1
vote
1answer
31 views

Having objects trajctories and directions how to find where objects traverse same path?

I have N objects that travel on some trajectories (unique for each object). At each agent curve point we can get object speed (direction). Having some distance ...
1
vote
2answers
79 views

Find triangel-area with Cavalieri's principle

A triangle is given by $A=(0,0), B=(5,1)$ and $C=(2,4)$. I already know $\lambda^2(\Delta ABC)=9$. Now I want to compute the area by using Cavalieri's principle. I know how to start when I have to ...
1
vote
2answers
131 views

Not able to solve $({\frac{1}{2}})^p + ({\frac{1}{3}})^p + ({\frac{1}{7}})^p - 1 = 0.$

I'm not able to solve $$({\frac{1}{2}})^p + ({\frac{1}{3}})^p + ({\frac{1}{7}})^p - 1 = 0.$$ If you put values of $p$ (like $\frac{1}{2}$ or 2) back in the equation it doesn't satisfy! So please ...
2
votes
1answer
116 views

Proving something is not differentiable

I am looking for confirmation so that I can be sure I understand what is being asked here. I need to show that the following function $f(x,y)$ is not differentiable at $(0,0)$ but that ...
0
votes
1answer
76 views

Conics in Complex Projective Spaces

I was reading classification of complex hyperquadrics, I am stuck in $\mathbb CP^2$ what is $X_0^2+X_1^2=0$ in $\mathbb CP^2$, ok in $\mathbb CP^1$ this represents just two points, my attempt if $X_1$ ...
0
votes
4answers
402 views

What is the subspace of $\mathbb{R}^3$ generated by $v_1= (2, -1,1)$ and $v_2= (1,2,3)$

What is the subspace of $\mathbb{R}^3$ generated by $v_1=(2, -1,1)$ and $v_2=(1,2,3)$? my options: $[\vec v_1,\vec v_2]=\{(x,y+1,x-y); x,y\in\mathbb R\}$ $[\vec v_1,\vec v_2]=\{(x,y,x+y); ...
6
votes
4answers
153 views

Determine if the expression is an integer.

I'm tryin to solve this: Determine $k$ such that $\frac{k^2-87}{3k+117}$ is an integer. I think chinese remainder theorem will be useful, but i don't see how.
14
votes
2answers
371 views

A Challenging Euler Sum $\sum\limits_{n=1}^\infty \frac{H_n}{\tbinom{2n}{n}}$

Recently, I encountered a strange series involving Harmonic Numbers and Binomial Coefficients both. According to Mathematica: $$\displaystyle \sum_{n=1}^\infty \frac{H_n}{\binom{2n}{n}} = ...
3
votes
4answers
134 views

Can a continuous topological function be one-many?

Can there exist a continuous topological mapping $f:X\to Y$ which is one-many? I ask this question because if such a mapping exists, then I can see potential contradictions in some theorems stated in ...
6
votes
2answers
263 views

Elementary Abelian $p$-Subgroups of $GL_n(\mathbb{F}_p)$

Let $p$ be a prime number. If $G$ is a finite group, an elementary abelian $p$-subgroup of $G$ of rank $r$ is any group $E\subset G$ such that $E\cong (\mathbb{Z}/p)^{\oplus r}.$ Let $\mathcal{E}_r$ ...
4
votes
1answer
381 views

splitting field of $(x^3-2)(x^3-3)$ over $\mathbb Q$

Question: What is the Galois group of $f(x)=(x^3-2)(x^3-3)$ over $\mathbb Q$, and what are the subfields which contain $\mathbb Q(\zeta_3)$? The roots of $f(x)$ are ...
2
votes
1answer
65 views

Linear Algebra dependent Eigenvectors Proof

Problem statement: Let $n \ge 2 $ be an integer. Suppose that A is an $n \times n$ matrix and that $\lambda_1$, $\lambda_2$ are eigenvalues of A with corresponding eigenvectors $v_1$, $v_2$ ...
0
votes
1answer
82 views

Inner product spaces of smooth functions

In the space $C^1([0,1])$ where each $f$ is an element of the space and $$||f||= \left(\int_0^1\left(|f|^2+|f'|^2\right)dx \right)^{1/2}$$ How can it be shown that $||\cdot||$ is a norm of the space?
2
votes
1answer
50 views

Identifying the constants and variables in statements

Please help me to identify the constants and variables in these statements. Thanks in advance. Ratio of the circumference of any circle to its diameter. Height of a boy on a given day. Height ...
1
vote
1answer
52 views

If $M$ is invariant under $L$ is it a vector space

I'm reading Linear Algebra by Peterson, and the excercise on p.31 reads as follows: Let $L: W \to V$ be a linear operator and $V$ a vector space over $\mathbb{F}$. Show that if $M \subset V$ is ...
1
vote
1answer
91 views

Connected Metric Space Exercise

Let $E$ be a connected metric space, in which the distance is not bounded. Show that in $E$ every sphere is nonempty.
2
votes
4answers
163 views

Finding root of equation

This question was asked in one of the enterance test of mathematics in India which is For the equation $1+2x+x^{3}+4x^{5}=0$, which of the following is true? (A) It does not possess any real root ...
5
votes
2answers
112 views

Prove that the class of well-founded sets is a proper class

Does anyone have an "elementary" proof of the following claim: If $A$ is a class such that $$(*)\qquad\forall x(x\subseteq A\to x\in A),$$ then $A$ is a proper class, i.e. $\forall y\ y\ne A$. ...
2
votes
1answer
93 views

How can I resolve this definte integral?

$$ \int (t^2-3)^3 t dt$$ $$ \int_a^b (t^2-3)^3 t dt$$ if $$ a = -1$$ $$ b=1$$ then $$ \frac 1 2\int (t^2-1)^3 2t dt$$ $$ \frac 1 2 \frac {u^{3+1}}{3+1}$$ $$ \frac 1 8 u^{4}$$ $$ \frac 1 8 ...
4
votes
1answer
149 views

Is the internal language of a topos complete, sound and effective?

The internal language of a topos is higher order intuitionistic typed logic. Now according to this article in wikipedia higher order classical logic with full semantics is never complete, sound or ...
3
votes
1answer
197 views

Consequence of Chinese Remainder Theorem

The following proof comes from J.P. May's online notes on Dedekind Rings. The question is as follows. Let $I$ be a non-zero ideal of a Dedekind domain $R$. Prove that there is an ideal $J$ such ...
2
votes
1answer
88 views

How to find the all elements of $\text{Aut}(\Bbb{D})$?

Use Schwarz lemma, we can verify that $$f(z)=e^{i\theta}\frac{\alpha-z}{1-\overline{\alpha}z}$$exhaust all automorphisms of the disc($\theta \in \Bbb{R},\alpha \in \Bbb{D}$). When I first see the ...
3
votes
0answers
82 views

Definition of a group representation

A representation of a group on a vector space $V$, is a group homomorphism $f: G \to GL(V)$, where $GL(V)$ is the general linear group. However, Wikipedia defines it as a map $G \times V \to V$ such ...
1
vote
1answer
125 views

Area bounded by a parametric curve

Be $\gamma: \mathbb{R}\rightarrow \mathbb{R}^3$ defined by: $$\gamma(\theta):= (\cos(\theta), \sin(2\theta), \cos(3\theta))$$ and be $$S:=\lbrace t\gamma (\theta): t\in [0,1], \hspace{0.1cm} ...
2
votes
2answers
91 views

Is $f$ a differentiable function?

Hello everyone I have this problem, Can somebody help me with this? $f:\mathbb{R}^2\rightarrow{}\mathbb{R}$ is defined by: $$f(x,y) = \left \{ \begin{matrix} ...
2
votes
1answer
42 views

Condition the tensor product of a vector is not singular

I am generating a matrix with a vector through outer product in my code, and resulting matrix is singular. Is there a condition I can use to check the vector and figure out where the singularity comes ...
3
votes
3answers
136 views

Characteristic polynomial of a matrix $A$ is $x^7$ and $\operatorname{rank}(A)=4$ and $\operatorname{rank}(A^2)=1$. Classify $A$.

I know how to do this problem the 'long' way. I was wondering if there was an easier, less computationally cumbersome way to do this. Here is the question: Let $A$ be a square matrix over $\mathbb ...
2
votes
1answer
232 views

At what times, t, does an airplane (vector defined) intersect with a radar beam (2D plane defined)?

A radar beam can be defined as a plane in $3D$ space. If the beam is moving such that the basis is $[1\text{ } 0 \text{ } \cos(wt)]^T$ and $[1 \text{ } \sin(wt) \text{ } 0]^T$ where $t=time$. ...
2
votes
0answers
147 views

Proving that a set is measurable and has a zero area

I want to prove that each of the following sets is measurable and has zero area. a) A set consisting of a single point. b) A set consisting of a finite number of points in a plane. c) The union of a ...
9
votes
1answer
266 views

Exercise 24.13 of T. Jech's *Set Theory*

Having struggled my way through most of chapter 24 of Jech's Set Theory, I'm stuck on the very last part of the very last question, 24.13: Let $I=I_{NS}$ be the nonstationary ideal on $\omega_1$, ...
5
votes
4answers
211 views

Finding another way of doing this integral $\sqrt{\frac{1-\sqrt{x}}{1+\sqrt{x}}}$

Problem : Integrate : $\sqrt{\frac{1-\sqrt{x}}{1+\sqrt{x}}}$ I have the solution : We can substitute $\sqrt{x}= \cos^2t$ and proceeding further, I got the the answer which is ...
0
votes
1answer
1k views

Finding the probability from a markov chain with transition matrix

Consider the Markov Chain with state space $S=\{v,w,x,y,z\}$, transition matrix below: $$\left[\begin{array}{cccccccccc} 0 & 0.4 & 0.6 & 0 & 0\\ 0 & 0.5 & 0.5 & 0 & ...
0
votes
1answer
66 views

Is there a quick way (a way) to do this integral

$f(x,y) = 2xy-y$ over the square $0-2$ $\int _0 ^2 \int _0 ^2 (\frac{2xy-y}{\sqrt{1+2x^2+2y^2-2x}} )dxdy$ My fallback is to write the square-root as (something to do with x)^2 + (some 'constant' ...
5
votes
5answers
727 views

Fixed point Exercise on a compact set

Let $K$ a compact normed space and $f:K\rightarrow K$ so that $$\|f(x)-f(y)\|<\|x-y\|\quad\quad\forall\,\, x, y\in K, x\neq y.$$ Prove that $f$ have a fixed point.
3
votes
1answer
89 views

How to prove " $¬\forall x P(x)$

I have a step but can't figure out the rest. I have been trying to understand for hours and the slides don't help. I know that since I have "not P" that there is a case where not All(x) has P... but ...
2
votes
1answer
141 views

Contraction Mapping

$$f(x)=\begin{pmatrix}1/4 & 0 & 1/2 \\ 0 & 1/3 & 0\\ -1/2 & 0 & 1/4 \end{pmatrix}\begin{pmatrix}x_1 \\ x_2\\ x_3 \end{pmatrix}, \qquad\forall x=\begin{pmatrix}x_1 \\ x_2\\ x_3 ...
2
votes
1answer
1k views

Orthogonality and linear independence

[Theorem] Let $V$ be an inner product space, and let $S$ be an orthogonal subset of $V$ consisting of nonzero vectors. Then $S$ is linearly independent. Also, orthogonal set and linearly ...
7
votes
3answers
321 views

Changing Summation Index Question

I'm sorry if this seems like a very novice question, but I am still relatively new to the world of discrete math ( still in 9th grade). I've been reviewing some of the concepts I learned in a ...
5
votes
2answers
252 views

Is This A Derivative?

I am in a little over my head. This all began with my reading how each level of pascals triangle adds to $2^n$, where n=row# starting with n=0. I then though, "wouldn't it be clever if the rows added ...
0
votes
1answer
170 views

Derivative of Trace of Matrix wrt parameters

I have the following function which I need to find the derivative of $$L=trace(\Sigma K^{-1})$$ where $K$ is a function of $\theta$ and $\Sigma$ is constant. If I'm correct what I need to do to find ...
1
vote
2answers
94 views

$\sup_{x\in A}x \sup_{y\in B}y=\sup_{x\in A,y\in B}xy$

Let $A$ and $B$ be two sets of nonnegtive numbers. Prove that $\sup_{x\in A}x \sup_{y\in B}y=\sup_{x\in A,y\in B}xy$. Thanks for your help.
0
votes
3answers
205 views

number theory: Let $m>n$ for $m,n\in\mathbb{Z}$, prove if $k$ divides $m$ and $k$ divides $n$ then $k$ divides $m\bmod{n}$

Let $m>n$ for $m,n\in\mathbb{Z}$, prove if $k$ divides $m$ and $k$ divides $n$ then $k$ divides $m\bmod{n}$. How should I approach this question? I only got $m=qk$ and $n=pk$ if ...
1
vote
2answers
65 views

How to compute $\mathbb{P}(\lambda X>4)$ directly?

Given a random variable $X$ which is exponentially distributed i.e. $X\sim E(\lambda)$. Calculate $\mathbb{P}(X-\frac{1}{\lambda}>\frac{3}{\lambda})$. My working: ...
0
votes
2answers
68 views

for $z\in\mathbb C-\{0\},~\dfrac{1}{1+nz}\to0.$

How to show that for $z\in\mathbb C-\{0\},~\dfrac{1}{1+nz}\to0.$ I've tried triangle inequality couldn't arrive at any conclusion. Please help me.
1
vote
2answers
80 views

Minimize the distance in the Euclidean space

The objective is to minimise the distance $d_{0}+d_{1}$. The points $c_{0}$ and $c_{1}$ are given. I need to locate the point $c$ which minimises the distance $d_{0}+d_{1}$. I have worked like this. ...
2
votes
0answers
19 views

Special case of the hodge decomposition theorem [duplicate]

I am trying to prove the following special case of the hodge decomposition theorem in differential geometry for a n component vector field $V_i$ in $\mathbb{R}^n$. any vector can be written as the ...
1
vote
4answers
54 views

Solve the following limit of the sequence

Another sequence limit I'm stuck. $$\lim_{n\rightarrow\infty}{\frac{\prod_{k=1}^n(2k-1)}{(2n)^n}}$$ Any idea ?
0
votes
2answers
113 views

Difficulty in Quadratic equation and realtion with irrational roots

One root of the quadratic equation $ax^2 +bx + c=0$ is $\dfrac{2}{\sqrt{3} + \sqrt{5}}$. If $\frac{c}{a}$ is rational, then how do we find the other root. the answer given is that the other root is ...
1
vote
2answers
67 views

Clarifying an example of limits?

For the function $\displaystyle h \sin \left(\frac{1}{h}\right)$ when it is evaluated at $h=0$, is it $0$ or is it undefined?
2
votes
3answers
2k views

The sum of two periodic functions need not be a periodic function

Let $f(x)=x-[x]$ and $g(x)=\tan x$. How could we see that $f(x)-g(x)$ is not a periodic function? This will show that the sum of two periodic functions need not be a periodic function. I hope ...

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