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H: Primitive of a rational function
Does there exists some simple criteria to know when the primitive of a rational function of $\mathbb{C}[z]$ is still a rational function?
In fact my question is more about the stability of this property. Let $P$ and $Q$ two coprime polynomials and let $A$ and $B$ two coprime polyno... |
H: Is there a compact contractible manifold?
Does there exist a compact connected manifold (without boundary), that has a trivial homotopy type?
AI: No. Closed manifold of dimension $n$ has $H^n(M;\mathbb Z/2)\cong\mathbb Z/2$. |
H: The equation $f(s)=a$ has a finite number of solution
Let $f$ be an analytic function. I am asking about this problem:
What are sufficient conditions (and possibly necessary conditions) in which the equation $$f(s)=a$$ has a finite number of solution with respect to $s$. Here $a≠0$.
AI: It depends on the domain of ... |
H: Distinguishable telephone poles being painted
Each of n (distinguishable) telephone poles is painted red, white, blue or yellow. An odd number are painted blue and an even number yellow. In how many ways can this be done?
Can some give me a hint how to approach this problem?
AI: Proposition$^{[1]}$ Fix $k\in\math... |
H: Is this equality in a double category true?
Caveat: This is a utterly trivial question from a person who always learned to manipulate diagrams in a double category "from the ground"; I'll be glad even if you simply address me to any source which gives precise rules of transformations for such diagrams (and this exp... |
H: A subgroup containing a kernel of a group homomorphism into an abelian group is a normal subgroup.
Let $ f \colon G \rightarrow H $ be a group homomorphism, where $ H $ is an abelian group.
If $\ker f \subset N $ for some $ N \subset G $, then $ N $ is a normal subgroup of $ G $.
I don't know how to start to prove ... |
H: Notes about evaluating double and triple integrals
I'm searching notes and exercises about multiple integrals to calculate volume of functions, but the information I find in internet is very bad. Can someone recommend me a book, pdf, videos, website... whatever to learn about this?
The exercise type I have to learn... |
H: Given $y=\arccos(x)$ find $\arcsin(x)$ in terms of y
Given that $y = \arccos x$, $ - 1 \le x \le 1\,and\,0 \le y \le \pi $, express $\arcsin x$ in terms of y.
The best I know how to do this is is:
$$\eqalign{
& \cos y = x \cr
& {\cos ^2}y + {\sin ^2}y = 1 \cr
& {\sin ^2}y = 1 - {\cos ^2}y \cr
& \si... |
H: Prove that a function f is continuous (2)
$$f:\mathbb{R} \rightarrow \mathbb{R}:$$
$$f(x) =
\begin{cases}
\cos\left(\frac{1}{x}\right) & \text{if $x\neq0$} \\
0 & \text{if $x=0$}
\end{cases}$$
Is the function $f$ continuous in $x=0$?
1) $\displaystyle\lim_{x \to 0^-} \cos\left(\frac{1}{x}\right)$
with the sequen... |
H: Need help with combinatorics question(probably cyclical permutation)
A human invites 6 of his friends to a meeting. In how many different arrangements they along with the human's wife can sit at a round table if the hosts and the wife always sit together?
Is this a cyclical permutation problem? Please explain.
AI: ... |
H: Are there any integer solutions to $\gcd(\sigma(n), \sigma(n^2)) = 1$ other than for prime $n$?
A good day to everyone!
Are there any integer solutions to $\gcd(\sigma(n), \sigma(n^2)) = 1$ other than for prime $n$ (where $\sigma = \sigma_1$ is the sum-of-divisors function)?
Note that, if $n = p$ for prime $p$ then... |
H: Is there a simple method to do LU decomposition by hand?
Today my professor in numerical analysis pointed out that in the exam we will probably have to do LU decomposition by hand.
I understand how the decomposition works theoretically, but when it comes actually getting my hands dirty, I'm never sure, if I'm writi... |
H: Upper bound for expression involving logarithms
Let $N = 2^p$ for some $p \in \mathbb{N}$. Find the smallest upper bound for $\frac{N}{2}\log\left(\frac{N}{2}\right) + \frac{N}{4}\log\left(\frac{N}{4}\right) + \ldots + 1$
I guess I could first rewrite this to $\frac{2^p}{2}\log\left(\frac{2^p}{2}\right) + \frac{2^p... |
H: Exponential practice exam question
Okay bear with me, this is one of those cumulative questions
The amount of a certain type of drug in the bloodstream t hours after it has been taken is given by the formula:
$$x = D{e^{ - {1 \over 8}t}}$$
where x is the amount of the drug in the bloodstream in milligrams and D is... |
H: Wikipedia's definition of isolated point.
Wikipedia defines an isolated point of a subset $S \subseteq X$ to be a point $x \in S$ such that there exists a neighborhood $U$ of $x$ not containing any other points of $S$. Furthermore, it claims that this is equivalent to saying $\{x\}$ is open in $X$.
Question: How i... |
H: Approximation of a harmonic function on the unit disc by harmonic polynomials.
Let $u$ be a real valued harmonic function on the open unit disc $D_1(0) \subseteq \mathbb{C}$. Show that there exists a sequence of real valued harmonic polynomials that converges uniformly on compact subsets of $D_1(0)$ to $u$.
Now, co... |
H: Proving $\alpha+(\beta+\gamma) = (\alpha+\beta)+\gamma$ for ordinals
I am following Jech's construction, by definition $\alpha+0 = \alpha, \alpha+(\beta+1)=(\alpha+\beta)+1$, and for limit $\beta$ we define $\alpha+\beta = \cup\{\alpha+\xi: \xi<\beta\}$.
Jech's proof of associativity just says "By induction on $\ga... |
H: Exponential function and uniform convergence of polynomials.
How can I prove that no sequence of polynomials converges uniformly to the exponential function?
Thanks in advance for any help.
AI: If a sequence $(f_n)$ of polynomials converges uniformly to $x\mapsto e^x$ on all of $\mathbb R$ (or, as the following arg... |
H: Finding the ratio of two persons time spent driving to a meeting
Mark and pat drive separately to a meeting. mark's average driving speed is $1/3rd$ greater than pat's and mark drives twice as many miles as pat.
What is ratio of number of hours mark spends driving to the meeting to the number of hours pat spends d... |
H: Confusion regarding direct sum decomposition of representations from Serre's book
Sorry if the question is dumb. I am trying to learn representation theory of finite groups from J.P.Serre's book by myself. In section 2.6 on canonical decomposition, he says that let V be a representation of a finite group G, $W_1,..... |
H: Formal grammar for the language $L = \{w\in\{a,b\}^*,\,w=xx,\,x=a^nb^na^mb^m,\,n\ge0,\,m\ge0\}$
What is the grammar of this language?
$$L = \{w\in\{a,b\}^*,\,w=xx,\,x=a^nb^na^mb^m,\,n\ge0,\,m\ge0\}$$
For example: $abab$, $abaabbabaabb$
AI: I think you need to do it in two steps: First a grammar to generate your $x$... |
H: Elementary lower bounds for $n^{1/n}$
I can show that
$n^{1/n} > 1+1/n$
for integer $n \ge 3$
by completely elementary means -
no logs, exponentials,
or calculus.
Are there better bounds that
can be proved in an elementary way?
Here is my proof:
The bound is equivalent to
$n > \frac{(n+1)^n}{n^n}$
or
$\frac{(n+1)^... |
H: Are real numbers also hyperreal? Are there hyperreal $\epsilon$ between $-a$ and $a$ for any positive real $a$?
The set of all hyper-real numbers is denoted by $R^*$. Every real number is a member of $R^*$, but $R^*$ has other elements too. The infinitesimals in $R^*$ are of three kinds: positive, negative and zer... |
H: What is the interpretation of the magnitude of a matrix?
Consider a vector $v$.
The magnitude of this vector (if it describes a position in euclidean space) is equal to the distance from the origin:
$$(v^Tv)^{\frac{1}{2}} = \sqrt{(v^Tv)}$$
that is, the square root of the dot product.
Suppose we compute this value f... |
H: Need explanation of passage about Lebesgue/Bochner space
From a book:
Let $V$ be Banach and $g \in L^2(0,T;V')$. For every $v \in V$, it holds that
$$\langle g(t), v \rangle_{V',V} = 0\tag{1}$$
for almost every $t \in [0,T]$.
What I don't understand is the following:
This is equivalent to $$\langle g(t), v(t... |
H: Inertial Frames of Refereence
I am told that in Newtonian mechanics, no coordinate system is "superior" to any other. Also, all inertial frames are in a state of constant, rectilinear motion with respect to one another.
So am I right to understand that "inertness of coordinate systems" is an equivalence relation on... |
H: Prove $x^{n}-5x+7=0$ has no rational roots
This question arises in STEP 2011 Paper III, question 2. The paper can be found here.
The first part of the question requires us to prove the result that if the polynomial
$$x^{n}+a_{n-1}x^{n-1}+a_{n-2}x^{n-2}+\ldots+a_{0}$$
where each of the $a_{n}$ are integers, has a ... |
H: derivative, the way to show in a graph
http://www.wolframalpha.com/input/?i=derivative+x%5E2%2C+x%5E2
Just reminding myself some math..
Is it ok to show the derivative in such a way like was shown in the link above for $x^2$ function?
A derivative is supposed to be for some point of function. So, then I should see ... |
H: Normalizing an exponential function
Given the equation $a^\frac yx + a^x=b$ is there a way to normalize this function into a form where $y=$...?
In short can I express $y$ in terms of $x$ if $a$ and $b$ are constants?
AI: First rewrite to $a^{y/x}=b-a^x$, then take the $\log$ on both sides to get $\frac{y}{x}\log(a... |
H: Concrete Mathematics Prerequisite Question
I've been very interested in the book Concrete Mathematics (Graham,Knuth,Patashnik) and I've been reading it for the past few weeks.
I'm at the chapter about Sums (Chapter 2), specificaly, the lesson about Finite and Infinite Calculus. My question is if any serious calculu... |
H: Find the maximum value of $r$ when $r=\cos\alpha \sin2\alpha$
Find the maximum value of $r$ when $$r=\cos\alpha \sin2\alpha$$
$$\frac{\rm dr}{\rm d\alpha}=(2\cos2\alpha )(\cos\alpha)-(\sin2\alpha)(\sin\alpha)=0 \tag {at maximum}$$
How do I now find alpha?
AI: We have
$$\cos\alpha\sin 2\alpha=\cos\alpha(2\sin\al... |
H: In diagonalization, can the eigenvector matrix be any scalar multiple?
One can decompose a diagonalizable matrix (A) into A = C D C^−1, where C contains the eigenvectors and D is a diagonal matrix with the eigenvalues in the diagonal positions. So here's where I get confused. If I start with a random eigenvector ma... |
H: Michelson-Morley Experiment
I've looked everywhere and I cannot find a complete derivation that includes the step I'm looking for lol...hopefully this will add another link for google.
So the full time for light to travel both directions with the ether then against as follows:
$$T = \frac{L}{(c+v)}+\frac{L}{(c-v)}\... |
H: Order of Permutation : If $\tau \in S_n$ has order $m$, then $\sigma \tau\sigma^{-1}$ has also order $m$.
I dont understand the following very simple statement:
If $\tau \in S_n$ has order $m$, then $\sigma \tau\sigma^{-1}$ has also order $m$.
The proof is:
Suppose $\tau$ has order $m$.
$(\sigma \tau \sigma^{-1})^... |
H: $\lim_{n\rightarrow\infty}\frac{a_{n+1}}{a_n} = L$ implies $\lim_{n\rightarrow\infty}a_n^{1/n}=L$
Let $\{a_n\}$ be a sequence of positive numbers. Prove that if $$\lim_{n\rightarrow\infty}\frac{a_{n+1}}{a_n} = L$$ then $$\lim_{n\rightarrow\infty}a_n^{1/n}=L.$$
The first condition means that for any $\epsilon$, th... |
H: Continuous right derivative implies differentiability
A book of mine says the following is true, and I am having some trouble proving it. (I've considered using the Lebesgue differentiation theorem and absolute continuity, as well as elementary analysis methods.)
Let $f: [0, \infty) \rightarrow \mathbb{R}$ be cont... |
H: Binomial coefficients equal to a prime squared
I am looking for some reading on when binomial coefficients are equal to $p^2$ for $p$ a prime. In general I imagine this is rare, as there are simply too many factors. Concretely, I am looking for pairs $(n, k)$ such that ${n + k - 1 \choose k} = p^2$.
AI: For the equ... |
H: Embedding into a morphism of distinguished triangles
Everything in this question happens in a triangulated category $\mathbf{D}$. I am trying to prove that in a diagram like this
$$
\newcommand{\ra}[1]{\!\!\!\!\!\!\!\!\!\!\!\!\xrightarrow{\quad#1\quad}\!\!\!\!\!\!\!\!}
\newcommand{\da}[1]{\left\downarrow{\scriptsty... |
H: Evaluating $\lim_{x\to1}\frac{\sqrt{x^2+3}-2}{x^2-1}$?
I tried to calculate, but couldn't get out of this:
$$\lim_{x\to1}\frac{x^2+5}{x^2 (\sqrt{x^2 +3}+2)-\sqrt{x^2 +3}}$$
then multiply by the conjugate.
$$\lim_{x\to1}\frac{\sqrt{x^2 +3}-2}{x^2 -1}$$
Thanks!
AI: You were right to multiply "top" and "bottom" by th... |
H: Find the values of the constants in the following identity $2x^3+3x^2-14x-5=(ax+b)(x+3)(x+1)+C$
I'm working through identities but I can't figure out how to get further than multiplying out the above to get :
$$2x^3+3x^2-14x-5=2ax^3+3ax^2+3ax+bx^2+3bx+bx+3b+C$$
can someone give me a hint on what to do next?
AI: Goo... |
H: Finding a maximum likelihood function
Let $X_{1},X_{2},\dots,X_{n}$
represent a random sample from a distribution with probability density function $$f(x;\theta)=\frac{x}{\theta}e^{-x^{2}/2\theta}\hspace{1em}x>0
$$
Find the maximum likelihood function.
I did:
$$L(\theta)=\prod_{i=1}^{n}f(x_{i};\theta)=\prod_{i=1... |
H: Polynomial lower bound on a sequence
Let $0<s<1$ and
$$a_i=\left(i+1\right)^s-(i)^s, \ i \in \mathbb{N}.$$
I'm trying to find a lower bound on $(a_i)_{i\in\mathbb{N}}$
of the form
$$ a_i \geq i^k \ \mbox{for large enough i}.$$
That is, it does not have to hold for small $i$, but only eventually.
Of course, we mus... |
H: Help with a different approach to extracting a polynomial equation from differences
It is well known that we can determine the degree of a polynomial can be found by finding when the differences are the same. i.e. if the second differences are the same, it is a polynomial of the 2nd degree and if the third differen... |
H: Finding a length of arc, what's wrong?
Find: $$ \int \sqrt{x^{2}+y^{2}}dl$$
$$L: x^{2}+y^{2}= Rx$$
(at image $p' = -R\cdot \sin(\phi)$ )
AI: You want to evaluate
$$\oint_L d\ell \sqrt{x^2+y^2}$$
where
$$\left ( x-\frac{R}{2}\right)^2+y^2=\frac{R^2}{4}$$
Parametrizing:
$$x(t) = \frac{R}{2} + \frac{R}{2} \cos{t}... |
H: Finding an Orthogonal Transformation with 2 given vectors
There are two possible orthogonal transformations of $\mathbb{R}^2$ that leave the origin fixed and send the point $(0,13)$ to $(5,12)$. Find their matrices and describe them geographically.
Can anyone explain to me how I'd go about working this out? Sorry b... |
H: Minimizing the distance between two boats.
the source of this problem is Stewart's Essential Calculus (Early Transcendentals) 2nd ed.
A boat leaves a dock at 2:00PM and travels due south at a speed of $20$ km/h. Another boat has been heading due east at $15$ km/h and reaches the same dock at 3:00 PM. I want to fin... |
H: Subobject classifiers as internalizations
I recently read the article on internalizations on nlab, but I am not quite sure what falls under that description.
Is it fair to say, that subobjects are internalizations of subsets and that the subobject classifier internalizes characteristic functions?
AI: Internalizatio... |
H: Impossibility to find the General Term of a sequence (or series)
Is there a way to formally show that the General Term of a sequence cannot be inferred from some set of given information? If so, how can one do that?
An example to illustrate what I mean is:
Find the General Term of the series
$$\frac{1}{2} + \le... |
H: Simplify trig function and calculate limit $\lim\limits_{x \to 0} \frac{\tan x-\sin x}{\sin^2 x}$
Using the fact that $\lim\limits_{x \to 0} \frac{\sin(x)}{x}=1$, please help me show that
$$\lim\limits_{x \to 0} \frac{\tan(x)-\sin(x)}{\sin^2(x)}=0.$$
Because I am not familar with L'Hôpital's rule and Taylor's The... |
H: Linear Transformation of a straight line
Let $L_{1}: x-y-2=0$ be a straight line in the x-y coordinate system. Find a coordinate system $(x_{1},y_{1})$ having its origin at $(0,0)$ and relative to which $L_{1}$ has equation $y_{1}= $constant.
I know I keep asking these stupid questions, but my final is tomorrow and... |
H: Bombing of Königsberg problem
A well-known problem in graph theory is the Seven Bridges of Königsberg. In Leonhard Euler's day, Königsberg had seven bridges which connected two islands in the Pregel River with the mainland, laid out like this:
And Euler proved that it was impossible to find a walk through the ci... |
H: Show that $\langle G^+\rangle=G$ in a directed group
Lemma 2.1.8 of Glass' Partially Ordered Groups states:
$G$ is a directed group if and only if $\langle G^+\rangle=G$ (where $G^+=\{x:x\geq1\}$)
This doesn't make any sense to me. For example, $(\mathbb Z,+)$ is a directed group (every two elements have an upper... |
H: radius of convergence for $\sum_{n=1}^{\infty}(n^{\frac1n}-1)x^n$
Find the radius of convergence $R$ for the power series $$\sum_{n=1}^{\infty}(n^{\frac1n}-1)x^n$$
Applying the ratio test, this becomes $\lim_{n\rightarrow\infty}\dfrac{(n+1)^{\frac{1}{n+1}}-1}{n^{\frac{1}{n}}-1}$, which is $\dfrac00$ and seems har... |
H: Absolute convergence condition for radius of convergence
Let $\sum_{n=0}^{\infty}a_n(x-t)^n$ be a power series. Let $X=\{|x-t|:\sum_{n=0}^{\infty}|a_n||x-t|^n$ converges$\}$. Let $R$ be the least upper bound of $X$ if $X$ is bounded and let $R=\infty$ if $X$ is unbounded. Prove that $R$ is the radius of convergenc... |
H: Confused by $\Re(z)$ and closed contours, why isn't it the integral 0?
Let the contour $\gamma$ be the triangle with vertices $\ 0, 1, 1+i $, taken anticlockwise.
As this is a closed contour and I understand $$\oint_\gamma z\ dz = 0 $$ as it is analytic inside the contour (and everywhere else).
However, how are... |
H: Can a vector space have more than one zero vector?
The question above is really it. The reason I ask is that my text says a vector space can have more than one zero vector (It's a true/false question: A vector space may have more than one zero vector). But if the zero vector in any space is unique, then it has only... |
H: Harnack's inequality Evans' PDE book
This is on page 33 of my edition, under the proof of Harnack's inequality. Let $V\subset \overline{V} \subset U$ with $\overline{V}$ compact. Let $r=\frac{1}{4}\text{dist}\left(V,\partial U\right)$. Let $x,y\in V$ s.t. $\left|x-y\right|\leq r$. It seems as though Evans uses
$$\i... |
H: Show $G=H_3H_5$
Let $G$ be a group and $H_3$ and $H_5$ normal subgroups of $G$ of index $3$ and $5$ respectively. Prove that every element $g\in G$ can be written in the form $g=h_3h_5$, with $h_3\in H_3$ and $h_5\in H_5$.
AI: Key facts:
The product $HN=\{hn:h\in H,n\in N\}$ is a subgroup of $G$ if $H$ and/or $N$ ... |
H: Differential Equation - Solve $y'=y\cot 2x, y (\frac{\pi}{4})=2$
Solve the following differential equation :
$$y'=y\cot 2x,\; y (\frac{\pi}{4})=2.$$
My approach :
$$\frac{dy}{dx}=y\cot 2x \Rightarrow \frac{dy}{y}=\cot 2x dx$$
Integrating both sides we get :
$$\Rightarrow \log y = \frac{\log|\sin 2x|}{2}+c$$
$$\R... |
H: How to think of $\vec{u}-\vec{v}$
Assume I have two vectors, $\vec{u}$ and $\vec{v}$. I know that I can think of their sum via Triangle or Parallelogram Law, but I'm having trouble knowing which way the vector would point depending on if it was $\vec{u}-\vec{v}$ or if it was $\vec{v}-\vec{u}$. Is there an easy way ... |
H: Likelihood Functon.
$n$ random variables or a random sample of size $n$ $\quad X_1,X_2,\ldots,X_n$ assume a particular value $\quad x_1,x_2,\ldots,x_n$ .
What does it mean? The set $\quad x_1,x_2,\ldots,x_n$ constitutes only a single value? or, $\quad x_1,x_2,\ldots,x_n$ are n values , that is, $X_1$ assumes th... |
H: Convergence of a series $\sum\limits_{n=1}^\infty\left(\frac{a_n}{n^p}\right)^\frac{1}{2}$
I've got a question about the convergence of a series during studying analysis.
If I know that a series of positive real numbers $$\sum_{n=1}^\infty a_n$$ converge, why does $$\sum_{n=1}^\infty\left(\frac{a_n}{n^p}\right)^\fr... |
H: $\epsilon$-$\delta$ proof that $\lim\limits_{x \to 1} \frac{1}{x} = 1$.
I'm starting Spivak's Calculus and finally decided to learn how to write epsilon-delta proofs.
I have been working on chapter 5, number 3(ii). The problem, in essence, asks to prove that
$$\lim\limits_{x \to 1} \frac{1}{x} = 1.$$
Here's how ... |
H: Linear Systems: Application
Zoltan(who is behind Mali) and Mali are 18 km apart and begin to walk at the same time. If they walk in the same direction they meet after 6h. If they walk towards each other they meet in 2h. Find their speeds
AI: Let $z$ be the speed of Zoltan, assumed constant, and let $m$ be the speed... |
H: single valued function in complex plane
Let
$$f(z)=\int_{1}^{z} \left(\frac{1}{w} + \frac{a}{w^3}\right)\cos(w)\,\mathrm{d}w$$
Find $a$ such that $f$ is a single valued function in the complex plane.
AI: $$
\begin{align}
\left(\frac1w+\frac{a}{w^3}\right)\cos(w)
&=\left(\frac1w+\frac{a}{w^3}\right)\left(1-\frac12w... |
H: point out the mistakes $1=2$?
Q:prove $1=2$ ?
method1:-
let us consider $x=1$
then $x=x^2$
$x-1 =(x^2)-1$
$x-1=(x-1)(x+1)$
$1=x+1$
finally
$1=2$
i have little confusion here, is the mistake is considering $x=1$ or else any thing other than this.
method2:-
$-2=-2$
$1-3=4-6$
$(1^2)-(\frac{2*1*3}{2})=(2^2)-(\frac{2*... |
H: Finding % of remaining students in the class would be girls?
In class of $120$ students, boys constitute $40$% of total. If $\dfrac 13^{rd}$ of boys and $4$ girls drop out of class to join a camp , what % of remaining students in the class would be girls ?
AI: Total number of students, $ T = 120 $. Number of boys, ... |
H: Computing $ \int_0^\infty \frac{\log x}{\exp x} \ dx $
I know that $$ \int_0^\infty \frac{\log x}{\exp x} = -\gamma $$ where $ \gamma $ is the Euler-Mascheroni constant, but I have no idea how to prove this.
The series definition of $ \gamma $ leads me to believe that I should break the integrand into a series and ... |
H: Closed Form of Recurrence Relation
I have a recurrence relation defined as:
$$f(k)=\frac{[f(k-1)]^2}{f(k-2)}$$
Wolfram Alpha shows that the closed form for this relation is:
$$
f(k)=\exp{(c_2k+c_1)}
$$
I'm not really sure how to go about finding this solution (it's been a few years...). Hints?
AI: Your recurrence r... |
H: evaluate the following limit on trigonometry
given that \begin{equation}
\lim_{y \rightarrow 0}
\frac{\sin y}{y}=1
\end{equation}
evaluate the following
\begin{equation}
\lim_{x \rightarrow 0}
\frac{2-2\cos^2 x-2 \cos x \sin ^2 x}{x^4}
\end{equation}
AI: $$\begin{align}
2-2\cos^2{x}-2\cos{x}\sin^2{x}
&=2(1-\cos^2{x... |
H: Help on showing a function is Riemann integrable
Problem Let $(a_k)_{k=1}^\infty$ be the sequence of values of $\mathbb{Q}\cap[0,1]$ (which is countable set). Let $g:\mathbb{R}\rightarrow\mathbb{R}$ by $g=\sum_{k=1}^\infty {1 \over k}\cdot \chi_{a_k}$, that is, $g(x)=0$ if $x\notin\mathbb{Q}\cap[0,1]$ and $g(a_k)={... |
H: Standard logic notation in mathematics
My profesor is always complaining that my proofs are very long and difficult to read because I never use notation, meaning I say everything in words. Tired of that I decided to study logic by myself and develop my proofs by using the methods of logic. The problem to me now is ... |
H: Change of Coordinate Matrix question.
I have this question and the wording is very confusing, I dont understand how to answer it. Any help will be greatly appreciated. I have tried answering it and I just dont know where to begin.
EDIT: I'm not just looking for an answer, I genuinely want to understand how it work... |
H: Euler's sum of divisors recurrence relation
Euler came up with following recurrence relation for the sum of divisors
$$\sigma(n) = \sigma(n−1) + \sigma(n−2) − \sigma(n−5) − \sigma(n−7) \dots$$
Since $\sigma(p) = p+1$, where $p$ is a prime number, we can use the recurrence relation to verify if a number is prime. It... |
H: For a given image $\mathbf X$, the equivalence class for pixels $p$ with labels $l$.
$\left[l\right]=\left\{p \in\mathbf X|\,p\sim l\right\}$.
This is taken from this paper on image segmentation, page $2$. I don't know how to interpret this, do they mean "all the pixels on image $\mathbf X$ that share a label $l$... |
H: Finding the cumulative distribution function of the minimum of a sample
The Weibull pdf is given as the following.
$$f(x) = \begin{cases} \frac{1}{\alpha}mx^{m-1}e^{-x^m/\alpha} \quad \text{if } x > 0 \\
0 \quad \text{else} \end{cases}$$
If a random sample of size $n$ is taken from a Weibull distribution, what is... |
H: Probability of adjacent seating
A homework question states:
A room holds two rows of six seats each. Two friends are assigned
randomly to the 12 seats. What is the probability that the 2 friends
sit in adjacent seats?
Note: Friends sitting behind friends don't count. Friends sitting
diagonally adjacent to ea... |
H: Another Error in Neukirch's Algebraic Number Theory?
I'm reading Neukirch's Algebraic Number Theory and trying to do the exercises. I think I may have found another error, but am not sure...
Exercise 7. In a noetherian ring $R$ in which every prime ideal is maximal, each descending chain of ideals $a_1 \supseteq... |
H: Does every infinite group have a maximal subgroup?
$G$ is an infinite group.
Is it necessary true that there exists a subgroup $H$ of $G$ and $H$ is maximal ?
Is it possible that there exists such series $H_1 < H_2 < H_3 <\cdots <G $ with the property that for every $H_i$ there exists $H_{i+1}$ such that $H_i < ... |
H: Question regarding the definition of direct sum decomposition of a representation
Please bear with me. I am trying to learn representation theory of finite groups from J.P. Serre's book by myself.
Here, the author has used the word 'representation' for the homomorphism $\rho : G\rightarrow GL(V)$, as well as the ve... |
H: Which ring homomorphisms preserve/reflect what?
Exams are coming up and I'm getting kind of desperate. So more now than ever, whatever help you're able to provide is much appreciated. In the abstract algebra exam I'm currently preparing for, there's a lot of focus on the following ring-theoretic concepts.
units
ir... |
H: smooth approximate parameterization to polygonal boundary
I can "almost" parameterize the boundary of a square using
$${\bf r}(t) = (\cos t)^{1/p} {\bf i} + (\sin t)^{1/p} {\bf j},$$
$0\leq t\leq 2 \pi$, and $p$ is odd. This parameterization is smooth (or at least $C^1$), and of course is the unit ball in the $L^p$... |
H: Number of ways to distribute 5 distinguishable balls between 3 kids such that each of them gets at least one ball
How many ways are there to distribute 5 distinguishable balls between 3 kids such that each of them gets at least one ball?
My approach is $ \binom{5}{3} 3! $ + $ \binom{2}{2} \binom{3}{2}2!$ which is... |
H: Why is $1 =1 $? why is it so why cant be $1 =$ something else?
It may sound stupid but why is $1=1$ or $n=n$ if thats the case does $1/0 = 1/0.$
AI: Because we define equals as an equivalence relation which needs to fulfill three properties
reflexive meaning $x\sim x$
symmetric $x\sim y \implies y\sim x$
transiti... |
H: Geometrical solution to a search problem
Given:
an infinite grid
a robot that can move to adiacent cells
you have available up(), down(), left(), right() and distanceToDestination() functions
destination coordinates and robot coordinates are unknown
To do
find an algorithm to move the robot to destination asa... |
H: How to integrate $\int \frac{1}{\cos(x)}\,\mathrm dx$
could you help me on this integral ?
$$\int \frac{1}{\cos(x)}\,\mathrm dx$$
Here's what I've started :
$$\int \frac{1}{\cos(x)}\,\mathrm dx = \int \frac{\cos(x)}{\cos(x)^2}\,\mathrm dx = \int \frac{\cos(x)}{1-\sin(x)^2}\,\mathrm dx$$
Now, I did : $u = \sin(x)$, ... |
H: Induced map between fundamental groups from covering map is injective
Question:
Let $f : X \to Y$ be a continuous map and let $x \in X$, $y \in Y$ be such that $f(x) = y$. Then there is an induced map $f_* : \pi_1(X, x) \to \pi_1(Y, y)$ such that $f_*([\gamma]) = [f \circ \gamma]$.
If $p$ is a covering map, show th... |
H: Find maximum value of $f(x)=2\cos 2x + 4 \sin x$ where $0 < x <\pi$
Find the maximum value of $f(x)$ where
\begin{equation}
f(x)=2\cos 2x + 4 \sin x \ \
\text{for} \ \ 0<x<\pi
\end{equation}
AI: $f(x)=2-4\sin^2(x)+4\sin(x)$. Let $t=\sin (x)$. Then $0<t\le 1$. We have a new function:
$$g(t)=-4t^2+4t+2$$.
So $t=\f... |
H: Finding $a$ s.t the cone $\sqrt{x^{2}+y^{2}}=za$ divides the upper half of the unit ball into two parts with the same volume
My friend gave me the following question:
For which value of the parameter $a$ does the cone
$\sqrt{x^{2}+y^{2}}=za$ divides $$\{(x,y,z):\,
x^{2}+y^{2}+z^{2}\leq1,z\geq0\}$$ into two part... |
H: Isometry and equivalence
Let $(X,d_1)$ and $(X,d_2)$ be two metric spaces on the same set $X$. Is there any relation between $d_1$ and $d_2$ being equivalent and $(X,d_1)$ and $(X,d_2)$ being isometric? If not, can anyone give examples where $d_1$ and $d_2$ are equivalent but $(X,d_1)$ and $(X,d_2)$ are not isometr... |
H: Eigenvalue calculation.
I am getting confused by this simple eigenvalue calculation.
Calculate the eigenvalues of $\begin{bmatrix} 5 & -2\\ 1 & 2\end{bmatrix}$.
Firstly, I row reduce it, to go from $\begin{bmatrix} 5 & -2\\ 1 & 2\end{bmatrix} \to \begin{bmatrix} 6 & 0\\ 1 & 2\end{bmatrix}$ by performing $R_1 \to ... |
H: Concerning the proof of the Cantor–Bernstein theorem
I've seen two proofs for the Cantor–Bernstein theorem which says that for two sets $X$ and $Y$ if $\#X \le \#Y$ and $\#Y \le \#X$ then $\#X=\#Y$, equivalently if we can find an injection from $X$ to $Y$ and $Y$ to $X$ then we can find a bijection between the two ... |
H: Map from a manifold to $[0,1]$
I am looking through a practice exam to prepare for an upcoming final and I am having through with this question.
Question: Let $M$ be a manifold, $p \in M$, and $U \subset M$ an open set containing $p$. Show there is a continuous function $f : M \to [0, 1]$ such that $f(p) = 1$ and $... |
H: Question about primary decompositions.
I am reading the book Introduction to commutative algebra by Atiyah and Macdonald. On page 50, Line -7, it is said that "if $f: A \to B$ and $\mathfrak{q}$ is a primary ideal in $B$, then $A/\mathfrak{q}^c$ is isomorphic to a subring of $B/\mathfrak{q}$". How to prove this res... |
H: What 's the upper limit of a binomial expansion with fractional power?
It's known that a binomial expansion can be writen as a sum,
$\displaystyle (a+b)^n=\binom{n}{0}a^n+ \binom{n}{1} a^{n-1}b+\binom{n}{2}a^{n-2}b^2+.....$
If the power, $n$, is a natural number, the last term of this sum will be $\displaystyle \bi... |
H: Random walk as a martingale?
Let $S_0$, $Z_1$, $Z_2$, $\ldots$ be independent random variables. $S_n=S_0+Z_1+\cdots+Z_n$, $n=0,1,2,\ldots$
$S_n$ is a random walk starting in a random point, $S_0$ I need to find out, when it is a martingale.
Is it enough to show, that $E(S_{n+1}|\mathcal F_n)=S_n$? Does "find out wh... |
H: Equivalence relation - Proof question
Prove that the relation, two finite sets are equivalent if there is a one-to-one correspondence between them, is an equivalence relation on the collection $S$ of all finite sets.
I'm sure I know the gist of how to do it, but I'm a beginner in proofs, and I'm not sure if I've wr... |
H: Question about zero-divisors and a quotient of a polynomial ring by an ideal in the book Introduction to commutative algebra by Atiyah and Macdonald.
I am reading the book the book Introduction to commutative algebra by Atiyah and Macdonald. I have two questions On Page 51.
On Line 5 of Page 51, it is said that th... |
H: Find the range of $ f(x)=9^x - 3^x+1$
Problem:-Find range of function $ f(x)=9^x - 3^x+1$, here the domain of $f$ is $\mathbb R$.
Solution: $ f(x)=9^x - 3^x+1$. Let $f(x)=y$. Then
$$ \begin{split}y&=9^x - 3^x+1\\
y&=3^{2x} - 3^x+1
\end{split}$$
Let $3^x= u$. Then $ y=u^2 - u+1$, so
$$ u^2 - u+1-y=0.$$
Am ... |
H: finding probability function, bills and dice
There are 2 white bills, and 4 green bills. We throw the dice, if it gives 6 - we take two bills, if it gives 1/2/3/4/5, we take one bill. X variable is the amount of white bills taken out. I need to find the probability function for X and draw the cumulative distributio... |
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