Direct formulation to Cholesky decomposition of a general nonsingular correlation matrix

We present two novel, explicit representations of Cholesky factor of a nonsingular correlation matrix. The first representation uses semi-partial correlation coefficients as its entries. The second, uses an equivalent form of the square roots of the differences between two ratios of successive determinants. Each of the two new forms enjoys parsimony of notations and offers a simpler alternative to both spherical factorization and the multiplicative partial correlation Cholesky matrix (Cooke et al 2011). Two relevant applications are offered for each form: a simple $t$-test for assessing the independence of a single variable in a multivariate normal structure, and a straightforward algorithm for generating random positive-definite correlation matrix. The second representation is also extended to any nonsingular hermitian matrix.Comment: Accepted to Statistics and Probability Letters, March 201

The starred Dixmier's conjecture

Dixmier's famous question says the following: Is every algebra endomorphism of the first Weyl algebra, $A_1(F)$, where $F$ is a zero characteristic field, an automorphism? Let $\alpha$ be the exchange involution on $A_1(F)$: $\alpha(x)= y$, $\alpha(y)= x$. An $\alpha$-endomorphism of $A_1(F)$ is an endomorphism which preserves the involution $\alpha$. Then one may ask the following question, which may be called the "$\alpha$-Dixmier's problem $1$" or the "starred Dixmier's problem $1$": Is every $\alpha$-endomorphism of $A_1(F)$ an automorphism?Comment: Revised proof in section

About Dixmier's conjecture

The well-known Dixmier conjecture asks if every algebra endomorphism of the first Weyl algebra over a characteristic zero field is an automorphism. We bring a hopefully easier to solve conjecture, called the $\gamma,\delta$ conjecture, and show that it is equivalent to the Dixmier conjecture. Up to checking that in the group generated by automorphisms and anti-automorphisms of $A_1$ all the involutions belong to one conjugacy class, we show that every involutive endomorphism from $(A_1,\gamma)$ to $(A_1,\delta)$ is an automorphism ($\gamma$ and $\delta$ are two involutions on $A_1$), and given an endomorphism $f$ of $A_1$ (not necessarily an involutive endomorphism), if one of $f(X)$,$f(Y)$ is symmetric or skew-symmetric (with respect to any involution on $A_1$), then $f$ is an automorphism