Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
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How would I evaluate this limit? I have no idea how to evaluate this limit. Wolfram gives $0$, and I believe this, but I would like to see how it is done. The limit is
$$\lim_{n\rightarrow\infty}\frac{x^n}{(1+x)^{n-1}}$$
assuming $x$ is positive. Thanks in advance.
| $\frac{x^n}{(1+x)^{n-1}} = \frac{x^n(1+x)}{(1+x)^{n}} = (\frac{x}{1+x})^{n}(1+x) = (\frac{x+1-1}{1+x})^{n}(1+x) = (1 - \frac{1}{1+x})^{n}(1+x)$
So taking the limit:
$\lim_{n\to\infty} \frac{x^n}{(1+x)^{n-1}} = \lim_{n\to\infty} (1 - \frac{1}{1+x})^{n}(1+x) = (1+x) * \lim_{n\to\infty} (1 - \frac{1}{1+x})^{n}$
Since $x>0... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/80779",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Solutions to Linear Diophantine equation $15x+21y=261$ Question
How many positive solutions are there to $15x+21y=261$?
What I got so far
$\gcd(15,21) = 3$ and $3|261$
So we can divide through by the gcd and get:
$5x+7y=87$
And I'm not really sure where to go from this point. In particular, I need to know how to tel... | 5, 7, and 87 are small enough numbers that you could just try all the possibilities. Can you see, for example, that $y$ can't be any bigger than 12?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/80822",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 7,
"answer_id": 1
} |
Help remembering a Putnam Problem I recall that there was a Putnam problem which went something like this:
Find all real functions satisfying
$$f(s^2+f(t)) = t+f(s)^2$$
for all $s,t \in \mathbb{R}$.
There was a cool trick to solving it that I wanted to remember. But I don't know which test it was from and google isn't ... | No idea. But I have a book called Putnam and Beyond by Gelca and Andreescu, and on page 185 they present a problem from a book called Functional Equations: A Problem Solving Approach by B. J. Venkatachala, from Prism Books PVT Ltd., 2002. I think the Ltd. means the publisher is British.
Almost, the publisher is (or ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/80932",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 3,
"answer_id": 2
} |
the name of a game I saw a two-player game described the other day and I was just wondering if it had an official name. The game is played as follows: You start with an $m \times n$ grid, and on each node of the grid there is a rock. On your turn, you point to a rock. The rock and all other rocks "northeast" of it a... | This game is called Chomp.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/80991",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Finding the Laurent expansion of $\frac{1}{\sin^3(z)}$ on $0<|z|<\pi$? How do you find the Laurent expansion of $\frac{1}{\sin^3(z)}$ on $0<|z|<\pi$? I would really appreciate someone carefully explaining this, as I'm very confused by this general concept! Thanks
| use this formula
$$\sum _{k=1}^{\infty } (-1)^{3 k} \left(-\frac{x^3}{\pi ^3 k^3 (\pi k-x)^3}-\frac{x^3}{\pi ^3 k^3 (\pi k+x)^3}+\frac{3 x^2}{\pi ^2 k^2 (\pi k-x)^3}-\frac{3 x^2}{\pi ^2 k^2 (\pi k+x)^3}-\frac{x^3}{2 \pi k (\pi k-x)^3}-\frac{x^3}{2 \pi k (\pi k+x)^3}+\frac{x^2}{(\pi k-x)^3}-\frac{x^2}{(\pi k+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/81105",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
} |
How can I prove $[0,1]\cap\operatorname{int}{(A^{c})} = \emptyset$?
If $A \subset [0,1]$ is the union of open intervals $(a_{i}, b_{i})$ such that each
rational number in $(0, 1)$ is contained in some $(a_{i}, b_{i})$,
show that boundary $\partial A= [0,1] - A$. (Spivak- calculus on
manifolds)
If I prove that ... | Since you say you want to prove it by contradiction, here we go:
Suppose that $x\in \mathrm{int}(A^c)\cap [0,1]$. Then there is an open interval $(r,s)$ such that $x\in (r,s)\subseteq A^c$. Every open interval contains infinitely many rational numbers, so there are lots and lots of rationals in $(r,s)$. However, since ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/81166",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Arithmetic error in Feller's Introduction to Probability? In my copy of An Introduction to Probability by William Feller (3rd ed, v.1), section I.2(b) begins as follows:
(b) Random placement of r balls in n cells. The more general case of [counting the number of ways to put] $r$ balls in $n$ cells can be studied in th... | Assume sampling with replacement, there are four possible balls for cell 1, four possible balls for cell 2, and four possible balls for cell 3. So there are 4^3=64 possibilities.
(Assuming sampling without replacement, there are four possible balls for cell 1, three for cell 2, and two for cell 3, for a total of 4*3*2=... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/81280",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 4,
"answer_id": 2
} |
Why are Gram points for the Riemann zeta important? Given the Riemann-Siegel function, why are the Gram points important? I say if we have $S(T)$, the oscillating part of the zeros, then given a Gram point and the imaginary part of the zeros (under the Riemann Hypothesis), are the Gram points near the imaginary part of... | One thing Gram points are good for is that they help in bracketing/locating the nontrivial zeroes of the Riemann $\zeta$ function.
More precisely, recall the Riemann-Siegel decomposition
$$\zeta\left(\frac12+it\right)=Z(t)\exp(-i\;\vartheta(t))$$
where $Z(t)$ and $\vartheta(t)$ are Riemann-Siegel functions.
$Z(t)$ is a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/81346",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Series around $s=1$ for $F(s)=\int_{1}^{\infty}\text{Li}(x)\,x^{-s-1}\,dx$
Consider the function $$F(s)=\int_{1}^{\infty}\frac{\text{Li}(x)}{x^{s+1}}dx$$ where $\text{Li}(x)=\int_2^x \frac{1}{\log t}dt$ is the logarithmic integral. What is the series expansion around $s=1$?
It has a logarithmic singularity at $s=1$,... | Note that the integral $F(s)$ diverges at infinity for $s\leqslant1$ and redefine $F(s)$ for every $s\gt1$ as
$$
F(s)=\int_2^{+\infty}\frac{\text{Li}(x)}{x^{s+1}}\mathrm dx.
$$
An integration by parts yields
$$
sF(s)=\int_2^{+\infty}\frac{\mathrm dx}{x^s\log x},
$$
and the change of variable $x^{s-1}=\mathrm e^t$ yield... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/81371",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 0
} |
Are sin and cos the only continuous and infinitely differentiable periodic functions we have? Sin and cos are everywhere continuous and infinitely differentiable. Those are nice properties to have. They come from the unit circle.
It seems there's no other periodic function that is also smooth and continuous. The onl... | "Are there any other well-known periodic functions?"
In one sense, the answer is "no". Every reasonable periodic complex-valued function $f$ of a real variable can be represented as an infinite linear combination of sines and cosines with periods equal the period $\tau$ of $f$, or equal to $\tau/2$ or to $\tau/3$, etc... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/81411",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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"answer_id": 2
} |
Factorization of zeta functions and $L$-functions I'm rewriting the whole question in a general form, since that's probably easier to answer and it's also easier to spot the actual question. Assume that we have some finite extension $K/F$ of number fields and assume that the extension is not Galois. Denote the Galois c... | Yes. The first thing to notice is that the Zeta function is the Artin L-function associated to the trivial representation of $Gal(E/K)$ i.e.
$$\zeta_K(s) = L(s, \mathbb{C}, E/K),$$
where $\mathbb{C}$ is endowed with the trivial action of $\mathrm{Gal}(E/K).$
Let $G = \mathrm{Gal}(E/F)$ and $H = \mathrm{Gal}(E/K).$ Now ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/81480",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Limits of integration for random variable Suppose you have two random variables $X$ and $Y$. If $X \sim N(0,1)$, $Y \sim N(0,1)$ and you want to find k s.t. $\mathbb P(X+Y >k)=0.01$, how would you do this? I am having a hard time finding the limits of integration. How would you generalize $\mathbb P(X+Y+Z+\cdots > k) ... | Hint: Are the random variables independent?
If so, you can avoid integration by using the facts
*
*the sum of independent normally distributed random variables has a normal distribution
*the mean of the sum of random variables is equal to the sum of the means
*the variance of the sum of independent random var... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/81531",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Divisibility by 4 I was asked to find divisibility tests for 2,3, and 4.
I could do this for 2 and 3, but for 4.
I could come only as far as:
let $a_na_{n-1}\cdots a_1a_0$ be the $n$ digit number.
Now from the hundredth digit onwards, the number is divisible by 4 when we express it as sum of digits.
So, the only par... | Note that $10a_1+a_0\equiv2a_1+a_0$ (mod 4). So for divisibility by 4, $a_0$ must be even and in this case $2a_1+a_0=2(a_1+\frac{a_0}{2})$. So, $a_1$ and $\frac{a0}{2}$ must be of same parity (means both are either even, or odd).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/81598",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Using recurrences to solve $3a^2=2b^2+1$ Is it possible to solve the equation $3a^2=2b^2+1$ for positive, integral $a$ and $b$ using recurrences?I am sure it is, as Arthur Engel in his Problem Solving Strategies has stated that as a method, but I don't think I understand what he means.Can anyone please tell me how I sh... | Yes. See, for example, the pair of sequences https://oeis.org/A054320 and https://oeis.org/A072256, where the solutions are listed. The recurrence is defined by $$a_0 = a_1 = 1; \qquad a_n = 10a_{n-1} - a_{n-2},\ n\ge 2.$$
As to how to go about solving this, there are many good references on how to do this, including W... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/81674",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
How many $p$-adic numbers are there? Let $\mathbb Q_p$ be $p$-adic numbers field. I know that the cardinal of $\mathbb Z_p$ (interger $p$-adic numbers) is continuum, and every $p$-adic number $x$ can be in form $x=p^nx^\prime$, where $x^\prime\in\mathbb Z_p$, $n\in\mathbb Z$.
So the cardinal of $\mathbb Q_p$ is contin... | The field $\mathbb Q_p$ is the fraction field of $\mathbb Z_p$.
Since you already know that $|\mathbb Z_p|=2^{\aleph_0}$, let us show that this is also the cardinality of $\mathbb Q_p$:
Note that every element of $\mathbb Q_p$ is an equivalence class of pairs in $\mathbb Z_p$, much like the rationals are with respect t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/81774",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 3,
"answer_id": 1
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Pumping lemma usage I need to know if my solution for a problem related with regular languages and pumping lemma is correct.
So, let $L = \{a^ib^jc^k \mid i, j,k \ge 0 \mbox{ and if } i = 1 \mbox{ then } j=k \}$
Now i need to use the pumping lemma to prove that this language is not regular. I wrote my proof like this:
... | You cannot choose $x$, $y$ and $z$. That is, the following statement does not help you prove that the language is not regular:
Since $q=p−1$, it implies that $x=a$, $y=b^q$ and $z=c^q$. It satisfies the propery $|xy| \le p$ and $|y|>0$.
The pumping lemma states that for every regular language $L$, there exists a str... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/81847",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Another quadratic Diophantine equation: How do I proceed? How would I find all the fundamental solutions of the Pell-like equation
$x^2-10y^2=9$
I've swapped out the original problem from this question for a couple reasons. I already know the solution to this problem, which comes from http://mathworld.wolfram.com/Pell... | You can type it into Dario Alpern's solver and tick the "step-by-step" button to see a detailed solution.
EDIT: I'm a little puzzled by Wolfram's three fundamental solutions, $(7,2)$, $(13,4)$, and $(57,18)$. It seems to me that there are two fundamental solutions, $(3,0)$ and $(7,2)$, and you can get everything else ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/81917",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
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"answer_id": 0
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Probability when I have one or 3 choices I'm wondering what is "better" to have in terms of profit:
Lets say we have 300 people come to your store and you have 3 products. Each individual is given (randomly) one product. If the individual likes the product he will buy it, but if he doesn't like the product, he will tur... |
is it better (more probable) to have 3 products or just one, in order to maximize the total number of sold products. Important to note is that customer doesn't decide which product he gets (he gets it randomly), he only decides if he likes it or not (50-50 chance he likes it).
Under the conditions you have described,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/81972",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 2
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Prove that $(1 - \frac{1}{n})^{-n}$ converges to $e$ This is a homework question and I am not really sure where to go with it. I have a lot of trouble with sequences and series, can I get a tip or push in the right direction?
| You have:
$$
x_n:=\left(1-\frac1n\right)^{-n} = \left(\frac{n-1}n\right)^{-n} = \left(\frac{n}{n-1}\right)^{n}
$$
$$
= \left(1+\frac{1}{n-1}\right)^{n} = \left(1+\frac{1}{n-1}\right)^{n-1}\cdot \left(1+\frac{1}{n-1}\right) = a_n\cdot b_n.
$$
Since $a_n\to \mathrm e$ and $b_n\to 1$ you obtain what you need.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/82034",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "18",
"answer_count": 1,
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Is there an easy way to determine when this fractional expression is an integer? For $x,y\in \mathbb{Z}^+,$ when is the following expression an integer?
$$z=\frac{(1-x)-(1+x)y}{(1+x)+(1-x)y}$$
The associated Diophantine equation is symmetric in $x, y, z$, but I couldn't do anything more with that. I tried several fact... | Since $$ \frac{(1-x)-(1+x)y}{(1+x)+(1-x)y} = \frac{ xy+x+y-1}{xy-x-y-1} = 1 + \frac{2(x+y) }{xy-x-y-1} $$
and $ 2x+2y < xy - x -y - 1 $ if $ 3(x+y) < xy - 1 .$ Suppose $ x\leq y$, then $ 3(x+y) \leq 6y \leq xy-1 $ if $ x\geq 7. $ So all solutions must have $0\leq x< 7 $ so it is reduced to solving $7$ simpler Diophant... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/82087",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Find the equation of the plane passing through a point and a vector orthogonal I have come across this question that I need a tip for.
Find the equation (general form) of the plane passing through the point $P(3,1,6)$ that is orthogonal to the vector $v=(1,7,-2)$.
I would be able to do this if it said "parallel to th... | vector equation of a plane is in the form : r.n=a.n,
in this case, a=(3,1,6), n=(1,7,-2).
Therefore,
r.(1,7,-2)=(3,1,6).(1,7,-2)
r.(1,7,-2)=(3x1)+(1x7)+(6x-2)
r.(1,7,-2)=3+7-12
r.(1,7,-2)=-2
OR
r=(x,y,z)
therefore,
(x,y,z).(1,7,-2)=-2
x+7y-2z=-2
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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"answer_id": 3
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A circle with infinite radius is a line I am curious about the following diagram:
The image implies a circle of infinite radius is a line. Intuitively, I understand this, but I was wondering whether this problem could be stated and proven formally? Under what definition of 'circle' and 'line' does this hold?
Thanks!
| There is no such thing as a circle of infinite radius. One might find it useful to use the phrase "circle of infinite radius" as shorthand for some limiting case of a family of circles of increasing radius, and (as the other answers show) that limit might give you a straight line.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/82220",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "27",
"answer_count": 8,
"answer_id": 1
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Product rule in calculus This is wonderful question I came across whiles doing calculus. We all know that $$\frac{d(AB)}{dt} = B\frac{dA}{dt} + A\frac{dB}{dt}.$$
Now if $A=B$ give an example for which
$$\frac{dA^2}
{dt} \neq 2A\frac{dA}{at}.$$
I have tried many examples and could't get an example, any help?
| let's observe function
$y=(f(x))^2$ , this function can be decomposed as the composite of two functions:
$y=f(u)=u^2$ and $u=f(x)$
So :
$\frac { d y}{ d u}=(u^2)'_u=2u=2f(x)$
$\frac{du}{dx}=f'(x)$
By the chain rule we know that :
$\frac{dy}{dx}=\frac{dy}{du}\cdot \frac{du}{dx}=2f(x)f'(x)$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/82266",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
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Solving the recurrence relation $A_n=n!+\sum_{i=1}^n{n\choose i}A_{n-i}$ I am attempting to solve the recurrence relation $A_n=n!+\sum_{i=1}^n{n\choose i}A_{n-i}$ with the initial condition $A_0=1$. By "solving" I mean finding an efficient way of computing $A_n$ for general $n$ in complexity better than $O(n^2)$.
I tri... | This isn’t an answer, but it may lead to useful references.
The form of the recurrnce suggests dividing through by $n!$ and substituting $B_n=\dfrac{A_n}{n!}$, after which the recurrence becomes $$B_n=1 + \sum_{i=1}^n\binom{n}i\frac{(n-i)!}{n!}B_{n-i}=1+\sum_{i=1}^n\frac{B_{n-i}}{i!}.$$
You didn’t specify an initial co... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/82315",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
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How to prove $\sum_{ d \mid n} \mu(d)f(d)=\prod_{i=1}^r (1-f(p_i))$? I have to prove for $n \in \mathbb{N}>1$ with $n=\prod \limits_{i=1}^r p_i^{e_i}$. $f$ is a multiplicative function with $f(1)=1$:
$$\sum_{ d \mid n} \mu(d)f(d)=\prod_{i=1}^r (1-f(p_i))$$
How I have to start? Are there different cases or can I prove i... | Please see Theorem 2.18 on page $37$ in Tom Apostol's Introduction to analytic number theory book.
The proof goes as follows:
Define $$ g(n) = \sum\limits_{d \mid n} \mu(d) \cdot f(d)$$
*
*Then $g$ is multiplicative, so to determine $g(n)$ it suffices to compute $g(p^a)$. But note that $$g(p^a) = \sum\limits_{d \mid ... | {
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"url": "https://math.stackexchange.com/questions/82379",
"timestamp": "2023-03-29T00:00:00",
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"question_score": "3",
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Sketch the graph of $y = \frac{4x^2 + 1}{x^2 - 1}$ I need help sketching the graph of $y = \frac{4x^2 + 1}{x^2 - 1}$.
I see that the domain is all real numbers except $1$ and $-1$ as $x^2 - 1 = (x + 1)(x - 1)$. I can also determine that between $-1$ and $1$, the graph lies below the x-axis.
What is the next step? I... | You can simplify right away with
$$
y = \frac{4x^2 + 1}{x^2 - 1} = 4+ \frac{5}{x^2 - 1} =4+ \frac{5}{(x - 1)(x+1)}
$$
Now when $x\to\infty$ or $x\to -\infty$, adding or subtracting 1 doesn't really matter hence that term goes to zero. When $x$ is quite large, say 1000, the second term is very small but positive henc... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/82443",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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A counterexample to theorem about orthogonal projection Can someone give me an example of noncomplete inner product space $H$, its closed linear subspace of $H_0$ and element $x\in H$ such that there is no orthogonal projection of $x$ on $H_0$. In other words I need to construct a counterexample to theorem about orthog... | Let $H$ be the inner product space consisting of $\ell^2$-sequences with finite support, let $\lambda = 2^{-1/2}$ and put
$$
z = \sum_{n=1}^\infty \;\lambda^n \,e_n \in \ell^2 \smallsetminus H
$$
Then $\langle z, z \rangle = \sum_{n=1}^\infty \lambda^{2n} = \sum_{n=1}^{\infty} 2^{-n} = 1$.
The subspace $H_0 = \{y \in... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/82499",
"timestamp": "2023-03-29T00:00:00",
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"question_score": "12",
"answer_count": 2,
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First Order Logic (deduction proof in Hilbert system)
Possible Duplicate:
First order logic proof question
I need to prove this:
⊢ (∀x.ϕ) →(∃x.ϕ)
Using the following axioms:
The only thing I did was use deduction theorem:
(∀x.ϕ) ⊢(∃x.ϕ)
And then changed (∃x.ϕ) into (~∀x.~ϕ), so:
(∀x.ϕ) ⊢ (~∀x.~ϕ)
How can I continue... | If the asterisks in your axioms mean that the axioms are to be fully universally quantified, so that they become sentences, and if your language has no constant symbols, then it will not be possible to make the desired deduction in your system. The reason is that since all the axioms are fully universally quantified, t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/82567",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Improper integral; exponential divided by polynomial How can I evaluate $$\int_{-\infty}^\infty {\exp(ixk)\over -x^2+2ixa+a^2+b^2} dx,$$ where $k\in \mathbb R, a>0$? Would Fourier transforms simplify anything? I know very little about complex analysis, so I am guessing there is a rather simple way to evaluate this? Tha... | Assume $b \neq 0$ and $k\neq 0$.
Write
$\dfrac{\exp(ixk)}{-x^2+2iax+a^2+b^2}= \dfrac{\exp(ixk)}{-(x-ia)^2+b^2}=\dfrac{\exp(i(x-ia)k)}{-(x-ia)^2+b^2} \exp(-ak)$ hence
the integral becomes $I=\int_{-\infty}^\infty \dfrac{\exp(i(x-ia)k)}{-(x-ia)^2+b^2} \exp(-ak)dx=\int_{-\infty-ia}^{\infty -ia} \dfrac{\exp(izk)}{-z^2+b... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/82642",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Proof of Convergence: Babylonian Method $x_{n+1}=\frac{1}{2}(x_n + \frac{a}{x_n})$ a) Let $a>0$ and the sequence $x_n$ fulfills $x_1>0$ and $x_{n+1}=\frac{1}{2}(x_n + \frac{a}{x_n})$ for $n \in \mathbb N$. Show that $x_n \rightarrow \sqrt a$ when $n\rightarrow \infty$.
I have done it in two ways, but I guess I'm not al... | For a):
The proof of convergence can be deduced from the question/answer LFT theory found in
Iterative Convergence Formulation for Linear Fractional Transformation with Rational Coefficients
Proof when $x_1^2 > a$
Note: If both $a$ and $x_1$ are rational numbers, then this solution is obtained without recourse to the ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/82682",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "22",
"answer_count": 7,
"answer_id": 5
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Differential Equation Breaks Euler Method Solving ${dy\over dx} = 2y^2$, $y(0)=2$ analytically yields $y(8)= -2/31$, but from using Euler's method and looking at the slope field, we see that $y(8)$ should be a really large positive answer. Why?
Differential equation:
$$\begin{align}
&\frac{dy}{dx}=2y^2\\
&\frac{dy}{... | As you found, the solution is $y={2\over 1-4x}$, which has a vertical asymptote at $x=1/4$. In the slope field, you should be able to convince yourself that such a function can indeed "fall along the slope vectors". The curve will shoot up to infinity as you approach $x=1/4$ from the left. To the right of $x=1/4$ the ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/82746",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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Locally compact nonarchimedian fields Is it true that if $F$ is a locally compact topological field with a proper nonarchimedean absolute value $A$, then $F$ is totally disconnected? I am aware of the classifications of local fields, but I can't think of a way to prove this directly.
| Yes: the non-Archimedean absolute value yields a non-Archimedean metric (also known as an ultrametric), and every ultrametric space is totally disconnected. In fact, every ultrametric space is even zero-dimensional, as it has a base of clopen sets.
Proof: Let $\langle X,d\rangle$ be an ultrametric space, meaning that $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/82796",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Function $f\colon 2^{\mathbb{N}}\to 2^{\mathbb{N}}$ preserving intersections and mapping sets to sets which differs only by finite number of elements Define on $2^{\mathbb{N}}$ equivalence relation
$$
X\sim Y\Leftrightarrow \text{Card}((X\setminus Y)\cup(Y\setminus X))<\aleph_0
$$
Is there exist a function $f\colon ... | Let $\{X_\alpha : \alpha\in\mathcal{A}\}\subset\mathbb{N}$ be an uncountable family of sets such that
$$
\alpha,\beta\in\mathcal{A},\quad\alpha\neq\beta\Rightarrow \text{Card}(X_\alpha\cap X_\beta)<\aleph_0
$$
Such a family does exist. Indeed for each irrational number $x\in\mathbb{I}$ consider sequence of rational ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/82855",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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how to find the parabola of a flying object how can you find the parabola of a flying object without testing it? what variables do you need? I want to calculate the maximum hight and distance using a parabola. Is this possible? Any help will be appreciated.
| I assume that you know that if an object is thrown straight upwards, with initial speed $v$, then its height $h(t)$ above the ground at time $t$ is given by
$$h(t)=vt-\frac{1}{2}gt^2,$$
where $g$ is the acceleration due to gravity. The acceleration is taken to be a positive number, constant since if our thrown object a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/82934",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Convergence of rationals to irrationals and the corresponding denominators going to zero If $(\frac{p_k}{q_k})$ is a sequence of rationals that converges to an irrational $y$, how do you prove that $(q_k)$ must go to $\infty$?
I thought some argument along the lines of "breaking up the interval $(0,p_k)$ into $q_k$ par... | Hint: For every positive integer $n$, consider the set $R(n)$ of rational numbers $p/q$ such that $1\leqslant q\leqslant n$. Show that for every $n$ the distance $\delta(n)$ of $y$ to $R(n)$, defined as $\delta(n)=\inf\{|y-r|\mid r\in R(n)\}$, is positive. Apply this to any sequence $(p_k/q_k)$ converging to $y$, showi... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Continuity at accumulation point Following was the homework question for my analysis class.
Given any sequence $x_n$ in a metric space $(X; d)$, and $x \in X$, consider the function $f : \mathbb N^\ast = \mathbb N \cup \{\infty\} \to X$ defined by $f(n) = x_n$, for all $n\in \mathbb N$, and $f(\infty) = x$.
Prove that... | It's not clear to me what your teacher meant by $\infty$ being the only accumulation point of $\mathbb N^*$, since $\mathbb N^*$ is yet to be equipped with a metric and it makes no sense to speak of accumulation points before that.
A metric on $\mathbb N^*$ satisfying the requirement is induced by mapping $\mathbb N^*$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/83044",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Zero divisor in $R[x]$ Let $R$ be commutative ring with no (nonzero) nilpotents. If $f(x) = a_0+a_1x+\cdots+a_nx^n$ is a zero divisor in $R[x]$, how do I show there's an element $b \ne 0$ in $R$ such that $ba_0=ba_1=\cdots=ba_n=0$?
| This is the case of Armendariz Rings, which I studied in last summer briefly. It is an interesting topic.
A ring $R$ is called Armendariz if whenever $f(x)=\sum_{i=0}^{m}a_ix^i, g(x)=\sum_{j=0}^{n}b_jx^j \in R[x]$ such that $f(x)g(x)=0$, then $a_ib_j=0\ \forall\ i,j$.
In his paper "A NOTE ON EXTENSIONS OF BAER AND P... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/83121",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "42",
"answer_count": 5,
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Fourier cosine series and sum help I have been having some problems with the following problem:
Find the Fourier cosine series of the function $\vert\sin x\vert$ in the interval $(-\pi, \pi)$. Use it to find the sums
$$ \sum_{n\: =\: 1}^{\infty}\:\ \frac{1}{4n^2-1}$$ and $$ \sum_{n\: =\: 1}^{\infty}\:\ \frac{(-1)^n}{4... | $f(x) = |\sin(x)| \quad \Rightarrow\quad f(x) = \left\{
\begin{array}{l l}
-\sin(x) & \quad \forall x \in [- \pi, 0\space]\\
\sin(x) & \quad \forall x \in [\space 0,\pi\space ]\\
\end{array} \right.$
The Fourier coefficients associated are
$$a_n= \frac{1}{\pi}\int_{-\pi}^\pi f(x) \cos(nx)\, dx = \frac{1}{\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/83176",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Integral of product of two functions in terms of the integral of the other function Problem:
Let $f$ and $g$ be two continuous functions on $[ a,b ]$ and assume $g$ is positive. Prove that $$\int_{a}^{b}f(x)g(x)dx=f(\xi )\int_{a}^{b}g(x)dx$$ for some $\xi$ in $[ a,b ]$.
Here is my solution:
Since $f(x)$ and $g(x)$ ar... | The integrals on both sides of the problem are well defined since $f$ and $g$ are continuous, and $g$ is positive so $ \displaystyle \int^b_a g(x) dx > 0.$ Thus there exists some constant $K$ such that $$ \int^b_a f(x) g(x) dx = K\int^b_a g(x) dx . $$
If $\displaystyle K > \max_{x\in [a,b]} f(x) $ then the left side i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/83246",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
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Characteristic polynomial equals minimal polynomial iff $x, Mx, \ldots, M^{n-1} x$ are linearly independent I have been trying to compile conditions for when characteristic polynomials equal minimal polynomials and I have found a result that I think is fairly standard but I have not been able to come up with a proof fo... | Here is a simple proof for the "only if" part, using rational canonical form.
For clarity's sake, I'll assume that $M$ is a linear map.
If $M$ is such that $p_M=c_M$, then it is similar to $F$, the companion matrix of $p_M$.i.e. There is a basis $\beta = (v_1, \dots ,v_n)$ for $V$ under which the matrix of $M$ is $F$. ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/83299",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
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What exactly happens, when I do a bit rotation? I'm sitting in my office and having difficulties to get to know that exactly happens, when I do a bit rotation of a binary number.
An example:
I have the binary number 1110. By doing bit rotation, I get 1101, 1011, 0111 ...
so I have 14 and get 7, 11, 13 and 14 again.
... | Interpret the bits as representing a number in standard binary representation (as you are doing). Then, bit rotation to the right is division by $2$ modulo $15$, or more generally, modulo $2^n-1$ where $n$ is the number of bits. Put more simply, if the number is even, divide it by $2$, while if it is odd, add $15$ an... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/83361",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Compact Sets in Projective Space Consider the projective space ${\mathbb P}^{n}_{k}$ with field $k$. We can naturally give this the Zariski topology.
Question: What are the (proper) compact sets in this space?
Motivation: I wanted nice examples of spaces and their corresponding compact sets; usually my spaces are Hau... | You are in for a big surprise, james: every subset of $\mathbb P^n_k$ is quasi-compact.
This is true more generally for any noetherian space, a space in which every decreasing sequence of closed sets is stationary.
However: the compact subsets of $\mathbb P^n_k$ are the finite sets of points such that no point is ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/83413",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
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A fair coin is tossed $n$ times by two people. What is the probability that they get same number of heads?
Say we have Tom and John, each tosses a fair coin $n$ times. What is the probability that they get same number of heads?
I tried to do it this way: individually, the probability of getting $k$ heads for each is... | As you have noted, the probability is
$$
p_n = \frac{1}{4^n} \sum_{k=0}^n \binom{n}{k} \binom{n}{k} = \frac{1}{4^n} \sum_{k=0}^n \binom{n}{k} \binom{n}{n-k} = \frac{1}{4^n} \binom{2n}{n}
$$
The middle equality uses symmetry of binomials, and last used Vandermonde's convolution identity.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/83489",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 3,
"answer_id": 0
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Whether $f(x)$ is reducible in $ \mathbb Z[x] $? Suppose that $f(x) \in \mathbb Z[x] $ has an integer root. Does it mean $f(x)$ is reducible in $\mathbb Z[x]$?
| No. $x-2$ is irreducible but has an integer root $2$.
If the degree of $f$ is greater than one, then yes. If $a$ is a root of $f(x)$, carry out synthetic division by $x-a$. You will get $f(x) = (x-a)g(x) + r$, and since $f(a) = 0$, $r=0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/83602",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Lambert series expansion identity I have a question which goes like this:
How can I show that $$\sum_{n=1}^{\infty} \frac{z^n}{\left(1-z^n\right)^2} =\sum_{n=1}^\infty \frac{nz^n}{1-z^n}$$ for $|z|<1$?
| Hint: Try using the expansions
$$
\frac{1}{1-x}=1+x+x^2+x^3+x^4+x^5+\dots
$$
and
$$
\frac{1}{(1-x)^2}=1+2x+3x^2+4x^3+5x^4+\dots
$$
Expansion:
$$
\begin{align}
\sum_{n=1}^\infty\frac{z^n}{(1-z^n)^2}
&=\sum_{n=1}^\infty\sum_{k=0}^\infty(k+1)z^{kn+n}\\
&=\sum_{n=1}^\infty\sum_{k=1}^\infty kz^{kn}\\
&=\sum_{k=1}^\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/83680",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 0
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Can't write context free grammar for language $L=\{a^n\#a^{n+2m}, n,m \geq 1\}$ Spent some time on this problem and seems like I am not able write context free grammar for language
$L=\{a^n\#a^{n+2m}, n\geq 1\wedge m \geq 1, n\in \mathbb{N} \wedge m\in\mathbb{N}\}$
I am sure I am missing something obvious, but can't f... | I think this is the solution:
$S \rightarrow aLaT$
$L \rightarrow aLa \mid \#$
$T \rightarrow Taa \mid aa$
This language is actually just $\{ a^n\#a^n \mid n \geq 1\} \circ (aa)^+$, where $\circ$ is the concatenation operator. Which is why this CFG is so easy to construct, as it is an easily expressible language follow... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/83740",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
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Dense subset of given space If $E$ is a Banach space, $A$ is a subset such that
$$A^{\perp}:= \{T \in E^{\ast}: T(A)=0\}=0,$$ then $$\overline{A} = E.$$
I don't why this is true. Does $E$ has to be Banach? Thanks
| Did you mean to say that $A$ is a vector subspace, or does $\overline{A}$ mean the closed subspace of $E$ generated by $A$? If $A$ were only assumed to be a subset and $\overline{A}$ means the closure, then this is false. E.g., let $A$ be the unit ball of $E$.
Suppose that the closed subspace of $E$ generated by $A... | {
"language": "en",
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What is a real world application of polynomial factoring? The wife and I are sitting here on a Saturday night doing some algebra homework. We're factoring polynomials and had the same thought at the same time: when will we use this?
I feel a bit silly because it always bugged me when people asked that in grade school.... | You need polynomial factoring (or what's the same, root finding) for higher mathematics. For example, when you are looking for the eigenvalues of a matrix, they appear as the roots of a polynomial, the "characteristic equation".
I suspect that none of this will be of any use to someone unless they continue their mathe... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/83837",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "70",
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A metric space in which every infinite set has a limit point is separable I am struggling with one problem.
I need to show that if $X$ is a metric space in which every infinite subset has a limit point then $X$ is separable (has countable dense subset in other words).
I am trying to use the result I have proven prior t... | Let $\langle X,d\rangle$ be a metric space in which each infinite subset has a limit point. For any $\epsilon>0$ an $\epsilon$-mesh in $X$ is a set $M\subseteq X$ such that $d(x,y)\ge\epsilon$ whenever $x$ and $y$ are distinct points of $M$. Every $\epsilon$-mesh in $X$ is finite, since an infinite $\epsilon$-mesh woul... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/83876",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
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Convergence of $\lim_{n \to \infty} \frac{5 n^2 +\sin n}{3 (n+2)^2 \cos(\frac{n \pi}{5})},$ I'm in trouble with this limit. The numerator diverges positively, but I do not understand how to operate on the denominator.
$$\lim_{n \to \infty} \frac{5 n^2 +\sin n}{3 (n+2)^2 \cos(\frac{n \pi}{5})},$$
$$\lim_{n \to \infty} ... | Let's make a few comments.
*
*Note that the terms of the sequence are always defined: for $n\geq 0$, $3(n+2)^2$ is greater than $0$; and $\cos(n\pi/5)$ can never be equal to zero (you would need $n\pi/5$ to be an odd multiple of $\pi/2$, and this is impossible).
*If $a_n$ and $b_n$ both have limits as $n\to\infty$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/83919",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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commuting matrices & polynomials 1 I need help on this problem:
Problem:
Find two 3x3 matrices, A and B that commute with each other;
and neither A is a polynomials of B nor B is a polynomial of A
| A=diag(1,1,2) en B is the matrix with rows [1,1,0\0,1,0\0,0,1].
Then AB=BA, B is not polynomial in A (B is not a diagonal matrix) en A is not polynomial in B. For any polynomial p of degree <3 with P(B)=A should have the property p(1)=1 (since p([1,1\0,1])=diag(1,1), so p(x)=1+(x-1)^2) and p(1)=2.
J. Vermeer
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Is $O(\frac{1}{n}) = o(1)$? Sorry about yet another big-Oh notation question, I just found it very confusing.
If $T(n)=\frac{5}{n}$, is it true that $T(n)=O(\frac{1}{n})$ and $T(n) = o(1)$? I think so because (if $h(n)=\frac{1}{n}$)
$$
\lim_{n \to \infty} \frac{T(n)}{h(n)}=\lim_{n \to \infty} \frac{\frac{5}{n}}{\frac... | If $x_n = O(1/n)$, this means there exists $N$ and $C$ such that for all $n > N$, $|x_n| \le C|1/n|$. Hence
$$
\lim_{n \to \infty} \frac{|x_n|}{1} \le \lim_{n \to \infty} \frac{C|1/n|}{1} = 0.
$$
This means if $x_n = O(1/n)$ then $x_n = o(1)$.
Conversely, it is not true though. Saying that $x_n = o(1)$ only means $x_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/84021",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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On the Origin and Precise Definition of the Term 'Surd' So, in the course of last week's class work, I ran across the Maple function surd() that takes the real part of an nth root. However, conversation with my professor and my own research have failed to produce even an adequate definition of the term, much less a goo... | An irrational root of rational number is defined as surd. An example is a root of (-1)
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/84075",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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Can we say a Markov Chain with only isolated states is time reversible? By "isolated", I mean that each state of this Markov Chain has 0 probability to move to another state, i.e. transition probability $p_{ij} = 0$ for $ i \ne j$. Thus, there isn't a unique stationary distribution.
But by definition, since for any st... | I do not think that constructing a markov chain with isolated states will give you a time irreversible markov chain.
Consider the case when you have one isolated state. Since, an isolated state can never be reached from any other state, your chain is actually a union of two different markov chains.
*
*A markov chai... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/84120",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Finding a correspondence between $\{0,1\}^A$ and $\mathcal P(A)$ I got this question in homework:
Let $\{0,1\}^A$ the set of all functions from A (not necessarily a finite set)
to $\{0,1\}$. Find a correspondence (function) between $\{0,1\}^A$ and
$\mathcal P(A)$ (The power set of $A$).
Prove that th... | I'll try to say this without all the technicalities that accompany some of the earlier answers.
Let $B$ be a member of $\mathcal{P}(A).$
That means $B\subseteq A$.
You want to define a function $f$ corresponding to the set $B$. If $x\in A$, then what is $f(x)$? It is: $f(x)=1$ if $x\in B$ and $f(x) = 0$ if $x\not\in ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/84180",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 3
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How determine or visualize level curves Let $f:\mathbb{C}\to\mathbb{C}$ given for $f(z)=\int_0^z \frac{1-e^t}{t} dt-\ln z$ and put $g(x,y)=\text{Re}(f(z))$. While using the computer, how to determine the curve $g(x,y)=0$?
Thanks for the help.
| Using Mathematica:
ContourPlot[With[{z = x + I y},
Re[EulerGamma - ExpIntegralEi[z]]] == 0,
{x, -20, 20}, {y, -20, 20}]
| {
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Check if point on circle is in between two other points (Java) I am struggling with the following question. I'd like to check if a point on a circle is between two other points to check if the point is in the boundary. It is easy to calculate when the boundary doesn't go over 360 degrees. But when the boundary goes ove... | From the question comments with added symbols
I have a circle with a certain sector blocked. Say for example the sector between $a = 90°$ and $b = 180°$ is blocked. I now want to check if a point $P = (x,y)$ in the circle of center $C = (x_0,y_0)$ of radius $r$ is in this sector or not to see if it is a valid point or... | {
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calculus textbook avoiding "nice" numbers: all numbers are decimals with 2 or 3 sig figs Many years ago, my father had a large number of older used textbooks.
I seem to remember a calculus textbook with a somewhat unusual feature, and I am wondering if the description rings a bell with anyone here.
Basically, this was ... | Though this is probably not the book you are thinking of, Calculus for the Practical Man by Thompson does this. It is, most famously, the book that Richard Feynman learned calculus from, and was part of a whole series of math books "for the practical man". The reason I do not think it is the particular book you are thi... | {
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If a topological space has $\aleph_1$-calibre and cardinality at most $2^{\aleph_0}$ must it be star-countable? If a topological space $X$ has $\aleph_1$-calibre and the cardinality of $X$ is $\le 2^{\aleph_0}$, then it must be star countable? A topological space $X$ is said to be star-countable if whenever $\mathscr{U... | Under CH the space is separable (hence, star countable ).
Proof(Ofelia). On the contrary, suppose that X is not separable .Under CH write $X = \{ x_\alpha : \alpha \in \omega_1 \}$ and for each $\alpha$ in $\omega_1$ define $U_\alpha$ = the complement of $cl ( { x_\beta : \beta \le \alpha } )$ . The family of the $U_\... | {
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Computational complexity of least square regression operation In a least square regression algorithm, I have to do the following operations to compute regression coefficients:
*
*Matrix multiplication, complexity: $O(C^2N)$
*Matrix inversion, complexity: $O(C^3)$
*Matrix multiplication, complexity: $O(C^2N)$... | In this work https://www.research-collection.ethz.ch/bitstream/handle/20.500.11850/153646/eth-6011-01.pdf?sequence=1&isAllowed=y two implementation possibilities (the Gaussian elimination alternative vs. using the QR decomposition) are discussed in pages 32 and 33 if you are interested in the actual cost DFLOP-wise.
| {
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proof of the Cauchy integral formula
$ D=D_{r}(c), r > 0 .$ Show that if $f$ is continuous in $\overline{D}$ and holomorphic in $D$, then for all $z\in D$: $$f(z)=\frac{1}{2\pi i} \int_{\partial D}\frac{f(\zeta)}{\zeta - z} d\zeta$$
I don't understand this question because I don't see how it is different to the spec... | If I understand correctly, $D_r(c)$ is the open ball centered in $c$ with radius $r$? If this is the case, the difference between the two is that above your $c$ is fixed, and in the special case your $c$ "moves" with the ball.
Fix $c$ then; we want to show that for every $z\in D_r(c)$ we have that
$$f(z)=\frac{1}{2\pi ... | {
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$\limsup $ and $\liminf$ of a sequence of subsets relative to a topology From Wikipedia
if $\{A_n\}$ is a sequence of subsets of a topological space $X$,
then:
$\limsup A_n$, which is also called the outer limit, consists of those
elements which are limits of points in $A_n$ taken from (countably)
infinitely man... | I must admit that I did not know these definitions, either.
*
*Yes, because if $x \in \liminf A_n$ you have a sequence $\{x_k\}$ with $x_k \in A_k$ and $x_k \rightarrow x$ and you can choose your subsequence $\{A_{n_k}\}$ to be your whole sequence $\{A_n\}$.
*For $X = \mathbb{R}$ take $A_n = \{0\}$ for all $n$. The... | {
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Solve $f(f(n))=n!$ What am I doing wrong here: ( n!=factorial )
Find $f(n)$ such that $f(f(n))=n!$
$$f(f(f(n)))=f(n)!=f(n!).$$
So $f(n)=n!$ is a solution, but it does not satisfy the original equation except for $n=1$, why?
How to solve $f(f(n))=n!$?
| The hypothesis is $f(f(n))=n!$. This implies that $f(n)!=f(n!)$ like you say, but unfortunately the converse is not true; you can't reverse the direction and say that a function satisfying the latter equation also satisfies $f(f(n))=n!$. For a similar situation, suppose we have $x=1$ and square it to obtain $x^2=1$; no... | {
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Algebraic Structures Question I am having problems understanding what this question is asking. any help would be appreciated. Thanks.
The dihedral group D8 is an 8 -element subgroup of the 24 -element symmetric group S4 .
Write down all left and right cosets of D8 in S4 and draw conclusions regarding normality of D8 in... | HINT:
Represent $D_8$ with your preferred notation. Perhaps it is the group generated by $(1234)$ and $(13)$. That's ok. Then write down the 8 elements. Then multiply each on the right and on the left by elements of $S_4$, i.e. write down the right and left cosets. You can just sort of do it, and I recommend it in orde... | {
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Complex equation solution. How can i resolve it? I have this complex equation $|z+2i|=| z-2 |$. How can i resolve it? Please help me
| The geometric way
The points $z$ that satisfy the equation are at the same distance of the points $2$ and $-2\,i$, that is, they are on the perpendicular bisector of the segment joining $2$ and $-2\,i$. This is a line, whose equation you should be able to find.
The algebraic way
When dealing vith equations with $|w|$, ... | {
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estimate the perimeter of the island I'm assigned a task involving solving a problem that can be described as follows: Suppose I'm driving a car around a lake. In the lake there is an island of irregular shape. I have a GPS with me in the car so I know how far I've driven and every turns I've made. Now suppose I also h... | I am going to take a stab at this, although I would like to hear a flaw in my argument.
If you have a reference point on the island where your camera is always pointed to then, since you know your exact travel path and distance from your path to the edge of the island, you can plot the shape of the island, then, findi... | {
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$f\colon\mathbb{R}\to\mathbb{R}$ such that $f(x)+f(f(x))=x^2$ for all $x$? A friend came up with this problem, and we and a few others tried to solve it. It turned out to be really hard, so one of us asked his professor. I came with him, and it took me, him and the professor about an hour to say anything interesting ... | UPDATE: This answer now makes significantly weaker claims than it used to.
Define the sequence of functions $f_n$ recursively by
$$f_1(t)=t,\ f_2(t) = 3.8 + 1.75(t-3),\ f_k(t) = f_{k-2}(t)^2 - f_{k-1}(t)$$
The definition of $f_2$ is rigged so that $f_2(3) = 3.8$ and $f_2(3.8) = 3^2- 3.8 = 5.2$.
Set $g_k=f_k(3)=f_{k-1}(... | {
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Vector Mid Point vs Mid Point Formula Given $OA=(2,9,-6)$ and $OB=(6,-3,-6)$. If $D$ is the midpoint, isit
$OD=((2+6)/2, (9-3)/2, (-6-6)/2)$?
The correct answer is
$OD=\frac{1}{2}AB=(2,-6,0)$
| Your first answer is the midpoint of the line segment that joins the tip of the vector $OA$ to the tip of the vector $OB$. The one that you call the correct answer is gotten by putting the vector $AB$ into standard position with, its initial end at the origin, and then finding the midpoint.
| {
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Can every group be represented by a group of matrices?
Can every group be represented by a group of matrices?
Or are there any counterexamples? Is it possible to prove this from the group axioms?
| Every finite group is isomorphic to a matrix group. This is a consequence of Cayley's theorem: every group is isomorphic to a subgroup of its symmetry group. Since the symmetric group $S_n$ has a natural faithful permutation representation as the group of $n\times n$ 0-1 matrices with exactly one 1 in each row and co... | {
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$f:[0,1] \rightarrow \mathbb{R}$ is absolutely continuous and $f' \in \mathcal{L}_{2}$ I am studying for an exam and am stuck on this practice problem.
Suppose $f:[0,1] \rightarrow \mathbb{R}$ is absolutely continuous and $f' \in \mathcal{L}_{2}$. If $f(0)=0$ does it follow that $\lim_{x\rightarrow 0} f(x)x^{-1/2}=0$... | Yes. The Cauchy-Schwarz inequality gives
$$f(x)^2=\left(\int^x_0 f^\prime(y)\ dy\right)^2\leq \left(\int^x_0 1\ dy\right) \left(\int^x_0 f^\prime(y)^2\ dy\right)=x\left(\int^x_0 f^\prime(y)^2\ dy\right).$$
Dividing by $x$ and taking square roots we get $$|f(x)/\sqrt{x}|\leq \left(\int^x_0 f^\prime(y)^2\ dy\right)^{1/2}... | {
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Solving $A(x) = 2A(x/2) + x^2$ Using Generating Functions Suppose I have the recurrence:
$$A(x) = 2A(x/2) + x^2$$ with $A(1) = 1$.
Is it possible to derive a function using Generating Functions? I know in Generatingfunctionology they shows show to solve for recurrences like $A(x) = 2A(x-1) + x$. But is it possible to s... | I am a little confused by the way you worded this question (it seems that you have a functional equation rather than a recurrence relation), so I interpreted it in the only way that I could make sense of it. If this is not what you are looking for, then please clarify in your original question or in a comment.
Let's as... | {
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Secret santa problem We decided to do secret Santa in our office. And this brought up a whole heap of problems that nobody could think of solutions for - bear with me here.. this is an important problem.
We have 4 people in our office - each with a partner that will be at our Christmas meal.
Steve,
Christine,
Mark,
Ma... | See this algorithm here: http://weaving-stories.blogspot.co.uk/2013/08/how-to-do-secret-santa-so-that-no-one.html. It's a little too long to include in a Stack Exchange answer.
Essentially, we fix the topology to be a simple cycle, and then once we have a random order of participants we can also determine who to get a ... | {
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Limits Involving Trigonometric Functions I should prove using the limit definition that
$$\lim_{x \rightarrow 0} \, x^{1/3}\cos(1/x) = 0.$$
I have a problem because the second function is much too complex, so I think I need transformation. And what form this function could have in case I will transform it?
| You can solve this problem with the Squeeze Theorem.
First, notice that $-1 \leq \cos(1/x) \leq 1$ (the cosine graph never goes beyond these bounds, no matter what you put inside as the argument).
Multiplying through by $x^{1/3}$, we get
$$
-x^{1/3} \leq x^{1/3}\cos(1/x) \leq x^{1/3}.
$$
Now, the Squeeze Theorem says... | {
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Zero Dim Topological group I have this assertion which looks rather easy (or as always I am missing something):
We have $G$ topological group which is zero dimensional, i.e it admits a basis for a topology which consists of clopen sets, then every open nbhd that contains the identity element of G also contains a clopen... | This is certainly false. Take $G=\mathbb{Q}$ with the standard topology and additive group structure. The topology is zero-dimensional since intersecting with $\mathbb{Q}$ the countably many open intervals whose endpoints are rational translates of $\sqrt 2$ gives a clopen basis for the topology on $G$. The trivial sub... | {
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The history of set-theoretic definitions of $\mathbb N$ What representations of the natural numbers have been used, historically, and who invented them? Are there any notable advantages or disadvantages?
I read about Frege's definition not long ago, which involves iterating over all other members of the universe; clea... | Maybe this book might be useful for you, too. I'll include a short quote from §1.2 Natural numbers.
Ebbinghaus et al.: Numbers, p.14
Counting with the help of number symbols marks the beginning of arithmetic.
Computation counting. Until well into the nineteenth
century, efforts were made to trace the idea of numbe... | {
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Geometry problem: Line intersecting a semicircle Suppose we have a semicircle that rests on the negative x-axis and is tangent to the y-axis.A line intersects both axes and the semicircle. Suppose that the points of intersection create three segments of equal length. What is the slope of the line?
I have tried numerou... | In this kind of problem, it is inevitable that plain old analytic geometry will work. A precise version of this assertion is an important theorem, due to Tarski. If "elementary geometry" is suitably defined, then there is an algorithm that will determine, given any sentence of elementary geometry, whether that senten... | {
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How to use the Extended Euclidean Algorithm manually? I've only found a recursive algorithm of the extended Euclidean algorithm. I'd like to know how to use it by hand. Any idea?
| The way to do this is due to Blankinship "A New Version of the Euclidean Algorithm", AMM 70:7 (Sep 1963), 742-745. Say we want $a x + b y = \gcd(a, b)$, for simplicity with positive $a$, $b$ with $a > b$. Set up auxiliary vectors $(x_1, x_2, x_3)$, $(y_1, y_2, y_3)$ and $(t_1, t_2, t_3)$ and keep them such that we alwa... | {
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Proof by Induction: Alternating Sum of Fibonacci Numbers
Possible Duplicate:
Show that $f_0 - f_1 + f_2 - \cdots - f_{2n-1} + f_{2n} = f_{2n-1} - 1$ when $n$ is a positive integer
This is a homework question so I'm looking to just be nudged in the right direction, I'm not asking for my work to be done for me.
The... | HINT $\ $ LHS,RHS both satisfy $\rm\ g(n+1) - g(n)\: =\: f_{2\:n},\ \ g(1) = 0\:.\:$ But it is both short and easy to prove by induction that the solutions $\rm\:g\:$ of this recurrence are unique. Therefore LHS = RHS.
Note that abstracting away from the specifics of the problem makes the proof both much more obvious ... | {
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The construction of function If $X$ is a paracompact Hausdorff space and $\{ {X_i}\} $ $i \ge 0$ are open subsets of $X$ such that ${X_i} \subset {X_{i + 1}}$ and $\bigcup\nolimits_{i \ge o} {{X_i}} = X$, can we find a continuous function $f$ such that $f(x) \ge i + 1$ when $x \notin {X_i}$ ?
| Yes, your function exists.
As $X$ is paracompact and Hausdorff you get a partition of unity subordinate to your cover $\{X_i\}$, that is, a family $\{f_i : X \rightarrow [0,1]\}_{i \geq 0}$ with $\text{supp}(f_i) \subset X_i$ and every $x \in X$ has a neighborhood such that $f_i(x) = 0$ for all but finitely many $i \ge... | {
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Reciprocal of a continued fraction I have to prove the following:
Let $\alpha=[a_0;a_1,a_2,...,a_n]$ and $\alpha>0$, then $\dfrac1{\alpha}=[0;a_0,a_1,...,a_n]$
I started with
$$\alpha=[a_0;a_1,a_2,...,a_n]=a_0+\cfrac1{a_1+\cfrac1{a_2+\cfrac1{a_3+\cfrac1{a_4+\cdots}}}}$$
and
$$\frac1{\alpha}=\frac1{[a_0;a_1,a_2,...,a_n]... | Coming in late: there's a similar approach that will let you take the reciprocal of nonsimple continued fractions as well.
*
*change the denominator sequence from $[b_0;a_0,a_1,a_2...]$ to $[0; b_0,a_0,a_1,a_2...]$
*change the numerator sequence from $[c_1,c_2,c_3,...]$ to $[1,c_1,c_2,c_3...]$
| {
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An unintuitive probability question Suppose you meet a stranger in the Street walking with a boy. He tells you that the boy is his son, and that he has another child. Assuming equal probability for boy and girl, and equal probability for Monty to walk with either child, what is the probability that the second child is ... | I don't think so. The probability of this is $\frac 1 2$ which seems clear since all it depends on is the probability that the child not walking with Monty is a boy. The probability is not $\frac 1 3$! The fact that he is actually walking with a son is different than the fact that he has at least one son! Let $A$ be th... | {
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How to find irreducible representation of a group from reducible one? I was reading this document to answer my question. But after teaching me hell lot of jargon like subgroup, normal subgroup, cosets, factor group, direct sums, modules and all that the document says this,
You likely realize immediately that this is ... | I do not believe that there is a straightforward way of doing what you want for complex representations. Probably the best way is to first compute the character table of the group. There are algorithms for that, such as Dixon-Schneider, but it is not something you can just sit down and program in an afternoon. Then you... | {
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How to find the GCD of two polynomials How do we find the GCD $G$ of two polynomials, $P_1$ and $P_2$ in a given ring (for example $\mathbf{F}_5[x]$)? Then how do we find polynomials $a,b\in \mathbf{F}_5[x]$ so that $P_1a+ P_2b=G$? An example would be great.
| If you have the factorization of each polynomial, then you know what the divisors look like, so you know what the common divisors look like, so you just pick out one of highest degree.
If you don't have the factorization, the Euclidean algorithm works for polynomials in ${\bf F}_5[x]$ just as it does in the integers, ... | {
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Infinite product of connected spaces may not be connected? Let $X$ be a connected topologoical space. Is it true that the countable product $X^\omega$ of $X$ with itself (under the product topology) need not be connected? I have heard that setting $X = \mathbb R$ gives an example of this phenomenon. If so, how can I pr... | Maybe this should have been a comment, but since I don't have enough reputation points, here it is.
On this webpage, you will find a proof that the product of connected spaces is connected (using the product topology). In case of another broken link in the future, the following summary (copied from here) could be usefu... | {
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Form of rational solutions to $a^2+b^2=1$? Is there a way to determine the form of all rational solutions to the equation $a^2+b^2=1$?
| I up-voted yunone's answer, but I notice that it begins by saying "if you know some field theory", and then gets into $N_{\mathbb{Q}(i)/\mathbb{Q}}(a_bi)$, and then talks about Galois groups, and then Hilbert's Theorem 90, and tau functions (where "$\tau$ is just the complex conjugation map in this case" (emphasis mine... | {
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"question_score": "3",
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"answer_id": 2
} |
Reference for "It is enough to specify a sheaf on a basis"? The wikipedia article on sheaves says:
It can be shown that to specify a sheaf, it is enough to specify its restriction to the open sets of a basis for the topology of the underlying space. Moreover, it can also be shown that it is enough to verify the sheaf ... | It is given in Daniel Perrin's Algebraic Geometry, Chapter 3, Section 2. And by the way, it is a nice introductory text for algebraic geometry, which does not cover much scheme theory, but gives a definition of an abstract variety (using sheaves, like in Mumford's Red book).
Added: I just saw that Perrin leaves most o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/86509",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
"answer_count": 5,
"answer_id": 1
} |
Proving an exponential bound for a recursively defined function I am working on a function that is defined by
$$a_1=1, a_2=2, a_3=3, a_n=a_{n-2}+a_{n-3}$$
Here are the first few values:
$$\{1,1,2,3,3,5,6,8,11,\ldots\}$$
I am trying to find a good approximation for $a_n$. Therefore I tried to let Mathematica diagonalize... | Your function $a_n$ is a classical recurrence relation. It is well-known that
$$a_n = \sum_{i=1}^3 A_i \alpha_i^n,$$
where $\alpha_i$ are the roots of the equation $x^3 = x + 1$. You can find the coefficients $A_i$ by solving a system of linear equations. In your case, one of the roots is real, and the other two are co... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/86569",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 1
} |
Numerically solve second-order ODE I want to solve a second-order ODE in the form of
$$ y^{''} = \frac{a (y^{'})^2}{b y^{'}+cy+d} $$ by numerical method (eg, solver ODE45), given initial condition of $y(0)$ and $y'(0)$. The results are wield and numbers go out of machinery bound. I guess the catch is that what is in th... | Discretizing the ODE by finite differences gives
$$\frac{y_2-2y_1 + y_0}{h^2} = \frac{a\left(\frac{y_1-y_0}{h}\right)^2}{b\left(\frac{y_1-y_0}{h}\right) + cy_1 + d},$$
or
$$y_2 = 2y_1 - y_0 + \frac{ah(y_1-y_0)^2}{b(y_1-y_0)+chy_1+dh}.$$
Here's C++ code I wrote which has no trouble integrating this ODE for $a=-10,b=c=d=... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/86633",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Homogeneous Fredholm Equation of Second Kind I'm trying to show that the eigenvalues of the following integral equation
\begin{align*}
\lambda \phi(t) = \int_{-T/2}^{T/2} dx \phi(x)e^{-\Gamma|t-x|}
\end{align*}
are given by
\begin{align*}
\Gamma \lambda_k = \frac{2}{1+u_k^2}
\end{align*}
where $u_k$ are the solutions ... | This problem was solved in Mark Kac, Random Walk in the presence of absorbing barriers, Ann. Math. Statistics 16, 62-67 (1945) https://projecteuclid.org/euclid.aoms/1177731171
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/86656",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Complexity substitution of variables in multivariate polynomials I want to substitute a variable with a number in multivariate polynomials. For example for the polynomial
$$
P = (z^2+yz^3)x^2 + zx
$$
I want to substitute $z$ with $3$.
I have intuition how to do that algorithmatic: I have to regard the coefficients in... | There are classical results (Ostrowski) on optimality of Horner's method and related evaluation schemes. These have been improved by Pan, Strassen and others. Searching on "algebraic complexity" with said authors should quickly locate pertinent literature.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/86751",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Subspaces in Linear Algebra Find the $\operatorname{Proj}_wv$ for the given vector $v$ and subspace $W$. Let $V$ be the Euclidean space $\mathbb{R}^4$, and $W$ the subspace with basis $[1, 1, 0, 1], [0, 1, 1, 0], [-1, 0, 0, 1]$
(a) $v = [2,1,3,0]$
ans should be - $[7/5,11/5,9/5,-3/5]$
My attempt at the solution was bas... | You can do it that way (though you must have an arithmetical error somewhere; the denominator of $3$ cannot be right), and the remaining piece is then simply to take $a[1, 1, 0, 1] + b[0, -1, 1, 0] + c[0 ,2, 0,3]$, forgetting the part perpendicular to $W$.
However, it is much easier to normalize your $[1,-2,2,1]$ to $n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/86864",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
One-to-one mapping from $\mathbb{R}^4$ to $\mathbb{R}^3$ I'm trying to define a mapping from $\mathbb{R}^4$ into $\mathbb{R}^3$ that takes the flat torus to a torus of revolution.
Where the flat torus is defined by $x(u,v) = (\cos u, \sin u, \cos v, \sin v)$.
And the torus of revolution by $x(u,v) = ( (R + r \cos u)\c... | Shaun's answer is insufficient since there are immersions which are not 1-1. For example, the figure 8 is an immersed circle. Also, the torus covers itself and all covering maps are immersions. http://en.wikipedia.org/wiki/Immersion_(mathematics)
Your parametrization of the torus of rotation is the the same as in http:... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/86930",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 1
} |
Calculating a Taylor Polynomial of a mystery function I need to calculate a taylor polynomial for a function $f:\mathbb{R} \to \mathbb{R}$ where we know the following
$$f\text{ }''(x)+f(x)=e^{-x} \text{ } \forall x$$ $$f(0)=0$$ $$f\text{ }'(0)=2$$
How would I even start?
| We have the following
$$f''(x) + f(x) = e^{-x}$$
and $f(0) = 0$, $f'(0) = 2$.
And thus we need to find $f^{(n)}(0)$ to construct the Taylor series.
Note that we already have two values and can find $f''(0)$ since
$$f''(0) + f(0) = e^{-0}$$
$$f''(0) +0 = 1$$
$$f''(0) = 1$$
So now we differentiate the original equation a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/86981",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 1
} |
Example of a real-life graph with a "hole"? Anyone ever come across a real non-textbook example of a graph with a hole in it?
In Precalc, you get into graphing rational expressions, some of which reduce to a non-rational. The cancelled factors in the denominator still identify discontinuity, yet can't result in vert... | A car goes 60 miles in 2 hours. So 60 miles/2 hours = 30 miles per hour.
But how fast is the car going at a particular instant? It goes 0 miles in 0 hours. There you have a hole!
It is for the purpose of removing that hole that limits are introduced in calculus. Then you can talk about instantaneous rates of change... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/87054",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 2
} |
Is it possible that in a metric space $(X, d)$ with more than one point, the only open sets are $X$ and $\emptyset$?
Is it possible that in a metric space $(X, d)$ with more than one point, the only open sets are $X$ and $\emptyset$?
I don't think this is possible in $\mathbb{R}$, but are there any possible metric sp... | One of the axioms is that for $x, y \in X$ we have $d(x, y) = 0$ if and only if $x = y$. So if you have two distinct points, you should be able to find an open ball around one of them that does not contain the other.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/87135",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Maximizing symmetric matrices v.s. non-symmetric matrices Quick clarification on the following will be appreciated.
I know that for a real symmetric matrix $M$, the maximum of $x^TMx$ over all unit vectors $x$ gives the largest eigenvalue of $M$. Why is the "symmetry" condition necessary? What if my matrix is not symme... | You can decompose any asymmetric matrix $M$ into its symmetric and antisymmetric parts, $M=M_S+M_A$, where
$$\begin{align}
M_S&=\frac12(M+M^T),\\
M_A&=\frac12(M-M^T).
\end{align}$$
Observe that $x^TM_Ax=0$ because $M_A=-M_A^T$. Then
$$x^TMx=x^T(M_S+M_A)x=x^TM_Sx+x^TM_Ax=x^TM_Sx.$$
Therefore, when dealing with something... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/87199",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 0
} |
Common internal tangent of two circles PA is the radius of a circle with center P, and QB is the radius of a circle with center Q, so that AB is a common internal tangent of the two circles, Let M be the midbout of AB and N be the point of line PQ so that line MN is perpendicular to PQ. Z is the point where AB and PQ i... | Since $BQ=10$, $AP=5$ and triangles $BQZ$ and $APZ$ are similar, we get $QZ=2PZ$. Because $PQ=17$, we get $PZ=17/3$ and $QZ=34/3$. Using the Pythagorean theorem, we get $BZ=16/3$ and $AZ=8/3$, and thus $AB=8$. Since $MZ=AB/6$, we get $MZ=8/6$ (and not 17/6 as you computed). Could you do the rest of the computation?... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/87251",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Proving Integral Inequality I am working on proving the below inequality, but I am stuck.
Let $g$ be a differentiable function such that $g(0)=0$ and $0<g'(x)\leq 1$ for all $x$. For all $x\geq 0$, prove that
$$\int_{0}^{x}(g(t))^{3}dt\leq \left (\int_{0}^{x}g(t)dt \right )^{2}$$
| Since $0<g'(x)$ for all $x$, we have $g(x)\geq g(0)=0$. Now let $F(x)=\left (\int_{0}^{x}g(t)dt \right )^{2}-\int_{0}^{x}(g(t))^{3}dt$. Then
$$F'(x)=2g(x)\left (\int_{0}^{x}g(t)dt \right )-g(x)^3=g(x)G(x),$$
where
$$G(x)=2\int_{0}^{x}g(t)dt-g(x)^2.$$
We claim that $G(x)\geq 0$. Assuming the claim, we have $F'(x)\ge... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/87305",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 2,
"answer_id": 0
} |
More Theoretical and Less Computational Linear Algebra Textbook I found what seems to be a good linear algebra book. However, I want a more theoretical as opposed to computational linear algebra book. The book is Linear Algebra with Applications 7th edition by Gareth Williams. How high quality is this? Will it prov... | I may be a little late responding to this, but I really enjoyed teaching from the book Visual Linear Algebra. It included labs that used Maple that I had students complete in pairs. We then were able to discuss their findings in the context of the theorems and concepts presented in the rest of the text. I think for man... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/87362",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "18",
"answer_count": 9,
"answer_id": 5
} |
Continued fraction: Show $\sqrt{n^2+2n}=[n; \overline{1,2n}]$ I have to show the following identity ($n \in \mathbb{N}$):
$$\sqrt{n^2+2n}=[n; \overline{1,2n}]$$
I had a look about the procedure for $\sqrt{n}$ on Wiki, but I don't know how to transform it to $\sqrt{n^2-2n}$.
Any help is appreciated.
EDIT:
I tried the fo... | HINT $\rm\ x = [\overline{1,2n}]\ \Rightarrow\ x\ = \cfrac{1}{1+\cfrac{1}{2\:n+x}}\ \iff\ x^2 + 2\:n\ x - 2\:n = 0\ \iff\ x = -n \pm \sqrt{n^2 + 2\:n} $
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/87526",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
} |
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