0
votes
0answers
5 views

Order of Rate of Growth

How would you put these functions in order of rate of growth from the greatest to the smallest? $f(x) = log_2 x$, $g(x) = x^x$, $h(x) = x^2 $, $k(x) = 2^x$ I took the derivatives and ended up with ...
2
votes
1answer
10 views

Evaluating limit (iterated sine function)

The limit is $$\lim_{x\rightarrow0} \frac{x-\sin_n(x)}{x^3},$$ where $\sin_n(x)$ is the $\sin(x)$ function composed with itself $n$ times: $$\sin_n(x) = \sin(\sin(\dots \sin(x)))$$ For $n=1$ the ...
0
votes
0answers
21 views

Is the reimann zeta function exactly pi(x)

Where pi(x) calculates the number of primes less than or equal to a certain x value. The pnt (prime number theorem) x/logx or more accurately x/logx-1 has been the most popular method for ...
0
votes
0answers
6 views

On singular points of parallels

Say $\gamma$ is a unit speed curve and its parallel is given by $$ p (t) = \gamma (t) + d n(t)$$ where $n$ is the unit normal vector. I read that as $d$ varies the singular points of $p$ trace the ...
0
votes
1answer
20 views

Rapid way to prove $ [e_{ij},e_{lk}]=\delta_{jl}e_{ik}-\delta_{ki}e_{lj} $

Let $e_{ij}$ denote the $n\times n$ matrix with entries all zero but the $(i,j)$th one, in which we put $1$. Let then $\delta_{ij}$ be the Kronecker Delta. Finally $[A,B]:=AB-BA$ is the commutator ...
0
votes
1answer
17 views

Let p be an odd prime number. Then show the following:

Let $p$ be prime, $p \geq 3$. Then show that $K_p$ is the union of $\frac{1}{2}(p-1)C_p$. I am once again at a loss for a starting point. Maybe just a small hint so I can work through this myself ...
3
votes
0answers
25 views

Do Electrical Engineering Researchers Usually Know Higher Math? e.g. Measure and Distribution Theory, Functional Analysis

I am an EE undergraduate student, and I recently took two courses in signals and systems. We were exposed to things like the Dirac delta (without mention of distributions), Fourier series (without ...
0
votes
0answers
15 views

Basic question related to dimension of intersection of two varities

Let $V$ and $W$ be irreducible varieties in $\mathbb{C}^n$. I have learned that intersection $V \cap W$ satisfies the following: $$ codim \ V + codim \ W \geq codim \ V \cap W. $$ I was wondering if ...
1
vote
2answers
16 views

On reparametrisation of curves (sorry for trivial question but I'm confused)

I'm confused about speed and reparametrisations of curves. To illustrate my confusion please let me elaborate using the simplest example I could think of: Let $\gamma : [0, 2 \pi ) \to \mathbb R^2$ ...
1
vote
0answers
13 views

Impact of random numbers on the eigen-values

How does the eigen-values of the following 3 diagonal matrix ($A$) changes adding random numbers $R_i$ (with the condition $|R_i| < m$). A is a square matrix defined as follows: For i = 1 to N $$ ...
0
votes
1answer
12 views

Find if a form is symmetric or skew-symmetric

Consider the set of all n × n matrices in R. Given the defined function Φ: $M$(n,n)× $M$(n,n) → R , which Φ(A,B) = $tr$(A$^T$JB) , where J is a skew-symmetric n × n matrix , define if Φ is a ...
0
votes
0answers
4 views

DTM that halts on a minimum one input

I am trying to show that this is Turing Recognizable. The language A, that is a DTM that halts on a minimum of one input. My justification is that of course it is because, you just need to check ...
0
votes
1answer
17 views

p-th root does not become a p-th power when adjoined?

Suppose $k$ is a number field of characteristic zero, $\zeta_p\in k$, and $u$ is not a $p$-th power in $k$. Show $\sqrt[p]{u}$ is not a $p$-th power in $k(\sqrt[p]{u})$. This seemingly simple ...
1
vote
1answer
16 views

Let $X_1$ and $X_2$ be two independent random variables each with probability density function $fX_i(x_i) = 1$, for $0 < xi < 1$ for $i = 1, 2$.

Find: (a) $E(X_1 X_2)$, and (b) $Var(X_1 X_2)$. Isn't (a) = zero, since this are independent? How do I go about (b)
2
votes
1answer
14 views

Let G be a simple graph with n vertices and m edges. Prove the following holds!!

Let G be a simple graph with n vertices and m edges. Prove the following holds using the Handshake Theorem: $$\frac{m}{\Delta} \leq \frac{n}{2} \leq \frac{m}{\delta}$$ where: $\Delta$ is the maximum ...
0
votes
1answer
13 views

Parallels of a parameterised curve if not unit speed

I just read that if $\gamma$ is a curve given in unit speed parametrisation then the parametrisation of a parallel curve is given by $$ p(s) = \gamma (s) + d n(s)$$ where $n$ is the unit normal to ...
0
votes
2answers
21 views

Explain the role of the numerator and denominator of a rational exponent such as $\left(\frac{27x^3}{8y^9}\right)^{-\frac{5}{3}}$

I know the answer, but I am kind of lost / confused when it comes to explaining the roles… Thanks in advance.
1
vote
0answers
14 views

Matrices derivative

I have a linear product of matrices, I did solve most of it, however, I stop at this component $(X^T W^T D W X)^{-1}$. Given that $X$ is $n \times p$ matrix and $D$ is $n\times n$ matrix. $W$ is a ...
0
votes
1answer
10 views

Parametric equations of a line

"Find the parametric equations of a line that passes through point $(1,1,0)$, parallel to plane $2x+3y+z=7$ and perpendicular to the line $\frac{x-1}{-2}= \frac{y}{3}=-z-2$" I don't know where to ...
0
votes
2answers
16 views

squaring both side for an absolute inequalites on only one side

this is about squaring both side for an absolute inequalities on only one side problem. For example: $|6-2x|< x+4$ when solved both by squaring both sides and by defining it$ -(x+4)<6-2x< ...
1
vote
2answers
13 views

Comparison between two exponentail random variables

A and B are exponentially distributed with parameter $\alpha$ and $\beta$. A and B race repeatedly. $N_b$ denotes the number of times B wins before A wins his first race. Find $P (N_b = n )$ for $n ...
3
votes
3answers
62 views

Calculating remainder of $666^{666}$ when divided by $1000$.

I want to calculate the remainder of $666^{666}$ when divided by $1000$. But for the usual methods I use the divisor is very big. Furthermore 1000 is not a prime, 666 is a zero divisor in ...
1
vote
1answer
22 views

What is the interactive explanation of a number to the power $\sqrt{-1}$

What happens when a number is multiplied with itself i times, i.e a number $n \in \mathbb{C}$, what is the explanation of $n^i$ ? I have tried a few by myself:- $e^i = cos \; 1 + i sin\; 1$ and $i^i$ ...
-4
votes
1answer
13 views

How can I find the area of this region? 11

Find the area of the region of the function y=x^2 +2, given [0,1].
0
votes
0answers
5 views

A property of Quasiconvex functions.

Let f be a strictly quasiconvex differentiable function and Df denote its gradient. Is the following implication true? :"Whenever f(y) < f(x), we also have (Df(x))'(y - x) < 0" . Suppose that f ...
1
vote
1answer
18 views

What is the probability of two random lines crossing in a unit square?

For the purposes of this question a random line is defined as the line connecting two random points inside the unit square, where a random point is found by generating two random numbers between 0 and ...
0
votes
0answers
5 views

Differential of square map in Lie group away from identity

I've looked everywhere for this specific example but couldn't find it. Probably simple but I only need it for a small application and my Lie theory is very rusty. Let $G$ be an arbitrary Lie group ...
1
vote
0answers
15 views

Abstract definition of a differential operator

In Kolar, Michor, and Slovak, a differential operator is said to be a rule transforming sections of a fibred manifold $Y \to M$ into sections of another fibred manifold $Y' \to M'$. Is this is a ...
1
vote
0answers
24 views

What's book that I should read?

When I read the chapter 4 of "Three Manifolds with Positive Ricci Curvature," I got stuck. I don't know what Fourier transform variable is, what derivative of second order nonlinear partial ...
-1
votes
0answers
14 views

Prove a certain function is discontinuity type I and integrable.

Let $x\in\mathbb{R}$ and let $m(x)$ be the unique integer minimizing the value $|x-m(x)|$, for $x \neq n/2$ for $n$ odd. Let $$ (x):=\begin{cases} x-m(x) & \mbox{ if } x\neq n/2, n \mbox{ odd ...
0
votes
2answers
25 views

Consider the vector space V = {(a, 1 + a) | a ∈ R} with irregular definitions of addition and multiplication

with addition and scalar multiplication defined by (a, 1 + a) ⊕ (b, 1 + b) = (a + b, 1 + a + b) k '*' (a, 1 + a) = (ka, 1 + ka), k ∈ R find a basis for V. I started off with taking the general ...
3
votes
0answers
9 views

Are there slight modifications to NP-complete problems which reduce them to P?

Recently I revisited the infinite harmonic series and its barely diverging sum, and how removing all the composite numbers from the sum still produces a divergent series (even more barely). In ...
0
votes
0answers
37 views

What is the opposite of $\colon\colon$?

For example: I have ten jelly beans, six red, three blue, and one green. $\text{red} \colon \text{blue} \colon \text{green} \colon\colon 6 \colon 3 \colon 1$ How would you write "$\text{red} ...
1
vote
1answer
52 views

Find roots of unity

Find the roots of $6z^5 + 15z^4 + 20z^3 + 15z^2 + 6z + 1 = 0.$ I know how to do this without the coefficients, but I do not know what to do in this problem. Thanks
2
votes
0answers
12 views

Calculating the rank of a Boolean matrix and Boolean matrix factorization

I am interesting in some sort of algorithm for calculating the Boolean rank of small $M \times N$ Boolean matrices. Just to be clear, by Boolean matrices I mean matrices with entries $0$ or $1$ where ...
2
votes
0answers
11 views

Why is the morphism induced by this linear system birational?

I have seen the following statement used in a few places, but I am not sure why it is true. Any explanation as to why it is (or is not) true would be appreciated. Let $P$ be a point of an ...
2
votes
1answer
54 views

How can you confirm that a problem is open?

I was reading an article on Wikipedia and I came across a list of two problems which they asserted to be open, but without citation. I have looked through some literature but not all, as I am afraid I ...
1
vote
1answer
20 views

Proof by contradiction - Predicates and quantifiers

Consider statement, For all integers, b,c,d, if x is a rational number such that $x^2+bx+c=d$, than x is an integer. a) express above statment in the form, $Q_1 b,c,d\in U_1 ( Q_2 x\in ...
0
votes
1answer
37 views

Least Common Denominator: $ \frac{\sqrt{x}}{x}+\frac{ln\ x}{2\sqrt{x}} $

$ \frac{\sqrt{x}}{x}+\frac{\ln \ x}{2\sqrt{x}} $ I have tried combining these two fractions; however, I keep getting stuck. $\frac{2\sqrt{x}}{2\sqrt{x}}\cdot\frac{\sqrt{x}}{x}+\frac{\ln\ ...
1
vote
2answers
38 views

Discontinuous maps taking compacts to compacts

It's commonly known that in general topology, a continuous map $f$ from a topological space $(X, \tau)$ to another topological space $(Y, \tau')$ will send every compact set to another compact set. ...
1
vote
0answers
29 views

How to give a rigorous proof of a fact about convex polygon?

I claim that there exists universal constants $0<\delta_1(m), \delta_2(m)<1$ such that for any convex polygon $P$ in $\mathbb{R}^n$ with $m$ faces, \begin{equation} \frac{\mathcal{H}^{n-1}(\{x ...
0
votes
1answer
12 views

Difficulty understanding the concept of writing an L-System?

I've recently tried my hand at L-Systems, but I'm having some difficulty wrapping my head around it. I watched this video on the subject which is pretty good, but I had a question around the 1:43 ...
1
vote
1answer
18 views

Best algorithm for finding permutations with constraint of average total value.

Let's assume I have a random number generator from 0-100 included (only integers) and I generate 5 numbers with it. I want to know the probability of hitting 80, 80, 80, 80, 80 with the constraint ...
2
votes
3answers
47 views

Show that $f(x)$ is bounded by $M$.

Let, $-\infty<a<b<\infty$. Suppose $f$ is continuous on $[a,b]$. Show that $f(x)$ bounded by $M$. We are supposed to use intermediate value theorem for this problem. But, I don't understand ...
5
votes
3answers
82 views

Perfect powers of successive naturals: Can you always reach a constant difference?

I was thinking about what happens if you take a sequence of consecutive squares, for example 1,4,9, 16. Taking the differences gives you another sequence, 7,5,3. And taking the differences between ...
0
votes
2answers
14 views

Orthonormal basis for the null space of almost-Householder matrix

A matrix $H$ is defined as: $$H = I - vv^T$$ where $v$ is a unit vector. What is the rank of $H$? What would be an orthonormal basis for the null space of $H$? How do we find the number of zero ...
1
vote
1answer
12 views

Optimize order of a list based on time to complete, probability of success

I'm a programmer participating in a coding challenge, but I'm not up on my advanced math. I'm currently working on a solution to a problem, and have a semi-functional program, but I'm still missing a ...
0
votes
0answers
12 views

dynamical system problem, pertubation

i came across this question $$ \left(\partial_{xx}-V''\left(u\right)\right)\left(u_{xx}-V'\left(u\right)\right)=\varepsilon $$ \begin{cases} u^{\left(k\right)}\left(\pm\infty\right)=0 & ...
-4
votes
1answer
29 views

Word problem to help me in my math class [on hold]

an estate valued at 124,104 is to be divided between two sons so that the older son receives twice as much as the younger son find each sons share of the estate
0
votes
1answer
19 views

Existence of Partials Imply the Existence of Gradient Vector?

Let $f$ be a scalar function of three variables. Then the gradient vector is defined by: I read here that the existence of partial derivatives at some point $(x_0, y_0, z_0)$ does not imply the ...

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