Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Solve the inequality $|x^2-2|<1$ I'm trying to solve inequalities that include absolute values. The way we solved these questions in class is different than what I am used to. The teacher would take the positive and negative possibilities of the inequality and treat them separately. I am trying to solve this as a union... | $$|x^2-2|<1\iff-1< x^2-2 <1$$
$$-1< x^2-2 <1\iff1<x^2<3$$
$$1<x^2<3\iff x^2-1>0\space \&\space x^2-3<0$$
$$x^2-1>0\space \&\space x^2-3<0 \iff x\in((-\infty,-1)\cup(1,\infty))\cap(-\sqrt 3,\sqrt 3)$$
$$ x\in((-\infty,-1)\cup(1,\infty))\cap(-\sqrt 3,\sqrt 3)\iff x\in(-\sqrt3,-1)\cup(1,\sqrt3)$$
| {
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"timestamp": "2023-03-29T00:00:00",
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Suppose the gcd (a,b) = 1 and c divides a + b. Prove that gcd (a,c) = 1 = gcd (b,c) I am lost.
So far...
If $\gcd (a,b) = 1$, by Bezout's Formula
$ax + by = 1$
If $c|(a+b)$, then
$cf = a+b$
Then,
$a (x-y) + cfy = 1$
$b(yx) + cfx = 1$
Am I on the right track? Any suggestions?
| If by contradiction, $(a,c)=(b,c)=m>1$ then $m|a,m|b$ and $m|c$, which means that there exist an $m>1$ that simultaneously divides both $a$ and $b$ which then implies that $gcd(a,b)\geq m$ which contradicts our assumption.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/513965",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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"answer_id": 5
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For which $a$ does the system of linear equations have a root Choose a possible $a$ such that the linear equations have a root
$$\begin{matrix} x+2y+3z=a \\
4x+5y+6z=a^2 \\
7x+8y+9z=a^3 \end{matrix}$$
Do I begin by finding the possible values of $a$ such that the system is consistent?
| Then first equation plus the third equation minus twice the second yields $a^3-2a^2+a=a(a-1)^2=0$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Geometric Brownian motion problem Here's the question:
Let $S(t)$, $t \geq 0$ be a Geometric Brownian motion process with drift parameter $\mu = 0.1$ and volatility parameter $\sigma = 0.2$. Find $P(S(3) < S(1) > S(0)).$
Is there something wrong with the following reasoning:
$P(S(3) < S(1) > S(0))=P(S(1)>S(3) \geq S(0)... | Tiny point: You are missing a "$)$".
Major point: $P(S(1)\gt S(3) \geq S(0))=P(S(1) \gt S(3))-P(S(3) \leq S(0))$ is wrong and similarly with the following equality. You should have something like $P(S(1)\gt S(3) \geq S(0)) =P(S(1) \gt S(3))-P(S(3) \leq S(0)) + P(S(0) \gt S(1) \geq S(3))$
Critical point: $S(0)$, $S(1... | {
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How does adding big O notations works can someone please explain how adding big O works.
i.e. $O(n^3)+O(n) = O(n^3)$
why does the answer turn out this way? is it because $O(n^3)$ dominates the whole expression thus the answer is still $ O(n^3)$
| The formal definition for equalities which contain $\;O(\cdot)\;$ and related notations, is that these notations are sets, and that such an equality holds if it holds for every function in each left hand side set, and for some function in each right hand side set.
(Anyone: Feel free to insert a reference; I don't have ... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Cardinality of Sets Proof I have a question as such:
Let $|A|$ denote the number of elements in A. For finite sets $A,B$,
prove that $|A\backslash B| = |A| - |A\cap B|$, where $\backslash$
stands for set difference.
Could someone show me how to prove it, please? I'm having difficulty working out a proof that feel... | Notice that $A\setminus B=A\cap B^c$ and we have $A=A\cap(B\cup B^c)=(A\cap B)\cup(A\cap B^c)$ and the sets in the last union are disjoint so
$$|A|=|A\cap B|+|A\cap B^c|$$
and we can conclude.
| {
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"timestamp": "2023-03-29T00:00:00",
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Question about compactly supported distribuions Let u be a distribution with compact support and let f be a Schwartz function: Is it true that the convolution of f with u is a Schwartz function?
| Yes. A distribution with compact support has finite order: it's just a finite collection of compactly supported Radon measures that get integrated against various derivatives of test functions. So, if you fix a Schwartz function $f$ and feed the translate $\tau_x f$ into the distribution, the value you get decays faste... | {
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Solutions of triangles $AA_1$, $BB_1$, $CC_1$ are the medians of triangle $ABC$ whose centroid is $G$. If points $A, C_1, G, B_1$ are concylic then prove that $2a^2= b^2 + c^2$.
Thanks
My try:-
$ar(GBC)=1/3ar(ABC)$
$\frac{1}{2}(GB.GC.\sin(\pi-A))=\frac{1}{3}(\frac{1}{2}bc\sin A)$
$GB.GC=\frac{1}{3}bc$
Now I can't thin... | It is really very simple. Using power of point property
$$BC_{1}.BA=BG.BB_{1}$$
$$\frac{c}{2}.c=\frac{2}{3}m_{b}.m_{b}$$
$$3c^2=4m^2_{b}$$
$$3c^2=2a^2+2c^2-b^2$$
$$2a^2=b^2+c^2$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Circular orientation of $n$ identical red balls and $n + 1$ identical black balls I encountered a question as follows:
In how many ways may $n$ identical red balls and $n + 1$
identical black balls be arranged in a circle (This number is called a Catalan number)?
While trying to analyze it
$1^{st}$ I considered linear... | The simple explanation: In a linear arrangement of $n$ red and $n+1$ black balls, there are ${2n+1 \choose n} = \frac{(2n+1)!}{n!(n+1)!}$ possibilities. Now consider put this line into a circle: $2n+1$ line patterns will produce the same circle since the line can start at any of the $2n+1$ points of the circle, so ther... | {
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Graph Entropy - What is it? I am having trouble getting some intuition as to what graph entropy measures. The definition that I have is that given a graph $G$, $H(G) = \min_{X,Y}I(X ;Y)$, where $X$ is a uniformly random vertex in $G$ and $Y$ is an independent set containing $X$. Also, $I(X; Y)$ is the mutual informatio... | $X$ is the source with maximal entropy (uniform distribution), and $Y$ is the set of distinguishable symbols (distinguishability is given by the edges). Graph entropy is trying to quantify the encoding capacity of such system for an arbitrary $Y$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Definition of the outer measure Let $X$ be a set. By definition, for every sequence of sets (disjoint or not), an outer measure $\theta:\mathcal{P}X\rightarrow [0,+\infty]$ is a monotic, countably subadditive (hence subadditive) function which vanishes at $0$.
We then have four possibilities:
*
*$A\cap B=\emptyset$... | It is not a case of non measurability: take the lebesgue outer measure and take $A = [0,1]$ and $B = [1,2]$.
$A \cup B = [0,2]$ and $A \cap B = \{1\} \neq \emptyset$ but clearly the outer measure of the union is the sum of the outer measure and both $A$ and $B$ will be measurable in the sense of Caratheodory (or in any... | {
"language": "en",
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Cantor’s diagonal proof revisited In his diagonal argument (although I believe he originally presented another proof to the same end) Cantor allows himself to manipulate the number he is checking for (as opposed to check for a fixed number such as $\pi$), and I wonder if that involves some meta-mathematical issues.
Le... | In the diagonal argument, a function $f$ from the set of sequences of real numbers to $\mathbb{R}$ is defined. We start from any sequence $S$ of real numebrs. Then it is shown that $f(S)$ is not an element in $S$. The formula is not "changing during the process"; the number we are searching for, $f(S)$, is well-defined... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Foundations of Forcing I am currently studying Forcing methods in order to understand some independence results and model's constructions.
Now I am interested on formalizing the main notions around forcing such as consistency, completeness, transitive models, well-founded relations, absoluteness, reflection principle, ... | Try A beginner's guide to forcing by Timothy Chow.
http://arxiv.org/abs/0712.1320
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
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"answer_id": 2
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Working out the difference in earnings I'm mathematically impaired/ignorant and trying to figure out the difference in earnings between my partner and I to work out a fair split of the bills.
So; I earn £2060 per month and partner earns £1650. As a percentage, how much more than her do I earn?
Therefore; If we had a mo... | You earn more than her : (2060-1650)/1650 * 100% = 24.85 % ,
You should pay for the mortgage : 850*2060/(2060+1650)=850*2060/3710 = 471.96 ,
Your gf should pay the rest
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/515158",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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What is a coset? Honestly. I just don't understand the word and keep running into highly technical explanations (wikipedia, I'm looking at you!). If somebody would be so awesome as to explain the concept assuming basic knowledge of group theory and high school algebra I would be delighted.
| If $H$ is a subgroup of $G$, then you can define a relation on $G$ by setting
$$
a\sim_H b\qquad\text{if and only if}\qquad ab^{-1}\in H
$$
It's a useful exercise in applying group laws proving that $\sim_H$ is an equivalence relation and that the equivalence class of $1\in G$ is
$$
[1]_{\sim_H}=H
$$
This equivalence r... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Find all primes of the form $n^n + 1$ less than $10^{19}$ Find all primes of the form $n^n + 1$ less than $10^{19}$
The first two primes are obvious: $n = 1, 2$ yields the primes $2, 5$. After that, it is clear that $n$ has to be even to yield an odd number.
So, $n = 2k \implies p = (2k)^{2k} + 1 \implies p-1 = (2k)^{k... | in $n^n+1$, it is algebraicly composite if n is not a power of 2. So you're left with which powers of 2. work.
In the case where n is not $1$or of the form $2^{2^m}$, one sees the power is a composite over two or more primes, and is thus algebraicly composite.
Thus, you just have to consider $1$ and those that come to... | {
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What's the probability of drawing every card at least from 82 cards, with replacement? What is the probability that if I draw 82 cards at random with replacemnt from a standard deck, every card is drawn at least once?
I've been banging my head against a wall for hours now, any help please.
I tried a smaller scale probl... | I found the answer: You use Stirling numbers of the Second Kind. So the answer ends up being 52!*S(82,52). Refer to this page: http://en.wikipedia.org/wiki/Stirling_numbers_of_the_second_kind
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Sum of closed and compact set in a TVS I am trying to prove: $A$ compact, $B$ closed $\Rightarrow A+B = \{a+b | a\in A, b\in B\}$ closed (exercise in Rudin's Functional Analysis), where $A$ and $B$ are subsets of a topological vector space $X$. In case $X=\mathbb{R}$ this is easy, using sequences. However, since I was ... | If $x\notin (A+B)$, then $A\cap(x-B)=\varnothing$. Since $(x-B)$ is closed, it follows from Theorem 1.10 in Rudin's book that there exists a neighborhood $V$ of $0$ such that $(A+V)\cap(x-B+V)=\varnothing$. Therefore $(A+B+V)\cap(x+V)=\varnothing$ and, in particular, $(A+B)\cap(x+V)=\varnothing.$ As $(x+V)$ is a neighb... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Problem in Algebra and Geometric sequence I need help on this one question which is in Algebra and on Geometric progression.
The question is as follows:
In a geometric sequence prove that:
$(b-c)^2 + (c-a)^2 + (d-b)^2 = (d-a)^2$.
Thanks,
Sudeep
| Let $$\frac dc=\frac cb=\frac ba=k\implies b=ak,c=bk=ak^2,d=ck=ak^3$$
$$(b-c)^2+(c-a)^2+(d-b)^2=a^2\{(k-k^2)^2+(k^2-1)^2+(k^3-k)^2\}=a^2(k^6-2k^3+1)=\{a(1-k^3)\}^2=(a-d)^2$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Find an angle of an isosceles triangle $\triangle ABC$ is an isosceles triangle such that $AB=AC$ and $\angle BAC$=$20^\circ$. And a point D is on $\overline{AC}$ so that AD=BC, , How to find $\angle{DBC}$?
I could not get how to use the condition $AD=BC$ , How do I use the condition to find $\angle{DBC}$?
EDIT 1: Wi... | I saw the following solution may years ago:
On side $AD$ construct in exterior equilateral triangle $ADE$. Connect $BE$.
Then $AB=AC, AE=BC, \angle BAE=\angle ABC$ gives $\Delta BAE =\Delta ABC$ and hence $AB=BE$.
But then
$$AB=BE, BD=BD, DA=DE \Rightarrow ADB =EDB$$
Hence $\angle ADB=\angle EDB$. Since the two angle... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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For the Compactness Theorem for Propositional Logic, show that the extension is not unique.
During the proof of the compactness theorem, from an arbitrary finitely satisfiable set $\Sigma$ of WFFs, we construct a finitely satisfiable set $\Delta\supseteq \sigma$ such that for every WFF $\alpha$, either $\alpha\in\De... | Hint. Suppose the language contains some unary predicate that is not mentioned in $\Sigma$ at all ...
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Cohomology of finite groups with finite coefficients I'm wondering if the group cohomology of a finite group $G$ can be made nontrivial with a nice choice of a finite $G$-module M. In other words, given a finite group $G$ and a number $n$, does there exist a finite $G$-module $M$ such that $H^n(G,M)$ is non-zero?
I wou... | Yes, for each $n\ge 0$ there is a $G$-module $M$ (depending on $n$) such that $H^n(G,M)\neq 0$ (provided $G\neq 1$ finite). .
Such an $M$ can be constructed by induction:
*
*First note that $H^i(G,F)=0$ for each free $\mathbb{Z}G$-module $F$ and all $i>0$ by Shapiro's lemma and Brown, VIII, 5.2.
*Next, show $H... | {
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"timestamp": "2023-03-29T00:00:00",
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Hard Integral $\frac{1}{(1+x^2+y^2+z^2)^2}$ Prove that $\displaystyle\int_{-\infty}^{\infty}\displaystyle\int_{-\infty}^{\infty}\displaystyle\int_{-\infty}^{\infty} \frac{1}{(1+x^2+y^2+z^2)^2}\, dx \, dy \, dz = \pi^2$
I tried substitution, trigonometric substitution, and partial fraction decomposition, but I can't sol... | An alternative is to overkill it with some measure theory. Unfortunately I don't know the names of the theorems and objects used (not in my language and not in english). If someone does, please edit my answer as you see fit.
Firstly note that $$\displaystyle\int \limits_{-\infty}^{\infty}\displaystyle\int \limits_{-\in... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/515948",
"timestamp": "2023-03-29T00:00:00",
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What is the most surprising result that you have personally discovered? This question is inspired by
my answer to this one:
Surprising identities / equations
In that question, people were asked about
the most surprising result
that they knew.
Almost all of them quoted someone
else's result.
I was one of the only ones
... | Well, "a long time ago" (1970s) it was not so clear that integrating restrictions of Eisenstein series on big groups against cuspforms on smaller reductive groups, or oppositely, etc., would do anything interesting... much less produce $L$-functions. The Rankin-Selberg example from 1939 was not necessarily clearly advo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/516001",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "38",
"answer_count": 21,
"answer_id": 10
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Why are they called orbits? When we study actions in group theory, we consider sets of the form
$$\text{Orb}_G(x)=\{gx\mid g\in G\} $$
that are called orbits. Although, the only reason I find convincing for that name is that in some sense the action of group over a set can be viewed as a dynamical system and thus the n... | You can think of the group action allowing you to move from one point to the next. In a "physical" sense, we are looking at where we can go in the set so we are looking at what elements we pass through on our way through.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/516057",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 1,
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centenes of $7^{999999}$
What is the value of the hundreds digit of the number $7^{999999}$?
Equivalent to finding the value of $a$ for the congruence $$7^{999999}\equiv a\pmod{1000}$$
| Use Euler's theorem: $7^{\phi (1000)} ≡ 1 \mod 1000 $.
By Euler's product formula: $\phi(1000) = 1000\cdot(1-\frac{1}{2})\cdot(1-\frac{1}{5})=400$
So $7^{400}≡1 \mod 1000 $.
$999999=400\cdot 2500-1$. So it suffices to find $7^{399}\mod 1000$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/516186",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Basic examples of monoids? What are some (simple/elementary) examples of noncommutative monoids with no additional structure? I'm having a hard time thinking of examples of "pure" monoids that aren't monoids simply because they are groups...
I've read this and this and some of this, but would like more examples that pr... | A generic answer is the monoid of all functions from a set $E$ into itself under the composition of functions. This example is generic since every monoid is isomorphic to a submonoid of such a monoid. In particular, take any set of functions from $E$ to $E$ and close under composition: you will get a monoid. See this l... | {
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How many n square can fit into a square of side N Suppose we have n small squares of equal sizes that has area w.
Suppose we have a fix square S of area A such that for area A, one area w < area A.
If square S's area A, length, and width are given,
and if the n small square length, width and area w is given,
and if n ... | I'm going to assume that all the squares are aligned (that is, each side of each square is vertical or horizontal), that the small squares are not to overlap (except possibly at their boundaries), and that the centers of the small squares are meant to form a square lattice (this takes care of the "equal spacing" requir... | {
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Is Dedekind completion of ${}^{\ast}\Bbb R$ a Archimedean field? Here's Theorem 1.2 on page 6, Martin Andreas Väth's Nonstandard Analysis(See here on googlebooks)
The Dedekind completion $\overline{X}$ of a totally ordered field $X$
is a complete Archimedean field with $\Bbb{Q}_{\overline{X}}$ as the canonical copy... | The Dedekind-completion is an order completion, and if the field is non-Arcihmedean then its Dedekind-completion is not a field at all.
To see this simply note that in the completion, there is a point $t$ which is the realization of the cut $R=\{x\mid\exists n\in\Bbb N. x<n\}$. And $t-1$ cannot exist.
| {
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Proving a subset is closed
Let $X$ be a metric space and let $A \subset X$ be an arbitrary subset. We define
$$A_r=\{x \in X : B_r(x) \subseteq A\}.$$ Prove that $A_r$ is closed for every radius $r$.
Maybe this is easy but I am totally stuck. First I've tried to prove it directly, i.e., take a convergent sequence... | The set $A_r$ is the complement of the set $\{y\in X\mid \exists u\notin A, y\in B_r(u) \}$. Can you show that this set is open?
| {
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"timestamp": "2023-03-29T00:00:00",
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The transform of a Hermitian matrix by a unitary matrix is Hermitian In the following document, p91 (or p4 of the PDF) , section 6.03, it is stated that "The transform of a hermitian matrix by a unitary matrix is hermitian." Apparently the proof is obvious, but not to me... could someone elaborate?
| A matrix is hermitian if $ A^\ast = A$ where $A^\ast$ is the conjugated and transposed of $A$.
Unitary matrices have the property that $U\cdot U^\ast= Id$ where Id is the identity. So in special we have $U^\ast =U^{-1}$.
Now we look at the transformed hermitian:
$$ U^{-1} A U=U^\ast A U$$
if conjugate and transpose... | {
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"timestamp": "2023-03-29T00:00:00",
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What square does not contain the middle? Consider the square $S = [-1,1]\times[-1,1]$. Suppose we put a smaller square inside it, which is rotated with an angle $\alpha$ relative to the large square. What is the largest such square that does not contain the origin in its interior?
When $\alpha=0$, the answer is obvious... | OK, this is my current attempt of an answer:
Consider a square that is pushed towards to top-right corner of $S$. The coordinates of this square are (where $c:=cos(\alpha)$ and $s:=sin(\alpha)$):
*
*T (top): $(1-dc,1)$
*R (right): $(1,1-ds)$
*B (bottom): $(1-ds,1-dc-ds)$
*L (left): $(1-dc-ds,1-dc)$
Due to symme... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/516799",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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An arithmetic sequence of numbers without certain prime factors I just attended a lecture at my school about prime numbers and the idea of being coprime and what kind of applications that has, and this question popped up in my head and I thought about it for a while and asked a few friends and wasn't quite sure what to... | In addition to what Patrick said, the only available terms that are the products of two primes $p, q$ (or $1$) where $p, q > p_n$. So for $4\#$ you would have $1, 11, 11 \times 11, 11 \times 13, 11 \times 17, 11 \times 19, 13, 13 \times 13, 17, 19, 23, 27,$ (along with the rest of primes less than $210$).
In its curren... | {
"language": "en",
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"source": "stackexchange",
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How to solve these series? Can anyone help me understand how to solve these two series? More than the solution I'm interested in understanding which process I should follow.
*
*Series 1:
$$
\sum_{i = 3}^{\infty} i * a^{i-1}, 0 < a < 1.
$$
*
*Series 2:
$$
\sum_{i = 3}^{\infty} i\sum_{k = 2}^{i-1} a^{i-k} * ... | Hint: For the first one if $\sum_{i = 3}^{\infty} x^{i}=f(x)$ then $\sum_{i = 3}^{\infty} i \times x^{i-1}=f'(x)$.
For the second one consider that:
$$
\sum_{k = 2}^{i-1} a^{i-k} b^{k-2}=a^{i-2}\frac{1-\frac{b^{i-2}}{a^{i-2}}}{1-\frac{b}{a}}= \frac{a^{i-2}-{b^{i-2}}}{1-\frac{b}{a}}
$$
Then you can decompose the seri... | {
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"source": "stackexchange",
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Vector equation for the tangent line of the intersection of $x^2 + y^2 = 25$ and $y^2 + z^2 = 20$ What is the vector equation for the tangent line of the intersection of $x^2 + y^2 = 25$ and $y^2 + z^2 = 20$ at the point $(3,4,2)$?
I think I should find a vector
$$
\gamma(t) = (x(t),y(t),z(t))
$$
that represents the in... | Hint: Maybe use $y=t,\ x=(25-t^2)^{1/2},\ z=(20-t^2)^{1/2}$ and restrict $t$ to make the radicals defined. I think the tangent line would then be possible except at the endpoints, for specific values of $t$ chosen.
ADDED: Actually $x$ and/or $z$ could be chosen as the negatives of the above radicals, and that would als... | {
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"source": "stackexchange",
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Predicates and Quantifiers? suppose that the domain of variable x is the set of people, and f(x) = "x is friendly" , t(x)= "x is tall" and a(x) = "x is angry". Write the statement using these predicates and any needed quantifiers.
1) some people are not angry
2) all tall people are friendly
3) No friendly people are an... | As others have said, you second and third answers are wrong -- but more worryingly, they are quite fundamentally wrong, not mere slips. So this suggests that you ought to be looking at some good text book that tells you about translation into predicate calculus notation. Lots of intro logic books do this (P-t-r Sm-th's... | {
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If a sequence of summable sequences converges to a sequence, then that sequence is summable. Let $(a_i)^n$ be a sequence of complex sequences each of which are summable (they converge). Then if they have a limit, the limit sequence $(b_i)$ is also summable. All under the sup norm for sequences.
Let $(a_i)^n$ sum to $... | (edited) This doesn't work, the example is pointwise convergent: Consider the alternating sequence $x=(-1,1,-1,1,\ldots)$ and let $x_n$ take the first $n$ terms of $x$ and be zero afterwards. $x_n$ converges pointwise to $x$, and is summable, but the (pointwise) limit $x$ is not summable.
Here is a uniform convergenc... | {
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In graph theory, what is the difference between a "trail" and a "path"? I'm reading Combinatorics and Graph Theory, 2nd Ed., and am beginning to think the terms used in the book might be outdated. Check out the following passage:
If the vertices in a walk are distinct, then the walk is called a path. If the edges in a... | You seem to have misunderstood something, probably the definitions in the book: they’re actually the same as the definitions that Wikipedia describes as the current ones.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "16",
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Big O estimate of simple while loop Give a big-O estimate for the number of operations, where an operation is an addition or a multiplication, used in this segment of an algorithm (ignoring comparisons used to test the conditions in the while loop).
i := 1
t := 0
while i ≤ n
t := t + i
i := 2i
My attempt:
n = 1 ... | $O(\log_2(n))$ and $O(\ln{n})$ are the same thing, since $\log_2$ and $\ln$ are related by the formula
$$\log_2{n} = \frac{\ln{n}}{\ln{2}} \approx 1.44 \ln{n}$$
The multiplicative constant is irrelevant for the Big O notation.
More precisely, we have the relations
$$1.44 \ln{n} \le \log_2{n} \le 1.45 \ln{n}$$
| {
"language": "en",
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"source": "stackexchange",
"question_score": "3",
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approximating a maximum function by a differentiable function Is it possible to approximate the $max\{x,y\}$ by a differentiable function?
$f(x,y)=max \{x,y\} ;\ x,y>0$
| Yes it is. One possibility is the following: Note that $\def\abs#1{\left|#1\right|}$
$$ \max\{x,y\} = \frac 12 \bigl( x+ y + \abs{x-y}\bigr), $$
take a differentiable approximation of $\abs\cdot$, for example $\def\abe{\mathop{\rm abs}\nolimits_\epsilon}$$\abe \colon \mathbb R \to \mathbb R$ for $\epsilon > 0$ given by... | {
"language": "en",
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Find an efficient algorithm to calculate $\sin(x) $ Suggest an efficient algorithm to determine the value of the
function $ \sin(x) $ for $ x \in [-4\pi, 4\pi] $.
You can use only Taylor series and $ +, -, *, /$.
I know, that $$\sin(x)=\sum_{n=0}^{\infty}(-1)^n\frac{x^{2n+1}}{(2n + 1)!}$$
but I can't find an efficient ... | Hint: The coefficients of the series are related to one another by the simple relation:
$$
t_{n+1}=\frac{x^{2(n+1)+1}}{(2(n+1)+1)!}=\frac{x^2}{(2n+2)(2n+3)}\frac{x^{2n+1}}{(2n+1)!}=
\frac{x^2}{(2n+2)(2n+3)}t_n
$$
(Here $t_n$ represents the coefficient of $x^{2n+1}$, multiplied by $x^{2n+1}$. So $\sin x=\sum_{n=1}^\in... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Are complete intersection prime ideals of regular rings regular ideals? Let $(R, \mathfrak{m})$ be a regular local ring and let $\mathfrak{p}$ be a prime ideal of $R$ which is a complete intersection, i.e. the minimal number of generators of $\mathfrak{p}$ equals its height $h$. Then by Macaulays theorem there is a sys... | Take $R=\mathbb{C}[x,y]_{(x,y)}$, and take $\mathfrak{p}=(x^2-y^3)$. $R/\mathfrak{p}$ is not a regular local ring, since it isn't integrally closed in its field of fractions.
| {
"language": "en",
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"source": "stackexchange",
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Congruences doubt!
What the rest of the division $2^{100}$ by $11$?
$$2^5=32\equiv10\equiv-1\pmod{11}\\(2^5)^{20}=2^{100}\equiv-1^{20}\;\text{or}\; (-1)^{20}$$??
| We have
\begin{align*}
2^{10} &= 2^5 \cdot 2^5 &\equiv (-1) \cdot (-1) &= (-1)^2 \pmod{11}\\
&\vdots\\
2^{100} &=2^5 \cdots 2^5 &\equiv (-1) \cdots (-1) &= (-1)^{20} \pmod{11}
\end{align*}
| {
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$\lim\inf a_n = 0$ as $n$ goes to infinity Let $(a_n)$ be a sequence of positive numbers such that for every $m$ in the natural numbers there is $n$ in the natural numbers such that $a_n = \frac{1}{m}$.
Prove $\lim\inf a_n = 0$ as $n$ goes to infinity.
I want to some suggestions on how to "approach" this problem becaus... | Hint: Prove that there is a subsequence converging to $0$.
| {
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What the rest of the division $1^6+2^6+...+100^6$ by $7$?
What the rest of the division $1^6+2^6+...+100^6$ by $7$?
$1^6\equiv1\pmod7\\2^6\equiv64\equiv1\pmod7\\3^6\equiv729\equiv1\pmod7$
Apparently all the leftovers are $one$, I thought of using Fermat's Little Theorem, however the $(7,7 k) = 7$, so you can not gener... | The number of multiples of $7$ from $1$ to $100$ is $\left\lfloor\frac{100}{7}\right\rfloor = 14$ so...
By Fermat's Little Theorem :
$7\mid 1^6+2^6+…+100^6 - 86 \implies 1^6+2^6+…+100^6 \equiv 86 \equiv 2 \pmod7 $
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Method of characteristics. Small question about initial conditions. Okay, so we're given a PDE
$$x \frac {\partial u} {\partial x} + (x+y) \frac{\partial u} {\partial y} = 1$$
with initial condition: $u(x=1,y)=y$
So $a=x, b=x+y, c=1$
$\Rightarrow$ characteristic equations: $$\frac{dx}{dt}=x, \frac{dy}{dt}=x+y, \frac... | $$
{{\rm d}y \over {\rm d}x} = 1 + {y \over x}
\tag{1}
$$
With the scaling $\tilde{x} = \mu x$ and $\tilde{y} = \nu\, y$, Eq. $\left(1\right)$
does not change its form whenever $\mu = \nu$ which is equivalent to
$\tilde{y}/\tilde{x} = y/x$. It means Ec. $\left(1\right)$ is simplified with the choice $y/x \equiv \phi\l... | {
"language": "en",
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For $f:\mathbb R^{<0}\to\mathbb R$, $f(x)=2x^2-3$, find the values of a for which $f(a)=f^{-1}(a)$ Okay, i've got the answer for this with some luck I guess, however i'm still left wondering specifically what this part of the question means:
"find the values of a for which $f(a)=f^{-1}(a)$"
My understanding of this is... | The values for which $f(x)=f^{-1}(x)$ must lie on the line $y=x$, since by the sheer definition of inverse function this is an axis of symmetry. This means that if a function and its inverse intersect, the points of intersection must lie on that line. Hence, you have to solve for which values of $x$ we have $f(x)=x$. S... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Locus of a point where two normals meet? Another exam question,
"Find the locus of a the point such that two of the normals drawn through it to the parabola $y^2=4ax$ are perpendicular to each other."
Does the locus mean the point of intersection of the two normals? I attempted to try to this by using the implicit der... | Find the locus of the point of intersection of two normals to a parabola which are at right angles to one another.
Solution:
The equation of the normal to the parabola y^2 = 4ax is
y = -tx + 2at + at^3. (t is parameter)
It passes through the point (h, k) if
k = -th + 2at + at^3 => at^3 + t(2a –... | {
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Quick method to find $|H \cap \overline{B} \cap \overline{K}|$ starting from $|H \cap B \cap K|$? Suppose sets $S, H, K, B$. Let
$$\begin{align}
&|S| = 100 \\
&|H| = 57 \\
&|K| = 77 \\
&|B| = 66 \\
&|H\cap B| = 30 \\
&|H \cap K| = 40 \\
&|B \cap K| = 50
\end{align}$$.
The question asks to find
*
*$|H \cap \overline... | I think that the quickest way is to draw a Venn diagram, calculate $|H\cap K\cap B|=20$, and fill in the cardinalities of the eight regions into which the diagram divides $S$. I very quickly get this:
I can now read off the answers; for instance, $|H\cap\overline{B}\cap\overline{K}|=7$. (In fact all three are $7$.)
| {
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Why is the modulus of a complex number $a^2+b^2$? Why is the modulus not $\sqrt{a^2-b^2}$? Carrying out standard multiplication this would be the result-why is this not the case? I know viewing the complex plane you can easily define the sum as being the distance to the points, but what meaning does $\sqrt{a^2-b^2}$ h... | I think you are not dealing with $i$ correctly in your multiplication. Note that $$(a + bi)(a-bi) = a^2 - (bi)^2 = a^2 - b^2i^2 = a^2 - b^2(-1) = a^2 + b^2.$$
| {
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Probability of 4 or fewer errors in 100,000 messages The probability of an error occurring in a message is 10^-5. The probability is independent for different messages. There are 100,000 messages sent. What is the probability that 4 or fewer errors occur?
| In principle, the number $X$ of errors in $100000$ messages has binomial distribution. But in this kind of situation (probability $p$ of an "error" small, number $n$ of trials large, $np$ of moderate size) it is standard to approximate the distribution of $X$ by using the Poisson distribution with parameter $\lambda=n... | {
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Geometric meaning of line equation in homogeneous coordinate In Euclidean space, a line's equation is $$ax + by + c = 0.$$ While in homogeneous coordinates,it can be represented with $$\begin{pmatrix}x &y &1\end{pmatrix}\begin{pmatrix}a\\ b\\ c\end{pmatrix} = 0.$$ I think the meaning of the homogeneous representation i... | *
*No. There is no notion of $\perp$ between a line and a point.
*No. Here the inner product is not a projection, but a measure ($\propto$) of the minimum distance between the objects. When coincident the minimum distance is zero.
| {
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Prove that the sequence $(a_n)$ defined by $a_0 = 1$, $a_{n+1} = 1 + \frac 1{a_n}$ is convergent in $\mathbb{R}$ I will post the exercise below:
Prove that the sequence $(a_n)$ defined by $a_0 = 1$, $a_{n+1} = 1 + \frac 1{a_n}$ for $n \in \mathbb N$ is convergent in $\mathbb R$ with the Euclidean metric, and determin... | Hint: If the limit $L$ exists, it must satisfy $L = 1 + \frac{1}{L}$, and so it cannot be 1. The solutions are the roots of the equation $L^2 - L - 1 = 0$, and so $L \in \{\frac{1+\sqrt{5}}{2}, \frac{1-\sqrt{5}}{2} \}$. That's where the golden ratio comes into play. Note also that the limit cannot be $\frac{1-\sqrt{5}}... | {
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"timestamp": "2023-03-29T00:00:00",
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Closed form for $\int \frac{1}{x^7 -1} dx$? I want to calculate:
$$\int \frac{1}{x^7 -1} dx$$
Since $\displaystyle \frac{1}{x^7 -1} = - \sum_{i=0}^\infty x^{7i} $, we have $\displaystyle(-x)\sum_{i=0}^\infty \frac{x^{7i}}{7i +1} $.
Is there another solution? That is, can this integral be written in terms of elementar... | Let u=x^2 and solve using u substitution.
=> 1/6 ln((x^6-1)/x^6)
| {
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"timestamp": "2023-03-29T00:00:00",
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Integral points on a circle Given radius $r$ which is an integer and center $(0,0)$, find the number of integral points on the circumference of the circle.
| You are looking for solutions to $m^2 + n^2 = r^2$ for a given $r$. Clearly $(\pm r, 0), (0, \pm r)$ are four solutions. For others, this is equivalent to finding Pythagorean triples with the same hypotenuse. You should be able to find a lot of references on this online.
In fact you can derive that, if the prime fact... | {
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Homework: Problem concerning first fundamental form Here's a strange problem in our differential geometry textbook.
At a point on surface $\mathbf{r}=\mathbf{r}(u,v)$, the equation $Pdudu+2Qdudv+Rdvdv=0$ determines two tangential directions. Prove that these two tangential directions are normal iff $$ER-2FQ+GP=0$$
w... | Consider $(1,a),\ (1,b)$ vectors on $uv$-plane. And ${\bf x}$ is a parametrication.
$$ d{\bf x}\ (1,a) \perp d{\bf x}\ (1,b) \Leftrightarrow ({\bf x}_u+a{\bf x}_v )\cdot ({\bf x}_u+b{\bf x}_v)=0\Leftrightarrow E+(a+b)F+abG =0 $$
And if $R=1,\ Q=-1/2(a+b),\ P=ab$, then note that $$ ( Pdudu +2 Qdudv+ Rdvdv )((1,a),(1,a)... | {
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Show that convolution of two measurable functions is well-defined Question:
Recall the definition of the convolution of $f$ and $g$ given by
$$(f*g)(x)=\int_{\mathbb{R}^d}f(x-y)g(y)dy.$$ If we only know that
$f$ and $g$ are measurable, can we show that $f*g$ is well defined for
a.e. $x$, that is, $f(x-y)g(y)$ ... | Hint: do that first when $f, g \ge 0$. Recall that if the integrand function has a sign, you can safely change the order of integration in a double integral. This is sometimes known as Tonelli's theorem.
Tonelli's theorem is easier than the closely related Fubini's theorem, which regards integrand functions which poss... | {
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algebra, equivalence relation regarding associates If f(x) ~ g(x) if and only if f and g are associates,
prove this is an equivalence relation
have tried to prove this both ways, struggling
| Well you need to show 3 things :
*
*Reflexivity : Take $u=1$
*Symmetry : If $u$ works in one direction, then $u^{-1}$ works in the other.
*Transitivity : If $f(x) = ug(x)$ and $g(x) = vh(x)$, then $f(x) = (uv)h(x)$, and $uv$ is a unit if $u$ and $v$ are.
| {
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Find the general solution of the given second order differential equation. Find the general solution of the given second order differential equation. $$4y''+y'=0$$
This was my procedure to solving this problem:
$\chi(r)=4r^2+r=0$
$r(4r+1)=0$
$r=0, -\frac14$
$y_1=e^{0x}, y_2=e^{-\frac14x}$
And this led to get the answer... | hint you can reduce the order by putting
$$y'=w$$
| {
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"url": "https://math.stackexchange.com/questions/519210",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
$E[X^4]$ for binomial random variable For a binomial random variable $X$ with parameters $n,p$, the expectations $E[X]$ and $E[X^2]$ are given be $np$ and $n(n-1)p^2+np$, respectively.
What about $E[X^4]$? Is there a table where I can look it up? Calculating it using the definition of expectation looks like a lot of wo... | Well, you can create a table if you know the moment generating function of $X$ i.e. $$M_X(t)=E[e^{tX}]$$
because $\frac{d^n}{dt^n}M_X(t)|_{t=0}=E[X^n].$
Hint: Show that $M_X(t)=(e^tp+(1-p))^n$ for binomial $X$ with parameters $n,p.$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/519337",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Find all couple $(x,y)$ for satisfy $\frac{x+iy}{x-iy}=(x-iy)$ I have a problem to solve this exercise, I hope someone help me.
Find all couple $(x,y)$ for satisfy $\frac{x+iy}{x-iy}=(x-iy)$
| $$\frac{z}{\bar{z}} = \bar{z}$$
Take a look at $|\cdot|$ of the two sides and you get $|z| = 1 (=x^2+y^2)$ for free.
Now expand with $z$ (since $z \neq 0$):
$$z^2 = \bar{z}$$
Now consider $\Re z^2 = x^2 - y^2$ and $\Re \bar z = x$ to get
$$x^2 - x =y^2$$
so
$$2x^2 - x = 1$$
which has solutions $x = \pm 1$ implying $y =... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/519423",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 6,
"answer_id": 4
} |
How to solve the equation $x^2=a\bmod p^2$ What is the standard approach to solve $x^2=a\bmod p^2$ or more general $x^n = a\bmod p^n$ ?
| The usual method for solving polynomial equations modulo $p^n$ is to solve it mod $p$, then use some method to extend a solution from mod $p$ to mod $p^2$, then to mod $p^3$, and so forth.
This can be done easily in an ad-hoc fashion: if you know that $f(a) = 0 \bmod p$, then you can make a new equation $f(a+px) = 0 \b... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/519535",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Explain why perpendicular lines have negative reciprocal slopes I am not sure how to explain this. I just know they have negative reciprocals because one one line will have a positive slope while the other negative.
| Assuming experience with algebra without calculus background. So, I would suggest keeping to the idea that slope, $m$, is equal to "rise over run." Given a line with slope, lets say $\frac{a}{b}$, that means it rises $a$ in the $y$ direction for every $b$ it goes in the positive $x$ direction in the plane. (I would als... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/519620",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "27",
"answer_count": 9,
"answer_id": 2
} |
the solution of Fredholm´s integral equation Be $\lambda \in \mathbb{R}$ such that $\left | \lambda \right |> \left \| \kappa \right \|_{\infty }(b-a)$.
Prove that the solution $f^*$ of the integral equation of Fredholm $$\lambda f -\int_{a}^{b}\kappa (x,y)f(y)dy=g(x)$$ for all $x\in [a,b]$ satisfies $$\left \| f^*... | Since $\lambda \ne 0$, we may write
\begin{align*}
f(x) - \frac{1}{\lambda}\int_{a}^{b} \kappa(x,y) f(y)\, \mathrm{d}y = \frac{g(x)}{\lambda}
\end{align*}
Let $\Im\colon C^{\infty}_{0} \to C^{\infty}_{0}$ be defined by
\begin{align*}
\Im[f](x):=\int_{a}^{b} \kappa(x,y) f(y) \, \mathrm{d}y
\end{align*}
Then by standard ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/519679",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Vertices of an equilateral triangle Question: show that the following three points in 3D space A = <-2,4,0>, B = <1,2,-1> C = <-1,1,2> form the vertices of an equilateral triangle.
How do i approach this problem?
| Find the distance between all the pairs of points
$$|AB|,|BC|,|CA|$$
and check if
$$|AB|=|BC|=|CA|$$
For example:
$$|A B| = \sqrt{(-2-1)^2 + (4-2)^2 + (0-(-1))^2} = \sqrt{14}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/519764",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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Finding eigenvalues. I'm working on the following problem:
Define $T \in L(F^n)$ (T an operator) by
$T(x_1,...,x_n) = (x_1+...+x_n,...,x_1+...+x_n)$
Find all eigenvalues and eigenvectors of $T$.
I've found that the eigenvalues of $T$ are $\lambda = 0$ and $\lambda = n$. Is there an easy way to prove that these are the ... | Try this: by direct computation,
$T^2 = nT, \tag{1}$
since every entry of $T^2$ is $n$.
So $m_T(x) = x^2 - nx$ is the minimal polynomial of $T$; every eigenvalue $\lambda$ of $T$ satisfies
$m_T(\lambda) = 0, \tag{2}$
so the only possibilities are $\lambda = 0$ and $\lambda = n$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/519862",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
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Use the Chinese remainder theorem to find the general solution of $x \equiv a \pmod {2^3}, \; x \equiv b \pmod {3^2}, \; x \equiv c \pmod {11}$ Help! Midterm exam is coming, but i still unable to solve this simple problem using the Chinese remainder theorem.
$$x \equiv a \pmod {2^3}, \quad x \equiv b \pmod {3^2}, \quad... | From the condition of the equation we have:
$$x \equiv a \pmod 8 \implies x = 8k + a$$
$$x \equiv b \pmod 9 \implies x = 9n + b$$
$$x \equiv c \pmod {11} \implies x = 11m + c$$
Now we have:
$$8k+a=9n+b$$
$$8k+a\equiv b \pmod 9$$
$$8k\equiv b-a \pmod 9$$
Having actual values would be easier to get congruence relation fo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/519930",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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Is ∑ with negative value solvable? Is it possible to have a negative value in sigma?
e.g.
$y = \Sigma_{k=0}^{k=-2} k \times 10$
Will this give the result $(0 \times 10) + (-1 \times 10) + (-2 \times 10) = -30 $?
Or will it be $\infty$ because $k$ will be increased with $1$ until it equals $-2$ (which is never).
Or s... | The concept of Sum has three basic definitions.
*
*Sum over (part of) a sequence
Given a unilateral sequence
$$
x_{\,0} ,\,x_{\,1} ,\, \cdots ,\;x_{\,n}
$$
we define a sum over a portion of it as
$$
\sum\limits_{k = \,a}^b {x_{\,k} }
$$
where it is understood that either $a$ and $b$ are integers and that $a \le b$... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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Induction proof that $n! > n^3$ for $n \ge 6$, and $\frac{(2n)!}{n! 2^n}$ is an integer for $n \ge 1$ Prove by induction that
(a) $n! > n^3$ for every $n \ge 6$.
(b) prove $\frac{(2n)!}{n!2^n}$ is an integer for every $n\geq 1$
I'm quite terrible with induction so any help would be appreciated.
| (a) $n! > n^3$ for every $n \geq 6$
For the induction base, we simple have to show that $6! \geq 6^3$. Hence, $6 * 5 * 4 * (3 * 2) * 1 \geq 6*6*6$. Hence $6 * 6 * (5 * 4) \geq 6^3$. This is evidently true, as $5*4 = 20 > 6$.
For the induction step, we have to show that $(n+1)! \geq (n+1)^3$ for $n\geq 6$. Now suppose $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/520235",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Little help with permutations There are 4 letters $A, B, C, D$ with repetitions permitted. These letters are used in a a 3 letter code (the order is important).
*
*Question 1:
How many different 3 letter codes can be made?
*Question 2:
If one code is chosen at random from the set of all possible codes, what is ... | Hint: How many codes satisfying the condition that they include two A and one D are there?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/520324",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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} |
Finding centre of ellipse using a tangent line? I need to determine the centre coordinates (a, b) of the ellipse given by the equation:
$$\dfrac{(x-a)^2}{9} + \dfrac{(y-b)^2}{16} = 1$$
A tangent with the equation $y = 1 - x$ passes by the point (0, 1) on the ellipse's circumference.
I'm guessing I have to find the im... | Hints: Follow, understand and prove the following
Since the point $\;(0,1)\;$ is on the ellipse then
$$\frac{a^2}9+\frac{(1-b)^2}{16}=1$$
Now differentiate implicitly:
$$\frac29(x-a)dx+\frac18(y-b)dy=0\implies \frac{dy}{dx}=-\frac{\frac29(x-a)}{\frac18(y-b)}=-\frac{16}9\frac{x-a}{y-b}$$
But we know that
$$-1=\left.\fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/520424",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
} |
Prove that $\sup[0,1] =1$ Alongside the question in the title, does it matter if the question is $(0,1)$ or $[0,1)$?
I know that it satisfies the first condition, $1$ is an upper bound but I am not sure where to go from there.
Thanks.
| Hint: To show that the supremum of $[0,1]$ is $1$, you need to show two things: 1) that it is an upper bound and 2) that there is no lower upper bound.
You say you've already proved (1); so, it comes down to (2). Can you show that if $x<1$, then $x$ is not an upper bound on $[0,1]$?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/520516",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Let $x_i$ be positive number satisfying $\sum x_i = 1$, what is $\sum ix_i$ Let $x_i$ be a positive number for each $i \in \{1, 2, 3 \dots \}$ such that $\sum_{i=1}^\infty x_i = 1$ is there a closed formula for
$$\sum_{i = 1}^\infty ix_i$$?
| There isn't a formula independent of the $x_i$. The series does not always converge (let $x_i=\frac{6}{\pi^{2}i^{2}}$), but can converge (let $x_{i}=\frac{1}{2^{i}}$). I would guess the only limitation on the value of the series is that it is greater than 1.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/520611",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Speed of object towards a point not in the object's trajectory? Trying to study for my mid-term, but I'm having slight difficulties understanding what I'm supposed to do in this one problem:
A batter starts running towards first base at a constant speed of 6 m/s. The distance between each adjacent plate is 27.5 m. Afte... | The first question is easy: the batter is running straight towards the first base, so he is approaching the first base with a speed of $6$ m/s. To answer the second question, try to find the function $f(t)$ of time that gives the distance to the third base. Then find the derivative at the point $t=\frac{20}{6}$, when t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/520676",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 2
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What am I doing wrong when trying to find a determinant of this 4x4 I have to find the determinant of this 4x4 matrix:
$
\begin{bmatrix}
5 & -7 & 2 & 2 \\
0 & 3 & 0 & -4 \\
-5 & -8 & 0 & 3 \\
0 & -5 & 0 & -6 \\
\end{bmatrix}
$
Here is my working which seems wrong according to the solutions. What am i doing wrong?:... | What you are doing wrong is precisely what the solution said you were doing wrong. The $2$ was alright, since that's the same as $2\cdot(-1)^{1+3},$ but the $-5$ was not, since $$-5\cdot(-1)^{2+1}=-5\cdot-1=5.$$ Keep in mind that we have an alternating sign factor as we move along a row/column, and that the starting si... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
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Equivalence class help I have a question that goes as follows:
Let d be a positive integer. Define the relation Rho on the integers Z as follows: for all m,n element of the integers.
m rho n if and only if d|(m-n)
Prove that rho is an equivalence relation. Then list its equivalence classes.
Now the first d that comes t... | There is a different relation for each $d$. What is being asked is "for all $d$, is the corresponding relation an equivalence relation?"
As for your proof, it is correct, but you may want to be clearer with some of the steps, depending on how familiar the intended audience is with divisibility.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/520806",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Rate of change of distance from particle (on a curve) to origin A particle is moving along the curve
$y = 2\sqrt{4x + 9}$
As the particle passes through the point
(4, 10)
its x-coordinate increases at a rate of
3 units per second. Find the rate of change of the distance from the particle to the origin at this insta... | The distance to the origin when the particle is at $(x,y)$ is given by $D(x,y)=\sqrt{x^2+y^2}$.
We want $\frac{dD}{dt}$ at a certain instant. I prefer to work with $D^2$. So we have
$$D^2=x^2+y^2.$$
Differentiate, using the Chain Rule. We have
$$2D\frac{dD}{dt}=2x\frac{dx}{dt}+2y\frac{dy}{dt}.\tag{1}.$$
We know that $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/520861",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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A couple has 2 children. What is the probability that both are girls if the eldest is a girl? This is another question like this one. And by the same reason, the book only has the final answer, I'd like to check if my reasoning is right.
A couple has 2 children. What is the probability that both are girls if the eldest... | An alternative viewpoint:
For the eldest child to be a girl, they must have had a girl first. Therefore the probability of there being two girls is the probability of having a second girl which is $\frac{1}{2}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/520968",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 0
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Independent set of formulas of the sentential logic A set $\Gamma$ of well formed formulas of the sentential logic is called independent if for each $\varphi\in\Gamma$, $\Gamma-\{\varphi\}\nRightarrow\varphi\\$.
1 when $\Gamma=\{\varphi\}$ is independent?
2 Is $\{A\rightarrow B, B\rightarrow C, C\rightarrow A\}$ indepe... | You haven't said whether $\implies$ is semantic or syntactic entailment, but the same goes either way. I'll assume you mean semantic entailment, but you can easily adjust the answer if you meant syntactic entailment.
*
*$\{\varphi\}$ is independent [on your definition] iff $\emptyset \nvDash \varphi$, i.e. iff $\va... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/521076",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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$3x\equiv7\pmod{11}, 5y\equiv9\pmod{11}$. Find the number which $x+y\pmod{11}$ is congruent to. Given that $3x\equiv7\pmod{11}, 5y\equiv9\pmod{11}$. Find the number which $x+y\pmod{11}$ is congruent to. I'm thinking $20\equiv9\pmod{11}$, But I am having trouble find a number $3x$ that is divisible by $3$? Is there a b... | 3x $\equiv$ 7 (mod 11) and 5y $\equiv$ 9 (mod 11)
3x $\equiv$ 18 (mod 11) by adding 11 to 7, then 5y $\equiv$ 20 (mod 11) by add 11 to 9.
x $\equiv$ 6 (mod 11) by dividing 3 to both sides, then y $\equiv$ 4 (mod 11) by dividing 5 to both sides.
Then, x + y $\equiv$ 10 (mod 11)
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/521144",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 1
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Symmetric Tridiagonal Matrix has distinct eigenvalues. Show that the rank of $ n\times n$ symmetric tridiagonal matrix is at least $n-1$, and prove that it has $n$ distinct eigenvalues.
| This is for tridiagonal matrices with nonzero off-diagonal elements.
Let $\lambda$ be an eigenvalue of $A\in\mathbb{R}^{n\times n}$ (which is symmetric tridiagonal with nonzero elements $a_{2,1},a_{3,2},\ldots,a_{n,n-1}$ on the subdiagonal). The submatrix constructed by deleting the first row and the last column of $A-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/521188",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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If $ac-bd=p$ and $ad+bc=0$, then $a^2+b^2\neq 1$ and $c^2+d^2\neq 1$? I'm trying to prove the following:
Let $a,b,c,d\in\Bbb{Z}$ and $p$ be a prime integer. If $ac-bd=p$ and $ad+bc=0$, prove that $a^2+b^2\neq 1$ and $c^2+d^2\neq 1$.
Actually I'm not even sure if this is correct. A proof or counter-example (in case th... | Consider $p^2 = p^2 + 0^2 = (ac-bd)^2 + (ad+bc)^2 = (a^2+b^2)(c^2+d^2)$. Then either one of $a^2+b^2$, $c^2+d^2$ equals $p^2$ and the other equals $1$ or both equal $p$. This follows from unique factorization of $\mathbb{Z}$.
Considering the first case, suppose wlog $a^2+b^2=1$ and $a=0$. Then $c^2+d^2=p^2$. Now if bot... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/521286",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Solve $4^{9x-4} = 3^{9x-4}$ I am having some trouble trying to solve
$$4^{9x-4} = 3^{9x-4}$$
I tried to make each the same base but then I'm becoming confused as to what to do next.
These are the steps I took:
$$\begin{align}
4^{9x-4} &= 3^{9x-4} \\
\log_4(4^{9x-4}) &= \log_4(3^{9x-4}) \\
\end{align}$$
Where do I go f... | So you thought @fasttouch was complex?
Adapted from Mathematica:
$$x = \frac{4\log\frac{4}{3}- 2 \pi ni}{9 \log \frac{4}{3}}, n \in \Bbb Z$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/521374",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 11,
"answer_id": 6
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$L_2$ is of first category in $L_1$ (Rudin Excercise 2.4b) We mean here $L_2$, and $L_1$ the usual Lebesgue spaces on the unit-interval. It is excercise 2.4 from Rudin. There's several ways to show that $L_2$ is nowhere dense in $L_1$.
But in (b) they ask to show that
$$\Lambda_n(f)=\int fg_n \to 0 $$
where $g_n = n$ ... | The simplest functions in $L_1 \setminus L_2$ are $f_\alpha \colon x \mapsto x^\alpha$ with $-1 < \alpha \leqslant -\frac12$.
Computing $\int fg_n$ for such an $f_\alpha$ yields
$$\begin{align}
\int f_\alpha g_n &= n\int_0^{n^{-3}} x^\alpha\,dx \\
&= \frac{n}{1+\alpha}n^{-3(1+\alpha)}\\
&= \frac{n^{-2-3\alpha}}{1+\alph... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/521447",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
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Iteration of an operator
Let $f_0(x)$ be integrable on $[0,1]$, and $f_0(x)>0$. We define $f_n$ iteratively by
$$f_n(x)=\sqrt{\int_0^x f_{n-1}(t)dt}$$
The question is, what is $\lim_{n\to\infty} f_n(x)$?
The fix point for operator $\sqrt{\int_0^x\cdot dt}$ is $f(x)=\frac{x}{2}$. But it's a bit hard to prove this... | Note: this is not a proof that the limit exists, but a computation of the limit if we know that it exists.
We know that $f(x)>0$ for $x>0$ and $f(0)=0$. We want to solve
$$
f(x)=\sqrt{\int_0^xf(t)\,dt},\quad 0\le x\le 1,
$$
that is,
$$
(f(x))^2=\int_0^xf(t)\,dt,\quad 0\le x\le 1.
$$
Derivate with respect to $x$ to obta... | {
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"url": "https://math.stackexchange.com/questions/521522",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Show that $\lim_{\delta \to 0}(1-\lambda \delta)^{1/\delta} = e^{-\lambda}$ My professor said that
$$\lim_{\delta \to 0}(1-\lambda \delta)^{t/\delta}=e^{-\lambda t}$$
can be shown with L'Hospital's rule. I don't know what he meant. What is the best way to show this (or, more simply, $\lim_{\delta \to 0}(1-\lambda \del... | Another approach: define
$$x:=\frac1\delta\implies \delta\to 0\implies x\to\infty$$
and our limit is
$$\left[\left(1-\frac\lambda x\right)^x\right]^t\xrightarrow[x\to\infty]{}(e^{-\lambda})^t=e^{-\lambda t}$$
We used above the basic
$$\lim_{x\to\infty}\left(1\pm\frac\lambda{f(x)}\right)^{f(x)}=e^{\pm\lambda}$$
for any... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/521605",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
} |
How to calculate conditional expectation only from the characteristic function I would like to calculate conditional expectation $E[X|A]$, where $A$ is a set, only from the characteristic function $\phi(\omega)$ of a random variable $X$. How can I do this?
Since the characteristic function describes the density functio... | The conditional expectation $E[X | A]$ will change depending on whether or not $A$ is independent from $X$. If independent, $E[X | A] = E[X]$, else $E[X | A]$ can have different values on $A$ and $A^c$. For example, if $X = 1_A$, then $E[1_A | A] = 1_A$, but if $B$ is a set independent from $A$, with $P(A) = P(B)$, the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/521689",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Show that $\sin(x^2)$ is integrable around $\infty$. I have to show that $f(x)=\sin(x^2)$ is integrable on $[1, \infty[$. This is French terminology, so "intégrable" specifically means that the integral of $|f|$ exists.
The only method I know is to compare it to functions of the form $\frac{1}{x^\alpha}$, but it's not ... | If you mean that
$$
\lim_{N\to\infty}\int_0^N\sin(x^2)\,\mathrm{d}x
$$
exists, then change variables $x\mapsto\sqrt{x}$ and integrate by parts:
$$
\begin{align}
&\lim_{N\to\infty}\int_0^N\sin(x^2)\,\mathrm{d}x\\
&=\int_0^1\sin(x^2)\,\mathrm{d}x
+\lim_{N\to\infty}\frac12\int_1^{N^2}\frac{\sin(x)}{x^{1/2}}\,\mathrm{d}x\\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/521747",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Is it possible to get a 'closed form' for $\sum_{k=0}^{n} a_k b_{n-k}$? This came up when trying to divide series, or rather, express $\frac1{f(x)}$ as a series, knowing that $f(x)$ has a zero of order one at $x=0$, and knowing the Taylor series for $f(x)$ (that is knowing the $b_i$ 's).
I write $$1=\frac1{f(x)}f(x) =... | If you know $f$ has a zero of order $1$ you can write it as $$f(x)=x\left(a_1+a_2x+a_3x^2+\ldots\right),$$
with $a_1\neq0$.
Then $\frac{1}{f}=\frac{1}{x}\frac{1}{a_1+a_2x+a_3x^2+\ldots}$.
To compute the series of $\frac{1}{a_1+a_2x+a_3x^2+\ldots}$ just apply long division of $1$ divided by $a_1+a_2x+a_3x^2+...$. This i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/521853",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
What did Newton and Leibniz actually discover? Most popular sources credit Newton and Leibniz with the creation and the discovery of calculus. However there are many things that are normally regarded as a part of calculus (such as the notion of a limit with its $\epsilon$-$\delta$ definition) that seem to have been dev... | This is actually quite a complicated question, since it spans two whole careers.
Some say calculus was not discovered by Newton and Leibniz because Archimedes and others did it first. That's a somewhat simple-minded view. Archimedes solved a whole slew of problems that would now be done by integral calculus, and his ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/521929",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "26",
"answer_count": 6,
"answer_id": 3
} |
Solving system of equations with R Suppose I have the following function :
$f(x,y,z,a)= \cos(ax) + 12y^2 - 9az$
and I want to solve the following syste of equations.
$ f(x,y,z,1)= 10 $,
$f(x,y,z,5)= 7 $,
and
$f(x,y,z,-3)= 17 $.
These are equivalent to
$\cos(x) + 12 y^2 - 9 z(1) = 10$,
$ \cos(5x) + 12y^2 - 9 z(5) ... | This is a partial answer; I can't give actual code, since I'm not familiar with R.
The standard way to solve this type of problem is to reformulate it as a nonlinear root-finding (or nonlinear optimization) problem and then use existing software or packages.
*
*First, write your equations as follows: \begin{align}
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/521996",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Complete and elementary proof that $(a^x - 1)/x $ converges as x goes to 0 Anybody who has taken a calculus course knows that
$$\lim_{x \to 0} \frac{a^x - 1}{x}$$
exists for any positive real number $a$, simply because the limit is by definition the derivative of the function $a^x$ at $x = 0$. However, for this argume... | The most common definition of $e$ is $$e:=\lim_{x\to0}\left(1+x\right)^{1/x}$$ although you often see it with $n=1/x$ and as $n\to\infty$.
Now $$\begin{aligned}\lim_{x\to0}\left(\frac{e^x-1}{x}\right)&=\lim_{x\to0}\left(\frac{\left(\lim_{y\to0}\left(1+y\right)^{1/y}\right)^x-1}{x}\right)\\
&=\lim_{x\to0}\left(\frac{\li... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/522077",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 5,
"answer_id": 1
} |
How prove this analysis function $a\le\frac{1}{2}$ let
$$f(x)=\begin{cases}
x\sin{\dfrac{1}{x}}&x\neq 0\\
0&x=0
\end{cases}$$
show that:there exsit $M>0,(x^2+y^2\neq 0)$ ,
$$F(x,y)=\dfrac{f(x)-f(y)}{|x-y|^{a}}|\le M \Longleftrightarrow a\le\dfrac{1}{2}$$
My try:
(1)if $a\le\dfrac{1}{2}$, then
$$\dfrac{f(x)-f(... | I think that this is just a partial answer or this is just an estimation for $a_0$ where $a\leq a_0$.
$ a_n=\frac{1}{2n\pi + \frac{\pi}{2}},\ b_n = \frac{1}{2n\pi - \frac{\pi}{2}}$
$f(a_n)- f(b_n) = a_n +b_n = \frac{4n\pi}{4n^2\pi^2 - (\pi/2)^2} \approx \frac{1}{n\pi}$ and $|a_n - b_n|^{a_0} \approx |\frac{\pi}{4n^2\p... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/522163",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
In the field $\mathbb{Z}_7[x]/\langle x^4+x+1\rangle$, find the inverse of $f(x)=x^3+x+3$. In the field $\mathbb{Z}_7[x]/\langle x^4+x+1\rangle$, find the inverse of $f(x)=x^3+x+3$.
I know how to find the inverses of elements within sets, rings, and fields. I know what to do if the field was just $\mathbb{Z}_7$, but th... | The computation here is the same in ${\mathbb Q}$ as in
${\mathbb Z}_7$.
You look for a solution of the form
$$
z=a+bx+cx^2+dx^3 \tag{1}
$$
You then have
$$
z(x^3+x+1)=dx^6+cx^5+(b+d)x^4+(a+c+3d)x^3+(b+3c)x^2+(a+3b)x+3a=Q(x) \tag{2}
$$
Next, divide the result by $x^4+x+1$ :
$$
Q(x)=(x^4+x+1)(dx^2+cx+b+d)+R(x) \tag{3}
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/522236",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Differentiation of function with chain rule the following expression is part of a function I have to differentiate:
$y = \tan^3(5x^4-7)$
I tried using the chain-rule, so:
$ y' = 3\tan^2(5x^4-7)\cdot(20x^3)$
is this correct?
| Set $t=5x^4-7,~~~u=\tan(t)$ so, you have $$y=u^3$$. Now use the following routine formulas:
$$\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dt}\cdot\frac{dt}{dx},~~~(\tan (t))'=1+\tan^2(t)$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/522414",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
What is the answer of this problem? Suppose that $f(x)$ is bounded on interval $[0,1]$, and for $0 < x < 1/a$, we have $f(ax)=bf(x)$. (Note that $a, b>1$). Please calculate $$\lim_{x\to 0^+} f(x) .$$
| We get $f\left(a^{n}x\right)=b^{n}f\left(x\right)$ for $0<x<a^{-n}$.
If the limit does not equal $0$ then there is a series with $x_{n}\in(0,a^{-n})$
with $f\left(x_{n}\right)\geq\varepsilon>0$
for each $n$. Then $f\left(a^{n}x_{n}\right)=b^{n}f\left(x_{n}\right)\geq b^{n}\varepsilon$.
This contradicts the boundedness... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/522525",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Where is $\operatorname{Log}(z^2-1)$ Analytic? $\newcommand{\Log}{\operatorname{Log}}$
The question stands as
Where is the function $\Log(z^2-1)$ analytic
, where $\Log$ stands for the principal complex logarithm. My understanding is that
The domain of analyticity of any function $f(z) = \Log\left[g(z)\right]$, wher... | If $2xy=0$ then either (a) $x=0$, in which case the other inequality becomes $-y^2-1\leq 0$ which is satisfied by all $y\in\mathbb{R}$, or (b) $y=0$, where the other inequality becomes $x^2 - 1 \leq 0$ which is satisfied by all $|x| \leq 1$.
These inequalities must both be satisfied together. You are describing the un... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/522579",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 1,
"answer_id": 0
} |
Geometric interpretation of the addition of linear equations in general form I have a very simple question: suppose I have two 2D linear equations in general form
$$ a_1x + b_1y + c_1 = 0$$
$$ a_2x + b_2y + c_2 = 0$$
I'd like to know what's the (intuitive) geometric interpretation of their addition and subtraction
$$ (... | In general, if you have a system of two linear equations whose solution is a line $L$ in $3$-space, you can visualize the general linear combination of the equations as giving another plane containing $L$. Think of this as different positions of a revolving door, pivoting around $L$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/522633",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 1
} |
Problem finding in simple algebra It is given,
$$x= \sqrt{3}+\sqrt{2}$$
How to find out the value of $$x^4-\frac{1}{x^4}$$/
The answer is given $40 \sqrt{6}$ but my answer was not in a square-root form
I have done in thsi way:
$$x+ \frac{1}{x}= 2 \sqrt{3}$$
Then,
$$(x^2)^2-\left(\frac{1}{x^2}\right)^2= \left(x^2 + \fra... | The idea you're having to change it to terms of $x^2$ isn't bad, but it seems a little overfancy. (Maybe I overlooked some economy about it, but I haven't seen the benefit yet.)
Why not just calculate it directly? (Hints follow:)
$x^2=3+2+2\sqrt{6}=5+2\sqrt{6}$
$x^4=(5+2\sqrt{6})^2=25+24+20\sqrt{6}=49+20\sqrt{6}$
$\dfr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/522712",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Perpendicular line intersection issues Do not downvote questions for being 'simple' to you. What one might find trivial another may find helpful. It is not in the spirit of SE. That being said,...
I have a line with the equation $y = -2.08x - 44$, and I must find the perpendicular equation, which will be $y \approx ... | Up to this point, you are correct: $\require{cancel}$
$$-2.5608\;x = 44$$
Dividing both sides of the equation by $-2.5608$ to solve for $x$ yields (or multiplying both sides by $\frac{1}{-2.5608}$) $$\dfrac{\cancel{-2.5608}\;x}{\cancel{-2.5608}} = \dfrac{44}{-2.5608} \iff x \approx \dfrac {44}{-2.5608} \approx -17.1821... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/522947",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
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