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Needless to say, most problems we encounter will not be so simple. Here's a slightly more complicated example: find $$\int 2x\cos(x^2)\,dx.$$ This is not a "simple'' derivative, but a little thought reveals that it must have come from an application of the chain rule. Multiplied on the "outside'' is $2x$, which is the derivative of the "inside'' function $\ds x^2$. Checking: $${d\over dx}\sin(x^2) = \cos(x^2){d\over dx}x^2 = 2x\cos(x^2),$$ so $$\int 2x\cos(x^2)\,dx=\sin(x^2)+C .$$

Even when the chain rule has "produced'' a certain derivative, it is not always easy to see. Consider this problem: $$\int x^3\sqrt{1-x^2}\,dx.$$ There are two factors in this expression, $\ds x^3$ and $\ds \sqrt{1-x^2}$, but it is not apparent that the chain rule is involved. Some clever rearrangement reveals that it is: $$\int x^3\sqrt{1-x^2}\,dx= \int(-2x)\left(-{1\over2}\right)(1-(1-x^2))\sqrt{1-x^2}\,dx.$$ This looks messy, but we do now have something that looks like the result of the chain rule: the function $\ds 1-x^2$ has been substituted into $\ds -(1/2)(1-x)\sqrt{x}$, and the derivative of $\ds 1-x^2$, $-2x$, multiplied on the outside. If we can find a function $F(x)$ whose derivative is $\ds -(1/2)(1-x)\sqrt{x}$ we'll be done, since then \eqalign{ {d\over dx}F(1-x^2) = -2xF'(1-x^2) &= (-2x)\left(-{1\over2}\right)(1-(1-x^2))\sqrt{1-x^2}\cr &=x^3\sqrt{1-x^2}\cr } But this isn't hard: \eqalignno{ \int-{1\over2}(1-x)\sqrt{x}\,dx&=\int -{1\over2}(x^{1/2}-x^{3/2})\,dx& (8.1.1)\cr &=-{1\over2}\left({2\over3}x^{3/2}-{2\over 5}x^{5/2}\right)+C\cr &=\left({1\over5}x-{1\over3}\right)x^{3/2}+C.\cr } So finally we have $$\int x^3\sqrt{1-x^2}\,dx=\left({1\over5}(1-x^2)-{1\over3}\right) (1-x^2)^{3/2}+C.$$

So we succeeded, but it required a clever first step, rewriting the original function so that it looked like the result of using the chain rule. Fortunately, there is a technique that makes such problems simpler, without requiring cleverness to rewrite a function in just the right way. It does sometimes not work, or may require more than one attempt, but the idea is simple: guess at the most likely candidate for the "inside function'', then do some algebra to see what this requires the rest of the function to look like.

One frequently good guess is any complicated expression inside a square root, so we start by trying $\ds u=1-x^2$, using a new variable, $u$, for convenience in the manipulations that follow. Now we know that the chain rule will multiply by the derivative of this inner function: $${du\over dx} = -2x,$$ so we need to rewrite the original function to include this: $$\int x^3\sqrt{1-x^2}=\int x^3\sqrt{u}{-2x\over-2x}\,dx= \int {x^2\over-2}\sqrt{u}\,{du\over dx}\,dx.$$ Recall that one benefit of the Leibniz notation is that it often turns out that what looks like ordinary arithmetic gives the correct answer, even if something more complicated is going on. For example, in Leibniz notation the chain rule is $${dy\over dx}={dy\over dt}{dt\over dx}.$$ The same is true of our current expression: $$\int {x^2\over-2}\sqrt{u}\,{du\over dx}\,dx= \int {x^2\over-2}\sqrt{u}\,du.$$ Now we're almost there: since $\ds u=1-x^2$, $\ds x^2=1-u$ and the integral is $$\int -{1\over2}(1-u)\sqrt{u}\,du.$$ It's no coincidence that this is exactly the integral we computed in (8.1.1), we have simply renamed the variable $u$ to make the calculations less confusing. Just as before: $$\int-{1\over2}(1-u)\sqrt{u}\,du= \left({1\over5}u-{1\over3}\right)u^{3/2}+C.$$ Then since $\ds u=1-x^2$: $$\int x^3\sqrt{1-x^2}\,dx=\left({1\over5}(1-x^2)-{1\over3}\right) (1-x^2)^{3/2}+C.$$ To summarize: if we suspect that a given function is the derivative of another via the chain rule, we let $u$ denote a likely candidate for the inner function, then translate the given function so that it is written entirely in terms of $u$, with no $x$ remaining in the expression. If we can integrate this new function of $u$, then the antiderivative of the original function is obtained by replacing $u$ by the equivalent expression in $x$.

Even in simple cases you may prefer to use this mechanical procedure, since it often helps to avoid silly mistakes. For example, consider again this simple problem: $$\int 2x\cos(x^2)\,dx.$$ Let $\ds u=x^2$, then $du/dx = 2x$ or $du = 2x\,dx$. Since we have exactly $2x\,dx$ in the original integral, we can replace it by $du$: $$\int 2x\cos(x^2)\,dx=\int \cos u\,du=\sin u +C = \sin(x^2)+C.$$ This is not the only way to do the algebra, and typically there are many paths to the correct answer. Another possibility, for example, is: Since $du/dx = 2x$, $dx=du/2x$, and then the integral becomes $$\int 2x\cos(x^2)\,dx=\int 2x\cos u\,{du\over 2x}=\int \cos u\,du.$$ The important thing to remember is that you must eliminate all instances of the original variable $x$.

Example 8.1.1 Evaluate $\ds\int(ax+b)^n\,dx$, assuming that $a$ and $b$ are constants, $a\not=0$, and $n$ is a positive integer. We let $u=ax+b$ so $du=a\,dx$ or $dx=du/a$. Then $$\int(ax+b)^n\,dx=\int {1\over a} u^n\,du={1\over a(n+1)}u^{n+1}+C= {1\over a(n+1)}(ax+b)^{n+1}+C.$$

Example 8.1.2 Evaluate $\ds\int \sin(ax+b)\,dx$, assuming that $a$ and $b$ are constants and $a\not=0$. Again we let $u=ax+b$ so $du=a\,dx$ or $dx=du/a$. Then $$\int\sin(ax+b)\,dx=\int {1\over a} \sin u\,du={1\over a}(-\cos u)+C= -{1\over a}\cos(ax+b)+C.$$

Example 8.1.3 Evaluate $\ds\int_2^4 x\sin(x^2)\,dx$. First we compute the antiderivative, then evaluate the definite integral. Let $\ds u=x^2$ so $du=2x\,dx$ or $x\,dx=du/2$. Then $$\int x\sin(x^2)\,dx=\int {1\over 2} \sin u\,du={1\over 2}(-\cos u)+C= -{1\over 2}\cos(x^2)+C.$$ Now $$\int_2^4 x\sin(x^2)\,dx=\left.-{1\over 2}\cos(x^2)\right|_2^4 =-{1\over 2}\cos(16)+{1\over 2}\cos(4).$$ A somewhat neater alternative to this method is to change the original limits to match the variable $u$. Since $\ds u=x^2$, when $x=2$, $u=4$, and when $x=4$, $u=16$. So we can do this: $$\int_2^4 x\sin(x^2)\,dx= \int_4^{16} {1\over 2} \sin u\,du=\left.-{1\over 2}(\cos u)\right|_4^{16} =-{1\over 2}\cos(16)+{1\over 2}\cos(4).$$ An incorrect, and dangerous, alternative is something like this: $$\int_2^4 x\sin(x^2)\,dx=\int_2^4 {1\over 2} \sin u\,du= \left.-{1\over 2}\cos (u)\right|_2^4= \left.-{1\over 2}\cos(x^2)\right|_2^4=-{1\over 2}\cos(16)+{1\over 2}\cos(4).$$ This is incorrect because $\ds\int_2^4 {1\over 2} \sin u\,du$ means that $u$ takes on values between 2 and 4, which is wrong. It is dangerous, because it is very easy to get to the point $\ds\left.-{1\over 2}\cos (u)\right|_2^4$ and forget to substitute $\ds x^2$ back in for $u$, thus getting the incorrect answer $\ds -{1\over 2}\cos(4)+{1\over 2}\cos(2)$. A somewhat clumsy, but acceptable, alternative is something like this: $$\int_2^4 x\sin(x^2)\,dx=\int_{x=2}^{x=4} {1\over 2} \sin u\,du= \left.-{1\over 2}\cos (u)\right|_{x=2}^{x=4}= \left.-{1\over 2}\cos(x^2)\right|_2^4=-{\cos(16)\over 2}+{\cos(4)\over2}.$$

Example 8.1.4 Evaluate $\ds\int_{1/4}^{1/2}{\cos(\pi t)\over\sin^2(\pi t)}\,dt$. Let $u=\sin(\pi t)$ so $du=\pi\cos(\pi t)\,dt$ or $du/\pi=\cos(\pi t)\,dt$. We change the limits to $\ds \sin(\pi/4)=\sqrt2/2$ and $\sin(\pi/2)=1$. Then $$\int_{1/4}^{1/2}{\cos(\pi t)\over\sin^2(\pi t)}\,dt= \int_{\sqrt2/2}^{1}{1\over \pi}{1\over u^2}\,du= \int_{\sqrt2/2}^{1} {1\over \pi}u^{-2}\,du= \left.{1\over \pi}{u^{-1}\over -1}\right|_{\sqrt2/2}^{1}= -{1\over\pi}+{\sqrt2\over\pi}.$$

## Exercises 8.1

Find the antiderivatives or evaluate the definite integral in each problem.

Ex 8.1.1 $\ds\int (1-t)^9\,dt$ (answer)

Ex 8.1.2 $\ds\int (x^2+1)^2\,dx$ (answer)

Ex 8.1.3 $\ds\int x(x^2+1)^{100}\,dx$ (answer)

Ex 8.1.4 $\ds\int {1\over\root 3 \of {1-5t}}\,dt$ (answer)

Ex 8.1.5 $\ds\int \sin^3x\cos x\,dx$ (answer)

Ex 8.1.6 $\ds\int x\sqrt{100-x^2}\,dx$ (answer)

Ex 8.1.7 $\ds\int {x^2\over\sqrt{1-x^3}}\,dx$ (answer)

Ex 8.1.8 $\ds\int \cos(\pi t)\cos\bigl(\sin(\pi t)\bigr)\,dt$ (answer)

Ex 8.1.9 $\ds\int {\sin x\over\cos^3 x}\,dx$ (answer)

Ex 8.1.10 $\ds\int\tan x\,dx$ (answer)

Ex 8.1.11 $\ds\int_0^\pi\sin^5(3x)\cos(3x)\,dx$ (answer)

Ex 8.1.12 $\ds\int\sec^2x\tan x\,dx$ (answer)

Ex 8.1.13 $\ds\int_0^{\sqrt{\pi}/2} x\sec^2(x^2)\tan(x^2)\,dx$ (answer)

Ex 8.1.14 $\ds\int {\sin(\tan x)\over\cos^2x}\,dx$ (answer)

Ex 8.1.15 $\ds\int_3^4 {1\over(3x-7)^2}\,dx$ (answer)

Ex 8.1.16 $\ds\int_0^{\pi/6}(\cos^2x - \sin^2x)\,dx$ (answer)

Ex 8.1.17 $\ds\int {6x\over(x^2 - 7)^{1/9}}\,dx$ (answer)

Ex 8.1.18 $\ds\int_{-1}^1 (2x^3-1)(x^4-2x)^6\,dx$ (answer)

Ex 8.1.19 $\ds\int_{-1}^1 \sin^7 x\,dx$ (answer)

Ex 8.1.20 $\ds\int f(x) f'(x)\,dx$ (answer)


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## You are here

• You are free to use any of the files of the SMART library of the SMAC toolbox for personal or academic use. The express permission of the authors is required for any commercial use.
• You can redistribute the SMART library of the SMAC toolbox and its manual without modification provided that it is for a non commercial purpose. Redistribution in any commercial form including CD-ROM or any other media is hereby forbidden, unless with the express written permission of the authors. Any of your software developments related to this download will be available under similar license conditions.
• Neither the authors nor ONERA accept any responsibility or liability with regard to this software that is licensed on an "as is" basis. There will be no duty on authors or ONERA to correct any errors or defects in the software.

### History

Current version of the SMART library: v1.7 released on 6 July 2021.
Older versions:

• v1.6 released on 12 October 2020
• v1.5 released on 4 October 2018
• v1.4 released on 11 July 2016
• v1.3 released on 12 November 2014
• v1.2 released on 18 June 2014
• v1.1 released on 30 April 2014
• v1.0 released on 21 March 2013

 [2] C. Roos, F. Lescher, J-M. Biannic, C. Doll and G. Ferreres, "A set of $\mu$-analysis based tools to evaluate the robustness properties of high-dimensional uncertain systems", in Proceedings of the IEEE Multiconference on Systems and Control, Denver, Colorado, September 2011, pp. 644-649. [3] C. Roos and J-M. Biannic, "A detailed comparative analysis of all practical algorithms to compute lower bounds on the structured singular value", Control Engineering Practice, 44:219-230, 2015.

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Posted by: holdenlee | September 3, 2012

## What is wrong with education? (Part I: Slowing down to speed up)

The US pours more and more money into the educational system every year–so that we spend upwards of \$10,000 per student per year. Yet our test scores have stayed flat relative to the rest of the world—for instance, the US still ranks 25th out of 34th countries in math test scores on the OECD. We call education “the great equalizer,” yet the students that aren’t graduating from high school are still those who come from inner city schools, and many of them are black. Every president seems to have some new educational policy–from No Child Left Behind to Common Core–that failed to create the promised change. And even for the students who’ve aced their classes there seems to be a widening gap–or perhaps a gap that has always been there–between getting the A’s and getting hired, and the reality of a weak correlation between their grades and their success in life.

What is wrong with education?

I like to start an essay with a “vision” of what we can do, but I refuse to do so here: there is no easy fix. If I give a vision, you, the reader, may strike it down as something that “sounds great in principle but won’t actually work.” Too many people have had their own “pet policies” on education that they adhere to religiously, simply because they came up with it. In lieu of a vision, I will say instead that to answer the question “what is wrong with education” we have to examine all aspects of the question:

• What are teachers doing; what are parents doing; what are students doing?
• How has society changed in the last half-century?
• How do we teach and certify our teachers?
• Are test scores actually a good metric for determining success?
• How relevant is what schools teach to the skills people actually need in their real lives?
• Where are all the good teachers going?
• What are the actual effects of policies like No Child Left Behind?

We need to reflect on how our own teachers taught us, what we liked and didn’t like. We need to read the statistics that are already out there, and opinions of people with many years of experience in education. This is why I am slowing down, and exploring these questions in a series of posts.

But much of what I want to say has already been said, by many other good teachers who can put it better than I. Most of what I learned about education is from Mathew Crawford, during my internship at Gliya this summer. Before I give my own perspective, I encourage you to take a look at what teachers themselves are saying, for instance Kenneth Bernstein (a.k.a. Teacherken; my AP government teacher, recently retired) blogs about education at dailykos; Mike Polinquin blogs about teaching math. Far more has been said then I have the time to read. If you’re interested, just search on the web, or go to the library and pick up a book about education (but do some research first to make sure it’s worth reading!).

Today I’d like to explore the question: How should we pace curriculum?

## Slowing down to speed up

I talked to a teacher who used to teach first grade (back when there wasn’t compulsory Kindergarten). In the past, she said, parents took responsibility for their children’s education. In parent-teacher conferences, she would say things like, Johnny is a little behind the other kids in school, but Johnny can catch up if he does some extra work on X, Y, and Z at home, and mostly, the parents would listen and give Johnny some extra help. And if at the end of the school year Johnny still wasn’t up to par, she would recommend that Johnny repeat first grade, or else the parents do a whole lot of extra work with Johnny over the summer.

Parents often trusted her and let their Johnny repeat first grade.

Although kids like Johnny are “held back” for one year, for the rest of their school life they are often in line with, if not ahead of, their peers. For instance, one of her “held back” students went on to become valedictorian in high school.

These kids aren’t dumber than their classmates. Maybe, they just weren’t quite ready at age 5, and they are at age 6.

Can you imagine a parent saying in this day and age, “Yes, by all means, let Johnny repeat 1st grade.”?

Now all the talk is about speeding up our children’s education. Accelerated tracks, the invasion of AP’s into a high school student’s schedule, loaded on with lots of extracurriculars to tack on to a model resume.

More likely: “Johnny is every bit as smart as his classmates, I see no reason why he shouldn’t go on to 2nd grade.”

We’re allergic to slowing down. Being “held back” has a bad ring to it. Just consider, for instance, the naming of the bill “No Child Left Behind.” Being held back is not the same as “being left behind.” It’s called “slowing down to speed up.”

To make a vast generalization, it seems a lot of people–parents and students–want good results NOW. They think some auditor is going to appear on their doorstep any minute and judge the students’ performance. It’s not surprising, since modern life seems so fast-paced: credit cards give us money now, pay later; science/technology is making rapid progress; we can text anyone via cell phones and check anything via high-speed Internet; all in all, our pace of life gives us a vibrant economy ${\,}^{dubious|discuss}$, increased convenience, the ability of a single person to be informed on a lot of topics.

In comparison, the educational system seems to have made disproportionately little progress.

You argue: of course we need accelerated programs. Didn’t you, Holden, do AP’s in high school, accelerated math/science/whatever? I’m not arguing against accelerated programs. In fact, they’ve done what they claimed for “smart” students–accelerated their education. But many times kids are pushed into these classes when they’re not quite ready. It’s not that they can’t succeed in the classes–but they might get more out of the class if they take it one or two years down the road.

If you stop a random person on the street, what are the chances that he or she will be able to recite the quadratic formula? Or, let’s stick to something more useful, read a graph or table in the New York Times? Surprisingly low.

If we push students on to algebra 1 without learning fractions, if we push them onto algebra 2 after they “pass” algebra 1 with a C or D, we are doing more than just wasting $1 \frac{\text{hour}}{\text{day}} \times 6 \text{ years}$ of a student’s life. Students will not pick up anything meaningful from the next subject if they did not learn the previous one, or if they passed it my rote memorization without complete understanding. Why, then are we pushing kids algebra-geometry-algebra 2-trig-precalculus if at the end they’ll forget what they learned in *algebra*? Why not slow down and make sure they actually learn the kind of math they’ll need in their daily lives—like how to read a graph or table in the New York Times?

Whether everyone actually needs to learn all these math classes is a separate post in itself (I argue not). But let’s ignore this point for now.

By pushing students before they’re ready, we’re doing more than wasting the students’ time with facts that they’ll forget. We’re teaching them how to feel hopeless. We’re teaching them to despise what they learn. We’re teaching these students that school is not giving them what they need for their lives. It only gets harder and harder for these students to catch up in each grade—and they don’t feel motivated to, because they don’t see how the algebra 2 they’re learning will be relevant to their lives in any way—and many of these students hit a point where they simply give up. Maybe they get in disciplinary trouble. Maybe they drop out. We say that college leads to a better life, but high school already feels like hell to them… and what will college be?

Don’t just take my word for it: http://www.usatoday.com/news/opinion/forum/story/2012-07-09/math-education-remedial-algebra/56118128/1

In future posts, I’ll explore ideas for how we can actually “slow down to speed up,” and another important question: How much of education should be standardized? How much freedom should we give teachers and students?


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## anonymous one year ago What is the midpoint of a line segment with the endpoints (-6, -3) and (9, -7)? A. (1, -4.5) B. (1.5, -5) C. (-4.5, 1) D. (-5, 1.5)

1. anonymous

What do you think is the answer?

2. anonymous

what formula do i use

3. anonymous

You use the midpoint formula, which is $$\huge (\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}) = (x, y)$$

4. anonymous

its B

5. anonymous

Correct!!!!


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# Question

A forensic chemist is given a white solid that is suspected of being pure cocaine (C17H21NO4, molar mass = 303.36 g/mol). She dissolves 1.22 + 0.01 g of the solid in 15.60 + 0.01 g of benzene. The freezing point is lowered by 1.32 + 0.04oC.
a. What is the molar mass of the substance? Assuming that the percentage of uncertainty in the calculated molar mass is the same as the percentage of uncertainty in the temperature change, calculate the uncertainty in the molar mass.
b. Could the chemist unequivocally state that the sub-stance is cocaine? For example, is the uncertainty small enough to distinguish cocaine from codeine (C18H21NO3, molar mass = 299.37)?
c. Assuming that the absolute uncertainties in the measurements of temperature and mass remain unchanged, how could the chemist improve the precision of her results?

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# University of web

In response to popular non-existent demand, we are publishing the most rigorous class that we require all first-year PhD students to briefly skim: Introduction to web3.

Of course, remember that I founded this institution with the goals of:

1. fomenting resentment against web3
2. perpetuating Big Tech, who rewards me handsomely with tungsten cubes to criticize any perceived threat to their continued dominance
3. doing well for a college course, which is what this is all actually for

TL;DR: web3 is overhyped, inefficient, and dangerous.

# Why is it called web3?

## web1.0

Archaeologists specializing in the prehistoric times of the 1990s agree that the World Wide Web began as universities and nerds hosting their own websites and mailservers. Despite the simplicity of this early version of the internet, many of the infrastructural components of the modern web were developed and iterated in a familiar alphabet soup: DNS, HTML, CSS, HTTP, etc.

## web2.0

Then we saw the emergence of tech companies that offered to ease the burden of having to run your own services. These companies would make it easy to connect with others and generate content for the world to see. You wouldn’t have to host your own website or mailserver anymore. You could just worry about producing content, and not the underlying infrastructure that was becoming increasingly complex as the internet was expanding.

In our current web2.0 era, we saw the rise of Facebook, Google, Instagram, Twitter, Reddit, Discord, WeChat, and so on. These platforms helped to faciliate an explosion in the internet’s global adoption. The user experience of communicating with others and finding information online was a lot easier and faster. The internet was no longer a niche thing.

Though these services have made our lives convenient, a lot of internet activity is now taking place under a few tech conglomerates. This has fueled concerns over privacy, censorship, and user autonomy.

## web3

web3 is a proposed set of technologies — largely centered around cryptocurrencies and the applications of crypto — that will define the next generation of the internet. In response to centralization of the internet that occurred under web2.0, proponents of web3 argue that a new internet built on immutable blockchain technology will improve privacy, avoid censorship, and create new economic opportunities.

web3 isn’t magic. It is merely a bundle of blockchain technologies working together so you can buy cryptocurrency. Nestled between two industries notoriously vulnerable to hype, crypto’s positioning at the center of finance and computer science can easily confound and disorient even the most technically-adept. So to fully explore the risks of web3, we need to start from the beginning.

# Is it effective?

## Bitcoin

Bitcoin was published in 2009 by the pseudonymous Satoshi Nakamoto, whose real identity remains the subject of speculation.

CAUTION – The following content distills a very technical topic

The basis of cryptocurrency is that the individual’s digital wallet is a kind of one-man show. You don’t need a banking apparatus to transact your crypto because each digital wallet has a unique address associated with it. You may have heard that crypto is like a digitized form of cash, which is a fairly appropriate way to describe it.

But unlike cash, crypto’s decentralized nature makes it resistant to inflation and intentional devaluation because it has no intrinsic governing body.

Decentralization is a buzzword that we’ll return to several times. In this context, decentralization refers to the lack of control any singular entity is supposed to have over the cryptocurrency as a whole. Whereas you’re probably familiar with the notion of a central bank dictating monetary policy – i.e., the US dollar and the Federal Reserve – crypto has no de jure authority.

### Proof-of-work

Though promising as described above, Bitcoin has fallen way short of its ambitions. Bitcoin relies upon a mechanism called “proof-of-work” to validate new nodes of transaction data added to the blockchain.

A blockchain is a digital ledger that cannot be edited once it is recorded. The Wikipedia page on blockchains is a good place to learn more.

This is a very computationally intensive process, which is to say that it requires an enormous amount of computing power. Bitcoin was designed to become more computationally intensive depending on the total computing power of the network. This was supposed to prevent a single actor from amassing a huge number of resources to dominate the blockchain, which is called a 51% attack.

a 51% attack is defined as a group of blockchain miners controlling the majority of the computing power available to the blockchain, and is therefore able to manipulate the blockchain. Imagine if a massive bunch of rogue voters decided to write-in their own candidate during an election, bypassing the typical party system.

So as the number of eager Bitcoin miners swelled in the past decade, the efficiency of the network plummeted. But as it turns out, these diminished returns are still marginally profitable, which makes mining exclusively favorable to miners who are willing to spend money on power-hungry mining rigs. Bitcoin alone consumes roughly the same amount of electricity as Argentina1, a country of over 45 million people.

The decentralized nature of Bitcoin paralyzes any attempt to improve the protocol. Imagine trying to change the way email works or switching the United States to the metric system; these endeavors are defeated by the inflexibility of the problem. Countless developers have attempted to “fork” the source code for Bitcoin and develop their own cryptocurrency to solve Bitcoin’s perceived flaws. Few of these offshoots have become successful, and most are generally indistinguishable from each other since creating a cryptocurrency is relatively trivial given the open-source nature of the field.

In fact, the issues plaguing Bitcoin are so endemic that within the crypto community, it is often regarded as inferior to other supposedly better currencies. Despite its poor reputation, Bitcoin remains one of the most popular crypto currencies because its skyrocketing price and notoriety in recent years makes it an attractive investment to many newcomers, who are inundated with advertisements to “invest in crypto today!”

## Ethereum

Ethereum is tapped to become the next big cryptocurrency superseding Bitcoin. Its blockchain is more versatile than that of Bitcoin’s, with the ability to handle and store arbitrary data beyond Ethereum transactions alone.

To this effect, people envision Ethereum being capable of being far more than a mere currency. In theory, it could store legal records and medical histories, although glaring complications make this far from trivial. Ethereum also advertises a consensus mechanism that would not require the insane power demands of Bitcoin: proof-of-stake.

### Proof-of-stake

Proof-of-stake is an alternative to proof-of-work whereby agents are issued transactions to validate depending on how much cryptocurrency they hold. Ethereum requires prospective validators to hold a measly 32 ETH to be eligible for this scheme. If you wanted to purchase 32 ETH sometime in the last year with fiat currency, it would have costed you a figure somewhere between $55,000 and$155,000 depending on your (un)lucky timing.

Despite proof-of-stake being designed to supposedly thwart the risk of a malicious actor by requiring that potential validators post some amount of collateral, this creates a system whereby only the most early adopters or wealthy individuals have the ability to participate in proof-of-stake. Given the typical American household has approximately $5,300 in savings,2 it is rare that a normal person just getting interested in crypto today can participate in proof-of-stake by idly investing money they likely don’t even have in a volatile, alien currency and running servers capable of handling Ethereum transactions. The odds are likely even less if we discard the assumption that our Ethereum dilettante isn’t from an developed economy. And while proof-of-stake is an order of magnitude more power efficient than proof-of-work, proof-of-stake still has an enormous energy demand. It also remains incredibly slow compared to established transaction systems, and inches Ethereum towards a level of centralization that cryptocurrencies ostensibly try to avoid. ## Transaction fees Cryptocurrency transactions require immense computing power to validate transaction data. To solve this, Bitcoin and Ethereum operate an auction system whereby users pay miners fees to process their transactions as quickly as possible. Let’s look in particular at Ethereum, which has exploded in popularity in the past few years. These fees, commonly known as “gas fees” within the Ethereum user base, have ranged sharply over the past few months: The more people trying to transact, the higher the fees become; a mechanism analogous to surge pricing on apps such as Uber or Lyft. These transaction fees make it implausible that mainstream crypto will be used to buy and sell goods and services, as they were originally intended. But unlike Uber and Lyft, ‘surge pricing’ is not a local event, and extends universally to the entire protocol. Imagine paying a higher ‘cryptocurrency processing fee’ at a local restaurant because people in a neighboring country are rushing to preorder movie tickets online. ## Paradox of centralization One of the most popularly cited advantages of web3 is decentralization. Decentralization in this case is the idea that the internet of the future will no longer be controlled by a handful of technology conglomerates that regulate and/or censor content If the internet was ‘decentralized,’ we would not have to worry about censorship. If we were to embed blockchain technology into the core of the internet, centralized power structures should eventually decay. But why exactly would this happen? It’s grandiose mental gymnastics. Centralization emerges in free markets because of economies of scale. And especially when Bitcoin and Ethereum have such high barriers to entry, the catalysts of centralization are even more apparent. Despite the hype surrounding decentralization, the cogs of web3 are ironically deeply centralized.3 We have: • Coinbase for trading and exchanging cryptocurrencies (89 million registered users) Un-fun fact: Mt. Gox used to handle upwards of 70% of all Bitcoin transactions in 2014. Then it suddenly shut down after people discovered that customers’ Bitcoins in Mt. Gox’s wallet had been systematically stolen and mismanaged for years. • OpenSea for trading NFTs • Metamask for Ethereum-based authentication • and company X to achieve basic function Y, etc. And most tragically, the blockchain is often so inefficient at scale that these organizations rely upon ordinary, boring technologies like SQL databases and C++ to handle transactions and user data. And where do these databases and code exist? On centralized computing resources like Amazon Web Services or Microsoft Azure. There are some genuinely effective implementations of a decentralized architecture. Folding@home, which uses volunteer computing power to simulate protein folding for medical research, is one of them. So is the Tor relay network, which enables millions of people to navigate the internet with anonymity and to evade heavy-handed electronic surveillance. Torrenting is an especially powerful example, because people attaining the contents of the torrent are simultaneously enabling that content to spread more easily. ## Renewable energy whataboutism Mining cryptocurrencies and validating blockchains consume an enormous amount of energy because their decentralized architecture demands a lot of redundant computational work. Crypto advocates defend cryptocurrency with two general stratagems: 1. Crypto is only as dirty as the energy it relies upon. This argument ignores the immense electronics waste that crypto generates — mining rigs are often several thousand dollars and consist of several high-end graphic cards originally designed for gaming. But more importantly, this argument shifts the blame of crypto’s wasted energy usage onto the world’s failure to adopt renewable energy. But cryptocurrencies do not accelerate the world’s shift to renewables, they instead delay it.The energy that goes into powering the Ethereum or Bitcoin network could have been used to power electric buses, tea kettles, and LED lightbulbs.4 In this sense, cryptocurrency is nullifying the effect of renewable energy, because it uses electrical capacity that could have gone to a far more productive cause. Granted, transaction speed and energy usage are loosely related, but the mere existence of a blockchain fundamentally demands energy. If the cost of electricity drops due to cheaper solar, wind, or nuclear power becoming readily available, then miners would simply scale up their activity in response. This is a classical demonstration of induced demand — the phenomenon where increased supply prompts increased demand. A city widens their highway (which takes great cost and time) to fight congestion; traffic instead increases to match the capacity of the road and congestion remains an issue. Eli Whitney thought his cotton gin would eliminate the need for slaves; instead, it made enslaved cotton production so efficient that plantations bought considerably more slaves to accomodate the throughput of the cotton gin. 2. The global banking industry consumes way more energy than crypto. People have different ideas of what the global banking industry is because the term itself is ambiguous. Is the global banking industry simply a collection of the largest banks, their subsidiaries, and payment processors such as Visa and Mastercard? Does it include the greater institutions that support the global banking industry, such as various national governments and the companies responsible for building millions of cash registers and payment keypads? Would it include the safeguards of international financial stability, such as the US military? The goalposts of this debate constantly shift and blur any attempt to answer the question of whether crypto or the global banking system is more efficient. If we were to adopt this whataboutism in the opposite direction, we can argue that crypto itself is complicit in the emissions of semiconductor manufacturers and rare-earth metal mining companies that provide the components and raw materials ultimately used to build mining rigs. What we do observe with certainty is that the gritty global banking system that the world relies on today provides financial services for nearly eight billion people transacting trillions of dollars per day. If crypto were to support the same global audience at its present stage of development, it’s doubtful that this system would be much more sustainable at scale. ## Crypto to the rescue of crypto? As we have seen, there are a bunch of fundamental mechanical problems with cryptocurrencies, whether they use proof-of-work or proof-of-stake. They spawn stratified power structures that are deeply antithetical to the spirit of the movement that promotes them. Because cryptocurrencies are open source, we have seen countless instances of developers forking a cryptocurrency and building their own implementation. This is especially easy when a nicely typeset $$\LaTeX$$ whitepaper is all you need to appear legitimate to gullible early adopters. The barriers to entry for a new cryptocurrency are so low that these derivative coins are often called shitcoins. But beyond those who are purely motivated by amusement or the hope of ‘getting rich quick,’ there are some developers who are working on hard-fought efforts to address the fundamental issues of cryptocurrencies and decentralized consensus protocols. You might hear of these technologies and movements – radix, sharding, DAOs, DeFi, layer 2. Here at the University of web3, we have decided to take a simple approach to these proposed technologies and ask ourselves: what problem are they truly solving? Is it too easy to confuse imitation for innovation, because many of these newer technologies ultimately only help catch crypto up to the performance and centralization of our established financial system? For instance, Ethereum proudly advertises that it will reduce its energy consumption by 99.95% once it fully adopts proof-of-stake. But as we have seen, proof-of-stake is a trade-off that further increases the centralization and barriers-to-entry of the Ethereum network. # Is it safe? web3 is built on technologies that are saturated with scams and fraud. ## Case study: non-fungible tokens (NFTs) In 2021, non-fungible tokens emerged as the keystone in a series of unfortunate events. A mechanism intended to somehow introduce scarcity to digital objects, NFTs bewilder any rational attempt to decipher their purpose. Non-fungible essentially means “non-substitutable” or “non-replicable”, and in this case means stamping unique serial numbers onto tokens hosted on the Ethereum blockchain. But this sense of uniqueness is completely lost upon the implementation of NFTs, which are little more than URLs pointing to an image file. If the URLs eventually no longer point to the artwork, then the NFT becomes an empty carcass since the blockchain can’t be retroactively edited. This implication of perpetual storage is a daunting responsibility to maintain, but we have already seen centralized services like NFT.STORAGE gladly advertise perpetual storage with no warning asterisks. The NFTs themselves are stored on “Filecoin storage providers,” which are basically just organizations with lots of hard drive server space who are hosting NFTs so long as they receive an appropriate return on investment. Becoming a Filecoin storage provider isn’t a trivial matter of installing some ordinary open-source software, it requires getting in touch with a sales team and building a power-hungry server, which again is antithetical to the whole “democratization of art” part of the NFT verbiage. From the Filecoin provider documentation, you’ll need: • 8+ core CPU • >128 gigabytes of RAM • one of these GPUs • a terabyte of NVMe-based SSD storage for caching • …and some other hard drives to serve as your actual server storage The CPU, RAM, GPU, and drives each cost more than a PlayStation 5.5 And this doesn’t begin to consider the cost of powering a +400 watt server6 that probably demands a faster internet connection than the one in a typical home. Those who can realistically afford and operate this kind of hardware belong to a technical minority, and their only incentive to hold your NFTs safely is their estimated return on investment. So now that we’ve addressed the worrying infrastructure of NFTs, let’s get back to the user side of things. Though the interface and user experience of NFT auction houses like OpenSea resemble that of an art auction, they are even more unscrupulous. Since cryptocurrency wallets are largely anonymous and at least pseudonymous,7 let’s see how we can synthesize some demand for a masterpiece in seven steps. 1. Alice, attracted to the NFT hype and possessing no artistic talent, mints an NFT for her ugly or stolen creation. 2. Alice publishes a page on OpenSea. 3. Alice observes no demand for her artwork. 4. Alice buys more Ethereum and deposits it into newly spawned wallets. 5. Alice uses her fresh wallets to bid on her worthless artwork and create the illusion of urgency and demand.8 6. Bob sees rapid interest and gets duped into buying Alice’s artwork. 7. Alice collects her profits. If Bob was swayed into buying the NFT, then Alice’s crypto would have gone back into her original wallet anyway. The big problem here is that both Alice and Bob need to spend their legitimate money to conduct this whole seven-act tragedy. The cost of minting the NFT is a volatile figure, but there are artists who have incurred net losses of several hundred dollars in various fees9 from minting their NFTs (which sometimes are merely stolen or lightly edited artwork). Meanwhile, they have provided upward pressure on the ETH:USD exchange rate and liquidity to cryptocurrency traders by having to buy Ethereum. ## Case study: rubber-hose cryptanalysis A bank offers many protections for its customers. If somebody steals your credit card, you can contact your bank’s fraud department to scrub the charges and you won’t have to pay anything. If your bank fails and you live in the United States, the federal government guarantees your savings are secure. “But wait! Consumer protections that guard against bank failure are an irrelevant counterargument against cryptocurrency’s safety, because cryptocurrencies operate without any need for banks! Why would you need to be protected from a bank robbery, if bank robbery can’t happen with crypto?” But in the cryptocurrency world, you are your own bank. There is nothing protecting you from poor memory, scammers, hackers, or people who are willing to beat you with a rubber hose and coerce you into revealing your wallet’s key. In fact, it is this complete lack of protection that forms the basis of why people created banks in the first place — so that their money could be safely kept away from them in a more secure location. If you put your account’s seed phrase on a flash drive and then forgot where that flash drive was, you’re SOL. There are horror stories of people who bought Bitcoin ten years ago, became millionaires unexpectedly, but are locked out of their newfound wealth because they can’t remember the decryption key to their hard drive, or they lost the storage device containing their wallet. Cryptocurrency transactions are not reversible. There isn’t any mechanism to revert a charge that you were misled into approving. This is one of the largest reasons why so many scams are conducted with cryptocurrency, because there is no law enforcement agency that can effectively halt a transaction or reverse it. # Is it wanted? At best, web3 developers have noble, if grandiose, motivations to improve financial equity or the freedom of the internet. At worst, it seems that web3 is nothing more than trying to lure normal people to buy cryptocurrency. This is the catch: by entering the web3 space, you provide liquidity to the Bitcoin and Ethereum market. As a striking figure, consider that users spent about$100 million in Ethereum gas fees to buy $200m in Bored Ape NFTs.10 The incentives in the web3 space are far removed from actually improving the quality of life issues we’ve seen with web2.0. As the price of Bitcoin and Ethereum has shot up by orders of magnitude in the last five years, we have seen countless news stories of random people who have become billionaires when considering their crypto net worth. The issue is that there are not enough buyers for these individuals to actually liquidate their holdings into a usable state of, say, US dollars.11 And this newfound wealth is tied to a highly speculative price. A price that is entirely contrived, and is tied to a currency that you cannot really use in a purposeful day-to-day manner. As such, the “whales” need somebody to buy into the crypto-hype, so they can wind their position down. a whale is a large holder of cryptocurrency who can manipulate a coin’s valuation through their sheer size This has been historically termed the “greater fool theory.” To stimulate demand for cryptocurrency, these bagholders need to create means for ordinary people to get involved. If you are particularly cynical, you can interpret the goals of NFTs in such a way. The greater fool theory posits that an already inflated asset can be resold at an even higher price, and generate profits for the original holder of said asset Remember the January 2021 GameStop fiasco? If you peer into the r/wallstreetbets threads, you’ll find countless users exhorting their fellow witless GME shareholders to “hold the line” and not sell their shares. All the while, those same users were quietly exiting their position and selling their shares to the next fool on the chopping block, such as the founder of University of web3. # So what’s next? The issue with web3, NFTs, or cryptocurrencies is that they fall way short of achieving their stated objectives. Cryptocurrencies are unable to solve the problems of the banking industry, web3 is unable to solve the problems of the tech industry, NFTs are unable to solve the problems of the art industry. This is because each of these industries suffers from dark patterns of human behavior, and web3 is not immune to such behavior. web3 does not prevent bad actors from exploiting and misleading others. The technologies surrounding web3 are instruments of human naivete and opportunism. Because of wild speculation and FOMO on behalf of venture capitalists eager to take a measured bet on such audacious claims, we have seen web3 startups explode in valuation despite delivering little value. Browse some of the technologies listed on this page and their landing pages, and try to convince yourself that you — as the potential user — are the intended audience rather than an investor. But there are still conceivable ways to dig us out of this mess. For instance, we can pin down the terrible externalities of crypto by instituting carbon taxes that will, among other things, disincentivize mining activity since its profitability is inextricably tied to the cost of electricity. But won’t this just cause miners to relocate to some country that doesn’t have a carbon tax? Sure, but the cost of shifting mining infrastructure is not cheap, and the margins of crypto mining are not particularly lucrative nowadays. This won’t outright halt emissions from cryptocurrency, but it may cause its inefficiencies to become so apparent that its users reconsider its utility, or at the very least hasten its transition to less polluting mechanisms. There is a direction that the internet is heading in, which unlike web3, does not need venture backing to sustainably grow or a minority of technocrat zealots to buy into. For better or worse, the actual web3 might be a network of tightening walled gardens and federations. Much of the internet has congealed into pools of user-produced content that are becoming increasingly inaccessible and myopic. Think of Reddit, Facebook, or Discord communities of thousands of people, whose activities are impossible to index on a search engine. Think of how the web became less navigable with artificial moats like login pages and paywalls, and is now being further entrenched with ideological barriers like echo chambers and content algorithms. An analogy we can use to express this trend is how our known universe is composed of galaxies that are drifting ever farther from each other, while each galaxy is crunching inward into a more compact form — the internet has created ever-tightening ingroups while separating people at large from each other. I myself know of two big ‘galaxies’ in the form of the Western internet and the Chinese internet, and we are seeing the schism of the alt-right internet taking hideous shape today. But I digress. At no point of this tectonic shift will a web3 of cryptocurrencies and NFTs have a naturally relevant seat. It will remain tomorrow as it lives on today: a messy dream. May 2022 Presented by Eric Cheng for 80-445 at Carnegie Mellon University. Opinions expressed are my own and do not necessarily reflect views of any affiliated people or organizations. No matter if you found reading this enjoyable, or feel as though I got something seriously wrong, I would be glad to talk further. Thank you to Simon Cullen for reading drafts of this. # Footnotes 1 According to researchers from the Cambridge Centre for Alternative Finance 2 According to the Federal Reserve’s Survey of Consumer Finances, American households had a median savings of$5,300 in 2019.

3

4

An interesting visualizer on Ethereum’s energy consumption

5

\$499 in the United States as of May 2022

6

Using the official supported GPU list and their average TDP, we can extrapolate the power requirements of the system at large.

7

See this Stack Exchange thread on the difference between anonymity and pseudonymity.

10

Data from this tweet by Will Papper


text:
# Image measure space

Is it possible to come up with a measure space $$(X, \mathcal{M}, \mu)$$ such that $$\{ \mu(E) : E \in \mathcal{M}\}=[0,+\infty]\setminus \Bbb Q^+$$ , where $$\Bbb Q^+$$ denotes positive rationals

I suspect yes, although I have no idea how to construct such a measure space like this

• @DonThousand if X is the empty set, then I suppose M is also empty, so {μ(E)∣E∈M} is empty and how could that be [0,+∞]\Q+?, I'm surely doing a bad reasoning – Robby S. Jan 13 at 20:38

There is no measure space $$(X, \mathcal{M}, \mu)$$ such that $$\{ \mu(E) : E \in \mathcal{M}\}=[0,+\infty]\setminus \Bbb Q^+$$ (where $$\Bbb Q^+$$ denotes the positive rational numbers).

Proof:

Let $$(X, \mathcal{M}, \mu)$$ be a measure space. Suppose that $$\{ \mu(E) : E \in \mathcal{M}\}=[0,+\infty]\setminus \Bbb Q^+$$.

By a result from Sierpinski, we know that the image of an non-atomic measure is an interval (see for instance, this article in Wikipedia). In fact, from Sierpinski's result, we can easy prove that: if the image of a measure $$\mu$$ does not contains any interval, then for every $$E \in \mathcal{M}$$ such that $$\mu(E)>0$$, $$E$$ contains (at least) one atom.

Since $$\{ \mu(E) : E \in \mathcal{M}\}=[0,+\infty]\setminus \Bbb Q^+$$, we can take a $$E \in \mathcal{M}$$ such that $$0<\mu(E)<+\infty$$. Let $$A \subseteq E$$ be an atom. We have that $$0<\mu(A) <+\infty$$.

Now take $$r\in \Bbb Q$$ such that $$\mu(A)< r < 2 \mu(A)$$. So, we have that $$0< r-\mu(A) < \mu(A)$$ Since $$\mu(A) \notin \Bbb Q$$ and $$r\in \Bbb Q$$, we have that $$r-\mu(A) \in (0, +\infty) \setminus \Bbb Q^+$$. So there is $$B \in \mathcal{M}$$, such that $$\mu(B) = r-\mu(A)$$.

Since $$A$$ is an atom and $$\mu(B) < \mu(A)$$, we have that $$\mu(A\cap B)=0$$. So $$\mu(A \cup B) = \mu(A) + \mu(B)= \mu(A) + r-\mu(A) = r \in \Bbb Q$$ Contradiction to the assumption that $$\{ \mu(E) : E \in \mathcal{M}\}=[0,+\infty]\setminus \Bbb Q^+$$.


text:
# Homework Help: Fourier Series help

1. Aug 4, 2011

### gtfitzpatrick

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

f(x) = $e^x$ for $-\pi/2 \leq x \leq \pi/2$

2. Relevant equations

3. The attempt at a solution

Having a bit of trouble getting this out.
This is a odd function right so $a_n = 0$ so i only have to find $b_n$ , right?

2. Aug 4, 2011

### lanedance

does the function repeat outside that interval?

the definition of odd is f(x) = -f(-x), does the function satisfy this?

3. Aug 4, 2011

### gtfitzpatrick

i'm not sure what you mean by this, but i assume it does as the function is periodic.

the definition of odd is f(-x) = -f(x) no???

i thought $e^{-x}$ = $-e^x$ but i guess im wrong so i'll work the whole lot out

4. Aug 4, 2011

### gtfitzpatrick

all the integrals below are over 0 to $\pi/2$

$a_0$ = $4/ \pi \int e^x$ dx = $4/ \pi (e^{\pi/2} - 1)$$a_0$ = $4/ \pi (e^{\pi/2} - 1)$

$a_n$ = $4/ \pi \int e^x cos(2nx)$ dx
$a_n$ = $4/ \pi(0-0)$ - $2/ n\pi n \int e^x sin(2nx)$dx
$a_n$ = $/n^2\pi (e^x (cosnx-cos0))$ - $1/ n^2\pi \int e^x cos(2nx)$ dx
$a_n$ = $e^x/ n^2\pi [ (-1)^2 - 1]$

am i going the right way with this?

5. Aug 4, 2011

### lanedance

yep this is the same as I wrote, but probably the way it is usually written

$f(-x) = e^{-x} \neq -e^x = -f(x)[/tex] so the function is not odd, hence you must calculate both an (cos terms) and bn (sin terms) the easiest way is to draw a sketch, it should eb pretty obvious whether it is symmetric (even) or anti-symmetric (odd) 6. Aug 4, 2011 ### lanedance why do you only integrate on the interval [itex] 0\leq x < \frac{\pi}{2}$?

isn't the problem defined on $-\frac{\pi}{2}\leq x < \frac{\pi}{2}$

7. Aug 5, 2011

### gtfitzpatrick

ahhh yes thanks lancedance, i was only integrating on half the interval and multiplying by 2, the same way that you would if it was an odd or even function! I should do it over the full interval!

A general question...if i wanted to get the half range fourier sine or cosine series for say f(x) = 2x if 0<x<1 ; f(x) = 2 if 1<x<2. This function can be extended over the y axis or across the origin as appropriate. But when im calculating the coefficients do i just integate over the 2 intervals and multiply be 2 in the way i would if i was dealing with and odd or even function? I'm thinking not.
?

8. Aug 7, 2011

### lanedance

I assume you mean it is known on the given interval and you a free to extend over the remaining x axis, the major importance is the fourier series is representative over the given interval

you have bit of leeway to play with here, generally it is eaisest to deifne the function outside 0<x<2 as a reflection in the y=0 axis and then repeating from there. That makes it an even function and halves the terms. However you could similarly come up with a scheme to make it odd or just repeat the function as is. There may be advantages to each in terms of evaluating the integrals or conitinuity which will leader to a faster convergence of terms.


text:
+0

# Geometric series

0
385
1

Hi, I start doing the task, but then I dont know what to do next. Here is:

1a+a2+a3=62

a1*a2*a3=1000

a1+a1*q+a1*q^2=62

a1*a1*q*a1*q^2=1000

a1+a1*q+a1*q^2=62

a1^3 * q^3=1000

Aug 23, 2017

#1
+22172
+2

Geometric series

Hi, I start doing the task, but then I dont know what to do next. Here is:

a1+a2+a3=62

a1*a2*a3=1000

$$\begin{array}{|rcll|} \hline a_1*a_2*a_3 &=& 1000 & | \quad a_1 = a \qquad a_2 = a*q \qquad a_3 = a*q^2 \\ a*(a*q)*(a*q^2) &=& 1000 \\ a^3*q^3 &=& 1000 \\ a^3*q^3 &=& 10^3 & | \quad \sqrt[3]{} \\ a*q &=& 10 \\ \mathbf{ a} &\mathbf{ =}&\mathbf{ \frac{10}{q} }\\\\ a_1+a_2+a_3 &=& 62 & | \quad a_1 = a \qquad a_2 = a*q \qquad a_3 = a*q^2 \\ a +a*q + a*q^2 &=& 62 \\ a *(1 + q + q^2) &=& 62 & | \quad \mathbf{a= \frac{10}{q}} \\ \frac{10}{q} *(1 + q + q^2) &=& 62 \\ \frac{1}{q} *(1 + q + q^2) &=& 6.2 \\ \frac{1}{q} +1+q &=& 6.2 \\ \frac{1+q^2}{q} &=& 5.2 \\ 1+q^2 &=& 5.2*q \\ q^2-5.2*q +1 &=& 0 \\ (q-5)*(q-0.2) &=& 0 \\\\ \mathbf{q = 5} &\mathbf{ \text{ or } }& \mathbf{q = 0.2 }\\\\ a =\frac{10}{5} && a =\frac{10}{0.2} \\ \mathbf{a = 2} &\mathbf{ \text{ or } }& \mathbf{a = 50} \\ \hline \end{array}$$

Geometric series:

1.

$$a=2 \quad q = 5 : \\ 2+2*5+2*5^2 = 62\ \checkmark \\ 2*(2*5)*(2*5^2) = 1000\ \checkmark$$

2

.$$a=50 \quad q = 0.2 : \\ 50 + 50*0.2 + 50*0.2^2 = 62 \ \checkmark \\ 50*(50*0.2)*(50*0.2^2) = 1000\ \checkmark$$

Aug 23, 2017

#1
+22172
+2

Geometric series

Hi, I start doing the task, but then I dont know what to do next. Here is:

a1+a2+a3=62

a1*a2*a3=1000

$$\begin{array}{|rcll|} \hline a_1*a_2*a_3 &=& 1000 & | \quad a_1 = a \qquad a_2 = a*q \qquad a_3 = a*q^2 \\ a*(a*q)*(a*q^2) &=& 1000 \\ a^3*q^3 &=& 1000 \\ a^3*q^3 &=& 10^3 & | \quad \sqrt[3]{} \\ a*q &=& 10 \\ \mathbf{ a} &\mathbf{ =}&\mathbf{ \frac{10}{q} }\\\\ a_1+a_2+a_3 &=& 62 & | \quad a_1 = a \qquad a_2 = a*q \qquad a_3 = a*q^2 \\ a +a*q + a*q^2 &=& 62 \\ a *(1 + q + q^2) &=& 62 & | \quad \mathbf{a= \frac{10}{q}} \\ \frac{10}{q} *(1 + q + q^2) &=& 62 \\ \frac{1}{q} *(1 + q + q^2) &=& 6.2 \\ \frac{1}{q} +1+q &=& 6.2 \\ \frac{1+q^2}{q} &=& 5.2 \\ 1+q^2 &=& 5.2*q \\ q^2-5.2*q +1 &=& 0 \\ (q-5)*(q-0.2) &=& 0 \\\\ \mathbf{q = 5} &\mathbf{ \text{ or } }& \mathbf{q = 0.2 }\\\\ a =\frac{10}{5} && a =\frac{10}{0.2} \\ \mathbf{a = 2} &\mathbf{ \text{ or } }& \mathbf{a = 50} \\ \hline \end{array}$$

Geometric series:

1.

$$a=2 \quad q = 5 : \\ 2+2*5+2*5^2 = 62\ \checkmark \\ 2*(2*5)*(2*5^2) = 1000\ \checkmark$$

2

.$$a=50 \quad q = 0.2 : \\ 50 + 50*0.2 + 50*0.2^2 = 62 \ \checkmark \\ 50*(50*0.2)*(50*0.2^2) = 1000\ \checkmark$$

heureka Aug 23, 2017


text:
### Theory:

When an electric current flows through an iron piece wounded by a coil, it behaves like a magnet. This phenomenon is known as electromagnetism.
A type of magnet that produces a magnetic field when an electric current flows through it is known as an electromagnet.
The electromagnetic poles are changed when the direction of the current is changed.

A simple electromagnetic setup
Electromagnetic train:
A good application of an electromagnet is an electromagnetic train. This train works on the principle of magnetic levitation. Hence, it is also known as a suspension train or a flying train. This train does not use diesel or petrol. The magnetic attraction and repulsion properties are used to run these super-fast trains.

An electromagnetic train

Maglev train:

The short form of 'Magnetic levitation' is 'Maglev'. The Maglev train is an electromagnetic train. With the help of the force of attraction or repulsion properties of magnets, any magnetic object can be moved. This train mainly works on the repulsion property of a magnet.

Schematic representation of magnets in the maglev train

The bottom of the train and the rail track is attached with the like poles of the magnets. This attachment of like poles is to repel each other and lift the train from the track for at least $$10\ cm$$.

Levitation of a maglev train

The sides of the track and the bottom sideway of the train is also fitted with the magnets to move the train. The flow of the electricity controls the motion of the train.

• The train travels at a high speed of $$300\ km\ per\ hour$$ and can even reach up to $$600\ km\ per\ hour$$ since there is no friction.


text:
# fitted.ppm

0th

Percentile

##### Fitted Conditional Intensity for Point Process Model

Given a point process model fitted to a point pattern, compute the fitted conditional intensity of the model at the points of the pattern, or at the points of the quadrature scheme used to fit the model.

Keywords
models, methods, spatial
##### Usage
## S3 method for class 'ppm':
fitted(object, \dots, type="lambda", dataonly=FALSE,
drop=FALSE, check=TRUE, repair=TRUE)
##### Arguments
object
The fitted point process model (an object of class "ppm")
...
Ignored.
type
String (partially matched) indicating whether the fitted value is the conditional intensity ("lambda") or the trend ("trend").
dataonly
Logical. If TRUE, then values will only be computed at the points of the data point pattern. If FALSE, then values will be computed at all the points of the quadrature scheme used to fit the model, including the point
drop
Logical value determining whether to delete quadrature points that were not used to fit the model.
check
Logical value indicating whether to check the internal format of object. If there is any possibility that this object has been restored from a dump file, or has otherwise lost track of the environment where it was originally compu
repair
Logical value indicating whether to repair the internal format of object, if it is found to be damaged.
##### Details

The argument object must be a fitted point process model (object of class "ppm"). Such objects are produced by the model-fitting algorithm ppm).

This function evaluates the conditional intensity $\hat\lambda(u, x)$ or spatial trend $\hat b(u)$ of the fitted point process model for certain locations $u$, where x is the original point pattern dataset to which the model was fitted.

The locations $u$ at which the fitted conditional intensity/trend is evaluated, are the points of the quadrature scheme used to fit the model in ppm. They include the data points (the points of the original point pattern dataset x) and other dummy'' points in the window of observation.

The argument drop is explained in quad.ppm. Use predict.ppm to compute the fitted conditional intensity at other locations or with other values of the explanatory variables.

##### Value

• A vector containing the values of the fitted conditional intensity or (if type="trend") the fitted spatial trend. Entries in this vector correspond to the quadrature points (data or dummy points) used to fit the model. The quadrature points can be extracted from object by union.quad(quad.ppm(object)).

##### References

Baddeley, A., Turner, R., Moller, J. and Hazelton, M. (2005). Residual analysis for spatial point processes (with discussion). Journal of the Royal Statistical Society, Series B 67, 617--666.

ppm.object, ppm, predict.ppm

• fitted.ppm
##### Examples
data(cells)
str <- ppm(cells, ~x, Strauss(r=0.15))
lambda <- fitted(str)

# extract quadrature points in corresponding order
plot(quadmarked)


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, 21.06.2019 16:10 pino40

# Find the least common denominator for these two rational expressions. у

### Another question on Mathematics

Mathematics, 21.06.2019 18:00
The given dot plot represents the average daily temperatures, in degrees fahrenheit, recorded in a town during the first 15 days of september. if the dot plot is converted to a box plot, the first quartile would be drawn at __ , and the third quartile would be drawn at __ link to graph: .
Mathematics, 21.06.2019 19:20
The area of a trapezoid with height z cm is 20 sq. cm. the lengths of the parallel sides are (2z + 3) cm and (6z – 1) cm. find the height of the trapezoid in your final answer, include all of the formulas and calculations necessary to solve for the height. ***show your work for the trapezoid***
Mathematics, 21.06.2019 19:30
If you can solve all of these i will give ! - 4% of 190 - 4% of 162.5 - 4% of 140 - a 4% increase from 155.1 - a 4% increase from 159.8


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The circular movement of electrified body

As shown in the above diagram, an object with mass $$m$$ and positive charge $$q$$ is in uniform circular motion with an angular velocity of $$w.$$ The gravitational field and the magnetic field with intensity $$B$$ are in the same direction. What is the radius $$r$$ of the circular orbit of the object?

×


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# Finding a polynomial function with given zeroes

• February 25th 2013, 08:17 PM
zsf1990
Finding a polynomial function with given zeroes
To be exact, "Find a polynomial function with real coefficients that has the given zeros." My instructor just glazed over this in lecture and I don't see how the book is getting its answers. What exactly is the process to doing this? An example set of given zeros:

$4, -3i$
• February 25th 2013, 08:41 PM
Prove It
Re: Finding a polynomial function with given zeroes
Quote:

Originally Posted by zsf1990
To be exact, "Find a polynomial function with real coefficients that has the given zeros." My instructor just glazed over this in lecture and I don't see how the book is getting its answers. What exactly is the process to doing this? An example set of given zeros:

$4, -3i$

For your polynomial function to have real coefficients, all nonreal solutions will occur as complex conjugates. So you know that x = 4, x = -3i and x = 3i will all be zeroes of the polynomial. That means a polynomial that works is

\displaystyle \begin{align*} y &= a\left( x - 4 \right) \left[ x - (-3i) \right] \left( x - 3i \right) \\ &= a \left( x - 4 \right) \left( x^2 + 9 \right) \end{align*}


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IMR Press / JIN / Volume 21 / Issue 5 / DOI: 10.31083/j.jin2105131
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Open Access Original Research
Adaptation of Optokinetic Reflex by Training with Different Frequency and Amplitude
Yong Gyu Kim2,3,4Sang Jeong Kim2,3,4,5
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1 Department of Otorhinolaryngology-Head and Neck Surgery, Konkuk University Medical Center, Research Institute of Medical Science, Konkuk University School of Medicine, 05030 Seoul, Republic of Korea
2 Department of Physiology, Seoul National University College of Medicine, 08826 Seoul, Republic of Korea
3 Department of Biomedical Sciences, Seoul National University College of Medicine, 08826 Seoul, Republic of Korea
4 Memory Network Research Center, Seoul National University College of Medicine, 08826 Seoul, Republic of Korea
5 Neuroscience Research Institute, Medical Research Center, Seoul National University College of Medicine, 03080 Seoul, Republic of Korea
*Correspondence: 20110552@kuh.ac.kr (Chang-Hee Kim)
J. Integr. Neurosci. 2022, 21(5), 131; https://doi.org/10.31083/j.jin2105131
Submitted: 29 March 2021 | Revised: 24 April 2022 | Accepted: 5 May 2022 | Published: 22 July 2022
This is an open access article under the CC BY 4.0 license.
Abstract

Background: Although the occurrence of optokinetic reflex (OKR) adaptation after OKR training is well established, the dynamic properties of OKR adaptation has not been fully studied. This study aimed to examine the difference in the amount of OKR adaptation according to OKR training protocols which have different frequency or amplitude of drum oscillation. Methods: Using C57BL/6N male mice, we induced OKR adaptation by 3 different categories of learning paradigm as follows: (1) Optokinetic drum oscillation for 60 min with same amplitude and different frequency. (2) Optokinetic drum oscillation for 60 min with same frequency and different amplitude. (3) Training with serial combination of different frequency or amplitude. Results: The results show that the amount of OKR adaptation was greater after OKR training with lower frequency or amplitude than that with higher frequency or amplitude. Conclusions: This finding may suggest that the retinal slip signal with lower-velocity OKR stimulation serves as more precise instructive signal for learning, leading to induction of more efficient training effect. Another interesting finding was that the OKR gain increase tended to be greater after training composed of sequential combination of decreasing frequency or amplitude than that composed of sequential combination of increasing frequency or amplitude. Furthermore, the OKR training with high frequency or amplitude eliminated a part of learning effects which have already formed by previous training. We postulate that the stimulation during training with high frequency or amplitude may implement a disturbing instruction for OKR learning when it is conducted in mice with increased OKR gain after previous OKR training.

Keywords
optokinetic reflex
cerebellum
motor learning


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×

# Trigonometry

If $$\cos^2(x) + \cos^4(x) = 1$$, prove that $$\tan^2(x) + \tan^4(x) = 1$$.

Note by Rishit Joshi
3 years, 1 month ago

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Divide both sides by $$\cos^2 (x)$$

$$1 + \cos^2 (x) = \sec^2 (x)$$

$$1 + \cos^2 (x) = \tan^2 (x) + 1$$

$$\cos^2 (x) = \tan^2 (x)$$

$$\frac {1}{ \sec^2 (x) } = \tan^2 (x)$$

$$\frac {1}{ \tan^2 (x) + 1 } = \tan^2 (x)$$

$$1 = \tan^2 (x) (\tan^2 (x) + 1 )$$

$$\tan^4 (x) + \tan^2 (x) = 1$$ · 3 years, 1 month ago

I was complete at the step $$cos^2(x) = tan^2(x)$$ . The next steps weren't required. You just had to replace $$cos^2(x)$$ by $$tan^2(x)$$ in the given equation. · 3 years, 1 month ago

can u explain me · 3 years, 1 month ago

IN THE FIRST STEP I DIDN 'T GOT HOW 1+COS^2X CAME CAN U EXPLAIN ME AGAIN PLSS · 3 years, 1 month ago

Let $$cos^2 x = t$$ $$\Rightarrow t^2 + t - 1 = 0$$ $$\Rightarrow t = \frac{\sqrt{5} - 1}{2}$$ and $$\frac{-\sqrt{5} - 1}{2}$$

Since ,$$0 \leq t \leq 1$$ $$\Rightarrow cos^2x = \frac{\sqrt{5} - 1}{2}$$ , $$\Rightarrow sec^2 x = \frac{2}{\sqrt{5} - 1}$$ = $$\frac{\sqrt{5} + 1}{2}$$ ,$$tan^2 x = sec^2x - 1 = \frac{\sqrt{5} - 1}{2} = cos^2x$$

$$\Rightarrow tan^2x + tan^4x = cos^2x + cos^4x =1$$ · 3 years, 1 month ago


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Intro to Moment of Inertia Video Lessons

Concept

# Problem: A uniform, thin, solid door has height 2.20 m, width 0.870 m, and mass 23.0 kg. (a) Find its moment of inertia for rotation on its hinges. (b) Is any piece of data unnecessary?

###### FREE Expert Solution

We get the moment of inertia from the principle of summing up the moment of inertia of infinitesimal mass elements.

Let the infinitesimal mass element be dm.

The moment of inertia of this mass element is dm•r2

Here, is the distance from the mass element to the axis about where we're considering the moment of inertia.

Therefore, the moment of inertia of the door on its hinges is given by:

$\overline{)\begin{array}{rcl}{\mathbf{I}}& {\mathbf{=}}& \mathbf{\int }{\mathbf{r}}^{\mathbf{2}}\mathbf{d}\mathbf{m}\end{array}}$

The horizontal position with respect to the hinge can be taken to be x.

We can also define a linear density λ = m/L where m is the mass of the door and L is the width of the door.

We see that dm = λdx

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###### Problem Details

A uniform, thin, solid door has height 2.20 m, width 0.870 m, and mass 23.0 kg.

(a) Find its moment of inertia for rotation on its hinges.

(b) Is any piece of data unnecessary?