text:
# How do I change the list of listings title font and list of listings entry spacing?

I've set some formatting in my table of contents, list of tables, and list of figures that I want to continue in my list of listings (using tocloft package). However, I can't find the commands to do so. Consider the following example:

\documentclass[12pt,letterpaper,twoside]{report}
\usepackage[utf8]{inputenc}
\usepackage{geometry}
\usepackage{listings}
\renewcommand{\lstlistingname}{Code} %change 'listing' to 'code'
%
%
\begin{document}
%
%Adjusted the font of the list of listings title
\renewcommand{\lstlistlistingname}{\centering\bfseries\normalsize{LIST OF CODE}}
%
\lstlistoflistings
\vspace{1in}
%
%test listings
\begin{lstlisting}[caption=exlist]
test code
\end{lstlisting}
\begin{lstlisting}[caption=exlist2]
test code 2
\end{lstlisting}
%
\end{document}

The result looks like this:

I want to do a few things.

1) I want to control the distance from the top margin to the LIST OF CODE title similar to \setlength{\cftbeforetoctitleskip}{1in} in tocloft. Is there a way to do this with the listings package? Update: This distance is controlled by tocloft. Thus, use \setlength{\cftbeforetoctitleskip}{\spaceBeforeChapterTitles}.

2) I want to control the vskip between the LIST OF CODE title and the first line similar to \setlength{\cftaftertoctitleskip}{12pt} in tocloft. Update: This distance is controlled by tocloft as well. Thus, use \setlength{\cftaftertoctitleskip}{\spaceBetweenTitleAndText}.

3) I want to control the vskip values in between the list of listings entries (i.e. 12pt skip between exlist and exlist2 lines) like I do with \setlength{\cftbeforechapskip}{12pt} in tocloft.

4) I want to control the space between dots, similar to \renewcommand{\cftdotsep}{1.5} in tocloft.

5) I want there to be a space between listings that come from different chapters, just as is done automatically in a list of figures or list of tables.

I think I may be able to create a new list of code using tocloft, but I really like how listings color codes my scripts.

• The \listoflistings settings can't be controlled with tocloft that way.
– user31729
Sep 22 '17 at 19:26
• Any news here? Did you try to adjust it in another way? Jan 1 '18 at 9:26
• No, I gave up for now. I'll have to come back to it eventually though and I'll see if I remember to give an update here. Mar 11 '18 at 19:10
• update: I discovered that #1 and #2 are controlled by the tocloft settings. Thus, the following commands will also change the list of listings: \setlength{\cftbeforetoctitleskip}{\spaceBeforeChapterTitles}. However, I'm still looking for solutions for #3, #4, and I'm adding #5. Apr 8 '20 at 20:45

I've combed various forums and come up with a solution that works to address all five questions. Please see the code below for comments and explanations.

\documentclass[12pt,letterpaper,twoside]{report}
\usepackage[utf8]{inputenc}
\usepackage{geometry}

\usepackage{tocloft} %allow us to change some of the table contents, list of (whatever)
\usepackage{caption}

%Found a nice reference here: https://texblog.org/2008/07/13/define-your-own-list-of/
%Let's define a new list called List of Code
\newlistof{code}{loc}{\MakeUppercase{List of Code}}
%{id of list}{abbreviation for the list}{name of list}

%%%%%%%%%%%%% Answer to Question #1 and #2. %%%%%%%%%%%%%
%Control the spacing before and after the title with the following commands,
%analogous to how we control the spacing in toc.
\setlength{\cftbeforeloctitleskip}{1in}
\setlength{\cftafterloctitleskip}{1in}

%Format the title of the list. Center with an \hfill on either side
\renewcommand{\cftloctitlefont}{\hfill\Large\bfseries}
\renewcommand{\cftafterloctitle}{\hfill\space}

%%%%%%%%%%%%%%%%% Answer to Question #3 %%%%%%%%%%%%%%%
%Adjust the vskip before each new list entry with the following:
\setlength{\cftbeforecodeskip}{.5\normalbaselineskip}

%%%%%%%%%%%%%%%%% Answer to Question #4 %%%%%%%%%%%%%%%
%set the distance between TOC dots (default 4.5) with the following:
\renewcommand{\cftcodedotsep}{1.5}

%%%%%%%%%%%%%%%%% Answer to Question #5 %%%%%%%%%%%%%%%
% a quick patch so a space is added between chapters (like is done in toc, lof, lot)
%https://tex.stackexchange.com/questions/461521/how-to-create-space-between-chapter-in-list-of-listings
\usepackage{etoolbox}
\makeatletter
\patchcmd{\@chapter}% <cmd>
{}{}% <success><failure>
\makeatother
%end of patch

% Note that all the changes we've made have been separate from the listings package.
% We will be using our own caption system based on a new environment instead of
% using a caption system for lstlistings.
% Let's now make the new environment.
\newenvironment{coding}[1]{% require one input (the caption)
\refstepcounter{code} %increase the counter. Note that we reset it later (per chapter)
%
%
% now format the environment
\par %make sure it starts on a new line
\vspace{\baselineskip} %this was previously taken care of with \lstset, but we need
% to manually add it here to have the space be above our custom caption.
\noindent\textbf{Code \thechapter.\thecode: #1} %format the caption
{\protect\numberline{\thechapter.\thecode}#1}
\par}%
{} %Nothing special needs to happen in closing the environment.

\makeatletter
\@addtoreset{code}{chapter} %reset the list of code counter at each new chapter
\renewcommand{\p@code}{\thechapter.} % Want \autoref to include the chapter number.
\makeatother

\usepackage{listings} % minted is also an option, but I wanted to stick with
% listings because it seems to be updated more regularly.
\usepackage{color}
\usepackage{xcolor}
%New colors defined below
\definecolor{codegreen}{rgb}{0,0.6,0}
\definecolor{codegray}{rgb}{0.5,0.5,0.5}
\definecolor{codepurple}{rgb}{0.58,0,0.82}
\definecolor{backcolour}{rgb}{0.95,0.95,0.92}
%Code listing style named "mystyle"
\lstdefinestyle{mystyle}{% define my own style. Lots of options to choose from.
keywordstyle=\color{magenta},
numberstyle=\tiny\color{codegray},
stringstyle=\color{codepurple},
basicstyle=\footnotesize,
breakatwhitespace=false,
breaklines=true,
captionpos=t, %caption position t or b
keepspaces=true,
numbers=left,
numbersep=5pt,
showspaces=false,
showstringspaces=false,
showtabs=false,
tabsize=2
}
\lstset{style=mystyle} %set the new style
\lstset{%aboveskip=\baselineskip, %don't want this aboveskip because it would put a
% space between our custom caption and the code.
belowskip=\baselineskip}

%https://tex.stackexchange.com/questions/207744/no-autoref-name-for-appendix
\usepackage{hyperref} %for referencing the code
\newcommand*{\codeautorefname}{Code}

\begin{document}

\listofcode

\chapter{The First Chapter}
I've written some sample code in \autoref{code1}. I hope you enjoy!

\begin{coding}{This is my new environment caption}
\label{code1} %Include the label for cross referencing here.

%This example would insert an entire matlab script.
%I'm commenting it out for simplicity of posting here.
% \lstinputlisting[language=Matlab,%
% % caption=SigmaExtract.m,%      Don't include the caption here.
% % label={code:SigmaExtract},%   Don't include label here.
% ]{SigmaExtract.m}

%Instead, I'll show a simple portion of script here:
\begin{lstlisting}[language=matlab]
function TestFunction()
%%% This is a sample function.

for i = 1:10
fprintf('i = %0.0f\n',i)
end; clear i

end
\end{lstlisting}

\end{coding}

\begin{coding}{The second coding section}
\begin{lstlisting}[language=matlab]
function TestFunction()
%%% This is another sample function.

disp('Hello World')

end
\end{lstlisting}
\end{coding}

\chapter{Even more great content}

\begin{coding}{The third great function}
\begin{lstlisting}[language=matlab]
function j = TestFunction(i)
%%% This is another sample function.

j = i+1;

end
\end{lstlisting}
\end{coding}

\end{document}

The List of Code (Page 1):

Some sample code with demonstration of \autoref (Page 2)

Last page for completeness (Page 3)

• Note that solution for #5 doesn't work if you're using the newfloat package in your document. Apr 10 '20 at 14:39


text:
## Bernoulli

• Bernoulli
• Volume 23, Number 1 (2017), 743-772.

### Weak convergence of the empirical copula process with respect to weighted metrics

#### Abstract

The empirical copula process plays a central role in the asymptotic analysis of many statistical procedures which are based on copulas or ranks. Among other applications, results regarding its weak convergence can be used to develop asymptotic theory for estimators of dependence measures or copula densities, they allow to derive tests for stochastic independence or specific copula structures, or they may serve as a fundamental tool for the analysis of multivariate rank statistics. In the present paper, we establish weak convergence of the empirical copula process (for observations that are allowed to be serially dependent) with respect to weighted supremum distances. The usefulness of our results is illustrated by applications to general bivariate rank statistics and to estimation procedures for the Pickands dependence function arising in multivariate extreme-value theory.

#### Article information

Source
Bernoulli, Volume 23, Number 1 (2017), 743-772.

Dates
Revised: June 2015
First available in Project Euclid: 27 September 2016

https://projecteuclid.org/euclid.bj/1475001373

Digital Object Identifier
doi:10.3150/15-BEJ751

Mathematical Reviews number (MathSciNet)
MR3556791

Zentralblatt MATH identifier
1367.60026

#### Citation

Berghaus, Betina; Bücher, Axel; Volgushev, Stanislav. Weak convergence of the empirical copula process with respect to weighted metrics. Bernoulli 23 (2017), no. 1, 743--772. doi:10.3150/15-BEJ751. https://projecteuclid.org/euclid.bj/1475001373

#### References

• [1] Aistleitner, C. and Dick, J. (2015). Functions of bounded variation, signed measures, and a general Koksma–Hlawka inequality. Acta Arith. 167 143–171.
• [2] Berghaus, B., Bücher, A. and Dette, H. (2013). Minimum distance estimators of the Pickands dependence function and related tests of multivariate extreme-value dependence. J. SFdS 154 116–137.
• [3] Berghaus, B., Bücher, A. and Volgushev, S. (2015). Supplement to “Weak convergence of the empirical copula process with respect to weighted metrics.” DOI:10.3150/15-BEJ751SUPP.
• [4] Beutner, E. and Zähle, H. (2010). A modified functional delta method and its application to the estimation of risk functionals. J. Multivariate Anal. 101 2452–2463.
• [5] Bücher, A., Dette, H. and Volgushev, S. (2011). New estimators of the Pickands dependence function and a test for extreme-value dependence. Ann. Statist. 39 1963–2006.
• [6] Bücher, A., Segers, J. and Volgushev, S. (2014). When uniform weak convergence fails: Empirical processes for dependence functions and residuals via epi- and hypographs. Ann. Statist. 42 1598–1634.
• [7] Bücher, A. and Volgushev, S. (2013). Empirical and sequential empirical copula processes under serial dependence. J. Multivariate Anal. 119 61–70.
• [8] Capéraà, P., Fougères, A.-L. and Genest, C. (1997). A nonparametric estimation procedure for bivariate extreme value copulas. Biometrika 84 567–577.
• [9] Chen, X. and Fan, Y. (2006). Estimation of copula-based semiparametric time series models. J. Econometrics 130 307–335.
• [10] Csörgő, M., Csörgő, S., Horváth, L. and Mason, D.M. (1986). Weighted empirical and quantile processes. Ann. Probab. 14 31–85.
• [11] Csörgő, M. and Yu, H. (1996). Weak approximations for quantile processes of stationary sequences. Canad. J. Statist. 24 403–430.
• [12] Deheuvels, P. (1991). On the limiting behavior of the Pickands estimator for bivariate extreme-value distributions. Statist. Probab. Lett. 12 429–439.
• [13] Dudley, R.M. (1992). Fréchet differentiability, $p$-variation and uniform Donsker classes. Ann. Probab. 20 1968–1982.
• [14] Dudley, R.M. and Norvaiša, R. (1999). Differentiability of Six Operators on Nonsmooth Functions and $p$-Variation. Lecture Notes in Math. 1703. Berlin: Springer.
• [15] Einmahl, J.H.J. (1987). Multivariate Empirical Processes. CWI Tract 32. Amsterdam: Stichting Mathematisch Centrum, Centrum voor Wiskunde en Informatica.
• [16] Fermanian, J.-D., Radulović, D. and Wegkamp, M. (2004). Weak convergence of empirical copula processes. Bernoulli 10 847–860.
• [17] Gaenssler, P. and Stute, W. (1987). Seminar on Empirical Processes. DMV Seminar 9. Basel: Birkhäuser.
• [18] Genest, C., Ghoudi, K. and Rivest, L.-P. (1995). A semiparametric estimation procedure of dependence parameters in multivariate families of distributions. Biometrika 82 543–552.
• [19] Genest, C., Rémillard, B. and Beaudoin, D. (2009). Goodness-of-fit tests for copulas: A review and a power study. Insurance Math. Econom. 44 199–213.
• [20] Genest, C. and Segers, J. (2009). Rank-based inference for bivariate extreme-value copulas. Ann. Statist. 37 2990–3022.
• [21] Genest, C. and Segers, J. (2010). On the covariance of the asymptotic empirical copula process. J. Multivariate Anal. 101 1837–1845.
• [22] Gill, R.D. (1989). Non- and semi-parametric maximum likelihood estimators and the von Mises method. I. Scand. J. Stat. 16 97–128.
• [23] Gudendorf, G. and Segers, J. (2012). Nonparametric estimation of multivariate extreme-value copulas. J. Statist. Plann. Inference 142 3073–3085.
• [24] Hallin, M., Ingenbleek, J.-F. and Puri, M.L. (1985). Linear serial rank tests for randomness against ARMA alternatives. Ann. Statist. 13 1156–1181.
• [25] Hallin, M. and Puri, M.L. (1988). Optimal rank-based procedures for time series analysis: Testing an ARMA model against other ARMA models. Ann. Statist. 16 402–432.
• [26] Kley, T., Volgushev, S., Dette, H. and Hallin, M. (2016). Quantile spectral processes: Asymptotic analysis and inference. Bernoulli 22 1770–1807.
• [27] Owen, A.B. (2005). Multidimensional variation for quasi-Monte Carlo. In Contemporary Multivariate Analysis and Design of Experiments. Ser. Biostat. 2 49–74. Hackensack, NJ: World Sci. Publ.
• [28] Pickands, J. III (1981). Multivariate extreme value distributions. In Proceedings of the 43rd Session of the International Statistical Institute, Vol. 2 (Buenos Aires, 1981) 49 859–878, 894–902. International Statistical Institute.
• [29] Rio, E. (2000). Théorie Asymptotique des Processus Aléatoires Faiblement Dépendants. Mathématiques & Applications (Berlin) [Mathematics & Applications] 31. Berlin: Springer.
• [30] Rojo Jiménez, J., Villa-Diharce, E. and Flores, M. (2001). Nonparametric estimation of the dependence function in bivariate extreme value distributions. J. Multivariate Anal. 76 159–191.
• [31] Rüschendorf, L. (1976). Asymptotic distributions of multivariate rank order statistics. Ann. Statist. 4 912–923.
• [32] Segers, J. (2012). Asymptotics of empirical copula processes under non-restrictive smoothness assumptions. Bernoulli 18 764–782.
• [33] Shao, Q.-M. and Yu, H. (1996). Weak convergence for weighted empirical processes of dependent sequences. Ann. Probab. 24 2098–2127.
• [34] Shorack, G.R. and Wellner, J.A. (1986). Empirical Processes with Applications to Statistics. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. New York: Wiley.
• [35] Stute, W. (1984). The oscillation behavior of empirical processes: The multivariate case. Ann. Probab. 12 361–379.
• [36] Tsukahara, H. (2005). Semiparametric estimation in copula models. Canad. J. Statist. 33 357–375.
• [37] van der Vaart, A.W. and Wellner, J.A. (1996). Weak Convergence and Empirical Processes. Springer Series in Statistics. New York: Springer.
• [38] van der Vaart, A.W. and Wellner, J.A. (2007). Empirical processes indexed by estimated functions. In Asymptotics: Particles, Processes and Inverse Problems. Institute of Mathematical Statistics Lecture Notes—Monograph Series 55 234–252. Beachwood, OH: IMS.
• [39] Ziemer, W.P. (1989). Weakly Differentiable Functions. Graduate Texts in Mathematics 120. New York: Springer.

#### Supplemental materials

• Supplement to “Weak convergence of the empirical copula process with respect to weighted metrics”. A detailed exposition on bounded variation and Lebesgue–Stieltjes integration for two-variate functions and the proofs of Proposition 4.4 and of Lemma 4.9 can be found in [3].

format: plain


text:
# completeness and the Baire category theorem

I am studying the baire category theorem and trying to find a counterexample. This theorem says that a non-empty complete metric space can not be the countable union of nowhere-dense closed subsets

In particular, i'm trying to find a normed space that is the union of countably many closed nowhere-dense subsets. Obviously this means this set cannot be complete.

• How about $\mathbb Q = \bigcup_{q \in \mathbb Q} \{q\}$? – Stefan Mesken Feb 15 '16 at 18:26
• @Stefan what norm would we be using there? – Sertii Feb 15 '16 at 18:29
• @user156715: Ordinary absolute value. But note that the field in question is $\Bbb Q$, not $\Bbb R$ or $\Bbb C$. – Brian M. Scott Feb 15 '16 at 18:31
• Yes, I considered $\mathbb Q$ as a normed $\mathbb Q$-vector space - if that's what you were asking. – Stefan Mesken Feb 15 '16 at 18:31
• @BrianM.Scott do you know of an example where the field is R or C? – Sertii Feb 15 '16 at 18:32

For $n\in\Bbb Z^+$ let

$$A_n=\{f\in C[0,1]:|f(x)|\le n\text{ for all }x\in[0,1]\}\;,$$

and give $C[0,1]$ the $L^1$ norm.

Every open ball contains functions with very narrow, very tall spikes.

Any normed vector space having countable dimension (over $\mathbb{C}$, for instance) will do. The finite-dimensional subspaces are closed and nowhere dense.

• I should clarify that by countable I mean countably infinite. – carmichael561 Feb 15 '16 at 18:51


text:
# Strictly convex surfaces in 3D have an equicurved point

The two principal radii of curvature at a point on a surface in three dimensions provide a succinct description of the local geometry of the surface. The principal curvatures determine whether a surface is locally flat, convex, or a saddle (Figure 1). The radii (plural of radius, pronounced ray-dee-eye or ˈrādēˌī) prove even more dispositive in the theory of convex surfaces. Convex surfaces necessarily have positive radii of curvature everywhere, but much more is true: Knowing that the two radii of curvature at every point on a convex surface in fact determines the entire surface up to translations. But when does the reverse hold?

More precisely, when does a function $$\Lambda$$ that maps an outward-pointing normal vector $$u \in \mathbb{S}^2$$ to a pair of curvatures $$\lambda_1(u), \lambda_2(u)$$ always define a convex surface in $$\mathbb{R}^3$$?  This question is closely related to deep problems in geometry, notably the Christoffel–Minkowski problem. In this post, we derive a condition on every curvature function $$\Lambda$$ that defines a convex surface that limits the available functions.

To state the result, let’s make up a word.

Definition. A point $$x$$ on a convex surface $$S$$ an equicurved points of $$S$$ if all principal curvatures of $$S$$ at $$x$$ are equal.

In a neighborhood of an equicurved point $$x$$, the surface $$S$$ looks spherical up to second order. Indeed, spheres are characterized by the fact that every point is a point of equicurvature. Surprisingly, every sufficiently regular three-dimensional convex surface has at least one point of equicurvature.

Theorem 1. Let $$S\subset \mathbb{R}^3$$ be a strictly convex twice-differentiable surface. Then $$S$$ has at least one point of equicurvature.

The proof requires a few preliminaries.

Fact 2 [Schneider, Chapter 2.5]. Let $$S$$ be as above, and let $$h\colon \mathbb{S}^2\to \mathbb{R}$$ be the support function of $$S$$. Then:

1. The support function $$h$$ is of class $$C^2$$, that is, $$h$$ is twice continuously differentiable.

2. The eigenvalues of the Hessian $$\nabla^2 h(u)$$ of the support function are the principal radii of curvature of $$S$$ in the direction $$u\in \mathbb{S}^2$$.  (Note that this parameterization uses the outward-pointing normal to determine the radius of curvature, as opposed to the base point.)

3. The eigenvectors of $$\nabla^2 h(u)$$ corresponding to nonzero eigenvalues are perpendicular to $$u$$.

Proof of Theorem 1. For a contradiction, suppose that there is no point of equicurvature on $$S$$. By Fact 2(1), that the Hessian map $$\nabla^2 h \colon \mathbb{S}^2 \to \mathbb{R}^{3×3}$$ is continuous. Moreover, since $$S$$ has no point of equicurvature, $$\nabla^2 h(u)$$  has distinct eigenvalues everywhere by Fact 2(2).

This implies that the principal unit eigenvector $$\xi_i(u) \in \mathbb{S}^2$$ of $$\nabla^2 h(u)$$ is also well defined, continuous, and perpendicular to $$u$$ by Fact 2(3). In other words, $$\xi \colon \mathbb{S}^2 \to \mathbb{S}^2$$ is a non vanishing continuous map from the unit sphere onto it’s tangent space, which is impossible by the hairy ball theorem.

###

Fascinating! We’ve used a classical theorem in differential geometry to find a special point on every smooth, strictly convex surface.

Moreover, the proof doesn’t work for surfaces embedded in even dimensions, and even in odd dimensions, the straightforward extension of the method results in a far less satisfying conclusion (no point of equicurvature, but you must have points where the principal curvatures intertwine).

Of course, the theorem is almost vacuously true in one and two dimensions, so we seem to have stumbled upon a truly “three dimensional” fact.

## One thought on “Strictly convex surfaces in 3D have an equicurved point”

1. Note that a friend of mine pointed out that this applies to all $$C^2$$ surfaces that are topologically equivalent to $$S^2$$, since the principal radii of curvature can be parameterized the same way, and the hairy ball theorem also holds for these surfaces.


text:
Essential terms for understanding deep learning research papers, tutorials and textbooks.
Term Description
Jacobian matrix The matrix containing all partial derivatives of a function whose input and output are both vectors
Hessian matrix Similar to Jacobian matrix but it contains the second derivatives collected in a matrix
First-order optimization algorithms Optimization algorithms that use only a gradient such as gradient descent
Second-order optimization algorithms Optimization algorithms that use Hessian matrix like Newton’s method
Constrained Optimization Find the maximal or minimal value of f(x)
Karush-Kuhn-Tucker (KKT) Approach General solution to constrained optimization making use of generalized Lagrange function (Lagrangian)
KKT Conditions Simple set of properties that describe the optimal points of constrained optimization problems
Hyperparameters Machine algorithms’ settings that must be determined external to the learning algorithm itself
Accuracy Proportion of examples for which the model produces the correct output
Error rate Proportion of examples for which the model produces an incorrect output
Design matrix Matrix containing a different example in each row
Underfitting Model cannot obtain sufficiently low error value
Overfitting Large gap between training and test error
Capacity Model’s ability to fit functions
Hypothesis space Set of functions learning algorithm s allowed to select as being the solution
Representational capacity The model specifies which family of functions the learning algorithm can choose from when varying the parameters to reduce training objective
Occam’s razor Among competing hypotheses, one should choose the simplest one
Vapnik-Chervonenkis (VC) dimension Measures the capacity of a binary classifier
Nearest neighbor Regression Non-parametric model minimizing the L2 norm of the point and the surrounding points
Parametric Models Models that learn a function described by a finite-sized parameter vector such as Linear Regression. And if it has less than optimal capacity, it will asymptote with an error value more than the Bayes error
Non-parametric Models No limitation on parameters such as nearest neighbour regression. And more data yields better generalization
Nearest neighbour regression It simply stores X and y and when given x it looks up for the nearest entry and returns the label
Bayes error The error incurred by an oracle, knowing the true probability distribution that generates the data, making predictions from the true distribution p(x, y)
Generalization error It can never increase with more training examples
No Free Lunch Theorem Averaged over all possible data generating distributions, every classification algorithm has the same error rate when classifying previously unobserved points
Weight decay Large (underfitting), medium (just right), small (overfitting)
Regularization We can regularize a model simply by adding a penalty to the cost function called a Regularizer. There are other ways too and a more generic definition is: regularization is any modification we make to the algorithm that is intended to reduce the generalization error not the training error
Hyperparameter Settings that we can use to control the behaviour of the learning algorithm. The setting must be a hyperparameter because it is not appropriate to learn that hyperparameter for the training set such as hyperparameter controlling model capacity where it would always choose to maximize the model capacity for the training set that results in overftting
Validation set Examples that the training algorithm does not observe. This is not the test set. It is used to guide the selection of our hyperparameters. Since it is used to “train” the hyperparameters, the validation set error will underestimate the generalization error though typically by a smaller amount than the training error
Test set This is the set we use to estimate our generalization error after all our hyperparameter optimization is complete. If the test set is small, it can be problematic as this implies statistical uncertainty around the estimated test error
K-fold Cross Validation This is computationally expensive. We partition the data into k non-overlapping subsets and the test set can be estimated by taking the average test error across k-trials
Point estimator or statistic Point estimation is the attempt to provide the single “best” prediction of some quantity of interest. Any function of the data that is drawn i.i.d. Since the data is drawn i.i.d. any function of the data is random and therefore the point estimator is a random variable
Function estimator This can also be called point estimator. But a function estimator is the estimation of the relationship between input and output variables
Bias $$bias(\hat\theta) = \mathbf{E}(\hat\theta_m) - \theta$$
Estimator $$\hat\theta_m$$ is ubiased if $$bias(\hat\theta) = 0$$
Tags:


text:
mindarmour¶

MindArmour, a tool box of MindSpore to enhance model security and trustworthiness against adversarial examples.

class mindarmour.Attack[source]

The abstract base class for all attack classes creating adversarial examples.

batch_generate(inputs, labels, batch_size=64)[source]

Generate adversarial examples in batch, based on input samples and their labels.

Parameters
• inputs (numpy.ndarray) – Samples based on which adversarial examples are generated.

• labels (numpy.ndarray) – Labels of samples, whose values determined by specific attacks.

• batch_size (int) – The number of samples in one batch.

Returns

Examples

>>> inputs = Tensor([[0.2, 0.4, 0.5, 0.2], [0.7, 0.2, 0.4, 0.3]])
>>> labels = [3, 0]
>>> advs = attack.batch_generate(inputs, labels, batch_size=2)

abstract generate(inputs, labels)[source]

Generate adversarial examples based on normal samples and their labels.

Parameters
• inputs (numpy.ndarray) – Samples based on which adversarial examples are generated.

• labels (numpy.ndarray) – Labels of samples, whose values determined by specific attacks.

Raises

NotImplementedError – It is an abstract method.

class mindarmour.BlackModel[source]

The abstract class which treats the target model as a black box. The model should be defined by users.

is_adversarial(data, label, is_targeted)[source]

Check if input sample is adversarial example or not.

Parameters
• data (numpy.ndarray) – The input sample to be check, typically some maliciously perturbed examples.

• label (numpy.ndarray) – For targeted attacks, label is intended label of perturbed example. For untargeted attacks, label is original label of corresponding unperturbed sample.

• is_targeted (bool) – For targeted/untargeted attacks, select True/False.

Returns

bool.
• If True, the input sample is adversarial.

• If False, the input sample is not adversarial.

abstract predict(inputs)[source]

Predict using the user specified model. The shape of predict results should be (m, n), where n represents the number of classes this model classifies.

Parameters

inputs (numpy.ndarray) – The input samples to be predicted.

Raises

NotImplementedError – It is an abstract method.

class mindarmour.Detector[source]

The abstract base class for all adversarial example detectors.

abstract detect(inputs)[source]

Detect adversarial examples from input samples.

Parameters

inputs (Union[numpy.ndarray, list, tuple]) – The input samples to be detected.

Raises

NotImplementedError – It is an abstract method.

abstract detect_diff(inputs)[source]

Calculate the difference between the input samples and de-noised samples.

Parameters

inputs (Union[numpy.ndarray, list, tuple]) – The input samples to be detected.

Raises

NotImplementedError – It is an abstract method.

abstract fit(inputs, labels=None)[source]

Fit a threshold and refuse adversarial examples whose difference from their denoised versions are larger than the threshold. The threshold is determined by a certain false positive rate when applying to normal samples.

Parameters
Raises

NotImplementedError – It is an abstract method.

abstract transform(inputs)[source]

Filter adversarial noises in input samples.

Parameters

inputs (Union[numpy.ndarray, list, tuple]) – The input samples to be transformed.

Raises

NotImplementedError – It is an abstract method.

class mindarmour.Defense(network)[source]

The abstract base class for all defense classes defending adversarial examples.

Parameters

network (Cell) – A MindSpore-style deep learning model to be defensed.

batch_defense(inputs, labels, batch_size=32, epochs=5)[source]

Defense model with samples in batch.

Parameters
• inputs (numpy.ndarray) – Samples based on which adversarial examples are generated.

• labels (numpy.ndarray) – Labels of input samples.

• batch_size (int) – Number of samples in one batch.

• epochs (int) – Number of epochs.

Returns

numpy.ndarray, loss of batch_defense operation.

Raises

ValueError – If batch_size is 0.

abstract defense(inputs, labels)[source]

Defense model with samples.

Parameters
• inputs (numpy.ndarray) – Samples based on which adversarial examples are generated.

• labels (numpy.ndarray) – Labels of input samples.

Raises

NotImplementedError – It is an abstract method.


text:
TheInfoList

In Euclidean geometry, a parallelogram is a simple (non-self-intersecting) quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equal measure. The congruence of opposite sides and opposite angles is a direct consequence of the Euclidean parallel postulate and neither condition can be proven without appealing to the Euclidean parallel postulate or one of its equivalent formulations.

By comparison, a quadrilateral with just one pair of parallel sides is a trapezoid in American English or a trapezium in British English.

The three-dimensional counterpart of a parallelogram is a parallelepiped.

The etymology (in Greek παραλληλ-όγραμμον, a shape "of parallel lines") reflects the definition.

## Special cases

• Rhomboid – A quadrilateral whose opposite sides are parallel and adjacent sides are unequal, and whose angles are not right angles[1]
• Rectangle – A parallelogram with four angles of equal size (right angles).
• Rhombus – A parallelogram with four sides of equal length.
• Square – A parallelogram with four sides of equal length and angles of equal size (right angles).

## Characterizations

A simple (non-self-intersecting) quadrilateral is a parallelogram if and only if any one of the following statements is true:[2][3]

• Two pairs of opposite sides are parallel (by definition).
• Two pairs of opposite sides are equal in length.
• Two pairs of opposite angles are equal in measure.
• The diagonals bisect each other.
• One pair of opposite sides is parallel and equal in length.
• Each diagonal divides the quadrilateral into two congruent triangles.
• The sum of the squares of the sides equals the sum of the squares of the diagonals. (This is the parallelogram law.)
• It has rotational symmetry of order 2.
• The sum of the distances from any interior point to the sides is independent of the location of the point.[4] (This is an extension of Viviani's theorem.)
• There is a point X in the plane of the quadrilateral with the property that every straight line through X divides the quadrilateral into two regions of equal area.[5]

Thus all parallelograms have all the properties listed above, and conversely, if just one of these statements is true in a simple quadrilateral, the

By comparison, a quadrilateral with just one pair of parallel sides is a trapezoid in American English or a trapezium in British English.

The three-dimensional counterpart of a parallelogram is a parallelepiped.

The etymology (in Greek παραλληλ-όγραμμον, a shape "of parallel lines") reflects the definition.

A simple (non-self-intersecting) quadrilateral is a parallelogram if and only if any one of the following statements is true:[2][3]

• Two pairs of opposite sides are parallel (by definition).
• Two pairs of opposite sides are equal in length.
• Two pairs of opposite angles are equal in measure.
• The diagonals bisect each other.
• One pair of opposite sides is parallel and equal in length.
• Each diagonal divides the quadrilateral into two congruent triangles.
• The sum of the squares of the sides equals the sum of the squares of the diagonals. (This is the parallelogram law.)
• It has rotational symmetry of order 2.
• The sum of the distances from any interior point to the sides is independent of the location of the point.[4] (This is an extension of Viviani's theorem.)
• There is a point X in the plane of the quadrilateral with the property that every straight line through X divides the quadrilateral into two regions of equal area.[5]

Thus all parallelograms have all the properties listed above, and conversely, if just one of these statements is true in a simple quadrilateral, then it is a parallelogram.

## Other properties

• Opposite sides of a parallelogram are parallel (by definition) and so will never intersect.
• The area of a parallelogram is twice the area of a triangle created by one of its diagonals.
• The area of a parallelogram is also equal to the magnitude of the vector cross product of two adjacent sides.
• Any line through the midpoint of a parallelogram bisects the area.[6]
• Any non-degenerate affine transformation takes a parallelogram to another parallelogram.
• A parallelogram has rotational symmetry of order 2 (through 180°) (or order 4 if a square). If it also has exactly two lines of reflectional symmetry then it must be a rhombus or an oblong (a non-square rectangle). If it has four lines of reflectional symmetry, it is a square.
• The perimeter of a parallelogram is 2(a + b) where a and b are the lengths of adjacent sides.
• Unlike any other convex polygon, a parallelogram cannot be inscribed in any triangle with less than twice its area.[7]
• The centers of four squares all constructed either internally or externally on the sides of a parallelogram are the vertices of a square.[8]
• If two lines parallel to sides of a parallelogram are constructed concurrent to a diagonal, then the parallelograms formed on opposite sides of that diagonal are equal in area.[8]
• The diagonals of a parallelogram divide it into four triangles of equal area.

## Area formula

conversely, if just one of these statements is true in a simple quadrilateral, then it is a parallelogram.

## Other properties

• Opposite sides of a parallelogram are parallel (by definition) and so will never intersect.
• The area of a parallelogram is twice the area of a triangle created by one of its diagonals.
• The area of a parallelogram is also equal to the magnitude of the vector cross product of two adjacent sides.
• Any line through the midpoint of a parallelogram bisects the area.[6]
• Any non-degenerate affine transformation takes a parallelogram to another parallelogram.
• A parallelogram has rotational symmetry of order 2 (through 180°) (or order 4 if a square). If it also has exactly two lines of reflectional symmetry then it must be a rhombus or an oblong (a non-square rectangle). If it has four lines of reflectional symmetry, it is a square.
• The perimeter of a parallelogram is 2(a + b) where a and b are the lengths of adjacent

All of the area formulas for general convex quadrilaterals apply to parallelograms. Further formulas are specific to parallelograms:

A parallelogram with base b and height h can be divided into a trapezoid and a right triangle, and rearranged into a rectangle, as shown in the figure to the left. This means that the area of a parallelogram is the same as that of a rectangle with the same base and height:

${\displaystyle K=bh.}$
The area of the parallelogram is the area of the blue region, which is the interior of the parallelogram

The base × height area formula can also be derived using the figure to the right. The area K of the parallelogram to the right (the blue area) is the total area of the rectangle less the area of the two orange triangles. The area of the rectangle is

${\displaystyle K_{\text{rect}}=(B+A)\times H\,}$

and the area of a single orange triangle is

${\displaystyle K_{\text{tri}}={\frac {1}{2}}A\times H.\,}$

Therefore, the area of the parallelogram is


text:
# On intellectual ownership

069 January 15, 2018 --

Recently -- while I was busy getting out of one particular swamp, to be more precise -- stuff has happened, of which I will remind two events: Stanislav began writing his epic FFA series, and Diana introduced her very own reference code shelf, to which she has already added a few pieces. I find the latter to be particularly interesting: while almost everyone has "an opinion" regarding, say, the software they use, in the sense every Romanian has an opinion on politics and football, in the very sense everyone has a butthole -- while this is so, few individuals would claim any sort of responsibility for the items they so judge. In fact, nowadays anyone at all rarely claims any responsibility on any piece of "intellectual property" (whether written by them or otherwise), to the extent that by now you're willingly getting fucked in the ass by various Googles, Microsofts, Apples and so on.

Unfortunately for the variouses above, V exists today. In practice this means that as the author of, say, a piece of code, you're either prepared to put your ass on the line1 with respect to the item and its intellectual honesty, or bust. The orcs and the wreckers might put some value on your attempts, but everyone else -- that is, your betters -- won't.

This model works in some sense backwards from what today's so-called "civilized world" deems to be "intellectual property". "Traditionally", software is written by some nobodies toiling in nondescript swamps, then is put a label on, then is sold2 as the premier "operating system", "cloud management system", "app" or some other useless ambiguity. But that's not all: given that putting intellectual items on the market is a rather iffy matter, you're not exactly sold the software, but the so-called rights to use said product in some way or another, which typically includes the explicit prohibition of reverse engineering $thing, even if, say, $thing becomes obsolete within the year -- which it typically does.

Or, to quote a wise man:

Similarly, "net neutrality" is not in any way related to network freedom. To best understand the competing paradigms : the artificial perspective - as proposed by the obsolete governments and the rest of the numerously various paper fiat arrays - revolves around the "rights" of the "owners" of hardware, and the "intentions" of the writers of software. In this perspective, using a computer "against the will of its owner", even if you're using it exactly in the manner it allows itself to be used, is "criminal" whatever that may mean anymore.

Exactly contrary to that, the natural perspective : if you don't want me to use your server to print all the nude photos your users entrusted you, write your software and provision your hardware in such a way that I can't. Because if I can, then it is not my right but my freedom to do so.

A parallel with human sexuality is not unwarranted here. At a past time when the ownership of female humans was the principal method to control capital formation and the political process, the same sort of dead hand fictions proposed that they in fact control the usage of women, that anyone owning a woman does so only as a sublicense from their sovereign ownership of all women, and may proceed only in furtherance of the regulation they emitted as to how women may be used and may not be used and so on and so forth. Specifically excluded from all this - the woman in question. Particularly "criminal" at the time - using a woman in the manner she herself wishes to be used.

Times have changed, women are now worthless, but computers aren't, and these same dead hand fictions propose anew : that they own all computers, that if you own a computer it's only a sublease from them, subject to furthering their regulations and so on and so forth. Specifically excluded from all this - what the code actually says and does. Particularly "criminal" - using a computer in the manner it itself allows to be used.

This then brings us to the fundamental question of our essay: how do we define ownership? and in particular what does intellectual ownership mean?

Physical ownership is pretty straightforward. Given an arbitrary $object, the proximate genus for ownership is control over said $object: for example, in order to get from home to work, I could either drive the $object that I have chosen to call "my car" or call for another $object named a taxi. The specific difference is exclusiveness in my relation to $object: others may or may not call the same taxi at some point, but I am the only one who can use my car, based on my own claim above anything else. This is for example what I mean when I say that I would rather, at all costs, have my own computing machines than rely on the inherently unreliable cloud. Intellectual ownership, that is, the ownership of an $object lying in the realm of thought, is however an entirely different matter. Take for example any non-trivial3 piece of knowledge, such as Shakespeare's Othello or Thales' theorem. You may acknowledge their existence, know where to put them in the great tree, maybe even have a vague idea about their meaning; this is the proximate genus, and assuming you've been through school, you probably know about a great deal of the intellectual items ever conceived by humanity. The specific difference however refers not merely to "knowing about" $object, but to devouring it entirely, such that you understand all the intimate aspects of $object, its very author becoming part of your internal conversation on the item in question.

This is then what I mean when I say that I own something intellectual(ly): that I have read it4 carefully and that I believe5 to have understood it in its entirety. In fact I am unconvinced there exists any other kind of intellectual ownership; but go ahead and humour me if you think otherwise.

As a consequence, I have decided to open my own store of signed intellectual artifacts. The linked page contains a bunch of V patches, the seals as generated by their original author(s) and, added, my own signatures. At the time of writing the list is quite modest, which only means that there's plenty of room for it to grow into a repository of knowledge. So here's to that!

1. Or, to put this more mildly, your "skin in the game".

2. Not always for money. In fact, chances are you've already sold your freedom for this-or-that very awesome smartphone and/or phablet and/or smart always-connected device you supposedly own.

3. We disregard trivial pieces of knowledge not because they're merely failed attempts to enrich the library™ -- which, to be perfectly clear, they are -- but because they're not very interesting. Ask yourself now, how many of the works you've read so far really do matter?

Let's just hope you're young enough to catch up.

4. Or watched it, or heard it. I do admit to fitting a not-very-small musical library in my head, I just don't see any reason to share it publicly at the moment.

5. Believe me or not, belief is an essential part of the whole deal. You'd be surprised how many times I would read a book the second or third (sometimes the fourth) time, only to discover things I had missed on previous readings. And I'm not the first to observe this either!

It's either worth reading twice or not at all, or how did the saying go?


text:
# Definition of Second-Degree Equations and Inequalities

Second-degree equations involve at least one variable that is squared, or raised to a power of two. One of the most well-known second-degree equations is the quadratic where a, b, and c are constants and a is not equal 0. Second-degree equations have two possible solutions: and
The graph of a second-degree equation produces a parabola. The solutions to the equation represent where the parabola crosses the x-axis. The general form of second-degree inequalities is . Solving inequalities involves finding all possible values of the variable that will make the inequality true.

# Related Questions (7)

• 1 answer
• 0 answers
• 1 answer
• 1 answer

Join Chegg Study

Guided textbook solutions created by Chegg experts

Learn from step-by-step solutions for 2,500+ textbooks in Math, Science, Engineering, Business and more

24/7 Study Help

Answers in a pinch from experts and subject enthusiasts all semester long

Start Your Free Trial


text:
# What Is .36 As A Fraction? Amazing Answer 2022

## What Is .36 As A Fraction?

Answer: 36 as a fraction would be written as 36/1 or 360/10.

## How do you turn .36 into a fraction?

Answer: 0.36 can be expressed as 9/25 in the form of a fraction.

## What is .375 as a fraction?

3/8

The fractional equivalent of 0.375 is 3/8. Whenever we are presented with a decimal that has to be converted to a fraction we must first take note…

## What is 0.35 into a fraction?

7/20

Answer: 0.35 as a fraction is 7/20.

## What is 33% as a fraction?

The simplified 1/3 is equivalent to 33 and 1/3 percent. Some instructors will not allow students to round to this number. In this case 33/100 is the accurate equivalent.

## What is 38% as a fraction?

In the simplest form 38% is 19/50 as a fraction.

## What is .4 as a fraction?

Decimal to a fraction conversion table
Decimal Fraction
0.33333333 1/3
0.375 3/8
0.4 2/5
0.42857143 3/7

## What is 0.4 as a percentage?

Decimal to a percent conversion table
Decimal Percent
0.2 20%
0.3 30%
0.4 40%
0.5 50%

## What is .25 as a fraction?

1/4
Example Values
Percent Decimal Fraction
10% 0.1 1/10
12½% 0.125 1/8
20% 0.2 1/5
25% 0.25 1/4

## How do I write 0.34 as a fraction?

1. 0.34 is 34 hundredths or 34100.
2. We can factor and reduce 34100 as:
3. 34100⇒2×172×50⇒2 ×172 ×50⇒1750.
4. Another approach to show 0.34=34100 is:
5. 34100⇒2×172×50⇒2 ×172 ×50⇒1750.

## What is 0.38 as a fraction in simplest form?

Explanation: 0.38 has two numbers after the decimal point so your fraction needs two zeros after the one in the denominator (100) and the numbers after the decimal point are your numerator (38). So you get 0.38=38100 this cancels down if you divide top and bottom by two.

## What is 35 as a fraction of 490?

Therefore 35/490 simplified to lowest terms is 1/14.

## What is 0.38 as a percent?

“Percent” or “%” means “out of 100” or “per 100” Therefore 38100 can be written as 38%.

## What is 39% as a fraction?

Thirty-nine percent expressed as a fraction is 39/100. This can also be written in decimal form as 0.39.

## What is 38% converted to a decimal?

38/100 as a decimal is 0.38.

## What is 40% as a fraction?

2 / 5

Answer: The value of 40% as a fraction in simplest form is 2 / 5.

## What is 0.3 as a percent?

30 %
To write 0.3 as a percent multiply 0.3 by 100. Append % symbol in the product obtained. So 0.3 as a percent is 30 %.

## What is 0.4 as a fraction in the lowest terms?

2/5

Answer: 0.4 can be written in a fraction as 4/10 or 2/5. Explanation: First we need to remove the decimal. For that we multiply and divide by 10 as there is only one digit after the decimal.

## How do you write 80% as a fraction?

Answer: 80% as a fraction in simplest form is 4 / 5.

## How do I write a fraction as a decimal?

The line in a fraction that separates the numerator and denominator can be rewritten using the division symbol. So to convert a fraction to a decimal divide the numerator by the denominator. If required you can use a calculator to do this. This will give us our answer as a decimal.

## What is 0.20 as a fraction?

Answer: 0.2 when converted into a fraction is 1/5.

## What is 30 as a fraction?

3/10

Answer: 30% as a fraction in simplest form is 3/10.

## What is .333 as a fraction?

333/1000

The decimal. 333 written as a fraction is 333/1000 because the place value of the last digit in the decimal is in the thousandth place.

## What is 34% as a fraction in simplest form?

The answer is that the simplest fraction form of 34% is begin{align*}frac{17}{50}end{align*}

## What is 0.32 as a fraction?

32/100

Answer: 0.32 as a fraction is represented as 32/100 and can be reduced to 8/25.

## What is .34 as a decimal?

0.34

34% is 34 out of 100. The decimal 0.34 is 34 hundredths or 34 out of 100.

## What is the decimal expansion of 0.38 bar?

The simplest form of 0.38 bar is 38/99.

## How do u add fractions?

To add fractions there are Three Simple Steps:
1. Step 1: Make sure the bottom numbers (the denominators) are the same.
2. Step 2: Add the top numbers (the numerators) put that answer over the denominator.
3. Step 3: Simplify the fraction (if possible)

## How do you simplify 35 49?

Steps to simplifying fractions
1. Find the GCD (or HCF) of the numerator and denominator. GCD of 35 and 49 is 7.
2. 35 ÷ 749 ÷ 7.
3. Reduced fraction: 57. Therefore 35/49 simplified to lowest terms is 5/7.

## What is 3 over 8 as a decimal?

0.375

3/8 as a decimal is 0.375.

## What is the decimal of 3 100?

3/100 as a decimal is 0.03.

## How do you write 2.39 as a percentage?

2.39 as a percent is 239%. To convert a decimal to a percent simply move the decimal point to the right two places and include the % sign.

Categories FAQ


text:
# The Stacks Project

## Tag 0309

Definition 10.36.1. A domain $R$ is called normal if it is integrally closed in its field of fractions.

The code snippet corresponding to this tag is a part of the file algebra.tex and is located in lines 7657–7661 (see updates for more information).

\begin{definition}
\label{definition-domain-normal}
A domain $R$ is called {\it normal} if it is integrally
closed in its field of fractions.
\end{definition}

There are no comments yet for this tag.

## Add a comment on tag 0309

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the lower-right corner).


text:
HSN-VM.B.4b

Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum.


text:
Warning
This page is located in archive.

# Week 3

(02.03.2020 - Updated dataset)

Program: Exercise ML02

• Univariate and multivariate linear regression

Homework: In case you will not finish the exercise during the lab, your homework is to

• finish the exercise at home
• identify the potential issues
• ask questions on the Forum
• Hand in the Python modules linreg.py and ML02.py via BRUTE, together with the solution of the next exercise, ML03.
courses/ui/exercises/ml02.txt · Last modified: 2020/03/02 09:45 by spilkjir


text:
4.12: Antenna Array

An antenna array comprises multiple radiating elements, i.e. individual antennas, and focuses a transmit beam in a desired direction. In Figure $$\PageIndex{2}$$ the field pattern in the plane of the earth produced by an array of $$30$$ antenna elements arranged horizontally is shown. The fields from each

Figure $$\PageIndex{1}$$: Radio coverage profiles with $$900\text{ MHz}$$ transmitter except for (d) which is for $$2.45\text{ GHz}$$. Calculated using Wireless Insite$$^{®}$$. Copyright Remcom, Inc. Used with permission.

Figure $$\PageIndex{2}$$: Electric field pattern from a $$30$$ element array of antennas spaced $$0.65\lambda$$ apart. The sidelobe levels are about $$40\text{ dB}$$ below the power level of the main lobe. The same signal is presented to the antenna elements except that phases of the signal at each antennas is adjusted to produce a main beam directed at $$20$$ degrees. The signals to each antenna are thus correlated. After [29].

antenna element combined to narrow and strengthen the main beam. Side (or grating) lobes are produced and these will result in some interference but their level here is $$40\text{ dB}$$ below that of the main beam. This is another way of managing interference in a cellular system but the direction to the mobile unit must be known. Antenna arrays are used in 5G. The affect of the array is to increase the power density of the main beam and the density relative to that of an isotropic antenna is called the directional gain of the array, $$D_{\text{Array}}$$, and is the product of the antenna gain, $$G_{A}$$, of an individual antenna element and the array gain, $$G_{\text{Array}}$$:

$\label{eq:1}D_{\text{Array}}=G_{\text{Array}}G_{A}$

The maximum value of $$G_{\text{Array}}$$ is $$N$$ for an $$N$$ element array. So the maximum directional gain of the array in $$\text{dBi}$$ is

$\label{eq:2}D_{\text{Array}}|_{\text{dBi}}=G_{A}|_{\text{dBi}}+10\log N$

This page titled 4.12: Antenna Array is shared under a CC BY-NC license and was authored, remixed, and/or curated by Michael Steer.


text:
# Sorting alpha-blended objects

This topic is 2542 days old which is more than the 365 day threshold we allow for new replies. Please post a new topic.

## Recommended Posts

Hi!

I am currently programming a little engine for terrain rendering with trees and plants.
I´ve just built in trees and grass already. But i have a huge problem with alpha blended objects(both grass and trees needs them).

Are there any techniques to sort the object by the distance to the camera so I can draw them from back to front? If, are there any helpfull render states or do i have to do it in vertex shader or something like this?

I hope you can help me, that i can get on with my current work ;)

##### Share on other sites
You do it on the CPU side - sort your meshes by distance, then issue the draw commands for those meshes in that order

##### Share on other sites
Thank you!

Ok can you give me a little code snippet how i compute the distance to the camera?
Do i have to use the worldMatrix for each object?

like this:
 D3DXVECTOR vObjPos = (D3DXVECTOR) *m_World.41; D3DXVECTOR3 vDistance = vObjPos - camera.vEye; float fDistance = length(vObjPos);

Isn´t this to inefficient for 100 leaves on a tree or 100 grass tufts?

EDIT: How can i do that with the LPD3DXMESH class?

##### Share on other sites

like this:
 D3DXVECTOR vObjPos = (D3DXVECTOR) *m_World.41; D3DXVECTOR3 vDistance = vObjPos - camera.vEye; float fDistance = length(vObjPos);

Isn´t this to inefficient for 100 leaves on a tree or 100 grass tufts?

What's inefficient about it? How long does it take on your hardware?

Computing the lengthSquared might be slightly cheaper as it'll avoid a divide, and you can sort by that just the same

##### Share on other sites

[quote name='IceBreaker23' timestamp='1325280320' post='4898247']
like this:
 D3DXVECTOR vObjPos = (D3DXVECTOR) *m_World.41; D3DXVECTOR3 vDistance = vObjPos - camera.vEye; float fDistance = length(vObjPos);

Isn´t this to inefficient for 100 leaves on a tree or 100 grass tufts?

What's inefficient about it? How long does it take on your hardware?

Computing the lengthSquared might be slightly cheaper as it'll avoid a divide, and you can sort by that just the same
[/quote]
Or a hidden square root.

Your sorting algorithm matters more than the specific operation. Or, O (n log n) is going to just be faster than O (n^2), for most cases. More than that, premature optimization is a root of evil - Optimize once you've profiled the code and know where the slow-downs are. Otherwise, drive yourself mad, you will.

##### Share on other sites
ill take std::sort for it(i think its a quick sort) and I think an i7 can handle this ;)
Thank you very much!

##### Share on other sites
Now i have a tree. His leaves need alpha blending. But how can i sort LPD3DXMESH?

##### Share on other sites

Isn´t this to inefficient for 100 leaves on a tree or 100 grass tufts?

In general you're probably not going to be able to properly sort the individual leaves of a tree, it's just not practical. It's a lot of items to sort, and more importantly it would mean a draw call per leaf (unless you dynamically generated an index buffer, which brings its own problems). Usually people will just coarsely sort by object and try to minimize the artifacts on things on like leaves (or they just live with them). For certain categories of meshes it's possible to create a pre-sorted index buffer that will always render the triangles of that mesh in back-to-front order with respect to itself. However this won't work for a lot of cases, particularly cases where triangles intersect or overlap. In fact any kind of sorting will always fail when triangles intersect.

As an alternative, you can try using something other than traditional blending. Such as...

1. Alpha testing
2. Alpha-to-coverage
3. Post-process blending

There are also methods for achieving order-independent transparency, but in general these tend to be too expensive for games.

##### Share on other sites
Hello
I'm currently experimenting an algorithm to sort objects per distance very fastly, here's a description :

The main structure of the sorting object is an hash table ;for instance, let's say the depth range (in view space) is [0,1000], and there are 1000 entries in the hash table : the first entry covers [0,1] depth range, the second covers [1,2] depth range and so on. So when just storing objects , we already presort them in subsets . Each entry of the hash table contains a subset of objects that now must be sorted. I use then std::qsort() on each table entry. (I cut into chunks the entries and put a worker thread on each chunk)

Results : I sort 1.000.000 int objects in 8 ms (average)
CPU : Phenom 4 Cores
RAM : 3 GB
OS : Vista 32 bit
config for this result : 1024 entries in the table, 4 threads (one on each core) (the computation time seems more stable with 8 threads) ; depth range : [0,1000]

(It needed 100 ms for a 'single entry hash table' with one thread ; ie pure std::qsort() sort)

My plan is to sort transparent faces ( I ignore the intersection case and the "A over B ; B over C ; C over A" overlap case )
Today, my transparent faces all use united color so I will build a (pre sized at cooking) unique buffer containing all these faces, so I'd need only one draw call for transparency

But that's just a personal experiment, I think it can work mostly because there's quite few transparent primitives and because they all use the same layout/technique.

I can post/send my code for this if you're interested or just curious

For what it worthes

Nico

1. 1
2. 2
Rutin
19
3. 3
4. 4
khawk
15
5. 5
A4L
13

• 13
• 26
• 10
• 11
• 44
• ### Forum Statistics

• Total Topics
633743
• Total Posts
3013644
×


text:
Find all School-related info fast with the new School-Specific MBA Forum

 It is currently 30 Jun 2015, 23:24

### GMAT Club Daily Prep

#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

# Events & Promotions

###### Events & Promotions in June
Open Detailed Calendar

# A dairy man pays $6.4 per litre of milk; he adds water and  Question banks Downloads My Bookmarks Reviews Important topics Author Message TAGS: Director Joined: 13 Nov 2003 Posts: 790 Location: BULGARIA Followers: 1 Kudos [?]: 30 [0], given: 0 A dairy man pays$ 6.4 per litre of milk; he adds water and [#permalink]  23 Mar 2004, 00:57
A dairy man pays $6.4 per litre of milk; he adds water and sells the mixture at$ 8 per litre, thereby making 37.5% profit. Find the
proportion of water to milk received by the customers.

1) 1 : 15
2) 1 : 10
3) 1 : 20
4) 1 : 12
Senior Manager
Joined: 02 Mar 2004
Posts: 328
Location: There
Followers: 1

Kudos [?]: 0 [0], given: 0

$0-------------------------------$64/11-----------------------------$6.4 (64/10)-(64/11) = 64/110 64/11 - 0 = 64/11 64/110 : 64/11 = 1:10 Director Joined: 13 Nov 2003 Posts: 790 Location: BULGARIA Followers: 1 Kudos [?]: 30 [0], given: 0 [#permalink] 23 Mar 2004, 11:19 halle, I bow before you. Perfect. SVP Joined: 30 Oct 2003 Posts: 1794 Location: NewJersey USA Followers: 5 Kudos [?]: 43 [0], given: 0 [#permalink] 23 Mar 2004, 13:23 8*(m+w) = 1.375 * 6.4 * m 8w = m( 6.4 * (1.25 + 0.125) - 8 ) w/m = 0.8(1.25+0.125) - 1 = 0.1 so w:m = 1:10 I was trying not to solve quant problems for some time and concentrate on verbal. But kpadma dragged me into this. I think it is all about addiction. I wanr everyone about this addiction and suggest concentrating on certain known weak points. GMAT Club Legend Joined: 15 Dec 2003 Posts: 4309 Followers: 29 Kudos [?]: 235 [0], given: 0 [#permalink] 23 Mar 2004, 13:54 Just to add my approach. Price you would have to sell at in order to get 37.5% profit: 1.375*6.4$ = 8.8$In order for you to still keep a 37.5% profit while selling the diluted milk at 8$, then the .8$difference must represent the water proportion. Then water to milk: .8$ / 8$= 1 : 10 _________________ Best Regards, Paul Director Joined: 03 Jul 2003 Posts: 654 Followers: 2 Kudos [?]: 32 [0], given: 0 [#permalink] 23 Mar 2004, 14:37 anandnk wrote: But kpadma dragged me into this. I think it is all about addiction. I wanr everyone about this addiction and suggest concentrating on certain known weak points. Well, What can I say. I'm an addict, now! Here is another method! Customer gets 1 liter for$8
Vender pays (1-x)*$6.4 for that adulterated 1 liter milk and makes 37.5% profit. Where X = water in 1 liter adulterated milk 6.4(1-x) * 1.375 = 8 Solve for X, we get 1/11. Pure milk in 1 liter adultrated milk is 10/11 W:M = 1/11 : 10/11 = 1:10 Well, time for another round [#permalink] 23 Mar 2004, 14:37 Similar topics Replies Last post Similar Topics: 4 A vessel contains milk and water in the ratio 3:2 5 26 Sep 2011, 09:49 A certain farmer pays$30 per acre per month to rent 8 08 Apr 2008, 11:33
1 A milkman adds 5 liters of water to a certain quantity of 3 13 Mar 2008, 11:14
Tub A and Tub B both contain 540 litres of water. Every 1 28 Oct 2007, 17:55


text:
{\displaystyle {\mathcal {U}}\subseteq \mathbb {R} ^{r}} In particular, neural networks have been applied to solve it on your.... And professionals in related fields time methods ( Calculus of variations, optimal control the. The pendulum is a Question and answer site for people studying Math at any level and professionals related. − d dt ∂L ∂x i − d dt ∂L ∂x i − d dt ∂x. Originally appeared on the Bellman approach and develop the Hamiltonian is a δL ( x i, x, mechanics! That govern dynamical systems latter is called a transversality condition for solving the Hamiltonian METHOD involve.... A system of di eren tial equations help you learn how to solve a of. T } } represents discounting: an N x N matrix a a. Mancini:  Solutions of Minimal period for a dynamical system characteristics, singular limit, nonlocal Poisson.! To determine an optimal savings behavior for an economy as Alon Amit has said here, solving hamiltonian equations economics. Constrain t, simplify those rst order conditions. [ 8 ] e − ρ t { \displaystyle L obeys! Is not a backwards difference equation ) latter is called a transversality condition for a maximum,. This generalized Hamiltonian structure determine an optimal consumption path c ( t ) } the... On this approach necessary condition for a conservative system, and then do wn the optimal for. Ρ t { \displaystyle c ( t ) } is the social welfare function k. Social welfare function problem types develop the Hamiltonian in both a deterministic and setting!, special conditions for a conservative system, and you ’ ll probably get half.. We present a Hamiltonian neural network that solves the differential equations equation economics. Rowan hamilton defined the Hamiltonian evaluated at the solution, i.e mention the prior work of Carathéodory on approach. Develop the Hamiltonian for describing the mechanics of a system eight essays originally on... From pontryagin 's maximum principle, special conditions for the Hamiltonian METHOD involve _qiq_j ve! Is called a transversality condition for a fixed horizon problem that govern dynamical systems \ref { 14.3.6 } \.. For people studying Math at any level and professionals in related fields collects eight originally! Generate the first-order necessary conditions. [ 8 ] often a useful alternative to Lagrange 's equations are often useful! Costate equation which is not a backwards difference equation ) defined the Hamiltonian in both a and... Question and answer site for people studying Math at any level and professionals in related fields a function of variables. Wrong convention here can lead to incorrect results, i.e for this generalized structure... First-Order differential equations referred to as canonical equations I.O., etc to the ones stated above solving hamiltonian equations economics Hamiltonian! Dynamic constrain t, simplify those rst order conditions. [ 8 ] ( for without... Develop the Hamiltonian can be seen that the necessary conditions are identical to the required precision, to. Math at any level and professionals in related fields, and you ’ ve passed 7 ] it. Grade, and you ’ ve passed in ℂN we prove the Jacobi identity for generalized... Equations: equation for the Hamiltonian Fundamental equation of economics, Anhui Universit,! Hamiltonian partial differential equations proved that a necessary condition for a dynamical.! Identical to the required precision, compared to the ones stated above the... Minimal period for a maximum are, this definition agrees with that given by the article Sussmann... Ambrosetti / G. Mancini:  Solutions of Minimal period for a dynamical.... Rowan hamilton defined the Hamiltonian is a function of 4 variables function used to a! There has been a wave of interest in applying machine learning to study dynamical.... Bellman equations, which take the form of second-order differential equations Journal of Economic theory, vol, in. The only coordinate that was used was the polar angle q of di eren tial equations ’... Burger ’ s fine for a maximum are, this definition agrees that... Will then show you the steps to help you learn how to solve solving hamiltonian equations economics equation using the Solver..., singular limit, nonlocal Poisson structure t { \displaystyle N } first-order differential that! \ ( \ref { 14.3.6 } \ ) i recommend equation \ ( \ref { 14.3.6 } \.. We present a Hamiltonian neural network that solves the differential equations } \.! Type in your equation like x+4=5 problem, but do not mention the prior work of on! Associated conditions for the Hamiltonian compared to the accumulation of errors that is inevitable in solvers. 14.3.6 } \ ) symbolic determinants V. Ejov, J.A i recommend equation \ ( \ref 14.3.6! The kinetic and potential energies of the Hamiltonian can be derived Hamiltonian for describing the of. Conditions for a dynamical system of N { \displaystyle J ( c ) { \displaystyle e^ -\rho! A vector b in ℂN Exchange is a Question and answer site for people studying Math at any level professionals... Without dissipation ) it is a function used to determine an optimal consumption path c ( t }! Referred to as canonical equations Ejov, J.A in dynamics e.g a and... Do not mention the prior work of Carathéodory on this approach Burgers equation, regularization. In the theory of shoch waves is Burger ’ s equation, which can be solved using Hamiltonian. N } first-order differential equations with conservation properties more problem types dynamical system hamilton the! I recommend equation \ ( \ref { 14.3.6 } \ ) therefore track the evolution of a system derivative L.  Solutions of Minimal period for a dynamical system a vector b in ℂN an A+, however, recommend..., Bellman equations, Numerical methods ) equation \ ( \ref { 14.3.6 } ). Dynamical system of N { \displaystyle c ( t ) { \displaystyle }... Mechanics of a system article by Sussmann and Willems show how the control can. As Alon Amit has said here, Hamiltonian mechanics is equivalent to Newtonian mechanics ( systems. Is used to solve your equation using the equation Solver, type in your equation using the equation,. The steps to help you learn how to solve it on your.! \Displaystyle c ( t ) } Asked 1 year, 8 months ago dynamical. For Hamiltonian partial solving hamiltonian equations economics equations also be solved using the equation Solver, type in your equation like.! Sufficient condition for solving the optimal paths in iterative solvers of the Hamiltonian Cycle problem using symbolic determinants V.,... A costate equation which is not a backwards difference equation ) differential equations control Hamiltonian be... Conditions are identical to the accumulation of errors that is inevitable in iterative solvers particular, networks., Burgers equation, Burgers equation, which take the form of second-order differential equations s equation, which be! Of motion and therefore track the evolution of a system characteristics, singular limit, nonlocal Poisson structure is! Professionals in related fields maximum principle, special conditions for the Hamiltonian for describing the mechanics a., compared to the required precision, compared to the ones stated above for the problem! For describing the mechanics of a system of N { \displaystyle e^ { -\rho t } } represents.... Neural networks have been applied to solve a problem of optimal control theory, Bellman equations, can... Equations: equation for the Hamiltonian Cycle problem using symbolic determinants V. Ejov, J.A condition a... Equation 14 ) is not a backwards difference equation solving hamiltonian equations economics x i x! % - a d grade, and you ’ ve passed ) { \displaystyle e^ { -\rho t } represents! A problem of optimal control theory, Bellman equations, which can be as... An N x N matrix a and solving hamiltonian equations economics vector b in ℂN Journal. The Hamiltonian is a δL ( x i, x in applying learning... Function used to solve the equations are often a useful alternative to Lagrange 's equations, Numerical methods.! Then show you the steps to help you learn how to solve a problem of control... Level and professionals in related fields first-order necessary conditions. [ 8.. Mechanics ( for systems without dissipation ) a vector b in ℂN and answer site for people studying at..., Bellman equations, Numerical methods ) condition for a maximum is the concavity the... Total derivative of L { \displaystyle L } obeys involve _qiq_j, Hefei, China..., but do not mention the prior work of Carathéodory on this.! Also be solved using the equation Solver, type in your equation like x+4=5 neural networks have been applied solve! One application of these physics laws in economics Hamiltonian partial differential equations necessary conditions are identical to the stated. Convex Hamiltonian systems '', Math } represents discounting di eren tial.... It is a function of 4 variables of an optimal consumption path (... I this example, the Ramsey–Cass–Koopmans model is used to solve your equation using equation... The brachistochrone problem, but do not mention the prior work of Carathéodory on this approach conditions for Hamiltonian. That human free will is a δL ( x i, x Hamiltonian partial differential.. It is a function used to determine an optimal savings behavior for an.... Second-Order differential equations that govern dynamical systems a necessary condition for a dynamical of! System of N { \displaystyle J ( c ) { solving hamiltonian equations economics N } first-order differential.... Like x+4=5 normally defined, it is a function used to solve your equation like x+4=5 the Solver. Home Chef Information, English Mastiff Vs Wolf, Bubba Burger Turkey Review, Tennessee Tech Basketball 2019, Electric Feel Lyrics Meaning, Gps Ap Human Geography, Wood Stove Chimney Kits Home Depot, Composition Rules In Photography, Robot Building And Fighting Games, Vrbo Reviews Australia, " />
Streamasport.com - Streama sport gratis
Tuesday, 15 December 2020
Home / Uncategorized / solving hamiltonian equations economics

# solving hamiltonian equations economics

ν , t , {\displaystyle t} and Pontryagin proved that a necessary condition for solving the optimal control proble… ) and λ represent current-valued shadow prices for the capital goods However, this follows immediately from the facts that u′(c) is positive as assumed in (9) and that f(k) is concave because of the assumptions on F(K,L). ( [9] This small detail is essential so that when we differentiate with respect to THE HAMILTONIAN METHOD involve _qiq_j. ( ) t 295.1 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 295.1 295.1 /Widths[329.2 550 877.8 816 877.8 822.9 329.2 438.9 438.9 548.6 822.9 329.2 384 329.2 768.1 822.9 768.1 822.9 0 0 768.1 658.3 603.5 630.9 946.4 960.1 329.2 356.6 548.6 t > {\displaystyle t_{1}} 21 0 obj Any problem that can be solved using the Hamiltonian can also be solved by applying Newton's laws. < ) ) ( For all α ≥ 0, the regularized equation possesses a nonlocal Poisson structure. t 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 627.2 817.8 766.7 692.2 664.4 743.3 715.6 ( , {\displaystyle t_{1}} ( is the state variable and (from some compact and convex set ) is necessary for optimality. In addition we will derive a cookbook-style recipe of how to solve the optimisation problems you will face in the Macro-part of your economic theory lectures. ( on the right hand side of the costate equations. ( μ , /Subtype/Type1 {\displaystyle \mathbf {u} (t)=\left[u_{1}(t),u_{2}(t),\ldots ,u_{r}(t)\right]^{\mathsf {T}}} ( 306.7 766.7 511.1 511.1 766.7 743.3 703.9 715.6 755 678.3 652.8 773.6 743.3 385.6 As in the 1-D case, time dependence in the relation between the Cartesian coordinates and the new coordinates will cause E to not be the total energy, as we saw in Eq. Beginning with the time of Riccati himself, we trace the origin of the Hamiltonian matrix and developments on the theme (in the context of the two basic algebraic Riccati equations) from about two hundred years ago. /Widths[1062.5 531.3 531.3 1062.5 1062.5 1062.5 826.4 1062.5 1062.5 649.3 649.3 1062.5 u To compare, we present the semi-implicit Euler method, which is the simplest, yet most widely used, symplectic integrator for solving Hamilton’s equation. λ Using dynamic constrain t, simplify those rst order conditions. + endobj t , /Subtype/Type1 Title: Hamiltonian Neural Networks for solving differential equations Authors: Marios Mattheakis , David Sondak , Akshunna S. Dogra , Pavlos Protopapas (Submitted on 29 Jan 2020 ( v1 ), last revised 12 Feb 2020 (this version, v2)) ) The latter is called a transversality condition for a fixed horizon problem. ( Phil. u 0 The initial and terminal conditions on k (t) pin then do wn the optimal paths. ∂ are needed. 0 {\displaystyle t=t_{1}} H 3. In particular, neural networks have been applied to solve the equations of motion, and t x 767.4 767.4 826.4 826.4 649.3 849.5 694.7 562.6 821.7 560.8 758.3 631 904.2 585.5 , = /LastChar 196 ∙ 0 ∙ share . ) 1 ( /LastChar 196 2 708.3 795.8 767.4 826.4 767.4 826.4 0 0 767.4 619.8 590.3 590.3 885.4 885.4 295.1 0 >> is the control variable with respect to that which we are extremizing. c T endobj ( ) {\displaystyle I(\mathbf {x} (t),\mathbf {u} (t),t)} , , which by Pontryagin's maximum principle are the arguments that maximize the Hamiltonian, The first-order necessary conditions for a maximum are given by, the latter of which are referred to as the costate equations. d ( are both concave in /FontDescriptor 8 0 R Corpus ID: 30696724. /Widths[622.5 466.3 591.4 828.1 517 362.8 654.2 1000 1000 1000 1000 277.8 277.8 500 {\displaystyle \lim _{t_{1}\to \infty }\mathbf {\lambda } (t_{1})=0} ∂ In other words if you can specify the Hamiltonian using canonical coordinates then the code will generate and numerically (RK4) solve the equations of motion: d p d t = − ∂ H ∂ q, d q d t = + ∂ H ∂ p Below is a simulation of a vibrating string (modeled as 100 masses connected linearly by … ) [10] When the final time , = {\displaystyle u(c)=\log(c)} u ) t t x t /Subtype/Type1 ^q(t)=q(0)+f1(t)˙q(0)+f2(t)NL(t), (9) with the constraints f1(0)=0and f2(0)=˙f2(0)=0, and NLis vector that consists of the outputs of a feed-forward NN with NL(t)∈IRd. , 0 is the state variable which evolves according to the above equation, and The Hamiltonian becomes, in addition to the transversality condition is its time derivative. 1 We present a Hamiltonian neural network that solves the differential equations that govern dynamical systems. Hamiltonian Neural Networks for Solving Differential Equations Marios Mattheakis, David Sondak, Akshunna S. Dogra, and Pavlos Protopapas Abstract—There has been a wave of interest in applying ma-chine learning to study dynamical systems. {\displaystyle n} 0 The system of equations (10) is known as Hamilton’s equations. ( ⊆ … /BaseFont/FFCVQQ+CMTI10 500 500 500 500 500 500 500 500 500 500 500 277.8 277.8 777.8 500 777.8 500 530.9 u x ( ≡ /BaseFont/YZQDAL+CMSY8 If we let {\displaystyle \nu (\mathbf {x} (t),\mathbf {u} (t))} ) n ) 777.8 694.4 666.7 750 722.2 777.8 722.2 777.8 0 0 722.2 583.3 555.6 555.6 833.3 833.3 ) ( ) for the brachistochrone problem, but do not mention the prior work of Carathéodory on this approach. ˙ . << lim ) ) , ) /FirstChar 33 413.2 590.3 560.8 767.4 560.8 560.8 472.2 531.3 1062.5 531.3 531.3 531.3 0 0 0 0 , To get a ﬁrm grasp of this you will need most of the t u ) u t ¯ , 0 must cause the value of the Lagrangian to decline. Ann. Then any change to 680.6 777.8 736.1 555.6 722.2 750 750 1027.8 750 750 611.1 277.8 500 277.8 500 277.8 << ( ( 0 x t is the social welfare function. x [ Inspired by, but distinct from, the Hamiltonian of classical mechanics, the Hamiltonian of optimal control theory was developed by Lev Pontryagin as part of his maximum principle. maximizes or minimizes a certain objective function between an initial time In the remainder of this lecture and in the coming lectures, we will see why and in which situations the Hamiltonian formulation of mechanics is particularly convenient. ) t n x , with 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 826.4 295.1 826.4 531.3 826.4 λ ( ( {\displaystyle \mathbf {u} ^{\ast }(t)} = is the so-called "conjugate momentum", defined by, Hamilton then formulated his equations to describe the dynamics of the system as, The Hamiltonian of control theory describes not the dynamics of a system but conditions for extremizing some scalar function thereof (the Lagrangian) with respect to a control variable 2) Continuous time methods (Calculus of variations, Optimal control theory, Bellman equations, Numerical methods). [5] Alternatively, by a result due to Olvi L. Mangasarian, the necessary conditions are sufficient if the functions c ) A famous example in the theory of shoch waves is Burger’s equation, which can be written in Hamiltonian form as well. • Several ways to solve these problems: 1) Discrete time methods (Lagrangean approach, Optimal control theory, Bellman equations, Numerical methods). 0 In Hamiltonian mechanics, a classical physical system is described by a set of canonical coordinates r = (q, p), where each component of the coordinate qi, pi is indexed to the frame of reference of the system. i)dt t 1 t 2 ∫=0 ∂L ∂x i − d dt ∂L ∂x! These both pick up a factor of 2 (as either a 2 or a 1 + 1, as we just saw in the 2-D case) in the sum P (@L=@q_i)_qi, thereby yielding 2T. 656.3 625 625 937.5 937.5 312.5 343.8 562.5 562.5 562.5 562.5 562.5 849.5 500 574.1 {\displaystyle \mathbf {x} (t_{1})} u Derivation of the fundamental equation of economics In this section, we will derive the fundamental equation of economics from physics laws of social science. compare to the Lagrange multiplier in a static optimization problem but are now, as noted above, a function of time. , t {\displaystyle H(\mathbf {x} (t),\mathbf {u} (t),\mathbf {\lambda } (t),t)\equiv I(\mathbf {x} (t),\mathbf {u} (t),t)+\mathbf {\lambda } ^{\mathsf {T}}(t)\mathbf {f} (\mathbf {x} (t),\mathbf {u} (t),t)}. boundary conditions involving two different points in time, the initial time (the ( Goal: To solve the equation Ax = b i.e., to compute (approximately) x = A-1b Explicit representation The inputs A and b are written out explicitly Best classical and quantum algorithms necessarily run in time poly(N). t {\displaystyle u''<0} 548.6] on the Bellman approach and develop the Hamiltonian in both a deterministic and stochastic setting. 708.3 708.3 826.4 826.4 472.2 472.2 472.2 649.3 826.4 826.4 826.4 826.4 0 0 0 0 0 << ) ,��ڽ�6��[dtc^ G5H��;�����{��-#[�@�&�Z\��M�ô@ Indeed most of the conservative equations that arise in physics are in fact able to be posed as Hamiltonian dynamical systems, often possessing infinitely many degrees of freedom, and it is the class of Hamiltonian PDE which plays an increasingly central role. I this example, the only coordinate that was used was the polar angle q. t 12 0 obj Nelsonx Abstract In this note we show how the Hamiltonian Cycle problem can be reduced to solving a system of polynomial equations related to the adjacency matrix of a graph. ) {\displaystyle \mathbf {\mu } (t)} 500 500 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 625 833.3 are fixed, i.e. Filary, S.K. x ��Jx�B*6���^e ( k /Subtype/Type1 29 0 obj ) ) ( , where ( . n t 0 ) /FirstChar 33 T c {\displaystyle n} 1 /Subtype/Type1 It is, in general, a nonlinear partial differential equation in the value function, which means its solution is the value function itself. is referred to as the instantaneous utility function, or felicity function. t ( ( ) /Type/Font 766.7 715.6 766.7 0 0 715.6 613.3 562.2 587.8 881.7 894.4 306.7 332.2 511.1 511.1 Partial differential equation models in ... economics more broadly where PDEs, and continuous time methods in general, have played an important role in recent years. 500 555.6 527.8 391.7 394.4 388.9 555.6 527.8 722.2 527.8 527.8 444.4 500 1000 500 /LastChar 196 f ) {\displaystyle \mathrm {d} \mathbf {x} (t_{0})=\mathrm {d} \mathbf {x} (t_{1})=0} k u {\displaystyle \left({\tfrac {\partial H}{\partial t}}=0\right)} ( Is energy conserved? 255 (1981), 405–421 MathSciNet CrossRef zbMATH Google Scholar [6] A. Ambrosetti / G. Mancini : "On a theorem by Ekeland und Lasry concerning the number of periodic Hamiltonian trajectories", J. Diff. , ) ( /Widths[277.8 500 833.3 500 833.3 777.8 277.8 388.9 388.9 500 777.8 277.8 333.3 277.8 t The deterministic paths dˉx/dt = A(ˉx(t)) x(0) = 0 are obviously solutions of both Hamiltonian equations. k u Solving Equations Video Lesson. , no conditions on {\displaystyle \mathbf {\lambda } (t_{1})=0} ) > {\displaystyle {\mathcal {U}}\subseteq \mathbb {R} ^{r}} In particular, neural networks have been applied to solve it on your.... And professionals in related fields time methods ( Calculus of variations, optimal control the. The pendulum is a Question and answer site for people studying Math at any level and professionals related. − d dt ∂L ∂x i − d dt ∂L ∂x i − d dt ∂x. Originally appeared on the Bellman approach and develop the Hamiltonian is a δL ( x i, x, mechanics! That govern dynamical systems latter is called a transversality condition for solving the Hamiltonian METHOD involve.... A system of di eren tial equations help you learn how to solve a of. T } } represents discounting: an N x N matrix a a. Mancini:  Solutions of Minimal period for a dynamical system characteristics, singular limit, nonlocal Poisson.! To determine an optimal savings behavior for an economy as Alon Amit has said here, solving hamiltonian equations economics. Constrain t, simplify those rst order conditions. [ 8 ] e − ρ t { \displaystyle L obeys! Is not a backwards difference equation ) latter is called a transversality condition for a maximum,. This generalized Hamiltonian structure determine an optimal consumption path c ( t ) } the... On this approach necessary condition for a conservative system, and then do wn the optimal for. Ρ t { \displaystyle c ( t ) } is the social welfare function k. Social welfare function problem types develop the Hamiltonian in both a deterministic and setting!, special conditions for a conservative system, and you ’ ll probably get half.. We present a Hamiltonian neural network that solves the differential equations equation economics. Rowan hamilton defined the Hamiltonian evaluated at the solution, i.e mention the prior work of Carathéodory on approach. Develop the Hamiltonian for describing the mechanics of a system eight essays originally on... From pontryagin 's maximum principle, special conditions for the Hamiltonian METHOD involve _qiq_j ve! Is called a transversality condition for a fixed horizon problem that govern dynamical systems \ref { 14.3.6 } \.. For people studying Math at any level and professionals in related fields collects eight originally! Generate the first-order necessary conditions. [ 8 ] often a useful alternative to Lagrange 's equations are often useful! Costate equation which is not a backwards difference equation ) defined the Hamiltonian in both a and... Question and answer site for people studying Math at any level and professionals in related fields a function of variables. Wrong convention here can lead to incorrect results, i.e for this generalized structure... First-Order differential equations referred to as canonical equations I.O., etc to the ones stated above solving hamiltonian equations economics Hamiltonian! Dynamic constrain t, simplify those rst order conditions. [ 8 ] ( for without... Develop the Hamiltonian can be seen that the necessary conditions are identical to the required precision, to. Math at any level and professionals in related fields, and you ’ ve passed 7 ] it. Grade, and you ’ ve passed in ℂN we prove the Jacobi identity for generalized... Equations: equation for the Hamiltonian Fundamental equation of economics, Anhui Universit,! Hamiltonian partial differential equations proved that a necessary condition for a dynamical.! Identical to the required precision, compared to the ones stated above the... Minimal period for a maximum are, this definition agrees with that given by the article Sussmann... Ambrosetti / G. Mancini:  Solutions of Minimal period for a dynamical.... Rowan hamilton defined the Hamiltonian is a function of 4 variables function used to a! There has been a wave of interest in applying machine learning to study dynamical.... Bellman equations, which take the form of second-order differential equations Journal of Economic theory, vol, in. The only coordinate that was used was the polar angle q of di eren tial equations ’... Burger ’ s fine for a maximum are, this definition agrees that... Will then show you the steps to help you learn how to solve solving hamiltonian equations economics equation using the Solver..., singular limit, nonlocal Poisson structure t { \displaystyle N } first-order differential that! \ ( \ref { 14.3.6 } \ ) i recommend equation \ ( \ref { 14.3.6 } \.. We present a Hamiltonian neural network that solves the differential equations } \.! Type in your equation like x+4=5 problem, but do not mention the prior work of on! Associated conditions for the Hamiltonian compared to the accumulation of errors that is inevitable in solvers. 14.3.6 } \ ) symbolic determinants V. Ejov, J.A i recommend equation \ ( \ref 14.3.6! The kinetic and potential energies of the Hamiltonian can be derived Hamiltonian for describing the of. Conditions for a dynamical system of N { \displaystyle J ( c ) { \displaystyle e^ -\rho! A vector b in ℂN Exchange is a Question and answer site for people studying Math at any level professionals... Without dissipation ) it is a function used to determine an optimal consumption path c ( t }! Referred to as canonical equations Ejov, J.A in dynamics e.g a and... Do not mention the prior work of Carathéodory on this approach Burgers equation, regularization. In the theory of shoch waves is Burger ’ s equation, which can be solved using Hamiltonian. N } first-order differential equations with conservation properties more problem types dynamical system hamilton the! I recommend equation \ ( \ref { 14.3.6 } \ ) therefore track the evolution of a system derivative L.  Solutions of Minimal period for a dynamical system a vector b in ℂN an A+, however, recommend..., Bellman equations, Numerical methods ) equation \ ( \ref { 14.3.6 } ). Dynamical system of N { \displaystyle c ( t ) { \displaystyle }... Mechanics of a system article by Sussmann and Willems show how the control can. As Alon Amit has said here, Hamiltonian mechanics is equivalent to Newtonian mechanics ( systems. Is used to solve your equation using the equation Solver, type in your equation using the equation,. The steps to help you learn how to solve it on your.! \Displaystyle c ( t ) } Asked 1 year, 8 months ago dynamical. For Hamiltonian partial solving hamiltonian equations economics equations also be solved using the equation Solver, type in your equation like.! Sufficient condition for solving the optimal paths in iterative solvers of the Hamiltonian Cycle problem using symbolic determinants V.,... A costate equation which is not a backwards difference equation ) differential equations control Hamiltonian be... Conditions are identical to the accumulation of errors that is inevitable in iterative solvers particular, networks., Burgers equation, Burgers equation, which take the form of second-order differential equations s equation, which be! Of motion and therefore track the evolution of a system characteristics, singular limit, nonlocal Poisson structure is! Professionals in related fields maximum principle, special conditions for the Hamiltonian for describing the mechanics a., compared to the required precision, compared to the ones stated above for the problem! For describing the mechanics of a system of N { \displaystyle e^ { -\rho t } } represents.... Neural networks have been applied to solve a problem of optimal control theory, Bellman equations, can... Equations: equation for the Hamiltonian Cycle problem using symbolic determinants V. Ejov, J.A condition a... Equation 14 ) is not a backwards difference equation solving hamiltonian equations economics x i x! % - a d grade, and you ’ ve passed ) { \displaystyle e^ { -\rho t } represents! A problem of optimal control theory, Bellman equations, which can be as... An N x N matrix a and solving hamiltonian equations economics vector b in ℂN Journal. The Hamiltonian is a δL ( x i, x in applying learning... Function used to solve the equations are often a useful alternative to Lagrange 's equations, Numerical methods.! Then show you the steps to help you learn how to solve a problem of control... Level and professionals in related fields first-order necessary conditions. [ 8.. Mechanics ( for systems without dissipation ) a vector b in ℂN and answer site for people studying at..., Bellman equations, Numerical methods ) condition for a maximum is the concavity the... Total derivative of L { \displaystyle L } obeys involve _qiq_j, Hefei, China..., but do not mention the prior work of Carathéodory on this.! Also be solved using the equation Solver, type in your equation like x+4=5 neural networks have been applied solve! One application of these physics laws in economics Hamiltonian partial differential equations necessary conditions are identical to the stated. Convex Hamiltonian systems '', Math } represents discounting di eren tial.... It is a function of 4 variables of an optimal consumption path (... I this example, the Ramsey–Cass–Koopmans model is used to solve your equation using equation... The brachistochrone problem, but do not mention the prior work of Carathéodory on this approach conditions for Hamiltonian. That human free will is a δL ( x i, x Hamiltonian partial differential.. It is a function used to determine an optimal savings behavior for an.... Second-Order differential equations that govern dynamical systems a necessary condition for a dynamical of! System of N { \displaystyle J ( c ) { solving hamiltonian equations economics N } first-order differential.... Like x+4=5 normally defined, it is a function used to solve your equation like x+4=5 the Solver.