0
votes
0answers
3 views

Find the Subgroup of $\mathbb Z_4 \times \mathbb Z_2$ (Joseph A. Gallian - Exercise - 8.22)

Find the Subgroup of $\mathbb Z_4 \times \mathbb Z_2$ that is not the form of $ H \times K$, where $H$ is a subgroup of $\mathbb Z_4$ and $ K$ is a subgroup of $\mathbb Z_2$ Order elements of ...
0
votes
0answers
5 views

converging subsequences of two metrics

if $d$ and $d'$ are two metrics on a space $X$, is it true that they induce the same topology if and only if they have the same converging sequences ?
0
votes
0answers
10 views

Why is there a subsequence of $(x_n)$ that converges to some point $y$ in $\mathbb R^p$?

A subset $A\subseteq\mathbb R^p$ is compact iff for every sequence $(x_n)$ in $A$ there is a subsequence $(x_{n_k})$ which converges to a point of $A$. I understand the whole proof of the above ...
0
votes
0answers
13 views

Rotation matrix

I'm finding different results for the 3D rotation matrix in the XY plane from different sources and I was hoping for someone to help clarify. In my "applications of vector calculus" book, the matrix ...
-1
votes
0answers
28 views

Greek School Exams-Calculus problem

Ok,so this problem was posed yesterday-along with 3 others of lesser difficulty-on the Greek national exams for the 3rd grade of Lyceum-the final class,that determines University success.The reaction ...
0
votes
0answers
19 views

Calculate the fifth root of the matrix

I have got the following matrix. How to start ?
0
votes
2answers
11 views

Show every chain has an upperbound?

Sometimes I feel like proofs like this are pointless. I mean, if we have a partially ordered subset, it seems automatically true that you have a max element. 1) Either you have an infinite sequence ...
0
votes
1answer
13 views

How to avoid rote learning and perform deep learning?

I saw this question on brillant's facebook and I didn't even thought of/figure out to use difference of squares to solve this question. All the while, I have been a C student for Maths and barely ...
0
votes
0answers
5 views

Definition of normal sets and compactness

I am struggling a little bit with this notion. In Conway's Functions of One Complex Variable, he offers the definition: A set $\mathscr F \subset C(G,\Omega)$ is "normal" if each sequence in ...
0
votes
0answers
7 views

One-Dimensional Jump-Diffusion Ito’s Formula

Let, $N_t$ be a Poisson process and let $X_t$ solve the SDE $d{X_t}=A_t dt +J_t dN_t$. Then, what is the correct Ito`s fórmula: i)$df(t,X_t)=(\frac{\partial f}{\partial t} + \frac{\partial ...
0
votes
2answers
41 views

How do I calculate $ \int_{1}^{3} x/(2-x) \;\mathrm{d}x$

$ \int_{1}^{3} \frac{x}{2-x} \;\mathrm{d}x$ $ \int_{1}^{2} \frac{x}{2-x} \;\mathrm{d}x$ + $ \int_{2}^{3} \frac{x}{2-x} \;\mathrm{d}x$ $u = 2-x$ $\lim_{e\to0} \left[ \int_{-e}^{1} \frac{2-u}{u} ...
0
votes
0answers
3 views

Characteristics and additional conditions for differential equation

I need to solve such a DE: $$(1+x^2)u_x+u_y=0$$ And then I need to draw its characteristics. The second part of the task says: Write three additional conditions such that this equation: Has one ...
1
vote
0answers
10 views

Example for the benefit from monotone convergence

I want to see a (preferably simple) example where I can apply monotone convergence to a sequence of functions $f_n$ but where I cant exchange limitation and integration in terms of the Riemann ...
0
votes
0answers
4 views

Extension Lemma for Functions on Submanifolds

The following lemma is my question. (cf GTM218, Introduction to Smooth manifold) I can prove (b) using partion of unity as follows: $Proof$ for any $p \in S$ choose a slice chart $W_p$ centered at ...
0
votes
1answer
15 views

An equality with inverse trigonometric functions

I've stumbled on the equality $$ \tan ^{-1}\left(\frac{3}{4}\right) \left(\pi -\tan ^{-1}\left(4 \sqrt{3}\right)\right)=4 \tan ^{-1}\left(\frac{2}{\sqrt{3}}\right) \cot ^{-1}(3). $$ Out of ...
3
votes
0answers
20 views

Geometric intuition for derivatives of basic trig functions

I was inspired by this question to try and come up with geometric proofs for the derivatives of basic trig functions--basically, those that have simple representations on the unit circle ($\sin, \cos, ...
5
votes
1answer
36 views

Given matrix $A$ such that $A^8+A^2=I$ prove that $A$ is diagonalized

Given matrix $A\in M_{n}(\mathbb{C})$ such that $A^8+A^2=I$ prove that $A$ is diagonalized. So let $p(x)=x^8+x^2-1$ and we know that $p(A)=0$. The next step would be to show that the algebric and ...
0
votes
1answer
9 views

CLIQUE to UNARY-CLIQUE reduction NP complete

Assume the following Language: UNARY-CLIQUE= {(G=(V,E),1^k) | G is an undirected graph and there is a clique of size k in G} I'm trying to determine whether this language belongs to NP complete or ...
0
votes
0answers
9 views

Determine the correct calculation to maintain visual exactness between two squares with different centers of rotation

I'm encountering this math problem while working on some CSS / JavaScript, so the coordinate system as well as dimensions are noted as such. I've been scratching my head at this for a while and ...
0
votes
0answers
4 views

Find the matrix belonging to the transformation S into a standard database space R3

Self-adjoint linear transformation S: IR3 -> IR3 has a dual own value 0, corresponding own subspace is a plane x + y - z = 0; and another inherent value of 1.
1
vote
2answers
35 views

Closed form of sum with binomial

I want to find closed form of the following expression : $$\sum\limits_{k=0}^{n} \binom{n}{k}\frac{(-1)^k}{2k+1}$$ I have no idea how to do it.
0
votes
2answers
12 views

methods of constructing a matrix from its null space span

I have a matrix of size $4\times3$ and its null-space span is $\{(1,2,3), (2,5,7)\}$. How can I find the original matrix? It is not obvious from the span which vectors are free.
1
vote
0answers
21 views

Calculating flux for a triangle

Find the flux of $\boldsymbol{\mathrm{F}}=x\boldsymbol{\mathrm{i}} +4y \boldsymbol{\mathrm{j}}$ outwards across the triangle with vertices at $(0,0),(2,0)$ and $(0,2)$. Solution: $10$ The ...
0
votes
0answers
17 views

canonical form of a pde

I'm really struggling with this one and I can't seem to find what's wrong with my approach. I am given a PDE in the form $$U_{xx} + x y U_{yy} = 0,$$ and I am supposed to bring it to its canonical ...
0
votes
3answers
43 views

Open sets and compact spaces

I am reading through Rudin's Principles of Mathematical Analysis and had a few related questions. First, Rudin defines an open set, $E$, as a set such that every point is an interior point. A point ...
0
votes
2answers
23 views

I'm struggling to find this transformation matrix

$T:\Bbb{P}_3 \to \Bbb{P}_3$ is a linear transformation such that: $T(-2 x^2)= 3 x^2 + 3 x,\\T(0.5 x + 4)= -2 x^2 - 2 x - 3,\\ T(2 x^2 - 1) = -3 x + 2 .$ With respect to these three input vectors, ...
3
votes
6answers
46 views

Understanding derivatives

I don't know if this is written somewhere else. I've looked all over the internet so apologies if this has already been covered. I'm doing Year 12 Maths in Australia for what it's worth. In our ...
0
votes
1answer
13 views

Largest $R$ value in domian $0<|z-1|<R$

Determine the largest real number $R>0$ such that the Laurent series of $$f(z)=\frac1{z-1} +\frac2{z-i}$$ about $z=1$ converges for $0<|z-1|<R$. The singularities are $1$ and $i$. But in the ...
0
votes
0answers
4 views

Closed form of parametric solutions of an ODE

I need to find the closed form of the following ODE: $y'+siny'=x$, $y=y(x)$. I have found these two parametric solutions: $x=t+sint$ and $y=\frac{t^2}{2}+tsint+cost$. Any idea?
2
votes
0answers
15 views

Homology of mapping cone

Let $f:X\to Y$ be a map, and $\text{cone}(f) = CX \sqcup_f Y$ its mapping cone. Let $(H_n)_{n\in \Bbb{Z}}, (\partial_n)_{n\in \Bbb{Z}}$ be a homology theory with values in the category of $R$-modules. ...
1
vote
4answers
22 views

$\vec{a}\times(\vec{a}\times\vec{R})-\vec{b}\times(\vec{b}\times\vec{R})$

I have $\vec{a}\times(\vec{a}\times\vec{R})-\vec{b}\times(\vec{b}\times\vec{R})$, my textbook says that this equals $((\vec{a}\times\vec{a})-(\vec{b}\times\vec{b}))\times\vec{R}=-(a^2-b^2)\vec{R}$. I ...
0
votes
2answers
13 views

Approximate summation of flooring function

I have this summation: $\sum_{k=0}^x \lfloor{\frac{k}{c}}\rfloor$ Do you have any ideas on any general expressions that can approximate this? P.S I know I can approximate it with a Fourier ...
0
votes
2answers
27 views

Is there such a thing as complex rational numbers and does it have the same properties as the usual complex numbers as extension of the real numbers?

I've been wondering if there is any use to defining a set that is isomorphic to $\mathbb{Q}^2$ (in the same way that $\mathbb{C}$ is isomorphic to $\mathbb{R}^2$). I immediately see a problem with ...
2
votes
2answers
79 views

Calculating an integral

Can somebody help me calculate the following integral: $$\int\limits_{1/3}^{3}\frac{\arctan(x)}{x^2-x+1}\;\mathrm{d}x$$ I have tried integration by parts, but I got stuck in it. Wolfram also didn't ...
0
votes
1answer
23 views

If you have a field isomorphism and the domain is algebraically closed then so is the image?

I know it makes sense because if they are isomorphic they are practically the same thing, but what would a proof look like?
1
vote
0answers
16 views

How to find the real polynomial, which, based on a given dot product, is the nearest to other polynomial

I have a subspace $$U := \{p \in P_2(\mathbb{R}): p'(0) = p'(1) =0\}$$ and a dot product: $$\langle p, q\rangle = p(-1)q(-1) + p(O)q(O) + p(l)q(l).$$ I would like to determine the shortest distance ...
0
votes
1answer
17 views

Laurent series in domain $|z|>0$

Find Laurent series, in powers of $z$, of $$f(z)=\frac{\sin(2z)}{z}$$ valid in the region $|z|>0$. The singularity is $0$ but $0$ isn't inside the region of the domain so what do you exactly ...
1
vote
1answer
12 views

Relationship between euclidean metric in sphere of radius $r$ and the unit sphere.

I want to show $g_r=r^2g_1$ where $g_1$ is the (Riemannian) metric in the unit sphere induced by its inclusion in $\mathbb{R}^n$ and $g_r$ is the metric in the sphere of radius $r$ also induced by ...
0
votes
2answers
26 views
1
vote
2answers
18 views

“isomorphic” normed spaces and reflexivity

Let X, Y be normed spaces and suppose that there exists an bijective isometry between them. And if X is reflexive, then it is intuitively clear that Y is reflexive also. But, when I tried to prove ...
1
vote
0answers
14 views

How to find that two adjacency matrices are equal

What is the easiest way to tell if these two graphs are isomorphic and how do I know which nodes in both graphs are the same. I've made the adjacency matrices but they are pretty big. I think I need ...
0
votes
3answers
49 views

Self-Study Linear Algebra book for a complete understanding

I recently took an introductory class on linear algebra (covered solving linear systems, determinants, eigenvectors, diagonalization, some vector spaces, basis and combinations, transformations etc.) ...
0
votes
1answer
11 views

Multidimensional convergence in probability

If I have a vector $X^n=(X^n_1,...,X^n_m)$ is it true that $ \mathbb{P}(X^n\geq\epsilon)\rightarrow 0$ if $ \mathbb{P}(X^n_i\geq\epsilon_i)\rightarrow 0\ \forall i =1,...,m$ As $n\rightarrow \infty$?
4
votes
1answer
19 views

Revolving a $k$-manifold around an axis gives a $(k+1)$-manifold

I want to solve the following problem from M. Spivak's Calculus on Manifolds: Let $\mathbb{K}^n=\{x \in \mathbb{R}^n:x^1=0 \text{ and }x^2>0,\dots,x^{n-1}>0\}$. If $M \subseteq \mathbb{K}^n$ ...
2
votes
4answers
67 views

Show that $P(x,y)=x^6+y^6$ is reducible over $\mathbb{R}$

Show that $P(x,y)=x^6+y^6$ is reducible over $\mathbb{R}$ In general , How do you show that a given polynomial is reducible over some field ?
1
vote
1answer
12 views

Algebric and geometric multiplicity and the way it affects the matrix

Given a matrix $A$. Suppose $A$ has $\lambda_1,\dots,\lambda_n$ eigenvalues each with $g_i$ geometric multiplicity and $r_1,\dots,r_n$ algebric multiplicity, $g_i\leq r_i$. Given this information ...
1
vote
0answers
10 views

does constant convexity assures global minimum

I have the following question: Consider a function $f:R^n \longrightarrow R$, s.t.: there is a point $x_0 \in R^n$ s.t. $\frac{\partial f}{\partial x^k} =0$ $\forall k$. the hessian matrix ...
0
votes
1answer
12 views

PDE of the form $x \partial_x T - y \partial_y T = F(x,y)$ where $F$ is a given function.

Is there a known solution, or technique, for solving the following PDE? $x \partial_x T - y \partial_y T = F(x,y)$ Here, $F$ is a given smooth function $\mathbb R^2 \to \mathbb R$, and $T: \mathbb ...
4
votes
3answers
53 views

Why does $\frac{1}{{\left\| {\left| {{A^{ - 1}}} \right|} \right\|}} \le \left\| {\left| B \right|} \right\|$?

Let $A,B \in {M_n}$ suppose that the following statements are true: $A$ is nonsingular, $A+B$ is singular, $\left\| {\left| . \right|} \right\|$ is matrix norm. Why is it true that: ...
4
votes
4answers
53 views

Doubt about the domain in logarithmic functions.

According to my book, the logarithmic function $$\log_{a}x=y$$ is defined if both $x$ and $a$ are positive and $x\neq 0$ and $a\neq 1$. So are these not correct? $$\log_{-3}9=2$$ $$\log_{-2}-8=3$$ ...

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