Irreducible Representations of Wreath Products of Association Schemes

The wreath product of finite association schemes is a natural generalization of the notion of the wreath product of finite permutation groups. We determine all irreducible representations (the Jacobson radical) of a wreath product of two finite association schemes over an algebraically closed field in terms of the irreducible representations (Jacobson radicals) of the two factors involved.     

A Theta Lift for Spin7

We establish an example of a functorial lift from generic cuspidal representations of a similitude group of the type A ₁×C ₂ to generic representations of Spin₇. Our construction uses the theta correspondence associated to the dual pair of the type (A ₁×C ₂ B ₃) inside E ₇. We also consider another theta correspondence associated to the dual pair of type (A ₁×C ₂ A ₁×A ₁) in D ₆ and show that these two pairs fit into a tower and the standard properties of a tower of theta correspondences hold.     

On Linear Representations of Analytic Moufang Loops

Mathematical Notes -     

Neural network representations of quantum many-body states

Machine learning is currently the most active interdisciplinary field having numerous applications;additionally,machine-learning techniques are used to research quantum many-body problems.In this study,we first propose neural network quantum states(NNQSs)with general input observables and explore a few related properties,such as the tensor product and local unitary operation.Second,we determine the necessary and sufficient conditions for the representability of a general graph state using normalized NNQS.Finally,to quantify the approximation degree of a given pure state,we define the best approximation degree using normalized NNQSs.Furthermore,we observe that some 7V-qubit states can be represented by a normalized NNQS,such as separable pure states,Bell states and GHZ states.     

Stability of a Special Class of $$q_{ij}$$ -CCR and Extensions of Higher-Dimensional Noncommutative Tori

The -algebras A_{{q i}}, generated by generalised quon commutation relations are considered. The nuclearity of these algebras is proved. It is shown that A_{{q i}}, is isomorphic to the extension of a higher-dimensional noncommutative torus. Irreducible representations of A_{{q i}}, are considered. It is shown that the Fock representation is faithful.     

Extension of statistical replacement to systems with time-correlated fluctuations

Nonlinear systems with correlated stochastic parameters are approximated by simpler systems. This method is an extension of an earlier version of statistical replacement and statistical linearization. The extended method is applicable to systems with correlated fluctuations. We show how this general method reduces to the earlier methods in special cases.     

Exponential decay of connectivities in the two-dimensional ising model

We prove some results concerning the decay of connectivities in the low-temperature phase of the two-dimensional Ising model. These provide the bounds necessary to establish, nonperturbatively, large-deviation properties for block magnetizations in these systems. We also obtain estimates on the rate at which the finite-volume, plus-boundary-condition expectation of the spin at the origin converges to the spontaneous magnetization.On leave from São Paulo University, Brazil.     

Coherent orthogonal polynomials

We discuss a fundamental characteristic of orthogonal polynomials, like the existence of a Lie algebra behind them, which can be added to their other relevant aspects. At the basis of the complete framework for orthogonal polynomials we include thus–in addition to differential equations, recurrence relations, Hilbert spaces and square integrable functions–Lie algebra theory.     

Factorizing multivariate time series operators

Motivated by problems occurring in the empirical identification and modelling of a n-dimensional ARMA time series X(t) we study the possibility of obtaining a factorization (I + a₁B + … + a_pB^p) X(t) = [Π_i=1^p (I ? α_iB)] X(t), where B is the backward shift operator. Using a result in [3] we conclude that as in the univariate case such a factorization always exists, but unlike the univariate case in general the factorization is not unique for given a₁, a₂,…, a_p. In fact the number of possibilities is limited upwards by $\text{(np)!}\text{(n!)}{}^{\mathrm{p}}$, there being cases, however, where this maximum is not reached. Implications for the existence and possible use of transformations which removes nonstationarity (or almost nonstationarity) of X(t) are mentioned.