1,282 research outputs found
The Segal conjecture for topological Hochschild homology of complex cobordism
We study the C_p-equivariant Tate construction on the topological Hochschild
homology THH(B) of a symmetric ring spectrum B by relating it to a topological
version R_+(B) of the Singer construction, extended by a natural circle action.
This enables us to prove that the fixed and homotopy fixed point spectra of
THH(B) are p-adically equivalent for B = MU and BP. This generalizes the
classical C_p-equivariant Segal conjecture, which corresponds to the case B =
S.Comment: Accepted for publication by the Journal of Topolog
Counting sets with small sumset, and the clique number of random Cayley graphs
Given a set A in Z/NZ we may form a Cayley sum graph G_A on vertex set Z/NZ
by joining i to j if and only if i + j is in A. We investigate the extent to
which performing this construction with a random set A simulates the generation
of a random graph, proving that the clique number of G_A is a.s. O(log N). This
shows that Cayley sum graphs can furnish good examples of Ramsey graphs. To
prove this result we must study the specific structure of set addition on Z/NZ.
Indeed, we also show that the clique number of a random Cayley sum graph on
(Z/2Z)^n, 2^n = N, is almost surely not O(log N).
Despite the graph-theoretical title, this is a paper in number theory. Our
main results are essentially estimates for the number of sets A in {1,...,N}
with |A| = k and |A + A| = m, for various values of k and m.Comment: 18 pages; to appear in Combinatorica, exposition has been improved
thanks to comments from Imre Ruzsa and Seva Le
On Artin algebras arising from Morita contexts
We study Morita rings \Lambda_{(\phi,\psi)}=\bigl({smallmatrix} A
&_AN_B_BM_A & B {smallmatrix}\bigr) in the context of Artin algebras from
various perspectives. First we study covariant finite, contravariant finite,
and functorially finite subcategories of the module category of a Morita ring
when the bimodule homomorphisms and are zero. Further we give
bounds for the global dimension of a Morita ring , regarded as
an Artin algebra, in terms of the global dimensions of and in the case
when both and are zero. We illustrate our bounds with some
examples. Finally we investigate when a Morita ring is a Gorenstein Artin
algebra and then we determine all the Gorenstein-projective modules over the
Morita ring with , where is an Artin algebra.Comment: 29 pages, revised versio
Symmetry as Bias: Rediscovering Special Relativity
This paper describes a rational reconstruction of Einstein's discovery of special relativity, validated through an implementation: the Erlanger program. Einstein's discovery of special relativity revolutionized both the content of physics and the research strategy used by theoretical physicists. This research strategy entails a mutual bootstrapping process between a hypothesis space for biases, defined through different postulated symmetries of the universe, and a hypothesis space for physical theories. The invariance principle mutually constrains these two spaces. The invariance principle enables detecting when an evolving physical theory becomes inconsistent with its bias, and also when the biases for theories describing different phenomena are inconsistent. Structural properties of the invariance principle facilitate generating a new bias when an inconsistency is detected. After a new bias is generated. this principle facilitates reformulating the old, inconsistent theory by treating the latter as a limiting approximation. The structural properties of the invariance principle can be suitably generalized to other types of biases to enable primal-dual learning
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