pinecone/core-2020-05-10-deduplication huggingface dataset:
Geometry of $\mathcal{P}\mathcal{R}$-semi-invariant warped product
  submanifolds in paracosymplectic manifold

The purpose of this paper is to study $\mathcal{P}\mathcal{R}$-semi-invariant
warped product submanifolds of a paracosymplectic manifold $\widetilde{M}$. We
prove that the distributions associated with the definition of
$\mathcal{P}\mathcal{R}$-semi-invariant warped product submanifold $M$ are
always integrable. A necessary and sufficient condition for an isometrically
immersed $\mathcal{P}\mathcal{R}$-semi-invariant submanifold of $\widetilde{M}$
to be a $\mathcal{P}\mathcal{R}$-semi-invariant warped product submanifold is
obtained in terms of the shape operator.Comment: 15 pages. arXiv admin note: text overlap with arXiv:1510.0204
The Critical Point Equation And Contact Geometry

In this paper, we consider the CPE conjecture in the frame-work of
$K$-contact and $(\kappa, \mu)$-contact manifolds. First, we prove that if a
complete $K$-contact metric satisfies the CPE is Einstein and is isometric to a
unit sphere $S^{2n+1}$. Next, we prove that if a non-Sasakian $ (\kappa, \mu)
$-contact metric satisfies the CPE, then $ M^{3} $ is flat and for $ n > 1 $, $
M^{2n+1} $ is locally isometric to $ E^{n+1}\times S^{n}(4)$.Comment: In the published version there was a sign error in Eq. (1.3). We have
  fixed it her
Synthetic foundations of cevian geometry, III: The generalized
  orthocenter

In this paper, the third in the series, we define the generalized orthocenter
$H$ corresponding to a point $P$, with respect to triangle $ABC$, as the unique
point for which the lines $HA, HB, HC$ are parallel, respectively, to $QD, QE,
QF$, where $DEF$ is the cevian triangle of $P$ and $Q=K \circ \iota(P)$ is the
$isotomcomplement$ of $P$, both with respect to $ABC$. We prove a generalized
Feuerbach Theorem, and characterize the center $Z$ of the cevian conic
$\mathcal{C}_P$, defined in Part II, as the center of the affine map $\Phi_P =
T_P \circ K^{-1} \circ T_{P'} \circ K^{-1}$, where $T_P$ is the unique affine
map for which $T_P(ABC)=DEF$; $T_{P'}$ is defined similarly for the isotomic
conjugate $P'=\iota(P)$ of $P$; and $K$ is the complement map. The affine map
$\Phi_P$ fixes $Z$ and takes the nine-point conic $\mathcal{N}_H$ for the
quadrangle $ABCH$ (with respect to the line at infinity) to the inconic
$\mathcal{I}$, defined to be the unique conic which is tangent to the sides of
$ABC$ at the points $D, E, F$. The point $Z$ is therefore the point where the
nine-point conic $\mathcal{N}_H$ and the inconic $\mathcal{I}$ touch. This
theorem generalizes the usual Feuerbach theorem and holds in all cases where
the point $P$ is not on a median, whether the conics involved are ellipses,
parabolas, or hyperbolas, and also holds when $Z$ is an infinite point. We also
determine the locus of points $P$ for which the generalized orthocenter $H$
coincides with a vertex of $ABC$; this locus turns out to be the union of three
conics minus six points. All our proofs are synthetic, and combine affine and
projective arguments.Comment: 34 pages, 7 figure
Deformation of minimal surfaces with planar curvature lines

Minimal surfaces with planar curvature lines are classical geometric objects,
having been studied since the late 19th century. In this paper, we revisit the
subject from a different point of view. After calculating their metric
functions using an analytical method, we recover the Weierstrass data, and give
clean parametrizations for these surfaces. Then, using these parametrizations,
we show that there exists a single continuous deformation between all minimal
surfaces with planar curvature lines. In the process, we establish the
existence of axial directions for these surfaces
Counting Arcs in Projective Planes via Glynn's Algorithm

An $n$-arc in a projective plane is a collection of $n$ distinct points in
the plane, no three of which lie on a line. Formulas counting the number of
$n$-arcs in any finite projective plane of order $q$ are known for $n \le 8$.
In 1995, Iampolskaia, Skorobogatov, and Sorokin counted $9$-arcs in the
projective plane over a finite field of order $q$ and showed that this count is
a quasipolynomial function of $q$. We present a formula for the number of
$9$-arcs in any projective plane of order $q$, even those that are
non-Desarguesian, deriving Iampolskaia, Skorobogatov, and Sorokin's formula as
a special case. We obtain our formula from a new implementation of an algorithm
due to Glynn; we give details of our implementation and discuss its
consequences for larger arcs.Comment: 19 pages, to appear in Journal of Geometr
Moving frames and the characterization of curves that lie on a surface

In this work we are interested in the characterization of curves that belong
to a given surface. To the best of our knowledge, there is no known general
solution to this problem. Indeed, a solution is only available for a few
examples: planes, spheres, or cylinders. Generally, the characterization of
such curves, both in Euclidean ($E^3$) and in Lorentz-Minkowski ($E_1^3$)
spaces, involves an ODE relating curvature and torsion. However, by equipping a
curve with a relatively parallel moving frame, Bishop was able to characterize
spherical curves in $E^3$ through a linear equation relating the coefficients
which dictate the frame motion. Here we apply these ideas to surfaces that are
implicitly defined by a smooth function, $\Sigma=F^{-1}(c)$, by reinterpreting
the problem in the context of the metric given by the Hessian of $F$, which is
not always positive definite. So, we are naturally led to the study of curves
in $E_1^3$. We develop a systematic approach to the construction of Bishop
frames by exploiting the structure of the normal planes induced by the casual
character of the curve, present a complete characterization of spherical curves
in $E_1^3$, and apply it to characterize curves that belong to a non-degenerate
Euclidean quadric. We also interpret the casual character that a curve may
assume when we pass from $E^3$ to $E_1^3$ and finally establish a criterion for
a curve to lie on a level surface of a smooth function, which reduces to a
linear equation when the Hessian is constant.Comment: 22 pages (23 in the published version), 3 figures; this version is
  essentially the same as the published on
Hopf hypersurfaces in spaces of oriented geodesics

A Hopf hypersurface in a (para-)Kaehler manifold is a real hypersurface for
which one of the principal directions of the second fundamental form is the
(para-)complex dual of the normal vector.
  We consider particular Hopf hypersurfaces in the space of oriented geodesics
of a non-flat space form of dimension greater than 2. For spherical and
hyperbolic space forms, the oriented geodesic space admits a canonical
Kaehler-Einstein and para-Kaehler-Einstein structure, respectively, so that a
natural notion of a Hopf hypersurface exists.
  The particular hypersurfaces considered are formed by the oriented geodesics
that are tangent to a given convex hypersurface in the underlying space form.
We prove that a tangent hypersurface is Hopf in the space of oriented geodesics
with respect to this canonical (para-)Kaehler structure iff the underlying
convex hypersurface is totally umbilic and non-flat.
  In the case of 3 dimensional space forms, however, there exists a second
canonical complex structure which can also be used to define Hopf
hypersurfaces. We prove that in this dimension, the tangent hypersurface of a
convex hypersurface in the space form is always Hopf with respect to this
second complex structure.Comment: 10 pages AMS-TE
Interacting Fermi liquid in three dimensions at finite temperature: Part
  I: Convergent Contributions

In this paper we complete the first step, namely the uniform bound on
completely convergent contributions, towards proving that a three dimensional
interacting system of Fermions is a Fermi liquid in the sense of Salmhofer. The
analysis relies on a direct space decomposition of the propagator, on a bosonic
multiscale cluster expansion and on the Hadamard inequality, rather than on a
Fermionic expansion and an angular analysis in momentum space, as was used in
the recent proof by two of us of Salmhofer's criterion in two dimensions.Comment: 85 pages, 13 figure
The Vlasov-Poisson system with radiation damping

We set up and analyze a model of radiation damping within the framework of
continuum mechanics, inspired by a model of post-Newtonian hydrodynamics due to
Blanchet, Damour and Schaefer. In order to simplify the problem as much as
possible we replace the gravitational field by the electromagnetic field and
the fluid by kinetic theory. We prove that the resulting system has a
well-posed Cauchy problem globally in time for general initial data and in all
solutions the fields decay to zero at late times. In particular, this means
that the model is free from the runaway solutions which frequently occur in
descriptions of radiation reaction
Localization for random perturbations of periodic Schroedinger operators
  with regular Floquet eigenvalues

We prove a localization theorem for continuous ergodic Schr\"odinger
operators $ H_\omega := H_0 + V_\omega $, where the random potential $ V_\omega
$ is a nonnegative Anderson-type perturbation of the periodic operator $ H_0$.
We consider a lower spectral band edge of $ \sigma (H_0) $, say $ E= 0 $, at a
gap which is preserved by the perturbation $ V_\omega $. Assuming that all
Floquet eigenvalues of $ H_0$, which reach the spectral edge 0 as a minimum,
have there a positive definite Hessian, we conclude that there exists an
interval $ I $ containing 0 such that $ H_\omega $ has only pure point spectrum
in $ I $ for almost all $ \omega $.Comment: 21 page
Remarks on nonlinear Schroedinger equations with harmonic potential

Bose-Einstein condensation is usually modeled by nonlinear Schroedinger
equations with harmonic potential. We study the Cauchy problem for these
equations. We show that the local problem can be treated as in the case with no
potential. For the global problem, we establish an evolution law, which is the
analogue of the pseudo-conformal conservation law for the nonlinear
Schroedinger equation. With this evolution law, we give wave collapse criteria,
as well as an upper bound for the blow up time. Taking the physical scales into
account, we finally give a lower bound for the blow up time.Comment: 16 pages, no figur
On the Algebra of Fluctuation in Quantum Spin Chains

We present a proof of the central limit theorem for a pair of mutually
non-commuting operators in mixing quantum spin chains. The operators are not
necessarily strictly local but quasi-local. As a corollary we obtain a direct
construction of the time evolution of the algebra of normal fluctuation for
Gibbs states of finite range interactions on a one-dimensional lattice. We show
that the state of the algebra of normal fluctuation satisfies the $\beta$-KMS
condition if the microscopic state is a $\beta$-KMS state
One non-relativistic particle coupled to a photon field

We investigate the ground state energy of an electron coupled to a photon
field. First, we regard the self-energy of a free electron, which we describe
by the Pauli-Fierz Hamiltonian. We show that, in the case of small values of
the coupling constant $\alpha$, the leading order term is represented by
$2\pi^{-1} \alpha (\Lambda - \ln[1 + \Lambda])$.
  Next we put the electron in the field of an arbitrary external potential $V$,
such that the corresponding Schr\"odinger operator $p^2 + V$ has at least one
eigenvalue, and show that by coupling to the radiation field the binding energy
increases, at least for small enough values of the coupling constant $\alpha$.
Moreover, we provide concrete numbers for $\alpha$, the ultraviolet cut-off
$\Lambda$, and the radiative correction for which our procedure works.Comment: final version, to appear in Ann. Henri Poincar
On the topology of vacuum spacetimes

We prove that there are no restrictions on the spatial topology of
asymptotically flat solutions of the vacuum Einstein equations in
(n+1)-dimensions. We do this by gluing a solution of the vacuum constraint
equations on an arbitrary compact manifold to an asymptotically Euclidean
solution of the constraints on R^n. For any compact manifold which does not
admit a metric of positive scalar curvature, this provides for the existence of
asymptotically flat vacuum spacetimes with no maximal slices. Our main theorem
is a special case of a more general gluing construction for nondegenerate
solutions of the vacuum constraint equations which have some restrictions on
the mean curvature, but for which the mean curvature is not necessarily
constant. This generalizes the construction [IMP] (gr-qc/0109045), which is
restricted to constant mean curvature data.Comment: 14 pages, v2 is a substantial revision of the previous version:
  superfluous condition removed from main theorem and applications to the
  existence of spacetimes with no maximal Cauchy surfaces adde
Generalized free fields and the AdS-CFT correspondence

Motivated by structural issues in the AdS-CFT correspondence, the theory of
generalized free fields is reconsidered. A stress-energy tensor for the
generalized free field is constructed as a limit of Wightman fields. Although
this limit is singular, it fulfils the requirements of a conserved local
density for the Poincar'e generators. An explicit "holographic" formula
relating the Klein-Gordon field on AdS to generalized free fields on Minkowski
space-time is provided, and contrasted with the "algebraic" notion of
holography. A simple relation between the singular stress-energy tensor and the
canonical AdS stress-energy tensor is exhibited.Comment: 25 page
An indefinite metric model for interacting quantum fields with
  non-stationary background gravitation

We consider a relativistic Ansatz for the vacuum expectation values (VEVs) of
a quantum field on a globally hyperbolic space-time which is motivated by
certain Euclidean field theories. The Yang-Feldman asymptotic condition w.r.t.
a "in"-field in a quasi-free representation of the canonic commutation
relations (CCR) leads to a solution of this Ansatz for the VEVs. A GNS-like
construction on a non-degenerate inner product space then gives local,
covariant quantum fields with indefinite metric on a globally hyperbolic
space-time. The non-trivial scattering behavior of quantum fields is analyzed
by construction of the "out"-fields and calculation of the scattering matrix. A
new combined effect of non-trivial quantum scattering and non-stationary
gravitational forces is described for this model, as quasi-free "in"- fields
are scattered to "out"-fields which form a non quasi-free representations of
the CCR. The asymptotic condition, on which the construction is based, is
verified for the concrete example of de Sitter space-time
Dissipative Transport: Thermal Contacts and Tunnelling Junctions

The general theory of simple transport processes between quantum mechanical
reservoirs is reviewed and extended. We focus on thermoelectric phenomena,
involving exchange of energy and particles. Entropy production and Onsager
relations are relevant thermodynamic notions which are shown to emerge from the
microscopic description. The theory is illustrated on the example of two
reservoirs of free fermions coupled through a local interaction. We construct a
stationary state and determine energy- and particle currents with the help of a
convergent perturbation series.
  We explicitly calculate several interesting quantities to lowest order, such
as the entropy production, the resistance, and the heat conductivity.
Convergence of the perturbation series allows us to prove that they are
strictly positive under suitable assumptions on the interaction between the
reservoirs.Comment: 55 pages; 2 figure
Charge Superselection Sectors for Scalar QED on the Lattice

The lattice model of scalar quantum electrodynamics (Maxwell field coupled to a complex scalar field) in the Hamiltonian framework is discussed. It is shown that the algebra of observables ${\cal O}({\Lambda})$ of this model is a $C^*$-algebra, generated by a set of gauge-invariant elements satisfying the Gauss law and some additional relations. Next, the faithful, irreducible and non-degenerate representations of ${\cal O}({\Lambda})$ are found. They are labeled by the value of the total electric charge, leading to a decomposition of the physical Hilbert space into charge superselection sectors. In the Appendices we give a unified description of spinorial and scalar quantum electrodynamics and, as a byproduct, we present an interesting example of weakly commuting operators, which do not commute strongly
Lattice QCD -- from quark confinement to asymptotic freedom

According to the present understanding, the observed diversity of the strong
interaction phenomena is described by Quantum Chromodynamics, a gauge field
theory with only very few parameters. One of the fundamental questions in this
context is how precisely the world of mesons and nucleons is related to the
properties of the theory at high energies, where quarks and gluons are the
important degrees of freedom. The lattice formulation of QCD combined with
numerical simulations and standard perturbation theory are the tools that allow
one to address this issue at a quantitative level.Comment: Plenary talk, International Conference on Theoretical Physics, Paris,
  UNESCO, 22--27 July 2002; TeX source, 15 pages, figures include
The New Cosmology: Mid-term Report Card for Inflation

Inflation has been the driving idea in cosmology for two decades and is a pillar of the New Cosmology. The inflationary paradigm has now passed its first round of significant tests, with two of its three basics predictions confirmed at about the 10% level. The Inflationary Paradigm has some of the truth. Over the next decade the precision of these tests, most of which involve measurements of CMB anisotropy and polarization, will improve 30 fold or more(!), testing inflation more sharply and possibly elucidating the underlying cause. Especially important in this regard is detecting the inflation-produced gravitational waves, either directly or through their CMB polarization signature. While inflation has by no means been verified, its successes have raised the bar for competitor theories: Any alternative must feature the two hallmarks of inflation: superluminal expansion and entropy production
Conformal Invariance in Percolation, Self-Avoiding Walks and Related
  Problems

Over the years, problems like percolation and self-avoiding walks have
provided important testing grounds for our understanding of the nature of the
critical state. I describe some very recent ideas, as well as some older ones,
which cast light both on these problems themselves and on the quantum field
theories to which they correspond. These ideas come from conformal field
theory, Coulomb gas mappings, and stochastic Loewner evolution.Comment: Plenary talk given at TH-2002, Paris. 21 pages, 9 figure
Dimensional Reduction and Crossover to Mean-Field Behavior for Branched
  Polymers

This article will review recent results on dimensional reduction for branched
polymers, and discuss implications for critical phenomena. Parisi and Sourlas
argued in 1981 that branched polymers fall into the universality class of the
Yang-Lee edge in two fewer dimensions. Brydges and I have proven in
[math-ph/0107005] that the generating function for self-avoiding branched
polymers in D+2 continuum dimensions is proportional to the pressure of the
hard-core continuum gas at negative activity in D dimensions (which is in the
Yang-Lee or $i \phi^3$ class). I will describe how this equivalence arises from
an underlying supersymmetry of the branched polymer model.
  -
  I will also use dimensional reduction to analyze the crossover of
two-dimensional branched polymers to their mean-field limit, and to show that
the scaling is given by an Airy function (the same as in [cond-mat/0107223]).Comment: 15 pages, 1 eps figur
Many Body Physics: Unfinished Revolution

The study of many body physics has provided a scientific playground of
surprise and continuing revolution over the past half century. The
serendipitous discovery of new states and properties of matter, phenomena such
as superfluidity, the Meissner, the Kondo and the fractional quantum hall
effect, have driven the development of new conceptual frameworks for our
understanding about collective behavior, the ramifications of which have spread
far beyond the confines of terrestrial condensed matter physics- to cosmology,
nuclear and particle physics. Here I shall selectively review some of the
developments in this field, from the cold-war period, until the present day. I
describe how, with the discovery of new classes of collective order, the
unfolding puzzles of high temperature superconductivity and quantum
criticality, the prospects for major conceptual discoveries remain as bright
today as they were more than half a century ago.Comment: Write up of talk presented at TH-2002, July 2002, Paris. Various
  corrections adde
Exact reformulation of the bosonic many-body problem in terms of
  stochastic wave functions: convergence issues

There exist methods to reformulate in an exact way the many-body problem of
interacting bosons in terms of the stochastic evolution of single particle wave
functions. For one such reformulation, the so-called simple Fock scheme, we
present an elementary derivation, much simpler than the original one.
Furthermore, we show that two other schemes, based on coherent states of the
matter field rather than on Fock states, lead to an infinite statistical
uncertainty in the continuous time limit. The simple Fock scheme is therefore,
up to now, the only one that was proved to lead to a convergent Monte Carlo
simulation scheme at all times.Comment: Proceedings of the Laser Physics Workshop held in Bratislava, July
  2002. Submitted to Laser Physic
Applications of Ideas from Random Matrix Theory to Step Distributions on
  "Misoriented" Surfaces

Arising as a fluctuation phenomenon, the equilibrium distribution of
meandering steps with mean separation $<\ell>$ on a "tilted" surface can be
fruitfully analyzed using results from RMT. The set of step configurations in
2D can be mapped onto the world lines of spinless fermions in 1+1D using the
Calogero-Sutherland model. The strength of the ("instantaneous",
inverse-square) elastic repulsion between steps, in dimensionless form, is
$\beta(\beta-2)/4$. The distribution of spacings $s< \ell>$ between neighboring
steps (analogous to the normalized spacings of energy levels) is well described
by a {\it "generalized" Wigner surmise}: $p_{\beta}(0,s) \approx a
s^{\beta}\exp(-b s^2)$. The value of $\beta$ is taken to best fit the data;
typically $2 \le \beta \le 10$. The procedure is superior to conventional
Gaussian and mean-field approaches, and progress is being made on formal
justification. Furthermore, the theoretically simpler step-step distribution
function can be measured and analyzed based on exact results. Formal results
and applications to experiments on metals and semiconductors are summarized,
along with open questions. (conference abstract)Comment: 7 pages, 2 figures; based on talk presented at TH-2002, UNESCO,
  Paris, July 2002; to be published in Ann. Henri Poincare
New applications of non-hermitian random matrices

We discuss recently discovered links of the statistical models of normal
random matrices to some important physical problems of pattern formation and to
the quantum Hall effect. Specifically, the large $N$ limit of the normal matrix
model with a general statistical weight describes dynamics of the interface
between two incompressible fluids with different viscousities in a thin plane
cell (the Saffman-Taylor problem). The latter appears to be mathematically
equivalent to the growth of semiclassical 2D electronic droplets in a strong
uniform magnetic field with localized magnetic impurities (fluxes), as the
number of electrons increases. The equivalence is most easily seen by relating
the both problems to the matrix model.Comment: 10 pages, 3 figures, Talk given at TH-2002, Paris, UNESCO, July 200