0
votes
0answers
3 views

Proove By Induction (Fibonacci Sequence)

Prove by PMI $\gcd(f_n,f_{n+1}) = 1$ for all natural numbers $n$. $f_n$ represents the Fibonacci sequence.
0
votes
0answers
5 views

Complex integrations

Calculate integrals $$\oint_{|z|=1} \frac{z^2 e^z}{2z+i} dz $$ and $$\oint_{|z|=2} \frac{e^z}{z^2+z} dz $$ These are simple integrals to do with cauchy integral theorem right? First one. ...
0
votes
0answers
8 views

Newton-Raphson with Exponentials

I'm having trouble getting initial values for x and y to be thrown into the Newton Raphson formulae, aka Xv1 and Yv1 respectively. Question; Show that the equation: 10e^-2x = 2x + 3x^2 has a root ...
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votes
0answers
20 views

Abstract Algebra and Chess

I am currently debtating to do a Mathmatics Paper on the comparison between the game of Chess and contemporary mathematics (namely group theory). I was wondering if this has potential to be a ...
0
votes
0answers
3 views

MR Function Question

I have a question about MR Function below: P = 1/Q^2 + 3Q + 1 Find the MR Function and Evaluate it at Q = 4 Are you guys able to elaborate? I've never done these types of questions before, am i ...
0
votes
1answer
7 views

Partial composition

Here is a simplification of my problem. I have the two following functions: $$ f : \mathbb R^{(m+p)} \rightarrow \mathbb R $$ $$ g : \mathbb R^n \rightarrow \mathbb R^m $$ with $m, p,n \in \mathbb ...
0
votes
0answers
4 views

Optimization with changing objective function

Is there any theory about (convex) optimization where the objective function is allowed to change during the optimization process? I have a problem where the objective function depends on some ...
1
vote
0answers
13 views

Tensor product of two linear map and its matrix representation

Suppose $T_1: \mathbb{R}^n\to\mathbb{R}^n$ be any linear map and wrt a basis $\{e_1,\dots,e_n\}$ the matrix of $T_1$ is $M$, aand $T_2:\mathbb{R}^m\to\mathbb{R}^m$ be another linear map whose matrix ...
2
votes
1answer
29 views

Prove that $\det(A) > 0$

Let $A \in \mathcal{M}_{n}(\mathbb{R})$ be a real $n \times n$ matrix such that : $A^{3} = A + \mathrm{I}_{n}$. Prove that $\det(A) > 0$. Here is what I tried : $X^{3}-X-1$ is a null polynomial ...
0
votes
0answers
8 views

Problem on Finding the rank from a Matrix which has a variable

$$ A = \begin{bmatrix} 1 & -1 & -2 & -3 \\ -2 & 1 & 7 & 2 \\ -3 & 3 & 6 & \alpha \\ 7 & -6 & -17 & -17 \end{bmatrix} $$ Find the rank when ...
3
votes
2answers
44 views

Solve $x^3 - x + 1 = 0$

Solve $x^3 - x + 1 = 0$, this cannot be done through elementary methods. Although, this is way out of my capabilities, I would love to see a solution (closed form only). Thanks!
0
votes
0answers
5 views

Question regarding double integrals

Regarding the Buffon's needle case for long needles of length $ l>t, $ (the distance between the parallel lines on the floor), we need to solve the integral $$ \int_{\theta=0}^{\frac{\pi}{2}} ...
0
votes
1answer
12 views

Analytic solution for a type of PDE systems

Peace be upon you, I have the following system of partial differential equations \begin{align*} \begin{cases} \frac{\partial}{\partial a}S(a,b,c,d)=f_1(a)\\ \frac{\partial}{\partial ...
0
votes
0answers
5 views

Test Hypothesis maximum likelihood

Let ${\theta _1},...,{\theta _n}$ be a sample of angular data where $0 \leqslant {\theta _i} \leqslant 2\pi $ $;i = 1,...,n$ Suggest why the statistic ${T_m} = \sum\limits_{i = 1}^n {\cos (m{\theta ...
0
votes
0answers
7 views

Duffing equation of forced spring motion

The motion of a forced spring is described by the equation $$x'' + \kappa x' +x-x^3 =\Gamma \cos(\omega t)$$ We wish to investigate the stability of solutions of this equation having the forcing ...
0
votes
0answers
21 views

Is there any easy way to see that elementary matrices commute in $\text {Mat}_{n \times n} (\mathbb F)$?

Is there any easy way to see that elementary matrices commute in $\text {Mat}_{n \times n} (\mathbb F)$ ? I've been trying to sketch a proof by induction, but it seems more complicated that it should ...
0
votes
0answers
2 views

2d spatio-time regression

I have this question which intuitively appears to be simple, but I couldn't find a solution to it. Imagine you have a ball moving in one direction (x) and that the measurements of the movement are ...
0
votes
1answer
12 views

Which of these sums is equal to $\mathbb R^4$?

I'm given the following sets: $$U=\{(0,a,b,a-b): a,b \in \mathbb{R}\} \\ V=\{(x,y,z,w): x=y, z=w\} \\ W=\{(x,y,z,w): x=y\}$$ I'm trying to determine which of the following is equal to $\mathbb R^4$: ...
0
votes
0answers
15 views

Can Legendre's theorem really help solve this equation: $ax^2+by^2=cz^2$?

let $a,b,c,x,y$ be non-zero positive integers such that $$\gcd(x,y,z)=1$$ $$ \gcd(x,a)>1$$$$ \gcd(y,b)>1$$ $$ \gcd(z,c)>1 $$ If $a,b,c$ are square-free, find all the non-trivial integral ...
1
vote
1answer
24 views

What are these tick marks after the x, y, and z called?

What are these marks called and what do they stand for? This is for a Affine Transformation.
2
votes
1answer
38 views

Is this number composite or prime: $2000^{2002} + 2000^{2000} + 1$?

Is this number composite or prime? $$2000^{2002} + 2000^{2000} + 1$$ I want to find an easy approach to this problem.
0
votes
0answers
3 views

A branch of the argument in the domain $\mathbb{C}$\ {$te^{it}|t\geq0$}

Usually most questions I come across on branches have domains similar to $\mathbb{C}$\ $[0,\infty)$, $\mathbb{C}$\ {$te^{i\alpha}|t\geq0$}. But the domain $\mathbb{C}$\ {$te^{it}|t\geq0$} is ...
0
votes
0answers
6 views

Question on Wedderburn components of $\mathbb C[G]$

$G$ is a finite group. Wedderburn's theorem says that $\mathbb{C}G\cong R_1 \times\cdot \cdot \cdot R_r$ as rings where $R_i=M_{n_i}(D^i)$ is the ring of $n_i\times n_i$ matrices over a division ring ...
0
votes
2answers
18 views

Solving a Complex Number polynomial problem

This is an example Complex equations problem, everything is well understood except --(ii) in the below solution. Please can anyone explain, how anyone could have guessed the expansion in (ii) of the ...
0
votes
3answers
25 views

How can I find fifth root of unity?

I have no idea to do this question, how can I find the fifth root of unity? Question : Find all the distinct fifth root of unity. Let $\alpha$ be a fifth root of unity such that $\alpha \ne 1$. ...
0
votes
0answers
4 views

Finite operator in banach space

Let $E$ be a Banach space and denote its dual space by $E^*$. Let $p \in [1, \infty)$ and $x : \mathbb{N}\rightarrow E$ be such that for every $\phi \in E^*$, $$\left( \sum_{n=1}^{\infty} \lvert ...
0
votes
0answers
14 views

solve this congruence please.

$7^{n+1} -(n+1)*7^{n} -1 ≡ 0 $ (mod 4) with the variable n as an exponent you can't use a modulo 4 table, which is why it bothers me a bit.i tried messing around with it and i got that this equation ...
1
vote
1answer
19 views

Tricky logarithm problem

I having a problem in this logarithm problem involving modulus- Solve for x |x-1|^((log(x))^2-2log(x))=|x-1|^3 Bases same so powers equal. If I take log x as a then I get the following quadratic- ...
0
votes
0answers
21 views

Evluating triple integrals via Spherical coordinates

Use Spherical coordinates to evaluate the triple integral $$\iiint_{\mathrm{x^2+y^2+z^2<z}}\sqrt{x^{2}+y^{2}+z^{2}}\, dV,$$ What i tried Converting $x^2+y^2+z^2<z$ to Spherical coordinates ...
1
vote
0answers
5 views

Straight forward derivation of the bch formula?

Im doing a project on rigid body dynamics and need to derive the bch formula, anyone know a simple yet complete derivation?
2
votes
2answers
23 views

Calculate an integral that has a sum within.

Im trying to calculate this integral: $\displaystyle \int_{0}^{\pi} \sum_{n=1}^{\infty} \frac{n \sin(nx)}{2^n}$ The only thing I have been able to do is switch the integral and the sum, and in the ...
0
votes
0answers
8 views

ODE -parabolic cylinder functions

How do we solve $\frac{d^2f}{dz^2} + \left(Az^2+Bz+C\right)f=0 \tag 1$ where $f(z),A,B,C$ are matrices of order $3 \times 3$.
0
votes
0answers
7 views

partition of convex n-gon with triangles.

A convex $n$-gon is partitioned into $n-2$ triangles with non-intersecting diagonals. For each vertex of the original polygon, odd number of the partitioning triangles share that vertex. Is it ...
0
votes
2answers
11 views

Quick way to determine existence of integral root of a polynomial in one variable

Suppose $p(x) \in \mathbb{Z}[x]$ and if there exist $b \in \mathbb{Z}$ s.t. $p(b)=0$, then $x-b|p(x)$. The other technique can be to put all $b \in \mathbb{Z}$. But this require to check every $b \in ...
0
votes
1answer
64 views

What is the formula to generate this number sequence : 1 , 7 , 14, 30

What is the formula to generate this number sequence : 1 , 7 , 14, 30 I'm sure this is very simple for you guys. But it's got me alittle stuck. Thanks To clarify, I'm not an advanced maths student. ...
1
vote
1answer
16 views

Are the normed spaces $ \mathbb{R}^{n^2} $ and $ M_n(\mathbb{R}) $ isometric?

Consider the spaces $ \mathbb{R}^{n^2} $ with euclidean norm and $ M_n(\mathbb{R}) $ of $n\times n$ matrices with the norm defined by $ \Vert A\Vert = \sup\limits_{\Vert x\Vert \le 1}\Vert Ax\Vert$. ...
0
votes
1answer
8 views

Connection between linear independence, non-/trivial and x solutions

I am having a hard time remembering which goes hand in hand with what. The math questions I get always include words like trivial etc. 1 solution no solution infinite amount of solutions And then ...
1
vote
0answers
14 views

how to find if a number has a representation in a powerbase format?

I better explain this problem with an example, $100$ would be represented as $983$ because $9^1 + 8^2 + 3^3$ is one hundred. So how to find the relation between the number $n$ and numbers that can be ...
0
votes
3answers
25 views

Solve in integers: $x^2 = y^2 + y + 1$

Solve this equation in integers: $$x^2 = y^2 + y + 1$$ I know $2$ ways to solve this. But they are not easy. Maybe there is some quick method.
0
votes
0answers
11 views

Check if the angle is constructable

To check if an angle $x$ is constructable do we have to use the formula $\cos{3 \theta}=4\cos^3{\theta}-3\cos{\theta}$ and find the minimum irreducible polynomial that $\cos{x}$ satisfies and if its ...
0
votes
0answers
7 views

How to show that the vertices of a convex hull are given by these specific subsets…

We work over $\mathbb{R}^N$. Let $V$ be the corners of the unit cube $[0,1]^N$, or equivalently the set of vectors whose coordinates take values $0$ or $1$. Let $d:\{0, \ldots, N\} \to \mathbb{R}_+$ ...
1
vote
2answers
20 views

For any integer a, if $6|(3−a)$, then $3| (a−2)$.

Prove: For any integer a, if $6|(3−a)$, then $3| (a−2)$. I've been trying to work this problem for a while, but missed a day of class and can't seem to work it out.
0
votes
0answers
3 views

Torsion free module over a PID is flat

Suppose a ring of integers $S$ is an extension of a ring of integers $R$ with $\mathfrak{q}$ a prime ideal in $S$ and $\mathfrak{p}=\mathfrak{q}^c$ in $R$. Is there a straightforward way of showing ...
1
vote
0answers
35 views

Integral Contest

Before you answer this OP, please read all the terms and conditions below. Thank you... Today I hold an unofficial little contest on brilliant.org. Now, I will hold it here on Math S.E. It's just for ...
0
votes
0answers
11 views

Inductive Property of Sets?

Why doesn't the set: $ {2,4,6,8,10,.....}$ have the inductive property. For example $ n = 2k$. So for every value of k you get a value of $n$. Plus $k+1$ is also present. So shouldn't this set have ...
0
votes
0answers
14 views

Why have we made a function to be many to one and not one to many?

We have allowed function to only relate many to one but not one to many. Why haven't we included sin(x) to be a function? Is it just for simplicity? Also, I've seen someone quote a function not even ...
0
votes
1answer
9 views

What is special about a transformation if the matrix of that transformation is symmetric?

If the matrix of a linear transformation T$\colon \mathbb{R}^{N} \rightarrow \mathbb{R}^{N}$ with respect to some basis is symmetric, what does it say about the transformation? Is there a way to ...
0
votes
1answer
15 views

find the differential dy

I know this is right is it not? I did the work and I am almost positive I did it right.
0
votes
0answers
10 views

f is continious and g is defined as the integral of f. Now how can I show that g'(1)=f(1)

Im trying to solve a question I had on my exam but Im not sure how I should go about solving it. I dont even know in what direction I should look if I want to solve it. I am hoping someone here has a ...
2
votes
0answers
17 views

Real Analysis Differential functions

I am currently working through an exercise set and I am a little stuck on the following question: For $a > 0$, define a function $f_a(x)= \begin{cases} x^a \sin(1/x), &x \ne 0\\0, &x=0 ...

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