0
votes
1answer
71 views

Algebra extension of modules question

Let $M$ be an extension of modules of $A$ by $B$ (modules), i.e. $A \leqslant M$ and $M/A \cong B$. Show that if $M$ is finitely generated then so is $B$. In the solution it just says: If $M = XR$ ...
2
votes
1answer
163 views

Exercise 2.2.1 in Charles A. Weibel's book An Introduction To Homological Algebra

How to prove that a chain complex is a projective object in $ {Ch} $ (chain complexes of $R$-modules) iff it is a split exact complex of projectives? A chain complex of projectives means a chain ...
2
votes
1answer
48 views

Connected spaces: is there a mistake in the example below from James Munkres' Topology?

A subspace $Y$ of $X$ is seperable if it can be written as the union of nonempty, disjoint, open sets in $Y$, neither containing the limit point of the other. $Y$ should then not be connected, yet ...
1
vote
0answers
26 views

Show that both these Prove $f$ is differentiable from the right at $0$

Let $f:[0,1) \rightarrow \mathbb{R}$ be continuous on [0,1) and differentiable on $(0,1)$. Suppose the limit $\lim_{x \to +0} f'(x)$ exists. Prove that $f$ is differentiable from the right at $0$. ...
1
vote
1answer
49 views

Metric space and closed sets (book misprint?)

I am not sure if there is a misprint in this corollary or if I am not getting the idea right. Corollary. Let $X$ be a metric space and let $A\subset X$. Then A is closed in $X$ iff: $$ ...
1
vote
0answers
51 views

Product of sequences proof

$(s_n)$ is a sequence with $\lim s_n = \infty$, and $(t_n)$ is a sequence with $\lim \sup t_n = t < 0$. Prove that $$\lim s_nt_n = -\infty$$ Proof (can someone please verify it?): Pick ...
1
vote
0answers
89 views

Proving irregularity using Myhill-Nerode theorem

I'm trying to prove that the following language is irregular using the Myhill-Nerode theorem $$ L = \{ w\space\epsilon \{a,b,c\}^* | \#_b(w) > (\#_a(w) + \#_c(w))! \} $$ While it's completely ...
4
votes
1answer
58 views

$\biggl ( \prod_p G_p \biggr) /\biggl( \bigoplus_p G_p\biggr)$ is divisible

Let $G$ be an abelian group, $p$ a prime, then $G_p$ is the $p$-primary component of $G$, i.e. $$G_p = \lbrace g \in G \ | \exists \ n \in \mathbb{N} \ , p^ng = 0\rbrace$$ I have to prove that ...
27
votes
6answers
2k views

Should I understand a theorem's proof before using the theorem?

I find myself embarrassed when using results in books. For example, there are so many results in Sobolev spaces that I think I would not be able to understand all of them. Yes, I could try to ...
4
votes
1answer
103 views

Find the maximum length of a line segment enclosed in a given area

$A = \{ (x, y) : x = u + v , y = v , (u^2) + (v^2) \le 1 \}$ . Then what is the maximum length of a line segment enclosed in this area? My friend suggested the answer $\sqrt{5}$, but I think it ...
1
vote
1answer
47 views

What is the remainder useful for when dividing a polynomial?

I'm studying AS maths, and am trying to connect my thoughts around polynomials and the factor and remainder theorems. I understand the factor theorem and its application: it helps me find roots of a ...
1
vote
2answers
60 views

writing $\ln(1+x)$ as power series

\begin{align*} \left[\ln\left(1+x\right)\right]' &= \frac{1}{1+x}\\ &= \frac{1}{1-(-x)}\\ ...
1
vote
1answer
40 views

How prove this $\sum_{k=1}^{p-1}r_{p}(k^2+k)=\frac{p^2-p}{2}$

let $r_{n}(m)$ be the remainder of the division of $m$ by $n$ ,and $p$ is a prime congruent to 7 modulo 8. show that $$\sum_{k=1}^{p-1}r_{p}(k^2+k)=\dfrac{p^2-p}{2}$$ my idea: let ...
3
votes
2answers
127 views

Is the arrow category of cartesian closed category cartesian closed?

in a cartesian closed category $\mathcal{C}$. if we have $f: A\to X$ and $g: B\to Y$ then because the functor $(\_)^A$ is continuous, we have $g':B^A\to Y^A$ composing $\text{id}\times f: Y^X\times ...
2
votes
1answer
31 views

Choosing two sets of values with the least difference between the averages of the sets

This is probably a simple Combination problem but I can't think of how to solve it in LibreOffice. I run multiplayer games along with seven of my friends, on two teams of 4 players each, for a total ...
2
votes
1answer
59 views

Find a Hermitian Matrix

We are given two column matrices A and B. Can we find a Hermitian matrix $H$ such that $ A_{4 \times 1} = H_{4 \times 4} B_{4\times 1} $ ? We tried to solve it by multiplying a $1\times4$ row matrix ...
0
votes
1answer
60 views

An interesting phenomenon of $C^*$-tensor product

On the algebraic tensor product space of $C^*$-algebra, I try to find an example whose maximal $C^*$-norm is not the minimal $C^*$-norm, but it seems as it is impossible to do this because the finite ...
2
votes
0answers
53 views

relation between of uniformly rotund in every direction and uniformly rotund and locally uniformly rotund

The norm of a Banach space $X$ is said to be uniformly rotund in every direction if $$\lim_{n→∞}\|x_n−y_n\|=0$$ whenever $$x_n,y_n∈SX$$ are such that $$\lim_{n→∞}\|x_n+y_n\|=2$$ and there is a $z∈X$ ...
1
vote
0answers
55 views

How to prove the zeroth homology in the long exact sequence of associated to (A,X,X/A) …

(I am working out of Hatcher. This is theorem 2.13.) I am brewing up some confusion about the long exact sequence of the homology groups of $A \subset X$ and $X / A$. (For (X,A) a good pair, which is ...
4
votes
0answers
117 views

Tangent developable of helix.

Let $T$ be union of tangent lines to helix $C=(\cos x, \sin x,x)$. 1) I want to prove that $T - C$ is a smooth manifold and find equation for $T$. 2) I want to find how many times a line can ...
6
votes
4answers
240 views

Tell whether $\dfrac{10^{91}-1}{9}$ is prime or not?

I really have no idea how to start. The only theorem considering prime numbers I know of is Fermat's little theorem and maybe its related with binomial theorem. Any help will be appreciated.
6
votes
1answer
111 views

Nonseparable $L^2$ space built on a sigma finite measure space

Is it possible to have a nonseparable $L^2$ Hilbert space for which the underlying measure space is sigma finite? I appreciate any example but prefer one built on the Borel sigma algebra of some ...
3
votes
1answer
147 views

Comparison of versions of the spectral theorem

Definition: (Resolution of Identity) A projection valued measure that is finitely additive and factors over intersections. Theorem 1: Let $A$ be a normal operator in $H$ a Hilbert Space where ...
3
votes
2answers
647 views

Lebesgue Integral, Riemann Integral and Integrals of all sorts

I've heard people refer to the Riemann integral as a "teaching integral" and in a sequence of an analysis course at my school (which is rarely offered) we discuss the mysterious Lebesgue integral. I ...
2
votes
2answers
59 views

Uniform Distribution Problem

Let $X$ be a random variable uniformly distributed in $[0,1]$, and let $Y$ be a RV uniformly distributed in $[X,1]$. I want to calculate the theoretical distribution of $Y$, any hints? I already tried ...
2
votes
1answer
142 views

Number of spanning trees by dividing graph into subgraphs

I'm trying to figure out the number of spanning trees that are in the graph depicted below, I am currently working on the idea of splitting up the graphs into subgraphs, because then I'm able to ...
2
votes
1answer
64 views

Calculating r(t) with line integrals

I have $F(x,y)$ equalling some $a \mathrm{i}+b\mathrm{j}+c\mathrm{k}$ is that all $r(t)$ is? What if all of $a,b,c$ are not in terms of $t$? Note: My $F(x,y)$ is a vector field. Or does it come ...
6
votes
2answers
571 views

How can I unfactorilize number?

Consider the equation $x! = y$ Say we know $y$ and were trying to find $x$: What method could I use to get $x$ (e.g. a closed formula)?
5
votes
4answers
251 views

rank($A$)=rank($A^T$) [duplicate]

Is there an elementary explanation of why the row-rank of a matrix equals its column-rank (without using adjoint maps, resp. lots of technical computations)? What is the geometric intuition behind ...
1
vote
0answers
49 views

Power series expansion of $e^{\frac{c}{2}(z-1/z)}$

Show that \begin{equation} e^{\frac{c}{2}(z-1/z)}=\sum_{n=-\infty}^{\infty}a_nz^n \end{equation} where \begin{equation} a_n:=\frac{1}{2\pi}\int_0^{2\pi}\cos(n\theta-c\sin(\theta))d\theta ...
1
vote
1answer
158 views

Inequality with pi

Prove that, for any sequences of real numbers $\{a_n\}$ and $\{b_n\}$, we have $$\sum_{m=1}^\infty\sum_{n=1}^\infty\frac{a_mb_n}{m+n}\le \pi\left(\sum_{m=1}^\infty ...
1
vote
1answer
108 views

Question about Big O Notation

I don't seem to understand big-O notation very well. If someone would explain it to me as well as explain how this problem would work Let f(n) = (3$^n$$^+$$^1$ - 3)/2. For each of the following ...
0
votes
1answer
37 views

Why a function satifies certain conditions is a norm?

Let $E$ a linear normed space over $\mathbb{R}$ and $n:E\to \mathbb{R}_+$ an application such that: $n(x)=0\Leftrightarrow x=0$, $n(\alpha x)= |\alpha|n(x)$ and the set $$A=\{ x\in E:n(x)\leq 1\}$$ is ...
1
vote
2answers
73 views

Question about proving subsets.

I need some help understanding the steps to take to prove subsets. Question: For each of the following universal statements regarding any three finite sets $X, Y$, and $Z$, determine whether it is ...
4
votes
2answers
88 views

Lebesgue Integrable functions and calculating the limit

$$ \lim_{n\rightarrow \infty} \int_{\frac 1 n }^1 \frac { 1+nx }{ (1+x)^n } \, dx $$ How can I solve this problem using Bounded convergence theorem?
0
votes
1answer
68 views

General Solution for Cosine (negative angles)

cos2(x+pi/3)=1/2 2(x+pi/3)=pi/3 x+pi/3=pi/6 x+2pi/6=pi/6 x=-pi/6 x=5pi/6 (is this step correct) ... ?? x = +/- pi/6 +kpi , k is a subset of Z x = +/- 5pi/6 +kpi , k is a subset of Z can someone ...
1
vote
1answer
35 views

Linear Optimization Study Material

I've recently enrolled in a linear optimization course, and it's been a while since I've taken linear algebra. I do not yet have access to the book for the course or I would skim it to see what I need ...
2
votes
1answer
33 views

Question about sums of limit superiors

Let $(s_n)$ and $(t_n)$ be sequences defined on $\mathbb{R}$. Prove that lim sup $s_n$ + lim sup $t_n \geq$ lim sup $(s_n+t_n)$ Proof (can someone please verify it?): Set $\alpha = $ lim ...
4
votes
3answers
124 views

Proof for inequality with $a,b,c,d$ with $d =\max(a,b,c,d)$

Let $a,b,c,d$ positive real numbers with $d= \max(a,b,c,d)$. Proof that $$a(d-c)+b(d-a)+c(d-b)\leq d^2$$ I believe that the GM-AM inequality with $n=4$ variables might be helpful. ...
0
votes
1answer
62 views

Cross-Validation

Does anyone understand the paragraph below? The paragraph comes from Cross-valiation explanation at wikipedia. "It can be shown under mild assumptions that the expected value of the MSE for the ...
0
votes
1answer
39 views

Integral Bessel recurrence relation

I want to show that $\int x^vJ_{v-1}(x)dx = x^vJ_v(x) + C$. Now I know the recurrence relations of the Bessel equation/function and the one I need to use is $x^vJ_v(x) = x^vJ_{v-1}(x)$ I'm just ...
4
votes
2answers
1k views

Sum of divergent and convergent sequence proof.

Suppose $(s_n)$ is a convergent sequence and $(t_n)$ diverges to $\infty$. Prove that lim $s_n+t_n = \infty$. Proof (can someone verify it?): Pick $N_1$ such that $\forall n > N_1$, ...
1
vote
1answer
57 views

Is integration/differentiation an inverse relation?

A: $\frac{d}{dx}(\sin(x)) = \cos(x)$, $\int(\cos(x)) = \sin(x) + C$ B: $\sin(\arcsin(x)) = x$ Both A and B are inverse relations? A goes full circle with a FUNCTION, and B goes full circle with a ...
1
vote
1answer
41 views

Probability question and how to approach it

Nine tiles are numbered 1, 2, 3, . . . , 9. Each of three players randomly selects and keeps three of the tiles, and sums those three values. Find the probability that all three players obtain an ...
1
vote
2answers
60 views

Definition of Cyclic subgroup

The above is a theorem from my book. What I don't understand is the second sentence when it says $b$ generates $H$ with $n/d$ elements. I thought that since $b = a^s$ generates $H$, it would ...
2
votes
1answer
1k views

Black Derman & Toy Model

The BDT model is given by $$d(\ln\,r)=\left(\theta(t)-\frac {d(\ln\sigma(t)}{dt}\ln r\right)\,dt+\sigma(t) \, dW.$$ How can one rewrite the BDT model as $dr=A\,dt+B\, dW$, using It$\hat o$?
1
vote
0answers
37 views

Minimizing the sum of vectors

I have this problem: Given a set of unit vectors $\{ \vec{v_i} \}$, I want to determine another set, $W$, the element of which are in $\{ \vec{v_i} \}$(repeating allowed), so that the module of the ...
0
votes
1answer
52 views

Borel sigma algebra in topology and R

This is a very basic question about which I am concerned. What is the difference between the Borel $\sigma$-algebra for a topological space and for $\mathbb R$? or they are same?
1
vote
0answers
78 views

Adjoint Representation of Lorentz Group

I'm thinking about the image under the adjoint representation $\mathrm{Ad}$ of the proper (identity connected component) Lorentz group $SO^+(1,3)$. Since this group has a trivial centre (it contains, ...
2
votes
6answers
205 views

Why does $\frac{4}{2} = \frac{2}{1}$?

I take for granted that $\frac{4}{2} = \frac{2}{1}$. Today, I thought about why it must be the case. My best answers amounted to $\frac{4}{2}=2$ and $\frac{2}{1}=2$; therefore ...

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