115 research outputs found
Comment on "Conjectures on exact solution of three-dimensional (3D) simple orthorhombic Ising lattices" [arXiv:0705.1045]
It is shown that a recent article by Z.-D. Zhang [arXiv:0705.1045] is in
error and violates well-known theorems.Comment: LaTeX, 3 pages, no figures, submitted to Philosophical Magazine.
Expanded versio
2D and 3D reconstructions in acousto-electric tomography
We propose and test stable algorithms for the reconstruction of the internal
conductivity of a biological object using acousto-electric measurements.
Namely, the conventional impedance tomography scheme is supplemented by
scanning the object with acoustic waves that slightly perturb the conductivity
and cause the change in the electric potential measured on the boundary of the
object. These perturbations of the potential are then used as the data for the
reconstruction of the conductivity. The present method does not rely on
"perfectly focused" acoustic beams. Instead, more realistic propagating
spherical fronts are utilized, and then the measurements that would correspond
to perfect focusing are synthesized. In other words, we use \emph{synthetic
focusing}. Numerical experiments with simulated data show that our techniques
produce high quality images, both in 2D and 3D, and that they remain accurate
in the presence of high-level noise in the data. Local uniqueness and stability
for the problem also hold
Optimal shapes of compact strings
Optimal geometrical arrangements, such as the stacking of atoms, are of
relevance in diverse disciplines. A classic problem is the determination of the
optimal arrangement of spheres in three dimensions in order to achieve the
highest packing fraction; only recently has it been proved that the answer for
infinite systems is a face-centred-cubic lattice. This simply stated problem
has had a profound impact in many areas, ranging from the crystallization and
melting of atomic systems, to optimal packing of objects and subdivision of
space. Here we study an analogous problem--that of determining the optimal
shapes of closely packed compact strings. This problem is a mathematical
idealization of situations commonly encountered in biology, chemistry and
physics, involving the optimal structure of folded polymeric chains. We find
that, in cases where boundary effects are not dominant, helices with a
particular pitch-radius ratio are selected. Interestingly, the same geometry is
observed in helices in naturally-occurring proteins.Comment: 8 pages, 3 composite ps figure
Tactical Voting in Plurality Elections
How often will elections end in landslides? What is the probability for a
head-to-head race? Analyzing ballot results from several large countries rather
anomalous and yet unexplained distributions have been observed. We identify
tactical voting as the driving ingredient for the anomalies and introduce a
model to study its effect on plurality elections, characterized by the relative
strength of the feedback from polls and the pairwise interaction between
individuals in the society. With this model it becomes possible to explain the
polarization of votes between two candidates, understand the small margin of
victories frequently observed for different elections, and analyze the polls'
impact in American, Canadian, and Brazilian ballots. Moreover, the model
reproduces, quantitatively, the distribution of votes obtained in the Brazilian
mayor elections with two, three, and four candidates.Comment: 7 pages, 4 figure
Equidistribution of Heegner Points and Ternary Quadratic Forms
We prove new equidistribution results for Galois orbits of Heegner points
with respect to reduction maps at inert primes. The arguments are based on two
different techniques: primitive representations of integers by quadratic forms
and distribution relations for Heegner points. Our results generalize one of
the equidistribution theorems established by Cornut and Vatsal in the sense
that we allow both the fundamental discriminant and the conductor to grow.
Moreover, for fixed fundamental discriminant and variable conductor, we deduce
an effective surjectivity theorem for the reduction map from Heegner points to
supersingular points at a fixed inert prime. Our results are applicable to the
setting considered by Kolyvagin in the construction of the Heegner points Euler
system
Quantum Phase Extraction in Isospectral Electronic Nanostructures
Quantum phase is not a direct observable and is usually determined by
interferometric methods. We present a method to map complete electron wave
functions, including internal quantum phase information, from measured
single-state probability densities. We harness the mathematical discovery of
drum-like manifolds bearing different shapes but identical resonances, and
construct quantum isospectral nanostructures possessing matching electronic
structure but divergent physical structure. Quantum measurement (scanning
tunneling microscopy) of these "quantum drums" [degenerate two-dimensional
electron states on the Cu(111) surface confined by individually positioned CO
molecules] reveals that isospectrality provides an extra topological degree of
freedom enabling robust quantum state transplantation and phase extraction.Comment: Published 8 February 2008 in Science; 13 page manuscript (including 4
figures) + 13 page supplement (including 6 figures); supplementary movies
available at http://mota.stanford.ed
Physics of the Riemann Hypothesis
Physicists become acquainted with special functions early in their studies.
Consider our perennial model, the harmonic oscillator, for which we need
Hermite functions, or the Laguerre functions in quantum mechanics. Here we
choose a particular number theoretical function, the Riemann zeta function and
examine its influence in the realm of physics and also how physics may be
suggestive for the resolution of one of mathematics' most famous unconfirmed
conjectures, the Riemann Hypothesis. Does physics hold an essential key to the
solution for this more than hundred-year-old problem? In this work we examine
numerous models from different branches of physics, from classical mechanics to
statistical physics, where this function plays an integral role. We also see
how this function is related to quantum chaos and how its pole-structure
encodes when particles can undergo Bose-Einstein condensation at low
temperature. Throughout these examinations we highlight how physics can perhaps
shed light on the Riemann Hypothesis. Naturally, our aim could not be to be
comprehensive, rather we focus on the major models and aim to give an informed
starting point for the interested Reader.Comment: 27 pages, 9 figure
Self-similarity of Mean Flow in Pipe Turbulence
Based on our previous modified log-wake law in turbulent pipe ‡flows, we invent two compound similarity numbers (Y;U), where Y is a combination of the inner variable y+ and outer variable , and U is the pure exect of the wall. The two similarity numbers can well collapse mean velocity profile data with different moderate and large Reynolds numbers into a single universal profile. We then propose an arctangent law for the buffer layer and a general log law for the outer region in terms of (Y;U). From Milikan’s maximum velocity law and the Princeton superpipe data, we derive the von Kármán constant = 0:43 and the additive constant B=6. Using an asymptotic matching method, we obtain a self-similarity law that describes the mean velocity profile from the wall to axis; and embeds the linear law in the viscous sublayer, the quartic law in the bursting sublayer, the classic log law in the overlap, the sine-square wake law in the wake layer, and the parabolic law near the pipe axis. The proposed arctangent law, the general log law and the self-similarity law have been compared with the high-quality data sets, with diffrent Reynolds numbers, including those from the Princeton superpipe, Loulou et al., Durst et al., Perry et al., and den Toonder and Nieuwstadt. Finally, as an application of the proposed laws, we improve the McKeon et al. method for Pitot probe displacement correction, which can be used to correct the widely used Zagarola and Smits data set
Quantum Simulation of Spin Models on an Arbitrary Lattice with Trapped Ions
A collection of trapped atomic ions represents one of the most attractive
platforms for the quantum simulation of interacting spin networks and quantum
magnetism. Spin-dependent optical dipole forces applied to an ion crystal
create long-range effective spin-spin interactions and allow the simulation of
spin Hamiltonians that possess nontrivial phases and dynamics. Here we show how
appropriate design of laser fields can provide for arbitrary multidimensional
spin-spin interaction graphs even for the case of a linear spatial array of
ions. This scheme uses currently existing trap technology and is scalable to
levels where classical methods of simulation are intractable.Comment: 5 pages, 4 figure
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