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H: Is the set of all rationals with denominators less than $10^6$ closed in $\mathbb{R}$?
I'm wondering if the set of all rationals with denominators less than $10^6$ is closed in the real number system. I think it is not, so that's what I've been trying to prove. I've tried looking at its complement and showing that ... |
H: A problem about invariant subspaces
I need help with the following problem:
Let $H\subseteq\mathbb R^3$ be a plane with cartesian equation $x=y,$ and let $r$ be the straight line generated by $(1,1,2)$.
Find an endomorphism $\phi$ of $\mathbb R^3$ such that $\phi(\mathbb R^3)=H$ and $\phi^2(\mathbb R^3)=r$.
I... |
H: Irreducible polynomial not attaining squares over finite field
Is it possible to construct an irreducible polynomial $f$ over $\mathbb{F}_{q}$ such that $f(x)$ is a non-square for any $x \in \mathbb{F}_{q}$?
I can prove the existence of irreducible polynomials (Euclid's argument), and I can construct polynomials wi... |
H: $f$ has an essential singularity in $z_0$. What about $1/f$?
Let $\Omega$ be a non-empty, open subset of $\mathbb C$. Consider an holomorphic function $f:\Omega \setminus \{z_0\} \to \mathbb C$ and suppose we know $z_0$ is an essential singularity of $f$.
I am wondering what we can say about the function $\tilde{f... |
H: Prove that an odd Collatz sequence will at some point have two consecutive even numbers
Is there an existent proof for this? It would be an important step to proving the conjecture in any case.
AI: Write the initial odd number in binary notation. The binary representation ends with a number of 1s, preceded by a 0 -... |
H: Fat Tail / Large Kurtosis Discrete Distributions?
All,
I'm wondering if there are any notable, basic discrete probability distributions with "fat/heavy tails" or a large kurtosis? I know the Geometric Distribution's excess kurtosis approaches 6, but I can't find any that are larger.
Are there any discrete distribu... |
H: Significant figures with a plus-minus
A question on my homework asks me to give the amount of significant figures of $2900±100$. Would this be one, two, or both?
AI: Well, if it had been $2900\pm 50$ then there would clearly have been 2 significant figures, namely the 2 and 9. But how do we account for the fact th... |
H: Differentiation from first principles of specific form.
I've been posed a question in which I'm to differentiate with respect to $x$ a function of the form $(x+a)^k$. I've successfully completed (matches the book's answer) the question by using the chain rule, however I cannot achieve the same result using the defi... |
H: Finding Asymptotes of Hyperbolas
To find a asymptote its either b2/a2 or a2/b2 depending on the way the equation is written.
With the problem
$$\frac{(x+1)^2}{16} - \frac{(y-2)^2}{9} = 1$$
The solutions the sheet I have is giving me is $3/4x - 3/4$ and $3/4 x + 5/4$
I thought it was just supposed to be $\pm 3/4x$.... |
H: Prove that there exists analytic $f$ such that $f(z) = 1/\bar{z}$ on the boundary
I'm doing some self-study in complex analysis, and came to the following question:
Let $D(a,1) \subset \mathbb{C}$ be the disk of radius $1$ with center at $a \in \mathbb{C}$, and let $\partial D(a,1)$ be the boundary of $D(a,1)$. P... |
H: What are some examples of non-identity bijections $f: X \to X$ such that $f^{-1} = f$
One example I can think of is $f: \mathbb{Z_2} \to \mathbb{Z_2}$ given by $f(1) = 0$ and $f(0) = 1$.
AI: In a sense, every such bijection is going to look the same. If $a,b \in X$, then we will either have things that look like $f... |
H: Exercise on compact $G_\delta$ sets
I'm having trouble proving an exercise in Folland's book on real analysis.
Problem: Consider a locally compact Hausdorff space $X$. If $K\subset X$ is a compact $G_\delta$ set, then show there exists a $f\in C_c(X, [0,1])$ with $K=f^{-1}(\{1\})$.
We can write $K=\cap_1^\infty U_... |
H: Understanding this summation identity
I'm currently reading a book in which part of the solution to the problem involve this identity:
$$\sum_{j=i+1}^{n}j = \sum_{j=1}^{n}j-\sum_{j=1}^{i}j$$
Which I cannot derive myself. The only thing I can do with it is this:
$$\sum_{j=i+1}^{n}j = \sum_{j=1}^{n}j+i = \sum_{j=1}^{... |
H: An increasing sequence $a_i \in \mathbb{N}$ such that $a_{j}-a_{i}\mid a_{j}-1$ for all $i
How to prove or disprove that there exists a set of natural numbers
$$a_{1}<a_{2}<\cdots$$ with
the property that, for every $i<j$, $$a_{j}-a_{i}\mid a_{j}-1\quad ?$$
I think it doesn't work for very big $a_{j}$'s and very sm... |
H: How many significant digits should be retained mid-calculation?
I always feel paranoid when dealing with long series of calculations on paper. How many significant digits should one use to reduce to negligible the probability of getting a wrong final result in a modestly long series of calculations? (I know that th... |
H: If $E$ and $F$ are subfields of a finite field $K$ and $E\cong F$, prove that $E = F$
If $E$ and $F$ are subfields of a finite field $K$ and $E\cong F$, prove that $E = F$.
A finite field is a simple extension of each of its subfields and $\mathbb{Z}_p$ is a subfield of every finite field. Hence $E\cong \mathbb{Z}... |
H: Is there a concrete description of the ideal $I$ such that $\mathbb{Q}[x]/I\cong\mathbb{Q}[\sqrt{2}+\sqrt{3}]$?
I know that in general if $R[u]$ is the ring obtained by adjoining an element $u$ to a ring $R$, then $R[u]\cong R[x]/I$ for some ideal $I$ such that $I\cap R=\{0\}$.
In a particular instance, I'm workin... |
H: Why is a strict $p$-ring whose residue ring is a field necessarily local?
Let $A$ be a strict $p$-ring. Recall that this means $A$ is $p$-adically separated and complete, $p:A\rightarrow A$ is injective, and $A/pA$ is a perfect $\mathbf{F}_p$-algebra. If $A/pA$ is a field, then $A$ is known to be a discrete valuati... |
H: proof for $ (\vec{A} \times \vec{B}) \times \vec{C} = (\vec{A}\cdot\vec{C})\vec{B}-(\vec{B}\cdot\vec{C})\vec{A}$
this formula just pop up in textbook I'm reading without any explanation
$ (\vec{A} \times \vec{B}) \times \vec{C} = (\vec{A}\cdot\vec{C})\vec{B}-(\vec{B}\cdot\vec{C})\vec{A}$
I did some "vector arithmet... |
H: Method to solve $xx'-x=f(t)$
I would like to resolve this differential equation:
$xx'-x=f(t)$
any suggestions (or any online texts on similar differential equation) please?
Thanks.
AI: This is an 'Abel equation of the second kind in the canonical form'. |
H: Continuous function from $(0,1)$ onto $[0,1]$
While revising, I came across this question(s):
A) Is there a continuous function from $(0,1)$ onto $[0,1]$?
B) Is there a continuous one-to-one function from $(0,1)$ onto $[0,1]$?
(clarification: one-to-one is taken as a synonym for injective)
I figured the answer to A... |
H: Why would anti-symmetric (0,2) tensor be traceless?
As it is, why would anti-symmetric (0,2) tensor be traceless? Is it because trace should allow any variable for its indices?
AI: Let $A_{ij}$ be (the entries of) a totally antisymmetric tensor. Then, its trace is
$$g^{j i} A_{i j} = g^{i j} A_{i j} = -g^{i j} A_{j... |
H: What's $T\left(n\right)$?
If $T\left( n \right) = 8T\left( n-1 \right) - 15T\left( n-2 \right); T\left(1\right) = 1; T\left( 2 \right) = 4$,
What's $T\left(n\right)$ ?
I use this method:
Let $c(T(n) - aT(n-1)) = T(n-1) - aT(n-2)$
from $T(n) = 8T(n-1) - 15T(n-2)$, we can get $\begin{cases}c = \frac{1}{3}\\a = 5\end... |
H: Mode for unique numbers
I'm trying to construct a rough statistical software and I'm confused about the following:
What is the mode if all numbers are unique?
What is the mode if 2 numbers have same (highest) frequency?
What I feel:
Should output nothing.
Should output the average of the 2.
Just as reference:
a... |
H: How to go about evaluating summation of something?
I am new to math and algorithms, I am trying to understand how does this equation:
$$|N| \le \sum\limits_{i = 1}^{h} 2^{i-1} $$
Come to a conclusion of:
$$|N| \le 2^h - 1$$ and then
$$h \ge log(|N|+1)$$
This is the evaluation of a height of a binary tree and N is t... |
H: What does it mean to be: "at least logarithmic"?
I am going through some CS basics and the documents says in places that:
We need the run time to be at least logarithmic.
What does that mean?
By def, log means:
The logarithm of a number is the exponent by which another fixed
value, the base, has to be raised... |
H: How to learn from proofs?
Recently I finished my 4-year undergraduate studies in mathematics. During the four years, I met all kinds of proofs. Some of them are friendly: they either show you a basic skill in one field or give you a better understanding of concepts and theorems.
However, there are many proofs whic... |
H: How to identify symmetric positive definite matrices?
I'm working on a project, implementing Successive over-relaxation (SOR) method (http://en.wikipedia.org/wiki/Successive_over-relaxation) using Python. SOR can only apply if given matrix is,
symmetric positive-definite (SPD)
OR
strictly or irreducibly dia... |
H: Product of two symmetry groups
Do there exist 2-sylow subgroups of $S_4\times S_3$ that are normal?
Do there exist 3-sylow subgroups of $S_4\times S_3$ that are normal?
Thank you for helping!
AI: Fact $1$: Suppose $G$ and $H$ are finite groups and $P_G$ and $P_H$ are Sylow $p$-subgroups of $G$ and $H$, respectively... |
H: Polynomial over characteristic two finite field of odd degree with certain image
I am trying to construct a polynomial $f \in \mathbb{F}_{2^k}$ of odd degree, such that $\forall x \in \mathbb{F}_{2^k} \exists \alpha \in \mathbb{F}_{2^k}$ such that $f(x)=\alpha ^2 -\alpha$.
Working with some normal basis $\{ \alpha... |
H: Idea behind factoring an operator?
Suppose if we have an operator $\partial_t^2-\partial_x^2 $ what does it mean to factorise this operator to write it as $(\partial_t-\partial_x) (\partial_t+\partial x)$ When does it actually make sense and why ?
AI: In the abstract sense, the decomposition $x^2-y^2=(x+y)(x-y)$ is... |
H: Compute Angle Between Quaternions (in Matlab)
I am working on a project where I have many quaternion attitude vectors, and I want to find the 'precision' of these quaternions with respect to each-other.
Without being an expert in this type of thing, my first thought is to find the angle between each (normalized) q... |
H: If $ u \in W^{2,3} ( \Omega ) $ then $u \in L^3 ( \Omega )$?
If $ u \in W^{2,3} ( \Omega ) $ then $u \in L^3 ( \Omega )$ ?
In wikipedia, the definition of Sobolev space is $$ W^{k,p} ( \Omega) = \{ u \in L^p ( \Omega)\mid D^{\alpha} u \in L^p , | \alpha| \leqslant k \},$$ where $ \Omega $ is an open set in $\mat... |
H: What's the difference between a curve and a graph?
My book states "... if $S$ is a level curve and $C$ is a curve in $S$ passing through a point $a$ ..."
What is a curve in $S$? Is it simply a subset of $S$?
From what I comprehend, a graph is usually a ordered set of the form
$ X = \{(a,f(a))\ |\ a \in \mathrm{dom... |
H: Quick question about comparing functions.
Suppose I have functions f and g; is there any theorem or technique that I can use to know if $f(x) \leq g(x)$ or if $f(x) \geq g(x)$ or neither in a given interval?
AI: I don't know if this is helpful and it doesn't solve your general problem, but as an example: if you hav... |
H: Summations manipulation: is this one right?
I've got a summation like this:
$\sum_{l=1}^L \sum_{i=1}^I p_l c_l^i = \sum_{l=1}^L \sum_{i=1}^I p_l w_l^i$
Is it right to bring $ p_l $ out of the symbol $ \sum_{i=1}^I $ such that:
$\sum_{l=1}^L p_l \sum_{i=1}^I c_l^i = \sum_{l=1}^L p_l \sum_{i=1}^I w_l^i$ ?
Thank you i... |
H: Exact DE with inital value
Solve the given initial value problem and determine at least approximately where the solution is valid $(2x - y) dx + (2y- x) dy = 0, y(1) = 3 $.
My Solution: Let $M=2x-y \Rightarrow M_y=-1$ and let $N=2y-x\Rightarrow N_x=-1$, therefore the given DE is exact. Let $\Psi_x=M=2x-y\Rightarr... |
H: Show that $\left(1+\dfrac{1}{n}\right)^n$ is monotonically increasing
Show that $U_n:=\left(1+\dfrac{1}{n}\right)^n$, $n\in\Bbb N$, defines a monotonically increasing sequence.
I must show that $U_{n+1}-U_n\geq0$, i.e. $$\left(1+\dfrac{1}{n+1}\right)^{n+1}-\left(1+\dfrac{1}{n}\right)^n\geq0.$$
I am trying to go a... |
H: Parametrise curve by angle and convex curves
Can one parametrise any closed curve by the angle its tangent makes to the $x$-axis? I seem to remember that this is only possible for convex curves. Could anyone tell me why, please?
Also is necessarily true that for convex curves, the mean curvature is always positive?... |
H: Spherical shell enclosing the Earth's surface
What is the minimum thickness of a spherical shell that encloses the Earth's surface?
The two spheres are concentric, the outer sphere encloses all the surface (to the highest mountain), and the inner sphere excludes all the surface (down to the deepest ocean trench).... |
H: Compute $\lim_{n\to\infty} \frac{\int_{0}^{1} f(x) \sin^{2n} (2 \pi x) \space dx}{\int_{0}^{1} e^{x^2}\sin^{2n} (2 \pi x) \space dx}$
Suppose that $f$ is continuous on $[0, 1]$. Then calculate the following limit:
$$\lim_{n\to\infty} \frac{\displaystyle\int_{0}^{1} f(x) \sin^{2n} (2 \pi x) \space dx}{\displaystyle... |
H: P, Q, R, S four points lie in a plane and PQ = PR = QR = PS then how many possible values of angle QSR can exist?
P, Q, R, S four points lie in a plane and PQ = PR = QR = PS then how many possible values of angle QSR can exist?
I think 2 values because PQRS is either a square or rhombus.
AI: Let $r = PQ = PR = QR =... |
H: linear transformation matrix
Any help on this linear transformation question is very much appreciated.
Let $V$ denote the real vector space $R^2$ and $\psi : V \rightarrow V$ be a real linear transformation such
that $\psi ((1, 0)) = (11, 8)$ and $\psi ((0, 1)) = (4, 3)$. Express the image $\psi ((x, y))$ of $(x, ... |
H: Proof that every normed vector space is a topological vector space
The topology induced by the norm of a normed vector space is such that the space is a topological vector space.
Can you tell me if my proof is correct? Of course we have to show that addition and scalar multiplication are continuous with respect to ... |
H: $(a+b)^\beta \leq a^\beta +b^\beta$ for $a,b\geq0$ and $0\leq\beta\leq1$
It seems that $(a+b)^\beta \leq a^\beta +b^\beta$ for $a,b\geq0$ and $0\leq\beta\leq1$. However, I could not prove this nor the same result for a general concave and increasing function (for which it might not hold). If the inequality is true,... |
H: Maximum-likelihood estimation for continuous random variable with unknown parameter
Let $X$ be a random variable with the unknown parameter $\lambda$ and the following pdf
$$f(t)=2\lambda t\cdot\mathrm e^{-\lambda t^2}\cdot\textbf{1}_{[0,\infty)}(t)$$
where $\textbf{1}_A(x)$ is an indicator function with
$$\... |
H: Check if a relation is a partial or total order and find minimum, maximum, minimal and maximal elements
Given $\mathbb Z^-=\{x\in \mathbb Z:x<0\}$ and $T = \mathbb Z^-\times \mathbb N$, let the binary relation $\odot$ be defined as follows:
$$\begin{aligned} (a,b) \odot (c,d) \Longleftrightarrow a \leq c \land b \m... |
H: Calculating new rotation matrix with its derivative given
I've got a skew-symmetric matrix representing gyroscope measurements, say $\Omega = [p,q,r]^T$, with $p$, $q$, $r$ being the angular velocities around $X$, $Y$ and $Z$ axes. I know my system's dynamics is:
$\dot{R} = R \Omega_\times$
with $\Omega_\times$ be... |
H: From an equality to a comparison
Let $X$ be a set. Let $0$ be an element of $X$.
For a function $P$ defined on tuples of $n$ elements of the set $X$ we know (for every tuples $f$ and $g$ each having $n$ elements)
$$\forall i \in n : ( f_i \neq 0 \wedge g_i \neq 0 ) \wedge P f = P
g \Rightarrow f = g.$$
Let $X$ be a... |
H: If $p, q$, and $r$ are relatively primes, then there exist integers $x$, $y$, and $z$ such that $px + qy + rz = 1$
True/False
If $p, q$, and $r$ are relatively primes, then there exist integers $x, y$, and $z$ such that $px + qy + rz = 1$
NOTE: $p, q$, and $r$ are positive primes.
AI: Let's take the more general qu... |
H: Stuck on space curves for vector valued functions
I'm working through the James Stewart Calculus text to prep for school. I'm stuck at this particular point.
How would you sketch the graph for the parametric equations:
$x = \cos t$, $y = \sin t$, and $z = \sin 5t$? I understand that if it were the case that $z=t$,... |
H: If $A^n$ has a free subset of $n+1$ elements, it has an infinite free subset.
I want to prove the following:
Let $A$ be a ring and $n$ a natural number. If the left $A$-module $A^n$ contains a free subset of $n+1$ elements, then $A^n$ already contains an infinite free subset.
Since we can embed $A^{n+1}$ into $A... |
H: Show that the limit of functions is continuous
Let $f_n$ be a sequence of not necessarily continuous functions $\mathbb{R} \rightarrow \mathbb{R}$ such that $f_n(x_n) \rightarrow f(x)$ whenever $x_n \rightarrow x$. Show that f is continuous.
What I am trying to do is to show that whenever we have $x \in \mathbb{R}... |
H: Closed form solution of Fibonacci-like sequence
Could someone please tell me the closed form solution of the equation below.
$$F(n) = 2F(n-1) + 2F(n-2)$$
$$F(1) = 1$$
$$F(2) = 3$$
Is there any way it can be easily deduced if the closed form solution of Fibonacci is known?
AI: Any of the standard methods for solving... |
H: Pythagorian quadruples
From my work on hyperelliptic equations I found how to get infinitely many solutions of the equation $a^4+b^4+c^2=d^4$. I call these solutions harmonic:
$$\begin{array}{rcccccl}
1^4 &+& 2^4 &+& 8^2 &=& 3^4\\
2^4 &+& 3^4 &+& 48^2 &=& 7^4\\
3^4 &+& 4^4 &+& 168^2 &=& 13^4\\
4^4 &+& 5^4 &+& 440^2... |
H: To show sum of residues of $f(z)$ over all poles is $0$
Let $p(z)$ and $q(z)$ be relatively prime polynomials with complex co-efficients so that $deg(q(z))\ge deg(p(z))+2$ and let $f(z)=p(z)/q(z)$. We need to show that the sum of residues of $f(z)$ over all poles is $0$
Well, I tried like this:
by Residue theorem: ... |
H: Is there easier way to calculate the limit of this function?
$$
\lim_{K\rightarrow\infty}\frac{(1-\epsilon)^K}{1+(1-\epsilon)^K}\frac{\sum_{i=1}^{\frac{K-1}{2}}\left(\begin{array}{l}
K \\
i \end{array}\right)\left[\left(\frac{2\epsilon-\epsilon^2}{(1-\epsilon)^2}\right)^i-\left(\frac{\epsilon}{1-\... |
H: Polynomial-related manipulation
My question is:
Factorize: $$x^{11} + x^{10} + x^9 + \cdots + x + 1$$
Any help to solve this question would be greatly appreciated.
AI: $$
\begin{align}
& {}\quad (x^{11} + x^{10}) + (x^9 + x^8)+(x^7+x^6)+(x^5+x^4)+(x^3+x^2 )+( x + 1)\\[8pt]
& =x^{10}(x+1)+x^8(x+1)+x^6(x+1)+x^4(x+1... |
H: Splitting field and subextension
Definition: Let $K/F$ be a field extension and let $p(x)\in F[x]$,
we say that $K$ is splitting field of $p$ over $F$ if $p$ splits
in $K$ and $K$ is generated by $p$'s roots; i.e. if $a_{0},...,a_{n}\in K$
are the roots of $p$ then $K=F(a_{0},...a_{n})$.
What I am trying to underst... |
H: Representation of Cyclic Group over Finite Field
The post Irreducible representations of a cyclic group over a field of prime order discusses the irreducible representations of a cyclic group of order $N$ over a finite field $\mathbb{F}_p$ where $N$ does not divide $p$.
Where can I find information about the irredu... |
H: integration of a continuous function $f(x) $ and $xf(x)$ is zero
Possible Duplicate:
Prove that $\exists a<b$ s.t. $f(a)=f(b)=0$ when $\int_0^1f(x)dx=\int_0^1xf(x)dx=0$
Suppose that $f:[0,1]\to \mathbb{R}$ is continuous, and that $$\int_{0}^{1} f(x)=\int_{0}^{1} xf(x)=0.$$
How does one prove that $f$ has at leas... |
H: Partition of Unity question
I am starting to read the book "Differential Forms in Algebraic Topology" by Bott and Tu.
In the proof of the exactness of the Mayer - Vietoris sequence (Proposition 2.3, page 22 - 23) a partition of unity $\{\rho_U,\rho_V\}$ subordinate to an open cover of two open sets $U,V$ is applie... |
H: Different Lifts of the Same Function
I'm just learning algebraic topology and have hit a problem I can't do. Lets say we have two topological spaces $X$ and $Y$ where $X$ is connected, and a continuous function $f:X \to Y$. Let $p:Y^\prime\to Y$ be a covering map of $Y$. Say $f_1^\prime$ and $f_2^\prime$ are two... |
H: Construction of special $\omega_1$-Aronszajn tree
Problem from Kunen II.40:
The definition is the following: An $\omega_1$-Aronszajn tree $T$ called special iff $T$ is the union of $\omega$ antichains.
Need to prove that $T$ is special iff there is a map $f: T \rightarrow \mathbb{Q}$ such that for $x,y \in T, x < y... |
H: Is the integral closure of a Henselian DVR $A$ in a finite extension of its field of fractions finite over $A$?
This question is related to the one here: A question related to Krull-Akizuki theorem
In the answers to that question, some examples are given of a discrete valuation ring $A$ and a finite (necessarily in... |
H: Percentages Issue
I am having a problem with the following question could you guys tell me what I am doing wrong?
When the tires of a taxicab are under-inflated, the cab's odometer will read $10\%$ over the true mileage. If the odometer of a cab with under-inflated tires read $m$ miles, what is the actual distance... |
H: Am I allowed to realize one object twice within one set-theory?
Say I consider a set theory with the Axioms of Extensionality and the Axiom of Pairing.
As I understand it, stating the axiom allows me to make a definition like
$$(a,b):=\{\{a\},\{a,b\}\}$$
and work with that $(a,b)$ in the context of my theory. Pairi... |
H: Product norm on infinite product space
Today I proved that if $V$ is a normed space with norm $\|\cdot\|$ then I can define a norm on $V \times V$ that induces the same topology as the product topology as follows: $\| (v,w) \|_{V \times V} = \|v\| + \|w\|$.
I think I can do the same for an infinite product $V^{\mat... |
H: Lying-over theorem without Axiom of Choice
This question is motivated by this and this.
Can the following proposition be proved without Axiom of Choice?
Proposition:
Let $k$ be a field.
Let $A$ and $B$ be commutative algebras without zero-divisors which are finitely generated over $k$. Suppose that $A$ is a subring... |
H: Set of finite subsets as vector space: Double dual?
Some problem I've found while thinking about duals of vector spaces:
Be $S$ an arbitrary set. Denote by $F(S)$ the set of finite subsets of $S$, and by $P(S)$ its power set.
Now it is easy to see that $F(S)$ forms a vector space over $\mathbb{Z}/2\mathbb{Z}$ if yo... |
H: Angles of a quadrilateral from a ratio.
I cant seem to find the angles here any suggestions ?
A quadrilateral has angles in the ratio 1:2:3 and a fourth angle that is 31 degrees larger than the smallest angle.What is the difference in degree between the middle two angles ?
AI: The angles in the quadrilateral sum ... |
H: support of a differential form on manifold
In the book "Differential forms in Algebraic Topology" by Bott and Tu, the support of a differential form $\omega$ on a manifold $M$ is defined to be "the smallest closed set $Z$ so that $\omega$ restricted to $Z$ is not $0$." (page 24).
I am a little confused, suppose we ... |
H: Proving $\frac{1-q}{q} - \frac{1-q^{x}}{xq^{x}} \geq 0 $
I am trying to find a nice way to verify that whenever $q \in (0,1)$ and $x \in (0,1)$, then $$\frac{1-q}{q} - \frac{1-q^{x}}{xq^{x}} \geq 0 .$$
AI: Take the derivative with respect to $q$ to get $$\frac{q^{1-x} - 1}{q^2}$$ which is negative for $q,x \in (0,1... |
H: Prove pseudoprime $N$ and base a must be relatively prime.
I would really appreciate some hints. Sorry if this is too easy. Thanks sincerely. Also, technically this is not homework but it is a problem from a textbook. Prove:
$a^{N-1} \not \equiv 1($mod$ \ N)$ if the gcd$(a,N)>1$. Where $a,N \in \mathbb{Z}$ and $N ... |
H: what is the physical importance of unitary group
What would be the physical importance of unitary group? By physical, I mean geometric intuiton and the usages in physics.
Also, how would special unitary group be used?
AI: In quantum mechanics, the time evolution of states is unitary. Also, many important symmetry o... |
H: Upper half plane is complete with the Lobatchevski metric
How do I show that the Upper half plane is complete with the Lobatchevski metric? I tried to use the fact that $M$ is complete iff the lengh of any divegert curve is unbounded,but did not get any results.thanks.
AI: Here's one possible approach:
If a Rieman... |
H: Prove that if $R$ is von Neumann regular and $P$ a prime ideal, then $P$ is maximal
Let $R$ be a commutative ring with $1\neq 0$. $R$ is said to be von Neumann regular if for all $a\in R$, there is some $x\in R$ such that $a^2x=a.$ Prove that if $R$ is von Neumann regular and $P$ a prime ideal, then $P$ is maximal... |
H: An integral domain whose every prime ideal is principal is a PID
Does anyone has a simple proof of the following fact:
An integral domain whose every prime ideal is principal is a principal ideal domain (PID).
AI: Here is a proof followed by conceptual elaboration, from my 2008/11/9 Ask an Algebraist post.
Let ... |
H: Find the radius of the smallest circle in terms of the side of the square.
The side length of the square is $a$. Two different quadrants are inscribed in it as follows. A small red circle is also inscribed between them.
Find the radius of the smallest circle (red one) in terms of '$a$' only.
AI: $r=\frac{3^2-2^2\sq... |
H: Every prime ideal is either zero or maximal in a PID.
$(1)$ Let $R$ be a commutative ring with $1\neq 0.$ If $R$ is a PID, show that every prime ideal is either zero or maximal.
In many books I have found the proof of the above statement where they show that
(2)Let $R$ be a commutative ring with $1\neq 0.$ If $... |
H: Motivation for adjoint operators in finite dimensional inner-product-spaces
Given a finite dimensional inner-product-space $(V,\langle\;,\rangle)$ and an endomorphism $A\in\mathrm{End}(V)$ we can define its adjoint $A^*$ as the only endomorphism such that $\langle Ax, y\rangle=\langle x, A^*y\rangle $ for all $x,y\... |
H: Are there "variables/unknowns" for operations?
We use letters for unknowns/variables:
$x^2=4$
Are there variables/unknowns for operations too?
$8 \star 7 $
With the $\star $ being any operation.
AI: Of course. For example, the Cayley-Hamilton theorem states that, if $a_nx^n+\cdots+a_1x+a_0$ is the characteristic p... |
H: $p$ is a $3$-digit prime, then there always exist $p$ consecutive composite numbers.
True/False
If $p$ is a $3$-digit prime, then there always exist $p$ consecutive composite numbers.
How to approach this?
AI: $P_k =(s + 1)! + k$ for $k = 2$ to $(s + 1)$
are $s$ consecutive positive integers
as $P_k$ is always di... |
H: Constructions of small set with big difference set
Does anyone know any constructions of a small set with a big difference set? Mathematically speaking:
Let $A\subseteq \mathbb{Z}$, such that $A-A=\mathbb{Z}_n$. Please give a sequence $(A_n)_{n\in \mathbb{N}}$ such that $|A_n|$ is small in terms of $n$.
AI: Here is... |
H: How can I properly isolate the variables for this differential equation?
I'm attempting to solve a problem involving this differential equation:
$$\frac{dy}{dx} = x^2y^2 + x^2 - y^2 - 1$$
Because this is a separable differential equation, I tend to split the equation between variables $x$ and $y$, and integrate bot... |
H: how to differentiate a single vector
I am trying to understand the basic rules of vector differentiation.
In a non-scalar expression where x is a column vector, is it valid to differentiate with respect to the kth element of x ?
For example, what would be the result of the following?
$ \partial/\partial x_k (x^T)... |
H: Question in do Carmo's book Riemannian geometry
This is a question on Do Carmo's book "Riemannian Geometry" (question 7 from chapter 7):
Let $f:M\to \bar{M}$ be a diffeomorphism beetwen two riemannian manifolds. Suppose $\bar{M}$ complete and that there is $c>0$ such that: $$|v|\geq c|df_pv|$$ for every $p\in M$ a... |
H: $f(x)$ is irreducible if and only if $f(x)$ does not have a root in $\mathbb{Z}/\mathbb{2Z}.$
Let $f(x)$ be a polynomial in $(\mathbb{Z}/\mathbb{2Z})[x]$ of degree $2$ or $3$. Prove that $f(x)$ is irreducible if and only if $f(x)$ does not have a root in $\mathbb{Z}/\mathbb{2Z}.$
I know that $f(x)$ is irreducible... |
H: In a field $F=\{0,1,x\}$, $x + x = 1$ and $x\cdot x = 1$
Looking for some pointers on how to approach this problem:
Let $F$ be a field consisting of exactly three elements $0$, $1$, $x$. Prove that
$x + x = 1$ and that $x x = 1$.
AI: Hint 1: We know that $1x=x$ and $0x=0$. But $x$ must have a multiplicative ... |
H: Solve the differential equation $y'=|x|$, $y(-1)=2$
Given the differential equation,
$y'=|x|$, $y(-1)=2$
I believe I understand how to solve the implicit solution but have questions about using the initial condition to solve the explicit solution. I have outlined my solution below in case there is a mistake I have... |
H: Proving a metric space to be compact
I have the following metric space: The set $X$ of all sequences with members from the set $\{1,2,\ldots, n\}$, together with the metric $$d(x,y)=\frac{1}{\min\{j\in\mathbb{N}:x_j\ne y_j\}}.$$ I wish to prove two things about this space:
1) $X$ is compact.
2) If $T:X\to X$ is d... |
H: Find functions family satisfying $ \lim_{n\to\infty} n \int_0^1 x^n f(x) = f(1)$
I wonder what kind of functions satisfy
$$ \lim_{n\to\infty} n \int_0^1 x^n f(x) = f(1)$$
I suppose all functions must be continuous.
AI: Your equation can be rewritten as
$$\lim_{n \to \infty} (n+1) \int_0^1 x^n (f(x) - f(1))\ dx = 0... |
H: Torque calculation, to achieve clean spin+tumble
Here's a pencil-like robotic spaceship carrying an experiment, it is a solid mass 100m long, 100 inches thick and weighs 1000kg. We're in deep solar space 100au above the sun.
Assume we can apply any platonic torque to the object.
Notice the global unchanging XYZ ax... |
H: Check if $(\mathbb Z_7, \odot)$ is an abelian group, issue in finding inverse element
Take $\mathbb Z_7$ and the operation $\odot$ defined on it as follows $\forall a,b \in \mathbb Z_7$:
$$\begin{aligned} a \odot b=a+b+3\end{aligned}$$
Check if $(\mathbb Z_7, \odot)$ is a group and in particular if it is an abelian... |
H: Is the adjoint representation of SO(4) self-dual?
The adjoint representation (over the complex numbers) of SO(4) is 6-dimensional. Is this representation self-dual?
Other than the adjoint representation and its dual, are there other irreducible 6-dimensional representations of SO(4) over the complex numbers?
AI: Th... |
H: Hölderian path connectedness
Let $X$ be a complete metric space and $\alpha \in (0,1)$. Suppose that for every $x,y \in X$ there exists $z \in X$ s.t.
$$
d(x,z) \le \frac{1}{2^\alpha}d(x,y), \qquad d(y,z) \le \frac{1}{2^\alpha}d(x,y).
$$
Then $X$ is Hölderian path-connected, i.e. for every $x,y \in X$ we can fi... |
H: Given that $x=\dfrac 1y$, show that $∫\frac {dx}{x \sqrt{(x^2-1)}} = -∫\frac {dy}{\sqrt{1-y^2}}$
Given that $x=\dfrac 1y$, show that $\displaystyle \int \frac 1{x\sqrt{x^2-1}}\,dx = -\int \frac 1{\sqrt{1-y^2}}\,dy$
Have no idea how to prove it.
here is a link to wolframalpha showing how to integrate the left sid... |
H: Dirichlet problem, uniqueness & counterexample
Help me please with this example:
Let's $q(x)$ be a continuous and constant sign function in $[0,1]$;
uniqueness of solution of Dirichlet problem to equation: $u''+qu=0$ depends on sign of $q$.
Prove the theorem of uniqueness in cases then it's true and give counterexa... |
H: Show that $f'' > 0$, $\lim_{x \to b^-} = \infty$ implies that $\lim_{x \to b^-} f'(x) = \infty$
Let $f$ be a continuous function on $[a,b)$, $f$ twice differentiable in $(a,b)$ so that $f''(x)>0$ for each $x \in (a,b)$. Prove that if $$\lim_{x\to b-}f(x) =\infty $$ then
$$ \lim _{x\to b-}f'(x)=\infty $$
AI: Suppo... |
H: Why isn't GL system of provability logic reflexive?
Formula $\square p \rightarrow p$ (axiom T; corresponding to reflexive modal frames) is interpreted as "if p is provable, then p", or more precisely: for all realizations (all substitutions for $p$), $PA \vdash Bew(\ulcorner p \urcorner) \rightarrow p$ (PA is syst... |
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