0
votes
0answers
8 views

The value of the perpendicular on a diagonal in a rectangle

How to calculate the perpendicular on the diagonal of a triangle with sides 2 and $\sqrt2$ ? I've already calculated the diagonal by using Pythagorean theorem which is $\sqrt 6$. Then I didn't know ...
0
votes
1answer
6 views

Compound propositions as assertions?

According to comments on my previous question, compound propositions are not assertions; i.e. the statement "$p \vee q$" does not mean "$p$ (is true) or $q$ (is true)", and it does not mean "$(p$ or $...
1
vote
0answers
7 views

Clarification for Manifold and Manifold with Boundary Definitions

I am reading Spivak's Calculus on Manifolds, and just need a bit of clarification when it comes to definitions. He defines a manifold as some space $M$ that satisfies: $(M)$: $\forall x \in M, \...
1
vote
1answer
14 views

Limit with Lambert-$W$ function

I have asked a similar question about this one particular limit: \begin{equation} A=\lim_{c\to 1}\exp\left[ -\left(\frac{1}{1-c}\right)\left(W_{0}\left[ B\left( 1+\frac{x}{rc}\right) \right]-W_{0}[B]\...
0
votes
1answer
15 views

Word problem for quadratic equation

Two water taps , One big , one small , are used to fill a rank of capacity 4 $m^3$ with water . Small water tap supplies water to the tank at a rate of p $m^3$ per min and the big water tap supplies ...
0
votes
0answers
16 views

Black are berries and maroon are cherries. Place 8 more cherries removing berries 1 from each row and each column. No of ways?

I tried to see it as a matrix where for a position (i,j) , i+j = 8, 9, 16 means you can't change that position. Any help?
1
vote
2answers
15 views

Find vectors u and v such that W = Span{u,v}

Let $W$ be the set of all vectors of the form $\begin{bmatrix}s-t\\2s+t\\0\\t\end{bmatrix}$ Find vectors $u$ and $v$ such that $W =$ Span{$u,v$} How can I do this? Any advice woulds be ...
0
votes
1answer
8 views

Brownian Motion-Independence of Increments

Consider a Brownian Motion $B(t)$ with $B(0)=0$. Suppose $s<t$. I read in a book that while $B(t)-B(s)$ is independent of the past, $2B(t)-B(s)$ or $B(t) - 2B(s)$ is not. Why is this the case? ...
0
votes
1answer
14 views

Counting number of cosets

Let $G = \big(\mathbb{Z}/n\mathbb{Z})^*$, that is the multiplicative group modulo $n$. For some $d$ coprime to $n$, let $H$ be a subgroup of $G$ generated by $d$. As $G$ is abelian, $H$ is normal in $...
-1
votes
0answers
10 views

prove or refute seriality in modal logic

prove or refute $\diamond A \vee \neg \diamond \Box A$ characterizes seriality? treat both directions separately. what about $\diamond A \vee \diamond \neg \Box A$ ?
0
votes
0answers
12 views

Null Laplace Transform

As the title says, if I had a real signed measure $\nu$ defined on Borel sets of $\mathbb{R}^m$ with Laplace Transform vanishing on every $m$-tuple, can I say that $\nu=0$?
0
votes
5answers
89 views

What is the sum of the series 1/3 + 2/9 + 3/27 + 4/81 + … [duplicate]

I remember solving this in highschool , but now I don't remember how to find sum of these kind of series . I want to find the sum of the general series Sum $\sum_{n=1}^{\infty} n .a^{-n} = ? $ ...
0
votes
0answers
15 views

Null spaces and their dimensions. how to decide the dimension of the null space

Here is a quote from a textbook. Within four dimensional space ofa ll possible vectors x the solutions to $$Ax=0$$ form a two dimensional subspace - the nullspace of A In this specific A we ...
0
votes
0answers
7 views

Trigonometric substitution [illustration / right triangle derivation]

Real quick: If I have the function $\int { \sqrt { { a }^{ 2 }-{ x }^{ 2 } } } dx$ I can easily substitute by setting x equal to $a\sin \theta$. But why actually is that? If I draw a right ...
2
votes
1answer
14 views

Calculating the Stokes Theorem

I was tasked with calculating $ \oint_{L}Fdr $ for when $F=xzi-j+yk$ (vetor form) and $$L = \begin{cases}z=5(x^2+y^2)-1 & \mbox{ } \mbox{} \\z=4 & \mbox{} \mbox{} \end{cases}$$ Using: ...
0
votes
0answers
4 views

Find radius of identical circles within a circular sector

The diagram shows sector $OCED$ of a circle, with centre $O$ and radius $R$. Two identical circles of radius $r$ are arranged within the sector such that they touch each other. The lines $\overline{OC}...
0
votes
1answer
17 views

Are there more non-perfect square numbers than perfect squares?

Can anything be said on this issue? I was wondering if one can find a mapping such that the cardinality of two sets of perfect and non-perfect squares can be compared. Not sure if it's a good question ...
0
votes
0answers
4 views

How to find a suitable function for Dulac's criteria in this example?

I have a system of odes $\dot{\mathbf{x}} = \mathbf{f(x)}$ where $\mathbf{x} \in \mathbb{R}^{2}$ and $\mathbf{f(x)}$ is defined below: $$\dot{x} = x- y - x^{3}, \qquad \dot{y} = x+y-y^{3}$$ I would ...
2
votes
0answers
12 views

Explaining and Integral involving the Divisor Function

In a 1973 paper by Martinet, Deshouilliers and Cohen, $A(x)$ is defined as $$A(x)=\lim_{N\to\infty}\frac{\#\{n\leq N\mid \frac{\sigma(n)}{n}≥x \}}{N}$$ where $\sigma(n)$ is the "sum-of-divisors" ...
0
votes
0answers
11 views

Coin Toss Normal Distribution

In 200 tosses of a coin, 115 heads and 85 tails were observed. In this problem we will compare using the normal, the normal with the continuity correction, and then using the binomial. For (a) -- (c),...
1
vote
1answer
25 views

What is $\lim_{t \rightarrow0} \frac{\Gamma(\alpha t)}{\Gamma(t)} (\Gamma$ is the Gamma function)?

What is $\lim_{t \rightarrow0} \frac{\Gamma(\alpha t)}{\Gamma(t)} (\Gamma$ is the Gamma function)? I took the numerator, used the change of coordinates $u = \alpha t$ and got that the limit was $\...
1
vote
0answers
8 views

Random walk on a connected graph

I am reading a book and I have a problem understanding why a relation holds. Assume that we have a time-homogeneous random walk on a connected graph $G=(V,E)$. For $o\in V$, the roundtrip from $o$ ...
0
votes
0answers
18 views

How to solve a non-linear system of equations

I have a question regarding the following system which looks like a Vandermonde system of equations except for the fact that (1) $A$is non-square (2) $A$ and $q$ are BOTH unknowns $$ \overbrace{\...
2
votes
0answers
15 views

Preserving equality between different branches of mathematics.

I'm taking an 'Intro to Higher Mathematics'-type course right now, were we learn about basic set theory, number theory, algebra, etc. and I had the following thought: Say you're trying to solve a ...
1
vote
1answer
60 views

How does one show that $\cos {\left (\ln 2 \right )}\approx \frac{10}{13}$?

How does one approximate the value of something like this? Apparently Euler found the value of $\large \frac{2^i+2^{-i}}{2}\large $ [which equals $\cos {\left (\ln 2 \right )}$] to be close to $\...
0
votes
3answers
29 views

Meaning of Vector Space over $\mathbb{R}$ being a Subspace of $\mathbb{R^R}$

$\mathscr{P(\mathbb{R})}$ is the set of all polynomials with coefficients in $\mathbb{R}$. How are below sentences related and why? (1) $\mathscr{P(\mathbb{R})}$ is a vector space over $\mathbb{R}...
0
votes
0answers
6 views

Spectrum of Kernel - Discrete orthogonal polynomials

Trying to solve a problem, I encounter a Kernel of the form $$K(m,n)= e^{-\frac{\beta}{4} (m+n+1)} \frac{2^{2+\frac{m+n}{2}}}{\sqrt{m! n!}} \frac{\sqrt{\pi}}{n-m} \left[ \frac{1}{\Gamma(-m/2)\Gamma(...
2
votes
0answers
35 views

Why are we interested in such things as $\zeta(3)\notin\mathbb{Q}$?

In 17xx Euler gived a formula for the real numbers $\zeta(2n),~n\ge 1,$ which showed the irrationality of $\zeta(2n)$. In 1975 Apéry showed $\zeta(3)\notin\mathbb{Q}$. Why are we interested in such ...
1
vote
3answers
41 views

Explain why the columns of a 3x4 matrix are linearly dependent

Explain why the columns of a $3 \times 4$ matrix are linearly dependent I also am curious what people are talking about when they say "rank"? We haven't touched anything with the word rank in our ...
0
votes
1answer
14 views

Maximum modulus theorem proof

I do not understand the proof for the maximum modulus theorem done with the open mapping theorem. Unfortunately my notes are a little bit cryptic. What I understand: Let $z_0$ be the maximum that is ...
-1
votes
0answers
11 views

Solving$ x(x-1)\ddot y-x\dot y+y=x(x-1)^2$ by using integrating factor

Solving $x(x-1)\ddot y-x\dot y+y=x(x-1)^2$ given the solution $y_1=x$ I set $y=vx$, have plugged and rearranged. I set my integrating factor to: $$e^\left({\int\frac{x-2}{x(x-1)}dx}\right)$$ ...
0
votes
1answer
15 views

I need a simple equation to measure a efficiency of attempts correction

I have a process where the user need correct an invalid information in your registry within a maximum number of attempts. The closer he gets this maximum number, the worse your rate. For example: ...
0
votes
4answers
47 views

What is the sum of the solutions of the equation: $x^2 - (3m-1)\,x + 2m+3=0$?

What is the sum of the solutions of the equation: $x^2 - (3m-1)x + 2m+3=0$ ?
0
votes
0answers
12 views

How can I find the measure of $B=\{(x,y,z) \in\mathbb R^3| \; x^2+y^2+4z^2 \le3, \;x^2-y^2+4z^2\le1, \; z\ge 0\}$?

$B=\{(x,y,z) \in\mathbb R^3| \; x^2+y^2+4z^2 \le3, \;x^2-y^2+4z^2\le1, \; z\ge 0\}$ The question is similar to that which I shared in another topic. Also here, the set is defined by an ellipsoid, ...
1
vote
0answers
27 views

What is $\sum_{i=0}^n \left\lfloor \sqrt{i}\right\rfloor \binom{n}{i}$?

Since both $\sum_{i=0}^n \left\lfloor \sqrt{i}\right\rfloor$ and $\sum_{i=0}^n \binom{n}{i}$ have simple closed-form evaluations, it is natural to consider the evaluation of the binomial sum $\sum_{...
0
votes
0answers
8 views

Reference: Gaussianity of linear functional of Gaussian process

My question is similar to this one, but I'm looking for a reference rather than derivation. I've been told, inserting my own commentary in square brackets, If you take $X$ in $C([a,b])$ [i.e., $X$...
0
votes
1answer
19 views

Is $\{(x,y) \in \mathbb R^2 \mid x^2 + x^3y^2 = 0\}$ compact in $(\mathbb R^2, \mathcal E_2)$?

Determine if $X = \{(x,y) \in \mathbb R^2 \mid x^2 + x^3y^2 = 0\}$ is compact in $(\mathbb R^2, \mathcal E_2)$, where $\mathcal E_2$ denotes the standard Euclidean topology. I know that $X$ is ...
0
votes
1answer
13 views

What value is added to a home when new landscape is added to the home's property

If I bought a new home for 400,000 and then invested with new landscaping for 19,000 what would be the new appraisal amount for my new home when added to my new plants, lawn etc in the first day after ...
0
votes
1answer
15 views

Grid walks binomial coefficient?

Here's a problem and my attempt to answer it: In this problem you will derive a binomial coefficient identity based on walks on a grid, starting, as usual, at the lower left corner and ending at the ...
1
vote
0answers
15 views

Factoring $x^5+B x^4+C x^3+D x^2+E x+F=(x^2+a x+b)(x^3+p x+q)$ over $\mathbb{Q}$

For a quntic polynomial to be reducible to the following form over $\mathbb{Q}$: $$x^5+B x^4+C x^3+D x^2+E x+F=(x^2+a x+b)(x^3+p x+q)$$ We need to match the coefficients ($a=B$ obviously, so we ...
0
votes
0answers
13 views

Module endomorphisms of simple modules where they do not commute by composition.

Is there an example of module homomorphisms $f,g : M\to M$ where $M$ is a simple $R$-module such that $f\circ g \ne g\circ f$ ?
0
votes
3answers
37 views

winding number example

Let $$\gamma:[a,b]\rightarrow \mathbb C $$ be a closed curve and$$ Int(y)=\{z\in\mathbb C-tra(\gamma): ind(z)\neq0\}, \\ Ext(y)=\{z\in\mathbb C-tra(\gamma): ind(z)=0\}, $$ where $ind(z)\;$is the ...
0
votes
0answers
10 views

Gödel Number and Recursive enumerable set

Assign a Gödel number to each non-true sentences of a system E over some standard arithmetic language. can one conclude system E is incomplete (incompleteness theorem)? Is the resulted set recursive ...
0
votes
2answers
13 views

Growth function and one misunderstanding point?!

I have a question about Growth and Asymptotic notation topic. My question is as follows: $2^n$ > $n^{log_2{(n)}}$ is True. anyone could say how we can deduce that this fact is true?
0
votes
0answers
4 views

Constructing specific hyperplane after injective mapping

Let $f: X \to Y$ be an injective continuous function where $X \subset \mathbb{R}^m$ is a nonempty compact set, and $Y \subset \mathbb{R}^n$. Let $\mathbf{y}^* = f\left(\mathbf{x}^*\right)$. If $E \...
1
vote
0answers
9 views

(B,N) pair and Steinberg idempotent

Let $q=p^f$ where $p$ is prime and $G$ be a finite group with a $(B,N)−$pair ($T=B\cap N$ and $W=N/T$), and assume that $B=UT$ with $U\triangleleft B$ and $U\cap T=1$. Define $$e=\dfrac{1}{[G:U]}\...
2
votes
1answer
35 views

$x_{n+m}\le \frac{x_n+x_{n+1}+\cdots+x_{n+m-1}}{m}$. Prove that this sequence has a limit.

Let $m \ge 2 -$ fixed positive integer. The sequence of non-negative real numbers $\{x_n\}_{n=1}^{\infty}$ is that for all $n\in \mathbb N$ $$x_{n+m}\le \frac{x_n+x_{n+1}+\cdots+x_{n+m-1}}{m}$$ ...
0
votes
0answers
7 views

Solving quasilinear PDE - 1D, time-dependant, convection

I have a task to solve the following quasilinear PDE (find $c(x,t)$): $$ c_x v + c_t = - v_x c $$ $c \in (0,20) , t \in (0, \infty)$ where I know function $v(x)$ to be $v(x) = \frac{3}{40}(1+\cos(\...
1
vote
2answers
25 views

What is the chance of randomly generating a given 10-character sentence?

Suppose we have an alphabet of the following allowed characters: the lowercase letters $a$ through $z$ (26) the uppercase letters $A$ through $Z$ (26) the numerals $0$ through $9$ (10) the common ...
0
votes
2answers
42 views

What is the largest of the five missing numbers?

This is Q28 from Australian Maths Competition 2014. A circle is surrounded by 6 other circles,in a hexagonal formation.The leftmost circle is 0,which the rightmost circle is 1000.Each of the five ...

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