text:
Nyquist Theorem

A slecture by ECE student Miguel Rodrigo Castellanos

Partly based on the ECE438 Fall 2014 lecture material of Prof. Mireille Boutin.

## Outline

1. Introduction
2. Proof
3. Example
4. Conclusion
5. References

## Introduction

The Nyquist theorem provides a sufficient condition for perfect reconstruction of a signal from its sampling. In this slecture I will give a proof of the Nyquist theorem and give an example demonstrating how perfect reconstruction is possible even when violating the Nyquist condition.

## Proof

Let the ideal sampling $x_s(t) \text{ of } x(t)$ be defined as

$x_s(t) := x(t)p_{\frac{1}{f_s}}(t)$,

where

$p_{\frac{1}{f_s}}(t) = \sum_{k = -\infty}^\infty \delta(t-\frac{k}{f_s})$.

Nyquist Theorem: A signal $x(t)$ that has the property $X(f) = 0$ for $|f| \ge f_M$ can be perfectly reconstructed from its sampling $x_s(t)$ if sampled at a rate $f_s > 2f_M$.

To prove that perfect reconstruction is possible, we must find an expression for $x(t)$ in terms of $x_s(t)$.

Given that $\mathcal{F}(x(t)) = X(f)$, we can find $X_s(f)$ using the convolution property.

\begin{align} X_s(f) &= X(f)*\mathcal{F}(p_{\frac{1}{f_s}})\\ &= X(f)*\mathcal{F}(\sum_{k = -\infty}^\infty \delta(t-\frac{k}{f_s}))\\ &= X(f)*f_s\sum_{k = -\infty}^\infty \delta(f-kf_s)\\ &= f_s\sum_{k = -\infty}^\infty X(f)*\delta(t-\frac{k}{f_s})\\ &= f_s\sum_{k = -\infty}^\infty X(f-kf_s)\\ \end{align}

Without loss of generality, we can assume that the signal $x(t)$ has the spectrum shown in the figure below. The shape of the graph of $X(f)$ does not matter because the only important feature of $X(f)$ is that $X(f) = 0$ for $|f| \ge f_M$.

We would like to determine what $X_s(f)$ looks like in order to find a way to reconstruct $x(t)$.

Since we have sampled at a rate $f_s > 2f_M$, the following inequalities hold:

$f_s > 2f_M \iff f_s - f_M > f_M \iff -f_s + f_M < f_M$.

Together, these inequalities, the graph of $X(f)$, and the expression for $X_s(f)$ in terms of $X(f)$ imply that $X_s(f)$ will have the spectrum shown in the figure below.

Notice that the spectrum of the ideal sampling of a signal is an amplitude scaled periodic repetition of the original spectrum. Since $x(t)$ is bandlimited and we have sampled at a rate $f_s > 2f_M$, the periodic repetitions of $X(f)$ do not overlap.

All the information needed to reconstruct $X(f)$ can be found in the portion of $X_s(f)$ that corresponds to $X(f)$ (shown in red). Therefore we can use a simple lowpass filter with gain $\tfrac{1}{f_s}$ and cutoff frequency $\tfrac{f_s}{2}$ to recover $X(f)$ from $X_s(f)$.

$X(f) = X_s(f)\left\{ \begin{array}{ll} \frac{1}{f_s}, & |f| \le \frac{f_s}{2}\\ 0, & \text{else} \end{array} \right.$

$\iff x(t) = x_s(t)*\text{sinc}(f_st)$

$\therefore$ We can perfectly reconstruct $x(t)$ from $x_s(t)$.

## Example

Though the Nyquist theorem states that perfect reconstruction is possible if we satisfy the Nyquist condition $(f_s > 2f_M)$, it is important to note that this condition is not necessary. The following example demonstrates how perfect reconstruction is sometimes possible even when undersampling.

Let the signal $x(t)$ have a spectrum $X(f)$ as seen in the figure below.

The Nyquist condition states that we should sample at a rate $f_s > 2(2a) = 4a$. Instead, let us sample at $f_s = 2a$.

As before, we have $x_s(t) = x(t)p_{\frac{1}{f_s}}(t)$ and $X_s(f) = f_s\sum_{k = -\infty}^\infty X(f-kf_s)$

$\implies X_s(f) = 2a\sum_{k = -\infty}^\infty X(f-2ka)$.

Therefore, $X_s(f)$ will have the spectrum shown in the figure below.

Notice that there is no aliasing in $X_s(f)$ even though $f_s < 4a$. In addition, the portion of $X_s(f)$ that corresponds to $X(f)$ (shown in red) can be recovered using a bandpass filter with gain $\tfrac{1}{2a}$ and cutoff frequencies $a \text{ and } 2a$.

## Conclusion

To summarize, the Nyquist theorem states that any bandlimited signal can be perfectly reconstructed from its sampling if sampled at a rate greater than twice its bandwidth $(f_s > 2f_M)$. However, the Nyquist condition is not necessary for perfect reconstruction as shown in the example above.

## References

[1] John G. Proakis, Dimitris G. Manolakis, "Digital Signal Processing with Principles, Algorithms, and Applications" 4th Edition,2006


text:
# Moving a particle from point A to B on a efficient way in GAMS

#### Frank Einstein

Hello everyone. I am triyng to calculate the route which takes less time to go from point A to point B in the presence of a constant flow (I. E. a simple version of Zermelo's navigation problem) using the GAMS software. However, if I put both points on a straight line and make the constant flow equal to zero, I don't get the results that I should.

Here it is my code: [Moderator: code tags added]
Code:
$set n 10 set j /0*%n%/; sets jlast(j) jnotlast(j); jlast(j)$(ord(j)=card(j))=yes;
jnotlast(j)=not jlast(j);
scalar

v aeroplane velocity /10/

u wind velocity in the x axis /0/

n number of intervals /%n%/

density density /1.225/
variable
gamma(j),
x(j),
y(j),
time,
objective;

positive variable
t(j)

step;

equation
diffx(j),
diffy(j),
obj;

* diffx[j]$(jnotlast(j)).. x[j+1]-x[j] =e=0.5*(t(j+1)-t(j))*(v*cos(gamma(j+1)) + v*cos(gamma(j)) ); * diffy[j]$(jnotlast(j)).. y[j+1]-y[j] =e=0.5*(t(j+1)-t(j))*(v*sin(gamma(j+1)) +  v*sin(gamma(j))   );

diffx[j]$(jnotlast(j)).. x[j+1]-x[j] =e=0.5*step*(v*cos(gamma(j+1))-u + v*cos(gamma(j))-u ); diffy[j]$(jnotlast(j)).. y[j+1]-y[j] =e=0.5*step*(v*sin(gamma(j+1)) +  v*sin(gamma(j))   );

obj.. time =e= n*step;

x.fx('0') = 1.0e-12;
x.fx('%n%') = 1.0e-12;
y.fx('0') = 1.0e-12;
y.fx('%n%') = 10;
t.fx('0')= 1.0e-12;

y.up(j) = 10;
y.lo(j) = 1.0e-12;
t.lo(j)=1.0e-12;
gamma.up(j)=pi;
gamma.lo(j)=0;

model brahstron1 /all/;

option
nlp=ipopt;
solve brahstron1 using nlp minimize time;
In this code I have defined two diferential equations, for x'=dx/dt and y'=dy/dt using the trapezoid rule. Here, gamma is our control variable and that's why it's free. Step is the difference between the times j and j+1, that is why time is n (number of intervals)*step(size of interval) and is what I seek to reduce.

Below this one can find the restrictions, these include the initial and final points for x and y, also the initial t=0.

I would spect as a result that the angle gamma became pi/2 and y going between 0 and 10, however, I find weird results, as Y being 5 on all the steps.

I have looked at this code for a while and I cannot find where my mistake is.

Can somebody please tell me what am I doing wrong?
Thank you very much for reading.

Last edited by a moderator:
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## Numerical methods for mean field games - Lecture 2: Monotone finite difference schemes Achdou, Yves | CIRM H

Post-edited

Research schools;Computer Science;Control Theory and Optimization;Partial Differential Equations

Recently, an important research activity on mean field games (MFGs for short) has been initiated by the pioneering works of Lasry and Lions: it aims at studying the asymptotic behavior of stochastic differential games (Nash equilibria) as the number $n$ of agents tends to infinity. The field is now rapidly growing in several directions, including stochastic optimal control, analysis of PDEs, calculus of variations, numerical analysis and computing, and the potential applications to economics and social sciences are numerous.
In the limit when $n \to +\infty$, a given agent feels the presence of the others through the statistical distribution of the states. Assuming that the perturbations of a single agent's strategy does not influence the statistical states distribution, the latter acts as a parameter in the control problem to be solved by each agent. When the dynamics of the agents are independent stochastic processes, MFGs naturally lead to a coupled system of two partial differential equations (PDEs for short), a forward Fokker-Planck equation and a backward Hamilton-Jacobi-Bellman equation.
The latter system of PDEs has closed form solutions in very few cases only. Therefore, numerical simulation are crucial in order to address applications. The present mini-course will be devoted to numerical methods that can be used to approximate the systems of PDEs.
The numerical schemes that will be presented rely basically on monotone approximations of the Hamiltonian and on a suitable weak formulation of the Fokker-Planck equation.
These schemes have several important features:

- The discrete problem has the same structure as the continous one, so existence, energy estimates, and possibly uniqueness can be obtained with the same kind of arguments

- Monotonicity guarantees the stability of the scheme: it is robust in the deterministic limit

- convergence to classical or weak solutions can be proved

Finally, there are particular cases named variational MFGS in which the system of PDEs can be seen as the optimality conditions of some optimal control problem driven by a PDE. In such cases, augmented Lagrangian methods can be used for solving the discrete nonlinear system. The mini-course will be orgamized as follows

1. Introduction to the system of PDEs and its interpretation. Uniqueness of classical solutions.

2. Monotone finite difference schemes

3. Examples of applications

4. Variational MFG and related algorithms for solving the discrete system of nonlinear equations
Recently, an important research activity on mean field games (MFGs for short) has been initiated by the pioneering works of Lasry and Lions: it aims at studying the asymptotic behavior of stochastic differential games (Nash equilibria) as the number $n$ of agents tends to infinity. The field is now rapidly growing in several directions, including stochastic optimal control, analysis of PDEs, calculus of variations, numerical analysis and ...

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## Recent advances in adaptative computation :proceedings of the international coference on recent...#May 24-28 Shi, Z. C. ; Chen, Z. ; Tang, T. ; Yu, D. | American Mathematical Society 2005

Congrès

- 386 p.
ISBN 978-0-8218-3662-0

Contemporary mathematics , 0383

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## Fast algorithms for structured matrices :theory and applications#AMS-IMS-SIAM joint summer research conference on fast algorithms in mathematics, computer science and engineering held at Mount Holyoke College#Aug. 5-9 Olshevsky, Vadim | American Mathematical Society 2003

Congrès

- 433 p.
ISBN 978-0-8218-3177-9

Contemporary mathematics , 0323

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matrice # transformation de Fourier # algorithme # interpolation rationnelle # décodage d'algorithme # algèbre de Lie # optimisation # résolution d'équation matricielle

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## Meshfree methods for partial differential equations V.Selected papers based on the presentations at the 5th international workshopBonn # august 17-19, 2009 Griebel, Michael ; Schweitzer, Marc Alexander | Springer 2011

Congrès

- viii; 263 p.
ISBN 978-3-642-16228-2

Lecture notes in computational science and engineering , 0079

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analyse numérique # méthodes sans maillage # équations différentielles #- Solutions numériques des équations différentielles # équations aux dérivées partielles # solutions numériques des EDP

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## European women in mathematics :proceedings of the 11th conference of EWM Dajani, Karma ; Von Reis, J. | Stichting Mathematisch Centrum 2005

Congrès

- 120 p.
ISBN 978-90-6196-527-5

CWI tract , 0135

Localisation : Collection 1er étage

femmes et mathématiques # nombre normal # structure de groupe modulaire # fraction continue # théorie métrique des fractions continues # entropie # dynamique symbolique # transformation de Fourier # ondelettes # stabilité des méthodes numériques # convergence des méthodes numériques # convergence

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## Nonlinear partial differential equations and related analysis :the emphasis year 2002-2003 program on ... held at Northwestern University#Sept. 2002 - July 2003 Chen, Gui-Qiang ; Gasper, George ; Jerome, Joseph W. | American Mathematical Society 2005

Congrès

- 323 p.
ISBN 978-0-8218-3533-3

Contemporary mathematics , 0371

Localisation : Collection 1er étage

opérateur aux dérivées partielles # EDP # théorème d'existence # comportement des solutions d'EDP # analyse numérique # EDP non-linéaire # équation elliptique # équation parabolique # équation de Navier-Stokes # loi de conservation hyperbolique # mécanique des fluides # fiabilité statistique # simulation # traitement d'image # onde de choc # limite libre # stabilité # limite singulière

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## Finite-difference methods theory and applications CFDM98. Vol. 1proceedings of the 2nd international conference on ... | National Academy of Sciences of Belarus 1998

Congrès

ISBN 978-985-6499-04-6

Localisation : Colloque 1er étage (BELA)

EDP # analyse numérique # modèle mathématique # méthode de différence finie # physique mathématique # problème aux limites de type multi-dimensionnel # problème de stabilité des méthodes numériques # problème à la frontière

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## Finite-difference methods theory and applications cfdm98. vol. 2Proceedings of the 2nd international conference on ... | National Academy of Sciences of Belarus 1998

Congrès

ISBN 978-985-6499-05-3

Localisation : Colloque 1er étage (BELA)

EDP # analyse numérique # modèle mathématique # méthode de différence finie # physique mathématique # problème aux limites de type multi-dimensionnel # problème de stabilité des méthodes numériques # problème à la frontière

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## Finite-difference methods theory and applications cfdm98. vol. 3Proceedings of the 2nd international conference on ... | National Academy of Sciences of Belarus 1998

Congrès

ISBN 978-985-6499-06-0

Localisation : Colloque 1er étage (BELA)

EDP # analyse numérique # modèle mathématique # méthode de différence finie # physique mathématique # problème aux limites de type multi-dimensionnel # problème de stabilité des méthodes numériques # problème à la frontière

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## High-order Magnus integrators for non-autonomous linear evolution equations Thalhammer, Mechthild | CIRM H

Multi angle

Research talks;Partial Differential Equations

The class of commutator-free Magnus integrators is known to provide a favourable alternative to standard interpolatory Magnus integrators, in particular for large-scale applications arising in the time integration of non-autonomous linear evolution equations. A high-order commutator-free Magnus integrator is given by a composition of several exponentials that comprise certain linear combinations of the values of the defining operator at specified nodes. Due to the fact that previously proposed commutator-free Magnus integrators of order five or higher involve negative coefficients in the linear combinations, severe instabilities are observed for spatially semi-discretised partial differential equations of parabolic type or for master equations describing dissipative quantum systems, respectively. In order to remedy this issue, two different approaches for the design of efficient Magnus integrators of orders four, five, and six are pursued: (i) the study of commutator-free Magnus integrators involving complex coefficients with positive real part, and (ii) the study of unconventional Magnus integrators that comprise in addition a single exponential involving a commutator. Numerical experiments for test equations of Schrödinger and parabolic type confirm that the identified novel Magnus integrators are superior to Magnus integrators previously proposed in the literature. The class of commutator-free Magnus integrators is known to provide a favourable alternative to standard interpolatory Magnus integrators, in particular for large-scale applications arising in the time integration of non-autonomous linear evolution equations. A high-order commutator-free Magnus integrator is given by a composition of several exponentials that comprise certain linear combinations of the values of the defining operator at ...

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## Optimized Schwarz waveform relaxation methods: theory and applications Omnes, Pascal | CIRM H

Multi angle

Research talks

We review Optimized Schwarz waveform relaxation methods which are space-time domain decomposition methods. The main ideas are explained on the heat equation, and extension to advection-diffusion equations are illustrated by numerical results. We present the Schwarz for TrioCFD project, which aims at using this kind of methods for the Stokes equations.

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## Inverse problems for linear PDEs using mixed formulations Münch, Arnaud | CIRM H

Multi angle

Research talks;Control Theory and Optimization;Partial Differential Equations

We explore a direct method allowing to solve numerically inverse type problems for hyperbolic type equations. We first consider the reconstruction of the full solution of the equation posed in $\Omega \times (0, T )$ - $\Omega$ a bounded subset of $\mathbb{R}^N$ - from a partial distributed observation. We employ a least-squares technic and minimize the $L^2$-norm of the distance from the observation to any solution. Taking the hyperbolic equation as the main constraint of the problem, the optimality conditions are reduced to a mixed formulation involving both the state to reconstruct and a Lagrange multiplier. Under usual geometric optic conditions, we show the well-posedness of this mixed formulation (in particular the inf-sup condition) and then introduce a numerical approximation based on space-time finite elements discretization. We show the strong convergence of the approximation and then discussed several examples for $N = 1$ and $N = 2$. The reconstruction of both the state and the source term is also discussed, as well as the boundary case. The parabolic case - more delicate as it requires the use of appropriate weights - will be also addressed. Joint works with Nicolae Cîndea and Diego Araujo de Souza. We explore a direct method allowing to solve numerically inverse type problems for hyperbolic type equations. We first consider the reconstruction of the full solution of the equation posed in $\Omega \times (0, T )$ - $\Omega$ a bounded subset of $\mathbb{R}^N$ - from a partial distributed observation. We employ a least-squares technic and minimize the $L^2$-norm of the distance from the observation to any solution. Taking the hyperbolic ...

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## Traffic flow models with non-local flux and extensions to networks Göttlich, Simone | CIRM H

Multi angle

Research talks

We present a Godunov type numerical scheme for a class of scalar conservation laws with nonlocal flux arising for example in traffic flow modeling. The scheme delivers more accurate solutions than the widely used Lax-Friedrichs type scheme and also allows to show well-posedness of the model. In a second step, we consider the extension of the non-local traffic flow model to road networks by defining appropriate conditions at junctions. Based on the proposed numerical scheme we show some properties of the approximate solution and provide several numerical examples. We present a Godunov type numerical scheme for a class of scalar conservation laws with nonlocal flux arising for example in traffic flow modeling. The scheme delivers more accurate solutions than the widely used Lax-Friedrichs type scheme and also allows to show well-posedness of the model. In a second step, we consider the extension of the non-local traffic flow model to road networks by defining appropriate conditions at junctions. Based on ...

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## Nonlinear free energy diminishing schemes for convection-diffusion equations: convergence and long time behaviour Chainais-Hillairet, Claire | CIRM H

Multi angle

Research talks

The aim of the talk is to introduce a nonlinear Discrete Duality Finite Volume scheme to approximate the solutions of drift-diffusion equations. The scheme is built to preserve at the discrete level even on severely distorted meshes the energy / energy dissipation relation. This relation is of paramount importance to capture the long-time behavior of the problem in an accurate way. To enforce it, the linear convection diffusion equation is rewritten in a nonlinear form before being discretized. This is a joint work with Clément Cancès (Lille) and Stella Krell (Nice). The aim of the talk is to introduce a nonlinear Discrete Duality Finite Volume scheme to approximate the solutions of drift-diffusion equations. The scheme is built to preserve at the discrete level even on severely distorted meshes the energy / energy dissipation relation. This relation is of paramount importance to capture the long-time behavior of the problem in an accurate way. To enforce it, the linear convection diffusion equation is ...

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## Numerical methods for mean field games - Lecture 1: Introduction to the system of PDEs and its interpretation. Uniqueness of classical solutions Achdou, Yves | CIRM H

Multi angle

Research schools;Computer Science;Control Theory and Optimization;Partial Differential Equations

Recently, an important research activity on mean field games (MFGs for short) has been initiated by the pioneering works of Lasry and Lions: it aims at studying the asymptotic behavior of stochastic differential games (Nash equilibria) as the number $n$ of agents tends to infinity. The field is now rapidly growing in several directions, including stochastic optimal control, analysis of PDEs, calculus of variations, numerical analysis and computing, and the potential applications to economics and social sciences are numerous.
In the limit when $n \to +\infty$, a given agent feels the presence of the others through the statistical distribution of the states. Assuming that the perturbations of a single agent's strategy does not influence the statistical states distribution, the latter acts as a parameter in the control problem to be solved by each agent. When the dynamics of the agents are independent stochastic processes, MFGs naturally lead to a coupled system of two partial differential equations (PDEs for short), a forward Fokker-Planck equation and a backward Hamilton-Jacobi-Bellman equation.
The latter system of PDEs has closed form solutions in very few cases only. Therefore, numerical simulation are crucial in order to address applications. The present mini-course will be devoted to numerical methods that can be used to approximate the systems of PDEs.
The numerical schemes that will be presented rely basically on monotone approximations of the Hamiltonian and on a suitable weak formulation of the Fokker-Planck equation.
These schemes have several important features:

- The discrete problem has the same structure as the continous one, so existence, energy estimates, and possibly uniqueness can be obtained with the same kind of arguments

- Monotonicity guarantees the stability of the scheme: it is robust in the deterministic limit

- convergence to classical or weak solutions can be proved

Finally, there are particular cases named variational MFGS in which the system of PDEs can be seen as the optimality conditions of some optimal control problem driven by a PDE. In such cases, augmented Lagrangian methods can be used for solving the discrete nonlinear system. The mini-course will be orgamized as follows

1. Introduction to the system of PDEs and its interpretation. Uniqueness of classical solutions.

2. Monotone finite difference schemes

3. Examples of applications

4. Variational MFG and related algorithms for solving the discrete system of nonlinear equations
Recently, an important research activity on mean field games (MFGs for short) has been initiated by the pioneering works of Lasry and Lions: it aims at studying the asymptotic behavior of stochastic differential games (Nash equilibria) as the number $n$ of agents tends to infinity. The field is now rapidly growing in several directions, including stochastic optimal control, analysis of PDEs, calculus of variations, numerical analysis and ...

Déposez votre fichier ici pour le déplacer vers cet enregistrement.

## Numerical methods for mean field games - Lecture 3: Variational MFG and related algorithms for solving the discrete system of nonlinear equations Achdou, Yves | CIRM H

Multi angle

Research schools;Computer Science;Control Theory and Optimization;Partial Differential Equations

Recently, an important research activity on mean field games (MFGs for short) has been initiated by the pioneering works of Lasry and Lions: it aims at studying the asymptotic behavior of stochastic differential games (Nash equilibria) as the number $n$ of agents tends to infinity. The field is now rapidly growing in several directions, including stochastic optimal control, analysis of PDEs, calculus of variations, numerical analysis and computing, and the potential applications to economics and social sciences are numerous.
In the limit when $n \to +\infty$, a given agent feels the presence of the others through the statistical distribution of the states. Assuming that the perturbations of a single agent's strategy does not influence the statistical states distribution, the latter acts as a parameter in the control problem to be solved by each agent. When the dynamics of the agents are independent stochastic processes, MFGs naturally lead to a coupled system of two partial differential equations (PDEs for short), a forward Fokker-Planck equation and a backward Hamilton-Jacobi-Bellman equation.
The latter system of PDEs has closed form solutions in very few cases only. Therefore, numerical simulation are crucial in order to address applications. The present mini-course will be devoted to numerical methods that can be used to approximate the systems of PDEs.
The numerical schemes that will be presented rely basically on monotone approximations of the Hamiltonian and on a suitable weak formulation of the Fokker-Planck equation.
These schemes have several important features:

- The discrete problem has the same structure as the continous one, so existence, energy estimates, and possibly uniqueness can be obtained with the same kind of arguments

- Monotonicity guarantees the stability of the scheme: it is robust in the deterministic limit

- convergence to classical or weak solutions can be proved

Finally, there are particular cases named variational MFGS in which the system of PDEs can be seen as the optimality conditions of some optimal control problem driven by a PDE. In such cases, augmented Lagrangian methods can be used for solving the discrete nonlinear system. The mini-course will be orgamized as follows

1. Introduction to the system of PDEs and its interpretation. Uniqueness of classical solutions.

2. Monotone finite difference schemes

3. Examples of applications

4. Variational MFG and related algorithms for solving the discrete system of nonlinear equations
Recently, an important research activity on mean field games (MFGs for short) has been initiated by the pioneering works of Lasry and Lions: it aims at studying the asymptotic behavior of stochastic differential games (Nash equilibria) as the number $n$ of agents tends to infinity. The field is now rapidly growing in several directions, including stochastic optimal control, analysis of PDEs, calculus of variations, numerical analysis and ...

Déposez votre fichier ici pour le déplacer vers cet enregistrement.

## Numerical solution of the shallow-water equations Wubs, F. W. | Centrum Voor Wiskunde En Informatica 1988

Ouvrage

- 115 p.
ISBN 978-90-6196-349-3

CWI tract , 0049

Localisation : Collection 1er étage

dérivation des approximations aux différences finies # méthode des différences finies # méthode des lignes # problème de valeur initiale # stabilité et convergence des méthodes aux différences # statistique # équation aux dérivées partielles

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## Riemann-Hilbert problems, their numerical solution, and the computation of nonlinear special functions Trogdon, Thomas ; Olver, Sheehan | Society For Industrial And Applied Mathematics 2016

Ouvrage

- xviii; 373 p.
ISBN 978-1-611974-19-5

Localisation : Ouvrage RdC (TROG)

problème de Riemann-Hilbert # système intégrable # fonction spéciale non-linéaire

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## Riemann solvers and numerical methods for fluid dynamics :a practical introduction Toro, Eleuterio F. | Springer 1999

Ouvrage

- 624 p.
ISBN 978-3-540-65966-2

Localisation : Ouvrage RdC (TORO)

analyse numérique # EDP # dynamique des fluides # équation de Euler # équation de construction de la chaleur # problème de Riemann # EDP hyperbolique # méthode de Godunov # méthode des différences finies # méthode numérique # stabilité et convergence des méthodes numériques # choix aléatoire

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## Galerkin finite element methods for parabolic problems Thomee, Vidar | Springer 1997

Ouvrage

- 192 p.
ISBN 978-3-540-63236-8

Springer series in computational mathematics , 0025

Localisation : Ouvrage RdC (THOM)

EDP # analyse numérique # convergence des méthodes numériques # erreur majorée # méthode de Galerkin # méthode des éléments finis # problème parabolique # stabilité

#### Filtrer

##### Codes MSC

Ressources Electroniques (Depuis le CIRM)

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text:
# Simulating an automotive load dump with a bank of charged capacitors?

I've been working on a data logging device that will eventually go into my personal vehicle. I've scoured the interwebs and built what, in theory, should be a fairly stout power supply capable to handle fast transients and undervoltage and the fabled load dump. My problem is, though... I want to actually test the power supply.

In previous questions that I've asked related to the design of the power supply portion, I've been told that the "right way" to simulate a load dump is with a proper, preofessional testing device. I'm curious, though... what's stopping me from rigging up something that is "close enough"?

My idea is simply this: get a low-voltage DC power source, such as an old PC power supply. Get a DC-DC converter to step that 12V up to something in the 80V range. Assemble a bank of capacitors to get me in the neighborhood of a theoretical 12V load dump.. which from what I can tell is in the hundreds of joules range. Once charged, simply connect it to the battery input of the device - which wouldn't be connected to an actual power source lest I want to potentially screw that power source up - and see if explodes, and if not, see if it works normally afterwards.

Does this sound like the dumbest idea you've ever heard? Is it a somewhat decent idea that simply needs a lot of planning and number checking to in order to work roughly as intended?

A load dump is what happens to an automobile electrical system when a large load (such as the headlights) switches off. The problem is that the charging system (primarily the alternator) has significant inductance, and any attempt to rapidly reduce the current draw results in an "inductive kick" that creates a large voltage spike on the 12V bus. This kick is the same phenomenon used to create the spark in the ignition system, just a different manifestation of it. The point is, any equipment that's attached to the 12V bus needs to be able to withstand these occasional 100-200V voltage spikes without damage.

Since load dump is primarily an inductive phenomenon, it would probably be easier to simulate it that way, too. You don't really need to simulate the full energy of an actual automotive load dump; you just need to create the same voltage waveform across the supply terminals of your device.

Put a largish inductor (L1, on the order of 1H, perhaps the primary of a large power transformer) in series with your device (i.e., connect the device to the power supply through the inductor). This represents the inductance of the automobile charging system.

Put a few µF of capacitance (C1) across (in parallel with) your device; this represents the distributed capacitance of the automobile wiring, and helps to limit the risetime of the load dump event. Make sure this capacitor is rated for a few hundred volts.

Put a 120Ω resistor (R1) in parallel with your device, too. This represents other static loads within the automobile, and will set an upper limit on the peak voltage that the load dump creates. (This resistor will be drawing 100 mA and dissipating 1.2W.)

Now, connect a low-value, high-power resistor (R2) across your device, in series with a switch (SW1). This represents the load that is going to get "dumped". The value of the resistor should be such that the DC current doesn't exceed the power supply's capability, and you can adjust the value of the resistor to change the current with respect to the value of the inductor to dump a specific amount of energy (0.5×I2×L). For example, if your inductor is 1H and your resistor is 12Ω (@ 12W), you'll be drawing 1A, and the stored energy will be 0.5 Joules.

Close the switch to "charge up" the inductor, then open it — there's your simulated load dump event. With these resistor values, the peak voltage will be on the order of 100-120V. You can use different combinations of resistor values to simulate different kinds of events. The ratio of R1 to R2 approximately determines the peak voltage of the spike (relative to the power supply voltage). Scale both resistors downward to simulate higher-current (higher-energy) events. Make the capacitor smaller to get faster risetimes; 1H and 1µF resonate at 160Hz, which gives you a fairly leisurely 1.5ms risetime (1/4 cycle). For example, changing C1 to 0.01µF would give you a risetime of about 150µs.

• Diagram is helpful - thanks! Could you elaborate a little more on how actually connecting and disconnecting the load dump resistor could generate an event comparable to the classical definition / numbers for a load dump? Sep 16 '12 at 2:14
• I've added more detail to the beginning and end of my answer; hopefully, that helps. Sep 17 '12 at 0:51
• It definitely does... and I'd say your answer hits the nail on the head. Do you have any advice on finding a suitable inductor for a decent price, though? Everything in the 0.5H and up range seems to be $10 or more. You mentioned a power transformed... what specs would I be looking for? Is the inductance of a transformer often quoted? Sep 17 '12 at 2:18 • No, but it isn't hard to measure the value of an inductor if you have a signal generator and an oscilloscope. Perhaps I should start another question that describes some procedures for doing that. Sep 17 '12 at 3:50 • I'll go with something commercially available for now.. I've found stuff in the$10 range but I was curious if there was a cheaper source. At any rate, I appreciate your continuing answers and information. Really appreciate it. :) Time to work up a prototype! :D Sep 17 '12 at 23:04


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# zbMATH — the first resource for mathematics

Bravely, moderately: a common theme in four recent works. (English) Zbl 1343.68113
Goldreich, Oded (ed.), Studies in complexity and cryptography. Miscellanea on the interplay between randomness and computation. In collaboration with Lidor Avigad, Mihir Bellare, Zvika Brakerski, Shafi Goldwasser, Shai Halevi, Tali Kaufman, Leonid Levin, Noam Nisan, Dana Ron, Madhu Sudan, Luca Trevisan, Salil Vadhan, Avi Wigderson, David Zuckerman. Berlin: Springer (ISBN 978-3-642-22669-4/pbk). Lecture Notes in Computer Science 6650, 373-389 (2011).
Summary: We highlight a common theme in four relatively recent works that establish remarkable results by an iterative approach. Starting from a trivial construct, each of these works applies an ingeniously designed sequence of iterations that yields the desired result, which is highly non-trivial. Furthermore, in each iteration, the construct is modified in a relatively moderate manner. The four works we refer to are
1.
the polynomial-time approximation of the permanent of non-negative matrices by M. Jerrum et al. [J. ACM 51, No. 4, 671–697 (2004; Zbl 1204.65044)];
2.
the iterative (Zig-Zag) construction of expander graphs by O. Reingold et al. [Ann. Math. (2) 155, No. 1, 157–187 (2002; Zbl 1008.05101)];
3.
the log-space algorithm for undirected connectivity by O. Reingold [“Undirected ST-connectivity in log-space”, in: Proceedings of the 37th annual ACM symposium on theory of computing, STOC 2005. New York, NY: ACM. 376–385 (2005; doi:10.1145/1060590.1060647)];
4.
and, the alternative proof of the PCP Theorem by I. Dinur [in: Proceedings of the 38th annual ACM symposium on theory of computing, STOC 2006. New York, NY: ACM Press. 241–250 (2006; Zbl 1301.68133)].

For the entire collection see [Zbl 1220.68005].

##### MSC:
 68Q25 Analysis of algorithms and problem complexity 05C85 Graph algorithms (graph-theoretic aspects) 68Q15 Complexity classes (hierarchies, relations among complexity classes, etc.) 68W25 Approximation algorithms
Full Text:
##### References:
 [1] Aleliunas, R., Karp, R.M., Lipton, R.J., Lovász, L., Rackoff, C.: Random walks, universal traversal sequences, and the complexity of maze problems. In: 20th FOCS, pp. 218–223 (1979) [2] Alon, N.: Eigenvalues and expanders. Combinatorica 6, 83–96 (1986) · Zbl 0661.05053 [3] Alon, N., Milman, V.D.: {$$\lambda$$} 1, Isoperimetric Inequalities for Graphs and Superconcentrators. J. Combinatorial Theory, Ser. B 38, 73–88 (1985) · Zbl 0549.05051 [4] Armoni, R., Ta-Shma, A., Wigderson, A., Zhou, S.: SL L 4/3. In: 29th STOC, pp. 230–239 (1997) · Zbl 0968.68118 [5] Arora, S., Lund, C., Motwani, R., Sudan, M., Szegedy, M.: Proof Verification and Intractability of Approximation Problems. JACM 45, 501–555 (1998); Preliminary version in 33rd FOCS (1992) · Zbl 1065.68570 [6] Arora, S., Safra, S.: Probabilistic Checkable Proofs: A New Characterization of NP. JACM 45, 70–122 (1998); Preliminary version in 33rd FOCS (1992) · Zbl 0903.68076 [7] Babai, L., Fortnow, L., Levin, L., Szegedy, M.: Checking Computations in Polylogarithmic Time. In: 23rd STOC, pp. 21–31 (1991) [8] Bellare, M., Goldreich, O., Sudan, M.: Free Bits, PCPs and Non-Approximability – Towards Tight Results. SICOMP 27(3), 804–915 (1998); Extended abstract in 36th FOCS (1995) · Zbl 0912.68041 [9] Ben-Sasson, E., Goldreich, O., Harsha, P., Sudan, M., Vadhan, S.: Robust PCPs of proximity, Shorter PCPs and Applications to Coding. In: 36th STOC, pp. 1–10 (2004); Full version in ECCC, TR04-021 (2004) · Zbl 1118.68071 [10] Ben-Sasson, E., Sudan, M.: Simple PCPs with Poly-log Rate and Query Complexity. In: ECCC, TR04-060 (2004) · Zbl 1192.68287 [11] Broder, A.: How hard is it to marry at random (On the approximation of the Permanent). In: 18th STOC, pp. 50–58 (1986) [12] Dinur, I.: The PCP Theorem by Gap Amplification. In: 38th STOC, pp. 241–250 (2006) · Zbl 1301.68133 [13] Dinur, I., Reingold, O.: Assignment-testers: Towards a combinatorial proof of the PCP-Theorem. In: 45th FOCS, pp. 155–164 (2004) · Zbl 1127.68031 [14] Feige, U., Goldwasser, S., Lovász, L., Safra, S., Szegedy, M.: Approximating Clique is almost NP-complete. JACM 43, 268–292 (1996); Preliminary version in 32nd FOCS (1991) · Zbl 0882.68129 [15] Goldreich, O.: Computational Complexity: A Conceptual Perspective. Cambridge University Press, Cambridge (2008) · Zbl 1154.68056 [16] Gaber, O., Galil, Z.: Explicit Constructions of Linear Size Superconcentrators. JCSS 22, 407–420 (1981) · Zbl 0487.05045 [17] Jerrum, M., Sinclair, A.: Approximating the Permanent. SICOMP 18, 1149–1178 (1989) · Zbl 0723.05107 [18] Jerrum, M., Sinclair, A., Vigoda, E.: A Polynomial-Time Approximation Algorithm for the Permanent of a Matrix with Non-Negative Entries. JACM 51(4), 671–697 (2004); Preliminary version in 33rd STOC pp. 712–721 (2001) · Zbl 1204.65044 [19] Linial, N., Samorodnitsky, A., Wigderson, A.: A Deterministic Strongly Polynomial Algorithm for Matrix Scaling and Approximate Permanents. Combinatorica 20(4), 545–568 (2000) · Zbl 0973.15004 [20] Lubotzky, A., Phillips, R., Sarnak, P.: Ramanujan Graphs. Combinatorica 8, 261–277 (1988) · Zbl 0661.05035 [21] Margulis, G.A.: Explicit Construction of Concentrators. Prob. Per. Infor. 9(4), 71–80 (1973) (in Russian); English translation in Problems of Infor. Trans., pp. 325–332 (1975) · Zbl 0312.22011 [22] Nisan, N.: Pseudorandom Generators for Space Bounded Computation. Combinatorica 12(4), 449–461 (1992) · Zbl 0759.68024 [23] Nisan, N.: $${\mathcal RL}\subseteq{\mathcal SC}$$ . Journal of Computational Complexity 4, 1–11 (1994) · Zbl 0806.68043 [24] Nisan, N., Zuckerman, D.: Randomness is Linear in Space. JCSS 52(1), 43–52 (1996) · Zbl 0846.68041 [25] Reingold, O.: Undirected ST-Connectivity in Log-Space. In: 37th STOC (2005) · Zbl 1192.68374 [26] Reingold, O., Vadhan, S., Wigderson, A.: Entropy Waves, the Zig-Zag Graph Product, and New Constant-Degree Expanders and Extractors. Annals of Mathematics 155(1), 157–187 (2001); Preliminary version in 41st FOCS, pp. 3–13 (2000) · Zbl 1008.05101 [27] Rozenman, E., Vadhan, S.: Derandomized Squaring of Graphs. In: Chekuri, C., Jansen, K., Rolim, J.D.P., Trevisan, L. (eds.) APPROX 2005 and RANDOM 2005. LNCS, vol. 3624, pp. 436–447. Springer, Heidelberg (2005) · Zbl 1142.05331 [28] Valiant, L.G.: The Complexity of Computing the Permanent. Theoretical Computer Science 8, 189–201 (1979) · Zbl 0415.68008 [29] Wittgenstein, L.: Tractatus Logico-Philosophicus (1922)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.


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# Calculate the DBE. Ian not sure how to do it.  Also how to calculate DBE of a charged species

Samyak Jain
333 Points
2 years ago
DBE stands for double bond equivalent. It is also called degree of unsaturation.
To calculate double bond equivalent of a compound if its structure is given, count total number of
$\pi$ bonds and rings in it.
In the given structure, number of $\pi$ bonds is 12 and that of rings is 8.
So, its DBE is 20.


text:
For Figure 2 compute the following limits a b (a) (b) For Figure 2 compute the following limits a b

For Figure 2 compute the following limits a b

(a)

(b)

For Figure 2 compute the following limits a b


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# How to prevent OnCollisionEnter2D calling a function again and again?

My current code makes a ball keep jumping whenever it hits the ground. But when it hits a vertical ground it keeps calling the jump function again and again. That is because OnCollisionEnter2D makes the isGrounded bool true as long as it is in touch with a vertical collider. I tried several ways for example this didn't work for me:

void FixedUpdate()
{

if(isGrounded == true){
Verticaljump();
isGrounded = false;
}
}
void OnCollisionEnter2D(Collision2D collision) {
if(collision.gameObject.CompareTag("H-Ground")){
isGrounded = true;
}
}

As shown in the picture the OncollisionEnter2D starts and keep adding force on y-axis.Is there any way I can call a function only once even if the gameObject keeps colliding with a surface.

OnCollisionExit2D also does not work. So how can I call a function only once as it collides with ground and even if it keeps colliding with the ground?

If I've understood your question correctly you have an issue in a situation where the ball touches the "ground" from a side?

Like there (consider = to be ground and o to be a ball)

     o=====

===========

One solution which comes to my mind is to add a small invisible object with rigidbody/collider on the bottom of the ball which will handle isGrounded situation. Then if the rest of the ball touches the ground you just do whatever else you want as grounded situation is already handled by other object (of course you will have to pass that isGrounded to the ball object itself.

Another solution is to check if the transform.position of the platform is actually below the transform.position of the ball. Like that

void OnCollisionEnter2D(Collision2D collision)
{
if (collision.gameObject.CompareTag("H-Ground") && (collision.gameObject.transform.position.y < this.transform.position.y))
{
isGrounded = true;
}
}

• The ground checker object doesn't need its own Rigidbody. Just a collider will do, and it can piggyback off of the parent object's body. A short raycast/circle cast is also a popular way to check for ground below only. Searching "Unity ground check" will turn up lots of examples. – DMGregory Feb 2 at 12:56
• @DMGregory thank you for the hint! – Pavel Stepanov Feb 2 at 13:03
• @PavelStepanov I the problem I am facing is a bit complicated to explain. I have edited my question so that it can be correctly understood. – Jacob Frye Feb 2 at 17:42


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# argument

1. ### Cantor's Diagonal Argument in a Nutshell

Cantor's Diagonal Argument in a Nutshell CDA allows every member of a list of infinite binary sequences to be counted except the last one. The last member doesn't exist. The list can be counted.
2. ### Cantor's Diagonal Argument is Wrong

This is a slight refinement (simplification and condensation) of my previous expositions which evolved in the quoted thread. I put it here because it was being buried, and because it addresses the subject directly.
3. ### About argument of complex numbers

Hi, Let, z, z{_{1}}, z{_{2}} be three complex numbers. How to prove the following two properties: (i) arg(z) - arg(-z) = \pm \pi (ii) \mid z{_{1}}+z{_{2 }}\mid=\mid z{_{1}}-z{_{2 }}\mid \Leftrightarrow arg(z{_{1}}) - arg(z{_{2}})=(\pi /2) Thx.
4. ### Cantors Diagonal Argument Automatically Fails

Cantors Diagonal Argument Aotomatically Fails Even if you "imagine" reaching the end of an infinite binary sequence, it doesn't matter since you always have more sequences than digit places no matter how many digit places there are, and CDA automatically fails. For Ex, two digit places give...
5. ### Cantor's Diagonal Argument and Infinity

If infinity means endless, you can never get to the end of CDA.
6. ### Cantor's Diagonal Argument

Suppose I have a set of all infinite sequences of binary digits. Either I know the digits of each sequence or I don't. If I don't, how do I know they are all different? If I do, I can arrange them in numerical (countable) order. The list has a smallest member 0, and I can decide which of any...
7. ### Oh Boy, A Crazy Challenge to Cantor's Argument

Let g(1) = 0 and g(0) = 1. For any infinite binary string $x = x_1x_2x_3\dots$, let $f( \,x) \, = g(x_1)g(x_2)g(x_3)\dots$. If for each element $x$ of a countable set $X$ of infinite binary strings, $f( \,x) \,$ is also in $X$, then let $X$ be considered â€˜balanced.â€™ Let V be a...
8. ### Modulus and Argument of z

Hello, I am studying complex numbers. I am a little confused since I see if you have $$z=re^{i\theta}$$ I see you can find $r$ by $$r = |z| = \sqrt{x^2+y^2}$$ and $$\theta = \arctan(x,y)$$ I have never seen trigonometric functions with more than one variable. Especially in this case where...
9. ### Cantors Diagonal Argument for n digits.

The number of binary sequences for n digits is always greater than n, for all n. Ex, n=2 10 01 11 00 11=00 is in the list. 00 01 10 11 01=10 is in the list.
10. ### Cantors Diagonal Argument, Logic

A \rightarrow NotA
11. ### Directional derivatives do not guarantee continuity : my argument

Hi, I've reached the conclusion that just because all of the directional derivatives at a point on a function f (as defined in the image attached) exist, that does not imply that the function is continuous at that point. In turn, that does not guarantee differentiability. Please let me know...
12. ### I don't see the significance of Cantor's diagonal argument

I know it shows that we can exhaust all of the naturals leaving out a real number by implementing a rule. But we can do the same thing with matching the naturals with rationals. For example, we can exhaust all elements of N by just matching each of them to each element in the infinite set of...
13. ### composite argument properties

sin(a+b) = sin(a)cos(b) + cos(a)sin(b) Of course there are more for cos etc. I am wondering if any of you fine gentlemen or ladies have any information on the history of this property: I.E: who discovered it and perhaps a location of their notes? I am also in search of a concise proof...
14. ### Reals are uncountable - without Cantor diagonal argument

Gist of proof: length of unit interval = 1, if the reals are countable, the length = 0, contradiction. Proof: Arrange reals into countable list. Let x be an arbitrarily number > 0. Cover first point on list by interval of length x/2, second point by interval of length x/4, third point by...
15. ### Cantors Diagonal Argument is Not Diagonal

In my view, Zylo's problem is that he thinks of infinite in terms of some sort of limit of finite cases.
16. ### Cantor's "Diagonal" Argument, Last Member

https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument By definition, Cantor's s is different from every member of the list, so it can only be at the end of the list. But the list is countably infinite and so has no end. So s doesn't exist and Cantor's "Diagonal" Argument fails.
17. ### Cantor's Diagonal Argument Fails by Inductiou

The only mathematical tool for dealing with infinity is Peano's axiom on Induction. For all n, there are more binary sequences than there are places. Cantor's diagonal construction is impossible. Example, for 2 places there are 4 sequences: 00 01 10 11
18. ### Cantor's Diagonal Argument and Square Arrays

In order for Cantor's construction to work, his array of countably infinite binary sequences has to be square. If si and sj are two binary sequences in the array, and each has countably infinite many digits, are they the same length? Ref: https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument
19. ### Cantor's Diagonal Argument and Binary Sequences

Countably Infinite Reversed Binary Sequence Definitions: Binary Number, Binary Sequence, Reversed Binary Number, Reversed Binary Sequences BN=p_{n}2^{n}+p_{n-1}2^{n-1}+...p_{0}2^{0}\\ BS=p_{n}p_{n-1}....p_{2}p_{1}p_{0}\\ RBN=p_{0}2^{0}+p_{1}2^{1}+p_{2}2^{2}+...p_{n}2^{n}\\...
20. ### Cantor's Diagonal Argument and Infinity

Cantor's argument fails because there is no natural number greater than every natural number.


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# What is the slope of any line perpendicular to the line passing through (29,36) and (57,57)?

Jun 2, 2018

$\text{perpendicular slope } = - \frac{4}{3}$

#### Explanation:

$\text{calculate the slope m using the "color(blue)"gradient formula}$

•color(white)(x)m=(y_2-y_1)/(x_2-x_1)

$\text{let "(x_1,y_1)=(29,36)" and } \left({x}_{2} , {y}_{2}\right) = \left(57 , 57\right)$

$m = \frac{57 - 36}{57 - 29} = \frac{21}{28} = \frac{3}{4}$

$\text{the slope of a line perpendicular to m is}$

•color(white)(x)m_(color(red)"perpendicular")=-1/m

${m}_{\text{perpendicular}} = - \frac{1}{\frac{3}{4}} = - \frac{4}{3}$


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#### Central angle. Arc length and circumference , Chords and secants (other) , Pigeonhole principle (finite number of poits, lines etc.) , Regular polygons

In a regular shape with 25 vertices, all the diagonals are drawn.
Prove that there are no nine diagonals passing through one interior point of the shape.

#### Equilateral triangle , Painting problems , Pigeonhole principle (finite number of poits, lines etc.)

A white plane is arbitrarily sprinkled with black ink. Prove that for any positive l there exists a line segment of length l with both ends of the same colour.

#### Pigeonhole principle (finite number of poits, lines etc.)

A square $ABCD$ contains 5 points. Prove that the distance between some pair of these points does not exceed $\frac{1}{2} AC$

#### Discrete geometry (other) , Pigeonhole principle (finite number of poits, lines etc.) , Regular polygons , Trapeziums (other)

In a regular 1981-gon 64 vertices were marked. Prove that there exists a trapezium with vertices at the marked points.

#### Pigeonhole principle (finite number of poits, lines etc.)

2022 points are selected from a cube, whose edge is equal to 13 units. Is it possible to place a cube with edge of 1 unit in this cube so that there is not one selected point inside it?

#### Chessboard colouring , Examples and counterexamples. Constructive proofs , Pigeonhole principle (finite number of poits, lines etc.)

What is the minimum number of 1×1 squares that need to be drawn in order to get an image of a 25×25 square divided into 625 smaller 1×1 squares?

#### Convex polygons , Pigeonhole principle (finite number of poits, lines etc.)

What is the minimum number of points necessary to mark inside a convex $n$-sided polygon, so that at least one marked point always lies inside any triangle whose vertices are shared with those of the polygon?

#### Covers , Inscribed and circumscribed polygons , Pigeonhole principle (finite number of poits, lines etc.)

A unit square contains 51 points. Prove that it is always possible to cover three of them with a circle of radius $\frac{1}{7}$.

#### Circle, sector, segment, etc , Pigeonhole principle (angles and lengths) , Pigeonhole principle (finite number of poits, lines etc.)

There are 25 points on a plane, and among any three of them there can be found two points with a distance between them of less than 1. Prove that there is a circle of radius 1 containing at least 13 of these points.

#### Geometry on grid paper , Painting problems , Pigeonhole principle (finite number of poits, lines etc.)

A ream of squared paper is shaded in in two colours. Prove that there are two horizontal and two vertical lines, the points of intersection of which are shaded in the same colour.

#### Lines, segments, rays, and angles (other) , Pigeonhole principle (finite number of poits, lines etc.) , Triangles (other)

Prove that no straight line can cross all three sides of a triangle, at points away from the vertices.

#### Incirlce and circumcircle of a triangle , Pentagons , Pigeonhole principle (finite number of poits, lines etc.) , Regular polygons

All the points on the edge of a circle are coloured in two different colours at random. Prove that there will be an equilateral triangle with vertices of the same colour inside the circle – the vertices are points on the circumference of the circle.

#### Partitions into pairs and groups bijections , Pigeonhole principle (finite number of poits, lines etc.) , Regular polygons

2001 vertices of a regular 5000-gon are painted. Prove that there are three coloured vertices lying on the vertices of an isosceles triangle.

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# Biogeochemistry of terrestrial soils as influenced by short-term flooding

Alfredo B. De-Campos, Chi-hua Huang and Cliff T. Johnston
Biogeochemistry
Vol. 111, No. 1/3 (November 2012), pp. 239-252
Stable URL: http://www.jstor.org/stable/23359740
Page Count: 14

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## Abstract

Many terrestrial soils in the US Midwest are temporally flooded during the spring. The effects of short-term flooding on biogeochemical processes that occur in these soils are not fully understood and are the subject of this study. To evaluate these processes we investigated the redox-induced changes in the soil solution for three-cultivated and three-uncultivated/forest soils with different organic matter concentrations. The soils were flooded for 1, 3, 7, and 14-days under anoxic conditions in a biogeochemical reactor. Samples were analyzed for Eh; pH; NO 3 - ; NH 4 + ; total dissolved Mn and Fe; soluble P; dissolved organic and inorganic carbon (DOC—DIC); and evolved CO 2 . We found strongly contrasting responses of the terrestrial soils to flooding. Reducing conditions were established quickly in the uncultivated and more slowly in the cultivated soils. Concomitant changes in pH were higher for the uncultivated soils. The uncultivated soils showed a higher increase in the amount of NH 4 + , P, Fe, Mn than the cultivated soils over the 14-day incubation. The total amount of carbon decomposed was much greater for the uncultivated soils with approximately 900 μg C (CO 2 + DOC + DIC) decomposed per gram of soil compared to a total decomposition of 240 μg C ${\mathrm{g}}_{\mathrm{s}\mathrm{o}\mathrm{i}\mathrm{l}}^{-1}$ for the cultivated soils indicating differences in the type of carbon decomposed. The rapid onset of reducing conditions for the uncultivated soils is attributed to a reactive carbon component that is either absent or occluded in the cultivated soils. This study demonstrates that the biogeochemically-induced changes in carbon dynamics in terrestrial soils are strongly influenced by short-term flooding and the history of soil management.

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1. ## Simplify the following

a) √3 x √15
b) √32 x √24
c) 2√5 x 5√2

2. Remember that $\sqrt{ab}=\sqrt{a}\times\sqrt{b}$

And that $\sqrt{x}\times\sqrt{x}=x$

1) $\sqrt{15}=\sqrt{3}\times\sqrt{5}$

so $\sqrt{15}\times\sqrt{3}$

$=\sqrt{3}\times\sqrt{3}\times\sqrt{5}$

How would this simplify?

2) $\sqrt{32}$

$=\sqrt{16}\times\sqrt{2}$

$=4\times\sqrt{2}$

And

$\sqrt{24}$

$=\sqrt{4}\times\sqrt{6}$

$=2\times\sqrt{6}$

Substitute these into the equation and then simplification should become clear.

3)Write this as:

$2\times5\times\sqrt{2}\times\sqrt{5}$

Simplifying from here should be straightforward.

Reply if you are still stuck and I'll try to be of more assistance.


text:
# image(LieAlgebraMap,LieSubSpace) -- make the image of a Lie subspace under a Lie algebra map

## Synopsis

• Function: image
• Usage:
I=image(f,S)
• Inputs:
• f, an instance of the type LieAlgebraMap,
• S, an instance of the type LieSubSpace, an instance of type LieSubSpace, a Lie subspace of the source of $f$
• Outputs:
• I, an instance of the type LieSubSpace, an instance of LieSubSpace, the image $f(S)$, a Lie subspace of the target of $f$

## Description

If $S$ is of type FGLieSubAlgebra, then image(f,S) is of type FGLieSubAlgebra. If $S$ is an instance of LieSubAlgebra, but not of FGLieSubAlgebra, then image(f,S) is of type LieSubAlgebra. Otherwise, image(f,S) is of type LieSubSpace.

 i1 : F=lieAlgebra({a,b,c,r3,r4,r42}, Weights => {{1,0},{1,0},{2,0},{3,1},{4,1},{4,2}},Signs=>{0,0,0,1,1,0}, LastWeightHomological=>true) o1 = F o1 : LieAlgebra i2 : D=differentialLieAlgebra{0_F,0_F,0_F,a c,a a c,r4 - a r3} o2 = D o2 : LieAlgebra i3 : I=lieIdeal{b c - a c,a b,b r4 - a r4} o3 = I o3 : FGLieIdeal i4 : S=lieIdeal{a c} o4 = S o4 : FGLieIdeal i5 : Q=D/I o5 = Q o5 : LieAlgebra i6 : f=map(Q,D) o6 = f o6 : LieAlgebraMap i7 : T=image(f,S) o7 = T o7 : LieSubAlgebra i8 : basis(5,T) o8 = {(b b b c), (c b c)} o8 : List


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# Mathematical Sciences Research Institute

Home > Scientific > Workshops > Hot Topics Workshops

# Current Hot Topic Workshops

No current hot topic workshops

1. # Hot Topics: Shape and Structure of Materials

Organizers: Myfanwy Evans (TU Berlin), LEAD Frank Lutz (TU Berlin), Dmitriy Morozov (Lawrence Berkeley National Laboratory), James Sethian (University of California, Berkeley), Ileana Streinu (Smith College)
Tangled honeycomb networks | and the Advanced Light Source at LBNL

The fascinating and complicated microstructures of materials that are now visible through advanced imaging techniques challenge the frontiers of characterisation and understanding. At the same time, developments in modern geometric and topological techniques are beginning to illuminate important features of material structures, while the microstructures themselves and the analysis and prediction of their macroscopic properties are inspiring new directions in pure and applied mathematics. In a collaboration with the Lawrence Berkeley National Laboratory (LBNL), this workshop aims at intensifying the interaction of mathematicians with material scientists, physicists and chemists on the structural description and design of materials.

Updated on May 01, 2018 01:55 PM PDT
2. # Hot Topics: Recent progress in Langlands Program

Organizers: Mark Kisin (Harvard University), Elena Mantovan (California Institute of Technology), LEAD Xinwen Zhu (California Institute of Technology)

The purpose of the workshop is to explain Vincent Lafforgue's ground breaking work, constructing the automorphic to Galois direction of the Langlands correspondence for function fields. There will also be a number of talks on more recent developments and related results.

Updated on Jul 12, 2018 10:55 AM PDT

1. # Hot Topics: The Homological Conjectures

Organizers: Bhargav Bhatt (University of Michigan), Srikanth Iyengar (University of Utah), Wieslawa Niziol (CNRS, Ecole Normale Superieure de Lyon), LEAD Anurag Singh (University of Utah)

The homological conjectures in commutative algebra are a network of conjectures that have generated a tremendous amount of activity in the last 50 years. They had largely been resolved for commutative rings that contain a field, but, with the exception of some low dimensional cases, several remained open in mixed characteristic --- until recently, when Yves André announced a proof of Hochster's Direct Summand Conjecture. The progress comes from systematically applying Scholze's theory of perfectoid spaces, which had already shown its value by solving formidable problems in number theory and representation theory. One of the goals of the workshop is to cover the ingredients going into the proofs of the Direct Summand Conjecture.

Updated on Mar 23, 2018 11:01 AM PDT
2. # Hot Topics: Galois Theory of Periods and Applications

Organizers: LEAD Francis Brown (All Souls College, University of Oxford), Clément Dupont (Université de Montpellier), Richard Hain (Duke University), Vadim Vologodsky ( Higher School of Economics)

Periods are integrals of algebraic differential forms over algebraically-defined domains and are ubiquitous in mathematics and physics. A deep idea, originating with Grothendieck, is that there should be a Galois theory of periods. This general principle provides a unifying approach to several problems in the theory of motives, quantum groups and geometric group theory.  This conference will bring together leading experts around this subject and cover topics such as the theory of multiple zeta values, modular forms, and motivic fundamental groups.

Updated on May 06, 2017 01:18 AM PDT
3. # Hot Topics: Cluster algebras and wall-crossing

Organizers: LEAD Mark Gross (University of Cambridge), Paul Hacking (University of Massachusetts, Amherst), Sean Keel (University of Texas, Austin), Lauren Williams (University of California, Berkeley)

Cluster algebras were introduced in 2001 by Fomin and Zelevinsky to capture the combinatorics of canonical bases and total positivity in semisimple Lie groups. Since then they have revealed a rich combinatorial and group-theoretic structure, and have had significant impact beyond these initial subjects, including string theory, algebraic geometry, and mirror symmetry. Recently Gross, Hacking, Keel and Kontsevich released a preprint introducing mirror symmetry techniques into the subject which resolved several long-standing conjectures, including the construction of canonical bases for cluster algebras and positivity of the Laurent phenomenon. This preprint reformulates the basic construction of cluster algebras in terms of scattering diagrams (or wall-crossing structures). This leads to the proofs of the conjectures and to new constructions of elements of cluster algebras. But fundamentally they provide a new tool for thinking about cluster algebras.

The workshop will bring together many of the different users of cluster algebras to achieve a synthesis of these new techniques with many of the different aspects of the subject. There will be lecture series on the new techniques, and other lecture series on connections with Lie theory, quiver representation theory, mirror symmetry, string theory, and stability conditions.

Updated on May 06, 2017 01:18 AM PDT
4. # Hot Topics: Kadison-Singer, Interlacing Polynomials, and Beyond

Organizers: Sorin Popa (University of California, Los Angeles), LEAD Daniel Spielman (Yale University), Nikhil Srivastava (University of California, Berkeley), Cynthia Vinzant (North Carolina State University)

In a recent paper, Marcus, Spielman and Srivastava solve the Kadison-Singer Problem by proving Weaver's KS2 conjecture and the Paving Conjecture. Their proof involved a technique they called the “method of interlacing families of polynomials” and a “barrier function” approach to proving bounds on the locations of the zeros of real stable polynomials. Using these techniques, they have also proved that there are infinite families of Ramanujan graphs of every degree, and they have developed a very simple proof of Bourgain and Tzafriri's Restricted Invertibility Theorem. The goal of this workshop is to help build upon this recent development by bringing together researchers from the disparate areas related to these techniques, including Functional Analysis, Spectral Graph Theory, Free Probability, Convex Optimization, Discrepancy Theory, and Real Algebraic Geometry.

Updated on May 06, 2017 01:18 AM PDT
5. # Hot Topics: Perfectoid Spaces and their Applications

Organizers: Sophie Morel (Princeton University), Peter Scholze (Universität Bonn), LEAD Richard Taylor (Institute for Advanced Study), Jared Weinstein (Boston University)

Since their introduction just two years ago, perfectoid spaces have played a crucial role in a number of striking advances in arithmetic algebraic geometry: the proof of Deligne's weight-monodromy conjecture for complete intersections in toric varieties; the development of p-adic Hodge theory for rigid analytic spaces;  a p-adic analogue of Riemann's classification of abelian varieties over the complex numbers; and the construction of Galois representations for torsion classes in the cohomology of many locally symmetric spaces (for instance arithmetic hyperbolic 3-manifolds). We will start the week with an exposition of the foundations of the theory of perfectoid spaces, with the aim of teaching novices to work with them. Then we will discuss their current and potential applications.

Updated on May 06, 2017 01:18 AM PDT
6. # Hot Topics: Surface subgroups and cube complexes

Organizers: Ian Agol* (University of California, Berkeley), Danny Calegari (University of Chicago), Ursula Hamenstädt (University Bonn), Vlad Markovic (California Institute of Technology)

Recently there has been substantial progress in our understanding of the related questions of which hyperbolic groups are cubulated on the one hand, and which contain a surface subgroup on the other. The most spectacular combination of these two ideas has been in 3-manifold topology, which has seen the resolution of many long-standing conjectures. In turn, the resolution of these conjectures has led to a new point of view in geometric group theory, and the introduction of powerful new tools and structures. The goal of this conference will be to explore the further potential of these new tools and perspectives, and to encourage communication between researchers working in various related fields.

Updated on May 06, 2017 01:18 AM PDT
7. # Hot Topics: Thin Groups and Super-strong Approximation

Organizers: Emmanuel Breuillard* (Universite Paris-Sud, Orsay), Alexander Gamburd (CUNY Graduate Center), Jordan Ellenberg (University of Wisconsin - Madison), Emmanuel Kowalski (ETH Zurich), Hee Oh (Brown University)

The workshop will focus on recent developments concerning various quantitative aspects of "thin groups". These are discrete subgroups of semisimple Lie groups which are both « big » (i.e. Zariski dense) and « small » (i.e. of infinite co-volume). This dual nature leads to many intricate questions. Over the past few years, many new ideas and techniques, arising in particular from arithmetic combinatorics, have been involved in the study of such groups, leading for instance to far-reaching generalizations of the strong approximation theorem in which congruence quotients are shown to exhibit a spectral gap (super-strong approximation).

Simultaneously and sometimes surprisingly, the study of thin groups turns out to be of fundamental importance in a variety of subjects, including equidistribution of homogeneous flows and lattice points counting problems, dynamics on Teichmuller space, the Bourgain-Gamburd-Sarnak sieve in orbit, and arithmetic or geometric properties of certain types of monodromy groups and coverings. The workshop will gather a variety of experts from group theory, number theory, ergodic theory and harmonic analysis to present the accomplishments to date to a broad audience and discuss directions for further study.

Updated on May 06, 2017 01:18 AM PDT
8. # Hot Topics: Kervaire invariant

Organizers: Mike Hill (University of Virginia), Michael Hopkins (Harvard University), and Douglas C. Ravanel* (University of Rochester)

This workshop will focus on the ideas surrounding the recent solution to the Arf-Kervaire invariant problem in stable homotopy theory by Mike Hill, Mike Hopkins and Doug Ravenel. There will be talks on relevant aspects of equivariant stable homotopy theory, including the norm functor and the slice tower. The pertinent parts of chromatic homotopy theory will be covered including formal groups and formal $A$-modules, the Hopkins-Miller theorem, finite subgroups of Morava stabilizer groups and Ravenel's 1978 solution to the analogous problem at primes bigger than 3. There will also be several talks by the organizers giving a detailed account of the proof of the main theorem. Finally there will be a discussion of the questions raised by the unexpected statement of the theorem.

Updated on May 06, 2017 01:18 AM PDT
9. # Hot Topics: Black Holes in Relativity

Organizers: Mihalis Dafermos (University of Cambridge) and Igor Rodnianski* (Princeton)

The mathematical study of the dynamics of the Einstein equations forms a central part of both partial differential equations and geometry, and is intimately related to our current physical understanding of gravitational collapse.

Updated on May 06, 2017 01:18 AM PDT
10. # Hot Topics: Contact structures, dynamics and the Seiberg-Witten equations in dimension 3

Organizers: Helmut Hofer, Michael Hutchings, Peter Kronheimer, Tom Mrowka and Cliff Taubes

This workshop will concentrate on recently discovered relationships between Seiberg-Witten theory and contact geometry on 3 dimensional manifolds. One consequence of these relationships is a proof of the Weinstein conjecture in dimension 3. Another is an isomorphism between the Seiberg-Witten Floer (co)homology and embedded contact homology, the latter a form of Floer homology that was defined by Michael Hutchings. The over arching plan is to introduce the salient features of both the contact geometry side of the story and the Seiberg-Witten side, and then discuss how they are related.

Updated on May 06, 2017 01:18 AM PDT
11. # Hot Topics: Minimal and Canonical Models in Algebraic Geometry

Organizers: Alessio Corti, Jean-Pierre Demailly, János Kollár, Shigefumi Mori

Updated on May 06, 2017 01:18 AM PDT
12. # CMI/MSRI Hot Topics Workshop: Modularity for GL(2) and Beyond

Organizers: Michael Harris, Mark Kisin, Kenneth Ribet, Richard Taylor, David Ellwood

This workshop is jointly funded by MSRI and the Clay Mathematics Institute.

Updated on May 06, 2017 01:18 AM PDT
13. # HOT TOPICS: Mathematical and Statistical Methods for Visualization and Analysis of High Dimensional Data

Organizers: Gunnar Carlsson, Susan Holmes, Persi Diaconis

Updated on May 06, 2017 01:18 AM PDT
14. # Hot Topics Workshop: The Modularity of Elliptic Curves, and Beyond

Organizers: Brian Conrad, Jean-Marc Fontaine, Barry Mazur, Ken Ribet (chair), Richard Taylor

Updated on May 06, 2017 01:17 AM PDT


text:
Tanearria L.

# Use the following information to determine sin(2x)? sin (x) = 2/3 and cos (x) is negative

Solve step by step to get the answer

By:


text:
6 thoughts
last posted April 1, 2016, 1:19 a.m.
0
get stream as: markdown or atom
0

# Update

Erik proposed this change to libraries@haskell.org:

"A better type signature for forM_" (continued here)

0

# Safer types

One solution to this is to have variants of the _ functions that are type-specialised to only accept () as the loop body's result:

traverse_' :: (Applicative f, Foldable t)
=> (a -> f ()) -> t a -> f ()
traverse_' = traverse_

for_' :: (Applicative f, Foldable t)
=> t a -> (a -> f ()) -> f ()
for_' = for_

mapM_' :: (Monad m, Foldable t)
=> (a -> m ()) -> t a -> m ()
mapM_' = mapM_

forM_' :: (Monad m, Foldable t)
=> t a -> (a -> m ()) -> m ()
forM_' = forM_

These make it explicit that the loop should have no result, and makes it a type error to accidentally introduce a non-() result.

0

# Checks and warnings?

GHC has a warning for this. Enabling -Wall (or -fwarn-unused-do-bind) will complain whenever a do block discards a value non-explicitly:

ghci> do putStrLn <$> getLine; return ()  Warning: A do-notation statement discarded a result of type ‘IO ()’ Suppress this warning by saying ‘_ <- (<$>) putStrLn getLine’ or by using the flag -fno-warn-unused-do-bind

However, forM_ defeats this check by discarding all the loop's result values regardless of type. One may intend to discard only (), but when a bug like the above slips in, forM_ will just as happily discard IO () or any other type too, and the checker will be none the wiser.

0

This error is easy to introduce, especially in more complex code.

It's insidious when it happens: there will often be no warning sign or hard failure, only strange results or inexplicable behaviour down the line due to the actions and effects that have silently gone "missing".

0

The problem is in the loop body: <$> applies HT.insert and abort to the result of the HT.lookup action, but the resulting IO actions are then simply discarded by forM_ as values, without being used or executed. This is similar to saying: putStrLn <$> getLine  -- has type IO (IO ())

This action yields a separate IO action as its result. Executing it and discarding its result will end up executing only the getLine, and not the putStrLn.

Correcting this requires using join, or equivalently using =<< as the application operator instead of <$>: putStrLn =<< getLine -- has type IO ()  This combines the two actions into a single action, as expected. 0 # The problem erikd on Freenode #haskell.au shares an interesting bug. The relevant code is: buildTable :: IO EvenCache buildTable = do ht <- HT.new forM_ pairs$ \ (k,v) ->
maybe (HT.insert ht k v) (const $abort k) <$>
HT.lookup ht k
return ht

Can you spot the error?


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Finally, I successfully installed Pytorch on my RaspberryPi 3B. I followed the instructions on How to install PyTorch v0.3.1 on RaspberryPi 3B and a blog post (in Chinese) 在 RaspberryPi 上编译 PyTorch. I am running latest raspian image 2018-04-18-raspbian-stretch and a self-compiled Python 3.6.5. If you are interested in compile Python 3.6.5 or above on your own, check out Installing Python 3.6 on Raspbian. In this post, I will write down steps to my successful installation, and share my compiled wheel file at the end.

First of all, we need to set up SWAP. Edit file /etc/dphys-swapfile, find the constant CONF_SWAPSIZE and change its value to at least 2048, which means sparing 2G for swap file system. I set it to 4096 (4G), because I use a 32G card. To activate the swap file system, do the following commands:

sudo /etc/init.d/dphys-swapfile stop
sudo /etc/init.d/dphys-swapfile start

Then, we need to install dependencies.

sudo apt-get install libopenblas-dev cython3 libatlas-dev \
m4 libblas-dev cmake
pip3 install --user pyyaml numpy

We shall now be ready to compile pytorch. We get pytorch from github, and choose a version (v0.4.0) to build.

git clone --recursive https://github.com/pytorch/pytorch
cd pytorch
git checkout tags/v0.4.0 -b build
git submodule update --init --recursive

Since Pi does not have a GPU that supports CUDA, we are going disable it. We also disable distributed computing.

export NO_CUDA=1
export NO_DISTRIBUTED=1

Suppose we are now in the pytorch folder, use the following command to build pytorch.

python3 setup.py build

This process will be long! On my Pi, it took at least 4 hours (I forgot to time it). I would recommend running the building process in the background:

date && python3 setup.py build && date &> message-build &

The above command will run the building process in the background and output building messages to a file named message-build. Also, date command is used to time the building process. When Pi is working hard to compile pytorch, don't run any other complicated work on it. Indeed, I will suggest not to interrupt it at all, including not to peek the building message in message-build frequently. It is a good idea to run the building process before going to sleep.

After a long enough time, check your message-build file to see if you have a successful build. We can look at the last row of the file:

tail -1 message-build

You should get a message about time from command date. If you do not get it, make sure to check running processes using pstree to see whether the building process has ended. Say we have successfully built pytorch. Since I wanted to get a wheel file for pytorch and installed it to the local user, I did not install it directly using sudo python3 setup.py install. Instead, I went to pytorch folder and did the following commands:

export NO_CUDA=1
export NO_DISTRIBUTED=1
pip3 install --user wheel
python3 setup.py bdist_wheel
cd dist
pip3 install --user torch-0.4.0a0+3749c58-cp36-cp36m-linux_armv7l.whl

Pytorch on RaspberryPi got! I share pytorch wheel file so that you can avoid the compilation. It should save you a lot of time. But make sure you are running Python 3.6 and have numpy installed. If so, use the following command to install pytorch.

pip install --user https://wormtooth.com/files/pytorch/torch-0.4.0a0+3749c58-cp36-cp36m-linux_armv7l.whl