we investigate buchbaum and eisenbud's construction of the second symmetric power $s_r(x)$ of a chain complex $x$ of modules over a commutative ring $r$. we state and prove a number of results from the folklore of the subject for which we know of no good direct references. we also provide several explicit computations and examples. we use this construction to prove the following ...

Below a (comparatively large) measure of non-Hermiticity Z = Z (crit) 0 > 0 of a PT symmetrically complexified square well, bound states are constructed non-numerically. All their energies prove real and continuous in the (Hermitian) limit Z → 0. Beyond the threshold Z (crit) 0 (and, in general, beyond Z (crit) m at m = 0, 1, . . .) the lowest two real energies (i.e., E2m and E2m+1) are shown t...

We investigate Buchbaum and Eisenbud's construction of the second symmetric power $s_R(X)$ of a chain complex $X$ of modules over a commutative ring $R$. We state and prove a number of results from the folklore of the subject for which we know of no good direct references. We also provide several explicit computations and examples. We use this construction to prove the following vers...

Let $p\geq 5$ be a prime, and $\mathfrak{p}$ prime of $\bar{\mathbb{Q}}$ above $p$. $g_1$ $g_2$ $\mathfrak{p}$-ordinary, $\mathfrak{p}$-distinguished $p$-stabilized cuspidal newforms nebentype characters $\epsilon_1, \epsilon_2$ respectively, weight $k\geq 2$, whose associated have level to Assume that the residual representations at are absolutely irreducible isomorphic. Then, imprimitive $p$-...