378 research outputs found
A Norm Compression Inequality for Block Partitioned Positive Semidefinite Matrices
Let be a positive semidefinite matrix, block partitioned as
A=\twomat{B}{C}{C^*}{D}, where and are square blocks. We prove the
following inequalities for the Schatten -norm , which are sharp
when the blocks are of size at least : and These bounds can be
extended to symmetric partitionings into larger numbers of blocks, at the
expense of no longer being sharp: and Comment: 24 page
A note on coloring vertex-transitive graphs
We prove bounds on the chromatic number of a vertex-transitive graph
in terms of its clique number and maximum degree . We
conjecture that every vertex-transitive graph satisfies and we
prove results supporting this conjecture. Finally, for vertex-transitive graphs
with we prove the Borodin-Kostochka conjecture, i.e.,
Extremes of the internal energy of the Potts model on cubic graphs
We prove tight upper and lower bounds on the internal energy per particle
(expected number of monochromatic edges per vertex) in the anti-ferromagnetic
Potts model on cubic graphs at every temperature and for all . This
immediately implies corresponding tight bounds on the anti-ferromagnetic Potts
partition function.
Taking the zero-temperature limit gives new results in extremal
combinatorics: the number of -colorings of a -regular graph, for any , is maximized by a union of 's. This proves the case of a
conjecture of Galvin and Tetali
High rank linear syzygies on low rank quadrics
We study the linear syzygies of a homogeneous ideal I in a polynomial ring S,
focusing on the graded betti numbers b_(i,i+1) = dim_k Tor_i(S/I, k)_(i+1). For
a variety X and divisor D with S = Sym(H^0(D)*), what conditions on D ensure
that b_(i,i+1) is nonzero? Eisenbud has shown that a decomposition D = A + B
such that A and B have at least two sections gives rise to determinantal
equations (and corresponding syzygies) in I_X; and conjectured that if I_2 is
generated by quadrics of rank at most 4, then the last nonvanishing b_(i,i+1)
is a consequence of such equations. We describe obstructions to this conjecture
and prove a variant. The obstructions arise from toric specializations of the
Rees algebra of Koszul cycles, and we give an explicit construction of toric
varieties with minimal linear syzygies of arbitrarily high rank. This gives one
answer to a question posed by Eisenbud and Koh about specializations of
syzygies.Comment: 16 pages, 3 figure
- β¦