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 |
|---|---|---|---|---|
1,314,992 | <p>What operations do I need to perform the following conversion?</p>
<p>$$
\frac{\partial ^2y}{\partial x\partial z} \mapsto \frac{\partial ^2y}{\partial x\partial t}
$$</p>
| ZanCoul | 109,553 | <p>It should be chain rule. From the way you wrote your question, $t$ depends on $z$. So then you should have
\begin{align*}
\frac{\partial^2 y }{\partial x \partial z} = \frac{\partial^2 y}{\partial x\partial t}\frac{\partial t}{\partial z}
\end{align*}</p>
|
2,962,377 | <p>Rewrite <span class="math-container">$f(x,y) = 1-x^2y^2$</span> as a product <span class="math-container">$g(x) \cdot h(y)$</span> (both arbitrary functions)</p>
<p>To make more clear what I'm talking about I will give a example.</p>
<p>Rewrite <span class="math-container">$f(x,y) = 1+x-y-xy$</span> as <span class... | Community | -1 | <p><span class="math-container">$$f(x,0)=1=h(0)\cdot h(y)$$</span>
and
<span class="math-container">$$f(0,y)=1=g(x)\cdot h(0)$$</span></p>
<p>so that both <span class="math-container">$g(x)$</span> and <span class="math-container">$h(y)$</span> are constant !?</p>
|
2,747,074 | <p>I've seen multiple ways on how to solve it online (and most likely the majority are wrong), but I don't know how to solve this fully so that I could get the full grade during my exam.</p>
<p>The question is as follows:</p>
<blockquote>
<p>Prove that if $\sum a_n$ is convergent with $a_n > 0$ for all $n$,
th... | José Carlos Santos | 446,262 | <p>Since <span class="math-container">$\lim_{n\to\infty}a_n=0$</span>, you have <span class="math-container">$a_n\leqslant1$</span> if <span class="math-container">$n$</span> is large enough. But then <span class="math-container">${a_n}^2\leqslant a_n$</span> if <span class="math-container">$n$</span> is large enough a... |
1,419,756 | <p>Let $A$ be a nonempty set in the metric space $(X,d)$ and, for $\epsilon>0$, define</p>
<p>$$A_\epsilon = \{x\in X: d(x,A) < \epsilon\}$$.</p>
<p>Then I want to prove that $A_\epsilon$ is open in $X$.</p>
<p>So what I have tried so far is that, well I want to prove that a set is open so I take $x \in A_\eps... | Arpit Kansal | 175,006 | <p><strong>Another Approach:</strong> First note that If $A$ is a non empty subset of a metric space $(X,d)$ then the function $f: X \to \mathbb R$ given by $f(x)=d(x,A)$ is uniformly continuous.</p>
<p>Because $$| f(x) - f(y) | = | d(x,A) - d(y,A) | \leq d(x,y)$$.</p>
<p>This means that $f$ is uniformly continuous (... |
110,709 | <p>By a closed geodesic, I mean a smooth periodic geodesic $\mathbb{R} \rightarrow (M,g)$. I will consider them up to geometric distinction.
This means that any two closed geodesics are equivalent if they have the same image in $(M,g)$. </p>
<p>Manifolds with constant curvature $\leq 0$,
by Cartan's theorem,
cannot ha... | Ian Agol | 1,345 | <p>I think if you take the metric on $\mathbb{R}^2$ obtained by rotating a curve which is $\sqrt{1-x^2}$ for $-1\leq x\leq 0$, and $x^2+1$ for $x\geq 0$ around the $x$-axis, then I think there will be a single closed contractible geodesic obtained by rotating the point $(0,1)$ around the $x$-axis.</p>
|
2,781,265 | <p>I am learning category theory as a hobby. In the book I am studying from, I was looking at an example of a functor. I want to understand what this functor looks like.</p>
<p>If $G$ is a group, a functor $F: G \to \mathbf{Set}$ picks out a set $A = F(\star)$, together with a homomorphism from $G$ to the group of per... | Community | -1 | <ol>
<li>Yes. It is common to use $*$ as the name of the object in a one-object category; more generally, it is often used as the name of the unique object contained in a singleton set. ($\star$ is less common, but is surely what's meant)</li>
<li>A functor must additionally preserve products of morphisms, as well as s... |
552,384 | <blockquote>
<p>Prove that</p>
<p>$$
\int_{0}^{\infty}\frac{e^{-bx}-e^{-ax}}{x}\,dx = \ln\left(\frac{a}{b}\right)
$$</p>
</blockquote>
<p><strong>My Attempt:</strong></p>
<p>Define the function $I(a,b)$ as</p>
<p>$$
I(a,b) = \int_{0}^{\infty}\frac{e^{-bx}-e^{-ax}}{x}\,dx
$$</p>
<p>Differentiate both side wit... | Mhenni Benghorbal | 35,472 | <p>Note that the following is a <a href="https://math.stackexchange.com/questions/294383/evaluate-int-0-infty-left-fracx-textex-texte-x/295326#295326">general technique that can handle much harder problems</a>. Recalling the Laplace transform </p>
<blockquote>
<p>$$ F(s) = \int_{0}^{\infty} f(x) e^{-sx} dx. $$</p>
<... |
1,013,776 | <p>I am looking at the following <strong>Theorem</strong>:</p>
<p>Let $\phi$ a type. We suppose that there is a set $Y$, such that $\forall x(\phi(x) \to x \in Y)$. Then, there is the set $\{ x: \phi(x) \}$.</p>
<p>and I try to understand its proof.</p>
<p>From the axiom shema of specification, there is the set $V=\... | Cameron Buie | 28,900 | <p>The idea, here, is to show that $V$ contains and is contained in the alleged set. One inclusion is clear, since $V$ is the intersection of the alleged set with $Y.$ For the reverse inclusion, use what you know about the type $\phi$ and the set $Y.$</p>
<p><strong>Added</strong>: On the one hand, suppose that $x\in ... |
3,764,342 | <p>Here since <span class="math-container">$\lim \frac{a_n}{a_{n+1}}=1.$</span> So no definite conclusion can be made about the nature of the sequence <span class="math-container">$\langle a_n\rangle$</span>.</p>
<p>So how can I can proceed to find the value of <span class="math-container">$a_1$</span> from the relatio... | Mushu Nrek | 743,552 | <p>Another way to get to series is through integration. The most known example is probably
<span class="math-container">$$
\sum_{n\geq 1} \dfrac{1}{n^2} = \int_0^1\int_0^1 \dfrac{1}{1 - xy}\,dx\,dy
$$</span>
which often serves as exercise for multiple integration and change of variables. Another example has recently be... |
2,368,771 | <blockquote>
<p>How to prove the function $$ f(z)=\exp\Big(\frac{z}{1-\cos z}\Big)$$ has an essential singularity at $z=0$ ?</p>
</blockquote>
<p>It's actually hard to express the Laurent series of $f(z)$ around $0$, because the power $\frac{z}{1-\cos z}$ itself is already in the series form (since $\cos z$ appears ... | Adayah | 149,178 | <p>It's sufficient to prove that the limit $$a = \lim_{z \to 0} \exp \left( \frac{z}{1-\cos z} \right)$$ does not exist, for we have the following trichotomy:</p>
<ul>
<li>if $z$ is a removable singularity, the limit exists and $a \in \mathbb{C},$</li>
<li>if $z$ is a pole, then $a = \infty$ (the complex infinity),</l... |
17,772 | <p>I have an algorithm that segments depth images using surface fitting. At the moment the algothim uses least squares polymonial fitting, but polynomials are not powerful enough to fit the shapes that are in these images. I replaced the explicit polynomial $z = f(x,y)$, with the implicit fitting problem $f(x,y,z) = 0$... | Tategami | 4,537 | <p>I finally have an example ready...</p>
<p>So we have an image of a sphere, and we take data points on it.</p>
<p><img src="https://imgur.com/4mKD9.png" alt="alt text"></p>
<p>Since this is a sphere it is reasonable to assume that we can fit a sphere to this part of the image. Here are the reults:</p>
<p><img src... |
255,230 | <p>A <em>complete linear hypergraph</em> is a <a href="https://en.wikipedia.org/wiki/Hypergraph" rel="nofollow noreferrer">hypergraph</a> $H=(V,E)$ such that </p>
<ol>
<li>$|e|\geq 2$ for all $e\in E$,</li>
<li>$|e_1\cap e_2|=1$ for all $e_1, e_2\in E$ with $e_1\neq e_2$, and</li>
<li>for all $v\in V$ we have $|\{e\in... | Pat Devlin | 22,512 | <p>Let $L$ be the number of edges in the configuration and $n$ the number of points. Then $n \leq {L \choose 2}$ and this bound is tight.</p>
<p><strong>There are configurations with that many points:</strong> Let the points correspond to the two element subsets of $\{1, 2, \ldots, k\}$. And let the edges correspon... |
1,740,032 | <p>In layman's terms, why would anyone ever want to change basis? Do eigenvalues have to do with changing basis? </p>
| Plutoro | 108,709 | <p>Changing basis allows you to convert a matrix from a complicated form to a simple form. It is often possible to represent a matrix in a basis where the only nonzero elements are on the diagonal, which is exceptionally simple. These diagonal elements will be the eigenvalues of the matrix. This is especially helpful i... |
1,740,032 | <p>In layman's terms, why would anyone ever want to change basis? Do eigenvalues have to do with changing basis? </p>
| djechlin | 79,767 | <p>It's used to prove the <a href="https://en.wikipedia.org/wiki/Master_theorem" rel="noreferrer">Master Theorem</a>. Diagonal matrices are very easy to take powers of ($diag(a_1,\ldots,a_n)^k = diag(a_1^k, \ldots, a_n^k)$) and we have $(PAP^{-1})^k = PA^kP^{-1}$. The Master Theorem relies on changing to a diagonal bas... |
1,740,032 | <p>In layman's terms, why would anyone ever want to change basis? Do eigenvalues have to do with changing basis? </p>
| Teodor | 333,240 | <h1>Structural dynamics with the Finite Element Method</h1>
<p>Computing the dynamic response of a structure is done with modal analysis.</p>
<p>The original vector space is displacements and rotations in each node. These are unsuited to dynamic response. The system of equations is therefore transformed into a new ve... |
24,660 | <p>Pro's and cons of number line model vs color counter model</p>
<p>When teaching multiplication to elementary schoolers, the "number line model" and "color counter model" are both widely used techniques. Can somebody help me to understand some of the pro's and cons for either model? Thank you</p>
| Steven Gubkin | 117 | <p>The color counter model is as follows: model an integer as a collection of positive and negative particles. I will use an ordered pair <span class="math-container">$(P,N)$</span> to denote the number of positive and negative chips present.</p>
<p>The interesting thing here is that different pairs can represent the ... |
365,166 | <p>I have trouble coming up with combinatorial proofs. How would you justify this equality?</p>
<p><span class="math-container">$$
n\binom {n-1}{k-1} = k \binom nk
$$</span>
where <span class="math-container">$n$</span> is a positive integer and <span class="math-container">$k$</span> is an integer.</p>
| André Nicolas | 6,312 | <p>We have a group of $n$ people, and want to count the number of ways to choose a committee of $k$ people with <strong>Chair</strong>. </p>
<p>For the left-hand side, we select the Chair first, and then $k-1$ from the remaining $n-1$ to join her.</p>
<p>For the right-hand side, we choose $k$ people, and select one o... |
365,166 | <p>I have trouble coming up with combinatorial proofs. How would you justify this equality?</p>
<p><span class="math-container">$$
n\binom {n-1}{k-1} = k \binom nk
$$</span>
where <span class="math-container">$n$</span> is a positive integer and <span class="math-container">$k$</span> is an integer.</p>
| Steven John | 73,144 | <p>Since k is on the right-hand side of the equation
equation
A mathematical statement that says two expressions have the same value; any number sentence with an = . , switch the sides so it is on the left-hand side of the equation
equation
A mathematical statement that says two expressions have the same value; any num... |
3,611,845 | <p>For the Stable Matching algorithm by Gale-Shapley, how do I prove that at most one man will get his worst choice? </p>
<p>My intuition is that I have to use contradiction. Assume that there are two men who will get their worst preferences: <span class="math-container">$M_1$</span> with <span class="math-container">... | Community | -1 | <p>Suppose all men have the same preferences and are indifferent over all women. In any stable match, any man who matches gets his worst choice.</p>
<p>Suppose all the men have strict preferences. Suppose there are <span class="math-container">$K$</span> men and <span class="math-container">$K+1$</span> women. All me... |
2,100,726 | <p><strong>Problem:</strong> In a binomial experiment with 45 trials, the probability of more than 25 successes can be approximated by $P(z > \frac{(25-27)}{3.29}$)</p>
<p>What is the probability of success of a single trial of this experiment? </p>
<p><strong>Choices:</strong> </p>
<ul>
<li>0.07</li>
<li>0.56</l... | Cettt | 397,540 | <p>Note that
$$
\lim_{x \to \infty} e^{\frac 1x}= e^0=1,
$$
and that $(0, \infty) \ni x \mapsto 1-e^{\frac 1x}$ is non-positive.
Therefore
$$
\lim_{x \to \infty} f(x) = - \infty.
$$
Also, $\lim_{x \to \infty} x = \infty$. Therefore by taking the limit $\lim_{x \to \infty} f(x) +x$ you have an expression of the type $... |
2,100,726 | <p><strong>Problem:</strong> In a binomial experiment with 45 trials, the probability of more than 25 successes can be approximated by $P(z > \frac{(25-27)}{3.29}$)</p>
<p>What is the probability of success of a single trial of this experiment? </p>
<p><strong>Choices:</strong> </p>
<ul>
<li>0.07</li>
<li>0.56</l... | Mark Viola | 218,419 | <p>Note that as $x\to \infty$, the asymptotic expansion of $e^{1/x}$ is </p>
<p>$$e^{1/x}=1+\frac1x+\frac1{2x^2}+O\left(\frac1{x^3}\right)$$</p>
<p>Hence, </p>
<p>$$\begin{align}
\lim_{x\to \infty}\left(\frac{1}{1-e^{1/x}}+x\right)&=\lim_{x\to \infty}\left(\frac{1}{1-\left(1+\frac1x+\frac1{2x^2}+O\left(\frac1{x^... |
2,100,726 | <p><strong>Problem:</strong> In a binomial experiment with 45 trials, the probability of more than 25 successes can be approximated by $P(z > \frac{(25-27)}{3.29}$)</p>
<p>What is the probability of success of a single trial of this experiment? </p>
<p><strong>Choices:</strong> </p>
<ul>
<li>0.07</li>
<li>0.56</l... | egreg | 62,967 | <p>In cases like these you can easily get tricked up, so it can be better to do $x=1/t$, whereupon the first limit becomes
$$
\lim_{x\to\infty}\frac{1}{1-e^{1/x}}=
\lim_{t\to0^+}\frac{1}{1-e^t}=-\infty
$$
because $1-e^t<0$ for $t>0$.</p>
<p>Also
$$
\lim_{x\to\infty}(f(x)+x)=
\lim_{t\to0^+}\left(\frac{1}{1-e^t}+\... |
2,708,902 | <p>I am looking for an inductive proof of the following.</p>
<blockquote>
<p>Let $\alpha \in S_n$ be a cycle of length $p$. Then $\alpha^p =1$.</p>
</blockquote>
<p>For example $\alpha = (123)$ then $\alpha^3 = 1$ it is easy to verify.</p>
<p><em>My attempt:</em> Induction on $p$: The base case is true for $p=1$. ... | David C. Ullrich | 248,223 | <p>No. A polynomial over $R$ is not the same thing as the function from $R$ to $R$ defined by a polynomial. In particular, if we assume that $r^3=0$ for every $r\in R$ then $$p(x)=2x^3+x^2$$and $$q(x)=x^2$$ define the same function, but (assuming that $2\ne0$ in $R$) they are different polynomials, and in fact $q$ is m... |
1,686,455 | <p>For $a_1, ... , a_2 > 0$, I need to prove that
$( \sum_{i=1}^{n} a_i)( \sum_{i=1}^{n} \frac{1}{a_i})\geq n^2 $.</p>
<p>When do we have inequality?</p>
<p>Should I do it by induction? any hints</p>
| Daniel Akech Thiong | 169,316 | <p>Hint: Take $y_{i} = \sqrt{a_{i}}$ and $x_{i} = \frac{1}{\sqrt{a_{i}}}$ in the Cauchy-Schwarz inequality. </p>
<p>Then $n^{2} = \left(\sum_{i=1}^{n}x_{i}y_{i} \right)^{2} \leq \left (\sum_{i=1}^{n}x_{i}^{2} \right) \cdot \left(\sum_{i=1}^{n} y_{i}^{2} \right)$</p>
|
77,246 | <p>What is the "right" analog in the orbifold case of a singular homology of a topological space?
We can not just take the homology of the underlying space, because it does not contain much information.
For example, is there any kind homology of orbifolds such that the first homology group is the abelianization of the ... | David Roberts | 4,177 | <p>If your orbifold is given to you as a proper etale Lie/topological groupoid, you can:</p>
<ul>
<li>take the nerve of this groupoid, to get a simplicial manifold/space, </li>
<li>take the levelwise singular set to get a bisimplicial set, </li>
<li>take the diagonal to get a simplicial set, </li>
<li>take geometric ... |
322,745 | <p>I'm faced with the following problem on primes. Does someone have any clue? Is it (a reformulation of) an open problem?</p>
<p>Let <span class="math-container">$d$</span> be a positive integer, <span class="math-container">$d\geq 2$</span>. By Dirichlet's theorem, there is an infinite set <span class="math-containe... | Robert Israel | 13,650 | <p>Yes, we should expect it.
For any even <span class="math-container">$d \ge 2$</span>, Dickson's conjecture implies that there are infinitely many primes <span class="math-container">$p \equiv 1 \bmod d$</span> such that <span class="math-container">$1 + p d$</span> is prime.</p>
<p>Of course, expecting and proving ... |
197,421 | <p>Consider the quotient group $\mathbb{Q}/\mathbb{Z}$ of the additive group of rational numbers. Then how to find the order of the element $2/3 + \mathbb{Z} $ in $\mathbb{Q}/\mathbb{Z}$.</p>
| Fredrik Meyer | 4,284 | <p>Supplementing Makoto's answer, we can generalize: the order of an element $p/q \in \mathbb{Q}/\mathbb{Z}$ is the least integer $n$ such that $(p/q)n$ is an integer. If the fraction is written in reduced form, this is the same as the denominator.</p>
|
4,293,199 | <p>let me preface by saying this <em>is</em> from a homework question, but the question is not to plot the decision boundary, just to train the model and do some predictions. I have already done that and my predictions <em>seem</em> to be correct.</p>
<p>But I would like to verify my results by plotting the decision bo... | MyMolecules | 962,751 | <p>Use <span class="math-container">$\cos (A+B)$</span> formula to show for a natural number <span class="math-container">$n$</span>,</p>
<p><span class="math-container">$$\cos nx + \cos \left(nx+\frac{2n\pi}{3}\right) + \cos \left(nx+\frac{4n\pi}{3}\right) = \begin{cases} 0 , \quad \text{when n is coprime to 3} \\ 3\c... |
4,293,199 | <p>let me preface by saying this <em>is</em> from a homework question, but the question is not to plot the decision boundary, just to train the model and do some predictions. I have already done that and my predictions <em>seem</em> to be correct.</p>
<p>But I would like to verify my results by plotting the decision bo... | Jyrki Lahtonen | 11,619 | <p>Swatting this fly with an oversized mallet.</p>
<p>Let
<span class="math-container">$$f(x)=\cos^5x+\cos^5\left( x+\frac{2\pi}{3}\right) + \cos^5\left( x+\frac{4\pi}{3}\right).$$</span>
Because cosine has period <span class="math-container">$2\pi$</span>, we easily see that <span class="math-container">$f(x)$</span> ... |
1,649,997 | <p>I'm having a hard time coming up with two unbounded sequences where their difference yields $0$ when $n\rightarrow\infty$. Any ideas?</p>
| Bernard | 202,857 | <p>Take $(\sqrt{n+1})$ and $(\sqrt n)$.</p>
|
2,468,781 | <p>The law of Universal Generalization states that: </p>
<p>P(c)</p>
<p>(x) P(x) </p>
<p>Now, I understand that this works only if c is <em>any random element</em> from the universe. Such arbitrary selection makes this rule mathematically valid. However, I do not understand how it holds true in practical examples. <... | skyking | 265,767 | <p>The rule requires $c$ not to be mentioned in the prerequisites of the theory or any assumptions made. This means that it is a symbol whose meaning hasn't been restricted in <strong>any</strong> way. It is this lack of restriction that makes it possible to generalize.</p>
<p>If $P(c)$ can be proved with a $c$ in suc... |
1,132,187 | <p>$x^2 = y^2 + xy + 5$, where $x$ and $y$ are natural numbers.</p>
<p>Here is what I have so far:</p>
<ol>
<li><p>$x \neq y$ (from the equation).</p></li>
<li><p>$x$ is always odd (using the equation and assuming $2$ cases - $y$ is odd or $y$ is even).</p></li>
<li><p>Solving for the equation as a quadratic in $y$, ... | Will Jagy | 10,400 | <p>I like Conway's graphical method for these, the quadratic form is $f(x,y) = x^2 - xy- y^2,$ and we are looking for the value $5.$ As you can see, these happen when $y,x$ are consecutive Lucas numbers, $x$ is the larger one. I will need to look this up, there is something about odd/even indices as well. Alright, look... |
2,391,505 | <p>Most ODE textbooks provide the following steps to the solution of a separable differential equation (here the exponential equation is used as an example):</p>
<p>$$\frac{dN}{dt}=-\lambda N(t) \Rightarrow \frac{dN}{N(t)}=-\lambda dt\Rightarrow \int\frac{1}{N}dN=-\lambda\int dt \Rightarrow ln\mid N\mid = -\lambda t+C... | eyeballfrog | 395,748 | <p>The "splitting of the derivative" is just a shorthand for u-substitution in the resulting integral. u-substitution is usually written as
$$
\int f(u(x))u'(x) dx = \int f(u)du
$$
but this statement in Leibniz notation is
$$
\int f(u(x))\frac{du}{dx} dx = \int f(u) du
$$
which is the justification for the formal algeb... |
2,998,811 | <p><strong>Exercise :</strong></p>
<blockquote>
<p>We consider a market of one period <span class="math-container">$(\Omega, \mathcal{F}, \mathbb P, S^0, S^1)$</span>, where the sample space <span class="math-container">$\Omega$</span> has a finite number of elements and the <span class="math-container">$\sigma-$</span... | RRL | 148,510 | <p>In your answer you proved the contrapositive and, hence, the forward implication that no arbitrage implies <span class="math-container">$a < S_0^1(1+r) < b$</span>.</p>
<p>To prove the converse we assume that <span class="math-container">$a < S_0^1(1+r) < b$</span> and show that there can be no possible... |
836,945 | <p>I am currently an undergraduate math/CS major with coursework done in Linear Algebra, Vector Calculus (that covered a significant amount of Real 1 material), Discrete Math, and about to take courses in Algebra and Real Analysis 1. I was wondering if there are any books about famous problems (Riemann, Goldbach, Colla... | Gerry Myerson | 8,269 | <p>Jeff Lagarias edited a book of papers about the Collatz problem. Some of the papers should be quite accessible to you. </p>
|
836,945 | <p>I am currently an undergraduate math/CS major with coursework done in Linear Algebra, Vector Calculus (that covered a significant amount of Real 1 material), Discrete Math, and about to take courses in Algebra and Real Analysis 1. I was wondering if there are any books about famous problems (Riemann, Goldbach, Colla... | Fred Daniel Kline | 28,555 | <p><a href="http://rads.stackoverflow.com/amzn/click/9812381597" rel="nofollow">The Goldbach Conjecture</a> by Yuan Wang. </p>
<p>From the book's description:</p>
<blockquote>
<p>A detailed description of a most important unsolved mathematical problem - the Goldbach conjecture. Raised in 1742 in a letter from Goldb... |
1,670,816 | <p>Given <span class="math-container">$a, b \in \Bbb R$</span>, consider the following large tridiagonal matrix</p>
<p><span class="math-container">$$M := \begin{pmatrix}
a^2 & b & 0 & 0 & \cdots \\
b & (a+1)^2 & b & 0 & \cdots & \\
0 & b & (a+2)^2 &am... | marty cohen | 13,079 | <p>If
$f(x)
=(x-1)(x-2)(x-3)(x-4)-10^{-6}x^6
$,
if $c$ is small,</p>
<p>$\begin{array}\\
f(4+c)
&=(3+c)(2+c)(1+c)(c)-10^{-6}(4+c)^6\\
&=6c(1+c/2)(1+c/3)(1+c)-10^{-6}4^6(1+c/4)^6\\
&\approx 6c(1+c/2+c/3+c)-(2/5)^6(1+6c/4)\\
&=6c(1+11c/6)-(2/5)^6(1+3c/2)\\
&=c(6+(3/2)(2/5)^6) +11c^2-(2/5)^6\\
&\a... |
4,092,182 | <p>Let <span class="math-container">$I$</span> be a interval and <span class="math-container">$f:I \rightarrow \mathbb{R}$</span> monotone and surjective prove that <span class="math-container">$f$</span> is continuous.</p>
<p>I tried using the definition of <span class="math-container">$\epsilon$</span>-<span class="m... | Jotabeta | 841,213 | <p>Given <span class="math-container">$X,Y$</span> arbitrary vector fields, we define the hessian of <span class="math-container">$f$</span> as the <span class="math-container">$(0,2)$</span> tensor given by
<span class="math-container">$$
H_f(X,Y)=X(Yf)-(\nabla_X Y)(f).
$$</span>
The expression in coordinates takes th... |
385,404 | <p>So, I have this question which is still troubling me:</p>
<blockquote>
<p>Find the value of $k$ such that the equation $2x^3 + 3x^2 + kx - 48 = 0$ has two solutions equal in value but opposite in sign.</p>
</blockquote>
<p>I've had numerous attempts at this, such as using simultaneous equations and the factor th... | asun | 76,794 | <p>well you can use comparison.</p>
<p>$2x^3+3x^2+kx−48=(2x+a)(x+b)(x-b) = (2x+a)(x^2-b^2) = (2x^3 +ax^2 - 2b^2x - ab^2)$</p>
<p>so a comparison </p>
<p>$2x^3+3x^2+kx−48 = (2x^3 +ax^2 - 2b^2x - ab^2)$</p>
<p>gives us a = 3, b = 4, k=-32</p>
<p>so answers the question would be
$(2x+3)(x-4)(x+4)$</p>
<p>sorry i am... |
1,958,616 | <p>Question: Let $z=x+iy $. Using complex notation, find an equation of a circle with radius $5$ and center $(3,-6) $.</p>
<p>The answer seems simple to derive, but I'm curious as to whether or not there is more that must be done to establish the equation than what I have accomplished.</p>
<p>The circle with the abov... | wgrenard | 142,968 | <p>In the complex plane, a circle of radius $r$ centered at the origin has the simple equation</p>
<p>$$|z| = 5$$</p>
<p>Where $z = x + iy$ and $|z|=\sqrt{x^2 + y^2}$. We can transform this equation from being centered at the origin, to being centered at the point $3-6i$ by doing the following:</p>
<p>$$|z - (3-6i)|... |
1,958,616 | <p>Question: Let $z=x+iy $. Using complex notation, find an equation of a circle with radius $5$ and center $(3,-6) $.</p>
<p>The answer seems simple to derive, but I'm curious as to whether or not there is more that must be done to establish the equation than what I have accomplished.</p>
<p>The circle with the abov... | John Wayland Bales | 246,513 | <p>You are doing it backwards. </p>
<p>Starting with the idea that the distance between $x+iy$ and $3-6i$ is $5$ we have</p>
<p>\begin{equation}
\vert x+iy - (3-6i)\vert=5
\end{equation}</p>
<p>or, rearranging terms</p>
<p>\begin{equation}
\vert x-3 +(y+6)i \vert=5
\end{equation}</p>
<p>thus</p>
<p>\begin{equatio... |
1,804,333 | <blockquote>
<p>How many divisors does $25^2+98^2$ have?</p>
</blockquote>
<hr>
<p>My Attempt:</p>
<p>Calculator is not allowed but using calculator I found $193\times53$ that means $8$ divisors and that $4$ of them are positive.</p>
| almagest | 172,006 | <p>The trick here is the useful identity $a^4+4b^4=(a^2+2ab+2b^2)(a^2-2ab+2b^2)$. In this case we have $a=5,b=7$, so we get the factorisation $(25+70+98)(25-70+98)=193\cdot53$.</p>
<p>It is now easy to check that 53,193 are prime, so there are as you say 4 divisors: 1, 53, 193 and the number itself.</p>
<p>It is usua... |
1,804,333 | <blockquote>
<p>How many divisors does $25^2+98^2$ have?</p>
</blockquote>
<hr>
<p>My Attempt:</p>
<p>Calculator is not allowed but using calculator I found $193\times53$ that means $8$ divisors and that $4$ of them are positive.</p>
| Taha Akbari | 331,405 | <p>$25^2+98^2=123^2-2 \times 98 \times 25=123^2-70^2=193 \times 53 $</p>
<p>I found this answer using almagest's answer thanks a lot for help.</p>
|
176,054 | <p>I have a friend who wants to study something applied to neurosciences. He is going to begin his grad studies in mathematics.
He asked me which areas of mathematics could be applied to neurosciences.
Since I don't know the answer, I thought mathoverflow would be the right place to ask.
I mean, there are many areas o... | Aaron Hoffman | 1,281 | <p>(just my two cents, but too long for a comment)</p>
<p>Does your friend care more about the tools he uses in his research (e.g. theorem-proving vs. developing models vs. using known mathematics to analyze data and/or models) or the community in which his work has impact (e.g. mathematicians vs. applied scientists).... |
4,136,532 | <p>I know it's a weird question.<br />
But this thing is confusing me.<br />
(x̅) : average μ<br />
</p>
<p>①<br />
∵ <span class="math-container">$\frac{1}{n}\sum\limits_{i=1}^n(x_i) = \bar{x}$</span><br />
∴ <span class="math-container">$\sum\limits_{i=1}^n(x_i) = {n}\bar{x}$</span><br />
</p>
<p>②<br />
∵ <s... | Siong Thye Goh | 306,553 | <p><span class="math-container">$$1+3=4$$</span>
<span class="math-container">$$2+2=4$$</span></p>
<p>We can't conclude that <span class="math-container">$1=2$</span>.</p>
|
142,734 | <p>What are some books that discuss elementary mathematical topics ('school mathematics'), like arithmetic, basic non-abstract algebra, plane & solid geometry, trigonometry, etc, in an insightful way? I'm thinking of books like Klein's <em>Elementary Mathematics from an Advanced Standpoint</em> and the books of the... | Mikhail Katz | 28,128 | <p>Stewart's book is an old favorite:</p>
<p>Stewart, Ian. From here to infinity. With a foreword by James Joseph Sylvester. The Clarendon Press, Oxford University Press, New York, 1996.</p>
<p>Stewart has many popularisation books some of which have reached best-seller status, e.g.:</p>
<p><a href="http://rads.stac... |
1,974,939 | <p>I don't understand red line parts. I dont know why the assumtion that the list in (1) contains every real number leads to the conclusion that the intersection n to infinite number In equal to empty set.
Please explain why... I will really appreciate it. </p>
<p><a href="https://i.stack.imgur.com/UtJHG.png" rel="no... | Berci | 41,488 | <p>Every real number is assumed to among the $x_n$'s, but we constructed the $I_k$ intervals so that $x_n\notin \bigcap_kI_k$ holds for all $n$ (because, by construction $x_n\notin I_{n+1}$).</p>
<p>So this intersection contains no real numbers. So it is empty.</p>
|
375,016 | <p>I just started reading the ABC of category theory using the appendix of a text, the first chapter of a text that I have never read, and above all (I found out now that they handle well the theory) the wikipedia pages. I want to know only this: in all three sources are given the definition of the category, and in all... | Simone | 49,584 | <p>I suggest you to read (at least parts of) the following nice paper:</p>
<p><a href="http://arxiv.org/pdf/0810.1279v2.pdf" rel="nofollow">http://arxiv.org/pdf/0810.1279v2.pdf</a></p>
<p>From the abstract:
"Questions of set-theoretic size play an essential role in category
theory, especially the distinction between ... |
1,273,086 | <p>Sum of three dice (six faced) throws is $15$. What is the probability that first throw was $4$?</p>
<p>The way I thought of solving this was...
- Given - sum of second and third throw is $11$
- Probability of getting first throw = $4$ is $1$ out of $6$, that is $\frac{1}{6}$</p>
<p>Is this correct?</p>
| aepound | 222,527 | <p>You can state this very nicely using <a href="http://en.wikipedia.org/wiki/Conditional_probability#As_an_axiom_of_probability" rel="nofollow">conditional probabilities</a>:</p>
<p>Following your start, I'd look at
$$
\begin{split}
P\left(\sum_{i=1}^3 x_i = 15, x_1 = 4\right) &= P\left(\sum_{i=1}^3 x_i = 15|x_1 ... |
28,395 | <p>I want to generate a table of Euclidean distances from the points in a list to a given point, e.g.:</p>
<pre><code>Table[EuclideanDistance[MyList[[i]], c], {i, 1, Length[MyList]}]
</code></pre>
<p>Moreover, when <code>EuclideanDistance</code> returns a value greater than $T$, I'd like to replace that value with a ... | bill s | 1,783 | <p>Here's a variation on Jens method that does the calculation by mapping the function <code>EuclideanDistance</code> onto the list and then clipping the result:</p>
<pre><code>myList = RandomReal[{-1, 1}, 10];
c = {0, 0}; t = 1; r = -1;
Clip[EuclideanDistance[#, c] & /@ myList, {0, 1}, {0, r}]
</code></pre>
|
907,154 | <p>Renata walks down an escalator that moves up and counts <span class="math-container">$150$</span> steps. Her sister Fernanda climbs the same escalator and counts <span class="math-container">$75$</span> steps. If the speed of Renata (in steps per time unit) is three times the speed of Fernanda, determine how many st... | Adriano | 76,987 | <p>Let:</p>
<ul>
<li>$x$ be the number of visible steps.</li>
<li>$y$ be the speed of the escalator (in steps per time unit).</li>
<li>$z$ be the speed of Fernanda (in steps per time unit), relative to the escalator.</li>
</ul>
<p>Consider Renata's trip. Notice that in the time that Renata takes to finish her trip, s... |
2,958,566 | <p>Expressing <span class="math-container">$\frac{t+2}{t^3+3}$</span> in the form <span class="math-container">$a_o+a_1t+...+a_4t^4$</span>, where <span class="math-container">$t$</span> is a root of <span class="math-container">$x^5+2x+2$</span>.</p>
<p>So i can deal with the numerator, but how do I get rid of the de... | Bill Dubuque | 242 | <p>As I show <a href="https://math.stackexchange.com/a/614369/242">here,</a> such results boil down to the first <span class="math-container">$\,2\,$</span> terms of the Binomial Theorem, viz.</p>
<p><span class="math-container">$\!\bmod m\!:\ c\equiv -a(b\!-\!1)\,$</span> so <span class="math-container">$\ \color{#0a... |
3,996,665 | <p>I am really confused about below questions. I am new both in this platform and below topics. I hope, I was clearly explain the questions.</p>
<p>For ∀n∈<span class="math-container">$\mathbb{Z}$</span><sup>+</sup></p>
<p>Based on <span class="math-container">$B_n$</span> = {n, n+1, n+2...}, <strong>B</strong> = {<spa... | Nikolay | 837,264 | <p>(i) to show that <span class="math-container">$\mathcal{B}=\{B_n\}$</span> is a base for some topology on some space <span class="math-container">$X$</span>, one should shows that <span class="math-container">$$\forall A, B \in \mathcal{B}\quad\exists C_1,\ldots,C_n\in\mathcal{B}\quad A\cap B = \bigcup_{i=1}^{n}C_n$... |
1,176,016 | <p>How do you show that if you have sets $B_1, B_2, \cdots ,B_n$ and a set $C$, then
$$(B_1\cap B_2, \cap \cdots B_n)\cup C= (B_1\cup C)\cap(B_2\cup C) \cap \cdots
\cap (B_n\cup C)\,?$$</p>
<p>Thanks</p>
| mvw | 86,776 | <p>Interpreting both sides as numbers
$$
e^x = 4x
$$
the equal sign might hold for certain values of $x$.</p>
<p>Interpreting both sides as functions
$$
\exp = 4 \mbox{ id}
$$
the equal sign is not valid.
The differentiation operator acts on functions, that is why the second interpretation has to be used, and explains... |
3,387,049 | <p>Say a deck of cards is dealt out equally to four players (each player receives 13 cards).</p>
<p>A friend of mine said he believed that if one player is dealt four-of-a-kind (for instance), then the likelihood of another player having four-of-a-kind is increased - compared to if no other players had received a four... | Ross Millikan | 1,827 | <p>You have to be careful what question you are asking. What your friend seems to be claiming is if you draw a hand of <span class="math-container">$13$</span> cards for one player and it has four of a kind, the chance of a second hand you draw from the rest of the deck having four of a kind is increased. My intuitio... |
3,489,898 | <p>In order to show that a metric space <span class="math-container">$(X, d)$</span> is not complete one may apply the definition and look for a Cauchy sequence <span class="math-container">$\{x_n\}\subset X$</span> which does not converge with respect to the metric <span class="math-container">$d$</span>. Now I have o... | José Carlos Santos | 446,262 | <p>It can be used when you are aware of the existence of a metric space <span class="math-container">$(Y,d^\ast)$</span> such that:</p>
<ol>
<li><span class="math-container">$X\subset Y$</span>;</li>
<li><span class="math-container">$(\forall x,x'\in X):d^\ast(x,x')=d(x,x')$</span>.</li>
</ol>
<p>In the example that ... |
4,335,896 | <blockquote>
<p><span class="math-container">$$\frac{1}{2\cdot4}+\frac{1\cdot3}{2\cdot4\cdot6}+\frac{1\cdot3\cdot5}{2\cdot4\cdot6\cdot8}+\frac{1\cdot3\cdot5\cdot7}{2\cdot4\cdot6\cdot8\cdot10}+\cdots$$</span>
is equal to?</p>
</blockquote>
<p><strong>My approach:</strong></p>
<p>We can see that the <span class="math-con... | epi163sqrt | 132,007 | <blockquote>
<p>We can write the series as
<span class="math-container">\begin{align*}
\color{blue}{\frac{1}{2}\sum_{n=1}^{\infty}\left(\frac{(2n)!\left(\frac{1}{4}\right)^n}{n!(n+1)!}\right)}
&=\frac{1}{2}\sum_{n=1}^{\infty}\frac{1}{n+1}\binom{2n}{n}\left(\frac{1}{4}\right)^n\tag{1}\\
&=\frac{1}{2}\sum_{n=1}^{... |
648,946 | <p>I want to find
$$ \lim_{x\to0} \frac{(e^{-x^2}-1)\sin x }{x\ln(1+x^2)}$$
using a Maclaurin series and not using the l'Hôpital's rule.</p>
<p>However I can't seem to get it right.</p>
<p>Thanks for any possible answers.</p>
| Ross Millikan | 1,827 | <p>Your first line is incorrect and should be deleted or the argument of $\log$ corrected to $(1+i\sqrt 3)$. The second line is correct if you are using the principal value of the complex logarithm. The rest is fine.</p>
|
3,064,943 | <p>Consider the senquence of iid r.v. <span class="math-container">$(Y_k)_{k\geq1}$</span> such that <span class="math-container">$\mathbb{P}(Y_k=1)=\mathbb{P}(Y_k=-1)=\frac{1}{2}$</span> and then consider the process <span class="math-container">$X=(X_k)_{n\geq1}$</span> such that <span class="math-container">$X_n=\su... | Kavi Rama Murthy | 142,385 | <p><span class="math-container">$EX_n^{2}=\sum_{k=1}^{n} var (\frac {Y_k} k)=\sum_{k=1}^{n} \frac 1 {k^{2}}$</span> which is bounded. It is well known that this implies uniform integrability and any uniformly integrable martingale converges almost surely. </p>
|
2,648,273 | <p>I am reading "Linear Algebra" by Takeshi SAITO.</p>
<p>Why $n \geq 0$ instead of $n \geq 1$?<br>
Why $K^0 = \{0\}$?<br>
Is $K^0 = \{0\}$ a definition or not?</p>
<p>He wrote as follows in his book:</p>
<p>Let $K$ be a field, and $n \geq 0$ be a natural number. </p>
<p>$$K^n = \left\{\begin{pmatrix}
... | edm | 356,114 | <p>It is a convention, which you can take as a definition.</p>
<p>Since $K^n$ is an $n$-dimensional vector space when $n$ is a positive integer, we would like to have $K^0$ to denote a zero-dimensional vector space, which would be $\{0\}$.</p>
|
4,048,158 | <p>Given data points <span class="math-container">$\{(x_i,f(x_i)\}_{i=0}^{m}$</span>, if we define divided difference recursively as:</p>
<p><span class="math-container">$$f[x_0,\cdots,x_{k+1}] = \frac{f[x_1,\cdots,x_{k+1}]-f[x_0,\cdots,x_k]}{x_{k+1}-x_0} \text{ with the definition that} f[x_0] = f(x_0)$$</span></p>
<p... | PierreCarre | 639,238 | <p>Consider adding the table entries one by one...</p>
<ol>
<li><p><span class="math-container">$p_0(x) = f_0$</span> interpolates the data on <span class="math-container">$x_0$</span>.</p>
</li>
<li><p>The interpolant on <span class="math-container">$x_0, x_1$</span> is given by
<span class="math-container">$$
p_1(x) ... |
2,727,598 | <p>Given p is a prime number greater than 2, and</p>
<p>$ 1 + \frac{1}{2} + \frac{1}{3} + ... \frac{1}{p-1} = \frac{N}{p-1}$ </p>
<p>how do I show, $ p | N $ ???</p>
<p>The previous part of this question had me factor $ x^{p-1} -1$ mod $p$. Which I think is just plainly $(x-1) ... (x-(p-1))$ </p>
| quasi | 400,434 | <p>Presumably, the intended problem was to show, assuming $p$ is an odd prime, that
$$1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{p-1} = \frac{N}{(p-1)!}$$
implies $p{\,\mid\,}N$.
<p>
It suffices to show the $\text{LHS}$ is congruent to $0$, mod $p$.
<p>
But mod $p$, the fractions
$$1, \frac{1}{2}, \frac{1}{3},..... |
476,955 | <p>Let $X$ be a set with 4 elements. Is it possible to have a topology on $X$ with 14 open sets?</p>
| MJD | 25,554 | <p>Suppose there are exactly two non-open sets, say $S$ and $T$, and that at least one of them, say $S = \{a\}$, is a 1-set. Each 1-set $\{a\}$ can be written as the intersection of 2-sets in three ways:</p>
<p>$$ \{a, b\}\cap \{a,c\} = \{a\}\\
\{a, b\}\cap \{a,d\} = \{a\}\\
\{a, c\}\cap \{a,d\} = \{a\}
$$</p>
<p>If... |
606,380 | <blockquote>
<p>If <span class="math-container">$abc=1$</span> and <span class="math-container">$a,b,c$</span> are positive real numbers, prove that <span class="math-container">$${1 \over a+b+1} + {1 \over b+c+1} + {1 \over c+a+1} \le 1\,.$$</span></p>
</blockquote>
<p>The whole problem is in the title. If you wanna h... | Community | -1 | <p>For my earlier comment: By expanding everything I mean, you can clear the denominator, write down everything in terms of symmetric polynomials, and try to use AM-GM to compare them.</p>
<p>On the other hand, there is also a one liner, similar to math110's solution:</p>
<p>$$\frac{1}{a+b+1} \leq \frac{2c+ab}{2(a+b+... |
606,380 | <blockquote>
<p>If <span class="math-container">$abc=1$</span> and <span class="math-container">$a,b,c$</span> are positive real numbers, prove that <span class="math-container">$${1 \over a+b+1} + {1 \over b+c+1} + {1 \over c+a+1} \le 1\,.$$</span></p>
</blockquote>
<p>The whole problem is in the title. If you wanna h... | nguyenhuyenag | 410,198 | <p>Using the Cauchy-Schwarz inequality we have
<span class="math-container">$$\frac{1}{1+a+b}=\frac{c+2}{(c+1+1)(1+a+b)} \leqslant \frac{c+2}{(\sqrt{c}+\sqrt{a}+\sqrt{b})^2}.$$</span>
Therefore
<span class="math-container">$$\frac{1}{1 + a + b} + \frac{1}{1 + b + c} + \frac{1}{1 + c + a} \leqslant \frac{a+b+c+6}{(\sqrt... |
1,678,687 | <p>Find the point $(x_0, y_0)$ on the line $ax + by = c$ that is closest to the origin. </p>
<p>According to this <a href="http://www.intmath.com/plane-analytic-geometry/perpendicular-distance-point-line.php" rel="nofollow noreferrer">source</a>, I thought that $\left( -\frac{ac}{a^{2}+b^{2}}, -\frac{bc}{a^{2}+b^{... | πr8 | 302,863 | <p>Calling $\bar v =(a,b), \bar x=(x,y)$, the line has equation $\bar v \cdot \bar x=c$.</p>
<p>Cauchy-Schwarz gives that $\vert \bar v\vert \cdot \vert \bar x\vert \ge c$, with equality when $\bar v, \bar x$ are linearly dependent.</p>
<p>Setting $\bar x = \lambda\bar v$ and plugging it into $\bar v \cdot \bar x=c$... |
1,678,687 | <p>Find the point $(x_0, y_0)$ on the line $ax + by = c$ that is closest to the origin. </p>
<p>According to this <a href="http://www.intmath.com/plane-analytic-geometry/perpendicular-distance-point-line.php" rel="nofollow noreferrer">source</a>, I thought that $\left( -\frac{ac}{a^{2}+b^{2}}, -\frac{bc}{a^{2}+b^{... | choco_addicted | 310,026 | <p>Since given question has <a href="/questions/tagged/multivariable-calculus" class="post-tag" title="show questions tagged 'multivariable-calculus'" rel="tag">multivariable-calculus</a>, I will use Lagrange multiplier. Define $f,g$ as
\begin{align}
f(x,y)&=x^2+y^2\\
g(x,y)&=ax+by-c=0,
\end{align}
Then... |
1,586,607 | <p>This is something I've been thinking about lately;</p>
<p>$$\int_0^1 \int_0^1 \frac{1}{1-(xy)^2} dydx$$
Solutions I've read involve making the substitutions: $x= \frac{sin(u)}{cos(v)}$ and $y= \frac{sin(v)}{cos(u)}$. This reduces the integral to the area of a right triangle with both legs of length $\frac{\pi}{2}$... | Jan Eerland | 226,665 | <p>If you haven't an idea, just start:</p>
<p>$$\int_{0}^{1}\int_{0}^{1}\frac{1}{1-(xy)^2}\space\text{d}y\text{d}x=$$
$$\int_{0}^{1}\left[\int_{0}^{1}\frac{1}{1-(xy)^2}\space\text{d}y\right]\text{d}x=$$ </p>
<hr>
<p>For the integrand $u=xy$ and $\text{d}u=x\space\text{d}y$.</p>
<p>This gives a new lower bound $u=x\... |
904,074 | <p>Is this Convergent or Divergent</p>
<p>$$\int_0^1 \frac{1}{\sin(x)}\mathrm dx $$</p>
<p>So little background to see if I am solid on this topic otherwise correct me please :) </p>
<p>To check for convergence I can look for a "bigger" function and compare if that is convergent, If yes then for sure the one in ques... | Lucian | 93,448 | <blockquote>
<p><em>Is this Convergent or Divergent</em> ?</p>
</blockquote>
<p>Yes! :-)</p>
<blockquote>
<p><em>I have no Idea WHAT kind of function to compare with:/</em></p>
</blockquote>
<p><strong>Hint:</strong> $\sin x\sim x$ for $x\to0$.</p>
|
2,845,451 | <p>I was reading a number theory book and it was stated that $d^k+(a-d)^k=a[d^{k-1}-d^{k-2}(a-d)+ . . .+(a-d)^{k-1}]$ for $k$ odd. How did they arrive at this factorization? Is there an easy way to see it?</p>
| J.G. | 56,861 | <p>It's slightly easier if we switch to $x=d,\,y=a-d$ so you want to verify $x^k+y^k=(x+y)(x^{k-1}-x^{k-2}+\cdots +y^{k-1})$ for odd $k$. You can now easily prove this by $k\mapsto k+2$ induction, or by taking it as the odd-$k$, $y=-z$ special case of $x^k-z^k=(x-z)\sum_{j=0}^{k-1}x^j z^{k-1-j}$. If you just want to pr... |
3,772,193 | <p>If I put the two vectors 1,0,0 and 0,1,0 next to each other</p>
<p><span class="math-container">$$\begin{pmatrix} 1 & 0 \\ 0 & 1 \\ 0 & 0 \end{pmatrix}$$</span></p>
<p>I can see that they are independent since I cannot write <span class="math-container">$(1,0,0)^t$</span> as <span class="math-container">... | José Carlos Santos | 446,262 | <p>That statement (“If you have a row of <span class="math-container">$0$</span>'s in your matrix, it's linearly dependent”) holds if you have <span class="math-container">$n$</span> vectors in <span class="math-container">$\Bbb R^n$</span>. That is not the case here: you have <strong>two</strong> vectors in <span clas... |
57,508 | <p>While learning commutative algebra and basic algebraic geometry and trying to understand the structure of results (i.e. what should be proven first and what next) I came to the following question: </p>
<p>Is it possible to prove that $\mathbb A^2-point$ is not an affine variety, if you don't know that the polynomia... | Sándor Kovács | 10,076 | <p>The abstract reason for this is that $\mathbb A^2$ satisfies Serre's property $(S_2)$ (which follows from being smooth) and hence every regular function is determined in codimension $1$. In other words, the point is that a regular function on $\mathbb A^2\setminus \{0\}$ is a rational function on $\mathbb A^2$ and t... |
2,939,058 | <p>The question to be solved is:</p>
<p><span class="math-container">$$ \lim_{n \to \infty} \left( \ \sum_{k=10}^{n+9} \frac{2^{11(k-9)/n}}{\log_2 e^{n/11}} \ - \sum_{k=0}^{n-1} \frac{58}{\pi\sqrt{(n-k)(n+k)}} \ \right)$$</span></p>
<p>The first thing that occured to me was to transform the limits into definite inte... | mathlove | 78,967 | <blockquote>
<p>If <span class="math-container">$\gcd(y,z)=1$</span>, is the biconditional "<span class="math-container">$yz$</span> is perfect <span class="math-container">$\iff D(y)D(z)=2s(y)s(z)$</span>" always true?</p>
</blockquote>
<p>Yes.</p>
<p>If <span class="math-container">$yz$</span> is perfect with <sp... |
2,048,962 | <blockquote>
<p><strong>Question:</strong> How would you solve this sinusoidal equation:</p>
<blockquote>
<p>Solve $5\cos(6x)+6=9$. Assume $n$ is an integer and the answers are in degrees.</p>
<ul>
<li><p>$-8.86+n\cdot 60$</p></li>
<li><p>$-3.54+n\cdot 60$</p></li>
<li><p>$3.54+n\cdot 60... | Ovi | 64,460 | <p>I don't believe there is a "nice" way to do it. Anyway the multiple choices all being terminating decimals should give you that hint.</p>
<p>Starting from $\cos(6x)=\dfrac 35$, <a href="https://www.wolframalpha.com/input/?i=(arccos%203%2F5)%2F6%20convert%20to%20degrees" rel="nofollow noreferrer">WolframAlpha</a> gi... |
1,406,325 | <p>If $p(x)$ and $q(x)$ are continuous functions for any $x$, can $y(x)=\sin(x^2)$ be solution of the diff equation $y''+p(x)y'+q(x)y=0$ in some interval $I=[a,b] $containing $0$? I think it is not as simple as replacing $\sin(x^2)$ into the equation and analyzing the obtained expression.</p>
| MathMajor | 113,330 | <p>We have $y = \sin (x^2)$, $y' = 2x \cos (x^2)$, and $y'' = 2 \cos(x^2) - 4x^2 \sin (x^2)$. Substituting,</p>
<p>$$ 2 \cos(x^2) - 4x^2 \sin (x^2) + p(x) 2x \cos (x^2) + q(x) \sin (x^2) = 0$$</p>
<p>Following A.G.'s solution given in the comments, setting $x=0$ in the equation gives $2=0$ if we assume $p(x)$ and $q(... |
1,406,325 | <p>If $p(x)$ and $q(x)$ are continuous functions for any $x$, can $y(x)=\sin(x^2)$ be solution of the diff equation $y''+p(x)y'+q(x)y=0$ in some interval $I=[a,b] $containing $0$? I think it is not as simple as replacing $\sin(x^2)$ into the equation and analyzing the obtained expression.</p>
| A.Γ. | 253,273 | <p>One can also use the standard argument about existence and uniqueness of Cauchy problem like <a href="https://en.wikipedia.org/wiki/Picard%E2%80%93Lindel%C3%B6f_theorem" rel="nofollow noreferrer">Picard–Lindelöf theorem</a>. By this theorem, a linear differential equation with continuous coefficients has a <em>uniqu... |
4,155,792 | <p>I have seen many math books, and some of them, very good books, that say that <span class="math-container">$0!=1$</span> 'by convention'. I think that <span class="math-container">$0!$</span> <em>must</em> be <span class="math-container">$1$</span> because it is the product of the empty set. That is, for <span class... | johnnyb | 298,360 | <p>I think if you are teaching factorial in a procedural way then trying to.explain the product of an empty set is pretty far outside the context, and simply saying it is a matter of convention is more cognitively in line with where the student is at.</p>
<p>When teaching at the board, I try to separate out <em>what</e... |
2,061,695 | <p>I'm examining function slope and determine relative extrema of function (local minimum and local maximum)</p>
<p>The example is as following:
$
y = x^8 e^{-5x}
$</p>
<p>To do this I have to determine derivative of this function, which is:
$$
y' = 8x^7e^{-5x}+x^8(-5e^{-5x}) = -e^{-5x}x^7(5x-8)
$$</p>
<p>Then accor... | Mark Fischler | 150,362 | <p>Notice that $e^{-5x}$ will never be zero, so you can divided through by that. Then it is easy to solve the equation, which is
$$
x^7(5x-8)=0
$$
The two solutions are $x=0$ and $x=\frac85$.</p>
|
1,968,267 | <p>When doing induction should you always try to put your final answer as the "<em>desired</em> " form? For example if: $$\sum^{n}_{k=1}(k+2)(k+4) = \frac{2n^{3} + 21n^{2} + 67n}{6}$$ we ought to give the final answer as $$\frac{2(k+1)^{3} + 21(k+1)^{2} + 67(k+1)}{6}?$$</p>
<p>I just expanded both the $\text{LHS}_{k+1... | MrYouMath | 262,304 | <p>Another method (maybe overkill :D): The sum $S(n)=\sum_{k=1}^nP(k)$ in which $P(k)$ is a polynomial of $l^{th}$ degree can be expressed by a polynomial in $n$ of degree $l+1$. For your problem $P(k)=(k+2)(k+4)$ and has degree 2. Hence the sum can be expressed as a polynomial $G(n)=a_3n^3+a_2n^2+a_1n+a_0$. Now calcul... |
176,600 | <p>The question is about factoring extremely large integers but you can have a look at <a href="https://stackoverflow.com/questions/11699464/mathematically-navigating-a-large-2d-numeric-grid-in-c-sharp">this question</a> to see the context if it helps. Please note that I am not very familiar with mathematical notation ... | Ross Millikan | 1,827 | <p>Are you given $N$? Then if the number to be factored is $A$, you have $A=N^k$ and only have to find $k$. Taking logs, $k=\frac {\log A}{\log N}$. So, yes, if you know $N$ it helps a lot.</p>
<p>If $N$ is prime, all the possible factors of $A$ are of the form $N^m$ for $0 \le m \le k$ so you have them. If $N$ is... |
176,600 | <p>The question is about factoring extremely large integers but you can have a look at <a href="https://stackoverflow.com/questions/11699464/mathematically-navigating-a-large-2d-numeric-grid-in-c-sharp">this question</a> to see the context if it helps. Please note that I am not very familiar with mathematical notation ... | Community | -1 | <p>If $N = 2$ (or any prime) and you can get $k$ as Ross indicated, then the divisors of $A$ are $\{1, N, N^2, \ldots, N^i, \ldots, N^{k} \}.$ If $N$ is a composite, then you will incur extra time complexity in computing the prime factorization of $N = \prod_{i = 1}^{\ell} p_i^{e_i}$ and the divisors of $A$ are all pos... |
1,372,997 | <p>This question originates from the 1996 Canada National Olympiad.</p>
<blockquote>
<p>Let $r_1, r_2, \dots, r_m$ be a given set of $m$ positive rational numbers such that</p>
<p>$\sum\limits^{m}_{k=1}{r_k} = 1 \tag{1}$</p>
<p>Define the function $f$ by </p>
<p>$f(n) = n − \sum\limits^{m}_{k=1}{\lflo... | Justin | 219,394 | <p>$f(n)=\sum_{k=1}^{m}(r_k n - \lfloor {r_k n}\rfloor)$</p>
<p>$\lfloor {r_k n}\rfloor \leq r_k n$</p>
<p>So, $f(n) \geq 0$</p>
<p>$r_k n - \lfloor {r_k n}\rfloor <1 $</p>
<p>$f(n) < \sum_{k=1}^{m}1$</p>
<p>So, $f(n) < m$</p>
<p>$r_k = p_k / q_k$ where $p_k < q_k$</p>
<p>If $n = q_1.q_2.....q_m - 1... |
1,321,704 | <p>For example: $3^\sqrt5$ versus $5^\sqrt3$</p>
<p>I tried to write numbers as this:</p>
<p>$3^{5^{\frac{1}{2}}}$ and then as
$3^{\frac{1}{2}^5}$</p>
<p>But this method gives the wrong answer because $a^{(b^c)} \ne a^{bc}$</p>
| user134824 | 134,824 | <p>Doing a problem like this takes a combination of knowledge about rules of exponentiation and skill at estimation.</p>
<p>First, raise both $3^{\sqrt 5}$ and $5^{\sqrt 3}$ to the power $\sqrt 5$, which gives the numbers $3^5=243$ and $5^{\sqrt{15}}$ using the rule $(a^b)^c = a^{bc}$. This has the effect of removing ... |
2,598,912 | <p>Let $f$ be some function in $L_{loc}^1(\mathbb{R})$ such that, for some $a \in \mathbb{R}$,</p>
<p>$$\int_{|x| \leq r} |f(x)|dx \leq (r+1)^a$$</p>
<p>for all $r \geq 0$. Show that $f(x)e^{-|tx|} \in L^1(\mathbb{R})$ for all $t \in \mathbb{R} \setminus \{0\}$. </p>
<p>I'm having a hard time finding use of the bou... | zhw. | 228,045 | <p>Hint: The given bound is a polynomial growth estimate. Exponential decay always wins over polynomial growth. To get the given estimate in play, integrate by parts.</p>
<p><strong>Added later</strong>: To see how integration by parts can work here, assume WLOG $f\ge 0$ and $t>0.$ The integral can be written</p>
... |
402,310 | <p>Vector math is something I find very interesting. However, we have never been told the link between vectors in physics (usually represented as arrows, e.g. a force vector) and in algebra (e.g. represented like a column matrix). It was really never explained well in classes.</p>
<p>Here are the things I can't wrap m... | Muphrid | 45,296 | <p>Physicists tend to emphasize a geometric interpretations of vectors, which mathematicians need not do. This is because one of the main uses of vectors in physics is to talk about the geometry of some system, while vectors in general can be used in more abstract senses (and perhaps with no geometric interpretation a... |
2,088,564 | <blockquote>
<p>Sum of binomial product $\displaystyle \sum^{n}_{r=0}\binom{p}{r}\binom{q}{r}\binom{n+r}{p+q}$</p>
</blockquote>
<p>Simplifying $\displaystyle \frac{p!}{r!\cdot (p-r)!} \cdot \frac{q!}{r!\cdot (q-r)!}\cdot \frac{(n+r)!}{(p+q)! \cdot (n+r-p-q)!}$.</p>
<p>Could some help me with this, thanks</p>
| epi163sqrt | 132,007 | <p>It is convenient to use the <em>coefficient of</em> operator $[t^r]$ to denote the coefficient of $t^r$ in a series. This way we can write e.g.
\begin{align*}
\binom{p}{r}=[t^r](1+t)^p
\end{align*}</p>
<blockquote>
<p>The following is valid
\begin{align*}
\sum_{r=0}^n\binom{p}{r}\binom{q}{r}\binom{n+r}{p+q}... |
637,699 | <p>We have a triangle $ABC$. Whats the value of angle $C$?</p>
<p>$$\sin^2(A)+\sin^2(B)-\sin^2(C)=1$$</p>
<p>I made a small java program and it gave me an answer. I want to know how to make it through other ways.</p>
| Lucian | 93,448 | <p>$$\sin^2 A+\sin^2 B-\sin^2(A+B)=\sin^2 A+\sin^2 B-(\sin A\cos B+\sin B\cos A)^2=$$</p>
<p>$$=\sin^2 A+\sin^2 B-(\sin^2A\cos^2B+\sin^2B\cos^2A+2\sin A\cos A\sin B\cos B)=$$</p>
<p>$$=\sin^2A\,(1-\cos^2B)+\sin^2B\,(1-\cos^2A)-2\sin A\cos A\sin B\cos B=$$</p>
<p>$$=2\sin^2A\sin^2B-2\sin A\cos A\sin B\cos B=1\iff$$</... |
17,610 | <p>Let $V$ be a finite dimensional highest weight representation of a (semi)-simple Lie algebra. For each $n\ge 0$ take $a_n$ to be the dimension of the space of invariant tensors in $\otimes^n V$.</p>
<p>In certain cases there is a formula for $a_n$. For example, for $V$ the two dimensional representation of $sl(2)$ ... | Martin Rubey | 3,032 | <p>The first question has a simple answer: somehow calculate the first few terms of your sequence, and feed your favorite guessing machine with them. I advertise the one built into FriCAS (because its authors are Waldek Hebisch and myself):</p>
<pre>
(1) -> guessPRec [1, 0, 1, 1, 4, 10, 35, 120, 455, 1792, 7413, 3178... |
17,610 | <p>Let $V$ be a finite dimensional highest weight representation of a (semi)-simple Lie algebra. For each $n\ge 0$ take $a_n$ to be the dimension of the space of invariant tensors in $\otimes^n V$.</p>
<p>In certain cases there is a formula for $a_n$. For example, for $V$ the two dimensional representation of $sl(2)$ ... | David E Speyer | 297 | <p>Finding the recurrence (and proving it is correct) can be done by the standard techniques for extracting the diagonal of a rational power series. </p>
<p>Let $\beta_1$, $\beta_2$, ..., $\beta_N$ be the weights of $V$. Let $\rho$ be half the sum of the positive roots and $\Delta = \sum (-1)^{\ell(w)} e^{w(\rho)}$ be... |
572,429 | <p>If A is invertible, prove that $\lambda \neq 0$, and $\vec{v}$ is also an eigenvector for $A^{-1}$, what is the corresponding eigenvalue?</p>
<p>I don't really know where to start with this one. I know that $p(0)=det(0*I_{n}-A)=det(-A)=(-1)^{n}*det(A)$, thus if both $p(0)$ and $det(0) = 0$ then $0$ is an eigenvalue... | BaronVT | 39,526 | <p>Remember that </p>
<p>If $v$ is an eigenvector of $A$ corresponding to eigenvalue $\lambda$, then
$$
Av = \lambda v.
$$</p>
<p>You want to show that, for some other number $\mu$,
$$
A^{-1}v = \mu v.
$$
So, starting with $Av = \lambda v$ again,
$$
A^{-1}A v = A^{-1} \lambda v = \lambda A^{-1} v
$$
and I'll leave th... |
575,069 | <p>Let $\mathbb R_n[x]$ the vector space of polynomials with degree less or equal $n$ and we consider the linear transformation $f$ defined by
$$\forall P\in \mathbb R_n[x]\quad f(P)=(x^2-1)P''+2xP'$$
I proved that $f$ has the spectrum
$$\mathrm{sp}(f)=\{k(k+1),\ k=0,\ldots,n\}$$
I'm stuck in this question: Prove that... | Lutz Lehmann | 115,115 | <p>You can compare, as said in the comments, by
<span class="math-container">$$
\frac1{n^2}<\frac1{n(n-1)}=\frac1{n-1}-\frac1n
\text{ so that } ~~
\sum_{n=1}^\infty\frac1{n^2}\le1+\sum_{n=2}^\infty \frac1{n-1}-\frac1n=2$$</span>
by telescoping the upper bound. A closer bound is
<span class="math-container">$$
\frac1... |
2,075 | <p>I've come across so many posts here and on the "main" math-SE site that voice complaints, frustrations, pet-peeves, grievances, or else are critical of another post/question, user, OP, etc. It is really an energy sapper! Certainly not a boost for morale.</p>
<p>Since I'm pretty new here, and feeling a bit ambival... | svenkatr | 2,641 | <p>One big difference between Math Stackexchange and Mathoverflow is that you don't need to be a mathematician or even a math graduate student to participate and contribute to math stackexchange. I am an electrical engineering grad student and I love this community. It gives me a chance to not only learn new things eve... |
804,310 | <p>The probability distribution of a discrete random variable x is
$$f (x)= \begin{pmatrix}3 \\ x \end{pmatrix} (1/4)^x (3/4)^{3-x} $$ Find the mean value of x.
Construct a cumulative distribution function for f (x).
i find out$$ P(X=o) = 0.421875$$ $$ P(X=1) = 0.421875$$ $$ P(X=2) = 0.140625$$ $$ P(X=3) = 0.015... | georg | 144,937 | <p>"The essence of the probability argument is that we have 3 independent trials, and on each trial, outcome 1 occurs with probability p and some other outcome with probability 1−p. help me for finding mean value"</p>
<p>I would say that
x = {0,1}, P(0) = 1-p, P(1) = p, E(x) =0*(1-p)+1*p = p</p>
<p>When $X = x_1+x_2+... |
3,947,632 | <p>It is almost obvious that we have to use derivative in order to find the answer. However, I couldn't see this one to be derivative of something.</p>
<p>It would be amazing if you share your thoughts on this problem. Also, I would be thankful if you would explain what was intuition behind your solution. Thanks!</p>
| Claude Leibovici | 82,404 | <p><em>An old trick !</em>
<span class="math-container">$$S=\sum_{k = 1}^{n}k^2 \, x^{k - 1}=\sum_{k = 1}^{n}\big[k(k-1)+k \big]\, x^{k - 1}$$</span> that is to say
<span class="math-container">$$S=\sum_{k = 1}^{n}k(k-1) \, x^{k - 1}+\sum_{k = 1}^{n}k \, x^{k - 1}$$</span>
<span class="math-container">$$S=x\sum_{k = 1}... |
1,703,140 | <p>Let $n$ be composite. If $a$ is coprime to $n$ such that $a^{n-1} \equiv 1 \pmod n$ then $a$ is called a Fermat liar. If $a^{n-1} \not\equiv 1 \pmod n$, then $a$ is a Fermat witness to the compositeness of $n$.</p>
<p>Assume the witnesses or liars are themselves prime. If two Fermat liars are multiplied together... | Win Vineeth | 311,216 | <p>(a)Fill the first $\lfloor\frac S9 \rfloor$ digits by $9$<br>Fill the next digit by $S\mod9$ <br>Fill the rest of the digits by $0$</p>
<p>(b)Fill the last $\lfloor\frac {S-1}9 \rfloor$ digits by $9$<br>Fill the previous digit by $S-1\mod9$ <br> Fill the rest of the digits by $0$<br> Add $1$ to the first digit.</p>... |
1,087,910 | <p>Erdős proved that if $f(n)$ is a monotone increasing function from the natural numbers to the positive reals, and $f(n)$ is completely multiplicative, then there exists some constant $C$ such that $f(n)=n^C$ for all $n$.</p>
<p>I have not been able to find a nice proof of this online and was hoping someone could pr... | Dr. Sonnhard Graubner | 175,066 | <p>hint: use that $$\frac{1}{x^2(x-1)^3}=3\, \left( x-1 \right) ^{-1}-{x}^{-2}+ \left( x-1 \right) ^{-3}-2\,
\left( x-1 \right) ^{-2}-3\,{x}^{-1}
$$</p>
|
3,369,129 | <p>I have the following formulas: </p>
<p><span class="math-container">$$S = 4 \times \tan\left(\frac{180^\circ-a}{2}\right) - \frac{π}{90^\circ} \times (180^\circ-a)$$</span></p>
<p>where:</p>
<p><span class="math-container">$$S \gt 0$$</span></p>
<p><span class="math-container">$$0^\circ \lt a \lt 180^\circ$$</... | albert chan | 696,342 | <p>Rewrite your expression, angle in radians</p>
<p><span class="math-container">$${S\over4} + {\pi \over 2} =\cot{a\over2} +{a\over2}$$</span>
<span class="math-container">$$a_{i+1} = 2\cot^{-1}({S \over 4} + {\pi-a_i \over 2})$$</span></p>
<p>Example, with <span class="math-container">$S=10, a_0=0$</span>, using <a... |
3,744,582 | <p>I was told that</p>
<blockquote>
<p>Two permutation matrices represent conjugate permutations iff they have same characteristic polynomial (where the conjugacy is considered only in <span class="math-container">$S_{n}$</span>).</p>
</blockquote>
<p>The first implication is clear to me i.e. permutation matrices repre... | ImBatman | 764,333 | <p>We'll first prove that the characteristic polynomial of a permutation matrix representing a permutation <span class="math-container">$\pi\in S_{n}$</span> having the cycle structure <span class="math-container">$1^{c_{1}}2^{c_{2}}...n^{c_{n}}$</span> is of the form <span class="math-container">$\prod_{j=1}^{n}(x^{j}... |
3,744,582 | <p>I was told that</p>
<blockquote>
<p>Two permutation matrices represent conjugate permutations iff they have same characteristic polynomial (where the conjugacy is considered only in <span class="math-container">$S_{n}$</span>).</p>
</blockquote>
<p>The first implication is clear to me i.e. permutation matrices repre... | Eric Wofsey | 86,856 | <p>If a matrix <span class="math-container">$A$</span> has eigenvalues <span class="math-container">$\lambda_1,\dots,\lambda_n$</span> (listed with algebraic multiplicity), then <span class="math-container">$A^k$</span> has eigenvalues <span class="math-container">$\lambda_1^k,\dots,\lambda_n^k$</span>, and so <span cl... |
11,172 | <p>As a TA who led calculus* 1 and 2 discussion section and holds office hour** in the previous year, I heard the following (wrong) arguments several times.</p>
<blockquote>
<ol>
<li><p><span class="math-container">$\displaystyle \lim_{x\to \infty} \sqrt{x+1}-\sqrt{x}=0$</span> because <span class="math-container">$\in... | Daniel Hast | 3,505 | <p>Often in courses like this, the problem is that students are almost exclusively asked to do problems that involve rote computation using procedures they've been taught, not problems that require reasoning about the concepts involved or even constructing their own examples. So, how about assigning problems that requi... |
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