1,036 research outputs found
Conformal Field Theories in Fractional Dimensions
We study the conformal bootstrap in fractional space-time dimensions,
obtaining rigorous bounds on operator dimensions. Our results show strong
evidence that there is a family of unitary CFTs connecting the 2D Ising model,
the 3D Ising model, and the free scalar theory in 4D. We give numerical
predictions for the leading operator dimensions and central charge in this
family at different values of D and compare these to calculations of phi^4
theory in the epsilon-expansion.Comment: 11 pages, 4 figures - references updated - one affiliation modifie
A discrete Laplace-Beltrami operator for simplicial surfaces
We define a discrete Laplace-Beltrami operator for simplicial surfaces. It
depends only on the intrinsic geometry of the surface and its edge weights are
positive. Our Laplace operator is similar to the well known finite-elements
Laplacian (the so called ``cotan formula'') except that it is based on the
intrinsic Delaunay triangulation of the simplicial surface. This leads to new
definitions of discrete harmonic functions, discrete mean curvature, and
discrete minimal surfaces. The definition of the discrete Laplace-Beltrami
operator depends on the existence and uniqueness of Delaunay tessellations in
piecewise flat surfaces. While the existence is known, we prove the uniqueness.
Using Rippa's Theorem we show that, as claimed, Musin's harmonic index provides
an optimality criterion for Delaunay triangulations, and this can be used to
prove that the edge flipping algorithm terminates also in the setting of
piecewise flat surfaces.Comment: 18 pages, 6 vector graphics figures. v2: Section 2 on Delaunay
triangulations of piecewise flat surfaces revised and expanded. References
added. Some minor changes, typos corrected. v3: fixed inaccuracies in
discussion of flip algorithm, corrected attributions, added references, some
minor revision to improve expositio
Functional Maps Representation on Product Manifolds
We consider the tasks of representing, analyzing and manipulating maps
between shapes. We model maps as densities over the product manifold of the
input shapes; these densities can be treated as scalar functions and therefore
are manipulable using the language of signal processing on manifolds. Being a
manifold itself, the product space endows the set of maps with a geometry of
its own, which we exploit to define map operations in the spectral domain; we
also derive relationships with other existing representations (soft maps and
functional maps). To apply these ideas in practice, we discretize product
manifolds and their Laplace--Beltrami operators, and we introduce localized
spectral analysis of the product manifold as a novel tool for map processing.
Our framework applies to maps defined between and across 2D and 3D shapes
without requiring special adjustment, and it can be implemented efficiently
with simple operations on sparse matrices.Comment: Accepted to Computer Graphics Foru
Discrete complex analysis on planar quad-graphs
We develop a linear theory of discrete complex analysis on general
quad-graphs, continuing and extending previous work of Duffin, Mercat, Kenyon,
Chelkak and Smirnov on discrete complex analysis on rhombic quad-graphs. Our
approach based on the medial graph yields more instructive proofs of discrete
analogs of several classical theorems and even new results. We provide discrete
counterparts of fundamental concepts in complex analysis such as holomorphic
functions, derivatives, the Laplacian, and exterior calculus. Also, we discuss
discrete versions of important basic theorems such as Green's identities and
Cauchy's integral formulae. For the first time, we discretize Green's first
identity and Cauchy's integral formula for the derivative of a holomorphic
function. In this paper, we focus on planar quad-graphs, but we would like to
mention that many notions and theorems can be adapted to discrete Riemann
surfaces in a straightforward way.
In the case of planar parallelogram-graphs with bounded interior angles and
bounded ratio of side lengths, we construct a discrete Green's function and
discrete Cauchy's kernels with asymptotics comparable to the smooth case.
Further restricting to the integer lattice of a two-dimensional skew coordinate
system yields appropriate discrete Cauchy's integral formulae for higher order
derivatives.Comment: 49 pages, 8 figure
Simplicity of extremal eigenvalues of the Klein-Gordon equation
We consider the spectral problem associated with the Klein-Gordon equation
for unbounded electric potentials. If the spectrum of this problem is contained
in two disjoint real intervals and the two inner boundary points are
eigenvalues, we show that these extremal eigenvalues are simple and possess
strictly positive eigenfunctions. Examples of electric potentials satisfying
these assumptions are given
On Equivalence of Duffin-Kemmer-Petiau and Klein-Gordon Equations
A strict proof of equivalence between Duffin-Kemmer-Petiau (DKP) and
Klein-Gordon (KG) theories is presented for physical S-matrix elements in the
case of charged scalar particles interacting in minimal way with an external or
quantized electromagnetic field. First, Hamiltonian canonical approach to DKP
theory is developed in both component and matrix form. The theory is then
quantized through the construction of the generating functional for Green
functions (GF) and the physical matrix elements of S-matrix are proved to be
relativistic invariants. The equivalence between both theories is then proved
using the connection between GF and the elements of S-matrix, including the
case of only many photons states, and for more general conditions - so called
reduction formulas of Lehmann, Symanzik, Zimmermann.Comment: 23 pages, no figures, requires macro tcilate
Equation of the field lines of an axisymmetric multipole with a source surface
Optical spectropolarimeters can be used to produce maps of the surface magnetic fields of stars and hence to determine how stellar magnetic fields vary with stellar mass, rotation rate, and evolutionary stage. In particular, we now can map the surface magnetic fields of forming solar-like stars, which are still contracting under gravity and are surrounded by a disk of gas and dust. Their large scale magnetic fields are almost dipolar on some stars, and there is evidence for many higher order multipole field components on other stars. The availability of new data has renewed interest in incorporating multipolar magnetic fields into models of stellar magnetospheres. I describe the basic properties of axial multipoles of arbitrary degree â and derive the equation of the field lines in spherical coordinates. The spherical magnetic field components that describe the global stellar field topology are obtained analytically assuming that currents can be neglected in the region exterior to the star, and interior to some fixed spherical equipotential surface. The field components follow from the solution of Laplaceâs equation for the magnetostatic potential
Comment on ``the Klein-Gordon Oscillator''
The different ways of description of the particle with oscillator-like
interaction are considered. The results are in conformity with the previous
paper of S. Bruce and P. Minning.Comment: LaTeX file, 5p
Association Between p.Leu54Met Polymorphism at the Paraoxonase-1 Gene and Plantar Fascia Thickness in Young Subjects With Type 1 Diabetes
OBJECTIVEâ In type 1 diabetes, plantar fascia, a collagen-rich tissue, is susceptible to glycation and oxidation. Paraoxonase-1 (PON1) is an HDL-bound antioxidant enzyme. PON1 polymorphisms have been associated with susceptibility to macro- and microvascular complications. We investigated the relationship between plantar fascia thickness (PFT) and PON1 gene variants, p.Leu54Met, p.Gln192Arg, and c.-107C>T, in type 1 diabetes
Solving the 3D Ising Model with the Conformal Bootstrap
We study the constraints of crossing symmetry and unitarity in general 3D
Conformal Field Theories. In doing so we derive new results for conformal
blocks appearing in four-point functions of scalars and present an efficient
method for their computation in arbitrary space-time dimension. Comparing the
resulting bounds on operator dimensions and OPE coefficients in 3D to known
results, we find that the 3D Ising model lies at a corner point on the boundary
of the allowed parameter space. We also derive general upper bounds on the
dimensions of higher spin operators, relevant in the context of theories with
weakly broken higher spin symmetries.Comment: 32 pages, 11 figures; v2: refs added, small changes in Section 5.3,
Fig. 7 replaced; v3: ref added, fits redone in Section 5.
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