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We study a bilinear multiplication rule on 2x2 matrices which is intermediate between the ordinary matrix product and the Hadamard matrix product, and we relate this to the hyperbolic motion group of the plane.
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Squared singular values of a product of s square random Ginibre matrices are asymptotically characterized by probability distribution P_s(x), such that their moments are equal to the Fuss-Catalan numbers or order s. We find a representation of the Fuss--Catalan distributions P_s(x) in terms of a combination of s hypergeometric functions of the type sF_{s-1}. The explicit formula derived here is exact for an arbitrary positive integer s and for s=1 it reduces to the Marchenko--Pastur distribution. Using similar techniques, involving Mellin transform and the Meijer G-function, we find exact expressions for the Raney probability distributions, the moments of which are given by a two parameter generalization of the Fuss-Catalan numbers. These distributions can also be considered as a two parameter generalization of the Wigner semicircle law.
By using the quasi-determinant the construction of Gelfand et al. leads to the inverse of a matrix with noncommuting entries. In this work we offer a new method that is more suitable for physical purposes and motivated by deformation quantization, where our constructed algorithm emulates the commutative case and in addition gives corrections coming from the noncommutativity of the entries. Furthermore, we provide an equivalence of the introduced algorithm and the construction via quasi-determinants.
A new Lax pair for the sixth Painleve equation $P_{VI}$ is constructed in the framework of the loop algebra $mathfrak{so}(8)[z,z^{-1}]$. The whole affine Weyl group symmetry of $P_{VI}$ is interpreted as gauge transformations of the corresponding linear problem.
Let $V$ be a finite dimensional inner product space over $mathbb{R}$ with dimension $n$, where $nin mathbb{N}$, $wedge^{r}V$ be the exterior algebra of $V$, the problem is to find $max_{| xi | = 1, | eta | = 1}| xi wedge eta |$ where $k,l$ $in mathbb{N},$ $forall xi in wedge^{k}V, eta in wedge^{l}V.$ This is a problem suggested by the famous Nobel Prize Winner C.N. Yang. He solved this problem for $kleq 2$ in [1], and made the following textbf{conjecture} in [2] : If $n=2m$, $k=2r$, $l=2s$, then the maximum is achieved when $xi_{max} = frac{omega^{k}}{| omega^{k}|}, eta_{max} = frac{omega^{l}}{| omega^{l}|}$, where $ omega = Sigma_{i=1}^m e_{2i-1}wedge e_{2i}, $ and ${e_{k}}_{k=1}^{2m}$ is an orthonormal basis of V. From a physicists point of view, this problem is just the dual version of the easier part of the well-known Beauzamy-Bombieri inequality for product of polynomials in many variables, which is discussed in [4]. Here the duality is referred as the well known Bose-Fermi correspondence, where we consider the skew-symmetric algebra(alternative forms) instead of the familiar symmetric algebra(polynomials in many variables) In this paper, for two cases we give estimations of the maximum of exterior products, and the Yangs conjecture is answered partially under some special cases.
We analyze the Moyal star product in deformation quantization from the resurgence theory perspective. By putting algebraic conditions on Borel transforms, one can define the space of ``algebro-resurgent series (a subspace of $1$-Gevrey formal series in $ihbar/2$ with coefficients in $C{q,p}$), which we show is stable under Moyal star product.
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