213,998 research outputs found
On Higher-Order Fourier Analysis over Non-Prime Fields
The celebrated Weil bound for character sums says that for any low-degree polynomial P and any additive character chi, either chi(P) is a constant function or it is distributed close to uniform. The goal of higher-order Fourier analysis is to understand the connection between the algebraic and analytic properties of polynomials (and functions, generally) at a more detailed level. For instance, what is the tradeoff between the equidistribution of chi(P) and its "structure"?
Previously, most of the work in this area was over fields of prime order. We extend the tools of higher-order Fourier analysis to analyze functions over general finite fields. Let K be a field extension of a prime finite field F_p. Our technical results are:
1. If P: K^n -> K is a polynomial of degree |K|^{-s} for some s > 0 and non-trivial additive character chi, then P is a function of O_{d, s}(1) many non-classical polynomials of weight degree < d. The definition of non-classical polynomials over non-prime fields is one of the contributions of this work.
2. Suppose K and F are of bounded order, and let H be an affine subspace of K^n. Then, if P: K^n -> K is a polynomial of degree d that is sufficiently regular, then (P(x): x in H) is distributed almost as uniformly as possible subject to constraints imposed by the degree of P. Such a theorem was previously known for H an affine subspace over a prime field.
The tools of higher-order Fourier analysis have found use in different areas of computer science, including list decoding, algorithmic decomposition and testing. Using our new results, we revisit some of these areas.
(i) For any fixed finite field K, we show that the list decoding radius of the generalized Reed Muller code over K equals the minimum distance of the code.
(ii) For any fixed finite field K, we give a polynomial time algorithm to decide whether a given polynomial P: K^n -> K can be decomposed as a particular composition of lesser degree polynomials.
(iii) For any fixed finite field K, we prove that all locally characterized affine-invariant properties of functions f: K^n -> K are testable with one-sided error
Higher-order Statistics of Weak Lensing Shear and Flexion
Owing to their more extensive sky coverage and tighter control on systematic
errors, future deep weak lensing surveys should provide a better statistical
picture of the dark matter clustering beyond the level of the power spectrum.
In this context, the study of non-Gaussianity induced by gravity can help
tighten constraints on the background cosmology by breaking parameter
degeneracies, as well as throwing light on the nature of dark matter, dark
energy or alternative gravity theories. Analysis of the shear or flexion
properties of such maps is more complicated than the simpler case of the
convergence due to the spinorial nature of the fields involved. Here we develop
analytical tools for the study of higher-order statistics such as the
bispectrum (or trispectrum) directly using such maps at different source
redshift. The statistics we introduce can be constructed from cumulants of the
shear or flexions, involving the cross-correlation of squared and cubic maps at
different redshifts. Typically, the low signal-to-noise ratio prevents recovery
of the bispectrum or trispectrum mode by mode. We define power spectra
associated with each multi- spectra which compresses some of the available
information of higher order multispectra. We show how these can be recovered
from a noisy observational data even in the presence of arbitrary mask, which
introduces mixing between Electric (E-type) and Magnetic (B-type) polarization,
in an unbiased way. We also introduce higher order cross-correlators which can
cross-correlate lensing shear with different tracers of large scale structures.Comment: 16 pages, 2 figure
Local criticality, diffusion and chaos in generalized Sachdev-Ye-Kitaev models
The Sachdev-Ye-Kitaev model is a -dimensional model describing
Majorana fermions or complex fermions with random interactions. This model has
various interesting properties such as approximate local criticality (power law
correlation in time), zero temperature entropy, and quantum chaos. In this
article, we propose a higher dimensional generalization of the
Sachdev-Ye-Kitaev model, which is a lattice model with Majorana fermions at
each site and random interactions between them. Our model can be defined on
arbitrary lattices in arbitrary spatial dimensions. In the large limit, the
higher dimensional model preserves many properties of the Sachdev-Ye-Kitaev
model such as local criticality in two-point functions, zero temperature
entropy and chaos measured by the out-of-time-ordered correlation functions. In
addition, we obtain new properties unique to higher dimensions such as
diffusive energy transport and a "butterfly velocity" describing the
propagation of chaos in space. We mainly present results for a
-dimensional example, and discuss the general case near the end.Comment: 1+37 pages, published versio
Reconstructing the cosmic Horseshoe gravitational lens using the singular perturbative approach
The cosmic horseshoe gravitational lens is analyzed using the perturbative
approach. The two first order perturbative fields are expanded in Fourier
series. The source is reconstructed using a fine adaptive grid. The expansion
of the fields at order 2 produces a higher value of the chi-square. Expanding
at order 3 provides a very significant improvement, while order 4 does not
bring a significant improvement over order 3. The presence of the order 3 terms
is not a consequence of limiting the perturbative expansion to the first order.
The amplitude and signs of the third order terms are recovered by including the
contribution of the other group members. This analysis demonstrates that the
fine details of the potential of the lens could be recovered independently of
any assumptions by using the perturbative approach.Comment: 22 pages 11 figure
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