0
votes
0answers
10 views

Semisimple modules and the radical

I don't need a proof, but can someone tell me whether it is true that for all $A$-modules $V$ we have that $V/\text{rad}V $ is semisimple, where we define $\text{rad} V$ as the intersection of all ...
-1
votes
0answers
7 views

Drop Payment -Actuarial Science

Vanessa is accumulating a $10,000 fund by depositing $100 at the end of each month, starting September 1, 2002. If the nominal interest rate on the fund is 12% convertible monthly until May 1, 2005, ...
2
votes
1answer
33 views

Strategy to find out set with nice subset structure

Let $A=\{0,1\}^n=\{(a_1,a_2,\ldots,a_n)\mid a_i\in\{0,1\}\}$. Let $B\subseteq A$ be such that if $(b_1,b_2,\ldots,b_n)\in B$,$ (c_1,c_2,\ldots,c_n)\in A$, and $c_i\leq b_i$ for all $i$, then ...
0
votes
3answers
25 views

Proving one limit is equal to another

Let ($x_s$) be a convergent sequence, where $x_s>0$ for all s, and $y_s$ be a sequence such that $$\displaystyle \Large y_s=\frac{s}{\frac{1}{x_1}+...+\frac{1}{x_s}}$$ Prove $\displaystyle \lim ...
0
votes
0answers
21 views

Error in Wielandt's book?! [on hold]

Wielandt, Exercise 5.2. Assume that the intransitive group $G$ has degree $n$ and minimal degree $n−1$. If no transitive constituent of $G$ has degree $1$, then they all are faithful and all except ...
0
votes
1answer
18 views

Differential equation: general solution for formula

I have following formula and I need the general solution: $$ \frac{d^{2}\theta}{d\xi ^{2}}-\mu ^{2}\cdot \theta =0 $$ EDIT Following solution was given: $$ \theta(\xi )=C_{1}\cdot exp(\mu \xi ...
7
votes
1answer
633 views

median of the F-distribution

Is the median of the F-distribution with m and n degrees of freedom decreasing in n, for any m? From experiments it looks like it might be, but I have been unable to prove it.
2
votes
0answers
7 views

About integral binary quadratic forms fixed by $\operatorname{GL_2(\mathbb Z)}$ matrices of order $3$

I am reading this paper of Manjul Bhargava and Ariel Shnidman, and I want to prove this claim, which appear at the first paragraph of Theorem $14$: Up to $\operatorname{SL_2}(\mathbb Z)$ ...
1
vote
2answers
45 views

Remainder of $3^7/8$

I read here that the remainder of $\frac{ab}{c}$ is equal to the remainder of $\frac{a}{c}\frac{b}{c}$ implying that the remainder of $\frac{a^b}{c}$ is equal to the remainder of $[\frac{a}{c}]^b$. ...
0
votes
0answers
25 views

Trouble simplfying quadratic indentity

I'm struggling to follow the derivation below: $u_j=\frac{\sigma^4+ \theta^2\delta_j^2\alpha_j^2+2\theta\sigma^2\delta_j\alpha_j}{\sigma^2+\delta_j^2\alpha_j^2} \leq \frac{\sigma^4+ ...
1
vote
1answer
163 views
+50

Fields of characteristic zero

Let $\mathcal{F}$ be the class of free fields of characteristic zero and let $X\neq \emptyset$ be a set. How would one show that there are no free fields in $\mathcal{F}$ over $X$? Also, how would ...
1
vote
1answer
19 views

A simple problem of graph theory.

There are how many $4$ vertices connected graphs not including a triangle? My try:I made 3 such graphs.Is it maximum possible number of such graphs or there are many others?Is there any formula in ...
0
votes
2answers
42 views

Is this sequence decreasing?

If a sequence $b_n>0$ and $b_n$ converges to $0$, can we say it is eventually decreasing? This problem bumps up when I am trying to something bigger. However, I am very unsure of this. If this is ...
0
votes
2answers
16 views

'A' transpose inverse equals to 'B' transpose

I searched everywhere but I could not find a solution to this problem. Let $A$ and $B$ be invertible matrices with $AB = I$. Show that
1
vote
0answers
6 views

The identity map $\Delta^n \rightarrow \Delta^n$ is a basis for $H_n(\Delta^n, \delta\Delta^n; R)$

Show that the identity map $\Delta^n \rightarrow \Delta^n$ is a basis for $H_n(\Delta^n, \delta\Delta^n; R)$. Here $\Delta^n$ is the n-simplex, and I know $\delta \Delta^n$ denotes its ...
0
votes
1answer
3 views

How to find the limits of integration to get the area for a loop of a lemniscate?

I know how to integrate the squared radius to get the equation that'll give me the area, like such for a lemniscate with $r^2=8\sin(2\theta)$ : $$1/2\int 8sin(2\theta) = 4 \int \sin(2\theta) = 4 * ...
0
votes
4answers
64 views

Is 1^2^3 = $1^{2^3}$ or $(1^2)^3$

Caret ^ signs can be used to describe the power of numbers. Is $1^{2^3} = 1^{(2^3)}$ or $(1^2)^3$ How do you calculate it? Do you start with $2^3$ and then do $1^8$ or do you start with $1^2$ ...
6
votes
1answer
38 views

Does this Infinite summation of Bessel function has a closed form?

The summation is $$\sum_{n>0}\frac{J_n^2(x)}{n}\sin\frac{2n\pi}{3}$$ I found a thread Infinite sum of Bessel Functions and a wiki article here may be helpful. However, I still cannot figure this ...
0
votes
0answers
8 views

Approximation of minimum among many binomials

We choose $k$ numbers independently from the binomial distribution $B(n,1/2)$, where we can think of $n$ as large. What is the expectation of the minimum of the $k$ numbers? Is there a good way to ...
3
votes
2answers
35 views

If $4x^2+5x+xy=4$ and $y(4)=-20$, find $y'(4)$ by implicit differentiation

If $4x^2+5x+xy=4$ and $y(4)=-20$, find $y'(4)$ by implicit differentiation. I implicitly differentiated the equation, but I don't see how I can use $y(4)=-20$ to my advantage.
1
vote
0answers
7 views

Is this application of the law of total probability correct?

Let us consider a counting process $N(t)_{t\geq0}$ which is neither Markovian nor Levy. Is it correct to write $$ \mathbb{P}(N(t)=j)=\int_{0}^{t}\mathbb{P}(N(t)=j, N(s)=i)ds $$ for $j\geq 1$ and ...
0
votes
2answers
21 views

Trigonometry: How to determine the Period

I'm still kinda confused with solving the period on the diagram above. Amplitude= $3$ Max = $3$ Min = $-3$ Period = ? $y=a\cos(bx+c)$ Value of $a$ = $3$ Value of $b$ = ? Value of $c$ = ?
0
votes
1answer
25 views

Let G be a group of order 17. What is the total no. of non-isomorphic subgroups of G?

I don't know how to find non-isomorphic subgroups of a group. Please explain in detail. Thanks.
0
votes
0answers
12 views

Prove concavity without testing the second derivative..

Consider a function $F(L)=(L-L^2a)T^{L-1}$, where $0<L<\frac{1}{a}$. The constants $a$ and $T$ may take values over $]0,1[$ and $[0.01,0.1]$, respectively. The first derivative of $F$: ...
0
votes
1answer
20 views

Normal distribution tail probability inequality

I am trying to show that $$P(X>t)\leq \frac{1}{2}e^\frac{-t^2}{2}$$ for $t>0$ where $X$ is a standard normal random variable. Perhaps this is simple. I have been starting with $$ ...
2
votes
1answer
20 views

How to prove or disprove this infinite sum of Bessel functions is zero

The sum is $$\sum_{n>0} \mathrm{i}^n\frac{J_n(x)}{n}\sin\frac{2\pi n}{3}\stackrel{?}{=}0$$ and $$\sum_{n>0} (\mathrm{-i})^n\frac{J_n(x)}{n}\sin\frac{2\pi n}{3}\stackrel{?}{=}0$$ I suspect they ...
6
votes
2answers
137 views

Words in the Category of Sets

I was wondering about free objects in different categories and the "words" in those categories. I think I have a generally good grasp on the idea, but I started to think about stranger free objects ...
0
votes
0answers
7 views

conditional expectation constant on part of partition

I have a question about conditional expectation, while looking for the answer here on stackexchange I noticed that there are a few different definitions used, so I will first give the definitions I ...
4
votes
3answers
129 views

Rigorous proof that surjectivity implies injectivity for finite sets

I'm trying to prove that, for a finite set $A$, if the map $f: A\rightarrow A$ is a surjection, then it's an injection. I've looked at this post: Surjectivity implies injectivity but the arguments ...
5
votes
0answers
81 views
+50

Concerning the classical normalized Eisenstein series

Earlier I asked this question. As of today, it has not been answered. Yet still, I have a follow-up question: In general, how does one express $E_4(\tau)$ and $E_6(\tau)$ in closed form for special ...
7
votes
1answer
61 views

Game replacing two numbers by mean

Alicia and Bart plays a game. Alicia first writes $100$ real numbers on the board. After that they move alternately; Bart goes first. In every move, the player chooses two numbers, erases them, and ...
5
votes
1answer
67 views

Simple proof that $\|p(A)\|\le \sup_{|z|\le 1}|p(z)|$ for polynomials $p$ and $\|A\| \le 1$.

Let $\mathcal{H}$ be a complex Hilbert space, and let $A$ be a bounded operator linear operator on $\mathcal{H}$ with $\|A\| \le 1$. It is known that $\|p(A)\|\le \sup_{|z|=1}|p(z)|$ for all complex ...
1
vote
0answers
25 views

How i could show that this inequality true or false: $|\zeta(s)| \leq |2^s \dfrac{1}{1-2^{-s}}|\leq {1}$ for $0< \sigma <1$?

Is this inequality true: $|\zeta(s)| \leq |2^s \dfrac{1}{1-2^{-s}}|\leq {1}$ for $0< \sigma <1$ ? note :$s=\sigma + it$, where $\sigma, t\in \mathbb{R}$. I would be interest for any replies ...
6
votes
1answer
124 views

Integral of Wiener Process and Central Limit Theorem

I am trying to solve the following exercise: (1) Given $W$ is a Wiener process, find a constant $M$ such that $\lim\limits_{t\to\infty} \frac{1}{t}\int_{0}^{t}\sin^2W_s ds=M$ (2) Then show ...
1
vote
1answer
23 views

How to show a Borel Operator Measure dilates to a Spectral Measure?

Does anyone know a simple proof of the following theorem stating that a positive Borel operator measure $P$ on $\mathbb{R}$ can be written as $V^{\star}EV$ for a Borel spectral measure $E$? ...
0
votes
1answer
27 views

Are the following quotient spaces finite dimensional?

If we take $\mathbb{F}[x]$ to be the set of all polynomials over the field $\mathbb{F}$, $E$ to be the subset of all such even polynomials, $N$ to be the set of these polynomials that have degree less ...
0
votes
1answer
42 views

Borel Measures: Discrete & Continuous? [on hold]

Here, the focus lies on discrete & continuous - not atomic & atomless!!! What is the rigorous definition for a Borel measure to be continuous? (The definition for discrete measure can be ...
0
votes
0answers
16 views

Borel Measures: Atoms (Summary)

Disclaimer: This is a summary of the discussions: Measure Atoms: Definition? Borel Measures: Discrete & Continuous? Borel Measures: Atoms vs. Point Masses Reference: Further results are ...
1
vote
3answers
27 views

How the find the smallest positive solution of $353x\equiv 254\mod 400$?

The method that I'm trying to follow is that x = 254 x 353$^{\phi(400)-1}$ where $\phi$ is the Euler's totient function. But how do we find the lowest possible solution?
0
votes
0answers
19 views

Is there a general formula for this set? Or can it be simplified?

I tried to compute some terms and i got to $I_4 = {0,2,6,8,18,20,24,28,54,56,60,62,72,74,78,80}$. The formula is given by recurrence, like this: $$ I_0 = \{0\} \text{ and }I_n = I_{n-1} \bigcup ...
1
vote
1answer
60 views

Calculate $\int_{-\pi}^{\pi}\prod_{n=1}^\infty \left(1-\frac{t^2}{n^2}\right) e^{-izt}dt$

Calculate $$\int_{-\pi}^{\pi}\prod_{n=1}^\infty \left(1-\frac{t^2}{n^2}\right) e^{-izt}dt$$ Any suggestions please? Thank you very much.
11
votes
1answer
2k views

6-letter permutations in MISSISSIPPI

How many 6-letter permutations can be formed using only the letters of the word, MISSISSIPPI? I understand the trivial case where there are no repeating letters in the word (for arranging smaller ...
0
votes
0answers
8 views

Uniform bound of the integral $ \int_{r}^{\infty}{(\frac{1}{\sinh s}\frac{\partial}{\partial s})^2 K_{2+i\sigma}(s) ds} $

Denote $K_{z}(s)=(\frac{s}{2})^{-z-\frac{1}{2}}J_{z+\frac{1}{2}}(s)$, Where $J_z$ is the standard Bessel function of order $z$. Now Set $$ g(\sigma)=\int_{r}^{\infty}{(\frac{1}{\sinh ...
0
votes
1answer
16 views

characteristic function differentiation

Let $\mu$ be a probability measure on $\mathbb{R}$. Then the characteristic function is: $$ \varphi: \mathbb{R} \rightarrow \mathbb{C} \;\;\ \varphi(t):=i\int_\mathbb{R} e^{itx}d\mu(x) $$ Prove with ...
17
votes
2answers
3k views

Will moving differentiation from inside, to outside an integral, change the result?

I'm interested in the potential of such a technique. I got the idea from Moron's answer to this question, which uses the technique of differentiation under the integral. Now, I'd like to consider ...
1
vote
0answers
19 views

Can I do the following when solving my integration??

I appreciate any feedback for my question. I have an integration as follows $$\int_{-\pi}^{\pi}\frac{1}{2\pi} \prod_i \frac{1}{1+ x_ig(\theta)} d\theta $$ I have that $g(\theta)$ is the defined as ...
2
votes
1answer
26 views

Group objects in category of $\mathcal{Set}$ are groups - How to proof?

Reading about group objects in categories, it's a fact that a group object is in the category of $\mathcal{Set}$ just a common group. I am trying to give an actual proof of this, but I'm a bit ...
12
votes
4answers
81 views

Evaluation of $\int_0^{\pi/4} \sqrt{\tan x} \sqrt{1-\tan x}\,\,dx$

How to evaluate the following integral $$\int_0^{\pi/4} \sqrt{\tan x} \sqrt{1-\tan x}\,\,dx$$ It looks like beta function but Wolfram Alpha cannot evaluate it. So, I computed the numerical value of ...
1
vote
0answers
25 views

How to solve the following integral

Do you have any idea how to solve the following integral: $$\int\limits_0^a {{e^{\large \left(- \frac{{by}}{{c - dy}} - ey\right)}}dy}$$ where $a$, $b$, $c$, $d$ and $e$ are constants? Thank you ...
1
vote
0answers
51 views

Relation I found: $ (\sum_{r=1}^{\infty}\frac{z(r)}{r})\times \int_0^\infty f(x) dx = \lim_{h \rightarrow 0} \sum_{i=0}^{n} f(k_ih)h$

I was fiddling with some maths and came up with an interesting relationship: $$ (\sum_{r=1}^{\infty}\frac{z(r)}{r})\times \int_0^\infty f(x) dx = \lim_{h \rightarrow 0} \sum_{i=0}^{n} f(k_ih)h$$ ...

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