290 research outputs found
Scaling of human behavior during portal browsing
We investigate transitions of portals users between different subpages. A
weighted network of portals subpages is reconstructed where edge weights are
numbers of corresponding transitions. Distributions of link weights and node
strengths follow power laws over several decades. Node strength increases
faster than linearly with node degree. The distribution of time spent by the
user at one subpage decays as power law with exponent around 1.3. Distribution
of numbers P(z) of unique subpages during one visit is exponential. We find a
square root dependence between the average z and the total number of
transitions n during a single visit. Individual path of portal user resembles
of self-attracting walk on the weighted network. Analytical model is developed
to recover in part the collected data.Comment: 6 pages, 7 figure
Non-additivity of Renyi entropy and Dvoretzky's Theorem
The goal of this note is to show that the analysis of the minimum output
p-Renyi entropy of a typical quantum channel essentially amounts to applying
Milman's version of Dvoretzky's Theorem about almost Euclidean sections of
high-dimensional convex bodies. This conceptually simplifies the
(nonconstructive) argument by Hayden-Winter disproving the additivity
conjecture for the minimal output p-Renyi entropy (for p>1).Comment: 8 pages, LaTeX; v2: added and updated references, minor editorial
changes, no content change
A forward-backward splitting algorithm for the minimization of non-smooth convex functionals in Banach space
We consider the task of computing an approximate minimizer of the sum of a
smooth and non-smooth convex functional, respectively, in Banach space.
Motivated by the classical forward-backward splitting method for the
subgradients in Hilbert space, we propose a generalization which involves the
iterative solution of simpler subproblems. Descent and convergence properties
of this new algorithm are studied. Furthermore, the results are applied to the
minimization of Tikhonov-functionals associated with linear inverse problems
and semi-norm penalization in Banach spaces. With the help of
Bregman-Taylor-distance estimates, rates of convergence for the
forward-backward splitting procedure are obtained. Examples which demonstrate
the applicability are given, in particular, a generalization of the iterative
soft-thresholding method by Daubechies, Defrise and De Mol to Banach spaces as
well as total-variation based image restoration in higher dimensions are
presented
From Low-Distortion Norm Embeddings to Explicit Uncertainty Relations and Efficient Information Locking
The existence of quantum uncertainty relations is the essential reason that
some classically impossible cryptographic primitives become possible when
quantum communication is allowed. One direct operational manifestation of these
uncertainty relations is a purely quantum effect referred to as information
locking. A locking scheme can be viewed as a cryptographic protocol in which a
uniformly random n-bit message is encoded in a quantum system using a classical
key of size much smaller than n. Without the key, no measurement of this
quantum state can extract more than a negligible amount of information about
the message, in which case the message is said to be "locked". Furthermore,
knowing the key, it is possible to recover, that is "unlock", the message. In
this paper, we make the following contributions by exploiting a connection
between uncertainty relations and low-distortion embeddings of L2 into L1. We
introduce the notion of metric uncertainty relations and connect it to
low-distortion embeddings of L2 into L1. A metric uncertainty relation also
implies an entropic uncertainty relation. We prove that random bases satisfy
uncertainty relations with a stronger definition and better parameters than
previously known. Our proof is also considerably simpler than earlier proofs.
We apply this result to show the existence of locking schemes with key size
independent of the message length. We give efficient constructions of metric
uncertainty relations. The bases defining these metric uncertainty relations
are computable by quantum circuits of almost linear size. This leads to the
first explicit construction of a strong information locking scheme. Moreover,
we present a locking scheme that is close to being implementable with current
technology. We apply our metric uncertainty relations to exhibit communication
protocols that perform quantum equality testing.Comment: 60 pages, 5 figures. v4: published versio
Number of Common Sites Visited by N Random Walkers
We compute analytically the mean number of common sites, W_N(t), visited by N
independent random walkers each of length t and all starting at the origin at
t=0 in d dimensions. We show that in the (N-d) plane, there are three distinct
regimes for the asymptotic large t growth of W_N(t). These three regimes are
separated by two critical lines d=2 and d=d_c(N)=2N/(N-1) in the (N-d) plane.
For d<2, W_N(t)\sim t^{d/2} for large t (the N dependence is only in the
prefactor). For 2<d<d_c(N), W_N(t)\sim t^{\nu} where the exponent \nu=
N-d(N-1)/2 varies with N and d. For d>d_c(N), W_N(t) approaches a constant as
t\to \infty. Exactly at the critical dimensions there are logaritmic
corrections: for d=2, we get W_N(t)\sim t/[\ln t]^N, while for d=d_c(N),
W_N(t)\sim \ln t for large t. Our analytical predictions are verified in
numerical simulations.Comment: 5 pages, 3 .eps figures include
Brownian Simulations and Uni-Directional Flux in Diffusion
Brownian dynamics simulations require the connection of a small discrete
simulation volume to large baths that are maintained at fixed concentrations
and voltages. The continuum baths are connected to the simulation through
interfaces, located in the baths sufficiently far from the channel. Average
boundary concentrations have to be maintained at their values in the baths by
injecting and removing particles at the interfaces. The particles injected into
the simulation volume represent a unidirectional diffusion flux, while the
outgoing particles represent the unidirectional flux in the opposite direction.
The classical diffusion equation defines net diffusion flux, but not
unidirectional fluxes. The stochastic formulation of classical diffusion in
terms of the Wiener process leads to a Wiener path integral, which can split
the net flux into unidirectional fluxes. These unidirectional fluxes are
infinite, though the net flux is finite and agrees with classical theory. We
find that the infinite unidirectional flux is an artifact caused by replacing
the Langevin dynamics with its Smoluchowski approximation, which is classical
diffusion. The Smoluchowski approximation fails on time scales shorter than the
relaxation time of the Langevin equation. We find the unidirectional
flux (source strength) needed to maintain average boundary concentrations in a
manner consistent with the physics of Brownian particles. This unidirectional
flux is proportional to the concentration and inversely proportional to
to leading order. We develop a BD simulation that maintains
fixed average boundary concentrations in a manner consistent with the actual
physics of the interface and without creating spurious boundary layers
Locally Perturbed Random Walks with Unbounded Jumps
In \cite{SzT}, D. Sz\'asz and A. Telcs have shown that for the diffusively
scaled, simple symmetric random walk, weak convergence to the Brownian motion
holds even in the case of local impurities if . The extension of their
result to finite range random walks is straightforward. Here, however, we are
interested in the situation when the random walk has unbounded range.
Concretely we generalize the statement of \cite{SzT} to unbounded random walks
whose jump distribution belongs to the domain of attraction of the normal law.
We do this first: for diffusively scaled random walks on having finite variance; and second: for random walks with distribution
belonging to the non-normal domain of attraction of the normal law. This result
can be applied to random walks with tail behavior analogous to that of the
infinite horizon Lorentz-process; these, in particular, have infinite variance,
and convergence to Brownian motion holds with the superdiffusive scaling.Comment: 16 page
Number of distinct sites visited by N random walkers on a Euclidean lattice
The evaluation of the average number S_N(t) of distinct sites visited up to
time t by N independent random walkers all starting from the same origin on an
Euclidean lattice is addressed. We find that, for the nontrivial time regime
and for large N, S_N(t) \approx \hat S_N(t) (1-\Delta), where \hat S_N(t) is
the volume of a hypersphere of radius (4Dt \ln N)^{1/2},
\Delta={1/2}\sum_{n=1}^\infty \ln^{-n} N \sum_{m=0}^n s_m^{(n)} \ln^{m} \ln N,
d is the dimension of the lattice, and the coefficients s_m^{(n)} depend on the
dimension and time. The first three terms of these series are calculated
explicitly and the resulting expressions are compared with other approximations
and with simulation results for dimensions 1, 2, and 3. Some implications of
these results on the geometry of the set of visited sites are discussed.Comment: 15 pages (RevTex), 4 figures (eps); to appear in Phys. Rev.
Multivariate Lagrange inversion formula and the cycle lemma
International audienceWe give a multitype extension of the cycle lemma of (Dvoretzky and Motzkin 1947). This allows us to obtain a combinatorial proof of the multivariate Lagrange inversion formula that generalizes the celebrated proof of (Raney 1963) in the univariate case, and its extension in (Chottin 1981) to the two variable case. Until now, only the alternative approach of (Joyal 1981) and (Labelle 1981) via labelled arborescences and endofunctions had been successfully extended to the multivariate case in (Gessel 1983), (Goulden and Kulkarni 1996), (Bousquet et al. 2003), and the extension of the cycle lemma to more than 2 variables was elusive. The cycle lemma has found a lot of applications in combinatorics, so we expect our multivariate extension to be quite fruitful: as a first application we mention economical linear time exact random sampling for multispecies trees
- âŠ