Artinian Gorenstein algebras with linear resolutions

Fix a pair of positive integers d and n. We create a ring R and a complex G of R-modules with the following universal property. Let P be a polynomial ring in d variables over a field and let I be a grade d Gorenstein ideal in P which is generated by homogeneous forms of degree n. If the resolution of P/I by free P-modules is linear, then there exists a ring homomorphism from R to P such that P tensor G is a minimal homogeneous resolution of P/I by free P-modules. Our construction is coordinate free

Bounds for the Multiplicity of Gorenstein algebras

We prove upper bounds for the Hilbert-Samuel multiplicity of standard graded Gorenstein algebras. The main tool that we use is Boij-S\"oderberg theory to obtain a decomposition of the Betti table of a Gorenstein algebra as the sum of rational multiples of symmetrized pure tables. Our bound agrees with the one in the quasi-pure case obtained by Srinivasan [J. Algebra, vol.~208, no.~2, (1998)]

Drone Shadow Tracking

Aerial videos taken by a drone not too far above the surface may contain the drone's shadow projected on the scene. This deteriorates the aesthetic quality of videos. With the presence of other shadows, shadow removal cannot be directly applied, and the shadow of the drone must be tracked. Tracking a drone's shadow in a video is, however, challenging. The varying size, shape, change of orientation and drone altitude pose difficulties. The shadow can also easily disappear over dark areas. However, a shadow has specific properties that can be leveraged, besides its geometric shape. In this paper, we incorporate knowledge of the shadow's physical properties, in the form of shadow detection masks, into a correlation-based tracking algorithm. We capture a test set of aerial videos taken with different settings and compare our results to those of a state-of-the-art tracking algorithm.Comment: 5 pages, 4 figure

Gorenstein Hilbert Coefficients

We prove upper and lower bounds for all the coefficients in the Hilbert Polynomial of a graded Gorenstein algebra $S=R/I$ with a quasi-pure resolution over $R$. The bounds are in terms of the minimal and the maximal shifts in the resolution of $R$ . These bounds are analogous to the bounds for the multiplicity found in \cite{S} and are stronger than the bounds for the Cohen Macaulay algebras found in \cite{HZ}.Comment: 20 page