0
votes
1answer
34 views

Integral Help!! Difficult Problem.

In $(x, y, z)$ space is considered the vector field $V(x,y,z)=(y^2 z, yx^2, ye^y)$ solid spatial region $Ω$ is given by the parameterization: $\left[ \begin{matrix} x \\ y \\ z \end{matrix} \right] ...
0
votes
0answers
9 views

Is there an atlas of Kirby diagrams of 4-manifolds?

We've defined an invariant of 4-manifolds (article) in terms of Kirby diagrams and I'm looking for a lot of manifolds to test it on. Now I'm not a big differential topologist myself, and coming up ...
0
votes
0answers
10 views

How to count the latin squares of order 4

a Latin square is an $n × n$ array filled with n different symbols, each occurring exactly once in each row and exactly once in each column. So, Assume that an integer like $4$ is given. How many ...
-2
votes
0answers
28 views

map extension in linear transformations

Let $V$ be a finite dimensional vector space over a field $F$ and $W$ be a subspace of $V$ If $T:W \to U$ is a linear map for some vector space $U$ over $F$ then how to prove there exists a ...
0
votes
0answers
13 views

A analytic function has a pole if $n ≤ |f(1/n)| ≤ n^{(3/2)}$ and $z^2 f(z)$ bounded

Let $f : \{z ∈ \mathbb{C} | 0 < |z| < 1 \} \to \mathbb{C}$ be analytic such that $n ≤ |f(1/n)| ≤ n^{(3/2)}$ for $n = 2, 3, . . ..$ Assume that $z^2f(z)$ bounded in $|z| < 1.$ Then I need ...
0
votes
1answer
18 views

Theorem : Let $x,y$ be words form $\mathbb{F}_{q}^{n}$. Then $d(x,y)=wt(x-y)$. In particular, $d(x,0)=wt(x)$ [on hold]

Error Correcting code prove : Theorem : Let $x,y$ be words form $\mathbb{F}_{q}^{n}$. Then $d(x,y)=wt(x-y)$. In particular, $d(x,0)=wt(x)$ This definition might be helpful to prove the theorem ...
0
votes
0answers
8 views

Non linear model, logistic regression exercise

Let $y_i$ follows $Bin(n_i,p_i)$ and for $p_i$ we consider the logit quadratic model: $\log\frac{p_i}{1-p_i}=\beta_0+\beta_1A_i+\beta_2(A_i-meanA)^2$ where $A_i$ is AGE_i during ith time. As you can ...
2
votes
2answers
277 views

convolution of compactly supported continuous function with schwartz class function is again a Schwartz class function?

Suppose $f$ is continuous function on $\mathbb R$ with compact support; and $g\in \mathcal{S}(\mathbb R),$ (Schwartz space) My Question is: Can we expect $f\ast g \in \mathcal{S(\mathbb R)}$ ? ...
0
votes
0answers
8 views

Prove that $G(n,m)\rightarrow \gamma (G)\leq n$.

In my Graph Theory subject, my professor give me this problem without other condition :Prove:$G(n,m)\rightarrow \gamma (G)\leq n$. Any HELP is very much appreciated. Thank you so much.
0
votes
1answer
15 views

linear transformations of same form

Let $m$ and $n$ be positive integers and $\mathbb{F}$ be a field . Let $f_1 , . . . , f_n$ be linear functionals on $\mathbb{F}^n$ . For any element $a$ in $\mathbb{F}^n$ ,define : $$T(a) = ( f_1(a) ...
1
vote
2answers
18 views

Is $f$ injective and surjective?

Let $f:\mathbb{N}\to \mathbb{Z}$ given by $\displaystyle f(a)=\frac{(-1)^a(2a-1)+1}{4}$. Is $f$ injective and surjective? I am having trouble bringing down the $a$ from the exponent. For injective, ...
1
vote
0answers
15 views

amusing applications of greens theorem

which application do you find most amusing and essential to the appreciation of Greens theorem suitable for a third semester calculus student ?
7
votes
2answers
173 views

Extend a function by convolution

Let $f \in \mathcal{C}^{\infty}(\mathbb{R})$ be a compactly supported function ($supp(f)\Subset\mathbb{R})$. I am wondering about the existence of a $g \in L^p(\mathbb{R})$, for some $p$, such that ...
2
votes
2answers
408 views

Is this notation for the set of limit points a standard notation?

Well, this doubt is probably silly. We have a standard notation for closure of a set $E$, we denote it $\bar{E}$ or $\operatorname{cl}{E}$ and we have a notation for the interior of a set $E$ we ...
-2
votes
1answer
66 views

can $P(\omega)$ be superatomic? [on hold]

A boolean algebra is superatomic if its every subalgebra has an atom. I´m trying to determine whether $P(\omega)$ can be superatomic.
1
vote
1answer
41 views

Cantor-set, product topology

We provide $\{0,1\}$ with the discrete topology and view the product space $X:=\{0,1\}^{\mathbb{N}}:=\prod_{i\in\mathbb{N}}\{0,1\}$ (the space of every 0-1-sequence), with $0\notin\mathbb{N}$ . Show ...
2
votes
2answers
24 views

Question about the constraint in Laplacian eigenmaps

When calculating Laplacian Eigenmaps, the original paper mentions about the constraint $$y^TDy=1$$ as "removes an arbitrary scaling factor in the embedding". My understanding is that it prevents $y$ ...
0
votes
1answer
37 views

Every morning the lecturer chooses pairs of students

There are 10 students in a class 7 males and 3 female. Every morning the lecturer chooses pairs of students in a random. X - numbets of teams, including a man and a woman (together) I thought ...
1
vote
1answer
27 views

$Tf = xf(x)$ is not compact in $L^2([0,1])$

I want to prove, in a rather elementary way, that $Tf = xf(x)$ is not compact in $L^2([0,1])$. I cannot find the appropriate bounded sequence whose image has no Cauchy sub-sequences. I have tried ...
0
votes
1answer
25 views

Convert ODE to polar coordinates.

$$k \frac{d}{dx}[A(x)\frac{dT(x)}{dx}] - hP(x)[T(x) - T] = 0 $$ What I had in mind was: $$x = rcosϴ, r = \frac{x}{cosϴ} , \frac{dr}{dx} = \frac{1}{cosϴ} $$ $$\frac{dA(x)}{dx} = ...
5
votes
0answers
40 views

Which prime factors of $8^{8^8}+1$ are known?

We have the partial factorization $$8^{8^8}+1=(2^{2^{24}}+1)\cdot (2^{2^{25}}-2^{2^{24}}+1)$$ The first factor is $F_{24}$. It is composite, but no prime factor is known. A prime factor of the ...
-1
votes
2answers
42 views

Find the mathematical expectation [on hold]

Find the expectation of $$f(x) = a(1+x)^{-(1+a)}, \quad x>0.$$ The answer given is $\frac{1}{a-1}$. I am not getting the answer. Please help.
1
vote
3answers
58 views

Combining error terms in Simpson's rule

My numerical analysis textbook (Burden and Faires) derives Simpson's rule as $$\begin{align} \int_{x_0}^{x_2}f(x)\,dx&=2hf(x_1)+\frac{h^3}{3}f''(x_1)+\frac{h^5}{60}f^{(4)}(\xi_1) ...
89
votes
25answers
8k views

Is there any integral for the Golden Ratio?

This is a curiosity. I was wondering about math important/famous constants, like $e$, $\pi$, $\gamma$ and obviously $\phi$. The first three ones are really well known, and there are lots of integrals ...
0
votes
2answers
16 views

A commutative unital ring is a field iff its only ideals are $0$ and $R$

A commutative ring $R$ with unity is a field if and only if its ideals are $0$ and $R$. How can I prove it?
4
votes
2answers
1k views

What will be the support of the convolution of two test functions.

If $g\in C^{\infty}_c$ defined on $\Bbb R^n$ and K is the support of function $g$. I want to find the support of $g_\epsilon$. Where $g_\epsilon$ is regularization of $g$. Regularization of $g$ is ...
0
votes
0answers
7 views

What is a metric to determine if a set of points were sampled from a curve?

Suppose we have a curve with its equation given as a spline. We also have a set of ordered $(x,y)$ coordinates. Is there a metric that would indicate whether the points were sampled from the spline? ...
0
votes
1answer
15 views

Multiple sequences of random variables that converge in probabilty

I'm struggling with this exercise: For each $k\in \mathbb{N}$, let $(X^{(k)}_n)_{n\in\mathbb{N}}$ be a sequence of real random variables converging to $0$ in probabilty as $n\to\infty$. Define for ...
0
votes
1answer
527 views

What is the fastest algorithm to calculate percentage change between two doubles

I would like to know what is the best solution to find the percentage change of a double compared to another one. For example, I have a variable called 'current', I want to compare it to a variable ...
1
vote
2answers
25 views

Convolution of two piecewise functions

$$\phi(t)=\begin{cases}1&t\in[0,1)\\0&\text{otherwise}\end{cases}$$ and $$\psi(t)=\begin{cases}1&t\in[0,1/2)\\-1&t\in[1/2,1)\\0&\text{otherwise}\end{cases}$$ I know that ...
0
votes
1answer
26 views

How can I tell if a matrix transformation is injective/surjective?

Determine whether or not $\mathbf v_1=(-2,0,0,2)$ or $\mathbf v_2=(-2,2,2,0)$ is in the kernel of the linear transformation $T:\mathbb R^4\to\mathbb R^3$ given by $T(\mathbf x)=A\mathbf x$ where ...
2
votes
1answer
31 views

Is there a book that teaches proofs from simple to intermediate level?

I am looking for a book that teaches proofs and the book has many exercises from very simple to more difficult? I have noticed with most math books, they seem to leave out pieces too soon before the ...
0
votes
1answer
33 views

How to do complicated problem without messy mind?

I have trouble when doing complicated problem, when I look at a problem with so much information. (e.g. deal with some concrete example such as proving a 'ugly' space is homeomorphic to another 'ugly' ...
0
votes
0answers
14 views

Power function exponential distribution

I am trying to find the power function for a test. I know that the power function is calculated by $\beta(0) = P_0(x \in R)$ where $R$ is the rejection region. What I know about this test is that $X ...
0
votes
4answers
24 views

Unseen Problem based on area of triangle

In $\triangle ABC$, $BD=2CD$ and $AE=ED$, prove that $6\triangle ACE=\triangle ABC$ If $A,X$ is joined such that $X$ is the mid point of $BC$ then: $\triangle ABX=\triangle AXC$ Also, $\triangle ...
1
vote
1answer
15 views

Complex Function Identities

Newcomer to Complex Analysis, I can't see any reason why these identities wouldn't hold, if taking multi valued log and exp the whole time. Am I correct?
0
votes
2answers
291 views

Function as a convolution product of other two

I need help with this: I have to prove that a function $f\in L_{2}(T)$ can be expressed as $f=g*h$ (convolution product) for some functions $g,h\in L_{2}(T)$ if and only if $(\hat{f}(n))_{n}\in ...
0
votes
0answers
21 views
+50

Reference for zero sum problems?

I am looking for books/ references which deal with the analysis of zero sum problems and weighted zero sum problems. I have found some articles on the internet, but they seem insufficient. Any ...
0
votes
0answers
13 views

Proof of the fixed point theorem in group action.

Fixed point theorem: Let $G$ be a $p$-group and let $S$ be a finite set on which $G$ operates. If the order of $S$ is not divisible by $p$, there is a fixed point for the operation of $G$ on $S$. ...
3
votes
0answers
30 views
+50

Show that the convex neighborhood in a Riemannian Manifold are subset contractibles

Show that the convex neighborhood in a Riemannian Manifold are subset contractibles (to any of their points). A subset $A$ of the differential manifold $M$ is contractible to the point $a\in A$ ...
0
votes
1answer
17 views

how to calculate the limit as $\lim_{s \to 0 }$ of this large equation?

I am having a hard time calculating this limit: $$\lim_{s \to 0 } \frac{-R_{4}}{R_{3}}\frac{sC_{2}\frac{R_{3}}{R_{3}+R_{4}}\frac{R_{5}+R_{6}}{R_{6}} } {\frac{s^2C_{1}C_{2}R_{1}R_{2}R_{5}}{R_{6}} ...
9
votes
5answers
79 views

Find the limit of $\frac{(n+1)^\sqrt{n+1}}{n^\sqrt{n}}$.

Find $$\lim_{n\to \infty}\frac{(n+1)^\sqrt{n+1}}{n^\sqrt{n}}$$ First I tried by taking $\ln y_n=\ln \frac{(n+1)^\sqrt{n+1}}{n^\sqrt{n}}=\sqrt{n+1}\ln(n+1)-\sqrt{n}\ln(n)$ which dose not seems to ...
2
votes
1answer
35 views

Defining bijections between sets

I am having troubles understanding how to properly define bijections between the sets say $X,Y,Z$ to show that 1) $(Z^X)^Y \cong Z^{(X\times Y)}$ In my notes it says that I can map, $F:X\times Y ...
0
votes
0answers
6 views

Bernstein polynomial in Banach

For a continuous function $f : [0,1] \to R$, there exists a sequence of polynomial functions: $P_n(x)=\sum_{k=0}^n C^k_n x^k(1-x)^{n-k} f(\frac{k}{n})$ (Bernstein's polynomes) which converges ...
3
votes
4answers
60 views

Show convexity of a function via inequalities

I am stuck with deriving the convexity of the function $$ f(x) = \sqrt{1 + x^2} $$ from first principles, that is I would like to show that for any $x,y \in \mathbb R$ and $\lambda \in (0,1)$ we ...
0
votes
0answers
6 views

Help understanding the proof of Trace Theorem given in Evans

I need help to understand the proof of the Trace Theorem given in Evans L.C. Partial differential equations (AMS, 1997): Asume $U$ is a bounded open set and that $\partial U$ is $C^1$. Then there ...
2
votes
1answer
61 views

Prove this inequality with $a+b+c=3$

Let $a,b,c>0$,and $a+b+c=3$,show that $$\dfrac{a}{2b^3+c}+\dfrac{b}{2c^3+a}+\dfrac{c}{2a^3+b}\ge 1$$ such Use Cauchy-Schwarz inequality we have ...
-2
votes
1answer
49 views

Do the $2$ modulus $3$ can be $-1$ or just $2$?

I need to calculate $2$ modulus $3$ as $2<3$ then the answer should be $2$ but instead in a math problem they use it as $-1$. Is this possible? thanks
0
votes
0answers
16 views

What is an upper bound for $\|E(X|\mathcal{A})-E(X)\|$?

Let $X$ be a random element in a Banach space with norm $\|\cdot\|$, and $\mathcal{A}$ be a $\sigma$-algebra. What is an upper bound for $\|E(X|\mathcal{A})-E(X)\|$? Existing results: It has been ...
0
votes
0answers
12 views

Proof: centers of circle tangent to circle and line lies on parabola

first please take a look at this: Given was a circle $c$ with center $A$ and ratio $r$, furthermore three lines $g$, $g1$, $g2$ with: $r = d(g, g1) = d(g, g2)$. Finally, two parabolas $p1$ (and ...

15 30 50 per page