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 |
|---|---|---|---|---|
1,021,631 | <p>Does anybody know how to solve the equation</p>
<p>$\mathbf{a} + \mathbf{b} \times \hat{\mathbf{v}} = c \hat{\mathbf{v}},$</p>
<p>where $\mathbf{a}$ and $\mathbf{b}$ are given real vectors, for the unit vector $\hat{\mathbf{v}}$ and the real number $c$?</p>
| JP McCarthy | 19,352 | <p>What you need is the <a href="http://en.wikipedia.org/wiki/Cauchy%E2%80%93Schwarz_inequality" rel="nofollow">Cauchy Schwarz Inequality</a>.</p>
|
1,045,941 | <p>Usually this is just given as a straight up definition in a calculus course. I am wondering how you prove it?
I tried using the limit definition, $$\lim\limits_{h\rightarrow 0} \dfrac{\log(x+h)-\log(x)}{h}$$
but this led to no developments.</p>
| Hernandez | 165,046 | <p>Set $y = \log_e(x).$ Rearranging, we get $e^{y} = x\,(*).$ Differentiating both sides implicitly with respect to $x$:
$$e^{y}\cdot\frac{dy}{dx} = 1.$$ It follows that $$\frac{dy}{dx} = \frac{1}{e^y}$$ and hence that the derivative is $\frac{1}{x}$ from $(*).$</p>
|
243,377 | <p>Consider a random walk on the real time, starting from $0$. But this time assume that we can decide, for each step $i$, a step size $t_i>0$ to the left or the right with equal probabilities. </p>
<p>To formalize this, we have $(X_n)_{n\geq 0}$ such that $X_0=0$, $Pr[X_n=t_n]=0.5$ and $Pr[X_n=-t_n]=0.5$ (hence st... | Ori Gurel-Gurevich | 1,061 | <p>The crucial requirement is that $\sum_{i=0}^\infty t_i^2 < \infty$. See <a href="https://en.wikipedia.org/wiki/Kolmogorov%27s_two-series_theorem">Kolmogorov's two-series theorem</a> and also the more general <a href="https://en.wikipedia.org/wiki/Kolmogorov%27s_three-series_theorem">Kolmogorov's three-series theo... |
1,926,383 | <p>Recently I baked a spherical cake ($3$ cm radius) and invited over a few friends, $6$ of them, for dinner. When done with main course, I thought of serving this spherical cake and to avoid uninvited disagreements over the size of the shares, I took my egg slicer with parallel wedges(and designed to cut $6$ slices at... | G Cab | 317,234 | <p>Let me recast <em>Lovsovs</em>' answer in another perspective.<br>
Consider at first to split an <strong>hemi-sphere</strong> into $n$ parts of equal volume.
Let designate as $h(k,n)$ the relative position of the $k$-th cut along the radius of the emisphere starting from the base circle
(corresponding to $k=0$).
$$... |
3,914,365 | <p>I'm pretty stuck with this exercise. Hope somebody can help me:</p>
<p>show that
<span class="math-container">$$
\sum_{n=2^{k}+1}^{2^{k+1}} \frac{1}{n} \geq\left(2^{k+1}-2^{k}\right) \frac{1}{2^{k+1}}=\frac{1}{2}
$$</span>
and use this to show that the harmonic series is divergent.</p>
<p>First I can't figure out ho... | Surb | 154,545 | <p>(I am summarizing in this answer the steps discussed with OP in the comments)</p>
<p>Let <span class="math-container">$k\geq 0$</span> be an integer.</p>
<p>First note that the sum <span class="math-container">$\sum_{n=2^{k}+1}^{2^{k+1}} \frac{1}{n}$</span> has <span class="math-container">$2^{k+1}-(2^{k}+1)+1= 2^{k... |
1,946,535 | <p>A Liouville number is an irrational number $x$ with the property that, for every positive integer $n$, there exist integers $p$ and $q$ with $q > 1$ and such that $0 < \mid x - \frac{p}{q} \mid < \frac{1}{q^n} $.</p>
<p>I'm looking for either hints or a complete proof for the fact that $e$ is not a Liouvil... | Thomas Andrews | 7,933 | <p>In continued fractions for $\alpha=[a_0,a_1,a_2,\dots,a_n,\dots]$ we have $$\left|\frac{P_n}{Q_n} -\alpha\right|>\frac{1}{2}\left|\frac{P_n}{Q_n}-\frac{P_{n+1}}{Q_{n+1}}\right|=\frac{1}{2Q_nQ_{n+1}}\geq\frac{1}{2Q_n^2(a_n+1)}$$.</p>
<p>So the fact that $Q_n\geq F_n$ (with $F_n$ the $n$th Fibonacci number) and $a... |
3,843,807 | <p>Let <span class="math-container">$d_1$</span> be a metric on <span class="math-container">$X$</span>. If there exists a strictly increasing, continuous, and subadditive <span class="math-container">$ f:\mathbb {R} \to \mathbb {R} _{+}$</span> such that <span class="math-container">$d_{2}=f\circ d_{1}$</span>. Then <... | Amirhossein | 456,050 | <p>I came up with a solution, and I will write it below; it seems the conditions of being subadditive is surplus. I think you needed the subadditivity if you wanted to show that <span class="math-container">$d_2$</span> is a metric. But when it is given, you won't need it.</p>
<p>Let <span class="math-container">$B^{1}... |
262,773 | <p>In the curve obtained with following</p>
<pre><code> Plot[{1/(3 x*Sqrt[1 - x^2])}, {x, 0, 1}, PlotRange -> {0, 1},
GridLines -> {{0.35, 0.94}, {}}]
</code></pre>
<p>how can one fill the top and bottom with different colors or patterns such that two regions are perfectly visible in a black-n-white printout?<... | Ulrich Neumann | 53,677 | <p>You have to plot it twice:</p>
<pre><code>Show[{
Plot[{1/(3 x*Sqrt[1 - x^2]), 1/(3 x*Sqrt[1 - x^2])}, {x, 0, 1},PlotRange -> {0, 1}, GridLines -> {{0.35, 0.94}, {}}, Filling -> {Top }, FillingStyle -> {Red }],
Plot[{1/(3 x*Sqrt[1 - x^2]), 1/(3 x*Sqrt[1 - x^2])}, {x, 0, 1},PlotRange -> {0, 1}, GridLine... |
262,773 | <p>In the curve obtained with following</p>
<pre><code> Plot[{1/(3 x*Sqrt[1 - x^2])}, {x, 0, 1}, PlotRange -> {0, 1},
GridLines -> {{0.35, 0.94}, {}}]
</code></pre>
<p>how can one fill the top and bottom with different colors or patterns such that two regions are perfectly visible in a black-n-white printout?<... | Bob Hanlon | 9,362 | <p>Using <a href="https://reference.wolfram.com/language/ref/RegionPlot.html" rel="nofollow noreferrer"><code>RegionPlot</code></a></p>
<pre><code>RegionPlot[
{0.35 <= x <= 0.94 && y > 1/(3 x*Sqrt[1 - x^2]),
0.35 <= x <= 0.94 && y < 1/(3 x*Sqrt[1 - x^2])},
{x, 0, 1}, {y, 0, 1},
Pl... |
1,268 | <p>There has long been debate about whether a first year undergraduate course in discrete mathematics would be better for students than the traditional calculus sequence. The purpose of this question is not to further that debate, but to inquire whether any text has tried to teach calculus by emphasizing probability as... | kcrisman | 1,608 | <p>This isn't exactly a full answer for you, but <a href="http://www.ams.org/bookstore-getitem/item=acalc" rel="nofollow">Approximately Calculus</a> has a large bit on the prime number theorem as a motivation, which was motivated by Gauss' thinking of prime distribution probabilistically and where using integrals doesn... |
4,204,282 | <p>How do I calculate <span class="math-container">$$\int_{-\infty}^{\infty} \frac{dw}{1+iw^3}$$</span> using complex path integrals?</p>
<p>I just need a hint on how to start, not the actual computation, because I need to understand how to deal with similar questions.</p>
<p><strong>Edit:</strong> Following @Tavish's ... | mjw | 655,367 | <p>Consider the integral <span class="math-container">$$\int_{-\infty}^\infty \frac{dw}{p(w)},$$</span> where <span class="math-container">$p(w)$</span> is a polynomial of degree <span class="math-container">$\ge 2$</span>. If <span class="math-container">$p(w)$</span> has no poles on the real axis (this can be dealt ... |
347,009 | <p>I am looking for a generalisation of a modular form that transforms as something like:</p>
<p><span class="math-container">$f(\frac{a \tau+b}{c \tau+d}) = (c \tau+d)^k c^k f(\tau)$</span></p>
<p>I understand this cannot be literally true, as the multiplier c^k is not a root of unity, but does something like this a... | Matt F. | 44,143 | <p>Here are three conjectures, in the hopes of showing that we can say something non-trivial.</p>
<p>[<strong>UPDATE</strong>: This is essentially all in a paper from Erdős, "On Sets of Distances of <span class="math-container">$n$</span> Points", from <a href="https://www.jstor.org/stable/2305092?seq=1" rel="nofollow... |
745,876 | <p>$$\exists! x : A(x) \Rightarrow \exists x : A(x)$$
Assuming that $A(x)$ is an open sentence. I'm new to abstract mathematics and proofs, so I came here to ask for some simplification. Thanks</p>
| Community | -1 | <p>The problem states that:</p>
<blockquote>
<p>If there exists exactly one $x$ such that $A(x)$ is true, then there exists at least one $x$ such that $A(x)$ is true.</p>
</blockquote>
<p>The proof should be trivial.</p>
|
4,230,782 | <blockquote>
<p>Compute <span class="math-container">$\frac 17$</span> in <span class="math-container">$\Bbb{Z}_3.$</span></p>
</blockquote>
<p>We will have to solve <span class="math-container">$7x\equiv 1\pmod p,~~p=3.$</span></p>
<ul>
<li>We get <span class="math-container">$x\equiv 1\pmod 3.$</span></li>
<li>Then <... | Sangchul Lee | 9,340 | <p>You are doing correct.</p>
<p>For a slightly more efficient approach, note that <span class="math-container">$7\mid 3^6-1$</span> by Fermat's little theorem. Then by noting that <span class="math-container">$3^6-1=7\cdot104$</span>, we have</p>
<p><span class="math-container">\begin{align*}
\frac{1}{7}
&= -\frac... |
1,792,402 | <p>I am trying to solve <a href="https://en.wikipedia.org/wiki/Mean_value_theorem" rel="nofollow">Mean Value theorem</a> but i ran into a road block trying to solve the question. So the question is:
Assuming</p>
<p>$$ \frac{f(b) − f(a)}{b − a} = f'(c) \quad \text{for some } c \in (a, b)$$</p>
<p>Let $f(x) = \sqrt x$ ... | Michael Hardy | 11,667 | <p>Since $f'(x) = \dfrac 1 {2\sqrt x}$, you need
$$
\frac{\sqrt 4 - \sqrt 0}{4-0} = \frac{f(4) - f(0)}{4-0} = \frac{f(b)-f(a)}{b-a} = f'(c) = \frac 1 {2\sqrt c}.
$$
If $\dfrac{\sqrt 4} 4 = \dfrac 1 {2\sqrt c}$ then what number is $c$?</p>
|
1,792,402 | <p>I am trying to solve <a href="https://en.wikipedia.org/wiki/Mean_value_theorem" rel="nofollow">Mean Value theorem</a> but i ran into a road block trying to solve the question. So the question is:
Assuming</p>
<p>$$ \frac{f(b) − f(a)}{b − a} = f'(c) \quad \text{for some } c \in (a, b)$$</p>
<p>Let $f(x) = \sqrt x$ ... | Martin Argerami | 22,857 | <p>$$
\frac1{2\sqrt c}=\frac{\sqrt 4-\sqrt 0}{4-0}=\frac24=\frac12.
$$
Then
$
\sqrt c=1
$, so $c=1$. </p>
|
2,513,509 | <p>If $f,g,h$ are functions defined on $\mathbb{R},$ the function $(f\circ g \circ h)$ is even:</p>
<p>i) If $f$ is even.</p>
<p>ii) If $g$ is even.</p>
<p>iii) If $h$ is even.</p>
<p>iv) Only if all the functions $f,g,h$ are even. </p>
<p>Shall I take different examples of functions and see?</p>
<p>Could anyone ... | egreg | 62,967 | <p>You may want to compute
$$
\lim_{x\to0}\frac{\arcsin x-\arctan x}{\sin x-\tan x} \tag{*}
$$
which your limit will be equal to, provided (*) exists.</p>
<p><sup>The limit of the sequence might exist also if the limit of the function doesn't. But if the limit of the function exists, also does the limit of the sequenc... |
267,753 | <p>A number of seemingly unrelated elementary notions can be defined uniformly with help of (iterated) Quillen lifting property
(a category-theoretic construction I define below) "starting" to a single (counter)example or a simple class of morphisms,
for example a finite group being nilpotent, solvable, p-group, a top... | user108780 | 108,780 | <p>To improve readability of iterated orthogonals, I write $C^l$ instead of $C^\perp$ and $C^r$ instead of $C^r$.</p>
<p>(i) $(\emptyset\longrightarrow \{*\})^r$, $(0\longrightarrow R)^r$, and $\{0\longrightarrow \Bbb Z\}^r$ are the classes of surjections in
Sets, R-modules, and Groups, resp,
(where $\{*\}... |
1,304,948 | <p>If I solve $Tx=0$ where $T$ is some square matrix
then if I multiply both sides by $T$ and solve for $T^2x=0$, will my x be the same?
In other words if I were to multiply to both sides of the equation $Tx=0$ to any order of $T$, will my x be the same? or at least satisfy the first equation?</p>
| Adren | 405,819 | <p>It's worth to note that if <span class="math-container">$V=W\oplus W^\perp$</span> (which is necessary true if <span class="math-container">$W$</span> is finite dimensional and, in particular, if <span class="math-container">$V$</span> is finite dimensional), we have <span class="math-container">$(W^\perp)^\perp=W$<... |
1,972,252 | <blockquote>
<p>If $v^TAv = v^TBv$ for all constant vectors $v$ and $A,B$ are matrices of size $n$ by $n$, is it true that $A=B$? </p>
</blockquote>
<p>I have thought about using basis vectors but cannot get system of equations uniquely. Is there a trick here?</p>
| H. H. Rugh | 355,946 | <p>If you assume that the matrices are symmetric then it is true. Otherwise not. In fact, $M=A-B$ may be any anti-symmetric matrix, i.e. $M^T=-M$.</p>
<p>If $M$ is symmetric on the other hand it is diagonaliable with real eigenvalues and in an orthonormal basis. Picking an eigenvector $v$ for eigenvalue $\lambda$ you ... |
2,375,023 | <p>So I have to find an interval (in the real numbers) such that it contains all roots of the following function:
$$f(x)=x^5+x^4+x^3+x^2+1$$</p>
<p>I've tried to work with the derivatives of the function but it doesn't give any information about the interval, only how many possible roots the function might have.</p>
| Angina Seng | 436,618 | <p>Can you find a positive number $N$ such that
$N^5>N^4+N^3+N^2+1$? If $|z|\ge N$ then one cannot
have $z^5=-z^4-z^3-z^2-1$.</p>
|
2,375,023 | <p>So I have to find an interval (in the real numbers) such that it contains all roots of the following function:
$$f(x)=x^5+x^4+x^3+x^2+1$$</p>
<p>I've tried to work with the derivatives of the function but it doesn't give any information about the interval, only how many possible roots the function might have.</p>
| Andreas | 317,854 | <p>EDITED for correct function.</p>
<p>$f(x) = 1+x^2+x^3+x^4+x^5$
is positive for x>0.</p>
<p>So consider $x < 0$:</p>
<p>One can write $f(x) = (1+x^2)(1+x^3) + x^4$ which is positive at least for $(1+x^3) >0$, i.e. $x > -1$.</p>
<p>$f'(x) = 2x+3x^2+4x^3+5x^4 = x(2+3x)+x^3(4+5x)$
so this is positive at lea... |
2,375,023 | <p>So I have to find an interval (in the real numbers) such that it contains all roots of the following function:
$$f(x)=x^5+x^4+x^3+x^2+1$$</p>
<p>I've tried to work with the derivatives of the function but it doesn't give any information about the interval, only how many possible roots the function might have.</p>
| Ovi | 64,460 | <p>The common factoring formula $x^n-1=(x-1)(x^{n-1}+x^{n-2}+ \cdots + x^2+1)$ tells us that your $f(x)$ is equivalent to $g(x)=\dfrac {x^6-1}{x-1}$ as long as $x \not = 1$. Setting $g(x)=0$ yields $x^6-1=0$. Clearly the only real solutions to this are $\pm1$. But only $-1$ is a zero of $g$ and of $f$, so $f$ has only ... |
1,929,977 | <p>Let $f\colon\mathbb R\to\mathbb R$ satisfy the Lipschitz condition: there exists $K\geq 0$ such that for all $x,y\in\mathbb R$, we have $|f(x)-f(y)|\leq K\cdot |x-y|$. Is it true that $f$ has one-sided derivatives everywhere? I.e., that the limits
$$\lim_{h\nearrow 0}\frac{f(x+h)-f(x)}{h}\quad\text{and}\quad\lim_{h\... | Hagen von Eitzen | 39,174 | <p>Let
$$ f(x)=\begin{cases}x\sin\frac1x&0<|x|<\frac1\pi\\0&\text{otherwise}\end{cases}$$
Then $f$ is Lipschitz, but none of the one-sided derivatives exists at $x=0$.</p>
|
2,538,553 | <p>For example in statistics we learn that mean = E(x) of a function which is defined as</p>
<p>$$\mu = \int_a^b xf(x) \,dx$$</p>
<p>however in calculus we learn that</p>
<p>$$\mu = \frac {1}{b-a}\int_a^b f(x) \,dx $$ </p>
<p>What is the difference between the means in statistics and calculus and why don't they gi... | Hilton Fernandes | 499,387 | <p>Maybe you can consider the second form of mean as a <strong>sample mean</strong>, analogue to $\bar{x} = \frac{1}{n} \sum_{i=1}^n x_i$ in the discrete case.</p>
<p>That is: the function $f(x)$ in this case would be the <strong>realization</strong> of a random variable.</p>
<p>Please consider the notion of a stocha... |
2,289,813 | <p>Let$\ f(x,y,z)=4xz -y^2 +z^2$ be a differentiable function, let$\ P=(0,1,1)$ and$\ Q=(1,3,2)$. Find$\ T$ such that $\ f(P)-f(Q) = \nabla f(T)(P-Q)$</p>
<p>What I did:</p>
<p>$\ f(P)=0$, $\ f(Q)=3$, $\ (P-Q)=(-1,-2,-1)$, $\nabla f(P-Q)=(-4z, 4y, -4x+2z)$</p>
<p>Let $\ T=(a,b,c)$ then$\ f(P)-f(Q) = \nabla f(T)(P-Q)... | DavidTomahawk | 430,305 | <p>Following from what I did:</p>
<p>Let $\ g(t)=(1, 3, 2)+t(-1, -2, -1) = (1-t, 3-2t, 2-t)$</p>
<p>$\nabla f(1-t, 3-2t, 2-t) = (-4t+8, 4t-6, -6t+8)$</p>
<p>$\ -3 = (-4t+8, 4t-6, -6t+8)(-1, -2, -1) = (4t-8, -8t+12, 6t-8)$ if and only if $\ t=1/2$, substitute$\ t$ in the function$\ g $ and </p>
<p>$\ T=g(t)=(\frac{1... |
1,031,464 | <p>I am supposed to simplify this:</p>
<p>$$(x^2-1)^2 (x^3+1) (3x^2) + (x^3+1)^2 (x^2-1) (2x)$$</p>
<p>The answer is supposed to be this, but I can not seem to get to it:</p>
<p>$$x(x^2-1)(x^3+1)(5x^3-3x+2)$$</p>
<p>Thanks</p>
| Aaron Maroja | 143,413 | <p>Try this</p>
<p>$$x(x^2-1)(x^3+1)\Bigg[3x(x^2-1)+2(x^3+1)\Bigg]$$</p>
|
734,203 | <blockquote>
<p>Find all vectors of $V_{3}$ which are perpendicular to the vector $(7,0,-7)$ and belong to the subspace $L((0,-1,4), (6,-3,0)$.</p>
</blockquote>
<p>As a note, this is an extra question of a long exercise, the vectors found above are replaced by results I found at the above questions.</p>
<p>Let $\o... | Supersingularity | 139,189 | <p>I misunderstood the question and will edit this later with (hopefully) a response.</p>
<p>Given any $x \in \mathbb{C}$, there exists $n$ possible $y \in \mathbb{C}$ such that $y^n = P(x)$ (possibly with multiplicity, if say $P(x) = 0$). However, determining whether the system has a solution (and how many) depends c... |
3,371,694 | <p>Let <span class="math-container">$ \forall n \in \mathbf{N}, ~ u_{n+1} = \frac{1}{\tanh^2(u_n)} - \frac{1}{u^2_n} $</span> with <span class="math-container">$ u_0 = a > 0 $</span>.</p>
<p>What is the limit of <span class="math-container">$(u_n)$</span> ?</p>
<p>I tried to find a fixed point of this sequence but... | Secret | 709,089 | <p>You have that <span class="math-container">$X = (-a, \sqrt{(b^{2} - a^{2})})$</span> and <span class="math-container">$Y = (a,-\sqrt{(b^{2} - a^{2})})$</span></p>
<p>Then <span class="math-container">$\|X-Y\| = \|(-2a, 2\sqrt{(b^{2} - a^{2})})\| = $</span> <span class="math-container">$\sqrt{4a^{2} + 4(b^{2} - a^{2... |
2,421,344 | <p>I'm studying the weak Mordell-Weil theorem in Silverman's "Arithmetic of elliptic curves" and I can't understand the proof of proposition 1.6 of chapter 8.</p>
<p>Here $K$ is a number field, $S$ a finite set of places containing the archimedean ones and $m>2$ an integer. I want to prove that the maximal abelian ... | Robert Z | 299,698 | <p>I think that here the eggs are drawn without replacement. Hence it should be an <a href="https://en.wikipedia.org/wiki/Hypergeometric_distribution#Combinatorial_identities" rel="nofollow noreferrer">Hypergeometric distribution</a>. So, if $3$ eggs are drawn then, for $0\leq k\leq 3$,
$$p_k:=P(\mbox{number of bad eg... |
3,460,204 | <p>If <span class="math-container">$X\subseteq \mathbb R^n$</span> and <span class="math-container">$x\in X$</span> is an isolated point, then <span class="math-container">$x$</span> is a cluster point of <span class="math-container">$X^c$</span>.</p>
<p>I don't know if that is true or not, can someone help me, please... | Adam Martens | 691,077 | <p><span class="math-container">$x\in X$</span> is isolated means that there exists open <span class="math-container">$U\ni x$</span> such that <span class="math-container">$U\cap X=\{x\}$</span>. But <span class="math-container">$U\subset \mathbb{R}^n$</span> is open implies there exits <span class="math-container">$r... |
3,781,968 | <p>Let <span class="math-container">$d\in\mathbb N$</span>, <span class="math-container">$U\subseteq\mathbb R^d$</span> be open and <span class="math-container">$M\subseteq U$</span> be a <span class="math-container">$k$</span>-dimensional embedded <span class="math-container">$C^1$</span>-submanifold of <span class="m... | supinf | 168,859 | <p><em>sketch of a counterexample</em>:</p>
<p>Let <span class="math-container">$d=2$</span>, <span class="math-container">$U=(-9,9)\times (-9,9)$</span> and let <span class="math-container">$M$</span> be the <span class="math-container">$1$</span>-dimensional submanifold
which is described by the circle with radius <s... |
2,111,734 | <p>a) Show that $f^2 - (f')^2 = 0$</p>
<p>I tried to solve by doing $f=\pm f'$ not sure what to do from here.</p>
| marwalix | 441 | <p>Consider $g=f^2-(f')^2$. One has</p>
<p>$$g'(x)=2f(x)f'(x)-2f''(x)f'(x)=2f'(x)\left(f(x)-f''(x)\right)=0$$</p>
<p>So $g$ is constant and by assumption $g(0)=0$ so $f^2-(f')^2=0$</p>
<p>This means $f'(x)=f(x)$ or $f'(x)=-f(x)$.</p>
<p>Solving these two ODE one gets $f(x)=f(0)e^x=0$ or $f(x)=f(0)e^{-x}=0$</p>
|
3,266,367 | <p>Find the solution set of <span class="math-container">$\frac{3\sqrt{2-x}}{x-1}<2$</span></p>
<p>Start by squaring both sides
<span class="math-container">$$\frac{-4x^2-x+14}{(x-1)^2}<0$$</span>
Factoring and multiplied both sides with -1
<span class="math-container">$$\frac{(4x-7)(x+2)}{(x-1)^2}>0$$</span>... | J. W. Tanner | 615,567 | <p>For <span class="math-container">$\dfrac{3\sqrt{2-x}}{x-1}$</span> to be defined, <span class="math-container">$x\le2$</span> and <span class="math-container">$x\ne1$</span>.</p>
<p>If <span class="math-container">$x<1,$</span> then the expression is negative (i.e., <span class="math-container">$<0$</span>), ... |
106,560 | <p>Mochizuki has recently announced a proof of the ABC conjecture. It is far too early to judge its correctness, but it builds on many years of work by him. Can someone briefly explain the philosophy behind his work and comment on why it might be expected to shed light on questions like the ABC conjecture?</p>
| grp | 26,145 | <p>[The answer below is a response to an earlier version of the question that was rather different in certain respects. Minhyong Kim's answer gives excellent insight into ideas that Mochizuki had back in 2000 and that provide essential building blocks for the more recent work. But I still believe that it is too prematu... |
106,560 | <p>Mochizuki has recently announced a proof of the ABC conjecture. It is far too early to judge its correctness, but it builds on many years of work by him. Can someone briefly explain the philosophy behind his work and comment on why it might be expected to shed light on questions like the ABC conjecture?</p>
| Marty | 3,545 | <p>I'll take a stab at answering this controversial question in a way that might satisfy the OP and benefit the mathematical community. I also want to give some opinions that contrast with or at least complement grp. Like others, I must give the caveats: I do not understand Mochizuki's claimed proof, his other work,... |
106,560 | <p>Mochizuki has recently announced a proof of the ABC conjecture. It is far too early to judge its correctness, but it builds on many years of work by him. Can someone briefly explain the philosophy behind his work and comment on why it might be expected to shed light on questions like the ABC conjecture?</p>
| Pasten | 26,218 | <p>I want to point out a bibliographical information that perhaps is not very well-known and can be taken as "evidence" for the possibility of applying anabelian geometry to the ABC conjecture successfully. However, I am not claiming that this is related in any sort of way to Mochizuki's work. </p>
<p>Here is the fact... |
106,560 | <p>Mochizuki has recently announced a proof of the ABC conjecture. It is far too early to judge its correctness, but it builds on many years of work by him. Can someone briefly explain the philosophy behind his work and comment on why it might be expected to shed light on questions like the ABC conjecture?</p>
| Vesselin Dimitrov | 26,522 | <p>Let me also try to give, in a modest complement to Minhyong Kim's great post, some additional remarks on Mochizuki's strategy. The idea that has led to the development of "Inter-universal Teichmuller theory for number fields" is certainly very beautiful, and was known to Mochizuki, along with the nature of the final... |
106,560 | <p>Mochizuki has recently announced a proof of the ABC conjecture. It is far too early to judge its correctness, but it builds on many years of work by him. Can someone briefly explain the philosophy behind his work and comment on why it might be expected to shed light on questions like the ABC conjecture?</p>
| user30304 | 30,304 | <p>NEW !! (2013-02-21)</p>
<p>A Panoramic Overview of Inter-universal Teichmüller
Theory
By
Shinichi Mochizuki</p>
<p><a href="http://www.kurims.kyoto-u.ac.jp/~motizuki/Panoramic%20Overview%20of%20Inter-universal%20Teichmuller%20Theory.pdf">http://www.kurims.kyoto-u.ac.jp/~motizuki/Panoramic%20Overview%20of%20Inter-u... |
320,019 | <p>Assume $Y$ is non negative random variable. Prove that $X+Y$ is stochastically greater than $X$ for any random variable $X$.</p>
<p>We have to prove there that $\Pr(X+Y > x) \geq \Pr(X>x) $ for all $x$</p>
| gnometorule | 21,386 | <p>Recalling that $A \subset B \Rightarrow P(A) \leq P(B)$, and that $B := \{ X > x -\epsilon \} \supset \{ X > x \} := A$ for any $\epsilon \geq 0$, and that by positivity of $Y$, $x - Y = x - \epsilon$ for some $\epsilon \geq 0$ (for every sample point $\omega$): </p>
<p>$$P(X+Y > x) = P(X > x - Y) \geq... |
2,591,976 | <p>If $f(x-1)=2x^2-10x+3$, find $f(x)$. I tried the problem and received the answer $f(x)=2x^2-14x-5$, is this right? </p>
| fleablood | 280,126 | <blockquote>
<p>$f(x)=2x^2-14x-5$ is this right?</p>
</blockquote>
<p>Let's see. $f(x-1) = 2(x-1)^2 - 14(x-1) - 5 = 2(x^2 -2x +1)-14x + 14 - 5$</p>
<p>$= 2x^2 - 4x +2 - 14x + 9 = 2x^2-18x +11$</p>
<p>No, that doesn't seem to be right.</p>
<p>.....</p>
<p>Instead try.</p>
<p>$x = (x+1) - 1$</p>
<p>Plug $(x+1)-... |
2,591,976 | <p>If $f(x-1)=2x^2-10x+3$, find $f(x)$. I tried the problem and received the answer $f(x)=2x^2-14x-5$, is this right? </p>
| Michael Rozenberg | 190,319 | <p>$$f(x)=2(x+1)^2-10(x+1)+3=2x^2-6x-5.$$</p>
|
2,486,334 | <p>What is the plane graph of $|z-1|+|z-5| < 4$ ?</p>
<p>What i know is that there is nothing for $y\geq4$ or $y \leq -4$ or $x \geq 5$ or $x \leq 1$.</p>
<p>Trying to let $z=x+y i$ such that $x,y \in \mathbb{R}$ did not help either.</p>
| jonsno | 310,635 | <p>Consider two points in cartesian plane, $A(1,0)$ and $B(5,0)$. You have to find a locus of point $P$ such that <em>sum of distance from these two points is less than</em> $4$. </p>
<p>$$PA+PB < 4$$</p>
<p>Since the points themselves are at a distance $4$, ie $AB = 4$ , you can never have $PA+PB < 4$, because... |
2,441,660 | <p>Suppose I have $n$ observations, which are all normally distributed with the same mean (which is unknown) but each has a different variance (the different variances could be called $v_1,...,v_n$ for example, which are all assumed to be known).</p>
<p>What is the best estimate of the mean? And further, how does one ... | N. F. Taussig | 173,070 | <p>We must find the number of solutions of the equation
<span class="math-container">$$x_1 + x_2 + x_3 = 9 \tag{1}$$</span>
in the positive integers subject to the restrictions that <span class="math-container">$x_1, x_2, x_3 \leq 6$</span>.</p>
<p>A particular solution of equation 1 in the positive integers correspon... |
2,232,531 | <p>I have a dash camera on my windshield, and I'm wondering if there is any way to determine my vehicle's speed using the video footage from the dash cam. Please help!</p>
| dantopa | 206,581 | <p>One strategy may be to use the dashed lines in the lane dividers.</p>
<p>A source like <a href="http://commons.bcit.ca/civil/edufacts/highway_paint.html" rel="nofollow noreferrer">this</a> can provide size and spacing. Then mean velocity $v$ can be expressed in terms of, $d$, distance (on the video) and time, $t$, ... |
2,232,531 | <p>I have a dash camera on my windshield, and I'm wondering if there is any way to determine my vehicle's speed using the video footage from the dash cam. Please help!</p>
| Corey Levinson | 299,500 | <p>The federal guideline for the white dashed lines on roads is 10 feet, and the empty spaces in between are 30 feet. If you measure the time it takes to move from the beginning of one white dashed line to the next, that is the time it takes your car to move 40 feet, which is a measure of speed.</p>
|
4,624,695 | <p>I have an exercise with text: With lines</p>
<p><span class="math-container">$$
p_1 \ldots \frac{x-1}{2}=\frac{y+1}{-1}=\frac{z-4}{1}, \quad p_2 \ldots \frac{x-5}{2}=\frac{y-1}{-1}=\frac{z-2}{1} .
$$</span>
Determine one plane <span class="math-container">$\pi$</span> with respect to which the lines <span class="mat... | GReyes | 633,848 | <p>Your function is odd with respect to <span class="math-container">$x=a$</span>. That is, the area from <span class="math-container">$0$</span> to <span class="math-container">$a$</span> and that from <span class="math-container">$a$</span> to <span class="math-container">$2a$</span> are equal in magnitude and have o... |
2,822,126 | <blockquote>
<p>If $0⩽x⩽y⩽z⩽w⩽u$ and $x+y+z+w+u=1$, prove$$
xw+wz+zy+yu+ux⩽\frac15.
$$</p>
</blockquote>
<p>I have tried using AM-GM, rearrangement, and Cauchy-Schwarz inequalities, but I always end up with squared terms. For example, applying AM-GM to each pair directly gives$$
x^2+y^2+z^2+w^2+u^2 ⩾ xw + wz + zy + ... | Cesareo | 397,348 | <p>It is quite easy to solve this problem with the Lagrange multipliers</p>
<p>Calling </p>
<p>$$
L(x,y,z,w,u,\lambda) = \frac 15 -(xw+wz+zy+yu+ux)+\lambda(x+y+z+w+u-1)
$$</p>
<p>The stationary points are the solutions for</p>
<p>$$
\nabla L = (L_x,L_y,L_z,L_w,L_u,L_{\lambda})=0
$$</p>
<p>or</p>
<p>$$
\lambda -u-... |
2,822,126 | <blockquote>
<p>If $0⩽x⩽y⩽z⩽w⩽u$ and $x+y+z+w+u=1$, prove$$
xw+wz+zy+yu+ux⩽\frac15.
$$</p>
</blockquote>
<p>I have tried using AM-GM, rearrangement, and Cauchy-Schwarz inequalities, but I always end up with squared terms. For example, applying AM-GM to each pair directly gives$$
x^2+y^2+z^2+w^2+u^2 ⩾ xw + wz + zy + ... | Michael Rozenberg | 190,319 | <p>Let $y=x+a$, $z=x+a+b$, $w=x+a+b+c$ and $u=x+a+b+c+d$.</p>
<p>Thus, $a$, $b$, $c$ and $d$ are non-negatives and we need to prove that:
$$(x+y+z+u+w)^2\geq5(xw+wz+zy+yu+ux)$$ or
$$(5x+4a+3b+2c+d)^2\geq5(x(x+a+b+c)+(x+a+b+c)(x+a+b)+(x+a+b)(x+a)+(x+a)(x+a+b+c+d)+(x+a+b+c+d)x)$$ or
$$(5x+4a+3b+2c+d)^2\geq5(5x^2+(8a+6b+... |
216,473 | <p>Let me begin with some background: I used to enjoy mathematics immensely in school, and wanted to pursue higher studies. However, everyone around me at that time told me it was a stupid area (that I should focus on earning as soon as possible), and so instead I opted for an engineering degree. While in college, I us... | Euler....IS_ALIVE | 38,265 | <p>As a grad student, I think that you should stick to engineering. The jobs are just so more plentiful.</p>
|
2,908,186 | <p>There's an old-school pocket calculator trick to calculate <span class="math-container">$x^n$</span> on a pocket calculator, where both, <span class="math-container">$x$</span> and <span class="math-container">$n$</span> are real numbers. So, things like <span class="math-container">$\,0.751^{3.2131}$</span> can be ... | Xiangxiang Xu | 86,009 | <p>For fixed $x > 0$ and $n$, let $t = 1/2^a \to 0$. Then we need to prove that
$$
\lim_{t \to 0} \left( n (x^t - 1) + 1 \right)^{1/t} = x^n.
$$
In fact, we have
$$
\ln \left[\lim_{t \to 0} \left( n (x^t - 1) + 1 \right)^{1/t}\right] =
\lim_{t \to 0} \frac{\ln (1 + n(x^t - 1))}{t} = \lim_{t \to 0} \frac{n(x^t - 1... |
256,138 | <p>I need to generate four positive random values in the range [.1, .6] with (at most) two significant digits to the right of the decimal, and which sum to exactly 1. Here are three attempts that do not work.</p>
<pre><code>x = {.15, .35, .1, .4}; While[Total[x] != 1,
x = Table[Round[RandomReal[{.1, .6}], .010], 4]];... | kglr | 125 | <pre><code>SeedRandom[1]
list = {##, 1 - +##} & @@@ Round[RandomPoint[Simplex[3], 10], .01];
Grid[Prepend[{"4-tuple", "total"}][{#, Total@#} & /@ list ],
Dividers -> {False, {True, True, {False}}}]
</code></pre>
<p><a href="https://i.stack.imgur.com/Axq8k.png" rel="nofollow noreferrer... |
256,138 | <p>I need to generate four positive random values in the range [.1, .6] with (at most) two significant digits to the right of the decimal, and which sum to exactly 1. Here are three attempts that do not work.</p>
<pre><code>x = {.15, .35, .1, .4}; While[Total[x] != 1,
x = Table[Round[RandomReal[{.1, .6}], .010], 4]];... | Joshua Schrier | 63,709 | <p>Building on @kglr's response—the way to do this is to incorporate some of the additional constraints into the definition of the region from which sampling occurs (it is a bit more complicated than a Simplex). We can define a region where the 3 variables go over the appropriate range. The implicit fourth variable m... |
256,138 | <p>I need to generate four positive random values in the range [.1, .6] with (at most) two significant digits to the right of the decimal, and which sum to exactly 1. Here are three attempts that do not work.</p>
<pre><code>x = {.15, .35, .1, .4}; While[Total[x] != 1,
x = Table[Round[RandomReal[{.1, .6}], .010], 4]];... | ubpdqn | 1,997 | <p>You could use<a href="https://reference.wolfram.com/language/ref/DirichletDistribution.html" rel="nofollow noreferrer"><code>DirichletDistribution</code></a></p>
<p>For example:</p>
<pre><code>rv = Select[{##, 1 - Total[{##}]} & @@@
RandomVariate[DirichletDistribution[{1, 1, 1, 1}], 100000],
Min[#] >... |
4,221,530 | <p>Assume terms <span class="math-container">$a_n$</span> and <span class="math-container">$a_{n+1}$</span> share some common factor <span class="math-container">$x$</span> so that <span class="math-container">$a_n = xm$</span> for some integer <span class="math-container">$m$</span> and <span class="math-container">$a... | Lion Heart | 809,481 | <p><span class="math-container">$$\angle BAM =\angle AFT = 120^\circ-90^\circ=30^\circ $$</span></p>
<p><span class="math-container">$$\angle ABM = \angle AMB = \angle FAT = \angle FTA = \frac{180^\circ-30^\circ}{2}=75^\circ$$</span></p>
<p><span class="math-container">$$\angle MAT = 120^\circ- (75^\circ+30^\circ)=15^\... |
3,441,438 | <p>I was wondering if anyone could get me started on solving the following:
<span class="math-container">$$(u')^2-u\cdot u''=0$$</span></p>
<p>I have tried letting <span class="math-container">$v=u'$</span>, but I don't seem to make progress with such a substitution.</p>
| user0102 | 322,814 | <p>Notice that, for a polynomial with real coefficients, if <span class="math-container">$z\in\mathbb{C}$</span> is a root, then <span class="math-container">$\overline{z}$</span> is also a root. At your case, <span class="math-container">$z = -2+i\sqrt{5}$</span> is known to be a root. Then divide the fourth degree po... |
554,825 | <p>Here's a question from my homework. First 2 questions I solved (but would appreciate any input you can give on my solution) and the last question I'm just completely stumped. It's quite complicated.</p>
<p>On the shelf there are 5 math books, 3 science fiction books and 2 thrillers (<strong>all of the books are di... | Sebastian | 25,717 | <p>The statement is correct if both $R$ and $S$ are local rings and if $f$ is a homomorphism of local rings, i.e. $f(\mathfrak{m}_{R}) \subset \mathfrak{m}_{S}$, where $\mathfrak{m}_{R}, \mathfrak{m}_{S}$ denotes the maximal ideal of $R$, $S$, respectively.</p>
<p>Proof: The set of non-units in $R$ is the maximal idea... |
112,394 | <p>Could someone clarify why the first of these <code>MatchQ</code> finds a match whereas the second does not? (I'm using version 10.0, in case that matters.)</p>
<pre><code>MatchQ[Hold[x + 2 y], Hold[x + 2 _]]
(*True*)
MatchQ[Hold[x + 2 y + 0], Hold[x + 2 _ + 0]]
(*False*)
</code></pre>
<p>EDIT: The conclusion belo... | M.R. | 403 | <p>You can put a <code>Verbatim</code> on the Plus:</p>
<pre><code>MatchQ[Hold[x + 2 y + 0], Hold[Verbatim[Plus][x, 2 _, 0]]] (* True *)
</code></pre>
<p>Another way:</p>
<pre><code>expr = Inactivate[x + 2 y + 1];
form = Inactivate[x + 2 _ + 1];
MatchQ[expr, IgnoringInactive@form] (* True *)
</code></pre>
|
4,014,756 | <p>I was reading the book "Quantum Computing Since Democritus".</p>
<blockquote>
<p>"The set of ordinal numbers has the important property of being well
ordered,which means that every subset has a minimum element. This is
unlike the integers or the positive real numbers, where any element
has another tha... | Evangelopoulos Phoevos | 739,818 | <p>Hint: show that <span class="math-container">$\prod C_i= \bigcap π_i^{-1}(C_i)$</span> where <span class="math-container">$π_i: \prod X_i \to X_i $</span> is the i-th projection</p>
|
2,133,870 | <p>I'm having a little trouble solving this contour integral. I've found the singularities to be at 0, 2, and $1\pm i$. I'm not quite sure how to evaluate the integral when there are two poles on the contour instead of just one. Any help is appreciated.</p>
| Jack D'Aurizio | 44,121 | <p>$$\text{PV}\int_{-\infty}^{+\infty}\frac{x}{(x-1)^4-1}\,dx = \text{PV}\int_{-\infty}^{+\infty}\frac{x+1}{x^4-1}\,dx = \text{PV}\int_{-\infty}^{+\infty}\frac{dx}{(x-1)(x^2+1)} $$
equals, by partial fraction decomposition,
$$ \frac{1}{2}\,\text{PV}\int_{-\infty}^{+\infty}\frac{dx}{x-1}-\frac{1}{2}\,\text{PV}\int_{-\in... |
1,791,849 | <p>On the <a href="https://en.wikipedia.org/wiki/Pentagon" rel="nofollow">Wikipedia Page about Pentagons</a>, I noticed a statement in their work saying that $\sqrt{25+10\sqrt{5}}=5\tan(54^{\circ})$ and $\sqrt{5-2\sqrt{5}}=\tan(\frac {\pi}{5})$</p>
<p>My question is: How would you justify that? My goal is to simplify ... | lulu | 252,071 | <p>Thanks to @N.F.Taussig for pointing out some errors.</p>
<p>Suppose you have a regular pentagon, with vertices $\{A,B,C,D,E\}$ (labelled cyclically so $A$ is neighbor to $B$ and $E$). Let us take each side length to be $1$. Let $X$ be the length of a diagonal, say $AC$. Let's first compute $X$. </p>
<p><a href... |
852,404 | <p>How to prove that there exist an infinite number of prime $n$ for which $n^2=p+8$ for some prime $p$?</p>
<p>Verification of the form $n^2=p+8$ where $n$ and $p$ are some $p$.</p>
<p>$$\begin{array}{|c|c|} \hline
n & n^2 = 8 + p \\
\hline
11 & 121 = 8 + 113 \... | Will Jagy | 10,400 | <p>No proof is available. There are no proofs that any polynomial in a single integer variable, of degree at least two, such as your $n^2 - 8,$ take on an infinite number of prime values. In comparison, $n^2 - 8 m^2$ certainly does, but this is a function of two integer variables. </p>
<p>The best known is $n^2 + 1$ ... |
1,905,186 | <blockquote>
<p>Let <span class="math-container">$R$</span> be a commutative Noetherian ring (with unity), and let <span class="math-container">$I$</span> be an ideal of <span class="math-container">$R$</span> such that <span class="math-container">$R/I \cong R$</span>. Then is it true that <span class="math-container"... | quid | 85,306 | <p>Assume that $R/I$ and $R$ are isomorphic. Let us denote the isomophism by $f:R/I \to R$. </p>
<p>Let $\pi:R \to R/I$ denote the usual map $x \mapsto x + I$.
This is a of course a surjective ring homomorphism. </p>
<p>The composition $f \circ \pi : R \to R$ is thus a surjective ring endomorphism (the composition o... |
1,905,186 | <blockquote>
<p>Let <span class="math-container">$R$</span> be a commutative Noetherian ring (with unity), and let <span class="math-container">$I$</span> be an ideal of <span class="math-container">$R$</span> such that <span class="math-container">$R/I \cong R$</span>. Then is it true that <span class="math-container"... | Aragogh | 665,469 | <p>Assume <span class="math-container">$a$</span> is a proper ideal. Suppose they were isomorphic. Then <span class="math-container">$\varphi: A \to A/a$</span> is some arbitrary isomorphism, and correspondingly <span class="math-container">$\varphi(a) := I_{1} \subset A/a$</span> is an ideal (proper inclusion as <span... |
496,178 | <p>Let $t$ be a positive real number. Differentiate the function</p>
<blockquote>
<p>$$g(x)=t^x x^t.$$</p>
</blockquote>
<p>Your answer should be an expression in $x$ and $t$.</p>
<p>came up with the answer </p>
<blockquote>
<p>$$(x/t)+(t/x)\ln(t^x)(x^t)=\ln(t^x)+\ln(x^t)=x\ln t+t\ln x .$$</p>
</blockquote>
<p... | Brian M. Scott | 12,042 | <p>I’m not sure just what you were doing; it looks as if you might have been trying to use logarithmic differentiation and getting a bit confused in the process. In any case there’s no need for anything fancy.</p>
<p>Since $g(x)$ is a product, your first step should be to use the product rule:</p>
<p>$$g'(x)=t^x(x^t)... |
1,465,989 | <p>I'm trying to find the column space of $\begin{bmatrix}a&0\\b&0\\c&0\end{bmatrix}$, which I think is $span\left(~\begin{bmatrix}a\\b\\c\end{bmatrix}~\begin{bmatrix}0\\0\\0\end{bmatrix}~\right)$. Since by definition a span needs to include the null vector, this is redundant. Would it also be correct to ... | Martin Argerami | 22,857 | <p>It is exactly as you say: in any vector space, the null vector belongs to the span of any vector. </p>
|
2,533,186 | <p>How would you find all pairs $(z,w)$ that satisfy $zw=-1$ </p>
<p>Im stuck on this problem and the best way I can think of would be to guess and check?</p>
| fleablood | 280,126 | <p>Well, $z\ne 0$ and $w = \frac {-1}z$ will do it. The question is what format do you want? </p>
<p>If you want it is $z = a+bi$ and $w= c+di$ where $c$ and $d$ are expressed in terms of $a$ and $b$.</p>
<p>Then $\frac {-1}{a+bi} = \frac {-1(a-bi)}{(a+bi)(a-bi)} = \frac {-a+bi}{a^2 + b^2} = -\frac {a}{a^2 + b^2} +... |
592,805 | <p>How to prove $$\limsup\limits_{n \to \infty} \frac{1}{\sqrt n}B(n) = +\infty$$ using Blumenthal zero-one law, where $(B(t))_{t \geq 0}$ is a Brownian motion?</p>
| Alex R. | 22,064 | <p>Here are some hints:</p>
<p>1) Use the fact that $aB_t$ has the same distribution as $B_{a^2t}$. This means that at any given time $n$, your quantity inside the supremum has distribution $B(1)$.</p>
<p>2) For any $a>0$, </p>
<p>$$\mathbb{P}(B(n)>a\sqrt{n} \mbox{ occurs infinitely often})\geq \limsup_{n\righ... |
4,000,459 | <p>How can we mathematically precisely argue that <span class="math-container">$$\lim_{n \to \infty} \left(1-\frac xn+\frac{x}{n^2}\right)^n = e^{-x}$$</span> holds?</p>
<p>So how can we bring</p>
<p><span class="math-container">$$1-\frac xn+\frac{x}{n^2} = 1- \frac{(n+1)x}{n^2} \approx 1 - \frac xn $$</span>
and
<span... | Z Ahmed | 671,540 | <p>When <span class="math-container">$L=\lim_{x\to a} f(x)^{g(x)}\to 1^{\infty}$</span>, then
<span class="math-container">$L=\exp[\lim_{x\to a} g(x) [f(x)-1]$</span>
<span class="math-container">$$K=\lim_{n \to \infty} \left(1-\frac xn+\frac{x}{n^2}\right)^n $$</span>
<span class="math-container">$$K=\exp[ \lim_{x \t... |
4,000,459 | <p>How can we mathematically precisely argue that <span class="math-container">$$\lim_{n \to \infty} \left(1-\frac xn+\frac{x}{n^2}\right)^n = e^{-x}$$</span> holds?</p>
<p>So how can we bring</p>
<p><span class="math-container">$$1-\frac xn+\frac{x}{n^2} = 1- \frac{(n+1)x}{n^2} \approx 1 - \frac xn $$</span>
and
<span... | CHAMSI | 758,100 | <p>Notice that <span class="math-container">$ \left(\forall y\in\left(0,1\right)\right) $</span> : <span class="math-container">$$ y\leq-\ln{\left(1-y\right)}\leq\frac{y}{1-y} $$</span></p>
<p>That can be proven either using the main value theorem,, or studiing some functions.</p>
<p>Thus, setting <span class="math-con... |
4,310,529 | <p>I need a formula that will give me all points of random ellipse and circle intersection (ok, not fully random, the center of circle is laying on ellipse curve)</p>
<p>I need step by step solution (algorithm how to find it) if this is possible.</p>
| David Quinn | 187,299 | <p>Referring to the standard equation of the ellipse, <span class="math-container">$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$</span>, choose the centre of the circle to correspond to the parameter value <span class="math-container">$\theta$</span>, so the equation of the circle with this centre and radius <span class="math-c... |
46,076 | <p>I'm developing a larger package which includes several subpackages. My problem is, that I can't introduce the symbols in the subpackages to the autocompletion by loading the main package, but by calling a subpackage.</p>
<p>Let's explain this with an example: I have a main package, called <code>main</code> which lo... | Mr.Wizard | 121 | <p>As <a href="https://mathematica.stackexchange.com/users/1356/oska">Öskå</a> notes you can <a href="http://reference.wolfram.com/mathematica/ref/Partition.html" rel="noreferrer"><code>Partition</code></a> your data and then <a href="http://reference.wolfram.com/mathematica/ref/Map.html" rel="noreferrer"><code>Map</co... |
3,851,650 | <p>We want to prove that the number of sequences <span class="math-container">$(a_1,...,a_{2n})$</span> such that
<span class="math-container">$$
• \text{ every } a_i \text{ is equal to} ±1;\\• a_1 + ··· + a_{2n} = 0;\\• \text{ every partial sum satisfies } a_1 + ··· + a_i > −2
$$</span>
is a Catalan number.</p>
<p... | Dr. Mathva | 588,272 | <p><em>Hint.</em> Catalan numbers <span class="math-container">$C_n$</span> can be used to count the number of "monotonic lattice paths along the edges of a grid with <span class="math-container">$n \times n$</span> square cells, which do not pass above the diagonal. A monotonic path is one which starts in the low... |
3,753,143 | <p>We spend a lot of time learning different theories (for instance, theory of differential forms, sobolev spaces, homology groups, distributions). Although (at least most parts of) these theories are very natural and understandable when we read them from books, they are very difficult to create at the first place: it ... | Favst | 742,787 | <p><a href="https://en.wikipedia.org/wiki/Fermat%27s_Last_Theorem" rel="nofollow noreferrer">Fermat's last theorem</a> is certainly an example of a problem that can be stated simply but led to magnificent efforts over hundreds of years and the development of much machinery before it was finally resolved. Wikipedia says... |
2,316,514 | <p>I want to calculate $\lim_{x \to 1}\frac{\sqrt{|x^2 - x|}}{x^2 - 1}$ . I tried to compute limit when $x \to 1^{+}$ and $x \to 1^{-}$ but didn't get any result . </p>
<p>Please help .</p>
<p>Note : I think it doesn't have limit but I can't prove it .</p>
| Nominal Animal | 318,422 | <p>The trick is that instead of using $t$ as a parameter along one axis, you use it as a free parameter, with $t = 0$ at the beginning of the curve, and $t = 1$ at the end of the curve, with $0 \le t \le 1$ specifying the points on the curve.</p>
<p>All cubic curves of a single parameter are of form
$$\begin{cases}
x(... |
4,563,725 | <p>Is it possible for two non real complex numbers a and b that are squares of each other? (<span class="math-container">$a^2=b$</span> and <span class="math-container">$b^2=a$</span>)?</p>
<p>My answer is not possible because for <span class="math-container">$a^2$</span> to be equal to <span class="math-container">$b$... | Torsten Schoeneberg | 96,384 | <p>As user Henry says in a comment, <span class="math-container">$a^2=b$</span> and <span class="math-container">$b^2=a$</span> imply <span class="math-container">$a^4=a$</span>, i.e. <span class="math-container">$a$</span> is a zero of the polynomial:</p>
<p><span class="math-container">$$p(x)=x^4-x$$</span></p>
<p>An... |
4,563,725 | <p>Is it possible for two non real complex numbers a and b that are squares of each other? (<span class="math-container">$a^2=b$</span> and <span class="math-container">$b^2=a$</span>)?</p>
<p>My answer is not possible because for <span class="math-container">$a^2$</span> to be equal to <span class="math-container">$b$... | Piquito | 219,998 | <p>There are the number of real solutions of the system
<span class="math-container">$$a^2-b^2=(a^2-b^2)^2-4a^2b^2\\2ab=4ab(a^2-b^2)\tag1$$</span> which come from
the equalities <span class="math-container">$$(a+bi)^2=(a^2-b^2)+2abi\\((a^2-b^2)+2abi)^2=(a^2-b^2)^2-4a^2b^2+4ab(a^2-b^2)i$$</span>
The solution of system <... |
3,267,216 | <p>An urn contains equal number of green and red balls. Suppose you are playing the following game. You draw one ball at random from the urn and note its colour. The ball is then placed back in the urn, and the selection process is repeated. Each time a green ball is picked you get 1 Rupee. The first time you pick a re... | Rohit Singh | 683,232 | <p>Apply componendo &dividendo in ratio of AM/GM and then u will get root A +root B ka whole square on top and root a -root B ka whole square in bottom,then u can easily get ratio.i am new so I am not able to upload image to show u</p>
|
369,435 | <p>My task is as in the topic, I've given function $$f(x)=\frac{1}{1+x+x^2+x^3}$$
My solution is following (when $|x|<1$):$$\frac{1}{1+x+x^2+x^3}=\frac{1}{(x+1)+(x^2+1)}=\frac{1}{1-(-x)}\cdot\frac{1}{1-(-x^2)}=$$$$=\sum_{k=0}^{\infty}(-x)^k\cdot \sum_{k=0}^{\infty}(-x^2)^k$$ Now I try to calculate it the following w... | Clayton | 43,239 | <p><strong>Hint:</strong> Another method to approach the problem (more by brute force than Andre's clever technique) is $$f(x)=\frac{1}{1+x+x^2+x^3}=\frac{1}{(x+1)(x^2+1)}.$$ Now use partial fraction decomposition and geometric series.</p>
|
37,380 | <p>I do have noisy data and want to smooth them by a Savitzky-Golay filter because I want to keep the magnitude of the signal. </p>
<p>a) Is there a ready-to-use Filter available for that? </p>
<p>b) what are appropriate values for m (the half width) and for the coefficients for 3000-4000 data points?</p>
| J. M.'s persistent exhaustion | 50 | <p>I'm just posting this to record for posterity <a href="http://chat.stackexchange.com/transcript/message/4829800#4829800">something I posted in the chatroom</a> a not-so-long time ago. As I noted there, the following routine will only do smoothing; <del>I had a more general routine for generating the differentiation ... |
2,523,570 | <p>$15x^2-4x-4$,
I factored it out to this:
$$5x(3x-2)+2(3x+2).$$
But I don’t know what to do next since the twos in the brackets have opposite signs, or is it still possible to factor them out?</p>
| Stella Biderman | 123,230 | <p>I think you made an arithmetic mistake. I got $$15x^2-4x-4=(5x+2)(3x-2)=5x(x-2)+2(3x-2).$$</p>
|
227,296 | <p>I happened to ponder about the differentiation of the following function:
$$f(x)=x^{2x^{3x^{4x^{5x^{6x^{7x^{.{^{.^{.}}}}}}}}}}$$
Now, while I do know how to manipulate power towers to a certain extent, and know the general formula to differentiate $g(x)$ wrt $x$, where $$g(x)=f(x)^{f(x)^{f(x)^{f(x)^{f(x)^{f(x)^{f(x)... | juan | 7,402 | <p>This is not exactly an answer. I try to give a meaning to $f(x)$.
Not as a limit. I will give a formal
solution as a power series (possibly not converging). </p>
<p>Consider a function of two parameters
$$u(t,x)=(t+1)x^{(t+2)x^{(t+3)x^{(t+4)x^{(t+5)x^{(t+6)x^{(t+7)x^{.{^{.^{.}}}}}}}}}}$$
so that $f(x)=u(0,x)$. ... |
3,960,189 | <p>An auto insurance company is implementing a new bonus system. In each month, if a policyholder does not have an accident, he or she will receive a $5 cashback bonus from the insurer. Among the 1000 policyholders, 400 are classified as low-risk drivers and 600 are classified as high-risk drivers. In each month, the p... | Alex | 38,873 | <p>OK here's the full solution:</p>
<p>Each risky customer is Binomial(<span class="math-container">$0.8$</span>), so the mean number of accident-free months is <span class="math-container">$\mathbf{E}X=np = 9.6$</span>. Bonus is a function of this rv, <span class="math-container">$^Y=cX$</span>, so <span class="math-c... |
2,078,142 | <p>Let $X$ be locally compact metric space which is $\sigma$-compact also. I want to show that $X=\bigcup\limits_{n=1}^{\infty}K_n$, where $K_n$'s are compact subsets of $X$ satisfying $K_n\subset K_{n+1}^{0}$ for all $n\in \mathbb N$. </p>
<p>I know that since $X$ is $\sigma-$compact, therefore there exists a sequenc... | PatrickR | 52,912 | <p>The desired property is called the existence of an <a href="https://en.wikipedia.org/wiki/Exhaustion_by_compact_sets" rel="nofollow noreferrer">exhaustion by compact sets</a> for the space <span class="math-container">$X$</span>. (see Wikipedia)</p>
<p>The answer of @bof holds in the context of locally compact Hausd... |
2,969,363 | <p>I have 3 points in space A, B, and C all with (x,y,z) coordinates, therefore I know the distances between all these points. I wish to find point D(x,y,z) and I know the distances BD and CD, I do NOT know AD.</p>
<p>The method I have attempted to solve this using is first saying that there are two spheres known on p... | Ross Millikan | 1,827 | <p>Knowing <span class="math-container">$|BD|$</span> and <span class="math-container">$|CD|$</span> is only enough to know that <span class="math-container">$D$</span> is on the intersection of two spheres, one around <span class="math-container">$B$</span> and one around <span class="math-container">$C$</span>. Assu... |
2,969,363 | <p>I have 3 points in space A, B, and C all with (x,y,z) coordinates, therefore I know the distances between all these points. I wish to find point D(x,y,z) and I know the distances BD and CD, I do NOT know AD.</p>
<p>The method I have attempted to solve this using is first saying that there are two spheres known on p... | Nominal Animal | 318,422 | <p>(This is not an answer to the stated question per se, but explains how to efficiently do trilateration.)</p>
<p>In 3D, you need four fixed points (that are not all on the same plane), and their distances to the unknown point, to exactly determine the location of the unknown point.</p>
<p>Three fixed points (that a... |
1,213,663 | <p>If you measure a task & it takes 3 seconds, then the next time you do the same task, it takes you 1 second, is the difference 200% or 67%? </p>
<p>Or would you say the difference is 200% because 3-1=2 or 200% better -- but the percentage of difference is 2/3 or 67%? I'm pretty sure I'm confusing something if ... | Robert Israel | 8,508 | <p>Permuting the rows of $M$ gives you $P M$ where $P$ is a permutation matrix.
The singular values of $PM$ are the square roots of the (nonzero) eigenvalues of $(PM)^* PM = M^* P^* P M = M^* M$, so they are the same as the singular values of $M$.</p>
|
1,691,825 | <p>My textbook goes from</p>
<p>$$\frac{\left( \frac{6\ln^22x}{2x} \right)}{\left(\frac{3}{2\sqrt{x}}\right)}$$</p>
<p>to:</p>
<p>$$\frac{6\ln^22x}{3\sqrt{x}}$$</p>
<p>I don't see how this is right. Could anyone explain?</p>
| Jan Eerland | 226,665 | <p>$$\frac{\frac{6\ln^2(2x)}{2x}}{\frac{3}{2\sqrt{x}}}=\frac{6\ln^2(2x)}{2x}\cdot\frac{2\sqrt{x}}{3}=\frac{6\ln^2(2x)}{x}\cdot\frac{\sqrt{x}}{3}=$$
$$\frac{2\ln^2(2x)}{x}\cdot\frac{\sqrt{x}}{1}=\frac{2\ln^2(2x)\sqrt{x}}{x}=\frac{2\ln^2(2x)}{\sqrt{x}}$$</p>
|
1,691,825 | <p>My textbook goes from</p>
<p>$$\frac{\left( \frac{6\ln^22x}{2x} \right)}{\left(\frac{3}{2\sqrt{x}}\right)}$$</p>
<p>to:</p>
<p>$$\frac{6\ln^22x}{3\sqrt{x}}$$</p>
<p>I don't see how this is right. Could anyone explain?</p>
| Michael Hoppe | 93,935 | <p>Just multiply denominator and numerator by $2x=2\sqrt x\cdot\sqrt x$ to obtain the given result.</p>
|
934,353 | <p>I am a high school student in Calculus, and we are finishing learning basic limits. I am reviewing for a big test tomorrow, and I could do all of the problems correctly except this one.</p>
<p>I have no idea how to solve the problem this problem correctly. I looked up the answer online, but I can't figure out how t... | Mohamed | 33,307 | <p>The formula :$$f(x)= \frac{1-\sqrt{1+x}}{x \sqrt{1+x}}$$is nont able to give the limit because it gives the indeterminate form $\frac00$.</p>
<p>Using the conjugate expression of $1-\sqrt{1+x}$, you get: $$f(x)=\frac{-x}{x \sqrt{1+x}(1+\sqrt{1+x})},$$
then: $$f(x)=\frac{-1}{ \sqrt{1+x}(1+\sqrt{1+x})}$$
and the lim... |
934,353 | <p>I am a high school student in Calculus, and we are finishing learning basic limits. I am reviewing for a big test tomorrow, and I could do all of the problems correctly except this one.</p>
<p>I have no idea how to solve the problem this problem correctly. I looked up the answer online, but I can't figure out how t... | user194150 | 194,150 | <p>$$
\displaylines{
f\left( x \right) = \frac{1}{{\sqrt {1 + x} }} \Rightarrow f'\left( x \right) = \frac{{ - 1}}{{2\left( {1 + x} \right)\sqrt {1 + x} }} \cr
f'\left( 0 \right) = \mathop {\lim }\limits_{x \to 0} \frac{{f\left( x \right) - f\left( 0 \right)}}{{x - 0}} \cr
= \mathop {\lim }\limits_{x \to 0} \f... |
33,303 | <p>Is it possible to attach certain pieces of code to certain controls in a Manipulate? For example, consider the following Manipulate</p>
<pre><code>Manipulate[
data = Table[function[x], {x, -Pi*10, Pi*10, Pi/1000}];
ListPlot[{x, data}, PlotRange -> {{start, stop}, Automatic}]
, {function, {Sin, Cos, Tan}}
, ... | Michael E2 | 4,999 | <p>Here's a fairly simple way to fix your <code>Manipulate</code> by applying <code>Dynamic</code> to <code>ListPlot</code>.</p>
<pre><code>Manipulate[
(* Beep[]; *)
data = function @ Range[-Pi*10., Pi*10, Pi/1000];
Dynamic @ ListPlot[data, PlotRange -> {{start, stop}, Automatic}],
{function, {Sin, Cos, Tan}},
... |
47,561 | <p>The Hilbert matrix is the square matrix given by</p>
<p>$$H_{ij}=\frac{1}{i+j-1}$$</p>
<p>Wikipedia states that its inverse is given by</p>
<p>$$(H^{-1})_{ij} = (-1)^{i+j}(i+j-1) {{n+i-1}\choose{n-j}}{{n+j-1}\choose{n-i}}{{i+j-2}\choose{i-1}}^2$$</p>
<p>It follows that the entries in the inverse matrix are all i... | Fedor Petrov | 4,312 | <p>It suffices to prove Schechter's formula for Cauchy matrix cited in Wikipedia (see link in Faisal's comment). We need to check $\sum_j b_{ij}a_{jk}=\delta_{ik}$, i.e.
$$
\frac{A(y_i)}{B'(y_i)}\sum_j \frac{f(x_j)}{A'(x_j)}=-\delta_{i,k},
$$
where $f(t)=B(t)/((t-y_i)(t-y_k))$. If $i\ne k$, then $f$ is just a polynomia... |
3,252,076 | <p>I'm doing some Galois cohomology stuff (specifically, trying to calculate <span class="math-container">$H^1(\mathbb{Q}_3,E[\varphi])$</span>, where <span class="math-container">$\varphi:E\to E'$</span> is an isogeny of elliptic curves), and it involves calculating <span class="math-container">$\mathbb{Q}_3(\sqrt{-6}... | nguyen quang do | 300,700 | <p>Your problem being local, why do you complicate it by bringing it back to a global one ? I'll keep your notation <span class="math-container">$K_v = \mathbf Q_3 (\sqrt {-6})$</span> and work locally.</p>
<p>For any <span class="math-container">$p$</span>-adic local field <span class="math-container">$K$</span> of d... |
1,091,653 | <p>Is this correct ?</p>
<p>$$
\frac{d}{dt} \left( \int_0^t \phi(t)dt \right) = \phi(t)
$$</p>
<p>If not, how can I recover $$ \phi(t) $$ knowing only $$ \int_0^t \phi(t)dt $$ ?</p>
| MathMajor | 113,330 | <p>(i) No, it is not correct. You probably mean
$$
\frac{\text{d}}{\text{d}t} \left( \int_0^t \phi(x) \, \text{d}x \right) = \phi(t)
.$$</p>
<p>This equality is also known as the Fundamental Theorem of Calculus, Part 1.</p>
<p>(ii) You can recover $\phi (t)$ by differentiating, as demonstrated above.</p>
|
1,964,139 | <p>$D_2n$ is not abelian. However, the group of rotations, denoted $R$, is. I've already shown that $R$ is a normal subgroup of $D_2n$; however I'm stuck at showing the quotient group is abelian.</p>
<p>I know if it is abelian, $xRyR=yRxR$ but I get stuck at $xRyR=(xy)R$.
But $x$ and $y$ are not necessarily commutativ... | Dietrich Burde | 83,966 | <p>Since $R=C_n$ is a subgroup of $D_{2n}$ of index $2$, it is a normal subgroup. The quotient $D_{2n}/C_n$ has exactly $2$ elements, because of
$$
|D_{2n}/C_n|=\frac{|D_{2n}|}{|C_n|}=\frac{2n}{n}=2.
$$
But every group of order $2$ is abelian (there is only one).</p>
|
1,553,440 | <p>Let , $g(x)=f(x)+f(1-x)$ and $f'(x)<0$ for all $x\in (0,1)$. Then , $g$ is monotone increasing in </p>
<p>(A) $(1/2,1)$.</p>
<p>(B) $(0,1/2)$</p>
<p>(C) $(0,1/2)\cup (1/2,1)$.</p>
<p>(D) none.</p>
<p>We have , $g'(x)=f'(x)-f'(1-x)$. Now $g'(x)>0$ if $f'(x)>f'(1-x)$. As $f'(x)<0$ so , $f$ is monotone... | Thomas Andrews | 7,933 | <p>Let $f_1(x)=-x^2$. Then $f_1'(x)=-2x<0$ and $g_1(x)=-x^2-(1-x))^2 = -2x^2+2x-1$ and $g_1'(x)=2-4x$ so $g(x)$ is decreasing when $x>\frac{1}{2}$.</p>
<p>On the other hand, if $f_2(x)=(1-x)^2$. Then $f_2'(x)=-2(1-x)<0$ and $g_2(x)=(1-x)^2+x^2=-g_1(x)$. So $g_2(x)$ is increasing precisely where $g_1(x)$ was n... |
1,523,329 | <p>Can cusps be considered points of inflection?</p>
<p>I'm getting conflicting information but my thought process is that cusps cannot be points of inflection?</p>
<p>Can points of inflection exist when there is a vertical tangent to the graph? Assume there is change in concavity and the function is continuous. </p... | abel | 9,252 | <p>no. look at the graph of $y = x^{2/3}.$ this has a cusp at $(0,0)$ but concave down on $(-\infty, \infty)$ and $(0,0)$ is certainly not a point of inflection. </p>
|
1,523,329 | <p>Can cusps be considered points of inflection?</p>
<p>I'm getting conflicting information but my thought process is that cusps cannot be points of inflection?</p>
<p>Can points of inflection exist when there is a vertical tangent to the graph? Assume there is change in concavity and the function is continuous. </p... | Carly Vollet | 989,151 | <p>I would not consider <span class="math-container">$f(x)=x^{2/3}$</span> to be concave down on <span class="math-container">$(-\infty,\infty)$</span>, but rather on <span class="math-container">$(-\infty, 0)\cup(0,\infty)$</span>. If we use the definition of concave down to be "A function <span class="math-conta... |
142,507 | <p>Let $\psi(x) := \sum_{n\leq x} \Lambda(n)$ where $\Lambda(n)$ is the Von-Mangoldt function.
I want to show that if $$ \lim_{x \rightarrow \infty} \frac{\psi(x)}{x} =1 $$ then also $$\lim_{x\rightarrow \infty} \frac{\pi(x) \log x }{x}=1.$$</p>
<p>I tried to play a little bit with $\psi$, what I want to show is that:... | Arturo Magidin | 742 | <p>Do you know how to prove that $\mathbb{R}$ is numerically equivalent to $\mathcal{P}(\mathbb{N})$? Show that $\mathbb{R}$ is numerically equivalent to $(0,1)$, then show (using binary representation; careful with the numbers with dual representation) that there is an embedding $(0,1)\hookrightarrow \mathcal{P}(\math... |
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