All Questions
0
votes
1answer
25 views
How to apply the second fundamental theorem to an itegration with respect to a different variable?
I am trying to figure out differentiate ( ∫p(t)q(s)ds from t0 to t) with respect to t.
I have tried to use the second fundamental theorem of calculus but get blocked.
How to apply the second ...
0
votes
2answers
26 views
Determine the numbers $n$ that are orders of elements of $\mathbb{Z}^3 / H$
Let $G=\mathbb{Z}^3/H$, where $H$ is a subgroup that has been generated by $(2,0,2), (6,6,6) $ and $(8,36,38)$.
How can I solve this problem? I don't know where to start.
A related question would ...
1
vote
3answers
49 views
Which function gets larger?
I am given two functions $f(x)$ and $g(x)$. I am supposed to figure out which function is eventually greater. I found $$\lim_{x\to\infty}\frac{f(x)}{g(x)} = \infty$$ that means the functions ...
0
votes
1answer
31 views
Dimension of $\text{Hom}(U,V)$
I read this question and do not understand: Dimension of Hom(U, V)
However, my question is more primitive - so regardless the complex discussion related to the two paper in the question and accepted ...
0
votes
0answers
19 views
What are the necessary conditions on a function $g$ to get that the function $f(g(s))$ has also infinitely many zeros?
Let $f$ be an analytic function with infinitely many zeros. Then:
Show that the function $f(s-1)$ has also infinitely many zeros. Generally, what are the necessary conditions on a function $g$ to ...
0
votes
2answers
23 views
How many elements are in the conjugacy class of $\tau \in S_9$?
Just one simple question:
Let $\tau =(56789)(3456)(234)(12)$.
How many elements does the conjugacy class of $\tau$ contain? How do you solve this exersie?
First step is to write it in disjunct ...
2
votes
1answer
50 views
Number of distinct points in $A$ is uncountable [duplicate]
How can one show:
Let $X$ be a metric space and $A$ is subset of $X$ be a connected set with
at least two distinct points then the number of distinct points in $A$
is uncountable.
0
votes
0answers
16 views
What is difference between Finite Different Method, Finite Element Method and Finite Volume Method for PDE?
Can you help me explain the basic difference between FDM, FEM and FVM?
What is the best and why?
Advantage and disadvantage of them?
Thank you so much.
Thi from Vietnam.
1
vote
3answers
48 views
$E$ is closed $\iff\partial E$ (boundary of set $E$) $\subseteq E$
I am studying topology of euclidean space from William Wade's text book.
I saw the question. But I cannot produce any idea. Please show me the solution way instructively and clearly. Thank you for ...
0
votes
1answer
17 views
How to determine if $G$ contains an element of order $k$?
I'm struggling with this kind of problem:
Given a group $G= (\mathbb{Z}/n\mathbb{Z})^*$ (which is the multiplicative modulo group), determine if the group contains an element of order $k$. What is ...
0
votes
1answer
20 views
Archimedean spiral: arc length of coil
Are the arc lengths of the coils - i.e. the parts 0-2pi, 2pi-4pi, etc. - in arithmetic progression?
0
votes
0answers
18 views
$f$ is holomorphic in $\Omega$ such taht $|f|$ is harmonic we need to show $f$ is constant.
$f$ is holomorphic in $\Omega$ such taht $|f|$ is harmonic we need to show $f$ is constant.
let $f=u(x,y)+iv(x,y)\Rightarrow |f|=\sqrt{u^2+v^2}$ and $\nabla^2|f|=0$ right? also I have ...
0
votes
1answer
44 views
show that f(x)=0 has no positive solution if f(0)=0
Let $f$ be twice differentiable function on $\mathbb R$. Given that $f''(x)>0,\; \forall x \in \mathbb R$, how to show that $f(x)=0$ has no positive solution if $f(0)=0$ and $f'(x)>0$?
1
vote
2answers
52 views
Easy way to compute the area between $f(x)=x$ and $g(x)=x^2\ln(x)$
Is there an easy to compute the area between $f(x)=x$ and $g(x)=x^2\ln(x)$ without refering to the Lambert W-function?
0
votes
0answers
9 views
Any reference to an algorithm for finding the largest empty circle on a sphere (with great-circle distance)?
Given a set $S$ of 2D points in the plane there are known algorithms for finding the largest empty circle ($LEC$) of the set of points.
The $LEC$ problem is stated in this way: find a $LEC$ whose ...
0
votes
0answers
21 views
closed-form solution for 1/tanh(x) - 1/x that can be evaluated at/near x=0?
I'm looking to evaluate $\frac{1}{\tanh x}-\frac{1}{x}$ over a range that includes x=0. Is there an alternate form that is both exact, and numerically stable at/near x=0? For now I'm using the Taylor ...
1
vote
1answer
19 views
$n$-th root of a compact self-adjoint operator
Let $ A\colon H\to H $ be a compact self-adjoint operator. Suppose $A$ is positive. Let $ n \geq 2 $. Prove that there is $B \colon H\to H $ bounded such $B^n = A$.
4
votes
1answer
48 views
List of explicit enumerations of rational numbers
A well-known mathematical fact is that the rational numbers are countable, i.e. there is a bijective function
$$f:\mathbb{N}\rightarrow \mathbb{Q}$$
I am interesting in making a list of all explicit ...
1
vote
0answers
19 views
Jacobian matrix in Stone-Weierstrass theorem
Let $f : \overline{\Omega} \to \mathbb{R}$ be a $C^1$ real-valued function where $\Omega \subset \mathbb{R}^n$ is a bounded open set and let $J_f$ be the Jacobian determinant of $f$. According to ...
1
vote
0answers
24 views
Is the product of non-separated schemes non-separated?
My question is the title, but let me be more specific: for schemes $X$ and $Y$ over $S$, with at least one non-separated over $S$, is it true that the fibered product $X\times_S Y$ is also not ...
2
votes
1answer
40 views
$M$ is compact iff $M$ is homeomorphic to a closed subset of $H^{\infty}$
(a) Let $M$ be a metric space. If there exists a countable subset $X$ of $M$ such that $\overline{X}=M$, $M$ is said to be separable. Prove that a compact metric space is separable.
(b) Let ...
0
votes
1answer
17 views
Coordinate-free definition of pseudotensors
How to define pseudotensors (particularly, pseudovectors) in a coordinate-free form? Can it be defined on a manifold (like a tensor field)?
Or may be the objects that physicists model via ...
2
votes
1answer
30 views
Prove: If in all subgraphs of $G$ there is a vertex of degree $<2$ then $G$ is a forest
I need help proving this:
Given a graph $G$, prove that if in all subgraphs of $G$ there is a vertex of degree less than $2$ ($1$ or $0$) then $G$ is a forest.
1
vote
1answer
16 views
If $L$ is a linear continuum in the order topology, then $L$ is connected.
From Munkres p.153:
Why does he begin with convex sets? Is it because if we know that convex sets are connected then we can write $L$ as a union of convex sets that have a point in common?
Why ...
2
votes
2answers
26 views
Let $f(x)=x^3+2x^2+1,\,\,g(x)=2x^2+x+2.$ Then over $\Bbb Z_3$…
I am stuck on the following problem:
Let $f(x)=x^3+2x^2+1,\,\,g(x)=2x^2+x+2.$ Then over $\Bbb Z_3$, show that $f(x)$ is irreducible ,but $g(x)$ is not.
Can someone explain how to tackle it? ...
1
vote
2answers
27 views
Random Variables from [0,1] - Integration Limits
I was wondering if someone could help me understand the first steps I should take for solving the next problem:
Let $U$, $V$ be random numbers chosen independently from the interval $[0, 1]$ with ...
1
vote
2answers
34 views
for $ F(x,y) = 10$, what is $ y'$?
For an input $x$ and output $y$ of a system it is know that $x,y$ always satisfy
$$ F(x,y) = 10 $$
At a certain point, $x=1$ and $y=1$. The question is how $y$ responds to a small decrease in $x$, ...
1
vote
3answers
51 views
Confusing Trigonometry Problem
Lets say at an intersection the words "STOP HERE" are painted on the road in red letters 2.5m high. It is important that drivers using this lane can read the letters. How can I find the angle ...
0
votes
0answers
21 views
borel-finite and non-negative measure
Let $\mu_y$ be a finite and non-negative measure on $\mathbb{R}$, and $\mu$ a non-decreasing measure on $\mathbb{R}$, such that for every fixed $y\in (a,b)$ with (a,b) NOT containing the origin
$$
...
0
votes
3answers
36 views
Index of a subgroup of $\mathbb{Z}\times\mathbb{Z}$
Let $p\in\mathbb{Z}$ be a prime and $u\in\mathbb{Z}$ be such that $u^2\equiv -1\pmod{p}$. Now define an additive subgroup $S$ of $\mathbb{Z}\times\mathbb{Z}$ by following, $$S:=\{ ...
0
votes
1answer
26 views
Why are projective schemes $\mathbb P_A^n$ over a ring not affine for $n>1$?
I recently posted a very similar question, but I hid the question I really wanted answered in it. I'm posting this to make that question explicit.
Let $A$ be a nonzero commutative ring with unit. ...
-1
votes
0answers
50 views
Want to convert Italian Mathematics Paper into English [closed]
I have a paper which is written in Italian. As I don't know the language, I need to translate it into English. I tried Google Translate but the resulting text came out illegible. Are there resources ...
1
vote
0answers
19 views
Simplify $\frac{(m+n-1)!}{m!n!}$ and a related polynomial.
Adopting the notation $[k] = x^k - x^{-k} $ and $[k]! = [2][3]...[k]$ (note that $[1]$ is omitted), and letting $m,n$ be two integers greater than $1$ such that $n>m$ and $gcd(m,n)=1$, would it be ...
1
vote
0answers
18 views
Solving a Compound Inequality Graph
What is the solution set of $\{x\mid x < -5\}\cap\{x \mid x > 5\}$?
A. all numbers less than $-5$ and greater than $5$;
B. the numbers between $-5$ and $5$;
C. the empty set;
D. all real ...
0
votes
1answer
25 views
A trouble about the topology of pointwise convergence $({\mathbb{R}}^M,\tau)$
Let $(M,d)$ be a separable metric space and $F=\{f_{\lambda}:M\longrightarrow\mathbb{R}:\lambda\in L \}\subset \mathbb{R}^M $ be an equicontinuous family of uniformly bounded functions on $M$.
How ...
0
votes
1answer
47 views
Sum of N numbers whose sum is M
In how many ways can we sum N nonnegative numbers (that is, taking values 1, 2, 3...) such that their sum is M? I found this problem doing convolution of series and combinatorics has never been my ...
3
votes
2answers
58 views
How to evaluate the integral $\int_{-\pi}^{\pi} \arctan(\pi^x)\,dx$
Evaluate the intergal:
$$\int_{-\pi}^{\pi} \arctan(\pi^x)\,dx.$$
Thank you
0
votes
1answer
25 views
Conceptually, how to deal with zero slope lines when using y=mx+b to drive other equations
I am (with wonderful help from this site) developing a number of VBA routines to drive some shape-related activity in Powerpoint. For example, I have a circle with a line segment that starts in the ...
2
votes
6answers
42 views
Let $p_n$ be the sequence defined by $p_n=\sum_{k=1}^n\frac{1}{k}$. Show that $p_n$ diverges even though $\lim_{n\to\infty}(p_n-p_{n-1})=0$ [duplicate]
I have tried this as :
$$p_n=\sum_{k=1}^n\frac{1}{k}=1+\frac{1}{2}+\frac{1}{3}+\ldots+\frac{1}{n-1}+\frac{1}{n}$$
...
1
vote
1answer
26 views
Gauß-Jordan algorithm - 'reading' the solution
Disclaimer: I'm not really sure how to do a proper coefficient-matrix in latex, if someone could edit it to look properly I'd be really thankful ;)
Given the following system of linear equations, ...
2
votes
1answer
32 views
Elementary Divisors
Let $\mathbb K$ be a field and $A\in M_n(\mathbb K)$. Consider a polynomial $p(x)\in\mathbb K[x]$.
How are the elementary divisors of $A$ and the elementary divisors of $p(A)$ related?
2
votes
2answers
72 views
Is this alternative definition of 'equivalence relation' well-known/useful/used?
I was puzzling over another question: Let $R$ be an equivalence relation on a set $A$, $a,b \in A$. Prove $[a] = [b]$ iff $aRb$. And this made me discover that $$(0) \; \langle \forall a,b :: aRb ...
2
votes
2answers
80 views
Is it possible to calculate sine by hand?
Without a calculator, how can I calculate the sine of an angle, for example 32(without drawing a triangle)?
0
votes
1answer
27 views
the distribution of the inverse of a standardized uniform variable
If $u$ is a standardized uniform variable, what is the mean and variance of $x=\frac{1}{u}$? What can be said about the distribution of $x$?
1
vote
0answers
14 views
Minimum ratio between surface and volume in a riemannian manifold
In an euclidean three - dimensional space the sphere is the geometric figure with the minimum ratio $R=\frac{S}{V}$ with $S=4\pi r^2$ and $V=\frac{4}{3}\pi r^3$, so we have: $$R=\frac{1}{3}r$$ where ...
0
votes
2answers
49 views
Meaning of $\bar{i}:=i+n\mathbb{Z}$ in Modular Arithmetic
I am starting to learn graph theory and ran into the following definition:
The set $\mathbb{Z}/n\mathbb{Z}$ of integers modulo $n$ is denoted by $\mathbb{Z}_n$; its elements are written as ...
3
votes
2answers
75 views
How prove this $\lim_{n\to\infty}\int_{0}^{\frac{\pi}{2}}\sin{x^n}dx=0$
show that
$$\lim_{n\to\infty}\int_{0}^{\dfrac{\pi}{2}}\sin{x^n}dx=0$$
I have see this similar problem
$$\lim_{n\to\infty}\int_{0}^{\dfrac{\pi}{2}}\sin^n{x}dx=0$$
poof:
$\forall ...
0
votes
2answers
36 views
Arrow impossibility theorem and social choice.
I have read the Arrow impossibility theorem in Foundations of Mathematical Economics(Michael Carter). It is just too difficult to understand.
So, does Arrow'theorem mean that there is always a ...
0
votes
1answer
17 views
Least Square Method for solving system of equations
So I am following this procedure through MathCad, but when I get to the bottom of page 3, he says I can use a built in command, which he doesn't include. So I am trying to figure out how to solve ...
0
votes
0answers
11 views
Unclear step building series out of a0, an, bn
I have a script that does a transformation step that is completely unclear to me.
Would be great if anyone could explain it to me in-depth. It seems like I don't know the math behind it at all.
Ok, ...
