0
votes
0answers
3 views

Connected-ness of the boundary of convex sets in $\mathbb R^n$ , $n>1$ , under additional assumptions of the convex set being compact or bounded

Is the boundary of every compact convex set in $\mathbb R^n$ , ($n>1 $ ) connected ? is it path connected ? What if we assume only that the convex set is bounded , is the boundary connected ( and ...
0
votes
0answers
2 views

Birational equivalent and isomorphic represenation of a subalgebra

here I am again with another exercise which gives me a hard time. Let $A$ be the subalgebra of $\mathbb{C}[t]$ of all polynomials $f(t)$ such that $f(1) = f(-1)$. Let X be an alebraic set such ...
0
votes
0answers
3 views

Dynamic programming approach for multidimensional problem

I use a dynamic programming approach to optimize the behaviour of individuals playing a game.I have one strategy matrix that describes the behaviour of individuals in situation 1, which depends on ...
0
votes
0answers
9 views

sum of perpendicular distances from the sides of a triangle.

I am trying to solve a problem and got stuck in the following:- P, A’, C’ are respectively points on the sides AC, CB, and AB of ⊿ABC. PA’ and PC’ are the perpendiculars to the sides of the ...
0
votes
2answers
31 views

Prove inequality $ab+bc+ca\ge 3,\ abc=1$

How can I prove \begin{equation*} ab+bc+ca\ge 3,~a,b,c \in\mathbb{R},~ a,b,c>0\ \end{equation*} and the product $abc=1$? I obtained only $(a+b+c)^2-(a^2+b^2+c^2)\ge6\ and \ ...
0
votes
1answer
19 views

Is there any demonstration for this Claim and what are it's application in mathematics?

I would like someone to proof me this claim and give me it's applications in mathematics if it's not a convention . claim :for all positive integer $n$ ,The ring $\frac{\mathbb{z}}{n\mathbb{z}}$ ...
1
vote
0answers
9 views

Lie groups in the mandelbrot set?

I was doing some reasearch on mandelbrot sets and accidently ran into a parabolic geometry section on wiki. I've never studied it but I saw a picture of anan "E8" group which looks almost exactly like ...
0
votes
0answers
11 views

polytope with 12 vertices and 48 edges

It seems like you can construct a polytope with 12 vertices, where each vertex connects to all the other vertices except 3. So there must be a totalt of 48 edges. (each of the 12 vertcies connects to ...
0
votes
0answers
17 views

Length of closed curve

How to find length of this closed curve? I dont know what limits should i take for the integral.
0
votes
0answers
6 views

About convergence in norm of the Fourier Transform

Duoandikoetxea's Fourier Analysis, on page 59 (Corollary 3.7) says that: \begin{equation} lim_{R \rightarrow \infty}||S_{R}f - f||_{p} = 0 \end{equation} for $1<p<\infty$, where $S_{R}f$ is ...
0
votes
1answer
20 views

Confusion with Summations

I am having a little bit of confusion regarding summations. I know that $$\sum_{i=m}^n a_i = a_{m}+a_{m+1}+\cdots +a_{n-1}+a_n$$ Here is my confusion. How do we interpret/decompose the following: ...
0
votes
0answers
3 views

Fourier transform square summable?

It is known that for a $L^1$ function $f: \mathbb{R} \rightarrow \mathbb{C}$ the Fourier transform vanishes at infinity and is continuous. Does this even mean that $(\hat{f}(n))_{n \in \mathbb{Z}}$ is ...
0
votes
0answers
3 views

Geodesic equation applied to halfplane model

I have learned some things regarding connections and geodesic. And I want to apply this knowledge to the exercise: show that the vertical lines in the halfplane model are geodesics. The metric is ...
1
vote
0answers
10 views

Looking for a “Guide for the Perplexed by Low-dimensional Topology”

The following excerpt is from pp. 56-57 of Loring Tu's (so far very enjoyable) textbook An Introduction to Manifolds (2nd ed.): One of the most surprising achievements in topology was John ...
0
votes
0answers
4 views

Decomposing the ring of integers of a number field as a sum of prime plus field fixed by ramification group

Let $L/K$ be a number field extension with ring of integers $\mathcal O_L/\mathcal O_K$. Let $Q$ be a prime ideal of $\mathcal O_L$ and ramification group $E = \{g \in ...
-2
votes
0answers
18 views

Im stuck on (b)

This is about changing fractions into a mixed expression. So I have to do divide them. But I don't know why (b) has to leave spaces
4
votes
2answers
20 views

Is it possible to have a sphere $S^m$ equidistant to sphere $S^n$ in $R^k$?

Is it possible to place a sphere $S^m$ and another sphere $S^n$ in Euclidean $k$-dimensional space $R^k$ in such a way that the distance from any point of the first sphere to any point of the second ...
0
votes
0answers
18 views

How to simplify the kronecker product of four product

Suppose that $A$ and $B$ are $N\times N$ matrices, and $I$ is a $m\times m$ identity matrix, then here comes the kronecker product $$ K_1 = (I\otimes A)\otimes(I\otimes B). $$ I now wonder how can we ...
1
vote
2answers
24 views

Is $(x^2,xy)$ a primary ideal in $k[x,y]$ for $k$ a field?

In Example of Page 52 in Atiyah's Introduction to Commutative Algebra $\mathfrak a = (x^2,xy)$ is not a primary ideal in $A = k[x,y]$ where $k$ is a field. I think, for any $z \in \mathfrak a$, ...
0
votes
0answers
10 views

Lines and planes-recursive formula

A family of $n$ lines is drawn in the plane such that each pair of lines cross and no $3$ dinstinct lines have a point in common Let $r(n)$ denote the number of regions into ...
0
votes
0answers
30 views

Euler's Equation [duplicate]

I'm new around here, I'm fourteen, and I am in Ninth Grade. Can somebody tell me what Euler's equation exactly is, and why it's important, and what we can use it for? The whole $e^{i\pi} = -1 $ thing. ...
0
votes
1answer
18 views

Solving second order nonhomogeneous linear equation

So i have the equation $$\frac{d^2y}{dt^2} + y = \sin(t)$$ I know the first step is to find the corresponding homogeneous equation, which i think would be: $$r^2+1=0$$ giving real roots and therefore ...
6
votes
1answer
38 views

which functions can be obtained as a composition of a continuous function with itself?

let $f(x)=x^2$, then $f(f(x))=x^4$, so $x^4$ is a continuous function from $\Bbb R$ to $\Bbb R$ which can be obtained as $f\circ f$ for a continuous $f\colon \Bbb R\to \Bbb R$. general example: for ...
2
votes
4answers
24 views

$A\backslash (B\cap C) = (A\backslash B)\cup (A\backslash C)$; only one inclusion seems to work

I encountered the following problem: $$A\backslash (B \cap C) = (A\backslash B)\cup(A\backslash C).$$ So I need to prove two things: $A\backslash (B\cap C) \subseteq (A\backslash B) \cup ...
2
votes
0answers
8 views

Prove that actions commute

I am trying to understand a proof from Kobayashi and Nomizu (foundations of differential geometry, p. 280). Suppose that we have Lie subalgebras $a<b<g$, with $g$ the Lie subalgebra of $SO(n)$ ...
0
votes
1answer
14 views

Integration limits of a Marginal Probability Density Function with a Triangle-Shaped Boundary

I have given a triangle shaped boundary $M$ of my probability density function in $\mathrm{R}^{2}$, with the limitations beeing: $$y = 0$$ $$y = x$$ $$y = 2-x$$ and a probability density function $$ ...
1
vote
1answer
27 views

suppose $f(x)$ is an entire function and everywhere $|f'(z)| \leq |z^2+1|$ and further $f(0) = f'(0) = 1$. Determine $f$

suppose $f(x)$ is an entire function and everywhere $|f'(z)| \leq |z^2+1|$ and further $f(0) = f'(0) = 1$. Determine $f$ I tried using liousville but i don't know if $f'(x)$ is an entire function, i ...
1
vote
3answers
15 views

Electrical Engineering (complex numbers)

Electrical Engineering ($j=i=\sqrt{-1}$): $$H_v(\omega)=\frac{R}{R+\frac{1}{j\omega C}}=\frac{j\omega CR}{1+Rj\omega C}$$ And we know that: $\omega_0=\frac{1}{RC}\Longleftrightarrow ...
-1
votes
0answers
14 views

Exponential problem for phase calculation. Find periodic t

Given $e^{2At{\pi}i} = - e^{2{\pi}(A-149)ti}\text{, where }A = 42.58\cdot10^6.$ Find periodic $t$.
1
vote
1answer
29 views

If $R[X]$ is ED then $R[X] $ is PID

Is this true and why. If $R[X]$ is ED then $R[X] $ is PID . Thanks for help.
1
vote
2answers
19 views

How many ways can you choose $4$ teams of $2$ from $8$ people.

How many ways can you choose $4$ teams of $2$ from $8$ people. My thoughts were that you have $8$ slots to be filled so you have $8!$ ways to arrange them but this overcounts by a factor of $2$ since ...
1
vote
0answers
23 views

Caccipoli Inequality

I am reading the book Elliptic Partial Differential Equation by Fanghua Lin and I got stuck at the lemma 1.36 ( the Caccipoli inequality). The conditions of this lemma are: $u\in C^1(B_1)$ ($B_1$ is ...
0
votes
0answers
18 views

Find the density of a ratio of random variables

$X$ has density $2x, 0 < x < 1,$ and $Y$ has density $1/10$ over $0 < y < 10$. $X$ and $Y$ are independent. I have to find (a) density of $Y/X$ (b) $E[Y/X]$ (c) $E[Y^2/X]$ I let $Z=Y/X,$ ...
0
votes
1answer
24 views

Exact functors preserve free modules?

Let $R$ be a principl ideal domain. Suppose that $F$ be an exact functor from the category of $R$-modules to itself. If $M$ is a free $R$-module, is $F(M)$ still a free $R$-module? I do not know how ...
0
votes
0answers
13 views

tensor product of polynomial algebra

Is $R[x] \otimes R[x]$ a free $R \otimes R$-module? Here $R$ is a $k$-algebra and $\otimes = \otimes_k$.
2
votes
0answers
13 views

Order of prime ideals over split primes in the class group.

Let $P$ be a prime ideal of $\mathcal O_K$ ($K$ a quadratic field) and let $P$ have norm $p$ where $p$ is a split prime. Is it possible for the ideal class $[P]$ to have order less than three? I feel ...
1
vote
0answers
7 views

Heat Transfer FEM 2D PDE Matlab

I'd love to know if this looks right in any way since I'm unfamiliar with Heattransfer. The Domain is correct. The heat transfer coefficient is 1. The Dirichlet BC is u(0,x) = 1 and u(y,1) = 0 ...
0
votes
2answers
18 views

finding the ratio when two other ratios are given

The ratio of incomes of a and b is 3:4 and ratio of their expenditure is 4:5 . what can be their possible ratio of savings 9:10 or 3:4 or 4:5 or 13:20. I don't find anything suitable in ...
0
votes
1answer
24 views

metric space: equivalence of several mertric.

I have two questions: Q1) Are all metric on a metric space are equivalent ? Q2) If not: Let $d_1,d_2$ two metric on $X$. If something has a property with a $d_1$ will it hold for $d_2$ too ? For ...
1
vote
1answer
22 views

prove or disprove Composition of linear transformations is one-one

Let $T$ and $F$ be 2 linear transformations from $\Bbb R^n \to \Bbb R^n $.Then prove or disprove $T \circ F=0 \to T$ is one-one. $|TF|$$=0$ $\implies$ $|T|$$|F|$$=0$ $\implies$ either $|T|$=$0$ ...
0
votes
1answer
55 views

Tricky question about binomial expansions.

State the binomial expansion of $(1+x)^n$ So I can do this $$(1+x)^n=\sum_{i=0}^{n} {n\choose i}x^i$$ Then given $n=2k$ is even. Derive an expression for $$\sum_{i=0}^{2k} (-1)^i{2k\choose i}$$ ...
-1
votes
0answers
6 views

How to find curvature between two lines in each four quadrants?

everyone plz help me by providing formula to calculate curvature of two lines in each quadrant. as this line in picture it could be in different direction too. so i have to calculate curve in every ...
1
vote
0answers
11 views

Functional Minimization of Exponential Decay

I would like to find a function $f$ that minimizes the functional: $$\ln(f(x))f(x)-\frac1x$$ over some range of $x > 0$. Is this a good application for functional calculus and the Euler-Lagrange ...
4
votes
3answers
46 views

Multiplicative Inverse Element in $\mathbb{Q}[\sqrt[3]{2}]$

So elements of this ring look like $$a+b\sqrt[3]{2}+c\sqrt[3]{4}$$ If I want to find the multiplicative inverse element for the above general element, then i'm trying to find $x,y,z\in\mathbb{Q}$ such ...
0
votes
1answer
5 views

finding the overall loss or profit percentage

A person sells two shirts for 880 each . he gets a 10% profit on one whereas 20% loss on the other. Find the overall profit or loss percentage I didn't understand how to find the ...
-2
votes
0answers
9 views

maple allvalues function?

I have two simultaneous equations that I have solved (for two parameters) using Maple. The solutions are themselves complicated expressions of yet another parameter. I had to use the ``allvalues'' ...
1
vote
2answers
42 views

Find two matrices $A$ and $B$ such that matrix $AB$ that is invertible but $BA$ is not.

I am trying to find two matrices $A$ and $B$ such that matrix $AB$ that is invertible but $BA$ is not. Have you got any ideas of easy examples? Thank you!
0
votes
0answers
25 views

The elliptic curve $y^2 = x^3 + 2015x - 2015$ over $\mathbb{Q}$

Consider the elliptic curve \begin{equation*} E: y^2 = x^3 + 2015x - 2015~\text{over}~\mathbb{Q}. \end{equation*} I want to prove that $|E(\mathbb{F}_7)| = 12$, that $|E(\mathbb{F}_{19})| = 19$ and ...
0
votes
1answer
10 views

Escape time for a not absorbing state

Let $X$ be a right-continuous Feller Dynkin process. For $r>0$ we define the $\{\mathcal{F}_t\}_t$ stopping time (which is called escape time) $$\eta_r=\inf\{t\geq 0: \|X_t -X_0\|\geq r\}$$ We have ...
0
votes
0answers
7 views

Half space representation of a convex polytope

We know that the half space representation of a polytope is given by: $Ax<b$. Consider a convex polytope in $\mathbb{R}^3_+$ with vertices given by the following set of points: ...

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