0
votes
0answers
3 views

What series of 'hyperpolyhedrons' do exist? Is there an effective way to derive their cross-sections by 3-d subspace?

There are two obvious series of 'hyperpolyhedrons'. 'Hyperoctahedron' with vertices $(\pm1,0...0), (0,\pm1,0,...0)...(0,...0,\pm1)$ and each vertex connected by an edge with each other vertex ...
0
votes
0answers
7 views

Problem in Linear algebra?

Let $V=V_0\oplus V_1$ is a $Z_2$-graded inner product vector space and $T:V\longrightarrow V$ is an odd linear map i.e, $T(V_0)\subseteq V_1 \,\text{and} \, T(V_1)\subseteq V_0$. If $T$ be ...
2
votes
2answers
28 views

No. of maps satisfying ϕ(ab)=ϕ(a)+ϕ(b)

How to show that there are finitely many maps $\phi:\mathbb{N}\cup\{0\}\to \mathbb{N}\cup\{0\}$ with the property that $\phi(ab)=\phi(a)+\phi(b)$ for all $a,b\in \mathbb{N}\cup\{0\}$.
0
votes
0answers
9 views

Diagonalization of a matrix with change of basis

I was trying to diagonalize a not really nice matrix doing first a change of basis but I noticed that the two characteristic polynomials I get are different. Original matrix and its characteristic ...
0
votes
0answers
8 views

A direct limit of pullbacks

Let an $R$-module $C$ be a direct limit of finitely presented $R$-modules $C_i$, and we have a short exact sequence as follows: $$0→A↪B\stackrel{\pi}→C→0.\qquad (S)$$ From each $C_i$ to the direct ...
0
votes
0answers
10 views

Problem on product space of Sorgenfrey line.

Let $(\mathbb R,\tau)$ be Sorgenfrey line, $(\mathbb R^2,\tau_1):=(\mathbb R,\tau)\times (\mathbb R,\tau)$. Let $L = \{(x, y) : x, y\in\mathbb R^2, x + y = 0\}$. Show that the line L is closed in ...
0
votes
0answers
9 views

Value of an iterated sum

I am interested in the number of function evaluations required to numerically evaluate an iterated integral of the form $$ \int_0^t \int_{t_1}^t \cdots \int_{t_{n-1}}^t f(t_1,\ldots,t_n) dt_n\cdots ...
1
vote
0answers
12 views

Find some probabilities given the probability tree

i've been practicing probability since it's not my strength, but i am doing that without a tutor or an official course, just books and videos. I was reading a problem, and i was capable of draw the ...
1
vote
1answer
17 views

Isomorphism between $E$ and $Lin(E)$ : infinite dimensional case.

Does $E$ and $Lin(E)$ where $Lin(E)=\{A : E \rightarrow E ∣ A \quad\text{is linear}\}$ are isomorph if $E$ infinite dimensional case ? I know that if $E$ is finite dimension the result is true ...
1
vote
0answers
8 views

Affine to linear like conversion of a concave function

Is the following true: $$\log \left( \frac{1}{f(x)+K}\right)\mathrm{is\;concave}\Longleftrightarrow \log \left( \frac{1}{f(x)}\right)\mathrm{is\;concave},$$ where $K\in\mathbb{R} $ and ...
0
votes
0answers
20 views

What would be the way to learn algorithms?

I want to know the mathematical theory associated algorithms. Story with a base of calculation, analytical geometry and linear algebra. What would be the previous topics to begin the study of ...
0
votes
0answers
7 views

Is it possible to use multiple time scale algorithm here?

Suppose a random sequence is being generated (the next term generated depends on the previous term, but we don't know any distribution) until we hit some specific number. We want to calculate the ...
-1
votes
0answers
24 views

How can I find the area of the shadow? 3

how can i find the surface area of circular coin placed 's' distance from the source(light producing source) which produces a shadow 'd' distance from the surface.the circular coin is placed between ...
1
vote
1answer
13 views

Expecation of uniform distriution with unknown parameter, given maximal (minimal) observation.

Let $x_i \text{ be} ~ i.i.d. ~ \sim Uni[0,\theta]$ $(\theta \text{ unknown})$. Denote $M_n = \max x_i$. So, through circumferential means, I can show that $E(x_1|M_n) = \frac{n+1}{2n} M_n$. The ...
0
votes
0answers
6 views

Ross probability models questions

I am studying for a course and have no professors to talk to live, so I hope some members here can be kind enough to help me. Rather than writing everything out, and splitting it up into different ...
2
votes
1answer
18 views

Proof that a random variable has exponential distribution.

Supose that $X_1$ is a continuous and positive (real) random variable with exponential distribution, namely $$P(X_1>t)=e^{-\lambda t}\quad t>0$$ Now suppose that $X_2$ is another continuous and ...
2
votes
0answers
24 views

Construction of a triangle

I need to construct a triangle with given information: $c = 6$, $h = 4$ and $\alpha - \beta = 30º$. I'll put approximate result for any clarification.
0
votes
2answers
13 views

What is the value of the unknown parameter so that the given area condition holds?

The graphs of $f(x) \colon= x^2$ and $g(x) \colon= cx^3$, where $c > 0$, intersect at the points $(0,0)$ and $(1/c, 1/c^2)$. What is the value of $c$---and how to compute this value---so that the ...
2
votes
1answer
15 views

Is $C(\mathbb R) \subset H_{s}^{loc}$ (localized Sobolev space)?

We put, Sobolev space $$H_{s}=H_{s}(\mathbb R)=\{f\in L^{2}(\mathbb R):[\int_{\mathbb R} |\hat{f}(\xi)|^{2}(1+|\xi|^{2})^{s}d\xi]^{1/2}<\infty \}.$$ If $U$ is an open set in $\mathbb R,$ the ...
-1
votes
1answer
12 views

Mean line from several lines

i want to find the average line from different line segments. I don't know how to do this and also don't know how to descripe it specifically. So I'll give you an example: If a have two lines with the ...
1
vote
0answers
13 views

A hereditarily Lindelöf minimal KC-space is sequential.

A space in which all compact subsets are closed is called $KC$ space. A space in which every infinite set contains an infinite subset with only a finite number of accumulation points is said to have ...
2
votes
0answers
12 views

Inequalities with $\|x-y\|$, $|\langle x,y\rangle|$, and $\sqrt{\|x\|^{2}+\|y\|^{2}}$ in a Hilbert space

Let $H$ be a Hilbert space, and let $\|x\|$ denote the norm of $x\in H$, and $\langle x,y\rangle$ denote the inner product of $x,y\in H$. For $x,y\in H$ let us denote $\alpha(x,y)=\|x-y\|$, ...
2
votes
0answers
79 views

Proof on why $0-1+2-3+4-\ldots\neq-1/4$

When reviewing my notes on series' convergence, I thought of applying a workaround on why $\sum_{n=0}^{\infty}(-1)^nn$ should or shouldn't be $-1/4$ (I recalled this page). I started by considering ...
0
votes
0answers
7 views

How to draw a border at a specific distance from a cylinder outline

I have a small cylinder (Cylinder A) with its minor radius of A. Minor radius measures the minor radius of the cylinder ellipse. I need to draw a border at a distance of X from the border of ...
1
vote
2answers
30 views

Showing that $\sum\limits_{k=2}^n {k\choose2} = {{n+1}\choose 3}$ for integers $n\geq 2$

I'm trying to prove that $\sum\limits_{k=2}^n {k\choose2} = {{n+1}\choose 3}$ for integers $n\geq 2$. I figured induction was the way to go, so I tried. This is what I've accomplished so far: Proved ...
1
vote
3answers
34 views

Solve the length AB (the dashed line)

Can someone show me how I can solve this? (Step by step example with solution appreciated a lot as I am currently practicing). EDIT: After a closer look, it looks as if this is an Isosceles ...
2
votes
2answers
32 views

If $\sin^2 \theta + 2\cos \theta – 2 = 0$, then find the value of $\cos^3 \theta + \sec^3 \theta$

If $\sin^2 \theta + 2\cos \theta – 2 = 0$, then find the value of $\cos^3 \theta + \sec^3 \theta$.
1
vote
2answers
17 views

A question on the Gelfand Transform.

There is a theorem that proves that if $A$ is a commutative C*-Algebra, the Gelfand map is an isometric *-isomorphism of $A$ onto $\hat{A}$ i.e. the spectrum of $A$. (Theorem 1.1 in Averson's "An ...
0
votes
2answers
53 views

Is it correct? $n^{(\log\,x)} = x^ {(\log\,n)} $?

Is it correct? $$n^{(\log\,x)} = x^ {(\log\,n)} $$ Can you proof and describe that, for any base? Please explain completely. Thank you.
-2
votes
1answer
34 views

Possible Numbers formed? [on hold]

6-digit numbers formed using three 3's and three 4's?
2
votes
1answer
13 views

A CFG Grammar for One Language

Suppose : $w_1,w_2 \in \{a,b\}^∗$ and $ L=\{w_1w_2 \mid w_1,w_2 \in \{a,b\}^* \land n_a(w_1)=n_b(w_2)\}$ $n_a$ is number of $a$'s and $n_b$ is number of $b$'s. This is a Entrance Exam question. I ...
1
vote
0answers
11 views

Counting the operations of a problem

I have a square matrix $A\in\mathbb{R}^{n\times n}$, it has a LU decomposition. $L$ and $U$ are triangular and $L$ has ones on the main diagonal. I'm counting the number of operations for ...
0
votes
0answers
6 views

Understanding edge correction with a 2nd order polynomial in Gaussian filter

I am trying to understand the following code from ImageJ: http://pastebin.com/tXfhNxqf The problem: When computing the gaussian kernel we use the gaussian function $$ f(x) = e^{-\dfrac{x^2}{2 ...
-1
votes
0answers
11 views

Ordered field - Adding inequalities (Looking for a proof)

In an ordered field, it is allowed to add two inequalities. Where can I find a detailed proof?
2
votes
2answers
37 views

Continuity almost everywhere

Let $$f(x) = \left\{\begin{array}{ll} 1 & \text{if }x\in\mathbb{Q},\\ 0 & \text{if }x\notin\mathbb{Q}. \end{array}\right.$$ $$g(x) = \left\{\begin{array}{ll} \frac{1}{n} & \text{if }x = ...
1
vote
2answers
21 views

Linear Transformation $T_{A}$ Is invertible $\iff$ A Is invertible

Let $T_{A}$ be the linear mapping corresponding to the matrix A, and $A \in F^{n*n}$ $T_{A}$ Is invertible $\iff$ there is $T_{A}^{-1}$ so $T_{A} \circ T_{A}^{-1}=I $ $T_{A} \circ T_{A}^{-1}(v)=v$ ...
0
votes
1answer
14 views

Principal angle and Euler form of cube root of unity.

The principal angles of the complex arguments are in the range $-\pi \le x\le \pi $. If so why does the comlex number $\omega =\frac{\left(-1-\sqrt{3}i\right)}{2}$ is represented by ...
1
vote
0answers
11 views

When may the bar resolution be truncated for calculating $\operatorname{hocolim}$?

Suppose $D$ is a small category and $F\colon D\to sSets$ a functor. I want to calculate the simplicial set $$\operatorname{hocolim}F$$ via the bar construction: $\operatorname{hocolim}F$ is the ...
1
vote
0answers
10 views

Uniform convergence with respect to a parameter

So this is just a notational question. Assume one has a sequence $f_n\to f$ uniformly, where $f_n,f:X\to Y$ for some metric/Banach spaces $X,Y$. Now suppose that $f_n$ and $f$ depend on a parameter ...
0
votes
1answer
26 views

Simplifying the sum $\sum_{i=1}^n \sum_{j=i+1}^n({(x_i-x_j)}^2+{(y_i-y_j)}^2)$

I am trying to evaluate the sum here , $$\sum_{i=1}^n \sum_{j=i+1}^n({(x_i-x_j)}^2+{(y_i-y_j)}^2)$$ How do this sum can be simplified to $$n\sum_{i=1}^n({x_i}^2+{y_i}^2) - ...
0
votes
1answer
15 views

How to calculate lower & upper quartiles?

I'm sure this has been asked many times before but it's confusing me a lot so hopefully someone can help! I am given this data set: 0.28 0.30 0.42 0.59 0.71 1.67 1.82 2.39 4.71 4.79 4.89 5.00 5.00 ...
0
votes
0answers
3 views

Regression with rounded dependent variable

I am looking at the following model: $c$ is a fixed vector in $\mathbb{R}_+^n$ and for any $x \in \mathbb{R}_+^n$ we obtain a value $y =[c^Tx]$, i.e. rounding $c^Tx$ to the nearest integer. I want ...
2
votes
2answers
61 views

Nested Radicals with multiplication

I think this one goes to section of nested radicals, I was trying to solve if for a couple of days now. Maybe you have some nice solution to this one. $$\sqrt{1\sqrt{2\sqrt{3\sqrt{4\sqrt{...}}}}}$$ ...
1
vote
3answers
96 views

Determine the cardinal of the following infinite set?

Me and my friends were debating the following question in set theory, it has been some time since we studied this subject and we don't remember if there are rules that determine the answer for our ...
2
votes
1answer
63 views

Evaluation of $\sum n a^n$ using telescoping property

Show that the series $$\sum\limits_{n=1}^{\infty}n a^n = \frac{a}{(a-1)^2} $$ for $|a|<1$ using the telescoping property. I know how to do this using other methods. But the exercise ...
0
votes
1answer
37 views

Calculate $\lim_{n\to\infty} \sum_{k=1}^{n} \Big(\frac{k}{n}\Big)^{\alpha k}$

Let $\alpha$ be a positive number. Calculate $$\lim_{n\to\infty} \sum_{k=1}^{n} \Big(\frac{k}{n}\Big)^{\alpha k}$$ Edit: I have deleted my attempt, it didn't seem to lead me anywhere and I discovered ...
0
votes
1answer
15 views

Inequality involving products

One is given two intervals $I_{a-\epsilon,b+\epsilon}$, $I_{a,b}$ of $\mathbb{R}^n$, and is asked to show that $\lambda(I_{a-\epsilon,b+\epsilon}) - \lambda(I_{a,b}) \leq c\epsilon$ for some constant ...
0
votes
0answers
20 views

Prove a result on transitive group actions.

Let $G$ be a group and $A$ & $B$ be two sets s.t. $G$ acts transitively on each of $A$ & $B$. Choose some $\alpha$ and $\beta$ in $A$ & $B$ respectively then prove that if $G=G_\alpha ...
0
votes
0answers
12 views

In an abelian category,every morphism can be written as composition of epi and mono.

Following Weibel's book on homological algebra, he states without proof that every morphism $f\colon A \to B$ can be written as composition of an epimorphism followed by a monomorphism. After many ...
0
votes
2answers
44 views

Compute $0.y_1y_2y_1y_2y_1y_2…$

Any tips? I'm taking $$Sn = \frac{y_1}{10} + \frac{y_2}{100} +\frac{y_1}{1000} + $$ and then I find $$\frac{S_n}{100}=...$$ I do $$Sn - \frac{S_n}{100}$$ and go on from there but I can't find the ...

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