# All Questions

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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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.
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### 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.
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### $E$ 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 ...
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### 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 ...
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### 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?
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### $f$ 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 ...
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### 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 $f'(x)>0$?
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### 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?
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### 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 ...
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### 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 ...
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### $n$-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 $B^n = A$.
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### 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 ...
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### 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 ...
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### 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 ...
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### $M$ 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 ...
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### 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 ...
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### 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.
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### 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 ...
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### 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? ...
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### 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 ...
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### 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 $x$, ...
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### 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 ...
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### 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. ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### How to evaluate the integral $\int_{-\pi}^{\pi} \arctan(\pi^x)\,dx$

Evaluate the intergal: $$\int_{-\pi}^{\pi} \arctan(\pi^x)\,dx.$$ Thank you
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### 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 ...
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### 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 : $$p_n=\sum_{k=1}^n\frac{1}{k}=1+\frac{1}{2}+\frac{1}{3}+\ldots+\frac{1}{n-1}+\frac{1}{n}$$ ...
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### 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, ...
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### 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?
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### 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 ...