314 research outputs found
Lookahead Pathology in Monte-Carlo Tree Search
Monte-Carlo Tree Search (MCTS) is an adversarial search paradigm that first
found prominence with its success in the domain of computer Go. Early
theoretical work established the game-theoretic soundness and convergence
bounds for Upper Confidence bounds applied to Trees (UCT), the most popular
instantiation of MCTS; however, there remain notable gaps in our understanding
of how UCT behaves in practice. In this work, we address one such gap by
considering the question of whether UCT can exhibit lookahead pathology -- a
paradoxical phenomenon first observed in Minimax search where greater search
effort leads to worse decision-making. We introduce a novel family of synthetic
games that offer rich modeling possibilities while remaining amenable to
mathematical analysis. Our theoretical and experimental results suggest that
UCT is indeed susceptible to pathological behavior in a range of games drawn
from this family
Direct measurement of molecular stiffness and damping in confined water layers
We present {\em direct} and {\em linear} measurements of the normal stiffness
and damping of a confined, few molecule thick water layer. The measurements
were obtained by use of a small amplitude (0.36 ), off-resonance
Atomic Force Microscopy (AFM) technique. We measured stiffness and damping
oscillations revealing up to 7 layers separated by 2.56 0.20
. Relaxation times could also be calculated and were found to
indicate a significant slow-down of the dynamics of the system as the confining
separation was reduced. We found that the dynamics of the system is determined
not only by the interfacial pressure, but more significantly by solvation
effects which depend on the exact separation of tip and surface. Thus `
solidification\rq seems to not be merely a result of pressure and confinement,
but depends strongly on how commensurate the confining cavity is with the
molecule size. We were able to model the results by starting from the simple
assumption that the relaxation time depends linearly on the film stiffness.Comment: 7 pages, 6 figures, will be submitted to PR
On the Complexity Landscape of Connected f-Factor Problems
Given an n-vertex graph G and a function f:V(G) -> {0, ..., n-1}, an f-factor is a subgraph H of G such that deg_H(v)=f(v) for every vertex v in V(G); we say that H is a connected f-factor if, in addition, the subgraph H is connected. A classical result of Tutte (1954) is the polynomial time algorithm to check whether a given graph has a specified f-factor. However, checking for the presence of a connected f-factor is easily seen to generalize Hamiltonian Cycle and hence is NP-complete. In fact, the Connected f-Factor problem remains NP-complete even when f(v) is at least n^epsilon for each vertex v and epsilon<1; on the other side of the spectrum, the problem was known to be polynomial-time solvable when f(v) is at least n/3 for every vertex v.
In this paper, we extend this line of work and obtain new complexity results based on restricting the function f. In particular, we show that when f(v) is required to be at least n/(log n)^c, the problem can be solved in quasi-polynomial time in general and in randomized polynomial time if c 1, the problem is NP-intermediate
Fermionic representations for characters of M(3,t), M(4,5), M(5,6) and M(6,7) minimal models and related Rogers-Ramanujan type and dilogarithm identities
Characters and linear combinations of characters that admit a fermionic sum
representation as well as a factorized form are considered for some minimal
Virasoro models. As a consequence, various Rogers-Ramanujan type identities are
obtained. Dilogarithm identities producing corresponding effective central
charges and secondary effective central charges are derived. Several ways of
constructing more general fermionic representations are discussed.Comment: 14 pages, LaTex; minor correction
An Algebraic Construction of Generalized Coherent States for Shape-Invariant Potentials
Generalized coherent states for shape invariant potentials are constructed
using an algebraic approach based on supersymmetric quantum mechanics. We show
this generalized formalism is able to: a) supply the essential requirements
necessary to establish a connection between classical and quantum formulations
of a given system (continuity of labeling, resolution of unity, temporal
stability, and action identity); b) reproduce results already known for
shape-invariant systems, like harmonic oscillator, double anharmonic,
Poschl-Teller and self-similar potentials and; c) point to a formalism that
provides an unified description of the different kind of coherent states for
quantum systems.Comment: 14 pages of REVTE
Completeness of Coherent States Associated with Self-Similar Potentials and Ramanujan's Integral Extension of the Beta Function
A decomposition of identity is given as a complex integral over the coherent
states associated with a class of shape-invariant self-similar potentials.
There is a remarkable connection between these coherent states and Ramanujan's
integral extension of the beta function.Comment: 9 pages of Late
What We Don't Know about BTZ Black Hole Entropy
With the recent discovery that many aspects of black hole thermodynamics can
be effectively reduced to problems in three spacetime dimensions, it has become
increasingly important to understand the ``statistical mechanics'' of the
(2+1)-dimensional black hole of Banados, Teitelboim, and Zanelli (BTZ). Several
conformal field theoretic derivations of the BTZ entropy exist, but none is
completely satisfactory, and many questions remain open: there is no consensus
as to what fields provide the relevant degrees of freedom or where these
excitations live. In this paper, I review some of the unresolved problems and
suggest avenues for their solution.Comment: 24 pages, LaTeX, no figures; references added, brief discussion of
relation to string theory added; to appear in Class. Quant. Gra
Beyond series expansions: mathematical structures for the susceptibility of the square lattice Ising model
We first study the properties of the Fuchsian ordinary differential equations
for the three and four-particle contributions and
of the square lattice Ising model susceptibility. An analysis of some
mathematical properties of these Fuchsian differential equations is sketched.
For instance, we study the factorization properties of the corresponding linear
differential operators, and consider the singularities of the three and
four-particle contributions and , versus the
singularities of the associated Fuchsian ordinary differential equations, which
actually exhibit new ``Landau-like'' singularities. We sketch the analysis of
the corresponding differential Galois groups. In particular we provide a
simple, but efficient, method to calculate the so-called ``connection
matrices'' (between two neighboring singularities) and deduce the singular
behaviors of and . We provide a set of comments and
speculations on the Fuchsian ordinary differential equations associated with
the -particle contributions and address the problem of the
apparent discrepancy between such a holonomic approach and some scaling results
deduced from a Painlev\'e oriented approach.Comment: 21 pages Proceedings of the Counting Complexity conferenc
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