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# The product of two lognormal random variables

Let $X_1$ and $X_2$ be two normal random variables. Write $X_1\sim N(\mu_1, \sigma^2_1)$ and $X_2\sim N(\mu_2, \sigma^2_2)$, to fix ideas.

Consider the corresponding log-normal random variables: $Z_1 = \exp(X_1)$, $Z_2 = \exp(X_2)$.

Question: what is the distribution of the product of the two random variables, i.e., the distribution of $Z_1Z_2$?

If the normal random variables $X_1, X_2$ are independent, or they have a bivariate normal distribution, the answer is simple: we have $Z_1Z_2 = \exp(X_1+X_2)$ with the sum $X_1+X_2$ normal, hence the product $Z_1Z_2$ is still lognormal.

But suppose that $X_1, X_2$ are generally $not$ independent, say with correlation $\rho$. What can we say about the distribution of $Z_1Z_2$?

• This might be useful: stats.stackexchange.com/questions/19948/… May 15 '16 at 17:20
• I doubt it though.. basically this question asks "if the marginals are normally distributed, can we say anything about their joint distribution?" And I don't think we can say much in general
– Ant
May 15 '16 at 19:27
• $Z_1 Z_2 = \exp(X_1 + X_2)$ in general, so your real question is whether $X_1+X_2$ is normal (which it will be if $X_1, X_2$ are bivariate normal with correlation $\rho$) May 15 '16 at 22:12
• If you don't have bivariate normality, merely specifying the correlation and the margins is not sufficient to pin down the bivariate distribution. May 16 '16 at 1:31

Using Dilips answer here, if $X$ and $Y$ are bi-variate normal and $X \sim N(\mu_1, \sigma_1^2)$ and $Y \sim N(\mu_2, \sigma_2^2)$ and the correlation between $X$ and $Y$ is $\rho$. Then

$$Cov(X,Y) = \rho \sigma_1 \sigma_2,$$

$$X + Y \sim N(\mu_1 + \mu_2, \sigma^2_1 + \sigma^2_2 + 2\rho\sigma_1 \sigma_2).$$

Thus $Z_1Z_2$ will also be a lognormal distribution with parameters $\mu_1 + \mu_2$ and $\sigma^2_1 + \sigma^2_2 + 2\rho\sigma_1 \sigma_2$.

• Add some qualification? This is true if the distribution of (X,Y) is bi-variate normal, but that the marginal distribution of X is normal and the marginal distribution of Y is normal does not imply the joint distribution is bi-variate normal... stats.stackexchange.com/questions/30159/… May 15 '16 at 17:41
• @MatthewGunn Interesting. I was blissfully unaware of this. As it stands, my answer does not address the question completely. I will wait a couple of hours before deleting it. Thanks. May 15 '16 at 17:44
• Do you have alot of data from this distribution? In such case you can plot it, or otherwise test if bivariate normal is a good distribution approximation. Otherwise, maybe you could estimate a copula and go from there! May 15 '16 at 18:36
• This is quite obvious, but you're missing the point: what happens if you don't know whether the two marginals have a bivariate distribution? May 16 '16 at 10:53
• Thanks! This amazing answer saved my day! May 26 '19 at 15:45


text:
Permutation
(1)
\begin{align} p=\left( \begin{array}{cccc} 1 & 2 & 3 & 4 \\ 2 & 3 & 4 & 1 \\ \end{array} \right) \end{align}
(2)
$$P:(1,2,3,4)→(4,1,2,3) not(2,3,4,1)$$

The marble in box 1 moves to box 2, the marble in box 2 moves to box 3, the marble in box 3 moves to box 4, and the marble in box 4 moves to box 1.
page revision: 2, last edited: 27 Dec 2013 09:34

format: plain


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# Definition:Properties of Algebraic Structures of Two Operations

## Definition

The purpose of this page is to gather into one place the various types of algebraic structure of two binary operations: $+, *$, and to classify them according to the properties they hold.

The properties are defined by the following key:

 $A0$ Closure of $+$ $A1$ Associativity of $+$ $A2$ Identity Element of $+$ $A3$ Inverses of $+$ $A4$ Commutativity of $+$ $M0$ Closure of $+$ $M1$ Associativity of $*$ $M2$ Identity Element of $*$ $M3$ Inverses of $*$ $M4$ Commutativity of $*$ $D$ Distributivity of $*$ over $+$ $NZD$ No (Proper) Zero Divisors $AC2$ Characteristic 2 for $+$ $AZ$ Zero Element of $+$ $D'$ Distributivity of $+$ over $*$ $MI$ Idempotence of $*$ $MZ$ Zero Element of $*$

$A0$ $A1$ $A2$ $A3$ $A4$ $M0$ $M1$ $M2$ $M3$ $M4$ $D$ $NZD$ $AC2$ $AZ$ $MI$ $MZ$ $D'$
Semiring $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$
Additive Semiring $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$
Rig $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$
Ring without associativity $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $(\checkmark)$ (see here)
Ring $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $(\checkmark)$ (see here)
Ring with Unity $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $(\checkmark)$ (see here)
Commutative Ring with Unity $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $(\checkmark)$ (see here)
Integral Domain $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $(\checkmark)$ (see here)
Division Ring $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $(\checkmark)$ (see here)
Field $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $(\checkmark)$ (see here)
Idempotent Ring $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$
Boolean Ring $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$
Boolean Algebra $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$

A checkmark in brackets: $(\checkmark)$ denotes that the property indicated can be derived from the others.


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## Buliga, Marius G.

Compute Distance To:
 Author ID: buliga.marius-g Published as: Buliga, Marius; Buliga, Marius G.; Buliga, M. External Links: ORCID
 Documents Indexed: 29 Publications since 1993 Co-Authors: 12 Co-Authors with 13 Joint Publications 166 Co-Co-Authors
all top 5

### Co-Authors

 16 single-authored 7 de Saxcé, Géry 6 Vallee, Claude 5 Constantine, Gregory M. 2 Bohnen, Nicolaas I. 2 Vodovotz, Yoram 1 Ballard, Patrick 1 Bartels, John D. 1 Clermont, Gilles 1 Constantinescu, Andrei 1 Ivanco, Larry S. 1 Melka, Richard F. 1 Moore, Robert Y.
all top 5

### Serials

 4 Mathematics and Mechanics of Solids 4 International Journal of Pure and Applied Mathematics 2 Revue Roumaine de Mathématiques Pures et Appliquées 2 Journal of Convex Analysis 1 Houston Journal of Mathematics 1 ZAMP. Zeitschrift für angewandte Mathematik und Physik 1 International Journal of Mathematics and Mathematical Sciences 1 Studii și Cercetări Matematice 1 Acta Applicandae Mathematicae 1 Complex Systems 1 Journal of Elasticity 1 Linear Algebra and its Applications 1 Comptes Rendus de l’Académie des Sciences. Série II 1 International Journal of Applied Mathematics 1 Analysis and Applications (Singapore) 1 Communications in Mathematical Analysis 1 Journal of Generalized Lie Theory and Applications 1 Annals of the Academy of Romanian Scientists. Mathematics and its Applications
all top 5

### Fields

 12 Mechanics of deformable solids (74-XX) 6 Calculus of variations and optimal control; optimization (49-XX) 5 Combinatorics (05-XX) 5 Differential geometry (53-XX) 4 Group theory and generalizations (20-XX) 4 Topological groups, Lie groups (22-XX) 4 Real functions (26-XX) 2 Partial differential equations (35-XX) 2 Global analysis, analysis on manifolds (58-XX) 2 Computer science (68-XX) 2 Mechanics of particles and systems (70-XX) 2 Biology and other natural sciences (92-XX) 1 Mathematical logic and foundations (03-XX) 1 Linear and multilinear algebra; matrix theory (15-XX) 1 Ordinary differential equations (34-XX) 1 Dynamical systems and ergodic theory (37-XX) 1 Geometry (51-XX) 1 General topology (54-XX) 1 Manifolds and cell complexes (57-XX) 1 Statistics (62-XX) 1 Relativity and gravitational theory (83-XX) 1 Systems theory; control (93-XX)

### Citations contained in zbMATH Open

19 Publications have been cited 83 times in 52 Documents Cited by Year
Energy minimizing brittle crack propagation. Zbl 0947.74055
Buliga, Marius
1999
Existence and construction of bipotentials for graphs of multivalued laws. Zbl 1133.49018
Buliga, M.; de Saxcé, G.; Vallée, C.
2008
Bipotentials for non-monotone multivalued operators: fundamental results and applications. Zbl 1217.49019
Buliga, Marius; de Saxcé, Géry; Vallée, Claude
2010
Non maximal cyclically monotone graphs and construction of a bipotential for the Coulomb’s dry friction law. Zbl 1186.49010
Buliga, Marius; de Saxcé, Géry; Vallée, Claude
2010
Four applications of majorization to convexity in the calculus of variations. Zbl 1160.49011
Buliga, Marius
2008
A symplectic Brezis-Ekeland-Nayroles principle. Zbl 1371.74051
Buliga, Marius; de Saxcé, Géry
2017
Dilatation structures. I: Fundamentals. Zbl 1163.53016
Buliga, Marius
2007
Infinitesimal affine geometry of metric spaces endowed with a dilatation structure. Zbl 1273.53025
Buliga, Marius
2010
Lower semi-continuity of integrals with $$G$$ -quasiconvex potential. Zbl 1036.74021
Buliga, M.
2002
Blurred maximal cyclically monotone sets and bipotentials. Zbl 1207.49019
Buliga, Marius; de Saxcé, Géry; Vallée, Claude
2010
Hamiltonian inclusions with convex dissipation with a view towards applications. Zbl 1378.70017
Buliga, Marius
2009
Dilatation structures in sub-Riemannian geometry. Zbl 1174.53020
Buliga, Marius
2008
Self-similar dilatation structures and automata. Zbl 1212.22005
Buliga, Marius
2009
Brittle crack propagation based on an optimal energy balance. Zbl 1150.74333
Buliga, Marius
2002
A characterization of sub-Riemannian spaces as length dilation structures constructed via coherent projections. Zbl 1214.51004
Buliga, Marius
2011
Blurred constitutive laws and bipotential convex covers. Zbl 1269.74008
de Saxcé, Géry; Buliga, Marius; Vallée, Claude
2011
Graphic lambda calculus. Zbl 1357.03045
Buliga, Marius
2013
The topological substratum of the derivative. I. Zbl 0900.57001
Buliga, Marius
1993
Inversion of residual stresses from surface stress measurements. Existence and uniqueness. (Reconstruction d’un champ de contraintes résiduelles à partir des contraintes mesurées sur des surfaces successives. Existence et unicité.) Zbl 0809.73017
Ballard, Patrick; Buliga, Marius; Constantinescu, Andrei
1994
A symplectic Brezis-Ekeland-Nayroles principle. Zbl 1371.74051
Buliga, Marius; de Saxcé, Géry
2017
Graphic lambda calculus. Zbl 1357.03045
Buliga, Marius
2013
A characterization of sub-Riemannian spaces as length dilation structures constructed via coherent projections. Zbl 1214.51004
Buliga, Marius
2011
Blurred constitutive laws and bipotential convex covers. Zbl 1269.74008
de Saxcé, Géry; Buliga, Marius; Vallée, Claude
2011
Bipotentials for non-monotone multivalued operators: fundamental results and applications. Zbl 1217.49019
Buliga, Marius; de Saxcé, Géry; Vallée, Claude
2010
Non maximal cyclically monotone graphs and construction of a bipotential for the Coulomb’s dry friction law. Zbl 1186.49010
Buliga, Marius; de Saxcé, Géry; Vallée, Claude
2010
Infinitesimal affine geometry of metric spaces endowed with a dilatation structure. Zbl 1273.53025
Buliga, Marius
2010
Blurred maximal cyclically monotone sets and bipotentials. Zbl 1207.49019
Buliga, Marius; de Saxcé, Géry; Vallée, Claude
2010
Hamiltonian inclusions with convex dissipation with a view towards applications. Zbl 1378.70017
Buliga, Marius
2009
Self-similar dilatation structures and automata. Zbl 1212.22005
Buliga, Marius
2009
Existence and construction of bipotentials for graphs of multivalued laws. Zbl 1133.49018
Buliga, M.; de Saxcé, G.; Vallée, C.
2008
Four applications of majorization to convexity in the calculus of variations. Zbl 1160.49011
Buliga, Marius
2008
Dilatation structures in sub-Riemannian geometry. Zbl 1174.53020
Buliga, Marius
2008
Dilatation structures. I: Fundamentals. Zbl 1163.53016
Buliga, Marius
2007
Lower semi-continuity of integrals with $$G$$ -quasiconvex potential. Zbl 1036.74021
Buliga, M.
2002
Brittle crack propagation based on an optimal energy balance. Zbl 1150.74333
Buliga, Marius
2002
Energy minimizing brittle crack propagation. Zbl 0947.74055
Buliga, Marius
1999
Inversion of residual stresses from surface stress measurements. Existence and uniqueness. (Reconstruction d’un champ de contraintes résiduelles à partir des contraintes mesurées sur des surfaces successives. Existence et unicité.) Zbl 0809.73017
Ballard, Patrick; Buliga, Marius; Constantinescu, Andrei
1994
The topological substratum of the derivative. I. Zbl 0900.57001
Buliga, Marius
1993
all top 5

### Cited by 73 Authors

 10 de Saxcé, Géry 7 Buliga, Marius G. 7 Vallee, Claude 4 Oueslati, Abdelbacet 3 Cao, Xiaodan 3 Hofacker, Martina 3 Matei, Andaluzia Cristina 3 Miehe, Christian 3 Mielke, Alexander 3 Visintin, Augusto 2 Atchonouglo, Kossi 2 Bomze, Immanuel M. 2 Oleaga, Gerardo E. 2 Placidi, Luca 2 Rossi, Riccarda 2 Selivanova, Svetlana V. 2 Welschinger, Fabian 1 Akbarzadeh, Farzaneh 1 Alicandro, Roberto 1 Armandnejad, Ali 1 Ballard, Patrick 1 Bamer, Franz 1 Ban, Michael 1 Ben Naceur, Ilyes 1 Bonaschi, Giovanni A. 1 Bouzid, Wassila 1 Braides, Andrea 1 Chaoufi, Jamal 1 Che, Haitao 1 Cheng, Long 1 Constantinescu, Andrei 1 Costea, Nicuşor 1 Csirik, Mihály 1 Dargush, Gary F. 1 Darrall, Bradley T. 1 Defranceschi, Anneliese 1 Dondl, Patrick Werner 1 Focardi, Matteo 1 Fortune, Danielle 1 Frenzel, Thomas 1 Gélébart, Lionel 1 Gelli, Maria Stella 1 Ghiba, Ionel-Dumitrel 1 Hentati, Hamdi 1 Jia, Yun 1 Kimura, Masato 1 Kondo, Djimédo 1 Lankeit, Johannes 1 Le Donne, Enrico 1 Lerintiu, Camelia 1 Maalej, Aref 1 Mainik, Andreas 1 Markert, Bernd 1 Milton, Graeme Walter 1 Neff, Patrizio 1 Negri, Matteo 1 Nguyen, An Danh 1 Niculescu, Constantin P. 1 Pan, Xintian 1 Park, Chung Hae 1 Rădulescu, Vicenţiu D. 1 Raptis, Theophanes E. 1 Savaré, Giuseppe 1 Schachinger, Werner 1 Shirafkan, Nima 1 Takaishi, Takeshi 1 Tsotsova, R. 1 Uchida, Gabriele 1 Varga, Csaba György 1 Vasiukov, Dmytro 1 Vitali, Enrico 1 Vodop’yanov, Sergeĭ Konstantinovich 1 Wang, Yiju
all top 5

### Cited in 37 Serials

 4 Journal of Elasticity 3 Computer Methods in Applied Mechanics and Engineering 3 Mathematics and Mechanics of Solids 2 Journal of Mathematical Analysis and Applications 2 Journal of the Mechanics and Physics of Solids 2 International Journal for Numerical Methods in Engineering 2 Acta Applicandae Mathematicae 2 Linear Algebra and its Applications 2 Continuum Mechanics and Thermodynamics 2 European Series in Applied and Industrial Mathematics (ESAIM): Control, Optimization and Calculus of Variations 2 Discrete and Continuous Dynamical Systems. Series S 1 International Journal of Engineering Science 1 Journal of Mathematical Physics 1 ZAMP. Zeitschrift für angewandte Mathematik und Physik 1 Chaos, Solitons and Fractals 1 Annali di Matematica Pura ed Applicata. Serie Quarta 1 Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie IV 1 Kybernetika 1 Proceedings of the American Mathematical Society 1 Siberian Mathematical Journal 1 Computational Mechanics 1 Journal of Global Optimization 1 Applied Mathematical Modelling 1 European Journal of Operational Research 1 Journal of Nonlinear Science 1 Calculus of Variations and Partial Differential Equations 1 Communications in Numerical Methods in Engineering 1 Doklady Mathematics 1 European Journal of Mechanics. A. Solids 1 Communications on Pure and Applied Analysis 1 Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie V 1 Analysis and Applications (Singapore) 1 International Journal of Fracture 1 Boundary Value Problems 1 Communications in Mathematical Analysis 1 Advances in Calculus of Variations 1 Advances in Applied Mathematics and Mechanics
all top 5

### Cited in 17 Fields

 34 Mechanics of deformable solids (74-XX) 17 Calculus of variations and optimal control; optimization (49-XX) 12 Partial differential equations (35-XX) 5 Numerical analysis (65-XX) 4 Operator theory (47-XX) 4 Differential geometry (53-XX) 3 Real functions (26-XX) 3 Global analysis, analysis on manifolds (58-XX) 3 Operations research, mathematical programming (90-XX) 2 Ordinary differential equations (34-XX) 2 General topology (54-XX) 2 Classical thermodynamics, heat transfer (80-XX) 1 Linear and multilinear algebra; matrix theory (15-XX) 1 Topological groups, Lie groups (22-XX) 1 Geometry (51-XX) 1 Computer science (68-XX) 1 Fluid mechanics (76-XX)


text:
# Visualize Evoked data¶

In this tutorial we focus on the plotting functions of mne.Evoked.

import os.path as op
import numpy as np
import matplotlib.pyplot as plt

import mne

# sphinx_gallery_thumbnail_number = 9

First we read the evoked object from a file. Check out Epoching and averaging (ERP/ERF) to get to this stage from raw data.

data_path = mne.datasets.sample.data_path()
fname = op.join(data_path, 'MEG', 'sample', 'sample_audvis-ave.fif')
evoked = mne.read_evokeds(fname, baseline=(None, 0), proj=True)
print(evoked)

Out:

[<Evoked  |  'Left Auditory' (mean, N=55), [-0.1998, 0.49949] sec, 376 ch, ~4.8 MB>, <Evoked  |  'Right Auditory' (mean, N=61), [-0.1998, 0.49949] sec, 376 ch, ~4.8 MB>, <Evoked  |  'Left visual' (mean, N=67), [-0.1998, 0.49949] sec, 376 ch, ~4.8 MB>, <Evoked  |  'Right visual' (mean, N=58), [-0.1998, 0.49949] sec, 376 ch, ~4.8 MB>]

Notice that evoked is a list of evoked instances. You can read only one of the categories by passing the argument condition to mne.read_evokeds(). To make things more simple for this tutorial, we read each instance to a variable.

evoked_l_aud = evoked[0]
evoked_r_aud = evoked[1]
evoked_l_vis = evoked[2]
evoked_r_vis = evoked[3]

Let’s start with a simple one. We plot event related potentials / fields (ERP/ERF). The bad channels are not plotted by default. Here we explicitly set the exclude parameter to show the bad channels in red. All plotting functions of MNE-python return a handle to the figure instance. When we have the handle, we can customise the plots to our liking.

fig = evoked_l_aud.plot(exclude=(), time_unit='s')

All plotting functions of MNE-python return a handle to the figure instance. When we have the handle, we can customise the plots to our liking. For example, we can get rid of the empty space with a simple function call.

fig.tight_layout()

Now we will make it a bit fancier and only use MEG channels. Many of the MNE-functions include a picks parameter to include a selection of channels. picks is simply a list of channel indices that you can easily construct with mne.pick_types(). See also mne.pick_channels() and mne.pick_channels_regexp(). Using spatial_colors=True, the individual channel lines are color coded to show the sensor positions - specifically, the x, y, and z locations of the sensors are transformed into R, G and B values.

picks = mne.pick_types(evoked_l_aud.info, meg=True, eeg=False, eog=False)
evoked_l_aud.plot(spatial_colors=True, gfp=True, picks=picks, time_unit='s')

Notice the legend on the left. The colors would suggest that there may be two separate sources for the signals. This wasn’t obvious from the first figure. Try painting the slopes with left mouse button. It should open a new window with topomaps (scalp plots) of the average over the painted area. There is also a function for drawing topomaps separately.

evoked_l_aud.plot_topomap(time_unit='s')

By default the topomaps are drawn from evenly spread out points of time over the evoked data. We can also define the times ourselves.

times = np.arange(0.05, 0.151, 0.05)
evoked_r_aud.plot_topomap(times=times, ch_type='mag', time_unit='s')

Or we can automatically select the peaks.

evoked_r_aud.plot_topomap(times='peaks', ch_type='mag', time_unit='s')

You can take a look at the documentation of mne.Evoked.plot_topomap() or simply write evoked_r_aud.plot_topomap? in your python console to see the different parameters you can pass to this function. Most of the plotting functions also accept axes parameter. With that, you can customise your plots even further. First we create a set of matplotlib axes in a single figure and plot all of our evoked categories next to each other.

fig, ax = plt.subplots(1, 5, figsize=(8, 2))
kwargs = dict(times=0.1, show=False, vmin=-300, vmax=300, time_unit='s')
evoked_l_aud.plot_topomap(axes=ax[0], colorbar=True, **kwargs)
evoked_r_aud.plot_topomap(axes=ax[1], colorbar=False, **kwargs)
evoked_l_vis.plot_topomap(axes=ax[2], colorbar=False, **kwargs)
evoked_r_vis.plot_topomap(axes=ax[3], colorbar=False, **kwargs)
for ax, title in zip(ax[:4], ['Aud/L', 'Aud/R', 'Vis/L', 'Vis/R']):
ax.set_title(title)
plt.show()

Notice that we created five axes, but had only four categories. The fifth axes was used for drawing the colorbar. You must provide room for it when you create this kind of custom plots or turn the colorbar off with colorbar=False. That’s what the warnings are trying to tell you. Also, we used show=False for the three first function calls. This prevents the showing of the figure prematurely. The behavior depends on the mode you are using for your python session. See http://matplotlib.org/users/shell.html for more information.

We can combine the two kinds of plots in one figure using the mne.Evoked.plot_joint() method of Evoked objects. Called as-is (evoked.plot_joint()), this function should give an informative display of spatio-temporal dynamics. You can directly style the time series part and the topomap part of the plot using the topomap_args and ts_args parameters. You can pass key-value pairs as a python dictionary. These are then passed as parameters to the topomaps (mne.Evoked.plot_topomap()) and time series (mne.Evoked.plot()) of the joint plot. For an example of specific styling using these topomap_args and ts_args arguments, here, topomaps at specific time points (90 and 200 ms) are shown, sensors are not plotted (via an argument forwarded to plot_topomap), and the Global Field Power is shown:

ts_args = dict(gfp=True, time_unit='s')
topomap_args = dict(sensors=False, time_unit='s')
evoked_r_aud.plot_joint(title='right auditory', times=[.09, .20],
ts_args=ts_args, topomap_args=topomap_args)

Sometimes, you may want to compare two or more conditions at a selection of sensors, or e.g. for the Global Field Power. For this, you can use the function mne.viz.plot_compare_evokeds(). The easiest way is to create a Python dictionary, where the keys are condition names and the values are mne.Evoked objects. If you provide lists of mne.Evoked objects, such as those for multiple subjects, the grand average is plotted, along with a confidence interval band - this can be used to contrast conditions for a whole experiment. First, we load in the evoked objects into a dictionary, setting the keys to ‘/’-separated tags (as we can do with event_ids for epochs). Then, we plot with mne.viz.plot_compare_evokeds(). The plot is styled with dict arguments, again using “/”-separated tags. We plot a MEG channel with a strong auditory response.

For move advanced plotting using mne.viz.plot_compare_evokeds(). See also Pandas querying and metadata with Epochs objects.

conditions = ["Left Auditory", "Right Auditory", "Left visual", "Right visual"]
evoked_dict = dict()
for condition in conditions:
fname, baseline=(None, 0), proj=True, condition=condition)
print(evoked_dict)

colors = dict(Left="Crimson", Right="CornFlowerBlue")
linestyles = dict(Auditory='-', visual='--')
pick = evoked_dict["Left/Auditory"].ch_names.index('MEG 1811')

mne.viz.plot_compare_evokeds(evoked_dict, picks=pick, colors=colors,
linestyles=linestyles, split_legend=True)

Out:

{'Left/Auditory': <Evoked  |  'Left Auditory' (mean, N=55), [-0.1998, 0.49949] sec, 376 ch, ~4.8 MB>, 'Right/Auditory': <Evoked  |  'Right Auditory' (mean, N=61), [-0.1998, 0.49949] sec, 376 ch, ~4.8 MB>, 'Left/visual': <Evoked  |  'Left visual' (mean, N=67), [-0.1998, 0.49949] sec, 376 ch, ~4.8 MB>, 'Right/visual': <Evoked  |  'Right visual' (mean, N=58), [-0.1998, 0.49949] sec, 376 ch, ~4.8 MB>}

We can also plot the activations as images. The time runs along the x-axis and the channels along the y-axis. The amplitudes are color coded so that the amplitudes from negative to positive translates to shift from blue to red. White means zero amplitude. You can use the cmap parameter to define the color map yourself. The accepted values include all matplotlib colormaps.

evoked_r_aud.plot_image(picks=picks, time_unit='s')

Finally we plot the sensor data as a topographical view. In the simple case we plot only left auditory responses, and then we plot them all in the same figure for comparison. Click on the individual plots to open them bigger.

title = 'MNE sample data\n(condition : %s)'
evoked_l_aud.plot_topo(title=title % evoked_l_aud.comment,
background_color='k', color=['white'])
mne.viz.plot_evoked_topo(evoked, title=title % 'Left/Right Auditory/Visual',
background_color='w')

## Visualizing field lines in 3D¶

We now compute the field maps to project MEG and EEG data to the MEG helmet and scalp surface.

To do this, we need coregistration information. See Head model and forward computation for more details. Here we just illustrate usage.

subjects_dir = data_path + '/subjects'
trans_fname = data_path + '/MEG/sample/sample_audvis_raw-trans.fif'

maps = mne.make_field_map(evoked_l_aud, trans=trans_fname, subject='sample',
subjects_dir=subjects_dir, n_jobs=1)

# Finally, explore several points in time
field_map = evoked_l_aud.plot_field(maps, time=.1)

Note

If trans_fname is set to None then only MEG estimates can be visualized.

Total running time of the script: ( 0 minutes 16.975 seconds)

Gallery generated by Sphinx-Gallery


text:
# 2020 AMC 10A Problems/Problem 24

## Problem

Let $n$ be the least positive integer greater than $1000$ for which$$\gcd(63, n+120) =21\quad \text{and} \quad \gcd(n+63, 120)=60.$$What is the sum of the digits of $n$?

$\textbf{(A) } 12 \qquad\textbf{(B) } 15 \qquad\textbf{(C) } 18 \qquad\textbf{(D) } 21\qquad\textbf{(E) } 24$

## Solution 1

We know that $gcd(63, n+120)=21$, so we can write $n+120\equiv0\pmod {21}$. Simplifying, we get $n\equiv6\pmod {21}$. Similarly, we can write $n+63\equiv0\pmod {60}$, or $n\equiv-3\pmod {60}$. Solving these two modular congruences, $n\equiv237\pmod {420}$ which we know is the only solution by CRT (Chinese Remainder Theorem used to so,be a system of MODULAR CONGURENCES). Now, since the problem is asking for the least positive integer greater than $1000$, we find the least solution is $n=1077$. However, we are have not considered cases where $gcd(63, n+120) =63$ or $gcd(n+63, 120) =120$. ${1077+120}\equiv0\pmod {63}$ so we try $n=1077+420=1497$. ${1497+63}\equiv0\pmod {120}$ so again we add $420$ to $n$. It turns out that $n=1497+420=1917$ does indeed satisfy the original conditions, so our answer is $1+9+1+7=\boxed{\textbf{(C) }18}$.

## Solution 2 (bashing)

We are given that $\gcd(63, n+120)=21$ and $\gcd(n+63,120) = 60$. This tells us that $n+120$ is divisible by $21$ but not $63$. It also tells us that $n+63$ is divisible by 60 but not 120. Starting, we find the least value of $n+120$ which is divisible by $21$ which satisfies the conditions for $n$, which is $1134$, making $n=1014$. We then now keep on adding $21$ until we get a number which satisfies the second equation. This number turns out to be $1917$, whose digits add up to $\boxed{\textbf{(C) } 18}$.

-Midnight

## Solution 3 (bashing but worse)

Assume that $n$ has 4 digits. Then $n = abcd$, where $a$, $b$, $c$, $d$ represent digits of the number (not to get confused with $a * b * c * d$). As given the problem, $gcd(63, n + 120) = 21$ and $gcd(n + 63, 120) = 60$. So we know that $d = 7$ (last digit of $n$). That means that $12 + abc \equiv0\pmod {7}$ and $7 + abc\equiv0\pmod {6}$. We can bash this after this. We just want to find all pairs of numbers $(x, y)$ such that $x$ is a multiple of 7 that is $5$ greater than a multiple of $6$. Our equation for $12 + abc$ would be $42*j + 35 = x$ and our equation for $7 + abc$ would be $42*j + 30 = y$, where $j$ is any integer. We plug this value in until we get a value of $abc$ that makes $n = abc7$ satisfy the original problem statement (remember, $abc > 100$). After bashing for hopefully a couple minutes, we find that $abc = 191$ works. So $n = 1917$ which means that the sum of its digits is $\boxed{\textbf{(C) } 18}$.

~ Baolan

## Solution 4

The conditions of the problem reduce to the following. $n+120 = 21k$ where $gcd(k,3) = 1$ and $n+63 = 60l$ where $gcd(l,2) = 1$. From these equations, we see that $21k - 60l = 57$. Solving this diophantine equation gives us that $k = 20a + 17$, $l = 7a + 5$ form. Since, $n$ is greater than $1000$, we can do some bounding and get that $k > 53$ and $l > 17$. Now we start the bash by plugging in numbers that satisfy these conditions. We get $l = 33$, $k = 97$. So the answer is $1917 \implies 1+9+1+7= \boxed{\textbf{(C) } 18}$.

Edited by ~fastnfurious1

## Solution 5

You can first find that n must be congruent to $6\equiv0\pmod {21}$ and $57\equiv0\pmod {60}$. The we can find that $n=21x+6$ and $n=60y+57$, where x and y are integers. Then we can find that y must be odd, since if it was even the gcd will be 120, not 60. Also, the unit digit of n has to be 7, since the unit digit of 60y is always 0 and the unit digit of 57 is 7. Therefore, you can find that x must end in 1 to satisfy n having a unit digit of 7. Also, you can find that x must not be a multiple of three or else the gcd will be 63. Therefore, you can test values for x and you can find that x=91 satisfies all these conditions.Therefore, n is 1917 and $1+9+1+7$=$\boxed{(C)18}$.-happykeeper

## Solution 6 (Reverse Euclidean Algorithm)

We are given that $\gcd(63, n+120) =21$ and $\gcd(n+63, 120)=60.$ By applying the Euclidean algorithm, but in reverse, we have $$\gcd(63, n+120) = \gcd(63, n+120 + 63) = \gcd(63, n+183) = 21$$ and $$\gcd(n+63, 120) = \gcd(n+63 + 120, 120) = \gcd(n+183, 120) = 60.$$

We now know that $n+183$ must be divisible by $21$ and $60,$ so it is divisible by $\text{lcm}(21, 60) = 420.$ Therefore, $n+183 = 420k$ for some integer $k.$ We know that $3 \nmid k,$ or else the first condition won't hold ($\gcd$ will be $63$) and $2 \nmid k,$ or else the second condition won't hold ($\gcd$ will be $120$). Since $k = 1$ gives us too small of an answer, then $k=5 \implies n = 1917,$ so the answer is $1+9+1+7 = \boxed{(C)18}.$

## Solution 7

$\gcd(n+63,120)=60$ tells us $n+63\equiv60\pmod {120}$. The smallest $n+63$ that satisfies the previous condition and $n>1000$ is $1140$, so we start from there. If $n+63=1140$, then $n+120=1197$. Because $\gcd(n+120,63)=21$, $n+120\equiv21\pmod {63}$ or $n+120\equiv42\pmod {63}$. We see that $1197\equiv0\pmod {63}$, which does not fulfill the requirement for $n+120$, so we continue by keep on adding $120$ to $1197$, in order to also fulfill the requirement for $n+63$. Soon, we see that $n+120\pmod {63}$ decreases by $6$ every time we add $120$, so we can quickly see that $n=1917$ because at that point $n+120\equiv21\pmod {63}$. Adding up all the digits in $1917$, we have $\boxed{(C)18}$.

-SmileKat32


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Article Contents
Article Contents

# Steady stratified periodic gravity waves with surface tension I: Local bifurcation

• In this paper we consider two-dimensional, stratified, steady water waves propagating over an impermeable flat bed and with a free surface. The motion is assumed to be driven by capillarity (that is, surface tension) on the surface and a gravitational force acting on the body of the fluid. We prove the existence of small-amplitude solutions. This is accomplished by first constructing a 1-parameter family of laminar flow solutions, $\mathcal{T}$, then applying bifurcation theory methods to obtain local curves of small amplitude solutions branching from $\mathcal{T}$ at an eigenvalue of the linearized problem.
Mathematics Subject Classification: Primary: 35C07, 35B32; Secondary: 35A16, 35R35, 76B45, 76B70.

 Citation:


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If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

# Picture graphs (pictographs) review

Picture graphs are a fun way to display data. This article reviews how to create and read picture graphs (also called pictographs).
A picture graph (or pictograph) is similar to a bar graph but way more fun. Here's one that shows how many sheep each shepherd has:
A picture graph where the y axis is labeled number of sheep and the x axis is labeled yuki, cam, justin, and kim. A picture of a sheep equals one sheep. There are four sheep above Yuki. There are two sheep above Cam. There are three sheep above Justing. There are five sheep above Kim.
This works well when we're dealing with small numbers, but what if shepherds had many more sheep?
We could make each sheep picture represent more than one sheep, like this:
A picture graph where the y axis is labeled number of sheep and the x axis is labeled Diane, Girish, and Willy. A picture of a sheep equals five sheep. There are four pictures of sheep above Diane. There are five pictures of sheep above Girish. There is one picture of Willy.
In this picture graph, Diane has 4, times, 5, equals, 20 sheep, Girish has 5, times, 5, equals, 25 sheep, and Willy has just 5 sheep.

## Practice

Problem 1
• Current
StudentNumber of bugs
Jen4
Lee6
Sara5
Create a picture graph of this data.

Want more practice like this? Check out this exercise and this more advanced exercise.

## Want to join the conversation?

• Sir, what is the real life application of finding mean, median, mode, range?
(or) why we have to find mean, median, mode, range?
• There are many applications of such things. As an adult, one must buy various items; at which point they may research how to obtain inexpensive yet high-quality goods. In many cases of doing research, ratings are involved. Ratings have a mean, a median, and a mode. Also, when researching variation in most anything, range is a factor along with mean, median, and mode.
• Can the scale have decimal values? For instance can 1 hot dog equals 1.5 hot dogs?
• yes , the scale can have any decimal values. However, you should reasonable scale so that the reading of your chart would be easy ;)
• what if there is 1,234,567,890,123,456,789,012,345,678,901,234,567,899 bugs and you need to graph them?
(1 vote)
• A pictorial graph would not be good in this case
• If all values are unique on a bar chart what will be the mode?
• If all numbers occur only once, there is no mode.
(1 vote)
• Using Charts and Graphs in Social Studies
are being used in what form in percentage
10%
23%
44%
77%
• With graphs like these, wouldn't it mean losing information if the data isn't exactly the value given. Like if I had 11 sheep, each sheep icon represents 5 sheep, and to represent my amount, 2 sheep were put.
(1 vote)
• That's very true. You kind of have to adjust the graph for the data given, e.g. in the case of 11, you could go with 1 sheep = 1 picture.
However, in some cases you could represent half the amount of sheep by slicing the picture in half, which is more visually distinguishable.
(E.g. 1 icon = 2 sheep -> 5 and 1/2 icons represent 11 sheep.)
(1 vote)
• Each medical intern was asked to chart how many patients per day each day of the week. By the end of the week, they had a chart to show which days were busiest.


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# Variation of parameters

1. Sep 13, 2007

### kasse

1. The problem statement, all variables and given/known data

x^2y''-2xy'+2y=x^3cosx

Find a general solution by using variation of parameters

2. The attempt at a solution

I already solved this one, but I have 4 questions:

1. I found the solutions x and x^2 to the homogeneous equation by inspection. Is this the only way to do it?

2. How about the solutions -x and -x^2? I first tried the solutions x^2 and -x^2, and that gave me 0 as the particular solution...Why can't I choose these two?

3. When calculating the Wronski, how do I know which solution is y1 and y2? I first chose y1 as x^2, and that gave me a different solution...

4. Why are there no constants of integration?

2. Sep 14, 2007

### HallsofIvy

Staff Emeritus
The homogeneous equation here is $x^2 y"- 2xy'+ 2=0$. That's an "equipotential" equation, also called an "Euler type" equation. You can get its general solution by "trying" y= xr as a solution. Also the substitution x= ln(t) changes the equation into a "constant coefficients" equation that is easy to solve.

The general solution to the homogeneous equation is Cx+ Dx2 where C and D can be any contstants. You can choose them to be -1, certainly. I don't know what you mean by "gave me 0 as the particular solution". y identically equal to 0 certainly doesn't satisfy the equation.

The Wronskian is a determinant. If you swap two columns, you just multiply the Wronskian by -1. That should do no more than multiply your previous solution by -1.

Because youdidn't put them in! Your general solution should be Cx+ Dx2+ a particular solution to the entire equation.

3. Sep 14, 2007

### HallsofIvy

Staff Emeritus
The homogeneous equation here is $x^2 y"- 2xy'+ 2=0$. That's an "equipotential" equation, also called an "Euler type" equation. You can get its general solution by "trying" y= xr as a solution. Also the substitution x= ln(t) changes the equation into a "constant coefficients" equation that is easy to solve.

The general solution to the homogeneous equation is Cx+ Dx2 where C and D can be any contstants. You can choose them to be -1, certainly. I don't know what you mean by "gave me 0 as the particular solution". y identically equal to 0 certainly doesn't satisfy the equation.

The Wronskian is a determinant. If you swap two columns, you just multiply the Wronskian by -1. That should do no more than multiply your previous solution by -1.

Because youdidn't put them in! Your general solution should be Cx+ Dx2+ a particular solution to the entire equation.

4. Sep 14, 2007

### kasse

Thanks!

Last edited: Sep 14, 2007
5. Sep 14, 2007

### kasse

The Wronski keeps confusing me, however.

W(x^2, x) = -x^2

W(x, x^2) = x^2

The particular solution depends on which one of these I choose.

6. Sep 17, 2007

Or not?


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# Things on a Heap

|
A Collection of Programming Ramblings by chjdev

# Docker overrules UFW!

Docker, by default, directly manipulates iptables in order to work its network magic. It thus completely circumvents UFW rules! In this post I show strategies to deal with this.

# Problem

While setting up my Docker server I noticed something weird. I started up all my containers to see if everything is building and running correctly, enabled ports 80 and 443 on my firewall, and everything worked fine. However, after disabling the ports again my server was still reachable. Even explicitly REJECT-ing the ports still kept them open! I’m using UFW as firewall to keep the base system as simple as possible.

Huh? What the hell is going on? Well, as we can see in the docs, Docker directly manipulates the systems iptables to forward trafffic to containers!

Well, I wasn’t aware that this completely circumvents rules set in UFW. UFW in turn (obviously) doesn’t show you the whole iptables state but only its subset of rules.

# Strategies

If you want to disable this behaviour you can disable it in your daemon.json (or via flags) and rely on the userland proxy:

Now you face a different problem though. Since Docker uses bridged networking by default you now lose the original source IP, which, depending on your use-case, can be quite important, e.g. to log it as part of the access log of your web servers.

To solve this issue you have two options: switch the networking stack to host either globally or the involved containers (mixed linking of containers doesn’t work), or bite the bullet and deal with the iptables behaviour and disable the userland proxy. I opted for the latter because of the security implications of host networking. According to the docs:

Note that this does not let the container reconfigure the host network stack — that would require –privileged=true — but it does let container processes open low-numbered ports like any other root process. It also allows the container to access local network services like D-bus. This can lead to processes in the container being able to do unexpected things like restart your computer.

Also it is apparently the more canonical approach, with discussion of disabling the userland-proxy by default.

You have a question or found an issue?


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# Boundary value problem, complex-variable methods

Methods for studying boundary value problems for partial differential equations in which one uses representations of solutions in terms of analytic functions of a complex variable.

Given a second-order elliptic equation

 (1)

where are analytic functions of the real variables in some domain of the -plane, , consider the following boundary value problem: Find a solution of equation (1), regular in a simply-connected domain , satisfying the boundary condition

 (2)

where , and the are linear operators mapping into , with completely continuous.

This problem includes as special cases the well-known classical boundary value problems of Dirichlet, Neumann, Poincaré, etc.

Using the formula for the general representation of solutions (see Differential equation, partial, complex-variable methods),

one reduces the problem to an equivalent boundary value problem for analytic functions:

 (3)

where are given Hölder-continuous functions, , is a completely-continuous operator, and the are linear operators.

Suppose that the finite simply-connected domain is bounded by a closed Lyapunov contour (see Lyapunov surfaces and curves) and that the -th derivative, , of the function (the latter is holomorphic in ), restricted to , is a function of class , . Then, assuming that the point is in , one can express as follows:

where is a real function of class , , and is a real constant; and are uniquely determined by .

Substituting these expressions into the boundary condition (3), one obtains a singular integral equation, equivalent to problem (2), for the unknown function :

, where is a completely-continuous operator.

A necessary and sufficient condition for the boundary value problem (2) to be normally solvable is that

 (4)

The Dirichlet problem () is always normally solvable. (Henceforth it is assumed throughout that condition (4) is satisfied.)

The index of the boundary value problem (2) is computed from the formula

where is the increment of the function when the contour is described once in the positive sense. The index of the Dirichlet problem is zero.

The homogeneous boundary value problem has a finite number of linearly independent solutions, where ; the inhomogeneous problem (2) has a solution if and only if

where is a complete system of linearly independent solutions of the associated homogeneous integral equation

The boundary value problem (2) has a solution, whatever the free term on the right, if and only if there exist exactly linearly independent solutions of the associated homogeneous problem . Consequently, if the homogeneous boundary value problem always has at least linearly independent solutions; if the inhomogeneous problem (2) is not solvable for an arbitrary free term on the right, but at least solvability conditions must be satisfied.

Necessary and sufficient conditions for the solvability of an inhomogeneous boundary value problem may be formulated in terms of the completeness of a certain system of functions. The kernel and system of functions may be constructed explicitly using the Riemann function of equation (1) and the coefficients of the boundary conditions. For example, let be some complete system of solutions in the basic domain of equation (1), and let . Then a necessary and sufficient condition for problem (2) to be solvable for any free right-hand side is that the system of functions be complete on the boundary.

Very complete results have been obtained for the following boundary value problem (the generalized Riemann–Hilbert problem): Find a solution of the equation

 (5)

which is continuous in and satisfies the boundary condition

 (6)

where are real functions of class , , with . The domain may be multiply connected. A problem of this type may be reduced to an equivalent singular integral equation; this yields a full qualitative analysis of the boundary value problem (6).

Suppose that the boundary of is the union of a finite number of simple closed curves satisfying the Lyapunov conditions. Since the forms of the equation and the boundary condition are preserved under conformal mapping, it may be assumed without loss of generality that is the unit circle with centre at , the latter being a point of , while are circles lying outside .

The index of problem (6) is defined as the integer equal to the increment of when the point describes once in the positive sense. The boundary condition can be reduced to the simpler form

where on , with , while are real constants, uniquely expressible in terms of and . The index of the adjoint problem

 (7)

is calculated by the formula .

Problem (6) has a solution if and only if

where is an arbitrary solution of the adjoint problem.

Let and be the numbers of linearly independent solutions of the homogeneous problems (6) and (7), respectively. Then

If , the homogeneous problem (6) has no non-trivial solutions. If , it has exactly linearly independent solutions, while the inhomogeneous problem (6) is always solvable. If , the inhomogeneous problem (6) is solvable if and only if

where is a complete system of solutions of the homogeneous problem (7). If and , then and all solutions of the homogeneous problem problem (6) are given by

where is a real constant and a continuous function on .

The above results completely characterize the problem in the simply-connected () and multiply-connected (, ) cases. The cases require special examination; they have also been worked out in considerable detail.

Boundary value problems of the type of the Poincaré problem have also been studied for equation (5).

For references see Differential equation, partial, complex-variable methods.

#### References

 [a1] I.N. Vekua, "Generalized analytic functions" , Pergamon (1962) (Translated from Russian) [a2] N.P. Vekua, "Systems of singular integral equations and some boundary value problems" , Moscow (1970) (In Russian) [a3] I.N. Vekua, "New methods for solving elliptic equations" , North-Holland (1968) (Translated from Russian)
How to Cite This Entry:
Boundary value problem, complex-variable methods. I.N. Vekua (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Boundary_value_problem,_complex-variable_methods&oldid=17473
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098


text:
## CryptoDB

### Paper: On Resilient Boolean Functions with Maximal Possible Nonlinearity

Authors: Yuriy Tarannikov URL: http://eprint.iacr.org/2000/005 Search ePrint Search Google It is proved that the maximal possible nonlinearity of $n$-variable $m$-resilient Boolean function is $2^{n-1}-2^{m+1}$ for ${2n-7\over 3}\le m\le n-2$. This value can be achieved only for optimized functions (i.~e. functions with an algebraic degree $n-m-1$). For ${2n-7\over 3}\le m\le n-\log_2{n-2\over 3}-2$ it is suggested a method to construct an $n$-variable $m$-resilient function with maximal possible nonlinearity $2^{n-1}-2^{m+1}$ such that each variable presents in ANF of this function in some term of maximal possible length $n-m-1$. For $n\equiv 2\pmod 3$, $m={2n-7\over 3}$, it is given a scheme of hardware implementation for such function that demands approximately $2n$ gates EXOR and $(2/3)n$ gates AND.
##### BibTeX
@misc{eprint-2000-11349,
title={On Resilient Boolean Functions with Maximal Possible Nonlinearity},
booktitle={IACR Eprint archive},
keywords={secret-key cryptography / boolean functions, stream ciphers, secret-key cryptography, implementation},
url={http://eprint.iacr.org/2000/005},
note={ yutaran@nw.math.msu.su, taran@vertex.inria.msu.ru 11026 received 10 Mar 2000},
author={Yuriy Tarannikov},
year=2000
}


text:
# latex listings python

I would use the minted package as mentioned from the developer Konrad Rudolph instead of the listing package.Here is why: listing package. Python code in LaTeX. Include source code in Latex with “Listings” « Blog on Latex Matters (tags: listings latex src) […] Reply. Then it's a good reason to buy me a coffee. Thank you ! ‘La’ is a front end to Don Knuth's typesetting program TeX. Below the fold, I'll present an example. Sympy; Buy Me A Coffee ! Code snippets The well-known LATEX command \verb typesets code snippets verbatim. … The listings package is a pure LaTeX implementation that works reasonably well, but the package appears to be orphaned and the latest documentation was written in 2007. It's fairly easy to use and there's good documentation available on how to use it. 其实就是使用Listings包，一个例子如下： 在正文前（\begin{document}之前）使用如下代码设置参数：\usepackage{listings}\usepackage{color}\definecolor{dkg. I have been running PyX and PGF/TikZ in parallel as I build up my skills in each, and this means I can more easily use them together. October 2008 at 15:39. Search for jobs related to Python3 latex or hire on the world's largest freelancing marketplace with 18m+ jobs. In LaTeX, you can easily reference almost anything that can be numbered, and have LaTeX automatically updating the numbering for you whenever necessary. While Javascript is not essential for this website, your interaction with the content will be limited. Calculations and figures may be next to the code that created them. It is simply a stand-alone program. Some features of pylatex are: We can access all the features of LaTeX in python using this module; We can make documents with fewer lines of code Search for jobs related to Python word to latex or hire on the world's largest freelancing marketplace with 18m+ jobs. Code listings in LaTeX “Pretty” code listings are sometimes considered worthwhile by the “ordinary” programmer, but they have a serious place in the typesetting of dissertations by computer science and other students who are expected to write programs. The Listings Package: LaTeX/Source Code Listings It's incredibly simple. make_env can be used to create an Environment that plays well with LaTex: Simple verbatim listings of programs are commonly useful, as well. LaTex, Python, and R are three popular languages that are especially useful in research and university-level work. Since code is adjacent to its output in the document, editing may be more efficient. Please check the size chart here (last photo) and state your size when you order. Search PyPI Search. Of course, this site is also created from reStructuredText sources using Sphinx! Thestring“Listing” in a listing’s caption can be changed. A simple Python highlighting style to be used with LaTeX. To do this, simply redeﬁne the macro \listingscaption,forexample: \renewcommand{\listingscaption}{Program code} \listoflistingscaption (Only applies when package option newfloat is not used.) Comments #1 Ken Starks, October 24, 2008 at 1:29 p.m. The goal of this library is to be easy but is also to provide an extensible interface between Python and latex. Python. 12. The following are 21 code examples for showing how to use sympy.latex().These examples are extracted from open source projects. This website was useful to you? Now think of LaTeX as La/TeX. Free resources: Overleaf, Sublime Text, and VSCode. It will give me the energy and motivation to continue this development. You can vote up the ones you like or vote down the ones you don't like, and go to the original project or source file by following the links above each example. PythonTeX is a LaTeX package that allows Python code in LaTeX documents to be executed and provides access to the output. nguyenminhhaivn. We use analytics cookies to understand how you use our websites so we can make them better, e.g. Python. Write LaTeX code to display the angle sum identity $$\cos(\alpha \pm \beta) = \cos \alpha \cos \beta \mp \sin \alpha \sin \beta$$ Write LaTeX code to display the indefinite integral $$\int \frac{1}{1 + x^2} \, dx = \arctan x + C$$ Write LaTeX code to display the Navier-Stokes Equation for Incompressible Flow To use, \usepackage{listings}, identify the language of the object to typeset, using a construct like: \lstset{language=Python}, then use environment lstlisting for inline code. Including Python code in LaTeX papers is very simple and convenient with the “listings” package. External files may be formatted using \lstinputlisting to process a given file in the form appropriate for the current language. Analytics cookies. Rekisteröityminen ja tarjoaminen on ilmaista. PyLaTeX 1.4.1 pip install PyLaTeX Copy PIP instructions. The problem, just as with Pygments, is how to use it meaningfully from within LaTeX and provide all the nice things that listings has. The commands to be used do not depend on what you are referencing, and they are: \label{marker} Used to … LaTeX differs from other programming languages in the respect that, instead of coding applications or programs, it typesets scientific and technical documents. silent56_th 2017-10-22 20:28:54 12740 收藏 9 分类专栏： python LaTeX 文章标签： python latex. In contrast, the minted package is not a pure LaTeX solution and relies on an external Python script, pygmentize, to typeset and fontify source code. 28. Black latex knickers with Black snakeskin pattern embossed latex detail. This makes possible reproducible documents that combine results with the code required to generate them. To use, \usepackage{listings}, identify the language of the object to typeset, using a construct like: \lstset{language=Python}, then use environment lstlisting for inline code. PyLaTeX is a Python library for creating and compiling latex documents. Section 2.5Special characters discusses this issue. Documentation of the package is part of the (awesome) LaTeX wikibook… First, include the package in your document: \documentclass{article} \usepackage{listings} \begin{document} \end{document} And then insert … Sides are adjustable with buckles. Knowledge base dedicated to Linux and applied mathematics. Skip to main content Switch to mobile version Help the Python Software Foundation raise \$60,000 USD by December 31st! A Python library for creating LaTeX files and snippets. Subsections. Matching Bra and garters are also Thanks for sharing this Kjell. The package is loaded by the following line: Joe Python. External files may be formatted using \lstinputlisting to process a given file in the form appropriate for the current language. PyTeX, or Py/TeX is you prefer, is to be a front end to TeX, written in Python. Get started with the official Dash docs and learn how to effortlessly style & deploy apps like this with Dash Enterprise. It's free to sign up and bid on jobs. To run the app below, run pip install dash, click "Download" to get the code and run python app.py. The official home of the Python Programming Language. Reply . It was originally created for the Python documentation, and it has excellent facilities for the documentation of software projects in a range of languages. Listings has support for Python out-of-the-box and has lots of options and features (check the manual to see what listings can do). I've tried with the latex listings package but wasn't able to produce something that looked as nice as the one below. Python highlighting in LaTeX. 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text:
# Maclaurin approximations

1. Jun 13, 2010

### estro

Suppose f(x) is differentiable 2 times, can I still use Maclaurin polynomial approximations and write:

$$f(x)=f(0)+f'(0)x+\frac {f''(0)x^2} {2!}+R_2(x)$$

If yes why? (Lagrange and Cauchy reminder theorem are using in this case the 3'rd order derivative...)

Last edited: Jun 13, 2010
2. Jun 13, 2010

### rock.freak667

Last edited by a moderator: Apr 25, 2017
3. Jun 13, 2010

### estro

I remember reading proof for a theoretical problem using this concept. (Can't remember where it was).

I hope I'll find it tomorrow and post it here.

Last edited: Jun 13, 2010
4. Jun 13, 2010

### rock.freak667

I think if you do it on a polynomial, you will just get back the polynomial.

5. Jun 13, 2010

### estro

Yes, I understand and know it.

I asked this question from theoretical point of view.
f(x) is unknown function we only have the numerical information about f(0), f'(0) and f''(0).

6. Jun 13, 2010

### rock.freak667

Yes I don't see why you can't use the series if you have those values.

7. Jun 13, 2010

### estro

But $$R_2(x)$$ is defined using $$f^{(3)} (x)$$ And we don't know if this function exists. [Lagrange Reminder Theorem]

Last edited: Jun 13, 2010
8. Jun 13, 2010

### DrRocket

You can always do that.
The difficulty comes the determination of a useful expression for $$R_2(x)$$. Probably the easiest way to determine such an expression is the integral form of the remainder which comes from integrating by parts repeatedly.The result is an expression in whch the remainder involves the integral of the third derivative. So you need at least three times differentiability to get a McClaurin polynomial of degree two with the usual integral form of the remainder.

$$f(x)-f(0)=\int_{0}^{x}f'(t)dt$$

= $$x f(0) - \int_{0}^{x}f''(t)(x-t) dt$$
.
.
.
= $$f(0) + f'(0)x + 1/2f''(0)x^2 + 1/2 \int_{0}^{x} f'''(t)(t-x)^2dt$$

Please excuse the sloppy Tex. I seem to be getting no correlation between the code, the preview and the post. The first expressions should be the remainder, not f and the expressions for the integrals are pretty much a mess. But I hope you can see through this disaster and recreate the proof.

Last edited: Jun 13, 2010
9. Jun 13, 2010

### estro

I'm not sure I understood your idea.
Basically if I knew that f(x) is differentiable 3 times, I would use Lagrange Reminder Theorem to determine $$R_2(x)$$. The problem arise what to do when I don't know such thing.

10. Jun 13, 2010

### DrRocket

The idea is that first, you can write any function as a quadratic polynomial plus a remainder. That is trivial, since the remainder is just the difference between the function and the polynomial. A Maclaurin expansion is useful only because there is a useful expression for the remainder -- which is "small" and therefore makes the approximation by the polynomial "close".

The problem is really one of finding a useful expression for the remainder. There are various forms for that remainder. What I showed (as best I could with problem that I was having with the Tex software) is a simple way to get the remainder by integrating by parts a couple of times. The process works iteratively and will produce a Maclaurin expansion of any degree that you like, so long as you can keep integrating by parts. That requires some level of differentiability -- it requires n+1 derivatives for an nth degree polynomial.

If you know nothiing whatever about the differentiability of the function then you are basically out of luck. That is because you have no good way to estimate the remainder.

Now, if you know something else about the function then depending on what that is one might be able to do something else -- I am not sure what at the moment.

Someone noted that a full blown Taylor series requires infinite differentiability. That is also true. In fact it requires more, It requires the function be not only infinitely differentiable but actually analytic, a much more restrictive condition. The point being that to approximate a function by a polynomial requires that you know something about the function.

What is the fundamental problem that you are trying to solve ?

11. Jun 13, 2010

### estro

Oh, I think I get it now, and you answered the next question I was about to ask.

I don't have any fundamental problem I'm trying to solve, just want to get idea and intuition with Taylor approximations and the Reminder Theorem of Lagrange.

I think because of this "analytic" thing I have so much trouble feeling this idea with intuition. (I don't have problems solving exercises in this topic in my book, But I don't "feel" this topic to the extend I want)

I will find myself over the moon if you can refer me to a good source about this topic.
(I'm self learner student with Calc 1-2 knowlage)

Last edited: Jun 13, 2010
12. Jun 13, 2010

### DrRocket

The first source is your calculus book, assuming that I understand what Calc 1-2 are.

I am not a big fan of most calculus books, but Spivak's book is pretty good.

The next level would be an introductory text on real analysis. There a couple that I like. Those are Elements of Real Analysis by Bartle and Principles of Mathematical Analysis by Rudin. These are generally considered junior-senior level books for mathematics majors.

Analytic functions are usually encountered in a class on complex analysis. It turns out that if a complex valued function is differentiable even one time, then it is differentiable infinitely often and in fact is analytic -- locally representable by a power series. So to see this you would need to read a book on complex analysis. There are several good texts on this subject, but a classic is Complex Analysis by Ahlfors. But to read it you may need a bit more than just Calc 1-2. It is considered at advanced undergraduate or beginnin graduate level book.

As far as the proof that I tried to sketch, I am sure that I have seen it in a book somewhere, but most texts take a slightly different tack and I am not sure of a ready reference off the top of my head. But in any case it is not difficult to work out for yourself -- just fill in the small blanks in what I posted (It is just integration by parts).

Basically what you get is a function minus some polynomial as the remainder, and at that point you have to know something to limit the size of the remainder. Knowing something about its differentiabillity gives you some knowledge of that sort. Without any knowledge at all, anything can happen. So the important part of such approximation theorems revolves around showing how some piece of information regarding the function limits the size of the remainder.

13. Jun 14, 2010

### estro

I'm from Israel, Calculus here is some sort of mix between "Real Analysis and "Calculus 2" tending heavily towards "Calculus 2 [In US]"

I'm too, they tend to concentrate more on solving very similar problems rather understanding theory.

I'll try to get my hands on one of these.

I have very long way to Complex Analysis.

I was never exposed to this approach, but I think now I get the idea and this is quite trivial.


text:
## Algebra 1

-$\frac{20}{9}$
$\frac{2x}{y^{2}z^{-1}}$, where x = 2, y = -3, z = -5 $\frac{2(2)}{-3^{2}-5^{-1}}$ $\frac{4}{9(-5^{-1})}$ Here, -5$^{-1}$ will be moved to the numerator and the -1 will change to simply 1: $\frac{(4)(-5^{1})}{9}$ = $\frac{(4)(-5)}{9}$ = $\frac{-20}{9}$


text:
Math1231 Partial Derivatives

A Partial Derivative is a multivariable function which for differentiation has all variables aside from your dF/d? as constant, and the differentiation is done in regards to ?

e.g.

(1)
\begin{align} \begin {array} {l l} F(x,y,z) & = x^3yz + 5z \\ \frac {\partial{F}}{\partial{x}} & = 3x^2yz \\ \frac {\partial{F}}{\partial{y}} & = x^3z \\ \frac {\partial{F}}{\partial{z}} & = x^3y + 5 \\ \end {array} \end{align}

## Second Partial Derivatives

With two variables there are 4 formats we can do this in.

(2)
\begin{align} \frac {\partial^2F}{\partial x^2} = \frac {\partial}{\partial x} \frac{\partial F}{\partial x} \\ \frac {\partial^2F}{\partial y^2} = \frac {\partial}{\partial y} \frac{\partial F}{\partial y} \\ \frac {\partial^2F}{\partial x\partial y} = \frac {\partial}{\partial x} \frac{\partial F}{\partial y} \\ \frac {\partial^2F}{\partial y\partial x} = \frac {\partial}{\partial y} \frac{\partial F}{\partial x} \\ \end{align}

e.g.

(3)
\begin{align} \begin {array} {r l} F(x,y) & = e^x sin(y) \\ \\ \frac {\partial F}{\partial x} & = e^xsin(y) \\ \frac {\partial F}{\partial y} & = e^xcos(y) \\ \\ \frac {\partial^2 F}{\partial x^2} & = e^xsin(y) \\ \frac {\partial^2 F}{\partial y^2} & = -e^xsin(y) \\ \frac {\partial^2 F}{\partial x\partial y} & = e^xcos(y) \\ \frac {\partial^2 F}{\partial y\partial x} & = e^xcos(y) \\ \end {array} \end{align}

## Chain Rules for Partial Derivatives

page revision: 10, last edited: 12 Aug 2011 00:48


text:
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text:
zbMATH — the first resource for mathematics

Armbruster, Dieter

Compute Distance To:
 Author ID: armbruster.dieter Published as: Armbruster, D.; Armbruster, Dieter
 Documents Indexed: 69 Publications since 1983, including 5 Books
all top 5

Co-Authors

 10 single-authored 8 Chossat, Pascal 8 Ringhofer, Christian A. 5 Dangelmayr, Gerhard 5 Herty, Michael Matthias 5 Oprea, Iuliana 3 Degond, Pierre 3 Kang, Yun 3 Nicolaenko, Basil 3 Smaoui, Nejib 3 Stone, Emily F. 2 Gel, Esma Senturk 2 Göttlich, Simone 2 Guckenheimer, John M. 2 Heiland, Randy W. 2 Holmes, Philip J. 2 Jo, Tae-Chang 2 Kempf, Karl G. 2 Kostelich, Eric J. 2 Marthaler, Daniel E. 2 Nagy, John D. 2 Thatcher, Andrea 1 Diaz-Rivera, Ivonne 1 Dong, Ming 1 George, Marguerite 1 Gilg, Brady 1 Güttinger, Werner 1 He, Fenglan 1 Ihrig, Edwin C. 1 Kaneko, Kunihiko 1 Kayser, Kirk 1 Kirby, Michael P. 1 Kirk, Vivien 1 Knapp, Stephan 1 Kredel, Heinz 1 Kuang, Yang 1 La Marca, Michael 1 Lai, Yingcheng 1 Lefeber, Erjen 1 Martin, Stephan 1 Mikhailov, Alexander S. 1 Motsch, Sébastien 1 Murakami, Junko 1 Neveling, M. 1 Oechsle, Rainer 1 Perdaen, Dominique 1 Rand, Richard H. 1 Rem, Bart 1 Taylor, Tom 1 Unver, Ali K. 1 Wang, Wenxu 1 Wang, Xinping 1 Wienke, Matthew 1 Young, Jon 1 Zhao, Lindu
all top 5

Serials

 8 Physica D 6 Chaos 3 SIAM Journal on Applied Mathematics 2 European Journal of Operational Research 2 Multiscale Modeling & Simulation 2 Kinetic and Related Models 2 Journal of Theoretical Biology 1 Mathematical Proceedings of the Cambridge Philosophical Society 1 ZAMP. Zeitschrift für angewandte Mathematik und Physik 1 Theoretical and Computational Fluid Dynamics 1 Computing 1 IEEE Transactions on Automatic Control 1 Mathematics of Operations Research 1 Operations Research 1 Proceedings of the London Mathematical Society. Third Series 1 Theoretical Computer Science 1 International Journal of Production Research 1 Journal of Symbolic Computation 1 Dynamics and Stability of Systems 1 European Journal of Mechanics. B. Fluids 1 Applied Mathematical Modelling 1 Journal of Non-Equilibrium Thermodynamics 1 SIAM Journal on Scientific Computing 1 Discrete and Continuous Dynamical Systems 1 Proceedings of the Royal Society of London. Series A. Mathematical, Physical and Engineering Sciences 1 Mathematical Biosciences and Engineering 1 Bulletin of the Institute of Mathematics. Academia Sinica. New Series 1 Applied Mathematical Sciences 1 NATO Science Series II: Mathematics, Physics and Chemistry 1 World Scientific Lecture Notes in Complex Systems 1 Journal of Biological Dynamics 1 Networks and Heterogeneous Media 1 International Journal of Biomathematics 1 Journal of Computational Dynamics
all top 5

Fields

 23 Dynamical systems and ergodic theory (37-XX) 18 Operations research, mathematical programming (90-XX) 17 Partial differential equations (35-XX) 14 Ordinary differential equations (34-XX) 14 Fluid mechanics (76-XX) 7 Computer science (68-XX) 6 Global analysis, analysis on manifolds (58-XX) 6 Biology and other natural sciences (92-XX) 5 Systems theory; control (93-XX) 4 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 3 Probability theory and stochastic processes (60-XX) 3 Numerical analysis (65-XX) 3 Classical thermodynamics, heat transfer (80-XX) 2 General and overarching topics; collections (00-XX) 2 Difference and functional equations (39-XX) 2 Geophysics (86-XX) 1 Combinatorics (05-XX) 1 Commutative algebra (13-XX) 1 Calculus of variations and optimal control; optimization (49-XX) 1 Manifolds and cell complexes (57-XX) 1 Statistics (62-XX) 1 Mechanics of particles and systems (70-XX) 1 Statistical mechanics, structure of matter (82-XX)

Citations contained in zbMATH Open

51 Publications have been cited 593 times in 437 Documents Cited by Year
Heteroclinic cycles and modulated travelling waves in systems with 0(2) symmetry. Zbl 0634.34027
Armbruster, Dieter; Guckenheimer, John; Holmes, Philip
1988
Perturbation methods, bifurcation theory and computer algebra. Zbl 0651.34001
Rand, Richard H.; Armbruster, Dieter
1987
Kuramoto-Sivashinsky dynamics on the center-unstable manifold. Zbl 0687.34036
Armbruster, Dieter; Guckenheimer, John; Holmes, Philip
1989
A model for the dynamics of large queuing networks and supply chains. Zbl 1107.90002
Armbruster, D.; Degond, P.; Ringhofer, C.
2006
A continuum model for a re-entrant factory. Zbl 1167.90477
Armbruster, Dieter; Marthaler, Daniel E.; Ringhofer, Christian; Kempf, Karl; Jo, Tae-Chang
2006
Coupled stationary bifurcations in nonflux boundary value problems. Zbl 0633.58011
Armbruster, D.; Dangelmayr, G.
1987
Control of continuum models of production systems. Zbl 1368.90056
La Marca, Michael; Armbruster, Dieter; Herty, Michael; Ringhofer, Christian
2010
Noisy heteroclinic networks. Zbl 1080.37510
Armbruster, Dieter; Stone, Emily; Kirk, Vivien
2003
Symmetries and dynamics for 2-D Navier-Stokes flow. Zbl 0899.76113
Armbruster, Dieter; Nicolaenko, B.; Smaoui, N.; Chossat, Pascal
1996
Kinetic and fluid model hierarchies for supply chains. Zbl 1078.90031
Armbruster, D.; Marthaler, D.; Ringhofer, C.
2004
Dynamics of a plant-herbivore model. Zbl 1140.92322
Kang, Yun; Armbruster, Dieter; Kuang, Yang
2008
Steady-state mode interactions in the presence of O(2)-symmetry and in non-flux boundary value problems. Zbl 0602.58039
Dangelmayr, G.; Armbruster, D.
1986
Reconstructing phase space from PDE simulations. Zbl 0764.35008
Kirby, Michael; Armbruster, Dieter
1992
Noise and $$O(1)$$ amplitude effects on heteroclinic cycles. Zbl 0971.37010
Stone, Emily; Armbruster, Dieter
1999
Symmetry and the Karhunen-Loève analysis. Zbl 0888.35008
Smaoui, Nejib; Armbruster, Dieter
1997
Thermalized kinetic and fluid models for reentrant supply chains. Zbl 1108.90006
Armbruster, D.; Ringhofer, C.
2005
Classification of Z(2)-equivariant imperfect bifurcations with corank 2. Zbl 0517.58027
Dangelmayr, G.; Armbruster, D.
1983
Bucket brigades with worker learning. Zbl 1137.90575
Armbruster, Dieter; Gel, Esma S.; Murakami, Junko
2007
Phase-space analysis of bursting behavior in Kolmogorov flow. Zbl 1194.76085
Armbruster, Dieter; Heiland, Randy; Kostelich, Eric J.; Nicolaenko, Basil
1992
Heteroclinic orbits in a spherically invariant system. Zbl 0739.58036
Armbruster, Dieter; Chossat, Pascal
1991
O(2)-symmetric bifurcation theory for convection rolls. Zbl 0625.76051
Armbruster, D.
1987
Cascading failures and the emergence of cooperation in evolutionary-game based models of social and economical networks. Zbl 1317.91005
Wang, Wen-Xu; Lai, Ying-Cheng; Armbruster, Dieter
2011
A scalar conservation law with discontinuous flux for supply chains with finite buffers. Zbl 1231.90154
Armbruster, Dieter; Göttlich, Simone; Herty, Michael
2011
Kinetic and fluid models for supply chains supporting policy attributes. Zbl 1165.90303
Armbruster, D.; Degond, P.; Ringhofer, C.
2007
Corank-two bifurcations for the Brusselator with non-flux boundary conditions. Zbl 0647.34037
Armbruster, D.; Dangelmayr, G.
1986
Evolution of uncontrolled proliferation and the angiogenic switch in cancer. Zbl 1259.92051
Nagy, John D.; Armbruster, Dieter
2012
Bifurcation theory and computer algebra: An initial approach. Zbl 0604.68042
Armbruster, D.
1985
Bucket brigades revisited: are they always effective? Zbl 1116.90034
Armbruster, Dieter; Gel, Esma S.
2006
Imperfection sensitivity of interacting Hopf and steady-state bifurcations and their classification. Zbl 0579.58017
Armbruster, D.; Dangelmayr, G.; Güttinger, W.
1985
Dispersal effects on a discrete two-patch model for plant-insect interactions. Zbl 1411.92249
Kang, Yun; Armbruster, Dieter
2011
Noise and seasonal effects on the dynamics of plant-herbivore models with monotonic plant growth functions. Zbl 1247.37091
Kang, Yun; Armbruster, Dieter
2011
A hyperbolic relaxation model for product flow in complex production networks. Zbl 1190.90040
Unver, Ali; Ringhofer, Christian; Armbruster, Dieter
2009
Networks of interacting machines. Production organization in complex industrial systems and biological cells. Zbl 1110.90001
Armbruster, Dieter (ed.); Kaneko, Kunihiko (ed.); Mikhailov, Alexander S. (ed.)
2005
Analyzing bifurcations in the Kolmogorov flow equations. Zbl 0808.35098
Armbruster, D.; Nicolaenko, B.; Smaoui, N.; Chossat, P.
1994
Kinetic models and intrinsic timescales: simulation comparison for a 2nd order queueing model. Zbl 1406.60126
Armbruster, Dieter; Wienke, Matthew
2019
Swarming in bounded domains. Zbl 1376.35096
Armbruster, Dieter; Motsch, Sébastien; Thatcher, Andrea
2017
Periodic orbits in a class of re-entrant manufacturing systems. Zbl 0977.90010
Diaz-Rivera, Ivonne; Armbruster, Dieter; Taylor, Tom
2000
Square and almost square symmetry in binary convection. Zbl 0727.76049
Armbruster, D.
1991
Structurally stable heteroclinic cycles in a system with O(3) symmetry. Zbl 0735.34035
Chossat, P.; Armbruster, D.
1991
Constructing universal unfoldings using Gröbner bases. Zbl 0629.68039
Armbruster, D.; Kredel, H.
1986
Kinetic models of conservative economies with need-based transfers as welfare. Zbl 1434.35247
Kayser, Kirk; Armbruster, Dieter; Herty, Michael
2020
Elastic and inelastic collisions of swarms. Zbl 1376.70026
Armbruster, Dieter; Martin, Stephan; Thatcher, Andrea
2017
Integrating release and dispatch policies in production models. Zbl 1345.35122
Armbruster, Dieter; Herty, Michael; Wang, Xinping; Zhao, Lindu
2015
Continuum models for interacting machines. Zbl 1203.90057
Armbruster, Dieter; Degond, Pierre; Ringhofer, Christian
2005
Control and synchronization in switched arrival systems. Zbl 1080.90519
Rem, Bart; Armbruster, Dieter
2003
Dynamics of polar reversals in spherical dynamos. Zbl 1044.86010
Chossat, Pascal; Armbruster, Dieter
2003
Towards analyzing the dynamics of flames. Zbl 0842.35125
Armbruster, Dieter; Stone, Emily; Heiland, Randy
1996
Topological constraints for explicit symmetry breaking. Zbl 0789.34035
Armbruster, D.; Ihrig, E.
1993
Analyzing spatio-temporal complexity. Zbl 0787.76034
Armbruster, D.
1993
Persistent heteroclinic orbits. Zbl 0687.58024
Armbruster, Dieter
1989
Entscheidbarkeit und Bestimmung der Rekursivität von Prozeduren in Pascal-ähnlichen Programmen. Zbl 0586.68018
Armbruster, Dieter
1985
Kinetic models of conservative economies with need-based transfers as welfare. Zbl 1434.35247
Kayser, Kirk; Armbruster, Dieter; Herty, Michael
2020
Kinetic models and intrinsic timescales: simulation comparison for a 2nd order queueing model. Zbl 1406.60126
Armbruster, Dieter; Wienke, Matthew
2019
Swarming in bounded domains. Zbl 1376.35096
Armbruster, Dieter; Motsch, Sébastien; Thatcher, Andrea
2017
Elastic and inelastic collisions of swarms. Zbl 1376.70026
Armbruster, Dieter; Martin, Stephan; Thatcher, Andrea
2017
Integrating release and dispatch policies in production models. Zbl 1345.35122
Armbruster, Dieter; Herty, Michael; Wang, Xinping; Zhao, Lindu
2015
Evolution of uncontrolled proliferation and the angiogenic switch in cancer. Zbl 1259.92051
Nagy, John D.; Armbruster, Dieter
2012
Cascading failures and the emergence of cooperation in evolutionary-game based models of social and economical networks. Zbl 1317.91005
Wang, Wen-Xu; Lai, Ying-Cheng; Armbruster, Dieter
2011
A scalar conservation law with discontinuous flux for supply chains with finite buffers. Zbl 1231.90154
Armbruster, Dieter; Göttlich, Simone; Herty, Michael
2011
Dispersal effects on a discrete two-patch model for plant-insect interactions. Zbl 1411.92249
Kang, Yun; Armbruster, Dieter
2011
Noise and seasonal effects on the dynamics of plant-herbivore models with monotonic plant growth functions. Zbl 1247.37091
Kang, Yun; Armbruster, Dieter
2011
Control of continuum models of production systems. Zbl 1368.90056
La Marca, Michael; Armbruster, Dieter; Herty, Michael; Ringhofer, Christian
2010
A hyperbolic relaxation model for product flow in complex production networks. Zbl 1190.90040
Unver, Ali; Ringhofer, Christian; Armbruster, Dieter
2009
Dynamics of a plant-herbivore model. Zbl 1140.92322
Kang, Yun; Armbruster, Dieter; Kuang, Yang
2008
Bucket brigades with worker learning. Zbl 1137.90575
Armbruster, Dieter; Gel, Esma S.; Murakami, Junko
2007
Kinetic and fluid models for supply chains supporting policy attributes. Zbl 1165.90303
Armbruster, D.; Degond, P.; Ringhofer, C.
2007
A model for the dynamics of large queuing networks and supply chains. Zbl 1107.90002
Armbruster, D.; Degond, P.; Ringhofer, C.
2006
A continuum model for a re-entrant factory. Zbl 1167.90477
Armbruster, Dieter; Marthaler, Daniel E.; Ringhofer, Christian; Kempf, Karl; Jo, Tae-Chang
2006
Bucket brigades revisited: are they always effective? Zbl 1116.90034
Armbruster, Dieter; Gel, Esma S.
2006
Thermalized kinetic and fluid models for reentrant supply chains. Zbl 1108.90006
Armbruster, D.; Ringhofer, C.
2005
Networks of interacting machines. Production organization in complex industrial systems and biological cells. Zbl 1110.90001
Armbruster, Dieter (ed.); Kaneko, Kunihiko (ed.); Mikhailov, Alexander S. (ed.)
2005
Continuum models for interacting machines. Zbl 1203.90057
Armbruster, Dieter; Degond, Pierre; Ringhofer, Christian
2005
Kinetic and fluid model hierarchies for supply chains. Zbl 1078.90031
Armbruster, D.; Marthaler, D.; Ringhofer, C.
2004
Noisy heteroclinic networks. Zbl 1080.37510
Armbruster, Dieter; Stone, Emily; Kirk, Vivien
2003
Control and synchronization in switched arrival systems. Zbl 1080.90519
Rem, Bart; Armbruster, Dieter
2003
Dynamics of polar reversals in spherical dynamos. Zbl 1044.86010
Chossat, Pascal; Armbruster, Dieter
2003
Periodic orbits in a class of re-entrant manufacturing systems. Zbl 0977.90010
Diaz-Rivera, Ivonne; Armbruster, Dieter; Taylor, Tom
2000
Noise and $$O(1)$$ amplitude effects on heteroclinic cycles. Zbl 0971.37010
Stone, Emily; Armbruster, Dieter
1999
Symmetry and the Karhunen-Loève analysis. Zbl 0888.35008
Smaoui, Nejib; Armbruster, Dieter
1997
Symmetries and dynamics for 2-D Navier-Stokes flow. Zbl 0899.76113
Armbruster, Dieter; Nicolaenko, B.; Smaoui, N.; Chossat, Pascal
1996
Towards analyzing the dynamics of flames. Zbl 0842.35125
Armbruster, Dieter; Stone, Emily; Heiland, Randy
1996
Analyzing bifurcations in the Kolmogorov flow equations. Zbl 0808.35098
Armbruster, D.; Nicolaenko, B.; Smaoui, N.; Chossat, P.
1994
Topological constraints for explicit symmetry breaking. Zbl 0789.34035
Armbruster, D.; Ihrig, E.
1993
Analyzing spatio-temporal complexity. Zbl 0787.76034
Armbruster, D.
1993
Reconstructing phase space from PDE simulations. Zbl 0764.35008
Kirby, Michael; Armbruster, Dieter
1992
Phase-space analysis of bursting behavior in Kolmogorov flow. Zbl 1194.76085
Armbruster, Dieter; Heiland, Randy; Kostelich, Eric J.; Nicolaenko, Basil
1992
Heteroclinic orbits in a spherically invariant system. Zbl 0739.58036
Armbruster, Dieter; Chossat, Pascal
1991
Square and almost square symmetry in binary convection. Zbl 0727.76049
Armbruster, D.
1991
Structurally stable heteroclinic cycles in a system with O(3) symmetry. Zbl 0735.34035
Chossat, P.; Armbruster, D.
1991
Kuramoto-Sivashinsky dynamics on the center-unstable manifold. Zbl 0687.34036
Armbruster, Dieter; Guckenheimer, John; Holmes, Philip
1989
Persistent heteroclinic orbits. Zbl 0687.58024
Armbruster, Dieter
1989
Heteroclinic cycles and modulated travelling waves in systems with 0(2) symmetry. Zbl 0634.34027
Armbruster, Dieter; Guckenheimer, John; Holmes, Philip
1988
Perturbation methods, bifurcation theory and computer algebra. Zbl 0651.34001
Rand, Richard H.; Armbruster, Dieter
1987
Coupled stationary bifurcations in nonflux boundary value problems. Zbl 0633.58011
Armbruster, D.; Dangelmayr, G.
1987
O(2)-symmetric bifurcation theory for convection rolls. Zbl 0625.76051
Armbruster, D.
1987
Steady-state mode interactions in the presence of O(2)-symmetry and in non-flux boundary value problems. Zbl 0602.58039
Dangelmayr, G.; Armbruster, D.
1986
Corank-two bifurcations for the Brusselator with non-flux boundary conditions. Zbl 0647.34037
Armbruster, D.; Dangelmayr, G.
1986
Constructing universal unfoldings using Gröbner bases. Zbl 0629.68039
Armbruster, D.; Kredel, H.
1986
Bifurcation theory and computer algebra: An initial approach. Zbl 0604.68042
Armbruster, D.
1985
Imperfection sensitivity of interacting Hopf and steady-state bifurcations and their classification. Zbl 0579.58017
Armbruster, D.; Dangelmayr, G.; Güttinger, W.
1985
Entscheidbarkeit und Bestimmung der Rekursivität von Prozeduren in Pascal-ähnlichen Programmen. Zbl 0586.68018
Armbruster, Dieter
1985
Classification of Z(2)-equivariant imperfect bifurcations with corank 2. Zbl 0517.58027
Dangelmayr, G.; Armbruster, D.
1983
all top 5

Cited by 691 Authors

 19 Armbruster, Dieter 18 Göttlich, Simone 17 Herty, Michael Matthias 17 Holmes, Philip J. 13 Knobloch, Edgar 11 Smaoui, Nejib 9 Palacios, Antonio 8 Ashwin, Peter 8 Chossat, Pascal 8 Golubitsky, Martin A. 8 Keimer, Alexander 7 Degond, Pierre 7 Kang, Yun 7 Lumley, John Leask 7 Rand, Richard H. 6 D’Apice, Ciro 6 Melbourne, Ian 6 Pflug, Lukas 6 Rashidi, Mohammad Mehdi 6 Stewart, Ian Nicholas 6 Stone, Emily F. 5 Aubry, Nadine 5 Castro, Sofia B. S. D. 5 Dangelmayr, Gerhard 5 Knapp, Stephan 5 Manzo, Rosanna 5 Postlethwaite, Claire M. 4 Bakhtin, Yuri Yu. 4 Berkooz, Gal 4 Blomgren, Peter 4 Dinarvand, Saeed 4 Dong, Ming 4 Moehlis, Jeff 4 Okamoto, Hisashi 4 Piccoli, Benedetto 4 Podvigina, Ol’ga Mikhaĭlovna 4 Ringhofer, Christian A. 4 Siddiqui, Abdul Majeed 3 Coller, B. D. 3 Cvitanović, Predrag 3 Dawes, Jonathan H. P. 3 Fujimura, Kaoru 3 Furter, Jacques-Elie 3 Gasner, Scott 3 Gutjahr, Walter J. 3 He, Fenglan 3 Jolly, Michael S. 3 Kevrekidis, Ioannis George 3 Kim, Sun-Chul 3 Kirk, Vivien 3 Kogut, Peter I. 3 Kolb, Oliver 3 Kreuzer, Edwin J. 3 Krstić, Miroslav 3 Labouriau, Isabel S. 3 Lauterbach, Reiner 3 Leugering, Günter 3 Mercader, Isabel 3 Porter, Jeff 3 Prat, Joana 3 Roberts, Anthony John 3 Sirovich, Lawrence 3 Titi, Edriss Saleh 3 Ziegler, Ute 3 Zribi, Mohamed 2 Aguiar, Manuela A. D. 2 Al-Jamal, Rasha H. 2 Amelina, N. O. 2 Asfaw, Manalebish Debalike 2 Aston, Philip J. 2 Bernoff, Andrew J. 2 Bhowmick, Amiya Ranjan 2 Bick, Christian 2 Borrero-Echeverry, Daniel 2 Budanur, Nazmi Burak 2 Castillo-Chavez, Carlos 2 Chang, Hsuehchia 2 Chen, Yushu 2 Chow, Shuinee 2 Colombo, Rinaldo M. 2 Coppola, Vincent T. 2 Corless, Robert Malcolm 2 Coron, Jean-Michel 2 Cox, Stephen M. 2 Crawford, John David 2 Dai, Shiqiang 2 de la Llave, Rafael 2 Denis, Fabrice 2 Dias, Ana Paula S. 2 Din, Qamar 2 Elezgaray, Juan 2 Field, Michael J. 2 Friedrich, Jan 2 Frouvelle, Amic 2 Fu, Shihua 2 Garavello, Mauro 2 Gatermann, Karin 2 Ghori, Q. K. 2 Gorman, Michael E. 2 Gottlieb, Oded ...and 591 more Authors
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Cited in 141 Serials

 61 Physica D 23 International Journal of Bifurcation and Chaos in Applied Sciences and Engineering 13 Physics of Fluids 12 Journal of Nonlinear Science 12 Chaos 11 Journal of Fluid Mechanics 10 Dynamical Systems 8 Journal of Mathematical Analysis and Applications 8 Journal of Symbolic Computation 8 Dynamics and Stability of Systems 8 Nonlinear Dynamics 8 Communications in Nonlinear Science and Numerical Simulation 7 European Journal of Operational Research 7 Networks and Heterogeneous Media 7 Journal of Theoretical Biology 6 Journal of Biological Dynamics 5 Journal of Computational Physics 5 Journal of Statistical Physics 5 Theoretical and Computational Fluid Dynamics 5 Applied Mathematics and Computation 5 Journal of Computational and Applied Mathematics 5 Journal of Differential Equations 5 M$$^3$$AS. Mathematical Models & Methods in Applied Sciences 5 Journal of Dynamics and Differential Equations 5 Mathematical Problems in Engineering 4 Computers & Mathematics with Applications 4 Computer Methods in Applied Mechanics and Engineering 4 Physics of Fluids, A 4 Nonlinear Analysis. Theory, Methods & Applications 3 Acta Mechanica 3 Zeitschrift für Angewandte Mathematik und Mechanik (ZAMM) 3 Automatica 3 SIAM Journal on Control and Optimization 3 Applied Mathematics and Mechanics. (English Edition) 3 International Journal of Production Research 3 SIAM Journal on Applied Mathematics 3 SIAM Journal on Mathematical Analysis 3 NoDEA. Nonlinear Differential Equations and Applications 3 European Journal of Mechanics. B. Fluids 3 Nonlinear Analysis. Real World Applications 3 Discrete and Continuous Dynamical Systems. Series B 3 SIAM Journal on Applied Dynamical Systems 3 Kinetic and Related Models 3 International Journal of Biomathematics 2 Archive for Rational Mechanics and Analysis 2 Computer Physics Communications 2 International Journal of Control 2 Journal of Mathematical Biology 2 Mathematical Proceedings of the Cambridge Philosophical Society 2 Nonlinearity 2 Theoretical Computer Science 2 Optimization 2 Numerical Methods for Partial Differential Equations 2 Applied Mathematical Modelling 2 International Journal of Computer Mathematics 2 Journal of Elasticity 2 Proceedings of the Royal Society of Edinburgh. Section A. Mathematics 2 SIAM Journal on Scientific Computing 2 Complexity 2 Discrete and Continuous Dynamical Systems 2 Abstract and Applied Analysis 2 Optimization and Engineering 2 Journal of Applied Mathematics 2 Multiscale Modeling & Simulation 2 Acta Numerica 1 American Mathematical Monthly 1 Artificial Intelligence 1 Bulletin of the Australian Mathematical Society 1 Computers and Fluids 1 Indian Journal of Pure & Applied Mathematics 1 Journal of Engineering Mathematics 1 Mathematical Biosciences 1 Physica A 1 Physics Letters. A 1 ZAMP. Zeitschrift für angewandte Mathematik und Physik 1 Chaos, Solitons and Fractals 1 Computing 1 Journal of Pure and Applied Algebra 1 Mathematics and Computers in Simulation 1 Meccanica 1 Naval Research Logistics 1 Nonlinear Analysis. Theory, Methods & Applications. Series A: Theory and Methods 1 Quarterly of Applied Mathematics 1 Ricerche di Matematica 1 Theoretical Population Biology 1 Ergodic Theory and Dynamical Systems 1 Systems & Control Letters 1 Operations Research Letters 1 Acta Applicandae Mathematicae 1 Applied Numerical Mathematics 1 Probability Theory and Related Fields 1 Computers & Operations Research 1 Mathematical and Computer Modelling 1 MCSS. Mathematics of Control, Signals, and Systems 1 European Journal of Applied Mathematics 1 Japan Journal of Industrial and Applied Mathematics 1 Dynamics and Control 1 Computational Mathematics and Mathematical Physics 1 Automation and Remote Control 1 Stochastic Processes and their Applications ...and 41 more Serials
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Cited in 36 Fields

 125 Dynamical systems and ergodic theory (37-XX) 123 Fluid mechanics (76-XX) 120 Partial differential equations (35-XX) 87 Ordinary differential equations (34-XX) 61 Biology and other natural sciences (92-XX) 54 Operations research, mathematical programming (90-XX) 50 Numerical analysis (65-XX) 35 Systems theory; control (93-XX) 24 Computer science (68-XX) 24 Classical thermodynamics, heat transfer (80-XX) 23 Global analysis, analysis on manifolds (58-XX) 19 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 18 Mechanics of particles and systems (70-XX) 17 Calculus of variations and optimal control; optimization (49-XX) 17 Probability theory and stochastic processes (60-XX) 15 Statistical mechanics, structure of matter (82-XX) 9 Difference and functional equations (39-XX) 9 Mechanics of deformable solids (74-XX) 4 Commutative algebra (13-XX) 4 Operator theory (47-XX) 4 Information and communication theory, circuits (94-XX) 3 History and biography (01-XX) 2 Algebraic geometry (14-XX) 2 Manifolds and cell complexes (57-XX) 2 Optics, electromagnetic theory (78-XX) 2 Geophysics (86-XX) 1 Combinatorics (05-XX) 1 General algebraic systems (08-XX) 1 Field theory and polynomials (12-XX) 1 Linear and multilinear algebra; matrix theory (15-XX) 1 Group theory and generalizations (20-XX) 1 Measure and integration (28-XX) 1 Sequences, series, summability (40-XX) 1 Integral equations (45-XX) 1 Statistics (62-XX) 1 Astronomy and astrophysics (85-XX)