0
votes
0answers
3 views

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 ...
0
votes
1answer
9 views

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?
0
votes
0answers
8 views

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? ...
0
votes
0answers
14 views

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 ...
0
votes
1answer
20 views

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 ...
0
votes
0answers
14 views

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 ...
1
vote
0answers
10 views

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 ...
0
votes
0answers
10 views

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 $f$ ...
1
vote
1answer
50 views

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.
0
votes
2answers
13 views

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 ...
0
votes
1answer
18 views

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." ...
0
votes
0answers
19 views

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 ...
0
votes
0answers
8 views

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: ...
0
votes
1answer
13 views

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" ...
0
votes
1answer
16 views

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 ...
0
votes
3answers
14 views

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 ...
0
votes
2answers
57 views

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 ...
1
vote
2answers
32 views

is a given expression an irreducible fraction

The following statement is pretty obvious: ...
1
vote
0answers
13 views

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 ...
0
votes
2answers
32 views

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
0
votes
0answers
31 views

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 ...
0
votes
3answers
29 views

How to find the values of m and b?

How do I find the values of m and b when: a) the function is continuous in $x = \pi$ b) the function can be derivated in $x =\pi$ $$y=\begin{cases} \sin x & x<\pi \\ mx+b & x\ge ...
0
votes
0answers
9 views

What is the superior strategy in opening cells here?

This is a problem I just completed for a quiz online. Was wondering if someone could help me reason this out. Suppose you receive a message and you are trying to decide whether Team Rocket or Team ...
0
votes
0answers
15 views

A problem on Matirices over Ring

Let $M_{m,n}(R)$ denote an $m\times n$ matrix with each entry over a commutative ring $R$, $m\leq 2\leq n$, and there is a matrix $\mathbf{B} = M_{m,n}(R)$. $\mathbf{B}\mathbf{s} = \mathbf{a}$, where ...
2
votes
1answer
10 views

Continuous Random Variables: Uniform

Problem: A person drives to work via a road with a single traffic signal. The light cycles, green for 45 seconds, red for 15 seconds – ignore yellow. Assume the person approaches the signal at a ...
0
votes
0answers
10 views

Maximum area enclosed by a string attached at fixed points

Two fixed points A and B have a string of length L attached between them. Supposing that the string does not intersect the line segment AB, then the string and AB will form a closed figure. What ...
0
votes
1answer
18 views

Closed 4-manifolds have CW-complex?

It looks like for compact 4-manifolds this question is open: When is a compact topological 4-manifold a CW complex? How about if we just consider closed 4-manifolds, does that have an answer/make the ...
0
votes
0answers
10 views

Can anyone give the equation of the inverse of radial projection from a tetrahedron to sphere?

$(x,y,z) \mapsto \bigg(\frac{x}{\sqrt{x^2+y^2+z^2}},\frac{y}{\sqrt{x^2+y^2+z^2}},\frac{z}{\sqrt{x^2+y^2+z^2}} \bigg)$ This is the equation of the radial projection. I need the inverse of this ...
0
votes
0answers
8 views

Best system ? Without martingale

I wonder what is the best system, if the percentage of winning bets is 52%. Bets have two options 1 or 2 and is rounded to the average odds 1.80, and the maximum wrong bets  in a  row is 8. So, with ...
0
votes
0answers
7 views

A question about conic sets of functionals.

The problem is the following. Let $(X,\|\cdot\|)$ be a normed space and let $C \subseteq X$ be a closed convex set with nonempty interior. Let be $x\in C$. I define to be a "normal cone to $C$ in ...
2
votes
1answer
79 views

Evaluate the limit $\lim \limits_{x \to \infty} \frac{1}{x(x+1)}$

How can I evaluate the limit $$\lim_{x \to \infty} \frac{1}{x(x+1)}$$
0
votes
1answer
17 views

having trouble with a characteristic polynomial and minimal polynomial question

Let T : V → V be a linear operator. Suppose α = {v1, v2} is a basis for V, and T(v1) = v2, T(v2) = −v1. a. Find [T]αα.(one alpha lower and one alpha upper) b. Let β = {v1 + v2, v2} be another basis ...
1
vote
1answer
19 views

Is the colimit of finite tensor products a tensor product?

Let $(R_\lambda)_{\lambda\in\Lambda}$ be a family of $A$-algebras. Atiyah & MacDonald defines the "tensor product" of the family as the direct limit of the tensor product of finite subfamilies. ...
-1
votes
1answer
12 views

Determine fraction length of fixed-point binary

How to determine fraction length of fixed-point binary so that distinct entries of a group of decimal numbers (for example: 1, 0.456, 0.444) remain distinct after converting them from decimal to ...
3
votes
3answers
45 views

What is the value of $ \int_{x}^{1} \arcsin \left( \frac{2t}{t^2+1} \right) \text{d}t $?

Is this result true? Wolfram doesn't seem to be able to evaluate the definite integral in the allowed time. $$ \int_{x}^{1} \arcsin \left( \dfrac{2t}{t^2+1} \right) \text{d}t = \dfrac{\pi}{2} - ...
2
votes
2answers
32 views

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 ...
0
votes
0answers
15 views

Covering Space of $\mathbb{C}-\{a,b\}$ via Multivalued Function

Consider the multivalued complex function $f(z)= \sqrt{z-a}+\sqrt{z-b}$, where $a\neq b$, defined in the domain $U=\mathbb{C}-\{a,b\}$. The graph $W$ of $f$ defines a regular covering space $W ...
1
vote
0answers
14 views

Number of ways distribute 12 identical action figures to 5 children

Need a little help with this problem. Use generating functions to determine the number of different ways 12 identical action figures can be given to five children so that each child receives at most ...
0
votes
1answer
28 views

How to calculate the sum of combinatorial numbers

For my work on an almost completely unrelated field I came across the following formula. I know that I have learned that all in high school. But since this is more than 15 years ago in which I never ...
0
votes
0answers
7 views

What is the maximum magnitude of $L_G \circ X$ for $X$ PSD?

Let $G$ be an undirected graph, let $L_G$ its graph Laplacian $D_G - A_G$, and let $X \succeq 0$ be a positive semidefinite matrix. What is the maximum value of $L_G \circ X$? Is this any easier (or ...
1
vote
1answer
20 views

Minimum of a potential function

I'm looking for extremes (minimum) of $$V = \frac{\alpha}{|\vec{r}_1-\vec{r}_2|} + \beta (\vec{r}_1 + \vec{r}_2)\cdot \vec{e}_z$$ where $\vec{r}_i = ...
1
vote
5answers
55 views

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 $$\lim_{x \to \infty} \frac{2^x+1}{2^{x+1}}$$
0
votes
2answers
21 views

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 ...
2
votes
0answers
22 views

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 $[0;1]$. ...
0
votes
1answer
11 views

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,
0
votes
0answers
77 views

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 ...
2
votes
0answers
36 views

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 $S$? ...
0
votes
1answer
9 views

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 ...
1
vote
1answer
15 views

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 ...
2
votes
1answer
21 views

Measuring sums of complex alternating series

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}} - ...

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