22,146 research outputs found
A Meshalkin theorem for projective geometries
Let M be a family of sequences (a_1,...,a_p) where each a_k is a flat in a
projective geometry of rank n (dimension n-1) and order q, and the sum of
ranks, r(a_1) + ... + r(a_p), equals the rank of the join a_1 v ... v a_p. We
prove upper bounds on |M| and corresponding LYM inequalities assuming that (i)
all joins are the whole geometry and for each k<p the set of all a_k's of
sequences in M contains no chain of length l, and that (ii) the joins are
arbitrary and the chain condition holds for all k. These results are q-analogs
of generalizations of Meshalkin's and Erdos's generalizations of Sperner's
theorem and their LYM companions, and they generalize Rota and Harper's
q-analog of Erdos's generalization.Comment: 8 pages, added journal referenc
Inside-Out Polytopes
We present a common generalization of counting lattice points in rational
polytopes and the enumeration of proper graph colorings, nowhere-zero flows on
graphs, magic squares and graphs, antimagic squares and graphs, compositions of
an integer whose parts are partially distinct, and generalized latin squares.
Our method is to generalize Ehrhart's theory of lattice-point counting to a
convex polytope dissected by a hyperplane arrangement. We particularly develop
the applications to graph and signed-graph coloring, compositions of an
integer, and antimagic labellings.Comment: 24 pages, 3 figures; to appear in Adv. Mat
Infrared singularities of QCD amplitudes with massive partons
A formula for the two-loop infrared singularities of dimensionally
regularized QCD scattering amplitudes with an arbitrary number of massive and
massless legs is derived. The singularities are obtained from the solution of a
renormalization-group equation, and factorization constraints on the relevant
anomalous-dimension matrix are analyzed. The simplicity of the structure of the
matrix relevant for massless partons does not carry over to the case with
massive legs, where starting at two-loop order new color and momentum
structures arise, which are not of the color-dipole form. The resulting
two-loop three-parton correlations can be expressed in terms of two functions,
for which some general properties are derived. This explains observations
recently made by Mitov et al. in terms of symmetry arguments.Comment: 7 pages, 1 figure; v2: minor changes, reference added; v3: note
added, correcting some statements regarding F1 and f2 in light of the recent
calculations in [45,46], references update
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