# All Questions

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### 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 ...
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### 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 ...
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### 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?
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### 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 ...
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### 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? ...
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### 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 ...
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### 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 ...
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### 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" ...
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### 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),...
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### 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., $X$...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $f\circ g \ne g\circ f$ ?
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### 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 ...
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### 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 ...
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### 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?
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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 $E \... 0answers 9 views ### (B,N) pair and Steinberg idempotent Let$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 $$e=\dfrac{1}{[G:U]}\... 1answer 35 views ### x_{n+m}\le \frac{x_n+x_{n+1}+\cdots+x_{n+m-1}}{m}. Prove that this sequence has a limit. Let m \ge 2 - fixed positive integer. The sequence of non-negative real numbers \{x_n\}_{n=1}^{\infty} is that for all n\in \mathbb N$$x_{n+m}\le \frac{x_n+x_{n+1}+\cdots+x_{n+m-1}}{m}$$... 0answers 7 views ### Solving quasilinear PDE - 1D, time-dependant, convection I have a task to solve the following quasilinear PDE (find c(x,t)):$$ 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(\...
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 ...