22,038 research outputs found
Characterization and computation of canonical tight windows for Gabor frames
Let be a Gabor frame for for given window .
We show that the window that generates the canonically
associated tight Gabor frame minimizes among all windows
generating a normalized tight Gabor frame. We present and prove versions of
this result in the time domain, the frequency domain, the time-frequency
domain, and the Zak transform domain, where in each domain the canonical
is expressed using functional calculus for Gabor frame operators. Furthermore,
we derive a Wiener-Levy type theorem for rationally oversampled Gabor frames.
Finally, a Newton-type method for a fast numerical calculation of \ho is
presented. We analyze the convergence behavior of this method and demonstrate
the efficiency of the proposed algorithm by some numerical examples
On Lerch's transcendent and the Gaussian random walk
Let be independent variables, each having a normal distribution
with negative mean and variance 1. We consider the partial sums
, with , and refer to the process as
the Gaussian random walk. We present explicit expressions for the mean and
variance of the maximum These expressions are in terms
of Taylor series about with coefficients that involve the Riemann
zeta function. Our results extend Kingman's first-order approximation [Proc.
Symp. on Congestion Theory (1965) 137--169] of the mean for .
We build upon the work of Chang and Peres [Ann. Probab. 25 (1997) 787--802],
and use Bateman's formulas on Lerch's transcendent and Euler--Maclaurin
summation as key ingredients.Comment: Published at http://dx.doi.org/10.1214/105051606000000781 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
Adoption as a Social Marker: Innovation Diffusion with Outgroup Aversion
Social identities are among the key factors driving behavior in complex
societies. Signals of social identity are known to influence individual
behaviors in the adoption of innovations. Yet the population-level consequences
of identity signaling on the diffusion of innovations are largely unknown. Here
we use both analytical and agent-based modeling to consider the spread of a
beneficial innovation in a structured population in which there exist two
groups who are averse to being mistaken for each other. We investigate the
dynamics of adoption and consider the role of structural factors such as
demographic skew and communication scale on population-level outcomes. We find
that outgroup aversion can lead to adoption being delayed or suppressed in one
group, and that population-wide underadoption is common. Comparing the two
models, we find that differential adoption can arise due to structural
constraints on information flow even in the absence of intrinsic between-group
differences in adoption rates. Further, we find that patterns of polarization
in adoption at both local and global scales depend on the details of
demographic organization and the scale of communication. This research has
particular relevance to widely beneficial but identity-relevant products and
behaviors, such as green technologies, where overall levels of adoption
determine the positive benefits that accrue to society at large.Comment: 26 pages, 10 figure
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