19,218 research outputs found
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
On Binary de Bruijn Sequences from LFSRs with Arbitrary Characteristic Polynomials
We propose a construction of de Bruijn sequences by the cycle joining method
from linear feedback shift registers (LFSRs) with arbitrary characteristic
polynomial . We study in detail the cycle structure of the set
that contains all sequences produced by a specific LFSR on
distinct inputs and provide a fast way to find a state of each cycle. This
leads to an efficient algorithm to find all conjugate pairs between any two
cycles, yielding the adjacency graph. The approach is practical to generate a
large class of de Bruijn sequences up to order . Many previously
proposed constructions of de Bruijn sequences are shown to be special cases of
our construction
Feynman integral relations from parametric annihilators
We study shift relations between Feynman integrals via the Mellin transform
through parametric annihilation operators. These contain the momentum space IBP
relations, which are well-known in the physics literature. Applying a result of
Loeser and Sabbah, we conclude that the number of master integrals is computed
by the Euler characteristic of the Lee-Pomeransky polynomial. We illustrate
techniques to compute this Euler characteristic in various examples and compare
it with numbers of master integrals obtained in previous works.Comment: v2: new section 3.1 added, several misprints corrected and additional
remark
- …