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Improved dispersion bounds for modified Fibonacci lattices [article]

Ralph Kritzinger, Jaspar Wiart
2020 arXiv   pre-print
The upper bound 2 is obtained by an explicit point construction - the well-known Fibonacci lattice.  ...  In this paper we find a modification of this point set such that its dispersion is significantly lower than the dispersion of the Fibonacci lattice.  ...  The motivation behindF m was to find a way to modify the Fibonacci lattice in such a way as to improve the constant of 2 in the limit.  ... 
arXiv:2007.02297v3 fatcat:mncs3q3u7zcv5lfwcerw2rux7a

Dispersion of the Fibonacci and the Frolov point sets [article]

V.N. Temlyakov
2017 arXiv   pre-print
It is shown how the optimal upper bounds for dispersion can be derived from the upper bounds for a new characteristic -- the smooth fixed volume discrepancy.  ...  It is proved that the Fibonacci and the Frolov point sets, which are known to be very good for numerical integration, have optimal rate of decay of dispersion with respect to the cardinality of sets.  ...  The author would like to thank the Erwin Schrödinger International Institute for Mathematics and Physics (ESI) at the University of Vienna for support.  ... 
arXiv:1709.08158v2 fatcat:kn6owbhzcbf4xmd2bmaimerjrm

Phonon excitations in quasicrystals

M. Quilichini
1997 Reviews of Modern Physics  
Recent progress concerning lattice dynamics in quasicrystals, both theoretical and experimental, is discussed.  ...  The theory deals with the general description, which differs from that for ordinary crystals, and with model calculations and rigorous results for one-, two-, and three-dimensional systems.  ...  A value of co for which there is an allowed solution is said to belong to the spectrum. F or a lattice periodic chain this is only the case when un is bounded.  ... 
doi:10.1103/revmodphys.69.277 fatcat:c5fggusmhrefxed57wissd4rfi

Fixed volume discrepancy in the periodic case [article]

V.N. Temlyakov
2017 arXiv   pre-print
The upper bounds for the r-smooth fixed volume periodic discrepancy for these sets are established.  ...  The author would like to thank the Erwin Schrödinger International Institute for Mathematics and Physics (ESI) at the University of Vienna for support.  ...  It was proved in [18] that for the Fibonacci point sets (d = 2) and the Frolov point sets we can improve the above upper bound to ≤ C(d, r)m −r (log mV ) d−1 , V ≥ c(r, d)/m, for the functions with the  ... 
arXiv:1710.11499v1 fatcat:q5sifas6ibdidiq2pvkgsd43qa

Connections between numerical integration, discrepancy, dispersion, and universal discretization [article]

Vladimir Temlyakov
2018 arXiv   pre-print
It was established recently that the new concept of fixed volume discrepancy is very useful in proving the upper bounds for the dispersion.  ...  Also, it was understood recently that point sets with small dispersion are very good for the universal discretization of the uniform norm of trigonometric polynomials.  ...  The upper bound in Theorem 7.1 combined with the trivial lower bound shows that the Fibonacci point set provides optimal rate of decay for the dispersion. (7. 2) The following Theorem 7.2 was derived in  ... 
arXiv:1812.04489v1 fatcat:keqkouzb7bc2lowsev2ut4o77a

A cavity-QED simulator of slow and fast scrambling [article]

J. Marino, A. M. Rey
2019 arXiv   pre-print
The model is realisable in a cavity QED system in the dispersive regime.  ...  The impact of initial conditions on the scrambling dynamics is attributed to the presence of a global conserved quantity, which critically slows down the evolution for initial states close to the poles  ...  However, by improving the quality of the mirrors it is feasible to improve it by one or even two orders of magnitude.  ... 
arXiv:1810.00866v2 fatcat:lh76oisfkvbgxbwdo5zjxbrtum

Atomic Structure of Decagonal Al-Cu-Rh Quasicrystal–Revisited: New Correction for Phonons

Radoslaw Strzalka, Ireneusz Buganski, Pawel Kuczera, Lucjan Pytlik, Janusz Wolny
2019 Crystals  
A significant improvement of the calculated atomic composition towards experimentally obtained and minor positional changes is also reported compared to the original paper.  ...  The results show the usefulness of investigating different corrective terms for diffraction data during a structure refinement.  ...  with pair-potentials for decagonal systems [17] or double-well potentials for model Fibonacci chains [18] .  ... 
doi:10.3390/cryst9020078 fatcat:ddpy7vitq5aorhj7ofc6oubrci

Connections between numerical integration, discrepancy, dispersion, and universal discretization

Vladimir Temlyakov
2019 SMAI Journal of Computational Mathematics  
It was established recently that the new concept of fixed volume discrepancy is very useful in proving the upper bounds for the dispersion.  ...  Also, it was understood recently that point sets with small dispersion are very good for the universal discretization of the uniform norm of trigonometric polynomials.  ...  Acknowledgements The author is grateful to the referees for many useful suggestions.  ... 
doi:10.5802/smai-jcm.58 fatcat:zexi4rixzjd6xhmyfuhgskpoby

Coupled nanopillar waveguides optical properties and applications

Dmitry N. Chigrin, Sergei V. Zhukovsky, Andrei V. Lavrinenko, Johann Kroha
2007 Physica Status Solidi (a) applications and materials science  
We present a systematic analysis of the optical properties of coupled nanopillar waveguides and discuss their possible applications for integrated optics.  ...  In such a waveguide, light confinement is due to the total internal reflection, while guided modes dispersion is strongly affected by the waveguide structure.  ...  In figure 4 the dispersion diagrams for CNPW with rods placed in the vertices of a triangular lattice are shown for W2, W3, W4 and W5 waveguides.  ... 
doi:10.1002/pssa.200776407 fatcat:xlr6ve57ozbzbpq5bra2efqr7m

The spectrum of 2+1 dimensional Yang-Mills theory on a twisted spatial torus

Margarita García Pérez, Antonio González-Arroyo, Mateusz Koren, Masanori Okawa
2018 Journal of High Energy Physics  
Here we made a more complete study and we are able to condense our results, obtained by non-perturbative lattice methods, into a simple expression which has important implications for the absence of tachyonic  ...  [17] might have to be modified accordingly.  ...  We will express most lattice computed quantities in units of λ L . Other choices are possible, for instance ref. [37] uses a mean-field improved coupling [43] .  ... 
doi:10.1007/jhep07(2018)169 fatcat:4j5nhsaifrfrtcium7ucgx5uq4

The square Thue–Morse tiling for photonic application

L. Moretti, V. Mocella
2008 Philosophical Magazine  
Differently from the traditional photonic quasicrystals based on the Penrose tiling, these structures were obtained by removing the lattice points from a square arrangement, following the inflation rules  ...  Therefore, since Fibonacci lattices have been demonstrated to have purely singular continuous energy spectrum, the associated electromagnetic states cannot be extended in the Bloch' sense, and can be classified  ...  However, electrons form bound states and photons do not.  ... 
doi:10.1080/14786430802047076 fatcat:rsm7bntd2ne3dpytjkuofxpigm

Nanoplasmonics of prime number arrays

Carlo Forestiere, Gary F. Walsh, Giovanni Miano, Luca Dal Negro
2009 Optics Express  
The study of prime number arrays of metal nanoparticles provides a novel strategy to achieve broadband enhancement and localization of plasmonic fields for the engineering of nanoscale nano-antenna arrays  ...  Army through the Natick Soldier Center (W911NF-07-D-001), the SMART Scholarship Program, the Air Force program "Deterministic Aperiodic Structures for on-chip nanophotonics and nanoplasmonics device applications  ...  Periodic (square lattice) and quasi-periodic Fibonacci plasmonic structures [3] are additionally investigated for comparison.  ... 
doi:10.1364/oe.17.024288 pmid:20052140 fatcat:gchve7ncfjbctfhlmdbr66w44m

Persistence of gaps in the interacting anisotropic Hofstadter model

Vieri Mastropietro
2019 Physical review B  
In other cases, instead, the bounds can be improved and the factorials cancel out; this is what happens in Lindstedt series for KAM tori.  ...  In order to show that they give a finite contribution one has to improve the estimate by the Diophantine property of α (5).  ... 
doi:10.1103/physrevb.99.155154 fatcat:w2vgru7bbbfffj3vw77nlz2sw4

Polygonal planforms and phyllotaxis on plants

P.D. Shipman, A.C. Newell
2005 Journal of Theoretical Biology  
Acknowledgements The authors are grateful to NSF Grant DMS 0202440 and NSF VIGRE Grant 9977116 for support and to Michael Ku¨cken for many useful discussions.  ...  thin lines) and the sets of active modes (bounded by thick lines) are shown.  ...  This, of course, is exactly the recipe for producing Fibonacci sequences.  ... 
doi:10.1016/j.jtbi.2005.03.007 pmid:16005308 fatcat:27kzazfo5zdz3jibkf2jarzpu4

On the problem of the relation between phason elasticity and phason dynamics in quasicrystals

G. Coddens
2006 European Physical Journal B : Condensed Matter Physics  
It is thus superfluous to call for a picture of "phason waves" in order to explain such data, especially as such "waves" violate many physical principles.  ...  We will show that the data reported call for a more detailed development of the elasticity theory of Jaric and Nelsson in order to determine the nature of small phonon-like atomic displacements with a  ...  This is obvious from the fact that the Fourier spectrum of QC * is spanned by three reciprocal lattice vectors, viz. the two reciprocal lattice vectors that span the Fourier spectrum of the Fibonacci lattice  ... 
doi:10.1140/epjb/e2006-00429-9 fatcat:qhykg5bkvfev7pideusbm4blu4
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