962 research outputs found
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 with a quasi-pure resolution
over . The bounds are in terms of the minimal and the maximal shifts in the
resolution of . 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
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