3,567 research outputs found
Categories, norms and weights
The well-known Lawvere category R of extended real positive numbers comes
with a monoidal closed structure where the tensor product is the sum. But R has
another such structure, given by multiplication, which is *-autonomous.
Normed sets, with a norm in R, inherit thus two symmetric monoidal closed
structures, and categories enriched on one of them have a 'subadditive' or
'submultiplicative' norm, respectively. Typically, the first case occurs when
the norm expresses a cost, the second with Lipschitz norms.
This paper is a preparation for a sequel, devoted to 'weighted algebraic
topology', an enrichment of directed algebraic topology. The structure of R,
and its extension to the complex projective line, might be a first step in
abstracting a notion of algebra of weights, linked with physical measures.Comment: Revised version, 16 pages. Some minor correction
Lyapunov exponents of random walks in small random potential: the lower bound
We consider the simple random walk on Z^d, d > 2, evolving in a potential of
the form \beta V, where (V(x), x \in Z^d) are i.i.d. random variables taking
values in [0,+\infty), and \beta\ > 0. When the potential is integrable, the
asymptotic behaviours as \beta\ tends to 0 of the associated quenched and
annealed Lyapunov exponents are known (and coincide). Here, we do not assume
such integrability, and prove a sharp lower bound on the annealed Lyapunov
exponent for small \beta. The result can be rephrased in terms of the decay of
the averaged Green function of the Anderson Hamiltonian -\Delta\ + \beta V.Comment: 42 pages, 3 figure
On Harder-Narasimhan filtrations and their compatibility with tensor products
We attach buildings to modular lattices and use them to develop a metric
approach to Harder-Narasimhan filtrations. Switching back to a categorical
framework, we establish an abstract numerical criterion for the compatibility
of these filtrations with tensor products. We finally verify our criterion in
three cases, one of which is new
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