# All Questions

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### Step Connected if and only if Connected

A space $X$ is step connected if given any open covering $\mathcal{U}$ of $X$ and any pair of points $p,q\in X$ there is a finite sequence $U_1,...,U_n$ of sets belonging to $\mathcal{U}$ so that ...
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### Determine the matrix of the reflections in the fol­lowing plane in $\Bbb R^3$.

$2x_1-2x_2-x_3= 0$ How would I go about approaching this problem?
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### primitive recursive predicate challenge

I see this question as a nice challenge on logic. Primitive Recursive Predicate Problem if P(x) and Q(x) be a primitive recursive predicate. which of the following is not a primitive recursive? ...
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### A city wants to encourage downtown

could you please help me with this ( part d ) A city wants to encourage downtown employees to use public transportation. Below is the time in minutes to get to work on one morning according to ...
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### Discrete Mathematics Function Proof

The question is as follows : Let $f:A\rightarrow B$ be a surjective function and let $C$ be a subset of $B$. Prove $f(f^{-1}(C)) = C$. I understand what the question is asking. It's basically ...
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### Probability notation P versus Pr

I have come across both $P(…)$ and $Pr(…)$ being used to represent probabilities. Is there any difference in the meaning of these notations, or are they just different shorthands? I seem to come by ...
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### Showing that a set is not infinite in measure

Suppose $f_n \geq 0$ for all $n \geq 1$, $f_n \to f$ a.e. on $[0, \infty)$ and there exists a constant $M>0$ such that $$\sup\limits_{n} \int_{E} f_n(x)dx \leq M \mu(E)$$ for each measurable ...
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### Property of linear growth function

Prove: Suppose function $f(X)$ has linear growth and there exists an $X^*$ such that $f(X)$ is monotonic for $|X|>X^*$. Then it is possible to define a function $g$ such that $g$ a rotation of $f$ ...
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### Are there infinitely many prime numbers?

I will post my own answer below. That should not deter others from answering. There are many ways to prove this.
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### Combining Functions Question

Question: If $f(x)=x^2-x+2$ and $g(x)=x-2$, find $h(x)$ such that $f(x)=g(h(x))$ I am not sure if I am on the right track here so far, I came to this mostly through guess and check, perhaps there is ...
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### Prove $\alpha: \mathbb{Z_n} \rightarrow \mathbb{Z_n}$ defined by $\alpha(s)=rs\mod{n}$ is an automorphism.

I'm working on proving the following claim: "Let $r \in U(n)$ and $\forall s \in \mathbb{Z_n}$, $\alpha: \mathbb{Z_n} \rightarrow \mathbb{Z_n}$ defined by $\alpha(s)=rs\mod{n}$ is an automorphism." ...
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### Is there a name for convex subsets that contain the origin?

Let $R$ denote a preordered commutative ring (e.g. take $R=\mathbb{R}$), and suppose $M$ is an $R$-module (e.g. take $M$ equal to the vector space $\mathbb{R}^2$). Question 0. Is there a name for ...
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### Transformation theorem, Cauchy distribution

I have derived the density for the ratio of two independent random variables,via the transformation formula: $V = X/Y$ and $U = X$ inversion yields: $Y = U/V$ och $X =U$ , the jacobian: ...
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### Formula for number of “root” nodes in a tree where Parent shares child nodes?

If I have a tree like this: {a},{b,c},{d,e,f},{g,h,i,j} in this case we have a total of 10 nodes. Is there any equation where given "10" I can calculate how many bottom nodes there are (answer: "4" ...
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### Differentiability: Partially Defined Functions

These ideas came to my mind while reading Lee's Introduction to Smooth Manifolds. (Cf. discussion on p. 45.) Also note that though I were able to resolve the first problem the second one is still ...
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### Definite integral fractional exponent in the denominator

I have come across this question and I cannot understand the step highlighted. I would have expected that the fractional exponents of the terms in the numerator would have a negative value after ...
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### Definition of dimension

Let us consider Euclidean space $\mathbb{R}^n$. We say it is $n$-dimensional because each vector in it is an $n$-tuple $(x_1,...,x_n)$. However, it is possible to represent this exact same space using ...
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### is a given expression an irreducible fraction

The following statement is pretty obvious: ...
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### Maximizing a particular integral / functional

I have a (probably simple) question whose answer seems obvious but I cannot prove it. It relates to the calculus of variations. Let $A = \int_a^bB(x)C(x)dx$. Find the scalar function $B(x)$ which ...
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### Proof by induction $\sum_{k=1}^{n}$ $k \binom{n}{k}$ $= n2^{n-1}$ for each natural number n

Prove by induction that $\sum_{k=1}^{n}$ $k \binom{n}{k}$ $= n2^{n-1}$ for each natural number n
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### What is this graph called?

I'm not sure what it's called. Update: It's from the NED. Also explain: The name of these 4 axes. I mean I guess Y and X are in the same position but what about the others? The ...
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### Real matrices with non-real eigenvalues

I know this covers a lot, so perhaps someone could redirect me to a helpful website. for a) I have no idea where to start on the proof, as I don't understand why this is true. for b) I also have ...
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### How to evaluate the limit $\lim_{x \to \infty} \frac{2^x+1}{2^{x+1}}$

How to evaluate the limit as it approaches infinity $$\lim_{x \to \infty} \frac{2^x+1}{2^{x+1}}$$
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### Subtraction in GF(2^8) Giving Incorrect Results

Let me preface this by stating that I'm not normally a math person, but I'm currently dabbling in finite fields to help wrap my head around certain cryptographic topics (specifically those based ...
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### How to scale “probabilities” to a given mean?

I have a set of scores $x_i$, $i=1,\ldots,N$ (mimicking probabilities, $0\le x_i\le 1$) and I want to transform them so that the result has a given mean $m$, while remaining in the interval $[0;1]$. ...
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### BINOMIAL PMF distribution [on hold]

X∼binomial(1,1/3) and Y∼binomial(2,1/2) Obtain PMF of W = XY + 1 How to approach and solve this problem. Can some one help me in this regard? Thanks,
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### If the Planck length exists, why doesn't it follow then that the world is one-dimensional? [on hold]

As I understand it, the planck length means that space itself as we preceive it is quantized. We think of space as 3-dimensional, right? But if there truly is a planck length, to me that shows that ...
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### Find the exact value of expression

Let $$S=\sqrt{4+\sqrt[3]{4+\sqrt[4]{4+\sqrt[5]{4+\sqrt[6]{4+\cdots}}}}}$$ Is it possible to write $S$ in terms of standard mathematical functions and operators? If yes, what is the exact value of $S$? ...
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### Curve sketching from derivative to the original

The graph of the derivative function f'(x) of a function f(x) is shown below. Determine: i) the intervals where f(x) is increasing; ii) the intervals where f(x) is decreasing; iii) the ...
### Sizes of Blocks of Consecutive Integers Divisible by at Least One Prime Less than or Equal to $r$.
Let $f(r)$ be the largest integer such that there exists a block of $f(r)$ consecutive integers each divisible by some prime that is less than or equal to $r$. For example, $f(2)=1$ because it is ...
Suppose we have functions $$f(x) = \sqrt{x}, \space g(f) = \frac{df}{dx}+\frac{d^2f}{dx^2}+\frac{d^3f}{dx^3}\space ...$$ Applying function f(x) to g(f) we get: g(f(x))=\frac{1}{2}x^{-\frac{1}{2}} - ...