|
Software installation/poster session/reception |
| Abstract: Software developers will offer their software for installation
on the laptops of the participants at various booths. |
| Daniel J. Bates (University of Minnesota Twin Cities) |
Introduction to Bertini: a software package for numerical algebraic geometry
|
| Abstract: Bertini is a new software package for computation in the field of numerical
algebraic geometry. Among other things, Bertini is capable of producing all
complex isolated solutions of a given polynomial as well as points on each
positive-dimensional irreducible component. Bertini makes use of adaptive
multiprecision, singular endgames, straight-line programs, and several other
useful tools. This talk will cover some of the capabilities of Bertini as well
as some of the implementation details. |
| Anna M. Bigatti (Università di Genova) |
CoCoALib, a C++ library for computations in commutative algebra |
| Abstract: For almost 20 years the CoCoA project has been conducting research into
computational commutative algebra, developing new algorithms and offering
implementations in the interactive program "CoCoA." Recently we took the
decision to rebuild the software from scratch with the specific aim of
making excellent implementations available to all researchers. The
implementations will be accessible in three distinct ways:
as a standalone interactive system, as a networked service (via an
OpenMath-like communications channel), as a C++ library, called
CoCoALib (free and open in the sense of the GPL).
CoCoALib, being the core of the project, is also its most evolved
part, and is the part that we shall look at most closely. |
| Allan Boardman (University of Salford) |
Radiation enhancement and radiation suppression by a left-handed
metamaterial |
| Abstract: Joint work with K. Marinov
(Photonics and Nonlinear Science Group, Joule Laboratory, Department of
Physics, University of Salford, Salford M5 4WT, UK).
It is shown that the perfect lens property of the left-handed metamaterials
can be exploited to control the radiation efficiency of an electromagnetic
radiation source (e.g. an antenna). In particular, the radiation
characteristics of two identical sources, in the focal planes of the lens
can be controlled depending on the relative phase difference between their
feeding voltages. When the feeding voltages are pi-out-of-phase the
resulting system behaves as a non-radiating configuration with a strong
electromagnetic field confined in the space between the lens and the
emitters and almost no electromagnetic radiation emitted. It is shown that
such a system can be used as a very sensitive detector since any object
disturbing the configuration of the electromagnetic fields inside the system
stimulates radiation. Even objects of subwavelength dimensions are able to
produce a substantial increase of the total power emitted by the system, and
thus their presence can be revealed. The finite-difference time-domain
(FDTD) numerical analysis performed allows a realistic system performance
evaluation to be made. It is shown that if a pair of identical sources
driven with in-phase feeding voltages are used in the same resonant
configuration this results in an increase of the radiation resistance of
each of the sources. The latter property can be useful for small antennas. |
| Nader Engheta (University of Pennsylvania) |
Metamaterials, plasmonics and optical nanocircuits |
| Abstract: Metamaterials, which are engineered composite media with unconventional electromagnetic and optical properties, can be formed by
embedding sub-wavelength inclusions as artificial molecules in host media in order to exhibit specific desired response
functions. They can have exciting characteristics in manipulating and processing RF, microwave, IR and optical signal
information. Various features of these media are being investigated and some of the fundamental concepts and theories and modeling
of wave interaction with a variety of structures and systems involving these material media are being developed. From our
analyses and simulations, we have found that the devices and components formed by these media may be ultracompact and
subwavelength, while supporting resonant and propagating modes. This implies that in such structures RF, microwave, IR and optical
signals can be controlled and reshaped beyond the diffraction limits, leading to the possibility of miniaturization of optical
interconnects and design and control of near-field devices and processors for the next generation of information technology. This
may also lead to nano-architectures capable of signal processing in the near-field optics, which has the potential for significant
size reduction in information processing and storage. Furthermore, the nanostructures made by pairing these media can be compact
resonant components, resulting in either enhanced wave signatures and higher directivity or in transparency and scattering
reduction. We are also interested in nano-optics of metamaterial structures that effectively act as lumped
nano-circuit-elements. These may provide nano-inductors, nano-capacitors, nano-resistors, and nanodiodes as part of field
nanocircuits in the optical regimes or optical-field nanoelectronics--, and can provide roadmaps to more complex nanocircuits
and systems formed by collection of such nanostructures. All these characteristics may offer various potential applications in
high-resolution near-field imaging and microscopy, enhancement or reduction of wave interaction with nano-particles and
nano-apertures, nanoantennas and arrays, far-field sub-diffraction optical microscopy (FSOM), nano-circuit-filters, optical data
storage, nano-beam patterning and spectroscopy, optical-molecular signaling and optical coupling and interfacing with cells, to
name a few. In this talk, we present an overview of the concepts, salient features, recent developments, and some of the
potential applications of these metamaterials and structures, and will forecast some futures ideas and directions in this area. |
| Alexander Figotin (University of California) |
Abnormal refraction of EM waves in periodic metamaterials |
| Abstract: Joint work with I Vitebskiy.
Wave propagation in spatially periodic media, such as photonic crystals,
can be qualitatively different
from any uniform substance. The differences are particularly pronounced
when the electromagnetic wavelength
is comparable to the minimal translation of the periodic structure. In
such a case, the periodic medium cannot
be assigned any meaningful refractive index. Still, such important
features as negative refraction and/or
opposite phase and group velocities for certain directions of light
propagation can be found in almost any
photonic crystal. The only reservation is that unlike hypothetical
uniform left-handed media, photonic crystals
are essentially anisotropic at frequency range of interest. Consider now
a plane wave incident on a semi-infinite photonic crystal. One can
assume, for instance, that in the case of positive refraction,
the normal components of the group and the phase velocities of the
transmitted Bloch wave have the same sign,
while in the case of negative refraction, those components have opposite
signs. What happens if the normal
component of the transmitted wave group velocity vanishes? Let us call
it a "zero-refraction" case.
At first sight, zero normal component of the transmitted wave group
velocity implies total reflection of the
incident wave. But we demonstrate that total reflection is not the only
possibility. Instead, the transmitted
wave can appear in the form of an abnormal grazing mode with huge
amplitude and nearly tangential group
velocity. This spectacular phenomenon is extremely sensitive to the
frequency and direction of propagation of
the incident plane wave. We also discuss some possible applications of
this effect.
REFERENCES:
- A. Figotin, and I. Vitebskiy. Phys. Rev. E68, 036609 (2003).
- J. Ballato, A. Ballato, A. Figotin, and I. Vitebskiy. Phys. Rev. E71,
(2005). |
| Anne Frühbis-Krüger (Universität Kaiserslautern) |
Resolution of singularities from the practical point of view
|
| Abstract: In the 1960s H.Hironaka proved the existence of a resolution
of singularities in arbitrary dimension over a field of
characteristic zero; but it was not before the late 1980s/early 1990s
that the first algorithmic approaches to this task appeared
which were at that time considered to be purely of theoretical
use. By now there are two independent implementations and in
this talk I would like to focus on the computational issues
and modifications to the algorithm which are necessary to keep
the very high time and memory consumption of the algorithm as
low as possible and hence make it usable for non-trivial examples.
|
| Rachel S. Goldman (University of Michigan) |
Directed seeding of three-dimensional metal-semiconductor
nanocomposites for negative index metamaterials |
| Abstract: Negative index of refraction materials (NIMs) are promising for
several applications including near-field imaging and steering of EM
radiation. Although NIMs have been demonstrated using hybrid
metamaterials at microwave frequencies, high losses and narrow
bandwidths are presently limiting their wide application. We are
developing a novel approach to fabricating low-loss high density NIM
semiconductor-metal nanocomposites, which consists of alternating
sequences of focused-ion beam nanopatterning of metallic droplet
arrays and film growth using molecular-beam epitaxy. We will
discuss the formation and ordering of Ga and In droplets and droplet
motifs on a variety of semiconductor surfaces. In addition, we will
discuss the extension of this approach to 3D. In particular,
information from scattering measurements of 1D and 2D droplet motifs
will be input into theoretical NIMs calculations to guide the
fabrication of 3D arrays of appropriate motifs. |
| Anand Gopinath (University of Minnesota Twin Cities), Jaewon Kim (University of Minnesota Twin Cities) |
Novel metamaterial using Cubic high dielectric resonators |
| Abstract: Simulations have been performed on a novel
metamaterial structure generated by periodic
placement of identical high dielectric cubic
resonators, in a low dielectric background. These
resonators have degenerate modes, which implies
that the TE and TM modes are resonant at the same
frequency. Negative index behavior is deduced
from these simulations near their resonant
frequency. The periodic cubic structure with
these high dielectric resonators results in a
metamaterial, without any plasmonic metallic
material, and should be low loss. |
| Daniel R. Grayson (University of Illinois at Urbana-Champaign) |
Macaulay 2, a software system for algebraic geometry |
| Abstract: Macaulay 2 supports research in algebraic geometry and commutative
algebra. Its versatile framework, based on Buchberger's algorithm for
computing Groebner bases, combined with an object-oriented interpreted language
supporting the introduction of new high-level mathematical types, allows
advanced algorithms to be coded. I'll demonstrate it, and I will describe
recent improvements aimed to support third-party development of code, including
the package system, the debugger, and the documentation processor. |
| Anthony Grbic (University of Michigan) |
Negative refractive index metamaterials based on transmission
lines |
| Abstract: This talk will describe negative refractive index metamaterials that are
based on transmission-line networks. It will focus on microwave structures
that consist of transmission lines loaded with reactive elements. Both
planar and volumetric negative refractive index metamaterials will be
presented and their operation explained. Finally, ways to push these
transmission-line based structures to optical frequencies using plasmonic
materials will be described. |
| Leslie F. Greengard (New York University) |
Simulation environments for electromagnetic scattering |
| Abstract: We will review the analytic and computational foundations of
Green's function-based methods for electromagnetic scattering,
including high order integral representations, fast solvers,
and quasi-periodicity. We will then discuss the development of
easy-to-use numerical simulation environments, and present some
applications to photonic crystals, random microstructures, and
negative index materials. |
| Raymond Hemmecke (Otto-von-Guericke-Universität Magdeburg) |
4ti2 — Computation of Hilbert bases, Graver bases, toric Gröbner bases
and more |
| Abstract: Hilbert bases, Graver bases and toric Gröbner bases are at the heart of
many problems arising in mathematics or in practice. In this talk we
present the main functionality and the algorithmic theory behind the
software package 4ti2. Furthermore, we present applications (theoretical
and computational) from various mathematical fields such as toric
algebra, integer programming or statistics.
Within this workshop, this talk is accompanied by a tutorial on 4ti2 by
Peter Malkin. |
| Evelyne Hubert (Institut National de Recherche en Informatique Automatique (INRIA)) |
Algebra & algorithms for differential elimination & completion |
Abstract: Differential algebra provides an algebraic viewpoint on nonlinear
differential systems.
The motivating questions for this talk are:
- How do we define the general solution of a nonlinear equations
- What are the conditions for a differential system to have a solution
- How do we measure the "degrees of freedom" for the solution set of a
differential system
Theory and algorithms for those are extensions of commutative algebra
(prime ideal decomposition, Hilbert polynomials) and Groebner bases
techniques.
The library diffalg in Maple supports this introduction to constructive
differential algebra.
It has been developed by F. Boulier (1996) and the speaker afterwards.
A recent extension of differential algebra to non-commutative derivations,
and its implementation in diffalg, allow to treat systems bearing on
differential invariants. |
| Zubin Jacob (Princeton University) |
Optical hyperlens : Far-field imaging beyond the
diffraction limit
|
| Abstract: Joint work with Leonid V. Alekseyev and Evgenii Narimanov.
We propose an approach to far-field optical imaging
beyond the
diffraction limit. The proposed system allows image
magnification,
is robust with respect to material losses and can be fabricated
by
adapting existing metamaterial technologies in a cylindrical
geometry.
|
| Anders Nedergaard Jensen (Aarhus University) |
Computing tropical varieties |
| Abstract: The tropical variety of a polynomial ideal I in n variables over Q is a
polyhedral complex in n-dimensional space. We may consider it as a
subfan of the Groebner fan of I. The polyhedral cones in the Groebner
fan can be computed using Groebner bases and by applying "Groebner walk"
techniques. This gives one method for computing the tropical variety of
I. We show how the method can be refined by applying a connectivity
result for tropical varieties of prime ideals and an algorithm for
constructing tropical bases of curves. The presented algorithms have
been implemented in the software package Gfan.
This is joint work with
T. Bogart, K. Fukuda, D. Speyer, B. Sturmfels and R. Thomas.
|
| Masakazu Kojima (Tokyo Institute of Technology) |
Parallel implementation of the polyhedral homotopy method for
polynomial systems |
| Abstract: The polyhedral homotopy method is known to be a powerful numerical method for
approximating all isolated solutions of a system of polynomial equations.
We discuss a parallel implementation of the polyhedral homotopy method,
a dynamic enumeration of all fine mixed cells which is used in constructing a family of
polyhedral homotopy functions and extensions of the Hornor Scheme to multivariate
polynomials for efficient evaluation of a system of polynomials and their partial
derivatives in the polyhedral homotopy method. |
| Akhlesh Lakhtakia (Pennsylvania State University) |
Twisted material gains |
| Abstract: A twisting and turning tale promises unimaginable
gains for the savvy investor of time and effort in metamaterials research.
|
| Grégoire Lecerf (Université Versailles/Saint Quentin-en-Yvelines) |
New recombination techniques for polynomial factorization
algorithms based on Hensel lifting |
| Abstract: Multivariate polynomial factorization algorithms are necessary
ingredients to several tasks in computational algebraic geometry such
as prime and primary decompositions. They are also extremely useful in
many places where they are not necessary but where they lead to major
speedups by splitting problems into several smaller ones.
In this talk I will present recent results concerning the
factorization reductions from several to two variables via Bertini's
theorem, and then from two to one variable. These reductions are based
on the very classical Hensel lifting strategy and new fast
recombination devices. Over a prime coefficient field, I will show
that the irreducible factorization of a bivariate polynomial of
bidegree (m,n) roughly reduces to the factorization of a univariate
polynomial of degree n, a Hensel lifting to precision m+1, and O~( m
n^2 ) arithmetic operations to recombine the lifted factors.
Finally we will report on practical performances of the new
algorithms, and on comparisons with other software. |
| Anton Leykin (University of Minnesota Twin Cities) |
D-modules for Macaulay 2 |
| Abstract: The package D-modules for Macaulay 2 implements the majority of
the now classical algorithms in the computational D-module theory. Based
on the ability of Macaulay 2 engine to compute Gröbner bases in
the Weyl algebra, the package provides, in particular, tools to work with
holonomic D-modules such as the algorithms for b-functions, localized
modules, restriction, etc. Amongst the applications there are computation
of the local cohomology modules, polynomial and rational solutions, and
A-hypergeometric systems.
(Joint work with Harry Tsai) |
| Jichun Li (University of Nevada) |
Error analysis of mixed finite element methods for
wave propagation in double negative metamaterials |
| Abstract: In this paper, we develop both semi-discrete and fully-discrete
mixed finite element methods for modeling wave propagation in three-dimensional
double negative metamaterials. Optimal error estimates are proved for Nedelec
spaces under the assumption of smooth solutions.
To our best knowledge, this is the first error analysis obtained for Maxwell's
equations when metamaterials are involved.
|
| Peijun Li (University of Michigan) |
A boundary integral method and adaptive treecode for the linear
Poisson-Boltzmann equation |
| Abstract: Joint work with Robert Krasny.
A boundary integral method (BIM) is developed for computing the
electrostatic potential of biomolecules governed by the linear
Poisson--Boltzmann equation (PBE). Compared with finite difference
method and finite element method, the BIM provides a rigorous
treatment on issues of the singular charges, the solute-solvent
interfaces, and the infinite domain associated with the PBE.
However, the BIM involves singular kernels. Their accurate
integration is an important issues. Rather than investing in the
development of complicated quadratures, we employ simple
regularization techniques to evaluate surface integrals with
regularized kernels. Furthermore, the high computational cost
incurred in the conventional BIM is reduced by using an adaptive
treecode algorithm based on Taylor approximation in Cartesian
coordinates, and necessary Taylor coefficients are computed by
recurrence relations. Numerical experiments are included to show the
efficiency and accuracy of the proposed method. |
| Robert P. Lipton (Louisiana State University) |
Optimization and control of energy transmission across
photonic crystal slabs |
| Abstract: A variational approach is developed for the design of defects
within a
two-dimensional lossless photonic crystal slab to create and
manipulate
the location of high Q transmission spikes within band gaps.
This phenomena is connected to the appearance of resonant
behavior within
the slab for certain crystal defects. The methodology is
applied to design
crystals constructed from circular dielectric rods embedded in
a
contrasting dielectric medium. This is joint work with Stephen
Shipman and
Stephanos Venakides.
|
| Natalia Litchinitser (University of Michigan) |
Nonlinear transmission in layered structures containing thin
film of negative index material |
| Abstract: Co-authors: Ildar R. Gabitov, Andrei I. Maimistov, and Vladimir M. Shalaev.
We investigate analytically and numerically nonlinear transmission in
a bilayer structure consisting of a slab of positive index material
with Kerr-type nonlinearity and a thin layer of negative index
material (NIM). We find that a sub-wavelength layer of NIM
significantly modifies the bistable nonlinear transmission
characteristics of the considered bilayer structure and leads to
nonreciprocal transmission with enhanced operational range,
potentially enabling novel photonic devices such as optical diodes.
The demonstrated high sensitivity of the nonlinear response of the
structure to the material parameters of NIMs suggests that optical
bistability in these structures has a strong potential for developing
new tools for NIM characterization. |
| Graeme Milton (University of Utah) |
Cloaking and opaque perfect lenses |
| Abstract: We show how a slightly lossy superlens of thickness d cloaks
collections of polarizable line dipoles or point dipoles or finite
energy dipole sources that lie
within a distance of d/2 of the lens. In the limit as the loss
in the lens tends to zero, these become essentially invisible
from the outside through the cancelling effects of localized resonances
generated by the interaction of the source and the superlens. The
lossless perfect Veselago lens has
attracted a lot of debate. It is shown that as time
progresses the lens becomes increasingly opaque
to any physical dipole source located within a distance d/2 from
the lens and which has been turned on at time t=0. Here a physical
source is defined as one which supplies a
bounded amount of energy per unit time. In fact the lens cloaks the source
so that it is not visible from behind the lens either. For sources which
are turned on exponentially slowly there is an exact correspondence
between the response of the perfect lens in the long time constant limit
and the response of lossy lenses in the low loss limit. This is joint
work with Nicolae Nicorovici and Ross McPhedran. |
| Bernard Mourrain (Institut National de Recherche en Informatique Automatique (INRIA)) |
SYNAPS, a library for symbolic-numeric computation |
| Abstract: SYNAPS (SYmbolic Numeric APplications) is a C++ library devoted to symbolic
and numeric computations. It provides data-structures for the manipulation of
basic algebraic objects, such as vectors, matrices (dense, sparse,
structured), univariate and multivariate polynomials. It contains solvers
for univariate and multivariate polynomials, including generalized normal form or subdivision solvers, tools for the manipulatiion of algebraic numbers, for the construction of resultants, ... In this talk, we will
describe shortly its design, its main features and functionalities, show some applications in computational geometry for algebraic curves and surfaces and explain how it can be embbeded in an interpreter such as mathemagix or a modeling tool such as axel.
|
| Evgenii Narimanov (Princeton University) |
"Optical Hyperspace": Negative refractive index and sub-wavelength
imaging in strongly anisotropic media |
| Abstract: We develop a new approach to negative index materials and
subwavelength imaging in the far field based on
strong anisotropy of the dielectric response. In contrast to
conventional negative refraction systems, our method does not rely on
magnetic resonance and does not require periodic patterning--leading
to lower losses and high tolerance to fabrication defects. |
| Jeremy Neal (Kent State University) |
Nanoparticle susceptibilities and the bianisotropic
formalism |
| Abstract: Since the spatial extent of nanoparticles is not negligible compared to
the wavelength of light, non-local effects may be expected in the
electric and magnetic response of nanoparticles at optical frequencies.
It has been suggested that such spatially non-local response may be
taken into account via the bianisotropic formalism for the constitutive
equations. We have calculated the susceptibilities of pairs of
nanowires as a function of orientation relative to the incident fields
using the discrete dipole approximation. We compare the results of our
simulations with predictions of the bianisotropic description, and
summarize our observations.
|
| Sia Nemat-Nasser (University of California, San Diego) |
Multifunctional composites with negative refractive index |
| Abstract: We outline recent achievements in creating structural composite
materials with controlled electromagnetic properties, as an integral
part of a multifunctional material system. The electromagnetic
response is tailored by incorporating within the material small
amounts of suitably configured, periodically distributed, electric
conductors to produce distributed electric inductance and
capacitance. The small-scale response of the conductors can be
homogenized to give overall macroscopic EM material properties at
wavelengths that are orders of magnitude larger than the dimensions
of the periodicity of the structure. Periodic arrays of inductive
elements such as thin straight wires, loop-wires, coils, and other
conductive thin metallic structures can modify the effective electric
permittivity and the effective magnetic permeability of a composite
and make it negative. I will discuss the process of design, analysis,
manufacturing, and measurement of such composites. In particular, I
will review the UCSD's work on the design, production, and
experimental characterization of a 2.7 mm thick composite panels
having negative refractive index between 8.4 and 9.2 GHz. I will
also examine our work on a flat lens having a gradient variation of
negative index of refraction that can focus in the 10GHz range,
showing excellent agreement with full-wave simulations. |
| Peter Palffy-Muhoray (Kent State University) |
Observation of increased transmission in sol-gel nanocomposites |
| Abstract: Nanocomposites made of Ag nanowires imbedded in a sol-gel host have been
morphologically and optically investigated. Sonication during solidification
significantly improved nanowire dispersal. The data from the nanocomposites
were compared to the data from pure sol-gels in order to determine the
effects of the nanowires. Reflectometry data at 1064 nm show that the
presence of ~5% nanowires (by volume) results in a decrease from 1.17 to
≈1.1 in the real part of the index of refraction accompanied by an increase
in the imaginary part. Transmission loss in the pure sol-gel is mainly due
to scattering from inhomogeneities, and the inclusion of nanowires (or the
process of doing so) results in a reduction of optical loss at VIS-NUV
wavelengths in several samples. |
| Chris Peterson (Colorado State University) |
Algebraic geometry and applications seminar: Examples of exact results from inexact methods |
| Abstract: In this talk, some examples will be presented that illustrate how
exact equations can often be recovered from numerically approximated
generic points on a variety.
|
| Viktor Podolskiy (Oregon State University) |
"Negative" nanophotonics: controlling diffraction limit and group velocity
in anisotropy-based NIMs |
| Abstract: We explore the perspectives of a new type of materials with negative index
of refraction - non-magnetic NIMs. In contrast to conventional NIMs, based
either on magnetism or on periodicity, our design is non-magnetic and relies
on the effective-medium response of anisotropic meta-materials in waveguide
geometries. Being highly-tolerable to fabrication defects, anisotropic
systems allow a versatile control over the magnitude and sign of effective
refractive index and open new ways to efficiently couple the radiation from
micro-scale optical fibers to nm-sized waveguides followed by
sub-diffraction light manipulation inside sub-critical waveguiding
structures. Specific applications include photonic funnels, capable of
transferring over 25% of radiation from conventional telecom fiber to the
spots smaller than 1/30-th of a wavelength, and NIM-based lenses with a
far-field resolution of the order of 1/10-th of a wavelength. We also
investigate the perspectives of active nanoscale NIMs and demonstrate that
material gain can not only eliminate problems associated with absorption,
but is also a powerful tool to control the group velocity from negative to
"slow" positive values. |
| Greg Reid (University of Western Ontario) |
Application of numerical algebraic geometry to partial differential equations |
| Abstract: In Numerical Algebraic Geometry, solution components of polynomial systems
are characterized by witness points. Such nice points are computed efficiently
by continuation methods.
In this talk, which is joint
work with Wenyuan Wu and Jan Verschelde, I will outline progress on extending
these
methods to Partial Differential Equations.
I will describe the jet geometric picture of PDE in their jet space, to which
the methods of Numerical Algebraic Geometry are applied.
In particular witness points in jet geometry are cut out by the intersection of
random
linear spaces with submanifolds in jet space (there will be nice pictures).
Applications to constrained mechanical
systems; overdetermined systems for symmetries of differential equations will
also be
described.
|
| Fabrice Rouillier (Institut National de Recherche en Informatique Automatique (INRIA)) |
On computing using FGb/RS software |
| Abstract: The goal of this lecture is to show how to use efficiently FGb and RS
software for solving various kinds of polynomial systems.
Such software can nowadays be used as a "black box" in many cases (J.
Gehrard's presentation) for computing/studying real roots of polynomial
systems, but the objective here is to show how to use them in extreme
situations.
We will introduce, step by step, several tricks and advanced options,
including the related mathematical background, and show how to solve some
examples presented in the tutorial which took place at IMA in September
(http://www.ima.umn.edu/2006-2007/T9.15-16.06/). |
| Fadil Santosa (University of Minnesota Twin Cities) |
Maximization of the quality factor of an optical resonator |
| Abstract: We consider resonance phenomena for the scalar wave equation in an
inhomogeneous medium. Resonance is a solution to the wave equation
which is spatially localized while its time dependence is harmonic
except for decay due to radiation. The decay rate, which is inversely
proportional to the qualify factor, depends on the material properties
of the medium. In this work, the problem of designing a resonator
which has high quality factor (low loss) is considered. The design
variable is the index of refraction of the medium.
Finding resonance in a linear wave equation with radiation boundary
condition involves solving a nonlinear eigenvalue problem. The
magnitude of the ratio between real and imaginary part of the
eigenvalue is proportional to the quality factor Q. The optimization
we perform is finding a structure which possesses an eigenvalue with
largest possible Q. We present a numerical approach for solving
this problem and describe results obtained by our method. |
| Frank-Olaf Schreyer (Universität des Saarlandes) |
Algebraic geometry and applications seminar: An experimental approach to numerical Godeaux surfaces |
| Abstract: A (numerical) Godeaux surface is a minimal surface X of general type with K_2=1 and p_g=0, hence also q=0 and H_1(X,Q)=0. So in some sense these are the surfaces of general type with smallest possible invariants. Godeaux constructed a family of such surfaces as quotients of a quintic hypersurface by a fixed point free action of Z_5. By the work of Miyaoka it is known that the torsion group T=H_1(X,Z) is a cyclic group of order at most 5. The surfaces with T=Z_d for d=3,4,5 have a moduli space which consists of one 8 dimensional component by work of Reid and Miyaoka. For T=Z_2 or T=0 much less is known. Existence of such surfaces was proved by Rebecca Barlow, by a complicated quotient construction. Traditionally there are two approaches to construct numerical Godeaux surfaces: Either by a Godeaux approach as a quotient of a simpler surface by a possibly non free group action, or by a Campedelli approach as a double plane branched along a curve with a specific configuration of singularities. In this talk I present a third approach based on homological algebra and computer algebra. |
| Vladimir Shalaev (Purdue University) |
Empowering Optical Metamaterials with Gain and Nonlinearities |
| Abstract: Metamaterials, i.e. artificial engineered structures with properties not available in nature are expected to open a gateway to unprecedented electromagnetic properties and functionality unattainable from naturally occurring materials. Negative-refractive index metamaterials create entirely new prospects for guiding light on the nanoscale, some of which may have revolutionary impact on present-day optical technologies. We review this new emerging field of metamaterials and recent progress in demonstrating a negative refractive index in the optical range, where applications can be particularly important. We also discuss strategies how to push the wavelength region of negative refractive index into the visible range by using plasmon resonant metal nanostructures. |
| Haiping Shen (New York University) |
A homogenization-based study of the scattering resonances of a
microstructured slab |
| Abstract: This poster studies the scattering resonance problem associated with a
waveguide consisting of an infinite slab with 2-D microstructure embedded
in a homogeneous material. The main goal is to understand how resonances
are affected by the presence of the microstructure in the slab. Our method
is similar to the prior work of S. Moskow, F. Santosa and M. Vogelius, as
the investigation concentrates on the first order correction to the
homogenized resonance. The outgoing radiation condition at infinity makes
the problem non-selfadjoint. Furthermore, there are boundary layers on the
edges of the slab, due to the presence of rapidly vaying coefficients in
the highest order term of the underlying equation. Our main result is a
formula for the first order correction. The formula indicates strong influence
of the way microstructure hits the edges of the slab. |
| Gennady Shvets (University of Texas) |
Homogenization theory of negative index materials in the optical
range |
| Abstract: The challenge in engineering negative index materials in the optical
frequency range involves designing sub-wavelength building blocks that
exhibit both
electric and magnetic activity. Achieving strong magnetic response is
particularly challenging because magnetic moment of a structure scales as
the square of the unit cell size. We address this challenge by employing
higher order (multipole) electrostatic resonances that have a non-vaishing
magnetic moment for a finite unite cell size. This approach provides a
natural starting point for a perturbation theory that uses the ratio
of the building block size to vacuum wavelength as the smallness
parameter. Perturbative calculation yields the effective parameters of the
metamaterial: effective epsilon and mu tensors. Those can be compared with
the effective parameters extracted from fully electromagnetic simulations.
Examples are given for two and three dimensional structures.
|
| William Stein (University of Washington) |
SAGE — Software for algebra and geometry experimentation |
| Abstract: The goal of SAGE is to create an optimal software environment for research
and experimentation in algebra, geometry, number theory, cryptography, and
related areas. The speaker started SAGE in 2005 by combining together the
very best of existing free software (e.g., Singular, PARI, GAP, Macaulay2,
Maxima, gfan, etc), creating interfaces to non-free software (e.g., MAGMA,
Maple, Mathematica), and beginning to fill in the gaps with new code. Now
many developers have joined him in working on filling these gaps and
making SAGE a polished and efficient piece of software. This talk will
demo SAGE, and explain how it works.
|
| Theodore L. Turocy (Texas A & M University) |
Towards a black-box solver for finite games: The Gambit system |
| Abstract: An intriguing fact about the celebrated Nash equilibrium concept in
finite
games is that it can be expressed mathematically in a variety of ways:
as a fixed point, a solution to a complementarity program, a (global)
minimizer, or a solution to a system of polynomial equations and
inequalities.
As a result, there are general-purpose algorithms to compute Nash
equilibria
on finite games. Many of these algorithms relate specifically to
well-studied areas of numerical programming, including linear
programming,
homotopy path-following, and solving polynomial systems.
Game theory appears in a broad range of disciplines, and is taught
broadly
at the undergraduate level. Students and practitioners alike need a
tool
to specify and analyze games without needing to worry about the
computational
details of finding equilibria. The Gambit system is an Open Source
(GPL)
set of software tools for specifying and analyzing games, with the goal
of packaging quality implementations of numerical codes from experts in
a convenient form for general use. This talk will introduce the Gambit
system and outline current implementations of algorithms for computing
equilibrium, with special emphasis on the role of a method for computing
all equilibria which uses the PHCPACK system as a backend. |
| Charles W. Wampler (General Motors Corporation) |
What should a software package for numerical algebraic geometry be? |
| Abstract: It has been approximately twenty years since the initial germination of
numerical continuation as an approach to finding solution points of systems
of polynomial equations and ten years since its flowering into numerical
algebraic geometry, which facilitates description of solution sets of any
dimension and operations on these, such as intersection and fiber products.
We will attempt to step back from the torrent of recent developments to
sketch out the big picture of what a general software package for numerical
algebraic geometry should be: what core functions are necessary, how these
can be organized into higher level algorithms, and how this might be wedded
to user interfaces that allow access by non-experts while still being open
to new advances in algorithms. |
| Wenyuan Wu (University of Western Ontario) |
Software for partial differential equations |
| Abstract: In this talk both symbolic and numerical software will be described for
overdetermined
systems of Partial Differential Equations.
The symbolic software is the rifsimp package which is available in distributed
Maple.
Examples and applications of the use of this software will be given.
The numeric-symbolic methods are implemented using
Jan Verschelde's PHCPack for numerical polynomial continuation together
with Maple programs. Again examples will be given.
This is joint work with Greg Reid, Jan verschelde and Allan Wittkopf and is a
partner
to the earlier talk Application of Numerical Algebraic Geometry to Partial
Differential
Equations given by Greg Reid.
|
| Arthur Yaghjian (US Air Force Research Laboratory) |
Plane-wave solutions to frequency-domain and time-domain scattering from negative permittivity and permeability slabs |
| Abstract: Plane-wave representations are used to formulate the exact solutions to frequency-domain and time-domain sources illuminating a magnetodielectric slab with complex permittivity and permeability. In the special case of a line source at z=0 a distance d2L but also divergent infinite fields in the region 2d2L and divergent in the region 2d
|
| Zhonggang Zeng (Northeastern Illinois University) |
APAtools: A Maple/Matlab toolbox for approximate polynomial
algebra |
| Abstract: This talk presents a Maple/Matlab toolbox for basic polynomial
computations
with exact or approximate data. The toolbox includes software for
computing
the approximate GCD, approximate factorization, dual basis and
multiplicity
identification, as well as numerical elimination in solving polynomial
systems. We shall present the underlying theory of approximate
polynomial
algebra, the main approach of the algorithms, and computational
results.
|
