973 research outputs found
Enumeration formulas for generalized q-Euler numbers
We find an enumeration formula for a -Euler number which is a
generalization of the -Euler number introduced by Han, Randrianarivony, and
Zeng. We also give a combinatorial expression for the -Euler number and
find another formula when for any integer . Special cases of our
latter formula include the formula of the -Euler number recently found by
Josuat-Verg\`es and Touchard-Riordan's formula.Comment: 21 pages, 12 figure
The condition number of join decompositions
The join set of a finite collection of smooth embedded submanifolds of a
mutual vector space is defined as their Minkowski sum. Join decompositions
generalize some ubiquitous decompositions in multilinear algebra, namely tensor
rank, Waring, partially symmetric rank and block term decompositions. This
paper examines the numerical sensitivity of join decompositions to
perturbations; specifically, we consider the condition number for general join
decompositions. It is characterized as a distance to a set of ill-posed points
in a supplementary product of Grassmannians. We prove that this condition
number can be computed efficiently as the smallest singular value of an
auxiliary matrix. For some special join sets, we characterized the behavior of
sequences in the join set converging to the latter's boundary points. Finally,
we specialize our discussion to the tensor rank and Waring decompositions and
provide several numerical experiments confirming the key results
On Degenerate Secant and Tangential Varieties and Local Differential Geometry
We study the local differential geometry of varieties with degenerate secant and tangential varieties. We show that the
second fundamental form of a smooth variety with degenerate tangential variety
is subject to certain rank restrictions. The rank restrictions imply a slightly
refined version of Zak's theorem on linear normality and a short proof of the
Zak-Fantecchi theorem on the superadditivity of multisecant defects. We show
there is a vector bundle defined over general points of whose fibers carry
the structure of a Clifford algebra. This structure implies additional
restrictions of the size of the secant defect. The Clifford algebra structure,
combined with further local computations, yields a new proof of Zak's theorem
on Severi varieties that is substantially shorter than the original. We also
prove local and global results on the dimension of the Gauss image of
degenerate tangential varieties, refining the results in [GH].Comment: Exposition altered according to the helpful recommendations of the
referee. AMSTe
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