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 |
|---|---|---|---|---|
488,498 | <p>Find $dy/dx$ in terms of $x$ and $y$ when
$$x^2y^3=7x−3y$$</p>
<p>Not sure how to start here, would be nice with some pointers </p>
| math110 | 58,742 | <p><strong>solution 1:</strong>
$$x^2y^3=7x-3y$$
$$\Longrightarrow 2xy^3+x^2\cdot 3y^2\cdot y'=7-3y'$$
$$\dfrac{dy}{dx}=y'=\dfrac{7-2xy^3}{3x^2y^2+3}$$</p>
<p><strong>solution 2:</strong></p>
<p>let
$$F(x,y)=x^2y^3-7x+3y$$
$$F'_{x}(x,y)=2xy^3-7$$
$$F'_{y}(x,y)=3x^2y^2+3$$
so
$$\dfrac{dy}{dx}=-\dfrac{F'_{x}(x,y)}{F'_{... |
3,786,797 | <p>I have two variables <span class="math-container">$x$</span> and <span class="math-container">$y$</span> which are measured at two time steps <span class="math-container">$0$</span> and <span class="math-container">$1$</span>. Available to me is the percent change from the first timestep to the second, relative to t... | Graham Kemp | 135,106 | <p> The first task is to decide were to <em>place</em> men and women so that at least one woman is placed between each man, (in doing so, decide which of these six spaces to put the remaining woman place: <span class="math-container">$\rm\_mw\_mw\_mw\_mw\_m\_$</span>).
<span class="math-container">$$\rm wmwmwmwmwm\\ ... |
2,592,581 | <p>Lets define the following series
$$S_n =\sum_{i=1}^{\infty} \frac 1 {i(i+1)(i+2)...(i+n)}.$$
I know that $S_0$ does not converge so let's suppose $n \in N$ and define $S_n$ for some $n$. We have $S_1=1$ , $S_2=\frac1 4$ , $S_3=\frac 1 {18}$ , $S_4=\frac 1{96}$ , $S_5=\frac 1{600}$ etc..<br/><br/>
the numerator in al... | Masacroso | 173,262 | <p>I will add another answer with a different approach (less elementary but also interesting). From <a href="https://www.cs.purdue.edu/homes/dgleich/publications/Gleich%202005%20-%20finite%20calculus.pdf" rel="nofollow noreferrer">finite calculus theory</a> your series can also be stated as <span class="math-container"... |
33,125 | <p>For some fixed $n$ define the quadratic form $$Q(x,y) = x^2 + n y^2.$$</p>
<p>I think that if $Q$ represents $m$ in two different ways then $m$ is composite.</p>
<p>I can prove this for $n$ prime. I was hoping someone could give me a hint towards proving this result for general $n$? Also would be interested in gen... | Bill Dubuque | 242 | <p>Below is Lucas' classic proof, from his <em>Theorie des nombres</em>, 1891, as described in section 215 of Mathews: <a href="http://books.google.com/books?id=iQ_vAAAAMAAJ&pg=PP1" rel="noreferrer">Theory of Numbers.</a>
<img src="https://i.stack.imgur.com/SzPeR.jpg" alt="enter image description here">
<img src="h... |
1,737,835 | <p>You have decided to buy candy for the trick-or-treaters and have estimated there will be 200 children coming to your door, and plan to give each children three pieces of candy. You have decided to offer Twix and 3 Musketeers. The cost of buying these two candies is<br>
$$C= 5T^2 + 2TM + 3M^2 + 800$$
Where T is... | Josse van Dobben de Bruyn | 246,783 | <p>Let $X \subseteq \mathbb{R}^n$ be any compact convex set that is symmetric about the origin and contains an open neighbourhood of $0$. Then we can define the <em><a href="https://en.wikipedia.org/wiki/Minkowski_functional" rel="nofollow noreferrer">Minkowski functional</a></em> $p_X : \mathbb{R}^n \to \mathbb{R}_{\g... |
3,911,610 | <p>Given the equation
<span class="math-container">$PV=nRT$</span>
if we evaluate <span class="math-container">$\frac{\partial{P}}{\partial{V}} * \frac{\partial{V}}{\partial{T}} * \frac{\partial{T}}{\partial{P}}$</span> through taking all three partials and multiplying, we get <span class="math-container">$-1$</span>. ... | mechanodroid | 144,766 | <p>Let <span class="math-container">$\{x_1, \ldots, x_n\}$</span> be a basis in the domain such that <span class="math-container">$\{Ax_1, \ldots, Ax_n\}$</span> is a basis in the codomain.</p>
<p>For injectivity, we wish to prove that <span class="math-container">$A$</span> has trivial kernel. Assume <span class="math... |
3,911,610 | <p>Given the equation
<span class="math-container">$PV=nRT$</span>
if we evaluate <span class="math-container">$\frac{\partial{P}}{\partial{V}} * \frac{\partial{V}}{\partial{T}} * \frac{\partial{T}}{\partial{P}}$</span> through taking all three partials and multiplying, we get <span class="math-container">$-1$</span>. ... | Ben Grossmann | 81,360 | <p>I assume that we consider finite dimensional spaces to keep the notation approachable. That said, the proof can be extended to the case of arbitrary dimension without too much effort.</p>
<p>Suppose that <span class="math-container">$T:V \to W$</span>. Let <span class="math-container">$\{v_1,\dots,v_n\}$</span> be a... |
3,701,175 | <p>If a function <span class="math-container">$h(x)$</span> satisfies:</p>
<p>exists partition <span class="math-container">$ P=\left\{ a_{0},a_{1},...,a_{n}\right\} $</span></p>
<p>of the interval [a,b], such that <span class="math-container">$ h $</span> is constant in the segment <span class="math-container">$(a_... | RRL | 148,510 | <p>If <span class="math-container">$f$</span> is <strong>Riemann integrable</strong>, then for any <span class="math-container">$\epsilon > 0$</span> there exists <span class="math-container">$P: a = x_0 < x_1< \ldots x_n = b$</span>, a partition of <span class="math-container">$[a,b]$</span>, such that</p>
<... |
3,809,026 | <p>I'm trying to solve this problem in the implicit differentiation section of the book I'm going through:</p>
<blockquote>
<p>The equation that implicitly defines <span class="math-container">$f$</span> can be written as</p>
<p><span class="math-container">$y = \dfrac{2 \sin x + \cos y}{3}$</span></p>
<p>In this probl... | aangulog | 597,826 | <p>If the base case doesn't work for the given range then the statment to prove is false.<br />
For example if you are given "Show that for every <span class="math-container">$\mathbb{N}$</span> the inequality <span class="math-container">$0<x-1$</span> holds" as for <span class="math-container">$x=1$</spa... |
309,841 | <p>Let $m$ be a positive integer. Prove that the dihedral group $D_{4m+2}$ of order $8m+4$ contains a subgroup of order 4.</p>
<p>Im fairly stumped as to where to start with this one, any input would be helpful.
Thanks</p>
| Vectornaut | 16,063 | <p>This is a really insightful question! You're pointing out that, in quantum mechanics, there are two different notions of "tensor product":</p>
<ul>
<li>The composition of physical systems (a physical notion).</li>
<li>The tensor product of Hilbert spaces (a mathematical notion).</li>
</ul>
<p>You say you can't see... |
449,617 | <p>I would like to know how do you solve summation expressions in an easy way (from my understanding).
I am computer science student analyzing for loops and finding it's time complexity.</p>
<p>e.g</p>
<p><strong>Code</strong>:</p>
<pre><code> for i=1 to n
x++
end for
</code></pre>
<p><strong>Summation</stro... | MJD | 25,554 | <p>Hint for #1: Suppose that $G_0$ is the purported graph, which contains any graph $H$ unless $H$ in turn contains $K_\omega$, the complete countable graph. Make a new graph, $G'_0$, by appending a new vertex $v_0$ to $G_0$ and adding edges from $v_0$ to every vertex of $G_0$. Then $G'_0$ does not contain $K_\omega$ (... |
63,601 | <p>I know that if a ring has a multiplicative identity, then the multiplicative identity must be unique. Are there simple-to-describe examples of rings with two (or more) multiplicative right-identities?</p>
| Salvo Tringali | 16,537 | <p>I was browsing some "What is...?" threads, and bumped in here: Let $n$ be a positive integer, and consider the subring of the <a href="https://en.wikipedia.org/wiki/Opposite_ring" rel="nofollow">opposite</a> of the ring of $n$-by-$n$ matrices over a commutative unital ring $\mathbb A = (A, +, \cdot)$ consisting of t... |
18,564 | <p>I want to get the eight points of intersection from the equations <code>2 Abs[x] + Abs[y] == 1</code> and <code>Abs[x] + 2 Abs[y] == 1</code>. To solve these equations, I tried</p>
<pre><code>Solve[{2 Abs[x] + Abs[y] == 1, Abs[x] + 2 Abs[y] == 1}, {x, y}]
</code></pre>
<p>but could only get the four points. I then... | rm -rf | 5 | <p>Your equations only have 4 points of intersections, not 8, so <code>Solve</code> is returning the correct result. To get the remaining 4, as you have indicated in the figure, you'll also have to consider the x and y axes. You can manually solve each curve with the appropriate axes (it seems like you don't care for t... |
1,250,322 | <p>This is the recurrence relation I am trying to solve:
\begin{align}
T(n) & = 2 \cdot T \left( \frac{n}{4} \right) + 16, \\
T(1) & = c.
\end{align}
I broke this down (i.e., solved this recurrence relation) to $ \sqrt{2} * c * n + 32 * \sqrt{2} * n - 32 $, which runs in tight bounds $ \Theta(n) $. Can you guys... | committedandroider | 170,019 | <p>I actually did most of the steps of solving this recurrence relation correctly. If anyone's having as similar issue, here is my mistake </p>
<p>My issue was that I simplified $2^{\frac{log_2n}{2}}$ as $2^{\frac12}*2^{log_2n}$, not $(2^{log_2n})^{\frac12}$. If I used the exponential rule correctly, I would have gott... |
1,160,741 | <p>Prove that continuous functions on $\mathbb{R}$ with period $1$ can be uniformly approximated by trigonometric polynomials, i.e. functions in the linear span of $\{e^{−2πinx}\}_{n \in \mathbb{Z}}$. Explain why $n \in \mathbb{N}$ is not enough.</p>
<p>I think I need to restrict the domain into a compact set and cons... | Cameron Williams | 22,551 | <p>The fundamental tenet of Stone-Weierstrass is an algebra of functions $\mathcal{A}$ on $X$ that is closed under complex conjugation, separate points and for every point $x\in X$ there is an $f\in\mathcal{A}$ such that $f(x)\neq 0$.</p>
<p>Your algebra satisfies the first property since $e^{-2\pi inx} = \overline{e^... |
1,368,815 | <p>Let $U\subset\mathbb{R}^n$ be an open set. I proved that </p>
<p>$$
H^0_{DR}(U):=\frac{\text{closed forms}}{\text{exact forms}}=\{f\in C^{\infty}(U):\,f\,\,\text{is locally constant} \}
$$</p>
<p>I have to show that $\dim H^0_{DR}=\text{number of connected components of}\,\, U$.</p>
<p>Here is my incomplete proof... | Michael Albanese | 39,599 | <p>A function $f : U \to \mathbb{R}$ is <em>locally constant</em> if for each $x \in U$, there is an open neighbourhood $V$ of $x$ such that $f|_V$ is constant. </p>
<p>Note that for any $y \in \mathbb{R}$, $f^{-1}(y)$ is open, so for any $A\subseteq \mathbb{R}$, $f^{-1}(A) = \bigcup_{y\in A}f^{-1}(y)$ is open. In par... |
962,301 | <p>$$(\sqrt 2-\sqrt 1)+(\sqrt 3-\sqrt 2)+(\sqrt 4-\sqrt 3)+(\sqrt 5-\sqrt 4)…$$</p>
<p>I have found partial sums equal to natural numbers. The first 3 addends sum to 1. The first 8 sum to 2. The first 15 sum to 3. When the minuend in an addend is the square root of a perfect square, the partial sum is a natural number... | C.S. | 95,894 | <p>So your series can be written as $$\sum_{n=1}^{\infty} (\sqrt{n+1}-\sqrt{n}) =\sum_{n=1}^{\infty} \frac{1}{\sqrt{n+1}+\sqrt{n}} \geq \frac{1}{2}\sum_{n=1}^{\infty}\frac{1}{\sqrt{n+1}}$$ and so by <strong><a href="http://en.wikipedia.org/wiki/Direct_comparison_test" rel="nofollow">Comparison Test</a></strong> diverge... |
52,194 | <p>Assuming you have a set of nodes, how do you determine how many connections are needed to connect every node to every other node in the set?</p>
<p>Example input and output:</p>
<pre><code>In Out
<=1 0
2 1
3 3
4 6
5 10
6 15
</code></pre>
| Xiang | 13,405 | <p>Here is what you want.$$\sum_{k=1}^{n-1}k=\frac{n(n-1)}{2}$$</p>
|
1,399,008 | <p>I'm asked to prove that $$\lim_{n \to \infty}\left(\frac{n}{n^2+1}+\frac{n}{n^2+4}+\frac{n}{n^2+9}+\cdots+\frac{n}{n^2+n^2}\right)=\frac{\pi}{4}$$ This looks like it can be solved with Riemann sums, so I proceed:</p>
<p>\begin{align*}
\lim_{n \to \infty}\left(\frac{n}{n^2+1}+\frac{n}{n^2+4}+\frac{n}{n^2+9}+\cdots+\... | davidlowryduda | 9,754 | <p>Yes, what you've written is correct.</p>
<p>I think your write-up would be cleaner if you explicitly mentioned the antiderivative
$$ \int \frac{1}{1 + x^2} dx = \arctan x,$$
which makes it completely clear where $\frac{\pi}{4}$ comes from at the end.</p>
|
13,478 | <p>This may be a silly question - but are there interesting results about the invariant: the minimal size of an open affine cover? For example, can it be expressed in a nice way? Maybe under some additional hypotheses?</p>
| David E Speyer | 297 | <p>I think there are some relevant comments in the work of Roth and Vakil, <em><a href="https://arxiv.org/abs/math/0406384" rel="nofollow noreferrer">The affine stratification number and the moduli space of curves</a></em>.</p>
|
2,839,823 | <p>I don't know where to ask, but I'm trying. I just thing we cannot do hocus pocus methods.</p>
<hr>
<p>Solve $3^x+4^x=5^x$</p>
<p>Okay, so my friend gave me this equation, and his solution. But I don't belive it holds. It comes here:</p>
<p>"Solution: </p>
<p>$3^x+4^x=5^x\Leftrightarrow \frac{3^x}{3^x}+\frac{4^x... | Alan | 175,602 | <p>If you're asking is this the only possible solution, the logical breakdown I see is at the step where he insists on the particular breakdown of -1 into that product. As far as I can see, you could do the same with any other two factors of -1. Not sure if that will always end up with x=2 or not, but it's an are... |
1,576,235 | <p>Take, for example: $3x + y + 2z = 6$.</p>
<p>Parameterized as: $ui + (6 -3u - 2v)j + vk$
restricted to the first octant</p>
| Ross Millikan | 1,827 | <p>A plane is closed. A half-space is closed. Finite intersections of closed sets are closed. So as a subset of $\Bbb R^3$ it is closed. If your definition of a closed surface is as in <a href="https://en.wikipedia.org/wiki/Surface#Closed_surfaces" rel="nofollow">Wikipedia</a>, a surface that is compact and withou... |
342,684 | <p>It would be useful to me to have a result of the following kind (which I would need to generalize, but this case is already interesting). Let <span class="math-container">$r<n$</span> be positive integers and let <span class="math-container">$\delta>0$</span> be a fixed constant such as 1/100. Does there exist... | Jyrki Lahtonen | 15,503 | <p>As an example I describe what is known about the BCH-codes for the purposes of this question. Just to give an idea of the type of results that are known. I suspect that they are not very useful for the full range of parameters here.</p>
<p>Let us specify a natural number <span class="math-container">$m$</span>, and... |
142,007 | <p>Assume: $p$ is a prime that satisfies $p \equiv 3 \pmod 4$</p>
<p>Show: $x^{2} \equiv -1 \pmod p$ has no solutions $\forall x \in \mathbb{Z}$.</p>
<p>I know this problem has something to do with Fermat's Little Theorem, that $a^{p-1} \equiv 1\pmod p$. I tried to do a proof by contradiction, assuming the conclusion... | Bill Dubuque | 242 | <p><strong>Hint</strong> <span class="math-container">$\bmod P\! = 4K\!+\!3\!:\;$</span> <span class="math-container">$\ \color{#c00}{X^{\large 2} \equiv -1} \;\Rightarrow\; 1\equiv X^{\large P-1} \equiv (\color{#c00}{X^{\large 2}})^{\large 2K+1}\equiv (\color{#c00}{-1})^{\large 2K+1} \equiv -1$</span></p>
<p><strong>A... |
142,007 | <p>Assume: $p$ is a prime that satisfies $p \equiv 3 \pmod 4$</p>
<p>Show: $x^{2} \equiv -1 \pmod p$ has no solutions $\forall x \in \mathbb{Z}$.</p>
<p>I know this problem has something to do with Fermat's Little Theorem, that $a^{p-1} \equiv 1\pmod p$. I tried to do a proof by contradiction, assuming the conclusion... | Oliver Kayende | 704,766 | <p>A solution to the given congruence would imply there exists an element of multiplicative order 4 in the finite field of size p which is false because (p-1)/2 is odd and therefore p-1 is not divisible by 4.</p>
|
142,007 | <p>Assume: $p$ is a prime that satisfies $p \equiv 3 \pmod 4$</p>
<p>Show: $x^{2} \equiv -1 \pmod p$ has no solutions $\forall x \in \mathbb{Z}$.</p>
<p>I know this problem has something to do with Fermat's Little Theorem, that $a^{p-1} \equiv 1\pmod p$. I tried to do a proof by contradiction, assuming the conclusion... | matqkks | 30,379 | <p>Isn't this easier than using FLT.
Consider a solution x then x is even or odd. Substituting this into x^2=-1 (mod p) leads to a contradiction in both cases.</p>
|
185,900 | <p>What is the <strong>fastest way</strong> to find the smallest positive root of the following transcendental equation:</p>
<p><span class="math-container">$$a + b\cdot e^{-0.045 t} = n \sin(t) - m \cos(t)$$</span></p>
<pre><code>eq = a + b E^(-0.045 t) == n Sin[t] - m Cos[t];
</code></pre>
<p>where
<span class="m... | Michael E2 | 4,999 | <p>You can solve by hand for to isolate the root. The following identity is helpful:</p>
<pre><code>n Sin[t] - m Cos[t] == Sqrt[m^2 + n^2] Cos[t - ArcTan[-m, n]]
</code></pre>
<p>Then you can figure out that the earliest to start your search is here:</p>
<pre><code>Solve[a + b E^(-45/1000 t) == Sqrt[m^2 + n^2], t, R... |
1,195,166 | <p>for a>0 can someone give me some clues to prove this inequality. </p>
<p><img src="https://i.stack.imgur.com/WFQA2.png" alt="enter image description here"></p>
<p>I found its derivative but I can't express the f(0) to use the Mean Value Theorem. Can you give me at least a point where to start at</p>
| Mark Bennet | 2,906 | <p>You should look at it as follows:</p>
<p>$$\sqrt {x+\delta x}-\sqrt x =\left(\sqrt {x+\delta x}-\sqrt x\right) \cdot \frac {\sqrt {x+\delta x}+\sqrt x }{\sqrt {x+\delta x}+\sqrt x}=\frac {\delta x}{\sqrt {x+\delta x}+\sqrt x }$$</p>
<p>Now the denominator is "large" compared with $\delta x$ and you don't need to g... |
643,658 | <p>Let <strong><em>K</em></strong> be a finite field, $F=K(\alpha)$ a finite simple extension of degree $n$, and $ f \in K[x]$ the minimal polynomial of $\alpha$ over $K$. Let $\frac{f\left( x \right)}{x-\alpha }={{\beta }_{0}}+{{\beta }_{1}}x+\cdots +{{\beta }_{n-1}}{{x}^{n-1}}\in F[x]$ and $\gamma={f}'\left( \alpha ... | Jyrki Lahtonen | 11,619 | <p>This is a standard fact of algebraic field extensions, and doesn't need the assumption that $K$ is finite.</p>
<p>Let $f(x)=\prod_{i=1}^n(x-\alpha_i),$ where $\alpha_1=\alpha$, so the conjugates of $\alpha$ are $\alpha_i, i=1,2,\ldots,n$. Let $0\le r\le n-1$. Consider the polynomial
$$
g_r(x)=\sum_{i=1}^n\frac{f(x)... |
459,124 | <p>Title says it all, I have an itch about series like this that seem to fall in the gray area where classical proofs that rational partial sums that converge too quickly must converge to transcendental don't seem to apply. This is such a special case, does any one know if the series</p>
<p>$$\sum_{i=0}^{\infty} \fra... | Theon Alexander | 165,460 | <p>It follows from Liouville's theorem. The partial sums converge too well.</p>
<p><a href="http://www-users.math.umn.edu/~garrett/m/mfms/notes_2013-14/04b_Liouville_approx.pdf" rel="nofollow">http://www-users.math.umn.edu/~garrett/m/mfms/notes_2013-14/04b_Liouville_approx.pdf</a></p>
<p>Assume that it is algebraic ... |
3,629,790 | <p>So I am a bit stuck on applying the limit definition for this problem. This is what I have so far: </p>
<p><span class="math-container">$\lvert x - c \rvert < \delta $</span> then <span class="math-container">$\lvert f(x) - l \rvert < \epsilon$$ $$\lvert x + 2 \rvert < \delta$</span> <span class="math-cont... | John Titor | 770,201 | <p>If <span class="math-container">$0<ε<2+$$\frac{1}4$$,then:
|f(x)-l|<ε$</span> <span class="math-container">$\Rightarrow$</span> <span class="math-container">$|x^2+x-5+3|<ε$</span> <span class="math-container">$\Rightarrow$$-ε<x^2+x-2<ε$</span> <span class="math-container">$\Rightarrow$</span> <spa... |
404,954 | <p>I've never seen phrases like "$\sqrt{5}$ people" or "a set with $\pi$ many elements". Are there sets with cardinality, say, $\frac{1}{2}$?</p>
<p><strong>Edit:</strong> As Brian M. Scott pointed out, the only real numbers that are cardinalities of sets are the non-negative integers. Could you explain why is it so?<... | MJD | 25,554 | <p>I am copying Brian M. Scott's comment here, because I think it answers the question completely and correctly:</p>
<blockquote>
<p>As the term <em>cardinality</em> is normally used, the only real numbers that are cardinalities of sets are the non-negative integers. The cardinalities of infinite sets, of course, ar... |
298,121 | <p>I was watching a video on functional programming and the speaker was introducing functional programming notation for functions:</p>
<p>". . . if <strong>g(a)</strong> is a function <strong>g</strong> with variable <strong>a</strong>, we can write it also as <strong>g a</strong> . . . now, this looks like <strong>g<... | Omar Antolín-Camarena | 1,070 | <p>Linear functions are often represented by matrices, in which case function application is the the same as multiplication of a matrix by a vector. If $f$ is a linear function, say, from $\mathbb{R}^m$ to $\mathbb{R}^n$, then there is an $n \times m$ matrix $A$ representing the function $f$ so that for any $v$ in $\ma... |
412,247 | <p>Let $X$ be a topological space. $X$ is connected if $X\neq U\cup V$ with open sets $U,V$ and $U\cap V=\emptyset$.</p>
<p>If you consider $A:=(0,1]\cup(2,3)\subset\mathbb R$, $A$ is not connected.</p>
<p>But how can you prove it? Clearly I have to find those open sets like above but how? Thanks!</p>
| Brian M. Scott | 12,042 | <p>Just take $U=(0,1]$ and $V=(2,3)$: those sets are both open in $A$. To see that $U$ is open in $A$, observe that it’s equal to $A\cap(0,2)$, where $(0,2)$ is open in $\Bbb R$.</p>
|
412,247 | <p>Let $X$ be a topological space. $X$ is connected if $X\neq U\cup V$ with open sets $U,V$ and $U\cap V=\emptyset$.</p>
<p>If you consider $A:=(0,1]\cup(2,3)\subset\mathbb R$, $A$ is not connected.</p>
<p>But how can you prove it? Clearly I have to find those open sets like above but how? Thanks!</p>
| Asaf Karagila | 622 | <p>Note that in $A$ the interval $(0,1]$ is open. It is open because $(0,1]=(0,1.1)\cap A$, so it is relatively open.</p>
<p>Therefore $A$ is the union of two nontrivial open sets, and therefore not connected.</p>
|
1,179,922 | <p>I would like to inquire whether there is a simple way to prove that a function is decreasing or not. For example how would I prove that the function</p>
<p>$$Y = (X^.5 - 1)/0.5$$ </p>
<p>is decreasing? I am not sure whether the negativity of the 2nd derivative solves this and would really like to understand the in... | Emily | 31,475 | <p>Intuitively, let's it's easier to go directly from the definition. Note that $\frac{(x^{1/2}-1)}{1/2} = 2 x^{1/2} - 2$. Note that $-2$ is constant. Now, consider $x_2 > x_1$ for any $x_2, x_1$ and compare $2x_2^{1/2}$ to $2x_1^{1/2}$:</p>
<p>$$\frac{2x_2^{1/2}}{2x_1^{1/2}} = \left(\frac{x_2}{x_1}\right)^{1/2}.$$... |
352,501 | <p>There are several things that confuse me about this proof, so I was wondering if anybody could clarify them for me.</p>
<blockquote>
<p><strong>Lemma</strong> Let <span class="math-container">$G$</span> be a group of order <span class="math-container">$30$</span>. Then the <span class="math-container">$5$</span>-Syl... | user113578 | 113,578 | <p>A group of order 30 has a normal 5-Sylow subgroup.</p>
<p>$|G|=2.3.5$</p>
<p>The divisors of $|G|$ are $1,2,3,5,6,10,15,30.$</p>
<p>$5|n_5-1\implies n_5=1,6\\3|n_3-1\implies n_3=1,10\\2|n_2-1\implies n_2=1,3,5,15.$</p>
<p>If possible let $n_5=6.$ Then $G$ has at least $5.6-5=25$ elements of order $5.$ Consequent... |
247,315 | <p>Please see my Edited version at the end of the post.</p>
<p>================================</p>
<p><a href="http://en.m.wikipedia.org/wiki/Cantor_set" rel="nofollow">http://en.m.wikipedia.org/wiki/Cantor_set</a></p>
<p>My definition of Cantor Set is just like that of wikipedia.</p>
<p>That is, $C=[0,1]\setminus... | Thibaut Dumont | 48,658 | <p>I don't have enough reputation to post a comment, so i'll write it here.</p>
<p>There's a little typo in the definition of the Cantor-set (first formula), the interval should be $\left(\frac{3k+1}{3^i},\frac{3k+2}{3^i}\right)$</p>
<p>I think you can prove it by induction using the recursive formula of the wikipedi... |
2,898,616 | <p>My Math teacher told me that $\frac{dx}{dy} = \frac{1}{\frac{dy}{dx}}$. I asked for a proof and he gave me the result by using the chain rule. I don't understand why this is the case for every function because the function $y=x^2$ doesn't have an inverse function. If this is the case then why is $\frac{dx}{dy}$ no... | Arnaldo | 391,612 | <p>If $y=x^2$ then $$\frac{dy}{dx}=2x$$
and $\frac{dy}{dx}$ is given by using implicity derivative w.r.t $y$:</p>
<p>$$1=2x\frac{dx}{dy}\to \frac{dx}{dy}=\frac{1}{2x}$$</p>
|
2,898,616 | <p>My Math teacher told me that $\frac{dx}{dy} = \frac{1}{\frac{dy}{dx}}$. I asked for a proof and he gave me the result by using the chain rule. I don't understand why this is the case for every function because the function $y=x^2$ doesn't have an inverse function. If this is the case then why is $\frac{dx}{dy}$ no... | Michael Hardy | 11,667 | <p>There is such a thing as a function being <b>locally</b> one-to-one, and you frequently, at least tacitly, rely on that idea in many of the implicit differentiation problems that get assigned in first-semester calculus.</p>
<p>To say that $f$ is locally one-to-one at a point $a$ in the domain of $f$ means that ther... |
2,898,616 | <p>My Math teacher told me that $\frac{dx}{dy} = \frac{1}{\frac{dy}{dx}}$. I asked for a proof and he gave me the result by using the chain rule. I don't understand why this is the case for every function because the function $y=x^2$ doesn't have an inverse function. If this is the case then why is $\frac{dx}{dy}$ no... | Christian Blatter | 1,303 | <p>The formula ${dx\over dy}={1\over dy/dx}$ as a general principle does not refer to a global situation, but to a window $W$ in the $(x,y)$-plane, centered at some point $(x_0,y_0)$, within which the "variables" $x$ and $y$ are dependent on each other in such a way that you can write $y=\phi(x)$, whereby $\phi(x_0)=y... |
2,198,462 | <p>Let $\mathbf{f} : \mathbb{R}^2 \rightarrow \mathbb{R}^2$ be given by:
$$\mathbf{f}(u, v) = (u − v,\sin(u) \sin(v)).$$</p>
<p>Let $(y, z) = (0, 3/4).$ Find the point $(a, b)$ with $0 < a < \frac{\pi}{2}$ such that $\mathbf{f}(a, b)=(y, z)$.</p>
<p>Then prove that the function $\mathbf{f}^{−1}$ such that $\mat... | Arnaldo | 391,612 | <p><strong>Hint for the second question:</strong></p>
<p>$$(a,b)=(u-v,\sin u \sin v)\\
u-v=a\to u=a+v \quad (1)\\
\sin u \sin v=b\to \sin (a+v)\sin v=b$$</p>
<p>So, using sum-product relation</p>
<p>$$\cos(a+2v)-\cos(a)=-2\sin(a+v)\sin v$$</p>
<p>so,</p>
<p>$$\cos(a+2v)-\cos(a)=-2b\to \cos(a+2v)=\cos(a)-2b\\
v=\f... |
119,492 | <p>First I define a function, just a sum of a few sin waves at different angular frequencies:</p>
<pre><code>ubdat = 50;
ws = 10*{2, 5, 10, 20, 40}
fn = Table[Sum[Sin[w*x], {w, ws}], {x, 0, ubdat, .001}];
pts = Length@fn
ListPlot[fn, Joined -> True, PlotRange -> {{0, 1000}, All}]
{20, 50, 100, 200, 400}
</code>... | david | 3,057 | <p>I've found that specifying the kernel filter length in <code>LowpassFilter</code> will give the better performance than leaving it unspecified. I'll show what I came up with. Note that I've modified your problem somewhat. I use <code>Fourier</code> with the <code>FourierParameters</code> set for "signal processing".... |
396,363 | <p>I have two random variables $X,Y$ which are independent and uniformly distributed on $(\frac{1}{2},1]$. Then I consider two more random variables, $D=|X-Y|$ and $Z=\log\frac{X}{Y}$. I would like to calculate both, the disitrbution functions $F_D(t), F_Z(t)$ and the the density functions $f_D(t),f_Z(t)$</p>
<p>To do... | alexlo | 47,612 | <p>Let's try for $D = |X-Y|$:
We have, for $t\geq 0$,
$$F_D(t) = P[D\leq t] = P[|X-Y|\leq t] = P[-t\leq X-Y \leq t] = P[-t+Y\leq X \leq t+Y] =\int_\mathbb{R}P[-t+y\leq X \leq t+y | Y=y]f_Y(y)dy =\int_\mathbb{R} P[-t+y\leq X \leq t+y]f_Y(y)dy= \\ = \int_\mathbb{R}\int_{-t+y}^{t+y} f_X(x)dx f_Y(y)dy= \dots$$
And you can ... |
3,107,194 | <p>Suppose <span class="math-container">$X$</span> is a compact metric space and <span class="math-container">$f: X \rightarrow \mathbb{R}$</span> is a function for which <span class="math-container">$f^{-1}([t,\infty))$</span> is closed for any real <span class="math-container">$t$</span>. Then <span class="math-conta... | José Carlos Santos | 446,262 | <p>You cannot know that, since it is not true. Suppose that <span class="math-container">$x\in\bigcap_{n=1}^\infty f^{-1}\bigl([n,\infty)\bigr)$</span>. Then <span class="math-container">$(\forall n\in\mathbb{N}):f(x)\geqslant n$</span>. But no real number is greater than or equal to <em>every</em> natural number.</p>
|
2,709,532 | <p>(True/False) If $S$ is a non-empty subset of $\mathbb{N}$, then there exists an element $m \in S$ such that $m\ge k$ for all $k \in S$.</p>
<p>So my reasoning is that the above statement is false. Since $S$ is a non-empty subset of $\mathbb{N}$, it may also be the case that $S=\mathbb{N}$, and so there may not be a... | egreg | 62,967 | <p>Your counterexample is perfectly good. No need to look for complicated ones: there is no maximum natural number, and that's all you need for deciding that the statement is false.</p>
<hr>
<p>For your own delight, you can prove that the property of having a maximum characterizes nonempty <em>finite</em> subsets of ... |
973,101 | <p>I am trying to find a way to generate random points uniformly distributed on the surface of an ellipsoid.</p>
<p>If it was a sphere there is a neat way of doing it: Generate three $N(0,1)$ variables $\{x_1,x_2,x_3\}$, calculate the distance from the origin</p>
<p>$$d=\sqrt{x_1^2+x_2^2+x_3^2}$$</p>
<p>and calculat... | elhuhdron | 501,509 | <p>What if you use the normal distribution method, as you propose, but use the respective radii as the variances? Then scale with d, as proposed. Empirically this does not look more bunched (at least to me) at "cigar poles" than a brute force selection from uniform distributed points in <span class="math-cont... |
3,930,373 | <p>Hi so im working on a question about finding the <span class="math-container">$rank(A)$</span> and the <span class="math-container">$dim(Ker(A))$</span> of a 7x5 Matrix. Without being given an actual matrix to work from.</p>
<p>I have been told that that the homogeneous equation <span class="math-container">$A\vec x... | Figaro | 829,712 | <p>The best way to interpret this limit as a complex number. Which can be written as: <span class="math-container">$$\underset{x\to0}{lim}\hspace{0.3cm}x^2i+0$$</span>
As always, you can separate both parts of the limit to get a more clear view of why the result is a real 0:
<span class="math-container">$$=\underset{x\... |
913,239 | <p>Given a circle with known center $c$, known radius $r$ and perimeter point $x$:
$$
(x - c_x)^2 + (y - c_y)^2 = r^2
$$
with a tangent line that also goes through a point $p$ lying outside the circle. How do I find the point $x$ at which the line touches the circle?</p>
<p>Given that the tangent line is orthogonal to... | RE60K | 67,609 | <p>Writing Chord of Contact for the circle ( which you wrote via vector approach ), where X and Y are coordinates of external point and $\alpha,\beta$ are the points on the circle where the tangent touches:
$$(\alpha - c_x) (\alpha - X) + (\beta - c_y) (\beta - Y) = 0$$
Also this cuts the circle at two different point... |
1,123,513 | <p>The question is$$\text{Is }\mathbb{R}^2\text{ a subspace of }\mathbb{C}^2?$$My first thing to think about it now is $$\text{Is }\mathbb{R}^2\text{ a subset of }\mathbb{C}^2?$$ I think no because what is $\mathbb{R}^2$ it's like all those pairs $(0,1)$ right? and what is $\mathbb{C}^2$ it's like euh $((a,b),(c,d))$ (... | hmakholm left over Monica | 14,366 | <p>Usually we say that $\mathbb R$ is a subset of $\mathbb C$, because any real number, such as $0$ or $2$ or $\pi$ is also a member of $\mathbb C$. There are <a href="https://math.stackexchange.com/questions/876195/what-is-an-advatage-of-defining-mathbbc-as-a-set-containing-mathbbr">various schools of thought</a> abou... |
258,893 | <p>I have a set of three moderately simple simultaneous equations that I'd like to simplify and eliminate a set of variables for (this is a simple example of a more general class of problem - but I'd like to get the simple example working before starting on the bigger ones). Asking Mathematica to <code>Eliminate</code>... | Michael E2 | 4,999 | <p>Setup:</p>
<pre><code>b1 = {{0, 0, 0}, {0, 0, 1}, {0, -1, 0}};
b2 = {{0, 0, -1}, {0, 0, 0}, {1, 0, 0}};
b3 = {{0, 1, 0}, {-1, 0, 0}, {0, 0, 0}};
sys = {xt1, xt2, xt3} == {x1, x2, x3} . MatrixExp[t1 b1] .
MatrixExp[t2 b2] . MatrixExp[t3 b3];
</code></pre>
<hr />
<p>Eliminating three variables from the three equa... |
3,276,958 | <p>Given two points <span class="math-container">$P_1(x_1,y_1)$</span> and <span class="math-container">$P_2(x_2,y_2)$</span> with <span class="math-container">$y_1>0$</span> and <span class="math-container">$y_2>0$</span> I need to find the parameters <span class="math-container">$a$</span> and <span class="math... | José Carlos Santos | 446,262 | <p>If <span class="math-container">$y_1=ab^{x_1}$</span> and <span class="math-container">$y_2=ab^{x_2}$</span>, then<span class="math-container">$$\frac{y_2}{y_1}=\frac{ab^{x_2}}{ab^{x_1}}=b^{x_2-x_1}.$$</span>So, take <span class="math-container">$b=\left(\dfrac{y_2}{y_1}\right)^{1/(x_2-x_1)}$</span>. And now <span c... |
2,369,133 | <p>I just found out that
if you want to get 1 with the fraction: $$\frac{5}{2}$$
Then you multiply it with: $$ \frac{2}{5} $$
Does anyone have a good way to think about this? </p>
| Dando18 | 274,085 | <p>This is called <a href="https://en.wikipedia.org/wiki/Multiplicative_inverse" rel="nofollow noreferrer">The Inverse Property of Multiplication</a>. Take $x$,
then </p>
<p>$$x \cdot \frac 1 x = 1$$</p>
<p>In your case of $x=\frac 5 2$,</p>
<p>$$ \frac 5 2 \cdot \frac{1}{\frac 5 2} = \frac 5 2 \cdot \frac 2 5 = 1 $... |
392,441 | <p>Assume that $f$ is infinitely differentiable. Let $\delta$ be the (Dirac) delta functional.</p>
<p>I know that $f\delta = f(0)\delta$, but I'm not sure how to derive the equation $f\delta′=f(0)\delta′−f′(0)\delta$.</p>
| user43286 | 116,305 | <p>you can also expand $f$ into Taylor series </p>
<p>$f(t)=f(0)+f'(0)t + \frac{f''(0)}{2!}t^2+\cdots$ </p>
<p>and consider </p>
<p>$(t \delta(t))' = 0 = \delta(t) + t\delta'(t)$</p>
<p>So we have </p>
<p>$t\delta'(t) =- \delta(t)$</p>
<p>all high order terms involving $t^n \delta'(t)=0, n\ge2$. </p>
<p>Finally<... |
1,899,740 | <p>Let $(\Omega,\Sigma,\nu)$ be a measurable space with $\nu$ a positive $\sigma$-additive measure on $\Sigma$ and le $f:\Omega\to \mathbb{R}$ be a function such that,
$$
\int_{\Omega}f(x)d\nu(x)=0
$$
Does this imply that $f=0\quad \nu-\ a.e.$ ?</p>
<p>Any hint will be appreciated :) thank you for your time !</p>
| Hasan Saad | 62,977 | <p>This result is true if and only if $f\geq0$ almost everywhere. Ant gave an example of where this result is fault when this condition is not true.</p>
<p>As for the proof when $f\geq0$, here goes:</p>
<p>$v(\{x\in\Omega|f(x)>0\})=\sum_2^\infty v(\{x\in\Omega|\frac{1}{n}\leq f(x)\leq\frac{1}{n-1}\})+v(\{x\in\Omeg... |
43,646 | <blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="https://math.stackexchange.com/questions/43531/havel-hakimi-theorem">Havel-Hakimi Theorem</a> </p>
</blockquote>
<p>Hi. I'm a beginner at graph theory, and I recently came across the Havel-Hakimi Theorem which is used to determine whether a seque... | Joseph Malkevitch | 1,369 | <p>Look at pages 44 and page 45 of the second edition of Doug West's book Introduction to Graph Theory for what I think is a clear account accompanied by a worked out example. </p>
|
732,132 | <blockquote>
<p>Suppose $P(x)$ is a polynomial of degree $n \geq 1$ such that $\int_{0}^{1}x^{k}P(x)\,dx = 0$ for $k = 1, 2, \ldots, n$. Show that $$\int_{0}^{1}\{P(x)\}^{2}\,dx = (n + 1)^{2}\left(\int_{0}^{1}P(x)\,dx\right)^{2}$$</p>
</blockquote>
<p>If we assume that $P(x) = a_{0}x^{n} + \cdots + a_{n - 1}x + a_{n... | r9m | 129,017 | <p>Since, $\displaystyle \int_0^1 x^kP(x)\,dx=\frac{a_0}{n+k+1}+\frac{a_1}{n+k}+\ldots+\frac{a_n}{k+1}=0$, for each $k=1,2,\ldots,n$</p>
<p>Then $f(x)=\dfrac{a_0}{n+x+1}+\dfrac{a_1}{n+x}+\ldots+\dfrac{a_n}{x+1}=\dfrac{Q(x)}{(n+x+1)\ldots(x+1)}$ (where, Q is a polynomial of degree atmost n), has $n$ zeros $x=1,2,\ldots... |
261,605 | <p>Applying the <a href="https://en.wikipedia.org/wiki/M%C3%BCntz%E2%80%93Sz%C3%A1sz_theorem" rel="nofollow noreferrer">Müntz–Szász theorem</a> on $[0,1]$ repeatedly, we can represent
$$
x= \sum_{n\geq 2} c_n x^n
$$
as a uniformly convergent series (<strong>edit:</strong> only over some subsequence, see edits below) o... | Kevin Buzzard | 1,384 | <p>[Sorry -- too long for a comment]. It has already been pointed out that if you want 0 in your domain then you'll have to use constant functions too. But even then the theorem does not say what you seem to think it says: it only says that we can approximate $f(x)=x$ arbitrarily well by such a power series -- and as t... |
100,957 | <p>I have this picture of small particles in a polymer film. I want to count how many particles in the figure, so that I can have a rough estimation of the particle density. But the image quality is poor, I had a hard time to do it.</p>
<p><a href="https://i.stack.imgur.com/ez50R.jpg" rel="noreferrer"><img src="https:... | yode | 21,532 | <p>get the binarize image</p>
<pre><code> img = Import["http://i.stack.imgur.com/0OJd7.jpg"];
binimg = LocalAdaptiveBinarize[img, 25];
</code></pre>
<p><a href="https://i.stack.imgur.com/Fbvri.png" rel="noreferrer"><img src="https://i.stack.imgur.com/Fbvri.png" alt="enter image description here"></a></p>
<p>The eff... |
990,467 | <p>I am having some trouble understanding these two questions. Any help is appreciated. Scanned questions are included at the end.</p>
<p>6) We are given the function $ f(x) =\frac{1 - 2x} {2x^2 - 3x - 2} $</p>
<p>6 a) Find the equation of the vertical asymptotes. Explain how.</p>
<p>For the above question how ... | Jasser | 170,011 | <p>It would be enough to find the intersection of two graphs $2^{\cos x}$ and $\sin x$ over interval $[0,2\pi ]$ and since the graph again repeats after this the other solutions cvan be found by just adding $2k\pi$ where $k=0,1,2...$ to the intersection points. Hence plotting the graph once will help you a lot.</p>
|
1,169,591 | <p>For all finite vector spaces and all linear transformations to/from those spaces, <strong>how can you prove/show from the definition of a linear transformation</strong> that all linear transformations can be calculated using a matrix.</p>
| pjs36 | 120,540 | <p>Each linear transformation $T: V \to W$ is completely determined by what it does to a set of basis vectors of $V$. </p>
<p>That is, suppose we have a linear transformation $T: V \to W$ where $V$ and $W$ are finite-dimensional vector spaces with bases $\mathcal{B}_V = \{e_1, e_2, \ldots, e_n\}$ and $\mathcal{B}_W = ... |
1,169,591 | <p>For all finite vector spaces and all linear transformations to/from those spaces, <strong>how can you prove/show from the definition of a linear transformation</strong> that all linear transformations can be calculated using a matrix.</p>
| Disintegrating By Parts | 112,478 | <p>If you have a linear transform $L : X \rightarrow Y$, where $X$ and $Y$ are finite dimensional linear spaces, then you choose a basis $\{ x_{i} \}_{i=1}^{n}$ of $X$ and a basis $\{ y_{j} \}_{j=1}^{m}$ of $Y$, and write
$$
Lx_{n} = \alpha_{1,n}y_{1}+\alpha_{2,n}y_{2}+\cdots+\alpha_{m,n}y_{m}.
$$
The co... |
111,343 | <p>Hallo,</p>
<p>I have the following question: Let $M$ smooth analytic manifold of dimension 4n. Assume furthermore that $M$ admits two foliations $A$, $B$, both with leaves of dimension 2n such that the leaves of $A$ are transvers to the leaves of $B$ at each point. Also the leaves of $A$ are $n$-dim complex manifol... | Robert Bryant | 13,972 | <p>The answer is 'no, in general': For example, if $(M^4,J)$ is <em>any</em> real-analytic almost complex $4$-manifold, one can easily construct (locally, in a neighborhood of any point of $M$) a pair $(A,B)$ of real-analytic, transverse foliations by pseudo-holomorphic curves, and these will satisfy your conditions. ... |
228,105 | <p>I want the last number after the "/". How can I do that?</p>
<pre><code>"http://arxiv.org/abs/math/0208009v1"
</code></pre>
<p>I just want <strong>0208009v1</strong></p>
<p>I want to import multiple links from the following:</p>
<pre><code>{{"http://arxiv.org/abs/math/0208009v1"}, \
{&q... | kglr | 125 | <pre><code>list = {{"http://arxiv.org/abs/math/0208009v1"},
{"http://arxiv.org/abs/0905.0227v1"},
{"http://arxiv.org/abs/0907.5143v2"},
{"http://arxiv.org/abs/math/0509348v1"},
{"http://arxiv.org/abs/math/0608711v2"},
{"http://arxiv.org/abs/math-ph/0002018... |
2,925,450 | <p>I am trying to solve the following problem: <span class="math-container">$|4-x| \leq |x|-2$</span>. I am trying to do it algebraically, but I'm getting a solution to the problem that makes no sense. I fail to see the error in my reasoning though. I hope to get an explanation where I went wrong.</p>
<p><span class="... | MathIsNice1729 | 274,536 | <p>For both $x$ and $4-x$ to be negative, $x<0$ and $x>4$, which is impossible.</p>
<p>As for the solution, the right side has to be positive, hence $| x | \geq 2$. Now, we solve it in $3$ parts.</p>
<p>In $( -\infty,-2 ]$, we have, $4+|x| \leq |x|-2$, which is not true.</p>
<p>In $(4,\infty )$, we have, $x-4 ... |
1,599,502 | <p>Solve for $x$ in exact value:</p>
<p>$\\3^{2x}-3^{x+2}+8=0$</p>
<p>I have tried substituting $3^x$ $=a$ but I didn't get anywhere.</p>
<p>$\\a^2-a^{1+\frac{2}{x}}+8=0$</p>
| Qiaochu Yuan | 232 | <p>"Unique irreducible module" means that there is a unique <em>isomorphism class</em> of irreducible modules; that is, every two irreducible modules are isomorphic. $\mathfrak{sl}_2$ does not have a unique irreducible module, since it has countably many nonisomorphic irreducible modules. </p>
|
4,645,582 | <h2><strong>Problem</strong></h2>
<p>Let <span class="math-container">$f$</span> be a continuous function, <span class="math-container">$f:[0,1]\to\mathbb{R}$</span> with <span class="math-container">$\int_{0}^1 (2x-1)f(x) dx = 0$</span>.</p>
<p>Prove that there exists a point c between <span class="math-container">$(0... | Morărescu Mihnea | 1,147,641 | <p>First, let's take a look at our conclusion:</p>
<p><span class="math-container">$\int_0^c(x-c)(f(x)-f(c))dx = 0 \Leftrightarrow \int_0^cxf(x)dx - \int_0^cxf(c)dx - \int_0^c cf(x)dx + \int_0^c cf(c)dx = 0 \Leftrightarrow \int_0^cxf(x)dx - f(c)\int_0^cxdx - c\int_0^cf(x)dx + c^2f(c) = 0 \Leftrightarrow \int_0^cxf(x)dx... |
4,535,957 | <blockquote>
<p>Please vote to close this question. It's really dumb as when I was reading the Peano axioms, axiom 8 didn't register. Don't waste your time reading this question.... I also cannot delete it (I have tried but it won't let me).</p>
</blockquote>
<p><span class="math-container">$1$</span> is defined as <sp... | Jay | 9,814 | <p>Axiom 8 in the Wikipedia list of axioms says that <span class="math-container">$0$</span> is not the successor of any natural number. Since <span class="math-container">$1 = S(0)$</span>, we have <span class="math-container">$1 \neq 0$</span>.</p>
|
4,535,957 | <blockquote>
<p>Please vote to close this question. It's really dumb as when I was reading the Peano axioms, axiom 8 didn't register. Don't waste your time reading this question.... I also cannot delete it (I have tried but it won't let me).</p>
</blockquote>
<p><span class="math-container">$1$</span> is defined as <sp... | 5xum | 112,884 | <p>You only need two of the axioms:</p>
<ol>
<li><em>Axiom 1</em>: <span class="math-container">$0$</span> is a natural number.</li>
<li><em>Axiom 8</em>: For every natural number <span class="math-container">$n$</span>, <span class="math-container">$S(n)=0$</span> is false.</li>
</ol>
<p>In particular, the statement o... |
2,557,608 | <p>Let $X$ be integer valued with Characteristic Function $\phi$. How to show that </p>
<p>$\ P(|X| = k)= \frac{1}{2\pi }\int _{-\pi} ^{\pi} e^{ikt} \phi(t) dt$</p>
<p>$P(S_n =k) =\frac{1}{2\pi }\int _{-\pi} ^{\pi} e^{ikt} (\phi(t))^n dt $</p>
| Jean Marie | 305,862 | <p>Using the formulas of half-angle substitution, also called Weierstrass substitution formulas,</p>
<p>$$\cos(u)=\dfrac{1-t^2}{1+t^2} , \ \sin(u)=\dfrac{2t}{1+t^2} \ \ \ \text{where} \ \ \ t:=\tan(\tfrac{u}{2}),$$</p>
<p>one has to prove, setting $a=\tan(\alpha/2)$ and $b=\tan(\beta/2)$, that :</p>
<p>$$2\dfrac{2a}... |
4,610,640 | <p>I have this equation :</p>
<p><span class="math-container">$$\frac{d\alpha}{dz} = - \frac{dr}{dz} * \frac{\tan(\alpha)}r $$</span></p>
<p>I searched for some similar examples but non of these equations was like this one.
I'm confused about this one. As far as I know, I used RK4 for equations like this : <span class=... | NameAlreadyUsed | 1,129,057 | <p>The way to proceed is to add and subtract the same quantity at the numerator:</p>
<p><span class="math-container">$\frac{s+1}{s+2}=\frac{s+1+(1-1)}{s+2}=\frac{s+2-1}{s+2}=\frac{s+2}{s+2}-\frac{1}{s+2}=1-\frac{1}{s+2}$</span></p>
<p>The procedure is justified by the fact that the two addends (1 and -1) cancel each ot... |
4,610,640 | <p>I have this equation :</p>
<p><span class="math-container">$$\frac{d\alpha}{dz} = - \frac{dr}{dz} * \frac{\tan(\alpha)}r $$</span></p>
<p>I searched for some similar examples but non of these equations was like this one.
I'm confused about this one. As far as I know, I used RK4 for equations like this : <span class=... | Ryszard Szwarc | 715,896 | <p>Perhaps it is more visible on substituting <span class="math-container">$x=s+2$</span> (hopefully it is legal in elementary school math course). Then
<span class="math-container">$${s+1\over s+2}={x-1\over x}$$</span>
Now it is hard to resist from dividing each term in the numerator by <span class="math-container">... |
3,560,378 | <p>Let <span class="math-container">$f: \mathbb{R} \to \mathbb{R}$</span> be a convex function. Let <span class="math-container">$x,y,z \in \mathbb{R}$</span> and <span class="math-container">$a,b,c \geq 0$</span> such that <span class="math-container">$a+b+c=1$</span>. We want to show that <span class="math-container"... | Riley | 565,701 | <p>If <span class="math-container">$c > 0$</span> then <span class="math-container">$b + c > 0$</span> so
<span class="math-container">$$ by + cz = (b + c)\left(\frac{b}{b + c}y + \frac{c}{b + c}z\right), $$</span>
and <span class="math-container">$a + (b + c) = 1$</span>. The motivation for this is that if you ... |
872,275 | <p>here's the question, how can I solve this:</p>
<p>$$\lim_{x \rightarrow \infty} x\sin (1/x) $$</p>
<p>Now, from textbooks I know it is possible to use the following substitution $x=1/t$, then, the ecuation is reformed in the following way </p>
<p>$$\frac{\sin t}{t}$$</p>
<p>then, and this is what I really can´t ... | 2'5 9'2 | 11,123 | <p>$$\lim_{x\to\infty}x\sin(1/x)$$ is like the limit of this sequence: $$1\sin(1/1), 10\sin(1/10), 100\sin(1/100),\ldots$$ where we have inserted $x$ marching along from $1$ to $10$ to $100$ on its merry way to $\infty$. This is almost literally the same as $$\frac{1}{1}\sin(1), \frac{1}{0.1}\sin(0.1), \frac{1}{0.01}\s... |
102,406 | <p>I have the following question: If $E$ is a ring spectrum, then a complex orientation of $E$ is an element of $E^2(\mathbb{C}P^{\infty})$ that is mapped to $1$ in $E^2(\mathbb{C}P^{1})$.
I have read that there is a natural orientation on $MU$ the spectrum of complex bordism. Can you tell me, how this orientation is d... | Akhil Mathew | 344 | <p>A complex orientation on a homotopy commutative ring spectrum $E$ is sometimes defined to be an element in $\widetilde{E}^2(\mathbb{CP}^\infty)$ which restricts to the canonical element of $\widetilde{E}^2(S^2) = \pi_0 E $. </p>
<p>In fact, there is another definition of an orientation, which generalizes better to ... |
102,406 | <p>I have the following question: If $E$ is a ring spectrum, then a complex orientation of $E$ is an element of $E^2(\mathbb{C}P^{\infty})$ that is mapped to $1$ in $E^2(\mathbb{C}P^{1})$.
I have read that there is a natural orientation on $MU$ the spectrum of complex bordism. Can you tell me, how this orientation is d... | Peter May | 14,447 | <p>I don't know how to comment rather than answer, but Craig, the equivalence is really not mysterious:
the zero section of the universal complex line bundle gives an equivalence from $\mathbb{C}P^{\infty}$ to the total space of the unit disk bundle $D$. The total space $S$ of the unit sphere bundle is really just $S^{... |
2,675,954 | <p>Fractals, when viewed as functions, are everywhere continuous and nowhere differentiable. Can this also be used as a definition for fractals?</p>
<p>i.e. Are all fractals everywhere continuous and nowhere differentiable? And also: Are all functions that are everywhere continuous and nowhere differentiable fractals?... | ancient mathematician | 414,424 | <p>Note that
$$
M:=
\begin{bmatrix}
yz-x^2&zx-y^2&xy-z^2\\
zx-y^2&xy-z^2&yz-x^2\\
xy-z^2&yz-x^2&zx-y^2
\end{bmatrix}
$$
is the matrix of $2\times 2$ cofactors of the matrix:
$$
N:=
\begin{bmatrix}
x & y & z\\
y & z & x\\
z& x & y \\
\end{bmatrix}.
$$</p>
<p>Then as usua... |
4,631,014 | <blockquote>
<p>Finding range of function</p>
</blockquote>
<blockquote>
<p><span class="math-container">$\displaystyle f(x)=\cos(x)\sin(x)+\cos(x)\sqrt{\sin^2(x)+\sin^2(\alpha)}$</span></p>
</blockquote>
<p>I have use Algebric inequality</p>
<p><span class="math-container">$\displaystyle -(a^2+b^2)\leq 2ab\leq (a^2+b^... | Yuri Negometyanov | 297,350 | <p>Let <span class="math-container">$p=\sin^2\alpha,\; y=\cos 2x,\;$</span> then
<span class="math-container">$$\cos x \sin x+\cos x\sqrt{\sin^2x+\sin^2\alpha\mathstrut}
=\dfrac{\pm\sqrt{1-y^2}\pm\sqrt{1-y^2+2p(1+y)\mathstrut}}2 = f(y),$$</span>
<span class="math-container">$$2f'(y) = \mp\dfrac y{\sqrt{1-y^2}}\pm\dfrac... |
227,732 | <p>Are there any general relations between the eigenvalues of a matrix $M$ and those of $M^2$?</p>
| Hagen von Eitzen | 39,174 | <p>Of course if $\lambda$ is an eigenvalue of $M$, then $\lambda^2$ is an eigenvalue of $M^2$:
$$M v =\lambda v\quad\Rightarrow\quad M^2v=M\lambda v=\lambda M v =\lambda^2 v.$$</p>
|
4,477,354 | <p>Fourier series for function <span class="math-container">$f(x)=c^x$</span>, <span class="math-container">$c\in\mathbb Z$</span>, <span class="math-container">$c>1$</span> on interval <span class="math-container">$(a,b)$</span>, where <span class="math-container">$a,b\in\mathbb R$</span>, <span class="math-contain... | Blitzer | 429,808 | <p>Lemma: For <span class="math-container">$x>0$</span> and <span class="math-container">$i=1,2,\dots$</span>,
<span class="math-container">$$x^i+i \geqslant ix+1$$</span>
Proof of lemma:</p>
<p>If <span class="math-container">$x \leqslant 1$</span> then
<span class="math-container">$$1+x+x^2+\cdots +x^{i-1} \leqsl... |
627,718 | <p>I need help evaluating the integrals in Fourier Series.</p>
<p>For example, for the function <span class="math-container">$\cos^{2}x$</span>, I can evaluate <span class="math-container">$a_0$</span>, <span class="math-container">$a_n$</span>, and <span class="math-container">$b_n$</span>, where <span class="math-con... | user71352 | 71,352 | <p>Hint:</p>
<p>$\cos^{2}(x)=\frac{1+\cos(2x)}{2}$ </p>
<p>and </p>
<p>$\cos(x)\cos(y)=\frac{1}{2}\big(\cos(x-y)-\cos(x+y)\big)$</p>
|
23,980 | <p>I am calculating huge data files with an external program. I would like then to import the data into Mathematica for analysis. The files are 2 columns and up to many millions of rows. </p>
<p>So for small data files I have just been using:</p>
<pre><code>dataTable = Import["data.txt", "Table"];
</code></pre>
... | BoLe | 6,555 | <p>Via @chuy and @rcollyer with sequential skipping and reading of records, and without backtracking.</p>
<pre><code>(
If[# != 1, Skip[s, Record, # - 1], Null];
ReadList[s, Number, 2]) & /@ Differences[p~Prepend~0]
</code></pre>
<p>List <code>p</code> of lines you want to read, sorted; eg. some random ten f... |
2,062,638 | <blockquote>
<p>Consider a curve $\gamma(t): [a,b] \to \mathbb{R}^n$. The curve
$$-\gamma(t)=\gamma(a+b-t) \,\,\,\,\,\,\,\,\,\,\,\,\,\, t \in [a,b]$$</p>
<p>is called the "reverse" curve (or path) of $\gamma(t)$.</p>
</blockquote>
<p>This definition is clear, but how is the "reverse" path defined in the follo... | Alex M. | 164,025 | <p>You want to map $a$ to $\gamma_2 (d)$, $b$ and $c$ to $\gamma_2 (c) = \gamma_1 (b)$ and $d$ to $\gamma_1 (a)$. Looking for a linear map $\varphi(t) = At + B$ taking $a$ and $b$ to $d$ and respectively $c$ we find $\begin{cases} Aa + B = d \\ Ab + B = c \end{cases}$, whence $A = \frac {d-c} {a-b}$ and $B = \frac {ac-... |
745,382 | <p>I'm interested in hypercomplexes, or number systems with many square roots of $-1$. Now, I know that <a href="http://en.wikipedia.org/wiki/Quaternion" rel="nofollow">quaternions</a> are non-commutative, but associative. I'm wondering if it's possible to have a number system with more "elements" (or square roots of... | hmakholm left over Monica | 14,366 | <p>Depends on what you mean by "number system". You can certainly construct the ring</p>
<p>$$ \mathbb R[X_1,X_2,\ldots, X_n]/\langle X_1^2-1, X_2^2-1, \ldots, X_n^2 - 1\rangle $$</p>
<p>in which $\prod_{i\in A} X_i$ is a square root of $-1$ for each of the $2^{n-1}$ odd-sized subsets $A\subseteq \{1,\ldots,n\}$. It ... |
2,722,170 | <p>I am a grade 11 student and I have to learn vectors for the IB exam. I know that to find the distance of a vector between two points in a 3 dimensional space for lets say point A and B, then you would use the formula
\begin{align*}
|AB|=\sqrt{(x_1-x_2)^2+(y_1-y_2)^2+(z_1-z_2)^2}.
\end{align*}
I am having difficulty... | DanielWainfleet | 254,665 | <p>Assuming $P=(x,0)\ne (0,0)$ and $\theta \not \in \{0,\pi /2\} $, whether you return to $P$ does not depend on $x$ but on the angle $\psi$ between the $X$-axis and the segment joining $P$ to the first point chosen on $L.$ Because we can multiply all distances by a positive constant without affecting the outcome.</p>
... |
131,320 | <p>Let $x=[a_0;a_1,a_2]$ be shorthand notation for the continued fraction
$$x=a_0+\frac{1}{a_1+\frac{1}{a_2}}.$$
Then every $x\in\mathbb{Q}$ can be represented as a finite continued fraction $[a_0;a_1,a_2,\dots,a_n]$, where $a_0\in\mathbb{Z}$ and $a_1,a_2,\dots,a_n\in\mathbb{N}$. Moreover, let
$$\begin{matrix}
p_0=a_0 ... | Did | 6,179 | <p>Write $x=[a_0;a_1,a_2,\ldots]$ as $x=F(a_0,a_1,a_2,\ldots)$, and let us first prove an auxiliary result;</p>
<blockquote>
<p>The function $(a_k)_{k\geqslant0}\mapsto F((a_k)_{k\geqslant0})$ is increasing with respect to its even-numbered arguments $a_{2k}$ and decreasing with respect to its odd-numbered arguments... |
131,320 | <p>Let $x=[a_0;a_1,a_2]$ be shorthand notation for the continued fraction
$$x=a_0+\frac{1}{a_1+\frac{1}{a_2}}.$$
Then every $x\in\mathbb{Q}$ can be represented as a finite continued fraction $[a_0;a_1,a_2,\dots,a_n]$, where $a_0\in\mathbb{Z}$ and $a_1,a_2,\dots,a_n\in\mathbb{N}$. Moreover, let
$$\begin{matrix}
p_0=a_0 ... | robjohn | 13,854 | <p>Given that
$$
\begin{array}{c}
p_0=a_0&\text{and}&q_0=1\\
p_1=a_0a_1+1&\text{and}&q_1=a_1\\
p_k=a_kp_{k-1}+p_{k-2}&\text{and}&q_k=a_kq_{k-1}+q_{k-2}\tag{1}
\end{array}
$$
notice that
$$
p_1q_0-p_0q_1=1\tag{2}
$$
and
$$
\begin{align}
p_{k+1}q_k-p_kq_{k+1}
&=(a_kp_k+p_{k-1})q_k-p_k(a_kq_k+q... |
217,580 | <p>Consider a compact manifold $M$ with boundary and corner. As an example, we could have the cube $\{(x_1, x_2,..x_n) \in \mathbb{R}^n : x_i \in [0,1]\}$. We could very well define the Laplacian $\Delta$ on such an $M$ with Dirichlet or Neumann boundary conditions. In such situations, are the eigenfunctions of the Lap... | Coffee | 40,090 | <p>Let me add some other references about elliptic regularity on singular manifolds:</p>
<p>1) B.-W. Schulze and his co-authors have developed a comprehensive calculus for elliptic pseudo-differential operators (in particular differential operators) on singular spaces, including elliptic regularity and Fredholm proper... |
3,507,927 | <p>Is there any simplification by the way of standard properties of Laplace Transform for this? <span class="math-container">$$L_{s} [t f(t) f'(t)]$$</span></p>
<p>where <span class="math-container">$f'(t)$</span> is <span class="math-container">$\frac{d}{dt} f(t)$</span></p>
<p>Alternately, is there a simplification... | GFauxPas | 173,170 | <p><span class="math-container">$$\int_0^{+\infty} f'(t)f(t) e^{-st} \, \mathrm d t = I(s)$$</span></p>
<p>Define:</p>
<p><span class="math-container">$$u(t)=f(t)e^{-st}$$</span>
<span class="math-container">$$u'(t)=f'(t)e^{-st}-s e^{-st} f(t)$$</span>
<span class="math-container">$$v'(t)=f'(t)$$</span>
<span class="... |
267,947 | <p>i want to use least square to find x and y that minimize the result of the following function for a series of points (xi,yi) -> (x1,y1), (x2,y2),...:</p>
<p>note: y = f(x)</p>
<pre><code>E(x,y) = SUM (y - ((mi*x) - (mi*xi)))^2
i --------------------
1 + |mi|^2
</code></pre>
<p>wh... | Ittay Weiss | 30,953 | <p>This is a very classical problem and you will find many available sources online. Anything elementary on parameter estimation will include an analysis of a problem very much like what you describe (if not identical).</p>
<p>You do need to keep in mind that there are basically two common approaches (one more common ... |
267,947 | <p>i want to use least square to find x and y that minimize the result of the following function for a series of points (xi,yi) -> (x1,y1), (x2,y2),...:</p>
<p>note: y = f(x)</p>
<pre><code>E(x,y) = SUM (y - ((mi*x) - (mi*xi)))^2
i --------------------
1 + |mi|^2
</code></pre>
<p>wh... | Qiaochu Yuan | 232 | <p>Here is an account of what I understand to be the Bayesian approach, which I learned from <a href="http://books.google.com/books/about/Probability_Theory.html?id=tTN4HuUNXjgC" rel="nofollow">Jaynes</a>. Having read Jaynes, I am currently deeply suspicious of non-Bayesian approaches. </p>
<p>We'll restrict our atten... |
2,652,662 | <blockquote>
<p>Let $f:[a,b] \to \mathbb{R}$ be a bounded function. Define $A_{f,x} : (0, \infty) \to \mathbb{R}$ by $$A_{f,x} (r) = \text{diam}\left( f((x-r,x+r) \cap [a,b])\right).$$ Show that for all $\epsilon >0$ the set $\{x\in [a,b] : \lim_{r\to 0^+} A_{f,x} (r)\geq \epsilon \}$ is a closed set.</p>
</blockq... | drhab | 75,923 | <p>Let $C_1,C_2,C_3$ take values in $\{G,B\}$ such that e.g. $\{C_1=G\}$ denotes the event that the firstborn child is a girl. To keep things more general let $p$ denote the probability on a girl.</p>
<p>1) To be found is: $$P(C_1=C_2=C_3=G\mid C_1=G)=P(C_2=C_3=G\mid C_1=G)=P(C_2=G\wedge C_3=G)=P(C_2=G)P(C_3=G)=p^2$$<... |
971,997 | <p>How do we find
$$\Re\left[\int_0^{\large\frac{\pi}{2}} e^{\Large e^{i\theta}}d\theta\right]$$</p>
<p>In the shortest and easiest possible manner?</p>
<p>I cannot think of anything good.</p>
| Random Variable | 16,033 | <p>The answer can be expressed in terms of the <a href="http://mathworld.wolfram.com/SineIntegral.html">sine integral</a>.</p>
<p>Let $z=e^{i \theta}$.</p>
<p>Then</p>
<p>$$ \begin{align} \text{Re} \int_{0}^{\pi /2} e^{e^{i \theta}} \ d \theta = \text{Re} \frac{1}{i} \int_{C} \frac{e^{z}}{z} \ dz \end{align}$$
wher... |
2,706,834 | <p>I have the next exercise: </p>
<p>Let $E$ be a normed space, $n\in \mathbb{N}$ and $\{x_1,\dots,x_n\}\in E$ a family of independent vectors. Show that for all $\alpha_1,\dots,\alpha_n\in \mathbb{K}$ there is a linear form $T\in E'$ such that $T(x_i)=\alpha_i$, for all $i=1,\dots,n$.</p>
<p>My solution:</p>
<p>I u... | user284331 | 284,331 | <p>Use the following version of Hahn-Banach Extension Theorem (<em>An Introduction to Banach Space Theory</em>, Robert Megginson, page 75):</p>
<blockquote>
<p>Suppose that $f_{0}$ is a bounded linear functional on a subspace $Y$ of a normed space $X$. Then there is a bounded linear functional $f$ on all of $X$ such... |
83,648 | <p><a href="http://reference.wolfram.com/language/ref/FullGraphics.html" rel="nofollow noreferrer"><code>FullGraphics</code></a> hasn't worked entirely for a long time and the situation appears to be getting worse instead of better. In <em>Mathematica</em> 10.0, 10.1, 11.3, 12.3 up to 13.1 a simple usage throws numero... | Alexey Popkov | 280 | <h2>News about <code>FullGraphics</code> and <code>AbsoluteOptions</code></h2>
<p>This CW answer is intended for collecting the ongoing information about changes in <code>FullGraphics</code> and <code>AbsoluteOptions</code> through the versions.</p>
<hr />
<blockquote>
<p>I wonder at this point if there is any indicati... |
2,250,546 | <p>I am not sure how to prove this but this is what I have so far.</p>
<p>$b= ak$ for some integer $k$ and
$c= bl$ for some integer $l$</p>
<p>Then........</p>
<p>Here is where I am stuck, any help would be appreciated.</p>
| Arthur | 15,500 | <p>... then $c = akl$, so $c^2 = bl\cdot akl = ab\cdot kl^2$</p>
|
2,250,546 | <p>I am not sure how to prove this but this is what I have so far.</p>
<p>$b= ak$ for some integer $k$ and
$c= bl$ for some integer $l$</p>
<p>Then........</p>
<p>Here is where I am stuck, any help would be appreciated.</p>
| dxiv | 291,201 | <p>Hint: prove first $m \mid n, \,p \mid q \implies m\cdot p \mid n \cdot q\,$, then combine the following:</p>
<ul>
<li><p>$a \mid b \implies ab \mid b^2$</p></li>
<li><p>$b \mid c \implies b^2 \mid c^2$</p></li>
</ul>
|
2,250,546 | <p>I am not sure how to prove this but this is what I have so far.</p>
<p>$b= ak$ for some integer $k$ and
$c= bl$ for some integer $l$</p>
<p>Then........</p>
<p>Here is where I am stuck, any help would be appreciated.</p>
| Jaideep Khare | 421,580 | <p>$$b=ak \tag1$$ </p>
<p>and </p>
<p>$$c=bl$$
from $(1)$</p>
<p>$$c=(ak)l \implies c^2=a^2k^2l^2$$</p>
<p>Since $b=ak$, we can write :</p>
<p>$$ c^2=a^2k^2l^2=a \cdot(ak)\cdot kl^2=ab \cdot kl^2$$</p>
<p>Since $$c^2= ab \times m ~~\text{where}~~ m=kl^2 \implies ab \mid c^2$$ </p>
|
2,784,531 | <p>Let <span class="math-container">$X = (X_1, X_2, \dots, X_n)$</span> be jointly Gaussian with mean vector <span class="math-container">$\mu$</span> and covariance matrix <span class="math-container">$\Sigma$</span>. Let <span class="math-container">$S$</span> be their sum.</p>
<p>I know that the distribution of eac... | jlewk | 484,640 | <p>Let A be a deterministic matrix of size <span class="math-container">$n\times n$</span> and let <span class="math-container">$v$</span> be a vector of size <span class="math-container">$n$</span>. The random vector <span class="math-container">$(AX, S)$</span> is jointly normal. The idea is to construct both</p>
<ol... |
1,072,561 | <p>can you help me with this quest?</p>
<p>About composition $f$ and vector space $\mathbf{V}=\mathbb{Z^4_2}$ we know the following:</p>
<p>$f \circ f = id_V$,$~~f $
$
\left(\begin{array}{ccc}
1\\
0\\
1\\
0\\
\end{array}\right)=
\left(\begin{array}{ccc}
1\\
1\\
0\\
0\\
\end{array}\right)
$,
$~~f $
$
\left(\begin{ar... | John McGee | 170,591 | <p>Create a matrix $M$ whose columns are the $b_{i}$, then $T(u)=Mu$ performs the transformation. It is now easy to show that $T$ has the properties of a linear transformation.</p>
|
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