0
votes
0answers
2 views

Every ordered field that has the least upper bound property is isomorphic to the real number system.

Okay, so here's a theorem from Rudin: "Every ordered field that has the least upper bound property is isomorphic to the real number system." Here's a definition: "Ordered fields are isomorphic if ...
0
votes
0answers
3 views

Find first positive perfect square in polynomial time

I have a quadratic. for example $$1x^2+6884x+3297$$ Is it possible to find the first perfect square in the series in polynomial time where both x and y are whole positive integers. In the above ...
0
votes
0answers
12 views

Mathematicians who didn't study mathematics in college or university

I would like to know mathematicians who were born after 1900 who didn't study mathematics in college or university. I posted a similar question recently but it was closed as opinion based. So I will ...
0
votes
0answers
8 views

Divisors and non divisors

How do i prove: The sum over the numbers that don't share a divisor with the divisors of n, including n it self, will be n-1. Example $n=12$: The numbers that share a divior are $2;3;4;6;12$. ...
0
votes
0answers
5 views

Automorphism group of a locally cyclic group

I am having difficulty proving that: The automorphism group of a locally cyclic group is commutative. Is there any easy way to prove it? Thanks.
0
votes
0answers
13 views

What's wrong with this proof of Schröder-Bernstein theorem?

In V. A. Zorich's Mathematical Analysis I there is an exercise to Analyze the following proof of the Schröder-Bernstein theorem: $({\rm card} X \leq {\rm card} Y) \land ({\rm card} Y \leq ...
0
votes
0answers
12 views

Lie Derivative Equals to Lie Bracket

I am reading the book Introduction to Smooth Manifold written by M.Lee. I am confusing with the concept of Lie derivative. We have $\mathcal{L}_XY=[X,Y]$. However we have ...
0
votes
1answer
7 views

What is the number of unique labeled connected graphs with N Vertices and K edges?

I've seen this question several times, and this one caught my attention. I'm now aware that there is no closed formula for this. My knowledge of graph theory is limited, and I wasn't able to find an ...
3
votes
1answer
12 views

$X \cap (Y \setminus Z) = (X \cap Y) \setminus (X \cap Z)$

As the title suggests, what is the easiest way to see that$$X \cap (Y \setminus Z) = (X \cap Y) \setminus (X \cap Z)?$$
0
votes
0answers
6 views

ASTC: Finding exact values.

Our teacher showed us this really dodgy way of finding exact values by drawing up the 4 ASTC (all stations to central diagram) quadrants and making a right angle to the x axis. So how would I do a ...
0
votes
0answers
8 views

Fractional Euler sums?

As we know, the classical linear double Euler sums is defined by $${S_{p,q}} = \sum\limits_{n = 1}^\infty {\frac{{{\zeta _n}\left( p \right)}}{{{n^q}}}} \;$$ where $p, q\ (q \ge 2)$ are positive ...
0
votes
1answer
17 views

$m \times n$ matrix gives rise to a well-defined map from $\mathbb{R}^n$ to $\mathbb{R}^m$?

As the title suggests, how do I see that an $m \times n$ matrix gives rise to a well-defined map from $\mathbb{R}^n$ to $\mathbb{R}^m$?
0
votes
0answers
21 views

Is this a valid proof of this math challenge problem?

From a fixed point P not in a given plane, three mutually perpendicular line segments are drawn terminating in the plane. Let a, b, c denote the lengths of the three segments. Show that ...
1
vote
0answers
10 views

product of subgroups and group G

Is there any example of two subgroups H and K of G whose product give G i.e. G = HK but none of which is normal in G
0
votes
0answers
6 views

Differentiability of parameter-dependent integrals when derivative exists only almost everywhere

This unanswered question asked in 2013 Differentiation under the Integral Sign (let's call this Q-zero) seems to be taken from this (or pdf ver.). The result on differentiation under the integral ...
0
votes
0answers
8 views

In which way are these logical statements similar to each other?

If x is even, then x is not divisible by 5. Every even integer is not divisible by 5. Alright so the original problem is for me to determine a counterexample if these are false. I already found a ...
0
votes
0answers
3 views

Frobenius Finite Rings/ Modules Isomorphism

So, i have some issues, and maybe someone can help me. A Finite Frobenius Ring is a Ring $R$ such that: $_RR\simeq {_R}\hat{R}$ (As R-Modules) Where ${_R}\hat{R}$ is the Module of Characters For ...
0
votes
0answers
6 views

Solve the following differential equation subject to the specified boundaries:

Solve the following differential equation subject to the specified boundaries: my answer please review my answer and correct it if it is wrong thanks
1
vote
0answers
22 views

About a matrix identity.

In a document named as "The Matrix Cook-Book" I saw two expressions of which I do not get any clue how they are derived. For $n = 3:$ $\det(I + A) = 1 + \det(A) + Tr(A) + 1/2\ Tr(A)^2 − 1/2\ ...
0
votes
3answers
26 views

$|(a,b)| = |\Bbb R|$ ? Cardinality of any open interval

I want to prove that any open interval $(a,b)$ has the same cardinality of the real numbers: $|(a,b)| = |\Bbb R|$. Do I have to find an function to prove it? Or is there a theorem to prove it ...
0
votes
1answer
13 views

Trigonometric Differential Equation 3

$(x\cos y-y\sin y)dy+(x\sin y+y\cos y)dx=0$ ATTEMPT: Rearranging the terms: $(x\cos ydy+y\cos ydx) -y\sin ydy+x\sin ydx=0$ Dividing by $\cos x$ we get: $(xdy+ydx)-y\tan ydy+x\tan ydx=0$ $ ...
0
votes
0answers
11 views

Linear Independence for functions defined by integration

Given that the set of functions $f_i(x,y)$, $i=1,\dots,n$ are linear independent for all $x\in \mathbb{R}$ and $y\in \mathbb{R}$, is the set of functions defined by $$ g_i(x) = \int_{y\in \mathbb{R} } ...
0
votes
1answer
18 views

How can I formed as below permutation problem

Hi I am writing a program and i encouraged the below permutation problem and need your help. There are 4 boxes: 3 of them have 2 balls The one box has 1 balls. The question is what is the ...
0
votes
1answer
11 views

Solving general linear ODE $\sum_{k=0}^n y^{(k)}=0$

Is there a way to solve this general linear ODE: $$\sum_{k=0}^n y^{(k)}=0$$ For the first few $n$ here are the solutions: $$\begin{array}{c|c} n & y \\ \hline 0 & 0 \\ 1 & c_1 e^x \\ 2 ...
0
votes
1answer
17 views

Anagrams contained within random strings

What is the probability that a random string of length $n$ will contain an anagram of a shorter string of length $k$? E.g., you generate a string of 50 random letters, repetitions allowed, what are ...
0
votes
1answer
31 views

If x is even, then x is not divisible by 5.

I have to provide a counterexample otherwise. So if one counterexample is enough, can I say x=10, because 10/5 = 2, thus x is not divisible by 5. Is this a justifiable answer?
0
votes
0answers
16 views

Probability distributions associated with Markov chain

Let's say I have a Markov chain, with all the transition probabilities known, and there's a cost associated with each transition. The cost for transitioning from node $a$ to node $b$ is given by the ...
0
votes
2answers
14 views

Calculating the number of ways to arrange a set of letters such that no two identical letters can occur consecutively

How can I find the number of ways in which the letters $$z,z,y,y,x,x,w,w,v,v$$ can be arranged so that two letters of the same kind never appear consecutively. I am not confident that my approach is ...
0
votes
1answer
14 views

Find a criterion for divisibility

Find a criterion such that $\displaystyle\sum_{i=1}^ni$ divides $\displaystyle\prod_{i=1}^ni^2$ for $n\in\mathbb N$. What I have done so far, $\displaystyle\sum_{i=1}^ni=\frac{n(n+1)}{2}$ and ...
0
votes
0answers
14 views

Simpsons Method of Order 4

I am trying to solve an ODE using Simpsons Method of order four. I don't know the corrector to use whether implicit or explicit. I need to correct for $y(x_{n+2})$ and $y(x_{n+1})$. Please help me ...
1
vote
1answer
41 views

Branch cut of $\sqrt{z}$

In my complex analysis book, the author defines the $\sqrt{z}$ on the slit plane $\mathbb{C}\setminus (-\infty,0]$. I understand this is done because $z^2$ is not injective on the entire complex ...
-3
votes
1answer
27 views

linear algebra (norm)

Can someone explain to me the following definition - $\|T\|$ := $ \sup \{\|T(v)\| : v \in \mathbb{R}^n, \|v\| = 1\}$ where $T$ is a linear transformation from $\mathbb{R}^n$ to $\mathbb{R}^m$ and ...
1
vote
0answers
12 views

Intuitive meaning of “Primal Dual Interior Point Method”

I am trying to understand how "Primal Dual Interior Point Method" works for nonlinear optimization. I have seen some examples already. Wikipedia has a very good example too. But I am still finding it ...
0
votes
1answer
38 views

Finding an integral.

Evaluate $$\!\int (x^5\sqrt{x} + x\sqrt[4]{x})\ \mathrm{d}x$$ My attempt: I tried to factor out a $\sqrt{x}$ and I got $$\sqrt{x}\int\! x^5+x\sqrt[3]{x} \ \mathrm{d}x$$ But here I cannot factor a ...
-3
votes
3answers
28 views

write an expression [on hold]

A word processor determines the width of the body of text on a page. The page is 11 inches wide and has two equal size margins of x inches on each side of the text. Write a formula that gives the ...
-1
votes
3answers
67 views

Is $\{\}$ equal to $\{ \{\} \}$?

Is $\emptyset$ equal to $\{\emptyset\}$? I know an emptyset contains no elements. So I feel like they would be equal. Can someone explain how they wouldn't be?
0
votes
0answers
16 views

Does this theorem for bases also hold for subbases?

Assume that we have a toological space $X$ with toplogy $\mathcal{T}$. If Y is a subspace of X, then $\mathcal{T}_Y=\{Y\cap U|U \in \mathcal{T}\}$ is a topology on Y (that it really is a topology, ...
1
vote
2answers
47 views

Analysis in $R^n$

I have to justify whether this statement is true or false - If a linear transformation(matrix) $||T||$ is non-invertible then $||T||$ = $0$. $||T||$ is the norm. Justification - $||T||$ is = $0$ iff ...
1
vote
1answer
24 views

Prove the convergence of sequences

Let $x_{n} = 0 $ if $n < 100 $ and $x_{n} = 1$ if $n \geq 100$ prove $x_n$ converges and find its limit. I started by letting $\epsilon > 0$, as per normal, and choosing $n \geq 100$ as well as ...
0
votes
0answers
16 views

When is the geometric Picard group $Pic(X_{\overline{K}})$ of finite type?

Let $X$ be a smooth proper geometrically connected variety over a field $K$ of characteristic 0. Let $\overline{K}$ denote an algebraic closure of $K$. What other conditions on $X$ are needed so ...
0
votes
0answers
26 views

Find all integers $m,n$ for which $m^2+n^2$ is a square and $\sqrt{\frac{2m^2+2}{n^2+1}}$ is rational

This is a repost of my old question here. The question is as follows: Find all integers m and n, such that $m^2 + n^2$ is a square and $\sqrt{\frac{2(m^2+1)}{n^2+1}}$ is rational. I have made no ...
2
votes
0answers
7 views

Minimal hypothesis for Jordan Chevalley decomposition to hold

When I look at different proofs or expositions of the Jordan-Chevalley decomposition of a matrix, the minimal hypothesis I usually found is about the perfection of the field over which such ...
-1
votes
0answers
24 views

How do you find the null space of an inconsistent system? [on hold]

For example, the augmented matrix: $$\left(\begin{array}{ccc|c} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right) $$
0
votes
0answers
17 views

Nth Homotopy Group Isomorphic to [T^n, X]

Following Spanier's book on algebraic topology chapter 1, section 6 about suspensions, I'm wondering about the following questions: 1) We know that $S^n$ is an H co-group for all $n\geq1$ because ...
-3
votes
4answers
62 views

Prove $\sin^2(\theta)+\cos^4(\theta)=\cos^2(\theta)+\sin^4(\theta)$

Prove $\sin^2(\theta)+\cos^4(\theta)=\cos^2(\theta)+\sin^4(\theta)$. I only know how to solve using factoring and the basic trig identities, I do not know reduction or anything of the sort, please ...
0
votes
0answers
26 views

Is every $\sigma$-algebra generated by a partition?

I know that every finite $\sigma$-algebra is generated by a finite partition, but is every infinite $\sigma$-algebra also generated by "kind of" partition? Can anyone help provide a explanation or ...
-1
votes
1answer
15 views

Find the position function from the piecewise-defined velocity function

I am getting stuck on a position function problem in my Diff Eq class. Problem 22 is shown on the right in the picture below. On the left is the answer. My work below shows that I get stuck ...
0
votes
0answers
18 views

About the Gibbs phenomenon for the Legendre polynomials as an orthogonal base of $L^2(-1,1)$

In a recent question, I proved that the Fourier-Legendre expansion of the function $f(x)=\text{sign}(x)$ over $(-1,1)$ is given by: $$2\sum_{m\geq ...
-1
votes
2answers
18 views

How to simplify this expression

How to simplify AB(A+B)(C+C), I tried but it did not seems to be coming out correctly not sure why.
0
votes
0answers
22 views

Tell whether the given statement is true or false. Explain your choice. [on hold]

All whole numbers are rational numbers & No irrational numbers are whole numbers

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