7 problems

Consider a reduced Gorenstein standard graded algebra with Krull dimension at least its socle dimension, and consider a generic Artinian reduction after passing to an appropriate f…

Let $\mathbbm{k}$ be any field of characteristic $p$, let $\mu$ be any $(d-1)$-dimensional cycle over $\mathbbm{k}$, and let $\mathbbm{k}[|\mu|]$ be the associated graded commutati…

Let $Y=\{0,1\}\times\{0,1\}$, let $\mathbf{c}$ be the coefficient vector defining the effective anisotropy $\phi$, and consider the loss … where…

Ellipsoids and spheres are compared as particles in static packings, where ellipsoids have anisotropic shapes and the packing uses an anisotropic accessible volume. Anisotropic acc…

Let $\alpha$ satisfy $-1<\alpha<-1/2$ in a generalized multicomponent Cuddeford system, and let $h(Q)$ be the function entering its distribution function. In the covered cases, the…

Consider a spherical stellar system with a consistent, positive phase-space distribution function. Let \gamma(r)=-d\text{{rm ln}, }\rho/d\text{{rm ln}, } r denote the density…

A spherical stellar system has density slope \gamma(r)=-d\text{{rm ln}, }\rho/d\text{{rm ln}, } r, anisotropy parameter $\beta(r)=1-\sigma_t^2/(2\sigma_r^2)$, and a positive…