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Register a new userSome Orthogonal Functions from Cauchy Determinants
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Lijing Wang
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We derive Legendre polynomials using Cauchy determinants with a generalization to power functions with real exponents greater than -1/2. We also provide a geometric formulation of Gram-Schmidt orthogonalization using the Hodge star operator.
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We evaluate Hankel determinants of matrices in which the entries are generating functions for paths consisting of up-steps, down-steps and level steps with a fixed starting point but variable end point. By specialisation, these determinant evaluations have numerous corollaries. In particular, one consequence is that the Hankel determinant of Motzkin prefix numbers equals 1, regardless of the size of the Hankel matrix.
As is well known, the common elementary functions defined over the real numbers can be generalized to act not only over the complex number field but also over the skew (non-commuting) field of the quaternions. In this paper, we detail a number of elementary functions extended to act over the skew field of Clifford multivectors, in both two and three dimensions. Complex numbers, quaternions and Cartesian vectors can be described by the various components within a Clifford multivector and from our results we are able to demonstrate new inter-relationships between these algebraic systems. One key relationship that we discover is that a complex number raised to a vector power produces a quaternion thus combining these systems within a single equation. We also find a single formula that produces the square root, amplitude and inverse of a multivector over one, two and three dimensions. Finally, comparing the functions over different dimension we observe that $ Cell left (Re^3 right) $ provides a particularly versatile algebraic framework.
We review some history and some recent results concerning Toeplitz determinants and their applications. We discuss, in particular, the crucial role of the two-dimensional Ising model in stimulating the development of the theory of Toeplitz determinants.
The accuracy and efficiency of ab-initio quantum Monte Carlo (QMC) algorithms benefits greatly from compact variational trial wave functions that accurately reproduce ground state properties of a system. We investigate the possibility of using multi-Slater-Jastrow trial wave functions with non-orthogonal determinants by optimizing identical single particle orbitals independently in separate determinants. As a test case, we compute variational and fixed-node diffusion Monte Carlo (FN-DMC) energies of a C$_2$ molecule. For a given multi-determinant expansion, we find that this non-orthogonal orbital optimization results in a consistent improvement in the variational energy and the FN-DMC energy on the order of a few tenths of an eV. Our calculations indicate that trial wave functions with non-orthogonal determinants can improve computed energies in a QMC calculation when compared to their orthogonal counterparts.
Let $O(2n+ell)$ be the group of orthogonal matrices of size $left(2n+ellright)times left(2n+ellright)$ equipped with the probability distribution given by normalized Haar measure. We study the probability begin{equation*} p_{2n}^{left(ellright)} = mathbb{P}left[M_{2n} , mbox{has no real eigenvalues}right], end{equation*} where $M_{2n}$ is the $2ntimes 2n$ left top minor of a $(2n+ell)times(2n+ell)$ orthogonal matrix. We prove that this probability is given in terms of a determinant identity minus a weighted Hankel matrix of size $ntimes n$ that depends on the truncation parameter $ell$. For $ell=1$ the matrix coincides with the Hilbert matrix and we prove begin{equation*} p_{2n}^{left(1right)} sim n^{-3/8}, mbox{ when }n to infty. end{equation*} We also discuss connections of the above to the persistence probability for random Kac polynomials.
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