expressible in terms of the Jacobi parameters, associated to its orthogonal polynomials, by means of diagrams involving only non crossing pair partitions (and singletons, in the non symmetric case). This means that, with our construction, the combinatorics of the momenta of any probability measure (with all moments) is reduced to that of a generalized Gaussian. This phenomenon we call {\it Gaussianization}. Finally we define, in terms of the Jacobi parameters, a new convolution among probability measures which we call {\it universal} because any probability measure (with all moments) is infinitely divisible with respect to this convolution. All these results are extended to the case of many (in fact infinitely many) variables

Exactness of the Fock space representation of the q-commutation relations

We show that for all q in the interval (-1,1), the Fock representation of the q-commutation relations can be unitarily embedded into the Fock representation of the extended Cuntz algebra. In particular, this implies that the C*-algebra generated by the Fock representation of the q-commutation relations is exact. An immediate consequence is that the q-Gaussian von Neumann algebra is weakly exact for all q in the interval (-1,1).Comment: 20 page

Parallel patterns determination in solving cyclic flow shop problem with setups

© 2017 Archives of Control Sciences 2017. The subject of this work is the new idea of blocks for the cyclic flow shop problem with setup times, using multiple patterns with different sizes determined for each machine constituting optimal schedule of cities for the traveling salesman problem (TSP). We propose to take advantage of the Intel Xeon Phi parallel computing environment during so-called 'blocks' determination basing on patterns, in effect significantly improving the quality of obtained results

Unbounded representations of qq-deformation of Cuntz algebra

We study a deformation of the Cuntz-Toeplitz $C^*$-algebra determined by the relations $a_i^*a_i=1+q a_ia_i^*, a_i^*a_j=0$. We define well-behaved unbounded *-representations of the *-algebra defined by relations above and classify all such irreducible representations up to unitary equivalence.Comment: 13 pages, Submitted to Lett. Math. Phy

Isomorphisms of Cayley graphs of surface groups

A combinatorial proof is given for the fact that the Cayley graph of the fundamental group Γg of the closed, orientable surface of genus g ≥ 2 with respect to the usual generating set is isomorphic to the Cayley graph of a certain Coxeter group generated by 4g elements

Solving the Frustrated Spherical Model with q-Polynomials

We analyse the Spherical Model with frustration induced by an external gauge field. In infinite dimensions, this has been recently mapped onto a problem of q-deformed oscillators, whose real parameter q measures the frustration. We find the analytic solution of this model by suitably representing the q-oscillator algebra with q-Hermite polynomials. We also present a related Matrix Model which possesses the same diagrammatic expansion in the planar approximation. Its interaction potential is oscillating at infinity with period log(q), and may lead to interesting metastability phenomena beyond the planar approximation. The Spherical Model is similarly q-periodic, but does not exhibit such phenomena: actually its low-temperature phase is not glassy and depends smoothly on q.Comment: Latex, 14 pages, 2 eps figure

The Energy Operator for a Model with a Multiparametric Infinite Statistics

In this paper we consider energy operator (a free Hamiltonian), in the second-quantized approach, for the multiparameter quon algebras: $a_{i}a_{j}^{\dagger}-q_{ij}a_{j}^{\dagger}a_{i} = \delta_{ij}, i,j\in I$ with $(q_{ij})_{i,j\in I}$ any hermitian matrix of deformation parameters. We obtain an elegant formula for normally ordered (sometimes called Wick-ordered) series expansions of number operators (which determine a free Hamiltonian). As a main result (see Theorem 1) we prove that the number operators are given, with respect to a basis formed by "generalized Lie elements", by certain normally ordered quadratic expressions with coefficients given precisely by the entries of the inverses of Gram matrices of multiparticle weight spaces. (This settles a conjecture of two of the authors (S.M and A.P), stated in [8]). These Gram matrices are hermitian generalizations of the Varchenko's matrices, associated to a quantum (symmetric) bilinear form of diagonal arrangements of hyperplanes (see [12]). The solution of the inversion problem of such matrices in [9] (Theorem 2.2.17), leads to an effective formula for the number operators studied in this paper. The one parameter case, in the monomial basis, was studied by Zagier [15], Stanciu [11] and M{\o}ller [6].Comment: 24 pages. accepted in J. Phys. A. Math. Ge

Meixner class of non-commutative generalized stochastic processes with freely independent values I. A characterization

Let $T$ be an underlying space with a non-atomic measure $\sigma$ on it (e.g. $T=\mathbb R^d$ and $\sigma$ is the Lebesgue measure). We introduce and study a class of non-commutative generalized stochastic processes, indexed by points of $T$, with freely independent values. Such a process (field), $\omega=\omega(t)$, $t\in T$, is given a rigorous meaning through smearing out with test functions on $T$, with $\int_T \sigma(dt)f(t)\omega(t)$ being a (bounded) linear operator in a full Fock space. We define a set $\mathbf{CP}$ of all continuous polynomials of $\omega$, and then define a con-commutative $L^2$-space $L^2(\tau)$ by taking the closure of $\mathbf{CP}$ in the norm $\|P\|_{L^2(\tau)}:=\|P\Omega\|$, where $\Omega$ is the vacuum in the Fock space. Through procedure of orthogonalization of polynomials, we construct a unitary isomorphism between $L^2(\tau)$ and a (Fock-space-type) Hilbert space $\mathbb F=\mathbb R\oplus\bigoplus_{n=1}^\infty L^2(T^n,\gamma_n)$, with explicitly given measures $\gamma_n$. We identify the Meixner class as those processes for which the procedure of orthogonalization leaves the set $\mathbf {CP}$ invariant. (Note that, in the general case, the projection of a continuous monomial of oder $n$ onto the $n$-th chaos need not remain a continuous polynomial.) Each element of the Meixner class is characterized by two continuous functions $\lambda$ and $\eta\ge0$ on $T$, such that, in the $\mathbb F$ space, $\omega$ has representation \omega(t)=\di_t^\dag+\lambda(t)\di_t^\dag\di_t+\di_t+\eta(t)\di_t^\dag\di^2_t, where \di_t^\dag and \di_t are the usual creation and annihilation operators at point $t$

On C*-algebras generated by pairs of q-commuting isometries

We consider the C*-algebras O_2^q and A_2^q generated, respectively, by isometries s_1, s_2 satisfying the relation s_1^* s_2 = q s_2 s_1^* with |q| < 1 (the deformed Cuntz relation), and by isometries s_1, s_2 satisfying the relation s_2 s_1 = q s_1 s_2 with |q| = 1. We show that O_2^q is isomorphic to the Cuntz-Toeplitz C*-algebra O_2^0 for any |q| < 1. We further prove that A_2^{q_1} is isomorphic to A_2^{q_2} if and only if either q_1 = q_2 or q_1 = complex conjugate of q_2. In the second part of our paper, we discuss the complexity of the representation theory of A_2^q. We show that A_2^q is *-wild for any q in the circle |q| = 1, and hence that A_2^q is not nuclear for any q in the circle.Comment: 18 pages, LaTeX2e "article" document class; submitted. V2 clarifies the relationships between the various deformation systems treate