17,499 research outputs found
Proof of the Umbral Moonshine Conjecture
The Umbral Moonshine Conjectures assert that there are infinite-dimensional
graded modules, for prescribed finite groups, whose McKay-Thompson series are
certain distinguished mock modular forms. Gannon has proved this for the
special case involving the largest sporadic simple Mathieu group. Here we
establish the existence of the umbral moonshine modules in the remaining 22
cases.Comment: 56 pages, to appear in Research in the Mathematical Science
On Cohen braids
For a general surface and an arbitrary braid from the surface
braid group we study the system of equations
where operation is deleting of
-th strand. We obtain that if or this system of
equations has a solution if and only if
The set of braids satisfying the last system of
equations we call Cohen braids. We also construct a set of generators for the
groups of Cohen braids. In the cases of the sphere and the projective plane we
give some examples for the small number of strands.Comment: 23 page
On a new class of summation formulae involving the Laguerre polynomial
By elementary manipulation of series, a general transformation involving the generalized hypergeometric function is established. Kummer’s first theorem, the classical Gauss summation theorem and the generalized Kummer summation theorem due to Lavoie et al. [Generalizations of Whipple’s theorem on the sum of a 3 F 2, J. Comput. Appl. Math. 72 (1996), pp. 293–300] are then applied to obtain a new class of summation formulae involving the Laguerre polynomial, which have not previously appeared in the literature. Several related results due to Exton have also been given in a corrected form
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On the number of additive permutations and Skolem-type sequences
Cavenagh and Wanless recently proved that, for sufficiently large odd n, the number of transversals in the Latin square formed from the addition table for integers modulo n is greater than (3.246)n. We adapt their proof to show that for sufficiently large t the number of additive permutations on [-t,t] is greater than (3.246)2t+1 and we go on to derive some much improved lower bounds on the numbers of Skolem-type sequences. For example, it is shown that for sufficiently large t ≡ 0$ or 3 (mod 4), the number of split Skolem sequences of order n=7t+3 is greater than (3.246)6t+3. This compares with the previous best bound of 2⌊n/3⌋
A combinatorial approach to knot recognition
This is a report on our ongoing research on a combinatorial approach to knot
recognition, using coloring of knots by certain algebraic objects called
quandles. The aim of the paper is to summarize the mathematical theory of knot
coloring in a compact, accessible manner, and to show how to use it for
computational purposes. In particular, we address how to determine colorability
of a knot, and propose to use SAT solving to search for colorings. The
computational complexity of the problem, both in theory and in our
implementation, is discussed. In the last part, we explain how coloring can be
utilized in knot recognition
Generalized Analogs of the Heisenberg Uncertainty Inequality
We investigate locally compact topological groups for which a generalized
analogue of Heisenberg uncertainty inequality hold. In particular, it is shown
that this inequality holds for (where is a
separable unimodular locally compact group of type I), Euclidean Motion group
and several general classes of nilpotent Lie groups which include thread-like
nilpotent Lie groups, -NPC nilpotent Lie groups and several low-dimensional
nilpotent Lie groups
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