5
votes
1answer
316 views

Finding the general solution of a quasilinear PDE

This is a homework that I'm having a bit of trouble with: Find a general solution of: $(x^2+3y^2+3u^2)u_x-2xyuu_y+2xu=0~.$ Of course this should be done using the method of characteristics but I'm ...
6
votes
0answers
159 views

What are 'weak' forms of Urysohn's lemma, which do not require choice?

Reference: http://web.mat.bham.ac.uk/C.Good/research/pdfs/horror.pdf It is well known that the original proof of Urysohn's lemma uses a choice principle. (DC) What are weak forms of the Urysohn's ...
2
votes
1answer
324 views

Commutation of limits of double sequence

Consider sequence $a_{nm}$ such that $a_{nm} \overset{n}{\rightarrow} a_m$ and $a_{im} \leq a_{jm} \;\;\forall\; i<j$. We also have $a_{m} \overset{m}{\rightarrow} a$ and $a_{p} \leq a_{q} ...
13
votes
9answers
3k views

Finding circumference without using $\pi$

If the area of a circle is $254.34\ldots\text{ cm}^2$ it has a diameter of $18\text{ cm}$, is it possible to find the circumference without using or making the irrational constant Pi ...
5
votes
2answers
129 views

Equivalence of intrinsic and extrinsic metrics of embedded manifolds.

Say a compact n-manifold $\mathcal{M}$ is embedded in $\mathbb{R}^m$, $m > n$. If $d_{\mathcal{M}}$ is the geodesic distance on $\mathcal{M}$, and $d$ the Euclidean distance in $\mathbb{R}^m$, ...
4
votes
2answers
184 views

Principal ideal in $\Bbb{Z}[\sqrt{15}]$

I want to show that $(3, \sqrt 15)$ is not a principal ideal in the ring $ R = \mathbb{Z}[\sqrt{15}]$ with norm $N(a + b \sqrt 15) = a^2 - 15b^2$. My attempt: Suppose $(3, \sqrt 15) = (x) $ Then $3 ...
2
votes
0answers
130 views

Mean value theorem implies first fundamental theorem of calculus?

My book states, that if we apply the "mean value theorem in integral form" on $f'$, then follows: $\int_a^b{f'} = (b-a)f'(c)$. If we compare this with the "canonical mean value theorem": $f(b) - f(a) ...
4
votes
1answer
113 views

Existence of a root in $k[x_1, \ldots, x_n]$

Prove the following: If $k$ is an algebraically closed field and $f(x_1, \ldots, x_n) \in k[x_1,\ldots, x_n]$ is non-zero, then there exists $(a_1, \ldots, a_n)\in k^n$ s.t $f(a_1, \ldots, a_n) = 0$. ...
14
votes
5answers
770 views

Notation Question: What does $\vdash$ mean in logic?

In a "math structures" class at the community college I'm attending (uses the book Discrete Math by Epp, and is basically a discrete math "light" edition), we've been covering some basic logic. I've ...
9
votes
1answer
233 views

Is the number of alternating primes infinite?

I'm not sure if the recreational-mathematics tag is appropriate, but this problem came up during a practice Putnam seminar so maybe? The problem: Say that a positive integer is alternating if ...
6
votes
0answers
197 views

Hints/Help studying an Abel Differential Equation

I want to know more than qualitative information about the Abel differential equation $\frac{dy}{dx}+y^3+x=0$. $\qquad ... \;(1)$ Since I don´t know how to solve this and as far as could see, this ...
2
votes
1answer
229 views

Closure of the set of functions with respect to the $\infty$-norm

I want to show that the set of functions $X=\{f\in C[0,1]:f(0)=0\}$ is closed with respect to the $\infty$-norm. Suppose $(f_n)\subset X$ is a convergent sequence, now I need to find some function ...
1
vote
1answer
256 views

Fibonacci Generating Function of a Complex Variable

So I'm doing work on the Fibonacci Numbers, and I came across this problem for the generating function for the recursive fibonacci numbers. I have two questions: 1. Why is it useful to use a ...
2
votes
1answer
76 views

Question on lagrange multipliers

Maximize $f(x,y,z) = x^4 + y^4 + z^4$ subject to $g(x,y,z) = x^2 + y^2 + z^2 = 1$ it is required that $$\partial_xf = \lambda \partial_xg$$ $$4x^3 = (2x) \lambda \implies x^2 = y^2 = z^2 = ...
5
votes
3answers
1k views

Cyclic sums — How do you use them?

Can someone give me an example of how cyclic sums are used? I don't really understand how they're used in problem-solving. For example, $$\sum_{a,b,c}a^2$$ Any help would be appreciated, and I'm not ...
1
vote
1answer
298 views

Metrics on Euclidean spaces

I vaguely remember a theorem that says that any two metrics on the Euclidean space $\mathbb{R}^n$ are equivalent in some sense, but probably not in the sense of metric equivalence: two metrics $d_1$ ...
6
votes
2answers
405 views

Distribution of points on a rectangle

Let $R$ be a rectangular region with sides $3$ and $4$. It is easy to show that for any $7$ points on $R$, there exists at least $2$ of them, namely $\{A,B\}$, with $d(A,B)\leq \sqrt{5}$. Just divide ...
4
votes
2answers
170 views

Inquality. $\sum_{k=1}^{n}{\frac{1}{\sqrt[k+1]{k}}} \geq \frac{n^2+3n}{2n+2}$

Help me to prove the following inequality(2)) without using Bernoulli's inequality. Prove that: 1) .$$\prod_{k=1}^{n}{\sqrt[k+1]{k}} \leq \frac{2^n}{n+1}.$$ 2) ...
2
votes
2answers
89 views

Geometric Distribution question

A motor insurance company has sold $150$ insurance policies. Let $N_{i}$ represent the number of claims made on policy $i$. You may assume that $N_1, N_2,..., N_{150}$ is a sequence of independent ...
1
vote
1answer
60 views

Ideas To Show this statement: $\prod_2^{2n+1}(1-k^{-2})=(n+1)/(2n+1)$

I would like to know an idea on how to show this: $\prod_{k=2}^{2n+1}$ $(1-{1\over k^2})$ = ${n+1\over 2n+1}$ $\forall$ $n\ge$ $2$. I already checked for $2$ and tried it by induction but I didn't ...
0
votes
2answers
86 views

Area of a function is the same as the area of the inverse function

The area of between the function $f(x)=x^2$ and the $x$-axis from $1\to a$ is the same as the area between $f^{-1}(x)$ and the $y$-axis from $1 \to b$ when $f(a)=b$ It says write two equations of $a$ ...
0
votes
0answers
373 views

Reliability and probability

A system comprises two subsystems operating in series. The first subsystem has two components that operate in parallel. The reliabilities of the components in both subsystem is $r$. Assume that the ...
5
votes
1answer
142 views

Aut($G$) $\simeq$ Aut($M$) $ \times$ Aut($N$)

A question in group theory: Let $ G = M \times N $ be the direct product of $ 2 $ normal subgroups. If $( | M | , | N | ) = 1 $ then Aut($G$) $\simeq$ Aut($M$) $ \times$ Aut($N$). I proved that ...
0
votes
3answers
103 views

Proof that if $p\equiv3\,\left(\mbox{mod 4}\right)$ then $p$ can't be written as a sum of two squares

I'd appreciate your help showing that if $p\equiv3\,\left(\mbox{mod 4}\right)$ then p can't be written as a sum of two squares. Thanks!
2
votes
1answer
178 views

proof that sum of ramification degrees is degree of morphism between curves?

If $X$, $Y$ are irreducible smooth projective curves over an algebraically closed field and $\alpha:X\rightarrow Y$ is a morphism, how do we prove that ...
5
votes
2answers
83 views

Tips on proving this convergence.

We have an inductively defined sequence $x_n=x_{n-1}+2y_{n-1}$ and $y_n=x_{n-1}+y_{n-1}$ where $x_n^2-2y_n^2=\pm 1$, where $x_0=1$ and $y_0=0$. I need to prove that the sequence ...
1
vote
1answer
108 views

Foundational problem with set theory notation, and writing proofs out in the language of set theory

Let $E \subseteq \mathcal{P}(X)$ be a family of sets, then there exists a smallest toplogy containing $E$, i.e. $$ O_E := \bigcap_{E \subseteq O, O \textrm{ is topology}} O. $$ And now I read about a ...
-1
votes
4answers
124 views

Another series convergence question

Does this series converge? $\displaystyle\sum\limits_{n=2}^\infty \frac{\sqrt{n+1}}{(2n^2-3n+1) (\ln n +(\ln n)^2)}$
2
votes
0answers
333 views

Frullani identity: justifying the application of Fubini theorem

Let \begin{align} f \in C^1 \text{ with } f(0) , f(\infty) \in \Bbb R \text{ and } b>a>0 \tag{1}. \end{align} Show \begin{align*} \int_0^\infty \frac{f(ax)-f(bx)}{x}dx = \left( f(0) - ...
1
vote
1answer
62 views

A certain combination of fourier coefficients of modular forms

Let $f$ be a modular form of weight 2 for the group $\Gamma_0(N)$, with Fourier expansion $$ \sum_{n\geq 0} a(n)\ q^n. $$ Let $d$ be an integer dividing $N$, and consider the Fourier series $\sum ...
4
votes
2answers
252 views

Another basic Ordinal arithmetic question: $\omega \cdot (\omega+1)=?$

I am getting confused about ordinal arithmetic I am trying to figure out what $\omega \cdot (\omega+1)$ is. The fact that the distributive law applies from the left side makes me think that the ...
4
votes
1answer
673 views

Order of an element in the factor group divides order or element

Let $N$ be a normal subgroup of a finite group $G$, and $a \in G$ is an element of order $o(a)$. Prove that the order $m$ of $aN$ in $G/N$ is a divisor of $o(a)$. Here what I did: ...
1
vote
1answer
133 views

Compactness and boundedness of multiplication operator .

I need help with the following problem, If $1\le p < \infty$ , $(m_k)_{k\in \mathbb N}$and $K \subset \ell^p(\mathbb N)$ and $T: \ell^p(\mathbb N) \to \ell^p(\mathbb N)$ , define multiplication ...
1
vote
3answers
65 views

Geometric distribution, tossing a die [duplicate]

Possible Duplicate: geometric distribution throwing a die Yesterday I posted a question which was answered but I disagree with the answer so I'd like to ask again so we can discuss it ...
1
vote
1answer
113 views

Uniform distribution

$X~U(0,1)$ Find $E[\sqrt {X}]$ and the probability density function of $Y$ defined as $Y=X^2$. I know that $E[Y]=E[g(X)] = \int_a^b g(x)f(x) dx$ I don't know why this question does not meet the ...
2
votes
1answer
109 views

Direct product of group order 2

If $P$ is group of order 2, how many subgroups (trivial and proper) has the group $P \times P \times P$? Labelling the elements of $P$ to be $e$ and $a$, list the proper subgroups.
0
votes
0answers
48 views

Vectors in $\mathbb R^2$ four statements

I have following 4 statements, is it true that all of them are true. 1) Two lines in the plane $\mathbb{R ^2}$ are orthogonal if and only if their normal vectors are orthogonal 2) If two lines ...
0
votes
1answer
83 views

Justification of taking limit under an integral

In the solution of a complex analysis problem I'm working through the following comment was made: $\displaystyle-\lim_{\epsilon \rightarrow 0} \int_{\pi}^{0} i e^{i\epsilon \displaystyle e^{i ...
2
votes
2answers
679 views

In how many ways can 5 different balls be packed in 5 different boxes such that 3 boxes are non-empty?

Well, I have $5$ labeled balls and $5$ labeled boxes. I need to pack boxes with balls, such that $3$ boxes are non-empty ($2$ boxes are empty). And how about if I need to pack $5$ unlabeled balls in ...
7
votes
3answers
192 views

Stein's lemma condition

(Apologies if I break some conventions, this is my first time posting!) I am working on proving Stein's characterization of the Normal distribution: for Z $\sim N(0,1)$ and some differentiable ...
0
votes
2answers
55 views

function continuous depending on the parameter

Find $a,c$ such that: $$f(x)= \begin{cases} a\frac{\exp(tgx)}{(1+\exp(tgx))} &\text{for }|x|<\pi/2 \\[2ex] \exp(cx)-2 &\text{for } |x|\ge\pi/2 \end{cases}$$ is continuous. How do I ...
1
vote
2answers
615 views

Surface Integral - Intersection of cone and circle

I am now looking at the following exercise: Calculate: $\int \int_S (2z^2 - x^2 - y^2) dS $ where $S $ is $ z=\sqrt{x^2+y^2} $ intersected with $x^2 + y^2 =2x$ (i.e. $ (x-1)^2 + y^2 =1 $ ) . Now, ...
0
votes
1answer
48 views

Continuous cosinus

Find $A$ such that: $f(x)=\lfloor{x}\rfloor \cos(Ax)$ and $ x \in \mathbb{R}$ is continuous on $\mathbb{R}$ We have to ensure continuity for integers. I think that A not exist.
15
votes
5answers
281 views

Evaluating $\sum_{k=1}^{\infty}\frac{\sin\left(k\theta\right)}{k^{2u+1}}$ with multiple integrals

I am trying to evaluate $$\sum_{k=1}^{\infty}\frac{\sin\left(k\theta\right)}{k^{2u+1}}\tag{$u\in\mathbb{N}$}$$ using some results I've got. I know that ...
2
votes
0answers
517 views

how integrate $\int\ln(1+\tan x)\,dx$ or $\int\ln(\sin x)\,dx$

I integrate by part I assume $dx=dv$ and $\ln(\sin x)= u$ or I compute by Maple but its answer wasn't clearly (Maple answer: ...
1
vote
1answer
127 views

Help with basic Ordinal arithmetic: what is $(\omega+2)\cdot \omega$

I am having trouble with some simple ordinal arithmetic. I am trying to figure out what $(\omega+2)\cdot \omega$ is. If you represent $\omega + 2$ as: $\omega+2 = 0_0 < 1_0 < 2_0 < ... < ...
2
votes
2answers
179 views

Lower bounds on numbers of arrangements of a Rubik's cube

Last night, a friend of mine informed me that there were forty-three quintillion positions that a Rubik's Cube could be in and asked me how many there were for my Professor's Cube (5x5x5). So I gave ...
0
votes
1answer
89 views

Isolate middle value from $3\times 3$ matrix

Not trained in math the solution to this problem is not immediately apparent, plus I am working on a larger problem which I'd rather get to. I'm trying to isolate the middle value from a $3\times 3$ ...
1
vote
1answer
147 views

Intuition Regarding Homotopic Spaces

I am just starting to do some algebraic topology (very basic stuff) so have obviously just been introduced to the notion of homotopys, contractible spaces and homotopic spaces. It's this last one that ...
3
votes
5answers
121 views

Two Multiple Integrals

Could anyone solve this two integrals? I would really appreciate. Thank you! $$\iiint\limits_D y^2 dx dy dz$$ where $D=\{(x,y,z)|y\geq 0 \text{ and } x^2+y^2+z^2 \leq 1\}$ Using variables change ...

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