2,092 research outputs found
Some determinants of path generating functions
We evaluate four families of determinants of matrices, where the entries are
sums or differences of generating functions for paths consisting of up-steps,
down-steps and level steps. By specialisation, these determinant evaluations
have numerous corollaries. In particular, they cover numerous determinant
evaluations of combinatorial numbers - most notably of Catalan, ballot, and of
Motzkin numbers - that appeared previously in the literature.Comment: 35 pages, AmS-TeX; minor corrections; final version to appear in Adv.
Appl. Mat
The L\^e numbers of the square of a function and their applications
L\^e numbers were introduced by Massey with the purpose of numerically
controlling the topological properties of families of non-isolated hypersurface
singularities and describing the topology associated with a function with
non-isolated singularities. They are a generalization of the Milnor number for
isolated hypersurface singularities.
In this note the authors investigate the composite of an arbitrary
square-free f and . They get a formula for the L\^e numbers of the
composite, and consider two applications of these numbers. The first
application is concerned with the extent to which the L\^e numbers are
invariant in a family of functions which satisfy some equisingularity
condition, the second is a quick proof of a new formula for the Euler
obstruction of a hypersurface singularity. Several examples are computed using
this formula including any X defined by a function which only has transverse
D(q,p) singularities off the origin.Comment: 14 page
Non-trivial 3-wise intersecting uniform families
A family of -element subsets of an -element set is called 3-wise
intersecting if any three members in the family have non-empty intersection. We
determine the maximum size of such families exactly or asymptotically. One of
our results shows that for every there exists such that if
and then the maximum size
is .Comment: 12 page
Quantum Knizhnik-Zamolodchikov Equation, Totally Symmetric Self-Complementary Plane Partitions and Alternating Sign Matrices
We present multiresidue formulae for partial sums in the basis of link
patterns of the polynomial solution to the level 1 U_q(\hat sl_2) quantum
Knizhnik--Zamolodchikov equation at generic values of the quantum parameter q.
These allow for rewriting and generalizing a recent conjecture [Di Francesco
'06] connecting the above to generating polynomials for weighted Totally
Symmetric Self-Complementary Plane Partitions. We reduce the corresponding
conjectures to a single integral identity, yet to be proved
Dynamical Supersymmetry Breaking in Intersecting Brane Models
In this paper we study dynamical supersymmetry breaking in absence of gravity
with the matter content of the minimal supersymmetric standard model. The
hidden sector of the theory is a strongly coupled gauge theory, realized in
terms of microscopic variables which condensate to form mesons. The
supersymmetry breaking scalar potential combines F, D terms with instanton
generated interactions in the Higgs-mesons sector. We show that for a large
region in parameter space the vacuum breaks in addition to supersymmetry also
electroweak gauge symmetry. We furthermore present local D-brane configurations
that realize these supersymmetry breaking patterns.Comment: 30 pages, 4 figures, pdflate
Polynomial Invariants for Arbitrary Rank Weakly-Colored Stranded Graphs
Polynomials on stranded graphs are higher dimensional generalization of Tutte
and Bollob\'as-Riordan polynomials [Math. Ann. 323 (2002), 81-96]. Here, we
deepen the analysis of the polynomial invariant defined on rank 3
weakly-colored stranded graphs introduced in arXiv:1301.1987. We successfully
find in dimension a modified Euler characteristic with
parameters. Using this modified invariant, we extend the rank 3 weakly-colored
graph polynomial, and its main properties, on rank 4 and then on arbitrary rank
weakly-colored stranded graphs.Comment: Basic definitions overlap with arXiv:1301.198
Short proofs of three results about intersecting systems
In this note, we give short proofs of three theorems about intersection
problems. The first one is a determination of the maximum size of a nontrivial
-uniform, -wise intersecting family for , which improves upon a recent result of
O'Neill and Verstra\"{e}te. Our proof also extends to -wise,
-intersecting families, and from this result we obtain a version of the
Erd\H{o}s-Ko-Rado theorem for -wise, -intersecting families.
The second result partially proves a conjecture of Frankl and Tokushige about
-uniform families with restricted pairwise intersection sizes.
The third result concerns graph intersections. Answering a question of Ellis,
we construct -intersecting families of graphs which have size larger
than the Erd\H{o}s-Ko-Rado-type construction whenever is sufficiently large
in terms of .Comment: 12 pages; we added a new result, Theorem 1
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