440 research outputs found
Minimal cubic cones via Clifford algebras
We construct two infinite families of algebraic minimal cones in . The
first family consists of minimal cubics given explicitly in terms of the
Clifford systems. We show that the classes of congruent minimal cubics are in
one to one correspondence with those of geometrically equivalent Clifford
systems. As a byproduct, we prove that for any , , there is
at least one minimal cone in given by an irreducible homogeneous cubic
polynomial. The second family consists of minimal cones in , ,
defined by an irreducible homogeneous polynomial of degree . These examples
provide particular answers to the questions on algebraic minimal cones posed by
Wu-Yi Hsiang in the 1960's.Comment: Final version, corrects typos in Table
Section Extension from Hyperbolic Geometry of Punctured Disk and Holomorphic Family of Flat Bundles
The construction of sections of bundles with prescribed jet values plays a
fundamental role in problems of algebraic and complex geometry. When the jet
values are prescribed on a positive dimensional subvariety, it is handled by
theorems of Ohsawa-Takegoshi type which give extension of line bundle valued
square-integrable top-degree holomorphic forms from the fiber at the origin of
a family of complex manifolds over the open unit 1-disk when the curvature of
the metric of line bundle is semipositive. We prove here an extension result
when the curvature of the line bundle is only semipositive on each fiber with
negativity on the total space assumed bounded from below and the connection of
the metric locally bounded, if a square-integrable extension is known to be
possible over a double point at the origin. It is a Hensel-lemma-type result
analogous to Artin's application of the generalized implicit function theorem
to the theory of obstruction in deformation theory. The motivation is the need
in the abundance conjecture to construct pluricanonical sections from flatly
twisted pluricanonical sections. We also give here a new approach to the
original theorem of Ohsawa-Takegoshi by using the hyperbolic geometry of the
punctured open unit 1-disk to reduce the original theorem of Ohsawa-Takegoshi
to a simple application of the standard method of constructing holomorphic
functions by solving the d-bar equation with cut-off functions and additional
blowup weight functions
Assessment of Axial Postural Abnormalities in Parkinsonism: Automatic Picture Analysis Software
BackgroundSoftware-based measurements of axial postural abnormalities in Parkinson's disease (PD) are the gold standard but may be time-consuming and not always feasible in clinical practice. An automatic and reliable software to accurately obtain real-time spine flexion angles according to the recently proposed consensus-based criteria would be a useful tool for both research and clinical practice. ObjectiveWe aimed to develop and validate a new software based on Deep Neural Networks to perform automatic measures of PD axial postural abnormalities. MethodsA total of 76 pictures from 55 PD patients with different degrees of anterior and lateral trunk flexion were used for the development and pilot validation of a new software called AutoPosturePD (APP); postural abnormalities were measured in lateral and posterior view using the freeware NeuroPostureApp (gold standard) and compared with the automatic measurement provided by the APP. Sensitivity and specificity for the diagnosis of camptocormia and Pisa syndrome were assessed. ResultsWe found an excellent agreement between the new APP and the gold standard for lateral trunk flexion (intraclass correlation coefficient [ICC] 0.960, IC95% 0.913-0.982, P < 0.001), anterior trunk flexion with thoracic fulcrum (ICC 0.929, IC95% 0.846-0.968, P < 0.001) and anterior trunk flexion with lumbar fulcrum (ICC 0.991, IC95% 0.962-0.997, P < 0.001). Sensitivity and specificity were 100% and 100% for detecting Pisa syndrome, 100% and 95.5% for camptocormia with thoracic fulcrum, 100% and 80.9% for camptocormia with lumbar fulcrum. ConclusionsAutoPosturePD is a valid tool for spine flexion measurement in PD, accurately supporting the diagnosis of Pisa syndrome and camptocormia
Order in glassy systems
A directly measurable correlation length may be defined for systems having a
two-step relaxation, based on the geometric properties of density profile that
remains after averaging out the fast motion. We argue that the length diverges
if and when the slow timescale diverges, whatever the microscopic mechanism at
the origin of the slowing down. Measuring the length amounts to determining
explicitly the complexity from the observed particle configurations. One may
compute in the same way the Renyi complexities K_q, their relative behavior for
different q characterizes the mechanism underlying the transition. In
particular, the 'Random First Order' scenario predicts that in the glass phase
K_q=0 for q>x, and K_q>0 for q<x, with x the Parisi parameter. The hypothesis
of a nonequilibrium effective temperature may also be directly tested directly
from configurations.Comment: Typos corrected, clarifications adde
On the Bohr inequality
The Bohr inequality, first introduced by Harald Bohr in 1914, deals with
finding the largest radius , , such that holds whenever in the unit disk
of the complex plane. The exact value of this largest radius,
known as the \emph{Bohr radius}, has been established to be This paper
surveys recent advances and generalizations on the Bohr inequality. It
discusses the Bohr radius for certain power series in as well as
for analytic functions from into particular domains. These domains
include the punctured unit disk, the exterior of the closed unit disk, and
concave wedge-domains. The analogous Bohr radius is also studied for harmonic
and starlike logharmonic mappings in The Bohr phenomenon which is
described in terms of the Euclidean distance is further investigated using the
spherical chordal metric and the hyperbolic metric. The exposition concludes
with a discussion on the -dimensional Bohr radius
Present status and advances in bladder pain syndrome: central sensitisation and the urinary microbiome
Zero Order Estimates for Analytic Functions
The primary goal of this paper is to provide a general multiplicity estimate.
Our main theorem allows to reduce a proof of multiplicity lemma to the study of
ideals stable under some appropriate transformation of a polynomial ring. In
particular, this result leads to a new link between the theory of polarized
algebraic dynamical systems and transcendental number theory. On the other
hand, it allows to establish an improvement of Nesterenko's conditional result
on solutions of systems of differential equations. We also deduce, under some
condition on stable varieties, the optimal multiplicity estimate in the case of
generalized Mahler's functional equations, previously studied by Mahler,
Nishioka, Topfer and others. Further, analyzing stable ideals we prove the
unconditional optimal result in the case of linear functional systems of
generalized Mahler's type. The latter result generalizes a famous theorem of
Nishioka (1986) previously conjectured by Mahler (1969), and simultaneously it
gives a counterpart in the case of functional systems for an important
unconditional result of Nesterenko (1977) concerning linear differential
systems. In summary, we provide a new universal tool for transcendental number
theory, applicable with fields of any characteristic. It opens the way to new
results on algebraic independence, as shown in Zorin (2010).Comment: 42 page
A p-adic analogue of Siegel's Theorem on sums of squares
Siegel proved that every totally positive element of a number field K is the sum of four squares, so in particular the Pythagoras number is uniformly bounded across number fields. The p ‐adic Kochen operator provides a p ‐adic analogue of squaring, and a certain localisation of the ring generated by this operator consists of precisely the totally p ‐integral elements of K . We use this to formulate and prove a p ‐adic analogue of Siegel's theorem, by introducing the p ‐Pythagoras number of a general field, and showing that this number is uniformly bounded across number fields. We also generally study fields with finite p ‐Pythagoras number and show that the growth of the p ‐Pythagoras number in finite extensions is bounded
Products of polynomials in many variables
We study the product of two polynomials in many variables, in several norms, and show that under suitable assumptions this product can be bounded from below independently of the number of variables.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/28840/1/0000675.pd
A quantum mechanical model of the Riemann zeros
In 1999 Berry and Keating showed that a regularization of the 1D classical
Hamiltonian H = xp gives semiclassically the smooth counting function of the
Riemann zeros. In this paper we first generalize this result by considering a
phase space delimited by two boundary functions in position and momenta, which
induce a fluctuation term in the counting of energy levels. We next quantize
the xp Hamiltonian, adding an interaction term that depends on two wave
functions associated to the classical boundaries in phase space. The general
model is solved exactly, obtaining a continuum spectrum with discrete bound
states embbeded in it. We find the boundary wave functions, associated to the
Berry-Keating regularization, for which the average Riemann zeros become
resonances. A spectral realization of the Riemann zeros is achieved exploiting
the symmetry of the model under the exchange of position and momenta which is
related to the duality symmetry of the zeta function. The boundary wave
functions, giving rise to the Riemann zeros, are found using the Riemann-Siegel
formula of the zeta function. Other Dirichlet L-functions are shown to find a
natural realization in the model.Comment: 42 pages, 12 figure
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