132,457 research outputs found
A note on quantum algorithms and the minimal degree of epsilon-error polynomials for symmetric functions
The degrees of polynomials representing or approximating Boolean functions
are a prominent tool in various branches of complexity theory. Sherstov
recently characterized the minimal degree deg_{\eps}(f) among all polynomials
(over the reals) that approximate a symmetric function f:{0,1}^n-->{0,1} up to
worst-case error \eps: deg_{\eps}(f) = ~\Theta(deg_{1/3}(f) +
\sqrt{n\log(1/\eps)}). In this note we show how a tighter version (without the
log-factors hidden in the ~\Theta-notation), can be derived quite easily using
the close connection between polynomials and quantum algorithms.Comment: 7 pages LaTeX. 2nd version: corrected a few small inaccuracie
Generalized Shortest Path Kernel on Graphs
We consider the problem of classifying graphs using graph kernels. We define
a new graph kernel, called the generalized shortest path kernel, based on the
number and length of shortest paths between nodes. For our example
classification problem, we consider the task of classifying random graphs from
two well-known families, by the number of clusters they contain. We verify
empirically that the generalized shortest path kernel outperforms the original
shortest path kernel on a number of datasets. We give a theoretical analysis
for explaining our experimental results. In particular, we estimate
distributions of the expected feature vectors for the shortest path kernel and
the generalized shortest path kernel, and we show some evidence explaining why
our graph kernel outperforms the shortest path kernel for our graph
classification problem.Comment: Short version presented at Discovery Science 2015 in Banf
Analyticity of homogenized coefficients under Bernoulli perturbations and the Clausius-Mossotti formulas
This paper is concerned with the behavior of the homogenized coefficients
associated with some random stationary ergodic medium under a Bernoulli
perturbation. Introducing a new family of energy estimates that combine
probability and physical spaces, we prove the analyticity of the perturbed
homogenized coefficients with respect to the Bernoulli parameter. Our approach
holds under the minimal assumptions of stationarity and ergodicity, both in the
scalar and vector cases, and gives analytical formulas for each derivative that
essentially coincide with the so-called cluster expansion used by physicists.
In particular, the first term yields the celebrated (electric and elastic)
Clausius-Mossotti formulas for isotropic spherical random inclusions in an
isotropic reference medium. This work constitutes the first general proof of
these formulas in the case of random inclusions.Comment: 47 page
On the asymptotic magnitude of subsets of Euclidean space
Magnitude is a canonical invariant of finite metric spaces which has its
origins in category theory; it is analogous to cardinality of finite sets.
Here, by approximating certain compact subsets of Euclidean space with finite
subsets, the magnitudes of line segments, circles and Cantor sets are defined
and calculated. It is observed that asymptotically these satisfy the
inclusion-exclusion principle, relating them to intrinsic volumes of polyconvex
sets.Comment: 23 pages. Version 2: updated to reflect more recent work, in
particular, the approximation method is now known to calculate (rather than
merely define) the magnitude; also minor alterations such as references adde
Cluster density functional theory for lattice models based on the theory of Mobius functions
Rosenfeld's fundamental measure theory for lattice models is given a rigorous
formulation in terms of the theory of Mobius functions of partially ordered
sets. The free-energy density functional is expressed as an expansion in a
finite set of lattice clusters. This set is endowed a partial order, so that
the coefficients of the cluster expansion are connected to its Mobius function.
Because of this, it is rigorously proven that a unique such expansion exists
for any lattice model. The low-density analysis of the free-energy functional
motivates a redefinition of the basic clusters (zero-dimensional cavities)
which guarantees a correct zero-density limit of the pair and triplet direct
correlation functions. This new definition extends Rosenfeld's theory to
lattice model with any kind of short-range interaction (repulsive or
attractive, hard or soft, one- or multi-component...). Finally, a proof is
given that these functionals have a consistent dimensional reduction, i.e. the
functional for dimension d' can be obtained from that for dimension d (d'<d) if
the latter is evaluated at a density profile confined to a d'-dimensional
subset.Comment: 21 pages, 2 figures, uses iopart.cls, as well as diagrams.sty
(included
q-deformed fermion oscillators, zero-point energy and inclusion-exclusion principle
The theory of Fermion oscillators has two essential ingredients: zero-point
energy and Pauli exclusion principle. We devlop the theory of the statistical
mechanics of generalized q-deformed Fermion oscillator algebra with inclusion
principle (i.e., without the exclusion principle), which corresponds to
ordinary fermions with Pauli exclusion principle in the classical limit . Some of the remarkable properties of this theory play a crucial role in the
understanding of the q-deformed Fermions. We show that if we insist on the weak
exclusion principle, then the theory has the expected low temperature limit as
well as the correct classical q-limit.Comment: 10 pages, Latex, submitted to Physical Review
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