46,867 research outputs found
On computing boundary functional sums
AbstractA method for solving enumeration problems is suggested. We consider the enumeration problems which are reducible to estimation of the sums of type T(X,f)=ΣAf(A) where f is so called boundary functional (BF) on X, and the summation is over all subsets of X (or over some special subfamily of 2X). An evolution of the n-cube, the percolation problem, the problem of computation of the matchings number and the independent sets number, the monotone Boolean functions number, the binary codes number and so on are among such problems. We show how to obtain asymptotics for T(X,f). In conclusion we give an example of application of the BF method to finding the number of independent sets in the bipartite graphs, induced by neighbouring levels of the n-cube
Zero-mode analysis of quantum statistical physics
We present a unified formulation for quantum statistical physics based on the
representation of the density matrix as a functional integral. We identify the
stochastic variable of the effective statistical theory that we derive as a
boundary configuration and a zero mode relevant to the discussion of infrared
physics. We illustrate our formulation by computing the partition function of
an interacting one-dimensional quantum mechanical system at finite temperature
from the path-integral representation for the density matrix. The method of
calculation provides an alternative to the usual sum over periodic
trajectories: it sums over paths with coincident endpoints, and includes
non-vanishing boundary terms. An appropriately modified expansion into
Matsubara modes provides a natural separation of the zero-mode physics. This
feature may be useful in the treatment of infrared divergences that plague the
perturbative approach in thermal field theory.Comment: 9 pages, 5 figure
Nonperturbative black hole entropy and Kloosterman sums
Non-perturbative quantum corrections to supersymmetric black hole entropy
often involve nontrivial number-theoretic phases called Kloosterman sums. We
show how these sums can be obtained naturally from the functional integral of
supergravity in asymptotically AdS_2 space for a class of black holes. They are
essentially topological in origin and correspond to charge-dependent phases
arising from the various gauge and gravitational Chern-Simons terms and
boundary Wilson lines evaluated on Dehn-filled solid 2-torus. These corrections
are essential to obtain an integer from supergravity in agreement with the
quantum degeneracies, and reveal an intriguing connection between topology,
number theory, and quantum gravity. We give an assessment of the current
understanding of quantum entropy of black holes.Comment: 35 pages; minor changes, JHEP versio
New Acceleration of Nearly Optimal Univariate Polynomial Root-findERS
Univariate polynomial root-finding has been studied for four millennia and is
still the subject of intensive research. Hundreds of efficient algorithms for
this task have been proposed. Two of them are nearly optimal. The first one,
proposed in 1995, relies on recursive factorization of a polynomial, is quite
involved, and has never been implemented. The second one, proposed in 2016,
relies on subdivision iterations, was implemented in 2018, and promises to be
practically competitive, although user's current choice for univariate
polynomial root-finding is the package MPSolve, proposed in 2000, revised in
2014, and based on Ehrlich's functional iterations. By proposing and
incorporating some novel techniques we significantly accelerate both
subdivision and Ehrlich's iterations. Moreover our acceleration of the known
subdivision root-finders is dramatic in the case of sparse input polynomials.
Our techniques can be of some independent interest for the design and analysis
of polynomial root-finders.Comment: 89 pages, 5 figures, 2 table
On the deterministic solution of multidimensional parametric models using the Proper Generalized Decomposition
This paper focuses on the efficient solution of models defined in high dimensional spaces. Those models involve numerous numerical challenges because of their associated curse of dimensionality. It is well known that in mesh-based discrete models the complexity (degrees of freedom) scales exponentially with the dimension of the space. Many models encountered in computational science and engineering involve numerous dimensions called configurational coordinates. Some examples are the models encoun- tered in biology making use of the chemical master equation, quantum chemistry involving the solution of the Schrödinger or Dirac equations, kinetic theory descriptions of complex systems based on the solution of the so-called Fokker–Planck equation, stochastic models in which the random variables are included as new coordinates, financial mathematics, etc. This paper revisits the curse of dimensionality and proposes an efficient strategy for circumventing such challenging issue. This strategy, based on the use of a Proper Generalized Decomposition, is specially well suited to treat the multidimensional parametric equations
Square lattice Ising model susceptibility: Series expansion method and differential equation for
In a previous paper (J. Phys. A {\bf 37} (2004) 9651-9668) we have given the
Fuchsian linear differential equation satisfied by , the
``three-particle'' contribution to the susceptibility of the isotropic square
lattice Ising model. This paper gives the details of the calculations (with
some useful tricks and tools) allowing one to obtain long series in polynomial
time. The method is based on series expansion in the variables that appear in
the -dimensional integrals representing the -particle contribution to
the isotropic square lattice Ising model susceptibility . The
integration rules are straightforward due to remarkable formulas we derived for
these variables. We obtain without any numerical approximation as
a fully integrated series in the variable , where , with the conventional Ising model coupling constant. We also
give some perspectives and comments on these results.Comment: 28 pages, no figur
A unified approach to structural change tests based on F statistics, OLS residuals, and ML scores
Three classes of structural change tests (or tests for parameter instability) which have been receiving much attention in both the statistics and econometrics communities but have been developed in rather loosely connected lines of research are unified by embedding them into the framework of generalized M-fluctuation tests (Zeileis and Hornik, 2003). These classes are tests based on F statistics (supF, aveF, expF tests), on OLS residuals (OLS-based CUSUM and MOSUM tests) and on maximum likelihood scores (including the Nyblom-Hansen test). We show that (represantives from) these classes are special cases of the generalized M-fluctuation tests, based on the same functional central limit theorem, but employing different functionals for capturing excessive fluctuations. After embedding these tests into the same framework and thus understanding the relationship between these procedures for testing in historical samples, it is shown how the tests can also be extended to a monitoring situation. This is achieved by establishing a general M-fluctuation monitoring procedure and then applying the different functionals corresponding to monitoring with F statistics, OLS residuals and ML scores. In particular, an extension of the supF test to a monitoring scenario is suggested and illustrated on a real-world data set.Series: Research Report Series / Department of Statistics and Mathematic
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