-1
votes
0answers
26 views

Trignometry, bearings yacht race question

In a yacht race each yacht has to sail around a set of 4 buoys, and then return to the start line in order to finish. We will assume that the buoys are just points and the start line is also a ...
0
votes
0answers
16 views

Need help about field decision; mathematics or physics? Who can be good at these? [on hold]

First of all, you may want to delete this question because it is not an mathematical question, but this question can be an opening door to thousands of mathematical question. Hello everybody, I need ...
0
votes
0answers
14 views

combine 4* 4 bytes values into 4 bytes and extract each value separately using one byte keys.

Here's my problem , I have 4 * 4 bytes i.e a,b,c,d and each one is 4 bytes length I want to generate x = function(a,b,c,d) , where x is 4 bytes and generate aKey,bKey,cKey,dKey where each key is 1 ...
0
votes
2answers
22 views

commutator (derived) subgroup of S3

how can i calculate it easily? i showed that the commutator group of S3 is generated by (123) in S3 using the fact that S3 is isomorphic to D6 and relation in D6 but that was tedious...are there any ...
0
votes
0answers
17 views

subgroups of the free product of a finite cyclic group and an infinite cyclic group.

How can I find all the subgroups of the free product $G = \mathbb{Z}*\mathbb{Z_2}$? I tried to answer this by looking at the subgroups of $\mathbb{Z}$ and $\mathbb{Z_2}$ separately. The subgroups of ...
4
votes
5answers
62 views

Matrix exponential: $\begin{pmatrix} 0 & 1 \\ -4 & 0 \end{pmatrix}$

It is asked to calculate $e^A$, where $$A=\begin{pmatrix} 0 & 1 \\ -4 & 0 \end{pmatrix}$$ I begin evaluating some powers of A: $A^0= \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\; ; ...
2
votes
0answers
30 views

How to find the period of $\cos(|\sin x|-|\cos x|)$?

My book did provide a rule as: If $f_1(x),f_2(x)$ are periodic functions with periods $T_1, T_2$ respectively, then we have $h(x)= f_1(x) + f_2(x)$ has period, as $\bullet$ LCM of $\{T_1, ...
-1
votes
0answers
11 views

Positive definite quadratic form . Riemannian manifolds

Does anybody know how to solve it? I've done a lot of tries but I didn't succeeded Let $H^n=\{ (x_0,x_1,...,x_n)\in \Re^{n+1}:x_0^2+x_1^2+...+x_n^2=-1,x_0>0\}$ and the symetric bilinear form ...
1
vote
0answers
9 views

Finding jordan normal form

Let be $T:\mathbb{R}^7\rightarrow \mathbb{R}^7$ Such that $(T-15I)^3=0$ and $\dim\text{Im}(T-15I)^2=2$ find the Jordan normal form of $T$ If $(T-15I)^3=0$ so the minimal polynomial can be ...
4
votes
1answer
36 views

Integrating $\frac{x^3}{(81-x^2)^2}$

I've been trying to figure out this integral for an hour or so now, but keep failing. I can't figure out where I go wrong: $$I = \int \frac{x^3}{(81-x^2)^2} dx$$ Let $x = 9sin\theta \implies dx = 9 ...
1
vote
0answers
16 views

On n! divided by a product of primes and related questions

We have the following Definition 1. For integers $n\geq 1$ we define $$f(n) = \begin{cases} 1, & \text{if $n=1$} \\[2ex] \frac{n!}{\prod_{p\leq n}p}, & \text{if $n>1$} ...
0
votes
0answers
24 views

conjectured generalization of euler's formula.

Given the elliptic modulus $k$ ,such that the complementary modulus is defined by $$k'\equiv \sqrt{1-k^2}$$,the jacobi amplitude $$\phi\equiv am(u|k)$$ and $K(k)$,is the complete elliptic integral of ...
0
votes
2answers
14 views

how to convert this into cnf $(P\vee Q) \leftrightarrow (P\wedge Q)$?

Given the statement $(P\vee Q) \leftrightarrow (P\wedge Q)$ How can we simplify the double implication to obtain a CNF ? Is there any logical equivalence which I can use?
0
votes
1answer
19 views

What should I do to tackle the following matrices calculation?

Through chapter 3 of Group Theory by Morton Hamermesh in part 3-6 (Equivalent representations; characters.) I stopped in some point. It's told "If we change the basis in the n-dimensional space $L$, ...
2
votes
0answers
32 views

Algebric equation problems

Find $a, b, c \in \Bbb Q$ such that: $\sqrt[3] a + \sqrt[3] b + \sqrt[3] c = \sqrt[3]{\sqrt[3] 2 - 1}$.
1
vote
0answers
16 views

Find the bigger possible order of element in the group $Z_2 \times Z_{36} \times Z_{10}$.Give an element in the group that has the order we found

Find the bigger possible order of element in the group $Z_2 \times Z_{36} \times Z_{10}$.Give an element in the group that has the order we found. How i can find the bigger order? i saw an example ...
1
vote
0answers
15 views

Standard and Adjoint representations of Lie algebra of SU(2)

I'm wondering whether the adjoint and the standard representations of su(2) (the lie algebra of SU(2)) are equivalent. I found this result for so(3) by showing that given the usual basis of so(3), F1, ...
-2
votes
2answers
45 views

What is the error in the following working?

$$\frac{\int_0^1 (1-x^{50})^{100}\mathrm{d}x}{\int_0^1(1-x^{50})^{101}\mathrm{d}x}$$ The question asks us to evaluate 5050 times the above fraction> To solve this i had made the following ...
3
votes
0answers
14 views

Trig substitution using reference triangles

Suppose we are doing a trig substitution and make some substition $x = a \sin \theta \equiv \sin \theta = \frac{x}{a}$ where the domain of x is $|x| \le a$ Then from the reference triangle we can ...
-5
votes
0answers
21 views

Standard of deviation [on hold]

If the number of trials be $n$ and the probability of occurrence be $p$ then the standard deviation with respect to $np$, a. $\sqrt{np}$ b. $\sqrt{np(1-p)}$ c. $\sqrt[4]{np}$ d. ...
0
votes
0answers
19 views

Consequences of exchangeability of random variables

Consider two random variables $X_i$ and $U_i$ respectively distributed as $F_{X_i}(\cdot)$ and $F_{U_i}(\cdot)$ for $i=1,...,N$. Let $X:=(X_1,...,X_N)$ and $U:=(U_1,...,U_N)$ be respectively ...
0
votes
1answer
19 views

Approximation of characteristic function by mollifiers

I have been asked to show that the Heaviside function $H := \chi_{[0,+ \infty)}$ does not admit weak derivative in $L^1_{loc}(\mathbb{R})$. Here's my reasoning: By definition the weak derivative of ...
4
votes
2answers
70 views

Given $a,b,c\ge1;abc\ge8$. Proving that $\sqrt{a^2-1}+\sqrt{b^2-1}+\sqrt{c^2-1}\ge 3\sqrt3$

Given $a,b,c\ge1;abc\ge8$. Proving that $$\sqrt{a^2-1}+\sqrt{b^2-1}+\sqrt{c^2-1}\ge 3\sqrt3$$ I have tried by using Jensen's inequality: We consider the inequality: ...
1
vote
1answer
17 views

Class $C^1$ function on a compact set

The problem is: Let g be of class $C^1$ on $\Delta$⊂$ℝ^n$ and K be a compact subset of Δ. Show that there is a number C such that |g(s)-g(t)|≤C|s-t| for every s,t∈K. I have proved that it is true ...
1
vote
0answers
7 views

Question regarding elementary distribution theory

Let $D'(I)$ be the space of distributions on an open interval $I$, and let $D(I)$ be the space of test functions on $I$. I got the following homework assignment: "Define $u\in D'(\mathbb{R})$ be ...
3
votes
1answer
33 views

Is $\left \{ x_{1}+y_{1},…, x_{n}+y_{n}\right \}$ a basis for $\mathbb{R}^{n}$?

Suppose $\left \{ x_{1},..., x_{n}\right \}$ and $\left \{ y_{1},..., y_{n}\right \}$ are two different bases for $\mathbb{R}^{n}$. Is $\left \{ x_{1}+y_{1},..., x_{n}+y_{n}\right \}$ also a basis for ...
-1
votes
1answer
31 views

Problem of Mathematical Induction [on hold]

Show that $n!\leq 2^{-n}(n+1)^n$ for all $n\in\mathbb{N}$ and equality holds if and only if $n=1$.
0
votes
0answers
14 views

Folland exercise 1.32

Here is a problems after the measure theory section. Suppose {$\alpha_j$} $\subset (0,1)$. a. $\prod $(1-$\alpha_j$) > 0 iff $\sum \alpha_j < \infty $. (Compare $\sum log(1- \alpha_j) to ...
-6
votes
0answers
18 views

What method we use when n = odd for evaluate the integral using simpson's rule ??? [on hold]

hi any one can tell me What method we use when n = odd for evaluate the integral using simpson's rule ??? plz help....
-1
votes
1answer
29 views

What's the summary probability of an event if it increases over time?

I'm having trouble calculating this one. Say there are two steps an event occurs with certain probability: 60% 70% What is the probability that an event occurs by the time second step is reached? ...
1
vote
0answers
12 views

Show the first Chern class of a $U(1)$ bundle is integral.

I am working from John Baez's book: "Gauge Fields, Knots and Gravity". So I will stick to the notation used in that book. I am stuck at exercise 122 of part II (page 283), it reads: Show that if ...
1
vote
0answers
27 views

Are there any solution for a,b,c,d such that $(a+bi)^{n}+(c+di)^{n}=2i$

Are there any solution for a,b,c,d such that $(a+bi)^{n}+(c+di)^{n}=2i$. With a,b,c,d,n are positive integer numbers and $a+bi, c+di$ are complex numbers . I just have started learning about comlex ...
-4
votes
2answers
66 views

Solve equation $x^3 - 3x = \sqrt{x + 2}$ [on hold]

Solve :  $$x^3-3x=\sqrt{x+2}$$ For any real number x.
3
votes
0answers
26 views

Explicit examples of (co)limit arguments in other fields

Over the past weeks, I have noticed that high level lecture notes in subjects like algebraic geometry, algebra, and algebraic topology often sketch proofs in the following form: Proof sketch ...
0
votes
2answers
43 views

Paradox or error in design?

Currently I'm writing a homework for my school. I've made an experiment built this way: There is a laser pointed at a half reflecting mirror which reflects 50% at a wall. The other half cross the ...
-1
votes
1answer
19 views

Good books to learn Combinatorial Game Theory?

I am currently doing my IB Diploma and we are supposed to make an extended essay on a subject of our choice- and i chose math. my research question is- "how to derive a perfect strategy to always win ...
-2
votes
1answer
23 views

Regarding Power series in complex analysis [on hold]

Suppose that I have a series $\sum_n^{\infty} \frac{z^n}{n}$.It is convergent for $|z|<1$. I want to know why the above series converges for $|z|=1$ except at $z=1$.
0
votes
0answers
14 views

Prove that if $|X|=\aleph _0$ then there exist a family of sets, $\mathcal{F}$, of subsets of $X$, s.t $|\mathcal{F}|=\aleph$ [duplicate]

Let $X$ be a set such that $|X|=\aleph _0$. I need to find a family of sets $\mathcal{F}$, of subsets of $X$ such that $|\mathcal{F}|=|\mathbb{R}|$. I saw a couple of examples of Specific X but I ...
0
votes
0answers
22 views

On some factorial inequalities

Denote $P_n$ to be product of primes at most $n$. What is the minimum value of $m$ such that $P_m\geq P_n^2$? What is the minimum value of $m$ such that $m!\geq n!^2$? What is the minimum value of ...
2
votes
1answer
29 views

Find the number of functions

How many functions $f : \{0,1\}^n \mapsto \{0,1\}$ have the equal number of function values $0$ and $1$? I have the answer to the question: $ \sum_{k=0}^{2^{n-1}} 2^{n-1}\binom{2n}{2k}\binom{2k}{k}$, ...
0
votes
0answers
15 views

Additive exponential function

I am trying to solve the following for $x$: $$ ax = b(P-P^x)$$ where $a$, $b$ are positive constants, and $P \in (0, 1)$. I already know that the left hand side is linear and the right hand side ...
0
votes
0answers
18 views

Property of nth root

I'm trying to prove the following result: "Let $x,y \geq 0$ be non-negative reals, an let $n,m \geq 1$ be positive integers. If $y=x^{1/n}$ then $y^n=x$." $x^{1/n}:=sup \{y \in \mathbb{R}: y \geq 0, ...
0
votes
3answers
39 views

Integrating trig substitution triangle equivalence

When we integrate certain integrals, such as $$\int \frac{x^2}{\sqrt{16-x^2}} dx$$ We can make a substitution like $x = 4 \sin \theta$ Then we can simplify the above integral to the following: $$8 ...
1
vote
2answers
14 views

Let $G$ group of order $pq$ where $p,q$ primes.Show that if $G$ contains normal groups $N$ and $K$ with $|N|=p$ and $|K|=q$ then is cyclic

Let $G$ group of order $pq$ where $p,q$ primes.Show that if $G$ contains normal groups $N$ and $K$ with $|N|=p$ and $|K|=q$ then is cyclic Any ideas or hints for showing this?
1
vote
1answer
13 views

checking definition of bounded linear function involves operator maps between different spaces

Let $H$ and $K$ be two Hilbert spaces. Let $T:K\to H$ be a bounded linear operator. Denote the inner products on $H$ and $K$ by $\langle\cdot,\cdot\rangle_H$, $\langle\cdot,\cdot\rangle_K$. Fix any ...
0
votes
3answers
37 views

Integrate $\frac{x^2}{\sqrt{16-x^2}}$ using trig substitution

During our integration of the following integral, using $x = 4 \sin \theta$ $$\int \frac{x^2}{\sqrt{16-x^2}} dx$$ We eventually come to the following point: $$\int \frac{16 {\sin ^2 \theta} }{4 ...
-3
votes
1answer
17 views

Graph software for representing nodes and functional relationships between them [on hold]

This may not be the best venue for this question, but I did not find an anatomy or medicine SE community, so I think mathematics (in particular, operations research) may be best. I briefly considered ...
-2
votes
2answers
17 views

Sample variance equation [on hold]

I'm studying statictics, I don't understand why the equation like this i attached. x bar is sample variance https://www.youtube.com/watch?v=D1hgiAla3KI : 5:08
2
votes
3answers
24 views

Linear Independence and Subset Relations

I've been reading the wikibook on Linear Algebra and in the section 'Linear Independence and Subset Relations' it defines the following lemma: Lemma 1.14: Any subset of a linearly independent ...
4
votes
4answers
39 views

Prove by contradiction $a,b,c>0$?

Suppose $a,b,c$ are real numbers such that $a+b+c>0$, $ab+bc+ca>0$, and $abc>0$. Prove by contradiction that $a,b,c>0$. I have tried to solving it case by case like: case $1$: ...

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