The document discusses and compares three methods for light scattering by small particles: the separation of variables method (SVM), the extended boundary condition method (EBCM), and the generalized point matching method (gPMM). All three methods use the same field expansions involving vector and scalar wave functions, but differ in important details of how the boundary conditions are applied. While the methods show generally close behavior, EBCM specifically can provide accurate results for some particle shapes but not others, possibly due to singularities in the analytic continuations of fields depending on the shape. The authors seek to understand these theoretical differences and limitations between the methods.
Light Scattering by Nonspherical Particlesavinokurov
This document discusses light scattering by non-spherical particles. It compares three methods for solving this problem: the separation of variables method, the extended boundary condition method, and the generalized point matching method. All three methods use the same field expansions, but differ in important details of how the boundary conditions are applied. The accuracy of the extended boundary condition method depends on the particle shape, working well for some shapes but not others. Near-field behavior and convergence are more complex for non-spherical particles compared to spheres. The concept of the Rayleigh hypothesis is also discussed.
We cover the inverses to the trigonometric functions sine, cosine, tangent, cotangent, secant, cosecant, and their derivatives. The remarkable fact is that although these functions and their inverses are transcendental (complicated) functions, the derivatives are algebraic functions. Also, we meet my all-time favorite function: arctan.
The document is a lecture on inverse trigonometric functions from a Calculus I class at New York University. It defines inverse trig functions like arcsin, arccos, and arctan and discusses their domains, ranges, and relationships to the original trig functions. It also provides examples of evaluating inverse trig functions at specific values.
Tensor Decomposition and its ApplicationsKeisuke OTAKI
This document discusses tensor factorizations and decompositions and their applications in data mining. It introduces tensors as multi-dimensional arrays and covers 2nd order tensors (matrices) and 3rd order tensors. It describes how tensor decompositions like the Tucker model and CANDECOMP/PARAFAC (CP) model can be used to decompose tensors into core elements to interpret data. It also discusses singular value decomposition (SVD) as a way to decompose matrices and reduce dimensions while approximating the original matrix.
This document discusses linear prediction analysis (LPC) for speech recognition. It begins by deriving the linear prediction equations and describing the autocorrelation method of LPC. It then interprets the LPC filter as a spectral whitener that flattens the spectrum of the prediction error. The document discusses alternative methods like covariance LPC and closed phase covariance LPC. It also describes alternative parameter sets that can represent the LPC filter, such as pole positions, reflection coefficients, and log area ratios.
The computation of automorphic forms for a group Gamma is
a major problem in number theory. The only known way to approach the higher rank cases is by computing the action of Hecke operators on the cohomology.
Henceforth, we consider the explicit computation of the cohomology by using cellular complexes. We then explain how the rational elements can be made to act on the complex when it originate from perfect forms. We illustrate the results obtained for the symplectic Sp4(Z) group.
Quantum key distribution with continuous variables at telecom wavelengthwtyru1989
This document outlines a study on quantum key distribution using continuous variables at telecom wavelengths. It discusses implementing quantum cryptography by encoding secret keys onto continuously varying properties like amplitude and phase of coherent laser pulses. The receiving party uses homodyne detection to measure the signals. While this allows higher key rates than single-photon methods, it requires precise optics and signal processing. The document covers analyzing the security and performance of such a system, including modeling noise and information leakage, and using error correction codes to reconcile the correlated data between parties. The goal is to establish a secret key that is secure against potential eavesdropping attacks.
Security of continuous variable quantum key distribution against general attackswtyru1989
This document summarizes a paper on the security of continuous-variable quantum key distribution (CVQKD) against general attacks. It describes several CVQKD protocols that use homodyne or heterodyne detection to encode and measure information on the quadratures of electromagnetic fields. The paper outlines a proof that CVQKD protocols are secure against general attacks by first making the protocols permutation invariant through a test that restricts the state to a finite-dimensional subspace, then applying the postselection technique from finite-dimensional systems. It proposes a test using heterodyne detection on some modes to bound the photon number and ensure the state is close to a finite-dimensional projection. Symmetries in phase space from transformations of phase shifts
Light Scattering by Nonspherical Particlesavinokurov
This document discusses light scattering by non-spherical particles. It compares three methods for solving this problem: the separation of variables method, the extended boundary condition method, and the generalized point matching method. All three methods use the same field expansions, but differ in important details of how the boundary conditions are applied. The accuracy of the extended boundary condition method depends on the particle shape, working well for some shapes but not others. Near-field behavior and convergence are more complex for non-spherical particles compared to spheres. The concept of the Rayleigh hypothesis is also discussed.
We cover the inverses to the trigonometric functions sine, cosine, tangent, cotangent, secant, cosecant, and their derivatives. The remarkable fact is that although these functions and their inverses are transcendental (complicated) functions, the derivatives are algebraic functions. Also, we meet my all-time favorite function: arctan.
The document is a lecture on inverse trigonometric functions from a Calculus I class at New York University. It defines inverse trig functions like arcsin, arccos, and arctan and discusses their domains, ranges, and relationships to the original trig functions. It also provides examples of evaluating inverse trig functions at specific values.
Tensor Decomposition and its ApplicationsKeisuke OTAKI
This document discusses tensor factorizations and decompositions and their applications in data mining. It introduces tensors as multi-dimensional arrays and covers 2nd order tensors (matrices) and 3rd order tensors. It describes how tensor decompositions like the Tucker model and CANDECOMP/PARAFAC (CP) model can be used to decompose tensors into core elements to interpret data. It also discusses singular value decomposition (SVD) as a way to decompose matrices and reduce dimensions while approximating the original matrix.
This document discusses linear prediction analysis (LPC) for speech recognition. It begins by deriving the linear prediction equations and describing the autocorrelation method of LPC. It then interprets the LPC filter as a spectral whitener that flattens the spectrum of the prediction error. The document discusses alternative methods like covariance LPC and closed phase covariance LPC. It also describes alternative parameter sets that can represent the LPC filter, such as pole positions, reflection coefficients, and log area ratios.
The computation of automorphic forms for a group Gamma is
a major problem in number theory. The only known way to approach the higher rank cases is by computing the action of Hecke operators on the cohomology.
Henceforth, we consider the explicit computation of the cohomology by using cellular complexes. We then explain how the rational elements can be made to act on the complex when it originate from perfect forms. We illustrate the results obtained for the symplectic Sp4(Z) group.
Quantum key distribution with continuous variables at telecom wavelengthwtyru1989
This document outlines a study on quantum key distribution using continuous variables at telecom wavelengths. It discusses implementing quantum cryptography by encoding secret keys onto continuously varying properties like amplitude and phase of coherent laser pulses. The receiving party uses homodyne detection to measure the signals. While this allows higher key rates than single-photon methods, it requires precise optics and signal processing. The document covers analyzing the security and performance of such a system, including modeling noise and information leakage, and using error correction codes to reconcile the correlated data between parties. The goal is to establish a secret key that is secure against potential eavesdropping attacks.
Security of continuous variable quantum key distribution against general attackswtyru1989
This document summarizes a paper on the security of continuous-variable quantum key distribution (CVQKD) against general attacks. It describes several CVQKD protocols that use homodyne or heterodyne detection to encode and measure information on the quadratures of electromagnetic fields. The paper outlines a proof that CVQKD protocols are secure against general attacks by first making the protocols permutation invariant through a test that restricts the state to a finite-dimensional subspace, then applying the postselection technique from finite-dimensional systems. It proposes a test using heterodyne detection on some modes to bound the photon number and ensure the state is close to a finite-dimensional projection. Symmetries in phase space from transformations of phase shifts
Adomian Decomposition Method for Certain Space-Time Fractional Partial Differ...IOSR Journals
This document presents an application of the Adomian Decomposition Method (ADM) to solve certain space-time fractional partial differential equations. It begins with an introduction to fractional calculus concepts and definitions. It then outlines the four cases of the ADM that are used to solve different types of space-time fractional PDEs. As an example, it presents the general steps of the direct ADM case to solve a linear space-time fractional PDE. The steps involve decomposing the unknown function into a series, determining the components recursively, and obtaining an approximate solution. Finally, several examples are solved to demonstrate the effectiveness of the ADM for fractional PDEs.
This document discusses properties of extremal black holes in N=8 supergravity. It notes that extremal black holes can be described as critical points of a black hole effective potential, where the scalar fields are stabilized at the event horizon regardless of their values at spatial infinity. The document outlines properties of electric-magnetic duality, radial evolution of scalar fields near black hole horizons, and attractor mechanisms in maximal supergravity. It also discusses classification of BPS black holes in terms of E6(6) and E7(7) orbits and expressions for Bekenstein-Hawking entropy.
This document summarizes research on sparse representations by Joel Tropp. It discusses how sparse approximation problems arise in applications like variable selection in regression and seismic imaging. It presents algorithms for solving sparse representation problems, including orthogonal matching pursuit and 1-minimization. It analyzes when these algorithms can recover sparse solutions and proves performance guarantees for random matrices and random sparse vectors. The document also discusses related areas like compressive sampling and simultaneous sparsity.
This document discusses using linear approximations to estimate functions. It provides an example estimating sin(61°) using linear approximations about a=0 and a=60°. When approximating about a=0, the estimate is 1.06465. When approximating about a=60°, the estimate is 0.87475, which is closer to the actual value of sin(61°) according to a calculator check. The document teaches that the tangent line provides the best linear approximation near a point, and its equation can be used to estimate function values.
A Decomposition-based Approach to Modeling and Understanding Arbitrary ShapesDavid Canino
Modeling and understanding complex non-manifold shapes is a key issue in shape analysis and retrieval. The topological structure of a non-manifold shape can be analyzed through its decomposition into a collection of components with a simpler topology. Here, we consider a representation for arbitrary shapes, that we call Manifold-Connected Decomposition (MC-decomposition), which is based on a unique decomposition of the shape into nearly manifold parts. We present efficient and powerful two-level representations for non-manifold shapes based on the MC-decomposition and on an efficient and compact data structure for encoding the underlying components. We describe a dimension-independent algorithm to generate such decomposition. We also show that the MC-decomposition provides a suitable basis for geometric reasoning and for homology computation on non- manifold shapes. Finally, we present a comparison with existing representations for arbitrary shapes.
High-order Finite Elements for Computational PhysicsRobert Rieben
The document discusses high order finite element methods for computational physics from Lawrence Livermore National Laboratory's perspective. It introduces the weak variational formulation of partial differential equations, finite element approximation using a Galerkin method, and the use of discrete differential forms and basis functions to represent solutions. The goal is to develop robust, modular software for solving multi-physics problems on massively parallel architectures.
Analysis and algebra on differentiable manifoldsSpringer
This chapter discusses integration on manifolds. It defines orientation of manifolds and orientation-preserving maps between manifolds. It presents Stokes' theorem and Green's theorem, which relate integrals over boundaries to integrals of differential forms. It introduces de Rham cohomology groups, which classify closed forms modulo exact forms. Examples are given of calculating the orientability and cohomology of manifolds like the cylinder, Möbius strip, and real projective plane.
Linear Programming and its Usage in Approximation Algorithms for NP Hard Opti...Reza Rahimi
The document provides an overview of linear programming and its usage in approximation algorithms for NP-hard optimization problems. It discusses linear programming formulations, the complexity classes P and NP, approximation algorithms, and two case studies on the minimum weight vertex cover problem and the MAXSAT problem. Randomized rounding techniques are used to generate approximation algorithms for these problems from their linear programming relaxations.
This document introduces tensors through examples. It defines a vector as a rank 1 tensor and a matrix as a rank 2 tensor. It then provides an example of a rank 3 tensor. The document discusses how to define an inner product between tensors and provides examples using vectors and matrices. It also gives an example of how derivatives of a function can produce tensors of different ranks. Finally, it introduces the concept of decomposing matrices into their symmetric and antisymmetric parts.
This are the notes of a seminar talk delivered in summer 2008 at Bonn. Let SU (2, 1) be the
moduli space of rank 2 bundles with a fixed determinant of rank 1 over a curve C of genus g ≥ 2.
This is a Fano manifold of Picard rank 1. We discuss the example g = 2 where SU (2, 1) is the
intersection of two quadrics in P5 . In this case the minimal rational curves are lines. There is a
very interesting class of rational curves on SU (2, 1), called Hecke curves, which are constructed
by extending a given bundle by torsion sheaves. In the case g ≥ 3 we will see that Hecke curves
have minimal anti-canonical degree (4) and that any rational curve passing through a generic
point is a Hecke curve.
EXPERT SYSTEMS AND SOLUTIONS
Project Center For Research in Power Electronics and Power Systems
IEEE 2010 , IEEE 2011 BASED PROJECTS FOR FINAL YEAR STUDENTS OF B.E
Email: expertsyssol@gmail.com,
Cell: +919952749533, +918608603634
www.researchprojects.info
OMR, CHENNAI
IEEE based Projects For
Final year students of B.E in
EEE, ECE, EIE,CSE
M.E (Power Systems)
M.E (Applied Electronics)
M.E (Power Electronics)
Ph.D Electrical and Electronics.
Training
Students can assemble their hardware in our Research labs. Experts will be guiding the projects.
EXPERT GUIDANCE IN POWER SYSTEMS POWER ELECTRONICS
We provide guidance and codes for the for the following power systems areas.
1. Deregulated Systems,
2. Wind power Generation and Grid connection
3. Unit commitment
4. Economic Dispatch using AI methods
5. Voltage stability
6. FLC Control
7. Transformer Fault Identifications
8. SCADA - Power system Automation
we provide guidance and codes for the for the following power Electronics areas.
1. Three phase inverter and converters
2. Buck Boost Converter
3. Matrix Converter
4. Inverter and converter topologies
5. Fuzzy based control of Electric Drives.
6. Optimal design of Electrical Machines
7. BLDC and SR motor Drives
The document discusses using permutations to study a classification problem on dividing curves in the solid torus. It introduces representing arclists on cutting disks as permutations and using permutations to check for tightness and the existence of bypasses. Permutations allow an algorithmic approach to identifying abstract bypasses without visualizing the geometry. The talk outlines representing arcs and arclists with permutations, using permutations to check for tightness and bypasses, and generating abstract bypass permutations.
The document discusses transformations in geometry. It defines a geometric transformation as a bijective mapping between two geometries that maps points to points and lines to lines. Reflections, rotations, and translations are provided as examples of geometric transformations in Euclidean plane geometry. It is shown that reflections, rotations, and translations are isometries that preserve distance, angle measure, and area. The composition of transformations is also discussed, and it is shown that the composition of isometries is again an isometry.
This document discusses noncommutative quantum field theory, where the coordinates do not commute. It begins by motivating noncommutativity from theories of quantum gravity and string theory. It then introduces the Moyal product to write actions for noncommutative fields. While Lorentz symmetry is broken, the actions are still invariant under a twisted Poincaré algebra. Representations are classified by mass and spin as in ordinary theories. The document considers both space-like and time-like noncommutativity, but argues that time-like noncommutativity poses challenges for perturbative unitarity.
Spectral Learning Methods for Finite State Machines with Applications to Na...LARCA UPC
The document summarizes a spectral learning method for probabilistic finite-state machines (FSMs). It introduces observable operator models that represent probabilistic transducers using conditional probabilities between inputs, outputs, and hidden states. A key contribution is a spectral algorithm that learns the parameters of these models from data in linear time, with theoretical PAC-style guarantees. Experimental results on synthetic data show the method outperforms baselines like HMMs and k-HMMs on learning tasks.
Discontinuous Petrov-Galerkin Methods for convection-dominated diffusion pro...Mohammad Zakerzadeh
This document discusses optimal trial and test spaces for the Petrov-Galerkin method applied to convection-dominated diffusion problems. It introduces the concept of optimal spaces that satisfy M=γ=1, resulting in a well-posed and well-conditioned problem with best error estimates. It shows that optimal test spaces can be constructed using Riesz representations to minimize residuals. The dual and primal approaches are presented to construct optimal finite dimensional subspaces that inherit the M=γ=1 property. Energy norm pairings induced by optimal norms on trial and test spaces are also discussed.
Topologically adaptable snakes, or simply T-snakes, are
a standard tool for automatically identifying multiple segments
in an image. This work introduces a novel approach
for controlling the topology of a T-snake. It focuses on the
loops formed by the so-called projected curve which is obtained
at every stage of the snake evolution. The idea is to
make that curve the image of a piecewise linear mapping
of an adequate class. Then, with the help of an additional
structure—the Loop-Tree—it is possible to decide in O(1)
time whether the region enclosed by each loop has already
been explored by the snake. This makes it possible to construct
an enhanced algorithm for evolving T-snakes whose
performance is assessed by means of statistics and examples.
This document discusses various unitary transforms that can be used to decompose images, including the discrete Fourier transform (DFT), discrete cosine transform (DCT), Karhunen-Loève transform (KLT), Hadamard transform, and wavelet transforms. Unitary transforms have desirable properties like energy conservation, orthonormal bases, and de-correlation of image elements. The KLT provides optimal energy compaction and de-correlation but relies on signal statistics. Practical transforms like the DCT approximate the KLT while having fast implementations and being signal-independent. Transforms are widely used for applications like image compression, feature extraction, and pattern recognition.
This document provides definitions and examples related to Fourier series and Fourier transforms. It defines the Fourier transform and inverse Fourier transform of a function f(x). It gives the Fourier integral representation of a function and provides an example of finding the Fourier integral representation of a rectangle function. It also defines Fourier sine and cosine integrals. Finally, it outlines some properties of Fourier transforms, including the modulation theorem and convolution theorem.
El documento describe el proceso de encendido de un generador, incluyendo la apertura de valvulas, el encendido de turbinas, y la medición de temperatura, presión y velocidad del oxígeno. La computadora digital controla las valvulas de agua, combustible y aire, y la velocidad de la turbina basado en las mediciones para asegurar que los niveles estén dentro de los parámetros correctos antes de continuar el proceso.
El documento describe un concurso de cometas alusivas al equipo de fútbol Deportes Tolima con el objetivo de incentivar a los hinchas a asistir en familia a los partidos. El concurso requiere sonido, apoyo policial, premios y difusión a través de redes sociales y páginas de hinchas. Se llevará a cabo el día del partido Deportes Tolima vs Cúcuta Deportivo en el estadio.
Adomian Decomposition Method for Certain Space-Time Fractional Partial Differ...IOSR Journals
This document presents an application of the Adomian Decomposition Method (ADM) to solve certain space-time fractional partial differential equations. It begins with an introduction to fractional calculus concepts and definitions. It then outlines the four cases of the ADM that are used to solve different types of space-time fractional PDEs. As an example, it presents the general steps of the direct ADM case to solve a linear space-time fractional PDE. The steps involve decomposing the unknown function into a series, determining the components recursively, and obtaining an approximate solution. Finally, several examples are solved to demonstrate the effectiveness of the ADM for fractional PDEs.
This document discusses properties of extremal black holes in N=8 supergravity. It notes that extremal black holes can be described as critical points of a black hole effective potential, where the scalar fields are stabilized at the event horizon regardless of their values at spatial infinity. The document outlines properties of electric-magnetic duality, radial evolution of scalar fields near black hole horizons, and attractor mechanisms in maximal supergravity. It also discusses classification of BPS black holes in terms of E6(6) and E7(7) orbits and expressions for Bekenstein-Hawking entropy.
This document summarizes research on sparse representations by Joel Tropp. It discusses how sparse approximation problems arise in applications like variable selection in regression and seismic imaging. It presents algorithms for solving sparse representation problems, including orthogonal matching pursuit and 1-minimization. It analyzes when these algorithms can recover sparse solutions and proves performance guarantees for random matrices and random sparse vectors. The document also discusses related areas like compressive sampling and simultaneous sparsity.
This document discusses using linear approximations to estimate functions. It provides an example estimating sin(61°) using linear approximations about a=0 and a=60°. When approximating about a=0, the estimate is 1.06465. When approximating about a=60°, the estimate is 0.87475, which is closer to the actual value of sin(61°) according to a calculator check. The document teaches that the tangent line provides the best linear approximation near a point, and its equation can be used to estimate function values.
A Decomposition-based Approach to Modeling and Understanding Arbitrary ShapesDavid Canino
Modeling and understanding complex non-manifold shapes is a key issue in shape analysis and retrieval. The topological structure of a non-manifold shape can be analyzed through its decomposition into a collection of components with a simpler topology. Here, we consider a representation for arbitrary shapes, that we call Manifold-Connected Decomposition (MC-decomposition), which is based on a unique decomposition of the shape into nearly manifold parts. We present efficient and powerful two-level representations for non-manifold shapes based on the MC-decomposition and on an efficient and compact data structure for encoding the underlying components. We describe a dimension-independent algorithm to generate such decomposition. We also show that the MC-decomposition provides a suitable basis for geometric reasoning and for homology computation on non- manifold shapes. Finally, we present a comparison with existing representations for arbitrary shapes.
High-order Finite Elements for Computational PhysicsRobert Rieben
The document discusses high order finite element methods for computational physics from Lawrence Livermore National Laboratory's perspective. It introduces the weak variational formulation of partial differential equations, finite element approximation using a Galerkin method, and the use of discrete differential forms and basis functions to represent solutions. The goal is to develop robust, modular software for solving multi-physics problems on massively parallel architectures.
Analysis and algebra on differentiable manifoldsSpringer
This chapter discusses integration on manifolds. It defines orientation of manifolds and orientation-preserving maps between manifolds. It presents Stokes' theorem and Green's theorem, which relate integrals over boundaries to integrals of differential forms. It introduces de Rham cohomology groups, which classify closed forms modulo exact forms. Examples are given of calculating the orientability and cohomology of manifolds like the cylinder, Möbius strip, and real projective plane.
Linear Programming and its Usage in Approximation Algorithms for NP Hard Opti...Reza Rahimi
The document provides an overview of linear programming and its usage in approximation algorithms for NP-hard optimization problems. It discusses linear programming formulations, the complexity classes P and NP, approximation algorithms, and two case studies on the minimum weight vertex cover problem and the MAXSAT problem. Randomized rounding techniques are used to generate approximation algorithms for these problems from their linear programming relaxations.
This document introduces tensors through examples. It defines a vector as a rank 1 tensor and a matrix as a rank 2 tensor. It then provides an example of a rank 3 tensor. The document discusses how to define an inner product between tensors and provides examples using vectors and matrices. It also gives an example of how derivatives of a function can produce tensors of different ranks. Finally, it introduces the concept of decomposing matrices into their symmetric and antisymmetric parts.
This are the notes of a seminar talk delivered in summer 2008 at Bonn. Let SU (2, 1) be the
moduli space of rank 2 bundles with a fixed determinant of rank 1 over a curve C of genus g ≥ 2.
This is a Fano manifold of Picard rank 1. We discuss the example g = 2 where SU (2, 1) is the
intersection of two quadrics in P5 . In this case the minimal rational curves are lines. There is a
very interesting class of rational curves on SU (2, 1), called Hecke curves, which are constructed
by extending a given bundle by torsion sheaves. In the case g ≥ 3 we will see that Hecke curves
have minimal anti-canonical degree (4) and that any rational curve passing through a generic
point is a Hecke curve.
EXPERT SYSTEMS AND SOLUTIONS
Project Center For Research in Power Electronics and Power Systems
IEEE 2010 , IEEE 2011 BASED PROJECTS FOR FINAL YEAR STUDENTS OF B.E
Email: expertsyssol@gmail.com,
Cell: +919952749533, +918608603634
www.researchprojects.info
OMR, CHENNAI
IEEE based Projects For
Final year students of B.E in
EEE, ECE, EIE,CSE
M.E (Power Systems)
M.E (Applied Electronics)
M.E (Power Electronics)
Ph.D Electrical and Electronics.
Training
Students can assemble their hardware in our Research labs. Experts will be guiding the projects.
EXPERT GUIDANCE IN POWER SYSTEMS POWER ELECTRONICS
We provide guidance and codes for the for the following power systems areas.
1. Deregulated Systems,
2. Wind power Generation and Grid connection
3. Unit commitment
4. Economic Dispatch using AI methods
5. Voltage stability
6. FLC Control
7. Transformer Fault Identifications
8. SCADA - Power system Automation
we provide guidance and codes for the for the following power Electronics areas.
1. Three phase inverter and converters
2. Buck Boost Converter
3. Matrix Converter
4. Inverter and converter topologies
5. Fuzzy based control of Electric Drives.
6. Optimal design of Electrical Machines
7. BLDC and SR motor Drives
The document discusses using permutations to study a classification problem on dividing curves in the solid torus. It introduces representing arclists on cutting disks as permutations and using permutations to check for tightness and the existence of bypasses. Permutations allow an algorithmic approach to identifying abstract bypasses without visualizing the geometry. The talk outlines representing arcs and arclists with permutations, using permutations to check for tightness and bypasses, and generating abstract bypass permutations.
The document discusses transformations in geometry. It defines a geometric transformation as a bijective mapping between two geometries that maps points to points and lines to lines. Reflections, rotations, and translations are provided as examples of geometric transformations in Euclidean plane geometry. It is shown that reflections, rotations, and translations are isometries that preserve distance, angle measure, and area. The composition of transformations is also discussed, and it is shown that the composition of isometries is again an isometry.
This document discusses noncommutative quantum field theory, where the coordinates do not commute. It begins by motivating noncommutativity from theories of quantum gravity and string theory. It then introduces the Moyal product to write actions for noncommutative fields. While Lorentz symmetry is broken, the actions are still invariant under a twisted Poincaré algebra. Representations are classified by mass and spin as in ordinary theories. The document considers both space-like and time-like noncommutativity, but argues that time-like noncommutativity poses challenges for perturbative unitarity.
Spectral Learning Methods for Finite State Machines with Applications to Na...LARCA UPC
The document summarizes a spectral learning method for probabilistic finite-state machines (FSMs). It introduces observable operator models that represent probabilistic transducers using conditional probabilities between inputs, outputs, and hidden states. A key contribution is a spectral algorithm that learns the parameters of these models from data in linear time, with theoretical PAC-style guarantees. Experimental results on synthetic data show the method outperforms baselines like HMMs and k-HMMs on learning tasks.
Discontinuous Petrov-Galerkin Methods for convection-dominated diffusion pro...Mohammad Zakerzadeh
This document discusses optimal trial and test spaces for the Petrov-Galerkin method applied to convection-dominated diffusion problems. It introduces the concept of optimal spaces that satisfy M=γ=1, resulting in a well-posed and well-conditioned problem with best error estimates. It shows that optimal test spaces can be constructed using Riesz representations to minimize residuals. The dual and primal approaches are presented to construct optimal finite dimensional subspaces that inherit the M=γ=1 property. Energy norm pairings induced by optimal norms on trial and test spaces are also discussed.
Topologically adaptable snakes, or simply T-snakes, are
a standard tool for automatically identifying multiple segments
in an image. This work introduces a novel approach
for controlling the topology of a T-snake. It focuses on the
loops formed by the so-called projected curve which is obtained
at every stage of the snake evolution. The idea is to
make that curve the image of a piecewise linear mapping
of an adequate class. Then, with the help of an additional
structure—the Loop-Tree—it is possible to decide in O(1)
time whether the region enclosed by each loop has already
been explored by the snake. This makes it possible to construct
an enhanced algorithm for evolving T-snakes whose
performance is assessed by means of statistics and examples.
This document discusses various unitary transforms that can be used to decompose images, including the discrete Fourier transform (DFT), discrete cosine transform (DCT), Karhunen-Loève transform (KLT), Hadamard transform, and wavelet transforms. Unitary transforms have desirable properties like energy conservation, orthonormal bases, and de-correlation of image elements. The KLT provides optimal energy compaction and de-correlation but relies on signal statistics. Practical transforms like the DCT approximate the KLT while having fast implementations and being signal-independent. Transforms are widely used for applications like image compression, feature extraction, and pattern recognition.
This document provides definitions and examples related to Fourier series and Fourier transforms. It defines the Fourier transform and inverse Fourier transform of a function f(x). It gives the Fourier integral representation of a function and provides an example of finding the Fourier integral representation of a rectangle function. It also defines Fourier sine and cosine integrals. Finally, it outlines some properties of Fourier transforms, including the modulation theorem and convolution theorem.
El documento describe el proceso de encendido de un generador, incluyendo la apertura de valvulas, el encendido de turbinas, y la medición de temperatura, presión y velocidad del oxígeno. La computadora digital controla las valvulas de agua, combustible y aire, y la velocidad de la turbina basado en las mediciones para asegurar que los niveles estén dentro de los parámetros correctos antes de continuar el proceso.
El documento describe un concurso de cometas alusivas al equipo de fútbol Deportes Tolima con el objetivo de incentivar a los hinchas a asistir en familia a los partidos. El concurso requiere sonido, apoyo policial, premios y difusión a través de redes sociales y páginas de hinchas. Se llevará a cabo el día del partido Deportes Tolima vs Cúcuta Deportivo en el estadio.
Este documento presenta una introducción general sobre seguridad informática. Explica que históricamente la seguridad no se ha considerado una prioridad, pero que el panorama ha cambiado radicalmente debido al aumento de vulnerabilidades, amenazas como ataques de ingeniería social, y la interconectividad generalizada. También discute los costos asociados con la seguridad y las principales tecnologías como firewalls, antivirus, IPS, VPN y autenticación única que son comunes en las organizaciones actuales.
La historia cuenta la vida de María, una mujer casada que mantiene un amor prohibido con Ricardo, un joven del pueblo. Ricardo enceguecido de celos asesina al marido de María, Damián. A pesar de no ser condenada por la ley, María es juzgada y rechazada por el pueblo, lo que la lleva a caer en una profunda tristeza y aislamiento que terminan con su muerte. El relato critica la moral hipócrita y la condena social que sufre María a manos de los lugareños.
La consolidación de la recaudación de impuestos internos y de comercio exterior en una sola entidad permitiría una gestión más eficiente con menos recursos y un mejor control de los contribuyentes. Varios países como Colombia, Guatemala, Honduras y Venezuela han unificado impuestos internos y aduanas, mientras que Argentina, Brasil y Perú también han unificado la seguridad social para buscar mayor eficiencia en la recaudación. República Dominicana ha tenido éxito al unificar las direcciones generales de impuestos internos, aunque la unificación requiere reform
El documento describe la historia de Internet, incluyendo a sus creadores principales como Vint Cerf, Robert Kahn, Louis Pouzin y Tim Berners-Lee. Explica que Internet es una red descentralizada de redes interconectadas que usa los protocolos TCP/IP para funcionar como una red global lógica. Además, menciona algunos avances recientes de Internet como las compras en línea y los diarios digitales.
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Theoretical and computational aspects of the SVM, EBCM, and PMM methods in light scatteringby small particles
1. Theoretical and computational aspects
of the SVM, EBCM, and PMM methods
in light scattering by small particles
V. B. Il’in1,2,3 V. G. Farafonov2 A. A. Vinokurov2,3
1 Saint-PetersburgState University, Russia
2 Saint-Petersburg State University of Aerospace Instrumentation, Russia
3 Pulkovo Observatory, Saint-Petersburg, Russia
12th Electromagnetic & Light Scattering Conference
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 1 / 46
2. Wave Equations and Functions
Maxwell equations
For time-harmonic fields
E(r, t) = E(r) exp(−iωt)
Helmholts equations for E(r), H(r)
∆E(r) + k 2 (r)E(r) = 0,
where k is the wavenumber
Solutions
Vector wave functions Fν (r)
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 2 / 46
3. Wave Equations and Functions
Additional condition div E(r) = 0 leads to:
Fν (r) = Ma (r) = rot(a ψν (r)),
ν
Fν (r) = Na (r) = rot rot(a ψν (r))/k,
ν
where a is a vector, ψν (r) are solutions to
∆ψν + k 2 ψν = 0.
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 3 / 46
4. Field/Potential Expansions
It looks natural to search for unknown fields as
E(r) = aν Fν (r),
ν
or equivalently
U, V (r) = aν ψν (r),
ν
where U, V are scalar potentials, e.g.
E = rot(bU) + rot rot(cV ).
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 4 / 46
5. Field/Potential Expansions
In all the methods vector/scalar wave functions are represented as:
in spherical coordinates (r , θ, ϕ):
m
ψν (r) = zn (r )Pn (θ) exp(imϕ),
in spheroidal coordinates (ξ, η, ϕ):
ψν (r) = Rnm (c, ξ)Snm (c, η) exp(imϕ),
where c is a parameter.
So, separation of variables is actually used in all 3 methods.
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 5 / 46
6. Field/Potential Expansions
In all the methods vector/scalar wave functions are represented as:
in spherical coordinates (r , θ, ϕ):
m
ψν (r) = zn (r )Pn (θ) exp(imϕ),
in spheroidal coordinates (ξ, η, ϕ):
ψν (r) = Rnm (c, ξ)Snm (c, η) exp(imϕ),
where c is a parameter.
So, separation of variables is actually used in all 3 methods.
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 5 / 46
7. Separation of Variables Method (SVM)
Field expansions are substituted in the boudary conditions
(Einc + Esca ) × n = Eint × n, r ∈ ∂Γ,
where n is the outer normal to the particle surface ∂Γ.
The conditions are mutiplied by the angular parts of ψν with different
indices and then are integrated over ∂Γ. This yelds the following
system:
A B xsca E inc
= x ,
C D xint F
where xinc , xsca , xint are vectors of expansion coefficients, A, . . . F —
matrices of surface integrals.
Generalised SVM1
1
see (Kahnert, 2003)
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 6 / 46
8. Extended Boundary Condition Method (EBCM)
Field expansions are substituted in the extended boundary condition:
−Einc (r ), r ∈ Γ− ,
rot n(r) × Eint (r)G(r , r)ds − . . . =
∂Γ Esca (r ), r ∈ Γ+ .
Due to linear independence of wave functions we get
0 Qs xsca I
= xinc ,
I Qr xint 0
where Qs , Qr are matrices, whose elements are surface integrals.
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 7 / 46
9. Generalized Point Matching Method (gPMM)
Residual of the standard boundary conditions
M
2
δ= Einc + Esca − Eint × n + . . . , r = rs ∈ ∂Γ.
s=1
Minimizing residual in the least squares sense gives
A B xsca E inc
= x ,
C D xint F
Sum in δ can be replaced with surface integral2
2
see (Farafonov & Il’in, 2006)
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 8 / 46
10. Comparison of gPMM and Integral gPMM
1 — PMM, M = N,
2 — gPMM, M = 2N,
3 — gPMM, M = 4N,
5 — iPMM, M = N,
6 — iPMM, M = 1.5N.
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 9 / 46
11. Key questions
1 EBCM, SVM, and PMM use the same field expansions. Does it result
in their similar behavior? How close are they?
[Yes and Generally close, but in important detail not.]
2 It is well known from numerical experiments that EBCM with a
spherical basis [being a widely used approach] gives high accuracy
results for some shapes, while for some others it cannot provide any
reliable results. Why? What can be said about theoretical applicability
of EBCM?
[We have some answers, but not anything is clear.]
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 10 / 46
12. Key questions
1 EBCM, SVM, and PMM use the same field expansions. Does it result
in their similar behavior? How close are they?
[Yes and Generally close, but in important detail not.]
2 It is well known from numerical experiments that EBCM with a
spherical basis [being a widely used approach] gives high accuracy
results for some shapes, while for some others it cannot provide any
reliable results. Why? What can be said about theoretical applicability
of EBCM?
[We have some answers, but not anything is clear.]
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 10 / 46
13. Key questions
1 EBCM, SVM, and PMM use the same field expansions. Does it result
in their similar behavior? How close are they?
[Yes and Generally close, but in important detail not.]
2 It is well known from numerical experiments that EBCM with a
spherical basis [being a widely used approach] gives high accuracy
results for some shapes, while for some others it cannot provide any
reliable results. Why? What can be said about theoretical applicability
of EBCM?
[We have some answers, but not anything is clear.]
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 10 / 46
14. Key questions
1 EBCM, SVM, and PMM use the same field expansions. Does it result
in their similar behavior? How close are they?
[Yes and Generally close, but in important detail not.]
2 It is well known from numerical experiments that EBCM with a
spherical basis [being a widely used approach] gives high accuracy
results for some shapes, while for some others it cannot provide any
reliable results. Why? What can be said about theoretical applicability
of EBCM?
[We have some answers, but not anything is clear.]
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 10 / 46
15. Singularities of analytic continuations of the fields
Obviously, a plane wave has no such singularities.
But a plane wave incident at a scatterer is known to produce scattered
field outside it and incident field inside it.
Generally, analytic continuations of both the scattered field (inside the
scatterer) and of the internal field (outside it) may have singularities
depending on the scatterer shape.
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 11 / 46
16. Singularities of analytic continuations of the fields
Obviously, a plane wave has no such singularities.
But a plane wave incident at a scatterer is known to produce scattered
field outside it and incident field inside it.
Generally, analytic continuations of both the scattered field (inside the
scatterer) and of the internal field (outside it) may have singularities
depending on the scatterer shape.
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 11 / 46
17. Singularities of analytic continuations of the fields
Obviously, a plane wave has no such singularities.
But a plane wave incident at a scatterer is known to produce scattered
field outside it and incident field inside it.
Generally, analytic continuations of both the scattered field (inside the
scatterer) and of the internal field (outside it) may have singularities
depending on the scatterer shape.
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 11 / 46
18. Singularities for a spheroid and Chebyshev particle
Spheroid Chebyshev particle
r (θ, ϕ) = r0 (1 + ε cos nθ)
d sca = a2 − b 2 , d sca = f (r0 , n, ε),
d int = ∞. d int = g (r0 , n, ε).
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 12 / 46
19. Near Field
1 In spherical coordinates
m
ψν (r , θ, ϕ) = zn (r )Pn (θ) exp(imϕ).
2 For r → 0 or ∞: zn (r ) behaves like r k .
3 Series U, V (r) = ν aν ψν (r) can be transformed into power series.
4 Radius of convergence is determined by the nearest singularity.
For scattered field: r ∈ (max d sca , ∞) .
For internal field: r ∈ 0, min d int .
5 In spheroidal coordinates, it is not that simple!
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 13 / 46
20. Near Field
1 In spherical coordinates
m
ψν (r , θ, ϕ) = zn (r )Pn (θ) exp(imϕ).
2 For r → 0 or ∞: zn (r ) behaves like r k .
3 Series U, V (r) = ν aν ψν (r) can be transformed into power series.
4 Radius of convergence is determined by the nearest singularity.
For scattered field: r ∈ (max d sca , ∞) .
For internal field: r ∈ 0, min d int .
5 In spheroidal coordinates, it is not that simple!
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 13 / 46
21. Near Field
1 In spherical coordinates
m
ψν (r , θ, ϕ) = zn (r )Pn (θ) exp(imϕ).
2 For r → 0 or ∞: zn (r ) behaves like r k .
3 Series U, V (r) = ν aν ψν (r) can be transformed into power series.
4 Radius of convergence is determined by the nearest singularity.
For scattered field: r ∈ (max d sca , ∞) .
For internal field: r ∈ 0, min d int .
5 In spheroidal coordinates, it is not that simple!
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 13 / 46
22. Near Field
1 In spherical coordinates
m
ψν (r , θ, ϕ) = zn (r )Pn (θ) exp(imϕ).
2 For r → 0 or ∞: zn (r ) behaves like r k .
3 Series U, V (r) = ν aν ψν (r) can be transformed into power series.
4 Radius of convergence is determined by the nearest singularity.
For scattered field: r ∈ (max d sca , ∞) .
For internal field: r ∈ 0, min d int .
5 In spheroidal coordinates, it is not that simple!
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 13 / 46
23. Near Field
1 In spherical coordinates
m
ψν (r , θ, ϕ) = zn (r )Pn (θ) exp(imϕ).
2 For r → 0 or ∞: zn (r ) behaves like r k .
3 Series U, V (r) = ν aν ψν (r) can be transformed into power series.
4 Radius of convergence is determined by the nearest singularity.
For scattered field: r ∈ (max d sca , ∞) .
For internal field: r ∈ 0, min d int .
5 In spheroidal coordinates, it is not that simple!
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 13 / 46
27. Spheroid convergence in the near field, a/b = 1.4
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 17 / 46
28. Spheroid convergence in the near field, a/b = 1.8
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 18 / 46
29. Spheroid convergence in the near field, a/b = 2.5
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 19 / 46
30. Spheroid convergence in the near field, a/b = 3.5
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 20 / 46
31. Rayleigh hypothesis
We use a generalized and simplified definition of the term as an
assumption that field expansions in terms of wave functions
converge everywhere up to the scatterer surface.
In spherical coordinates and for spherical basis:
max d sca < min r (θ, ϕ) and max r (θ, ϕ) < min d int .
As field expansions are substituted in the boundary conditions,
Rayleigh hypothesis seems to be required to be valid.
However, we know that the methods provide accurate results when
Rayleigh hypothesis is not valid.
Do we realy need Rayleigh hypothesis to be valid?
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 21 / 46
32. Rayleigh hypothesis
We use a generalized and simplified definition of the term as an
assumption that field expansions in terms of wave functions
converge everywhere up to the scatterer surface.
In spherical coordinates and for spherical basis:
max d sca < min r (θ, ϕ) and max r (θ, ϕ) < min d int .
As field expansions are substituted in the boundary conditions,
Rayleigh hypothesis seems to be required to be valid.
However, we know that the methods provide accurate results when
Rayleigh hypothesis is not valid.
Do we realy need Rayleigh hypothesis to be valid?
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 21 / 46
33. Rayleigh hypothesis
We use a generalized and simplified definition of the term as an
assumption that field expansions in terms of wave functions
converge everywhere up to the scatterer surface.
In spherical coordinates and for spherical basis:
max d sca < min r (θ, ϕ) and max r (θ, ϕ) < min d int .
As field expansions are substituted in the boundary conditions,
Rayleigh hypothesis seems to be required to be valid.
However, we know that the methods provide accurate results when
Rayleigh hypothesis is not valid.
Do we realy need Rayleigh hypothesis to be valid?
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 21 / 46
34. Rayleigh hypothesis
We use a generalized and simplified definition of the term as an
assumption that field expansions in terms of wave functions
converge everywhere up to the scatterer surface.
In spherical coordinates and for spherical basis:
max d sca < min r (θ, ϕ) and max r (θ, ϕ) < min d int .
As field expansions are substituted in the boundary conditions,
Rayleigh hypothesis seems to be required to be valid.
However, we know that the methods provide accurate results when
Rayleigh hypothesis is not valid.
Do we realy need Rayleigh hypothesis to be valid?
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 21 / 46
35. Rayleigh hypothesis
We use a generalized and simplified definition of the term as an
assumption that field expansions in terms of wave functions
converge everywhere up to the scatterer surface.
In spherical coordinates and for spherical basis:
max d sca < min r (θ, ϕ) and max r (θ, ϕ) < min d int .
As field expansions are substituted in the boundary conditions,
Rayleigh hypothesis seems to be required to be valid.
However, we know that the methods provide accurate results when
Rayleigh hypothesis is not valid.
Do we realy need Rayleigh hypothesis to be valid?
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 21 / 46
36. Infinite linear systems analysis
a11 a12 · · · x1 b1
a21 a22 · · · x2 b2
=
. . .. . .
.
. .
. . .
. .
.
Kantorovich & Krylov (1958)
Regular systems: ρi = 1 − ∞ |aik | > 0, (i = 1, 2, . . .).
k=1
Theorem. If ∃K > 0 : |bi | < K ρi , (i = 1, 2, . . .), then
regular system is solvable,
it has the only solution,
solutions of truncated systems converge to it.
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 22 / 46
37. Infinite linear systems analysis
a11 a12 · · · x1 b1
a21 a22 · · · x2 b2
=
. . .. . .
.
. .
. . .
. .
.
Kantorovich & Krylov (1958)
Regular systems: ρi = 1 − ∞ |aik | > 0, (i = 1, 2, . . .).
k=1
Theorem. If ∃K > 0 : |bi | < K ρi , (i = 1, 2, . . .), then
regular system is solvable,
it has the only solution,
solutions of truncated systems converge to it.
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 22 / 46
38. Infinite linear systems analysis
a11 a12 · · · x1 b1
a21 a22 · · · x2 b2
=
. . .. . .
.
. .
. . .
. .
.
Kantorovich & Krylov (1958)
Regular systems: ρi = 1 − ∞ |aik | > 0, (i = 1, 2, . . .).
k=1
Theorem. If ∃K > 0 : |bi | < K ρi , (i = 1, 2, . . .), then
regular system is solvable,
it has the only solution,
solutions of truncated systems converge to it.
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 22 / 46
39. Analysis of EBCM, gSVM and gPMM systems
gPMM
System has positively determined matrix and hence has always the only
solution.
EBCM
System is regular and satisfies solvability condition if
max d sca < min d int .
gSVM
There is no such condition for SVM.
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 23 / 46
40. Analysis of EBCM, gSVM and gPMM systems
gPMM
System has positively determined matrix and hence has always the only
solution.
EBCM
System is regular and satisfies solvability condition if
max d sca < min d int .
gSVM
There is no such condition for SVM.
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 23 / 46
41. Analysis of EBCM, gSVM and gPMM systems
gPMM
System has positively determined matrix and hence has always the only
solution.
EBCM
System is regular and satisfies solvability condition if
max d sca < min d int .
gSVM
There is no such condition for SVM.
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 23 / 46
47. Convergence of results, Chebyshev particle, n = 5, ε = 0.07
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 29 / 46
48. Convergence of results, Chebyshev particle, n = 5, ε = 0.14
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 30 / 46
49. Convergence of results, Chebyshev particle, n = 5, ε = 0.21
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 31 / 46
50. Paradox of the EBCM
EBCM solutions converge even if field expansions used in the
boundary conditions diverge.
How is this possible?
Let’s consider Pattern Equation Method3 .
Search for far field pattern in terms of angular parts of ψν (as r → ∞)
As the patterns are defined only in the far field zone, one does not
need convergence of any expansions at scatterer boundary.
We found that the infinite systems arisen in EBCM coincide with
those arisen in the PEM.
When Rayleigh hypothesis is not valid, EBCM is not
mathematically correct, but its applicability is extended in the
far field due to lucky coincindence with PEM.
3
see works by Kyurkchan and Smirnova
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 32 / 46
51. Paradox of the EBCM
EBCM solutions converge even if field expansions used in the
boundary conditions diverge.
How is this possible?
Let’s consider Pattern Equation Method3 .
Search for far field pattern in terms of angular parts of ψν (as r → ∞)
As the patterns are defined only in the far field zone, one does not
need convergence of any expansions at scatterer boundary.
We found that the infinite systems arisen in EBCM coincide with
those arisen in the PEM.
When Rayleigh hypothesis is not valid, EBCM is not
mathematically correct, but its applicability is extended in the
far field due to lucky coincindence with PEM.
3
see works by Kyurkchan and Smirnova
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 32 / 46
52. Paradox of the EBCM
EBCM solutions converge even if field expansions used in the
boundary conditions diverge.
How is this possible?
Let’s consider Pattern Equation Method3 .
Search for far field pattern in terms of angular parts of ψν (as r → ∞)
As the patterns are defined only in the far field zone, one does not
need convergence of any expansions at scatterer boundary.
We found that the infinite systems arisen in EBCM coincide with
those arisen in the PEM.
When Rayleigh hypothesis is not valid, EBCM is not
mathematically correct, but its applicability is extended in the
far field due to lucky coincindence with PEM.
3
see works by Kyurkchan and Smirnova
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 32 / 46
53. Paradox of the EBCM
EBCM solutions converge even if field expansions used in the
boundary conditions diverge.
How is this possible?
Let’s consider Pattern Equation Method3 .
Search for far field pattern in terms of angular parts of ψν (as r → ∞)
As the patterns are defined only in the far field zone, one does not
need convergence of any expansions at scatterer boundary.
We found that the infinite systems arisen in EBCM coincide with
those arisen in the PEM.
When Rayleigh hypothesis is not valid, EBCM is not
mathematically correct, but its applicability is extended in the
far field due to lucky coincindence with PEM.
3
see works by Kyurkchan and Smirnova
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 32 / 46
54. Paradox of the EBCM
EBCM solutions converge even if field expansions used in the
boundary conditions diverge.
How is this possible?
Let’s consider Pattern Equation Method3 .
Search for far field pattern in terms of angular parts of ψν (as r → ∞)
As the patterns are defined only in the far field zone, one does not
need convergence of any expansions at scatterer boundary.
We found that the infinite systems arisen in EBCM coincide with
those arisen in the PEM.
When Rayleigh hypothesis is not valid, EBCM is not
mathematically correct, but its applicability is extended in the
far field due to lucky coincindence with PEM.
3
see works by Kyurkchan and Smirnova
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 32 / 46
55. Equivalence of the EBCM and gSVM systems
It was generally shown earlier (e.g., Schmidt et al., 1998).
We have strictly demonstrated that the matrix of EBCM infinite
system can be transformed into the matrix of gSVM system and vice
versa.
Qs = i CT B − AT D , Qr = i FT B − ET D ,
where A = ASVM , B = BSVM , . . .
If in iPMM residual ∆ = 0, then
APMM = AT∗ A + CT∗ A, . . .
Hence, iPMM infinite system is also equivalent to gSVM system.
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 33 / 46
56. Equivalence of the EBCM and gSVM systems
It was generally shown earlier (e.g., Schmidt et al., 1998).
We have strictly demonstrated that the matrix of EBCM infinite
system can be transformed into the matrix of gSVM system and vice
versa.
Qs = i CT B − AT D , Qr = i FT B − ET D ,
where A = ASVM , B = BSVM , . . .
If in iPMM residual ∆ = 0, then
APMM = AT∗ A + CT∗ A, . . .
Hence, iPMM infinite system is also equivalent to gSVM system.
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 33 / 46
57. Equivalence of the EBCM and gSVM systems
It was generally shown earlier (e.g., Schmidt et al., 1998).
We have strictly demonstrated that the matrix of EBCM infinite
system can be transformed into the matrix of gSVM system and vice
versa.
Qs = i CT B − AT D , Qr = i FT B − ET D ,
where A = ASVM , B = BSVM , . . .
If in iPMM residual ∆ = 0, then
APMM = AT∗ A + CT∗ A, . . .
Hence, iPMM infinite system is also equivalent to gSVM system.
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 33 / 46
58. Equivalence of the EBCM and gSVM systems
It was generally shown earlier (e.g., Schmidt et al., 1998).
We have strictly demonstrated that the matrix of EBCM infinite
system can be transformed into the matrix of gSVM system and vice
versa.
Qs = i CT B − AT D , Qr = i FT B − ET D ,
where A = ASVM , B = BSVM , . . .
If in iPMM residual ∆ = 0, then
APMM = AT∗ A + CT∗ A, . . .
Hence, iPMM infinite system is also equivalent to gSVM system.
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 33 / 46
59. Truncation of infinite systems
For truncated systems the proof of equivalence is not correct.
For EBCM and iPMM we have regular systems.
For gSVM we couldn’t prove that systems are regular.
Infinite EBCM and gSVM systems are equivalent, but
truncated are not.
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 34 / 46
60. Truncation of infinite systems
For truncated systems the proof of equivalence is not correct.
For EBCM and iPMM we have regular systems.
For gSVM we couldn’t prove that systems are regular.
Infinite EBCM and gSVM systems are equivalent, but
truncated are not.
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 34 / 46
61. Truncation of infinite systems
For truncated systems the proof of equivalence is not correct.
For EBCM and iPMM we have regular systems.
For gSVM we couldn’t prove that systems are regular.
Infinite EBCM and gSVM systems are equivalent, but
truncated are not.
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 34 / 46
62. Truncation of infinite systems
For truncated systems the proof of equivalence is not correct.
For EBCM and iPMM we have regular systems.
For gSVM we couldn’t prove that systems are regular.
Infinite EBCM and gSVM systems are equivalent, but
truncated are not.
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 34 / 46
74. Conclusions
1 Methods are very similar, but have key differencies.
2 Methods applicability ranges are affected by singularities.
3 Rayleigh hypothesis is required for near field computations.
4 EBCM has solvability condition for far field.
5 Infinite matrices of the methods’ systems are equivalent.
6 Truncated matrices are not.
7 Different methods are good for different particles.
8 Systems are ill-conditioned from the beginning.
9 Because of solvability condition EBCM doesn’t provide accurate
results for layered scatterers.
Il’in, Farafonov, Vinokurov (Russia) ELS-XII 46 / 46