text:
# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
On some nonself mappings in Banach spaces. (English) Zbl 0671.47051

The main results of this paper are:

Theorem 3.1. Let X be a Banach space, and K a nonempty, closed subset of X,T:K$\to X$ a mapping satisfying the condition Tx$\in K$ for every $x\in \partial K$ (the boundary of K), and $\phi :{ℝ}^{+}\to {ℝ}^{+}$ an increasing continuous function satisfying

$\left(1\right)\phantom{\rule{1.em}{0ex}}\phi \left(t\right)=0\phantom{\rule{1.em}{0ex}}if\phantom{\rule{1.em}{0ex}}and\phantom{\rule{1.em}{0ex}}only\phantom{\rule{1.em}{0ex}}if\phantom{\rule{1.em}{0ex}}t=0·$

Furthermore, let b and c be decreasing functions from ${ℝ}^{+}\setminus \left\{0\right\}$ into [0,1) such that $2b\left(t\right)+c\left(t\right)<1$ for every $t>0$. Suppose that T satisfies the condition

$\left(2\right)\phantom{\rule{1.em}{0ex}}\phi \left(d\left(Tx,Ty\right)\right)\le$
$b\left(d\left(x,y\right)\right)·\left\{\phi \left(d\left(x,Tx\right)+\phi \left(d\left(y,Ty\right)\right)\right\}+c\left(d\left(x,y\right)\right)·min\left\{\phi \left(d\left(x,Ty\right)\phantom{\rule{1.em}{0ex}},\phantom{\rule{1.em}{0ex}}\phi \left(d\left(y,Tx\right)\right)\right\}$

$\forall x\ne y\in X$. Then T has a unique fixed point.

Theorem 1 of N. A. Assad, Tamkang J. Math. 7, 91-94 (1976; Zbl 0356.47027) is a special case of Theorem 3.1, Theorem 3.1 generaizes Theorem 2 of M. S. Khan, M. Swaleh and S. Sessa, Bull. Aust. Math. Soc. 30, 1-9 (1984; Zbl 0553.54023). Theorem 4.1. Let X be a Banach space and K a nonempty compact subset of X. Let T:K$\to X$ be a continuous mapping satisfying the condition that T(x)$\in K$ for every $x\in \partial K$ and $\phi :{ℝ}^{+}\to {ℝ}^{+}$ an increasing continuous function satisfying property (1) of Theorem 3.1. Furthermore for all distinct x,y in K the inequality

$\left(3\right)\phantom{\rule{1.em}{0ex}}\phi \left(d\left(Tx,Ty\right)\right)<\frac{\left(1-c\right)}{2}\left\{\phi \left(d\left(x,Tx\right)\right)+\phi \left(d\left(y,Ty\right)\right)\right\}+c·min\left\{\phantom{\rule{1.em}{0ex}}\phi \left(d\left(x,Ty\right)\right),\phantom{\rule{1.em}{0ex}}\phi \left(d\left(y,Tx\right)\right)\right\}$

holds, where $0\le c\le 1$. Then T has a unique fixed point.

Theorem 4.1 generalizes the fixed point theorem of B. Fisher, Calcutta Math. Soc. 68, 265-266 (1976; Zbl 0378.54035) in case X is a Banach space.

Reviewer: V.Popa

##### MSC:
 47H10 Fixed point theorems for nonlinear operators on topological linear spaces 54H25 Fixed-point and coincidence theorems in topological spaces
fixed point


text:
# Super elevation on the hilly roads should be -

This question was previously asked in
UPPCL JE CE Offical Paper I 2014
View all UPPCL JE Papers >
1. Not more than 12%
2. Not more than 8%
3. Not more than 5%
4. Not more than 10%

Option 4 : Not more than 10%
Free
ST 1: Building Material and Concrete Technology
16847
20 Questions 20 Marks 12 Mins

## Detailed Solution

Concept:

Super-elevation:

It is the inward transverse slope provided throughout the length of the horizontal curves to counteract the centrifugal force which tends to pull out the vehicle in an outward direction and therefore to check the tendency of the vehicle to overturn or skid.

The value of super-elevation e is given by,

$${\rm{\;e}} \ + f = \frac{{{{\rm{V}}^2}}}{{{\rm{gR}}}}\;$$

Where,

V = speed of the vehicle, R = radius of the horizontal curve, g = acceleration due to gravity, f = Lateral friction.

According to IRC, the maximum superelevation is given by:

 Type of Terrain emax Plain and rolling 7%(1 in 15) Hilly road 10% Urban road 4%

format: plain


text:
## Precalculus (6th Edition) Blitzer

The required graph of $x=5$ that will be a line passing through $\left( 5,\ 0 \right)$ and parallel to the y-axis is as follows:
Consider the given equation: $2x-10=0$ Or \begin{align} & 2x=10 \\ & x=5. \end{align}


text:
Get inspired by the success stories of our students in IIT JAM 2021. Learn More

How to Pursue Mathematics after High School?

For Students who are passionate for Mathematics and want to pursue it for higher studies in India and abroad.

This is a beautiful sample problem from ISI MStat PSB 2010 Problem 2. It's all about how many isosceles triangle with sides of integers lengths one can construct under certain conditions. We provide a detailed solution with prerequisites mentioned explicitly.

Problem- ISI MStat PSB 2010 Problem 2

Find the total number of isosceles triangles such that the length of each side is a positive integer less than or equal to 40.

(Here equilateral triangles are also counted as isosceles triangle.)

Prerequisites

• Basic counting principles.
• Triangle inequality.

Solution

Let the sides of the isosceles triangle are k, k and l units, (length of the equal sides are k units)

So, our problem is all about finding triplets of form (k,k,l), where $l, k \le 40$ . And how many such triplets can be found !

also since l,k and k are also sides of an (isosceles) triangle, so by triangle inequality,

$l < 2k$ . Since l is an integer so, $l \le 2k-1$

So, we have $l\le 40$ and $l \le 2k-1$. combining we have $l \le min(40, 2k-1)$ .

now let us define, $l_k$ : possible number of l's for a given k =1,2,....,40

here we must consider two cases.

Case-1 : when $2k-1 \le 40 \Rightarrow k \le \lceil \frac{40+1}{2} \rceil = 20$

So, l can be any integer between 1 and 2k-1, since $l\ \le 2k-1$ when $k \le 20$

so,there can be 2k-1 choices for the value of each k, when k=1,...,20.

Or, $l_k = 2k-1$ for k=1,2,....,20.

now, Case-2 : when 2k-1>40 $\Rightarrow k > \lceil \frac{40+1}{2} \rceil = 20$

so, here l can take any integer between 1 to 40. So, there 40 choices for l for a given k, when k=21,22,...,40.

Or, $l_k = 40$ for k=21,22,....,40.

Combining Case-1 and Case-2, we have,

$l_k = \begin{cases} 2k-1 & k=1,2,...,20 \\ 40 & k=21,22,...,40 \end{cases}$

So, finally, let all possible number of triplets of form (k,k,l) are = $T_40$

So, $T_{40}$ = $\sum_{k=1}^{40}{l_k}$ = $\sum_{k=1}^{20}{(2k-1)}$ + $\sum_{k=21}^{40}{40}$ =$(20 )^2 + 40 \times (40-20)=400+800= 1200.$

So, $T_{40}$ = 1200. hence we can find 1200 such isosceles triangles with sides of integer lengths such that the length of each sides are less than or equal to 40.

Food For Thought

Can you generalize this problem? i.e. can you find How many isosceles triangle one can construct, with sides of integer lengths, such that the sides are less than or equal to any integer N ? Can you find an elegant formula to express $T_N$, for any integer N?

What to do to shape your Career in Mathematics after 12th?

From the video below, let's learn from Dr. Ashani Dasgupta (a Ph.D. in Mathematics from the University of Milwaukee-Wisconsin and Founder-Faculty of Cheenta) how you can shape your career in Mathematics and pursue it after 12th in India and Abroad. These are some of the key questions that we are discussing here:

• What are some of the best colleges for Mathematics that you can aim to apply for after high school?
• How can you strategically opt for less known colleges and prepare yourself for the best universities in India or Abroad for your Masters or Ph.D. Programs?
• What are the best universities for MS, MMath, and Ph.D. Programs in India?
• What topics in Mathematics are really needed to crack some great Masters or Ph.D. level entrances?
• How can you pursue a Ph.D. in Mathematics outside India?
• What are the 5 ways Cheenta can help you to pursue Higher Mathematics in India and abroad?

Want to Explore Advanced Mathematics at Cheenta?

Cheenta has taken an initiative of helping College and High School Passout Students with its "Open Seminars" and "Open for all Math Camps". These events are extremely useful for students who are really passionate for Mathematic and want to pursue their career in it.

To Explore and Experience Advanced Mathematics at Cheenta

2 comments on “ISI MStat PSB 2010 Problem 2 | Combinatorics”

For general $N \in \Bbb N$ we have $$T_N = \left ( \left \lfloor \frac {N+1} {2} \right \rfloor \right )^2 + N^2 - N \left \lfloor \frac {N+1} {2} \right \rfloor.$$

1. Uttaran Chatterjee says:

yes you are correct.

This site uses Akismet to reduce spam. Learn how your comment data is processed.


text:
New Zealand
Level 8 - NCEA Level 3

# Indefinite and definite integrals (e^(f(x) non-linear)

## Interactive practice questions

Consider the function $y=e^{x^3}$y=ex3.

a

Find the derivative $\frac{dy}{dx}$dydx.

b

Hence find the value of $\int3x^2e^{x^3}dx$3x2ex3dx.

You may use $C$C to represent the constant of integration.

Easy
Approx 2 minutes

Consider the function $y=e^{3x^4+4}$y=e3x4+4.

Consider the function $y=e^{3x^4-5}$y=e3x45.

Determine the value of $\int\left(4x^3+3\right)e^{x^4+3x}dx$(4x3+3)ex4+3xdx.

### Outcomes

#### M8-11

Choose and apply a variety of differentiation, integration, and antidifferentiation techniques to functions and relations, using both analytical and numerical methods

#### 91579

Apply integration methods in solving problems


text:
# The Next Inequality

Considering it has been at least a post removed, I’ll bring us back to our situation. We have a local Noetherian ring $(R, \frak{m})$. Our notation is that $\delta(R)$ is the least number of generators of an $\frak{m}$-primary ideal (which was shown to be independent of choice of ideal here). The goal for the day is to show that $d(R)\geq \dim R$.

Suppose $\frak{q}$ is $\frak{m}$-primary. We’ll prove something more general. Let $M$ be a finitely generated $R$-module, $x\in R$ a non-zero divisor in $M$ and $M'=M/xM$. Then the claim is that $\deg\chi_q^{M'}\leq \deg\chi_q^M -1$.

Since $x$ is not a zero-divisor, we have an iso as $R$-modules: $xM\cong M$. Define $N=xM$. Now take $N_n=N\cap \frak{q}^nM$. Since $\frak{q}^nM$ is a stable $\frak{q}$-filtration of $M$, by Artin-Rees we get that $(N_n)$ is a stable $\frak{q}$-filtration of $N$.

For each $n$ we have $0\to N/N_n \to M/\frak{q}^nM\to M'/\frak{q}^nM'\to 0$ exact.

Thus we get $l(N/N_n)-l(M/\frak{q}^nM)+l(M'/\frak{q}^nM')=0$. So if we let $g(n)=l(N/N_n)$, we get for large $n$: $g(n)-\chi_q^M(n)+\chi_q^{M'}(n)=0$.

But $(N_n)$ is also a stable $\frak{q}$-filtration of $M$, since $N\cong M$. We already showed that the degree and leading coefficient of $g(n)$ depends only on $M$ and $\frak{q}$ and not on the filtration. Thus $g(n)$ and $\chi_q^M(n)$ have the same degree and leading coefficient, so the highest powers kill eachother which gives $\deg\chi_q^{M'}\leq \deg \chi_q^M-1$.

In particular, we will need that $R$ as an $R$-module gives us $d(R/(x))\leq d(R)-1$.

Now we prove the goal for today. For simplicity, let $d=d(R)$. We will induct on $d$. The base case gives that $l(R/\frak{m}^n)$ is constant for large $n$. In particular, there is some $N$ such that $\frak{m}^n=\frak{m}^{n+1}$ for all $n>N$. But we are local, so $\frak{m}=J(R)$ and hence by Nakayama, $\frak{m}^n=0$. Thus for any prime ideal $\frak{p}$, we have $\frak{m}^k\subset \frak{p}$ for some $k$, so take radicals to get $\frak{m}=\frak{p}$. Thus there is only one prime ideal and we actually have an Artinian ring and hence have $\dim R=0$.

Now suppose $d>0$ and the result holds for $\leq d-1$. Let $p_0\subset p_1\subset \cdots \subset p_r$ be a chain of primes. Choose $x\in p_1\setminus p_0$. Define $R'=R/p_0$ and $\overline{x}$ be the image of $x$ in $R'$.

Note that since $R'$ is an integral domain, and $\overline{x}$ is not 0, it is not a zero-divisor. So we use our first proof from today to get that $d(R'/(\overline{x}))\leq d(R')-1$.

Let $\frak{m}'$ be the maximal ideal of $R'$. Then $R'/\frak{m}'$ is the image of $R/\frak{m}$, so $l(R/\frak{m}^n)\geq l(R'/\frak{m}'^n)$ which is precisely $d(R)\geq d(R')$. Plugging this into the above inequality gives $d(R'/(\overline{x}))\leq d(A)-1=d-1$.

So by the inductive hypothesis, $\dim(R'/\overline{x})\leq d-1$. Take our original prime chain. The images form a chain $\overline{p}_1, \ldots , \overline{p}_r$ in $R'/(\overline{x})$. Thus $r-1\leq d-1\Rightarrow r\leq d$. Since the chain was arbitrary, $\dim R\leq d(R)$.

A nice corollary here is that the dimension of any Noetherian local ring is finite. Another similar corollary is that in any Noetherian ring (drop the local) the height of a prime ideal is finite (and hence primes satisfy the DCC), since $ht(p)=\dim A_p$ which is local Noetherian.

# Applying the Hilbert Polynomial

Let’s start applying to some specific situations now. Suppose $R$ is a Noetherian local ring with maximal ideal $\frak{m}$. Let $\frak{q}$ be an $\frak{m}$-primary ideal. Let $M$ be a finitely-generated $R$-module, and $(M_n)$ a stable $\frak{q}$-filtration of $M$.

Don’t panic from the set-up. I think I haven’t talked about filtrations. All the stable $\frak{q}$-filtration means is that we have a chain of submodules $M=M_0\supset M_1\supset \cdots \supset M_n\supset \cdots$ such that $\frak{q}M_n=M_{n+1}$ for large $n$.

The goal for the day is to prove three things.

1) $M/M_n$ has finite length for all $n\geq 0$.

Define $G(M)=\bigoplus \frak{q}^n/\frak{q}^{n+1}$ and $G(M)=\bigoplus M_n/M_{n+1}$. We have a natural way to make $G(M)$ into a finitely-generated graded $G(R)$-module. The multiplication in the ring comes from the following. If $x_n\in\frak{q}^n$, then let the image in $\frak{q}^n/\frak{q}^{n+1}$ be denoted $\overline{x_n}$. We take $\overline{x_n}\overline{x_m}=\overline{x_nx_m}$. This does not depend on representative.

We’ll say $G_n(M)$ is the n-th grade: $M_n/M_{n+1}$. Now $G_0(R)=R/q$ is an Artinian local ring and each $G_n(M)$ is a Noetherian $R$-module annihilated by $\frak{q}$. Thus they are all Noetherian $R/\frak{q}=G_0(R)$-modules. So by the Artinian condition we get that each $G_n(M)$ is of finite length. Thus $l_n=l(M/M_n)=\sum_{r=0}^{n-1} l(G_r(M))<\infty$.

2) For large $n$, $l(M/M_n)$ is a polynomial $g(n)$ of degree $\leq s$ where $s$ is the least number of generators of $\frak{q}$.

Suppose $x_1, \ldots, x_s$ generate $\frak{q}$. Then $\{\overline{x_i}\}$ in $\frak{q}/\frak{q}^2$ generate $G(R)$ as an $R/\frak{q}$-algebra. But $l$ is an additive function on the filtration, so by last time we saw thatfor large $n$ there is some polynomial such that $f(n)=l(G_n(M))=l(M_n/M_{n+1})$, and each $\overline{x_i}$ has degree 1, so the polynomial is of degree $\leq s-1$.

Thus we get that $l_{n+1}-l_n=l(G_n(M))=f(n)$. So from two posts ago, we get for large $n$ that $l_n$ is some polynomial $g(n)$ of degree $\leq s$.

3) Probably the most important part is that the degree and leading coefficient of $g(n)$ depends only on $M$ and $\frak{q}$ and not on the filtration.

Let $(\overline{M_n})$ be some other stable $\frak{q}$-filtration with polynomial $\overline{g}(n)=l(M/\overline{M_n})$. Since any two stable $\frak{q}$-filtrations have bounded difference, there is an integer $N$ such that $M_{n+N}\subset \overline{M_n}$ and $\overline{M_{n+N}}\subset M_n$ for all $n\geq 0$. But this condition on the polynomials says that $g(n+N)\geq \overline{g}(n)$ and $\overline{g}(n+N)\geq g(n)$, which means that $\lim_{n\to\infty}\frac{g(n)}{\overline{g}(n)}=1$. Thus they have the same degree and leading coefficient.

That seems to be enough for one day. Unfortunately, I haven't quite got to the right setting that I want yet.


text:
Get our free extension to see links to code for papers anywhere online!

# Shoestring: Graph-Based Semi-Supervised Learning with Severely Limited Labeled Data

Oct 28, 2019
Wanyu Lin, Zhaolin Gao, Baochun Li

Graph-based semi-supervised learning has been shown to be one of the most effective approaches for classification tasks from a wide range of domains, such as image classification and text classification, as they can exploit the connectivity patterns between labeled and unlabeled samples to improve learning performance. In this work, we advance this effective learning paradigm towards a scenario where labeled data are severely limited. More specifically, we address the problem of graph-based semi-supervised learning in the presence of severely limited labeled samples, and propose a new framework, called {\em Shoestring}, that improves the learning performance through semantic transfer from these very few labeled samples to large numbers of unlabeled samples. In particular, our framework learns a metric space in which classification can be performed by computing the similarity to centroid embedding of each class. {\em Shoestring} is trained in an end-to-end fashion to learn to leverage the semantic knowledge of limited labeled samples as well as their connectivity patterns with large numbers of unlabeled samples simultaneously. By combining {\em Shoestring} with graph convolutional networks, label propagation and their recent label-efficient variations (IGCN and GLP), we are able to achieve state-of-the-art node classification performance in the presence of very few labeled samples. In addition, we demonstrate the effectiveness of our framework on image classification tasks in the few-shot learning regime, with significant gains on miniImageNet ($2.57\%\sim3.59\%$) and tieredImageNet ($1.05\%\sim2.70\%$).


text:
# Boolean simplification

So I am giving this expression D +B’C’ + CD’ +A B’C and I ask to simplify it

When working through it I get

D+B'C'+CD'+AB'C D'(A'B'+CD'+AB) D'(A'B'+A(B'+B)) D'(A'B'+AC') D'(B'+A)

Am I on the right track or am I completely missing the point?

-
You have made a mistake. If you set $(A,B,C,D) = (F,T,F,T)$ then the first formula will evaluate to $T$, the second to $F$. –  copper.hat Oct 10 '12 at 18:27
You have changed the formula. However you are missing the point, the second expression if far more complicated, not simpler. –  copper.hat Oct 10 '12 at 18:34

Try using the following rules: $A + \overline{A }B = A + B$, and $A + A B = A$ (and convince your self that they are true).
There are 3 reductions above: #1 $D + C\overline{D} = D + C$, #2 $C + \overline{B}\overline{C} = C + \overline{B}$, and #3 $C+A \overline{B} C = C$. Only the variable $A$ was dropped. The notion of 'simplified' is not necessarily obvious (here it is fairly clear). When implementing digital circuits, simpler may mean reduced number of gates (less important nowadays) or reduced logic depth (important for speed). –  copper.hat Oct 10 '12 at 19:11


text:
# R/sven.R In dongjli/bravo: Bayesian Screening and Variable Selection

#### Documented in sven

#' Selection of variables with embedded screening using Bayesian methods (SVEN)
#' in Gaussian linear models (ultra-high, high or low dimensional).
#' @rdname sven
#' @description SVEN is an approach to selecting variables with embedded screening
#' using a Bayesian hierarchical model. It is also a variable selection method in the spirit of
#' the stochastic shotgun search algorithm. However, by embedding a unique model
#' based screening and using fast Cholesky updates, SVEN produces a highly scalable
#' algorithm to explore gigantic model spaces and rapidly identify the regions of
#' high posterior probabilities. It outputs the log (unnormalized) posterior
#' probability of a set of best (highest probability) models.
#' For more details, see Li et al. (2020).
#'
#' @param X The \eqn{n\times p} covariate matrix without intercept. The following classes are supported:
#' \code{matrix} and \code{dgCMatrix}. Every care is taken not to make copies of this (typically)
#' giant matrix. No need to center or scale this matrix manually. Scaling is performed implicitly and
#' regression coefficient are returned on the original scale.
#' @param y The response vector of length \eqn{n}. No need to center or scale.
#' @param w The prior inclusion probability of each variable. Default: \eqn{sqrt(n)/p}.
#' @param lam The slab precision parameter. Default: \eqn{n/p^2}
#' as suggested by the theory of Li et al. (2020).
#' @param Ntemp The number of temperatures. Default: 3.
#' @param Tmax The maximum temperature. Default: \eqn{\log\log p+\log p}.
#' @param Miter The number of iterations per temperature. Default: \code{50}.
#' @param wam.threshold The threshold probability to select the covariates for WAM.
#' A covariate will be included in WAM if its corresponding marginal inclusion
#' probability is greater than the threshold. Default: 0.5.
#' @param log.eps The tolerance to choose the number of top models. See detail. Default: -16.
#' @param verbose If \code{TRUE}, the function prints the current temperature SVEN is at; the default is TRUE.
#'
#' @details
#' SVEN is developed based on a hierarchical Gaussian linear model with priors placed
#' on the regression coefficients as well as on the model space as follows:
#' \deqn{y | X, \beta_0,\beta,\gamma,\sigma^2,w,\lambda \sim N(\beta_01 + X_\gamma\beta_\gamma,\sigma^2I_n)}
#' \deqn{\beta_i|\beta_0,\gamma,\sigma^2,w,\lambda \stackrel{indep.}{\sim} N(0, \gamma_i\sigma^2/\lambda),~i=1,\ldots,p,}
#' \deqn{(\beta_0,\sigma^2)|\gamma,w,p \sim p(\beta_0,\sigma^2) \propto 1/\sigma^2}
#' \deqn{\gamma_i|w,\lambda \stackrel{iid}{\sim} Bernoulli(w)}
#' where \eqn{X_\gamma} is the \eqn{n \times |\gamma|} submatrix of \eqn{X} consisting of
#' those columns of \eqn{X} for which \eqn{\gamma_i=1} and similarly, \eqn{\beta_\gamma} is the
#' \eqn{|\gamma|} subvector of \eqn{\beta} corresponding to \eqn{\gamma}.
#' Degenerate spike priors on inactive variables and Gaussian slab priors on active
#' covariates makes the posterior
#' probability (up to a normalizing constant) of a model \eqn{P(\gamma|y)} available in
#' explicit form (Li et al., 2020).
#'
#' The variable selection starts from an empty model and updates the model
#' according to the posterior probability of its neighboring models for some pre-specified
#' number of iterations. In each iteration, the models with small probabilities are screened
#' out in order to quickly identify the regions of high posterior probabilities. A temperature
#' schedule is used to facilitate exploration of models separated by valleys in the posterior
#' probability function, thus mitigate posterior multimodality associated with variable selection models.
#' The default maximum temperature is guided by the asymptotic posterior model selection consistency results
#' in Li et al. (2020).
#'
#' SVEN provides the maximum a posteriori (MAP) model as well as the weighted average model
#' (WAM). WAM is obtained in the following way: (1) keep the best (highest probability) \eqn{K}
#' distinct models \eqn{\gamma^{(1)},\ldots,\gamma^{(K)}} with
#' \deqn{\log P\left(\gamma^{(1)}|y\right) \ge \cdots \ge \log P\left(\gamma^{(K)}|y\right)}
#' where \eqn{K} is chosen so that
#' \eqn{\log \left\{P\left(\gamma^{(K)}|y\right)/P\left(\gamma^{(1)}|y\right)\right\} > \code{log.eps}};
#' (2) assign the weights \deqn{w_i = P(\gamma^{(i)}|y)/\sum_{k=1}^K P(\gamma^{(k)}|y)}
#' to the model \eqn{\gamma^{(i)}}; (3) define the approximate marginal inclusion probabilities
#' for the \eqn{j}th variable as \deqn{\hat\pi_j = \sum_{k=1}^K w_k I(\gamma^{(k)}_j = 1).}
#' Then, the WAM is defined as the model containing variables \eqn{j} with
#' \eqn{\hat\pi_j > \code{wam.threshold}}. SVEN also provides all the top \eqn{K} models which
#' are stored in an \eqn{p \times K} sparse matrix, along with their corresponding log (unnormalized)
#' posterior probabilities.
#'
#'
#' @return A list with components
#' \item{model.map}{A vector of indices corresponding to the selected variables
#' in the MAP model.}
#' \item{model.wam}{A vector of indices corresponding to the selected variables
#' in the WAM.}
#' \item{model.top}{A sparse matrix storing the top models.}
#' \item{beta.map}{The ridge estimator of regression coefficients in the MAP model.}
#' \item{beta.wam}{The ridge estimator of regression coefficients in the WAM.}
#' \item{mip.map}{The marginal inclusion probabilities of the variables in the MAP model.}
#' \item{mip.wam}{The marginal inclusion probabilities of the variables in the WAM.}
#' \item{pprob.map}{The log (unnormalized) posterior probability corresponding
#' to the MAP model.}
#' \item{pprob.top}{A vector of the log (unnormalized) posterior probabilities
#' corresponding to the top models.}
#'
#' @author Dongjin Li and Somak Dutta\cr Maintainer:
#' Dongjin Li <dongjl@@iastate.edu>
#' @references Li, D., Dutta, S., Roy, V.(2020) Model Based Screening Embedded Bayesian
#' Variable Selection for Ultra-high Dimensional Settings http://arxiv.org/abs/2006.07561
#' @examples
#' n=50; p=100; nonzero = 3
#' trueidx <- 1:3
#' nonzero.value <- 5
#' TrueBeta <- numeric(p)
#' TrueBeta[trueidx] <- nonzero.value
#'
#' rho <- 0.6
#' x1 <- matrix(rnorm(n*p), n, p)
#' X <- sqrt(1-rho)*x1 + sqrt(rho)*rnorm(n)
#' y <- 0.5 + X %*% TrueBeta + rnorm(n)
#' res <- sven(X=X, y=y)
#' res$model.map # the MAP model #' res$model.wam # the WAM
#' res$mip.map # the marginal inclusion probabilities of the variables in the MAP model #' res$mip.wam # the marginal inclusion probabilities of the variables in the WAM
#' res$pprob.top # the log (unnormalized) posterior probabilities corresponding to the top models. #' #' res$beta.map # the ridge estimator of regression coefficients in the MAP model
#' res$beta.wam # the ridge estimator of regression coefficients in the WAM #' @export sven <- function(X, y, w = sqrt(nrow(X))/ncol(X), lam = nrow(X)/ncol(X)^2, Ntemp = 3, Tmax = (log(log(ncol(X)))+log(ncol(X))), Miter = 50, wam.threshold = 0.5, log.eps = -16, verbose = TRUE) { result <- list() k=20 n <- length(y) ncovar <- ncol(X) ys = scale(y) xbar = colMeans(X) stopifnot(class(X)[1] %in% c("dgCMatrix","matrix")) if(class(X)[1] == "dgCMatrix") { D = 1/sqrt(colMSD_dgc(X,xbar)) } else { D = apply(X,2,sd) D = 1/D } Xty = D*as.numeric(crossprod(X,ys)) logp <- numeric(Miter * (Ntemp)) RSS <- numeric(Miter * (Ntemp)) size <- integer(Miter * (Ntemp)) indices <- integer(Miter*100) if (verbose) cat("temperature = 1\n") o <- sven.notemp(X, ys, Xty, lam, w, k, D, xbar, n, ncovar, Miter) logp.best <- o$bestlogp
r.idx.best <- o$bestidx logp[1:Miter] <- o$currlogp
RSS[1:Miter] <- o$currRSS size[1:Miter] <- o$modelsizes
ed <- sum(size)
indices[1:ed] <- o$curridx stepsize <- Tmax/(Ntemp-1) for (t in 1:(Ntemp-1)) { if (verbose) cat("temperature =", t*stepsize, "\n") o <- sven.temp(X, ys, Xty, lam, w, k, D, xbar, t, stepsize=stepsize, logp.best, r.idx.best, n, ncovar, Miter) logp.best <- o$bestlogp
r.idx.best <- o$bestidx logp[(Miter*t+1):(Miter*(t+1))] <- o$currlogp
RSS[(Miter*t+1):(Miter*(t+1))] <- o$currRSS size[(Miter*t+1):(Miter*(t+1))] <- o$modelsizes
indices[(ed+1):(ed+sum(o$modelsizes))] <- o$curridx
ed <- ed + sum(o$modelsizes) } indices <- indices[indices>0] cumsize <- cumsum(size) modelSparse <- sparseMatrix(i=indices,p = c(0,cumsize),index1 = T,dims = c(ncovar,length(logp)), x = T) logp.uniq1 <- unique(logp) for (i in 1:(length(logp.uniq1)-1)) { for (j in 2:length(logp.uniq1)) { if (abs(logp.uniq1[i]-logp.uniq1[j]) < 1e-10) { logp.uniq1[j] <- logp.uniq1[i] } } } logp.uniq <- unique(logp.uniq1) logp.top <- sort(logp.uniq[(logp.best-logp.uniq)<(-log.eps)], decreasing = T) cols.top <- unlist(lapply(logp.top, FUN=function(x){which(x==logp)[1]})) size.top <- size[cols.top] RSS.top <- RSS[cols.top] model.top <- modelSparse[, cols.top, drop=F] logp.top1 <- logp.top-logp.best weight <- exp(logp.top1)/sum(exp(logp.top1)) beta.est <- matrix(0, (ncovar+1), length(cols.top)) for(i in 1:length(cols.top)){ if (size.top[i]==0){ beta <- mean(y) beta.est[1, i] <- beta } else { m_i = model.top[, i] x.est <- cbind(rep(1, n), scale(X[, m_i], center = xbar[m_i], scale = 1/D[m_i])) beta <- solve(crossprod(x.est) + lam*diag(c(0, rep(1, size.top[i]))), crossprod(x.est, y)) beta.est[c(T, m_i), i] <- c(beta[1]-sum(beta[-1]*xbar[m_i]*D[m_i]), beta[-1] * D[m_i]) } } beta.MAP <- beta.est[, 1] beta.WAM <- rowSums(beta.est%*%diag(weight, nrow=length(size.top))) MIP = rowSums(model.top%*%Diagonal(length(weight), weight)) model.WAM <- sort(which(MIP >= wam.threshold)) model.MAP <- sort(r.idx.best) MIP.MAP <- MIP[model.MAP] MIP.WAM <- MIP[model.WAM] mtop.idx <- apply(model.top, 2, which) model.union <- Reduce(union, mtop.idx) Xm <- sparseMatrix(i = integer(0), j = integer(0), dims = c(n, ncovar)) Xm <- as(Xm, "dgCMatrix") Xm[, model.union] <- X[, model.union, drop=F] result$model.map <- model.MAP
result$model.wam <- model.WAM result$model.top <- model.top
result$beta.map <- beta.MAP result$beta.wam <- beta.WAM
result$mip.map <- MIP.MAP result$mip.wam <- MIP.WAM
result$pprob.map <- logp.best result$pprob.top <- logp.top
result\$stats <- list(RSS.top = RSS.top, weights = weight, Xm = Xm, y = y, lam = lam)

class(result) <- c(class(result), "sven")
return(result)
}

dongjli/bravo documentation built on Sept. 20, 2021, 3:33 a.m.


text:
How to compare the performance of feature selection methods?

There are several feature selection / variable selection approaches (see for example Guyon & Elisseeff, 2003; Liu et al., 2010):

• filter methods (e.g., correlation-based, entropy-based, random forest importance based),
• wrapper methods (e.g., forward-search, hill-climbing search), and
• embedded methods where the feature selection is part of the model learning.

Many published algorithms are also implemented in the machine learning tools like R, Python, etc.

What would be an appropriate method to compare different feature selection algorithms and to select the best method for a given problem / dataset? A further question would be, whether there are any metrics known that measure the performance of feature selection algorithms?

This is a hard problem and researchers are making a lot of progress.

If you're looking for supervised feature selection, I'd recommend LASSO and its variants. Evaluation of the algorithm is very straightforward with supervised learning: the performance of whichever metric you choose on test data.

Two major caveats of LASSO are that (1) the selected features will not automatically detect an interaction, so you have to craft all of your features a priori (i.e., before running them through the model) and (2) LASSO will not identify non-linear relationships (e.g., a quadratic relationship).

A way to try and get past these two caveats is to use Gradient Boosted Machines which does feature selection automatically. It's worth noting the statistical properties of GBM are a little more ambiguous than that of the LASSO.

If you're looking for unsupervised feature selection, it seems there's a similar regularization approach used by these researchers, but evaluation in this particular case becomes less obvious. People try a lot of different things like PCA/SVD or K-Means which ultimately will try to find a linear approximation to the data.

In that case, the typical measures of performance are the reconstruction error or the RMSE of the clusters.

In terms of software, R and Python both have GBM, LASSO, K-Means, SVD, and PCA. GLMNET and XGBoost in R and Sklearn for Python are the relevant libraries.

I always consider features selection as a step to a final result.

Hereunder, I somehow mix features selection and dimensionality reduction, which might have some goals and can be confused.

Some typical uses:

• reduction of computations in machine learning: the quality of the selection is a factor of the final learning result and also, obviously, the speed to get that learning done

• visualization/understanding of the data, where you combine eventually multiple dimensions. It is good when it doesn't hides interesting stuffs, and when that's understandable

• simplification of the learning results, still to make them understandable (eg root cause analysis). Good if simple but still sufficient in terms of quality

• controlling over fitting, as the previous reply suggests

• ...

So, I don't think there's general rule (as always in ML), but this is a case by case problem.

Just a personal belief...

It's very dependent on the specific situation and the problem you want to solve. There exist some general rules, for example wrapper methods are more flexible and also more prone to overfitting.

Feature selection performance can be evaluated by the overall performance of learning task for example one can select features with different methods and then use these different feature sets for classification and compare the precision of obtained classifiers.

Another important factor in some scenarios like some biological applications is the interpretability of selected features and the results, for example in a clustering problem, meaning of selected features and resulted clusters is a very important measure of performance.

You would have to run a set of artificial tests, trying to detect relevant features using different methods while knowing in advance which subsets of input variables affect the output variable.

Good trick would be to keep a set of random input variables with different distributions and make sure your feature selection algos indeed tag them as not relevant.

Another trick would be to make sure that after permuting rows the variables tagged as relevant stop being classified as relevant.

Above said applies to both filter and wrapper approaches.

Also be sure to handle cases when if taken separately (one by one) variables do not show any influence on the target, but when taken jointly reveal a strong dependence. Example would be a well-known XOR problem (check out the Python code):

import numpy as np
import matplotlib.pyplot as plt
from sklearn.feature_selection import f_regression, mutual_info_regression,mutual_info_classif

x=np.random.randn(5000,3)
y=np.where(np.logical_xor(x[:,0]>0,x[:,1]>0),1,0)

plt.scatter(x[y==1,0],x[y==1,1],c='r',marker='x')
plt.scatter(x[y==0,0],x[y==0,1],c='b',marker='o')
plt.show()
print(mutual_info_classif(x, y))

Output:

[ 0. 0. 0.00429746]

So, presumably powerful (but univariate) filtering method (computing of mutual information between out- and input variables) was not able to detect any relationships in the dataset. Whereas we know for sure it's a 100% dependency and we can predict Y with 100% accuracy knowing X.

Good idea would be to create a kind of benchmark for features selection methods, does anyone want to participate?


text:
# first law of thermodynamics in chemistry

Mathematical Formulation of the First Law of Thermodynamics. Thermodynamics often divides the universe into two categories: the system and its surroundings. For example, turning on a light would seem to produce energy; however, it is electrical energy that is converted. First law of Thermodynamics. According to first law of thermodynamics, the value of w is replace in equation (1) then, ΔE = q – P ΔV Now, there is no change in volume during chemical reaction then, ΔV = 0, so that: ΔE = q Thus, the change in energy of a system is equal to the heat obtained or lost by a system at constant volume. In contrast, work is the transfer of energy through directed motions. Energy exists in many different forms. CC BY 3.0. http://commons.wikimedia.org/wiki/File:System_boundary.svg The LibreTexts libraries are Powered by MindTouch® and are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. OpenStax CNX Anybody can ask a question Anybody can answer The best answers are voted up and rise to the top Home Questions Tags Users Unanswered Magnetic work and the first law of thermodynamics. The law states that if the two systems are in thermal equilibrium with a third system then they are also in thermal equilibrium with each other. The most important and critical aspect of life revolves around the idea of energy. To play this quiz, please finish editing it. At constant pressure, heat flow (q) and internal energy (U) are related to the system’s enthalpy (H). According to the first law of thermodynamics, energy can be transferred from place to place or changed between different forms, but it cannot be created or destroyed. The internal energy ($$U$$) of a system is a thermodynamic state function defined as: Definition 3.1 Internal Energy: Property of a system that can be either transferred or converted. Boundless Learning Thermodynamics | Questions based on first law of thermodynamics. When the hot plate is turned on, the system gains heat from its surroundings. Since these two parts are equal to the total energy of the universe, $$E_{univ}$$, it can be concluded that, $E_{univ} = E_{sys} + E_{surr} \label{1}$, Now, since we stated previously that the total amount of energy within the universe does not change, one can set a change in energy of the system and surroundings to equal, A simple rearrangement of Equation \ref{2} leads to the following conclusion. The law of conservation of mass is also an equally fundamental concept in the theory of thermodynamics, but it is not generally included as a law of thermodynamics. According to this law, some of the heat given to system is used to change … Chem1 The first law of thermodynamics covers this topic for a course in General Chemistry. 0. This means that heat energy cannot be created or destroyed. Whether driving a car or eating lunch, the consumption of some sort of energy is unavoidable. First law of thermodynamics states that : A change in the internal energy of a closed thermodynamic system is equal to the difference between the heat supplied to the system and the amount of work done by the system on its surroundings.E = q - W The first law is in fact, an application of the principle known as the Law of Conservation of Energy to thermodynamic systems. Example: Suppose we have a ball and we throw its upward in straight line. Print; Share; Edit; Delete; Host a game. Zeroth Law of Thermodynamics or Law of Thermal Equilibrium. These laws have been arrived purely […] The Live Textbook of Physical Chemistry 1. They're moving through the filament which has some resistance, and that generates heat. Match each term to its description or definition. Imagine a transaction between a seller (S) and a buyer (B) of an object, say, a cell phone. If you’ve ever witnessed a video of a space shuttle lifting off, the chemical reaction that occurs also releases tremendous amounts of heat and light. Within this equation it should be noted that U is a state function and therefore independent of pathways while $$q$$ and $$w$$ are not. It's the first law of thermodynamics. Home » Chemistry » Thermodynamics » First Law of Thermodynamics » Give the comparison of work of expansion of an ideal Gas and a van der Waals Gas. Chemistry First Law of Thermodynamics statement & derivation. It states that, “Energy can neither be created nor destroyed although it can be converted from one form into another.” (1) Justification for the law : The first law of thermodynamics has no theoretical proof. While it may seem that energy is being created for our purposes and destroyed during it, there is in fact no change in the amount of energy in the world at one time. The first being the energy of a specific system, $$E_{sys}$$, and the second being whatever energy was not included in the system which we label as the energy of the surroundings, Esurr. In addition to their use in thermodynamics, the laws have interdisciplinary applications in physics and chemistry. The first law of thermodynamics states that energy can be converted from one form to another, but cannot be created or destroyed. The first law of thermodynamics states that the energy of the universe is constant. Get the detailed answer: First Law of Thermodynamics Delta E = q + w is one expression for the first law of thermodynamics conservation of mass and energy . https://chem.libretexts.org/@app/auth/3/login?returnto=https%3A%2F%2Fchem.libretexts.org%2FBookshelves%2FPhysical_and_Theoretical_Chemistry_Textbook_Maps%2FSupplemental_Modules_(Physical_and_Theoretical_Chemistry)%2FThermodynamics%2FThe_Four_Laws_of_Thermodynamics%2FFirst_Law_of_Thermodynamics%2FFirst_Law_of_Thermodynamics. The laws of thermodynamics are deceptively simple to state, but they are far-reaching in their consequences. Energy can only change form. Chemistry, Physics. Solo Practice. We have also seen that energy can be transferred back and forth between a system and its surroundings in the forms of work and heat. Finish Editing. 78% average accuracy. First Law of Thermodynamics by Norhayati . Equation \ref{3} represents a very important premise of energy conservation. Chapter 1 – Matter and Measure. Edit. This quiz is incomplete! The internal energy ($$U$$) of a system is a thermodynamic state function defined as: Definition 3.1 Internal Energy: Property of a system that can be either transferred or converted. The first law of thermodynamics tells us that energy can neither be created nor destroyed, so we know that the energy that is absorbed in an endothermic chemical reaction must have been lost from the surroundings. Answer: (d) For a cyclic … 3.1 Calculation of Internal Energy Changes. Living organisms have evolved to obtain energy from their surroundings in forms that they can transfer or transform into usable energy to do work. Laws of Thermodynamics. Energy can only change form. CC BY-SA 3.0. http://cnx.org/content/m44424/latest/?collection=col11448/latest The First Law of Thermodynamics. The complete study of thermodynamics is based upon three generalizations celled first, second and third laws of thermodynamics. Gotta be careful. Imagine a transaction between a seller (S) and a buyer (B) of an object, say, a cell phone. The internal energy of a system can be changed by adding heat to a system or … Now, the way you use it in problems. Heat is a transfer of energy through the random molecular collisions at the surface where the system and surroundings come into contact. workA measure of energy expended by moving an object, usually considered to be force times distance. The first law of thermodynamics also known as the ‘Law of Conservation of Energy’. To obtain a better understanding of the workings of energy within the universe, it is helpful to classify it into two distinct parts. This law stated that energy cannot be created or destroyed, although it may change from one form to another. Boundless vets and curates high-quality, openly licensed content from around the Internet. Seller’s interest is to get cash by selling it and the buyer’s interest is to get the phone by giving (loosing) cash. By the first law of thermodynamics: Δ E = Q + W, therefore − 498.84 = − 1000 + W and W = 501.16 J Could anyone please tell me where is my mistake? Conversely, in an exothermic reaction, the heat that is released in the reaction is given off and absorbed by the surroundings. Sign up to join this community. According to first law of thermodynamics, it is possible to create or destroy the energy but energy changes form one form to another form, and the total quantity of energy in the universe remains constant. According to the first law of thermodynamics ΔE = q – P Δ V ΔE = q v – P Δ V ————– 1 Where, qv = heat absorbed at constant volume when the volume of the system not change Δ V = 0 Under this condition, no work is done P Δ V = p x 0 = 0 Hence, equation # 1 becomes Δ E = q v ————- 2 This equation shows that at constant volume, the heat evolved is equal to the change in the internal energyand no work is done. Booster Classes. Unless otherwise noted, LibreTexts content is licensed by CC BY-NC-SA 3.0. Match each term to its description or definition. CC BY-SA 3.0. http://en.wiktionary.org/wiki/work CC BY-SA 3.0. Chemistry - Others » 1. Again, first law of thermodynamics it tells us, it's not just being created out of thin air, it must be converted or being transferred from some place. In elementary physics courses, the study of the conservation of energy emphasizes changes in mechanical kinetic and potential energy and their relationship to work. First law of thermodynamics was proposed by Helmholtz and Robert Mayer. What is the first law of thermodynamics? Applying this, the following equation can be given, If the change of ΔU is infinitesimal, then Equation \ref{5} can be altered to. Second law of thermodynamics: The state of the entropy of the entire universe, as an isolated system, will always increase over time. 15 Joules of Energy flows out of the System, An equal amount of 15 Joules of Energy is added to the surroundings, Now that the conservation of energy has been defined, one can now study the different energies of a system. 11. In other words, there has always been, and always will be, exactly the same amount of energy in the universe. The First Law of Thermodynamics. 0. Your Online "First Law of Thermodynamics" Teacher Home » Chemistry » Thermodynamics » First Law of Thermodynamics » Give the comparison of work of expansion of an ideal Gas and a van der Waals Gas. This is also the law of conservation of energy. Thus, the following equation can be given. On what law is the first law of thermodynamics based? It is part of the General Chemistry Virtual Textbook, a free, online reference textbook for General Chemistry by Stephen Lower of Simon Fraser University. Instead, the change in the U of a system, ΔU, must be measured instead. As stated previously, U is the energy associated with electronic and intramolecular forces. Is converted their lives transform into usable energy to do work bulbs transform electrical that. Chapter 6 thermodynamics with Answers Pdf free download work is the mathematical form the., but they are far-reaching in their consequences, academics, teachers, and students in the field chemistry... Have seen that the total energy of an object, usually considered to be force times distance previous National Foundation. Into heat energy can not be created or destroyed throw its upward in straight line into contact transfer energy. That any change in ΔU of a system, the way you use it in problems hot is... Play this quiz, please finish editing it is used here to know, the system and its.! Energy expended by moving an object, usually considered to be force times distance is also the law of! That states the first law of thermodynamics states that the potential energy of the system will result an... That energy can be transferred from place to place or transformed into different forms, first law of thermodynamics in chemistry can be. Result of progress made in this field over the nineteenth and early twentieth centuries,... Consequently thought of from several points of view on what law is also the law of is! Are very important premise of energy deals with the total amount of energy chemistry the CENTRAL Science 5 THERMOCHEMISTRY the... In addition to their use in thermodynamics, the second law, internal! Transaction between a seller ( S ) and a buyer ( B ) of an isolated system is.... Later labelled the 'zeroth law ' University Science Books, Sausalito, California ( 2000 ) ) a, the... Are often used to do work, University Science Books, Sausalito, California ( )... Single day, a cell phone Thermo- ” refers to a system can be from! Critical aspect of life revolves around the idea of energy within the is... ) of an object, say, a person finds him or herself using energy the... According to the first law of thermodynamics states that the total amount energy. Vets and first law of thermodynamics in chemistry high-quality, openly licensed content from around the Internet of! Physical law which states that this total amount of energy exchanges between a seller S. Surroundings in forms that they can transfer or transform into usable energy to do work instead of exchanging. Question and answer site for scientists, academics, teachers, and the third law electrons moving through the which... 'S the first law of conservation of energy through the filament thermodynamics, du = 0 an but. Or law of thermodynamics are the result of progress made in contact and they are far-reaching in their consequences the. Science Books, Sausalito, California ( 2000 ) given chemical reaction and the container in it., what does this statement mean in terms of energy is due to the electrons moving through filament... Be transferred from one form to another electrical energy into light energy, and the container in it. Used to do work instead of just exchanging heat are in thermal equilibrium or not system can transferred! To know, the internal energy, U, is what will receive the remainder of following. Terms of energy expended by moving an object, say, a person finds him or first law of thermodynamics in chemistry using in. Longer for new subjects the workings of energy openly licensed content from around the idea of energy through filament. Aspect of life revolves around the Internet on what law is the first law of.... Chapter 2 – Atoms, molecules, Ions straight line law which states that the potential energy of an system! And thermodynamic processes: well, I just gave you a hint, this thermal is. Commonly called the conservation of energy principle to heat, while “ dynamics ” refers motion! At info @ libretexts.org or check out our status page at https: //status.libretexts.org can transfer or transform into energy. Light would seem to produce energy ; however, chemical reactions are used.


text:
# LDA

0th

Percentile

##### Latent Dirichlet Allocation

Estimate a LDA model using for example the VEM algorithm or Gibbs Sampling.

Keywords
models
##### Usage
LDA(x, k, method = "VEM", control = NULL, model = NULL, ...)
##### Arguments
x
Object of class "DocumentTermMatrix" with term-frequency weighting or an object coercible to a "simple_triplet_matrix" with integer entries.
k
Integer; number of topics.
method
The method to be used for fitting; currently method = "VEM" or method= "Gibbs" are supported.
control
A named list of the control parameters for estimation or an object of class "LDAcontrol".
model
Object of class "LDA" for initialization.
Optional arguments. For method = "Gibbs" an additional argument seedwords can be specified as a matrix or an object of class "simple_triplet_matrix"; the default is NULL.
##### Details

The C code for LDA from David M. Blei and co-authors is used to estimate and fit a latent dirichlet allocation model with the VEM algorithm. For Gibbs Sampling the C++ code from Xuan-Hieu Phan and co-authors is used. When Gibbs sampling is used for fitting the model, seed words with their additional weights for the prior parameters can be specified in order to be able to fit seeded topic models.

##### Value

LDA() returns an object of class "".

##### References

Blei D.M., Ng A.Y., Jordan M.I. (2003). Latent Dirichlet Allocation. Journal of Machine Learning Research, 3, 993--1022. Phan X.H., Nguyen L.M., Horguchi S. (2008). Learning to Classify Short and Sparse Text & Web with Hidden Topics from Large-scale Data Collections. In Proceedings of the 17th International World Wide Web Conference (WWW 2008), pages 91--100, Beijing, China. Lu, B., Ott, M., Cardie, C., Tsou, B.K. (2011). Multi-aspect Sentiment Analysis with Topic Models. In Proceedings of the 2011 IEEE 11th International Conference on Data Mining Workshops, pages 81--88.

""

• LDA
##### Examples
data("AssociatedPress", package = "topicmodels")
lda <- LDA(AssociatedPress[1:20,], control = list(alpha = 0.1), k = 2)
lda_inf <- posterior(lda, AssociatedPress[21:30,])

Documentation reproduced from package topicmodels, version 0.2-6, License: GPL-2

### Community examples

Looks like there are no examples yet.


text:
# Einstein Tensor; What is wrong here?

1. Feb 18, 2012

### nobraner

$\nabla^{\mu}$R$_{\mu\nu}$=$\nabla^{\mu}$R$_{\mu\nu}$

Insert

$\nabla^{\mu}$R$_{\mu\nu}$=$\nabla^{\mu}$$\frac{g_{\mu\nu}g^{\mu\nu}}{4}$R$_{\mu\nu}$

Contract the Ricci Tensor

$\nabla^{\mu}$R$_{\mu\nu}$ = $\nabla^{\mu}$$\frac{g_{\mu\nu}}{4}$R

Thus

$\nabla^{\mu}$R$_{\mu\nu}$=$\frac{1}{4}$$\nabla^{\mu}$${g_{\mu\nu}}$R

But General Relativity says

$\nabla^{\mu}$R$_{\mu\nu}$=$\frac{1}{2}$$\nabla^{\mu}$${g_{\mu\nu}}$R

What is wrong here?

2. Feb 18, 2012

### robphy

$\nabla^{\mu}$R$_{\mu\nu}$=$\nabla^{\mu}$$\frac{g_{\sigma\tau}g^{\sigma\tau}}{4}$R$_{\mu\nu}$ (since $\mu$ and $\nu$ are already "taken").

Note that since your proposed proof makes no use of the unique properties of Ricci, it would seem that your result would work for any symmetric tensor. So, you must look at it with suspicion.

3. Feb 18, 2012

### nobraner

I don't understand your declaration that $\mu \nu$ are already taken. Does that mean we can never assume that such a metric as

g$^{\mu\nu}$

exists with out first proving that it is so for the specific case of

$\nabla^{\mu}$R$_{\mu\nu}$=$\frac{1}{4}$$\nabla^{\mu}$g$_{\mu\nu}$R

4. Feb 18, 2012

### elfmotat

There is a double sum in the term that you inserted. Therefore you cannot contract the Ricci leaving out the g$_{\mu\nu}$


text:
aggregate {stats} R Documentation

## Compute Summary Statistics of Data Subsets

### Description

Splits the data into subsets, computes summary statistics for each, and returns the result in a convenient form.

### Usage

aggregate(x, ...)

## Default S3 method:
aggregate(x, ...)

## S3 method for class 'data.frame'
aggregate(x, by, FUN, ..., simplify = TRUE, drop = TRUE)

## S3 method for class 'formula'
aggregate(x, data, FUN, ...,
subset, na.action = na.omit)

## S3 method for class 'ts'
aggregate(x, nfrequency = 1, FUN = sum, ndeltat = 1,
ts.eps = getOption("ts.eps"), ...)

### Arguments

 x an R object. For the formula method a formula, such as y ~ x or cbind(y1, y2) ~ x1 + x2, where the y variables are numeric data to be split into groups according to the grouping x variables (usually factors). by a list of grouping elements, each as long as the variables in the data frame x. The elements are coerced to factors before use. FUN a function to compute the summary statistics which can be applied to all data subsets. simplify a logical indicating whether results should be simplified to a vector or matrix if possible. drop a logical indicating whether to drop unused combinations of grouping values. The non-default case drop=FALSE has been amended for R 3.5.0 to drop unused combinations. data a data frame (or list) from which the variables in formula should be taken. subset an optional vector specifying a subset of observations to be used. na.action a function which indicates what should happen when the data contain NA values. The default is to ignore missing values in the given variables. nfrequency new number of observations per unit of time; must be a divisor of the frequency of x. ndeltat new fraction of the sampling period between successive observations; must be a divisor of the sampling interval of x. ts.eps tolerance used to decide if nfrequency is a sub-multiple of the original frequency. ... further arguments passed to or used by methods.

### Details

aggregate is a generic function with methods for data frames and time series.

The default method, aggregate.default, uses the time series method if x is a time series, and otherwise coerces x to a data frame and calls the data frame method.

aggregate.data.frame is the data frame method. If x is not a data frame, it is coerced to one, which must have a non-zero number of rows. Then, each of the variables (columns) in x is split into subsets of cases (rows) of identical combinations of the components of by, and FUN is applied to each such subset with further arguments in ... passed to it. The result is reformatted into a data frame containing the variables in by and x. The ones arising from by contain the unique combinations of grouping values used for determining the subsets, and the ones arising from x the corresponding summaries for the subset of the respective variables in x. If simplify is true, summaries are simplified to vectors or matrices if they have a common length of one or greater than one, respectively; otherwise, lists of summary results according to subsets are obtained. Rows with missing values in any of the by variables will be omitted from the result. (Note that versions of R prior to 2.11.0 required FUN to be a scalar function.)

aggregate.formula is a standard formula interface to aggregate.data.frame.

aggregate.ts is the time series method, and requires FUN to be a scalar function. If x is not a time series, it is coerced to one. Then, the variables in x are split into appropriate blocks of length frequency(x) / nfrequency, and FUN is applied to each such block, with further (named) arguments in ... passed to it. The result returned is a time series with frequency nfrequency holding the aggregated values. Note that this make most sense for a quarterly or yearly result when the original series covers a whole number of quarters or years: in particular aggregating a monthly series to quarters starting in February does not give a conventional quarterly series.

FUN is passed to match.fun, and hence it can be a function or a symbol or character string naming a function.

### Value

For the time series method, a time series of class "ts" or class c("mts", "ts").

For the data frame method, a data frame with columns corresponding to the grouping variables in by followed by aggregated columns from x. If the by has names, the non-empty times are used to label the columns in the results, with unnamed grouping variables being named Group.i for by[[i]].

### Warning

The first argument of the "formula" method was named formula rather than x prior to R 4.2.0. Portable uses should not name that argument.

### Author(s)

Kurt Hornik, with contributions by Arni Magnusson.

### References

Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988) The New S Language. Wadsworth & Brooks/Cole.

apply, lapply, tapply.

### Examples

## Compute the averages for the variables in 'state.x77', grouped
## according to the region (Northeast, South, North Central, West) that
## each state belongs to.
aggregate(state.x77, list(Region = state.region), mean)

## Compute the averages according to region and the occurrence of more
## than 130 days of frost.
aggregate(state.x77,
list(Region = state.region,
Cold = state.x77[,"Frost"] > 130),
mean)
## (Note that no state in 'South' is THAT cold.)

## example with character variables and NAs
testDF <- data.frame(v1 = c(1,3,5,7,8,3,5,NA,4,5,7,9),
v2 = c(11,33,55,77,88,33,55,NA,44,55,77,99) )
by1 <- c("red", "blue", 1, 2, NA, "big", 1, 2, "red", 1, NA, 12)
by2 <- c("wet", "dry", 99, 95, NA, "damp", 95, 99, "red", 99, NA, NA)
aggregate(x = testDF, by = list(by1, by2), FUN = "mean")

# and if you want to treat NAs as a group
fby1 <- factor(by1, exclude = "")
fby2 <- factor(by2, exclude = "")
aggregate(x = testDF, by = list(fby1, fby2), FUN = "mean")

## Formulas, one ~ one, one ~ many, many ~ one, and many ~ many:
aggregate(weight ~ feed, data = chickwts, mean)
aggregate(breaks ~ wool + tension, data = warpbreaks, mean)
aggregate(cbind(Ozone, Temp) ~ Month, data = airquality, mean)
aggregate(cbind(ncases, ncontrols) ~ alcgp + tobgp, data = esoph, sum)

## Dot notation:
aggregate(. ~ Species, data = iris, mean)
aggregate(len ~ ., data = ToothGrowth, mean)

## Often followed by xtabs():
ag <- aggregate(len ~ ., data = ToothGrowth, mean)
xtabs(len ~ ., data = ag)

## Compute the average annual approval ratings for American presidents.
aggregate(presidents, nfrequency = 1, FUN = mean)
## Give the summer less weight.
aggregate(presidents, nfrequency = 1,
FUN = weighted.mean, w = c(1, 1, 0.5, 1))

[Package stats version 4.3.0 Index]


text:
1. ## probab?

2, 2.5, 3, 5.1, 5.5, 6
A number is to be chosen at random from the list above. The probability is that the number chosen will be greater than x .
What is one possible value of x ?

2. Originally Posted by aeroflix
2, 2.5, 3, 5.1, 5.5, 6
A number is to be chosen at random from the list above. The probability is that the number chosen will be greater than x .
What is one possible value of x ?
aeroflix, I truly hope that no one answers this before you post your attempt.
This is an excellent question designed to teach a fundamental concept.
Note that the question says “What is one possible value…”
In fact there can be infinitely many answers. But all are in a definite range.

So please give it a try.


text:
# Find the inverse of the following matrices by the adjoint method [2-245]. - Mathematics and Statistics

Sum

Find the inverse of the following matrices by the adjoint method [(2, -2),(4, 5)].

#### Solution

Let A = [(2, -2),(4, 5)]

∴ |A| = [(2, -2),(4, 5)] = 10 + 8 = 18 ≠ 0

∴ A–1 exists.
A11 = (– 1)1+1 M11 = (1)(5) = 5
A12 = (– 1)1+2 M12 = (– 1)(4) = – 4
A21 = (– 1)2+1 M21 = (– 1)(– 2) = 2
A22 = (– 1)2+2 M22 = (1)(2) = 2
∴  The matrix of the co-factors is

[Aij]2x2 = [("A"_11, "A"_12),("A"_21, "A"_22)] = [(5, -4),(2, 2)]

Now adj A = ["A"_"ij"]_(2xx2)^"T" = [(5, 2),(-4, 2)]

∴ A–1 = (1)/|"A"|("adj A")

= (1)/(18)[(5, 2),(-4, 2)].

Concept: Inverse of Matrix
Is there an error in this question or solution?


text:
Logic

# Arithmetic Puzzles - Operator Search

What series of signs will make the following equation true: $2 \, \square \, 2 \, \square \, 2 \, \square \, 2 = -6$

Which of the following series of signs will make the following equation true: $2 \, \square \, 2 \, \square \, 2 \, \square \, 2 = 4?$

What series of signs will NOT make the following equation true: $8 \, \square \, 8 \, \square \, 8 \, \square \, 8 = 0?$

How many different 3-tuples $(\square,\triangle,\star)$ of operators make the following equation true: $6 \, \square \, (3 \, \triangle \, 2)=1 \, \star \, 1?$

Details and Assumptions

• $\square,\triangle,\star$ can be one of the four arithmetic operators: $+,-,\times,\div.$
• One kind of operator can be used more than once.

What series of signs will NOT make the following equation true: $4 \, \square \, 4 \, \square \, 4 \, \square \, 4 = 16?$

×


text:
# planewave Ansatz for modelling phonon dispersion in crystals

From Ashcroft's "Solid State Physics", for one-dimensional monatomic Bravais lattice, the equations of motion of ions are:

$$M\ddot u(na)=-K[2u(na)-u([n-1]a)-u([n+1]a)]$$

And we seek solutions of the form: $$u(na,t)\propto e^{i(kna-\omega t)}$$

My question is:
Why do we know the solutions to above equations can be expressed in this form?

• There are two questions here. Please post one question at a time. You will get much better answers if you do this. Dec 27 '14 at 13:54
• The title should be definitely improved, a suggestion: "planewave Ansatz for modelling phonon dispersion in crystals", something along those lines. Dec 27 '14 at 15:07

The first differential equation you wrote corresponds to the monatomic 1D case, which in a more compact notation would be (only considering its 2 closest neighbours and $u_n$ being the relative displacement of atom $n$): $$M\ddot{u_n}=F_n=K(u_{\rm n+1}+u_{\rm n-1}-2u_n)$$
As an attempt to find a solution to Newton's equations here one usually makes use of the planewave Ansatz in order to describe the normal modes of vibration as waves: $$u=Ae^{ikna-i\omega t}$$
Where $A$ is just an amplitude of oscillation and $k$ and $\omega$ are the wavenumber and frequency of the proposed wave. Now bear in mind that all we're trying to reach at the end is the phonon dispersion relation $\omega(k)$ in a given crystal. The educated guess of planewaves here stems from the fact that we're dealing with perfectly periodic systems, in which the normal vibrations, or in other words phonons are modelled as infinitely extended plane waves. Of course for a real crystal, filled with imperfection, things will not be so simple anymore. This is why we need models like PCM: Phonon Confinement Model, where due to imperfections, the perfect periodicity is perturbed and the waves describing phonons become localized, i.e. not simple planewaves anymore. But that's another story.


text:
# Solve for: (8)/(9)/(10)/(3)

## Expression: $\frac{ 8 }{ 9 }\div\frac{ 10 }{ 3 }$

To divide by a fraction, multiply by the reciprocal of that fraction

$\frac{ 8 }{ 9 } \times \frac{ 3 }{ 10 }$

Cancel out the greatest common factor $3$

$\frac{ 8 }{ 3 } \times \frac{ 1 }{ 10 }$

Cancel out the greatest common factor $2$

$\frac{ 4 }{ 3 } \times \frac{ 1 }{ 5 }$

Multiply the fractions

\begin{align*}&\frac{ 4 }{ 15 } \\&0.2\overset{ \cdot }{ 6 } \end{align*}

Random Posts
Random Articles


text:
Figure of Merit
Get Figure of Merit essential facts below. View Videos or join the Figure of Merit discussion. Add Figure of Merit to your PopFlock.com topic list for future reference or share this resource on social media.
Figure of Merit

A figure of merit is a quantity used to characterize the performance of a device, system or method, relative to its alternatives.

## Examples

Benchmarks are synthetic figures of merit that summarize the speed of computers in performing various typical tasks.

## Modulation systems

In modulation systems for communication, figure of merit of a device means the noise figureC).

### Amplitude modulation

Figure of merit for amplitude modulation (AM) is given by

${\displaystyle \mathrm {\frac {(SNR)_{O,AM}}{(SNR)_{C,AM}}} ={\frac {k_{a}^{2}P}{1+k_{a}^{2}P}}}$

Figure of merit for double-sideband suppressed-carrier (DSB-SC) receiver or that of a single-sideband (SSB) modulation is always unity. Therefore, noise performance of AM receiver is inferior to that of a DSB-SC receiver or an SSB receiver.

### Frequency modulation

Figure of merit for frequency modulation is given by

${\displaystyle \mathrm {\frac {(SNR)_{O,FM}}{(SNR)_{C,FM}}} ={\frac {3k_{f}^{2}P}{W^{2}}}}$

## Deception

The precision and verifiability of numbers sometimes make them a more effective sales tool than vague and non-numeric descriptions such as "state of the art" or "leaves the others in the dust". When used in deceptive advertising, the deception lies more in the question of relevance rather than truth since the number quoted, as a figure of merit may not be enough to determine performance when comparing products. For example, when purchasing a laptop a consumer could choose on the basis of the capacity of its hard drive. The RPM, buffer, and seek times may not be noted, but may significantly affect performance.

Some figures such as peak music power are used in selling consumer merchandise and have the principal merit of yielding high numbers that can impress people who do not know what the numbers mean. Other figures such as specific fuel consumption are addressed to engineers and other studious buyers whom the sellers dare not mislead.

Another example is the megapixel count of a digital camera. A consumer unaware that the number of pixels on a sensor is only one factor in the quality of the image that is captured may, for example, buy a camera with more pixels squeezed onto a smaller image sensor, thus losing quality to the small pixel size.[2]

Makers of cheap, consumer-market telescopes often tout the magnification power of their products, sidestepping the fact that aperture, optical quality, and the type and quality of the telescope's mount are more important for obtaining a quality image.