327 research outputs found
On Polynomial Approximations Over Z/2^kZ*
We study approximation of Boolean functions by low-degree polynomials over the ring Z/2^kZ. More precisely, given a Boolean function F:{0,1}^n -> {0,1}, define its k-lift to be F_k:{0,1}^n -> {0,2^(k-1)} by F_k(x) = 2^(k-F(x)) (mod 2^k). We consider the fractional agreement (which we refer to as gamma_{d,k}(F)) of F_k with degree d polynomials from Z/2^kZ[x_1,..,x_n].
Our results are the following:
* Increasing k can help: We observe that as k increases, gamma_{d,k}(F) cannot decrease. We give two kinds of examples where gamma_{d,k}(F) actually increases. The first is an infinite family of functions F such that gamma_{2d,2}(F) - gamma_{3d-1,1}(F) >= Omega(1). The second is an infinite family of functions F such that gamma_{d,1}(F) = 1/2 + Omega(1).
* Increasing k doesn\u27t always help: Adapting a proof of Green [Comput. Complexity, 9(1):16--38, 2000], we show that irrespective of the value of k, the Majority function Maj_n satisfies gamma_{d,k}(Maj_n) <= 1/2+ O(d)/sqrt{n}. In other words, polynomials over Z/2^kZ for large k do not approximate the majority function any better than polynomials over Z/2Z.
We observe that the model we study subsumes the model of non-classical polynomials, in the sense that proving bounds in our model implies bounds on the agreement of non-classical polynomials with Boolean functions. In particular, our results answer questions raised by Bhowmick and Lovett [In Proc. 30th Computational Complexity Conf., pages 72-87, 2015] that ask whether non-classical polynomials approximate Boolean functions better than classical polynomials of the same degree
On the Polyphase Decomposition for Design of Generalized Comb Decimation Filters
Generalized comb filters (GCFs) are efficient anti-aliasing decimation
filters with improved selectivity and quantization noise (QN) rejection
performance around the so called folding bands with respect to classical comb
filters.
In this paper, we address the design of GCF filters by proposing an efficient
partial polyphase architecture with the aim to reduce the data rate as much as
possible after the Sigma-Delta A/D conversion. We propose a mathematical
framework in order to completely characterize the dependence of the frequency
response of GCFs on the quantization of the multipliers embedded in the
proposed filter architecture. This analysis paves the way to the design of
multiplier-less decimation architectures.
We also derive the impulse response of a sample 3rd order GCF filter used as
a reference scheme throughout the paper.Comment: Submitted to IEEE TCAS-I, February 2007; 11 double-column pages, 9
figures, 1 tabl
Beyond the periodic orbit theory
The global constraints on chaotic dynamics induced by the analyticity of
smooth flows are used to dispense with individual periodic orbits and derive
infinite families of exact sum rules for several simple dynamical systems. The
associated Fredholm determinants are of particularly simple polynomial form.
The theory developed suggests an alternative to the conventional periodic orbit
theory approach to determining eigenspectra of transfer operators.Comment: 29 pages Latex2
Moving Atom-Field Interaction: Correction to Casimir-Polder Effect from Coherent Back-action
The Casimir-Polder force is an attractive force between a polarizable atom
and a conducting or dielectric boundary. Its original computation was in terms
of the Lamb shift of the atomic ground state in an electromagnetic field (EMF)
modified by boundary conditions along the wall and assuming a stationary atom.
We calculate the corrections to this force due to a moving atom, demanding
maximal preservation of entanglement generated by the moving atom-conducting
wall system. We do this by using non-perturbative path integral techniques
which allow for coherent back-action and thus can treat non-Markovian
processes. We recompute the atom-wall force for a conducting boundary by
allowing the bare atom-EMF ground state to evolve (or self-dress) into the
interacting ground state. We find a clear distinction between the cases of
stationary and adiabatic motions. Our result for the retardation correction for
adiabatic motion is up to twice as much as that computed for stationary atoms.
We give physical interpretations of both the stationary and adiabatic atom-wall
forces in terms of alteration of the virtual photon cloud surrounding the atom
by the wall and the Doppler effect.Comment: 16 pages, 2 figures, clarified discussions; to appear in Phys. Rev.
Steady streaming confined between three-dimensional wavy surfaces
We present a theoretical and numerical study of three-dimensional pulsatile confined flow between two rigid horizontal surfaces separated by an average gap h, and having three-dimensional wavy shapes with arbitrary amplitude σ h where σ ∼ O(1), but long-wavelength variations λ, with h/λ 1. We are interested in pulsating flows with moderate inertial effect arising from the Reynolds stress due to the cavity non-parallelism. We analyse the inertial steady-streaming and the second harmonic flows in a lubrication approximation. The dependence of the three-dimensional velocity field in the transverse direction is analytically obtained for arbitrary Womersley numbers and possibly overlapping Stokes layers. The horizontal dependence of the flow is solved numerically by computing the first two pressure fields of an asymptotic expansion in the small inertial limit. We study the variations of the flow structure with the amplitude, the channel’s wavelength and the Womersley number for various families of three-dimensional channels. The steady-streaming flow field in the horizontal plane exhibits a quadrupolar vortex, the size of which is adjusted to the cavity wavelength. When increasing the wall amplitude, the wavelengths characterizing the channel or the Womersley number, we find higher-order harmonic flow structures, the origin of which can either be inertially driven or geometrically induced. When some of the channel symmetries are broken, a steady-streaming current appears which has a quadratic dependence on the pressure drop, the amplitude of which is linked to the Womersley number
Chiral dynamics in a magnetic field from the functional renormalization group
We investigate the quark-meson model in a magnetic field using the exact
functional renormalization group equation beyond the local-potential
approximation. Our truncation of the effective action involves anisotropic wave
function renormalization for mesons, which allows us to investigate how the
magnetic field distorts the propagation of neutral mesons. Solving the flow
equation numerically, we find that the transverse velocity of mesons decreases
with the magnetic field at all temperatures, which is most prominent at zero
temperature. The meson screening masses and the pion decay constants are also
computed. The constituent quark mass is found to increase with magnetic field
at all temperatures, resulting in the crossover temperature that increases
monotonically with the magnetic field. This tendency is consistent with most
model calculations but not with the lattice simulation performed at the
physical point. Our work suggests that the strong anisotropy of meson
propagation may not be the fundamental origin of the inverse magnetic
catalysis.Comment: 37 pages, 10 figures. v2: References added, the version published in
JHE
Physical meaning of the radial index of Laguerre-Gauss beams
The Laguerre-Gauss modes are a class of fundamental and well-studied optical
fields. These stable, shape-invariant photons - exhibiting circular-cylindrical
symmetry - are familiar from laser optics, micro-mechanical manipulation,
quantum optics, communication, and foundational studies in both classical
optics and quantum physics. They are characterized, chiefly, by two modes
numbers: the azimuthal index indicating the orbital angular momentum of the
beam - which itself has spawned a burgeoning and vibrant sub-field - and the
radial index, which up until recently, has largely been ignored. In this
manuscript we develop a differential operator formalism for dealing with the
radial modes in both the position and momentum representations, and - more
importantly - give for the first time the meaning of this quantum number in
terms of a well-defined physical parameter: the "intrinsic hyperbolic momentum
charge".Comment: 12 pages, 4 figures, comments encourage
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