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H: Fast multiplication of orthogonal matrices
Given $A,B\in SO(3)$, direct matrix multiplication computes $C=AB$ with 27 multiplies. The group $SO(3)$ is a $3$-dimensional manifold. This suggests that direct matrix multiplication, which thinks of elements of $SO(3)$ as 9-dimensional, is not optimal. What is the minima... |
H: Special orthogonal matrix uniquely determined by $n-1 \times n-1$ entries?
For example, consider the specific question: Given $a_{11},a_{12},a_{21},a_{22}$ does that uniquely determine
$A=\begin{bmatrix} a_{11}&a_{12}&a_{13} \\ a_{21}&a_{22}&a_{23} \\ a_{31}&a_{32}&a_{33} \end{bmatrix}$
where $A\in SO(3)$.
AI: Hint... |
H: Show that $a^{q-1} \equiv 1 \pmod{pq}$
Assume that $p$ and $q$ are distinct add primes such that $p-1\mid q-1$. If $\gcd(a,pq)=1$ ,show that: $$a^{q-1} \equiv 1 \pmod{pq}$$
I have tried as follows:
$$a^{q-1} \equiv 1 \pmod{q} \quad \text{and} \quad a^{p-1} \equiv 1 \pmod{p}$$
$$\implies a^{(q-1)(p-1)} \equiv 1 \p... |
H: Understanding Formula for Sampling without Ordering and without Replacement
Alright, so I've been working through a couple combinatorics problems and I'm having trouble understanding the underlying reason for why a formula is written in a certain way. So here's the problem:
Suppose that r flags of different colors... |
H: How does he get a perfect swap numerator and denominator.
I'm going through a exercise, in which all the answers are given, but the tutor makes a step and I can't follow at all. A massive jump with no explanation.
Here is the question:
$\lim_{x \to 2} \frac{\frac{1}{2}-\frac{1}{x}}{x-2}$
He then simplifies:
$ \frac... |
H: Show that for all $\lambda \geq 1~$ $\frac{\lambda^n}{e^\lambda} < \frac{C}{\lambda^2}$
Show that for any $n \in \mathbb N$ there exists $C_n > 0$ such that for all $\lambda \geq 1$
$$ \frac{\lambda^n}{e^\lambda} < \frac{C_n}{\lambda^2}$$
I can see that both sides of the inequality have a limit of $0$ as $\lambda... |
H: Giving a hypersurface by higher degree polynomials
Suppose a hypersurface in $\mathbb{P}^n$ is given by an equation $F(Z) = 0$. It is easy to show that polynomials $Z_i F$ ($i = 0,\ldots,n$) give the same hypersurface, but I have trouble demonstrating that $K[Z_0,\ldots,Z_n]/I \cong K[Z_0, \ldots, Z_n]/I'$ where $I... |
H: What are the zero divisors of $C[0,1]$?
Suppose you have a ring $(C[0,1],+,\cdot,0,1)$ of continuous real valued functions on $[0,1]$, with addition defined as $(f+g)(x)=f(x)+g(x)$ and multiplication defined as $(fg)(x)=f(x)g(x)$. I'm curious what the zero divisors are.
My hunch is that the zero divisors are precis... |
H: How to "stretch" a procedural half-sphere texture on X and/or Y axis
I've implemented an Objective-C function to display the "height" of a half-sphere, with "1.0" being "full-height" and "0.0" being "no-height"
The sphere currently has a few parameters:
Center (x,y: real coordinates)
Radius (r: real value)
These ... |
H: Arcwise connected part of $\mathbb R^2$
Here's a question that I share:
Show that if $D$ is a countable subset of $\mathbb R^2$ (provided with its usual topology) then $X=\mathbb R^2 \backslash D $ is arcwise connected.
AI: HINT: Not only is $\Bbb R^2\setminus D$ arcwise connected, but you can connect any two poin... |
H: Constructing a holomorphic function with some specific points zero/nonzero
Given $n \in \mathbb{Z}$, is it possible to construct a holomorphic function
$f : \mathbb{C} \rightarrow \mathbb{C}$ such that $f(n) \neq 0$, but
for any integer $m \neq n$ we have $f(m)=0$?
This is actually a homework problem in algebra whi... |
H: If an element has a unique right inverse, is it invertible?
Suppose $u$ is an element of a ring with a right inverse. I'm trying to understand why the following are equivalent.
$u$ has at least two right inverses
$u$ is a left zero divisor
$u$ is not a unit
If $v$ and $w$ are distinct right inverse of $u$, then $... |
H: Solving a biquadratic $x^{4}-2x^3 + x^2 - 2x +1 =0$
How do I find the roots of $$x^{4}-2x^3 + x^2 - 2x +1 =0$$
I am not able to find any roots by trial and error.
AI: Divide throughout by $x^2$.
Then you have $x^{2}-2x + 1 - \frac{2}{x} + \frac{1}{x^2} = 0$
You can re-write as $x^{2}+\frac{1}{x^2} - 2(x+\frac{1}{x}... |
H: Find the square root of the polynomial
My question is:
Find the square root of the polynomial-
$$\frac{x^2}{y^2} + \frac{y^2}{x^2} - 2\left(\frac{x}y + \frac{y}x\right) + 3$$
AI: Let $t=\dfrac xy +\dfrac yx$.
Then $t^2=\dfrac {x^2}{y^2} +\dfrac{y^2}{x^2}+2$.
Your "polynomial" becomes finally an actual polynomial:
$... |
H: Angles of triangle inside a cricle
In the figure shown if area of circle with center o is 100pi and CA has length of 6 what is length of AB ?
I looked around on the web and cant seem to get an idea of what the angles AOC and OCA inside the triangle would be. Any suggestions on how I would go about determining ... |
H: Understanding adjoint functors
To understand adjoint functors I tried to look at an example. Can you tell me if the following is correct?
Before I give the example I'd like to recap the definition: Given two categories $C,D$ and two functors $F: C \to D$ and $G: D \to C$ we say that $F$ and $G$ are adjoint if we ca... |
H: To show $A\implies B$, is that sufficient to show for all $C$ s.t. $C\implies A$ then $C\implies B$
my question is in the title:
to show $A\implies B$ is it enough to show for any $C$ such that $C\implies A$
we have $C\implies B$?
AI: Yes but that doesn't make it easier since you could choose $C = A$. |
H: Spanning a vector with no zero coordinates
Given a complex square matrix with 1-s on the main diagonal (and arbitrary values elsewhere), do its columns span a vector with no zero coordinate?
Clarification: What I'm asking is, given a complex matrix with 1-s on the main diagonal (and arbitrary values elsewhere), doe... |
H: non-trivial common zero of polynomials
The following situation occurs in a proof that I would like to understand: we have polynomials $F_1,\ldots, F_N$ in $k[X_1,\ldots,X_M]$, where $k$ is of characteristic zero and algebraically closed. The polynomials $F_i$ are homogeneous of positive degree. And we have $M>N$.
N... |
H: Deriving the characteristic function for $N(0,2)$
Could someone please help me with an easy derivation of the characteristic function for a $N(0,2)$ distribution? Or a link to somewhere it is done.
AI: Once you know the characteristic function of $N(0,1)$, you can deduce the corresponding for $N(m,\sigma^2)$ for ea... |
H: Galois group of $x^6 + 3$ isomorphic to a copy of $S_3$ inside $S_6$
I have seen the the thread here related to the computation of the Galois group of the same polynomial. However, my question is not about the computation itself but about the group presentation of the Galois group. I will explain.
I have determine... |
H: abel summable implies convergence
Prove that:
If $\sum c_n$ is Abel summable to $s$ and $c_n=O(\frac{1}{n})$ , then $\sum c_n $ converges to $s$.
"A series of complex number $\sum_{n=0}^{\infty} c_n $ is said to be Abel summable to $s$ if for every $0 \le r <1$ ,the series $A(r)=\sum_{k=0}^{\infty} c_kr^k$ converg... |
H: Parametric representation of rectangular form in terms of parameters $\rho$ & $\theta$
I need to represent the cone $z=\sqrt{3x^2+3y^2}$ parametrically in terms of $\rho$ and $\theta$ where $(\rho,\theta,\phi)$ are spherical coordinates.
Attempt. I tried using: $$x=\rho\sin\phi\cos\theta \\y=\rho\sin\phi\sin\thet... |
H: Find a maximum of complex function
I am trying to find a simple method that does not use the tools of advanced differential calculus to find following maximum, whose existence is justified by the compactness of the close ball $\Delta$ of $\mathbb C$ and continuity of the function $f:z \mapsto |z^3 + 2iz|$ fro... |
H: Every $k$ vertices in an $k$ - connected graph are contained in a cycle.
Let $G$ be a $k$-connected graph. Meaning, $G$ has no fewer than $k$ vertices, and for every set of $k-1$ or fewer vertices, if we remove them from $G$, the graph stays connected (Of course, $G$ itself is also connected).
I want to prove that ... |
H: a Function with several periods
A periodic function is given by $ f(x+nT)=f(x) $, with 'n' an integer and T the period.
My question is if we can define a non-constant function with several periods; by that, I mean
$ f(x+T_{i})=f(x) $ with $ i=1,2,3,4,\dots $ a set of different numbers.
For example, a function that ... |
H: In mean value theorem, does the mean value vary continuously?
Let $f\colon\mathbb R\to\mathbb R$ be continuously differentiable and let's say, for simplicity, that $f(0)=0$. Then by mean value theorem it's
$$f(x)=f'(\xi)\cdot x \,\text{ for some } \xi \in (0, x)$$
What I wondered is: What can we tell about the $\xi... |
H: Can't understand this solution.
I came across a problem which was already present on the internet.
If an arc with a length of $12\pi$ is $\frac{3}{4}$ of the circumference of the circle, what is the
shortest distance between the endpoints of the arc?
According to a certain site the solution is something like t... |
H: Solving a quadratic Inequality
My question is:
Solve $$9x-14-x^2>0$$
My answer is: $2 < x < 7$
Though I know my answer is right, I want to know in what ways I can solve it and how it can be graphically represented.
Thank you.
AI: Let's rearrange the inequality to get:
$x^2 - 9x + 14 < 0$
i.e.
$(x-7)(x-2) < 0$.
N... |
H: What's the name of this operator?
Let $f,g$ be functions in $C^A$ and $C^B$ respectively.
Let $\boxtimes:C^A \times C^B \to (C\times C)^{A \times B}$ s.t.
$f\boxtimes g(a,b)=(f(a),g(b))$
It seems not the tensor product, nor Cartesian product. Then can we call it direct product? But it seems the term 'direct produc... |
H: some uniform continuous functions
We need to find which are uniform continuous (UC) on a) $(0,1)$ and b) $(0,\infty)$.
I have done, could you confirm me, if I am wrong any where?
$\frac{1}{(1-x)}$
$\frac{1}{(2-x)}$
$\sin x$
$\sin(1/x)$
$x^{1/2}$
$x^3$
1) is not UC on a) because limit does not exist when $x\r... |
H: Semicontinuity problem
If $A \subset \mathbb{R}^n$, is that claim true?
$$\chi_A \text{ is LSC} \Longleftrightarrow A\text{ is open}$$
And then how can I prove it?
($\chi_A$ is characteristic function : if $x \in A$ than $\chi_A =1$ otherwise zero.)
AI: True. In fact
\begin{equation}
\chi_{A} \mbox{is LSC} \ \Lef... |
H: a non separable metric space
Let $X$ be a metric space with discrete metric whose points are the positive integers. We have to show $C(X,\mathbb{R})$ is non separable. Well, what I have to do is to show $C(X,\mathbb{R})$ has no countable dense subset. I have no idea how to show that It has no countable as well as ... |
H: lower enveloper and upper enveloper
Let see $A \subset \mathbb{R}^n$, Define $\overline{f}(x) = \limsup_{y\to x} f(y)$
and $\underline{f}(x) = \liminf_{y\to x} f(y)$
$$\underline{\chi_A} =\chi_{A^o}, \text{ }\overline{\chi_A} =\chi_{A^-} $$
I wanna prove that. How can I approach?
AI: If $x \in A^{\circ}$ there ex... |
H: Limit of a function tending to zero.
If $F(t)$ is twice differentiable at $x$ and $$G(h)=\max_{t\in(0,h)}\left[\frac{F'(x+t)-F'(x-t)}{2t}-F''(x)\right],$$ where $x$ is fixed; then how can we show that $\displaystyle\lim_{h\to 0}G(h)=0$.
AI: Hint:
$$\frac{F'(x+t)-F'(x-t)}{t}=\frac{F'(x+t)-F'(x)}{t}+\frac{F'(x-t)-F'... |
H: Show that $\frac{(3^{77}-1)}{2}$ is odd and composite
The question given to me is:
Show that $\large\frac{(3^{77}-1)}{2}$ is odd and composite.
We can show that $\forall n\in\mathbb{N}$:
$$3^{n}\equiv\left\{
\begin{array}{l l}
1 & \quad \text{if $n\equiv0\pmod{2}$ }\\
3 & \quad \text{if $n\equiv1\pmod{... |
H: Is $C^1(A)$ a Banach space?
Let $A \subset \mathbb R$ and consider the space $C^1(A)$. I am asked to prove that $( C^1(A), \Vert \cdot \Vert_{C^1(A)})$ is a Banach space, where
$$
\Vert f(x) \Vert_{C^1(A)} = \sup_{x \in A} \vert f(x) \vert + \sup_{x \in A} \vert f'(x) \vert
$$
First question: $A$ should be compact... |
H: mathematical notation for a logical statement
The proof of the statement below is a homework question, however I did not tag this question as such since I don't need the actual proof: I have already proved the statement wring and don't need a solution; my question here is strictly in regards to the mathematical not... |
H: Probability; can't understand the maths
For a random variable $x$, define a probability distribution $p[x=n]=c (3^n/n!)$ when $x=0, 1, 2, \dots$ and $p(x)=0$ otherwise. Find the value of $c$.
My professor provided the solution
$$
\sum_{x=0}^\infty \ c\frac{3^n}{n!}=1
$$
so $c\;e^3 = 1$.
I can't understand why the ... |
H: Different ways of computing $\sqrt{1+\sqrt{1+\sqrt{1+\cdots}}}$
Possible Duplicate:
On the sequence $x_{n+1} = \sqrt{c+x_n}$
I am wondering how many different solutions one can get to the following question:
Calculate $\sqrt{1+\sqrt{1+\sqrt{1+\cdots}}}$
Post your favorite solution please.
AI: If
$$\phi = \sqr... |
H: How do I show that this function is always $> 0$
Show that $$f(x) = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} +
\frac{x^4}{4!} > 0 ~~~ \forall_x \in \mathbb{R}$$
I can show that the first 3 terms are $> 0$ for all $x$:
$(x+1)^2 + 1 > 0$
But, I'm having trouble with the last two terms. I tried to show that the foll... |
H: Solution to $x=1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\ldots}}}$
Possible Duplicate:
Why does this process, when iterated, tend towards a certain number? (the golden ratio?)
Please post your favorit solution to the following
Compute $x=1+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{1+\ldots}}}$
Thank you
AI: Denote by $x=1+\f... |
H: $\ell_1$ distance of a point to a convex polygon
Let us have a set of $n$ points, $x_1, x_2, \ldots, x_n \in \mathbb{R}^d$, that form a convex polytope. And let us have a single point $x \in \mathbb{R}^d$ that is outside of the polytope. How can I compute the $\ell_1$ distance of the point to the polytope?
I wanted... |
H: Computing: $\lim\limits_{n\to\infty}\left(\prod\limits_{k=1}^{n} \binom{n}{k}\right)^\frac{1}{n}$
I try to compute the following limit:
$$\lim_{n\to\infty}\left(\prod_{k=1}^{n} \binom{n}{k}\right)^\frac{1}{n}$$
I'm interested in finding some reasonable ways of solving the limit. I don't find any easy approach. Any ... |
H: Pigeon principle question: Nine points in a diamond
A diamond (a parallelogram with equal sides) is given, and its sides are 2 cm long. The sharp angels are 60 degrees. If there are nine points inside the diamond, prove that there must be two of them so that the distance between them is at most 1 cm.
Ideas where to... |
H: Big Oh notation Question in calculus
In my text book, they state the following:
$$\begin{align*}f(x) &= (\frac{1}{x} + \frac{1}{2}) (x-\frac{1}{2}x^2+\frac{1}{3}x^3+O(x^4))-1& ,x \rightarrow 0\\&= 1-\frac{1}{2}x+\frac{1}{3}x^2+\frac{1}{2}x-\frac{1}{4}x^3+O(x^3)-1& ,x \rightarrow 0 \end{align*}$$
However, when I cal... |
H: When was the significance of $i$ first noticed?
Complex analysis is an entire field of mathematics that focuses on the use of the complex constant $i$. When was the significance of $i$, an imaginary number, first noticed?
If I did not know some of the uses of complex analysis, I would likely believe, being the lay... |
H: How to evaluate $\lim\limits_{t\to 0} \frac{e^{-1/t}}{t}$?
How can I evaluate
\[
\lim_{t\to 0} \frac{e^{-1/t}}{t}\quad ?
\]
I tried to use L'Hôpital's rule but it didn't help me. Any hints are welcome. Thanks.
AI: Note that as $t \to 0$, $\exp(-1/t)$ tends 'faster' to $0$ than $1/t$ tends to $\infty$. To make this ... |
H: Summing the series $\sum u_{n}, u_{n}=\frac{a+n-1}{(a+1)...(a+n)}$
I am trying to sum the series
$$ \sum u_{n}$$
where $$ u_{n}=\frac{a+n-1}{\prod_{j=1}^n (a+j)}$$
$$ a>0$$
We have:
$$ \frac{a+n-1}{\prod_{j=1}^n (a+j)}=\sum_{k=1}^n\frac{b_k}{a+k} $$
$$ b_{k}=\frac{n-k-1}{\prod_{j=1,j\neq k}^n (j-k)}$$
$$ \sum_{n=1... |
H: A particular case of Truesdell's unified theory of special functions
I'm reading through Clifford Truesdell's "An essay toward a unified theory of special functions", Princeton Univ. Press, 1948. All his exposition is based on the functional equation
$$\frac{\partial}{\partial z}\mathrm F(z,\alpha)=\mathrm F(z,\alp... |
H: Automorphism on integers
Is multiplying by a constant m (integer) on group of set of all integers on addition an automorphism?
If so why does the 2nd example in http://en.wikipedia.org/wiki/Automorphism says that the unique non trivial automorphism is negation?
AI: Automorphism is a permutation of a set which resp... |
H: Is there a simple way of arriving at this solution?
Suppose we are given the matrix $$\begin{pmatrix}x'\\y'\end{pmatrix}=\begin{pmatrix}\cos(\omega t)& -\sin(\omega t)\\\sin(\omega t)& \cos(\omega t)\end{pmatrix}\begin{pmatrix}x\\y\end{pmatrix}$$
In other words the new coordinate system is a rotating coordinate sys... |
H: Rotating about a point?
This is part of a program. I am trying to take an image at a given pair of coordinates and move it in a circular path around another given pair of coordinates.
The program is structured so that the image is redrawn 30 times per second, and with each redraw I need to update its position. That... |
H: Banach-Tarski theorem without axiom of choice
Is it possible to prove the infamous Banach-Tarski theorem without using the Axiom of Choice?
I have never seen a proof which refutes this claim.
AI: The Banach-Tarski theorem heavily uses non-measurable sets. It is consistent that without the axiom of choice all sets a... |
H: Two-sided ideal generated by subset of not commutative ring
Let $R$ be a non-commutative ring, and $S\subset R$ some subset. Let $I_S$ be the smallest two-sided ideal of $R$ such that $I_S\supseteq S$. Is it true that (if $R$ is unital ring) $I_S$ consist only of elements of the form:
$$\sum_{s\in S}\left(\sum_{k=... |
H: Essentially continuous a.e. characteristic function
I'm reading a few papers and trying to understand why one paper is supposed to be a strengthening of the other. The bulk of the papers didn't have to do with what follows, but at one point they imply that an a.e. essentially continuous characteristic function is a... |
H: Show that $\operatorname{int}(A \cap B)= \operatorname{int}(A) \cap \operatorname{int}(B)$
It's kind of a simple proof (I think) but I´m stuck!
I have to show that $\operatorname{int} (A \cap B)=\operatorname{int} (A) \cap \operatorname{int}(B)$.
(The interior point of the intersection is the intersection of the in... |
H: Find $A \in M_{2}(\mathbb{Z})$ such that $M_{2}(\mathbb {Z})=\{\sum a_{i}A^{i} : a_{i} \in \mathbb{Z}\}$
Question: Does there exist $A \in M_{2}(\mathbb{Z})$ such that every element of $M_{2}(\mathbb{Z})$ can be represented as a linear combination of powers of $A$ with integer coefficients? In other words,
$$\exist... |
H: Non-zero prime ideals in the ring of all algebraic integers
Let $\mathcal{O}$ be the ring of all algebraic integers: elements of $\mathbb{C}$ which occur as zeros of monic polynomials with coefficients in $\mathbb{Z}$.
It is known that $\mathcal{O}$ is a Bezout domain: any finitely generated ideal is a principal id... |
H: gradient flow and what is, for example, $L^2$ gradient?
Am I right that the gradient flow of a functional $E$ is
$$f_t = -\nabla E(f).$$
Solving this for $f$ gives you a minimiser of $E$ in some way?
Here the $\nabla$ denotes the gradient or the first variation or Gateaux derivative or whatever is appropriate.
Wha... |
H: Kaplansky's theorem of infinitely many right inverses in monoids?
There's a theorem of Kaplansky that states that if an element $u$ of a ring has more than one right inverse, then it in fact has infinitely many. I could prove this by assuming $v$ is a right inverse, and then showing that the elements $v+(1-vu)u^n$ ... |
H: question about cofinality and function
In a paper, I want to prove a result that seems to me general.
Let $g:\delta\longrightarrow cf(\lambda)$ where $\delta$ is an ordinal less than $\lambda^+$ and $\lambda$ a cardinal. Suppose that $\forall i<cf(\lambda)$, $g^{-1}[i]$ is not cofinal in $\delta$. Does one have $cf... |
H: Represent the following set of points in the XY plane
Represent the following set of points in the XY plane:
{ (x,y) | |x|=1 }
{ (x,y) | |x| is less than or = 1 }
AI: HINT: Note that your set imposes no restriction at all on the $y$-coordinate. If you find a point in the set, every other point with the same $y$-coo... |
H: Prove that $\mathcal{W}(\mathbb{R})$ is not metrizable.
A colection $\mathcal{V}$ of open sets in a topological space $X$ is called a Fundamental System of Open Neighborhoods (FSON) of a point $x\in X$ when:
$\forall\ V\in\mathcal{V}$ we have that $x\in V.$
If $A\subset X$ is open set containing $x$ then $\exists... |
H: Building the integers from scratch (and multiplying negative numbers)
Now I understand that what I am about to ask may seem like an incredibly simple question, but I like to try and understand math (especially something as fundamental as this) at the deepest level possible. And for the life of me, I can't shake thi... |
H: Solving a quadratic?
Lets say you have a quadratic that you factor into root form. To solve for the roots, you let the $y$ value be $0$:
$0 = (x_1-h)(x_2-k)$
Following this you would divide both sides by one of the multipliers.
This would leave you with $x_1-h =0$ and therefore $x_1 =h$ and $x_2-k = 0$ and therefo... |
H: Derivative of an implicit function
I am asked to take the derivative of the following equation for $y$:
$$y = x + xe^y$$
However, I get lost. I thought that it would be
$$\begin{align}
& y' = 1 + e^y + xy'e^y\\
& y'(1 - xe^y) = 1 + e^y\\
& y' = \frac{1+e^y}{1-xe^y}
\end{align}$$
However, the text book gives me a di... |
H: Blow-up along an ideal sheaf
Let $k^2=\operatorname{Spec} \; k[x,y]$ where $k$ is an algebraically closed field. Let $\mathcal{I}$ be the ideal sheaf defined by $(x,y)$. Then
$$
Bl_{\mathcal{I}}k^2
$$
is covered by two open charts $\operatorname{Spec} \; k[x, y/x] \cup \operatorname{Spec}\; k[y,x/y]$.
Q1: Why c... |
H: Get point on ellipse from point and angle
I have the bounds (x from, x to ...) of an ellipse (and thus its radius and center), x and y of a point A that is in the ellipse and an angle. I want to get point B to which the angle points (from point A) and that lies on the ellipse.
This is what I can do:
center_x = (x_f... |
H: Proving that $S_n$ has order $n!$
I have been working on this exercise for a while now. It's in B.L. van der Waerden's Algebra (Volume I), page $19$. The exercise is as follows:
The order of the symmetric group $S_n$ is $n!=\prod_{1}^{n}\nu$. (Mathematical induction on $n$.)
I don't comprehend how we can logicall... |
H: How does graph theory describe a sequence or line or path of nodes?
I have a dataset of pairs of map coordinates, and I suspect that they could be connected to make a path. However, I'm not sure what the end points are, or if the coordinates actually make a path. I'm using the python networkx library, but it assu... |
H: Groups where all elements are order 3
I am a student trying to learn some abstract algebra this summer, and I recently proved (as an exercise) that if $G$ is a group where every element has order 2, then $G$ is abelian. I was wondering could we make a similar conclusion about groups where every element has order 3,... |
H: Proof that every metric space is homeomorphic to a bounded metric space
I have tried to show that every metric space $(X,d)$ is homeomorphic to a bounded metric space. My book gives the hint to use a metric $d'(x,y)=\mbox{min}\{1,d(x,y)\}$.
If we can show that $d(x,y) \le c_1 \cdot d'(x,y)$ with $c_1$ some positiv... |
H: Evaluate the integral $H(y)=\int_{z=1}^{\infty} \frac{1}{z^4+zy}\,dz$
$y\geq0$ define $$H(y)=\int_{z=1}^{\infty} \frac{1}{z^4+zy}\,dz$$ Show that $H$ is a continuous function of $y$ and show $\lim\limits_{y \to +\infty}H(y)=0$.
AI: This is kind of a brute force method where we explicitly find the function $f(a)$... |
H: Showing inequality for harmonic series.
I want to show that $$\log N<\sum_{n=1}^{N}\frac{1}{n}<1+\log N.$$ But I don't know how to show this.
AI: I think I wrote this up somewhere on this website but anyways here she is again.
From the figure, you can see that the area under the blue-curve is bounded below by the ... |
H: Finding the derivative of a function?
Could someone explain how I would find the derivative of the following function? I am completely lost:
$$f(x) = e^{i(x!^{\log x})}$$
AI: For complex $\,z\,$ with $\,\operatorname{Re}(z)>0\,$ , $$\Gamma'(z)=\int_0^\infty t^{z-1}e^{-t}\log t\,dt$$ so $$\left(e^{ix!^{\log x}}\righ... |
H: How to draw pictures of prime spectra
In Atiyah-MacDonald's Commutative Algebra, they give in Exercise 16 of Chapter 1 the instruction:
Draw pictures of Spec($\mathbb{Z}$), Spec($\mathbb{R}$),
Spec($\mathbb{C}[x]$), Spec($\mathbb{R}[x]$), Spec($\mathbb{Z}[x]$).
What exactly do they have in mind? I can enumerate... |
H: If $z$ is the unique element such that $uzu=u$, why is $z=u^{-1}$?
I'm trying to figure out why an element $u$ in some ring is invertible with inverse $z$ if any only if
$uzu=u$ and $zu^2z=1$
OR
$uzu=u$ and $z$ is the unique element meeting this condition.
Clearly, both conditions follow if $u$ is a unit with ... |
H: Does $\tan(1/z)$ has a Laurent series convergent?
Well, I have no idea: Does $\tan(1/z)$ have a Laurent series convergent on $0<|z|<R$?
AI: In short, the answer is no. The function $\tan(1/z)$ has poles at $z=\frac{1}{\pi/2+n\pi}$, which means that it is not analytic on the annulas you mentioned for any $R$. |
H: Is this an odd function?
I have this function:
\begin{equation*}
f(x)=%
\begin{cases}
1 &x\in\left[ -\pi,-\pi/2\right[ \\
-1 &x\in\left[ -\pi/2,0\right[\\
1 & x\in\left[ 0,\pi/2\right[ \\
-1 & x\in\left[ \pi/2,\pi\right]\\
\end{cases}
\end{equation*}
First I thought that it was odd, but th... |
H: LSC function Problem
$I$ : finite index set, $f_i$ is LSC for each $i \in I$
I wanna prove below. $$\min_i f_i \text{ is LSC}$$
And is there some example of that does not extend to infinite $I$?
AI: We use properties of $\liminf$. First, it's enough to show the result when $I$ has two elements (then use induction o... |
H: plotting the following set of points in the XY plane
Represent the following set of points in the XY plane :
$$\{ ( x , y ) \; | \; |x| + |y| = 1 \}$$
What i got:
1) if $x > 0, y > 0 : x = 1 - y$
2) if $x > 0, y < 0 : x = 1 + y$
3) if $x < 0, y > 0 : x = y - 1$
4) if $x < 0, y < 0 : x = -y -1$
Any help to solve thi... |
H: Prove that $(p-1)! \equiv (p-1) \pmod{1+2+3+\cdots+(p-1)}$
Given a prime number $p$ , establish the congruence:
$$(p-1)! \equiv (p-1) \pmod{1+2+3+\cdots+(p-1)}$$
I have proceeded like this:
$$\begin{align*}&(p-1)! \equiv (-1) \pmod{p} \quad \quad \quad \text{by Wilson's Theorem}\\
&(p-1)! \equiv 0 \pmod{\frac{p... |
H: Nature of the series $ \sum u_{n}, u_{n}=n!\prod_{k=1}^n \sin\left(\frac{x}{k}\right) $
Is the series $$ \sum u_{n}$$
$$ u_{n}=n!\prod_{k=1}^n \sin\left(\frac{x}{k}\right)$$
$$ x\in]0,\pi/2] $$
convergent or divergent?
We have:
$$ u_{n}\leq n!\prod_{k=1}^n \frac{x}{k}$$
$$ u_{n}\leq x^n$$
If $0<x<1$ the series is ... |
H: Can I change the order of the double integration?
Let $f \in C_0^\infty $, $g \in L^1 $ . Then $$ \int_{\mathbb R^n} \int_{\mathbb R^n} f(x-y)g(y) dy dx = \int_{\mathbb R^n}\int_{\mathbb R^n} f(x-y)g(y) dx dy $$holds? If so, why?
($f,g : \mathbb R^n \to \mathbb R $)
AI: This depends on the finiteness of the integr... |
H: Continuous function a.e.
I'm not sure these two statement are not same thing.
$$ "f equals a continuous function a.e." & "f is continuous a.e."$$
The concept is too much abstract, so I wanna find some counter examples.
a function $f$ and a continuous function $g$ s.t. $f=g$ a.e
and $f$ is not continuous a.e.
a fun... |
H: Equivalent of $ I_{n}=\int_0^1 \frac{x^n \ln x}{x-1}\mathrm dx, n\rightarrow \infty$
I would like to show that
$$ I_{n}=\int_0^1 \frac{x^n \ln x}{x-1}\mathrm dx \sim_{n\rightarrow \infty} \frac{1}{n}$$
Using the change of variable $u=x^n$:
$$ I_{n}=\frac{1}{n^2} \int_0^1 \frac{u^{1/n} \ln u}{u^{1/n}-1} \mathrm du=\... |
H: Quotient of two free abelian groups of the same rank is finite?
Let $A,B$ be abelian groups such that $B\subseteq A$ and $A,B$ both are free of rank $n$. I want to show that $|A/B|$ is finite, or equivalently that $[A:B
]$ (the index of $B$ in $A$) is finite.
For example, if $A=\mathbb{Z}^n$ and $B=(2\mathbb{Z})^n$... |
H: Why $C_0^\infty$ is dense in $L^p$?
Why $C_0^\infty$ is dense in $L^p$?
Would you give me a simple proof or the outline of the proof?
AI: The outline (the proof isn't simple, at least not according to my understanding of simple):
Let $f \in L^p$. Then there is a sequence of simple functions $s_n \in L^p$ that conv... |
H: Finding the equation of the holomorphic function from its components
Let $f(z)=u(x,y)+ i v(x,y)$ be a holomorphic function. I often find it difficult to deduce the form of $f$ as a function of $z=x+i y$ only (for example: $f=\sin x \cosh y + i \cos x \sinh y$ cas be written as $f(z)=\sin z$).
However, my book state... |
H: Cauchy-Product of non-absolutely convergent series
While grading some basic coursework on analysis, I read an argument, that a Cauchy product of two series that converge but not absolutely can never converge i.e. if $\sum a_n$, $\sum b_n$ converge but not absolutely, the series $\sum c_n$ with $$c_n= \sum_{k=0}^n a... |
H: Riemann Integral Problem
I'm studying Riemann Integral on measure theory class.
There is a function $f : [a,b] \to \mathbb{R}$
that is increasing or decreasing.
Is that function f is Riemann integrable? And if then, what are appropriate step functions?
And is that function continuous a.e.?
AI: Yes, a monotonic fun... |
H: Geodesics that self-intersect at finitely many points
Notations
$M$ will denote a smooth manifold and $\nabla$ an affine connection on it. A smooth curve $\gamma\colon I \to M$ will be called a geodesic if it is $\nabla$-parallel along itself, that is $\nabla_{\dot{\gamma}(t)}\dot{\gamma}=0$ for every $t \in I$. A ... |
H: If $ f \in C_0^\infty$, then is $f$ uniformly continuous?
If $ f \in C_0^\infty=\{ g: g\in C^\infty, \lim_{|x|\rightarrow \infty}g(x)=0\}$, then is $f$ uniformly continuous on $\mathbb R$?
($ f : \mathbb R \to \mathbb R $)
AI: HINTs
A continuous function on a compact interval is uniformly continuous.
$\lim_{|x| \t... |
H: plotting the following set of points in the XY plane 2 :
Represent the following set of points in the $XY$-plane
$$\left\{ (x,y) \big| (x-|x|)^2 + (y-|y|)^2 \leq 4 \right\}$$
Any help to solve this problem would be greatly appreciated.
Thank you.
AI: As I answered another question of your about absolutes, you have ... |
H: How to make derivative operation in matrix space?
\begin{equation}\frac{d}{d\theta}\frac{1}{2}(\theta^TX - y)^2 = 0\end{equation} where, $X$ is $m $ on $ n$ matrix, $y$ is $m$-dimensional vector, $\theta$ is n-dimensional vector.
I can solve this equation, but only intuitively, because of I know solution from lec... |
H: Quick way to check if a polynomial of degree $> 3$ is irreducible?
What's the easiest way to check if a polynomial of degree > 3 is irreducible in $\mathbb{Z}_2[x]$?
I want to find out if $x^7+x^6+1$ is irreducible in $\mathbb{Z}_2[x]$.
If a quadratic polynomial factors, it must be a product of two linear factors,... |
H: Prove that 16, 1156, 111556, 11115556, 1111155556… are squares.
I'm 16 years old, and I'm studying for my exam maths coming this monday. In the chapter "sequences and series", there is this exercise:
Prove that a positive integer formed by $k$ times digit 1, followed by $(k-1)$
times digit 5 and ending on one 6,... |
H: Relation between sides and angles.
Is this phrase safe to consider in general:
Equal sides of a polygon have corresponding equal angles
if not how would you refine or correct it
Example of a corresponding angle would be
Edited:
For example suppose you ignore the fact that the above triangle is an equilateral ... |
H: For which $n$, $G$ is abelian?
My question is:
For Which natural numbers $n$, a finite group $G$ of order $n$ is an abelian group?
Obviouslyو for $n≤4$ and when $n$ is a prime number, we have $G$ is abelian. Can we consider any other restrictions or conditions for $n$ to have the above statement or the group its... |
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