The link to the eight queens puzzle, however, is a good start for solving your problem on any board (go down to the "Exercise in algorithm design" section).
For instance, you can use a recursive algorithm to find all the ways to place p pawns at least distance D apart on a given board, by...
I'm afraid the ideas in the previous post won't be too helpful in solving your problem for D>1 (for D=1 with p pawns and any board with n spaces the answer is n choose p), but what will do the trick is setting up a recurrence relation with initial conditions and either solving it in general or...
This is not the matrix I described above, but it should be obvious that continuing the construction below for an infinite matrix will make each row sum -1 and each column sum 1.
\begin{array}{cccccccccc}
1 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 &...
Consider the infinite matrix where for each nonnegative k and each nonnegative j less than 2^k, row j+2^k has entry 1 in column k+1 and entry -1.5 in column k+2 and entry 0 elsewhere. Then each row sum is -.5 while each column sum is positive.
The problem with your "proof" is that it would not be a contradiction to have a matrix in which every row sum is positive and every column sum is negative.
Even if you do not care about showing absolute convergence for your end result, your "proof" absolutely requires it for there to be a contradiction when the sum is changed by having its terms reordered.
The series in your last post is absolutely convergent, as are each of the row sums in your matrix; thus they have a unique sum no matter the order of summation. The series including all terms in your matrix is, however, not absolutely convergent.
There is indeed an error. When an alternating sum is not absolutely convergent, there is no guarantee that a rearrangement of the terms will not yield a different sum. Thus nothing is proved when you assume the conjecture but get differing sums when you rearrange the terms.
For example...
This is just a standard exercise from enumerative combinatorics.
Label the urns 1 through k. How many ways can we place the M balls in the k urns so that each urn gets a ball? Line up the balls and urns in a straight line, with lower-labeled urns to the left of higher-labeled urns and with...
I'm a grad student at a pure math program, I only took the physics intro class as an undergrad, and I don't think many other people in my program have taken more physics than that, if any at all. The only time in a math class I felt physics classes would have helped was solving certain problems...
It does not necessarily carry out to infinity; induction only tells you your proposition holds for all natural numbers n.
Consider the proposition P(n) that the sum 1+2+...+n is finite. P(1) is true, and if P(n) holds, then 1+2+...+n is finite, so 1+2+...+n+(n+1) is also finite, so P(n+1)...
It is generally accepted that Fermat did not have a correct proof himself. He famously wrote his conjecture in the margin of a copy of a text he was reading, but wrote that even though he had a proof the margin was too small for him to write it there. He did have a proof for exponents equal to...
One easy way to get more comfortable with the solution to the Monty Hall problem is to imagine that there are 100 doors, only one of which has a car behind it. You pick a door, then the host opens 98 doors, none of which lead to a car. Any argument I've seen for the theory that no advantage is...
If it makes you feel any better about your chances, remember that being female will probably give your application a bit of a boost. (Full disclosure: I'm in grad school for math, not physics, but I'd imagine physics departments like to diversify their pool of grad students too.)
It holds if A,B,C are nonnegative, but play around with some negative numbers and you'll find a counterexample. (Hint: Does the inequality always hold if max{A+B,C}=A+B, or if max{A+B,C}=C? Pick the case you can't prove easily and search for counterexamples there.)
If n statements are true, then "at most k statements are true" would be false for k<n and true for k between n and 99 inclusive. Thus we must have n=99-(n-1) which yields n=50.
No one gives 2 shizz about the writing score. I know great writers who got your score and bad writers who got a 6. I think part of the score is actually based on how many "GRE words" you are able to work into your essays.
You definitely need to split this into paragraphs, and I have fixed a few grammatical errors below.
As a first generation immigrant I have had to face many setbacks. Without an immediate family to lean on, I joined the work force at an early age. For several years, I worked as a janitor...
From your stats it appears you have no chance for admission into any PhD programs, but I think you could get into an unfunded masters program somewhere if you can find a couple of recommendation writers. Be aware that many masters programs do not require the subject GRE.
If you do want to...
If A and B are tied for first and C has the next best score, then A and B are tied for first and C is third since there are two better than C (look at sports standings/rankings, this is common). There is no reason 2nd place has to be filled.
Pertaining more to the issue you for some reason...
If all colleges except for Sucky State University of Sucking were tied for number 1 with a perfect score and Sucky State was the only college to receive a zero score, would you call Sucky State the second best college in the country?
When you calculate how many ways you can get zero 1s or 5s you count as though you are rolling the dice one at a time and order of results matter (ie, rolling 2,2,4,4,4 is different from rolling 4,4,4,2,2). This is fine as long as you are consistent throughout, but when you count how many ways...
Now it seems to be kind of working.
Given any finite list X(n)_0 = (x_0, x_1, ..., x_n) of at least 2 integers, we define the first difference iteration of X(n)_0 to be the list X(n)_1 = (x_1-x_0, x_2-x_1, ..., x_n-x_{n-1}), and we inductively define the (j+1)st difference iteration of X(n)_0...
My tex is not working properly, so I have separated what should be in tex but left out the delineating commands so that someone may hopefully be able to present this post with the proper tex.
Given any finite list X(n)_0 = (x_0, x_1, ..., x_n) of at least 2 integers, we define the first...
Your original assertion of negation is correct, but your final statement is incorrect: for a given x, you must only show there is a neighborhood of x outside of which infinitely many x_n lie. The sequence 0,1,0,1,0,1,0,1,... does not converge to 0, but any neighborhood of 0 contains infinitely...
Wayne, your original equation is wrong but your revised one is correct. To verify that the original is wrong, see the following counterexample.
Let a(w)=w and b(w)=w+1 for real w, and let t=1 and u=2. Then
{w: t<a(w)} = (1,inf) does not intersect {w: b(w)<u} = (-inf,1) while {w: t<a(w)<u} =...
You are right, the subject test is much more important and you should spend all of your time studying for that.
As for the general, without studying at all you should get an 800 on the math section (it's similar to the SAT/ACT math section) and your scores for verbal/writing really don't matter...
For a first course in the following topics, I especially liked the following books:
"Complex Variables and Applications" by Brown and Churchill
"Elementary Number Theory and its Applications" by Rosen
"Contemporary Abstract Algebra" by Gallian
"Fourier Series and Boundary Value Problems" by...
The number of partitions of any integer m into j parts equals the number of partitions of m with largest part j (easy bijective proof using Ferrer's diagrams). Thus the number of partitions of 2n into n parts equals the number of partitions of 2n with largest part n. The bijection between...
Me again (from the OP's previous thread on the subject).
If you want the $100,000 prize I suggest you hurry because it looks like you've got some competition:
https://www.physicsforums.com/showthread.php?t=301139
If you are given the equation A^x+B^y=C^z, and P is any prime dividing C, then yes, P divides A^x+B^y, as you state, but that does not automatically imply that P divides both A and B; once again, consider the case x=y=z=1 and A=2,B=3,C=5. Nowhere in your proof do you use the fact that x,y,z>2...
Your statements about factorization are vague and mostly untrue. For instance, for any positive a,b,c and any integer n>1, a^n+b^n=c^n can be written as a^n=c^n-b^n=(c-b)(c^(n-1)+c^(n-2)b+...+cb^(n-1)+b^(n-1))
The logic you employ in the "proof" doesn't seem to use anything specific about...
One way to derive the formula for the derivative of cot(x) is to rewrite it as 1/tan(x) (or cos(x)/sin(x)) and use the quotient rule. The mistake you probably made in getting -1/tan^2(x) was to treat cot(x) as the composition of the functions y=1/u(x) and u(x)=tan(x) and then forgetting to...
You are right; you will need more coursework. You should consider studying abroad for a semester at one of the following mathematics-only programs: Penn State MASS, Budapest Semesters in Mathematics, Math in Moscow.
Unfortunately you will not be competitive for admission into any top schools with a math GPA below 3.7 (really, below 3.8). It would also be in your best interest (in terms of making yourself a better applicant) to stay 4 years as an undergrad, but if there are prohibitive circumstances out of...
I think you might have some shot at grad school (I assume you are applying for PhD programs). Are you absolutely positive you won't be able to retake the subject GRE? For most schools, the subject GRE is such that a bad score can keep you out but a good score can't get you in. If you could...
If you are trying to get into a PhD program, even if they say you don't need a math subject gre score, they actually mean that you do. If you are just going for a masters, then you could probably get away without sending in your score. Remember, math PhD admissions are pretty tough anywhere...
I too have always struggled with timed tests. One problem I found I had was that, especially on hard tests, when I got to a problem I knew how to do I would spend more time than I needed checking/double checking my solution was perfect, rather than saving that time to work on the harder...
There is probably no way you could get into any math PhD program with your score (unless, possibly, you are a woman or underrepresented minority, but even in this case you would still want to at least try to bring your score off the floor). There are, however, masters programs that do not...
in discrete math we like to say 0^0=1 because for any x, x^0 is an instance of the empty product, which logically should be the multiplicative identity (since any number is itself multiplied by the empty product), which is 1. Thus once we see the 0 in the exponent we don't even look to see what...
I did go to Carleton. It's a great school and I had a wonderful time (I did math, not physics). Carleton is switching toward having everyone live on campus, so you would have to check their website to find those rates. You will find that, while Northfield itself is pretty cheap, attending...