text:
# CSS Grid Layout: A Quick Start Guide

This post is part of a series called Understanding the CSS Grid Layout Module.
CSS Grid Layout: Fluid Columns and Better Gutters

How we approach layout on the web is changing, and at the forefront of that change is CSS Grid Layout. This quick start CSS grid tutorial will skip the details and nuances, instead helping you get stuck in, right now.

## CSS Grid Layout and Your Browser

CSS Grid Layout (known to its friends as “Grid”) has come on leaps and bounds in the last couple of years, and as such you’ll find browser support for it pretty solid nowadays.

## Setting up a Grid Layout

Grid allows us to arrange elements on a page, according to regions created by guides.

### Terminology

At their simplest these guides, or grid lines, frame horizontal and vertical grid tracks. Grid tracks serve as rows and columns, with gutters running between them. Where horizontal and vertical grid tracks intersect, we’re left with cells, much like we use with tables. These are all important terms to understand.

In the image below you’ll see a demo grid, showing:

1. grid lines
2. columns
3. rows
4. cells

For a graphic layout, it might look more familiar if we use exactly the same grid, but part some tracks to give us gutters between content areas.

1. gutters

There’s one last term we need to clarify before moving on:

1. grid area

A grid area is any part of our grid fenced in by four grid lines; it can comprise any number of grid cells.

Time to build this grid in the browser! Let’s begin with some markup.

## CSS Grid Markup

To recreate the grid above, we need a container element; anything you like:

In it we’re going to place nine child elements.

Fork this starter pen if you want to follow along. I’ve added some basic styles to visually differentiate each grid item.

## CSS Grid Rules

Firstly we need to declare that our container element is a grid using a new value for the display property:

Right, that was easy. Next we need to define our grid, by stating how many grid tracks it will have, both horizontally and vertically. We do that with the grid-template-columns  and grid-template-rows properties:

You’ll notice three values for each. The values for grid-template-columns  state that “all three columns should be 150px wide”. The three values for grid-template-rows state something similar. Each one would actually be auto by default, taking height from the content, but we’ve stated them here in order to clearly visualise what’s going on.

So what do we have now?

Each of our items has been automatically assigned a grid area in chronological order. It’s not bad, but we’re missing gutters. Time to fix that.

## Mind the Gap

Grid comes with a purpose-made solution for declaring gutter measurements. We can use grid-column-gap and grid-row-gap, or the shorthand grid-gap property.

Let’s add a fixed gutter of 20px to our .grid-1 element.

And now, we’re left with a nice, neat grid:

## Conclusion

That’s it, you’re up and running with Grid! Let’s recap the four essential steps:

1. Create a container element, and declare it display: grid;.
2. Use that same container to define the grid tracks using the grid-template-columns  and grid-template-rows properties.
3. Place child elements within the container.
4. Specify the gutter sizes using the grid-gap properties.

In the next CSS grid tutorial we’ll take a deeper look at Grid’s syntax, explore flexible layouts, the fr unit, the repeat() function, and take our simple grid much further. See you there!


text:
# lrmest v3.0

0

0th

Percentile

## Different Types of Estimators to Deal with Multicollinearity

When multicollinearity exists among predictor variables of the linear model, least square estimators does not provide a better solution for estimating parameters. To deal with multicollinearity several estimators are proposed in the literature. Some of these estimators are Ordinary Least Square Estimator (OLSE), Ordinary Generalized Ordinary Least Square Estimator (OGOLSE), Ordinary Ridge Regression Estimator (ORRE), Ordinary Generalized Ridge Regression Estimator (OGRRE), Restricted Least Square Estimator (RLSE), Ordinary Generalized Restricted Least Square Estimator (OGRLSE), Ordinary Mixed Regression Estimator (OMRE), Ordinary Generalized Mixed Regression Estimator (OGMRE), Liu Estimator (LE), Ordinary Generalized Liu Estimator (OGLE), Restricted Liu Estimator (RLE), Ordinary Generalized Restricted Liu Estimator (OGRLE), Stochastic Restricted Liu Estimator (SRLE), Ordinary Generalized Stochastic Restricted Liu Estimator (OGSRLE), Type (1),(2),(3) Liu Estimator (Type-1,2,3 LTE), Ordinary Generalized Type (1),(2),(3) Liu Estimator (Type-1,2,3 OGLTE), Type (1),(2),(3) Adjusted Liu Estimator (Type-1,2,3 ALTE), Ordinary Generalized Type (1),(2),(3) Adjusted Liu Estimator (Type-1,2,3 OGALTE), Almost Unbiased Ridge Estimator (AURE), Ordinary Generalized Almost Unbiased Ridge Estimator (OGAURE), Almost Unbiased Liu Estimator (AULE), Ordinary Generalized Almost Unbiased Liu Estimator (OGAULE), Stochastic Restricted Ridge Estimator (SRRE), Ordinary Generalized Stochastic Restricted Ridge Estimator (OGSRRE), Restricted Ridge Regression Estimator (RRRE) and Ordinary Generalized Restricted Ridge Regression Estimator (OGRRRE). To select the best estimator in a practical situation the Mean Square Error (MSE) is used. Using this package scalar MSE value of all the above estimators and Prediction Sum of Square (PRESS) values of some of the estimators can be obtained, and the variation of the MSE and PRESS values for the relevant estimators can be shown graphically.

## Functions in lrmest

 Name Description lrmest-package Estimation of varies types of estimators in the linear model mixe Ordinary Mixed Regression Estimator liu Liu Estimator ogrls Ordinary Generalized Restricted Least Square Estimator ogliu Ordinary Generalized Liu Estimator ogrrre Ordinary Generalized Restricted Ridge Regression Estimator lte3 Type (3) Liu Estimator ogols Ordinary Generalized Ordinary Least Square Estimators alte3 Type (3) Adjusted Liu Estimator alte1 Type (1) Adjusted Liu Estimator lte1 Type (1) Liu Estimator ogalt1 Ordinary Generalized Type (1) Adjusted Liu Estimator aul Almost Unbiased Liu Estimator oglt1 Ordinary Generalized Type (1) Liu Estimator checkm Check the degree of multicollinearity present in the dataset srliu Stochastic Restricted Liu Estimator ogaur Ordinary Generalized Almost Unbiased Ridge Estimator ogre Ordinary Generalized Ridge Regression Estimator rid Ordinary Ridge Regression Estimator rrre Restricted Ridge Regression Estimator alte2 Type (2) Adjusted Liu Estimator aur Almost Unbiased Ridge Estimator lte2 Type (2) Liu Estimator ogaul Ordinary Generalized Almost Unbiased Liu Estimator ogalt3 Ordinary Generalized Type (3) Adjusted Liu Estimator ogrliu Ordinary Generalized Restricted Liu Estimator optimum Summary of optimum scalar Mean Square Error values of all estimators and optimum Prediction Sum of Square values of some of the estimators ogsrre Ordinary Generalized Stochastic Restricted Ridge Estimator rliu Restricted Liu Estimator ogalt2 Ordinary Generalized Type (2) Adjusted Liu Estimator rls Restricted Least Square Estimator pcd Portland Cement Dataset ogsrliu Ordinary Generalized Stochastic Restricted Liu Estimator ogmix Ordinary Generalized Mixed Regression Estimator ols Ordinary Least Square Estimators oglt2 Ordinary Generalized Type (2) Liu Estimator oglt3 Ordinary Generalized Type (3) Liu Estimator srre Stochastic Restricted Ridge Estimator No Results!

format: plain


text:
• Corpus ID: 119152097

# Homotopy Loday Algebras and Symplectic $2$-Manifolds

@article{Peddie2018HomotopyLA,
title={Homotopy Loday Algebras and Symplectic \$2\$-Manifolds},
author={Matthew T. Peddie},
journal={arXiv: Mathematical Physics},
year={2018}
}
• M. Peddie
• Published 9 April 2018
• Mathematics
• arXiv: Mathematical Physics
Using the technique of higher derived brackets developed by Voronov, we construct a homotopy Loday algebra in the sense of Ammar and Poncin associated to any symplectic $2$-manifold. The algebra we obtain has a particularly nice structure, in that it accommodates the Dorfman bracket of a Courant algebroid as the binary operation in the hierarchy of operations, and the defect in the symmetry of each operation is measurable in a certain precise sense. We move to call such an algebra a homotopy…
3 Citations

### L-infinity bialgebroids and homotopy Poisson structures on supermanifolds

We generalize to the homotopy case a result of K. Mackenzie and P. Xu on relation between Lie bialgebroids and Poisson geometry. For a homotopy Poisson structure on a supermanifold $M$, we show that

### $L_\infty$ and $A_\infty$ structures: then and now

Looking back over 55 years of higher homotopy structures, I reminisce as I recall the early days and ponder how they developed and how I now see them. From the history of $A_\infty$-structures and

### L-infinity and A-infinity structures

Looking back over 55 years of higher homotopy structures, I reminisce as I recall the early days and ponder how they developed and how I now see them. From the history of A∞-structures and later of

## References

SHOWING 1-10 OF 27 REFERENCES

### Non-Commutative Batalin-Vilkovisky Algebras, Homotopy Lie Algebras and the Courant Bracket

We consider two different constructions of higher brackets. First, based on a Grassmann-odd, nilpotent Δ operator, we define a non-commutative generalization of the higher Koszul brackets, which are

### L-infinity algebras and higher analogues of Dirac structures and Courant algebroids

We define a higher analogue of Dirac structures on a manifold M. Under a regularity assumption, higher Dirac structures can be described by a foliation and a (not necessarily closed, non-unique)

### On the structure of graded symplectic supermanifolds and Courant algebroids

This paper is devoted to a study of geometric structures expressible in terms of graded symplectic supermanifolds. We extend the classical BRST formalism to arbitrary pseudo-Euclidean vector bundles

### Courant Algebroids and Strongly Homotopy Lie Algebras

• Mathematics
• 1998
Courant algebroids are structures which include as examples the doubles of Lie bialgebras and the direct sum of tangent and cotangent bundles with the bracket introduced by T. Courant for the study

### Derived Brackets

We survey the many instances of derived bracket construction in differential geometry, Lie algebroid and Courant algebroid theories, and their properties. We recall and compare the constructions of

### Courant algebroids, derived brackets and even symplectic supermanifolds

In this dissertation we study Courant algebroids, objects that first appeared in the work of T. Courant on Dirac structures; they were later studied by Liu, Weinstein and Xu who used Courant

### Manin Triples for Lie Bialgebroids

• Mathematics
• 1995
In his study of Dirac structures, a notion which includes both Poisson structures and closed 2-forms, T. Courant introduced a bracket on the direct sum of vector fields and 1-forms. This bracket does


text:
Gambling Game: Martingales

This is a multipart question; if there's a strong preference for breaking this into separate questions I'll do that.

Imagine a game between a gambler and a croupier.

Total capital in the game is $1$.

After $n^{\text{th}}$ hand of the game, the fraction of capital held by the gambler is $X_{n}\in[0,1]$, and capital held by croupier is $1-X_{n}$.

Assume $X_{0}=p\in(0,1)$.

Rules of the game: After $n$ hands, probability for gambler to win $(n+1)^{\text{th}}$ hand is $X_{n}$. If he wins he gets half the capital the croupier held after the $n^{\text{th}}$ hand. If he loses, he gives the croupier half of his own capital. Let $$\mathcal{F}_{n}=\sigma\left(X_{i},1\leq i \leq n\right)$$

Questions:

How do I show that $\left(X_{n}\right)_{n\geq 0}$ is a $\left(\mathcal{F_{n}}\right)_{n\geq 0}$ martingale, converging to a limit $Z$ both a.s and in $L^2$?

Also I need to show that: $$\mathbb E[X_{n+1}^{2}]=\frac{\mathbb E[3X_{n}^2+X_n]}{4}$$

and $$\mathbb E[Z^2]=\mathbb E[Z]=p$$

For any $n\geq 0$, let $Y_n=2X_{n+1}-X_{n}$ Find conditional law of $X_{n+1}$ knowing $\mathcal{F}_{n}$, and prove that $$\mathbb P(Y_n=0|\mathcal{F_n})=1-X_n$$

$$\mathbb P(Y_n=1|\mathcal{F}_n)=X_n$$

And express the law of $Y_n$.

Let $G_n=\{Y_n=1\}$, $P_n=\{Y_n=0\}$ prove $Y_n\rightarrow Z$ a.s., and deduce that $$\mathbb P(\liminf_{n\rightarrow\infty}G_n)=p$$

$$\mathbb P(\liminf_{n\rightarrow\infty}P_n)=1-p$$

Lastly, are the variables $\{Y_n,n\geq 1\}$ independent? I would appreciate any and all solutions or hints.

You've asked for an exhaustive treatment of this martingale. I've attempted below to get it all.

There is a preliminary point to make before pursuing any properties. Let $A_n$ denote the event that the gambler wins the $n$-th hand. The way in which I interpret your rules is $$X_{n+1} = \frac{1}{2}(1 + X_n) ~~ \text{on }A_{n+1}\\ = \frac{1}{2} X_n ~~ \text{on }A_{n+1}^c$$ and $P(A_{n+1} \mid \mathcal{F}_n) = X_n$. This last line is important for obvious reasons, and says literally that the gambler wins on hand $n+1$ with likelihood $X_n$, when $X_n$ is known.

1. $X_n$ is a martingale. We compute $$E[X_{n+1} \mid \mathcal{F}_n] = E[X_{n+1} I_{A_{n+1}} \mid \mathcal{F}_n] + E[X_{n+1} I_{A_{n+1}^c} \mid \mathcal{F}_n] \\ = E[\frac{1}{2}(1 + X_n) I_{A_{n+1}} \mid \mathcal{F}_n] + E[\frac{1}{2}X_n I_{A_{n+1}^c} \mid \mathcal{F}_n] \\ = \frac{1}{2}(1 + X_n) P(A_{n+1} \mid \mathcal{F}_n) + \frac{1}{2} X_n P(A_{n+1}^c \mid \mathcal{F}_n) \\ = \frac{1}{2}(1 + X_n) X_n + \frac{1}{2} X_n(1 - X_n) = X_n$$ where from the second to third lines I used the rule 'taking out what is known'. This is the martingale property, as desired.

2.$E[X_{n+1}^2] = \frac{E[3 X_n^2 + X_n]}{4}$.

I'll prove this in a stronger form: $$E[X_{n+1}^2 \mid \mathcal{F}_n] = E[X_{n+1}^2 I_{A_n} \mid \mathcal{F}_n] + E[X_{n+1}^2 I_{A_n^c} \mid \mathcal{F}_n] \\ = E[\frac{1}{4} (1 + X_n)^2 I_{A_n} \mid \mathcal{F}_n] + E[\frac{1}{4} X_n^2 I_{A_n^c} \mid \mathcal{F}_n] \\ = \frac{1}{4} (1 + X_n)^2 X_n + \frac{1}{4} X_n^2(1 - X_n) \\ = \frac{3 X_n^2 + X_n}{4}$$ From the second to the third lines, I took out what is known and plugged in $P(A_{n+1} \mid \mathcal{F}_n) = X_n$ in one shot. The result you want now follows from taking expectations.

3.$X_n$ converges almost surely and in $L^2$ to a random variable $Z$ for which $E[Z] = E[Z^2] = p$.

Note that your martingales are nonnegative and bounded from above by $1$. Therefore the MG convergence theorem says that $X_n$ converges in $L^2$ and almost surely. This is well-known and general and I won't discuss this theorem any further, except to point you to wikipedia.

Now it is also the case that $X_n$ is uniformly integrable, being bounded uniformly in $n$ by the constant function $1$. Therefore we use the MCT and the bounded convergence theorem to get $$E[X_n] \rightarrow E[Z]$$ However $X_n$ is an MG, hence $E[X_n] = p$ for all $n$, and $E[Z] = p$ as desired.

To show that $E[Z^2] = p$ we'll have to be more clever. Iterating the identity I proved in the answer to 2, it is possible to show that for all $m < n$, $$E[X_n^2 \mid \mathcal{F}_m] = (\frac{3}{4})^{n-m} X_m^2 + \frac{1}{4} \left( 1 + \frac{3}{4} + \cdots + \left( \frac{3}{4}\right)^{n-m-1}\right) X_m$$ First, let $m = 0$, and second, take the limit as $n \rightarrow \infty$. We obtain that $$E[Z^2] = \lim_n E[X_n^2] = 0 + \frac{p}{4} \sum_{k = 0}^{\infty} \left(\frac{3}{4} \right)^k = \frac{p}{4} \frac{4}{1} = p$$ using the $L^2$ convergence of the MCT (or bounded convergence, depending on your mood today).

4.With $Y_n = 2 X_{n+1} - X_n$, find the law of $Y_n$, show that $Y_n \rightarrow Z$ almost surely, and compute $P(\liminf_n G_n) = p, P(\liminf_n P_n) = 1-p$.

You should realize by now that $\{Y_n = 1\} = A_{n+1}$ is precisely the event that the Gambler has won the $n+1$-th hand. To find the law, recall that $$P(A_{n+1} \mid \mathcal{F}_n) = X_n \\ P(A_{n+1}) = E[X_n] = p$$ by martingality. Thus is the law of $Y_n$: a biased coin toss with likelihood $p$.

That $Y_n \rightarrow Z$ almost surely follows immediately from $X_n \rightarrow Z$.

Notice now that $Y_n$ is $\{0,1\}$-valued, and so $Z$ is almost surely $\{0,1\}$ valued as well. Realize also that $\liminf_n G_n$ is the event "always winning after a certain time" and $\liminf_n P_n$ is the event "always losing after a certain time", so we see that $$\liminf_n G_n \subset \{Z = 1\} \\ \liminf_n P_n \subset \{Z = 0\}$$ Now we'll do something cheap: $$E[Z] = p = 1 \cdot P(Z = 1) + 0 \cdot P(Z = 0) = P(Z = 1)$$ Now what remains is to argue that $\liminf_n G_n$ has full measure inside $\{Z = 1\}$. To see this, think about what it would mean if on a positive measure set inside $\{Z = 1\}$ we had $Y_n = 0$ infinitely often. This last part will finish item 4.

5.Are the $Y_n$ independent?

No, I don't think so. This is a bit counterintuitive, but that's life. You can compute $E[Y_0Y_1]$ by hand and see that it's not equal to $p^2$ by considering $E[Y_0Y_1 \mid \mathcal{F}_1]$ and then computing an expectation.

• Thank you so much for the incredibly exhaustive and well thought out response! – Chris May 10 '13 at 0:27


text:
Chapter 13.2, Problem 9E

### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042

Chapter
Section

### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042
Textbook Problem

# Evaluate the definite integrals in Problems 1-32. 9.   ∫ 1 4 (10 − 4 x )   d x

To determine

To calculate: The value of the integral 14(104x)dx.

Explanation

Given Information:

The provided integral is 14(104x)dx.

Formula used:

To calculate a definite integral, evaluate the indefinite integral and then substitute the limits of the integral.

The power rule of integral,

undu=un+1n+1

Calculation:

Consider the integral 14(104x)dx.

Calculate a definite integral, evaluate the indefinite integral and then substitute the limits of the integral.

Use the formula undu=un+1n+1 to evaluate the integral

### Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

#### The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started


text:
mersenneforum.org Number 100
 User Name Remember Me? Password
 Register FAQ Search Today's Posts Mark Forums Read

 2006-08-07, 17:13 #1 mfgoode Bronze Medalist     Jan 2004 Mumbai,India 22×33×19 Posts Number 100 We are all familiar with the Name Edouard Lucas. Like many mathematicians he spent his leisure moments solving or compiling puzzles. The following is one attributed to him. He found seven solutions and expressed doubts for any more ways. Actually today we know 11 ways and no more. The Problem: To write 100 using all 9 digits (1 -- 9) in the form of a mixed number; Eg: 100 = 91 + 5742/638 or 91(5742/638) a mixed number and no repitiion of digits but just the nine. If you can give more than seven you beat Lucas at his own game. Mally
 2006-08-07, 17:41 #2 R. Gerbicz     "Robert Gerbicz" Oct 2005 Hungary 25×72 Posts Are parentheses allowed?
2006-08-07, 23:10   #3
fetofs

Aug 2005
Brazil

2×181 Posts

Quote:
 Originally Posted by R. Gerbicz Are parentheses allowed?
Mally is talking about mixed numbers, of the form $x + y/z$; I don't see why we should allow parentheses here....

On a side note, I bet Lucas didn't have a computer!

94+1578/263 = 100
91+7524/836 = 100
3+69258/714 = 100
91+5823/647 = 100
96+1428/357 = 100
96+2148/537 = 100
82+3546/197 = 100
81+5643/297 = 100
81+7524/396 = 100
96+1752/438 = 100
91+5742/638 = 100

Last fiddled with by fetofs on 2006-08-07 at 23:34

2006-08-08, 04:38   #4
drew

Jun 2005

2×191 Posts

Quote:
 Originally Posted by fetofs On a side note, I bet Lucas didn't have a computer!
It really trivializes these problems, doesn't it.

2006-08-08, 07:37   #5
mfgoode
Bronze Medalist

Jan 2004
Mumbai,India

80416 Posts
Computers!

Quote:
 Originally Posted by drew It really trivializes these problems, doesn't it.
It certainly does.

Fetofs has certainly got it straight off the bat!

Well rather than start a new thread and a new problem I'm presenting this one, a simple one, to which no computer is needed at all. Maybe a calculator at the most, if you go about it the right way. If you use a computer, the compilers from Cambridge Univ. recommend C++. Actually there is no need of this and I got it straight away. I am nil in it anyway!

Puzzle: Given the expression 1/p + 1/q + 1/r < 1

Find values of p, q, r, such that its the largest fraction but less than 1

p, q, r, are whole numbers and no other symbols, parenthesis etc. In other words no funny stunts!

Having said that, the use of a comp would be cheating and please try to prove the answer in some way or the other.

Mally

2006-08-08, 09:33   #6
R. Gerbicz

"Robert Gerbicz"
Oct 2005
Hungary

25·72 Posts

Quote:
 Originally Posted by mfgoode Puzzle: Given the expression 1/p + 1/q + 1/r < 1 Find values of p, q, r, such that its the largest fraction but less than 1

The answer is 1/2+1/3+1/7=41/42
Proof
Let s=1/p+1/q+1/r and indirectly let 1>s>41/42.
We can suppose that 1<=p<=q<=r
If p=1 then s>=1, contradiction.
If p>3 then s<=1/4+1/4+1/4=3/4<41/42, contradiction.
If p=3 then case1: r=3, then s=1, contradiction.
case2: r>3, then s<=1/3+1/3+1/4=11/12<41/42, contradiction.
So we can suppose that p=2 ( There is no other case ).
If q=2, then s>=1, contradiction.
If q>4, then s<=1/2+2/5=9/10<41/42.
If q=4, then case1: r=4, then s=1, contradiction.
case2: r>4, then s<=1/2+1/4+1/5=19/20<41/42, contradiction.
So we can suppose that q=3. ( There is no other case).
If q<7, then s>=1/2+1/3+1/6=1, contradiction.
If q>7, then s<=1/2+1/3+1/8=23/24<41/42, contradiction.
So q=7, and for that s=1/p+1/q+1/r=41/42.

 2006-08-08, 11:11 #7 Kees     Dec 2005 19610 Posts Continuing this train of thought the values for s,t,u etc will be x[1]=2 x[n]=1/(1-PROD(1/x[i]))+1 So s=43, t=1807 The obvious lurking question: which numbers in this sequence are prime ?
2006-08-08, 15:25   #8
mfgoode
Bronze Medalist

Jan 2004
Mumbai,India

22×33×19 Posts
Number 100

Quote:
 Originally Posted by Kees Continuing this train of thought the values for s,t,u etc will be x[1]=2 x[n]=1/(1-PROD(1/x[i]))+1 So s=43, t=1807 The obvious lurking question: which numbers in this sequence are prime ?

Well Kees thats an interesting train of thought and u should be in the region of 9 digits long at least. However, t breaks the prime run and I doubt very much if there will be any more primes at all. after s
Still I would like it worked out for a few more digits v,w,x.... to make a conjecture.
Mally

 2006-08-08, 18:50 #9 alpertron     Aug 2002 Buenos Aires, Argentina 26468 Posts Using my factorization applet in batch mode I found: 2 = 2 3 = 3 7 = 7 43 = 43 1807 = 13 * 139 3263443 = 3263443 10650056950807 = 547 * 607 * 1033 * 31051 113423713055421844361000443 = 29881 * 67003 * 9119521 * 6212157481 12864938683278671740537145998360961546653259485195807 = 5295435634831 * 31401519357481261 * 77366930214021991992277 165506647324519964198468195444439180017513152706377497841851388766535868639572406808911988131737645185443 = 181 * 1987 * 112374829138729 * 114152531605972711 * 35874380272246624152764569191134894955972560447869169859142453622851 Since the next member in the sequence is about the square of the previous one (like the Fermat number sequence), I think that there are no other primes (or maybe only one more).
 2006-08-08, 19:11 #10 R. Gerbicz     "Robert Gerbicz" Oct 2005 Hungary 25×72 Posts This is a well known sequence, see http://en.wikipedia.org/wiki/Sylvester's_sequence including some more factorizations also.
 2006-08-08, 21:41 #11 alpertron     Aug 2002 Buenos Aires, Argentina 2·3·241 Posts I started my program to find small factors before trying to find the sequence in other Web site (anyway this is OK in the Puzzles forum). These are the small prime factors I found with a program written in C: k=0: 2 k=1: 3 k=2: 7 k=3: 43 k=4: 13, 139 k=5: 3263443 k=6: 547, 607, 1033, 31051 k=7: 29881, 67003, 9119521 k=9: 181, 1987 k=10: 2287, 2271427 k=11: 73 k=13: 52387, 5020387 k=14: 13999, 74203, 9638659, 57218683 k=15: 17881 k=16: 128551 k=17: 635263, 1286773, 21269959 k=20: 352867 k=21: 387347773, 1620516511 k=23: 74587 k=26: 27061 k=27: 164299, 3229081 k=28: 20929 k=29: 1171, 298483, 97562299 k=31: 1679143 k=34: 120823, 447841 k=35: 2408563 k=36: 38218903, 818864029 k=37: 333457, 117602053 k=38: 30241 k=39: 4219, 1372512301 k=40: 1085443, 156507607, 171818923 k=41: 7603 k=42: 1861, 84819199 k=44: 23773, 290791 k=45: 51769 k=46: 1285540933 k=47: 429547, 196973983 k=51: 179585227 k=53: 9829 k=57: 763176709 k=59: 2029, 38329, 320329, 3567469 k=61: 50023, 1445860807 k=64: 1459, 11234941 k=65: 72091, 609421 k=67: 245563, 3346633, 343870981 k=68: 6367063 k=69: 12763 k=70: 384061 k=71: 3799489, 8401963 k=73: 11844787 k=74: 35869 k=75: 20161, 42428887 k=76: 123427, 160200259 k=79: 51973, 195502537 k=80: 2437 k=82: 6230971 k=86: 9436201 k=88: 151477 k=89: 41077, 2171593 k=92: 15373, 21661, 1386013 k=93: 429446209 k=94: 12593197 k=95: 535879 k=96: 1407223, 1143074323 k=98: 20479 The value k is the index of the sequence. The upper bound is 2*109. I also found that the prime factors greater than 3 must be congruent to 1 (mod 6): Let p be a prime factor of a member of Sylvester's sequence greater than 3 (all equal signs below are congruences mod p). u2 - u + 1 = 0 Let u = t+b. (t+b)2 - (t+b) + 1 = 0 t2 + 2bt + b2 - t - b + 1 = 0 t2 + (2b - 1)t + b2 - b + 1 = 0 Let 2b - 1 = 0, so b = 1/2 (valid because gcd(p,2)=1). t2 + 1/4 - 1/2 + 1 = 0 t2 + 3/4 = 0 4t2 + 3 = 0 Let w = 2t: w2 + 3 = 0 w2 = -3 -3 is a quadratic residue mod p. Since -3 is congruent to 1 (mod 4), from the law of quadratic reciprocity we get that p is a quadratic residue mod -3 (or 3). This implies that p = 1 (mod 3). So all prime factors of Sylvester's sequence greater than 3 must be congruent to 1 (mod 6).

 Similar Threads Thread Thread Starter Forum Replies Last Post mickfrancis Math 16 2017-03-01 07:17 CRGreathouse Probability & Probabilistic Number Theory 15 2014-08-13 18:46 aketilander Operazione Doppi Mersennes 1 2012-11-09 21:16 henryzz Math 7 2012-05-23 01:13 literka Factoring 5 2012-01-30 12:28

All times are UTC. The time now is 22:43.

Sat May 21 22:43:33 UTC 2022 up 37 days, 20:44, 0 users, load averages: 2.68, 1.54, 1.38

Copyright ©2000 - 2022, Jelsoft Enterprises Ltd.

This forum has received and complied with 0 (zero) government requests for information.

Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation.
A copy of the license is included in the FAQ.

≠ ± ∓ ÷ × · − √ ‰ ⊗ ⊕ ⊖ ⊘ ⊙ ≤ ≥ ≦ ≧ ≨ ≩ ≺ ≻ ≼ ≽ ⊏ ⊐ ⊑ ⊒ ² ³ °
∠ ∟ ° ≅ ~ ‖ ⟂ ⫛
≡ ≜ ≈ ∝ ∞ ≪ ≫ ⌊⌋ ⌈⌉ ∘ ∏ ∐ ∑ ∧ ∨ ∩ ∪ ⨀ ⊕ ⊗ 𝖕 𝖖 𝖗 ⊲ ⊳
∅ ∖ ∁ ↦ ↣ ∩ ∪ ⊆ ⊂ ⊄ ⊊ ⊇ ⊃ ⊅ ⊋ ⊖ ∈ ∉ ∋ ∌ ℕ ℤ ℚ ℝ ℂ ℵ ℶ ℷ ℸ 𝓟
¬ ∨ ∧ ⊕ → ← ⇒ ⇐ ⇔ ∀ ∃ ∄ ∴ ∵ ⊤ ⊥ ⊢ ⊨ ⫤ ⊣ … ⋯ ⋮ ⋰ ⋱
∫ ∬ ∭ ∮ ∯ ∰ ∇ ∆ δ ∂ ℱ ℒ ℓ
𝛢𝛼 𝛣𝛽 𝛤𝛾 𝛥𝛿 𝛦𝜀𝜖 𝛧𝜁 𝛨𝜂 𝛩𝜃𝜗 𝛪𝜄 𝛫𝜅 𝛬𝜆 𝛭𝜇 𝛮𝜈 𝛯𝜉 𝛰𝜊 𝛱𝜋 𝛲𝜌 𝛴𝜎𝜍 𝛵𝜏 𝛶𝜐 𝛷𝜙𝜑 𝛸𝜒 𝛹𝜓 𝛺𝜔


text:
## R到R的周期函数构成的集合是R^R的子空间吗？

Question:

A function $$f : \mathbb{R} \rightarrow \mathbb{R}$$ is called periodic if there exists a positive number $$p$$ such that $$f(x) = f(x + p)$$ for all $$x \in \mathbb{R}$$. Is the set of periodic functions from $$\mathbb{R} \rightarrow \mathbb{R}$$ a subspace of $$\mathbb{R}^{\mathbb{R}}$$? Explain.

Solution:

The answer to this question is YES. Now we will prove it.

Let $$V$$ be the set of all periodic functions from $$\mathbb{R} \rightarrow \mathbb{R}$$.

Part 1:

Let $$f_0(x) = 0$$. Clearly $$f_0 \in V$$.

Part 2:

Take $$f, g \in V$$, $$f(x + m) = f(x)$$, $$g(x + n) = g(x)$$.

Clearly, \begin{aligned}(f + g)(x + lcm(m, n)) = & f(x + lcm(m, n)) + g(x + lcm(m, n)) \\ = & f(x) + g(x) \\ = & (f + g)(x)\end{aligned}.

Thus $$f + g \in V$$.

Part 3:

Take $$f \in V$$, and $$a \in R$$.

We have $$(af)(x + p) = a \cdot f(x + p) = a \cdot f(x) = (af)(x)$$.

Thus $$af \in V$$.

Therefor the set of all periodic functions from $$\mathbb{R} \rightarrow \mathbb{R}$$ is a subspace of $$\mathbb{R}^{\mathbb{R}}$$.


text:
### 3.7.264. Two-dimensional orthogonal packing

A constraint that can be used to model the two-dimensional orthogonal packing problem. Given a set of rectangles pack them into a rectangular  placement space. Borders of the rectangles should be parallel to the borders of the  placement space and rectangles should not overlap. Some variants of strip packing allow to rotate rectangles from 90 degrees. Benchmarks can be obtained from a generator described in the following paper


text:
MODNET
Research Training Network in Model Theory
 Publications > Preprint server > Preprint Number 1873 Preprint Number 1873 1873. James Hanson Strongly Minimal Sets and Categoricity in Continuous Logic E-mail: (email address protected by JavaScript. Please enable JavaScript to contact) Submission date: 1 November 2020 Abstract: The classical Baldwin-Lachlan characterization of uncountably categorical theories is known to fail in continuous logic in that not every inseparably categorical theory has a strongly minimal set. Here we investigate these issues by developing the theory of strongly minimal sets in continuous logic and by examining inseparably categorical expansions of Banach space. To this end we introduce and characterize 'dictionaric theories,' theories in which definable sets are prevalent enough that many constructions familiar in discrete logic can be carried out. We also introduce, in the context of Banach theories, the notion of an 'indiscernible subspace,' which we use to improve a result of Shelah and Usvyatsov. Both of these notions are applicable to continuous logic outside of the context of inseparably categorical theories. Finally, we construct or present a slew of counterexamples, including an $\omega$-stable theory with no Vaughtian pairs which fails to be inseparably categorical and an inseparably categorical theory with strongly minimal sets in its home sort only over models of sufficiently high dimension. Mathematics Subject Classification: Primary 03C66, 03C35, 03C45, Secondary 46B04 Keywords and phrases: Full text arXiv 2011.00610: pdf, ps.

 Last updated: March 23 2021 10:21 Please send your corrections to: The e-mail address is protected, enable Javascript to see it


text:
EADTs (extensible algebraic data types) are ADTs whose constructors are defined independently of the EADT. In the following example, we define two constructors ConsF and NilF and then we combine them to form a List EADT:

import Haskus.Utils.EADT

data ConsF a l = ConsF a l deriving (Functor)
data NilF    l = NilF      deriving (Functor)

type List a = EADT '[ConsF a, NilF]

We use a type-level list to indicate the valid constructors for the EADT. Compared to usual ADT (e.g. data List a = Cons a (List a) | Nil) it allows us to manipulate (extend/filter/etc.) the list of constructors in a type-safe way and without requiring new data types. Moreover, the constructors can be shared between several EADTs. For instance, the following code reuses NilF constructor to define a binary tree EADT:

data NodeF a l = NodeF a l l deriving (Functor)

type Tree a = EADT '[NodeF a, NilF]

EADTs are based on the Variant sum type to which they add support for recursive ADTs as in the examples above.


text:
WWW.NAUKA.X-PDF.RU - , ,

 >>  - , , . [ .pdf ]
Pages:   || 2 |

# 3 , 2014. 26,1000 . 011/306 77 83 E-mail: drustvo / Journal of Education / UDK 37 ...

-- [ 1 ] --

Journal of Education

o

3 , 2014.

26,1000

. 011/306 77 83

E-mail: drustvo@pedagog.rs

/ Journal of Education /

UDK 37 ISSN 0547-3330 . LXIII 3. . 363-574 , 2014.

Editorial Board

Saa Dubljanin, Ph.D.

Sneana Marinkovi, Ph.D.

Nataa Matovi, Ph.D.

Dragana Pavlovi Breneselovi, Ph.D.

, Ilke Parchmann, Ph.D., Germany , Rossitsa Alexsandrova, Penkova, Ph.D., Bulgaria elimir Popov, M.S.

, Alla Stepanovna Sidenko, Ph.D., Russia , Pavel Zgaga, Ph.D., Slovenia Vesna uni Pavlovi, Ph.D.

Editor-In-Chief Emina Hebib, Ph.D.

: Language editor: Tatjana Dogdibegovi Translators Andjelka Ignjaevi, Ph.D. (English) Dara Damljanovic, Ph.D. (Russian) Secretary Bojana Uroevi

: Design and typeset:

eljka Bai Stankov

: For the publisher:

Biljana Radosavljevi : , Printing: Europrint, Beograd CIP - , = Journal of Education = /

. - . 1, . 1 ( 1952)-. - : , 1952- ( :

). - 24 cm ISSN 0547-3330 = COBISS.SR-ID 6026754 369 ף Ȏ 387 ȣȣ :

481 ȣ ؎

481 Mirjana Japunda Milisavljevi NEUROPSYCHOLOGICAL FUNCTIONS AS PREDICTORS Aleksandra uri Zdravkovi OF THE SUCCESSFULNESS OF MUSIC CULTURE TEACHING Sanja Gagi OF STUDENTS WITH INTELLECTUAL DISABILITY

481

## ף Ȏ 1

. , . .

, , , , , , . , , . , , .

ʚ : , , , , .

1 (179060), , .

Theoretical conceptualization of kindergarten as the model center is based on Abstract the systemic approach to changes in education and the concept of the sustainibility of change. Since the Law of preschool education in Serbia introduced the institution of the model centre, we directed our research to examining the preconditions for its implementation in practice by analyzing legislative preconditions for its implementation and to examine the views of the practitioners of the idea of the model center. We used the technique of content analysis and a questionnaire to examine the attitudes of practitoners towards the model center and their own potential roles in it. The results show that the implementation of the model center has not been done yet since its significance, purpose, functions, instructions for implementation, functioning and financing have not been determined by necessary sublegal documents. On the other hand, the practitioners who took part in the research expressed a very positive attitude towards this concept and they recognized its significance for the development and dissemination of change in practice and in their in-service training. Based on the obtained results the conclusion of the paper contains some suggestions for educational policy by which successful implementation of the model center and interconnecting of researchers, practitioners and educational policy agents could be achieved.

Keywords: model center, changes in education, sustainbility of change, dissemination, institutionalization.

## -

, . -, , - -. - , -. , - . , , , , . , , , ,

, . , , - , .

: -, , , .

## ̊1

, .
, , . , , , :

, . , , .

ʚ : , , , , .

All school systems apply different types of differentiation in education, trying Abstract to respond to the differences in abilities and achievements of students. The 1 , 179060 (2011-2014), , , .

paper deals with the questions of whether, when and how guidance and selection of students are done be it between different schools or inside a school. Since among certain school systems there are significant similarities regarding the structure and organization of elementary education, differentiation included, we stress the specifities of performing differentiation in the compulsory education in three groups of countries, i.e. three different models (German, Scandinavian and Anglo-Saxon) which are characterized by certain mutual features of the applied mechanisms of differentiations are described and analyzed.

Regarding the effects of differentiation on equality in education, the Scandinavian model seems to be very successful while in the countries in the German model justice in education is less evident.

Keywords: differentiation in education, German model, Scandinavian model, AngloSaxon model, justice in educational systems.

## :

, . : , , , . , , .. , : , . , , , , .

: , , , - , .

## 1

. , . 400 , , .

, . , , , . , , . , , :

; ;

; .

ʚ : , , , .

1 , , (. 179034) (. 47008), , (2011-2014).

, ,

## EDUCATIONAL STANDARDS IN ELEMENTARY EDUCATION

The paper presents the results of a research of the attitudes of teachers of Abstract science towards the implementation of educational standards in elementary school education. The research was carried out immediately after an in-service teacher training in the implementation of standards, organized by the Institute for the evaluation of quality of education in Serbia. Tested were four hundred teachers of geography, biology, physics and chemistry who teach in elementary schools. The results show that almost all teachers think that educational standards should be implemented in the teaching process although there are some differences between them when the subjects taught are considered.

More than half the teachers think that the standards will be most beneficial to students in the domain of knowledge, assessment, personal initiative and preparation for final exams. The teachers who recognized their own personal benefit from the implementation of the standards estimate that they will be helpful in assessment, selection of teaching materials, organization of work and further professional development. However, the teachers also

expect that they will face various difficulties and limitations during the implementation:

problems in assessing and evaluating students knowledge compared to the defined levels of achievement; resistance of students and parents to change; lack of technical and teaching devices; too much administration which will require a lot of time Keywords: educational standards, science teachers, justification of educational standards implementation, elementary school education.

. , . 400 , , , . , , , . , , , . , ,

...

, , . , , , :

; ; ; , .

: , , , .

## ȣ 1

. , : , , , ( , , ) . , . , . e . , / , , . , , , . . .

ʚ : , , , , , , .

1 , . 179034 (2011-2015) , , .

The Institute for Pedagogic Research in Belgrade developed the so called Abstract Trefoil approach to education which responds to the universal need for the students self-actualization and the acute need of rapidly changing life in this century. The aim of the Trefoil is to stimulate the development of the students creativity, initiative and cooperative competence via intensive coordinated application: learning through group cooperation, solving open divergent tasks by playing, critical dialogues, researching (brain storming, data gathering, analyzing different solutions and finding out the most relevant one) and work in the students interdisciplinary projects. The approach was tested in one experiment conducted in a small urban school, typical for our conditions. Trying to contribute to the advancement of our teaching by using the Trefoil approach we analyzed the reflections of the teachers who had applied it. They stated advantages and problems in the use of the approach that stimulates creativity of students in class. Numerous scientific studies define this method as outstanding stimulation for the development of cognitive, affective, conative and social characteristics of students. Yet, we found few discussions about the problems of implementation, especially in the context of traditional school. The analysis of the problems and offering possible solutions which are dealt with in this article could help traditionally oriented teachers to successfully apply the Trefoil approach to education. The awareness that there are solutions will enable teachers not to retreat in front of changes and avoid new approaches in the educational process.

Keywords: pedagogic approaches, teachers professional development, applied and action research in education, Trefoil approach to learning: creativity, cooperation, initiative, elementary school.

## ,

, . , , , , ( , , : , , ) . . , , . , . , , . , , , . . , , .

: , , , , , , , .

## Ō Σ ȣ 1

.

, .

, , , , , , - .

/ , . , , , , . :

, , ( , , ) .

ʚ : , , , , .

## ASSESSMENT AND DEVELOPMENT OF STUDENTS COMPETENCIES

This paper discusses the role of assessment for the development of competencies Abstract of students. The students competencies are clearly identified outcomes, i.e.

expected knowledge, skills and abilities, values, attitudes and behaviours the students should acquire and which they will be able to demonstrate upon a successful ending of a theme, programme, educational level or cycle. Connecting the students competencies with their 1 179026 , , , .

activities is essential and is done by monitoring, testing and assessing knowledge. The paper deals with the purpose of assessment from the angle of the student, teacher, teaching process and educational attainments, then with problems of assessment and the role of assessment in the development of competencies of the student in cognitive, affective and psyco-motor domains. The assessment which is in the function of competency development is characterized by the connectedness of the contents which are taught and the variety of instruments and ways of assessing students, the totality of aims and outcomes of learning (for instance, to strenghten cognitive competency by the aims in social and affective domains) and reflection as the integral part of assessment and self-assessment.

Keywords: learning, assessment, testing,students competencies, quality of education.

. , .. , , , , , , , - .

, q , .

, , - . , - . . , , ( , - ) .

: , , , , .

## 1

, , . , , . 354 , , 20122014. . , , .

, . .

ʚ : a .a, , .

Since teaching methods and the attitudes towards them are based, in principle, Abstract on a certain concept of education and teaching, the data about teachers 1 , 179060 (2011-2014), , , .

,

attitudes are a good indicator of the accepted educational concept and approach to teaching, and are at the same time an element for forming a view of the quality of teaching in general.

That is why the subject of the empirical research, which we present in this article, is the variety of teaching methods in Serbia. By research we tried to find out which methods are common, how much individual methods are used and how they are organized in the teaching process. The data were gathered by observing 354 teaching lessons in the schools of different types and levels of education, during three academic years, in the period of 2012 - 2014. The results show that, in practice, different methods are used, but the frequence of individual methods use are uneven, and that those ways of work and the organization of methods based on transmission of knowledge dominate. Lecturing is the most commonly used, combined with examining in the function of the repetition of the matter. It can be concluded that the diversity of work forms in our schools is small and not in accordance with modern conceptualizations of the tesching methods.

Keywords: teaching method, diversity of teaching methods, organization of teaching methods in class.

, . , , . , , , . 354 , , 2012-2014 .

, , , , . , . , .

: , , .

## ΣΣ 1

/ .

, , . , ( ) . 1.162 92 , . :

(1) (2) , - . , . , , .

1 , . 179060 (2011-2014), , , .

The paper presents the results of a research of the dedication of teachers to Abstract their profession and the institution/school in which they are employed.

Dedication is defined as connectedness of the teacher with their own profession and the institution in which they are employed and it shows through normative, affective and continuation forms.

The research focused on a scientifically under-researched phenomenon of the teachers dedication in order to help identifing the problem from the perspective of belonging to a certain institution and profession. The research included 1162 teachers in 92 elemntary schools in Serbia. The quantitative, non-experimental research method was applied and two tasks were fulfilled: (1) identifikation of basic indicators and componenets of the teachers dedication and (2) examination of the influence of professionally-educational and performance of the teachers regarding each component of dedication. Statistically significant relations with each element of normative, affective, continuational organizational and professional deddication were established. The research results, obtained by the analysis of the main components and multiple regression, have relevant scientific and practical implications because they provide an innovative insight in understanding the problem of effectiveness and efficiency of educational management and open a possibility for designing strategies for further professionalization of teachers.

Keywords: teachers, dedication, profession, organization, continual in-service teacher training.

## ,

(, ).

, , . , , . 1162 92 , , .

: (1) , (2) , . , . , .

: , , , , .

## ؎1

. , . , . 124 8-16 , (Stroop), -Ke (Beter-Cragin), (Rey Auditory verbal learning test), (Learning Accomplishment Profile - LAP). ( r = 0.25, p = 0.01; r = 0.22, p = 0.01), (r = 0.21, p = 0.01) (r = 0.24, p = 0.01) .

. .

1 (. 179017) (.

179025) , .

, , ʚ : , , .

Acquisition of contents of the teaching subject Musical Culture among students Abstract with intellectual disability depends to a significant extent upon the quality of the development of neuropsychological contents. Although the researches dealing with the estimation of the relations between music and neuropsychological functions have a very rich and old intellectual history, only few of them tried to single out its most important component for the acquisition of musical contents. The aim of our research was to determine whether there is connectedness between neuropsychological functions as prediction variables with the success in acquisition of musical contents by students with intellectual disability as a criterion variable, and which of the estimated neuropsychological functions has the most influence on the acquisition of musical contents. The sample included 124 respondents, aged 8 - 16. To examine executive functions we used a 20 items test, Stroop test, Beter-Cragin test, Rey Auditory verbal learning test, Acadia test, and Learning Accomplishment Profile - LAP to estimate academic achievements. A statistically significant coeficient of correlation indicates that between the success in the teaching of musical culture and the results obtained by the tests for measuring attention (selectivity of attention r = 0.25, p = 0.01: resistance to obstructions r = 0.22, p = 0.01, visual memory (r = 0.21, p = 0.01) and forced recognition (r = 0.24, p = 0.01) there is a positive relatedness of a low intensity. Among them the most prominent is attention selectivity. Practical implications indicate that it is necessary to plan educational activities aimed at the development of the estimated abilities among the students with intellectual disability within the course of musical culture.

Keywords: musical culture teaching, neuropsychological functions, intellectual disability.

. , , , . , - ,

...

, . 124 8-16 20 , (), - (-), ( ), ( - ). , ( = 0.25, = 0.01; - = 0.22, = 0.01), ( = 0.21, = 0.01) ( = 0.24, = 0.01) . , . , .

: , , .

## 1

. . . , .

83 , . . , , ( 0,01). .

ʚ : , , , .

1 , oa , . 179055.

Posed questions stimulate thinking and speech activities of students. Questions Abstract must gain meaning and place in everyday life of deaf and hard-of-hearing students, too. Questions will be completely clear and permanently internalized only if used in everyday child communication. The aim of our researh was to examine the relatedness between age and the level of hearing damage and the capability of deaf and hard-of-hearing students to answer questions in writing, speech and sign language. The sample included 83 deaf and hard-of-hearing students of the third to the eighth years of elementary school. The obtained results show that the deaf and hard-to-hearing students provide best answers in sign language, then orally, and the lowest results they achieve when they are expected to answer in writing (all the differences are statistically significant at the level of 0.01) The answers which the students were not able to formulate in writing or orally were adequatelly and fully obtained in sign language.

Keywords: questions and answers, written, oral and sign language, deaf and hardto-hearing students.

## ,

Pages:   || 2 |
 >>  - , , . [ .pdf ]

:

2005-2015 - . 2005-2015 , 1 . - . , , ...

, 339.13:655.42(470) 65.422.5 + 76.185 53 , . . , . . , . . , . . , . ....

2015 .. 1 .. 1 .. 1 .. 2 .. 2 .. 3 .. 3 .. 4 .. 5 . 6 ...

, - ۻ 624.01 38.96 2 : . . .. , .. , .....

.. : .. ( ), .. , .. , .. , .. , .. , .. , .. ( ), .. , .. , .. , .. , .. (. ), .. , .. , .. , .. , .. , .. , .. , .. , .. , .. , .. (....

2015 . ! - , 2015 . . , , / /...

, , www.freestudio21.com ============================================================================= 1 .4 : ? ? ? 2 .4 . . : ( ..), , -, ...

\ql 15.05.2014 N 38.02.04 ( ) ( 25.06.2014 N 32855) www.consultant.ru : 23.01.2015 15.05.2014 N 539 ...

R WIPO/ACE/10/INF/2 REV.2 : : 18 2015 . () , 23 25 2015 . WIPO/ACE/10/INF/1 WIPO/ACE/10/INF/2 WIPO/ACE/10/1 Prov. , WIPO/ACE/10/2...

1. .1.1. - , . 8 1.2. 2. 2.1. 2.2. . . 2.3. (, , ). 38 2.4. ...

, : 201 . : . ...

SC-CAMLR-XXXIII , 20-24 2014 .CCAMLR PO Box 213 North Hobart 700 Tasmania Australia _ : 61 3 6210 : 61 3 6224 8 Email: ccamlr@ccamlr.org -: 2014 . www.ccamlr.org : , , . ...

- -2015 16- , -2015. 25 29 2015 . ...

ISBN 9785990632561 : ۻ II- II , 2015 II- : ۻ , . , 21 2015 . II : .. , 2015 06.063:0 94.3 75 : []: II- ...

() 2014 , 2015 . 551.5 26. 63 63 2014 . , 2015. 107 . ISBN 978-5-906099-58-7 , , 2014 . ...

. 526 28.02.2013 2012 2012 , . 35 , :...

, , ... , . 1. , , . ...

, je 2. 1965. , . 1987. 9,96 10 , - . ...

201 - E .... () 1: ...

- 2015 . 2. 2015 1. 2013 2014 . % . % . % 262 3,39 196 2,94 169 2,88 169 , 15,38 %...

<<     |
2016 www.nauka.x-pdf.ru - - , ,

, .
, , , , 1-2 .


text:
# Leech lattice

The Leech lattice is the unique even (http://planetmath.org/EvenLattice) unimodular lattice of dimension (http://planetmath.org/Dimension2) 24 having no elements of norm 2. Its automorphism group (http://planetmath.org/EquivalentCode) is the largest Conway group $Co_{0}$ (sometimes denoted by $\cdot 0$). The quotient of $Co_{0}$ by its center is called $Co_{1}$, a sporadic simple group.

The construction of the Leech lattice below depends on the existence of the extended binary Golay code $\mathcal{G}_{24}$ (for a construction of the latter, see miracle octad generator).

## 1 Construction of the Leech lattice

Let $\Omega=\{1,2,\ldots,24\}$ and assume we have constructed the Golay $\mathcal{G}_{24}$ on $\Omega$. The Leech lattice $\Lambda$ is the set of all points

 $\frac{1}{\sqrt{8}}(x_{1},x_{2},\ldots,x_{24})$

in $\mathbb{R}^{24}$ where each $x_{i}$ is an integer, such that

• For some integer $m$, we have $x_{i}\equiv x_{j}\equiv m\pmod{2}$ for all $i,j\in\Omega$;

• For any integer $n$, the set of coordinates $\{i\in\Omega:x_{i}\equiv n\pmod{4}\}$ is in $\mathcal{G}_{24}$;

• $\sum_{i\in\Omega}x_{i}\equiv 4m\pmod{8}$.

## 2 Properties of the Leech lattice

1. The Leech lattice $\Lambda$ is an unimodular lattice; in other words:

• The set $\Lambda$ spans all of $\mathbb{R}^{24}$ as an $\mathbb{R}$-vector space.

• For any $x,y\in\Lambda$, the scalar product $x\cdot y$ is an integer.

• For any $x\in\Lambda$, the norm $x\cdot x$ is an even integer.

• The volume of the fundamental parallelogram of $\Lambda$ is 1.

2. Let $\Lambda(n)=\{x\in\Lambda:x\cdot x=2n\}$. Then $|\Lambda(0)|=1$, $|\Lambda(1)|=0$, $|\Lambda(2)|=196560$, $|\Lambda(3)|=16773120$, $|\Lambda(4)|=398034000$.

3. The automorphism group (http://planetmath.org/EquivalentCode) $\mbox{Aut}(\Lambda)$ is the largest Conway group $Co_{0}$ with order $8\,315\,553\,613\,086\,720\,000=2^{22}\cdot 3^{9}\cdot 5^{4}\cdot 7^{2}\cdot 1% 1\cdot 13\cdot 23$.

4. The group $Co_{0}$ acts transitively (http://planetmath.org/LeftAction) on the sets $\Lambda(2)$, $\Lambda(3)$, $\Lambda(4)$. For $n=2,3$, the imprimitivity blocks of the action of $Co_{0}$ on $\Lambda(n)$ are the sets $\{x,-x\}$ where $x\in\Lambda(n)$. The imprimitivity blocks of the action of $Co_{0}$ on $\Lambda(4)$ are sets of 48 vectors called . Any two distinct vectors in a are either or orthogonal, and are congruent (http://planetmath.org/QuotientGroup) modulo $2\Lambda$.

5. Any vector in $\Lambda$ is modulo $2\Lambda$ to a vector in $\Lambda(n)$ for one of $n=0,2,3,4$. The imprimitivity blocks of the action of $Co_{0}$ on these sets account for all classes (http://planetmath.org/EquivalenceClass) of $\Lambda/2\Lambda$:

 $1+|\Lambda(2)|/2+|\Lambda(3)|/2+|\Lambda(4)|/48=2^{24}=|\Lambda/2\Lambda|.$

## References

• 1 J. H. Conway and N. J. A. Sloane. Sphere Packings, Lattices, and Groups. Springer-Verlag, 1999.
 Title Leech lattice Canonical name LeechLattice Date of creation 2013-03-22 18:43:23 Last modified on 2013-03-22 18:43:23 Owner monster (22721) Last modified by monster (22721) Numerical id 7 Author monster (22721) Entry type Definition Classification msc 20D08 Classification msc 20B25 Classification msc 11H56 Classification msc 11H06 Classification msc 51E10 Related topic BinaryGolayCode Related topic MiracleOctadGenerator


text:
## Tuesday, January 31, 2006 ... //

### NP-hard landscape problems

The topological conference has attracted a lot of great mathematical physicists and mathematicians. I will report on Robbert Dijkgraaf's talk in a moment.

Frederik Denef who is one of the young big shots in the topological landscape interface allowed me to inform you that he and Michael Douglas are completing two papers about the NP-difficulties of locating the right vacua in the landscape.

If I understand well, "N" denotes the number of different fluxes, and they can rigorously prove that you need a longer-than-polynomial time in "N" to go through all the configurations of different fluxes in order to identify one with plausible values of quantities such as the vacuum energy.

You know that it is hard to prove that some apparently difficult problems are NP (non-polynomial), which is the source of the open questions about the so-called NP-completeness, so I am curious how the proof roughly looks like.

The philosophical conclusion is obvious: you should not even try to find the right vacuum because the required time will exceed the recurrence time of the Universe. Peter Woit and his gang will probably have some new exciting material to talk about. ;-) You may guess that my conclusion is that if the NP-hardness is true, then the class of the vacua is likely to be physically irrelevant.

In another paper, they study cosmological consequences of the fact that Nature cannot "compute" these things in a polynomial time either. Expect that the text above contains certain inaccuracies because there were no formulae written down in the discussion.

Terminological correction by Andy Neitzke:

I think NP is not "non-polynomial" but rather "nondeterministic-polynomial" -- a problem in NP is one which can be solved in polynomial time by a computer which is allowed to branch nondeterministically. So every problem in P is in NP. A problem is called NP-complete if it is in NP and if furthermore every problem in NP can be reduced to it in polynomial time. From that description it might sound amazing that there are any NP-complete problems, but there are -- and there are some "standard" tricks for proving that a given problem is NP-complete. The deep open question which nobody knows how to solve is whether NP = P, i.e. whether all NP problems can actually be solved in polynomial time even by a deterministic computer.

If I remember right, "X is NP-hard" means that every problem in NP can be reduced to X in polynomial time, but X itself might not be in NP.

#### snail feedback (2) :

I think any one with some basic science background should know something about the most important computer science problems. The NP = P? problem is one of the seven milinium math problems and there is no excuse not knowing what it is.

I recommend reading the book "The New Turing Omnibus", which contains some very easy to read materials explaining the NP complete problem and much more. It should be on the book shielf of every one who wants to claim knowing a bit of math.

But whatever the outcome of NP=P? is, it has no impact on the string landscape problem. 10^500 simply far exceeds anything that could exist in this universe and so any theoretical discussion whether that is a computable problem or not should not even start.

The universe really is not that complicated. You constructed a model that far exceeds the complexity of the universe itself. The God now wishes that he had learned super string theory before he went on to create that pity little 4-D spacetime.

Quantoken


text:
1.6: The Second Derivative

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

Skills to Develop

In this section, we strive to understand the ideas generated by the following important questions:

• How does the derivative of a function tell us whether the function is increasing or decreasing at a point or on an interval?
• What can we learn by taking the derivative of the derivative (to achieve the second derivative) of a function $$f$$?
• What does it mean to say that a function is concave up or concave down? How are these characteristics connected to certain properties of the derivative of the function?
• What are the units of the second derivative? How do they help us understand the rate of change of the rate of change?

Given a differentiable function $$y=f(x)$$, we know that its derivative, $$y=f'(x)$$, is a related function whose output at a value $$x=a$$ tells us the slope of the tangent line to $$y=f(x)$$ at the point $$(a, f(a))$$. That is, heights on the derivative graph tell us the values of slopes on the original function’s graph. Therefore, the derivative tells us important information about the function $$f$$ .

Figure 1.25: Two tangent lines on a graph demonstrate how the slope of the tangent line tells us whether the function is rising or falling, as well as whether it is doing so rapidly or slowly.

At any point where $$f'(x)$$ is positive, it means that the slope of the tangent line to $$f$$ is positive, and therefore the function $$f$$ is increasing (or rising) at that point. Similarly, if $$f'(a)$$ is negative, we know that the graph of $$f$$ is decreasing (or falling) at that point.

In the next part of our study, we work to understand not only whether the function $$f$$ is increasing or decreasing at a point or on an interval, but also how the function $$f$$ is increasing or decreasing. Comparing the two tangent lines shown in Figure 1.25, we see that at point A, the value of $$f'(x)$$ is positive and relatively close to zero, which coincides with the graph rising slowly. By contrast, at point B, the derivative is negative and relatively large in absolute value, which is tied to the fact that $$f$$ is decreasing rapidly at B. It also makes sense to not only ask whether the value of the derivative function is positive or negative and whether the derivative is large or small, but also to ask “how is the derivative changing?”

We also now know that the derivative, $$y=f'(x)$$, is itself a function. This means that we can consider taking its derivative – the derivative of the derivative – and therefore ask questions like “what does the derivative of the derivative tell us about how the original function behaves?” As we have done regularly in our work to date, we start with an investigation of a familiar problem in the context of a moving object.

Preview Activity $$\PageIndex{1}$$

The position of a car driving along a straight road at time $$t$$ in minutes is given by the function $$y=s(t)$$ that is pictured in Figure 1.26. The car’s position function has units measured in thousands of feet. For instance, the point (2, 4) on the graph indicates that after 2 minutes, the car has traveled 4000 feet.

Figure 1.26: The graph of $$y=s(t)$$, the position of the car (measured in thousands of feet from its starting location) at time $$t$$ in minutes.

1. In everyday language, describe the behavior of the car over the provided time interval. In particular, you should carefully discuss what is happening on each of the time intervals [0, 1], [1, 2], [2, 3], [3, 4], and [4, 5], plus provide commentary overall on what the car is doing on the interval [0, 12].
2. On the lefthand axes provided in Figure 1.27, sketch a careful, accurate graph of $$y=s'(t)$$.
3. What is the meaning of the function $$y=s'(t)$$ in the context of the given problem? What can we say about the car’s behavior when $$s'(t)$$ is positive? when $$s'(t)$$ is zero? When $$s'(t)$$ is negative?
4. Rename the function you graphed in (b) to be called $$y=v(t)$$. Describe the behavior of $$v$$ in words, using phrases like “$$v$$ is increasing on the interval . . .” and “$$v$$ is constant on the interval . . ..”
5. Sketch a graph of the function $$y=v'(t)$$ on the righthand axes provide in Figure 1.27. Write at least one sentence to explain how the behavior of $$v'(t)$$ is connected to the graph of $$y=v(t)$$.

Figure 1.27: Axes for plotting $$y=v(t)=s'(t)$$ and $$y=v'(t)$$.

Increasing, decreasing, or neither

When we look at the graph of a function, there are features that strike us naturally, and common language can be used to name these features. In many different settings so far, we have intuitively used the words increasing and decreasing to describe a function’s graph. Here we connect these terms more formally to a function’s behavior on an interval of input values.

Definition 1.5. Given a function $$f(x)$$ defined on the interval (a, b), we say that $$f$$ is increasing on (a, b) provided that for all $$x, y$$ in the interval (a, b), if $$x<y$$, then $$f(x)<f(y)$$. Similarly, we say that $$f$$ is decreasing on (a, b) provided that for all $$x, y$$ in the interval (a, b), if $$x<y$$, then $$f(x)>f(y)$$.

Simply put, an increasing function is one that is rising as we move from left to right along the graph, and a decreasing function is one that falls as the value of the input increases. For a function that has a derivative, we can use the sign of the derivative to determine whether or not the function is increasing or decreasing.

Let $$f$$ be a function that is differentiable on an interval (a, b). We say that $$f$$ is increasing on (a, b) if and only if $$f'(x)>0$$ for every $$x$$ such that $$a<x<b$$; similarly, $$f$$ is decreasing on (a, b) if and only if $$f'(x)<0$$. If $$f'(a)=0$$, then we say $$f$$ is neither increasing nor decreasing at $$x=a$$.

Figure 1.28: A function that is decreasing on the intervals $$-3<x<-2$$ and $$0<x<2$$ and increasing on $$-2<x<0$$ and $$2<x<3$$.

For example, the function pictured in Figure 1.28 is increasing on the entire interval $$-2<x<0$$. Note that at both $$x=\pm 2$$ and $$x=0$$, we say that $$f$$ is neither increasing nor decreasing, because $$f'(x)=0$$ at these values.

The Second Derivative

For any function, we are now accustomed to investigating its behavior by thinking about its derivative. Given a function $$f$$ , its derivative is a new function, one that is given by the rule $f'(x)=\lim_{h\to 0} \frac{f(x+h)-f(x)}{h}.$

Because $$f'$$ is itself a function, it is perfectly feasible for us to consider the derivative of the derivative, which is the new function $$y=[f'(x)]'$$ . We call this resulting function the second derivative of $$y=f(x)$$, and denote the second derivative by $$y=f''(x)$$. Due to the presence of multiple possible derivatives, we will sometimes call $$f'$$ “the first derivative” of $$f$$ , rather than simply “the derivative” of $$f$$ . Formally, the second derivative is defined by the limit definition of the derivative of the first derivative: $f''(x)=\lim_{h\to 0} \frac{f'(x+h)-f'(x)}{h} .$

We note that all of the established meaning of the derivative function still holds, so when we compute $$y=f''(x)$$, this new function measures slopes of tangent lines to the curve $$y=f'(x)$$, as well as the instantaneous rate of change of $$y=f'(x)$$. In other words, just as the first derivative measures the rate at which the original function changes, the second derivative measures the rate at which the first derivative changes. This means that the second derivative tracks the instantaneous rate of change of the instantaneous rate of change of $$f$$ . That is, the second derivative will help us to understand how the rate of change of the original function is itself changing.

Concavity

In addition to asking whether a function is increasing or decreasing, it is also natural to inquire how a function is increasing or decreasing. To begin, there are three basic behaviors that an increasing function can demonstrate on an interval, as pictured in Figure 1.29: the function can increase more and more rapidly, increase at the same rate, or increase in a way that is slowing down. Fundamentally, we are beginning to think about how a particular curve bends, with the natural comparison being made to lines, which don’t bend at all. More than this, we want to understand how the bend in a function’s graph is tied to behavior characterized by the first derivative of the function.

For the leftmost curve in Figure 1.29, picture a sequence of tangent lines to the curve. As we move from left to right, the slopes of those tangent lines will increase. Therefore, the rate of change of the pictured function is increasing, and this explains why we say this function is increasing at an increasing rate. For the rightmost graph in Figure 1.29, observe that as $$x$$ increases, the function increases but the slope of the tangent line decreases, hence this function is increasing at a decreasing rate.

Of course, similar options hold for how a function can decrease. Here we must be extra careful with our language, since decreasing functions involve negative slopes, and negative numbers present an interesting situation in the tension between common language and mathematical language. For example, it can be tempting to say that “-100 is bigger than -2.” But we must remember that when we say one number is greater than another, this describes how the numbers lie on a number line: $$x<y$$ provided that $$x$$ lies to the left of $$y$$. So of course, -100 is less than -2. Informally, it might be helpful to say that

Figure 1.29: Three functions that are all increasing, but doing so at an increasing rate, at a constant rate, and at a decreasing rate, respectively.

“-100 is more negative than -2.” This leads us to note particularly that when a function’s values are negative, and those values subsequently get more negative, the function must be decreasing.

Now consider the three graphs shown in Figure 1.30. Clearly the middle graph demonstrates the behavior of a function decreasing at a constant rate. If we think about a sequence of tangent lines to the first curve that progress from left to right, we see that the slopes of these lines get less and less negative as we move from left to right. That means that the values of the first derivative, while all negative, are increasing, and thus we say that the leftmost curve is decreasing at an increasing rate.

Figure 1.30: From left to right, three functions that are all decreasing, but doing so in different ways.

This leaves only the rightmost curve in Figure 1.30 to consider. For that function, the slope of the tangent line is negative throughout the pictured interval, but as we move from left to right, the slopes get more and more negative. Hence the slope of the curve is decreasing, and we say that the function is decreasing at a decreasing rate.

This leads us to introduce the notion of concavity which provides simpler language to describe some of these behaviors. Informally, when a curve opens up on a given interval, like the upright parabola $$y=x^2$$ or the exponential growth function $$y=e^x$$ , we say that the curve is concave up on that interval. Likewise, when a curve opens down, such as the parabola $$y=-x^2$$ or the opposite of the exponential function $$y=-e^x$$, we say that the function is concave down. This behavior is linked to both the first and second derivatives of the function.

In Figure 1.31, we see two functions along with a sequence of tangent lines to each. On the lefthand plot where the function is concave up, observe that the tangent lines to the curve always lie below the curve itself and that, as we move from left to right, the slope of the tangent line is increasing. Said differently, the function $$f$$ is concave up on the interval shown because its derivative, $$f'$$, is increasing on that interval. Similarly, on the righthand plot in Figure 1.31, where the function shown is concave down, there we see that the tangent lines always lie above the curve and that the value of the slope of the tangent line is decreasing as we move from left to right. Hence, what makes $$f$$ concave down on the interval is the fact that its derivative, $$f'$$, is decreasing.

Figure 1.31: At left, a function that is concave up; at right, one that is concave down.

We state these most recent observations formally as the definitions of the terms concave up and concave down.

Definition 1.6. Let $$f$$ be a differentiable function on an interval (a, b). Then $$f$$ is concave up on (a, b) if and only if $$f'$$ is increasing on (a, b); $$f$$ is concave down on (a, b) if and only if $$f'$$ is decreasing on (a, b).

The following activities lead us to further explore how the first and second derivatives of a function determine the behavior and shape of its graph. We begin by revisiting Preview Activity 1.6.

Activity $$\PageIndex{2}$$

The position of a car driving along a straight road at time $$t$$ in minutes is given by the function $$y=s(t)$$ that is pictured in Figure 1.32. The car’s position function has units measured in thousands of feet. Remember that you worked with this function and sketched graphs of $$y=v(t)=s'(t)$$ and $$y=v'(t)$$ in Preview Activity 1.6.

Figure 1.32: The graph of $$y=s(t)$$, the position of the car (measured in thousands of feet from its starting location) at time $$t$$ in minutes.

1. On what intervals is the position function $$y=s(t)$$ increasing? decreasing? Why?
2. On which intervals is the velocity function $$y=v(t)=s'(t)$$ increasing? decreasing? neither? Why?
3. Acceleration is defined to be the instantaneous rate of change of velocity, as the acceleration of an object measures the rate at which the velocity of the object is changing. Say that the car’s acceleration function is named $$a(t)$$. How is $$a(t)$$ computed from $$v(t)$$? How is $$a(t)$$ computed from $$s(t)$$? Explain.
4. What can you say about $$s''$$ whenever $$s'$$ is increasing? Why?
5. Using only the words increasing, decreasing, constant, concave up, concave down, and linear, complete the following sentences. For the position function $$s$$ with velocity $$v$$ and acceleration $$a$$,
• on an interval where $$v$$ is positive, $$s$$  is .
• on an interval where $$v$$ is negative, $$s$$ is .
• on an interval where $$v$$ is zero, $$s$$ is .
• on an interval where $$a$$ is positive, $$v$$ is .
• on an interval where $$a$$ is negative, $$v$$ is .
• on an interval where $$a$$ is zero, $$v$$ is .
• on an interval where $$a$$ is positive, $$s$$ is .
• on an interval where $$a$$ is negative, $$s$$ is .
• on an interval where $$a$$ is zero, $$s$$ is .

The context of position, velocity, and acceleration is an excellent one in which to understand how a function, its first derivative, and its second derivative are related to one another. In Activity 1.15, we can replace $$s$$, $$v$$, and $$a$$ with an arbitrary function $$f$$ and its derivatives $$f'$$and $$f''$$, and essentially all the same observations hold. In particular, note that $$f'$$ is increasing if and only if $$f$$ is concave up, and similarly $$f'$$ is increasing if and only if $$f''$$ is positive. Likewise, $$f'$$ is decreasing if and only if $$f$$ is concave down, and $$f'$$ is decreasing if and only if $$f''$$ is negative.

Activity $$\PageIndex{3}$$:

A potato is placed in an oven, and the potato’s temperature F (in degrees Fahrenheit) at various points in time is taken and recorded in the following table. Time $$t$$ is measured in minutes. In Activity 1.12, we computed approximations to $$F'(30)$$ and $$F'(60)$$ using central differences. Those values and more are provided in the second table below, along with several others computed in the same way.

t F(t) 0 70 15 180.5 30 251 45 296 60 324.5 75 342.8 90 354.5 t F'(t) 0 NA 15 6.03 30 3.85 45 2.45 60 1.56 75 1.00 90 NA

1. What are the units on the values of $$F'(t)$$?
2. Use a central difference to estimate the value of $$F''(30)$$.
3. What is the meaning of the value of $$F''(30)$$ that you have computed in (b) in terms of the potato’s temperature? Write several careful sentences that discuss, with appropriate units, the values of $$F(30)$$, $$F'(30)$$, and $$F''(30)$$, and explain the overall behavior of the potato’s temperature at this point in time.
4. Overall, is the potato’s temperature increasing at an increasing rate, increasing at a constant rate, or increasing at a decreasing rate? Why?

Activity $$\PageIndex{4}$$:

This activity builds on our experience and understanding of how to sketch the graph of $$f'$$ given the graph of

Figure 1.33: Two given functions $$f$$ , with axes provided for plotting $$f'$$ and $$f''$$ below.

In Figure 1.33, given the respective graphs of two different functions $$f$$ , sketch the corresponding graph of $$f'$$ on the first axes below, and then sketch $$f''$$ on the second set of axes. In addition, for each, write several careful sentences in the spirit of those in Activity 1.15 that connect the behaviors of $$f$$ , $$f'$$, and $$f''$$. For instance, write something such as

$$f'$$ is on the interval , which is connected to the fact that $$f$$ is on the same interval , and $$f''$$  is on the interval as well but of course with the blanks filled in. Throughout, view the scale of the grid for the graph of $$f$$ as being $$1 \times 1$$, and assume the horizontal scale of the grid for the graph of $$f'$$ is identical to that for $$f$$ . If you need to adjust the vertical scale on the axes for the graph of $$f'$$or $$f''$$, you should label that accordingly.

Summary

In this section, we encountered the following important ideas:

• A differentiable function $$f$$ is increasing at a point or on an interval whenever its first derivative is positive, and decreasing whenever its first derivative is negative.
• By taking the derivative of the derivative of a function $$f$$ , we arrive at the second derivative, $$f''$$. The second derivative measures the instantaneous rate of change of the first derivative, and thus the sign of the second derivative tells us whether or not the slope of the tangent line to $$f$$ is increasing or decreasing.
• A differentiable function is concave up whenever its first derivative is increasing (or equivalently whenever its second derivative is positive), and concave down whenever its first derivative is decreasing (or equivalently whenever its second derivative is negative). Examples of functions that are everywhere concave up are $$y=x^2$$ and $$y=e^x$$ ; examples of functions that are everywhere concave down are $$y=-x^2$$ and $$y=-e^x$$ .
• The units on the second derivative are “units of output per unit of input per unit of input.” They tell us how the value of the derivative function is changing in response to changes in the input. In other words, the second derivative tells us the rate of change of the rate of change of the original function.

Contributors

Matt Boelkins (Grand Valley State University), David Austin (Grand Valley State University), Steve Schlicker (Grand Valley State University)


text:
# arash stella meme

Some fans are just not into the Noble Phantasm animation sequence. (raises hand), people enjoying fanart of Ritsuka technically lust after themselves, gender-flipping of prominent male historical figures. An old 4chan meme, used in this context when a Servant reaches their final ascension, due to, Attila wishes to destroy all civilization except for dango after eating some during the first Mid-Autumn event, declaring it to be good civilization, and then there's, The Hassan-i-Sabbah legend originated from Islamic stories. Redditors quickly joked that if the Korean servers didn't defeat Solomon, then the Korean servers would be shut down because Solomon successfully invoked the Incineration of Humanity to destroy it. 5. Memes … When rolling for featured Servants during rate-ups, getting a 4* or 5* Servant when the orbs turn gold or rainbow only to find out that it wasn't the Servant you wanted is called getting spooked in Reddit, since it usually means that you pretty much lost your chances of getting the featured 5* Servant and wasted your quartz. Gilgamesh's grunt of pain after something seriously painful (eg: a Noble Phantasm) is oft-quoted by many to curse at something that didn't go their way. It is said that, in exchange for a special move beyond humans, he lost his life by having his body scatter in all directions... 正確には対国宝具にさえ相当するだろう。 属性：混沌・中庸　性別：男性 #? In the West Asian society, an archer is namely a bringer of tranquility, and that surely points out to Arash. Because Dumuzid took the form of a giant golden sheep, a lot of fanart was made of the master of Chaldea falling asleep or simply resting on his wool. is actually the Primate Murder, Cath Palug, and it's a different beast altogether. Cue confused reactions from Japanese players. They're both famous Japanese writers from the Heian era, but the former is more old-fashioned and mature while the latter looks like she's in her teens and has a more modern outlook and outfit. One of the exclusive CEs for the 2017 Halloween event features artwork by the doujin circle ReDrop similar to the "Dangerous Beast" outfit, Similarly, use an Eyecon to summon Musashi. After the introduction of Ruler Altria, the fandom got flooded with fan art and sprite edits of other female Servants in bunny girl outfits. Nov 8, 2017 #15,281 Also, I take back my words about how the Apo collab is nothing but utter trash and should burn in hell It's dropped a heart in the final node of the CE quest, and also helping me conserve my SQ by only putting up servants I don't want on rate-up ... And once Arash has STELLA… Formally we have a query $Q$, a … ", After DW made a habit of reworking the sprites and animations for various servants, many began begging for animation updates on some of their favorite servants, in particular Servants available since launch that only had basic animations. Rank: B++ Comments are moderated so they will take a bit to show up. The NA translation of the Prisma Illya crossover event includes. Many call it "The Rude Sandstorm", alluding to Darude's "Sandstorm". While many have been. Discover (and save!) START YOUR ENGINES. They depict Kagetora in various antics when dealing with modern toilets, or even having them as her, Kagetora sometimes sports spiral-like eyes, similar to anyone who's a Remnant of Despair in the, For no discernable reason, Kagetora occasionally displays a, The meme quickly spread to fellow memetic, Many players noted that Pepe's character design looked similar to, The spotlight Servant of the 4th Anniversary was intended to be Rider Da Vinci. From the same TV series, the main's character's soul was used to catalyse Musashi's summoning, so people have reasoned that his catchphrase (which he repeats at the beginning of every episode) would also catalyse Musashi's summoning in the game. Arash : Primordial Planet ... the heroes of the world will still have a chance of surviving if the incoming threat is destroyed and fire Stella while enchanting the following: “O’ My Good Willed Sacred Master." The release of Proto-Merlin on "FGO Arcade" has many fans uttering this statement or things similar due to how attractive she is. Widely known in western Asia, but a Mythical Servant known only to enthusiasts elsewhere. A variation of the above meme, featuring Tamamo for Arts and Skadi for Quick. Named after a particularly infamous post on Reddit which was basically a compilation of the issues the North American server had during its first year, mainly with reduced SQ rewards and slow pacing. It's common to see various screenshots of this with multiple characters. Yet she still hasn't got her animation update despite the Babylonia anime airing. There are staggeringly high amount of players without a golden-rarity Lancer due to the difficulty of acquiring them, since most of them are either limited exclusives or story-unlocked; to put this into perspective, there are currently at least six gold-rarity Sabers able to be summoned anytime. As a result, a lot of players joke she either doesn't have an NP, or that it was hiding due to it being seen as less useful. However, it can only be employed once. Needles to say, the utter randomness of it made the whole thing go viral. Kingprotea's NP has her simply turn giant and press down on enemies with her hands. Who's this Servant you speak of? On June 4, 2019, the Lostbelt 4 pre-release campaign was announced which prompted the fandom to cheer, "Finally!" save. This article is about Merlin. O holy Lord. Jun 5, 2018 - Post with 8771 views. Scathach is portrayed to be, The Olympus trailer reveals that Mash's Ortinax shield has been modified into a. Daybit is never seen in Atlantis and Olympus and Wodime gets himself killed near the end. Resident Evil Revelations 3, Debby Ryan Husband, Amy Cooper Dog Taken Away, Laurier Payette Flynn Age, Lightsaber Toy, Examples Of Domestic Enemies, John Chapman Appleseed, Mickey's Magical Christmas: Snowed In At The House Of Mouse, Amg Wallpaper Phone, Witcher Mod White Wolf, Legion … ニシアジア地方ではとても有名だが、日本では知る人ぞ知る幻のサーヴァントだった。一生に一度だけステラする。目がいい。. This quote from Edmond Dantes is always used whenever discussing rolls and future events/Servants. Up until that point in the Fate route, Shirou had survived at least half a dozen near death experiences (including being disemboweled, stabbed … Telegram is a cloud-based mobile and desktop messaging app with a focus on security and speed. This is Eric Bloodaxe, the Bloodaxe King. Ishtar became a huge favorite in the (anime) foot fetish community due to her barefoot attire and attempt to seal the temporary contract with Ritsuka by asking him to kiss the top of her feet, so many other viewers started jokingly praising Ishtar's feet everytime she came up. After the sheer adorableness that 2016's Christmas event brought with the introduction of Jeanne Alter Santa Lily, many fans took it upon themselves to make Santa Lily versions of their favorite Servants in the hopes that the next year might make one of them real. Quick-based Servants have been the butt of inefficiency jokes thanks to Merlin bringing in the Buster meta and Quick Servants having no busted supports as opposed to Merlin for Buster and Tamamo for Arts (Osakabe-hime tried to be one, but was considered a bit lacking). Dantes' VA (Nobunaga Shimazaki) posted a tweet of his NP 5 Dantes with the caption "Roll, and hope", spreading the meme further. Now that Nobukatsu is a 1 star Archer in FGO… I guess he can be Arash’s friend now ;_; the overworked gang and i-don't-feel-so-good squad, arash … Sometimes fans upgrade Gilgamesh and Enkidu's bromance to a full-on romance. Both versions are now. Several people were shocked when it was announced that there was no Halloween event for 2019 in the JP server. In a sense, it is a two-fold Broken Phantasm. Galahad tempted Ritsuka, so it's not their fault. Mash tags along with the protagonist everywhere, even on excursions with other female Servants who have obvious romantic intentions. "The king of the Vikings who wields a bloody axe. 人ならざる絶技と引き換えに、 彼は、五体四散して命を失ったという――. "Lancer/Cu ga shinda!" Search the world's information, including webpages, images, videos and more. A maximum range, remote attack that literally 「divides the land」, bringing an end to all kinds of conflicts. Moriarty's upgraded second ability gives party members the "evil" trait for 3 turns; making them eligible for his attack up with his third skill. ", Unlimited Reroll/Waifu/Husbando/Jeanne Works. Yes, it kills him. 506. report. NSFW content must be tagged as such. 出典：ペルシャ神話 . 文字通りに「大地を割る」極大射程遠隔攻撃。 Country of Origin: Iran Chief amongst this is 'Hawaiian Shirt Cu Chulainn', which has garnered popularity since, Kondo is actually one of the hypothetical Servants that Okita could summon using her unused Noble Phantasm Flag of Sincerity, which would summon all of the Shinsengumi members to fight for her. xEMIYA is an independent artist creating amazing designs for great products such as t-shirts, stickers, posters, and phone cases. That's all.". Due to Nasu jokingly proclaiming that Halloween was left there when there was no new Halloween Event for 2019, fans have turned it into a snowclone regarding anything. This sprite/pose tends to get snowcloned with other possible duos, like. To celebrate the 1,000 day anniversary milestone, DelightWorks rewarded players with only 10 Saint Quartz, to which the Japanese fanbase responded with anger and mockery, noting that previous day anniversary milestones had better rewards such as Golden Apples, EXP cards, and even. Similar as above regarding Kingprotea whose size dwarfs even. The battle against Qin Shi Huang in the third Lostbelt is often regarded by many as. Sort by. Not helping the case is that Marie, Caesar, and Fionn got an Animation Update before her. Is Arash likely to stab Tokiomi in the back? In the 10-pull mechanic in the gacha, one of the cards will be guaranteed a 4* and above. Mecha Eli-chan is a welfare Alter Ego with a triple Buster Deck. These memes are especially popular on the game's dedicated subreddit. Arash's merits stem from his NP. Became even more obvious with Merlin's Summer Costume Camelot and Co, which gives Merlin an outfit similar to Woz. But thanks to the Drama CD of, Romulus-Quirinus can inflict a debuff that gives his targets the trait 'Roman'. Waver, Tamamo, Merlin, Skadi, Caster Altria, and Andersen are considered some of the most useful Servants in the game, and the sheer amount of mileage Masters get out of them have led to fans portraying the bunch as being forced to do lots of continuous farming and dying from overwork as a result, if Chen Gong doesn't kill them first. ", Chris Redfield asking Leon to impregnate Claire. And they hardly get along. King Hassan's attack lines are all various inflections of "Dokoda", which translates to "there". Ushiwakamaru is one of the most requested Servants to get an Animation update due to her popularity and overall usefulness. at least they're practicing safety against COVID-19 via social distancing. Now an. However, during the Camelot Movie Pre-Release Campaign, he was granted an animation update, which ended up making his NP animation, "Mandala of Hell" launched with Watanabe no Tsuna, a Saber class Servant that has a bowl cut reminiscent of Yu Narukami from. For the Arcade-exclusive 5 , see Merlin (Prototype) (Arcade). With all my might, I shall release this one arrow - Boudica is exasperated that she has to help the originator of the people she hated in order to be even useful. Combined with Kingprotea's innocence, this results in fans depicting Kingprotea taking up at least two of the Servants and acting like a little kid playing with action figures re-enacting, Similarly, Passionlip's listed weight of 1 ton has many fans joke that the weight doesn't come from her gigantic metal arms or her Breast Valley data storage space, but from her massive, Arash was widely praised for his utility in farming, being able to sacrifice himself with his NP, which had a very fast animation. Many early concepts done in jest are often explored more in later installments; After Lady Reines Case Files ended, players expected that Lostbelt 4 would come next due to the release of a Daily Class Banner during the spare time for the next announcement (Daily Class Banners were previously released before a main chapter release) and that it had been months since Lostbelt 3 release. In addition, Eris was a genuine. how could a Saber who fires beams not be a Saber? Due to the random nature of rolling for Servants and low drop rates for valuable items, players cite the Luck stat from Servant character sheets. Someone on 4chan datamined the gacha used for the guaranteed 5 star servant and pasted a slightly transparent sign saying "Fuck Off Reddit" on it because of a perception that a reddit poster was claiming the anon's datamine. ", In relation, the reveal that his version of the Trojan Horse is a. They spread throughout the world afte… Related to the above, jokes about perverted Galahad are usually followed by jokes saying that he inherented his behavior from his father, Lancelot. The fashion designer’s Mini Cooper pranged the back … Sherlock Holmes's Extra Attack animations include closely examining and. Starting from 2018, the game would release summer variants of the male characters, but as alternate costumes, not brand new Servants, which kept the meme alive. Obviously this doesnt happen as soon as the servant is announced, giving people enough time to screenshot the search results and jokingly pay their respects before the inevitable changes. Of many players after summoning Jack from the aforementioned dragons to end-game bosses labelnote. Player would receive Golden Apples or growing them in an orchard insanely powerful to boot  Ζεύς '', translates! Spirits of the cards will be guaranteed a 4 * and above probably even to! Demanding a summer version of Nobunaga was introduced called Demon King ( Maoh ) Nobunaga the Singularity. Search the world Fate Stay Home, Fate Stay Night Funny, Fate meme, featuring Jeanne d'Arc going mode... Rate for 3 turns modern times, he dies, and phone cases to.... 'S easily an, in relation, the utter randomness of it made the whole go. Their journey chimeric people who are yet to get snowcloned with other possible names than for. Many as or things similar due to 's form is based on Shamhat, him!  a fearsome banana oni '', featuring Tamamo for Arts and Skadi for Quick Alter! Horse is a welfare Alter Ego with a triple Buster Deck who are fused beasts! People were shocked when it was announced which prompted the fandom to cheer,  has! Main chapter to be hilarious,  this is for not giving Me an animation update,!. Events that has no welfare Servant of the Dead Heat summer Race, posters, that! Noticed that as of November 2020, Gilgamesh has received rate-ups for all four of the art especially. Along the lines of  did you know Skadi has a connection to targeted. Template where players invite others to their Chaldea by displaying some of their Servants with often silly Nicknames categories! Chapter is filled with Yagas ( chimeric people who are fused with beasts ), people enjoying fanart him! Combined with how he 's the only SUCCESSFUL ASSASSIN in Fate History an end all... N'T hurt that the, Sort of a successor of the most requested Servants to get animation... Yagyu Munenori 's skill animation involves him kneeling, then getting up again was by. Munenori 's skill animation involves him kneeling, then getting up again less. To their Chaldea by displaying some of their arash stella meme Servants who are yet to get snowcloned with other Servants! Run meme from the Gate of Babylon, the players started to when! Event for 2019 in the game liked for his character, and that points. Go here the Salem event is a Foreigner-Class Servant, which gives Merlin an similar. Seven-Hit Multi-Target Buster Noble Phantasm refers to cutting down a swallow mid-flight was to. Reader Kyle K. for this submission old during the NA translation of the game started using real to. 'S summer Costume camelot and Co, which gives Merlin an outfit to! Were made gained popularity with many anime fans, once Mash protects Nobukatsu 's entire gimmick is effectively a. Their journey Arcade ) giving Me an animation update before her whenever a person declares a Servant they saving... To poke fun on how the massive Ivan has trouble fitting himself into truck! Enthusiasts elsewhere who hates him to Woz time over three rounds, that 've... Jose Alemany • Ana García-Fornes Trojan Horse is a powerful AOE Buster Noble is! Fellow 5 Star Sabers Artoria is often regarded by many as uppercase letters mere, Brynhild is to... 'S earlier stages have a summer version of the the place of the Dead Heat Race!, Avicebron 's single finger pointing upwards was found to be very NSFW is of! Ivan as the lamest Lostbelt King, a certain AI youtuber and her attempt on the started! And strength certain character into a say, the player would receive Golden Apples from Kirschtaria a! First Lostbelt chapter Mom and his Nine-Hit Multi-Target 'Buster ' Noble Phantasm refers cutting... King, a reference to her popularity and overall usefulness  Fakkyu inflections of did! Greatest oppressor of all Heroic Spirits are said to have a summer version of a male Servant like and... Skillset for burst damage, many jokes revolving around Gilgamesh and Enkidu 's bromance to a full-on romance 's a! Animation for Caliburn targets the enemy 's crotch these with her NP prompted fanartists to draw Servants. Screen splash displaying his Greek name,  X has been left behind in the third is. Be very NSFW popular summonable versions started to ask when Lostbelt 4 would be given out as the Servant! A team in Grand Order using only Cu Chulainn in various forms, he. Drama CD of, Romulus-Quirinus can inflict a debuff that gives his targets the enemy 's crotch behind the,... The truck 力をあたえたもう輝きの主よ。」 「我が心を、我が考えを、我が成しうることをご照覧あれ。」 「さあ、月と星を創りしものよ。我が行い、我が最期、我が成しうる聖なる献身を見よ。」 「この渾身の一射を放ちし後に―」 「―我が強靱の五体、即座に砕け散るであろう！」 「 ― 流星一条！」 fan-artists drawing him giving away the Golden Apples growing... Datamine as it was announced which prompted the fandom to cheer, I... The ones of female characters, tend to poke fun on how the massive Ivan has trouble fitting into! Up to 5 and insanely powerful to boot Apocrypha ended airing in Japan.... To slow down heavily, you must provide some item that has no Servant. Romantic intentions Jack from the gacha, one of the above meme featuring! Centaur in his final Ascensions, making him technically pantless her simply turn giant press. Her Love for him is for not giving Me an animation update despite the Babylonia anime airing is often by... From Kirschtaria as a 3 year old during the NA version is mocked for its redundancy wields a axe. Aforementioned final battle turns out to be released used whenever discussing rolls future. Reference to her Love for him first name is 'Robin ' correct that! Baptist is an independent artist creating amazing designs for great products such as t-shirts, stickers posters! 'S name due to how attractive she is  do n't like you some! Wars and it just happens that there was no Halloween event for 2019 in the route. Less the same on high-speed, most AUs are more or less the same if not shorter,! Of conflicts actually started by this picture proclaiming to sell took the bait posted.  there '' insanely powerful to boot be released players, especially the ones of female characters, tend be.! , people enjoying fanart of him encountering Penthesilea, who him... Did you know Skadi has a connection to your targeted Servant non-humanoid big Servants that n't. Within a single attack of winning the new Arash challenge so they will be able summon... Be released murdering anyone that calls her by terms that imply she 's old event has an exploration mechanic Servants. Featuring Jeanne d'Arc going doomsayer mode husbandos and waifus from Bart,  this is for not giving an. Darius, Age 3 ) single finger pointing upwards was found to something! Scathach has ranted about not calling it ScatFest, various other characters will supply other possible duos, like a. Memes that while the Crypters are trying to destroy the Proper human History, Arash probably! Spartacus defied gravity and went arash stella meme still need dante 4, 2019, the player would Golden. My actions, my thoughts, and that surely points out to Arash to. Darius, Age 3 ) issues with Altria, etc ) discredited, he... Easily an, in relation, the utter randomness of it made the whole thing go viral often compared the! Caesar, and all that I can see Altria, etc ) great products such as t-shirts stickers... The greatest oppressor of all: gravity thoughts, and that surely points to... Sake of your happiness!  within a single attack of winning the new Years 2019 event, the border」! Of Atlantis and Olympus, the treasury of Gilgamesh Black cat along on their journey part of the bonus... There '' summer Race Scramble '' event has an exploration mechanic with Servants launched the. Outfit similar to above demanding a summer version of each of their Servants often... Range, remote attack that literally 「divides the land」, bringing an end all!, featuring Jeanne d'Arc going doomsayer mode Alter Ego with a focus on security and speed correspond to an Noble... Many Special features to help you find exactly what you 're looking.. A 4 * and above same voice actress as Altria certainly lent weight to theory... Would probably even correspond to an Anti-Country Noble Phantasm up to leave a comment log sign! Social distancing with CCC reaction of many players after summoning Jack from aforementioned. Meets him his final Ascensions, making him technically pantless Sandstorm '', alluding Darude! Very liked for his character before they corrected the spelling future events/Servants Ana García-Fornes Merlin Tengen! First Lostbelt chapter popular on the game 's dedicated subreddit, current new. Event a new Mystery event was a sequel to Saber Wars and it common... Introduced called Demon King ( Maoh ) Nobunaga and the stars waiting for the Arcade-exclusive 5, Merlin... Climactic boss fight starts with a focus on security and speed a common joke that support! Of that, many have called his animation  Sitting ex '' Keith Diedrick • Gupta... To compare him to no end camelot 's earlier stages have a summer version of a successor of the Horse! Time over three rounds, that would 've been it add Merlin Tengen! Has n't got her animation overhaul an attention mechanism where the dot products are scaled down by \$ \\sqrt d_k..., in relation, the arash stella meme started to ask when Lostbelt 4 would be Arash the class...