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H: Partitions of a simple function, doesn't make sense.
Let $(\Omega, \mathcal{F})$ be a measurable space and let $A_i,B_j$ be in $\mathcal{F}$ such that $\Omega = \uplus_{i=1}^n A_i = \uplus_{j=1}^m B_j$, ie they're finite disjoint unions. Let $s$ be a simple function and $s = \sum_{i=1}^n \alpha_i 1_{A_i}$ and $s =... |
H: A positive harmonic function on the punctured plane is constant
Let $f(z)$ be a positive harmonic function on $\mathbb{C}\backslash
\{0\}$. Prove that $f(z)$ is constant.
I have no idea to prove this statement.
AI: Add a constant so it is bounded away from zero. It is still harmonic. But now the reciprocal is bo... |
H: Solve $\cos x>1/2$ for $-180
Solve
$$\cos x>\frac12\quad\text{for}\quad-180^{\circ}<x<180^{\circ}\;.$$
Hey guys, Ive got the solution to this question although I cant seem to figure out how the textbook did it. Can someone please explain this question thanks
AI: Recall that, in degrees, we know that $$\cos(\pm 60^... |
H: Is there a standard notation for the set of negative integers and zero?
There is $\mathbb Z^-$, $\mathbb Z^*$, $\mathbb Z^+$. There does not seem to be a notation for negative integers and zero.
AI: If you wrote $\Bbb Z_{\le 0}$, I’m quite sure that you’d be understood, and I have seen this notation. |
H: If $f,g$ are analytic in the unit disk, and $|f|^2+|g|^2=1$, then $f,g$ constant.
I need to prove that if $f,g$ are analytic in the unit disk, and $|f|^2+|g|^2=1$ for all $z$ in the unit disk, then $f,g$ are constant.
This is an exercise question so it should not be very hard, but I don't know where to start. Any h... |
H: Classic optimization - fence problem
I'm taking a class in the fall and need to dust off my $10$-year-old calculus skills, particularly optimization. I'm attempting to remember how to tackle the classic fence problem, i.e. how to calculate the dimensions of a field so that the cost of fencing is minimized. Here's... |
H: A homomorphism between two A-algebras.
A homomorphism between two algebras is described here. I want to describe a homomorphism $f:A[x_1,x_2,\dots,x_n]\to R$, where $R$ is an A-algebra. $A$ is a ring.
Obviously, $A[x_1,x_2,\dots,x_n]$ is an A-algebra.
The article says that if $A$ and $B$ are two algebras over $K$... |
H: Calculate Danny allowances in 2010.
In 2011, Danny received an allowance of $ 260. This amount is seeing an increase of 30% compared with the previous year. Calculate Danny allowances in 2010.
Calculation:
= 30/100 x 260
= 7800/100
= 78
= 260-78
answer = 183
=> Refer to my exercise book, the answer is 200. Are my c... |
H: If $M\prec (L_{\omega_1},\in)$, then $M=L_\alpha$ for some $\alpha$.
This is exercise 13.17 of Jech's Set Theory:
If $M\prec (L_{\omega_1},\in)$, then $M=L_\alpha$ for some $\alpha.$
[Show that $M$ is transitive. Let $X\in M$. Let $f$ be the $<_L$-least mapping of $\omega$ onto $X$. Since $f$ is definable in $(L... |
H: Find $F_{n}$ in : $F_{n} +2F_{n-1} + ... + (n+1)\cdot F_{0} = 3^{n}$
I'm stuck with the question for a while : Find in $$F_{n} +2F_{n-1} + ... + (n+1)\cdot F_{0} = 3^{n}$$
the element $F_{n}$ .
Placing $n-1$ instead on $n$ results in :
$$F_{n-1} +2F_{n-2} + ... + (n-1+1)\cdot F_{0} = 3^{n-1}$$
$$ F_{n-1} +2F_{n-2}... |
H: Homeomorphism between subsets of $\mathbb{R}^n$
How to establish a homeomorphism between the "closed unit ball minus origin" and the "complement of open unit ball"? In other words, how to establish a homeomorphism between $X$ and $Y$, where $X=\{x\in\mathbb{R}^n;\;0<|x|\leq 1\}$ and $Y=\{y\in\mathbb{R}^n;\;|y|\geq ... |
H: Simplify $(x - 1/2)^2$
Not sure what subject this is.
again the question is
$$\left(x-\frac12\right)^2$$
as a multiple choice question, i chose the answer:
$$x^2+\frac1x$$
Thank you.
AI: Assuming that in this case the (not terribly well-defined) term simplify actually means expand:
$$\left(x-\frac12\right)^2=\le... |
H: Calculate the average sales per day in 2010.
In 2005, Company K was 730 000 units of the car market. Total sales is 50% of the total cars sold in 2010. Calculate the average sales per day in 2010.
Calculation:
Total sales in 2010 = $50\% \times 730 000 = 365000$
Average sales per day in 2010 = $\frac{365000}{365} ... |
H: tough algebric problem?
I wanted to know how can i prove that if
$xy+yz+zx=1$, then
$$ \frac{x}{1+x^2}+\frac{y}{1+y^2}+\frac{z}{1+z^2}
= \frac{2}{\sqrt{(1+x^2)(1+y^2)(1+z^2)}}$$
I did let $x=\tan A$, $y=\tan B$, $z=\tan C$
given $xy+yz+zx =1$ we have $\tan A \tan B+ \tan B \tan C+\tan C \tan A=1$
$\tan C(... |
H: Limit theorem of extreme values
What does it mean for a cumulative density function of a standardized variable to tend to a limiting value? This is main part of the problem I don't understand too well. Below is the problem I am trying to solve, and my current progress in doing so:
Problem
Let $U_1,\dots,U_n$ be in... |
H: Simple confusion about $\vartheta(x)$
By definition $\vartheta(x) \leq \psi(x). \hspace{20mm}(1) $
and also we know
$\frac{\psi(x)}{x} \sim \frac{\vartheta(x)}{x},\hspace{45mm}(2) $
Littlewood's result that, for large x, successively
$\pi(x) - Li(x) < -K\frac{\sqrt{x}\log\log\log x}{\log x},$
$\pi(x) - Li(x) > K \f... |
H: Are the Gödel's incompleteness theorems valid for both classical and intuitionistic logic?
I am studying an undergraduate text about math logic.
The proofs of the two Gödel's incompleteness theorems are not completely formal: they are admittedly simpler that the real proofs.
For what I understood, I deduce the two ... |
H: Difference between the behavior of a sequence and a function in product and box topology on same set
Let $\prod_{\alpha \in J} X_{\alpha}$ is the product of typologies. Consider product topology on the set. Then any function $f : A \rightarrow \prod_{\alpha \in J} X_{\alpha}$ will be continuous on the space $\prod_... |
H: What is the family / equation of this function?
I'm making up a function and I want to figure out the equation for it so that I can define it continuously. Right now I'm using ExcelGoogle Spreadsheets to define it on a point-by-point basis. I have a feeling it's a type of exponential/logarithmic/power function but ... |
H: Is $3 \ge 1$ or is it just $3 > 1$?
Well, probably this might seem a really simple question (and it might be so too!), but off late me and my friends have been debating quite hard over this question.
Is $3 \ge 1$ or is it just $3 > 1$?
Actually it all started here: if functions $f$ and $g$ satisfy $f>g$ then do t... |
H: Solving $ F_{n} = \sum_{i=1}^{n-1} (F_{i}\cdot F_{n-i}) $?
I need to find $F_{n}$ in :
$$ F_{n} = \sum_{i=1}^{n-1} (F_{i}\cdot F_{n-i}) , F_0 = 0 , n>=2 $$
This equation screams convolution , I think , but I find it as a quite long solution sometimes.
Here, I first tried to play with the indexes :
$$ F_{n-1} =... |
H: Characterization of the field $\mathbb{Z}/2\mathbb{Z}$
Let $R \neq 0$ be a ring which may not be commutative and may not have an identity.
Suppose $R$ satisfies the following conditions.
1) $a^2 = a$ for every element $a$ of $R$.
2) $ab \neq 0$ whenever $a \neq 0$ and $b\neq 0$.
Is $R$ isomorphic to the field $\mat... |
H: Given $a > b+c$, $e>d+f$, and $i>g+h$, can the quantity $a(ei-hf) + b(-di+fg) - c(dh+eg)$ ever be zero?
Given positive reals $a > b+c$, $e>d+f$, and $i>g+h$, can the quantity
$a(ei-hf) + b(-di+fg) - c(dh+eg)$ ever be zero?
AI: Consider $\det\left(\begin{array}{ccc} a & b & c\\ d & e & f \\ g & h & i \end{array}\ri... |
H: How to show that there is no condition that can meet two inequalities?
Here's an excerpt from Spivak's Calculus, 4th Edition, page 96:
If we consider the function
$$
f(x)= \left\{ \matrix{0, x \text{ irrational} \\
1, x \text{ rational}} \right.
$$
then, no matter what $a$ is, $f$ does not approach an number $l... |
H: Show that the arc length of a curve is invariant under rigid transformation.
Show that the arc length of a curve is invariant under rigid transformation.
The curve here is in $\mathbb R^3$, and the definition of arc length is $\int^b_a||\bf r'$$(t)||dt$. This theorem appears in my book without proof, can somebod... |
H: Show that $\frac {\sin(3x)}{ \sin x} + \frac {\cos(3x)}{ \cos x} = 4\cos(2x)$
Show that
$$\frac{\sin(3x)}{\sin x} + \frac{\cos(3x)}{\cos x} = 4\cos(2x).$$
AI: $$\dfrac{\sin(3x)}{\sin x} + \frac{\cos(3x)}{\cos x}$$
$$\dfrac{\sin(3x)\cos x + \cos(3x)\sin x}{\cos x{\sin x}}$$
$$\dfrac {\sin (3x+x)}{\sin x\cos x}$$
$... |
H: Homework: Smooth mapping $f$ satisfying $f\circ f=f$
This is an exercise in Mathematical Analysis by Zorich, in the subsection 12.1.
Let $f:\mathbb{R}^n\rightarrow\mathbb{R}^n$ be a smooth mapping satisfying condition $f\circ f=f$.
$\quad$a) Show that the set $f(\mathbb{R}^n)$ is a smooth surface in $\mathbb{R}^n$... |
H: Full Adder boolean Algebra simplification
I have an expression here from the Full Adder circuit, used for binary addition. One equation used to make it work, is this one:
$$C = xy + xz + yz \tag{1}$$
Now, the book transforms this equation into this:
$$C = z(x'y + xy') + xy \tag{2}$$
In the immediate step, the do th... |
H: If a $k$-algebra is finitely generated, then does $k$ also have to be a finitely generated field?
Let $k$ be a field, and $A$ be a finitely generated $k$-algebra. Then does $k$ also have to be a finitely generated field?
Motivation: Let $A$ be generated by $\{a_1,a_2,\dots,a_n\}$, and $k$ be generated by $\{k_1,k_2... |
H: Simple counterexample to sampling theorem
I know this has to be wrong, but can't see what is wrong with it:
Take a simple sinusoid. It crosses zero every half cycle. Sample it at double its frequency. If the samples coincide with the zero-crossings, you get all samples at zero. In that case, you cannot know the sin... |
H: $F(x) =\int \limits_{\tan x}^{\cot x}\sqrt{1+t^2}\,\mathrm{d}t\,\,$ then $F'( \pi/4) =?$
$F(x) =\int \limits_{\tan x}^{\cot x}\sqrt{1+t^2}\,\mathrm{d}t\,\,$
then $F'(\pi/4) =?$
$$F(x) =\int \limits_{\tan x}^{\cot x}\sqrt{1+t^2}\,\mathrm{d}t$$
$$F(x) =-\int \limits_{0}^{\cot x}\sqrt{1+t^2}\,\mathrm{d}t + \int \l... |
H: Convex Function of Two Vectors
Let $f:\mathbb{R}^M\times\mathbb{R}^N\rightarrow\mathbb{R}$ be a mapping such that for $\mathbf{Y}\in\mathbb{R}^N$ constant, $f(\mathbf{X}, \mathbf{Y})$ is a convex function of $\mathbf{X}$ and for $\mathbf{X}\in\mathbb{R}^M$ constant, $f(\mathbf{X}, \mathbf{Y})$ is a convex function... |
H: Why not number some definitions in Rudin's book?
I'm reading Rudin's Principle of Math Analysis
Why some definitions about functions in Chapter 8, there are no the number marks, and the boldfaced words Definition?
This chapter only have 22 marked theorem and definitions and...
For example
Section 3
We define
$$\be... |
H: $\lim \limits_{h\to 0} \frac{e^h-1}{h}=1$ and the relation with $\sin'(x)=\cos(x)$
$$e^x=\sum _{n=0}^{\infty } \frac{x^n}{n!}$$
$$\begin{align*}\lim_{h\to 0} \frac{ e^{(x+h)}-e^x}{h}=e^x\color{blue}{\lim_{h\to 0} \frac{e^h-1}{h}}=e^x\tag{1}\end{align*}$$
How obvious about the following limit?
I think it is obvious... |
H: A function of the inverse function of a set contained in a set.
I'm doing an intro course on set theory and need to prove:
Suppose that
$$
f : X \rightarrow Y, A \subset X, B \subset Y
$$
Prove that:
a) $ f(f^{-1}(B)) \subset B $ and give an example where equality does not hold
b) $ A \subset f^{-1}(f(A)) $ and gi... |
H: Why $\cos (x)>0$?
$$\begin{align*}\cos '(x)=-\sin (x), \sin '(x)=\cos (x)\tag{1}\end{align*}$$
We assert that there exist positive numbers $x$ such that $\cos (x)=0$. For suppose this is not so. Since $\cos (0)=1$, it then follows that $\color{Green}{\cos (x)>0}$ for all $x>0$,...
Why $\cos (x)>0$?
AI: Because... |
H: Struggling with explanation of transitive closure
I'm reading a book about structures on collections, chapter equivalence relations, and I try to get through the explanation of transitive closure. They use the following example:
$A$ is a collection of people. In $A$ the relation $R$ is defined by $xRy$ if $x$ is a ... |
H: When is $\mathfrak{S}_n \times \mathfrak{S}_m$ a subgroup of $\mathfrak{S}_p$?
Inspired by another question, I wondered when $\mathfrak{S}_n \times \mathfrak{S}_m$ is isomorphic to a subgroup of $\mathfrak{S}_p$. Eliminating the obvious cases, the question becomes:
Let $n,m,p>1$ be such that $\mathfrak{S}_n \times... |
H: Solving a recurrence relation with floors and comparing it with other complexity classes
The problem that I am struggling with is the recurrance relation
$T(n) = \lfloor(T(n/2))\rfloor + \lfloor(log \space n)\rfloor$ Where $T(1) = 1$
I am supposed to answer true/false to each of the following (along with explanat... |
H: Prove: the set of zeros of a continuous function is closed.
Prove: the set of zeros of a continuous function is closed.
And should the function on a closed interval?
AI: Hint: No matter what the domain of the function is (as long as it has a topology — no restriction on "closed interval", in particular):
The inver... |
H: finding the basis of an eigenspace
"Given the matrix $A=\left[\begin{array}{cc}6 & 6\\-2 & -1\end{array}\right].$
"The roots of the characteristic polynomial of $A$ are: $$\begin{align}\det(A-tI)=0 &\Leftrightarrow \det\left[\begin{array}{cc}6-t & 6\\-2 & -1-t\end{array}\right]=0\\ &\Leftrightarrow 6-5t+t^2=0\\ &... |
H: Picard method to solve a diferential equation
I have to obtain by Picard method the solution to this problem.
$$x'=x+t, x(0)=x_0$$
doing $$x_j=x_0 + \int_{0}^{t} f(s,x_{j-1}(s))ds$$
i have obtain $$x_{j}=x_0(1+...+\dfrac{t^j}{j!})+\dfrac{t^{j+1}}{j+1!}$$
but i dont know how to continue because the first part i thin... |
H: Intervals where a function is convex and/or concave
I find myself in need of the solution of the following problem for my work. An help is appreciated.
Let $a$ be a real such that $0 \le a \le 1$. For what real values of $y$ is the function
$$
f(x) = a^{x^y}
$$
1) always concave for all $x \ge 0$.
2) always convex... |
H: Side-Splitter Theorem, which sides are proportionally
I'm currently wondering about the side splitter theorem.
Lets say I have this triangle. Now the side splitter theorem says, that if a line is parallel to a side of a triangle and intersects the two other sides, then this line divides those two sides proportiona... |
H: Prove that $2^n < \binom{2n}{n} < 2^{2n}$
Prove that $2^n < \binom{2n}{n} < 2^{2n}$. This is proven easily enough by splitting it up into two parts and then proving each part by induction.
First part: $2^n < \binom{2n}{n}$. The base $n = 1$ is trivial. Assume inductively that some $k$ satisfies our statement. The ... |
H: Determine convergence or divergence $\sum_1^\infty \sin \frac{1}{n}$
$$\sum_1^\infty \sin \frac{1}{n}$$
So now I konw that to evaluate this I can just look at the limit as it reaches infinity. I see that it would result in 1 over 0, but it approaches 0 so I could that that it approaches sin0 which is 0 so doesn't t... |
H: How to calculate closed form expression for r-simplex numbers
Hello,
I've gotten far on this exercise, with the following insight:
Here is a matrix of examples (vertical-axis is n=1,2,3,4,5,6,7,8; horizontal-axis is k=1,2,3,4)
1: 1 1 1 1
2: 2 3 4 5
3: 3 6 10 15
4: 4 10 20 35
5: 5 15 35 70
6:... |
H: How do lambda calculus most basic definitions work?
Good afternoon,
I'm trying to learn lambda calculus, and I do understand the notation (it's not hard, $f=\lambda a_1.\cdots\lambda a_n.x=\lambda a_1\cdots a_n.x\Leftrightarrow f(a_1;\cdots;a_n)=x$), but not what is defined by it. For an example of something I don'... |
H: Proving (or disproving) that the sine and cosine of integers are always unique
Can it be proven that
$ \forall x, y \in \mathbb{Z} \left( \sin(x) = \sin(y) \iff x = y\right)$
? Or disproven, of course. And likewise with cosine?
Since sine and cosine have periods of $2\pi$, for $x$ and $y$ radians to have the same l... |
H: Can I prove Pythagoras' Theorem using that $\sin^2(\theta)+\cos^2(\theta)=1$?
In any right-angled triangle, the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares whose sides are the two legs (the two sides that meet at a right angle)... |
H: Do we assert the existence of set theory when reasoning about L-structures?
In model theory, if L is a first order language, by the definition of a L-structure $\mathcal{M}$ it is partly given by a non-empty set $M$ called the universe or domain of $\mathcal{M}$.
From where did we get this set? I know how I could d... |
H: Create range in base of ratio
I have a big problem to calculate votes with my excel documents. I have some field for a project: - minimum estimate in hours (before I start a project I tell how many hours I think to do for this project) - maximum estimate in hours(before I start a project I tell how many hours maxim... |
H: Dividing polynomial by binomial using remainder theorem
An A level text book claims that one can find the quotient by first:
1.) Setting up an identity, $f(x)≡ Q(x)(divisor) + remainder$
2.) Finding the coefficients
However, another A level text book says, "Note. This theorem gives a (simple) method for evaluating ... |
H: How many times the counter gets incremented?
In the loop:
int counter=0;
for(i=0;i<10;i++)
for(j=i+1;j<10;j++)
for(k=j+1;k<10;k++)
for(l=k+1;l<10;l++)
for(m=l+1;m<10;m++)
counter++;
The answer for this counter is 10C5. i Actually don't understand how it is 10C5 ?
10c5= 10*9*8*7*6/(5!).
Is this five factorial ... |
H: If $P$ is a prime ideal of an integral domain $D$, then is $D$ equal to its localization at $P$?
I refer to this article on the localization of integral domains.
Let $D$ be an integral domain, and $P$ a prime ideal of $D$. $$D_P=\{ab^{-1}\mid a\in D,b\notin P\}$$
Let us suppose $P\subset D$. This means that $1\not... |
H: Summation Series for fractions
I have a summation series of the form: $n + n/2 + n/4 + n/8 + n/16 +\ldots + 1$. Is there a formula for this series? Basically, the denominators are powers of 2.
AI: There is : it is the sum of the $\log n + 1$ first terms of a geometric series:
$$
\sum_{i=0}^{\log n} \frac{n}{2^i} =... |
H: Lifting linear solutions mod $p$ to mod $p^2$
I've always known the famous Hensel's lemma in number theory which allows us to lift solutions of an equation $f(x) \equiv 0 \pmod p$ to solutions modulo $p^n$ under non-degeneracy.
What about the following problem : if I start with a linear system of equations of the ... |
H: Conjuncting two independent statements
Suppose there are two statements, $A$ and $B$ that are independent. As far as I know one needn't to prove $A$ or $B$ either, it is enough to generate $C = A \land B$, and then proving $C$ shows $A$ and $B$ are both true. If my understanding is correct, we are practically gener... |
H: Arranging $6$ orange marbles and $6$ red marbles in a circle.
I would like to count the number of ways to arrange $6$ orange marbles and $6$ red marbles in a circle, if rotations of a given arrangement are considered the same (and marbles of the same color are identical).
I realize this can be done by considering... |
H: Equality involving $\sum_n \sin(\gamma_n \log x)/\gamma_n$
This is I think an algebra confusion about an equality of Littlewood,
$$\frac{\psi(x) - x}{\sqrt{x}} = -2\sum_{1}^{\infty}\frac{\sin( \gamma_n\log x)}{\gamma_n} + O(1).\hspace{20mm}(1)$$
He refers the reader to "equivalent formulas" in Landau, and for this... |
H: Every topological space $X$ has the initial topology with respect to the family of continuous functions from $X$ to the Sierpiński space.
I am currently reading about initial topologies w.r.t. the Sierpiński space, and on Wikipedia I read the following
Every topological space $X$ has the initial topology with resp... |
H: Why not extending to the whole disk implies have a zero
For any complex polynomial $p(z)$ of order $m$, we showed earlier that on a circle $S$ of sufficiently large radius $r$ in the plane,
$$\frac{p(z)}{|p(z)|}\quad \text{and}\quad \frac{z^m}{|z^m|}=\left(\frac{z}{r}\right)^m$$
are homotopic maps of $S \to S^... |
H: nth roots of negative numbers
Disclaimer: I know what complex numbers are.
Let $x,\space n\in\Bbb R$
What is the complex algebraic solution to $\sqrt[n]{-x}$?
Could I have a 'general' formula and a walk through on how to accomplish this.
I know about roots of Unity such that:
$$\large\sqrt[n]{\pm 1}=\pm e^{(2\pi ki... |
H: A point $(x,y)$ moves so that its distance from the line...
A point $(x,y)$ moves so that its distance from the line $x=5$ is twice as great as its distance from the line $y=8$. Find an equation of the path of the point.
I got the two equations:
$$x-2y+11=0 \tag{1}$$
$$x+2y-21=0 \tag{2}$$
The book claims that a sin... |
H: Why the degree of $p/|p|$ is zero on $\partial W^\prime?$
Let $W$ be a smooth compact region in $\mathbb{C}$ whose boundary contains no zeros of the polynomial $p$. $p$ has only finitely many roots, $z_0, \dots, z_n$ in $W$. Around each $z_i$, circumscribe a small closed disk $D_i$, making the disks all disjoin... |
H: Proving : $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\ge4(a+b+c)$
For $a,b,c > 0$ and $ab+bc+ca+2abc=1$, how to prove that:
$$\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\ge4(a+b+c) \, ?$$
AI: The condition is equivalent to $$\frac{a}{a + 1} + \frac{b}{b + 1} + \frac{c}{c + 1} = 1$$ then take $x = \frac{a}{a + 1}$, $y = \frac{b}{b... |
H: "Tricks" for solving the determinant of a matrix
I've got an exam coming up, and I'm worried that (as I usually tend to do) I will mess up with some minor algebraic calculation in a large problem like finding the determinant of a 4x4 or larger matrix. I'm likely to be asked to use Laplace Expansion, and I'd like to... |
H: Maclaurin polynomial of $\ln(1+x)$
My book is demanding that I show that the Maclaurin polynomial for $\ln(1+x)$ is
$$T_n (x) = x - \frac{x^2}{2} + \frac{x^3}{3}+ \cdots + (-1)^{n-1} \frac{x^n}{n}$$
I don't think this is true at all actually. Following the given formula for finding this
$$f(a) + \frac{f'(a)}{1!}(x)... |
H: Is it possible to build a circle with quadratic Bézier curves?
i'm searching for a curve type with a minimum of functionality and maximum of usability. I run into quadratic Bézier curves and i wonder, if its possible to draw a circle with it.
AI: No, you can only produce some good approximations for sufficiently sm... |
H: counting the number of elements in a conjugacy class of $S_n$
I want to know if there is some systematic way (using some combinatorial argument)
to find the number of elements of conjugacy classes of $S_n$ for some given $n$.
For example, let's consider $S_5$. If the representative for the conjugacy class
is an $m$... |
H: Plus or minus? Is there a canonical orientation, like counterclockwisely?
When $X$ also happens to be a submanifold of $Y$, then, as in the mod $2$ case, we define its intersection number with $Z, I(X, Z)$, to be the intersection number of the inclusion map of $X$ with $Z$. If $X \pitchfork Z$, then $I(X, Z)$ is ... |
H: Do groups, rings and fields have practical applications in CS? If so, what are some?
This is ONE thing about my undergraduate studies in computer science that I haven't been able to 'link' in my real life (academic and professional). Almost everything I studied I've observed be applied (directly or indirectly) or h... |
H: Extension of Fermat's little theorem with Carmichael numbers
I'm a bit confused about the nature of one of my homework problems. It is requesting an explanation for why a congruence holds for $a^n \equiv a \;(\!\!\!\mod n)$ for a composite $n$, however this congruence is not always true from my understanding as hav... |
H: Combinatorics: 7 character passwords
I am currently reviewing probability using Probability, Statistics, and Stochastic Processes by Peter Olofsson, and I have a dispute over the answer to problem #16 on page 67.
The question is as follows:
"You are asked to select a password for a Web site. It must consist of five... |
H: Telescoping series sum
I am suppose to find $\sum_{n = 3}^\infty \frac{1}{n(n-1)}$
I am suppose to rewrite it as a telescoping series, but that isn't really defined so I don't know how to do that so I just copied the wikipedia page and get
$$\frac{-1}{n} + \frac{1}{n-1}$$
Ok whatever, I try and find the sum and i s... |
H: Prove : $P(n | \text{ number of parts $\le m$}) = P(n | \text{ all parts $\le m$})$
I'm trying to prove both sides of :
$$P(n | \text{ number of parts $\le m$}) = P(n | \text{ all parts $\le m$}).$$
First side: Given a partition where all parts $\le m$, we can build
a Ferrer's graph where
$$\lambda_1 \ge \lambda_... |
H: Geometric series sum $\sum_2^\infty e^{3-2n}$
$$\sum_2^\infty e^{3-2n}$$
The formulas for these things are so ambiguous I really have no clue on how to use them.
$$\frac {cr^M}{1-r}$$
$$\frac {1e^2}{1-\frac{1}{e}}$$
Is that a wrong application of the formula and why?
AI: Note $e^{3-2n}=e^3(e^{-2})^n$ so
$$\sum_{n=2... |
H: Rounding to the nearest term in a geometric progression
Consider the following progression:
$$R(i)=5\cdot10^{\frac{i}{30}}$$
where $i$ is ith number within the progression.
I would like to devise an equation that will round input value to the nearest number from this progression.
For example 5th, 6th, and 7th numbe... |
H: Multilingual hedge fund - Puzzle
I'm having difficulty finding the solution for the following problem:
A hedge fund has 70 employees. For any two employees $X$ and $Y$ there is a language that $X$ speaks but $Y$ does not, and there is a language that $Y$ speaks but $X$ does not. At least how many different languag... |
H: Show the solution of this ODE has to be the trivial one
I'm trying to solve this question:
Let $f$ be continuous in $\Omega=\{(t,x);|t|\leq a,|x|\leq b\}$. If
$f(t,x)\lt 0$ when $tx\gt0$ and $f(t,x)\gt 0$ when $tx\lt0$. Show that $x'=f(t,x), x(0)=0$ has $\varphi=0$ as the unique solution.
I'm completely confuse... |
H: Limit comparison test $\sum_2^\infty \frac{n}{\sqrt{n^3 + 1}}$
$$\sum_{n=2}^\infty \frac{n}{\sqrt{n^3 + 1}}$$
How do I pick a $b_n$? I don't understand the strategy to that, my books just have waves that part and picks perfect ones that work out for their easily contruscted examples but for this one I pick somethin... |
H: Leibniz test is wrong?
Maybe I am reading the mathese wrong but according to my book: If $a_n$ is positive and decreasing and $\,\displaystyle\lim_{n\to\infty} a_n = 0,\,$ then the alternating series converges.
So for example if I have $\,\displaystyle\sum_1^\infty \frac{1}{n},\,$ I know that diverges so $\,\displa... |
H: Absolute convergence $\frac{(-1)^n}{n \ln n}$
$$\sum_2^\infty \frac{(-1)^n}{n \ln n}$$
So I claim that it is absolutely convergent because it only coverges when it is an absolute value. This is wrong but I don't see how.
AI: What is written below is really the Cauchy Condensation Test, without mention of Cauchy, an... |
H: Example of an open set $A\subset\mathbb{R}^n$ and a point $a\in \mathbb{R}^n\setminus A$ which is an isolated point of $\partial A$
Could someone give me an example of an open set $A\subset\mathbb{R}^n$ and a point $a\in \mathbb{R}^n\setminus A$ such that $a$ is an isolated point of the boundary of $A$?
AI: Try $A=... |
H: Books Arranged on a Shelf: 5 must be to the left, and 2 must be to the right
I'm told that there are five distinct books of category $A$, three distinct books of category $B$, and two of category $C$.
I'm then asked the following question:
In how many ways can these books be arranged on a shelf if all five categor... |
H: What textbook should I get to self-learn Calculus?
I did not have the option to take calculus during high school. I would like to pick up this subject during my free time.
I am a software engineer. I would like to improve my understanding of maths.
What textbook should I buy to understand Calculus.
I found "Calcu... |
H: A simple question of Littlewood-Paley decomposition.
Let $\{f_k(x)\}_{k=0}^\infty$ be a Littlewood-Paley decompositon, that is,
$$ f_k \in C_c^\infty $$
$$ \sum_{k=0}^\infty f_k (x) = 1,$$
$$ \text{supp} f_0 \subset \{ |x| \leq 2 \},$$
$$ \exists f \in C_c^\infty \; \text{such that}\; \text{supp} f \subset \{ 2... |
H: Quotient of unitary group by special unitary group
Can someone help me, I don't understand the following question.
"Using without proof, the homomorphism theorem, or otherwise, show that
$U(n)/SU(n)$ is isomorphic to $U(1)$."
Here, $U(n)$ is the unitary group while $SU(n)$ is the special unitary group.
AI: Conside... |
H: (CHECK) Cardinality of Terms in the Expansion of a Product of Multinomials
QUESTION: How many terms are there in the expansion of
$$(x+y)(a+b+c)(e+f+g)(h+i)$$
I'd like some help with this one, but I'd also like to discuss a method of generalization on the problem, namely how should one compute the cardinality o... |
H: Proof of Noether's Normalization theorem.
As stated here, Noether's Normalization Theorem states:
Suppose that $R$ is a finitely generated integral domain over a field $K$. Then there exists an algebraically independent subset $\{y_1,y_2,y_3,\dots,y_r\}\in R$ such that $R$ is integral over $K[y_1,y_2,y_3,\dots,y_r]... |
H: Triangle of Multinomial Coefficients
What is the "Triangle Of Multinomial Coefficients" seen here: http://oeis.org/A036038 (OEIS: A036038)
I can see that the diagonals of this triangle are just factorials... for example the last number in each row is as follows: 1,2,6,24,120 and that's just 1!, 2!, 3!, 4!, 5!
And t... |
H: Proof that every cauchy sequence converges in $\mathbb R^k$
I'm having a hard time understanding this proof (the portion in bold).
I know $E_N$ is bounded but how is the finite set
$\{x_1, \ldots, x_{n-1}\}\,$ bounded?
(Is it because every finite set in $\mathbb R^k$ is bounded?)
I didn't get the last sentence at ... |
H: Adding points of an elliptic curve over a finite field
I'm a bit confused with how fractions are handled with adding points of elliptic curves over finite fields. Below is an example from the text which I am trying to understand:
The part that particularly confuses me is
If I think about this the same way that ... |
H: Show that $f=g$ a.e on $[a,b]$ implies thats $f=g$ on $[a,b]$.
I'm reading Real Analysis by Royden (4th edition).
The problem I'm working on is in the title, were $f$ and $g$ are continuous functions on $[a,b]$.
What I know so far is that the set $\{x\,:\,f(x)\neq g(x)\}$ has measure zero. Also, I realized that $(f... |
H: Calculate trig limit of type $\frac{0}{0}$ without L'Hopital
I am trying to figure out the solution to this Limit without using L'Hopital.
$$
\lim \limits_{x \to \pi} \frac {(\tan (4x))^2 } {(x - \pi )^2}
$$
Any help would be greatly appreciated.
AI: One could evaluate the Taylor series expansion of the numerator ... |
H: What's the meaning of random variables $X_i^2(A)$
What's the meaning of random variables $X_i^2(A)$
For example:
Consider we are doing Bernoulli trials, $\omega =\{A, \text{not} A\}$ with $P(A)=p$ and $P(\text{not} A)=1-p=q$, Given $n$ independent random variables $X_1,X_2,\text{...},x_n$, each taking
$$\begin{ali... |
H: On the surface of the moon
On the surface of the moon, acceleration due to gravity is approximately 5.3 feet per second squared. Suppose a baseball is thrown upward from a height of 6 feet with an initial velocity of 15 feet per second.
A) Dertimne the maximum height attained by the baseball.
I have to show all wo... |
H: An Isomorphism of Rings
Let $R$ be the ring of Quaternions over $\Bbb{Z}_{(3\Bbb{Z})}$ ($\Bbb{Z}$ localized in $3\Bbb{Z}$). Is it true that $\frac{R}{J(R)}$ can be represented as $M_2(\Bbb{Z}_3)$ ?
($\Bbb{Z}_3$ is the integers modulo 3).
AI: $\def\ZZ{\mathbb{Z}}$Let us write, for every ring $R$, $Q(R)$ for the q... |
H: An elementary question about cardinal arithmetic 2
I put this one in other question that I had. But I'm not sure if that is permitted here. So, better open other question though is related to the last one.
Claim 1: If $X$ and $Y$ are finite sets, the set of all the functions from $X$ to $Y$ is finite.
Proof: Let $\... |
H: 2 Different integrals of $\int \left ( \tan{x}\right ) ^3 dx $.
My friend asked me why this function has 2 different integrals. I'm very confused.
\begin{align}
\int \left ( \tan{x}\right ) ^3 dx &=\int \left ( \tan{x} \right )^2 \tan{x}dx \\
&=\int \left ( \sec^2 {x} -1 \right ) \tan{x} dx \\
&=\int \tan{x} \left ... |
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