6,583 research outputs found
MUBs, polytopes, and finite geometries
A complete set of N+1 mutually unbiased bases (MUBs) exists in Hilbert spaces
of dimension N = p^k, where p is a prime number. They mesh naturally with
finite affine planes of order N, that exist when N = p^k. The existence of MUBs
for other values of N is an open question, and the same is true for finite
affine planes. I explore the question whether the existence of complete sets of
MUBs is directly related to the existence of finite affine planes. Both
questions can be shown to be geometrical questions about a convex polytope, but
not in any obvious way the same question.Comment: 15 pages; talk at the Vaxjo conference on probability and physic
Cauchy, infinitesimals and ghosts of departed quantifiers
Procedures relying on infinitesimals in Leibniz, Euler and Cauchy have been
interpreted in both a Weierstrassian and Robinson's frameworks. The latter
provides closer proxies for the procedures of the classical masters. Thus,
Leibniz's distinction between assignable and inassignable numbers finds a proxy
in the distinction between standard and nonstandard numbers in Robinson's
framework, while Leibniz's law of homogeneity with the implied notion of
equality up to negligible terms finds a mathematical formalisation in terms of
standard part. It is hard to provide parallel formalisations in a
Weierstrassian framework but scholars since Ishiguro have engaged in a quest
for ghosts of departed quantifiers to provide a Weierstrassian account for
Leibniz's infinitesimals. Euler similarly had notions of equality up to
negligible terms, of which he distinguished two types: geometric and
arithmetic. Euler routinely used product decompositions into a specific
infinite number of factors, and used the binomial formula with an infinite
exponent. Such procedures have immediate hyperfinite analogues in Robinson's
framework, while in a Weierstrassian framework they can only be reinterpreted
by means of paraphrases departing significantly from Euler's own presentation.
Cauchy gives lucid definitions of continuity in terms of infinitesimals that
find ready formalisations in Robinson's framework but scholars working in a
Weierstrassian framework bend over backwards either to claim that Cauchy was
vague or to engage in a quest for ghosts of departed quantifiers in his work.
Cauchy's procedures in the context of his 1853 sum theorem (for series of
continuous functions) are more readily understood from the viewpoint of
Robinson's framework, where one can exploit tools such as the pointwise
definition of the concept of uniform convergence.
Keywords: historiography; infinitesimal; Latin model; butterfly modelComment: 45 pages, published in Mat. Stu
Looking backward: From Euler to Riemann
We survey the main ideas in the early history of the subjects on which
Riemann worked and that led to some of his most important discoveries. The
subjects discussed include the theory of functions of a complex variable,
elliptic and Abelian integrals, the hypergeometric series, the zeta function,
topology, differential geometry, integration, and the notion of space. We shall
see that among Riemann's predecessors in all these fields, one name occupies a
prominent place, this is Leonhard Euler. The final version of this paper will
appear in the book \emph{From Riemann to differential geometry and relativity}
(L. Ji, A. Papadopoulos and S. Yamada, ed.) Berlin: Springer, 2017
The geometry of nonlinear least squares with applications to sloppy models and optimization
Parameter estimation by nonlinear least squares minimization is a common
problem with an elegant geometric interpretation: the possible parameter values
of a model induce a manifold in the space of data predictions. The minimization
problem is then to find the point on the manifold closest to the data. We show
that the model manifolds of a large class of models, known as sloppy models,
have many universal features; they are characterized by a geometric series of
widths, extrinsic curvatures, and parameter-effects curvatures. A number of
common difficulties in optimizing least squares problems are due to this common
structure. First, algorithms tend to run into the boundaries of the model
manifold, causing parameters to diverge or become unphysical. We introduce the
model graph as an extension of the model manifold to remedy this problem. We
argue that appropriate priors can remove the boundaries and improve convergence
rates. We show that typical fits will have many evaporated parameters. Second,
bare model parameters are usually ill-suited to describing model behavior; cost
contours in parameter space tend to form hierarchies of plateaus and canyons.
Geometrically, we understand this inconvenient parametrization as an extremely
skewed coordinate basis and show that it induces a large parameter-effects
curvature on the manifold. Using coordinates based on geodesic motion, these
narrow canyons are transformed in many cases into a single quadratic, isotropic
basin. We interpret the modified Gauss-Newton and Levenberg-Marquardt fitting
algorithms as an Euler approximation to geodesic motion in these natural
coordinates on the model manifold and the model graph respectively. By adding a
geodesic acceleration adjustment to these algorithms, we alleviate the
difficulties from parameter-effects curvature, improving both efficiency and
success rates at finding good fits.Comment: 40 pages, 29 Figure
Vacuum structure and string tension in Yang-Mills dimeron ensembles
We numerically simulate ensembles of SU(2) Yang-Mills dimeron solutions with
a statistical weight determined by the classical action and perform a
comprehensive analysis of their properties. In particular, we examine the
extent to which these ensembles capture topological and confinement properties
of the Yang-Mills vacuum. This further allows us to test the classic picture of
meron-induced quark confinement as triggered by dimeron dissociation. At small
bare couplings, spacial, topological-charge and color correlations among the
dimerons generate a short-range order which screens topological charges. With
increasing coupling this order weakens rapidly, however, in part because the
dimerons gradually dissociate into their meron constituents. Monitoring
confinement properties by evaluating Wilson-loop expectation values, we find
the growing disorder due to these progressively liberated merons to generate a
finite and (with the coupling) increasing string tension. The short-distance
behavior of the static quark-antiquark potential, on the other hand, is
dominated by small, "instanton-like" dimerons. String tension, action density
and topological susceptibility of the dimeron ensembles in the physical
coupling region turn out to be of the order of standard values. Hence the above
results demonstrate without reliance on weak-coupling or low-density
approximations that the dissociating dimeron component in the Yang-Mills vacuum
can indeed produce a meron-populated confining phase. The density of
coexisting, hardly dissociated and thus instanton-like dimerons seems to remain
large enough, on the other hand, to reproduce much of the additional
phenomenology successfully accounted for by non-confining instanton vacuum
models. Hence dimeron ensembles should provide an efficient basis for a rather
complete description of the Yang-Mills vacuum.Comment: 36 pages, 17 figure
ARPES and NMTO Wannier Orbital Theory of LiMoO - Implications for Unusually Robust Quasi-One Dimensional Behavior
We present the results of a combined study by band theory and angle resolved
photoemission spectroscopy (ARPES) of the purple bronze,
LiMoO. Structural and electronic origins of its unusually
robust quasi-one dimensional (quasi-1D) behavior are investigated in detail.
The band structure, in a large energy window around the Fermi energy, is
basically 2D and formed by three Mo -like extended Wannier orbitals,
each one giving rise to a 1D band running at a 120 angle to the two
others. A structural "dimerization" from to gaps
the and bands while leaving the bands metallic in the gap, but
resonantly coupled to the gap edges and, hence, to the other directions. The
resulting complex shape of the quasi-1D Fermi surface (FS), verified by our
ARPES, thus depends strongly on the Fermi energy position in the gap, implying
a great sensitivity to Li stoichiometry of properties dependent on the FS, such
as FS nesting or superconductivity. The strong resonances prevent either a
two-band tight-binding model or a related real-space ladder picture from giving
a valid description of the low-energy electronic structure. We use our extended
knowledge of the electronic structure to newly advocate for framing
LiMoO as a weak-coupling material and in that framework can
rationalize both the robustness of its quasi-1D behavior and the rather large
value of its Luttinger liquid (LL) exponent . Down to a temperature of
6K we find no evidence for a theoretically expected downward
renormalization of perpendicular single particle hopping due to LL fluctuations
in the quasi-1D chains.Comment: 53 pages, 17 Figures, 6 year
Wallis on Indivisibles
International audienceThe present chapter is devoted, first, to discuss in detail the structure and results of Wallis's major and most influential mathematical work, the Arithmetica Infinitorum ([51]). Next we will revise Wallis's views on indivisibles as articulated in his answer to Hobbes's criticism in the early 1670s. Finally, we will turn to his discussion of the proper way to understand the angle of contingence in the first half of the 1680s. As we shall see, there are marked differences in the status that indivisibles seem to enjoy in Wallis's thought along his mathematical career. These differences correlate with the changing context of 17th-century mathematics from the 1650s through the 1680s, but also respond to the different uses Wallis gave to indivisibles in different kinds of texts—purely mathematical, openly polemical, or devoted to philosophical discussion of foundational matters
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