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• The Global ETD Search service is a free service for researchers to find electronic theses and dissertations. This service is provided by the Networked Digital Library of Theses and Dissertations.
Our metadata is collected from universities around the world. If you manage a university/consortium/country archive and want to be added, details can be found on the NDLTD website.
1

#### Code constructions and code families for nonbinary quantum stabilizer code

Ketkar, Avanti Ulhas 01 November 2005 (has links)
Stabilizer codes form a special class of quantum error correcting codes. Nonbinary quantum stabilizer codes are studied in this thesis. A lot of work on binary quantum stabilizer codes has been done. Nonbinary stabilizer codes have received much less attention. Various results on binary stabilizer codes such as various code families and general code constructions are generalized to the nonbinary case in this thesis. The lower bound on the minimum distance of a code is nothing but the minimum distance of the currently best known code. The focus of this research is to improve the lower bounds on this minimum distance. To achieve this goal, various existing quantum codes are studied that have good minimum distance. Some new families of nonbinary stabilizer codes such as quantum BCH codes are constructed. Diﬀerent ways of constructing new codes from the existing ones are also found. All these constructions together help improve the lower bounds.
2

#### Code constructions and code families for nonbinary quantum stabilizer code

Ketkar, Avanti Ulhas 01 November 2005 (has links)
Stabilizer codes form a special class of quantum error correcting codes. Nonbinary quantum stabilizer codes are studied in this thesis. A lot of work on binary quantum stabilizer codes has been done. Nonbinary stabilizer codes have received much less attention. Various results on binary stabilizer codes such as various code families and general code constructions are generalized to the nonbinary case in this thesis. The lower bound on the minimum distance of a code is nothing but the minimum distance of the currently best known code. The focus of this research is to improve the lower bounds on this minimum distance. To achieve this goal, various existing quantum codes are studied that have good minimum distance. Some new families of nonbinary stabilizer codes such as quantum BCH codes are constructed. Diﬀerent ways of constructing new codes from the existing ones are also found. All these constructions together help improve the lower bounds.
3

#### Color Segmentation on FPGA for Automatic Road Sign Recognition

Zhao, Jingbo January 2012 (has links)
No description available.
4

#### A New Algorithm for Finding the Minimum Distance between Two Convex Hulls

Kaown, Dougsoo 05 1900 (has links)
The problem of computing the minimum distance between two convex hulls has applications to many areas including robotics, computer graphics and path planning. Moreover, determining the minimum distance between two convex hulls plays a significant role in support vector machines (SVM). In this study, a new algorithm for finding the minimum distance between two convex hulls is proposed and investigated. A convergence of the algorithm is proved and applicability of the algorithm to support vector machines is demostrated. The performance of the new algorithm is compared with the performance of one of the most popular algorithms, the sequential minimal optimization (SMO) method. The new algorithm is simple to understand, easy to implement, and can be more efficient than the SMO method for many SVM problems.
5

#### Classification of image pixels based on minimum distance and hypothesis testing

Ghimire, Santosh January 1900 (has links)
Master of Science / Department of Statistics / Haiyan Wang / We introduce a new classification method that is applicable to classify image pixels. This work was motivated by the test-based classification (TBC) introduced by Liao and Akritas(2007). We found that direct application of TBC on image pixel classification can lead to high mis-classification rate. We propose a method that combines the minimum distance and evidence from hypothesis testing to classify image pixels. The method is implemented in R programming language. Our method eliminates the drawback of Liao and Akritas (2007).Extensive experiments show that our modified method works better in the classification of image pixels in comparison with some standard methods of classification; namely, Linear Discriminant Analysis (LDA), Quadratic Discriminant Analysis (QDA), Classification Tree(CT), Polyclass classification, and TBC. We demonstrate that our method works well in the case of both grayscale and color images.
6

#### Critical knots for minimum distance energy and complementary domains of arrangements of hypersurfaces

Hager, William George 01 July 2010 (has links)
In this thesis, we will discuss two separate topics. First, we find a critical knot for an knot energy function. A knot is a closed curve or polygon in three space. It is possible to for a computer to simulate the flow of a knot to its minimum energy conformation. There is no guarantee, however, that a true minimizer exists near the computer's alleged minimizer. We take advantage of both the symmetry of the minimizer and the symmetry invariance of the energy function to prove that there is a critical point of the energy function near the computer's minimizer. Second, we will discuss how to determine the number of complementary domains of arrangements of algebraic curves in 2-space and ellipsoids in 3-space. In each of these situations, we supply equations that provide an upper bound for the number of complementary domains. These upper bounds are applicable even when the exact intersections between the curves or surfaces are unknown.
7

#### Parametric classification and variable selection by the minimum integrated squared error criterion

This thesis presents a robust solution to the classification and variable selection problem when the dimension of the data, or number of predictor variables, may greatly exceed the number of observations. When faced with the problem of classifying objects given many measured attributes of the objects, the goal is to build a model that makes the most accurate predictions using only the most meaningful subset of the available measurements. The introduction of [cursive l] 1 regularized model titling has inspired many approaches that simultaneously do model fitting and variable selection. If parametric models are employed, the standard approach is some form of regularized maximum likelihood estimation. While this is an asymptotically efficient procedure under very general conditions, it is not robust. Outliers can negatively impact both estimation and variable selection. Moreover, outliers can be very difficult to identify as the number of predictor variables becomes large. Minimizing the integrated squared error, or L 2 error, while less efficient, has been shown to generate parametric estimators that are robust to a fair amount of contamination in several contexts. In this thesis, we present a novel robust parametric regression model for the binary classification problem based on L 2 distance, the logistic L 2 estimator (L 2 E). To perform simultaneous model fitting and variable selection among correlated predictors in the high dimensional setting, an elastic net penalty is introduced. A fast computational algorithm for minimizing the elastic net penalized logistic L 2 E loss is derived and results on the algorithm's global convergence properties are given. Through simulations we demonstrate the utility of the penalized logistic L 2 E at robustly recovering sparse models from high dimensional data in the presence of outliers and inliers. Results on real genomic data are also presented.
8

#### Minimum Distance Estimation in Categorical Conditional Independence Models

One of the oldest and most fundamental problems in statistics is the analysis of cross-classified data called contingency tables. Analyzing contingency tables is typically a question of association - do the variables represented in the table exhibit special dependencies or lack thereof? The statistical models which best capture these experimental notions of dependence are the categorical conditional independence models; however, until recent discoveries concerning the strongly algebraic nature of the conditional independence models surfaced, the models were widely overlooked due to their unwieldy implicit description. Apart from the inferential question above, this thesis asks the more basic question - suppose such an experimental model of association is known, how can one incorporate this information into the estimation of the joint distribution of the table? In the traditional parametric setting several estimation paradigms have been developed over the past century; however, traditional results are not applicable to arbitrary categorical conditional independence models due to their implicit nature. After laying out the framework for conditional independence and algebraic statistical models, we consider three aspects of estimation in the models using the minimum Euclidean (L2E), minimum Pearson chi-squared, and minimum Neyman modified chi-squared distance paradigms as well as the more ubiquitous maximum likelihood approach (MLE). First, we consider the theoretical properties of the estimators and demonstrate that under general conditions the estimators exist and are asymptotically normal. For small samples, we present the results of large scale simulations to address the estimators' bias and mean squared error (in the Euclidean and Frobenius norms, respectively). Second, we identify the computation of such estimators as an optimization problem and, for the case of the L2E, propose two different methods by which the problem can be solved, one algebraic and one numerical. Finally, we present an R implementation via two novel packages, mpoly for symbolic computing with multivariate polynomials and catcim for fitting categorical conditional independence models. It is found that in general minimum distance estimators in categorical conditional independence models behave as they do in the more traditional parametric setting and can be computed in many practical situations with the implementation provided.
9

#### A brief survey of self-dual codes

Oktavia, Rini 2009 August 1900 (has links)
This report is a survey of self-dual binary codes. We present the fundamental MacWilliams identity and Gleason’s theorem on self-dual binary codes. We also examine the upper bound of minimum weights of self-dual binary codes using the extremal weight enumerator formula. We describe the shadow code of a self-dual code and the restrictions of the weight enumerator of the shadow code. Then using the restrictions, we calculate the weight enumerators of self-dual codes of length 38 and 40 and we obtain the known weight enumerators of this lengths. Finally, we investigate the Gaborit-Otmani experimental construction of selfdual binary codes. This construction involves a fixed orthogonal matrix, and we compare the result to the results obtained using other orthogonal matrices. / text
10

#### Deconvolution in Random Effects Models via Normal Mixtures

Litton, Nathaniel A. 2009 August 1900 (has links)
This dissertation describes a minimum distance method for density estimation when the variable of interest is not directly observed. It is assumed that the underlying target density can be well approximated by a mixture of normals. The method compares a density estimate of observable data with a density of the observable data induced from assuming the target density can be written as a mixture of normals. The goal is to choose the parameters in the normal mixture that minimize the distance between the density estimate of the observable data and the induced density from the model. The method is applied to the deconvolution problem to estimate the density of $X_{i}$ when the variable $% Y_{i}=X_{i}+Z_{i}$, $i=1,\ldots ,n$, is observed, and the density of $Z_{i}$ is known. Additionally, it is applied to a location random effects model to estimate the density of $Z_{ij}$ when the observable quantities are $p$ data sets of size $n$ given by $X_{ij}=\alpha _{i}+\gamma Z_{ij},~i=1,\ldots ,p,~j=1,\ldots ,n$, where the densities of $\alpha_{i}$ and $Z_{ij}$ are both unknown. The performance of the minimum distance approach in the measurement error model is compared with the deconvoluting kernel density estimator of Stefanski and Carroll (1990). In the location random effects model, the minimum distance estimator is compared with the explicit characteristic function inversion method from Hall and Yao (2003). In both models, the methods are compared using simulated and real data sets. In the simulations, performance is evaluated using an integrated squared error criterion. Results indicate that the minimum distance methodology is comparable to the deconvoluting kernel density estimator and outperforms the explicit characteristic function inversion method.

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# Roller's problem on median groups

At the end of his dissertation Poc Sets, Median Algebras and Group Actions, Martin Roller asks

A group $G$ is called median if it acts freely and transitively on a median algebra. This is equivalent to saying that the Cayley graph with respect to a certain set of generators is a cubing. [...] Characterize all presentations of median groups.

Do you know any work on the subset? The only examples I am able to imagine are of the form $$\langle x_1, \ldots, x_n \mid [x_i,x_j]=1 \ \text{and} \ x_k^2=1 \ \text{for some} \ 1 \leq i,j,k \leq n \rangle.$$ Are there other simple examples?

• Right-angled Artin groups. Given a (simplicial) graph $\Gamma$, define $$A(\Gamma)= \langle \text{vertices of \Gamma} \mid [u,v]=1, \ (u,v) \in E(\Gamma) \rangle.$$ Then the Cayley graph of $A(\Gamma)$ with respect to the generating set given by the previous presentation is median graph, or if you prefer, it is naturally the one-skeleton of a CAT(0) cube complex.
• Right-angled Coxeter groups. The same comment holds for the group $$\langle \text{vertices of \Gamma} \mid u^2=1 \ (\text{u vertex}), \ [u,v]=1 \ (\text{u,v adjacent}) \rangle$$
• LOG groups. Given an oriented graph $\Gamma$ whose edges are labelled by its vertices (such that any vertex labels at most one edge), one can define the group $$G(\Gamma)= \langle \text{vertices of \Gamma} \mid cac^{-1}=b, \ (a \mid c \mid b) \in E(\Gamma) \rangle$$ where $(a \mid c \mid b) \in E(\Gamma)$ means that there exists an oriented edge from $a$ to $b$ which is labelled by $c$. In his paper On the realization of Wirtinger presentations as knot groups, Rosebrock determines precisely when the square complex associated to the above presentation is nonpositively curved.
• Right-angled mock reflection groups. In his paper Right-angled mock reflection and mock Artin groups, Scott classifies explicitely all the groups acting vertex-simply-transitively on CAT(0) cube complexes with $\mathbb{Z}_2$ edge-stabilisers.
There are compact nonpositively-curved square complexes $X$ so that $X$ has a single vertex and the universal cover of $X$ is a CAT(0) square complex isomorphic to the product of two trees, but $\pi_1X$ is not of the form you've mentioned, and doesn't even have a finite-index subgroup admitting such a presentation. For example, there is this paper of Wise.

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# Answering Fuzzy Conjunctive Queries over Finitely Valued Fuzzy Ontologies

11 Aug 2015Stefan BorgwardtTheofilos MailisRafael PeñalozaAnni-Yasmin Turhan

Fuzzy Description Logics (DLs) provide a means for representing vague knowledge about an application domain. In this paper, we study fuzzy extensions of conjunctive queries (CQs) over the DL $\mathcal{SROIQ}$ based on finite chains of degrees of truth... (read more)

PDF Abstract


text:
# Are there other definitions of Jacobi polynomials?

While I am reading "On some dual integral equations occurring in potential problems with axial symmetry" by C. J. Tranter, Quarterly Journal of Mechanics & Applied Math (1950) p. 414, the author stated that link to the article $$\mathcal{F}_m (\alpha,\gamma,x) = {}_2F_1 (-m,\alpha+m;\gamma;x) \, ,$$ where $\mathcal{F}$ is Jacobi polynomial. From Wikipedia, I have found another definition which reads $$\mathcal{F}_m (\alpha,\gamma,x) = \frac{(\alpha+1)_m}{m!} {}_2F_1 (-m, 1+\alpha+\gamma+m;\alpha+1;\tfrac{1}{2} (1-x))\, ,$$ with $(\alpha+1)_m$ being the Pochhammer's symbol. Apparently the two definitions are not equivalent.

I was wondering whether other definitions of Jacobi polynomials have also been introduced in the literature. Any clarification is highly appreciated.

Thanks.

R


text:
# Position margin notes relative to left margin in plain TeX

I want to add margin notes in the Tufte style to my Plain TeX document. (I'm using eplain, if that matters.) I know that if I were using LaTeX I could just use tufte-latex, but as a thorough contrarian I enjoy coding these things myself.

I'm almost there but my current macro will place text relative to the current location when I invoke it, instead of the margin. In my MWE, the note is placed smack dab in the middle of things. But if you move the invocation of \sidenote around, the side note may appear closer to the left margin. How can I reposition my sidenote text relative to the left (or right) margin, instead of the current location? (I used the left margin in my MWE.)

The thing I really do like about my macro is that the baselines of the first line of the side note and the main body text to which it refers align. I want to keep that. And any other suggestions about my macro are welcome.

MWE:

\input eplain
\leftmargin=2in

\font\sidenotefont=      pxi at8.5pt

\long\def\sidenote#1{\llap{\smash{\vtop{%
\parindent=0pt\hsize=1.5in\rightskip=.87in\parfillskip=0pt\leftskip=-1in plus1fil%
\baselineskip=10pt\sidenotefont #1}}}}

When we are working at a difficult task and strive after a good thing, we are
fighting a righteous battle, the direct reward of which is that we are kept from
\sidenote{This note has been brought to you by the letter pi and
the number e. Please make a note of it. The reason for its existence is the
great lengths I went to do things.}much evil. As we advance in life it becomes more and more difficult, but in
fighting the difficulties the inmost strength of the heart is developed. Indeed
life is a have to defend and protect ourselves, and with a cheerful and brace
spirit we must battle; we plan and calculate in order to make progress.

\bye

-
Would an \insert-based solution work for you? i.e., output routine. –  morbusg Dec 8 '12 at 10:04
@morbusg, I was looking at \insert yesterday. I was concerned that I wouldn't be able to easily get the alignment right; sidenotes must be beside the text the're commenting on. Do you know if this can be done? –  EfForEffort Dec 8 '12 at 12:58
Ah, yes, it gets very difficult then; I know because I have tried (and failed). I think \pagetotal could somehow be used for those calculations, but the problems (again: at least for me) arise when there are more than one sidenote on a page. –  morbusg Dec 8 '12 at 20:36

If lines don't have unusual depth, this could be a starting point:

\input eplain
\leftmargin=2in

\font\mainfont=pxr at 10pt
\font\sidenotefont=pxi at 8.5pt

\long\def\sidenote#1{%
\parindent=0pt
\hsize=1.7in
\parfillskip=0pt
\leftskip=0pt plus1fil
\baselineskip=10pt\sidenotefont\vglue-\ht\strutbox #1}}\kern1em}}}

\mainfont

When we are working at a difficult task and strive after a good thing, we
are fighting a righteous battle, the direct reward of which is that we are
kept from \sidenote{This note has been brought to you by the letter pi and
the number e. Please make a note of it. The reason for its existence is the
great lengths I went to do things.}much evil. As we advance in life it becomes
more and more difficult, but in fighting the difficulties the inmost strength
of the heart is developed. Indeed life is a have to defend and protect ourselves,
and with a cheerful and brace spirit we must battle; we plan and calculate in
order to make progress.

\bye

The key is \vadjust that adds the material to the enclosing vertical list.

-
This is great, thanks! Now to learn what strutbox is and how I can adjust it for different font sizes.... –  EfForEffort Dec 8 '12 at 14:37
@D.Bueno: You might want to look at exercise 14.28 of the TeXbook (and it's answer in the Appendix A). The solution is very similar to egreg's proposal and Knuth's explanations are always nice, so it's a good read. –  Mafra Dec 8 '12 at 23:08
@D.Bueno: For that, you could use LaTeX's definition of the strutbox; it uses .7\baselineskip for height, and .3\baselineskip for depth, instead of “hard-coded” values as in plain. Also you might want to set \parskip=0pt to avoid possible “drifting” of baselines between the main column content and the sidenote column (plain's default is 0pt plus 1pt). –  morbusg Dec 11 '12 at 6:26
@morbusg: Thanks! These are good suggestions. –  EfForEffort Dec 11 '12 at 19:59

After reading the appropriate exercise in the TeXBook (14.28), here is my \strut-based solution. Thanks to @Mafra for suggesting the exercise.

\def\strutdepth{\dp\strutbox}
\def\simplesidenote#1{%
\def\lsidenote#1{\vtop to \strutdepth{%
\baselineskip\strutdepth%
\vss\llap{\vtop to 0pt{%
\leftskip=0pt plus1fill \rightskip=8pt%
\parfillskip=0pt \parindent=0pt%
\hsize=1.7in%
\eightpoint\it #1}}\null}}

Note that \eightpoint is just a macro to adjust the font size.

-
Heh, I think I suggested exercise 14.28 :-) –  Mafra Dec 12 '12 at 23:22
lol. Indeed, you did. Sorry about that! I've corrected it in the answer. –  EfForEffort Dec 12 '12 at 23:30


text:
#### Vol. 8, No. 1, 2014

 Download this article For screen For printing
 Recent Issues
 The Journal Cover Editorial Board Editors' Addresses Editors' Interests About the Journal Scientific Advantages Submission Guidelines Submission Form Subscriptions Editorial Login Contacts Author Index To Appear ISSN: 1944-7833 (e-only) ISSN: 1937-0652 (print)
The Tannakian formalism and the Langlands conjectures

### David Kazhdan, Michael Larsen and Yakov Varshavsky

Vol. 8 (2014), No. 1, 243–256
##### Abstract

Let $H$ be a connected reductive group over an algebraically closed field of characteristic zero, and let $\Gamma$ be an abstract group. In this note, we show that every homomorphism of Grothendieck semirings $\varphi :{K}_{0}^{+}\left[H\right]\to {K}_{0}^{+}\left[\Gamma \right]$, which maps irreducible representations to irreducible, comes from a group homomorphism $\rho :\Gamma \to H\left(K\right)$. We also connect this result with the Langlands conjectures.

##### Keywords
Tannaka duality, Langlands conjectures
##### Mathematical Subject Classification 2010
Primary: 11R39
Secondary: 11F80, 17B10, 18D10
##### Milestones
Received: 2 September 2012
Revised: 20 August 2013
Accepted: 19 September 2013
Published: 20 April 2014
##### Authors
 David Kazhdan Einstein Institute of Mathematics Hebrew University Givat Ram 91904 Jerusalem Israel Michael Larsen Department of Mathematics Indiana University Rawles Hall Bloomington, IN 47405-5701 United States Yakov Varshavsky Einstein Institute of Mathematics Hebrew University Givat Ram 91904 Jerusalem Israel


text:
The Universe of Discourse

Sat, 05 Oct 2013

PIGS in SPACE!!!
Today I was at the Pittsburgh Perl Workshop and I gave a talk because that's my favorite part of a workshop, getting to give a talk.

The talk is about Moonpig, the billing system that Rik Signes and I wrote in Perl. Actually it's about Moonpig as little as possible because I didn't think the audience would be interested in the details of the billing system. (They are very interesting, and someone who is actually interested in a billing system, or in a case study of a medium-sized software system, would enjoy a three-hour talk about the financial architecture of Moonpig. But I wasn't sure that person would be at the workshop.) Instead the talk is mostly about the interesting technical underpinnings of Moonpig. Here's the description:

Moonpig is an innovative billing and accounting system that Rik Signes and I worked on between 2010 and 2012, totaling about twenty thousand lines of Perl. It was a success, and that is because Rik and I made a number of very smart decisions along the way, many of which weren't obviously smart at the time.

You don't want to hear about the billing and accounting end of Moonpig, so I will discuss that as little as possible, to establish a context for the clever technical designs we made. The rest of the talk is organized as a series of horrible problems and how we avoided, parried, or mitigated them:

• Times and time zones suck
• Floating-point arithmetic sucks
• It sucks to fix your mangled data after an automated process fails
• Testing a yearlong sequence of events sucks
• It sucks to have your automated test accidentally send a bunch of bogus invoices to the customers
• Rounding errors suck
• Relational databases usually suck
• Modeling objects in the RDB really really sucks
• Perl's garbage collection sucks
• OO inheritance sucks
Moonpig, however, does not suck.

Some of the things I'll talk about will include the design of our web API server and how it played an integral role in the system, our testing strategies, and our idiotically simple (but not simply idiotic) persistent storage solution. An extended digression on our pervasive use of Moose roles will be deferred to the lightning talks session on Sunday.

Much of the design is reusable, and is encapsulated in modules that have been released to CPAN or that are available on GitHub, or both.

I went to sleep too late the night before, slept badly, and woke up at 6:30 and couldn't go back to sleep. I spent an hour wandering around Oakdale looking for a place that served breakfast before 8 AM, and didn't find one. Then I was in a terrible mood. But for this talk, that was just right. I snarled and cursed at all the horrible problems and by the end of the talk I felt pretty good.

Slides are available here. Video may be forthcoming.

Share and enjoy.

[ Addendum 20131217: I wrote up the talk contents in detail and posted them here. ]

[ Addendum 20131217: The video of the talk is available on YouTube, and has been since October, but I forgot to mention it. Unfortunately, the sound quality is poor; I tend to wander around a lot when I talk, and that confuses the microphone. Many thanks to Dan Wright for the video and especially for putting on PPW. ]

Offensive stuff in classes
Many years ago I gave a short talk about the internals of the Perl regex engine. One continuing theme through the class was vomit. I handed out custom-made barf bags before the talk. There's one point in the engine where if the internal compiled format is invalid, the engine dies with the helpful error message Internal urp. I had a slide about that bit of code, illustrated with a picture of two people vomiting. (At least in 2001, that was harder to find on the Internet than you would have thought.)

Shortly before I gave the talk, I got cold feet. I asked a bunch of conference attendees if they thought I should take the picture out. All but one set yes, I should. The one dissenter was Gurusamy Sarathy, at that time the Pumpking, the Perl release manager. "Take it out?" he said incredulously. "Are you crazy?"

I left it in. I don't think it was an error.

I wonder, though, if I would have gotten away with it now. People are a lot more on edge about unprofessional-seeming and potentially offensive conference presenations. They also seem to be more common. When I hear about conference presentations with pictures of porn stars, or "humorous" presentations about projects to "titstare", I wonder where these morlocks came from; I've never seen anything like that at a conference and I would have been surprised and offended to see it ten years ago.

On the other hand, maybe I'm a hypocrite. One year at OSCON I did present a slide that used a still from a porno:

# The Biggest Mistake You Can Make

• I'm going to start big

• What is the biggest mistake you can make when making a conference presentation?

• Forgetting to go to the bathroom beforehand

The porno in question was a full bladder fetish video.

Were there people offended by this then? I don't know; I hope not. Nobody complained, but that doesn't prove anything. I can certainly imagine people finding this extremely offensive today. But I still think it's hilarious. Would I do it again? I don't know, but probably not.

From the same talk, was the illustration I used on a slide about "bullshit". I don't know if that's more or less offensive. On the one hand, the picture is a lot more visually disgusting. There is no mistaking what is happening; in the omoroshi picture you might not realize what is happening. (At least one participant commented on my "clever visual pun", explaining "well, of course she's just bowing". "Huh," I said, "that would have been clever.") On the other hand, if you do take the omoroshi picture at face value, it could be understood as a powerfully misogynist image of the humiliation of a woman by men. On the other other hand, I could have used an identical image that wasn't from an actual porno and it could have been understood the same way. But then I would have had an ironclad defense: No, it's just a picture of a woman standing in a room. But it could be the same picture. So I'm not sure. Regardless, I will apologize to anyone who wants an apology; although I considered the image carefully at the time, I hoped that it would not offend anyone, but nevertheless recognized the possibility and chose to risk offense anyway.

Another conceivably offensive slide image I used once is below:

These are two-headed sperm; it's possible that this is an image of human semen. I can imagine someone complaining bitterly about this. Meng Wong told me once that if anyone ever did, I should soothe them by claiming that it was gorilla semen; then it would be all right. That does seem like it might work, although I'm not sure why it should.

But none of that is really what I wanted to discuss; I was only thinking about it today because because I was writing a new talk in which I discussed the following piece of code:

        #!/usr/bin/perl

use DateTime;
use DateTime::Duration;

my $then = DateTime->new( month => 4, day => 2, year => 1969, hour => 0, minute => 0, second => 0); my$now = DateTime->now();
my $elapsed =$now - $then; print$elapsed->in_units('seconds'), "\n";

This looks like it ought to calculate and print out the number of seconds between midnight 1969-04-02 and the current time. Instead, because of a series of bad decisions by the designers of DateTime::Duration, it calculates that value as a number of months, days, minutes, and nanoseconds, and then prints out the leftover nanoseconds divided by one billion; typically the answer is a number on the order of 30. In this case it's obvious that something peculiar is going on, but in general you might not realize that this code is not doing what you (or anyone else) wants; you might test it on two times with only a few seconds difference, see that it produces the correct answer, and fail to realize that it breaks completely for any interval that crosses a minute boundary.

My first idea was to claim that this code prints out FUCK YOUR MOTHER, suggesting that although the object knows how much time has elapsed, it refuses to tell you, because it is a spiteful and foulmouthed object. But people have pointed out that sometimes kids attend these conference talks and don't want to hear FUCK YOUR MOTHER (or their parents don't want them to hear it) and anyway, FUCK YOUR MOTHER is sexist and I would prefer to offend everyone without resorting to racism or sexism. I don't mind offending people in general, but it is unfair to do it from a position of power, and to direct it against people with less power than I have.

I spent quite a while thinking about this. Before I committed the slides I struck out FUCK YOUR MOTHER and replaced it with UP YOUR NOSE, EPSTEIN (a catchphrase from the 1970s sitcom Welcome Back, Kotter) and then SIT ON IT, CUNNINGHAM (a catchphrase from the 1970s sitcom Happy Days). Both of these, having aired on prime-time network television in the 1970s, can be presumed to be sufficiently innofensive for a conference talk in 2013.

But they didn't seem to be quite what I wanted, and for a while I considered CHINGA A TU MADRE (Mexican slang for FUCK YOUR MOTHER) which has advantages and disadvantages. I think it's funnier if the DateTime::Duration module turns out to be a Spanish-speaking punk. And some of the people who might have been offended won't understand the Spanish. On the other hand, some of the people who might have thought the joke was funny won't understand the Spanish. And the sexism is the same as in English.

A friend suggested that I just go with GO FUCK YOURSELF, but I decided that I could probably find something funnier if I went in a scatalogical direction. Scatology is safer than sex, because sex is a forbidden adult subject, but everybody poops. Four-year-olds can, and do, make poop jokes and find them hilarious.

The first thing I thought of was I'M SHITTING IN YOUR MOUTH which I do think is funny, but perhaps too physically offensive; it's a really unpleasant image.

But Google search for the Mexican version of FUCK YOUR MOTHER had led me to Wikipedia's page on Spanish profanity, which is a gold mine. Reading through it, I knew, as soon as I saw ME CAGO EN LA LECHE, that I'd found the right phrase.

It means "I shit in your milk!"

That seems to me to lie at the perfect intersection of what I want. It displays the program's obnoxiously unhelpful attitude, it should be no more offensive to anyone than to anyone else, and it conjures u pa mental image whose unpleasantness is strongly tempered by its surreality.

[ Addendum: I should mention that this accusation against DateTime::Duration is false, and it would be completely unfair, except that the desired answer here is so perfectly at the center of DateTime::Duration's purported area of expertise, the answer actually given by DateTime::Duration is so utterly and manifestly useless, and there is no way to get it to produce that answer you actually wanted. If you ask me how many seconds have elapsed since April 2, 1969, and I say "43", can't you reasonably claim that I might as well have said "¡Me cago en la leche!"


text:
What is the solution for $2y''+3y'+5y=7x^2+9x+11$ thanks.

Note by Ce Die
4 months ago

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$2y'' + 3y' + 5y = 7x^2 + 9x + 11$

The homogeneous equation is:

$2y'' + 3y' + 5y = 0$

The general solution to the homogeneous equation has the form:

$y = c \, e^{\alpha x}$

Plugging this back into the H equation yields a complex conjugate solution pair:

$2 \alpha^2 \, y + 3 \alpha y + 5y = 0 \\ 2 \alpha^2 + 3 \alpha + 5 = 0 \\ \alpha_1 = \frac{-3 - \sqrt{31} i}{4} \\ \alpha_2 = \frac{-3 + \sqrt{31} i}{4}$

The solution to the H equation is:

$y_H = c_1 e^{\alpha_1 x} + c_2 e^{\alpha_2 x}$

Now for the particular solution associated with the the polynomial, it is evident that this must be a second order polynomial:

$y_P = A x^2 + B x + C$

Plugging into the differential equation:

$2(2A) + 3(2A x + B) + 5(A x^2 + B x + C) = 7x^2 + 9x + 11 \\ 5A x^2 + (6A + 5B)x + 4A + 3B + 5C = 7x^2 + 9x + 11 \\ 5A = 7 \implies A = \frac{7}{5} \\ 6A + 5B = 9 \implies B = \frac{3}{25} \\ 4A + 3B + 5C = 11 \implies C = \frac{126}{125}$

The particular solution is:

$y_P = \frac{7}{5} x^2 + \frac{3}{25} x + \frac{126}{125}$

The total solution is:

$y = y_H + y_P = c_1 e^{\alpha_1 x} + c_2 e^{\alpha_2 x} + \frac{7}{5} x^2 + \frac{3}{25} x + \frac{126}{125} \\ \alpha_1 = \frac{-3 - \sqrt{31} i}{4} \\ \alpha_2 = \frac{-3 + \sqrt{31} i}{4}$

To solve for constants $c_1$ and $c_2$, solve initial-value equations for the function and its derivative

End of first part

$2y'' + 3y' + 5y = 7x^3 + 11x^2 + 13x + 9$

The homogeneous solution is identical to the previous problem. The particular solution has the form:

$y_P = A x^3 + B x^2 + Cx + D$

Plugging into the differential equation:

$2(6 A x + 2B) + 3(3 A x^2 + 2 B x + C) + 5(A x^3 + B x^2 + Cx + D) = 7x^3 + 11x^2 + 13x + 9 \\ 5A = 7 \implies A = \frac{7}{5} \\ 9A + 5B = 11 \implies B = -\frac{8}{25} \\ 12A + 6B + 5C = 13 \implies C = -\frac{47}{125} \\ 4B + 3C + 5D = 9 \implies D = \frac{1426}{625}$

The total solution is therefore:

$y = y_H + y_P = c_1 e^{\alpha_1 x} + c_2 e^{\alpha_2 x} + \frac{7}{5} x^3 -\frac{8}{25} x^2 -\frac{47}{125} x + \frac{1426}{625} \\ \alpha_1 = \frac{-3 - \sqrt{31} i}{4} \\ \alpha_2 = \frac{-3 + \sqrt{31} i}{4}$

- 4 months ago

Thank you and also how to solve this $2y’’+3y’+5y=7x^3+11x^2+13x+9$ ? Thanks

- 3 months, 3 weeks ago

The process is the same as for the other problem. The only difference is that the particular solution is a third order polynomial instead of a second order polynomial.

- 3 months, 3 weeks ago

Please I give me a solution for this

- 3 months, 3 weeks ago

Ok, I will post it a bit later

- 3 months, 3 weeks ago

thank you i hope to see it

- 3 months, 3 weeks ago

- 3 months, 3 weeks ago

thank you so much you enlight me! :)

- 3 months, 2 weeks ago

You're welcome

- 3 months, 2 weeks ago

- 3 months ago

Since the general procedure is the same as for the other two, I'll leave this one to you

- 3 months ago

Is this correct if 4th degree is $y'' = 8Ax^2+6Bx+2C$, $y'=4Ax^3+3Bx^2+2Cx+D$ and $y = Ax^4+Bx^3+Cx^2+Dx+E$? I hope its right. Thanks

- 3 months ago

Correct, except that the first term in your $y''$ equation should have a coefficient of $12$ instead of $8$

- 3 months ago

$y''=12Ax^2+6Bx+2C$ is this correct?

- 3 months ago

Yes, it is

- 3 months ago

for $x^5$ the $y_{p}$ is $y''=15Ax^3+12Bx^2+6Cx+2D$? This this correct Sir?

- 3 months ago

Look again at the coefficient on your $x^3$ term

- 3 months ago

I think its 18?

- 3 months ago


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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeJan 6th 2011

have expanded the single sentence at differential geometry to something like a paragraph, indicating how differential geometry is the “higher geometry modeled on the pre-geometry $\mathcal{G} = CartSp$

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeAug 30th 2011

HELP:

at differential geometry I want under Related concepts a square table as in the introduction of Sharpe’s book.

I managed to produce parts of it (have a look) but I have not been able to insert the obvious vertical arrows without destroying the outcome. I don’t really know that table syntax. Can anyone help?

• CommentRowNumber3.
• CommentAuthorDavid_Corfield
• CommentTimeAug 31st 2011

I’m afraid I don’t know how to fix the table, but I’d like to ask a question about its appearance there. Are we to think of $CartSp$ as the pregeometry corresponding to Euclidean and then Riemannian geometry, and for there to be other pregeometries for the Kleinian homogeneous spaces and associated Cartan geometries? Or do the latter arise from imposing further conditions on the geometry emerging from $CartSp$?

• CommentRowNumber4.
• CommentAuthorzskoda
• CommentTimeAug 31st 2011
• (edited Aug 31st 2011)

Scary entry so far. Do you know a differential geometer who can read it ?

Technically

the geometry modeled on the pre-geometry

is puzzling, as Lurie explains that there are many geometries corresponding to the same pre-geometry.

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeAug 31st 2011

I’d like to ask a question about its appearance there. Are we to think of $CartSp$ as the pregeometry corresponding to Euclidean and then Riemannian geometry, and for there to be other pregeometries for the Kleinian homogeneous spaces and associated Cartan geometries?

I have to admit that I don’t fully buy into the idea that “Cartan geometry generalizes Riemannian geometry”. I rather think that the useful point of view is: “Cartan geometry is a special case of principal bundles over Riemannian spaces”.

Accordingly, I’d tend to answer your question with: “No.”

But I am open for discussion. Maybe I am missing something here.

• CommentRowNumber6.
• CommentAuthorzskoda
• CommentTimeAug 31st 2011

I rather think that the useful point of view is: “Cartan geometry is a special case of principal bundles over Riemannian spaces”.

This is possible/likely, though I am not fully sure, if the unrolling the frame in Cartan’s moving frame business corresponds exactly to a special kind of charting of a principal bundle, or there are adiditonal global phenomena implied there. Who is an expert ?

• CommentRowNumber7.
• CommentAuthorzskoda
• CommentTimeAug 31st 2011

What is the difference intended between the entry differential geometry and the entry generalized smooth space ? Both are now about a type of a space, but differential geometry is a subject and not a type of a space. In particular, differentiable manifolds can be studied in differential geometry, differential topology and analysis on manifolds. As my teacher in symplectic geometry Y-G. Oh used to say, the analysis on manifolds is mainly concerned with the maps out of a manifold (functions on a manifold), while differential geometry mainly with geometric objects in a manifold, that is maps into a manifold (e.g. curves).

• CommentRowNumber8.
• CommentAuthorUrs
• CommentTimeAug 31st 2011

This is possible/likely, though I am not fully sure, if the unrolling the frame in Cartan’s moving frame business corresponds exactly to a special kind of charting of a principal bundle, or there are adiditonal global phenomena implied there. Who is an expert ?

Not sure what you want an expert to say here? The definition of Cartan connection is simple enough. It’s just a principal connection with a simple extra constraint on it.

The question seems to be more philosophical. Of a principal bundle we usually think of as being extra structure over the base manifold. Should we change that point of view only because we impose an extra constraint?

• CommentRowNumber9.
• CommentAuthorUrs
• CommentTimeAug 31st 2011
• (edited Aug 31st 2011)

What is the difference intended between the entry differential geometry and the entry generalized smooth space ?

These two entries weren’t created in parallel, I think. But to my mind, the entry “differential geometry” should say what differential geometry is about. In a useful generalized sense, this involves also generalized smooth spaces, and so there should be an entry focused on these. No?

• CommentRowNumber10.
• CommentAuthorUrs
• CommentTimeAug 31st 2011
• (edited Aug 31st 2011)

Concerning #8:

In the very interesting foreword to Sharpe’s book (listed at Cartan geometry) Chern recalls that Cartan geometry was mainly forgotten by the community after Ehresmann produced the definition of principal connections. Chern lists two reasons:

1. Ehresmann was easier to read “at that time”, he says;

2. Ehresmann’s definition was more general anyway.

Chern then does come back to Cartan’s original point of view and emphasizes how it is useful. But I think there is a small bit of truth here to the saying: “A concept which has been forgotten – and rightly so.” :-)

Not to get me wrong, I think that the notion of Cartan connections (and of solder forms and vielbeins and so on) is important. But I think it is to be regarded a special case of Ehresmann theory / principal connections. Not as a stand-alone theory.

• CommentRowNumber11.
• CommentAuthorDavid_Corfield
• CommentTimeAug 31st 2011

I wonder if there’s any abstract nonsense way of characterising Cartan theory as a special case of Ehresmann theory.

Are there cases where you sympathise with the point of view that something may be lost when a theory is subsumed within another. Probability theorists complained of their theory being seen as a mere chapter of measure theory.

• CommentRowNumber12.
• CommentAuthorUrs
• CommentTimeAug 31st 2011

I wonder if there’s any abstract nonsense way of characterising Cartan theory as a special case of Ehresmann theory.

I have thought about this a bit for the case of the Poincaré group. Because there it amounts to a famous open problem in quantum gravity: in the first order formulation of gravity (mandatory if spinors are to be included) the field content is an $Iso(d,1)$-connection. The metric itself is encoded in the $\mathbb{R}^{d+1}$-summand of the connection: the vielbein. Or rather, in general that vielbein encodes a symmetric tensor, not, however, necessarily a non-degenerate one: a Riemannian metric.

Now, the problem is this: classically we would tend to demand that indeed we have a non-degenerate 2-tensor, hence a metric. This demand is equivalent to requiring that the given $Iso(d,1)$-connection is in fact an $(O(d,1) \hookrightarrow Iso(d,1))$-Cartan connection.

But is this a good constraint to impose when looking at the quantum theory? Should not the path integral maybe be over all vielbein fields? Who says that fundamentally there must be a non-degenerate metric tensor?

Who actually says that the correct model of gravity by $Iso(d,1)$-geometry needs to have a non-degenerate vielbein? It’s not clear to me that it has to. The theory works much more smoothly without an extra constraints. Extra constraints are generally a source of pain in quantum theory.

• CommentRowNumber13.
• CommentAuthorUrs
• CommentTimeAug 31st 2011

I have decided to slightly change that little table at differential geometry. It now looks roughly like this:

local model global geometry
Euclidean geometry Riemannian geometry
Klein geometry Cartan geometry

I’ll copy this over the the other three entries, too.

• CommentRowNumber14.
• CommentAuthorzskoda
• CommentTimeAug 31st 2011

Not sure what you want an expert to say here? The definition of Cartan connection is simple enough.

The Cartan connection is just a basic input datum in a more complex method of studying geometry, the Cartan’s moving frame method in which one kind of scrolls out the charts to get geometry. I think this method is more than just charting a principal bundle.

• CommentRowNumber15.
• CommentAuthorUrs
• CommentTimeAug 31st 2011

I thought the definition of Cartan connection includes the idea of “moving frames”. Is this not just the special case of a $O(n) \hookrightarrow Iso(\mathbb{R}^n)$-Cartan connection? The moving frame is nothing but the vielbein that is part of this. No?

• CommentRowNumber16.
• CommentAuthorzskoda
• CommentTimeAug 31st 2011

In a useful generalized sense, this involves also generalized smooth spaces, and so there should be an entry focused on these. No?

I think not. The entries for differential manifolds, orbifolds, polyfolds, differential stacks, smooth spaces, generalized smooth spaces etc. are object (rather than subject) pages which should exist on their own. Differential geometry should be different from those in the same sense as the entry topology is different from topological space or topological structure. So I would plan the chapters like classical differential geometry of curves, surfaces, the difference between internal and external geometry, then the geometry of manifolds, and manifolds with structure, and of smooth bundles with structure, then links to generalizations like generalized smooth space, differentiable stacks etc. All subfields like symplectic geometry, Riemannian geometry, Finsler geometry, contact geometry, generalized complex geometry could be inside.

Next as I said, the differential geometry is not decided by being ABOUT manifolds but rather about specific geometric aspect fo manifolds, differing from analysius on manifolds and differing from differential topology. Wikipedia is more along this line, though too short: differential geometry.

• CommentRowNumber17.
• CommentAuthorzskoda
• CommentTimeAug 31st 2011
• (edited Aug 31st 2011)

The Cartan’s METHOD Of moving frames is NOT the same as moving frame. (As car race is not the same as a car.) The essence is in certain kind of charting by rolling out the frame from a bundle to the manifold, and I am not sure what nontrivial global aspects come out of that rolling.

• CommentRowNumber18.
• CommentAuthorUrs
• CommentTimeAug 31st 2011
• (edited Aug 31st 2011)

David,

one more comment re #12:

here is an important case to keep in mind:

quantum gravity is well understood in dimension 3. There the Einstein-Hilbert action functional is equivalent to the $Iso(2,1)$-Chern-Simons action functional (for a certain choice invariant polynomial). So one can quantize this. But not as a theory of Cartan connections, but as a theory of principal $Iso(2,1)$-connections. The metric can become degenerate. The Cartan-geometry condition can be violated.

I’ll try to write more about this at Chern-Simons gravity later.

• CommentRowNumber19.
• CommentAuthorDavid_Corfield
• CommentTimeAug 31st 2011

re #18 That’s a telling piece of evidence.

• CommentRowNumber20.
• CommentAuthorUrs
• CommentTimeAug 31st 2011
• (edited Aug 31st 2011)

Yes. On page 7 of Witten’s old article “2 + 1 DIMENSIONAL GRAVITY AS AN EXACTLY SOLUBLE SYSTEM” (sorry for the caps) you can find the argument sketched why 3d quantum gravity cannot make sense if one imposes invertibility of the vielbein.

This means: 3d quantum gravity does not make sense if we regard the gravitational field as an $Iso(2,1)$- Cartan geometry/Cartan connection. We have to take it to be more generally to be a principal connection.

Since the 4d case will of course not be simpler than the 3d case, it seems unlikely that the sitation will change there.

• CommentRowNumber21.
• CommentAuthorzskoda
• CommentTimeAug 31st 2011

Urs, I asked above why do you say “the” geometry based on pregeometry of Cart if Lujrie says that more than one geometry (say n-cat truncations) can correspond to the same pregeometry.

• CommentRowNumber22.
• CommentAuthorUrs
• CommentTimeAug 31st 2011
• (edited Aug 31st 2011)

Sorry, I had missed that.

why do you say “the” geometry based on pregeometry of Cart if Lujrie says that more than one geometry (say n-cat truncations) can correspond to the same pregeometry.

Because of prop 3.4.10 in Structured Spaces:

the $n$-truncated geometric envelopes $\mathcal{G}_n$ of a pre-geometry $\mathcal{T}$ are such that the $\mathcal{G}_n$-structured $\infty$-toposes are precisely those $\mathcal{G}_\infty$-structured $\infty$-toposes (where $\mathcal{G}_\infty := \mathcal{G}$ is the full geometric envelope) whose structure sheaf happens to be $n$-truncated.

So this gives a way to characterize certain full sub-$\infty$-categories of all $\mathcal{G}$-structured $\infty$-toposes. No new information, no new “geometry” in the non-technical sense of the word. Just a sub-case of the full geometry induced by $\mathcal{T}$.

• CommentRowNumber23.
• CommentAuthorzskoda
• CommentTimeAug 31st 2011

Good, thanks.

• CommentRowNumber24.
• CommentAuthorzskoda
• CommentTimeSep 1st 2011

I have revised differential geometry inserting a title which more traditionally describes its scope and then putting the $n$POV into a separate section (which could eventually maybe be partly moved into generalized smooth space with a link and shorter description).

• CommentRowNumber25.
• CommentAuthorUrs
• CommentTimeMar 24th 2021


text:
# Chapter 9 - Sequences and Series - Get Ready! - Page 561: 5

f(1)=3$\frac{1}{3}$ f(2)=7$\frac{1}{3}$ f(3)=11$\frac{1}{3}$ f(4)=15$\frac{1}{3}$

#### Work Step by Step

f(x)=4x-$\frac{2}{3}$ f(1)=4(1)-$\frac{2}{3}$ =3$\frac{1}{3}$ f(2)=4(2)-$\frac{2}{3}$ =7$\frac{1}{3}$ f(3)=4(3)-$\frac{2}{3}$ =11$\frac{1}{3}$ f(4)=4(4)-$\frac{2}{3}$ =15$\frac{1}{3}$ *You just plug the new number into each of the x's in the original function and then simplify to get your answer.

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.


text:
# Bayes' Theorem/General Result

## Theorem

Let $\Pr$ be a probability measure on a probability space $\struct {\Omega, \Sigma, \Pr}$.

Let $\set {B_1, B_2, \ldots}$ be a partition of the event space $\Sigma$.

Then, for any $B_i$ in the partition:

$\condprob {B_i} A = \dfrac {\condprob A {B_i} \map \Pr {B_i} } {\map \Pr A} = \dfrac {\condprob A {B_i} \map \Pr {B_i} } {\sum_j \condprob A {B_j} \map \Pr {B_j} }$

where $\ds \sum_j$ denotes the sum over $j$.

## Proof

Follows directly from the Total Probability Theorem:

$\ds \map \Pr A = \sum_i \condprob A {B_i} \map \Pr {B_i}$

and Bayes' Theorem:

$\condprob {B_i} A = \dfrac {\condprob A {B_i} \map \Pr {B_i} } {\map \Pr A}$

$\blacksquare$

## Source of Name

This entry was named for Thomas Bayes.


text:
# Tag Info

40

What you are calling a focal point is the end of the umbra, the point at which the umbra changes to the antumbra. In a total solar eclipse, that point is below rather than above the surface of the Earth. An annular eclipse occurs when that point is above the surface of the Earth. The Moon appears to be smaller than the Sun, leaving an unbroken ring of the ...

15

As the Moon's eccentric orbit around the Earth brings it nearer and farther, current solar eclipses can be total or annular. A few in between are hybrid eclipses: total along the midday part of the path, and annular near the sunrise or sunset end. This happens mostly due to the Earth's surface curvature and partly due to the Moon's orbital eccentricity. ...

11

"Harvest moon" just means the full moon at around the time of the equinox on September 21. The full moon is low in the sky during summer (June and July in the Northern Hemisphere) and high during the winter. So September is a month when the full moon is much higher in the sky than the moon in August. In the past this allowed harvests to continue after dark....

8

Almost certainly Jupiter, if the following are true: The title of the image: Inked20190904_194204.LI.jpg gives the accurate date and time of September 4 2019, 19:42:04. The orientation of the Moon in the image is such that the illuminated portion is on the right side of the image. The circled object is a planet or star, and brighter than any other nearby ...

6

Because the Moon phase changes relatively slowly (especially at the Full Moon), a similar Moon phase can be seen from all places around the world. For example, the Full Moon in September 2019 occurs at 4:33 UT. If the Moon is visible from your location at that time, then you can see the Moon when it is exactly "Full". If Moon is not visible until 5 hours ...

6

I think they did make a mistake. The moon waxes after a new moon and wanes before a new moon. To wax means to grow bigger or stronger (and is cognate with "waist"!). To wane means to become smaller or weaker (cognate with "waste" and "vaccum"). The moon that sets shortly after sunset in the evening is a waxing moon. The moon that rises shortly before ...

6

There are a couple of atmospheric phenomena that can create rings around the moon. Corona are caused by water droplets diffracting the moon's light, they are fairly small, and close to the moon. They are coloured but not as brightly as a rainbow. Our eyes are also not so good at seeing colours in dim light. The 22-degree halo is a larger ring, as big as ...

5

Millimeter and sub-mm observations (110-300 GHz = 2.7-1.0 mm) are sensitive to the thermal emission and provide a brightness temperature of airless bodies like the Moon or asteroids. The radio emission is from just below the surface/skin of the body (down to $\sim10\lambda$ so ~1 cm in this case) and so is also sensitive to the thermal inertia and ...

4

Is it possible that the shadow of the moon is a single dot during solar eclipse? No, because the Moon is not round. Yes. I have to modify my answer after reading @IlmariKaronen's comment(s) below. Mathematically the umbra could collapse to a line segment, arc, or other short 1D shape, or to two or more points, but these are not likely for a randomly ...

3

Is seeing the apollo moon Landers via earth telescope that hard? Yes, it still is. If the resolution of an interferometric telescope is really 0.001 arcseconds, that translates to 4.85E-09 radians. At a perigee of 362,600 km that's 1.8 meters. The top part of the Lunar Modules is about 4.2 meters wide which is a little more than double the resolution, so ...

3

There are no other known natural satellites orbiting the earth. Most of the objects orbiting the earth are man-made and largest object other than the Moon orbiting the earth is the International Space Station which orbits the earth at a height of about 400km from the surface and is about 100m wide. It may be odd to you that there are no other natural ...

2

In the absence of any detailed information, such as @Uhoh asks for, this is just a guess, but here goes. I'm assuming you live at mid northern latitude. The moon rises between south east and north east. If we consider full moons, then during the summer, the full moon will rise in the south east. As we move into the autumn and winter, the full moon ...

2

Let's double-check @barrycarter's conclusion: Ignoring the rest of the solar system and even the Moon's non-negligible mass, the velocity of the Moon's motion around the Earth is on average roughly 1 km/sec per $v=\sqrt{GM_E/a_M}$ (the vis-viva equation). The Earth is about 12760 km in diameter, so the Moon traverses the Earth's projection in about 3.5 ...

2

First of all any body revolving around a planet is known as its satellite, not as an "asteroid". And at this time Earth only has one natural satellite which is the Moon (but there are few theories suggesting that Earth once had two moons but they merged into each other due to gravitational pull between them and formed a single moon). Asteroids are the ...

1

Temporary's have come and gone. https://en.wikipedia.org/wiki/Claimed_moons_of_Earth#Temporary_satellites

Only top voted, non community-wiki answers of a minimum length are eligible


text:
Ashburn, VA

Ashburn, VA

# How much does Mathnasium cost?

$250 –$400average cost per month
$395 –$1,000+summer program cost (8 to unlimited sessions)

Get free estimates for your project or view our cost guide below:

## $395 –$1,000+summer program cost (8 to unlimited sessions)

Get free estimates for your project or view our cost guide below:
Written by
Tom Grupa

## Mathnasium cost

Mathnasium costs $250 to$400 per month or $25 to$75 per session, depending on the location and whether you book a 3, 6, or 12-month plan. Mathnasium summer programs cost $395 for eight sessions or$1,000+ for unlimited sessions. Some Mathnasium locations charge $100 to$150 for evaluation and enrollment fees.

Most locations offer both online and in-person tutoring sessions.

Mathnasium cost
Service Average cost
Evaluation and enrollment fees $100 –$150
Cost per session $25 –$75
Cost per month $250 –$400
Summer programs (8 sessions to unlimited) $395 –$1,000

## Background on Mathnasium

Larry Martinek graduated college in 1974 and began teaching in schools in the Los Angeles area. To his horror, he realized that all he was doing was repair work with the students. As a teacher, he found it frustrating that he was working with kids who were well below grade level by the time they got to high school. In his role, his efforts were largely centered around getting students up to the grade level they should be at instead of taking them further than that grade level.

After the birth of his son Nic in 1980, Larry looked for ways to discuss math ideas with Nic that were beyond his years. Larry decided to change the language around teaching math, and his vision became the mission statement for Mathnasium: “Teach children math in a way that makes sense to them.”

Tragically, in May of 1999, Nic passed away after falling asleep at the wheel while driving home. As a living memorial to their son, Larry and his wife wanted to take all the creative ways Larry had conveyed math topics to Nic to a broader audience. Three years later, in 2002, education industry leaders Peter Markovitz and David Ullendorff made Larry and his curriculum the driving force of the Mathnasium Method.

Los Angeles schools began teaching directly from his material instead of their textbooks, and after seeing tremendous success in test scores, his method became an entire curriculum spanning pre-K to twelfth grade.

## Why do math tutoring?

The reasons why parents elect to seek out a math tutor for their children can vary widely. The most common thread in most parents’ hearts and minds is to either see improved grades for their kids who are not doing so well, or beyond average grades for those who are.

The reasons for the low grades are varied and often include one or more of the following conditions:

### Lack of understanding

Even with every possible factor in the students’ favor, they could still have issues with their learning style being different to the one-size-fits-all approach in most school systems, or their brains just may not be suited to getting math concepts outside a system in the way Mathnasium Method can.

### Future minded

In reality, not every student will excel in math. Some will not need it once they leave high school, and they will choose vocations that don’t require technical excellence in math on a daily basis. These students tune out what they feel is not mission critical in the pursuit of the dream.

### Poor curriculum

School politics, the educational budget, the geographic location, the school principal, and parental involvement are some of the factors which influence the math curriculum in a school. Each and every school district having a different mix of these factors, either at a city level or statewide, will determine the quality of the curriculum offered.

### Class to teacher ratio

Local and city budgetary constraints are at the root of the worsening child to teacher ratios over the last fifteen to twenty years. With schools on land of a limited size, or lacking the funds to build extra classrooms, this trend is likely to continue beyond where we are today.

Some would argue it’s already in breach of a number that is fair to all students. If we take the 80/20 rule and assume that 80 percent of students are not doing the best in any topic, then that 80 percent is competing for a portion of the teacher’s time and attention.

In a 2015 report from the National Center for Education Statistics, classroom sizes for elementary school classes ranged in size from 16.6 in Vermont to 27.4 in Utah, which yields a national average of 21.1. For high school classroom sizes in the US, the numbers ranged in size from 18.7 in Alaska to 34.5 in Nevada, which yields a national average of 26.8. Alaska, North Dakota, Wyoming, Vermont, and Maine were the only states with fewer than 20 students per class, and the remaining states had approx. 22. These numbers all shift over time.

In a similar report based on the rest of the world, the class size goes from 17.3 in Belgium to 35.5 in Singapore, yielding an average of 24.1 per teacher. While classroom size is only one contributing factor that potentially inhibits students’ chances of excelling in their studies, it is important to be aware of this factor.

### Working parents

For many kids today, their parents grew up in an era where only one parent worked outside the home. With many families either making a career choice to have the mother return to work after the birth of a child, or current economic pressures dictating the need for two incomes, it is often true that without that extra layer of support for homework, etc., that many kids have to work harder to get through school with good grades.

### Parents’ knowledge

Not all parents are math geniuses, and they may not have done well in math themselves. As a result, when kids come home with math homework, parents can find it difficult to be of much assistance to the child.

### Ineffective teacher

We all appreciate teachers in what is considered to be a fairly thankless profession. They turn up every day to teach the nation’s next generation—our kids. As with any profession, there will be those who are gifted and love what they do, and the majority of their kids will excel in their studies.

There are also those who are in the wrong job, are not great communicators, have little patience, or are simply jaded. Sadly, not every school can only have teachers from the first group.

### Child lacks confidence or self-esteem

Maybe your kid had a bad teacher, or any number of factors that resulted in a few failed exams, and it has taken a hit on your child’s self-confidence. Now he or she just sits in class without asking questions out of fear the question is too simple.

### Child lacks motivation

Closely tied to lacking confidence or self-esteem, this can also be because of mean kids at school or bullies. In cases like this, children often have higher priorities on their minds—like staying safe or away from certain kids rather than excelling in their studies.

### Change of schools

Often, as is common here in the US, families can find themselves moving house to pursue better career opportunities in a different city or state. The impact to the children can go either way—they will either go from a school where they are surrounded by the wrong crowd to one where there are very few kids to influence them negatively, or the reverse.

In conjunction with that, kids could go from a school where they had great teachers, including ones who really made efforts with them, to a school with a high percentage of burned-out teachers.

### Homework struggle

This is a biggie. At a certain point, we find our young ones morphing into teens. They are grasping for the right path as they sit at the intersection of independence and become their own person, with us slowly letting go. Right in that period of transition is the constant shout down the hall, “Did you do your homework yet?”

With any number of combined factors that could hold them back, and with all the possible distractions keeping them from opening the books, if they don’t put in the work, it’s likely they will have to catch up with the assistance of a tutor at some point.

## Why the Mathnasium Method?

Mathnasium is different from many other tutoring options available because they specialize in math. Using proprietary curriculum, materials, and a custom lesson plan designed from the results of ongoing assessments; and led by the instructor whose focus is to ensure the student masters the material; students will see improved grades, have a better understanding and confidence in math, and have a better retention of math concepts and formulas.

### Understanding

Mathnasium is based on the core belief that every child can succeed at math when taught in a way that makes sense; and when children understand the subject, they will excel beyond the grade averages of most students in public or private school settings.

The approach creates excitement in students about math, as they finally understand it and become passionate about it.

A strong understanding of the building blocks of math is the best way to help a student feel excited about learning while also setting them up for broad future success.” says Kathryn Kistler, the Mathnasium center director at the Westlake location in Austin. Texas.

According to Oak Park, IL, Mathnasium director, Jana Frank, with its focus on number sense, Mathnasium has provided her the perfect forum through which she can offer best practices in mathematical instruction and learning.

### Custom plans

The method begins with unique initial assessments to accurately discover where students are at individually, and a custom plan is created to get them to where they want to or need to be. The learning plan is designed to master the concepts as identified by taking their current level into consideration and beginning there.

The cycle begins with an assessment, a custom plan, teaching for understanding, and then they complete the circle again. The students and their families will see assessment scores increase, and as their progress grows, their confidence follows.

Children’s confidence about math is critical, and when they see that they can do the work, they gain that confidence. This will put an end to the struggle, both in school and at home, when it’s time for homework, because the intimidation and frustration will be gone.

### Caring environment

The Mathnasium Method teachers are referred to as “caring people who teach.” Because there is no assigned homework, the specially trained instructors are with students when the learning takes place, in an encouraging, caring environment.

John Van Horn of Mathnasium in Plainfield, IL, says “We enjoy helping students understand math. We love watching a student catch a concept and having "the light come on."

### Passion

After researching the Mathnasium Method, Leslie Chiu, the Mathnasium Center Director In Astoria Queens, NY, instantly knew this is how all children would come understand math and wanted to spread the word, so she started her own Mathnasium center. She says the method goes beyond teaching formulaic structure and testing by helping Mathnasium students master number facts, build computational skills, improve number sense, go beyond understanding to passion, and develop a love for math.

Pasadena Mathnasium’s asst. center director, Andrew Lukman, thought math was just another subject that he was good at throughout his childhood, but after taking his first calculus class, his eyes opened to how "cool" math can be.

## Case Studies

EyeCues Educational Systems has been conducting research on the Mathnasium Method since 2004. The results of this research is available on the Mathnasium website, and its findings, which show improvement in student performance on standards-based tests in 20 sessions or fewer, are grouped into three areas:

• Understanding – 82% of parents report an improvement in their child’s math skills and understanding.

• Attitude – 85% of parents report improved attitude toward math after attending Mathnasium.

• School progress – 88% of students saw improvement in their school grades.

## Why not offer online classes?

Unless measures have been taken to control the internet access in the house, the student may have access to a myriad of distractions similar to those in an unsupervised classroom. Additionally, the Mathnasium classroom setting typically has between one to five students per instructor, which gives students the focus to hear other students’ questions and the instructor’s answers, thus making the Mathnasium setting the ideal solution.

## In summary

Each student’s understanding and skills in math can grow beyond what the vast majority of traditional teaching methods accomplish in our schools today. For any students learning math, when the topic makes sense, they will see progress; and as that understanding grows, the student gets excited about the subject. Many will grow to love it, and some will even consider turning their love for it into a career.

Related articles


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The skewness of a probability distribution

In this post, we discuss how to calculate the moment coefficient of skewness and also discuss some issues surrounding the notion of skewness.

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Looking at graphs

One informal but useful way of checking the skewness of a distribution is to look at the density curve (or a histogram). Consider the following density functions.

Figure 1

Figure 2

The density curve in Figure 1 has a longer tail to the right than to the left. The example in Figure 1 is a distribution that is skewed to the right. It is also said to be positively skewed since its coefficient of skewness is positive. The density curve in Figure 2 has a longer tail to the left than to the right. The example in Figure 2 is a distribution that is skewed to the left. It is also said to be negatively skewed since the skewness coefficient is negative. If a density curve looks the same to the left and to the right (such as the bell curve for the normal distribution), then it is a symmetric distribution and the skewness coefficient is zero.

The distribution in Figure 1 is a right skewed distribution (the longer tail is on the right). It is a gamma distribution with mean 2 and median approximately 1.678347. The mode (the highest peak) is at x = 1. The distribution in Figure 2 is a left skewed distribution (the longer tail is on the left) with mean and median approximately 0.909 and 0.9213562, respectively. The mode is at 0.95.

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In the distribution for Figure 1, we can say that “mode < median < mean". In the distribution for Figure 2, we can say that "mean < median < mode". A common conception is that these simple rules characterize all skewed distribution, i.e., the mean is to the right of the median, which in turn is to the right of the mode in a right skewed distribution and that the mean is to the left of the median, which in turn is to the left of the mode in a left skewed distribution. Such rules are certainly easy to remember and are stated in some statistics textbooks. In the above two figures, this rule of thumb is certainly true. It turns out that this rule of thumb does not hold in many instances. The above two graphs are "textbook" demonstrations of skewness. They are gamma distributions and beta distributions and they behave well according to the usual notion of how skewed distributions should look like. In a later section of this post, we will discuss this issue in greater details. First we define the coefficient of skewness.

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Pearson moment coefficient of skewness

The measure of skewness defined here is called the Pearson moment coefficient of skewness. This measure provides information about the amount and direction of the departure from symmetry. Its value can be positive or negative, or even undefined. The higher the absolute value of the skewness measure, the more asymmetric the distribution. The skewness measure of symmetric distributions is, or near, zero.

To help put the definition of skewness in context, we first define raw moments and central moments of a random variable $X$. The $k$th raw moment of $X$ is $E(X^k)$, the expected value of the $k$th power of the random variable $X$. The first raw moment is the mean of the random variable and is usually denoted by $\mu$.

The $k$th central moment of a random variable $X$ is $E[(X-\mu)^k]$, the expected value of the $k$th power of the deviation of the variable from its mean. The moment $E[(X-\mu)^k]$ is usually denoted by $\mu_k$. The second central moment is usually called the variance and is denoted by $\sigma^2$. The square root of $\sigma^2$, $\sigma$, is the standard deviation.

The ratio of the standard deviation to the mean, $\displaystyle \frac{\sigma}{\mu}$, is called the coefficient of variation.

The ratio of the third central moment to the cube of the standard deviation is called Pearson’s moment coefficient of skewness (or the coefficient of skewness) and is denoted by $\gamma_1$.

$\displaystyle \gamma_1=\frac{E[ (X-\mu)^3 ]}{\sigma^3}=\frac{\mu_3}{\sigma^3} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (1)$

The skewness in (1) can be expanded to derive a version that can be calculated more easily:

\displaystyle \begin{aligned} \gamma_1&=\frac{E[ (X-\mu)^3 ]}{\sigma^3} \\&=\frac{E(X^3)-3 \mu E(X^2)+3 \mu^2 E(X)-\mu^3}{\sigma^3} \\&=\frac{E(X^3)-3 \mu [E(X^2)-\mu E(X)]-\mu^3}{\sigma^3} \\&=\frac{E(X^3)-3 \mu \sigma^2-\mu^3}{\sigma^3} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (2) \\&=\frac{E(X^3)-3 \mu \sigma^2-\mu^3}{(\sigma^2)^{\frac{3}{2}}} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (3) \end{aligned}

The last version (3) is in terms of the first raw moment $\mu$, the second central moment $\sigma^2$ and the third raw moment $E(X^3)$. Essentially, the coefficient $\gamma_1$ can be obtained via (3) by first computing the first three raw moments.

Even though kurtosis is not the focus of this post, we would like to state it to complete the the brief discussion on moments. The ratio of the fourth central moment to the fourth power of the standard deviation, $\displaystyle \gamma_2=\frac{\mu_4}{\sigma^4}$, is called the kurtosis.

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Examples

In this section, we discuss the skewness in two familiar families of continuous distributions – gamma and beta. We also demonstrate how exponentiation can affect skewness.

Example 1Gamma Distribution
The following is the probability density function of the gamma distribution.

$\displaystyle f(x)=\frac{\beta^\alpha}{\Gamma(\alpha)} \ x^{\alpha-1} \ e^{-\beta x} \ \ \ \ \ \ \ \ \ x>0$

where $\Gamma(\cdot)$ is the gamma function, and $\alpha$ and $\beta$ are parameters such that $\alpha>0$ and $\beta>0$. The number $\alpha$ is the shape parameter and the number $\beta$ here is the rate parameter. Figure 1 shows the gamma distribution with $\alpha=2$ and $\beta=1$. When $\alpha=1$, we obtain the exponential distribution. When $\beta=\frac{1}{2}$ and $\alpha=\frac{k}{2}$ where $k$ is a positive integer, we obtain the chi square distribution with $k$ degrees of freedom.

Let $X$ be a random variable with the above gamma density function. The raw moments $E(X^k)$, where $k=1,2,3,\cdots$, are:

$\displaystyle E(X^k)=\frac{(\alpha+k-1)(\alpha+k-2) \cdots \alpha}{\beta^k}$

Using the first two raw moments to calculate the variance as well as the third moment, the following calculates the moment coefficient of skewness, based on the form in (3):

$\displaystyle \gamma_1=\frac{\displaystyle \frac{(\alpha+2)(\alpha+1)\alpha}{\beta^3}-3 \frac{\alpha}{\beta} \frac{\alpha}{\beta^3}-\frac{\alpha^3}{\beta^3}}{\biggl( \displaystyle \frac{\alpha}{\beta^2} \biggr)^{\frac{3}{2}}}=\frac{2}{\sqrt{\alpha}}$

The above calculation shows that the rate parameter $\beta$ has no effect on skewness. The example in Figure 1 has $\alpha=2$, giving a coefficient of skewness of $\sqrt{2}$ = 1.414213562. In general, the gamma distribution is skewed positively. However, the gamma distribution becomes more and more symmetric as the shape parameter $\alpha \rightarrow \infty$. The following graph the gamma densities for $\alpha=1, 2, 3, 5, 6$ and $\beta=1$.

Figure 3

In Figure 3, the light blue density with $\alpha=1$ is an exponential distribution. The red one with $\alpha=2$ is the density in Figure 1. With $\alpha=6$, the gamma density already looks very symmetric (the dark blue).

On the other hand, as the shape parameter $\alpha \rightarrow 0$, the gamma distribution becomes increasingly positively skewed. When $\alpha=\frac{1}{n}$, $\gamma_1=2 \sqrt{n}$. When $n \rightarrow \infty$, $\gamma_1 \rightarrow \infty$.

Example 2Beta Distribution
The following is the PDF of a beta distribution:

$\displaystyle f(x)=\frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha) \Gamma(\beta)} \ x^{\alpha-1} \ (1-x)^{\beta-1} \ \ \ \ \ \ \ \ \ \ \ \ 0

where $\Gamma(\cdot)$ is the gamma function, and $\alpha$ and $\beta$ are parameters such that $\alpha>0$ and $\beta>0$. Both $\alpha$ and $\beta$ are shape parameters.

In the beta family of distributions, the skewness can range from positive to negative. If the $\alpha$ parameter dominates (i.e. $x$ is to a higher power and $1-x$ is to a small power in the density function), then the beta distribution has a negative skew (skewed to the left). This is because the function $x^n$ has left skew and the function $(1-x)^n$ has right skew. Then the skewness of the beta distribution follows the one that dominates. If the $\beta$ parameter dominates, the beta distribution is skewed to the right. If both parameters are roughly equal, the beta distribution is close to symmetric. For example when $\alpha=20$ and $\beta=2$, the beta distribution is left skewed (its density curve is in Figure 2). As in the gamma case, the skewness of the beta distribution has a close form. The following formula confirms the intuition about the skewness of the beta distribution (found here).

$\displaystyle \gamma_1=\frac{2(\beta-\alpha) \ \sqrt{\alpha+\beta+1}}{(\alpha+\beta+2) \ \sqrt{\alpha \ \beta}}$

Thus the beta distribution with $\alpha=20$ and $\beta=2$ has skewness coefficient -1.137431317. The following figure further demonstrates the role of the shape parameters play in changing the skewness of the beta distribution.

Figure 4

In Figure 4, as the $\alpha$ parameter goes from 2 to 20, the skewness goes from 1.137431317 to 0.659393193 to 0 to -0.659393193 to -1.137431317.

Example 3Exponentiation
Symmetric distributions have zero coefficient of skewness. Raising a symmetric distribution to a positive power can produce a skewed distribution. For example, let $X$ be the standard normal random variable (mean 0 and variance 1). Let $Y=X^2$. Then $Y$ has a chi-square distribution with 1 degree of freedom, which means that it is a gamma distribution with $\alpha=\frac{1}{2}$ and $\beta=\frac{1}{2}$. According to Example 1 above, the skewness coefficient is $\frac{2}{\sqrt{0.5}}=2 \sqrt{2}=2.828$. Thus squaring a standard normal distribution produces a very strongly positively skewed distribution.

Example 4Exponentiation
When raising a positively skewed distribution to positive power can produce a more strongly positively skewed distribution. For example, let $X$ be an exponential random variable. Example 1 shows that exponential distributions have skewness coefficient 2. We show that the coefficient of skewness for $Y=X^2$ is approximately 6.619.

The density function for the exponential random variable $X$ is $f(x)=\beta e^{-\beta x}$ where $\beta>0$ is the rate parameter. It can be shown that the raw moments of $X$ is:

$\displaystyle E(X^k)=\frac{k!}{\beta^k} \ \ \ \ \ \ \ \ \ \ \ k=1,2,3,\cdots$

Then the first three moments of $Y$ are:

$\displaystyle E(Y)=E(X^2)=\frac{2}{\beta^2}$

$\displaystyle E(Y^2)=E(X^4)=\frac{24}{\beta^4}$

$\displaystyle E(Y^3)=E(X^6)=\frac{720}{\beta^6}$

With the first two raw moments, calculate the variance of $Y$. Then compute $\gamma_1$ via formula (3).

$\displaystyle \gamma_1=\frac{\displaystyle \frac{720}{\beta^6}-3 \ \frac{2}{\beta^2} \ \frac{20}{\beta^4}-\frac{8}{\beta^6}}{\biggl( \displaystyle \frac{20}{\beta^4} \biggr)^{\frac{3}{2}}}=\frac{74}{5 \sqrt{5}}=6.61876$

Example 5Exponentiation
Raising a left skewed distribution to a positive power can produce a distribution that is less left skewed. The use of increasing exponents eventually produces a positively skewed distribution. Let $X$ be the beta random variable with $\alpha=5$ and $\beta=1$. The density function for $X$ is $f(x)=5x^4$ where $0. Using the formula shown in Example 2 above, the coefficient of skewness is

$\displaystyle \gamma_1=\frac{2(1-5) \sqrt{5+1+1}}{(5+1+2) \sqrt{5}}=-1.183215957$

We wish to calculate the coefficient of skewness for $X^2$. To do that, it will be helpful to have a formula for the raw moments of $X$. It is easy to verify that:

$\displaystyle E(X^k)=\frac{5}{5+k} \ \ \ \ \ \ \ \ \ k=1,2,3,\cdots$

The first three moments of $Y=X^2$ are:

$\displaystyle E(Y)=E(X^2)=\frac{5}{7}$

$\displaystyle E(Y^2)=E(X^4)=\frac{5}{9}$

$\displaystyle E(Y^3)=E(X^6)=\frac{5}{11}$

Via formula (3), the following is the coefficient of skewness for $Y=X^2$.

$\displaystyle \gamma_1=\frac{\displaystyle \frac{5}{11}-3 \ \frac{5}{7} \ \frac{20}{441}-\frac{125}{7^3}}{\biggl( \displaystyle \frac{20}{441} \biggr)^{\frac{3}{2}}}=\frac{-18}{11 \sqrt{5}}=-0.731804065$

In this example, squaring the beta distribution with skewness -1.1832 produces a distribution a negatively skewed distribution but with a smaller skew. Let’s raise $X$ to higher powers. The following shows the results:

$\displaystyle X^3 \ \ \ \ \ \ \gamma_1=\frac{-2 \sqrt{11}}{7 \sqrt{5}}=-0.423782771$

$\displaystyle X^4 \ \ \ \ \ \ \gamma_1=\frac{-2 \sqrt{13}}{17 \sqrt{5}}=-0.189700182$

$\displaystyle X^5 \ \ \ \ \ \ \gamma_1=6 \sqrt{2}=8.485281374$

Raising the beta distribution with $\alpha=5$ and $\beta=1$ to higher powers eventually produces a positively skewed distribution. This is an interesting example, though this observation probably should not be taken as a rule.

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Counterexamples

All the examples discussed previously are good “textbook” examples in that they help build intuition on how skewness behaves in familiar distributions. However, it is also easy to take the wrong lessons from these examples. The above examples can serve as good introduction to the topic of skewness. It is also important to attempt to provide a caveat that some of the commonly drawn lessons are not appropriate in all circumstances.

As indicated earlier, one wrong lesson from Figure 1 and Figure 2 is that a density curve such as Figure 1 may suggest that “mode < median < mean" for a right skewed distribution and that Figure 2 may suggest that "mean < median < mode" for a left skewed distribution. In both Figure 1 and Figure 2, the mean is further out in the long tail than the median. In certain textbooks, these two observations are even stated as characterizations of right skew and left skew. Such a rule of thumb is easy to state and easy to apply. For some students, such rule provides a lot of clarity about how skewness should work. For such students, checking for skewness is simply a matter of finding the relative position of the mean and median (e.g. in such thinking, if mean is greater than the median, then it is a right skew).

Any discussion of skewness should point out that the simple rule described in the above paragraph, though useful in many situations, is imperfect and may not apply outside of certain familiar distributions. For a good discussion on this issue, see this article.

We highlight one example found in the article mentioned above. This example demonstrates a clear violation of the common misconception indicated above. The following is the density function of the example.

$\displaystyle f_p(x)=\left\{\begin{matrix} \displaystyle (1-p)\biggl(1+\frac{1-p}{2p} x \biggr)&\ \ \ \ \ \ \frac{-2p}{1-p} \le x \le 0 \\{\text{ }}& \\ (1-p) e^{-x}&\ \ \ \ \ \ x>0 \end{matrix}\right.$

where $0. To facilitate the discussion, let $X_p$ be the random variable whose PDF is $f_p(x)$ defined above. The above density function is a juxtaposition of a triangular density and an exponential density. This triangular-exponential distribution has positive coefficient of skewness when $0. Yet within this range for $p$, the mean can be made to be on either side of the median. We consider three cases where $p=0.7$, $p=0.6$ and $p=0.9$.

Example 6
First the case $p=0.7$.

$\displaystyle f_{0.7}(x)=\left\{\begin{matrix} \displaystyle 0.3\biggl(1+\frac{3}{14} x \biggr)&\ \ \ \ \ \ \frac{-14}{3} \le x \le 0 \\{\text{ }}& \\ 0.3 e^{-x}&\ \ \ \ \ \ x>0 \end{matrix}\right.$

The following is the graph of the density curve $f_{0.7}(x)$. The right tail is long since the exponential distribution is on the right side. However, the left side is heavier (with 70% of the weight on the triangle on the left side).

Figure 5

The following shows the results for the density function $f_{0.7}(x)$.

$\displaystyle E(X_{0.7})=\frac{-21.3}{27}=-0.7889$

$\displaystyle \text{median of } X_{0.7} = -0.722613478$

$\displaystyle E(X_{0.7}^2)=\frac{254.4}{81}=3.140740741$

$\displaystyle Var(X_{0.7})=\frac{1835.91}{3^6}$

$\displaystyle E(X_{0.7}^3)=\frac{-2152.2}{405}$

$\displaystyle \gamma_1=\frac{111906.63}{5 1835.91^{1.5}}=0.284517335$

The calculation confirms the positive skew (0.2845), which is a moderately strong positive skewness. Note that the mean is to the left of the median. Both the mean and median are to the left of the mode (at x = 0). In Figure 5, the right side is infinitely long, thus a positively skewed distribution (and is confirmed by the calculation of $\gamma_1$). According to the common notion of how right skew should work, the mean should be further out on the right tail. But this is not the case. The mean is further out on the left side than the median. The violation of the common conception of skewness can occur when one tail is long but the other side is heavier.

Example 7
Now the case $p=0.6$.

$\displaystyle f_{0.6}(x)=\left\{\begin{matrix} \displaystyle 0.4\biggl(1+\frac{1}{3} x \biggr)&\ \ \ \ \ \ -3 \le x \le 0 \\{\text{ }}& \\ 0.4 e^{-x}&\ \ \ \ \ \ x>0 \end{matrix}\right.$

The following is the graph of the density curve $f_{0.6}(x)$. The right tail is long since the exponential distribution is on the right side. The left side is still heavy but a little less heavier than in the previous example (with 60% of the weight on the triangle on the left side).

Figure 6

The following shows the results for the density function $f_{0.6}(x)$.

$\displaystyle E(X_{0.6})=-0.2$

$\displaystyle \text{median of } X_{0.6} = -0.261387212$

$\displaystyle E(X_{0.6}^2)=1.7$

$\displaystyle Var(X_{0.6})=1.66$

$\displaystyle E(X_{0.6}^3)=0.78$

$\displaystyle \gamma_1=0.834128035$

The density curve $f_{0.6}(x)$ has a stronger positive skew than the previous example as there is a little more weight on the exponential side (the right side). Even though the mean in this case is to the right of the median, both the mean and median are not on the right tail but on the left triangular side (the heavier side). In any case, the mean is definitely not further out on the longer tail (the right tail) as the common rule of thumb would suggest.

Both Example 6 and Example 7 are right skewed distributions that do not conform to the common expectation about right skewed distributions. The following example will dispel the notion about the direction of the skew.

Example 8
Here we use $p=0.9$ so that there is a still a long right tail but 90% of the weight is on the other side.

$\displaystyle f_{0.9}(x)=\left\{\begin{matrix} \displaystyle 0.1\biggl(1+\frac{1}{18} x \biggr)&\ \ \ \ \ \ -18 \le x \le 0 \\{\text{ }}& \\ 0.1 e^{-x}&\ \ \ \ \ \ x>0 \end{matrix}\right.$

The overall shape of the $f_{0.9}(x)$ is similar to Figure 5 and Figure 6. The following shows the results for the density function $f_{0.9}(x)$.

$\displaystyle E(X_{0.9})=-5.3$

$\displaystyle E(X_{0.9}^2)=48.8$

$\displaystyle Var(X_{0.9})=20.71$

$\displaystyle E(X_{0.9}^3)=-524.28$

$\displaystyle \gamma_1=-0.489285839$

Because there is so little weight on the right tail, the skewness is actually negative (-0.48928). Here we have a right skewed looking distribution that is actually skewed to the left!

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Remarks

Examples 5 through 7 demonstrate that when one tail is long but the other side is heavy, the common conception of right skew and left skew do not apply. The common conception, as discussed earlier, is that the both the mean and the median are located in the longer tail and that the mean is further out in the long tail than the median. The article mentioned earlier is easy to read and gives a fuller discussion of the issues when dealing with the notion of skewness. The common conception of skewness can be easily violated in discrete distributions, especially when the weights on both sides of the median are not equal. All the above examples are unimodal distributions. According the quoted article, bimodal or multimodal distributions can be problematic too.

Of course, the caveat presented here is not meant to discourage anyone from discussing the common conception about skewness. The common conception about the locations of mode, mean and median conveys useful intuition and we should continue to focus on it. But the common rule of thumb should definitely be not be presented as gospel truth as some textbooks had done. Instead, it should be pointed out that the common rule of thumb is imperfect and it would be helpful to have a discussion why the rule is imperfect.

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Practice problems

Practice problems to reinforce the calculation are found in the companion blog to this blog.

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$\copyright \ \text{2015 by Dan Ma}$


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# Custom generators

The graherator package already provides several node, edge and weight generators respectively. However, the collection is by far not complete and it does not aim to be. Therefore the user interface enables the definition of custom generators for each of the three steps (see basic graph generation vignette). A generator is a simple function myGenerator = function(graph, …). The first argument needs to be a grapherator object. More arguments are possible, but optional. Each generator must return a named list. The first field must be either coords (matrix of node coordinates) for node generators, adj.mat (logical matrix defining the adjacency matrix) or weights (numeric matrix of weights or list of numeric matrices) for weight generators respectively. The second field is named generator and contains a short string representation of the generator, e.g., DEG for Delauney Edge Generator.

In the following we demonstrate the definition of a custom edge generator, namely the Threshold Edge Generator. Given a graph and a positive numeric argument threshold this edge generator computes the pairwise Euclidean distances between the nodes and only keeps the edges whose distance is below the threshold value.

library(grapherator)

addEdgesThreshold = function(graph, threshold, ...) {
n = getNumberOfNodes(graph)
coords = getNodeCoordinates(graph)

# compute distances
dists = as.matrix(dist(coords, method = "euclidean", ...))

# no loops allowed
}
set.seed(1) # reproducability
plot(g)$pl.coords Due to smart internals the custom generators automatically work for intra- or intercluster linking without any adaptations. In the next example, we create a clustered network, apply the threshold edge generator for each cluster separately and connect cluster centers utilizing another edge generator. set.seed(1) # reproducability g = graph(lower = 0, upper = 100) g = addNodes(g, n = 5, generator = addNodesUniform) g = addNodes(g, n = 25, by.centers = TRUE, generator = addNodesGrid, lower = c(0, 0), upper = c(15, 15)) g = addEdges(g, generator = addEdgesThreshold, threshold = 13, type = "intracluster") g = addEdges(g, generator = addEdgesDelauney, type = "intercenter") # weight generation is skipped here plot(g)$pl.coords


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# Fossils and their Environments

Fossils are the remains or traces of once living organisms. Our introduction to paleontology (the study of fossils), will be limited to those forms that are commonly preserved and geologically useful. For Historical Geology purposes, we will be concerned with two aspects of paleontology: (1) fossils as indicators of relative age (using the Principle of Floral and Faunal Succession. fossils as indicators of environment. All organisms are dependent on their environments for survival. Some organisms, however, can survive only in restricted environments. Consequently, we can apply the principle of uniformitarianism to fossils to interpret the environment in which they lived. Modern fish, for example, tend to live in water. Ancient fish, then, probably lived in water as well. We will divide the environments into two basic categories-- the terrestrial (land-based) environment, and the marine (ocean) environment. Because marine fossils are most commonly preserved, the marine environments are shown in figure1.

Shallow marine bottom-dwelling organisms such as corals represent the benthic environment. Similarly, some species of fish indicate the neritic environment (the shallow seas covering the continental shelves). Other organisms such as radiolarians and some diatoms are characteristic of the oceanic or deep ocean environment. Still other organisms inhabit the ocean in both the neritic and oceanic areas and simply suggest a pelagic environment.

Marine fossils can be further described in terms of their life habits. These habits are shown in Figure A2.  Corals, for example tend to be attached to the bottom and are thus referred to as benthic (bottom dwelling) sessile (fixed). Lobsters, however, move along the ocean bottom and although still benthic, are also considered vagrant. Swimming and floating organisms are referred to as nektonic and planktonic, respectively. In our example above, some fish might represent the nektonic pelagic environment.
Geology IN