2
votes
0answers
2 views

What is the algebraic structure of $\Bbb Q_p/\Bbb Z_p$?

I am curious about the algebraic structure of $\Bbb Q_p/\Bbb Z_p$. Is there any result in this direction? Thanks!
1
vote
0answers
7 views

Only five solvable quintic equations of the form $x^5+ax^2+b=0$? What are their solutions?

According to Wikipedia there is only five solvable quintic equations of the form $x^5+ax^2+b=0$ (up to a scaling constant $s$). $$x^5-2s^3x^2-\frac{s^5}{5}=0 $$ $$ x^5-100s^3x^2-1000s^5=0 $$ $$x^5-...
0
votes
0answers
2 views

The discrete Laplacian

I am working on the $d$-dimensional integer lattice. Let $S$ be a random walk with increment distribution $p$. Given the distribution $p$ we can define the discrete Laplacian just as in Wikipedia is ...
0
votes
0answers
4 views

How to prove that probability for different initial conditions to yield similar trajectory is very small?

For $\epsilon > 0$, suppose $f$ is a chaotic function. Then, for any trajectory $\mathbf{x,y}$ how can I prove that we have $Probability\{f(\mathbf{x}) = f(\mathbf{y}) \}$ $\le \epsilon$ where $\...
0
votes
0answers
7 views

Find a point along the line created by two points based on distance

Let's say I have two people on two points on a 2 dimensional plane. Person B is 400 units away from Person A. But Person A wants to be always 950 units away from Person B, preferably moving back in a ...
0
votes
0answers
9 views

Improper rotation matrx in $2D$

The following is the related problem: Improper Rotations in Even Dimensions I want the simpler explanation. An improper rotation is rotation, followed by reflection in the plane perpendicular ...
1
vote
0answers
20 views

Where does the premise of this idea come from?

Let $x$ , $y$ be positive real numbers. Prove the inequality $$x^ y + y^x \ge 1$$ This is the solution provided by my textbook: Where does this first idea (proving that $a^b \ge \frac{a}{a+ b - ...
0
votes
1answer
7 views

Elimination and exchanging rows

Solve by elimination, exchanging rows when necessary $$ v + w = 0\\ u + v = 0\\ u + v + w = 1\\ $$ Which permutation matrix is required? answer is $$ P= \begin{bmatrix} 0 & 1 & 0 \\ 1 ...
-1
votes
0answers
10 views

A complex no. as the limit of integration

I have come across an integral equation, integral (f(x)dx) limits from 0 to i=sqrt(-1) and f(x) is a real function. Now, I know the value of integral(f(x)dx) limits from 0 to real number a. Also, f(x) ...
0
votes
0answers
20 views

The $n$-th root of $(1+q^n)^2$

Let $0<q<1$ be rational. I am suspecting that $\sqrt[n]{(1+q^n)^2}$ is irrational. Can someone please help me to prove or to disprove this? $n=1$ and $n=2$ are simple cases. I am interested ...
1
vote
1answer
15 views

Are there at least $3$ groups of order $16$ that has element of order $8$?

Are there at least $3$ groups of order $16$ that has element of order $8$? I know that probably the simplest way of doing this problem is looking at the element structure of the abelian groups of ...
-1
votes
1answer
17 views

Simmetric group exercise of an exam

In $S_5$ let be $\sigma=(12435)$ and $\tau=(25)(34)$, and $H= <\sigma, \tau>$. Show that $N(<\sigma>) = H$ N.B. $N(<\sigma>)$ is normalizer in $H$, not in S5. Deduce that $H=&...
1
vote
1answer
12 views

Proving the congruence $((p-1)\ /\ 2)!^2 \equiv (p-1)!\ (\textrm{mod}\ p)$

We are given that $p$ is a prime congruent to $1$ modulo $4$. The proof for the congruence $$\left(\frac{p-1}{2}\right)!^2 \equiv (p-1)!\ (\textrm{mod}\ p)$$ is argued as follows: Proof. Since we ...
-3
votes
0answers
9 views

Producing factors L and U

Apply elimination to produce the factors L and U for the matrices below. $$ A= \begin{bmatrix} 3 & 1 & 1 \\ 1 & 3 & 1 \\ 1 & 1 & 3 \\ \end{bmatrix} $$ and $$ A= \begin{...
1
vote
1answer
13 views

Converting Fractional Coordinates to Cartesian

I'm confused about what I am reading online - different sites tell me different answers. Lets say I have a point pair in fractional coordinates, [xf,yf,zf]. I know that to convert them to their ...
-2
votes
0answers
15 views

Existence of (non) complete metric on an interval

I am stuck with this problem. Can anyone help me out? Thank you in advance. Which one of these are correct. a) (0,1) with the usual topology admits a metric which is complete. b) (0,1) with the ...
1
vote
0answers
4 views

Clarification on a concept involving Hall subgroups

Suppose that $G$ is a finite group and $H\leq G$. If $Q \in$ Hall$_{\pi}(G)$ such that $Q \cap H \in$ Hall$_{\pi}(H)$. Is true that $Q \leq H$ when Hall$_\pi(H)$ $\subseteq$ Hall$_\pi(G)$
1
vote
1answer
14 views

What is the maximum length continuity of function $f$?

Let $f(x) = \int_{.25}^x {(\frac{t}{{{t^2} - 1}})} (\cos \frac{1}{{\sqrt t }})dt$. What is the maximum length continuity of function $f$?
-1
votes
0answers
12 views

Can i say that ( U(Zm) , * ) is isomorphic to ( Zk , +)

Can i say that ( U(Zm) , * ) is isomorphic to ( Zk , +) where k=phi(m) or to Za x Zb x Zc x....where abc..=k with the right combination of a,b,c... ?
0
votes
3answers
32 views

Why are the solutions of the equation different? : $x=2 => x^2=4 => x=±2$

If I define the variable $x$ as $x=2$, then $x^2=4$. But the solutions of $x^2=4$ are $±2$(two solutions). I defined what the variable $x$ is, then why are the solutions for the equation $x^2=4$ two, ...
-1
votes
2answers
22 views

question from logarithms

Simplify without using tables $$\frac{\log25+\log625}{\log5}$$
1
vote
0answers
20 views

combination problem on Lego blocks

Working on some interesting combination problems related to Lego blocks. For example, this one. Confusion is, I often see two dimensions (e.g. height and width in below problem) mentioned to calculate ...
-2
votes
1answer
22 views

Suppose {an} is a sequence, and M is a fixed positive integer. We define a new sequence {bn} by bn = aM+n. Suppose {an} → L. Show that {bn} → L.

Suppose {an} is a sequence, and M is a fixed positive integer. We define a new sequence {bn} by bn = aM+n. (so the new sequence is the old one ‘shifted’ by M terms.) Exercise 1. Suppose {an} → L. Show ...
-1
votes
1answer
11 views

Finding a point between 2 moving points colinearly , given 2 moving points and distance.

A------B---------------C A and C are moving points that can move anywhere A = (xa, ya), C = (xc, yc) B (xb, yb) is a point between A and C colinearly With one condition that distance of AB = ...
2
votes
0answers
12 views

Factory inspections on a budget

A factory inspector is testing the efficiency of $n$ machines. To pass the inspection, each machine is required to run at or above a certain standard efficiency. The inspector can measure the ...
0
votes
0answers
7 views

Zeros in pivot position

When zero appears in a pivot position, $$ A = LU $$ is not possible. What do we have to do here to make A=LU possible then? Do we have to find a specific P (permutation matrix) for A and continue ...
0
votes
0answers
8 views

Can't understand one step in the Vitali Covering Lemma

I am having a slight doubt in understanding the Vitali Covering Lemma in the infinite case. So I have a collection $\mathcal C$ of balls with the ball $B$ having radius $|B|$. Then, first define $d_0=...
0
votes
1answer
17 views

How to find a modulus equation?

Let $x$ and $q$ be an integer.Also, $a$ and $b$ are integers. We know the two modulus equations. i) $x \equiv y$ mod $p^a -1$ ii) $x \equiv z$ mod $p^b -1$ Then how to find $x$? Can we find $x$ ...
0
votes
1answer
29 views

Solving vector equations

Using vector method solve the two following problems, $p \bar{x}+\bar{x}(\bar{x}.\bar{b})=\bar{a}\times \bar{b}+\bar{c}$ $\bar{x}\times \bar{a}+(\bar{x}.\bar{b})\bar{c}=\bar{d}$ How to solve $\bar{...
4
votes
2answers
36 views

$p^2$ misses 2 primitive roots

When I Checked primitive roots of some primes P, I found this following phenomenon: $14$ is a primitive root of prime $29$, but it's not primitive root of $29^2$ $18$ is a primitive root of prime $...
2
votes
1answer
25 views

Logarithmic function in complex number

Show that: $$\cos[i\log(2+\sqrt3)]=2$$ I attempted by taking$(2+\sqrt3)$ into trigonometrical form but i am stuck Please help me out.
0
votes
0answers
10 views

Predicate Logic - Archimedes' Library

Problem: Our class takes a field-trip to Archimedes’ Library. Before entering the library, your tour guide makes you notice the sign on the main doors which reads: “Observe the Rule of Archimedes’ ...
0
votes
1answer
9 views

Iteration: Approximation and Errors, finding all possible iterative arrangements

I am looking at a relatively simple problem to reiterate: $x^4=e^x$ I've found 5 different possible forms 1: $x_{r+1}=\frac{e^x}{x^3}$ 2: $x_{r+1}=(\frac{e^x}{x^2})^{0.5}$ 3: $x_{r+1}=(\frac{e^x}...
0
votes
0answers
16 views

show that $h|_{(-\infty,p]} \equiv 0$ and $h|_{[q,\infty]} \equiv 1$ is lipschitz

In Lemma $6.4$, the author states that the function $h:\mathbb{R} \rightarrow \mathbb{R}$ defined by $h|_{(-\infty,p]} \equiv 0$ and $h|_{[q,\infty]} \equiv 1$ is lipschitz, where $p,q \in \mathbb{R}$ ...
0
votes
0answers
15 views

Potential theory, potentials and harmonic functions

In the development of potential theory we mostly study harmonic functions. However I found some paper, which present potential theory as the study of potentials. Are potentials harmonic functions?
3
votes
2answers
23 views

What is common notation for “disjoint union of copies of $\mathbb{R}$”?

I'm looking at a question out of Lee's Smooth Manifolds: Show that a disjoint union of uncountably many copies of $\Bbb{R}$ is locally Euclidian and Hausdorff but not second countable. My ...
1
vote
0answers
30 views

Watches on a table

From Peter Winkler's 'Mathematical puzzles', taken from an All USSR Mathematical Competition, 1976: 50 accurate watches lie on a table. Prove that there exists a moment in time when the sum of the ...
1
vote
0answers
31 views

Is this what universal covering spaces are used for?

From the perspective of real analysis, we have: $$\int_{-1}^{1}\frac{1}{1+x^2} = \mathrm{tan}^{-1}(1)-\mathrm{tan}^{-1}(-1) = 2\mathrm{tan}^{-1}(1) = 2 \cdot \frac{\pi}{4} = \frac{\pi}{2}$$ Something ...
1
vote
1answer
35 views

Volume of solid lies under $z=x^2+y^2$

Find the volume of solid lies under $z=x^2+y^2$ above $x$-$y$ plane and inside the cylinder $x^2+y^2=2x$. I know, for volume we have to us $V=\iiint { \mathrm dx\mathrm dy\mathrm dz}$ but i was not ...
1
vote
2answers
21 views

order of multiple quantifiers

Problem: For a, b, c, d restricted to the universe of positive integers, explain why ∀a ∃b ∀c ∃d a/b = c/d is true, but ∀a ∃d ∀c ∃b a/b = c/d is false. I understand that the order of ...
4
votes
3answers
45 views

How to integrate $\frac{dx}{(x-p)\sqrt {(x-p)(x-q)}} $?

How to integrate $\frac{dx}{(x-p)\sqrt {(x-p)(x-q)}} $ ? I tried substituting $x=1/t$ but that's making it more complicated.Any suggestions?
0
votes
4answers
41 views

Reduce Number of digit

I have 24 digit number like 281564785148270616103860.I want to reduce this number with any optimization technique to 12 digit number . So i can use this to decode with the same technique.Any answer ...
0
votes
1answer
13 views

Is there any approach to computer vision that doesn't make use of geometry?

I've long been interested in applying my background in functional analysis (especially wavelets) and other related areas to actually create something with "real world" value (not that I don't enjoy ...
2
votes
2answers
42 views

Where to learn the algebra behind the use of differential operators in calculus

Algebra with differential operators is often used as a shortcut in calculus problems. I have previously asked about manipulations such as these: $$\int x^5e^xdx =\frac1Dx^5e^x=e^x\frac{1}{1+D}x^5=e^x(...
-3
votes
2answers
53 views

How to solve $x^3+y^3=z^4$?

Any three positive integers that are greater than 2 satisfying $x^3+y^3=z^4$ . You can not have two or more of the same number.
0
votes
2answers
23 views

Give a geometric comparison of the solutions to

Give a geometric comparison of the solutions to $x_1 + 3x_2 - 5x_3 = 4$ $x_1 +4x_2 -8x_3 = 7$ $-3x_1 -7x_2 + 9x_3 = -6$ and $x_1 + 3x_2 -5x_3 = 0$ $x_1 +4x_2 -8x_3 = 0$...
1
vote
0answers
26 views

What is the difference between derevative w.r.t a vector and directional derivative?

Say we have a scalar-valued function $f: \mathbb R^3 \rightarrow \mathbb R$, such that: $$f(\mathbf x) = \mathbf x^T\mathbf a$$ $\mathbf x$ and $\mathbf a$ are two vectors. The derivative of $f$ ...

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