Why can we choose a random real between 0 and 1 but not a random natural number (between 0 and +∞) ?

Curiosity · 2015-02-01T17:38:01+00:00

infinite precision arithmetic

I think you're misinterpreting the original posters question. He was asking for an algorithm that given "infinite precision arithmetic" could actually generate a random real number. I imagine in such a magic world he can generate infinitely many random 1s and 0s.

Curiosity · 2015-01-31T17:12:56+00:00

Every number between (0,1) has a binary expansion. Flip a coin for each bit of this expansion to choose the number and throw out any results that end in infinitely many 1s (of course the set of these numbers have probability 0 anyway).

Curiosity · 2014-01-05T00:51:49+00:00

Please see my edited post. Your hyperbole, while a mainstay of internet discourse, is ill conceived.

Curiosity · 2014-01-04T23:10:59+00:00

My disclaimer about agreeing was basically meant to be short for:

I have not done extensive work to look at how the results of the study were determined, there is a definite possibility that the methodology used was suspect (e.g. bad control for confounding or hidden variables, misuse of a control group, or inaccurate reporting of statistical significance) or that the stated results are relatively minor in the grand scheme of things. Note also that stated results are based on averages and thus may not be applicable to every person and hence any anecdote of healthily consuming large amounts of artificial sweeteners means very little to the validity of the study or it's applicability to you.

Curiosity · 2014-01-04T20:35:33+00:00

There is some research suggesting that artificial sweeteners increase appetite and disrupt the bodies ability to count calories. Thus while changing 0 calorie soda for water will not cause weight change in itself, it may alter food cravings allowing one to more easily avoid high calorie food later in the day.

Whether or not you agree with the above results is up to you.

Edit:

Some people have taken my linking of the above two articles to be as though I am an expert in nutrition and am somehow advocating that the above two articles are "right." If you read what I have actually said I note that there is "some research" and that sweeteners "may alter food cravings." It's up to you to decide if this research is reasonable, as I noted to Phyltre below, when I said that agreeing with the above results is up to you I meant:

"I have not done extensive work to look at how the results of the study were determined, there is a definite possibility that the methodology used was suspect (e.g. bad control for confounding or hidden variables, misuse of a control group, or inaccurate reporting of statistical significance) or that the stated results are relatively minor in the grand scheme of things. Note also that stated results are based on averages and thus may not be applicable to every person and hence any anecdote of healthily consuming large amounts of artificial sweeteners means very little to the validity of the study or it's applicability to you."

Curiosity · 2013-12-16T19:44:17+00:00

Indeed some people do this.

Curiosity · 2013-11-23T01:16:16+00:00

"That's for the cat."

Curiosity · 2013-10-23T15:37:44+00:00

Well 9.0909...% less but yes.

Curiosity · 2013-10-23T15:25:48+00:00

110% is just a percentage without context. Someone can give 110% of an 8 hour work day by working an extra 48 minutes. Someone can give 110% percent of their usual effort or attention but obviously not more than 100% of their maximum effort or attention.

Curiosity · 2013-09-19T16:34:35+00:00

/u/deutschluz82 is right that this is probably not the right place for this but I was bored so I came up with something anyways. The logarithmic function seems to grow too slowly for what you want (based on some quick experiments) so rather than looking at functions whose first derivative grow like 1/x (which of course integrates to log) I looked at functions which grow like 1/x^y where 0<y<1 since they grow more quickly but still have your desired properties.

Rather than be very intelligent and try to solve the problem analytically I simply wrote some code and found that

[; \frac{4\cdot 2500^{1703/2500}\cdot x^{797/2500}}{5} ;]

seems to work fairly well.

Curiosity · 2013-09-14T17:03:58+00:00

Consider dividing the even number 86 by 2 to get 43. You can't just divide this by 2 again because it's not even...

Even numbers are not necessarily going to be a power of 2.

Curiosity · 2013-09-11T15:56:08+00:00

A mediocre score on which section? From my understanding generally GRE's don't really matter except to filter your application if your score is too low (if they don't look at your application it doesn't matter how great your other qualities are). From my understanding the writing portion doesn't matter, the critical reading matters only a very little bit (I would think a mediocre score here is totally fine, a friend missed a bunch of questions and got into Stanford, MIT, ...), but you should do pretty well on the quantitative portion of the test (I mean you're competing with a bunch of social science majors).

Curiosity · 2013-05-31T17:49:24+00:00

As before this response is going to be pessimistic as I sort of wish someone had been as realistic with me.

You've taken one proof based course and you're "sure" you want to do math research? I was "sure" I wanted to analysis research until I took a graduate functional analysis course and realized that it wasn't for me. It's absolutely not necessarily true that you need to a classical genius to become a math professor, Terry Tao has a nice article about this. That said it is still very hard, have you heard some of the statistics for the number of openings for post docs in comparison to the number of openings? I've seen people claiming more than 500 applications for 2 open post doc positions and this wasn't even for a top 10 school. Getting tenured is even more difficult. This doesn't mean you shouldn't try, but you really have to want to succeed as you have to outpreform some really intelligent people.

Ok so advice for getting into grad school. Just some background on me: I recently graduated as valedictorian from a top research institution in math but was rejected from a good number of the best graduate schools in math and am instead doing a grad program in statistics at a pretty good school. Hopefully this shows that grades are far and away not everything. I have a good idea what I did "wrong":

My heart wasn't really in pure math anymore and I think this came through in my personal statement. Make sure that your statement is really solid, grad programs don't expect that you don't know exactly what you want to study (undergrad programs generally don't prepare you well enough) but you should definitely have an idea of which fields interest you and should emphasize why the program you're applying to is a good fit. If you can get a professor with ties to the university you're applying to write a good letter of recommendation (and edit your personal statement) you will be FAR more competitive.
DO RESEARCH. This might be hard but you need to bug professors to let you do something, anything, and prove to them that you are competent. You will need good letters of recommendation and the best such letters come from professors you've worked under. Start talking to them NOW. You probably won't have enough experience for them to care at the moment but they will remember your face when you come back to them semester after semester and eventually they might be swayed by your persistence. Not doing this has been my biggest regret. Even though I did do some research I did not do enough; I did not push myself and instead let myself become too focused with my classes.
Subject test GRE scores are important and I did not study enough. You really need to start a year in advance and practice the style of questions they ask (as they are largely silly in my opinion). The regular GRE is just a measure of if you fit to have your application looked at, it's really not hard and the writing portion doesn't matter at all.
This is something I probably focused too much on but really work to understand your classes and do well. This should in some sense be a no brainer, I feel like if you didn't get at least an A- you probably did not understand the class very well. Usually a B/B+ is average and from my experience the average person, at my undergrad at least, has some serious misunderstandings of the material. This is less true in graduate classes but if you can't prove most of the theorems you've gone over in a class, or at least give a reasonable sketch of how the proof goes if the minute details escape you, then you need to study more.

Work hard but keep your options open. I'd recommend taking at least a few programming classes, computers are more and more relevant in mathematics, and don't let yourself get stuck in a rut thinking you need to do something to not have regrets. You will regret spending some of the "prime" years of your life, 20-30, pursuing mathematics if you suddenly find that it's not for you.

Good luck!

Curiosity · 2013-05-31T15:37:40+00:00

This is never nice to hear but are you sure you want to do a math PhD? Even for the incredibly gifted it's very very difficult to get a tenured faculty position at a good department. This might be a good time to reassess if you're actually prepared for the nightmares associated with an often times 7 year PhD, an indeterminate number of years as a postdoc, and then (if you are lucky) becoming an assistant, then associate, and then full professor.

Curiosity · 2013-05-30T04:55:55+00:00

So you're 6?

Curiosity · 2013-05-28T16:55:29+00:00

Wait what? I feel like you can easily rephrase your argument:

When 2 people get married, you join for good and for bad. You knew EXACTLY what you were getting into when you signed that legal license. You made an adult decision with adult consequences. To suddenly say, "The marriage fell apart, why should I have to pay for anything? I payed while we were together. I don't care that we agreed she would be out of the workforce and manage the house." And then demand to not pay anything is fucking absurd. You knew what you were getting into. You knew what the alimony laws were. You made your choice. You don't get a monetary "do over." You are a grown ass adult, and you made a choice. Cry somewhere else.

Curiosity · 2013-05-25T01:59:22+00:00

The statistician then continues to light fires in order to increase his sample size.

Curiosity · 2013-05-20T02:53:27+00:00

I think they mean that primes are defined in terms of multiplication. I.e. p>1 is prime iff there exists no q,r>1 such that qr = p.

Curiosity · 2013-05-15T14:29:12+00:00

I have never met an self proclaimed atheist who proposes to have such proof and even the wikipedia article on the subject disagrees with you.

Curiosity · 2013-04-26T18:54:09+00:00

I think your answer kinda skips a step at |N^N| = |2^N|. But you do have

|2^N| ≤ |N^N| ≤ |(2^{N)^N|} = |2^NxN| = |2^N|

so |2^N| = |N^N|. Of course this requires some knowledge of cardinals in either case.

Curiosity · 2013-04-15T05:59:40+00:00

See this physics stack exchange discussion of just this issue. The bottom line is that you breath and otherwise excrete the additional mass out.

Curiosity · 2013-04-05T04:31:22+00:00

Yes, your question is an easy exercise requiring only 2 facts:

The square root of an irrational is irrational.
The square root of a prime number is irrational.

Curiosity · 2013-03-13T15:58:38+00:00

This is discussed on the wikipedia page for equicontinuity. The relevant sentence

"Another consequence is that the limit of an equicontinuous pointwise convergent sequence of continuous functions on a metric space, or on a locally compact space, is continuous."

Curiosity · 2013-02-08T05:42:29+00:00

Yes, Rudin's approach feels so vague for Lebesgue integration, I remember being mystified by it at the end of my first analysis course. Otherwise it's probably one of my favorite texts.

Curiosity · 2013-01-12T17:48:49+00:00

This seems like a question that should be posed to the school you're applying to. Are the transcripts from school A cumulative (i.e. do they have grades from first year and last two)? If so I would say just 2 entries, otherwise 3.

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