The document discusses Hermite integrators and Riordan arrays. It provides:
1) An overview of a general form for the correctors of a family of 2-step Hermite integrators that achieve 2(p+1)-th order accuracy by directly calculating up to the p-th order derivative of the force.
2) Details on constructing Hermite integrators by solving a linear equation to determine the coefficients, with an example of a 6th order integrator.
3) An outline of the proof for the general form of the coefficients, which involves setting up a differential recurrence relation and solving a linear system using LU decomposition, with the proof of the inverse matrices later shown using Riordan arrays.
Convergence methods for approximated reciprocal and reciprocal-square-rootKeigo Nitadori
This document discusses convergence methods for approximating reciprocal (1/x) and reciprocal-square-root (1/√x) values using polynomials expanded in a Taylor series. It presents the general forms for reciprocal and reciprocal-square-root approximations up to eighth order. For reciprocal-square-root, the well-known Newton-Raphson method and its optimized form for fused multiply-add hardware are described. Examples for other functions like reciprocal-cube-root are also provided. Higher order methods are noted to require more registers for coefficient storage.
Solving the energy problem of helium final reportJamesMa54
The document discusses solving the ground state energy of a helium atom. It involves computing the Hamiltonian and overlap matrices (H and S) of the system by representing the wavefunction as a linear combination of basis functions. Computing the entries of H and S requires evaluating triple integrals over the internal coordinates of the atom. The main work is to derive a general closed form for these integrals. This involves repeatedly using integration by parts to reduce the exponents in the integrands, yielding sums of terms that can be directly evaluated or fed into computational software for further analysis. Solving these integrals is the crucial step to enable determining the ground state energy by solving the eigenvalue problem Hc = λSc.
The Solovay-Kitaev Theorem guarantees that for any single-qubit gate U and precision ε > 0, it is possible to approximate U to within ε using Θ(logc(1/ε)) gates from a fixed finite universal set of quantum gates. The proof involves first using a "shrinking lemma" to show that any gate in an ε-net can be approximated to within Cε using a constant number of applications of gates from the universal set. This is then iterated to construct an approximation of the desired gate U to arbitrary precision using a number of gates that scales as the logarithm of 1/ε.
Fast and efficient exact synthesis of single qubit unitaries generated by cli...JamesMa54
The document describes a presentation on an algorithm for exact synthesis of single qubit unitaries generated by Clifford and T gates. The algorithm reduces the problem of implementing a unitary to the problem of state preparation. It then uses a series of HT gates to iteratively decrease the smallest denominator exponent of the state entries until it reaches a base case that can be looked up. The algorithm runs in time linear in the initial smallest denominator exponent and provides an optimal sequence of H and T gates for implementing the input unitary exactly.
Trilinear embedding for divergence-form operatorsVjekoslavKovac1
The document discusses a trilinear embedding theorem for divergence-form operators with complex coefficients. It proves that if matrices A, B, C are appropriately p,q,r-elliptic, then there is a bound on the integral of the product of the gradients of the semigroups associated with the operators. The proof uses a Bellman function technique and shows the relationship to the concept of p-ellipticity. It generalizes previous work on bilinear embeddings to the trilinear case.
Using blurred images to assess damage in bridge structures?Alessandro Palmeri
Faster trains and augmented traffic have significantly increased the number and amplitude of loading cycles experienced on a daily basis by composite steel-concrete bridges. This higher demand accelerates the occurrence of damage in the shear connectors between the two materials, which in turn can severely affect performance and reliability of these structures. The aim of this talk is to present the preliminary results of theoretical and experimental investigations undertaken to assess the feasibility of using the envelope of deflections and rotations induced by moving loads as a practical and cost-effective alternative to traditional methods of health monitoring for composite bridges. Both analytical and numerical formulations for this dynamic problem are presented and the results of a parametric study are discussed. A novel photogrammetric approach is also introduced, which allows identifying vibration patterns in civil engineering structures by analysing blurred targets in long-exposure digital images. The initial experimental validation of this approach is presented and further challenges are highlighted.
Quantitative norm convergence of some ergodic averagesVjekoslavKovac1
The document summarizes quantitative estimates for the convergence of multiple ergodic averages of commuting transformations. Specifically, it presents a theorem that provides an explicit bound on the number of jumps in the Lp norm for double averages over commuting Aω actions on a probability space. The proof transfers the structure of the Cantor group AZ to R+ and establishes norm estimates for bilinear averages of functions on R2+. This allows bounding the variation of the double averages and proving the theorem.
The document discusses Hermite integrators and Riordan arrays. It provides:
1) An overview of a general form for the correctors of a family of 2-step Hermite integrators that achieve 2(p+1)-th order accuracy by directly calculating up to the p-th order derivative of the force.
2) Details on constructing Hermite integrators by solving a linear equation to determine the coefficients, with an example of a 6th order integrator.
3) An outline of the proof for the general form of the coefficients, which involves setting up a differential recurrence relation and solving a linear system using LU decomposition, with the proof of the inverse matrices later shown using Riordan arrays.
Convergence methods for approximated reciprocal and reciprocal-square-rootKeigo Nitadori
This document discusses convergence methods for approximating reciprocal (1/x) and reciprocal-square-root (1/√x) values using polynomials expanded in a Taylor series. It presents the general forms for reciprocal and reciprocal-square-root approximations up to eighth order. For reciprocal-square-root, the well-known Newton-Raphson method and its optimized form for fused multiply-add hardware are described. Examples for other functions like reciprocal-cube-root are also provided. Higher order methods are noted to require more registers for coefficient storage.
Solving the energy problem of helium final reportJamesMa54
The document discusses solving the ground state energy of a helium atom. It involves computing the Hamiltonian and overlap matrices (H and S) of the system by representing the wavefunction as a linear combination of basis functions. Computing the entries of H and S requires evaluating triple integrals over the internal coordinates of the atom. The main work is to derive a general closed form for these integrals. This involves repeatedly using integration by parts to reduce the exponents in the integrands, yielding sums of terms that can be directly evaluated or fed into computational software for further analysis. Solving these integrals is the crucial step to enable determining the ground state energy by solving the eigenvalue problem Hc = λSc.
The Solovay-Kitaev Theorem guarantees that for any single-qubit gate U and precision ε > 0, it is possible to approximate U to within ε using Θ(logc(1/ε)) gates from a fixed finite universal set of quantum gates. The proof involves first using a "shrinking lemma" to show that any gate in an ε-net can be approximated to within Cε using a constant number of applications of gates from the universal set. This is then iterated to construct an approximation of the desired gate U to arbitrary precision using a number of gates that scales as the logarithm of 1/ε.
Fast and efficient exact synthesis of single qubit unitaries generated by cli...JamesMa54
The document describes a presentation on an algorithm for exact synthesis of single qubit unitaries generated by Clifford and T gates. The algorithm reduces the problem of implementing a unitary to the problem of state preparation. It then uses a series of HT gates to iteratively decrease the smallest denominator exponent of the state entries until it reaches a base case that can be looked up. The algorithm runs in time linear in the initial smallest denominator exponent and provides an optimal sequence of H and T gates for implementing the input unitary exactly.
Trilinear embedding for divergence-form operatorsVjekoslavKovac1
The document discusses a trilinear embedding theorem for divergence-form operators with complex coefficients. It proves that if matrices A, B, C are appropriately p,q,r-elliptic, then there is a bound on the integral of the product of the gradients of the semigroups associated with the operators. The proof uses a Bellman function technique and shows the relationship to the concept of p-ellipticity. It generalizes previous work on bilinear embeddings to the trilinear case.
Using blurred images to assess damage in bridge structures?Alessandro Palmeri
Faster trains and augmented traffic have significantly increased the number and amplitude of loading cycles experienced on a daily basis by composite steel-concrete bridges. This higher demand accelerates the occurrence of damage in the shear connectors between the two materials, which in turn can severely affect performance and reliability of these structures. The aim of this talk is to present the preliminary results of theoretical and experimental investigations undertaken to assess the feasibility of using the envelope of deflections and rotations induced by moving loads as a practical and cost-effective alternative to traditional methods of health monitoring for composite bridges. Both analytical and numerical formulations for this dynamic problem are presented and the results of a parametric study are discussed. A novel photogrammetric approach is also introduced, which allows identifying vibration patterns in civil engineering structures by analysing blurred targets in long-exposure digital images. The initial experimental validation of this approach is presented and further challenges are highlighted.
Quantitative norm convergence of some ergodic averagesVjekoslavKovac1
The document summarizes quantitative estimates for the convergence of multiple ergodic averages of commuting transformations. Specifically, it presents a theorem that provides an explicit bound on the number of jumps in the Lp norm for double averages over commuting Aω actions on a probability space. The proof transfers the structure of the Cantor group AZ to R+ and establishes norm estimates for bilinear averages of functions on R2+. This allows bounding the variation of the double averages and proving the theorem.
On maximal and variational Fourier restrictionVjekoslavKovac1
Workshop talk slides, Follow-up workshop to trimester program "Harmonic Analysis and Partial Differential Equations", Hausdorff Institute, Bonn, May 2019.
Toward an Improved Computational Strategy for Vibration-Proof Structures Equi...Alessandro Palmeri
This presentation has been delivered at the 15th World Conference on Earthquake Engineering in Lisbon (Portugal) on 28th September 2012, and shows some preliminary results on the dynamic analysis on non-linear viscoelastic structures.
The document discusses minimum spanning trees (MST) and two algorithms for finding them: Prim's algorithm and Kruskal's algorithm. It begins by defining an MST as a spanning tree (connected acyclic graph containing all vertices) with minimum total edge weight. Prim's algorithm grows a single tree by repeatedly adding the minimum weight edge connecting the growing tree to another vertex. Kruskal's algorithm grows a forest by repeatedly merging two components via the minimum weight edge connecting them. Both algorithms produce optimal MSTs by adding only "safe" edges that cannot be part of a cycle.
A Note on the Derivation of the Variational Inference Updates for DILNTomonari Masada
This document describes the derivation of the variational inference updates for the Dirichlet-logistic normal model. It begins by defining the joint distribution and obtaining a lower bound on the log evidence using Jensen's inequality. It then examines each term in the lower bound, applying integrals and derivations. This results in an expression for the lower bound involving parameters of the model distributions. The document concludes by stating assumptions made about some of the distributions and providing the update equations for variational distributions q(Cmn) and q(Zmk).
This document summarizes optimization techniques for matrix factorization and completion problems. Section 8.1 introduces the matrix factorization problem and considers minimizing reconstruction error subject to a nuclear norm penalty. Section 8.2 discusses properties of the nuclear norm, including relationships to the trace norm and Frobenius norm. Section 8.3 provides performance guarantees for matrix completion when the underlying matrix is exactly low-rank. Section 8.4 describes proximal gradient methods for optimization, including updates that involve singular value thresholding. The document concludes by discussing an extension of these methods to dictionary learning and alignment problems.
This document summarizes research on norm-variation estimates for ergodic bilinear and multiple averages. It begins by motivating the study of ergodic averages and their convergence properties. Previous results are discussed that provide pointwise convergence and norm estimates for certain cases. The document then presents new norm-variation estimates obtained by the authors for bilinear and multiple ergodic averages over general measure-preserving systems. These estimates bound the number of jumps in the L2 norm as the averages converge. Finally, analogous results are discussed for bilinear averages on R2 and Z2, linking the estimates to established bounds for singular integrals.
Phase diagram at finite T & Mu in strong coupling limit of lattice QCDBenjamin Jaedon Choi
This document summarizes the derivation of an effective free energy for QCD at strong coupling using a mean field approximation with 1 flavor staggered fermion. Key steps include:
1) Performing a path integral over spatial link variables to obtain quark propagators.
2) Introducing auxiliary bosonic fields using a Hubbard-Stratonovich transformation to obtain a bilinear form in quark fields.
3) Applying a mean field approximation to the auxiliary fields.
4) Exactly integrating over temporal links, quark and auxiliary baryon fields to obtain an effective free energy in terms of the auxiliary meson field.
5) Analyzing the effective free energy to determine the QCD phase diagram as functions of temperature and
This document summarizes Andrew Hone's talk on reductions of the discrete Hirota (discrete KP) equation. Plane wave reductions of the discrete Hirota equation yield Somos-type recurrence relations. Reductions of the discrete Hirota Lax pair give scalar Lax pairs with spectral parameters. Certain reductions produce periodic coefficients, leading to cluster algebra structures. Reductions of the discrete KdV equation are also considered, giving bi-Hamiltonian structures.
The document summarizes research on magnetic monopoles in noncommutative spacetime. It begins by motivating noncommutative spacetime as a way to incorporate quantum gravitational effects. It then shows that attempting to quantize spacetime by imposing noncommutativity of coordinates leads to inconsistencies when trying to define a Wu-Yang magnetic monopole in this framework. Specifically, the potentials describing the monopole fail to simultaneously satisfy Maxwell's equations and transform correctly under gauge transformations when expanded to second order in the noncommutativity parameter. This suggests the Dirac quantization condition cannot be satisfied in noncommutative spacetime. Possible reasons for this failure and directions for future work are discussed.
The document describes Johnson's algorithm for finding shortest paths between all pairs of vertices in a sparse graph. It discusses how the algorithm uses reweighting to compute new edge weights that preserve shortest paths while making all weights nonnegative. It shows how Dijkstra's algorithm can then be run on the reweighted graph to find shortest paths between all pairs of vertices. The key steps are: (1) adding a source node and zero-weight edges, (2) running Bellman-Ford to compute distances from the source, (3) using these distances to reweight the edges while preserving shortest paths, resulting in nonnegative weights.
Solving TD-DFT/BSE equations with Lanczos-Haydock approachClaudio Attaccalite
This document discusses methods for solving the time-dependent density functional theory (TD-DFT) and Bethe-Salpeter equation (BSE) using the Lanczos-Haydock approach. Specifically:
1. The Casida approach rewrites the TD-DFT/BSE equations in a basis of electron-hole pairs, but this matrix can be very large.
2. The Tamm-Dancoff approximation reduces the matrix size by only considering positive electron-hole pairs.
3. The Lanczos-Haydock approach allows rewriting the dielectric function in terms of the ground state and eigenstates of the BSE Hamiltonian, which can then be solved iteratively without directly diagonalizing
Gamma Function mathematics and history.
Please send comments and suggestions for improvements to solo.hermelin@gmail.com. Thanks.
More presentations on different subjects can be found on my website at http://www.solohermelin.com.
Second part of Matrices at undergraduate in science (math, physics, engineering) level.
Please send comments and suggestions to solo.hermelin@gmail.com.
For more presentations visit my website at
http://www.solohermelin.com.
The document discusses algorithms for finding the minimum spanning tree of a graph. It begins by defining what a spanning tree is - a subset of edges that connects all vertices using the fewest possible edges. It then explains Kruskal's and Prim's algorithms for finding the minimum spanning tree, which is the spanning tree with the lowest total edge weight. Kruskal's algorithm sorts the edges by weight and builds the tree by adding edges that do not create cycles. Prim's algorithm grows the tree from an initial vertex by always adding the lowest weight edge that connects to a new vertex. Pseudocode is provided for both algorithms.
Response Surface in Tensor Train format for Uncertainty QuantificationAlexander Litvinenko
We apply low-rank Tensor Train format to solve PDEs with uncertain coefficients. First, we approximate uncertain permeability coefficient in TT format, then the operator and then apply iterations to solve stochastic Galerkin system.
(Neamen)solution manual for semiconductor physics and devices 3edKadu Brito
This document contains solutions to problems from Chapter 1 of Semiconductor Physics and Devices: Basic Principles, 3rd edition. The problems calculate properties of semiconductor unit cells such as number of atoms, density, and volume percentage occupied by atoms. Lattice structures including simple cubic, body-centered cubic, face-centered cubic and diamond are considered. Properties of common semiconductors such as silicon and gallium arsenide are also calculated.
Kittel c. introduction to solid state physics 8 th edition - solution manualamnahnura
1. The document discusses crystallographic planes and directions in a cube, the Miller indices of planes with respect to primitive axes, and the spacing between dots projected onto different planes of a crystal structure.
2. Key concepts from crystallography such as Miller indices, primitive lattice vectors, reciprocal lattice vectors, and the first Brillouin zone are defined. Calculations of interplanar spacing and lattice parameters are shown for simple cubic and face-centered cubic lattices.
3. Binding energies, cohesive energies, and equilibrium properties are calculated and compared for body-centered cubic and face-centered cubic crystal structures. Approximations made in describing crystal binding using Madelung energies and pair potentials are
This document discusses two methods for balancing flexible rotors: the N method and the (N+2) method.
- The N method uses N balancing planes to balance a rotor up to and including the Nth critical speed. It aims to satisfy the condition A=0 from equation (13).
- The (N+2) method requires two additional balancing planes (for a total of N+2 planes) to balance the same speed range as the N method. It aims to satisfy both the conditions A=0 and B=0 from equations (13a) and (13b).
- While the two methods differ for a finite number of balancing planes N, in the limiting case of an infinite
On maximal and variational Fourier restrictionVjekoslavKovac1
Workshop talk slides, Follow-up workshop to trimester program "Harmonic Analysis and Partial Differential Equations", Hausdorff Institute, Bonn, May 2019.
Toward an Improved Computational Strategy for Vibration-Proof Structures Equi...Alessandro Palmeri
This presentation has been delivered at the 15th World Conference on Earthquake Engineering in Lisbon (Portugal) on 28th September 2012, and shows some preliminary results on the dynamic analysis on non-linear viscoelastic structures.
The document discusses minimum spanning trees (MST) and two algorithms for finding them: Prim's algorithm and Kruskal's algorithm. It begins by defining an MST as a spanning tree (connected acyclic graph containing all vertices) with minimum total edge weight. Prim's algorithm grows a single tree by repeatedly adding the minimum weight edge connecting the growing tree to another vertex. Kruskal's algorithm grows a forest by repeatedly merging two components via the minimum weight edge connecting them. Both algorithms produce optimal MSTs by adding only "safe" edges that cannot be part of a cycle.
A Note on the Derivation of the Variational Inference Updates for DILNTomonari Masada
This document describes the derivation of the variational inference updates for the Dirichlet-logistic normal model. It begins by defining the joint distribution and obtaining a lower bound on the log evidence using Jensen's inequality. It then examines each term in the lower bound, applying integrals and derivations. This results in an expression for the lower bound involving parameters of the model distributions. The document concludes by stating assumptions made about some of the distributions and providing the update equations for variational distributions q(Cmn) and q(Zmk).
This document summarizes optimization techniques for matrix factorization and completion problems. Section 8.1 introduces the matrix factorization problem and considers minimizing reconstruction error subject to a nuclear norm penalty. Section 8.2 discusses properties of the nuclear norm, including relationships to the trace norm and Frobenius norm. Section 8.3 provides performance guarantees for matrix completion when the underlying matrix is exactly low-rank. Section 8.4 describes proximal gradient methods for optimization, including updates that involve singular value thresholding. The document concludes by discussing an extension of these methods to dictionary learning and alignment problems.
This document summarizes research on norm-variation estimates for ergodic bilinear and multiple averages. It begins by motivating the study of ergodic averages and their convergence properties. Previous results are discussed that provide pointwise convergence and norm estimates for certain cases. The document then presents new norm-variation estimates obtained by the authors for bilinear and multiple ergodic averages over general measure-preserving systems. These estimates bound the number of jumps in the L2 norm as the averages converge. Finally, analogous results are discussed for bilinear averages on R2 and Z2, linking the estimates to established bounds for singular integrals.
Phase diagram at finite T & Mu in strong coupling limit of lattice QCDBenjamin Jaedon Choi
This document summarizes the derivation of an effective free energy for QCD at strong coupling using a mean field approximation with 1 flavor staggered fermion. Key steps include:
1) Performing a path integral over spatial link variables to obtain quark propagators.
2) Introducing auxiliary bosonic fields using a Hubbard-Stratonovich transformation to obtain a bilinear form in quark fields.
3) Applying a mean field approximation to the auxiliary fields.
4) Exactly integrating over temporal links, quark and auxiliary baryon fields to obtain an effective free energy in terms of the auxiliary meson field.
5) Analyzing the effective free energy to determine the QCD phase diagram as functions of temperature and
This document summarizes Andrew Hone's talk on reductions of the discrete Hirota (discrete KP) equation. Plane wave reductions of the discrete Hirota equation yield Somos-type recurrence relations. Reductions of the discrete Hirota Lax pair give scalar Lax pairs with spectral parameters. Certain reductions produce periodic coefficients, leading to cluster algebra structures. Reductions of the discrete KdV equation are also considered, giving bi-Hamiltonian structures.
The document summarizes research on magnetic monopoles in noncommutative spacetime. It begins by motivating noncommutative spacetime as a way to incorporate quantum gravitational effects. It then shows that attempting to quantize spacetime by imposing noncommutativity of coordinates leads to inconsistencies when trying to define a Wu-Yang magnetic monopole in this framework. Specifically, the potentials describing the monopole fail to simultaneously satisfy Maxwell's equations and transform correctly under gauge transformations when expanded to second order in the noncommutativity parameter. This suggests the Dirac quantization condition cannot be satisfied in noncommutative spacetime. Possible reasons for this failure and directions for future work are discussed.
The document describes Johnson's algorithm for finding shortest paths between all pairs of vertices in a sparse graph. It discusses how the algorithm uses reweighting to compute new edge weights that preserve shortest paths while making all weights nonnegative. It shows how Dijkstra's algorithm can then be run on the reweighted graph to find shortest paths between all pairs of vertices. The key steps are: (1) adding a source node and zero-weight edges, (2) running Bellman-Ford to compute distances from the source, (3) using these distances to reweight the edges while preserving shortest paths, resulting in nonnegative weights.
Solving TD-DFT/BSE equations with Lanczos-Haydock approachClaudio Attaccalite
This document discusses methods for solving the time-dependent density functional theory (TD-DFT) and Bethe-Salpeter equation (BSE) using the Lanczos-Haydock approach. Specifically:
1. The Casida approach rewrites the TD-DFT/BSE equations in a basis of electron-hole pairs, but this matrix can be very large.
2. The Tamm-Dancoff approximation reduces the matrix size by only considering positive electron-hole pairs.
3. The Lanczos-Haydock approach allows rewriting the dielectric function in terms of the ground state and eigenstates of the BSE Hamiltonian, which can then be solved iteratively without directly diagonalizing
Gamma Function mathematics and history.
Please send comments and suggestions for improvements to solo.hermelin@gmail.com. Thanks.
More presentations on different subjects can be found on my website at http://www.solohermelin.com.
Second part of Matrices at undergraduate in science (math, physics, engineering) level.
Please send comments and suggestions to solo.hermelin@gmail.com.
For more presentations visit my website at
http://www.solohermelin.com.
The document discusses algorithms for finding the minimum spanning tree of a graph. It begins by defining what a spanning tree is - a subset of edges that connects all vertices using the fewest possible edges. It then explains Kruskal's and Prim's algorithms for finding the minimum spanning tree, which is the spanning tree with the lowest total edge weight. Kruskal's algorithm sorts the edges by weight and builds the tree by adding edges that do not create cycles. Prim's algorithm grows the tree from an initial vertex by always adding the lowest weight edge that connects to a new vertex. Pseudocode is provided for both algorithms.
Response Surface in Tensor Train format for Uncertainty QuantificationAlexander Litvinenko
We apply low-rank Tensor Train format to solve PDEs with uncertain coefficients. First, we approximate uncertain permeability coefficient in TT format, then the operator and then apply iterations to solve stochastic Galerkin system.
(Neamen)solution manual for semiconductor physics and devices 3edKadu Brito
This document contains solutions to problems from Chapter 1 of Semiconductor Physics and Devices: Basic Principles, 3rd edition. The problems calculate properties of semiconductor unit cells such as number of atoms, density, and volume percentage occupied by atoms. Lattice structures including simple cubic, body-centered cubic, face-centered cubic and diamond are considered. Properties of common semiconductors such as silicon and gallium arsenide are also calculated.
Kittel c. introduction to solid state physics 8 th edition - solution manualamnahnura
1. The document discusses crystallographic planes and directions in a cube, the Miller indices of planes with respect to primitive axes, and the spacing between dots projected onto different planes of a crystal structure.
2. Key concepts from crystallography such as Miller indices, primitive lattice vectors, reciprocal lattice vectors, and the first Brillouin zone are defined. Calculations of interplanar spacing and lattice parameters are shown for simple cubic and face-centered cubic lattices.
3. Binding energies, cohesive energies, and equilibrium properties are calculated and compared for body-centered cubic and face-centered cubic crystal structures. Approximations made in describing crystal binding using Madelung energies and pair potentials are
This document discusses two methods for balancing flexible rotors: the N method and the (N+2) method.
- The N method uses N balancing planes to balance a rotor up to and including the Nth critical speed. It aims to satisfy the condition A=0 from equation (13).
- The (N+2) method requires two additional balancing planes (for a total of N+2 planes) to balance the same speed range as the N method. It aims to satisfy both the conditions A=0 and B=0 from equations (13a) and (13b).
- While the two methods differ for a finite number of balancing planes N, in the limiting case of an infinite
(i) The document discusses a computational approach using finite element method to analyze the dynamic stability of pile structures subjected to periodic loads.
(ii) It develops the governing Mathieu-Hill type eigenvalue equation to determine stability and instability regions for different ranges of static and dynamic load factors.
(iii) Key steps involve discretizing the pile into finite elements, developing element stiffness, mass and stability matrices, and assembling them to solve the eigenvalue problem and analyze dynamic stability conditions for the pile structure.
This document contains information about data structures and algorithms taught at KTH Royal Institute of Technology. It includes code templates for a contest, descriptions and implementations of common data structures like an order statistic tree and hash map, as well as summaries of mathematical and algorithmic concepts like trigonometry, probability theory, and Markov chains.
1) The document presents a wavelet collocation method for numerically solving nth order Volterra integro-differential equations. It expands the unknown function as a series of Chebyshev wavelets of the second kind with unknown coefficients.
2) It states and proves a uniform convergence theorem that establishes the convergence of approximating the solution using truncated Chebyshev wavelet series expansions.
3) The paper demonstrates the validity and applicability of the proposed method through some illustrative examples of solving integro-differential equations using the Chebyshev wavelet collocation approach.
This chapter discusses discrete image transforms. It introduces linear transformations and unitary transforms. The discrete Fourier transform (DFT) and discrete cosine transform (DCT) are presented as examples of unitary transforms. The DFT represents an image as a sum of sinusoidal basis images, while the DCT uses cosine basis images. Other transforms discussed include the discrete sine transform (DST), Hartley transform, and Hadamard transform. Orthogonal transforms preserve image properties while changing the representation basis.
This chapter discusses the stability and behavior of columns under compressive loads. It begins by introducing columns and their goals of studying stability, critical load, effective length, and the secant formula. It then covers the stability of structures and Euler's formula for pin-ended columns. Subsequent sections extend Euler's formula to other end conditions, discuss eccentric loading and the secant formula, and address design of columns under centric and eccentric loads using empirical equations and allowable stress or load resistance factor approaches.
The document discusses the Z-transform, which is a tool for analyzing and solving linear time-invariant difference equations. It defines the Z-transform, provides examples of common sequences and their corresponding Z-transforms, and discusses properties such as the region of convergence. Key topics covered include the difference between difference and differential equations, properties of linear time-invariant systems, and mapping between the s-plane and z-plane.
This document discusses Fourier series and integrals. It begins by explaining Fourier series using sines, cosines, and exponentials to represent periodic functions. Square waves are given as examples that can be expressed as infinite combinations of sines. Any periodic function can be expressed as a Fourier series. Fourier series are then derived for specific examples, including a square wave, repeating ramp, and up-down train of delta functions. Cosine series are also discussed. The document concludes by deriving the Fourier series for the delta function.
SolutionsPlease see answer in bold letters.Note pi = 3.14.docxrafbolet0
Solution
s:
Please see answer in bold letters.
Note pi = 3.1415….
1. The voltage across a 15Ω is as indicated. Find the sinusoidal expression for the current. In addition, sketch the v and i waveform on the same axis.
Note: For the graph of a and b please see attached jpg photo with filename 1ab.jpg and for c and d please see attached photo with filename 1cd.jpg.
a. 15sin20t
v= 15sin20t
By ohms law,
i = v/r
i = 15sin20t / 15
i = sin20t A
Computation of period for graphing:
v= 15sin20t
i = sin20t
w = 20 = 2pi*f
f = 3.183 Hz
Period =1/f = 0.314 seconds
b. 300sin (377t+20)
v = 300sin (377t+20)
i = 300sin (377t+20) /15
i = 20 sin (377t+20) A
Computation of period for graphing:
v = 300sin (377t+20)
i = 20 sin (377t+20)
w = 377 = 2pi*f
f = 60 Hz
Period = 1/60 = 0.017 seconds
shift to the left by:
2pi/0.017 = (20/180*pi)/x
x = 9.44x10-4 seconds
c. 60cos (wt+10)
v = 60cos (wt+10)
i = 60cos (wt+10)/15
i = 4cos (wt+10) A
Computation of period for graphing:
let’s denote the period as w sifted to the left by:
10/180*pi = pi/18
d. -45sin (wt+45)
v = -45sin (wt+45)
i = -45sin (wt+45) / 15
i = -3 sin (wt+45) A
Computation of period for graphing:
let’s denote the period as w sifted to the left by:
45/180 * pi = 1/4*pi
2. Determine the inductive reactance (in ohms) of a 5mH coil for
a. dc
Note at dc, frequency (f) = 0
Formula: XL = 2*pi*fL
XL = 2*pi* (0) (5m)
XL = 0 Ω
b. 60 Hz
Formula: XL = 2*pi*fL
XL = 2 (60) (5m)
XL = 1.885 Ω
c. 4kHz
Formula: XL = 2*pi*fL
XL = = 2*pi* (4k)(5m)
XL = 125.664 Ω
d. 1.2 MHz
Formula: XL = 2*pi*fL
XL = 2*pi* (1.2 M) (5m)
XL = 37.7 kΩ
3. Determine the frequency at which a 10 mH inductance has the following inductive reactance.
a. XL = 10 Ω
Formula: XL = 2*pi*fL
Express in terms in f:
f = XL/2 pi*L
f = 10 / (2pi*10m)
f = 159.155 Hz
b. XL = 4 kΩ
f = XL/2pi*L
f = 4k / (2pi*10m)
f = 63.662 kHz
c. XL = 12 kΩ
f = XL/2piL
f = 12k / (2pi*10m)
f = 190.99 kHz
d. XL = 0.5 kΩ
f = XL/2piL
f = 0.5k / (2pi*10m)
f = 7.958 kHz
4. Determine the frequency at which a 1.3uF capacitor has the following capacitive reactance.
a. 10 Ω
Formula: XC = 1/ (2pifC)
Expressing in terms of f:
f = 1/ (2pi*XC*C)
f = 1/ (2pi*10*1.3u)
f = 12.243 kΩ
b. 1.2 kΩ
f = 1/ (2pi*XC*C)
f = 1/ (2pi*1.2k*1.3u)
f = 102.022 Ω
c. 0.1 Ω
f = 1/ (2pi*XC*C)
f = 1/ (2pi*0.1*1.3u)
f = 1.224 MΩ
d. 2000 Ω
f = 1/ (2pi*XC*C)
f = 1/ (2pi*2000*1.3u)
f = 61.213 Ω
5. For the following pairs of voltage and current, indicate whether the element is a capacitor, an inductor and a capacitor, an inductor, or a resistor and find the value of C, L, or R if insufficient data are given.
a. v = 55 sin (377t + 50)
i = 11 sin (377t -40)
Element is inductor
In this case voltage leads current (ELI) by exactly 90 degrees so that means the circuit is inductive and the element is inductor.
XL = 55/11 = 5 Ω
we know the w=2pif so
w= 377=2pif
f= 60 Hz
To compute for th.
The document outlines the syllabus for the first semester M.Tech exam in computational structural mechanics, covering topics like static and kinematic indeterminacy, flexibility and stiffness methods, finite element analysis of beams, frames and trusses, and numerical techniques for solving systems of equations. It lists 10 questions, asking students to solve structural analysis problems using different analytical methods, perform structural modeling, and carry out structural design computations. Short notes may also be asked on topics related to matrix operations and structural analysis algorithms.
Fast parallelizable scenario-based stochastic optimizationPantelis Sopasakis
Fast parallelizable scenario-based stochastic optimization: a forward-backward LBFGS method for stochastic optimal control problems with global convergence rate guarantees. (Talk at EUCCO 2016, Leuven, Belgium).
2d beam element with combined loading bending axial and torsionrro7560
The document discusses beam theory and finite element modeling of beams and frames. It provides information on modeling beams using one-dimensional beam elements with cubic shape functions. The formulation describes defining the element stiffness matrix and calculating the element's contribution to the global structural stiffness matrix and force vector based on applied loads. Boundary conditions and sample problems are presented to demonstrate the element modeling approach.
Wang-Landau Monte Carlo simulation is a method for calculating the density of states function which can then be used to calculate thermodynamic properties like the mean value of variables. It improves on traditional Monte Carlo methods which struggle at low temperatures due to complicated energy landscapes with many local minima separated by large barriers. The Wang-Landau algorithm calculates the density of states function directly rather than relying on sampling configurations, allowing it to overcome barriers and fully explore the configuration space even at low temperatures.
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Is ellipse really a section of cone. The question intrigued me for 20 odd years after leaving high school. Finally got the proof on a cremation ground. Only thereafter I came to know of Dandelin spheres. But this proof uses only bare basics within the scope of high school course of Analytical geometry.
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Similar to A block-step version of KS regularization (20)
Hermite integrators and 2-parameter subgroup of Riordan groupKeigo Nitadori
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Higher order derivatives for N -body simulationsKeigo Nitadori
This document discusses higher order derivatives that are useful for N-body simulations. It presents formulas for calculating higher order derivatives of power functions like y=xn, and applies this to derivatives of gravitational force f=mr-3. Specifically:
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ESA/ACT Science Coffee: Diego Blas - Gravitational wave detection with orbita...Advanced-Concepts-Team
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1. A block-step version
of KS regularization
Keigo Nitadori
KS regularization
Hermite integrator
KS block-step
Summary
A block-step version of KS regularization
at Steller N-body Dynamics, Sexten
Keigo Nitadori
Co-Design Team, Exascale Computing Project
RIKEN Advanced Institute for Computational Sciencd
September 10, 2014
2. A block-step version
of KS regularization
Keigo Nitadori
KS regularization
Hermite integrator
KS block-step
Summary
Acknowledgement
My sincerest thanks go to Sverre Aarseth and IoA for inviting
me 7 times, 07, 08, 09, 10, 11, 12, and 13, in which the most
important and creative collaboration on the development of
NBODY6/GPU was done.
3. A block-step version
of KS regularization
Keigo Nitadori
KS regularization
Hermite integrator
KS block-step
Summary
Abstract
I talk about a new future implemented in NBODY6/GPU:
A block-step version of KS regularized binary.
I To eliminate serial bottleneck
I Very accurate with fully conservational Hermite integrator
Amazingly, it is already working!
Public code is available as nbody6b and nbody7b
4. A block-step version
of KS regularization
Keigo Nitadori
KS regularization
Hermite integrator
KS block-step
Summary
Topics
1. Brief introduction of Kustaanheimo–Stiefel (KS)
regularization (with Hamilton’s quaternion numbers)
2. A very accurate variant of Hermite integrator for harmonic
oscillators (and regularized binaries)
3. Block stepping in real time t
5. A block-step version
of KS regularization
Keigo Nitadori
KS regularization
Hermite integrator
KS block-step
Summary
3 steps of regularization
1. Time transformation. dt = krkd, (hereafter, [˙] = d
dt [ ],
d [ ] = krk[˙])
and [ ]0 = d
I r() is already a harmonic oscillator (+bias ).
I But equation of motion r00 = f (r; r0) is numerically
dangerous.
2. Write the equation of motion in conserved quantities.
ex. r00 = f (E;e; r) in Sparing–Burdeet–Heggie
regularization.
I Perturbation to E and e are integrated independently.
3. Coordinate transformation from r to u. Finally, it becomes
a harmonic oscillator, u00 = 12
hu.
I h is an energy integral and available as h(u;u0), but should
be integrated independently.
6. A block-step version
of KS regularization
Keigo Nitadori
KS regularization
Hermite integrator
KS block-step
Summary
Levi-Civita transformation in a complex plane
1.5
1
0.5
0
-0.5
-1
-1.5
harmonic
Kepler
F1
F2
-1.5 -1 -0.5 0 0.5 1 1.5
Ellipse of harmonic oscillator:
u = A cos + iB sin (1)
Ellipse of Kepler motion:
r =u2
=
A2 B2
2 +
A2 + B2
2
cos 2 + i2AB sin 2
=a(e + cos ) + ia
p
1 e2 sin
with
= 2; a =
A2 + B2
2
; e =
A2 B2
A2 + B2
(2)
7. A block-step version
of KS regularization
Keigo Nitadori
KS regularization
Hermite integrator
KS block-step
Summary
Hamilton’s quaternion
I The complex numbers algebra can be applied to the
two-dimensional geometry
I We can add, subtract, multiply, and divide 2D vectors
I What about on 3D vectors?
I According to a letter Hamilton wrote later to his son
Archibald:
Every morning in the early part of October 1843, on my coming
down to breakfast, your brother William Edward and yourself
used to ask me: Well, Papa, can you multiply triples? Whereto
I was always obliged to reply, with a sad shake of the head, No,
I can only add and subtract them.
I Finally, he discovered that quadruple numbers can be applied to
the arithmetics of 3D vectors
8. A block-step version
of KS regularization
Keigo Nitadori
KS regularization
Hermite integrator
KS block-step
Summary
cont.
Introduce a new imaginary unit j that anti-commutes with i,
e.g. ij = ji, and let k = ij. Then,
i2 = j2 = k2 = ijk = 1
or
ij = ji = k; jk = kj = i; ki = ik = j
defines the (non-commutative) quaternion algebra.
A quaternion is
H 3 q = s + ix + jy + kz (s;x;y; z 2 R)
s is referred to as scalar and ix + jy + kz vector.
10. A block-step version
of KS regularization
Keigo Nitadori
KS regularization
Hermite integrator
KS block-step
Summary
Usage of quaternion
A unit quaternion rotates a vector.
rrot = r ¯; (3)
with a unit quaternion and a vector r = ix + jy + kz.
Example:
ek=2(ix + jy + kz)ek=2 =ek (ix + jy) + kz
=ek (x + ky)i + kz (4)
iek=2 = ek=2i; jek=2 = ek=2j; kek=2 = ek=2k
= ek=2
= cos
2 + k sin
2
rotated a vector about the z-axis
by an angle .
11. A block-step version
of KS regularization
Keigo Nitadori
KS regularization
Hermite integrator
KS block-step
Summary
Let’s rotate the orbit
Let Kepler orbit on the xy-plane
x + iy = u2; (5)
with LC coordinate u = u0 + iu1. By multiplying j from right,
we move the orbit on the yz-plane,
(x + iy)j = jx + ky = u2j = uj¯u: (6)
With arbitrary rotational quaternions and
20. A block-step version
of KS regularization
Keigo Nitadori
KS regularization
Hermite integrator
KS block-step
Summary
Comparison of LC and KS
LC KS
12
12
algebra complex, C quaternion, 12
12H
phys. coord. r = x + iy r = ix + jy + kz
reg. coord. u = u0 + iu1 u = u0 + iu1 + ju2 + ku3
transformation r = u2 r = u u
¯eq. of motion u00 = hu + juj2fu u00 = hu kuk2fu
I f is a perturbation vector
I Original 2 2 real matrix formulation is possible for LC
I Original 4 4 real matrix or 2 2 complex matrix (Pauli
matrices) formulation is possible for KS
21. A block-step version
of KS regularization
Keigo Nitadori
KS regularization
Hermite integrator
KS block-step
Summary
A very accurate Hermite integrator
(Accidentally) it turn out that the following symmetric
corrector form
ucorr =uold +
2
(u0
corr + u0
old)
2
12
(u00
new u00
old);
u0
corr =u0
old +
2
(u00
new + u00
old)
2
12
(u000
new u000
old); (9)
after P(EC)n convergence 5 of 6 orbital elements (except for the
phase) conserve in machine accuracy.
If you use another form for the position
ucorr =uold +
2
(u0
corr + u0
old)
2
10
(u00
new u00
old) +
3
120
(u000
new + u000
old);
(10)
this property is lost.
22. A block-step version
of KS regularization
Keigo Nitadori
KS regularization
Hermite integrator
KS block-step
Summary
Why it works
Consider a harmonic oscillator ¨x = !x, and let vi = ˙xi and
fi = ¨xi.
An analytical solution for one step t writes,
!x1
v1
!
=
cos!t sin!t
sin!t cos!t
!
!x0
v0
!
: (11)
It is equivalent to
v1 v0
x1 x0
!
=
tan(!t=2)
!t=2
!
t
2
f1 + f0
v1 + v0
!
: (12)
A second order integrator approximates this as
v1 v0
x1 x0
!
=
t
2
f1 + f0
v1 + v0
!
: (13)
The dierence is only a small factor on the stepsize t.
23. A block-step version
of KS regularization
Keigo Nitadori
KS regularization
Hermite integrator
KS block-step
Summary
Fourth- and sixth-order version
The 4th-order Hermite integrator is equivalent to
v1 v0
x1 x0
!
=
*..
,
1
1 13
!t
2
2
+//
-
t
2
f1 + f0
v1 + v0
!
; (14)
and a 6th-order one
v1 v0
x1 x0
!
=
*..
,
1 1
15
!t
2
2
1 25
!t
2
2
+//
-
t
2
f1 + f0
v1 + v0
!
: (15)
The factor to t is approaching to
tan(!t=2)
!t=2
!
, of exact
solution.
Phase error shared among 4 waves of KS
) 5 of 6 orbital elements conserve
24. A block-step version
of KS regularization
Keigo Nitadori
KS regularization
Hermite integrator
KS block-step
Summary
Phase correction
If one likes to conserve also the phase, emphasize the step-size
t.
x1 =x0 +
c2t
2
(v1 + v0);
x1 =x0 +
c4t
2
(v1 + v0)
t2
12
(f1 f0);
x1 =x0 +
c6t
2
(v1 + v0)
t2
10
(f1 f0) +
c6t3
120
(˙f1 + ˙f0); (16)
(so on the velocity), with
c2 =
tan
; c4 =
tan
1
1
3
2
!
; c6 =
tan
1 25
2
1 1
15 2
; (17)
and = !t=2. Now the results agree with the analitic
trigonometric functions in machine accuracy.
25. A block-step version
of KS regularization
Keigo Nitadori
KS regularization
Hermite integrator
KS block-step
Summary
Block stepping
I Usually, fixed , 30 step/orbit is enough for weakly
perturbed binary.
I Then, real step t =
Z +
dt
d
d =
Z +
krkd,
becomes a varying real (non-quantized) number.
I This prevents parallelization, we hope t restricted to
2n (n 2 Z) (McMillan, 1986; Makino 1992).
26. A block-step version
of KS regularization
Keigo Nitadori
KS regularization
Hermite integrator
KS block-step
Summary
Truncation procedure
From given natural step nat, we compute a truncated step
bs ( nat) for which the corresponding step tbs = 2n obeys
the block-step criterion.
nat !
integrate
tnat !
truncate
tbs !
solve
bs (18)
We integrate
tnat =
X6
n=1
thni (nat)n
n!
;
thni =
dn
dn t
!
: (19)
After truncation tnat ! tbs, we solve bs from
tbs +
X6
n=1
thni (bs)n
n! = 0; (20)
by using of Newton–Raphson iterations (unique solution
exists).
27. A block-step version
of KS regularization
Keigo Nitadori
KS regularization
Hermite integrator
KS block-step
Summary
Derivatives of physical time t
I After a 4th-order Hermite integration, up to 3rd derivative
of force (5th derivative of coordinate) is available
I Up to 6th derivative of time is available
t0 =u u;
t00 =2(u u0);
t000 =2(u u00 + u0 u0);
th4i =2(u u000) + 6(u0 u00);
th5i =2(u uh4i) + 8(u0 u000) + 6(u00 u00);
th6i =2(u uh5i) + 10(u0 uh4i) + 20(u00 u000): (21)
I tnext is explicitly available at the end of this step
I Predictor only, no corrector is applied for t
28. A block-step version
of KS regularization
Keigo Nitadori
KS regularization
Hermite integrator
KS block-step
Summary
Summary
I The block-step version of KS is working in Sverre’s code
I Some Hermite integrators turned out to integrate harmonic
oscillators very accurately with P(EC)n iterations
I Regularization is not acrobatics, nor a black magic
(at least for two-body).