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Commutators and the Uncertainty Principle

Commutators and the uncertainty principle are central to quantum mechanics. Using quaternions in these roles has already been established by others (Horwitz and Biedenharn, Annals of Physics, 157:432, 1984). The first proof of the uncertainty principle I saw relied solely on the properties of complex numbers, not on physics! In this notebook I will repeat that analysis, showing how commutators and an uncertainty principle arise from the properties of quaternions (or their subfield the complex numbers).

Commutators

Any quaternion can be written in a polar form.

This is identical to Euler's formula except that the imaginary unit vector i is replaced by the normalized 3-vector. The two are equivalent if j = k = 0. Any quaternion could be the limit of the sum of an infinite number of other quaternions expressed in a polar form. I hope to show that such a quaternion mathematically behaves like the wave function of quantum mechanics, even if the notation is different.

To simplify things, use a normalized quaternion, so that q q = 1. Collect the normalized 3-vector together with I = V/(V V)^.5.

The angle s/(q* q)^.5 is a real number. Any real number can be viewed as the product of two other real numbers. This seemingly irrelevant observation lends much of the flexibility seen in quantum mechanics :-) Here is the rewrite of q.

The unit vector I could also be viewed as the product of two quaternions.For classical quantum mechanics, this additional complication is unnecessary. It may be required for relativistic quantum mechanics, so this should be kept in mind.

A point of clarification on notation:the same letter will be used 4 distinct ways.There are operators, A hat, which act on a quaternion wave function by multiplying by a quaternion, capital A. If the operator A hat is an observable, then it generates a real number, (a, 0), which commutes with all quaternions, whatever their form. There is also a variable with respect to a component of a quaternion, a_i, that can be used to form a differential operator.

Define a linear operator A hat that multiplies q by the quaternion A.

If the operator A hat is an observable, then the quaternion A is a real number, (a, 0). This will commute with any quaternion. This equation is functionally equivalent to an eigenvalue equation, with A hat as an eigenvector of q and (a, 0) as the eigenvalue. However, all of the components of this equation are quaternions, not separate structures such as an operator belonging to a group and a vector. This might make a subtle but significant difference for the mathematical structure of the theory, a point that will not be investigated here.

Define a linear operator B hat that multiplies q by the quaternion B. If B hat is an observable, then this operator can be defined in terms of the scalar variable a.

Operators A and B are linear.

Calculate the commutator [A, B], which involves the scalar a and the derivative with respect to a.

The commutator acting on a quaternion is equivalent to multiplying that quaternion by the normalized 3-vector I.

The Uncertainty Principle

Use these operators to construct things that behave like averages (expectation values) and standard deviations.

The scalar a--generated by the observable operator A hat acting on the normalized q--can be calculated using the Euclidean product.

It is hard to shuffle quaternions or their operators around. Real scalars commute with any quaternion and are their own conjugates. Operators that generate such scalars can move around. Look at ways to express the expectation value of A.

Define a new operator A' based on A whose expectation value is always zero.

Define the square of the operator in a way designed to link up with the standard deviation.

An identical set of tools can be defined for B.

In the section on bracket notation, the Schwarz inequality for quaternions was shown.

The Schwarz inequality applies to quaternions, not quaternion operators. If the operators A' and B' are surrounded on both sides by q and q*, then they will behave like scalars.

The left-hand side of the Schwarz inequality can be rearranged to form a commutator.

The right-hand side of the Schwarz inequality can be rearranged to form the square of the standard deviation operators.

Plug both of these back into the Schwarz inequality, stripping the primes and the q's which appear on both sides along the way.

This is the uncertainty principle for complementary observable operators.

Connections to Standard Notation

This quaternion exercise can be mapped to the standard notation used in physics

One subtlety to note is that a quaternion operator is anti-Hermitian only if the scalar is zero. This is probably the case for classical quantum mechanics, but quantum field theory may require full quaternion operators. The proof of the uncertainty principle shown here is independent of this issue. I do not yet understand the consequence of this point.

To get to the position-momentum uncertainty equation, make these specific maps

The product of the squares of the standard deviation for position and momentum in the x-direction has a lower bound equal to half the expectation value of the commutator of those operators. The proof is in the structure of quaternions.

Implications

There are many interpretations of the uncertainty principle. I come away with two strange observations. First, the uncertainty principle is about quaternions of the form q = Exp[a b I]. With this insight, one can see by inspection that a plane wave Exp[((Et - P.X)/hbar I], or wave packets that are superpositions of plane waves, will have four uncertainty relations, one for the scalar Et and another three for the three-part scalar P.X. This perspective should be easy to generalize.

Second, the uncertainty principle and gravity are related to the same mathematical properties. This proof of the uncertainty relation involved the Schwarz inequality. It is fairly straightforward to convert that inequality to the triangle inequality. Finding geodesics with quaternions involves the triangle inequality. If a complete theory of gravity can be built from these geodesics (it hasn't yet been done :-) then the inequalities may open connections where none appeared before.


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## Archive for the ‘Sportz’ Category

### The NBA Finals!!1!

June 4, 2009

Tonight, the anticipated NBA Finals begin.

Orlando Magic vs. Los Angeles Lakers.

There is so much going on, and i’ll break down EVERYTHING that is important.

If its not on here, don’t worry about it.

Speaking of all those Q’s and not worrying about it, the first big topic is Kobe’s inability to get a title without Shaq.

They won 3 together, and Shaq left and won a 4th with Miami.

Kobe got straight up dommed last year in the finals, and is “looking for revenge”.

Psh.

Apparently winning a title without Shaq isn’t important to him, he just wants another.

Even Shaq twittered on the subject: “I am saying it today and today only, I want kobe bryant to get number four, spread da word.”

That is a direct quote, not me trying to use ebonics over the computer.

So now that Shaq’s pulling for him he’s automatically going to win right?

No.

Kobe ALWAYS tries to do the opposite of what Shaq says.

Shaq says trade me, Kobe says stay in LA.

etc.

Shaq says win title, Kobe says i’ll throw it.

Knowing that, Orlando will win the series.

2. Jameer Nelson coming back?!?!

He probably will but play scarce minutes, and I don’t think he should come back.

Stan Van Man has said he doesn’t think him coming back will tinker with the chemistry.

He won’t demolish it, but it will change.

Rafer “Reefer” Alston has been playing well and Anthony Johnson has been doing everything a 50-year-old in disguise can. Not to mention Courtney Lee using Rip Hamilton’s old mask to tear up the game.

Throwing another guard in changes the rotation, even for a little.

They get rest, but a guy who hasn’t played in a few months takes their spot.

If it isn’t broken, don’t fix it, right?

Don’t try to fix the Magic.

They’ll repeat next season with Jameer at the point.

Orlando is a pretty streaky team usually, because if the 3-ball is off, they lose.

But almost every game this postseason they have been on.

Can they keep it up?

Why not, it’s a magical run right now. (Get it?)

I love Mickael Pietrus.

He’s an all-around G.

When lil wayne asks what G is, the answer is Mickael Pietrus.

Just look at the name folks!!!!

is he a fan of 30's as well??

He can even throw punches with the best of them.

Golden State Warriors' Mickael Pietrus of France, wears oversized boxing gloves during a film shoot on media day.

How punny, and I even used the exact caption from the picture on the internet.

Go Mag.

P.S. Fun fact, the Magic have beaten the Boston Celtics and Cleveland Cavaliers.

If they beat the Fakers they will have beaten three 60-win teams in one postseason, which has never been done before.

History is on their side.

### Zack Greinke, out of this world

May 5, 2009

This kid is amazing.

He turned my attention to him after his 2nd and 3rd starts which increased his scoreless streak over 30 games and had an amazing ERA, WHIP and K’s.

Now, he sits 6 starts in, at 6-0, 54 k’s, 0.4 ERA, .84 WHIP, and 6 hits per 9 innings.

Everyone knows how important wins, era, and k’s are.

But WHIP is actually an underrated stat along with h/9.

WHIP is walks and hits per inning pitched.

So basically, Zack lets less than one person on per inning pitched.

He’s had 3 complete games, with 2 shutouts, one of which came last night against a powerful Chicago white sox team.

He’a number one in baseball, including hitters and pitchers according to yahoo’s fantasy stats.

45 innings pitched, 2 earned runs.

Unheard of.

But the amazing thing is the backstory of Zack.

He had an article written about him in SI not too long ago, after he turned mine and the rest of the world’s heads towards him.

He came into the game as skilled as a young player can be, and dominated in his rookie season.

He became the royals pitcher of the year as a rookie.

It’s not really saying much, because its the royals, but a rookie doing that is still awesome.

Then he became unamused with the game.

It was boring to him, dominating at such a young age.

He even didn’t like pitching as much as hitting.

He felt like he was forced to become a pitcher.

He one time promised he’d throw a 50-mph curveball in the next inning he pitched.

50 mph is amazingly slow, kids in 5th grade can get that high with a fastball.

He threw it to Dmitri Young (what a perfect person to throw it to), and it looped in there slow as possible.

50 on the dot.

He eventually had problems mentally and took a year or two off of baseball.

He had severe depression issues and finally sorted them out after a while.

He came back about 3 years later and continued his domination.

Now he puts the royals on his back, and is capable of making his own baseball league.

Like a blitz, but baseball.

He’d be the only one that didn’t cheat, and i’m sure he’d still dom the whole league.

He and nelson cruz should go together into the ZNBL.

Zack-Nelson Baseball League.

Where Fun Happens.

I can see it now.

What a stud.

This is actually the exact pitch where zack threw the 50-mph curveball.

But wait!?! Isn’t Dmitri’s bat broken?

Yes, it is. But Dmitri is so cocky he heard the comment from zacky and assumed he could hit that pitch with a broken bat so he tried.

And failed miserably.

"hey next time i'm gonna throw a 100 mph change-up." "Is that possible?" "Don't question me stupid catcher"

What a rough guy.

Don’t mess with Zacky, or as he shall now be called, the Royal(e) with Cheese.

Pronounced roy-al.

This is a graph.

It’s basically the red and pink dots versus the blue and green.

I hope the blue and green ones win, that’d be cool.

Actually this is some odd pitch chart that shows the break of all his pitches.

It’s good to have a graph like this, just in case you uneducated fools didn’t know.

See what i did there??????????????^________^

just in case.

as if he were named Justin Case.

Or Just Incase.

how ironic.

April 14, 2009

Watch.
Learn.
Live.
Be.
Jussi.
Jokinen.
36.

### What the Hell Rocco Baldelli?!?

April 11, 2009

So Rocco Baldelli “was” my idol.

He, cliff floyd and mo vaughn.

All-star lineup, i know.

But now that list is just down to two men.

Cliff and Mo.

Rocco signed by his own will with the Red Sox.

Nobody in their right mind would do that.

Not after being the only team that beat the BoSux in the playoffs.

He’s now a sellout and has no soul.

He is even on the Rhode Island “mount rushmore” thing that Sportscenter did over the summer.

That was cool, but Rocco didn’t get any tv time so i was upset.

Then to learn my number 1 idol is now on my 2nd least favorite team?

I might die.

I don’t need a future mvp(rocco) learning from a past mvp(dustin tool pedroia) how to “hit the high fastball on mlb 09 the show”.

What a terrible commercial.

I don’t want Rocco to stoop to that level.

He doesn’t need to be “scrappy” or “pesky”.

He needs to be Rocco, which was a beast.

I thought i could at leas trust in Rocco, but now i don’t know what to do.

I can’t trust baseball anymore.

at least he left the tag on the bottom of the hat.

He doesn’t look very happy to be a sellout.

He probably already lost his soul anyway.

this is the way things should be:

how can you not want this??

It almost looks like he’s conquering the skyline of Tampa.

But the skyline is just a net for batting practice.

So it works on two levels.

I want the old, smarter rocco back.

Not a soul-less sellout.

### Alex Roidriguez

April 8, 2009

Ughhhh.

I’d rather die than seeing that…

I hope he comes back to the yanks and sucks real bad.

Because the yankees are like the baseball version of the spurs.

Red Sox too.

So now all yankees and red sox have no souls, no excuses.

I would defend Paps because i kindof like him

but if you’ve seen the young man you know for a fact he has no soul.

He’s just like the devil, and he takes your soul.

With that stare, he penetrates your defense and then takes your soul.

And because he’s the devil, on top of that he’ll get you out.

proof.

this next picture isn’t scary at first, but once you realize what’s happening, it’s deathly scary.

Look at his face. And then look at him sucking the soul out of his own baby before it's even alive.

Full sized just to catch the deetz.

What a devil

and i like how this talked about hating on a-rod to how paps is a devil.

I’m just able to go on that tangent 24/7.

### Opening Day Starters

April 8, 2009

So baseball season has began, along with the fantasy season that everyone loves.

And many pitchers that were in the Cy Young running were trashed in their first outing.

Tim Lincecum, CC Sabathia, Cliff Lee, Roy Halladay(to an extent), James Shields, and Brandon Webb.

Last year Cliff Lee and Brandon Webb jumped off to great undefeated starts.

Roy Halladay was consistently beastly all season.

CC picked up the slack after his trade, and Shields picked up around mid-season.

Tim Lincecum was just Tim Lincecum, so he was purely amazing all season.

But this year all of them have ERAs that are off the charts, but mostly because they have had only one bad start.

Don’t fret fans, all of these pitchers, especially the “Say-K Kid” will have stellar seasons.

Not all will be Cy Young seasons, but all will top-tier pitchers as expected.

Fantasy owners: keep them, unless they all go 0-5 or have a 2-2 record with an insanely high ERA.

USA should not worry that Lincecum had one bad outing, he will be back to his goofy windup/dominating self sooner than expected.

He might pitch on 1 day’s rest just to prove it to you ignorant people.

They can only go up from here.

Diggin' the fam Cliff. Keep the bitty in good shape.

### Manu is Gone-u

April 8, 2009

Manu Ginobli.

Hurt.

Best words to ever be uttered by anything but a cow.

Get it? Cows have utters.

But it’s not good because a person is hurt, it’s good because a spur is hurt.

Essentially, when you go to the spurs, you lose your soul.

So then, you aren’t a person, and i can be happy that you are hurt.

So there you have it, spurs aren’t human, and now they can’t make it past the 2nd round, if they even make it through the first.

I am going to go on a limb and say they won’t make the playoffs.

And yes, you’re going to ask “hey didn’t they clinch a spot and are currently at the 2 or 3 seed? There’s no way they can get out!!”

But david stern will say that they are a now even more boring team with out Girl-u and will automatically put Charlotte in their spot.

Isn’t charlotte in the east?

Yeah, but aren’t the grizzlies in memphis, which is further east than cleveland?

Aren’t the raptors in a totally different country?

Aren’t the OKC Thunder gonna be amazing in a year or two with the possibility of getting Blizzy Blake Griffin?

All yesses(whoaaaaaa new word).

Charlotte will be the 2 seed, and with the power of Gerald Wallace, they will make it to the finals against cleveland, and then it doesn’t matter who wins, i will be happy.

Why Spurs?? Why have you taken my soul?

### Steve Nash’s shoes

April 5, 2009

And no, he’s not following the wonderful Starbury and even Al Harrington fad.

He’s not giving sweet shoes away for a low price (i can’t really say that al’s are that sweet though)

But he is using Nike’s “Trash Talk” shoes that are made all 100% of recycled goods.

And yes, the “trash talk” name is legit.

It’s wordplay good enough for fantastik sports or even jason mraz.

I said it, mhm.

But they actually look awesome for shoes that are made of recycled goods.

Here is a picture of them:

These are the shoes. They are pretty tizzy.

Now he did wear these in the All-star games so you may be asking why i’m writing this so late.

But he just started re-wearing them in honor of NBA Green Week.

I also like this idea.

I’m no hippy, but i do enjoy saving the planet in any easy way posible.

It’s good that the NBA is giving this attention too.

Now the only way that Nash could have gotten this idea is from Goran Dragic and Louis Amundson helping him out with the idea.

There’s really no other explanation.

Try and give me one, comment or something.

It’s not going to be as good as Dragic and Amundson.

Irony: do you think a child laborer made Nash’s shoes?

probably…

### Jay Cutler

April 2, 2009

word on the street is jay cutler is on the market.

i actually read that on the road by my house, crazy huh?!?

so yeah, jay cut-her (that’s the best i had on short notice, sorry) is now “on the market”

says trent dilfer.

who is a bald, chunky, big, former qb.

i’m shocked the guy was a qb because he’s so large.

but what do i know about qb size?

everything.

Jay is a perfect fit in dallas because he is a qb that can actually lead a team and win games in the playoffs, or even get his team to the playoffs.

I know they didn’t make it last season, but hey neither did dallas, so why can’t he work?

he actually pushed his team to half a game out of the playoffs, whereas the ‘boys choked on their chance to make it in by getting dominated.

he’s defffff an upgrade from Tony “Turnover” Romo

and no, not an apple turnover, a turnover like fumbles, pics, and being stupid.

Jay had a better qb rating, and had to have less turnovers.

My campaign is to now get Jay because he can throw the ball really far!

omg!!

but seriously, he’s probably a better qb than tony already, and will be much better in the future.

woopsies

first pic under jay cutler.

but thats not the one i’m talking about

although i was gunna full-screen that but i didn’t want his muscles to be too big.

"This throw is goin to land in France. I hope they know how to catch there."

Jay was also in south park, and tony wasn’t.

Jay also looks sweet in every pic besides the one espn had for him in fantasy football last season, so we can still have a “pretty boy” leading “america’s team”.

Although he’s pretty, and good.

No hate to tony romo, Jay is just an upgrade.

Like BeYonCeE.

March 24, 2009

As we all know Adam Dunn basically tried to lose the WBC for US (the United States)

he was probably paid off, and i don’t blame him for accepting.

He’s so bad he can’t make money from teams or incentives from having a high enough average.

The only incentive possible for him to get is strike out less than $\infty$.

Awwwwww shoot, bringin back infinity guyz.

But seriously, he strikes out more than Ryan Howard and every NL pitcher besides Tim Lincecum.

Err, he’s not even a player.

He’s a hitter.

a 1-tool “player”.

a 1-tool guy?

It only fits because he’s a tool.

I don’t even know what to call him.

Besides sucky.

When he’s at the plate the ball ends up in two places: the catcher’s glove because of a strikeout, or the stands because of a homerun.

I don’t like that, not at all.

He can’t throw, but he plays in the outfield, so that’s a problem.

He’s slower than a 13 year olds curveball in the LLWS.

He makes poor fielding choices when he’s not in the outfield.

I think he belongs in the LLWS with the skill set he has.

He gave up 3 sac flys in the semis against japan because he can’t throw from the outfield to home plate, let alone the cut off man 10 feet away from him.

I’ll give you a picture of how bad he is as evidence.

I’ll probably have to explain it for you to understand.

The funny thing about this picture is the fact that he was out sliding into second after he hit the ball out of the park.

What does that caption mean?

It means he so slow, that after he hit a homerun against arizona, he slid into second late because the team had enough time to go into the stands, try and get the ball from the unlucky fan that caught it, negotiated with him, went into a full-scale debate with evidence and strategies I can’t even explain (I see you Mr. Batik), eventually paid him off, and got it and even ran it to second base.

Yes, he was late after that play.

Why does everyone have to be paid off?

Greedy americans…

"Yessssss! I screwed America and got paid extra for it! I'm such a genius."

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My Math Forum Coin flipping game, approximation of one milion flips

 March 21st, 2011, 02:15 AM #2 Senior Member   Joined: May 2008 From: York, UK Posts: 1,300 Thanks: 0 Re: Coin flipping game, approximation of one milion flips The expected value is certainly correct - i.e. if you were to perform the million-coin-toss experiment a large number of times then the mean number of heads recorded would approach 500,000. As for the average difference between the recorded result and the 'perfect' 500k/500k split, from memory (you should check) it will be of the order of $\sqrt N$ or 1000. In other words, if you think of your experiment in the number line sense the average mean distance away from the origin would approach something on the order of 1000. However, since this is a simple binomial distribution with a very large N you will get a good approximation by the Normal approximation with mean 500000 and variance 250000. Using this to calculate the probability of returning a loss (no. of tails < 400000) gives a result more unlikely than finding Elvis riding the Loch Ness Monster.
 March 21st, 2011, 02:26 AM #3 Newbie   Joined: Mar 2011 Posts: 2 Thanks: 0 Re: Coin flipping game, approximation of one milion flips thanks man (bow) really appreciate
March 21st, 2011, 06:10 AM   #4
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Re: Coin flipping game, approximation of one milion flips

Quote:
 Originally Posted by mr_kaktus The absolute value (distance from 0 point) will become greater and greater, but ratio of number of flips and number of heads/tails will be closer and closer to 1/2.
It's worth mentioning that, if the coin is truly fair, you expect the distance from 0 to decrease to 0 or 1 infinitely often, even as the expected value of the absolute distance from 0 increases.

 Tags approximation, coin, flipping, flips, game, milion

### standard deviation of heads or tails

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It is September in southeastern Mali, and Louka Coulibaly is standing in the shade of a squat, concrete building, giving instructions to a dozen men and women perched on a wobbly wooden bench. Coulibaly, a local medical supervisor, hands out nylon backpacks, each filled with bags of pills, plastic cups and a porcelain mortar and pestle that the women pause to admire. By noon, the men and women are packing up and heading back to their respective villages on foot, bicycle and motorcycle.

The following day, they and about 1,400 other health workers throughout the region will set up shop in public spaces: under the shade of mango trees, in one-room schools, at market stands and in district health centres. They will mix and mash the pills with the mortar and pestle, dissolve them in water in a cup, and hand the bitter dandelion-coloured liquid to about 164,000 children.

The effort is part of a broad campaign to prevent malaria by providing African children with drugs usually used to treat the disease. Nearly 1.2 million healthy children from parts of Mali, Togo, Chad, Niger, Nigeria and Senegal received these drugs during the rainy season — from around July to November — when malaria usually ravages the population.

The countries' governments are deploying this intervention — known as seasonal malaria chemoprevention, or SMC — with financial support from the United States, the United Nations and the medical aid organization Médecins sans Frontières (MSF), also called Doctors Without Borders. Next year, many plan to expand the campaigns, and other countries hope to launch their own, encouraged by recommendations from the World Health Organization (WHO).

Preventive use of anti-malarial drugs is not new: tourists routinely swallow them when travelling. But public-health officials have long instructed people living in regions where the disease is endemic to refrain from taking drugs prophylactically, in part because of concerns that the parasite that causes malaria will develop resistance when many people take the medicine on a long-term basis.

That risk has not disappeared. In fact, scientists fully expect SMC to encourage widespread drug resistance. No one knows when, exactly, but it could happen within as few as five years. Until then, SMC has the power to prevent 8.8 million cases and 80,000 deaths each year if implemented in regions with high rates of seasonal malaria. That is considered a powerful enough benefit to justify losing the drugs.

“Life is a risk,” says Coulibaly, a Malian hired by MSF to train local health workers. “And if you don't take risks, you don't win.”

The project is designed to forestall drug resistance as long as possible, and to work in concert with mosquito nets and other preventive methods. Supporters hope that the combination will significantly suppress malaria, so that even if resistance eventually spreads, the caseload should be smaller and manageable with other treatments. But SMC will not be as successful if funding and infrastructure falter — and so far, programmes have had a shaky start. Still, advocates say that the challenges can be overcome. “SMC is feasible,” says Estrella Lasry, technical adviser for malaria chemoprevention at MSF. “But it requires organization, a series of steps and money to back it.”

When intentions backfire

Previous attempts at large-scale malaria chemoprevention offer lessons on what not to do. In the 1950s, David Clyde, a malaria researcher with the British Colonial Medical Service, administered the drug pyrimethamine to villagers in Tanzania. At the time, pyrimethamine had a strong track record of clearing the parasite. But with any drug, there is a slim chance that some strains of parasite will be resistant and will survive to infect others — a chance that increases when many people take the medicine in an area where the parasites are abundant and circulate year-round.

Clyde's experiment drove this concept home: malaria rates dropped at first, but after five months, 37% of infections in the village no longer responded to the drug1. Eight years later, pyrimethamine resistance had spread: up to 40% of infections within 25 kilometres of the original intervention site were unresponsive.

In the Debela village in southeastern Mali, a mother gives her baby drugs meant to stave off malaria. Credit: Amy Maxmen

The 1960s brought more lessons — this time, when scientists tried adding the drug chloroquine to table salt. Clinical trials had shown2 that the salt drastically lowered malaria rates. But when the tactic was scaled up and the salt was distributed to markets in Guyana and Brazil, people consumed only what met their tastes. Others opted for untreated salt when they could, because the chloroquine made their skin itch. As a result, many people carried sub-therapeutic levels of the drug — not enough to reduce the malaria burden, but enough to promote resistance. “The salt campaigns were a disaster,” says Christopher Plowe, a malariologist at the University of Maryland School of Medicine in Baltimore.

Governments and aid organizations mostly shelved chemoprevention programmes after that, but resistance continued to grow — albeit slowly — as people used drugs to treat malaria infections. Between 1960 and 2000, chloroquine resistance crept around the globe and the malaria death toll steadily rose. That trend started to reverse around 2005, after the widespread adoption of the drug artemisinin, derived from Chinese sweet wormwood (Artemisia annua). Today, artemisinin-based drugs are the gold standard for treating malaria.

Second chance

Alassane Dicko, a malariologist at the University of Bamako in Mali, was a graduate student in Plowe's laboratory in 2001, when he started to think seriously about reviving chemoprevention. As a child, Dicko had lost his older brother and his best friend to malaria. Later, as a medical student working in hospitals, he was distraught at the number of children he saw dying. “You really feel it,” he says. “If we want to do anything for this country in terms of health, we need to stop malaria first.”

Dicko suggested that older antimalarials might be repurposed for prevention in places where resistance to them is not yet widespread. By using drugs seasonally, only in uninfected children and in combination rather than alone, he hoped to avoid some of the mistakes of the past. With drug combinations, parasites need to acquire several mutations to survive. These mutations usually come at a cost to the parasite, so removing the selective pressure of the drugs during the dry season would give parasites still sensitive to the treatment a chance to outcompete resistant ones.

Credit: SOURCE: MSF/MALARIA CONSORTIUM/M. CAIRNS, LSHTM/UNICEF/PMI/D. KONÉ, MALI NATL PROG. FOR THE FIGHT AGAINST MALARIA

Dicko proposed using a mixture of sulphadoxine and pyrimethamine called SP, which was known to be relatively safe over the long term. In 2002, his team treated 130 children with SP for two months in a placebo-controlled trial in Mali3. The treatment reduced malaria by 68%.

Other West African scientists followed the study. Among them was Badara Cissé, a Senegalese researcher then pursuing his doctorate with malariologist Brian Greenwood at the London School of Hygiene and Tropical Medicine. Greenwood had been considering chemoprevention since the 1980s, and he and Cissé immediately grasped the potential in Dicko's approach. In 2004, they began a trial in Senegal to test three monthly doses of SP plus artesunate, an artemisinin derivative. Compared with the placebo group, nearly nine out of ten malaria cases were averted4.

With a US$4.5-million grant from the Bill & Melinda Gates Foundation in 2008, Cissé and his colleagues launched an as-yet-unpublished, 3-year clinical trial to study SP with another drug, amodiaquine (to preserve the efficacy of artemisinin). They treated nearly 200,000 children under 10 years old and found that they had 83% fewer cases of malaria than controls, says Cissé. Smaller trials in other African nations reported similar findings. These are impressive numbers, especially given how recalcitrant malaria has been to preventive measures. No vaccine has ever proved fully effective against the disease, for example. And the one that is closest to approval — RTS,S — has shown disappointing results in ongoing clinical trials, with less than a 50% reduction in cases (see Nature 502, 271–272; 2013). Resisting the critics SMC raised some concerns that slowed its adoption. Some health officials suggested that natural, partial immunity to the parasite — built up as a child survives multiple bouts of malaria — would be compromised. Others fretted about the potential side effects of taking the drugs regularly. But the loudest complaints were about losing the drugs to resistance. In a cramped office in a makeshift building at the University of Dakar, Cissé explains how he was frustrated by the deliberations among public-health officials as malaria waged war on Senegal's children. He slumps in a chair that seems much too small for him and asks, “Isn't it selfish to sit in our offices with air conditioning, saying that we should save these drugs?” He recalls a single night, 20 years ago, when he watched five children die of malaria. There was nothing he could do to save them. “If this happened to you, you would not be debating about the fear of losing a drug,” he says. In 2012, SMC finally won over most officials. The Cochrane Collaboration — an international group based in Melbourne, Australia, that specializes in evidence assessment — analysed results from trials in Senegal, Mali, Burkina Faso, Ghana and Gambia, and concluded5 that SMC could prevent more than three-quarters of malaria cases in places where the disease struck seasonally. In the trials, the signs of side effects, resistance and reduced immunity were all minimal. According to another report6, nearly 21 million children in these regions stood to gain from SMC each year. And prevention is cheaper than treatment. Each month, chemoprevention costs$1.50 per child, which pales in comparison to the costs of travel and medical care for a child who falls ill. In November 2012, the WHO published SMC-implementation guidelines that enabled countries to apply for funds from international organizations7.

Slow start

Implementation has been a challenge, however. Mamadou Lamine Diouf, the drug-procurement manager for Senegal's National Malaria Control Program, says that the rollout there was supposed to reach nearly 600,000 children each month, starting in July and August. But he and the US agency footing the bill for the medicine had underestimated how much time it would take to get these older drugs manufactured anew and assessed by various organizations. By early November, health workers had managed to reach only 53,000 children. “We are learning by doing,” says Diouf. “Now we know that if we don't master this long supply chain, nothing will be possible.”

Drug delays set back chemoprevention pilots in northern Nigeria by a month. Togo's campaign did not start until September. Burkina Faso's project failed to launch when funds came up short. And the size of Mali's intended intervention dropped after a coup d'état and an invasion by al-Qaeda affiliates last year sent the nation into disarray.

Still, with the lessons learned, supporters say that they will be better prepared next year (see 'A million ounces of prevention'). In March, some countries plan to apply for funding from the Global Fund to Fight AIDS, Tuberculosis and Malaria. Scott Filler, a disease coordinator at the Global Fund, which is based in Geneva, Switzerland, says, “There are not many things that can prevent malaria in 75% of children, so we will fully support it when countries come to us.”

If we just roll this out without surveillance, we risk repeating all of the mistakes made in the past

As the programmes continue, researchers will keep watch to see if resistance to the drugs mounts. Randomly selected people who come to hospitals to be treated for malaria in Mali, Chad and Niger will have a spot of their blood smeared on filter paper, placed in a ziplock bag and shipped to a laboratory in Bamako, where Dicko and his colleagues will look for mutations associated with resistance to SP and amodiaquine. The University of Dakar will conduct similar tests.

For the campaigns to have a long-lasting effect, chemoprevention must work faster than the parasites acquire resistance. Supporters hope that the treatments will destroy most malaria parasites over the next several years, driving down infection rates and keeping them down even when resistance begins to spread.

Ramanan Laxminarayan, director of the Center for Disease Dynamics, Economics and Policy, a health-policy think tank in Washington DC, is sceptical. He predicts that imperfect implementation will prevent campaigns from having the benefits seen in clinical trials, and that the disease will bounce back in the end.

Importantly, says Paul Milligan, a malaria researcher at the London School of Hygiene and Tropical Medicine, funding agencies must support follow-up evaluations to catch unintended effects such as increased vulnerability to malaria in children who outgrow the interventions. Plowe adds: “If we just roll this out without surveillance, we risk repeating all of the mistakes made in the past.”

Yet surveillance and drug resistance mean little to the mothers who congregate in a small village in the Koutiala region of Mali just after sunrise in September. Awa Damale, 25 years old and clad in an embroidered aqua dress and matching headscarf, arrives by donkey cart with her four children and two from another family. Five of the children swallow their medicine, but one of Damale's sons has felt ill this week. He tests positive for malaria and gets a referral to the nearest clinic. SMC is for prevention only.

The boy's illness may be a sign that the drugs he took last month are not 100% effective — or that he did not swallow all of the medicine — but his condition does not dampen Damale's enthusiasm. It is the first time this year that one of her children has had malaria. Before the intervention, she constantly juggled working on the farm with caring for sick children. She does not want to hear about the possibility of the programme drying up or the drugs losing potency years down the road. Most of her children are healthy now, and that is what matters most.


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# velocity of a particle varies with time as v=t^2-t-2. average speed of the particle in time interval from t=0 to t=4sec?

Eshan
3 years ago
Dear student,

For average speed we need to know the total distance travelled by particle in 4 seconds.

velocity of body is zero at t=2 seconds and the velocity reverses its direction.

Distance travelled upto 2 seconds=

Distance travelled from t=2 to t=4 =

Hence total distance travelled by body=

Hence average speed=3m/s


text:
# (Framed) environment blank spaces

Could be a very beginner question, but still. Considering the following:

\usepackage{framed}
\usepackage{tikz}
\usepackage{listings}
...
\newcommand{\some}{\textcolor{red}{\lstinline{some}}}
\newcommand{\other}{\textcolor{blue}{\lstinline{other}}}
...
\begin{document}
....
\begin{framed}
\some \other
\end{framed}
....
\end{document}

Which goes into this:

Why is there no white space between some and other, when there is a blank space there in the source? How would it be possible to make it consider the blank space in the output?

• Because TeX gobbles white space after commands. Use \some\ \other, for example to insert an explicit space character – user31729 Mar 22 '18 at 9:26
• You could use the xspace package as well. And well be punished by some certain Italian Professor for using it. I will be punished for suggesting it ;-) – user31729 Mar 22 '18 at 9:33
• Thanks! What about passing some math as argument to a new command? Considering this \newcommand{\evenmore}[1]{\textcolor{blue}{\lstinline{#1}}} with \evenmore{$\sim$}? – Mihai Mar 22 '18 at 9:39


text:
# What's the term for "the supremum of constants $\alpha$ such that a function is $\alpha$-Hölder continuous"?

The $\alpha$-Hölder norm of a function $f(x)\colon I \to X$ where $I=[0,T]$ and $X$ is some Banach space with norm $\|\cdot\|$ is:

$$\|f(t)\|_{\alpha}\colon=\sup_{s \neq t \in I}\frac{\|f(t)-f(s)\|}{|t-s|^{\alpha}}$$

If $\|f(x)\|_{\alpha}<\infty$ we say $f(x)$ is $\alpha$-Hölder continuous. What is interesting is that if $f(x)$ is $\alpha$-Hölder continuous, then it is $\beta$-Hölder continuous for $0<\beta<\alpha$. Hölder continuity is integral in understanding how rough a function is. If a function is $1$-Hölder continuous, it is Lipschitz and thus a.e. differentiable. If a function is $1+\epsilon$-Hölder continuous it is constant.

Because of these things and other technical considerations we are interested in the quantity:

$$\sup\{\alpha>0 \colon \|f(x)\|_{\alpha} <\infty\}$$

My question is simple, is there a name for this quantity? I often use "the Hölder norm is $\frac{1}{2}$" for example but this is not quite right even if it is understood in context.

• Do you mean the (Hölder) exponent, see wiki? Aug 20 '15 at 5:59
• @mickep Hölder exponent just means $\alpha$, I want to know what the supremum is called.
– user223391
Aug 20 '15 at 6:00
• @mickep I want to find the supremum of Hölder exponents such that a function is that exponent Hölder continuous.
– user223391
Aug 20 '15 at 6:02
• Thank you for clearifying (for me). Aug 20 '15 at 6:24

It is sometimes referred to as Hölder exponent, see for example the abstract in this paper.

This terminology is also used in Signal and Image Multiresolution Analysis.

I would use this terminology with caution since on the Wikipedia page, for example, it appears that if $f$ is Hölder continuous with parameter $\alpha$, then $\alpha$ is a Hölder exponent for $f$. (there is no mention of the maximality of $\alpha$.)

Also, all the problems I have encountered where phrased in the following way:

• prove that $f$ is Hölder continuous for $\alpha < \beta$,

• prove that $\beta$ is the XXX for $f$.


text:
# CSePub Home Book Store Instructors Independent Authors Abo Robert has inherited some money from his great-grandfather....

###### Question:

CSePub Home Book Store Instructors Independent Authors Abo Robert has inherited some money from his great-grandfather. He will get $5,006.00 when he graduates college (at age 22) and an additional$11253 00 atthe age of 25 Heiscurrently 20 years old Robert will invest each cashflow in an account that pays 600% APR what wll be he future value of this account if he keeps it invested until he turms 30? Answer Format: Currency: Round to: 2 decimal places Enter Answer Here Subrnit Answer ← Prev Problem t 섀 Problems Next Problem Contact Us Pwany Plicy Temms

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text:
show · cmf.twist all knowls · up · search:

Associated to each newform $f$ and primitive Dirichlet character $\psi$, there is a unique newform $g:=f\otimes\psi$, the twist of $f$ by $\psi$, that satisfies $a_n(g)=\psi(n)a_n(f)$ for all integers $n\ge 1$ coprime to $N$ and the conductor of $\psi$. The newforms $f$ and $g$ are then twist equivalent. When $g$ is a Galois conjugate of $f$, it is said to be an inner twist.

The newform orbit $[g]$ is a twist of the newform orbit $[f]$ by the character orbit $[\psi]$ if some $g\in [g]$ is a twist of $f$ by some $\psi$ in $[\psi]$. This may occur with multiplicity.

Twist equivalence is an equivalence relation. The twist class of a newform or newform orbit is its equivalence class under this relation.

In the LMFDB each twist class is identified by the label of its minimal twist.

Authors:
Knowl status:
• Review status: reviewed
• Last edited by Andrew Sutherland on 2020-02-06 08:03:53
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# How to deal with problems by changing the variables in wavefunctions?

I found a problems where a particle was moving under a potential $$V(r) = kr^2$$ and with a wave function $$\psi(r,t)$$. If the wave function is changed to $$\psi(ar ,t)$$, the ratio of average kinetic energy of final and initial is asked. Now if I take $$\psi$$ to be linear and take the $$a$$ out and then find kinetic energy the answer comes out to be wrong. In the solution they have taken $$ar$$ to be some dummy variable and then changed r to the dummy variable and then integrated it. Is taking $$\psi$$ linear wrong? And also what is the use of the potential given. Is it just to distract?

• Why would you expect $\psi$ to be linear in $r$? – J. Murray Mar 27 at 20:29
• It looks like they're trying to prove the virial theorem. In that case you should get the same answer if you take $\psi$ to be linear (after all, they prove it to be true for any function of $ar$ so it should work for the specific case of it being lienar). So in general your method should get the correct answer, but is just not a general result. – Alex Gower Mar 27 at 21:06

I think to answer your question, the point of the $$V(r) = kr^2$$ is not to distract, but because in one of the integrals you will need to make a change of variables $$R = ar$$ and you will get an $$a^{-2}$$ factor from the $$r^2$$.

For the broader picture, this technique is often used to show the virial theorem from the variational method in quantum mechanics (and alot of this method will be identical to what is wanted in your question).

The general proof of the virial theorem for a quadratic harmonic oscillator potential $$V(r) = kr^2$$ should go something like this:

Suppose an exact eigenstate of the system is $$\psi(\vec{r})$$ and consider a variational state $$\psi(a\vec{r})$$ (where we will set $$a=1$$ later but keep it general for now).

The expectation of the potential energy of the state in d-dimensions is given by:

$$\langle PE \rangle_{var} = \langle \psi|V(r)|\psi\rangle = \frac{\int d^d\vec{r} |\psi(a\vec{r})|^2 kr^2 }{\int d^d \vec{r} |\psi(a\vec{r})|^2}$$

(using the fact that $$\psi(a\vec{r})$$ will not be normalised so we must divide by its norm square in order to normalise the 2 $$\psi$$'s in the expectation value).

Using a change of variables $$\vec{R} = a\vec{r}$$ we get:

$$\langle PE \rangle_{var} = a^{-2} \frac{\int d^d\vec{R} |\psi(a\vec{R})|^2 kR^2 }{\int d^d \vec{r} |\psi(a\vec{R})|^2} = a^{-2} \langle PE \rangle_{exact}$$

(where we have seen that the fraction is equal to $$\langle PE \rangle_{exact}$$ since it's nothing but the usual expectation of potential energy for $$\psi(\vec{r})$$ but using the symbol $$\vec{R}$$ instead of $$\vec{r}$$)

Similarly we can show $$\langle KE \rangle_{var} = a^2 \langle KE \rangle_{exact}$$.

So $$\langle E \rangle_{var} = a^2 \langle KE \rangle_{exact} + a^{-2} \langle PE \rangle_{exact}$$

Therefore differentiating with respect to $$a$$ to find the optimum value of $$a$$ which minimises the total energy.

$$2a \langle KE \rangle_{exact} -2a^{-3} \langle PE \rangle_{exact} = 0$$

But we know $$a=1$$ is optimal so:

$$\langle KE \rangle_{exact} = \langle PE \rangle_{exact}$$

Try writing out the S.E.

$$$$\frac{-\hbar^2}{2m} \Delta\psi+kr^2\psi=E\psi$$$$

although cumbersome, it may be helpful to just solve the equation outright. In general it is not a good idea to assume the wavefunction is linear (this would usually make the wavefunction impossible to normalize over).

• Actually what they did was take the new wave function and operated on it using the H operator excluding the potential term which gives the final kinetic energy. I got the answer this way but I was confused because my first approach was to take it a linear function and take constant term out but I didn't understand why it was wrong. – aakash Mar 27 at 20:48


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# Finding the approximate distance between molecules in a gas

I have a problem to understand why the below derivation can be used to find the distance between molecules:

This example is an approximation. We are supposed to find the average distance between the molecules in air, and the first approximation is to assume that air only consists of $N_2$ (I know it is only about 78% $N_2$, but this is only an approximation).

The atomic number of $N$ is 14, which gives the molecular mass

$$2 \cdot14 \cdot u \approx 46\cdot10^{-27} \, \text{kg}$$

where $u$ is representing the atomic mass unit: $u\approx 1.66 \cdot 10^{-27} \, \text{kg}$.

The density of air is given as: $\rho_\text{air} \approx 1.3 \, \text{kg}/\text{m}^3$.

The volume occupied by 1 molecule can therefore be written as

$$V_\text{molecule} = \frac{m_{N_2}}{\rho_\text{air}} \approx 35 \cdot 10^{-27} \, \text{m}^3 \, .$$

Then we make the simplification that the volume occupied by one molecule can be seen as a cube, which then gives the average distance between the molecules to be

$$d_\text{molecules} \approx (V_\text{molecule})^{\frac{1}{3}} \approx 3 \cdot 10^{-9} = 3 \, \text{nm} \, .$$

Since a typical size of a molecule is about $1.5 \, \text{nm}$, it means that there are much space to compress the gas-molecules.

My problem is that I don't understand how we can say that the volume we found is the occupied volume, and not the actual volume of particles. This was used to show that there are much distance between the particles, causing it easier to compress, but I don't understand why the answer $3 \, \text{nm}$ is not the actual "diameter" of the cube of particles. The actual answer for the size of the molecule was $\approx 1.5 \, \text{nm}$, and this was used to show that the we had about $1.5 \, \text{nm}$ of void space between the particles.

• I was completely lost while reading your question... Can you please rephrase what causes you trouble? Are you interested why the gas is occupying larger space then the close-packaged molecules? – Chaosit Aug 25 '15 at 7:37
• Yes, I don't understand why we are not finding the volume of the molecules (without the empty space) when we are taking the mass of the molecule divided by the density. – David Aug 25 '15 at 8:15
• The (mass) density of air is the mass of air per unit volume of air. That volume includes both the molecules and the empty space. – John1024 Aug 25 '15 at 8:19
• Yes, that is true. I now understand why we find the volume of the molecule. Thank you. – David Aug 25 '15 at 8:55

Molecules are not that large.

The density of air at 1 Atm and 20°C is $2.5\times 10^{25}$ m$^{-3}$. So, the average spacing, using your teacher's method, is about $(2.5\times 10^{25})^{-1/3} = 3.4$ nm.

The radius of molecular nitrogen is 0.2 nm. So, the diameter is about 0.4 nm. The means that only about $(0.4/3.4)^3 = 0.0016$ of the volume of air is filled with molecules. Air is overwhelmingly empty space.

## Alternate calculation

Let's calculate this again without using the molecules-on-a-grid approach.

Using the Van De Waals radius, the volume of a nitrogen molecule is

$$V = \frac{4}{3}\pi r^3 = \frac{4}{3}\pi (0.2\times 10^{-9} \text{m})^3 = 4\times 10^{-29} \text{ m}^3$$

To find how much of air is occupied by molecules, we can multiply the number of molecules per unit volume by the volume of a molecule:

$$n \times V = 2.5\times 10^{25} \text{ m}^{-3} \times 4\times 10^{-29} \text{ m}^3 = 0.001 = 0.1\%$$

In other words, only 0.1% of the volume of air consists of molecules. The remaining 99.9% is empty space.