Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
What is the difference between topological and metric spaces? What is the difference between a topological and a metric space?
| A metric space gives rise to a topological space on the same set (generated by the open balls in the metric). Different metrics can give the same topology. A topology that arises in this way is a metrizable topology. Using the topology we can define notions that are purely topological, like convergence, compactness, c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/21313",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "31",
"answer_count": 5,
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Alchemist's problem Consider an alchemist that has many ($N$) sorts of ingredients in his possession. Initial amounts of each of the ingredients is expressed by vector $C^0=(C_1, C_2, \dots, C_N)$.
Alchemist knows several ($M$) recipes of ingredient transmutation, expressed as a set of recipes: $R=\{ R^1, R^2, \dots, ... | Since the market prices are fixed and $V^L = C^0 \cdot v + S^0 \cdot v + S^1 \cdot v + \ldots + S^L \cdot v$, you can also assign a value $R^i \cdot v$ to each recipe and use Dijkstras algorithm.
The exact implementation works as follows:
*
*A vertex in the graph is represented by some choice of $n \leq N$ recipes. ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/21348",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Arrangement of six triangles in a hexagon You have six triangles. Two are red, two are blue, and two are green. How many truly different hexagons can you make by combining these triangles?
I have two possible approachtes to solving this question:
*
*In general, you can arrange $n$ objects, of which $a$ are of type o... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "2",
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Determine which of the following mappings F are linear I'm having a really hard time understanding how to figure out if a mapping is linear or not. Here is my homework question:
Determine which of the following mappings F are linear.
(a) $F: \mathbb{R}^3 \to \mathbb{R}^2$ defined by $F(x,y,z) = (x, z)$
(b) $F: \mathbb{... | To check if a mapping is linear in general, all you need is verify the two properties.
*
*$f(x+y) = f(x) + f(y)$
*$f(\alpha x) = \alpha f(x)$
The above two can be combined into one property: $f(\alpha x + \beta y) = \alpha f(x) + \beta f(y)$
Edit
For instance, if we want to show say $F(x) = f(x_1,x_2,x_3) = x_1 -... | {
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"timestamp": "2023-03-29T00:00:00",
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Using congruences, show $\frac{1}{5}n^5 + \frac{1}{3}n^3 + \frac{7}{15}n$ is integer for every $n$ Using congruences, show that the following is always an integer for every integer
value of $n$:
$$\frac{1}{5}n^5 + \frac{1}{3}n^3 + \frac{7}{15}n.$$
| Lets show that $P(n)=3n^5+5n^3+7n$ is divisible by 15 for every $n$. To do this, we will show that it is divisible by $3$ and $5$ for every $n$.
Recall that for a prime $p$, $x^p\equiv x \pmod{p}$. (Fermat's Little Theorem) Then, looking modulo 5 we see that
$$P(n)\equiv 3n^5+7n\equiv 3n+7n=10n\equiv 0.$$
Now looking... | {
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Is there a definition of determinants that does not rely on how they are calculated? In the few linear algebra texts I have read, the determinant is introduced in the following manner;
"Here is a formula for what we call $detA$. Here are some other formulas. And finally, here are some nice properties of the determinant... | Let $B$ a basis of a vector space $E$ of dimension $n$ over $\Bbbk$. Then $det_B$ is the only $n$-alternating multilinear form with $det_B(B) = 1$.
A $n$-multilinear form is a map of $E^n$ in $\Bbbk$ which is linear for each variable.
A $n$- alternated multilinear form is a multilinear form which verify for all $i,j$
$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/21614",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "61",
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"answer_id": 2
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Can contractible subspace be ignored/collapsed when computing $\pi_n$ or $H_n$? Can contractible subspace be ignored/collapsed when computing $\pi_n$ or $H_n$?
Motivation: I took this for granted for a long time, as I thought collapsing the contractible subspace does not change the homotopy type. Now it seems that this... | Let me note a general fact: if the inclusion $A \hookrightarrow X$ (for $A$ a closed subspace) is a cofibration, and $A$ is contractible, then the map $X \to X/A$ is a homotopy equivalence. See Corollary 5.13 in chapter 1 of Whitehead's "Elements of homotopy theory."
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/21705",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 3,
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Meaning of $\mathbf{C}^{0}$? My book introduces $\mathbf{C}^{\infty}$ as subspace of $F(\mathbb{R},\mathbb{R})$ that consists of 'smooth' functions, that is, functions that are differentiable infinitely many times. It then asks me to tell whether or not $\mathbf{C}^{0}$=$(f\in(\mathbb{R},\mathbb{R})$ such that $f$ is ... | Typically, $C^{0}(\Omega)$ denotes the space of functions which are continuous over $\Omega$. The higher derivatives may or may not exist.
$C^{\infty}(\Omega) \subset C^{0}(\Omega)$ since if the function is infinitely "smooth" it has to continuous.
Typically, people use the notation $C^{(n)}(\Omega)$ where $n \in \math... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/21767",
"timestamp": "2023-03-29T00:00:00",
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Can we construct a function $f:\mathbb{R} \rightarrow \mathbb{R}$ such that it has intermediate value property and discontinuous everywhere?
Can we construct a function $f:\mathbb{R} \rightarrow \mathbb{R}$ such that it has intermediate value property and discontinuous everywhere?
I think it is probable because we ca... | Sure. The class of functions satisfying the conclusion of the Intermediate Value Theorem is actually vast and well-studied: such functions are called Darboux functions in honor of Jean Gaston Darboux, who showed that any derivative is such a function (the point being that not every derivative is continuous).
A standar... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/21812",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "32",
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Solving short trigo equation with sine - need some help! From the relation $M=E-\epsilon\cdot\sin(E)$, I need to find the value of E, knowing the two other parameters. How should I go about this?
This is part of a computation which will be done quite a number of times per second. I hope there's a quick way to get E out... | I assume $\epsilon$ is a small quantity and propose one of the following:
(a) Write your equation in the form $E=M+\epsilon \sin(E)=: f(E)$ and consider this as a fixed point problem for the function $f$. Starting with $E_0:=M$ compute numerically successive iterates $E_{n+1}:=f(E_n)$; these will converge to the desire... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/21864",
"timestamp": "2023-03-29T00:00:00",
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Chain rule for multi-variable functions So I have been studying the multi-variable chain rule. Most importantly, and this is what I must have overlooked, is it's not always clear to me how to see which variables are functions of other variables, so that you know when to use the chain rule. For example, if you have:
$... | If we have an explicit function $z = f(x_1,x_2,\ldots,x_n)$, then
$$\displaystyle \frac{dz}{dt} = \frac{\partial z}{\partial x_1} \frac{dx_1}{dt} + \frac{\partial z}{\partial x_2} \frac{dx_2}{dt} + \cdots +\frac{\partial z}{\partial x_n} \frac{dx_n}{dt}$$
If we have an implicit function $f(z,x_1,x_2,\ldots,x_n) = 0$, t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/21915",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Statistics: Predict 90th percentile with small sample set I have a quite small data set (on the order of 8-20) from an essentially unknown system and would like to predict a value that will be higher than the next number generated by the same system 90% of the time. Both underestimation and overestimation are problemat... | This is where the technique of "Bootstrap" comes in extremely handy. You do not need to know anything about the underlying distribution.
Your question fits in perfectly for a good example of "Bootstrap" technique. The bootstrap technique would also let you determine the confidence intervals. Bootstrap is very elementar... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/21959",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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What this kind probability should be called? I have $m$ continues integer points on a line, randomly uniform select $n$ points from the $m$ point without replacement. Order the points ascendingly.
Let the random variable $A_i$ is the position (coordination on the line) of the $i$th point. So, $$P(A_i=k)=\frac{{k-1\cho... | Edit: See Didier's comment below. The binomial coefficients are "upside down" and so what's written below is meaningless. It is worthwhile, however, to see which tools are used to obtain tail estimates on the hypergeometric distribution, to get some ideas. Perhaps all they do is use Stirling's approximation and integra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/22016",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Simultaneous equations, trig functions and the existence of solutions Came across this conundrum while going over the proof that
$$A \cdot \sin(bx) + B \cdot \cos(bx) = C \cdot \sin(bx + k)$$
for some numbers $C$ and $k$. ($A$, $B$ and $b$ are known.)
The usual method is to expand the RHS using the compound angle ident... | In general, you don't. You know that all of the solutions of the pair of equations you started with are solutions of the single equation you ended up with (barring division-by-zero issues), but you generally don't know the converse. In this case, the reason you can get away with the converse is that you can choose $C$.... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Prove that if $A^2=0$ then $A$ is not invertible Let $A$ be $n\times n$ matrix. Prove that if $A^2=\mathbf{0}$ then $A$ is not invertible.
| Well I've heard that the more ways you can prove something, the merrier. :) So here's a sketch of the proof that immediately came to mind, although it may not be as snappy as some of the other good ones here:
Let's prove the contrapositive, that is if $A$ is invertible then $A^2 \neq 0$.
If $A$ is invertible then we ca... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/22195",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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"answer_id": 6
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Reaching all possible simple directed graphs with a given degree sequence with 2-edge swaps Starting with a given simple, directed Graph G, I define a two-edge swap as:
*
*select two edges u->v and x->y such that (u!=x) and (v!=y) and (u!=y) and (x!=v)
*delete the two edges u->v and x->y
*add edges u->y and x->v
... | The question is whether a triple swap is necessary or not. One of the examples in the paper is the directed cycle between three nodes (i->j), (j->k), (k->i). Obviously, another graph with the same degree sequence is the one in which all directions are reversed: (i <- j), (j <- k), (k <- i). It is, however, not possible... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/22272",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Algorithm complexity in for loop I have an algorithm and I would like to know how many times each line is called.
There I wrote which lines I understand and some lines is left.
j := 1 ---------------------- 1 time
while j < n do --------------- n times
x := B[j] ---------------- n-1 times
k := j ----------... | Hint: You enter the for loop n-1 times as shown from the line above. Then how many loops to you do? It should be something like n-j, but you have to figure out the ends-there may be a +1 or -1 or something. Does your definition of for do the loop with i=n or not? so the for will be executed 2*(n-1)*(something like ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/22358",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Proof by contradiction: $r - \frac{1}{r} =5\Longrightarrow r$ is irrational? Prove that any positive real number $r$ satisfying:
$r - \frac{1}{r} = 5$ must be irrational.
Using the contradiction that the equation must be rational, we set $r= a/b$, where a,b are positive integers and substitute:
$\begin{align*}
&\fra... | To complete your solution, note that you can, without loss of generality, set $a$ and $b$ to be coprime. So you have $a^2=b^2+5ab=b(b+5a)$. Hence $a$ divides $b(b+5a)$. Euclid's lemma now tells you that $a$ divides $b+5a$ (because $a$ and $b$ are coprime). But then $a$ must divide $b$, which is contradiction with the f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/22423",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to Prove the following Function Properties Definition: F is a function iff F is a relation and $(x,y) \in F$ and $(x,z) \in F \implies y=z$.
I'm reading Introduction to Set Theory by Monk, J. Donald (James Donald), 1930 and i came across a theorem 4.10.
Theorem 4.10
(ii)$0:0 \to A$, if $F : 0 \to A$, then $F=... | Consider the function $F\colon 0\to A$, suppose there is some $\langle x,y\rangle\in F$. This means that $x\in dom F$, since we have $dom F = 0$ then $x\in 0$ which is a contradiction. Therefore there are no ordered pairs in $F$, from the fact that it is a function we know that there are not other elements in $F$.
If s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/22473",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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What are the steps to solve this simple algebraic equation? This is the equation that I use to calculate a percentage margin between cost and sales prices, where $x$ = sales price and $y$ = cost price:
\begin{equation}
z=\frac{x-y}{x}*100
\end{equation}
This can be solved for $x$ to give the following equation, which c... | $$ z = 100 \cdot \frac{x-y}{x}$$
$$ zx = 100(x-y)$$
$$zx - 100x = -100y$$
$$x(z-100) = -100y$$
$$x = -\frac{100y}{z-100}$$
Then divide both numerator and denominator by $-100$ to get $$x = \frac{y}{1-(\frac{z}{100})}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/22560",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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"answer_id": 1
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Cartesian product set difference I know how to handle the 2d case: http://www.proofwiki.org/wiki/Set_Difference_of_Cartesian_Products
But I am having trouble simplifying the following:
Let $X=\prod_{1}^\infty X_i, A_i \subset X_i$
How can I simplify/rewrite $X - (A_1 \times A_2 \times \cdots A_n \times X_{n+1} \times X... | Try writing
$$\prod_{k=n+1}^{\infty} X_k = X'$$
then you want the difference of
$$(X_1 \times X_2 \times \cdots \times X_n \times X') - (A_1 \times A_2 \times\cdots \times A_n \times X')$$
You can use the rule that you linked inductively to this difference. Then note that in some parts of the expression you wil... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/22607",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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What kind of matrices are orthogonally equivalent to themselves? A matrix $A \in R^{n\times n}$ is said to be orthogonally equivalent to $B\in R^{n\times n}$ if there is an orthogonal matrix $U\in R^{n\times n}$, $U^T U=I$, such that $A=U^T B U$. My question is what kind of matrices are orthogonally equivalent to thems... | The family of matrices $U^{T}BU$, where $B$ is a fixed, positive definite matrix $\mathbb{R}^{n\times n}$, and $U$ varies over the orthogonal group $O(n)$, is obtaining by rigidly rotating and reflecting the eigenvectors of $B$. The matrix $B$ is invariant under such a transformation iff its eigenspaces are preserved.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/22660",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Simple limit, wolframalpha doesn't agree, what's wrong? (Just the sign of the answer that's off) $\begin{align*}
\lim_{x\to 0}\frac{\frac{1}{\sqrt{4+x}}-\frac{1}{2}}{x}
&=\lim_{x\to 0}\frac{\frac{2}{2\sqrt{4+x}}-\frac{\sqrt{4+x}}{2\sqrt{4+x}}}{x}\\
&=\lim_{x\to 0}\frac{\frac{2-\sqrt{4+x}}{2\sqrt{4+x}}}{x}\\
&=\... | Others have already pointed out a sign error. One way to avoid such is to first simplify the problem by changing variables. Let $\rm\ z = \sqrt{4+x}\ $ so $\rm\ x = z^2 - 4\:.\:$ Then
$$\rm \frac{\frac{1}{\sqrt{4+x}}-\frac{1}{2}}{x}\ =\ \frac{\frac{1}z - \frac{1}2}{z^2-4}\ =\ \frac{-(z-2)}{2\:z\:(z^2-4)}\ =\ \frac{-1}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/22704",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Nondeterministic Finite Automata to Deterministic Finite Automata? I am unfamiliar with the general process of converting NFA to DFA. I have general understanding of the theory, but I don't have the method established. Please help explain the process required to transform an NFA to DFA. Thank you.
| Suppose the original NFA had state set $S$, initial state $q \in S$, and accepting states $F \subset S$. The DFA is going to keep track of what possible states the NFA could get into reading the input so far. Therefore each state of the DFA corresponds to a subset of $S$, viz. the possible states the NFA could get into... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/22749",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Number of inner nodes in relation to the leaf number N I am aware that if there is a bifurcating tree with N leaves, then there are (N-1) internal nodes (branching points) with a single root node. How is this relationship proved?
Best,
| Here is an approach considering a directed binary tree:
Let there be $k$ internal nodes (Note that we consider the root to be an internal node as well). Since we consider a binary tree, the $k$ internal nodes contributes 2 edges each and thus we have $2k$ many edges in the tree that implies there is a total degree of $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/22788",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 2
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Questions about determining local extremum by derivative
*
*Second derivative test in
Wikipedia says that:
For a real function of one variable:
If the function f is twice
differentiable at a stationary point
x, meaning that $\ f^{\prime}(x) = 0$
, then:
If $ f^{\prime\prime}(x) < 0$ then $f$
has a local maximum at $... | Actually, the continuity of the partials is not needed, twice total differentiability at the point is sufficient, so the one and the higher dimensional cases are totally analogous. But the existence of the second order partials is insufficient for twice total differentiability, so the Hessian is not necessarily the sec... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/22859",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 1
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Paying off a mortgage twice as fast? My brother has a 30 year fixed mortgage. He pays monthly. Every month my brother doubles his principal payment (so every month, he pays a little bit more, according to how much more principal he's paying).
He told me he'd pay his mortgage off in 15 years this way. I told him I thoug... | Let's look at two scenarios: two months of payment $P$ vs. one month of payment $2P$. Start with the second scenario. Assume the total amount to be payed is $X$ and the rate is $r > 1$, the total amount to be payed after one month would be $$r(X-2P).$$ Under the first scheme, the total amount to be payed after two mont... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/22886",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 4,
"answer_id": 3
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Geometrical construction for Snell's law? Snell's law from geometrical optics states that the ratio of the angles of incidence $\theta_1$ and of the angle of refraction $\theta_2$ as shown in figure1, is the same as the opposite ratio of the indices of refraction $n_1$ and $n_2$.
$$
\frac{\sin\theta_1}{\sin \theta_2} ... | Yes, right. When entering a denser medium light slows down.. Adding another answer keeping only to the construction method. Drawn on Geogebra, removed the axes and grid to trace a ray inside a medium of higher refractive index $\mu$.
Choose an arbitrary point X such that
$$ \mu= \dfrac{XP}{XQ} \left( =\dfrac{n_2}{n_1}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/22945",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "20",
"answer_count": 8,
"answer_id": 7
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Limit of function of a set of intervals labeled i_n in [0,1] Suppose we divide the the interval $[0,1]$ into $t$ equal intervals labeled $i_1$ upto $i_t$, then we make a function $f(t,x)$ that returns $1$ if $x$ is in $i_n$ and $n$ is odd, and $0$ if $n$ is even.
What is $\lim_{t \rightarrow \infty} f(t,1/3)$?
What is ... | There is no limit for any $0<x<1$.
(a) $f(t, 1/3) = 1$ for $t$ of the form $6n+1$ or $6n+2$ and $0$ for $t$ of the form $6n+4$ or $6n+5$, and on the boundary in other cases. For example $\frac{2n}{6n+1} < \frac{1}{3} < \frac{2n+1}{6n+1}$ and $\frac{2n+1}{6n+4} < \frac{1}{3} < \frac{2n+2}{6n+4} .$
(b) $f(t, 1/2) = 1$ f... | {
"language": "en",
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A basic question about finding ideals of rings and proving that these are all the ideals I am a student taking a "discrete maths" course. Teacher seems to jump from one subject to another rapidly and this time he covered ring theory, Z/nZ, and polynomial rings.
It is hard for me to understand anything in his class, and... | Since $\mathbb{Z}/6\mathbb{Z}$ is finite, it is not difficult to try to find all ideals: you've got $\{0\}$ and you've got $\mathbb{Z}/6\mathbb{Z}$. Suppose the ideal contains $a\neq 0$. Then it must also contain $a+a$, $a+a+a$, and so on. Check the possibilities.
No, there generally is no mapping between $F[X]/(p(x))... | {
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Question about total derivative If $z=f(x,y)$, then total derivative is $\mathrm{d}z=\frac{\partial f}{\partial x}\mathrm{d}x+\frac{\partial f}{\partial y}\mathrm{d}y$. If $\mathrm{d} z=0$, how do you show that $z$ is a constant?
| If $df = 0$, then $\frac{\partial f}{\partial x} \ dx = -\frac{\partial f}{\partial y} \ dy$. I guess one could solve for $f(x,y)$ to get $f(x,y) = g(x-y)$ since $(f_x+f_y)g(x-y) = g'(x-y)+(-1)g'(x-y) = 0$ identically for some $g \in C^1$.
| {
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Covering and Cycles Let $G = (V, E)$ and $G' = (V', E')$ be two graphs, and let $f: V \rightarrow V'$ be a surjection. Then $f$ is a covering map from $G$ to $G'$ if for each $v \in G$, the restriction of $f$ to the neighbourhood of $v$ is a bijection onto the neighbourhood of $f(v) \in V'$ in $G'$.
My question (homew... | If $u,v\in V$ and $(u,v)\in E$ then since $u$ is in the neighborhood of $v$ then the condition on the local bijection gives you that $(f(v),f(u))\in E'$.
Suppose C is a cycle in G. Take the subgraph C' of G' with the vertices $f(v)$ such that $v\in C$ and the edges $(f(v),f(u))$ such that $(v,u)\in C$. If $u\in C$ the... | {
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Question on proof in "Primer on MCGs" This is a question about the proof of Proposition 1.4 in Farb and Margalit's "Primer on Mapping Class Groups" (in v. 5.0, it is on page 37 in the PDF, which you can download here). The proposition states
Let $\alpha$ be a non-nullhomotopic simple closed curve on the (hyperbolic) ... | For (1): They mean pick any element of the conjugacy class, and look at the corresponding $\phi$. It doesn't matter which one you look at because being primitive is a conjugacy invariant.
| {
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Prove that if $a^{k} \equiv b^{k} \pmod m $ and $a^{k+1} \equiv b^{k+1} \pmod m $ and $\gcd( a, m ) = 1$ then $a \equiv b \pmod m $ My attempt:
Since $a^{k} \equiv b^{k}( \text{mod}\ \ m ) \implies m|( a^{k} - b^{k} )$ and $a^{k+1} \equiv b^{k+1}( \text{mod}\ \ m ) \implies m|( a^{k+1} - b^{k+1} ) $
Using binomial id... | *
*$a\equiv b\pmod{n} \quad\textrm{and}\quad b\equiv c\pmod{n}\Rightarrow a\equiv c\pmod{n}$
*$a\equiv b\pmod{n}\Leftrightarrow b\equiv a\pmod{n}$
*$a\equiv b\pmod{n}\Rightarrow ac\equiv bc\pmod{n}$ for any $c$
The list can be enough to get $$a^kb\equiv a^ka\pmod{m}.$$
I would like to mentioned a theorem I just lea... | {
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How to solve 700 = 7x + 6y for a range of values? Last time I did any maths was A-Level (some time ago!).
This is a programming/layout problem where I need to display 7 items across a page, with 6 gaps between them. I have a fixed width, but need to determine values of x and y that are integers where x > y.
This 'feels... | $$700 = 7x + 6y\implies y = \frac{-7(x - 100)}{6}$$
Experimentation in a spreadsheet shows that [integer] $x,y$ values increase/decrease by amounts corresponding to their opposite coefficients. For example, a valid $x$-value occurs only every $6$ integers and a resulting $y$-value occurs every $7$ integers. Here is a s... | {
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Isomorphism in coordinate ring Let $x_{1},x_{2},...,x_{m}$ be elements of $\mathbb{A}^{n}$, where $\mathbb{A}^{n}$ is the n-affine space over an algebraically closed field $k$. Now define $X=\{x_{1},x_{2},...,x_{m}\}$. Why is the coordinate ring $A(X)$, isomorphic to $\oplus_{j=1}^{m} k = k^{m}$?
| For each $i$, $A(\{x_i\}) = k[x]/I(x_i) \cong k$, so each $I(x_i)$ is a maximal ideal of
$k[x]$. I assume the points $x_1,\ldots,x_n$ are distinct, from which it follows easily
that the ideals $I(x_1),\ldots,I(x_n)$ are distinct maximal ideals. Thus they are pairwise comaximal and the Chinese Remainder Theorem -- see... | {
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Question on Solving a Double Summation $$ \sum_{i=0}^{n-2}\left(\sum_{j=i+1}^{n-1} i\right) $$
Formulas in my book give me equations to memorize and solve simple questions like $$ \sum_{i=0}^{n} i $$ ... However, For the question on top, how would I go about solving it by hand without a calculator? WolfRamAlpha seems t... | Since the sum $$\sum_{j=i+1}^{n-1} i $$ does not depend on $j$ we see $$\sum_{j=i+1}^{n-1} i = i\cdot\sum_{j=i+1}^{n-1} 1= i(n-1-i) $$
Then you have to find $$\sum_{i=0}^{n-2} \left( i(n-1-i)\right)=\sum_{i=0}^{n-2} \left( (n-1)i-i^2)\right)=(n-1) \sum_{i=0}^{n-2} i- \sum_{i=0}^{n-2} i^2$$
Can you solve it from here?
H... | {
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Number Theory - Proof of divisibility by $3$ Prove that for every positive integer $x$ of exactly four digits, if the sum of digits is divisible by $3$, then $x$ itself is divisible by 3 (i.e., consider $x = 6132$, the sum of digits of $x$ is $6+1+3+2 = 12$, which is divisible by 3, so $x$ is also divisible by $3$.)
Ho... | Actually, this is true for an integral number with any digits. The proof is quite easy. Let's denote the integral number by $\overline{a_n a_{n-1} \ldots a_1}$. If the sum of its digits $\sum_{i=1}^n{a_i}$ is divisible by 3, then $\sum_{i=1}^n{(1+\overline{9...9}_{i-1})*a_i}$ is too. Here $\overline{9...9}_{i-1}$ denot... | {
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Largest known integer Does there exist a property which is known to be satisfied by only one integer, but such that this property does not provide a means by which to compute this number? I am asking because this number could be unfathomably large.
I was reading Conjectures that have been disproved with extremely larg... | One can easily generate "conjectures" with large counterexamples using Goodstein's theorem and related results. For example, if we conjecture that the Goodstein sequence $\rm\:4_k\:$ never converges to $0$ then the least counterexample is $\rm\ k = 3\ (2^{402653211}-1) \approx 10^{121210695}\:$. For much further discus... | {
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Example for Cyclic Groups and Selecting a generator In Cryptography, I find it commonly mentioned:
Let G be cyclic group of Prime order q and with a generator g.
Can you please exemplify this with a trivial example please!
Thanks.
| In fact if you take the group $(\mathbb{Z}_{p},+)$ for a prime number $p$, then every element is a generator.
*
*Take $G= \{a^{q}=e, a ,a^{2}, \cdots, a^{q-1}\}$. Now $|G|=q$ and $G = <a>$, which means that $G$ is generated by $a$.
| {
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Numerical approximation of an integral I read a problem to determine the integral $\int_1^{100}x^xdx$ with error at most 5% from the book "Which way did the bicycle go". I was a bit disappointed to read the solution which used computer or calculator. I was wondering whether there is a solution to the problem which does... | Because your integrand grows so fast the whole integral is dominates for $x\approx 100$. We can write $x^x = \exp[ x \ln(x)]$ and then expanding $x \ln(x) = 100 \ln(100) + [1+ \ln(100)] (x- 100) + \cdots$ around $x = 100$ (note that it is important to expand inside the exponent). The integral can therefore be estimated... | {
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How exactly do differential equations work? My textbook says that solutions for the equation $y'=-y^2$ must always be 0 or decreasing. I don't understand—if we're solving for y', then wouldn't it be more accurate to say it must always be 0 or negative. Decreasing seems to imply that we're looking at a full graph, even ... | *
*Remember if $y'(x) > 0$, $y(x)$ is increasing; if $y'(x) < 0$, $y(x)$ is decreasing; if $y'(x) = 0$ then $y(x)$ is constant. In our case $y'(x) \le 0$ which means $y(x)$ is always constant or decreasing.
*You can verify yourself: if $y(x) = 0$ for all $x$, then $y'(x) = 0$ so it is true that $y' = -y^2$ therefore ... | {
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If $a\mathbf{x}+b\mathbf{y}$ is an element of the non-empty subset $W$, then $W$ is a subspace of $V$ Okay, so my text required me to actually prove both sides; The non-empty subset $W$ is a subspace of a $V$ if and only if $a\mathbf{x}+b\mathbf{y} \in W$ for all scalars $a,b$ and all vectors $\mathbf{x},\mathbf{y} \in... | Consider $a=b=1$ and $a=-b=1$.
| {
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Graph coloring problem (possibly related to partitions) Given an undirected graph I'd like to color each node either black or red such that at most half of every node's neighbors have the same color as the node itself. As a first step, I'd like to show that this is always possible.
I believe this problem is the essenc... | Hint: Start with a random colouring and try to increase the number of edges which have differently coloured endpoints.
Spoiler:
Pick a node which has more than half of it's neighbour of the same colour as itself and flip it's colour. Now show that, as a result, the number of edges with different coloured endpoints in... | {
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Stock Option induction problem Can anyone help me solve this problem. I have no idea where to even start on it. Link inside stock option problem
| You are asked to prove that
$$
\int_{ - \infty }^\infty {V_{n - 1} (s + x)dF(x)} > s - c,
$$
for all $n \geq 1$. For $n=1$, substituting from the definition of $V_0$, you need to show that
$$
\int_{ - \infty }^\infty {\max \lbrace s + x - c,0\rbrace dF(x)} > s - c.
$$
For this purpose, first note that
$$
\max... | {
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Is $2^{340} - 1 \equiv 1 \pmod{341} $? Is $2^{340} - 1 \equiv 1 \pmod{341} $?
This is one of my homework problem, prove the statement above. However, I believe it is wrong. Since $2^{10} \equiv 1 \pmod{341}$, so $2^{10 \times 34} \equiv 1 \pmod{341}$ which implies $2^{340} - 1 \equiv 0 \pmod{341}$
Any idea?
Thanks,... | What you wrote is correct. $$2^{340}\equiv 1\pmod {341}$$ This is smallest example of a pseudoprime to the base two. See Fermat Pseudo Prime.
| {
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Big $O$ vs Big $\Theta$ I am aware of the big theta notation $f = \Theta(g)$ if and only if there are positive constants $A, B$ and $x_0 > 0$ such that for all $x > x_0$ we have
$$
A|g(x)| \leq |f(x)| \leq B |g(x)|.
$$
What if the condition is the following:
$$
C_1 + A|g(x)| \leq |f(x)| \leq C_2 + B |g(x)|
$$
where... | If $g(x)$ and $f(x)$ tends to $\infty$, then there is a value $x_0$ such that for $x > x_0$, $g(x)$ and $f(x)$ are strictly positive. Therefore, if $-C \leq f(x) - g(x) \leq C$, then for $x > x_0$, we have
$$ \frac{-C}{g(x)} \leq \frac{f(x)}{g(x)} -1 \leq \frac{C}{g(x)}. $$
Taking limits, you see that
$$ \lim_{x \to \... | {
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Some questions about $\mathbb{Z}[\zeta_3]$ I am giving a talk on Euler's proof that $X^3+Y^3=Z^3$ has no solutions in positive integers. Some facts that I believe to be true are the following. For some I give proof. Please verify that my reasoning is correct and make any pertinent comments. I use the notation $\zet... | b) does not follow from a). By definition, for $K$ a number field, $\mathcal{O}_K$ is the set of algebraic integers in $K$. You have shown, at best, that the ring of algebraic integers in $K = \mathbb{Q}(\zeta)$ contains $\mathbb{Z}[\zeta]$, but you have not shown that this is all of $\mathcal{O}_K$.
Here is a complet... | {
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Prove that $f(n) = 2^{\omega(n)}$ is multiplicative where $\omega(n)$ is the number of distinct primes
Prove that $f(n) = 2^{\omega(n)}$ is multiplicative where $\omega(n)$ is the number of distinct primes.
My attempt:
Let $a = p_1p_2\cdots p_k$ and $b = q_1q_2\cdots q_t$
where $p_i$ and $q_j$ are prime factors, and ... | Hint: Notice $$2^{\omega(a)}\times 2^{\omega(b)}=2^{\omega(a)+\omega(b)}$$ so try to relate $\omega(a)+\omega(b)$ and $\omega(ab)$.
To simplify things, you need only show it holds for $a=p^r$, $b=q^t$ where $q,p$ are prime numbers.
| {
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Find the average of a collection of points (in 2D space) I'm a bit rusty on my math, so please forgive me if my terminology is wrong or I'm overlooking extending a simple formula to solve the problem.
I have a collection of points in 2D space (x, y coordinates). I want to find the "average" point within that collectio... | There are different types of averages. Only the average of numbers is unambigious. The average you are looking for depends on what you want to use it for. If you take the avg. x and y coordinates separately, that will give you the center of mass.
| {
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Stronger than strong-mixing I have the following exercise:
"Show that if a measure-preserving system $(X, \mathcal B, \mu, T)$ has the property that for any $A,B \in \mathcal B$ there exists $N$ such that
$$\mu(A \cap T^{-n} B) = \mu(A)\mu(B)$$
for all $n \geq N$, then $\mu(A) = 0$ or $1$ for all $A \in \mathcal B$"
No... | Hint: what happens if $A=T^{-N}B$?
| {
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Is there a gap in the standard treatment of simplicial homology? On MO, Daniel Moskovich has this to say about the Hauptvermutung:
The Hauptvermutung is so obvious that it gets taken for granted everywhere, and most of us learn algebraic topology without ever noticing this huge gap in its foundations (of the text-book... | One doesn't need to explicitly compare with singular homology to get homotopy invariance (although that is certainly one way to do it), and one certainly doesn't need the Hauptvermutung (thankfully, since it is false in general). Rather, as you say, one can simplicially approximate a continuous map between simplicial... | {
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Why isn't $\mathbb{CP}^2$ a covering space for any other manifold? This is one of those perhaps rare occasions when someone takes the advice of the FAQ and asks a question to which they already know the answer. This puzzle took me a while, but I found it both simple and satisfying. It's also great because the proof d... | Euler characteristic is multiplicative, so (since $\chi(P^2)=3$ is a prime number) if $P^2\to X$ is a cover, $\chi(X)=1$ and $\pi_1(X)=\mathbb Z/3\mathbb Z$ (in particular, X is orientable). But in this case $H_1(X)$ is torsion, so (using Poincare duality) $\chi(X)=1+\dim H_2(X)+1>1$.
| {
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Cross product: problem with breakdown of vector to parallel + orthogonal components Vector $\vec{a}$ can be broken down into its components $\vec{a}_\parallel$ and $\vec{a}_\perp$ relative to $\vec{e}$.
*
*$\vec{a}_\parallel = (\vec{a}\vec{e})\vec{e}$
and
*
*$\vec{a}_\perp = \vec{e} \times (\vec{a} \times \ve... | To get the projection along $\vec{e}$ i.e. $\vec{a_{||}}$, you need to project your vector $a$ along the unit vector in the direction of $\vec{e}$. Similarly, if you want the component of $\vec{a}$ perpendicular to $\vec{e}$, $\vec{a_{\perp}} = \frac{\vec{e}}{||\vec{e}||}_2 \times \left( \vec{a} \times \frac{\vec{e}}{|... | {
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Elementary proof that $\mathbb{R}^n$ is not homeomorphic to $\mathbb{R}^m$ It is very elementary to show that $\mathbb{R}$ isn't homeomorphic to $\mathbb{R}^m$ for $m>1$: subtract a point and use the fact that connectedness is a homeomorphism invariant.
Along similar lines, you can show that $\mathbb{R^2}$ isn't homeom... | Consider the one point compactifications, $S^n$ and $S^m$, respectively. If $\mathbb R^n$ is homeomorphic to $R^m$, their one-point compactifications would be, as well. But $H_n(S^n)=\mathbb Z$, whereas $H_n(S^m)=0$, for $n\ne m,0$.
| {
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Find radius of the Smallest Circle that contains this figure A two dimensional silo shaped figure is formed by placing a semi-circle of diameter 1 on top of a unit square, with the diameter coinciding with the top of the square. How do we find the radius of the smallest circle that contains this silo?
| Draw the line of length 1.5 that cuts both the square and the half circle into two identical pieces. (Starting from the middle of the base of the square, go straight up) Notice that the center of the larger circle must lie somewhere on this line. Say that the center point is distance $x$ from the base of the square,... | {
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Nasty examples for different classes of functions Let $f: \mathbb{R} \to \mathbb{R}$ be a function. Usually when proving a theorem where $f$ is assumed to be continuous, differentiable, $C^1$ or smooth, it is enough to draw intuition by assuming that $f$ is piecewise smooth (something that one could perhaps draw on a p... | The Wikipedia article http://en.wikipedia.org/wiki/Pompeiu_derivative gives one example of how bad a non-continuous derivative can be.
One can show that any set whose complement is a dense intersection of countably many open sets is the point of discontinuities for some derivative. In particular a derivative can be di... | {
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Equality of polynomials: formal vs. functional Given two polynomials $A = \sum_{0\le k<n} a_k x^k$ and $B =\sum_{0\le k<n} b_k x^k$ of the same degree $n$, which are equal for all $x$, is it always true that $\ a_k = b_k\ $ for all $0\le k<n?$. All Coefficients and $x$ are complex numbers.
Edit:
Sorry, formulated the q... | The answer is in general no. If the ground field is infinite,then it is true. In general it is not TRUE. In the polynomial algebra ${\mathbb{Z}/2\mathbb{Z}}[X]$ consider the polynomials $X^2$ and $X$. But they are different in ${\mathbb{Z}/2\mathbb{Z}}[X]$.
| {
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More help with implicit differentiation Given $(5x+5y~)^3= 125x^3+125y^3$, find the derivative.
Using the chain rule and power rule, I came up with
$3(5x+5y)^2 \cdot (\frac{d}{dx}5x+\frac{dy}{dx}5y)= 3 \cdot 125x^2 +3 \cdot 125y^2$
Now, the derivative of $5x~$ is 5, but what about the derivative of $5y~$?
I know that... | This seems settled; I'll address how a problem like this would be handled in Mathematica (unfortunately the functionality doesn't work in Wolfram Alpha).
You'll first want to express your given in the form $f(x,y)=0$, like so:
expr = (5x + 5y)^3 - 125x^3 - 125y^3;
The key is to remember that Mathematica supports two s... | {
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Cardinality of sets of functions with well-ordered domain and codomain I would like to determine the cardinality of the sets specified bellow. Nevertheless, I don't know how to approach or how to start such a proof. Any help will be appreciated.
If $X$ and $Y$ are well-ordered sets, then determine the cardinality of:
... | *
*The cardinality of the set of functions from $X$ to $Y$ is the definition of the cardinal $Y^X$.
*The number of order-preserving functions from $X$ to $Y$, given that well-orders of each set have been fixed, depends on the nature of those orders. For example, there are no such orders in the case that the order ty... | {
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proof of inequality $e^x\le x+e^{x^2}$ Does anybody have a simple proof this inequality
$$e^x\le x+e^{x^2}.$$
Thanks.
| Consider $f(x) = x+e^{x^2}-e^x$. Then $f'(x) = 1+ 2xe^{x^2}-e^x$. Find the critical points. So the minimum of $f(x)$ is $0$ which implies that $e^x \leq x+e^{x^2}$.
| {
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Proving that $\gcd(ac,bc)=|c|\gcd(a,b)$ Let $a$, $b$ an element of $\mathbb{Z}$ with $a$ and $b$ not both zero and let $c$ be a nonzero integer. Prove that $$(ca,cb) = |c|(a,b)$$
| Let $d = (ca,cb)$ and $d' = |c|(a,b)$. Show that $d|d'$ and $d'|d$.
| {
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Guessing a subset of $\{1,...,N\}$ I pick a random subset $S$ of $\{1,\ldots,N\}$, and you have to guess what it is. After each guess $G$, I tell you the number of elements in $G \cap S$. How many guesses do you need?
| This can be solved in $\Theta(N/\log N)$ queries. First, here is a lemma:
Lemma: If you can solve $N$ in $Q$ queries, where one of the queries is the entire set $\{1,\dots,N\}$, then you can solve $2N+Q-1$ in $2Q$ queries, where one of the queries is the entire set.
Proof: Divide $\{1,\dots,2N+Q-1\}$ into three sets,... | {
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Homogeneous topological spaces Let $X$ be a topological space.
Call $x,y\in X$ swappable if there is a homeomorphism $\phi\colon X\to X$ with $\phi(x)=y$. This defines an equivalence relation on $X$.
One might call $X$ homogeneous if all pairs of points in $X$ are swappable.
Then, for instance, topological groups are h... | Googling
"topological space is homogeneous"
brings up several articles that use the same terminology, for example this one. It is also the terminology used in the question Why is the Hilbert cube homogeneous?. The Wikipedia article on Perfect space mentions that a homogeneous space is either perfect or discrete. T... | {
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Trig identity proof help I'm trying to prove that
$$ \frac{\cos(A)}{1-\tan(A)} + \frac{\sin(A)}{1-\cot(A)} = \sin(A) + \cos(A)$$
Can someone help me to get started? I've done other proofs but this one has me stumped! Just a start - I don't need the whole proof. Thanks.
| I would try multiplying the numerator and denominator both by (for the first term) $1+\tan{(A)}$ and for the second $1+\cot{(A)}$. From there it should just be a little bit of playing with Pythagorean identities ($\sin^2{(A)}+\cos^2{(A)}=1$, $\tan^2{(A)}+1=\sec^2{(A)}$, and $1+\cot^2{(A)}=\csc^2{(A)}$) and writing $\t... | {
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Is it standard to say $-i \log(-1)$ is $\pi$? I typed $\pi$ into Wolfram Alpha and in the short list of definitions there appeared
$$ \pi = -i \log(-1)$$
which really bothered me. Multiplying on both sides by $2i$:
$$ 2\pi i = 2 \log(-1) = \log(-1)^2 = \log 1= 0$$
which is clearly false. I guess my error is $\log 1 = ... | The (principal value) of the complex logarithm is defined as $\log z = \ln |z| + i Arg(z)$.
Therefore,
$$\log(-1) = \ln|-1| + i Arg(-1) = 0 + i \pi.$$
and then, one simply gets
$$
-i \log(-1) = -i (i \pi) = \pi.
$$
| {
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Squaring across an inequality with an unknown number This should be something relatively simple. I know there's a trick to this, I just can't remember it.
I have an equation
$$\frac{3x}{x-3}\geq 4.$$
I remember being shown at some point in my life that you could could multiply the entire equation by $(x-3)^2$
in order... | That method will work, but there's actually a simpler more general way. But first let's finish that method. $\:$ After multiplying through by $\rm\: (x-3)^2\: $ (squared to preserve the sense of the inequality) we obtain $\rm\ 3\:x\:(x-3) \ge\: 4\:(x-3)^2\:.\:$ Putting all terms on one side and factoring out $\rm\ x-3\... | {
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In a unital $R$-module $M$, if $\forall M_1\!\lneq\!M\;\;\exists M_2\!\lneq\!M$, such that $M_1\!\cap\!M_2\!=\!\{0\}$, then $M$ is semisimple PROBLEM: Let $R$ be a ring with $1$ and $M$ be a unital $R$-module (i.e. $1x=x$). Let there for each submodule $M_1\neq M$ exist a submodule $M_2\neq M$, such that $M_1\cap M_2=\... | Let $M$ be your module, and let $M_1$ be a submodule. Consider the set $\mathcal S$ of all submodules $N$ of $M$ such that $M_1\cap N=0$, and order $\mathcal S$ by inclusion. It is easy to see that $\mathcal S$ satisfies the hypothesis of Zorn's Lemma, so there exists an element $M_2\in\mathcal S$ which is maximal.
We... | {
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$T(1) = 1 , T(n) = 2T(n/2) + n^3$? Divide and conquer $T(1) = 1 , T(n) = 2T(n/2) + n^3$? Divide and conquer, need help, I dont know how to solve it?
| Use Akra-Bazzi which is more useful than the Master Theorem.
Using Akra-Bazzi, I believe you get $$T(x) = \theta(x^3)$$
You can also use the Case 3 of Master theorem in the wiki link above. (Note: That also gives $\theta(x^3)$.)
| {
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Why every polynomial over the algebraic numbers $F$ splits over $F$? I read that if $F$ is the field of algebraic numbers over $\mathbb{Q}$, then every polynomial in $F[x]$ splits over $F$. That's awesome! Nevertheless, I don't fully understand why it is true. Can you throw some ideas about why this is true?
| Consider some polynomial $$x^n = \sum_{i=0}^{n-1} c_i x^i,$$ where the $c_i$ are algebraic numbers. Thus for each $i$ we have a similar identity $$c_i^{n_i} = \sum_{j=0}^{n_i-1} d_{i,j} c_i^j,$$ where this time the $d_{i,j}$ are rationals.
Suppose that $\alpha$ is a root of the original polynomial. By using the above i... | {
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Factorial of 0 - a convenience? If I am correct in stating that a factorial of a number ( of entities ) is the number of ways in which those entities can be arranged, then my question is as simple as asking - how do you conceive the idea of arranging nothing ?
Its easy to conceive of a null element in the context of a... | There is exactly one way to arrange nothing: the null arrangement. You've misused your chair analogy: when you arrange null humans, you do it on null chairs, and there is exactly one way to do this.
Perhaps the following alternate definition will make things clearer. Suppose I have $n$ cards labeled $1, 2, ... n$ in o... | {
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How to prove that square-summable sequences form a Hilbert space? Let $\ell^2$ be the set of sequences $x = (x_n)_{n\in\mathbb{N}}$ ($x_n \in \mathbb{C}$) such that $\sum_{k\in\mathbb{N}} \left|x_k\right|^2 < \infty$, how can I prove that $\ell^2$ is a Hilbert space (with dot-product $\left(x,y\right) = \sum_{k\in\math... | This is more of an addendum for later, when you dig deeper into functional analysis:
Complex analysis tells us, that every holomorphic function can be represented by its Taylor series, locally. Actually, the space of square summable complex numbers is, as a Hilbert space, isomporph to all holomorphic functions on the ... | {
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How to compute the following formulas? $\sqrt{2+\sqrt{2+\sqrt{2+\dots}}}$
$\dots\sqrt{2+\sqrt{2+\sqrt{2}}}$
Why they are different?
| Suppose that the first converges to some value $x$. Because the whole expression is identical to the first inner radical, $\sqrt{2+\sqrt{2+\sqrt{2+\cdots}}}=x=\sqrt{2+x}$ and solving for $x$ gives $x=2$. Of course, I haven't justified that it converges to some value.
The second can be thought of as starting with $\sq... | {
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Intuitive Understanding of the constant "$e$" Potentially related-questions, shown before posting, didn't have anything like this, so I apologize in advance if this is a duplicate.
I know there are many ways of calculating (or should I say "ending up at") the constant e. How would you explain e concisely?
It's a rat... | Professor Ghrist of University of Pennsylvania would say that e^x is the sum of the infinite series with k going from zero to infinity of (x^k)/k!. If you are interested in Euler's number then you should not miss his Calculus of a Single Variable Course on Coursera
| {
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Why eliminate radicals in the denominator? [rationalizing the denominator] Why do all school algebra texts define simplest form for expressions with radicals to not allow a radical in the denominator. For the classic example, $1/\sqrt{3}$ needs to be "simplified" to $\sqrt{3}/3$.
Is there a mathematical or other reaso... | The form with neither denominators in radicals nor radicals in denominators and with only squarefree expressions under square-root signs, etc., is a canonical form, and two expressions are equal precisely if they're the same when put into canonical form.
When are two fractions equal? How do you know that $\dfrac{51}{68... | {
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What is the most general mathematical object that defines a solution to an ordinary differential equation? What is the most general object that defines a solution to an ordinary differential equation? (I don't know enough to know if this question is specific enough. I am hoping the answer will be something like "a fu... | (All links go to Wikipedia unless stated otherwise.)
Elaborating on joriki's answer: The most general spaces where it does make sense to talk about differential equations are certain classes of topological vector spaces, it is for example rather straight forward to formulate the concept of a differential equation in B... | {
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Formula to Move the object in Circular Path I want to move one object (dot) in circular path.
By using x and y position of that object.
Thanks.
| There are a few ways to choose from, but a nice one that doesn't require per-step trig functions (so can be calculated by a computer very quickly) is the midpoint circle algorithm.
Otherwise, you can use x=cos(theta)*radius, y=sin(theta)*radius for 0 < theta < 360.
| {
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Center of gravity of a self intersecting irregular polygon I am trying to calculate the center of gravity of a polygon.
My problem is that I need to be able to calculate the center of gravity for both regular and irregular polygons and even self intersecting polygons.
Is that possible?
I've also read that: http://paulb... | I think your best bet will be to convert the self-intersecting polygon into a set of non-self-intersecting polygons and apply the algorithm that you linked to to each of them. I don't think it's possible to solve your problem without finding the intersections, and if you have to find the intersections anyway, the addit... | {
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Graph - MST in O(v+e) G=(v,e) , with weight on the edges than can be only a or b (when $a
I need to find MST of the graph in O(v+e).
I think to put all the edges in array, and than scanning the array. first only check about a, and after about b. The algorithm is like Kruskal's: check about evey edge if it doesnt form ... | The running time of the algorithm of Kruskal is dominated by the sorting time of the edges according to their weights. In your case you can do this in linear time.
The rest of Kruskal's algorithm also runs in linear time. So you get linear running time after all.
| {
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Disjoint convex sets that are not strictly separated Question 2.23 out of Boyd & Vanderberghe's Convex Optimization:
Give an example of two closed convex sets that are disjoint but cannot be strictly separated.
The obvious idea is to take something like unbounded sets which are disjoint but approach each other in the... | Take $X = \{(x,y) \mid xy\geq 1, x,y>0\}$ and $Y = \{(x,y) \mid x\leq 0\}$.
| {
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Proving continuous image of compact sets are compact How to prove: Continuous function maps compact set to compact set using real analysis?
i.e. if $f: [a,b] \rightarrow \mathbb{R}$ is continuous, then $f([a,b])$ is closed and bounded.
I have proved the bounded part. So now I need some insight on how to prove $f([a,b]... | Lindsay, what you need is the intermediate value theorem, its proof is given in wikipedia.
| {
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Why is the generalized quaternion group $Q_n$ not a semi-direct product? Why is the generalized quaternion group $Q_n$ not a semidirect product?
| How many elements of order 2 does a generalized quaternion 2-group have? How many elements of order 2 must each factor in the semi-direct product have?
Note that dicyclic groups (generalized quaternion groups that are not 2-groups) can be semi-direct products. The dicyclic group of order 24 is a semi-direct product o... | {
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How can I re-arrange this equation? I haven't used my algebra skills much for years and they seem to have atrophied significantly!
I'm having real trouble working out how to re-arrange a formula I've come across to get $x$ by itself on the left hand side. It looks like this:
$\frac{x}{\sqrt{A^{2}-x^{2}}}=\frac{B+\sqrt{... | I would start by multiplying the numerator and denomenator on the right by $E-\sqrt{F+G\sqrt{A^2-x^2}}$:
$$\frac{x}{\sqrt{A^2 - x^2}} = \frac{\left(B + \sqrt{C + Dx}\right)\left(E - \sqrt{F + G\sqrt{A^2 - x^2}}\right)}{E^2-F+G\sqrt{A^2 - x^2}}$$
It may also help the manipulation to set $y = \sqrt{A^2 - x^2}$ for a whil... | {
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Question regarding Hensel's Lemma Hensel's Lemma
Suppose that f(x) is a polynomial with integer coefficients, $k$ is an integer with $k \geq 2$, and $p$ a prime.
Suppose further that $r$ is a solution of the congruence $f(x) \equiv 0 \pmod{p^{k-1}}$. Then,
If $f'(r) \not\equiv 0 \pmod{p}$, then there is a unique inte... | You go off track at the word "Hence". If $f'(3)\equiv 0\pmod 5$ and $f(3)\equiv 0 \pmod{25}$ (I assume you've done this correctly; I didn't check), that means that $x\equiv 3,8,13,18,23 \pmod{25}$ are all solutions modulo 25. You only verified that there are no solutions modulo 125 which are 3 modulo 25. There may stil... | {
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A sufficient condition for $U \subseteq \mathbb{R}^2$ such that $f(x,y) = f(x)$ I have another short question. Let $U \subseteq \mathbb{R}^2$ be open and $f: U \rightarrow \mathbb{R}$ be continuously differentiable. Also, $\partial_y f(x,y) = 0$ for all $(x,y) \in U$. I want to find a sufficient condition for $U$ such ... | It's enough for $U$ to have the property that, whenever $(x,y_1)$ and $(x,y_2)$ are in $U$, so is the line segment between them. This can be proved by applying the mean value theorem to the function $g(y)=f(x,y)$.
It is not enough for $U$ to be connected. For example, take $U=\mathbb{R} \setminus \{(x,0)|x \le 0 \}$. L... | {
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Is it Variation? Counting elements Lets assume that we have a element, which can have value from 1 till n. (let's set it on 20 to make it easier)
And we have the Set, that consists of object, which consists of three elements $\langle e_1, e_2, e_3 \rangle$.
We have also one rule regarding to objects in the set: $e_1 \g... | If the first number is $k$, and the second number is $j$, where $j \leq k$ then the last number has $j$ choices.
So the number of favorable cases is $$\sum_{k=1}^n \sum_{j=1}^k j = \sum_{k=1}^n \frac{k(k+1)}{2} = \frac{n(n+1)(n+2)}{6}$$
In general, if you have elements from $1$ to $n$ and want to choose an $m$ element... | {
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Non-associative, non-commutative binary operation with a identity Can you give me few examples of binary operation that it is not associative, not commutative but has an identity element?
| Here's an example of an abelian group without associativity, inspired by an answer to this question. Consider the game of rock-paper-scissors: $R$ is rock, $P$ is paper, $S$ is scissors, and $1$ is fold/draw/indeterminate. Let $\ast$ be the binary operation "play".
\begin{array}{r|cccc}
\ast & 1 & R & P & S\\
\hline
... | {
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How to check if derivative equation is correct? I can calculate the derivative of a function using the product rule, chain rule or quotient rule.
When I find the resulting derivative function however, I have no way to check if my answer is correct!
How can I check if the calculated derivative equation is correct? (ie I... | For any specific derivative, you can ask a computer to check your result, as several other answers suggest.
However, if you want to be self-sufficient in taking derivatives (for an exam or other work), I recommend lots of focused practice. Most calculus textbooks include answers to the odd-numbered problems in the bac... | {
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Embedding of finite groups It is well known that any finite group can be embedded in Symmetric group $S_n$, $GL(n,q)$ ($q=p^m$) for some $m,n,q\in \mathbb{N}$. Can we embed any finite group in $A_n$, or $SL(n,q)$ for some $n,q\in \mathbb{N}$?
| Yes.
The symmetric group $Sym(n)$ is generated by $\{(1,2), (2,3),\ldots, (n−1,n)\}$. You can embed $Sym(n)$ into $Alt(n+2)$ as the group generated by $\{(1,2)(n+1,n+2), (2,3)(n+1,n+2), …, (n−1,n)(n+1,n+2)\}$. This embedding takes a permution $\pi\in Sym(n)$ and sends it to $\pi⋅(n+1,n+2)^{\text{sgn}(\pi)}$, where $\... | {
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"answer_id": 2
} |
Equation of the complex locus: $|z-1|=2|z +1|$ This question requires finding the Cartesian equation for the locus:
$|z-1| = 2|z+1|$
that is, where the modulus of $z -1$ is twice the modulus of $z+1$
I've solved this problem algebraically (by letting $z=x+iy$) as follows:
$\sqrt{(x-1)^2 + y^2} = 2\sqrt{(x+1)^2 + y^2... | Just to add on to Aryabhata's comment above. The map $f(z) = \frac{1}{z}$ for $ z \in \mathbb{C} -\{0\}$, $f(0) = \infty$ and $f(\infty) = 0$ is a circle preserving homeomorphism of $\bar{\mathbb{C}}$. To see this, one needs to prove that it is continuous on $\bar{\mathbb{C}}$, and since $f(z)$ is an involution proving... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/27199",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
"answer_count": 3,
"answer_id": 2
} |
Prove a 3x3 system of linear equations with arithmetic progression coefficients has infinitely many solutions How can I prove that a 3x3 system of linear equations of the form:
$\begin{pmatrix}
a&a+b&a+2b\\
c&c+d&c+2d\\
e&e+f&e+2f
\end{pmatrix}
\begin{pmatrix}
x\\ y\\ z
\end{pmatrix}
=\begin{pmatrix}
a+3b\\
c... | First, consider the homogeneous system
$$\left(\begin{array}{ccc}
a & a+b & a+2b\\\
c & c+d & c+2d\\\
e & e+f & e+2f
\end{array}\right)\left(\begin{array}{c}x\\y\\z\end{array}\right) = \left(\begin{array}{c}0\\0\\0\end{array}\right).$$
If $(a,c,e)$ and $(b,d,f)$ are not scalar multiples of each other, then the coef... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/27259",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Understanding a proof by descent [Fibonacci's Lost Theorem] I am trying to understand the proof in Carmichaels book Diophantine Analysis but I have got stuck at one point in the proof where $w_1$ and $w_2$ are introduced.
The theorem it is proving is that the system of diophantine equations:
*
*$$x^2 + y^2 = z^2$$
... | $u^2$ and $v^2$ are $m$ and $n$, respectively, which are coprime. Then since $(u^2+v^2)+(u^2-v^2)=2u^2$ and $(u^2+v^2)-(u^2-v^2)=2v^2$, the only factor that $u^2+v^2$ and $u^2-v^2$ can have in common is a single factor of $2$. Since their product is the square $w^2$, that leaves the two possibilities given.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/27309",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 2,
"answer_id": 0
} |
Seeking a textbook proof of a formula for the number of set partitions whose parts induce a given integer partition Let $t \geq 1$ and $\pi$ be an integer partition of $t$. Then the number of set partitions $Q$ of $\{1,2,\ldots,t\}$ for which the multiset $\{|q|:q \in Q\}=\pi$ is given by \[\frac{t!}{\prod_{i \geq 1} ... | These are the coefficients in the expansion of power-sum symmetric functions in terms of augmented monomial symmetric functions. I believe you will find a proof in:
Peter Doubilet. On the foundations of combinatorial theory. VII. Symmetric functions through the theory of distribution and occupancy. Studies in Appl. Mat... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/27359",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 1,
"answer_id": 0
} |
How many countable graphs are there? Even though there are uncountably many subsets of $\mathbb{N}$ there are only countably many isomorphism classes of countably infinite - or countable, for short - models of the empty theory (with no axioms) over one unary relation.
How many isomorphism classes of
countable models... | I assume you mean by countable graph one that is countably infinite. I will also assume that your relation can be an arbitrary binary relation and not just symmetric since you seem to be interested in that case.
In this case there are uncountably many. For, a special case of a binary relation is a total order. We do no... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/27398",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 8,
"answer_id": 3
} |
Eigenvalues of the differentiation operator I have a linear operator $T_1$ which acts on the vector space of polynomials in this way:
$$T_1(p(x))=p'(x).$$
How can I find its eigenvalues and how can I know whether it is diagonalizable or not?
| Take the derivative of $a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$ (with $a_n\neq 0$), and set it equal to $\lambda a_nx^n+\cdots+\lambda a_0$. Look particularly at the equality of the coefficients of $x^n$ to determine what $\lambda$ must be. Once you know what the eigenvalues are, consider which possible diagonalized l... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/27446",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 3,
"answer_id": 0
} |
What are good books to learn graph theory? What are some of the best books on graph theory, particularly directed towards an upper division undergraduate student who has taken most the standard undergraduate courses? I'm learning graph theory as part of a combinatorics course, and would like to look deeper into it on m... | I learned graph theory from the inexpensive duo of Introduction to Graph Theory by Richard J. Trudeau and Pearls in Graph Theory: A Comprehensive Introduction by Nora Hartsfield and Gerhard Ringel. Both are excellent despite their age and cover all the basics. They aren't the most comprehensive of sources and they do ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/27480",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "130",
"answer_count": 16,
"answer_id": 10
} |
fair value of a hat-drawing game I've been going through a problem solving book, and I'm a little stumped on the following question:
At each round, draw a number 1-100 out of a hat (and replace the number after you draw). You can play as many rounds as you want, and the last number you draw is the number of dollars you... | Your expected return if you draw a number on the last round is 49.5 (because it costs a dollar to make the draw). On round N-1, you should keep what you have if it is greater than 49.5, or take your chances if it is less. The expected value if N=2 is then $\frac {51}{100}\frac {100+50}{2} -1 + \frac {49}{100}49.5=61.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/27524",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 3,
"answer_id": 0
} |
zeroes of holomorphic function I know that zeroes of holomorphic functions are isolated,and I know that if a holomorphic function has zero set whic has a limit point then it is identically zero function,i know a holomorphic function can have countable zero set, does there exixt a holomorphic function which is not ident... | A holomorphic function on a connected open set that is not identically zero cannot have uncountably many zeros. Open subsets of $\mathbb{C}$ are $\sigma$-compact, so if $G$ is the domain, then there is a sequence $K_1,K_2,\ldots$ of compact subsets of $G$ such that $G=K_1\cup K_2\cup\cdots$. (It is not hard to constr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/27546",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 4,
"answer_id": 2
} |
Krylov-like method for solving systems of polynomials? To iteratively solve large linear systems, many current state-of-the-art methods work by finding approximate solutions in successively larger (Krylov) subspaces. Are there similar iterative methods for solving systems of polynomial equations by finding approximate ... | Sort of, the root finding problem is equivalent to the eigenvalue problem associated with the companion matrix. Nonsymmetric eigenvalue methods such as "Krylov-Schur" can be used here.
Notes:
*
*The monic polynomials are extremely ill-conditioned and thus a better conditioned polynomial basis is mandatory for modera... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/27598",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
"answer_count": 1,
"answer_id": 0
} |
extend an alternative definition of limits to one-sided limits The following theorem is an alternative definition of limits. In this theorem, you don't need to know the value of $\lim \limits_{x \rightarrow c} {f(x)}$ in order to prove the limit exists.
Let $I \in R$ be an open interval, let $c \in I$, and let $f: I-{... | Assume that $\lim_{x \to c^+} f(x) = g$ exists. Then for any $\varepsilon > 0$ there exists $\delta > 0$ such that if $0 < x - c < \delta$ then $|f(x) - g| < \varepsilon$ and if $0 < y - c < \delta$ then $|f(y) - g| < \varepsilon$. Hence
$$
|f(x) - f(y)| \leq |f(x) - g| + |f(y) - g| < 2\varepsilon
$$
for $0 < x - c <... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/27662",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 2
} |
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