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    Length of parallel curve

    Hi, I found this problem in Do Carmos "differential geometry of curves and surfaces". it asks to show that the length of a parallel curve B to A given by: B=A-rn where r is a positive constant, and n is the normal vector, and A is a closed convex plane curve, positively oriented. is...
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    Imaginary parts of roots of unity

    Hi all, What happens when we take the product of the imaginary parts of all the n-roots of unity (excluding 1)? I read somewhere that we get n/(2^(n-1)). How can we prove this? Thanks!
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    Splitting Fields

    I was thinking about this, finding the splitting field of x^4-2 in Q[x] over Q is standard enough... but would much be different is i wanted the splitting field over F_5? (field with 5 elements) would it just be F_5(2^(1/4), i) analogously to the Q case? or do any of the arguments break down...
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    Similarity classes for matrices

    Homework Statement How would we find number of similarity classes for a nxn matrix over the field Fp (cyclic of order p) for n=2,3,4? Homework Equations A and B are similar iff they have the same monic invariant factors The Attempt at a Solution I would say 4 classes for n=2, since...
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    Torsion-free modules

    Hi all, I came across this problem in a book and I can`t seem to crack it. It says that if we have an integral domain R and M is any non-principal ideal of R, then M is torsion-free of rank 1 and is NOT a free R-module. Why is this true? cheers
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    Connectedness of coordinates with one rational point

    Hi all, i found this problem in a topology book, but it seems to be of an analysis flavour. I'm stumped. Show that the collection of all points in R^2 such that at least one of the coordinated is rational is connected. My gut says that it should be path-connected too (thus connected), but...
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    Sequence of closed sets

    I was reading about the Nested sphere theorem and a thought occurred. if you have a sequence of decreasing closed sets whose diameter goes to zero in the limit, we can show that the intersection of all these sets is a single point. my idea was to show this using nested sphere theorem if we...
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    Origin of the term seperable

    Origin of the term "seperable" I was just curious as to why out of all properties of metric spaces (ie compactness, closure, etc), i dont know how the term seperable makes sense intuitively.... is there an origin to this term? Just curious. cheers
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    Supposedly easy arcsine problem

    Homework Statement find x if sin(x)=1/5 Homework Equations x=arcsin(1/5)= ??something in closed form maybe?? The Attempt at a Solution This actually isnt my homework. A student im tutoring in differential calculus came to me with this problem and im convinced it is a typo. Just for...
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    Interesting problem

    Heres something i came across in a book but theres no solution... take two inspectors in a factory (they cant talk to each other), and they inspect a series of products and they deem them defective or not. So the results would be (D,N), (N,N) ..... where each coordinate is each inpectors...
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    Equivalent metrics

    Homework Statement Show that d and p are equivalent metrics on X where p=d(x,y)/(1+d(x,y)) Homework Equations ive proved already that p is indeed a metric too (if d is a metric). The Attempt at a Solution I believe im supposed to use the Lipschitz condition where there exits...
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    Analysis Inequality

    hey guys, i came across this inequality in analysis and am not sure how to prove it. Any ideas? It's not homework, im just curious.. Let a1, a2, . . . , an be strictly positive real numbers. Show that a1 + a2 + · · · + an−1 + an <= ((a1)^2)/a2 + ((a2)^2)/a3) +...+ ((an)^2)/a1 cheers
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    Volume Problem

    Homework Statement What would be the most efficient way to find the volume of the solid x^4+y^4+z^4=1? Homework Equations The Attempt at a Solution Cylindrical and spherical coordinates end up messy with integrals that cannot be computed by hand. Im at a loss to find something...
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    A pair of inequalities from analysis

    Homework Statement Prove ln(1+x)>=(x)/Sqrt(1+x) Prove (x-1)^2>=x((ln(x))^2) For x>0 Homework Equations The Attempt at a Solution I have tried using MVT, but i only end up with more inequalities that i cannot seem to prove... Another idea that works but i cannot prove exactly...
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    Cauchy sequence

    How could i show that the sequence (An)= (1+(1/sqrt(2))+(1/Sqrt(3))+...+(1/sqrt(n))-2sqrt(n))) is Cauchy? Thanks in advance!
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    Chasing Problem

    Heres a problem that could use some light. Say person A at (0,1) is following a person B at (0,0). Person B moves horizontally with a constant speed v (towards the positive x direction). Person A also moves with speed v, but always in the direction pointing at person B (to chase him). How can we...
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    Average value of a function sent to infinity

    I was wondering if it would be possible to find the average value of a function with the only condition that x is element of R. For example, could we say that f(x)=4 has an average value of 4 since no matter what values we give for a or b in the integral from a to b of f(x)/(b-a) (b is not equal...
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    Integration Y= (1/(e^x + 1))

    I seem to be having trouble with this antiderivative can i please see a solution? Y= (1/(e^x + 1)) I tried substituting e^x + 1 as u, but that left me with the quest to find an antiderivative of 1/(u^2 - u). i feel like im missing something silly here. thanks
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    Solution yields 120 possibilities

    what would a solution to the following problem look like? How many "words" (distinct orderings of letters) can you make of the word BANANAS in which no As are beside each other. This may turn out to be a simple counting problem, and i apologize if i waste anyones time. My solution yields 120...
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    More fun

    how many consecutive zeros are at the end of 52! ?
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    Number of possible scenarios of tic-tac-toe

    I was debating with a friend whether it is easy to know the number of possible scenarios of tic-tac-toe possible. what do we think?
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    Max Min application

    Heres an intersting problem i would like to see max-min based solutions for. Fine the maximum length of a bar to be transported horizontally around a 90 degree corner. That is, from a corridor of width a to a corridor of width b. ----------- --------l l l l thanks
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    Solving cubics

    Hiya. Is anyone familiar with solving cubics? Currently, i am able to to solve cubics where q^2 - p^3>0 however, i cannot when it is negative or equal to 0. what is the procedure? ps. the p and q are from cardanos method. im assuming anyone who is familiar with the method knows what they...
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    Derive most trigonometric identities from the addition formulas

    In the same way that it is possible to derive most trigonometric identities from the addition formulas, what is the way that the difference of sines and cosines formulas were derived, such as \sin{a}-\sin{b}=2\cos{\frac{a+b}{2}}\sin{\frac{a-b}{2}} thanks, im trying to avoid as much...
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    Fun math proof

    Any nice proofs for this? 2\sqrt{n+1}-2\sqrt{n}<\frac{1}{\sqrt{n}}<2\sqrt{n}-2\sqrt{n-1} I hope the tex came out alright. have fun! ps. n is any natural number.
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    Impacts of Chaos Theory

    The question i would like to raise is what exactly are the implications of chaos theory on the way science is developping? More specifically, is there any trouble in changing framesets from a deterministic view of the universe to a perhaps more chaotic or probabilistic one? If so, what changes...
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    Mathematica Mathematical induction

    It seems to me that sometimes mathematical induction can be a lazy alternative to a real proof. I am bothered by the fact that induction does not tell you why something works, nor can it help in the actual derivation of formulae. Am i right in throwing it away so quickly? It just seems...