611 research outputs found
Entropy of Operator-valued Random Variables: A Variational Principle for Large N Matrix Models
We show that, in 't Hooft's large N limit, matrix models can be formulated as
a classical theory whose equations of motion are the factorized
Schwinger--Dyson equations. We discover an action principle for this classical
theory. This action contains a universal term describing the entropy of the
non-commutative probability distributions. We show that this entropy is a
nontrivial 1-cocycle of the non-commutative analogue of the diffeomorphism
group and derive an explicit formula for it. The action principle allows us to
solve matrix models using novel variational approximation methods; in the
simple cases where comparisons with other methods are possible, we get
reasonable agreement.Comment: 45 pages with 1 figure, added reference
Circuit complexity, proof complexity, and polynomial identity testing
We introduce a new algebraic proof system, which has tight connections to
(algebraic) circuit complexity. In particular, we show that any
super-polynomial lower bound on any Boolean tautology in our proof system
implies that the permanent does not have polynomial-size algebraic circuits
(VNP is not equal to VP). As a corollary to the proof, we also show that
super-polynomial lower bounds on the number of lines in Polynomial Calculus
proofs (as opposed to the usual measure of number of monomials) imply the
Permanent versus Determinant Conjecture. Note that, prior to our work, there
was no proof system for which lower bounds on an arbitrary tautology implied
any computational lower bound.
Our proof system helps clarify the relationships between previous algebraic
proof systems, and begins to shed light on why proof complexity lower bounds
for various proof systems have been so much harder than lower bounds on the
corresponding circuit classes. In doing so, we highlight the importance of
polynomial identity testing (PIT) for understanding proof complexity.
More specifically, we introduce certain propositional axioms satisfied by any
Boolean circuit computing PIT. We use these PIT axioms to shed light on
AC^0[p]-Frege lower bounds, which have been open for nearly 30 years, with no
satisfactory explanation as to their apparent difficulty. We show that either:
a) Proving super-polynomial lower bounds on AC^0[p]-Frege implies VNP does not
have polynomial-size circuits of depth d - a notoriously open question for d at
least 4 - thus explaining the difficulty of lower bounds on AC^0[p]-Frege, or
b) AC^0[p]-Frege cannot efficiently prove the depth d PIT axioms, and hence we
have a lower bound on AC^0[p]-Frege.
Using the algebraic structure of our proof system, we propose a novel way to
extend techniques from algebraic circuit complexity to prove lower bounds in
proof complexity
Quasi-symmetric functions and the KP hierarchy
Quasi-symmetric functions show up in an approach to solve the
Kadomtsev-Petviashvili (KP) hierarchy. This moreover features a new
nonassociative product of quasi-symmetric functions that satisfies simple
relations with the ordinary product and the outer coproduct. In particular,
supplied with this new product and the outer coproduct, the algebra of
quasi-symmetric functions becomes an infinitesimal bialgebra. Using these
results we derive a sequence of identities in the algebra of quasi-symmetric
functions that are in formal correspondence with the equations of the KP
hierarchy.Comment: 16 page
Algebraic Structures and Stochastic Differential Equations driven by Levy processes
We construct an efficient integrator for stochastic differential systems
driven by Levy processes. An efficient integrator is a strong approximation
that is more accurate than the corresponding stochastic Taylor approximation,
to all orders and independent of the governing vector fields. This holds
provided the driving processes possess moments of all orders and the vector
fields are sufficiently smooth. Moreover the efficient integrator in question
is optimal within a broad class of perturbations for half-integer global root
mean-square orders of convergence. We obtain these results using the
quasi-shuffle algebra of multiple iterated integrals of independent Levy
processes.Comment: 41 pages, 11 figure
From m-clusters to m-noncrossing partitions via exceptional sequences
Let W be a finite crystallographic reflection group. The generalized Catalan
number of W coincides both with the number of clusters in the cluster algebra
associated to W, and with the number of noncrossing partitions for W. Natural
bijections between these two sets are known. For any positive integer m, both
m-clusters and m-noncrossing partitions have been defined, and the cardinality
of both these sets is the Fuss-Catalan number. We give a natural bijection
between these two sets by first establishing a bijection between two particular
sets of exceptional sequences in the bounded derived category for any
finite-dimensional hereditary algebra.Comment: 25 pages: v2: added new section (section 8
Algebraic structures in stochastic differential equations
We define a new numerical integration scheme for stochastic differential equations
driven by Levy processes with uniformly lower mean square remainder than that
of the scheme of the same strong order of convergence obtained by truncating the
stochastic Taylor series. In doing so we generalize recent results concerning stochastic
differential equations driven by Wiener processes. The aforementioned works
studied integration schemes obtained by applying an invertible mapping to the
stochastic Taylor series, truncating the resulting series and applying the inverse
of the original mapping. The shuffle Hopf algebra and its associated convolution
algebra play important roles in the their analysis, arising from the combinatorial
structure of iterated Stratonovich integrals. It was recently shown that the algebra
generated by iterated It^o integrals of independent Levy processes is isomorphic to
a quasi-shuffle algebra. We utilise this to consider map-truncate-invert schemes for
Levy processes. To facilitate this, we derive a new form of stochastic Taylor expansion
from those of Wagner & Platen, enabling us to extend existing algebraic encodings
of integration schemes. We then derive an alternative method of computing
map-truncate-invert schemes using a single step, resolving diffculties encountered
at the inversion step in previous methods
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