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H: A question about the convergence of the integral
Define $u : (-1,1) \to \Bbb R$ by $$ u(x) = \left( \log \frac{1}{1-x} \right)^{\alpha} ( 1/2 \leqslant x < 1 ), \;\;\;\;\;u(x) = (\log 2 )^{\alpha} \;( -1 < x \leqslant 1/2 ) $$
If $0 < \alpha < 1/2$ then I want to prove that $$ \int_{-1}^1 (1-x^2 ) | u'(x) |^2 dx ... |
H: Probability density function of a quotient of two normal random variables
I have the following expression:
$$R=\frac{\sigma_1^2\nu_1(t)-\sigma_2^2\nu_2(t)}{\sigma_1^2\nu_1(t)+\sigma_2^2\nu_2(t)}$$
where:
$$[\nu_1(t),\nu_2(t)]$$ are two independent normally distributed random variables.
My question is: how can I fin... |
H: Sum of n different positive integers is less than 100. What is the greatest possible value for n?
I am stumped on the following question:
The sum of n different positive integers is less than 100. What is the greatest possible value for n?
a) 10, b) 11, c) 12, d) 13, e) 14
The answer is d).
Any idea on how to s... |
H: construct $\{a_n\}$, for which exists $\{n_k\}$ and $\sum_{n=1}^{n_k}a_nx^n$ converges uniformly to $f$, where $f\in C[0,1]$, $f(0)=0$
Construct the sequence $\{a_n\}\subset\mathbb R$ such, that for every $f\in C[0,1]$ with $f(0)=0$, there exists a sequence $\{n_k\}$ for which
\begin{equation}
\lim_{k\to\infty}\sup... |
H: How to find rate of depreciation in this problem?
The value of a machine is estimated to be 27,000 at the end of 1994 and 21,870 at the beginning of 1997. Supposing it depreciates at a constant rate per year of it's value at the beginning of the year, calculate:
1) Rate of depreciation
2) The value of the machine a... |
H: How many rectangles can fit in a polygon with n-sides?
I am trying to write an algorithm to solve a problem I have. I have a few ideas of what the algorithm might be like but I am posting to see if anyone else has a better more efficient solution or any helpful suggestions. For context, this algorithm is going to b... |
H: Is the problem of calculating the induced norm *difficult*?
Is the problem of calculating the induced norm of a linear operator (in a finite or infinite-dimensional space) generally a difficult one ?
And by difficult I mean, that there are no closed formulas or no general procedure that always yields the induced no... |
H: How to find the original population if it increases with a constant rate?
The population increases by 5% every year. What was the population in 1982, if in 1985 it was 1,85220?
My working:
population in 1985 = 1,85,220
rate=5%
time = 3yrs
( A=P(1-R/100)^n )
therefore, population in 1982 = 185... |
H: (ZF) Every nonempty perfect set in $\mathbb{R}^k$ is uncountable.
This is the part of proof in Rudin PMA p.41
Let $P(\subset \mathbb{R})$ be a perfect set. Since $P$ has limit points, $P$must be infinite. Suppose that $P$ is countable. Then, we can denote the points of $P$ by $x_1, x_2,...$.
Let $V_1$ be any neighb... |
H: How many int. values of n will the expression be greater than 1
How would I solve this problem:
How many integer values of n will the expression $4n+7$ be an integer greater than 1 and less than 200 a)48 b)49 c)50 d)51 e)52 ? Ans 50
I am trying to solve this by using inequalities and doing something like this
$... |
H: Show the result of the following infinite sum, based on a binomial random variable conditioned on a Poisson random variable
$$\sum_{n=0}^\infty \binom{n}{k}p^k(1-p)^{n-k}\frac{\lambda^ne^{-\lambda}}{n!} = \frac{(\lambda p)^ke^{-\lambda p}}{k!}$$
That is, given a random variable $X$ with Poisson distribution $X \sim... |
H: Using Simpson's Rule to approximate $\int_0^3{\sqrt{9-x^2}dx}$
I have to use Simpson's rule:
$$\int_a^b f(x) \, dx \approx S_n = \frac{1}{3}h[f(a)+4f(x_1)+2f(x_2)+4f(x_3)+2f(x_4)+\cdots+4f(x_{n-1})+f(b)]$$
when $n=6$ to approximate the integral:
$$\int_0^3{\sqrt{9-x^2}dx}$$
to four decimal places.
I've gotten $$f(0... |
H: Are bounded open regions in $\mathbb{R}^n$ determined by their boundary?
Let $U$ and $V$ be two bounded open regions in $\mathbb{R}^n$, and let us further assume that their topological boundaries are nice enough that they are homeomorphic to finite simplicial complexes.
Assume $\partial U$ is homeomorphic to $\pa... |
H: What is smallest possible integer k such that $1575 \times k$ is perfect square?
I wanted to know how to solve this question:
What is smallest possible integer $k$ such that $1575 \times k$ is a perfect square?
a) 7, b) 9, c) 15, d) 25, e) 63. The answer is 7.
Since this was a multiple choice question, I guess ... |
H: What function has a graph that looks like this?
I delete my file which I used to produce this graph. Does anybody have some idea how to produce it again?
Thanks for a while.
AI: For example :
$$f(x,y)=\sqrt{x^2+y^2}\sin(8 \arctan(y/x))$$ |
H: Is there a Cantor-Schroder-Bernstein statement about surjective maps?
Let $A,B$ be two sets. The Cantor-Schroder-Bernstein states that if there is an injection $f\colon A\to B$ and an injection $g\colon B\to A$, then there exists a bijection $h\colon A\to B$.
I was wondering whether the following statements are tru... |
H: Generating function with binomial coefficients
I want to derive formula for generating function $$\sum_{n=0}^{+\infty}{m+n\choose m}z^n$$ because it is very often very useful for me. Unfortunately I'm stuck:
$$ f(z)=\sum_{n\ge 0}{m+n\choose n}z^n= \\ \sum_{n\ge 1}{m+n-1\choose n-1}z^n+\sum_{n\ge 0}{m+n-1\choose n}z... |
H: Different ways to wedge spaces
I am not a topologist, so please excuse me if the question is trivial.
Suppose I am given three nice, path-connected spaces. Then I can think of two ways to wedge these spaces: join all three at a common basepoint, or have one space in the "middle" with two basepoints (so I guess ther... |
H: Distribution function and $L^p$ spaces.
I saw the result below without a proof and I would like to see it.
Result:
Let $g$ a nonnegative and measurable function in $\Omega$ and $\mu_{g} $ its distribuction function, i.e.,
\begin{equation}
\mu_{g}(t)= |\{x\in \Omega : g(x)>t\}|, t>0.
\end{equation}
Let $\eta... |
H: Finitely generated module over $\mathbb{Q}$.
Consider the following problem.
Problem: Suppose $M$ is a module over $\mathbb{Q}[x]$ such that $M$ is finitely generated over $\mathbb{Q}$. Prove that there is a non-zero polynomial $p(x) \in \mathbb{Q}[x]$ and a non-zero $m \in M$ such that $p(x)*m = 0$.
My attempt: Si... |
H: Blocks in sequences from {1,...,k}
In Lecture Notes on Enormous integers Harvey M. Friedman introduces
"... longest finite sequence $x_1,...,x_n$ from $\{1,...,k\}$ such that for
no i < j <= n/2 is $x_i,...,x_{2i}$ a subsequence of $x_j,...,x_{2j}$.
For k ≥ 1, let n(k) be the length of this longest finite
sequence.... |
H: basic question on integral schemes
If $X$ is a (reduced) scheme and $P$ is a point of $X$ (not necessarily closed) such that the local ring $\mathcal{O}_{X,P}$ is a regular domain, then must there exist an open affine neighborhood $U = \text{Spec }A$ of $P$ such that $A$ is an integral domain?
I'm almost certain th... |
H: Solving $5^n > 4,000,000$ without a calculator
If $n$ is an integer and $5^n > 4,000,000.$ What is the least possible value of $n$? (answer: $10$)
How could I find the value of $n$ without using a calculator ?
AI: \begin{eqnarray}
& 5^n &>& 4.000.000\\
\Leftrightarrow & 5^n &>& 5^6 \cdot 2^8 \\
\Leftrightarrow &... |
H: Geometric difference between two actions of $GL_n(\mathbb{C})$ on $G\times \mathfrak{g}^*$
Let $G=GL_n(\mathbb{C})$.
Scenerio 1: Let $G$ act on $T^*(G)=G\times \mathfrak{g}^*$ by
$$
g.(x,y)=(gx,y).
$$
Scenerio 2: Let $G$ act on $T^*(G)=G\times \mathfrak{g}^*$ by
$$
g.(x,y)=(xg^{-1},gyg^{-1}).
$$
Is there a ge... |
H: I want to prove $ \int_0^\infty \frac{e^{-x}}{x} dx = \infty $
How can I prove this integral diverges?
$$ \int_0^\infty \frac{e^{-x}}{x} dx = \infty $$
AI: $$
\int_{0}^{\infty}\frac{e^{-x}}{x}= \int_{0}^{1}\frac{e^{-x}}{x}+\int_{1}^{\infty}\frac{e^{-x}}{x} \\
> \int_{0}^{1}\frac{e^{-x}}{x} \\
> e^{-1}\int_{0}^{1}\... |
H: Proof that both $ \int_0^\infty | \ell_0 (x) |^2 e^{-x} dx$ and $\int_0^\infty | \ell_1 (x) |^2 e^{-x} dx $ diverge
Define $ \ell_0 (x)$ by $$ \ell_0 (x) := \int_1^x \frac{e^{\zeta}}{\zeta} d \zeta $$ and define $\ell_1 (x)$ by $\ell_1 (x) := (1-x) \ell_0 (x) + x \ell_0 ' (x).$ Then I want to prove that $$ \int_0^... |
H: Constructing a local nested base at a point in a first-countable space
I am trying to prove the following:
Let $X$ be a first countable space and $x$ a member of $X$. Prove that there is a local nested basis $\{S_n\}_{n=1}^\infty$ at $x$.
Since $X$ is first countable there is a countable local base $\mathcal{B}... |
H: A calculation involving two circles
I have two circles, one of which is completely within the other. They do not touch, but are not necessarily concentric. I am given the sum of their circumferences, and the difference in their areas (ie, the area of the space inside the outer circle and outside the inner circle). ... |
H: Properties and identities of $\text{ord}_{p}(n)$
$\mathrm{ord}_{p}(a+b)\ge\mathrm{min}(\mathrm{ord}_{p}a,\mathrm{ord}_{p}b)$ with equality holding if $\mathrm{ord}_{p}a\ne \mathrm{ord}_{p}b$. is a the statement that prompted this question.
It was found in Ireland & Rosen's Elements of Number Theory (precurser to th... |
H: The orientation of quotient manifold
If $T$ is a torus and $\mathbb Z_2$ acts on it by $(z_1,z_2)\rightarrow(z^{-1}_1,-z_2)$, then is $T/\mathbb Z_2$ orientable?
AI: This argument is morally the same as Matt E's, but interpreted in de Rham cohomology. We denote the quotient by $X$; observe that $X$ inherits the str... |
H: Definition and example of a partition
Partition of a Set is defined as "A collection of disjoint subsets of a given set. The union of the subsets must equal the entire original set." For example, one possible partition of $(1, 2, 3, 4, 5, 6 )$ is $(1, 3), (2), (4, 5, 6).$ Rudin, while defining integral on page $12... |
H: Differentiation continuous iff domain is finite dimensional
Let $A\subset C([0,1])$ a closed linear subspace with respect to the usual supremum norm satisfying $A\subset C^1([0,1])$. Is $D\colon A\rightarrow C([0,1]), \ f\rightarrow f'$ continuous iff $A$ is finite dimensional?
If $A$ is finite dimensional $D$ i... |
H: A question on the compact subset
This is an exercise from a topological book.
Let $X$ is Hausdorff and $K$ is a compact subset of $X$. $\{U_i:i=1,2,...,k\}$ is the open sets of $X$ which covers $K$. How to prove that there exist compact subsets of $X$: $\{K_i:i=1,2,...,k\}$ such that $K=\cup^k_{i=1}K_i$ and for an... |
H: L'hospital rule for two variable.
How to use L'hospital rule to compute the limit of the given function
$$\lim_{(x,y)\to (0,0)} \frac{x^{2}+y^{2}}{x+y}?$$
AI: There is no L'Hopital's Rule for multiple variable limits. For calculating limits in multiple variables, you need to consider every possible path of approa... |
H: The metrizable space may be not locally compact
My text book said:
Not every metrizable space is locally compact.
And it lists a counterexample as following: The subspace $Q=\{r: r=\frac pq; p,q \in Z\}$ of $R$ with usual topology, i.e., $Q$ is the set of all rationals. It said: for any open ball of any point $... |
H: $\int_{-\infty}^\infty e^{ikx}dx$ equals what?
What would $\int\limits_{-\infty}^\infty e^{ikx}dx$ be equal to where $i$ refers to imaginary unit? What steps should I go over to solve this integral?
I saw this in the Fourier transform, and am unsure how to solve this.
AI: $\int\limits_{-\infty}^\infty e^{ikx}dx$ i... |
H: the rank of an interesting matrix
Let $A$ be a square matrix whose off-diagonal entries $a_{i,j} \in (0,1)$ when $i \neq j$. The diagonal entries of $A$ are all 1s. I am wondering whether $A$ has a full rank.
AI: If
$$A=\begin{pmatrix}
1 & \frac34 & \frac12 &\frac34 \\
\frac34 & 1 & \frac34 & \frac12 \\
\frac12 &\f... |
H: Proof of existence of zero point
In the proof of existence of zero point, $f(x)$ is continuous in $[a,b]$, where $f(a)<0$ and $f(b)>0$.
It is shown on the proof process in the textbook that when we define a set $V$ as follows:
$$V=\{x |f(x)<0,x\in[a,b]\},$$
so, there exists the supremum for $V$. Take $\xi=\sup V$... |
H: Quaternions and Rotations
Two of the interesting achievements in Mathematics are Classification of platonic solids, and also classification of finite groups acting on the unit sphere in $\mathbb{R}^3$, and they are very nicely connected to each other. These objects also enter in the classification of finite subgrou... |
H: Vector Taylor series
From pg. 35 of Classical Electrodynamics 3rd edition, Jackson,
$$\begin{aligned} \nabla^{2} \Phi_{a}(\mathbf{x}) &=-\frac{1}{4 \pi \epsilon_{0}} \int \rho\left(\mathbf{x}^{\prime}\right)\left[\frac{3 a^{2}}{\left(r^{2}+a^{2}\right)^{5 / 2}}\right] d^{3} x^{\prime} \end{aligned}$$
"Choose R such... |
H: How to evaluate $\int \frac{\mathrm dx}{\sqrt[3]{\tan\,x}}$?
Please show me the steps of the following integration. I got an answer in Wolfram, but I need steps..
$$\int \frac{\mathrm dx}{\sqrt[3]{\tan\,x}}$$
AI: We try the substitution $t^3 = \tan^2 x$. Therefore, $3t^2 dt = 2 \tan x \sec^2 x dx$, giving us $\frac... |
H: How can I generate the binary representation of any real number?
In p. 30 of Baby Rudin, I find a reference to the fact that the binary representation of a real number implies the uncountablity of the set of real numbers. But I have two questions:
Does every real number have a binary representation? If yes, how do... |
H: holomorphic function on the complex plane
let $ f(z) \in Hol(\mathbb{C}) $ holomorphic function such that for each $ z_0 \in \mathbb{C} $ there exists $ N(z_0) $ such that $ f^{(N(z_0))}(z_0) = 0 $
Prove: that f is a polynom
AI: Modifying Davide's argument but avoiding Baire's category theorem: Let $\overline{\math... |
H: Arc direction in given point
I have an arc with a given center, start angle, end angle, and radius.
I want to draw an arrow showing the arc direction in the arc middle point. What is the easiest way to calculate this direction (it does not need to be precise, it's only a representation issue)?
I've found the arc mi... |
H: Integral:$ \int^\infty_{-\infty}\frac{\ln(x^{2}+1)}{x^{2}+1}dx $
How to evaluate:
$$ \int^\infty_{-\infty}\frac{\ln(x^{2}+1)}{x^{2}+1}dx $$
Maybe we can evaluate it using the well-known result:$\int_{0}^{\frac{\pi}{2}} \ln{\sin t} \text{d}t=\int_{0}^{\frac{\pi}{2}} \ln{\cos t} \text{d}t=-\frac{\pi}{2}\ln{2}$
But h... |
H: Tips for an adult to learn math -- from the beginning.
First let me start with I am an adult and I can't do simple maths. I some how got through all of my math courses in University (after several attempts) but I honestly couldn't tell you how...
I cannot do these:
Add/Subtract with decimals
Add/Subtract fractions... |
H: Complex polynomial and the unit circle
Given a polynomial $ P(z) = z^n + a_{n-1}z^{n-1} + \cdots + a_0 $, such that
$\max_{|z|=1} |P(z)| = 1 $
Prove: $ P(z) = z^n $
Hint: Use cauchy derivative estimation
$$ |f^{(n)} (z_0)| \leq \frac{n!}{r^n} \max_{|z-z_0|\leq r} |f(z)| $$ and look at the function $ \frac{P(z... |
H: What's wrong with this conversion?
I need to calculate the following limes:
$$
\lim_{n\rightarrow\infty} \sqrt{\frac{1}{n^2}+x^2}
$$
My first intuition was that the answer is $x$, but after a bit of fiddling with the root I got thoroughly confused. I know that below conversion goes wrong somwhere, but where?
$$
\li... |
H: Is the product of two sets well-defined if one is empty
Let $X$ be a set. What is $X\times \emptyset$ supposed to mean? Is it just the empty set?
AI: And more can be said: a cartesian product is empty if and only if one of the two factors is empty. |
H: How to conclude $\Re $ is zero?
I'm in a Hilbert space $H$ and for $z,v, h \in H$ and $t \in \mathbb C$ I have
$$ \|z\|^2 \leq \|h−(tv+y)\|^2 = \|z−tv\|^2 =\|z\|^2 −2\Re(t⟨v,z⟩)+|t|^2\|v\|^2$$
According to my notes it follows from this that $\Re(t⟨v,z⟩) = 0$ for all $t$. How does that follow? I can't seem to show ... |
H: Show that if X is a discrete random variable such that for nonnegative integers m,n, $P(X>m+n|X>m)=P(X>n)$, then X is geometric
Here's my attempt, and I'm not sure if it is even close to the right direction.
If $P(X>m+n|X>m)=P(X>n)$ then if F is the cumulative distribution function of X, we have
$$\frac{P(X>m+n)}{... |
H: How to measure the volume of rock?
I have a object which is similar to the shape of irregular rock like this
I would like to find the volume of this. How to do it?
If I have to find the volume, what are the things I would need. eg., If it is cylindrical, I would measure length and diameter. But, it is irregularly ... |
H: Is this a correct definition of a line integral?
This comes from the beginning chapter on line integrals in the book Mathematical Methods for Science Students:
Suppose $y=f(x)$ is a real single-valued monotonic continuous function of $x$ in some interval $x_1<x<x_2$. Then if $P(x,y)$ and $Q(x,y)$ are two real sing... |
H: Residue of $\frac{\tan(z)}{z^3}$
What is the easiest way to calculate the residue of $\dfrac{\tan(z)}{z^3}$ at zero? I could either use the line integral theorem, or expand it out as a series. Is there a right way to do it?
AI: You don't have to calculate the series expansion. The $z^{-1}$-coefficient in the series... |
H: Is there a version of the Gershgorin circle theorem that is suitable for nearly triangular matricies?
The Gershgorin circle theorem, http://en.wikipedia.org/wiki/Gershgorin_circle_theorem, gives bounds on the eigenvalues of a square matrix, and works well for nearly diagonal matrices.
For a triangular matrix, how... |
H: what is $c$ in Mandelbrot set?
The Mandelbrot Set is an extremly complex object that shows new structure at all magnifications. It is the set of complex numbers $c$ for which the iteration indicated nearby remains bounded.
$$z_0=c$$
$$z_{n+1}=z_n^2+c$$
what is $c$ in Mandelbrot set?
isn't $c$ complex number
AI: Wh... |
H: Equivalents norms in Sobolev Spaces
I know that this is classical but I have never do the calculations to show that the norms in the sobolev space $W^{k,p}(\Omega)$
\begin{equation}
\|u\|_{k,p,\Omega}= \Bigl(\int_{\Omega} \sum_{|\alpha| \le k}|D^{\alpha} u |^{p} dx \Bigr)^{1/p}
\end{equation}
and
\begin{equation}... |
H: Two forms of quantified conditional statement: equivalent?
There seems to be two forms of the conditional statement in predicate logic.
$$\forall x\,(P(x)\Rightarrow Q(x))$$
versus
$$(\forall x\in S)\Rightarrow Q(x)$$
$$S=\{x:P(x)\}$$
Are these equivalent? I'm a bit confused in the second form because It looks lik... |
H: Correct interpretation of $E \subseteq V^{(2)}$
I'm trying to learn more about graph theory, but I'm getting confused by the initial definition:
"A graph $G = (V, E)$ is an ordered pair of finite sets. Elements of V are called vertices or nodes, and elements of $E \subseteq V^{(2)}$ are called edges or arcs. We re... |
H: How to disprove there exists a real number $x$ with $x^2 < x < x^3$
I realize that the only method is to show various cases:
I must test for $x > 1$, $x < -1$, $0 \leq x \leq 1$, and $-1\leq x \leq0$.
But even with this, I don't understand how to inject the properties of these four distinct possible $x$'s into the ... |
H: modified $\sum{k{n \choose k}}$ closed form expression
There is probably something stupidly simple I'm missing, but I'm trying to find a closed form for:
$$
2\sum_{k=1}^{(n-1)/2} k \, {n \choose k} \hspace{1cm} (n\textrm{ is odd})
$$
Anyone know how to do this?
I've figured out that since $n$ is odd,
$$
2+2\sum_{... |
H: What digit appears in unit place when $2^{320}$ is multiplied out
Is there a way to answer the following preferably without a calculator
What digit appears in unit place when $2^{320}$ is multiplied out ? a)$0$ b)$2$ c)$4$ d)$6$ e)$8$
---- Ans(d)
AI: Quickly look at the last digit of $2^n$, for $n=1$, $2$,... |
H: How to prove that a series expansion of pi has converged to a certain accuracy?
Wikipedia has a great article on methods for calculating pi with arbitrary precision, using for example Machin's infinite series expansion:
$\frac{\pi}{ 4} = 4$ arccot $5 - $arccot $ 239 $
where
arccot $x = \frac{1}{x} - \frac{1}{3 x^3}... |
H: How to find this moment generating function
I am trying to find the moment generating function of a random variable $X$, which has probability density function given by
$$f_{X}\left( x\right) =\dfrac {\lambda ^{2}x} {e^{\lambda x}}$$
Where $x>0$ and $λ>0$.
The standard approach to doing this i believe is to follo... |
H: Can the range of this operator be closed?
Given an operator $T:H\rightarrow L$, where $H$ is a finite-dimensional Hilbert space and $L$ an infinitedimensional one, is the range of $T$, $T(H)$, a closed set in $L$ ?
I know that the image doesn't have to be closed if $H$ were infinite, but I'm not sure in this case.
... |
H: Equality in rng with no zero divisors.
I'm working on this problem, but I'm missing some manipulation.
Suppose $R$ is a rng without zero divisors and has elements $a$ and $b\neq 0$ such that $ab+kb=0$ for some $k\in\mathbb{N}$ (that is,
$ab+\underbrace{b+b+\cdots+b}_{k\ \textrm{times}}=0$.) I'm trying to show that... |
H: Finding the order of another element of a group
If in a group $G$, $a^5 = e$ , $e$ is the identity element of $G$. If $a,b \in G$ and $aba^{-1}=b^2$ then find the order of b
I have got no clue how to go forward with it, except that perhaps we can construct a cyclic group of order 5 with a and thus the order of $G$ ... |
H: Cocountable fibers
Let $C$ be an uncountable set.
Can we construct a set $A \subseteq C^2$ such that it has a cocountable number of cocountable horizontal fibers, and a cocountable number of countable vertical fibers?
AI: Let $C = \omega_1$ and $<$ be the ordinal ordering of $C$. Since $C = \omega_1$, it is the lea... |
H: Extension of an operator defined on (not necessarily closed) subspaces
Is there always an extension of an operator $T:U\rightarrow W$, defined on (not necessarily closed) subspaces of the infinitedimensional Hilbert spaces $H\supseteq U,L\supseteq W$, to the operator $$T':cl(U)\rightarrow cl(W),$$ where $cl$ denote... |
H: Integral Galois Extensions (Lang)
I have trouble understanding an argument in the proof of Proposition 2.5, p. 342, of Lang's Algebra. The setup of my question is the following:
Let $A$ be integrally closed in its quotient field $K$ and $B$ be its integral closure in a finite Galois extension $L$ of $K$, with group... |
H: Solution space to a functional equation
This question comes from my attempts at understanding an example presented by Bill Gasarch on his blog. The example is of a continuous strictly increasing function whose derivative is zero almost everywhere. The example is apparently discussed in the book Probability and Meas... |
H: Simple inequality for measures
Let $(X,\mathscr B_X)$ and $(Y,\mathscr B_Y)$ be two measure spaces and $(Z,\mathscr B_Z)$ be their product space. Consider two finite measures (not necessarily product measures) $\mu,\nu$ on $(Z,\mathscr B_Z)$. Suppose that for any $A\in \mathscr B_X$ and for any $B\in \mathscr B_X$ ... |
H: Spivak Exercise involving operator norm
The exercise as stated:
If $T:\mathbb{R}^{m}\to \mathbb{R}^{n}$ is a linear transformation, show that there is a number $M$ such that $|T(h)|\leq M\cdot |h|$ for all $h\in \mathbb{R}^{n}$.
Hint: Estimate $|T(h)|$ in terms of $|h|$ and the other entries in the matrix of $T$.
B... |
H: Fast algorithms for calculating the Möbius inversion
Recall the Möbius inversion formula: if we have two functions $f,g \colon \mathbf{N} \to \mathbf{N}$ such that
$$g(n) = \sum_{k=1}^n f\left(\left\lfloor \frac{n}{k} \right\rfloor\right)$$
holds for every $n \in \mathbf{N}$, then the values of $f$ can be recovered... |
H: Cartesian products of families in Halmos' book.
I'm studying some set theory from Halmos' book. He introduces the generalization of cartesian products by means of families. However, I can't understand what is going on. I get the first introduction "The notation..." to "... one-to-one correspondence". What I'm havin... |
H: Computing $\frac{d^k}{dx^k}\left(f(x)^k\right)$ where $k$ is a positive integer
Does anyone know a formula for the derivative $$\frac{d^k}{dx^k}\left(f(x)^k\right)$$ where $k$ is some positive integer?
I started trying to work it out but it got messy.
AI: Apply Faà di Bruno's formula to get
$$\frac{d^n}{dx^n}(g(x... |
H: Fundamental matrix and exponential of matrix using Laplace Transform
I'm trying to work out how to find $$\exp(At)$$ for a system of linear differential equations $$x'=Ax.$$
I know that the solution is a fundamental matrix of the system such that $$\exp(At)=I$$
at time $0$.
What is the method for solving this usi... |
H: Zariski topology on prime $\mathrm{Spec}$ of a ring $R$
Let $R$ be a commutative unital ring. Let $\mathrm{Spec}(R) = \{ \mathfrak p \subset R \mid \mathfrak p \text{ a prime ideal of } R \}$. We define a set $C$ to be closed in this space if and only if there is an ideal $I$ such that $C(I) = \{\mathfrak p \mid I ... |
H: Is the function $\theta(a,b) = a-2ab+b$ a bijection from $\{0,1\}\times\mathbb{N}$ to $\mathbb{Z}$?
Consider the function $\theta:\{0,1\}\times\mathbb{N}\rightarrow\mathbb{Z}$ defined as $\theta(a,b) = a-2ab+b$. Is this function bijective?
For injective, I tried doing the contrapositive by supposing $\theta(a,b)=\t... |
H: Please explain how to do this proof (involving functions)
Suppose that $f:A \longrightarrow B$ and $g:B \longrightarrow C$ are functions.
If $g \circ f$ is onto and $g$ is one-to-one, then prove that $f$ is onto.
How do I go about proving this?
From $g \circ f$ is onto, I know that there exists an $a \in A$ such ... |
H: Prove by induction: $2^n = C(n,0) + C(n,1) + \cdots + C(n,n)$
This is a question I came across in an old midterm and I'm not sure how to do it. Any help is appreciated.
$$2^n = C(n,0) + C(n,1) + \cdots + C(n,n).$$
Prove this statement is true for all $n \ge 0$ by induction.
AI: Marvis's hint suffices. Anyways:
Ba... |
H: Higher dimensional analogue of an arc of a circle
What is the higher dimensional analogue for the arc of a circle?
I'd like to work with the set of all points lying within a certain distance of a given point on an n-sphere, and I'd like to describe these sets by the (generalised) solid angles which they subtend.
AI... |
H: Combinatorics question: Prove 2 people at a party know the same amount of people
I recently had an assignment and got this question wrong, was wondering what I left out.
Prove that at a party where there are at least two people, there are two people who know the same number of other people there. Be sure to use the... |
H: Increasing Sequence of Rationals
Let $x$ be any real number. Construct a sequence $x_n$ of rational numbers such that
$$x = \sup\{x_n : n \in \mathbb{N} \}.$$
I was trying $x_n = [ 10^n x ]/10^n$, but is it actually monotone increasing?
If so, how to prove it analytically?
Thanks for any help.
AI: Yes, your sequen... |
H: Approximating an $L^2$ function in the Riemann sense
$\newcommand{\R}{\Bbb R}$
Consider the Lebesgue measure in $\R$ and the following proposition:
P. For each representative of a function class $f\in L^2[0,1]$ there is a sequence of continuous functions $(f_n)_{n\in\Bbb N}$ such that:
$|f_n-f|$ is Riemann integr... |
H: Why isn't this square root $+$ or $-$?
I was tasked with proving the identity $\tan(\frac x 2) = \dfrac {\sin(x)}{1+\cos(x)}$
I used the quotient identity for tangent and the half angle identities for sine and cosine to get $ \pm \dfrac {\sqrt{\dfrac {1-\cos(x)}{2}}}{\sqrt{\dfrac {1-\cos(x)}{2}}}$
which I reduced t... |
H: Some exact sequence of ideals and quotients
I saw an exact sequence of ideals
$$0 \rightarrow I \cap J\rightarrow I \oplus J \rightarrow I + J \rightarrow 0$$In this sequence, maps are ring homomorphisms or module homomorphisms?
And how the above sequence yield the exact seqeunce
$$0 \rightarrow R/I \cap J\righta... |
H: Are there sub sigma algebras of the borel sigma algebra on the real line that are not sigma finite?
Im wondering if all sub sigma algebras of sigma finite measure space are also sigma finite?
AI: One example is $\{\emptyset,\mathbb R\}$.
Another is the $\sigma$-algebra of countable or cocountable sets. |
H: $\lim_{n\to\infty}n^{n\cdot a_{n}}=1$ as a sufficient condition for $\sum a_{n}<\infty$
[Not HW]
Let $\left(a_{n}\right)_{n=1}^{\infty}$
a sequence of real numbers such that, for all $ n\in\mathbb{N}$
, $0<a_{n}\leq\frac{1}{2}$
.
Why the following statements don't imply the convergence of $\sum_{n=1}^{\infty}a_... |
H: How to determine units in a partial differential equation
How do we determine the units used in a differential equation? Yes, in theory a PDE has nothing to do with units, but I'm interested in this question from a modeling point of view. By units, I mean the following. In an ordinary differential equation, find... |
H: complex function with real values on the real interval
let $ B(0,1) = \{ z\in \mathbb{C} | |z|<1\} $
and $ f $ be an holomorphic function on $ B(0,1) $ such that $ f(z)\in\mathbb{R} \iff z\in\mathbb{R} $
Prove: $ f $ has at most 1 root in $ B(0,1) $
i think this exercise requires rouche theorem or the argument prin... |
H: Finding the number of combinations of two dates from n dates where one precedes the other?
Suppose that I am given n consecutive dates for buying and selling shares . So what are the number of ways of choosing a pair of buy date and sell date such that buy date always precede sale date ?
AI: Pick any two distinct d... |
H: Number of line segments intersecting diagonals are divided into in a convex polygon
I am currently self-studying introductory combinatorics, and I don't fully get an example in the book.
The question was as follows:
If no three diagonals of a convex decagon meet at the same point, into how many line segments are ... |
H: How to show the subset $Y$ is closed discrete?
The example is following:
Let $Y=\{(0,y):y \in R\}$. Let $E \subset R^2$, i.e., the subset of the real plane, and $E=Y\cup \{ (\frac 1n, \frac k{n^2}): n\in Z^+, k \in Z\}$. The topology on $E$ is this:
The point $(\frac 1n, \frac k{n^2})$ is open;
$\{U_n(y_0): n=1,2,... |
H: What is the $\tau$ symbol in the Bourbaki text?
I'm reading the first book by Bourbaki (set theory) and he introduces this logical symbol $\tau$ to later define the quantifiers with it. It is such that if $A$ is an assembly possibly contianing $x$ (term, variable?), then $\tau_xA$ does not contain it.
Is there a r... |
H: Question about inverse limit
I'm puzzled by the definition of inverse limit in this Wolfram article.
I thought if an object was defined by a universal property it meant that the object is unique up to unique isomorphism. This would mean that in the article I linked, $\alpha$ should be a unique isomorphism not just ... |
H: $I\cdot J$ principal implies $I$ and $J$ principal?
Let $R$ be a Noetherian domain, and let $I$ and $J$ be two ideals of $R$ such that their product $I\cdot J$ is a non-zero principal ideal. Is it true that $I$ and $J$ are principal ideals ? This seems an easy question to settle, but I can't find an answer.
Any ide... |
H: Is the set of dyadic rationals a field?
I recently learned that the dyadic rationals is the set of rational numbers of the form
$$\frac{p}{2^q}$$
where $p$ is an integer and $q$ is greater than or equal to zero.
I think the set of dyadic rationals is not a field. Here's why:
One of the requirements for a set to be... |
H: Explanation of matrix elements containing integers modulo a prime
From Cormen et all:
The elements of a matrix or vectors are numbers from a number system, such as the real numbers , the complex numbers , or integers modulo a prime .
What do they mean by integers modulo a prime ? I thought real numbers and comple... |
H: Question on statistics
I have a kind of weird question. But this wont be a harder one.
Actually, i feel it is incomplete. I don't have much experience on statistics. But some advance user will be able to understand this (may be guess the incomplete parts). I'll post all the information i have. Please try to give an... |
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