qid int64 1 4.65M | question large_stringlengths 27 36.3k | author large_stringlengths 3 36 | author_id int64 -1 1.16M | answer large_stringlengths 18 63k |
|---|---|---|---|---|
785,188 | <p>I found a very simple algorithm that draws values from a Poisson distribution from <a href="http://www.akira.ruc.dk/~keld/research/javasimulation/javasimulation-1.1/docs/report.pdf" rel="nofollow">this project.</a></p>
<p>The algorithm's code in Java is:</p>
<pre><code>public final int poisson(double a) {
... | Zook | 135,276 | <p>From <a href="http://en.wikipedia.org/wiki/Poisson_distribution" rel="nofollow">wikipedia:</a></p>
<blockquote>
<p>The Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time and/or space if these events occur wit... |
193,522 | <p>Suppose $\mathbb P$ is a countably closed forcing, and $\mathbb Q$ is a c.c.c. forcing that adds reals. Is $\mathbb P$ still proper in $V^{\mathbb Q}$?</p>
| Victoria Gitman | 5,984 | <p>I have a very partial answer to the question. I suspect that the argument given in the proof of Theorem 3.5 in my paper <a href="http://arxiv.org/abs/0801.4368" rel="nofollow">Proper and piecewise proper families of reals</a> generalizes to show that if $\mathbb P$ is countably closed and has size $\omega_1$, then i... |
193,522 | <p>Suppose $\mathbb P$ is a countably closed forcing, and $\mathbb Q$ is a c.c.c. forcing that adds reals. Is $\mathbb P$ still proper in $V^{\mathbb Q}$?</p>
| Goldstern | 14,915 | <p>Here is the short version of a negative answer. Let $P$ be the collapse of $\omega_2$ to $\omega_1$ with countable conditions. Fix a tree $(N_\eta:\eta\in 2^{<\omega})$ of quite different models, increasing along each branch. Specifically, make sure that along each branch in $2^\omega$ the union of the models wi... |
1,365,489 | <p>What is the value of the following expression?</p>
<p>$$\sqrt[3]{\ 17\sqrt{5}+38} - \sqrt[3]{17\sqrt{5}-38}$$</p>
| MY USER NAME IS A LIE | 209,888 | <p>Here's the thing about this question: It is a trick.For most numbers of the form $a+b\sqrt 5$ the cube root is a mess. But some such numbers have nice cube roots and when they do we can find them.</p>
<p>The one thing you have to know is that if $x$ is any number of the form $a+b\sqrt5$ then $x^2, x^3$, etc. also h... |
3,511,445 | <p>Well it is the problem from jmo odisha. It is of 5 marks
I tried a lot of ways but I can't get the answer. Only elementary mathematics is allowed.</p>
| Edison Valentim Lopes | 737,955 | <p><strong>Aos usuários de língua <em>portuguesa</em></strong></p>
<p><strong>Questão</strong>: Encontre os valores "<span class="math-container">$x$</span>", "<span class="math-container">$y$</span>" e "<span class="math-container">$z$</span>"
se</p>
<p><span class="math-container">$\left\{\begin{array}{lcr}(x-4)(y-... |
446,456 | <p>Educators and Professors: when you teach first year calculus students that infinity isn't a number, how would you logically present to them $-\infty < x < +\infty$, where $x$ is a real number?</p>
| Anonymous Computer | 128,641 | <p>Just think of $\infty$ and $-\infty$ as beyond our number system; beyond the scope of our knowledge of numbers. Remember that $\infty$ and $-\infty$ are simply IDEAS, not actual numbers. If there is a number line that goes on forever and ever (technically ALL number lines go on forever actually), then $\infty$ would... |
738,743 | <p>The following equation,
$$(\partial_x + i\partial_y)u - c(\partial_x+i\partial_y)au=0$$</p>
<p>($a=a(x,y)$ and $\partial_x=\frac{\partial}{\partial x}$)</p>
<p>with solution,
$$u=\exp(ca)f(x+iy)$$</p>
<p>where $f$ and $g$ are arbitrary entire functions, a is some scalar function and $c$ is a scalar.</p>
<p>How c... | Cameron Williams | 22,551 | <p>I would solve this via characteristics and a variant of integrating factors. (It took me a while to realize that your operator on the second term is the $\bar{\partial}$ derivative of $a$, not $au$.) Let's multiply our PDE by $\exp(-ca)$. Then our PDE becomes</p>
<p>$$0 = \exp(-ca)(\partial_x+i\partial_y)u-c\exp(-c... |
738,743 | <p>The following equation,
$$(\partial_x + i\partial_y)u - c(\partial_x+i\partial_y)au=0$$</p>
<p>($a=a(x,y)$ and $\partial_x=\frac{\partial}{\partial x}$)</p>
<p>with solution,
$$u=\exp(ca)f(x+iy)$$</p>
<p>where $f$ and $g$ are arbitrary entire functions, a is some scalar function and $c$ is a scalar.</p>
<p>How c... | Felix Marin | 85,343 | <p>$\newcommand{\+}{^{\dagger}}
\newcommand{\angles}[1]{\left\langle\, #1 \,\right\rangle}
\newcommand{\braces}[1]{\left\lbrace\, #1 \,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\, #1 \,\right\rbrack}
\newcommand{\ceil}[1]{\,\left\lceil\, #1 \,\right\rceil\,}
\newcommand{\dd}{{\rm d}}
\newcommand{\down}{\... |
1,866,931 | <p>I would like to see a proof to this fact.</p>
<blockquote>
<p>If $A$ is an invertible matrix and $B \in \mathcal{L}(\mathbb{R}^n,\mathbb{R}^n)$, that is an bounded linear opertor in $\mathbb{R}^n$. Then, if there holds
$$
\|B-A\| \|A^{-1}\| <1,
$$
we have that B is invertible.</p>
</blockquote>
<p>Moreov... | Henno Brandsma | 4,280 | <p>You should start with $(x,y) \in R$ and $(y,z) \in R$ and from these two statements show that $(x,z) \in R$. This shows transitivity.</p>
<p>Now, the two statements show that indeed $(x,z) \in R \circ R$ by definition (using $y$ as intermediate).</p>
<p>We have the assumption that $R \circ R \subseteq R$, so we kn... |
3,193,288 | <p>i have the following question. Let <span class="math-container">$\phi_1$</span> and <span class="math-container">$\phi_2$</span> fundamental system solutions on an interval <span class="math-container">$I$</span> for the second order equation
<span class="math-container">$$
y''+a(x)y= 0.
$$</span>
Prove that there ... | Robert Lewis | 67,071 | <p>Given the equation</p>
<p><span class="math-container">$y'' + a(x)y = 0, \tag 1$</span></p>
<p>with Wronskian</p>
<p><span class="math-container">$W[y_1, y_2] = y_1y_2' - y_1'y_2, \tag 2$</span></p>
<p>we have</p>
<p><span class="math-container">$W' = (y_1y_2' - y_1'y_2)' = y_1'y_2' + y_1y_2'' - y_1''y_2 - y_1'... |
1,268,598 | <p>I thought the <code>|</code> symbol meant "divides by", but in set theory it seems that it means "such that." However, I thought we wrote "such that" as <code>:</code>.</p>
<p>Can anybody elaborate?</p>
| Daniel W. Farlow | 191,378 | <p>Usually, in elementary number theory and the like, the symbol "$\mid$", which is typeset by <code>\mid</code>, means "divides." That is, $2\mid 4$ means "$2$ divides $4$," whereas something like $2\not\mid 3$ would mean "$2$ does not divide $3$." The difference between <code>:</code> and <code>|</code> in set theory... |
3,942,512 | <p>If X and Y are independent binomial random variables with identical parameters n and p, calculate the conditional expected value of X given X+Y = m.</p>
<p>The conditional pmf turned out to be a hypergeometric pmf, but I'm a but unclear on how to relate that back into finding E[X|X+Y=m]</p>
| Guillerme | 1,007,094 | <p>This can be easily answered if you know the characterization of global dimension by the Ext functor and the cohomology of the cyclic group. For these, see sections 4.1 and 6.2 of Weibel's book <em>An Introduction to Homological Algebra</em>.</p>
<p>Using this, we know that there are infinite indices <span class="mat... |
1,217,175 | <p><strong>Here's the question:</strong></p>
<p>Is the following true or false?</p>
<p>There is a function $f: \mathbb R \to \mathbb R$ that satisfies the following condition:</p>
<p>For every $a \in \mathbb R $ and $ \epsilon \gt 0 $ there is $\delta \gt 0$ such that $\left| f(x)-f(a) \right| \lt \epsilon \implies ... | layman | 131,740 | <p>The problem with the equation isn't as mysterious as you might think. When we take the square root of any number, if the square root is a real number, it is positive. For example, $\sqrt{4} = 2$. A lot of people think that $\sqrt{4} = \pm 2$ since "well, you are getting the number that, when squared, gives you 4"... |
794,875 | <p>Let $\{v_1, v_2,....,v_n\}$ be the standard basis for $\mathbb R^n$.Prove for any two $m\times n$ matrices that their linear transformations are equal if and only if the two matrices are equal. I know what two linear transformations need to be equal (same basis, domain and codomain), but how do I show that?</p>
| EPS | 133,563 | <p>You need to make two observations: (1) A linear transformation is completely determined by its values over basis elements. (2) The columns of the associated matrix to a linear transformation are precisely the values of the transformation at basis elements.</p>
|
794,875 | <p>Let $\{v_1, v_2,....,v_n\}$ be the standard basis for $\mathbb R^n$.Prove for any two $m\times n$ matrices that their linear transformations are equal if and only if the two matrices are equal. I know what two linear transformations need to be equal (same basis, domain and codomain), but how do I show that?</p>
| usermath | 147,693 | <p>Let $\mathbb B_1=\{v_1, v_2,....,v_n\}$ be the standard basis for $\mathbb R^n$.
As you are interested in $m\times n$ matrix consider $\mathbb B_2=\{u_1, u_2,....,u_m\}$ as the standard basis for $\mathbb R^m$.
Say $T$ and $S$ be two linear transformation from $\mathbb R^n$ to $\mathbb R^m$.</p>
<p>First assume $[T... |
898,543 | <p>I have the random vector $(X,Y)$ with density function $8x^{2}y$ for $0 < x < 1$, $0 < y < \sqrt{x}$ I am trying to find the marginal distributions of $X$ and $Y$. For $X$ this seems to be simply the integral $\int_{0}^{\sqrt{x}}8x^{2}y = 4x^{3}$, which is also the given solution, and follows the general... | Darth Geek | 163,930 | <p>Let $P(L)$ be the probability of a person is left-handed and $P(B)$ the probability that a person is blue-eyed.</p>
<p>Then we know:</p>
<p>$$\frac{1}{7} = P(L|B) = \frac{P(L\cap B)}{P(B)}$$</p>
<p>$$\frac{1}{3} = P(B|L) = \frac{P(L\cap B)}{P(L)}$$</p>
<p>$$\frac{4}{5} = 1 - P(L\cup B) = 1 + P(L \cap B) - P(L) -... |
373,578 | <p><a href="http://en.wikipedia.org/wiki/Quiver_%28mathematics%29" rel="noreferrer">Quivers</a> are directed graphs where loops and multi-arrows are allowed. And we can talk about representations of quivers by assigning each vertex a vector space and each arrow a homomorphism. Moreover, Gabriel gives <a href="http://en... | Aaron | 28,841 | <ol>
<li><p><strong>Morita Equivalence</strong> This is a supplement to the aspect of quiver representations mentioned in <a href="https://math.stackexchange.com/a/373644">Alistair's answer</a>. Every associative finite dimensional $k$-algebra $A$ is Morita equivalent to a path algebra $kQ/I$ (this is another Gabriel's... |
1,687,336 | <p>I've been searching through the internet and through SE to find something to help me understand generating functions, but I haven't found anything that would solve my problem with them.</p>
<p>I understand that </p>
<p>$$\frac1{1-x}=\sum_{n\ge 0}x^n\;,\tag{1}$$</p>
<p>gives the sequence $(1, 1, 1, 1,...) $ becaus... | GEdgar | 442 | <p>You are correct,
$$
\frac{4}{1-x^3} = \sum_{n=0}^\infty 4x^{3n} =
4 + 4x^3 + 4x^6 + 4x^9 + \dots
$$
But the formula you doubt is also right, since all other terms are zero.</p>
|
1,687,336 | <p>I've been searching through the internet and through SE to find something to help me understand generating functions, but I haven't found anything that would solve my problem with them.</p>
<p>I understand that </p>
<p>$$\frac1{1-x}=\sum_{n\ge 0}x^n\;,\tag{1}$$</p>
<p>gives the sequence $(1, 1, 1, 1,...) $ becaus... | J. Bush | 317,724 | <p>You are correct in your thinking, but are missing a key point. $$4x^{3 \times 0} + 4x^{3 \times 1} + 4x^{3 \times 2} + 4x^{3 \times 3}... = 4x^{3 \times 0} + 0x^1+0x^2+ 4x^{3 \times 1} +0x^4+0x^5+ 4x^{3 \times 2} + 0x^7+0x^8+4x^{3 \times 3}$$ This is because adding 0 to an equation does not change the sum.</p>
|
2,130,776 | <p>Let $(L,R)$ be a pair of adjoint functor. </p>
<p>How to show that the commutativity of the left diagram induces the commutativity of the right one?</p>
<p><a href="https://i.stack.imgur.com/lOffl.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/lOffl.jpg" alt="diagram"></a></p>
| Pece | 73,610 | <p>You probably know that $L : \mathfrak C \to \mathfrak D$ is left adjoint to $R : \mathfrak D \to \mathfrak C$ if an only if it exists a natural bijection
$$ \mathfrak C (A,R(X)) \stackrel {\alpha_{A,X}} {\overset\sim\to} \mathfrak D (L(A), X). $$</p>
<p>Now what does it mean that this $\alpha$ is natural? It is a t... |
3,394,378 | <p>I am stuck with this Precalculus problem about polynomial functions. The problem:</p>
<blockquote>
<p>Consider <span class="math-container">$f(x)=x^2+ax+b$</span> with <span class="math-container">$a^2-4b>0$</span>. Let <span class="math-container">$\alpha$</span> and <span class="math-container">$\beta$</span> b... | Calvin Lin | 54,563 | <p><span class="math-container">$ f(x) = ( x - \alpha) ( x - \beta) = (x - c + 1)( x - c - 1) = (x-c)^2 - 1$</span></p>
<p>Hence, the minimum value is -1. </p>
|
3,394,378 | <p>I am stuck with this Precalculus problem about polynomial functions. The problem:</p>
<blockquote>
<p>Consider <span class="math-container">$f(x)=x^2+ax+b$</span> with <span class="math-container">$a^2-4b>0$</span>. Let <span class="math-container">$\alpha$</span> and <span class="math-container">$\beta$</span> b... | Community | -1 | <p>In terms of the roots <span class="math-container">$\alpha$</span> and <span class="math-container">$\beta$</span> of a monic quadratic the minimum value is <span class="math-container">$-(\frac{\alpha -\beta}{2})^2$</span>. This is <span class="math-container">$-1$</span> for your quadratic.</p>
|
2,617,235 | <p>Given a triangle $\Delta$ABC, how to draw any inscribed equilateral triangle whose vertices lie on different sides of $\Delta$ABC?</p>
| Jean Marie | 305,862 | <p>(See figure and Matlab program below). We are going to show how to capture the issue in all its generality, and thus build a program giving as many inscribed equilateral triangles $UVW$ as desired, the key observation being an affine relationship between the evolution speeds of vertices $U$ and $V$ (and $W$). </p>
... |
1,807,683 | <p>In his "Classification of (n-1)-connected 2n-dimensional manifolds and the discovery of exotic spheres", Milnor observes that since his exotic 7-spheres admit a Morse function with only two critical points, they are diffeomorphic to two 7-disks glued along a diffeomorphism $g : S^6 \to S^6$, which can't be isotopic ... | Cheerful Parsnip | 2,941 | <p>Look under "Twisted Spheres" in the Wikipedia article: <a href="https://en.wikipedia.org/wiki/Exotic_sphere" rel="nofollow">https://en.wikipedia.org/wiki/Exotic_sphere</a>. For n>6, all diffeomorphisms not isotopic to the identity give an exotic sphere. Edit: As Mike Miller points out and the article as well, all ex... |
1,807,683 | <p>In his "Classification of (n-1)-connected 2n-dimensional manifolds and the discovery of exotic spheres", Milnor observes that since his exotic 7-spheres admit a Morse function with only two critical points, they are diffeomorphic to two 7-disks glued along a diffeomorphism $g : S^6 \to S^6$, which can't be isotopic ... | archipelago | 67,907 | <p>Grumpy Parsnip answered the question, but let me summarize the situation and add some other aspects of the story.</p>
<p>Denote by </p>
<ul>
<li>$Diff^\partial(D^n)$ the group of diffeomorphisms of the $n$-disc that are the identity on a neighborhood of the boundary, by</li>
<li>$Diff^+(S^n)$ the group of orientat... |
1,003,379 | <p>I've been working problems all day so maybe I'm just confusing myself but in order to do this. I have to the take the integral along each contour $C_1-C_4$. My issue is how to convert to parametric functions in order to this so that I can integrate</p>
<p><img src="https://i.stack.imgur.com/HWRoM.jpg" alt="enter im... | neptun | 192,385 | <p>I don't know if you are familiar with or allowed to use residues, but if you calculate the residue of the pole in zero (its value is $1$), note that it is the only pole and use the residue formula*, you get $2\pi i$.</p>
<p>* that for residues $a_k$ inside $\gamma$, we have $\int_\gamma f(z) \ \text{d}z = 2\pi i\su... |
1,003,379 | <p>I've been working problems all day so maybe I'm just confusing myself but in order to do this. I have to the take the integral along each contour $C_1-C_4$. My issue is how to convert to parametric functions in order to this so that I can integrate</p>
<p><img src="https://i.stack.imgur.com/HWRoM.jpg" alt="enter im... | Kevin Sheng | 150,297 | <p>I'm not sure if you are allowed to use the Cauchy Integral Formula, but consider $f(z) = 1$ which is analytic in and on the region bounded by your curve. Then you have $2\pi if(0) = \int_C\frac{1}{z}dz = 2\pi i$.</p>
|
345,589 | <p>I am interested in proving that the family of functions
$$\{f_{\omega}: \mathbb{C}^n\rightarrow\mathbb{C}, f_\omega(z) = \exp(i\langle \omega, z \rangle): \omega \in \mathbb{C}^n\},$$
where $\langle \cdot,\cdot\rangle$ is the usual hermitian dot product, is $\mathbb{C}$-linearly independent.</p>
<p>In the case $n=1... | Georges Elencwajg | 3,217 | <p>a) Changing your notation slightly, we must prove that if $L_i:\mathbb C^n\to \mathbb C $ are a finite set of mutually distinct linear forms, then the entire functions $\exp (L_i):\mathbb C^n\to \mathbb C$ are linearly independant. </p>
<p>b) If we could find $u\in \mathbb C^n$ such that the numbers $L_i(u)\in... |
690,621 | <p>Consider the Quotient ring $\mathbb{Z}[x]/(x^2+3,5)$. </p>
<p>Solution: I first tried to take care of $(5)$ in the above ring. Therefor we can consider $\mathbb{Z_5}[x]/(x^2+3)$. Now and interesting point to note here is $(5) \subset (x^2+3)$. So, we can consider $\mathbb{Z_5}[x]/(5)$. But this is just $\mathbb{Z... | Emily | 31,475 | <p>An equivalence class is the collection of all things such that $a \sim b$.</p>
<p>Imagine that $\sim$ means "lives in the same county as." Then, for two residents of Smith County, Alice and Bob, $\textrm{Alice} \sim \textrm{Bob}$. Eve, who lives in Franklin County, does not belong to the same equivalence class as ... |
690,621 | <p>Consider the Quotient ring $\mathbb{Z}[x]/(x^2+3,5)$. </p>
<p>Solution: I first tried to take care of $(5)$ in the above ring. Therefor we can consider $\mathbb{Z_5}[x]/(x^2+3)$. Now and interesting point to note here is $(5) \subset (x^2+3)$. So, we can consider $\mathbb{Z_5}[x]/(5)$. But this is just $\mathbb{Z... | David | 119,775 | <p>Here is an example which I find useful.</p>
<p>Let $S$ be the set of all locations on the earth's surface and define
$$X\sim Y\quad\hbox{iff}\quad
\hbox{clocks at $X$ and $Y$ show the same time}\ .$$
For an example from Australia (you can supply one from your own country), we have
$$\hbox{Sydney}\sim\hbox{Melbour... |
1,791,837 | <p>Recently I have been very fascinated by the claim and impact of Godel's incompleteness theorem. To understand the proof given by Godel, I felt the need to read an introductory book in logic to begin with. I have started reading the book titled "A Mathematical Introduction to Logic" by Herbert Enderton. As mentioned ... | Pedro | 23,350 | <p>You can imitate the argument that shows $\pi_1(S^n)=0$ if $n>1$ by removing a point! Try it!</p>
|
2,067,794 | <p>Let $A$ be a ring and $u,v \in A^\times$. When do we have that $u + v \in A^\times$?</p>
<p>I think that A is needed to be an integral domain. For example consider $\mathbb{Z/6}$. Both $1$ and $5$ is a unit but their sum $1+5=0$ is not a unit.</p>
| Pawel | 355,836 | <p>I do not think there is such a condition. For any unital ring such that $1 \neq 0,$ $1$ and $-1$ are units, but $1+(-1)=0,$ which is not a unit.</p>
|
1,968 | <p>We're evaluating the feasibility of <strong>sponsoring a member of the math community to speak at a conference in 2011</strong>.</p>
<p>Speaking is a relatively big "ask", so this needs to be planned many months in advance. Let's get started! </p>
<p>We'd like the community to establish <strong>where</strong> ...<... | Eric Naslund | 6,075 | <p>The AMS Calendar provides a large list of conferences: <a href="http://www.ams.org/meetings/calendar/mathcal" rel="nofollow">http://www.ams.org/meetings/calendar/mathcal</a>
I am sure others with more experience could point out the best choices.</p>
<p>Here in Canada, the largest conferences are the annual <a href... |
1,968 | <p>We're evaluating the feasibility of <strong>sponsoring a member of the math community to speak at a conference in 2011</strong>.</p>
<p>Speaking is a relatively big "ask", so this needs to be planned many months in advance. Let's get started! </p>
<p>We'd like the community to establish <strong>where</strong> ...<... | Willie Wong | 1,543 | <p>Just to throw in some ideas about <em>relevancy</em>:</p>
<p>One possible idea is, of course, to see if there are any mathematical results which grew out of a question/answer on Math.SE. I know such has happened for MathOverflow, so it shouldn't be too farfetched that something similar happens here. </p>
<p>Simila... |
1,241,864 | <p>I would like to know if there is formula to calculate sum of series of square roots $\sqrt{1} + \sqrt{2}+\dotsb+ \sqrt{n}$ like the one for the series $1 + 2 +\ldots+ n = \frac{n(n+1)}{2}$.</p>
<p>Thanks in advance.</p>
| Aditya Agarwal | 217,555 | <p><a href="http://ramanujan.sirinudi.org/Volumes/published/ram09.pdf" rel="noreferrer">http://ramanujan.sirinudi.org/Volumes/published/ram09.pdf</a>
<a href="http://arxiv.org/pdf/1204.0877.pdf" rel="noreferrer">http://arxiv.org/pdf/1204.0877.pdf</a>
<br> Refer to the docs.
Simple as that!</p>
|
1,241,864 | <p>I would like to know if there is formula to calculate sum of series of square roots $\sqrt{1} + \sqrt{2}+\dotsb+ \sqrt{n}$ like the one for the series $1 + 2 +\ldots+ n = \frac{n(n+1)}{2}$.</p>
<p>Thanks in advance.</p>
| Claude Leibovici | 82,404 | <p>The definition of harmonic numbers is $$H_p^{(-a)}=\sum_{i=1}^p i^a $$ When $a$ is not a positive integer, there is no closed form but, as Yves Daoust commented, there are quite nice expansions.</p>
<p>For example, if $n=\frac 12$ as in the post, you have $$H_p^{\left(-\frac{1}{2}\right)}=\frac{2 p^{3/2}}{3}+\frac{... |
1,954,470 | <p>The question defines $x \in \mathbb{R}$ where x>0 and a sequence of integers with $a_0: [x], a_1=[10^1(x-a_0)]$ until $a_n=[10^n(x-(a_0+10^{-1}a_1+ ... + 10^{-n}a_{n-1}))]$. I want to prove that $0 \leq a_n \leq 9$ for each $n \in \mathbb{N}$.</p>
<p>I am completely stumped at what to do. I feel like the Archimdea... | fleablood | 280,126 | <p>Your intuition is perfectly correct.</p>
<p>Let's start with a stating and review of the archimedean property, which I will prove at the end of this post..</p>
<p>In $\mathbb R$</p>
<p>a) for $x > 0$ and $y$ there exists an $n; n \in \mathbb Z$ so that $nx > y$. </p>
<p>a') for $x > 0$ and $y$ there ex... |
3,805,745 | <p>I am working my way through a linear algebra book and would appreciate some help verifying my proof.</p>
<p><strong>Prove that <span class="math-container">$|u \cdot v| = |u | |v |$</span> if and only if one vector is a scalar multiple of the
other.</strong></p>
<p><strong>PROOF:</strong></p>
<p>Let <span class="mat... | Siong Thye Goh | 306,553 | <p>It is not right.</p>
<p>I don't see any attempt to check what happens if <span class="math-container">$u$</span> is not a multiple of <span class="math-container">$v$</span>.</p>
<p>It would be helpful if you can use <span class="math-container">$u.v = \|u\|\|v\|\cos \theta$</span>.</p>
|
2,764,073 | <p>I recently was working on a question posted in an AP calculus BC multiple choice sheet which asked:</p>
<p>Let f(x) be a positive, continuous deceasing function. If $\int_1^∞ f(x)dx$ = 5, then which of the following statements must be true about the series $\sum_1^∞f(n)$?</p>
<p>(a) $\sum_1^∞f(n)$ = 0</p>
<p>(b) ... | user | 505,767 | <p>Note that since the function is decreasing</p>
<ul>
<li>$f(1)>\int_1^2 f(x)dx, \quad f(2)>\int_2^3 f(x)dx, ... \implies
\sum_1^\infty f(n)> \int_1^\infty f(x)dx=5$</li>
</ul>
<p>and the series converges since</p>
<ul>
<li>$f(2)<\int_1^2 f(x)dx, \quad f(3)<\int_2^3 f(x)dx, ... \implies
\sum... |
2,708,071 | <p>Question: Suppose $|x_n - x_k| \le n/k^2$ for all $n$ and $k$.Show that $\{x_n\}_{n=1}^{\infty}$ is cauchy. </p>
<p>Attempt : To prove this, I have to find $M \in N$ that for $\varepsilon >0$, $n/k^2 < \varepsilon$ for $n,k \ge M$. </p>
<p>Let $\varepsilon > 1/M$. </p>
<p>Then, $n/k^2 \le M/M^2$ (#) $= ... | Ovi | 64,460 | <p>To prove a sequence is Cauchy we have to study the behavior of terms of the form $|x_n-x_k|$, where $n, k$ are greater than some positive integer $N$.</p>
<p>Let's fix some $N$ to get a better feeling for what is happening. Say $N=10$, and we concerned ourselves with terms of the form $|x_n-x_k|$, where $n, k \ge 1... |
2,452,172 | <p>Let $(a_n)_{n \geq 1}$ be a decreasing sequence of positive reals. Let $s_n = a_1 + a_2 + ... + a_n$ and </p>
<p>\begin{align}
b_n = \frac{1}{a_{n+1}} - \frac{1}{a_n}, n \geq 1
\end{align}</p>
<p>Prove that if $(s_n)_{n \geq 1}$ is convergent, then $(b_n)_{n \geq 1}$ is unbounded.</p>
<p>My attempt: If $\lim_{n \... | Oliver Díaz | 121,671 | <p>This is an old problem but I think a more detailed solution is worth having.</p>
<p>Suppose (1) <span class="math-container">$0<a_n\searrow0$</span> and (2) <span class="math-container">$\sum_na_n<\infty$</span>, and
(3) <span class="math-container">$B:=\sup_n\Big(\frac{1}{a_{n+1}}-\frac{1}{a_n}\Big)<\infty... |
2,312,096 | <p>How do I compute the integration for $a^2<1$,
$$\int_0^{2\pi} \dfrac{\cos 2x}{1-2a\cos x+a^2}dx=? $$
I think that:
$$\cos2x =\dfrac{e^{i2x}+e^{-2ix}}{2},
\qquad\cos x =\dfrac{e^{ix}+e^{-ix}}{2}$$
But I cannot. Please help me.</p>
| Jack D'Aurizio | 44,121 | <p>We have $1-2a\cos x+a^2 = (1-a e^{ix})(1-a e^{-ix})$ hence
$$\frac{1}{1-2a\cos x+a^2} = \left(\sum_{m\geq 0}a^m e^{mix}\right)\cdot\left(\sum_{n\geq 0}a^n e^{-nix}\right)\tag{1}$$
and since for every $a,b\in\mathbb{Z}$ we have
$$\int_{0}^{2\pi}e^{aix}e^{bix}\,dx = 2\pi\cdot\delta(a+b) \tag{2} $$
it follows that
$$ ... |
2,701,128 | <p>I have read multiple definitions so far but something is not clicking.</p>
<p>My most naive understanding is that $|G:H|$ is a "number" (could be infinite) that represents how many times $H$ is in $G$.</p>
<p>But even this doesn't seem fully correct.</p>
<p>I would like a general non-formal explanation and perhap... | P Vanchinathan | 28,915 | <p>Your naive understanding is roughly correct, especially for the finite groups.</p>
<p>Here are two examples in the infinite case in the same setup (you should draw pictures while reading this).</p>
<p>The set of all nonzero complex numbers, is a group under multiplication of complex numbers, call it $G$. This can ... |
949,052 | <p>I am having a hard time understanding the definitions of dimensions. Dimension of a finite-dimensional vector space is defined as the length of any basis of the vector space. The definition seems to be easy to understand. But I need examples to have a better understanding of dimensions so if anyone can explain me mo... | David P | 49,975 | <p>From a geometry perspective, we can look at $\mathbb{R}, \mathbb{R}^2, \mathbb{R}^3$. A basis is any minimal set of vectors in which "directions" on how to arrive at any point in the space is possible.</p>
<p>$$\mathbb{R} = \text{span}\{1\}$$</p>
<p>A one-dimensional space is a line. Think of this as being able to... |
2,917,742 | <p>$\left\{ 1 + \left( \frac { d y } { d x } \right) ^ { 2 } \right\} ^ { \frac { 3 } { 2 } } = \frac { d ^ { 2 } y } { d x ^ { 2 } }$</p>
<p>what is the degree and order for above equation </p>
<p>well according to my knowledge the order be should $2$</p>
<p>and degree should be $\frac { 3 } { 2 }$</p>
<p>is my an... | Chris | 593,513 | <p>From what I can remember, the <em>order</em> of an ODE is the order of the highest-order derivative, and the <em>degree</em> is the power of the highest-order derivative.</p>
<p>So in your example, the <strong>order should be 2 and the degree should be 2/3</strong>; since if we raise the whole equation to the power... |
82,279 | <p>Consider a ten-digit sequence of positive integers 0 - 9.<br>
The 1st, 4th, and 5th digits are either 7 or 9<br>
3rd and tenth digits are either 2 or 4.<br>
Somewhere in the phone number are 2 zeros, and the sum of all the digits equate to 42.<br>
What are all possible such sequence of 10 digits??</p>
<p>Just from ... | David Mitra | 18,986 | <p>1) False.
$f/g$ would indeed be differentiable; but, not neccesarily increasing.
For example, take $$f(x)=e^x+1,{\rm\ and\ } g(x)=e^{2x}+1.$$ Then $\bigl(f/g\bigr)(0)=1$ but $\bigl(f/g\bigr)(\ln (2)) =3/5$ (or look at Arturo's nicer argument above).</p>
<p>2) True. If $f$ has period $P$ and is differentiable, write... |
242,773 | <p>Did I tackle this implicit differentiation correctly?</p>
<p>$$5x^2+3xy-y^2=5$$</p>
<p>$$10x+3x\dfrac{dy}{dx}+3y-2y\dfrac{dy}{dx}=0$$
$$\dfrac{dy}{dx}(y-2y)=-10x-3z-3y$$</p>
<p>$$\dfrac{dy}{dx}=\dfrac{-10x-3x-3y}{y-2y}$$</p>
| Mikasa | 8,581 | <p>Beside to @Andre's answer, if you are familiar to partial differentiation then you can use the following rule: $$y'=-\frac{F_x(x,y)}{F_y(x,y)}$$ where in we assume our relation can be written as $F(x,y)=0$. Note that $y$ should be a function of $x$ at this rule.</p>
|
242,773 | <p>Did I tackle this implicit differentiation correctly?</p>
<p>$$5x^2+3xy-y^2=5$$</p>
<p>$$10x+3x\dfrac{dy}{dx}+3y-2y\dfrac{dy}{dx}=0$$
$$\dfrac{dy}{dx}(y-2y)=-10x-3z-3y$$</p>
<p>$$\dfrac{dy}{dx}=\dfrac{-10x-3x-3y}{y-2y}$$</p>
| Fly by Night | 38,495 | <p>Unfortunately not. There seems to be a mistake in your second line. Let me take you through the steps:</p>
<p>$$5x^2+3xy-y^2 = 5 \, ,$$
$$10x + 3y + 3x\frac{dy}{dx} - 2y\frac{dy}{dx} = 0 \, ,$$
$$(3x-2y)\frac{dy}{dx} = -10x - 3y \, , $$
$$(2y-3x)\frac{dy}{dx} = 10x + 3y \, , $$
$$\frac{dy}{dx} = \frac{10x+3y}{2y-3x... |
561,921 | <p>So far I have,
$$
\lim_{x\to 1} \frac{\frac{x}{\sqrt{x^2+1}} - \frac{1}{\sqrt{1^2+1}}}{x-1}=\lim_{x\to 1} \frac{\frac{x}{\sqrt{x^2+1}} - \frac{1}{\sqrt{2}}}{x-1}
$$</p>
<p>I have no idea how to keep going with this, every way I try I get stuck and can't do anything with it. </p>
| aaa | 103,780 | <p>Hint: Use L'Hopital's Rule. Take the derivatives of both the numerator and denominator, then substitute the limit value.</p>
|
145,612 | <p>Why are isosceles triangles called that — or called anything? Why is their class given a name? Why did they find their way into the <em>Elements</em> and every single elementary geometry text and course ever since? Did no one ever ask himself, "What use is this, or why is it interesting?"?</p>
<p>Here are som... | Ronald | 27,884 | <p>Equal sides occur commonly, e.g. in radii of a circle. </p>
<p>Using these equal sides to identify equal angles is useful and leads to various theorems about circles. In this sense, isosceles triangles are perhaps more useful than equilateral triangles. I am not an expert, but I could recommend that you read The El... |
1,474,067 | <p>A silly example:</p>
<p>$\exists x (P (x, x)) \leftrightarrow \exists x\forall x (P (x, x))$</p>
<p>Intuition tells me that, because we're dealing with the same variable, the Exists on the right side is of no importance, so that side of the equation would be equivalent to $\forall x (P (x, x))$.</p>
<p>Now, conce... | Graham Kemp | 135,106 | <p>Although confusing, the nested quantifiers are well formed. The inner most quantifier binds all occurrences of the entity $x$ within its scope, leaving none free to be bound by the outermost quantifier.</p>
<p>For clarity, the tokens in the statement $\exists x \big(P (x, x)\big) \leftrightarrow \exists x\fo... |
4,147,856 | <p>If <span class="math-container">$(a_n)_{n\ge 0}$</span> is a sequence of complex numbers and <span class="math-container">$\lim_{n\to \infty} a_n=L\neq 0$</span> then
<span class="math-container">$$\lim_{x\to 1^-} (1-x)\sum_{n\ge 0} a_n x^n=L$$</span></p>
<p>EDIT: The initial question had a typo from the place I'd s... | Dosidis | 194,169 | <p>This isn't true. Take <span class="math-container">$a_n = L$</span>. Then you're claiming that <span class="math-container">$\lim_{x\to 1^-} \sum_{n\ge 0} L x^n=1-L$</span>.</p>
|
11,916 | <p>In <a href="https://mathoverflow.net/questions/11845/theory-mainly-concerned-with-lambda-calculus/11861#11861">Theory mainly concerned with lambda-calculus?</a>, F. G. Dorais wrote, of the idea that the lambda-calulus defines a domain of mathematics:</p>
<blockquote>
<p>That would never stick unless there's anoth... | B. Bischof | 348 | <p>There is some work in using monoidal category theory in functional programming in languages like Haskell. In addition to simply using the ideas and implementing them, they develop new frameworks for the theory, new applications of the theorems, and presentations of the theory. I say this is a branch of mathematics, ... |
2,727,320 | <p>I've seen a proof of the uniqueness of <span class="math-container">$q,r$</span> in <span class="math-container">$a=bq+r$</span> when <span class="math-container">$a,b$</span> are given. The proof is as follows:</p>
<blockquote>
<p>Suppose <span class="math-container">$a=bq+r=bq'+r'$</span> with the obvious assumpti... | Alex Jones | 350,433 | <p>A higher slope should correspond to a steeper line. Thus, if $\Delta Y$ increases while $\Delta X$ stays the same, the line gets steeper (visualizing this on the $XY$-plane), so we should expect the slope to increase as well. If the slope was $\Delta X/\Delta Y$, the slope would decrease, and thus a word like "flatn... |
2,727,320 | <p>I've seen a proof of the uniqueness of <span class="math-container">$q,r$</span> in <span class="math-container">$a=bq+r$</span> when <span class="math-container">$a,b$</span> are given. The proof is as follows:</p>
<blockquote>
<p>Suppose <span class="math-container">$a=bq+r=bq'+r'$</span> with the obvious assumpti... | user326210 | 326,210 | <p>When you're graphing a function like $y=f(x)$, the slope of the function $f$ means "when you change the input $x$ by an amount, how much does the output $y=f(x)$ change?"</p>
<p>To measure that, you take the ratio of the change in output $\Delta y$ and divide it by the change in the input $\Delta x$.</p>
<p>For ex... |
64,613 | <p>EDIT: I meant to have the coefficients reversed, showing:
$$\frac{n}{n-1}(1-(1-x)^n)^n + (1-x)^{n-1} \leq 1$$
This version should be true.. but still trying to prove it...</p>
<p>ORIGINAL:
Is it possible to show:
$$(1-(1-x)^n)^n + \frac{n}{n-1}(1-x)^{n-1} \leq 1$$ for $0<x<1$ and $n\geq 2$ (and $n$ is an ... | Aryabhata | 1,102 | <p>This might be false for every positive integer $n \ge 2$.</p>
<p>I believe we can show this for $\displaystyle x \gt 1 - \frac{1}{n-1}$ by using <a href="http://en.wikipedia.org/wiki/Bernoulli%27s_inequality" rel="nofollow">Bernoulli's inequality</a> on the first term on the left side.</p>
<p>Using Bernoulli's.</p... |
3,941,815 | <p><span class="math-container">$$\sqrt[4]{16} = ?$$</span></p>
<p><span class="math-container">$$16 = |16|(\cos(0)+i\sin(\pi))$$</span>
the equality above holds.</p>
<p><span class="math-container">$$\sqrt[4]{16} = |\sqrt[4]{16}|(\cos(0 + \frac{2k\pi}{4})+i\sin(\pi+\frac{2k\pi}{4})) $$</span>
using De Moivre"s fo... | user859786 | 859,786 | <p>Because starting at <span class="math-container">$\sin (\pi)$</span> as opposed to <span class="math-container">$\sin(0)$</span>, you’re basically multiplying the imaginary part with <span class="math-container">$-1$</span>.</p>
|
2,333,702 | <p>Firstly, I have opened the brackets and solved using both compositions of trig and inverse trig functions and using right triangle (the results were the same):$$\arcsin(\frac{40}{41})=\gamma$$$$\frac{40}{41}=\sin\gamma.$$ Coming from the Pythagoras theorem, the adjacent side to $\gamma^\circ$ is 9, so $\cos(\gamma)=... | reuns | 276,986 | <p>My thoughts : </p>
<p>Let $K= \mathbb{Q}(\sqrt{m},2^{1/3},\zeta_3)$,$[K:\mathbb{Q}] = 12$ the Galois closure of the two fields. </p>
<p>Find $\sigma_1 \in Gal(K/\mathbb{Q})$ such that $\sigma|_{\mathbb{Q}(2^{1/3},\zeta_3)}$ has order $3$ and $\sigma_1|_{\mathbb{Q}(\sqrt{m})}$ has order $1$. Also let $\rho \in Gal(... |
1,397,576 | <p>To me there is a hierarchy where vectors $\subset$ sequences $\subset$ functions $\subset$ operators</p>
<ul>
<li><p>All vectors are sequences, but not all sequences are vectors because
sequences are infinite dimensional</p></li>
<li><p>All sequences are functions, but not all functions are sequences
because functi... | John Gowers | 26,267 | <ul>
<li><p><em>All vectors are sequences</em> <strong>False</strong> To a mathematician, a vector is an element of a vector space, and therefore need not be a sequence. For example, the set of all continuous real valued functions on <span class="math-container">$[0,1]$</span> forms a vector space, and its elements ar... |
1,601,575 | <p>$1/\sin50^\circ + √3/\cos50^\circ=4$</p>
<p>I have tried it as:
LHS
$(\cos50+√3 \sin50)/\sin50\cos50$</p>
<p>$(2\cos50+2√3 \sin50)/2\sin50\cos50$</p>
<p>$(2\cos50+2√3 \sin50)/(\sin100)$</p>
<p>Now, whats next??</p>
| Deepak | 151,732 | <p>Hints:</p>
<p>$a\sin \theta + b \cos \theta = \sqrt {a^2 + b^2}\sin(\theta + \arctan {\frac ba})$</p>
<p>$\sin \theta = \sin (180^{\circ} - \theta)$</p>
|
1,601,575 | <p>$1/\sin50^\circ + √3/\cos50^\circ=4$</p>
<p>I have tried it as:
LHS
$(\cos50+√3 \sin50)/\sin50\cos50$</p>
<p>$(2\cos50+2√3 \sin50)/2\sin50\cos50$</p>
<p>$(2\cos50+2√3 \sin50)/(\sin100)$</p>
<p>Now, whats next??</p>
| Michael Hardy | 11,667 | <p>\begin{align}
& \frac 1 {\sin50^\circ} + \frac{\sqrt 3}{\cos50^\circ} = 2 \left( \frac {1/2} {\sin50^\circ} + \frac{\sqrt 3/2}{\cos50^\circ} \right) = 2\left( \frac{\sin30^\circ}{\sin50^\circ} + \frac{\cos 30^\circ}{\cos50^\circ} \right) \\[10pt]
= {} & 2\cdot \frac{\sin30^\circ\cos50^\circ + \cos30^\circ\s... |
1,601,575 | <p>$1/\sin50^\circ + √3/\cos50^\circ=4$</p>
<p>I have tried it as:
LHS
$(\cos50+√3 \sin50)/\sin50\cos50$</p>
<p>$(2\cos50+2√3 \sin50)/2\sin50\cos50$</p>
<p>$(2\cos50+2√3 \sin50)/(\sin100)$</p>
<p>Now, whats next??</p>
| lab bhattacharjee | 33,337 | <p>$$\dfrac{\sin3A}{\sin x}+\dfrac{\cos3A}{\cos x}=2\cdot\dfrac{\sin(3A+x)}{\sin2x}$$</p>
<p>If $\sin2x=\sin(3A+x),2x=n180^\circ+(-1)^n3(A+x)$ where $n$ is any integer</p>
<p>If $n=2m$(even), $x=m360^\circ+3A$</p>
<p>If $n=2m+1$(odd) $x=\dfrac{(2m+1)180^\circ-3A}3=(2m+1)60^\circ-A$</p>
<p>Here $A=10^\circ\implies ... |
255,483 | <p>How to transform this infinite sum</p>
<p>$$1+\sum_{i\geq1}\frac{x^i}{(1-x)(1-x^2)\cdots(1-x^i)}$$</p>
<p>to an infinite product</p>
<p>$$\prod_{i\geq1}\frac{1}{1-x^i}$$</p>
| anon | 11,763 | <p>Combinatorical arguments are a nice way to manipulate generating functions (and vice-versa); we can make convergence a non-issue by working over formal polynomial series rings if need be.</p>
<p>The $q$-Pochhammer symbol is defined by $(a,q)_n:=(1-a)(1-aq)\cdots(1-aq^{n-1})$. The inverse of $(q,q)_n$ is a generatin... |
920,050 | <p>The answer is $\frac1{500}$ but I don't understand why that is so. </p>
<p>I am given the fact that the summation of $x^{n}$ from $n=0$ to infinity is $\frac1{1-x}$. So if that's the case then I have that $x=\frac15$ and plugging in the values I have $\frac1{1-(\frac15)}= \frac54$.</p>
| Pauly B | 166,413 | <p>The formula gives from $n=0$ to infinity, but you are asked to sum from $n=4$ to infinity. In this case, you take the terms from $n=0$ to infinity using the formula (which you determined is $\frac54$), and get rid of the extra terms. In this case, we don't need the terms when $n=0,1,2,3$, so we can get rid of those ... |
2,871,105 | <p>I am trying to prove the following statement, but starting to doubt its correctness.</p>
<p>Suppose that $H$ is a Hausdorf topological space (I am formulating generally, though my specific case is $H=S'(R)$ - a space of tempered distributions). </p>
<p>Suppose I have a set of nested subsets $\Omega_i \subseteq H$ ... | jeanmfischer | 34,904 | <p><strong>The space of distributions is not even a sequential space (indeed not first countable) so what you are trying to prove has little hope of beeing true.</strong> </p>
<p>If $x \in \cap_i \bar{\Omega_i}$, then for each $i$, there is a sequence $(y_{i,n})_n$ in $\Omega_i$ that converges to $x$, since $x \in \ba... |
4,588,868 | <p>As the title suggests, this is a college entrance exam practice problem from Japan, it is as follows:</p>
<blockquote>
<p>Given a scalene triangle <span class="math-container">$\triangle ABC$</span>, prove that it is a right triangle if <span class="math-container">$\sin(A)\cos(A)=\sin(B)\cos(B)$</span></p>
</blockq... | Oscar Lanzi | 248,217 | <p>Your method is correct.</p>
<p>If you can use trigonometric identities, proceed as follows:</p>
<p><span class="math-container">$2\sin(A)\cos(A)=2\sin(B)\cos(B)$</span></p>
<p><span class="math-container">$\sin(2A)=\sin(2B)$</span></p>
<p>Thus <span class="math-container">$2A$</span> and <span class="math-container"... |
4,588,868 | <p>As the title suggests, this is a college entrance exam practice problem from Japan, it is as follows:</p>
<blockquote>
<p>Given a scalene triangle <span class="math-container">$\triangle ABC$</span>, prove that it is a right triangle if <span class="math-container">$\sin(A)\cos(A)=\sin(B)\cos(B)$</span></p>
</blockq... | Blue | 409 | <p>Here's an identity-free approach:</p>
<p><a href="https://i.stack.imgur.com/3NvA8.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/3NvA8.png" alt="enter image description here" /></a></p>
<p>Writing <span class="math-container">$D$</span> and <span class="math-container">$E$</span> for the feet of ... |
2,477,137 | <p>$\left(1+3+5...+(2n+1)\right ) + \left(3.5+5+6.5+...+(\frac{7+3n}{2})\right)=105$ </p>
<p>It is the equation that I did not understand how to find $n.$</p>
| Overflowian | 211,748 | <p>By the spectral theorem $A^2 = \int_{\mathbb{R}} \lambda^2 dE_\lambda$ you want to show that $A^2 = \int_{\mathbb{R}} \lambda \ dF_\lambda$.</p>
<p>Now, $\int_{\mathbb{R}} \lambda^2 \ dE_{\lambda} = \int_{\mathbb{R}_-} \lambda ^2\ dE_{\lambda} +\int_{\mathbb{R}_+} \lambda^2 \ dE_{\lambda} $
(notice that $0\ \... |
24,975 | <p><img src="https://i.stack.imgur.com/aoRn2.png" alt="1/2 of PI"></p>
<p>I was wondering - how would I specify the interval (the amount that <code>n</code> increases each time) between terms? Is that possible? What if I want it to increase by, say, 2, each time instead of one.</p>
<p>In Python it would be <code>n +=... | Marc Rasi | 7,782 | <p>You could do it two different ways. You could use a substack, like this</p>
<p>$$ \prod_{\substack{n=1 \\ 2 \mid n}}^\infty \frac{4n^2}{4n^2 -1} .$$</p>
<p>Or you could also let $n = 2m$ (or $n = 2m - 1$ if you want to start at $1$) and take the product as $m$ goes from $1$ to $\infty$, like this</p>
<p>$$ \prod_... |
24,975 | <p><img src="https://i.stack.imgur.com/aoRn2.png" alt="1/2 of PI"></p>
<p>I was wondering - how would I specify the interval (the amount that <code>n</code> increases each time) between terms? Is that possible? What if I want it to increase by, say, 2, each time instead of one.</p>
<p>In Python it would be <code>n +=... | J. J. | 3,776 | <p>There are some different ways mathematicians deal with this. For simple cases like this one, the most popular way is to write the product as</p>
<p>$$\prod_{n=1}^\infty \frac{4(2n)^2}{4(2n)^2 - 1}.$$</p>
<p>For a more general case, one can use the notation</p>
<p>$$\prod_{n \in S} \frac{4n^2}{4n^2 - 1},$$</p>
<p... |
4,261,763 | <p>I was working with this problem:</p>
<p><a href="https://i.stack.imgur.com/AgVpu.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/AgVpu.png" alt="enter image description here" /></a></p>
<p><span class="math-container">$f(x) = 1, x \in(-\infty,0)\cup(0,\infty)\\
f(x) = -1, x = 0 $</span></p>
<p>Th... | PierreCarre | 639,238 | <p>A simple example is</p>
<p><span class="math-container">$$f(x)=
\begin{cases}
-1, & x < 0\\
1, & x \ge 0
\end{cases},
$$</span></p>
<p>which is not continuous at <span class="math-container">$x=0$</span>, but <span class="math-container">$|f(x)|= 1, x \in \mathbb{R}$</span>. More generally, just take any ... |
2,141,406 | <p>I want to show that the function $f : \mathbb{R}\times \mathbb{R} \to \mathbb{R}$ defined by </p>
<p>$ f(x,y)=
\begin{cases}
\frac{xy}{x^{2}+y^{2}},& \text{if } (x,y)\neq (0,0)\\ 0, & \text{otherwise}
\end{cases}
$</p>
<p>is continuous on $\mathbb{R} \times \mathbb{R} \setminus{(0,0)}$.... | Thomas Rasberry | 265,575 | <p>The tangent function reports a number between $(-\infty, \infty)$ in accordance with the angle input from the domain. So the formula $\tan(\tan^{-1}(x))$ reports the tangent of the angle between $(\frac{-\pi}{2},\frac{\pi}{2})$ whose tangent is $x,$ and hence will always return $x.$ </p>
<p>On the other hand, $\tan... |
2,141,406 | <p>I want to show that the function $f : \mathbb{R}\times \mathbb{R} \to \mathbb{R}$ defined by </p>
<p>$ f(x,y)=
\begin{cases}
\frac{xy}{x^{2}+y^{2}},& \text{if } (x,y)\neq (0,0)\\ 0, & \text{otherwise}
\end{cases}
$</p>
<p>is continuous on $\mathbb{R} \times \mathbb{R} \setminus{(0,0)}$.... | Travis Willse | 155,629 | <p>This is an object lesson in why $\tan^{-1}$ is a misleading and hence poor notation!</p>
<p>We know that the restriction of the function $\tan$ to the interval $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ is bijective, so it is invertible. We denote this inverse, which we often called the <em>arctangent</em> by
$$\... |
75,862 | <blockquote>
<p>In quadilateral $ABCD$ (usual clockwise or anticlockwise naming), $AB=16\sqrt{2}$ cm, $CD=10$ cm, $DA=8.5$ cm, $\angle D = 120^\circ $ and $\angle ACB = 45^\circ$. How to find $\angle ABC$?</p>
</blockquote>
<p><a href="http://testfunda.com/examprep/learningresources/smsqod/cat-sms-question-of-the-... | Beni Bogosel | 7,327 | <p>My guess is:</p>
<ul>
<li><p>find $AC$ applying cosine law in triangle $ADC$;</p></li>
<li><p>apply sine theorem in triangle $ABC$: $$ \frac{AB}{\sin \angle ACB}=\frac{AC}{\sin\angle ABC}$$</p></li>
</ul>
<p>From the last relation you should be able to find $\sin B$ and then $B$. I think that your side lengths are... |
4,508,558 | <p>I'm trying to make sure that I have correctly proved Munkres' Lemma 2.1, which is left to the reader. The lemma states:</p>
<blockquote>
<p>Let <span class="math-container">$f: A \to B$</span>. If there are functions <span class="math-container">$g: B \to A$</span> and <span class="math-container">$h: B \to A$</span... | zwim | 399,263 | <p>You can solve it algebraically using the inverse transformations:</p>
<p><span class="math-container">$x=a^{-1}\circ c\circ b^{-1}=\begin{pmatrix}1&2&3&4\\3&1&2&4\end{pmatrix}\circ\begin{pmatrix}1&2&3&4\\1&3&4&2\end{pmatrix}\circ\begin{pmatrix}1&2&3&4\\2&am... |
375,549 | <p>I need to solve this recurrence equation with the help of Generating Functions in Combinatorics.</p>
<p>Given:
$$f(0) = 0 , f(1) = 1, f(n) = 10f(n-1) - 25f(n-2) \forall n \geq 2$$</p>
<p>So I said the following:</p>
<p>$$f(n) = \sum_{n=2}^{\infty} {10(n-1)x^n} - \sum_{n=2}^{\infty} {25(n-2)x^n}$$</p>
<p>Is that ... | vonbrand | 43,946 | <p>Use Wilf's techniques from "generatingfunctionology". Write:
$$
f(n + 2) = 10 f(n + 1) - 25 f(n) \qquad f(0) = 0, f(1) = 1
$$
Define:
$$
F(z) = \sum_{n \ge 0} f(n) z^n
$$
From the recurrence:
$$
\frac{F(z) - f(0) - f(1) z}{z^2} = 10 \frac{F(z) - f(0)}{z} - 25 F(z)
$$
Thus:
$$
F(z) = \frac{z}{(1 - 5 z)^2}
= - \... |
97,131 | <p>I have the following problem:</p>
<p>I have a convex hull $\Omega$ defined by a set of n-dimensional hyperplanes $S = [(n_1,d_1), (n_2,d_2),...,(n_k,d_k)]$ such that a point $p \in \Omega$ if $n_i^T p \geq d_i \quad \forall (n_i,d_i) \in S $. Now I have a "joining" hyperplane $(n_{k+1},d_{k+1})$ and I want to know ... | Igor Rivin | 11,142 | <p>See <a href="https://mathoverflow.net/questions/70068/conic-hulls-and-cones">this question</a>, in particular Ken Clarkson's answer.</p>
|
48,746 | <p>Let's assume that I have some particular signal on the finite time interval which is described by function <span class="math-container">$f(t)$</span>. It could be, for instance, a rectangular pulse with amplitude <span class="math-container">$a$</span> and period T; Gauss function with <span class="math-container">$... | m_goldberg | 3,066 | <p>You have a lot of points, so generating two interpolation functions from them, one for the bottom of the surface and the other for the top, should give a smooth surface of revolution. If the default plot isn't smooth enough, you can always increase <code>PlotPoints</code>.</p>
<pre><code>max = Max[First /@ points];... |
637,898 | <p>I have to find the number of irreducible factors of $x^{63}-1$ over $\mathbb F_2$ using the $2$-cyclotomic cosets modulo $63$.</p>
<p>Is there a way to see how many the cyclotomic cosets are and what is their cardinality which is faster than the direct computation?</p>
<p>Thank you.</p>
| Matt E | 221 | <p>Direct computation is pretty fast in this case.</p>
<p>The poly. $x^{63} -1 $ factors as cyclotomic polys $\Phi_d$ for $d \mid 63$, so
$d = 1, 3, 9, 7, 21, 63.$</p>
<p>The corresponding degrees of $\Phi_d$ are $\phi(d)$: $1, 2, 6, 6, 12, 36$.</p>
<p>To compute how $\Phi_d$ factors over $\mathbb F_2$, you have to... |
2,265,203 | <p>I was reading the paper</p>
<p><a href="https://aimsciences.org/journals/pdfs.jsp?paperID=1058&mode=full" rel="nofollow noreferrer">Dynamical models of tuberculosis and their applications</a> </p>
<p>by Castillo-Chavez, Song B. and it says </p>
<blockquote>
<p>" it is clear that the matrix
$$
D_xf=
\begin... | GDumphart | 124,970 | <p>"simple" is an uncommon term that refers to <a href="https://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors#Algebraic_multiplicity" rel="nofollow noreferrer">an eigenvalue of algebraic multiplicity $1$</a>. And "zero" means that the eigenvalue is $0$, of course. This implies that exactly one eigenvalue of the gi... |
304,209 | <p>I am trying to learn weak derivatives. In that, we call <span class="math-container">$\mathbb{C}^{\infty}_{c}$</span> functions as test functions and we use these functions in weak derivatives. I want to understand why these are called <em>test functions</em> and why the functions with these properties are needed. I... | Damien L | 59,825 | <p>Suppose you want to find the solution of a differential equation, $f'' = gf$ for example.</p>
<p>Take any solution $f$ of this equation, then if you take any function $\psi \in \rm C^{\infty}_c$ it is true to write $$f'' = gf \Longrightarrow \psi f'' = \psi gf \Longrightarrow \int \psi f'' = \int \psi g f \Longrig... |
801,464 | <p>Let $2^n-1$ be a prime number. If $1<i<n$, I need to prove that $2^n-1$ does not divide $1+2^{{2i}}$. Any comment would be appreciated. </p>
| David | 119,775 | <p>The following proves more than you want.</p>
<p><strong>Lemma</strong>. If $n>1$ is odd and $m=2^n-1$ and $i$ is a positive integer then $m\not\mid 1+2^{2i}$.</p>
<p><strong>Proof</strong>. If $m\mid 1+2^{2i}$ then modulo $m$ we have
$$2^{2i}\equiv-1\quad\Rightarrow\quad 2^{2in}\equiv(-1)^n\quad\Rightarrow\qu... |
1,425,519 | <p>I'm trying to solve <a href="http://poj.org/problem?id=2140" rel="nofollow">this problem</a> on POJ and I thought that I had it. Since I can't figure out what's wrong with my code, I'd like to test it against a huge list of correct answers. This will make my code much easier to debug.</p>
<p>If you don't want to go... | Jack D'Aurizio | 44,121 | <p>Assume that $N$ is the sum of $k$ consecutive integers:
$$ N = a+(a+1)+\ldots+(a+k-1).\tag{1}$$
It follows that:
$$ N = ka + \frac{k(k-1)}{2}\tag{2}$$
that is equivalent to:
$$ 2N=k(2a+k-1) \tag{3}$$
so there is a solution iff $k$ is a divisor of $2N$ and $\frac{2N}{k}$ has the opposite parity of $k$.</p>
<p>That ... |
2,356,813 | <p>Let $f:[0,\infty)\to\mathbb R$ be a function in $C^2$ such that
$\lim_{x\to\infty} (f(x)+f'(x)+f''(x)) = a.$
Prove that $\lim_{x\to\infty} f(x)=a$</p>
| Redsbefall | 97,835 | <p>Regards @merow . If $ \lim_{x \rightarrow \infty} f(x) $ exists, here is one argument : $ \lim_{x \rightarrow \infty}f(x) = \lim_{x \rightarrow \infty}f(x+h) = \lim_{x \rightarrow \infty}f(x-h) $ for any finite value $h$.</p>
<p>First we have
$$ \lim_{x \rightarrow \infty}f(x) + \lim_{x \rightarrow \infty} \left[... |
70,582 | <p>For which n can $a^{2}+(a+n)^{2}=c^{2}$ be solved, where $a,b,c,n$ are positive integers?
I have found solutions for $n=1,7,17,23,31,41,47,79,89$ and for multiples of $7,17,23$...
Are there infinitely many prime $n$ for which it is solvable? </p>
| Angela Pretorius | 15,624 | <p>$2a^{2}+2na+n^{2}=c^{2}$ --> $a=-\frac{-n+\sqrt{2c^{2}-n^{2}}}{2}$ --> there are solutions iff $x^{2}+n^{2}=2c^{2}$ has solutions --> find the set of the squares of all integers 0 in the set such that $y=2x-n$ then there is a primitive pythagorean triple with a difference of n between legs, and also for any multiple... |
2,158,636 | <p>I am trying to find the value of $$\sum_{k=1}^{\infty}{\frac{1}{(k+2)(k+3)}}$$</p>
<p>I do not believe it is geometric, it cannot be divided into two fractions that both converge, but it definitely does converge, and, according to WolframAlpha, to $\frac{1}{3}$. Any way I can easily show this with no more than basi... | A. Salguero-Alarcón | 405,514 | <p>Note that</p>
<p>$$\frac{1}{k+2}-\frac{1}{k+3}=\frac{(k+3)-(k+2)}{(k+2)(k+3)}=\frac{1}{(k+2)(k+3)}$$</p>
<p>so</p>
<p>$$\begin{align}\sum_{k=1}^n \frac{1}{(k+2)(k+3)}&=\vphantom{\cfrac1{\cfrac11}}\sum_{k=1}^n \left(\frac{1}{k+2}-\frac{1}{k+3}\right)\\&=\left(\frac{1}{3}-\color{#4488dd}{\frac{1}{4}}\right)... |
3,849,851 | <p>During my research work I found a non-linear differential equation <span class="math-container">$y'''+y^2y'=0$</span>. Now I am stuck here. Please help me solve this.</p>
| user577215664 | 475,762 | <p><span class="math-container">$$y'''+y^2y'=0$$</span>
Note that <span class="math-container">$y=C$</span> is a solution of the differential equation. Integrate to reduce the order:
<span class="math-container">$$y''+\dfrac 13y^3=C$$</span>
Multiply by <span class="math-container">$2y'$</span>:
<span class="math-conta... |
4,565,584 | <blockquote>
<p>Let <span class="math-container">$X = (-1,1)^{\Bbb N}$</span> have the product topology. Is the subset <span class="math-container">$(0,1)^{\Bbb N}$</span> open?</p>
</blockquote>
<p>To consider whether <span class="math-container">$(0,1)^{\Bbb N}$</span> is open I know that it is if I can find a basic ... | Arthur | 15,500 | <p>Your space <span class="math-container">$X$</span> is the product of infinitely many copies of <span class="math-container">$(-1, 1)$</span>, so for the basic open in <span class="math-container">$X$</span>, <em>that</em> is the interval that almost all of the <span class="math-container">$V_n$</span> must be equal.... |
11,175 | <p>This question is a follow-up from <a href="https://mathematica.stackexchange.com/questions/11171/functional-programming-and-do-loops">here</a>. I have a function that generates a list of correlations between some random variables:</p>
<pre><code>varparams = Table[i, {i, 0.1, 0.9, 0.1}];
var[i_] :=
RandomChoice[{... | Mark McClure | 36 | <p>I guess you could use <code>Check</code> to check for the error message and return some symbol or string indicating the failure</p>
<pre><code>corrcheck[i_, j_, n_] := Quiet[Check[
Table[BlockRandom[SeedRandom[k];
Correlation[var[i], var[j]]], {k, n}],
"bad",Correlation::zerosd]];
</code></pre>
<p>Now</p>
... |
2,099,555 | <p>let $x_0$ = 2, $y_0 = 1$; $x_{n+1} = \frac{x_n + y_n}{2}$, $y_{n+1} = \frac{2}{x_{n+1}}, n = 0, 1, 2...$.</p>
<p>Need to show the sequences converge to $\sqrt 2$. I wish I could add more in terms of my attempted solution but I have really gotten anywhere; I haven't worked on problems like this in a while, and afte... | S.C.B. | 310,930 | <p>Note that you are attempting to calculate the limit of $x_{n}$ where $$x_{n}=\frac{1}{2} \left(x_{n-1}+\frac{2}{x_{n-1}} \right)$$</p>
<p>This is known as the <a href="https://en.wikipedia.org/wiki/Methods_of_computing_square_roots#Babylonian_method" rel="nofollow noreferrer">Babylonian Method</a>, and you can prov... |
2,099,555 | <p>let $x_0$ = 2, $y_0 = 1$; $x_{n+1} = \frac{x_n + y_n}{2}$, $y_{n+1} = \frac{2}{x_{n+1}}, n = 0, 1, 2...$.</p>
<p>Need to show the sequences converge to $\sqrt 2$. I wish I could add more in terms of my attempted solution but I have really gotten anywhere; I haven't worked on problems like this in a while, and afte... | Simply Beautiful Art | 272,831 | <p>We have directly the following:</p>
<p>$$x_{n+1}=\frac{x_n+\frac2{x_n}}2$$</p>
<p>So $y_n$ is really irrelevant. Now, if $L=\lim\limits_{n\to\infty}x_n$, then</p>
<p>$$L=\frac{L+\frac2L}2\implies L\stackrel?=\pm\sqrt2$$</p>
<p>Applying AM-GM inequality, we see that</p>
<p>$$x_{n+1}>\sqrt{x_n\frac2{x_n}}=\sq... |
386,799 | <blockquote>
<p>P1086: For a closed surface, the positive orientation is the one for which the normal vectors point outward from the surface, and inward-pointing normals give the negative orientation.</p>
<p>P1087: If <span class="math-container">$S$</span> is a smooth orientable surface given in parametric form by a v... | Eric Towers | 123,905 | <p>Answer one: If you make the wrong choice, you get minus the correct answer. If you make the correct choice, you get the correct answer. It is trivial to correct <em>a posteriori</em> and computationally intractable to correct <em>a priori</em>. As an example: what is the correct choice on the surface of the Kle... |
2,519,623 | <p>How do I calculate the side B of the triangle if I know the following:</p>
<p>Side $A = 15 \rm {cm}
;\beta = 12^{\circ}
;\gamma= 90^{\circ}
;\alpha = 78^{\circ}
$</p>
<p>Thank you.</p>
| drhab | 75,923 | <p>For $i=1,2,3$ let $E_i$ denote the event that the number of the $i$-th drawn ball is smaller than $17$. </p>
<p>Then you are looking for: $$1-P(E_1\cap E_2\cap E_3)=1-P(E_1)P(E_2\mid E_1)P(E_3\mid E_1\cap E_2)=1-\frac{16}{20}\frac{15}{19}\frac{14}{18}$$</p>
|
1,437,979 | <p>The given equation is $\dfrac{d^2y}{dx^2}+y=f(x)$.
I know that the C.F. is $~a\sin{x}+b\cos{x}$ but i stuck on P.I.. For non-homogeneous eautions, the theorem stating the methods to find P.I.s are not helpful for me in this case. The answer given is $~y(x)=a\sin{x}+b\cos{x}+\int_{0}^{x}f(t)\sin{(x-t)}dt$. How can ... | MrYouMath | 262,304 | <p>Let us condider $y_h$ as the homogenous solution to your ODE. In order to find the particular solution you will have to plug in </p>
<p>$$y=c(x)y_h=c(x)[a\sin(x)+b\cos(x)]$$ into your ODE. Then solve for $c(x)$ and resubstitute into $y$ to get your general solution.</p>
|
2,549,891 | <p>Suppose that $n$ different letters are sent to $n$ different addresses on the same street, one to each address. A drunk mailman randomly delivers the letters to the $n$ addresses on the street, one to each address. What is the expected number of letters that were received at correct addresses? Find the probability t... | Hw Chu | 507,264 | <p>The expected value is 1 for any $n$. We use induction on $n$ to prove this.</p>
<p>The base cases are straightforward. Now suppose the assertion is true for $n \leq k$, and we need to see the case $n = k+1$.</p>
<p>Case 1: The first letter is correctly sent. The chance is $\frac{1}{k+1}$. For the rest $k$ letters,... |
3,521,382 | <p>Let <span class="math-container">$X$</span> be a metric space with inner product space. Suppose that there is a sequence, <span class="math-container">$ \lbrace x_{n} \rbrace $</span>, in <span class="math-container">$X$</span> such <span class="math-container">$\lim_{n \to \infty}\|x_{n}\|=\|x\|$</span> and such <... | Masacroso | 173,262 | <p>HINT: remember that</p>
<p><span class="math-container">$$
\|a-b\|^2=\|a\|^2+\|b\|^2-\langle a,b \rangle-\langle b,a \rangle
$$</span></p>
|
2,392,114 | <p>It is possible to rewrite the equation $x^3+ax^2+bx+c=0$ as $y^3+3hy+k=0$ by setting $y=x+a/3$</p>
<p>How do you find the coefficient h in the equation $y^3+3hy+k=0$?</p>
| Donald Splutterwit | 404,247 | <p>\begin{eqnarray*}
x^3+ax^2+bx+c=\left( \color{red}{x+\frac{a}{3}} \right)^3+\left( \color{blue}{b-\frac{a^2}{3}} \right) \left( \color{red}{x+\frac{a}{3}} \right)+\left( \color{orange}{c-\frac{ab}{3}+\frac{2a^3}{27}} \right)=0.
\end{eqnarray*}</p>
|
266,124 | <p>A palindrome is a number or word that is the same when read forward
and backward, for example, “176671” and “civic.” Can the number obtained by writing
the numbers from 1 to n in order (n > 1) be a palindrome?</p>
| VCZ | 49,092 | <p>The answer is no - there's a variety of ways to prove this. For example, consider the number $k$ in the list with the most 0's, say $m$ of them. Clearly, $k$ must consist of a single digit followed by $m$ 0's, otherwise there would be a number before it with more 0's. Now we have two cases:</p>
<ul>
<li>Case 1: $k$... |
4,527,880 | <p>Suppose</p>
<p><span class="math-container">$$ R = \begin{bmatrix} A & B\\ C & D\end{bmatrix} $$</span></p>
<p>is a <span class="math-container">$2 \times 2$</span> block matrix of real numbers, where <span class="math-container">$A$</span> and <span class="math-container">$D$</span> are squared diagonal mat... | peek-a-boo | 568,204 | <p>Any linear map between finite-dimensional normed vector spaces is bounded. In the case of <span class="math-container">$L:\Bbb{R}^n\to\Bbb{R}^m$</span>, this follows easily from the Heine-Borel theorem (which implies compactness of the unit spheres). The general case follows by equivalence of norms to reduce it to t... |
3,475,893 | <p>Is <span class="math-container">$ \mathbb{Q} \times \mathbb{Q[i]}$</span> an integral domain ?</p>
<p>My attempt : I know that <span class="math-container">$ \mathbb{Q} \times \mathbb{Q}$</span> is not integral domain take <span class="math-container">$(0,1) \times (1,0) =( 0,0)$</span></p>
<p>But im confuse... | badjohn | 332,763 | <p>The same example works. Here's another <span class="math-container">$(0, i) \times (1, 0) = (0, 0)$</span>.</p>
|
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