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 |
|---|---|---|---|---|
678,768 | <p>"Let $A$, $B$ be two infinite sets. Suppose that $f: A \to B$ is injective. Show that there exists a surjective map $g: B \to A$"</p>
<p>I am not sure how to go about this proof, I am trying to gather information to help me, and deduce as much as I can:
. Since $f$ is injective we know that $|A| \leq |B|$. </p>
<p... | Martín-Blas Pérez Pinilla | 98,199 | <p>Definition of $g$:</p>
<p>If $y=f(x)\in F(A)$, $g(y)=x$.</p>
<p>If $y\in B\backslash f(A)$, $g(y)=$ arbitrary.</p>
<p>The function is well-defined because $f$ is injective.</p>
|
1,843,274 | <p>Good evening to everyone. So I have this inequality: $$\frac{\left(1-x\right)}{x^2+x} <0 $$ It becomes $$ \frac{\left(1-x\right)}{x^2+x} <0 \rightarrow \left(1-x\right)\left(x^2+x\right)<0 \rightarrow x^3-x>0 \rightarrow x\left(x^2-1\right)>0 $$ Therefore from the first $ x>0 $, from the second $ x... | Siong Thye Goh | 306,553 | <p>It seems to me that you are only considering two roots for the cubic polynomial. The polynomial has three roots, $-1,0,1$ and they are all simple root.</p>
<p>Hence the polynomials is positive on $(1,\infty)$. (it is positive as it goes to positive infinity).</p>
<p>The sign changes at the roots.</p>
<p>it is neg... |
777,863 | <p>Does there exists an $f:\mathbb{R}\rightarrow \mathbb{R}$ differentiable everywhere with $f'$ discontinuous at some point?</p>
| David | 119,775 | <p>Yes there does exist such a function, for example
$$f(x)=\cases{x^2\sin(1/x)&if $x\ne0$\cr 0&if $x=0$.\cr}$$
By normal differentiation rules we have
$$f'(x)=2x\sin(1/x)-\cos(1/x)$$
if $x\ne0$, and for $x=0$ we use the definition:
$$f'(0)=\lim_{h\to0}\frac{f(h)-f(0)}{h-0}=\lim_{h\to0}h\sin(1/h)=0\ .$$
So $f$ ... |
184,699 | <p>First, we make the following observation: let $X: M \rightarrow TM $ be a vector
field on a smooth manifold. Taking the contraction with respect to $X$ twice gives zero, i.e.
$$ i_X \circ i_{X} =0.$$
Is there any "name" for the corresponding "homology" group that one can define
(Kernel mod image)? Has this "homo... | Matthias Ludewig | 16,702 | <p>I believe that the answer is something in between "not really" and "kind of" and was indicated by Qiaochu Yan.</p>
<p>In Wittens famous paper "Morse Theory and Supersymmetry", he considers operators of the form $\sigma d + \alpha$ for a one-form $\alpha$ and a coefficient $\sigma$ with only non-degenerate critical ... |
2,970,787 | <blockquote>
<p>Find <span class="math-container">$\lim_{x\to -\infty} \frac{(x-1)^2}{x+1}$</span></p>
</blockquote>
<p>If I divide whole expression by maximum power i.e. <span class="math-container">$x^2$</span> I get,<span class="math-container">$$\lim_{x\to -\infty} \frac{(1-\frac1x)^2}{\frac1x+\frac{1}{x^2}}$$</... | Dr. Sonnhard Graubner | 175,066 | <p>Hint: Write <span class="math-container">$$\frac{x^2\left(1-\frac{1}{x}\right)^2}{x\left(1+\frac{1}{x}\right)}$$</span></p>
|
2,970,787 | <blockquote>
<p>Find <span class="math-container">$\lim_{x\to -\infty} \frac{(x-1)^2}{x+1}$</span></p>
</blockquote>
<p>If I divide whole expression by maximum power i.e. <span class="math-container">$x^2$</span> I get,<span class="math-container">$$\lim_{x\to -\infty} \frac{(1-\frac1x)^2}{\frac1x+\frac{1}{x^2}}$$</... | user | 505,767 | <p><strong>HINT</strong></p>
<p>We have that</p>
<p><span class="math-container">$$\lim_{x\to -\infty} \frac{(x-1)^2}{x+1}=\lim_{x\to -\infty} (x-1)\cdot \frac{x-1}{x+1}$$</span></p>
|
3,078,707 | <p>The above question is the equation <span class="math-container">$(2.4)$</span> of the following paper:</p>
<p><a href="http://www.jmlr.org/papers/volume6/tsuda05a/tsuda05a.pdf" rel="nofollow noreferrer">MATRIX EXPONENTIATED GRADIENT UPDATES</a>.</p>
<p>Let <span class="math-container">$M$</span> and <span class="m... | David K | 139,123 | <p>Here's a construction that I think is relatively easy to implement.</p>
<p>Find the point <span class="math-container">$P$</span> where the tangent at <span class="math-container">$(-c,d)$</span> intersects the <span class="math-container">$y$</span> axis.
Suppose the coordinates of that point are <span class="math... |
192,020 | <p>I suspect this is a duplicate, but I can't seem to find what I'm looking for.</p>
<p>A routine problem I have is the following.</p>
<p>I have a set of data in three (or two, or more) lists:</p>
<pre><code>l1={a1, a2, a3}
l2={b1, b2, b3, b4}
l3={{c1, c2, c3, c4}, {d1, d2, d3, d4}, {e1, e2, e3, e4}}
</code></pre>
... | Roman | 26,598 | <p>This will generate a nested list, in accordance with <code>l3</code>:</p>
<pre><code>MapThread[Append, {Outer[List, l1, l2], l3}, 2]
</code></pre>
<blockquote>
<p>{{{a1, b1, c1}, {a1, b2, c2}, {a1, b3, c3}, {a1, b4, c4}}, {{a2, b1, d1}, {a2, b2, d2}, {a2, b3, d3}, {a2, b4, d4}}, {{a3, b1, e1}, {a3, b2, e2}, {a3,... |
341,202 | <p>The divisibility rule for $3$ is well-known: if you add up the digits of $n$ and the sum is divisible by $3$, then $n$ is divisible by three. This is quite helpful for determining if really large numbers are multiples of three, because we can recursively apply this rule:</p>
<p>$$1212582439 \rightarrow 37 \rightar... | Brian M. Scott | 12,042 | <p>HINT: Suppose that you have a four-digit number $n$ that is written $abcd$. Then</p>
<p>$$\begin{align*}
n&=10^3a+10^2b+10c+d\\
&=(999+1)a+(99+1)b+(9+1)c+d\\
&=(999a+99b+9c)+(a+b+c+d)\\
&=3(333a+33b+3c)+(a+b+c+d)\;,
\end{align*}$$</p>
<p>so when you divide $n$ by $3$, you’ll get </p>
<p>$$333a+33b... |
341,202 | <p>The divisibility rule for $3$ is well-known: if you add up the digits of $n$ and the sum is divisible by $3$, then $n$ is divisible by three. This is quite helpful for determining if really large numbers are multiples of three, because we can recursively apply this rule:</p>
<p>$$1212582439 \rightarrow 37 \rightar... | Arthur | 15,500 | <p>How about induction?</p>
<p>It is obviously true for the one-digit numbers $3, 6$ and $9$, so we have our base case (really, just the case $3$ is all it takes, but I like to be on the safe side when it comes to induction).</p>
<p>Now, let's say that we have a number divisible by $3$, and let's call it $n$. We can ... |
341,202 | <p>The divisibility rule for $3$ is well-known: if you add up the digits of $n$ and the sum is divisible by $3$, then $n$ is divisible by three. This is quite helpful for determining if really large numbers are multiples of three, because we can recursively apply this rule:</p>
<p>$$1212582439 \rightarrow 37 \rightar... | lab bhattacharjee | 33,337 | <p>Using Property$\#10$ of <a href="http://mathworld.wolfram.com/Congruence.html" rel="nofollow noreferrer">this</a> (<a href="https://math.stackexchange.com/questions/188657/why-an-bn-is-divisible-by-a-b"> indirect Proof</a>),</p>
<p>as $10\equiv1\pmod9,10^r\equiv1^r\equiv1$ for integer $r\ge0$</p>
<p>$$\implies\sum... |
1,687,147 | <blockquote>
<p>A category <span class="math-container">$\mathsf C$</span> consists of the following three mathematical entities:</p>
<ul>
<li><p>A class <span class="math-container">$\operatorname{ob}(\mathsf{C})$</span>, whose elements are called objects;</p>
</li>
<li><p>A class <span class="math-container">$\hom(\m... | SixWingedSeraph | 318 | <p>To say an object "consists of" followed by a list of entities means that entities are the data types you have to specify to describe the object.</p>
<p>"Consists of" in the case of the definition of category means that to define a category, you have to specify three mathematical objects: Two classes and a binary op... |
2,987,994 | <p>I'm looking for definite integrals that are solvable using the method of differentiation under the integral sign (also called the Feynman Trick) in order to practice using this technique.</p>
<p>Does anyone know of any good ones to tackle?</p>
| Community | -1 | <p>Another example is in evaluating
<span class="math-container">$$\displaystyle \int_0^\infty \dfrac{\cos xdx}{1+x^2}$$</span></p>
<p>by first considering
<span class="math-container">$$I\left(a\right)=\int_{0}^{\infty}\frac{\sin\left(ax\right)}{x\left(1+x^{2}\right)}dx,\,a>0$$</span> we have <span class="math-con... |
2,386,182 | <p>Let $f:[a,b]\to \mathbb{R}$ is an increasing function and for any $y\in[f(a),f(b)],$ there exists a $\xi\in[a,b]$ such that $f(\xi)=y.$ Show that $f(.)$ is continuous on $[a,b]. $</p>
<p>Here is my argument, but I got stuck on the last part.</p>
<p>My goal is to show $$\lim_{n\to \infty}f(x_0+\xi_n)=\lim_{n\to\inf... | Community | -1 | <p>From what I hear jj sylvester pretty much invented invariant theory. Maybe take a look at some of his work. He was apparently a pretty good mathematician. .. well, upon looking it up it appears Cayley gets the credit: the idea was suggested to him by an elegant paper of Boole. Here is the <a href="https://en... |
474,568 | <p>In some books I've seen this symbol $\dagger$, next to some theorem's name, and I don't know what it means. I've googled it with no results which makes me suspect it's not standard.</p>
<p>Does anybody know what it means? One example I'm looking at right now is in a probability book, next to a section about Sitrlin... | RSh | 56,827 | <p>It depends on the author choice.
Some books are using it to show the kind of a specific problem (Harder, Article related, etc.)
Some Books are using it to mark some chapters/sections as an independent one (or in your book as a historical section)
you should refer to the book's preface to know the author mean.
(Gener... |
300,753 | <p>Revision in response to early comments. Users of set theory need an <em>implementation</em> (in case "model" means something different) of the axioms. I would expect something like this:</p>
<blockquote>
<p>An <em>implementation</em> consists of a "collection-of-elements" $X$, and a relation
(logical pairing)... | Giorgio Mossa | 14,969 | <p>If by "<em>the domain of $x \in A$</em>" you mean the objects you can put in $x$ and $A$ then the answer is everything.
This is due to the fact that in set theories such as ZFC and NBG all objects are set/classes (they have all the same type).</p>
<p>I am assuming your thinking $x \in A$ as an operation that associ... |
300,753 | <p>Revision in response to early comments. Users of set theory need an <em>implementation</em> (in case "model" means something different) of the axioms. I would expect something like this:</p>
<blockquote>
<p>An <em>implementation</em> consists of a "collection-of-elements" $X$, and a relation
(logical pairing)... | Burak | 33,039 | <p>Perhaps, your confusion may be resolved by realizing that we do <strong>not</strong> define what a set is, using the axioms of ZFC. Sets are to us like points are to Euclid. Sets are the primitive objects that we are going to work with.</p>
<p>Let me take a Platonist approach to elaborate. When you set up your axio... |
3,809,699 | <p>While we define norm on a vector space, we consider only real or complex vector field. But can we generalize this norm on a vector space over an arbitrary field ? I think this can be done, but we have to define a suitable modulus function on the ground field to be meaningful in the property ||cx||=|c|||x||, what |c|... | Community | -1 | <p>We can (at least sometimes) even define a norm on a module over a ring, a notion which generalizes that of a vector space.</p>
<p>One interesting example is the Gaußian integers, <span class="math-container">$\Bbb Z[i]$</span>. An arbitrary element looks like <span class="math-container">$a+bi, a,b\in\Bbb Z$</span>... |
3,482,476 | <p>For an arbitrary <span class="math-container">$0\leqslant x \leq\frac{\pi^2}6$</span>, can we write <span class="math-container">$x$</span> in the form
<span class="math-container">$$
x = x_0+\sum_{j\in S\subset\mathbb N\setminus\{0\}} \frac1{j^2}, \tag 1
$$</span></p>
<p>where <span class="math-container">$x_0\in... | Zarrax | 3,035 | <p>The sum <span class="math-container">$\sum_{n=2}^{\infty} {1 \over n^2}$</span> is less than <span class="math-container">$1$</span>, so if <span class="math-container">$r$</span> satisfies <span class="math-container">$\sum_{n=2}^{\infty} {1 \over n^2} < r < 1$</span> you won't be able to express <span class=... |
1,142,624 | <p>Find a generating function for $\{a_n\}$ where $a_0=1$ and $a_n=a_{n-1} + n$</p>
| kobe | 190,421 | <p>Multiply both sides of the recurrence by $x^n$ and sum over all $n\ge 1$ to get </p>
<p>\begin{align}\sum_{n \ge 1} a_n x^n &= \sum_{n \ge 1} a_{n-1}x^n + \sum_{n\ge 1} nx^n\\
\sum_{n\ge 1} a_nx^n &= x\sum_{n\ge 1} a_{n-1}x^{n-1} + x\sum_{n\ge 1} nx^{n-1}\\
\sum_{n\ge 1} a_n x^n &= x\sum_{n\ge 0} a_n x^... |
1,814,216 | <p>I was trying to show that $\sin(x)$ is non-zero for integers $x$ other than zero and I thought that this result might emerge as a corollary if I managed to show that the result in question is true. </p>
<p>I think it's possible to demonstrate this by looking at the power series expansion of $\sin(x)$ and assuming t... | egreg | 62,967 | <p>You have $\sin m=\sin n$ if and only if $m=n+2k\pi$ or $m=\pi-n+2k\pi$ (for some integer $k$.</p>
<p>In the first case, if $k\ne0$, you get $\pi=(m-n)/(2k)$ is rational. In the second case, $\pi=(m+n)/(2k+1)$.</p>
<p>Since $\pi$ is irrational, the only possibility is $k=0$ and $m=n$.</p>
<hr>
<p>You can't prove ... |
2,702,726 | <p>Find the absolute minimum and maximum values of,</p>
<p>$$f(x) = 2 \sin(x) + \cos^2 (x) \text{ on } [0, 2\pi]$$</p>
<p>What I did so far is</p>
<p>$$f'(x) = 2\cos(x) -2 \cos(x) \sin(x)$$</p>
<p>Could someone please help me get started?</p>
| DeepSea | 101,504 | <p><strong>hint:</strong> $\cos^2x = 1 -\sin^2x \implies f(x) = 2-\left(1-\sin x\right)^2$. Then use the fact that $-1 \le \sin x \le 1$ to obtain the min and max.</p>
|
2,431,027 | <p>I am asking about changing the limits of integration. </p>
<p>I have the following integral to evaluate - </p>
<p>$$\int_2^{3}\frac{1}{(x^2-1)^{\frac{3}{2}}}dx$$ using the substitution $x = sec \theta$. </p>
<p>The problem states</p>
<p><strong>Use the substitution to change the limits into the form $\int_a^b$ w... | egreg | 62,967 | <p>For simplicity, write $a=\arccos\frac{1}{2}$ and $b=\arccos\frac{1}{3}$. One can take care of them at the end.</p>
<p>It is true that $a=\pi/3$, but it's definitely wrong that $b=\pi/3$. Actually, $b\approx1.230959$, but you won't need that.</p>
<p>The integral becomes
$$
\int_a^b-\frac{1}{(\sec^2\theta-1)^{3/2}}\... |
1,694,159 | <p>I am prepping for my mid semester exam, and came across with the following question:</p>
<blockquote>
<p>Find the closed form for the sum $\sum_{j=k}^n (-1)^{j+k}\binom{n}{j}\binom{j}{k}$, using the assumption that $k = 0, 1,...n$ and $n$ can be any natural number.</p>
</blockquote>
<p>So what I have done is to ... | mathlove | 78,967 | <p>Note that </p>
<p>$$\frac{1}{(n-m-k)!\ m!}=\frac{1}{(n-k)!}\cdot \frac{(n-k)!}{(n-m-k)!\ m!}=\frac{1}{(n-k)!}\binom{n-k}{m}$$
Then, you'll have
$$\frac{n!}{k!\ (n-k)!}\sum_{m=0}^{n-k}(-1)^m\cdot 1^{n-k-m}\cdot\binom{n-k}{m}$$</p>
|
1,694,159 | <p>I am prepping for my mid semester exam, and came across with the following question:</p>
<blockquote>
<p>Find the closed form for the sum $\sum_{j=k}^n (-1)^{j+k}\binom{n}{j}\binom{j}{k}$, using the assumption that $k = 0, 1,...n$ and $n$ can be any natural number.</p>
</blockquote>
<p>So what I have done is to ... | Yakov Shklarov | 222,048 | <p>Here is a cleaner way to solve it, without mucking around with factorials.</p>
<p>It's useful to know the "trinomial revision" identity,
$$\binom{x}{n}\binom{n}{m} = \binom{x}{m}\binom{x-m}{n-m},$$
which holds for all reals $x$ and integers $m,n$. The combinatorial interpretation (under a condition that certain val... |
1,514,094 | <p>Given three points on the $xy$ plane on $O(0,0),A(1,0)$ and $B(-1,0)$.Point $P$ is moving on the plane satisfying the condition $(\vec{PA}.\vec{PB})+3(\vec{OA}.\vec{OB})=0$<br>
If the maximum and minimum values of $|\vec{PA}||\vec{PB}|$ are $M$ and $m$ respectively then prove that the value of $M^2+m^2=34$</p>
<hr>... | Macavity | 58,320 | <p><strong>Hint:</strong> Let $P(x) = x^5-20x^4+bx^3+cx^2+dx+e$. If $P$ has all real roots, then $P'''(x)$ must have two real roots..</p>
<p>$\implies 60x^2-480x+6b$ has real roots $\implies b \le 160$</p>
|
373,510 | <p>I was wondering if there is some generalization of the concept metric to take positive and negative and zero values, such that it can induce an order on the metric space? If there already exists such a concept, what is its name?</p>
<p>For example on $\forall x,y \in \mathbb R$, we can use difference $x-y$ as such... | Abel | 71,157 | <p>If we have a function $\delta$ such that</p>
<ol>
<li>$\delta(x,y)=0$ if and only if $x=y$</li>
<li>$\delta(x,y) = -\delta(y,x)$</li>
<li>If $\delta(x,y)\geq 0$ and $\delta(y,z)\geq 0$, then $0\leq \delta(x,z) \leq \delta(x,y)+\delta(y,z)$.</li>
</ol>
<p>then $d(x,y) = |\delta(x,y)|$ clearly defines a metric. Furt... |
1,349 | <p>In this question here the OP asks for hints for a problem rather than a full proof.</p>
<p><a href="https://math.stackexchange.com/questions/14477">Proof of subfactorial formula $!n = n!- \sum_{i=1}^{n} {{n} \choose {i}} \quad!(n-i)$</a></p>
<p>Now, while I would like to respect that request, I also feel that questi... | Jonas Meyer | 1,424 | <p>Good idea. I searched meta.stackoverflow.com to see if this had already come up, and I found <a href="https://meta.stackexchange.com/questions/1191/add-markdown-support-for-hidden-until-you-click-text-aka-spoilers/71396#71396">this</a>, showing that it was recently implemented. I'm going to try it:</p>
<blockquote... |
74,108 | <p>Background: I was trying to convert a MATLAB code (fluid simulation, SPH method) into a <em>Mathematica</em> one, but the speed difference is huge.</p>
<p>MATLAB code:</p>
<pre class="lang-matlab prettyprint-override"><code>function s = initializeDensity2(s)
nTotal = s.params.nTotal; %# particles
h = s.params.... | ybeltukov | 4,678 | <p>You miss that many Mathematica functions are <a href="http://reference.wolfram.com/language/ref/Listable.html" rel="noreferrer">Listable</a>. It allows you to write a fast and clear code</p>
<pre><code>init2[distance_] := uniMass Total[liuQuartic[distance, h] UnitStep[2 h - distance], {2}]
h = 0.1;
uniMass = 1.0;
... |
185,478 | <blockquote>
<p>How do I solve for $x$ in $$x\left(x^3+\sin(x)\cos(x)\right)-\big(\sin(x)\big)^2=0$$</p>
</blockquote>
<p>I hate when I find something that looks simple, that I should know how to do, but it holds me up. </p>
<p>I could come up with an approximate answer using Taylor's, but how do I solve this? </... | Christian Blatter | 1,303 | <p>Using the identity $\cos x=1-2\sin^2(x/2)$ and introduccing the function ${\rm sinc}(x):={\sin x\over x}$ we can rewrite the given function $f$ in the following way:
$$f(x)=x^2\left(x^2\left(1-{1\over2}{\rm sinc}(x){\rm sinc}^2(x/2)\right)+{\rm sinc}(x)\bigl(1-{\rm sinc}(x)\bigr)\right)\ .\qquad(*)$$
Now ${\rm sin... |
4,134,734 | <p>I'm trying to verify that a certain function of two variables <span class="math-container">$F(x,y)$</span> satisfies the conditions of a joint CDF. Showing that each condition holds has been fairly straightforward except, that is, for the condition that</p>
<p><span class="math-container">$a<b,c<d\implies F(b,... | Raskolnikov | 3,567 | <p>This is also called the 2-increasing property and if your function is continuously twice differentiable, you can equivalently check that</p>
<p><span class="math-container">$$\frac{\partial^2 F}{\partial x \partial y} \geq 0 \; .$$</span></p>
<p>However, your function is not on the transitions from one "block&q... |
2,986,647 | <p>If I know coordinates of point <span class="math-container">$A$</span>, coordinates of circle center <span class="math-container">$B$</span> and <span class="math-container">$r$</span> is the radius of the circle, is it possible to calculate the angle of the lines that are passing through point A that are also tange... | saz | 36,150 | <p>I don't see that Itô's formula is of much use for this problem (in particular, since there is the sinularity at zero which means that you would need to work with local times). It's much easier to use the fact that we know the transition density of <span class="math-container">$(B_t)$</span>.</p>
<hr>
<p>By the ver... |
4,149,355 | <p>Are there any stochastic processes <span class="math-container">$(X_t)_{t \in \mathbb{R}^d}$</span> such that</p>
<ol>
<li>almost surely paths are continuous but nowhere differentiable and</li>
<li>sampling of <span class="math-container">$n$</span> points <span class="math-container">$X_{t_n}$</span> on a path can ... | user1337 | 62,839 | <p>Cauchy-Schwarz works, as you suspected. Define
<span class="math-container">$$\mathbf{u}=(x,y) \\
\mathbf{v}=(1,1) $$</span>
We have
<span class="math-container">$$\mathbf{u} \cdot \mathbf{v} \leq \|\mathbf{u} \| \| \mathbf{v}\|, $$</span>
or</p>
<p><span class="math-container">$$x+y \leq \sqrt{2}.$$</span></p>
|
4,149,355 | <p>Are there any stochastic processes <span class="math-container">$(X_t)_{t \in \mathbb{R}^d}$</span> such that</p>
<ol>
<li>almost surely paths are continuous but nowhere differentiable and</li>
<li>sampling of <span class="math-container">$n$</span> points <span class="math-container">$X_{t_n}$</span> on a path can ... | Community | -1 | <p>Let us solve</p>
<p><span class="math-container">$$\begin{cases}x^2+y^2=1,\\x+y=\sqrt2.\end{cases}$$</span></p>
<p>Subracting the second multiplied by <span class="math-container">$\sqrt2$</span>,</p>
<p><span class="math-container">$$\left(x-\frac1{\sqrt2}\right)^2+\left(y-\frac1{\sqrt2}\right)^2-1=1-2$$</span>
giv... |
3,581,707 | <p>I'm struggling with the following proof and hope some of you can help me:</p>
<p>By <span class="math-container">$H$</span> we denote a real Hilbert space and let be <span class="math-container">$T: H \rightarrow H$</span> be a compact, self-adjoint and linear operator. </p>
<p>(i) Show that <span class="math-cont... | Disintegrating By Parts | 112,478 | <p>Suppose that <span class="math-container">$\lambda=\sup_{\|x\|=1}\langle Tx,x\rangle$</span>. Then <span class="math-container">$[x,y]=\langle (\lambda I-T)x,y\rangle$</span> satisfies the properties of an inner product, except that it may not be strictly positive. It is non-negative. Regardless, the Cauchy-Schwarz ... |
2,924,380 | <p><span class="math-container">$\sum_{k=0}^{n}{k\binom{n}{k}}=n2^{n-1}$</span></p>
<p><span class="math-container">$n2^{n-1} = \frac{n}{2}2^{n} = \frac{n}{2}(1+1)^n = \frac{n}{2}\sum_{k=0}^{n}{\binom{n}{k}}$</span></p>
<p>That's all I got so far, I don't know how to proceed</p>
| Subhasis Biswas | 389,992 | <p>Take the function <span class="math-container">$(x+1)^n = 1^nx^0 {n \choose 0} \ + 1^{n-1}x^1{ n \choose 1} \ + 1^{n-2}x^2{ n \choose 2}+... + 1^{1}x^{n-1}{ n \choose {n-1}}+\ + 1^{0}x^n{ n \choose n} $</span></p>
<p>Differentiating both sides: <span class="math-container">$n(x+1)^{n-1} = \sum_{k=0}^{n}k{n \choose... |
4,027,604 | <p>Let <span class="math-container">$f(x)$</span> be continuous on <span class="math-container">$[a,b]$</span> and <span class="math-container">$F(x)=\frac{1}{x-a}\int_a^xf(t)dt$</span></p>
<p>Proof: The functions <span class="math-container">$F(x)$</span> and <span class="math-container">$f(x)$</span> have the same mo... | Arthur | 15,500 | <p>The integrand <span class="math-container">$f(x)$</span> is a constant (we're integrating with respect to <span class="math-container">$t$</span>). And the integral of a constant is equal to the product of that constant with the width of the interval you're integrating over, which is <span class="math-container">$x-... |
2,605,546 | <p>Simple question but not sure why for, $ f = \frac{\lambda}{2}\sum_{j=1}^{D} w_j^2$
$$\frac{\partial f}{\partial wj}= \lambda w_j$$</p>
<p>I would have thought the answer would be $\frac{\partial f}{\partial wj}= \lambda \sum_{j=1}^{D} w_j^2$</p>
<p>Since we get the derivative of $w_j^2$ which is $2w_j$, pull out t... | John B | 301,742 | <p>Take for example $D=2$. Then
$$
f=\frac\lambda2(w_1^2+w_2^2)
$$
and indeed
$$
\frac{\partial f}{\partial w_1}=\frac\lambda22w_1=\lambda w_1
$$
and
$$
\frac{\partial f}{\partial w_2}=\frac\lambda22w_2=\lambda w_2.
$$
The case of arbitrary $D$ is analogous.</p>
|
676,171 | <blockquote>
<p>Prove that if $F$ is a field, every proper prime ideal of $F[X]$ is maximal.</p>
</blockquote>
<p>Should I be using the theorem that says an ideal $M$ of a commutative ring $R$ is maximal iff $R/M$ is a field? Any suggestions on this would be appreciated. </p>
| Bill Dubuque | 242 | <p><strong>Hint</strong> $\ $ Polynomial rings over fields enjoy a (Euclidean) division algorithm, hence every ideal is principal, generated by an element of minimal degree (= gcd of all elements). But for principal ideals: contains $\!\iff\!$ divides, i.e. $\rm\: (a)\supseteq (b)\!\iff\! a\mid b.\:$ Thus, having no pr... |
3,294,402 | <p>Prove that it is impossible for three consecutive squares to sum to another perfect square.
I have tried for the three numbers <span class="math-container">$x-1$</span>, <span class="math-container">$x$</span>, and <span class="math-container">$x+1$</span>.</p>
| Anand | 687,233 | <p>The smallest solution you may think of:</p>
<p>Note that <span class="math-container">$(x-1)^2+x^2+(x+1)^2=3x^2+2\equiv 2\pmod{3}$</span>, but <span class="math-container">$$y^2\equiv 0,1\pmod{3}$$</span>for all integers <span class="math-container">$y$</span>. Therefore, <span class="math-container">$(x-1)^2+x^2+(... |
849,038 | <p>Im once again struggling to see the equivalence of two definitions. In my abstract algebra book (abstract algebra by beachy and blair) it says that elements $d_1,d_2,....,d_n \in D$ where $D$ is a UFD is said to be relatively prime in $D$ if there is no irreducible element $p \in D$ such that $p \mid a_i$ for $i=1,2... | Claude Leibovici | 82,404 | <p>Let us suppose that you fit a model $Z = a +b X+cY$ based on $N$ data points $(X_i,Y_i,Z_i)$. The so-called normal equations are $$\sum _{i=1}^N Z_i= N a + b\sum _{i=1}^N X_i+ c\sum _{i=1}^N Y_i$$ $$\sum _{i=1}^N X_iZ_i= a\sum _{i=1}^N X_i + b\sum _{i=1}^N X_i^2+ c\sum _{i=1}^N X_iY_i$$ $$\sum _{i=1}^N Y_iZ_i= a\s... |
673,334 | <p>I used below pseudocode to generate a discrete normal distribution over 101 points.</p>
<pre><code>mean = 0;
stddev = 1;
lowerLimit = mean - 4*stddev;
upperLimit = mean + 4*stddev;
interval = (upperLimit-lowerLimit)/101;
for ( x = lowerLimit + 0.5*interval ; x < upperLimit; x = x + interval) { ... | Batman | 127,428 | <p>y isn't a probability distribution.</p>
|
2,907,771 | <p>I tried coming up with a proof of compactness of $[0,1]$ in $\mathbb{R}$ and thought of the following method. please let me know if it is correct or how it could be made more correct.</p>
<p>For any open cover of $[0,1]$ there exists an $\mathbb{\epsilon}$ such that $[0,\mathbb{\epsilon})$ is contained in one open... | hmakholm left over Monica | 14,366 | <p>When you say</p>
<blockquote>
<p>form a new increasing sequence $(0,l_1,l_2,\ldots,l_m,\ldots)$ similar to the previous method.</p>
</blockquote>
<p>you're sweeping quite a deal under the carpet. What if it keeps going wrong and you habe to keep backtracking? In general arguments that simply state "continue this... |
199,695 | <p>I believe the answer is $\frac12(n-1)^2$, but I couldn't confirm by googling, and I'm not confident in my ability to derive the formula myself.</p>
| Belgi | 21,335 | <p>Another way to calculate what Matt said it to this: number the vertices from $1$ to $n$, and consider the graph with $n$ vertices but with no edges.</p>
<p>Take the first vertice: it has vertics to the other $n-1$ vertices, connect those $n-1$ vertices</p>
<p>Take the second vertice: it has vertics to the other $n... |
287,859 | <p>Prove that $\lim\limits_{x\rightarrow+\infty}\frac{x^k}{a^x} = 0\ (a>1,k>0)$.</p>
<p>P.S. This problem comes from my analysis book. You may use the definition of limits or invoke the Heine theorem for help. <em>It means the proof should only use some basic properties and definition of limits rather than more ... | L. F. | 56,837 | <p>$$x>0:$$</p>
<p>$$\frac{a^x}{x^k}=\sum_{n=0}^{\infty}\frac{(x\ln a)^n}{x^kn!}$$</p>
<p>$$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\;\;\;\;\;\;=\frac{1}{x^k}+\frac{\ln a}{x^{k-1}}+\cdots+\frac{(\ln a)^k}{k!}+\frac{x(\ln a)^{k+1}}{(k+1)!}+\cdots$$</p>
<p>$$\;>\frac{x(\ln a)^{k+1}}{(k+1)!}$$</p>
<p>$$\Righta... |
1,570,193 | <p>Let $X$ be a random variable and $\;f_X(x)=c6^{-x^2}\;\forall x\in\Bbb R$ its pdf. What I'm trying to compute is $\sqrt {Var(X)}$. I've got that $c=\sqrt{\frac{\ln(6)}{\pi}}$ for $f_X(x)$ to be a pdf and also that $\Bbb E(X)=0$. So my problem reduces to compute $\Bbb E(X^2)$ where</p>
<p>$$\Bbb E(X^2)=\sqrt{\frac{\... | Jonathan Lamar | 204,429 | <p>You are correct about the identity $1=u(1_{\mathbb{K}})$. For the second part, this should follow from the distributivity of tensor product over direct sum and the fact that, as an element of $\overline{C}$, $x$ has no nonzero part in $\mathbb{K}1$.</p>
|
1,902,878 | <p>If $a^x=bc$, $b^y=ca$ and $c^z=ab$, prove that: $xyz=x+y+z+2$.</p>
<p>My Approach;
Here,</p>
<p>$$a^x=bc$$
$$a={bc}^{\frac {1}{x}}$$</p>
<p>and,</p>
<p>$$b={ca}^{\frac {1}{y}}$$
$$c={ab}^{\frac {1}{z}}$$</p>
<p>I got stopped from here. Please help me to continue </p>
| Ng Chung Tak | 299,599 | <p><strong>Hint:</strong></p>
<blockquote class="spoiler">
<p> \begin{align*} x &= \frac{\log b+\log c}{\log a} \\[7pt] (B+C)(C+A)(A+B) &= BC(B+C)+CA(C+A)+AB(A+B)+2ABC \end{align*}</p>
</blockquote>
|
4,486,131 | <p><span class="math-container">$F=Q(\sqrt{2i})$</span>,then
Which one of the following is not true (Duet-2017 Q.26)</p>
<p>1.<span class="math-container">$\sqrt{2}\in F$</span></p>
<p>2.<span class="math-container">$i \in F$</span></p>
<p>3.<span class="math-container">$x^8-16=0$</span> has a solution in <span class="... | Mr.Gandalf Sauron | 683,801 | <p>First take a look at what <span class="math-container">$\sqrt{i}$</span> can be. From elementary complex analysis you have <span class="math-container">$\sqrt{2i}=\sqrt{2}(\frac{1}{\sqrt{2}}+i\frac{1}{\sqrt{2}})=1+i$</span> .</p>
<p>Note that we are only considering <span class="math-container">$\sqrt{i}=(\frac{1}{\... |
246,114 | <p>A Latin Square is a square of size <strong>n × n</strong> containing numbers <strong>1</strong> to <strong>n</strong> inclusive. Each number occurs once in each row and column.</p>
<p>An example of a 3 × 3 Latin Square is:</p>
<p><span class="math-container">$$
\left(
\begin{array}{ccc}
1 & 2 & 3 \\
3 &... | flinty | 72,682 | <p>If you do a <a href="https://resources.wolframcloud.com/FunctionRepository/resources/BacktrackSearch/" rel="noreferrer"><code>"BacktrackSearch"</code></a> then it will take forever for <span class="math-container">$n=5$</span> but use less memory. There are 161280 <span class="math-container">$5\times5$</s... |
246,114 | <p>A Latin Square is a square of size <strong>n × n</strong> containing numbers <strong>1</strong> to <strong>n</strong> inclusive. Each number occurs once in each row and column.</p>
<p>An example of a 3 × 3 Latin Square is:</p>
<p><span class="math-container">$$
\left(
\begin{array}{ccc}
1 & 2 & 3 \\
3 &... | Neil Strickland | 80,069 | <p>We can say that a latin square is standard if the first row is [1,...,n] and the first column is also [1,...,n]. Any latin square arises uniquely as follows: take a standard latin square, apply an arbitrary permutation to the full set of n columns, then apply an arbitrary permutation to the last (n-1) rows. Thus, ... |
246,114 | <p>A Latin Square is a square of size <strong>n × n</strong> containing numbers <strong>1</strong> to <strong>n</strong> inclusive. Each number occurs once in each row and column.</p>
<p>An example of a 3 × 3 Latin Square is:</p>
<p><span class="math-container">$$
\left(
\begin{array}{ccc}
1 & 2 & 3 \\
3 &... | chyanog | 2,090 | <pre><code>ClearAll[next];
next =
With[{Part = Compile`GetElement},
Compile[{{perms, _Integer, 2}, {rows, _Integer, 2}},
Table[
Append[rows, p],
{p, Select[perms,
Catch[Do[If[#[[j]] == rows[[i, j]], Throw@False],
{i, Length@rows}, {j, Length@#}]; True] &]
}
], Compila... |
379,669 | <p>So I was exploring some math the other day... and I came across the following neat identity:</p>
<p>Given $y$ is a function of $x$ ($y(x)$) and
$$
y = 1 + \frac{\mathrm{d}}{\mathrm{d}x} \left(1 + \frac{\mathrm{d}}{\mathrm{d}x} \left(1 + \frac{\mathrm{d}}{\mathrm{d}x} \left(1 + \frac{\mathrm{d}}{\mathrm{d}x} \left(1... | Lucian | 93,448 | <p>If the operator is meant to derive what follows, then we have $$y(x) = \lim_{n \to \infty} n! \cdot y^{(n)}(x) \qquad,\qquad \forall\ x \in X$$</p>
<p>since the derivative of a constant is always $0$ , and the derivative of a sum is the sum of derivatives. However, if multiplication is meant, with the “last” term o... |
2,368,813 | <p>I'm in a bit of trouble I want to calculate the cross ratio of 4 points $ A B C D $ that are on a circle </p>
<p>Sadly "officially" it has to be calculated with A B C D as complex numbers and geometers sketchpad ( the geomerty program I am used to) don't know about complex numbers</p>
<p>Now I am wondering
The cro... | MvG | 35,416 | <p>Cross ratios are invariant under projective transformations, and projective transformations of the projective complex line $\mathbb{CP}^1$ are Möbius transformations. You can use a Möbius transformation to map the unit circle to the real line. (A circle inversion would do the same for the circle itself, although it ... |
3,354,684 | <p>I was trying to prove following inequality:</p>
<p><span class="math-container">$$|\sin n\theta| \leq n\sin \theta \
\text{for all n=1,2,3... and } \
0<\theta<π $$</span></p>
<p>I succeeded in proving this via induction but I didn't get "feel" over the proof. Are there other proof for this inequality?</p>
| Theo Bendit | 248,286 | <p>You can show that <span class="math-container">$|\sin(x)|$</span> is subadditive, i.e.
<span class="math-container">$$|\sin(x + y)| \le |\sin(x)| + |\sin(y)|.$$</span>
To prove this, simply expand the left side:
<span class="math-container">\begin{align*}
|\sin(x + y)| &= |\sin(x)\cos(y) + \sin(y)\cos(x)| \\
&am... |
4,027,889 | <p>I'm trying to graph <span class="math-container">$f(x,y)=\ln(x)-y$</span>, however, I am not sure how as all of my tools are refusing to graph it.</p>
<p>Can you please help me?</p>
<p>Thanks</p>
| DMcMor | 155,622 | <p>Note that there is no way to plot this in 2D because you would need an equation of the form <span class="math-container">$f(x,y) = 0$</span>, and what you have is <span class="math-container">$f(x,y) = \ln(x) - y$</span>, which can be thought of as <span class="math-container">$f(x,y) = z$</span>. That means you ne... |
3,299,492 | <p>Is there any nice characterization of the class of polynomials can be written with the following formula for some <span class="math-container">$c_i , d_i \in \mathbb{N}$</span>? Alternatively, where can I read more about these? do they have a name?
<span class="math-container">$$c_1 + \left( c_2 + \left( \dots (c_k ... | Kavi Rama Murthy | 142,385 | <p>It converges for <span class="math-container">$x=0$</span> but not for any other value of <span class="math-container">$x$</span>. </p>
|
3,426,907 | <p>I'm not a mathematician, but a programmer who loves solving math puzzles, so please forgive me if I don't use the correct terms.</p>
<p>Imagine there are 100 lotteries with 100 tickets each. The lotteries have no connection at each other and all lotteries have exactly 1 price. I buy 1 ticket from each lottery so in... | Eric Towers | 123,905 | <p>Regarding 2: Are you familiar with the <a href="https://en.wikipedia.org/wiki/Topologist%27s_sine_curve" rel="nofollow noreferrer">topologist's sine curve</a>, <span class="math-container">$T$</span>, and its closure? </p>
<p><span class="math-container">$T \smallsetminus \{(0,0)\}$</span> is path connected, but i... |
2,546,497 | <p>987x ≡ 610 (mod 1597)</p>
<p>Is this correct way of applying little Fermat's theorem for linear congruences? Does it make any sense? If not could someone advice a bit.</p>
<p>Since gcd(987,1597)=1</p>
<p>-> 987ˆ1597-1 ≡ 1 (mod 1597)</p>
<p>-> 987ˆ1596 ≡ 1 (mod 1597)</p>
<pre><code>610 ≡ 610 (1597)
987ˆ1596 * 6... | vrugtehagel | 304,329 | <p>Yes the order matters, but we still use $c(n,r)$ (more commonly denoted ${n\choose r}$) because we essentially want to find the number of ways to pick $r$ zeroes out of a string with length $n$ and change those to $1$'s, resulting in the amount of strings with exactly $r$ ones.</p>
|
1,285,416 | <p>Is there an easy way to see that the polynomial $x^2 + 3x + 10$ is irreducible modulo 29 without having to go through each element 0,1,..,28 and check for roots?</p>
| André Nicolas | 6,312 | <p>Probably not. But I can suggest a harder way. The congruence $x^2+3x+10\equiv 0$ can be rewritten as $4x^2+12x+40\equiv 0$, and then as $(2x+3)^2+31\equiv 0$ and then as $(2x+3)^2 \equiv -2$. So the issue is whether $-2$ is a quadratic residue of $29$. Note that $-1$ is a QR of $29$. But $2$ is an NR of $29$, since ... |
1,285,416 | <p>Is there an easy way to see that the polynomial $x^2 + 3x + 10$ is irreducible modulo 29 without having to go through each element 0,1,..,28 and check for roots?</p>
| Rolf Hoyer | 228,612 | <p>One way to show it isn't a root is to complete the square.
$$
x^2+3x+10
\equiv x^2 -26x + 155
= (x-13)^2 -14 \pmod{29}
$$
Now, to show that there is no root is equivalent to showing that $14$ is not a quadratic residue mod 29. Using quadratic reciprocity and the Lagrange symbol, we compute
$$
\binom{14}{29} = \bin... |
1,285,416 | <p>Is there an easy way to see that the polynomial $x^2 + 3x + 10$ is irreducible modulo 29 without having to go through each element 0,1,..,28 and check for roots?</p>
| Mark Bennet | 2,906 | <p>Note that $$x^2+3x+10\equiv x^2+32x+10 = (x+16)^2-246$$ So the real question is whether $246\equiv 14\equiv 72 \bmod 29$ is a square modulo $29$ i.e. whether $6^2\times 2$ is a square modulo $29$, i.e. whether $2$ is a square. But $2$ is a square modulo a prime $p$ only if $p\equiv \pm 1 \bmod 8$</p>
|
1,285,416 | <p>Is there an easy way to see that the polynomial $x^2 + 3x + 10$ is irreducible modulo 29 without having to go through each element 0,1,..,28 and check for roots?</p>
| Bill Dubuque | 242 | <p>One doesn't need quadratic reciprocity, only the simple idea behind <a href="http://en.wikipedia.org/wiki/Euler%27s_criterion" rel="nofollow">Euler's Criterion</a></p>
<p>$ {\rm mod}\ 29\!:\ 0 \equiv 4f(x) \equiv {(2x\!+\!3)^2}\!+31\,\Rightarrow \color{#0a0}{(2x\!+\!3)^2\equiv -2}.\ $ But $\,\color{#0a0}{-2}\,$ is ... |
3,267,550 | <p>Considering the system <span class="math-container">$x_{k+1}=Ax_k+Bu_k$</span> with quadratic cost </p>
<p><span class="math-container">$J^* = \min x_N^T S x_N + \sum_{k=0}^{N-1} x_k^T Qx_k+u^T_kRu_k$</span></p>
<p>where <span class="math-container">$Q,S\succeq 0, R\succ 0$</span>. The optimal state feedback is fo... | Kwin van der Veen | 76,466 | <p>The inequality </p>
<p><span class="math-container">$$
Q + A^\top P_{k+1} A - P_{k} - A^\top P_{k+1} B \left(R + B^\top P_{k+1} B\right)^{-1} B^\top P_{k+1} A \succeq 0 \tag{1}
$$</span></p>
<p>can also be seen as an equality plus some unknown positive semi-definite term <span class="math-container">$X_k=X_k^\top\... |
4,642,566 | <p>For <span class="math-container">$x, y ∈$</span> <span class="math-container">$\mathbb{R}$</span>, let <span class="math-container">$x△y = 2(x + y)$</span>. Then <span class="math-container">$△$</span> is a binary operation on <span class="math-container">$\mathbb{R}$</span>.</p>
<p>Show that there is no identity el... | Parcly Taxel | 357,390 | <p>The operator is easily seen to be commutative, so we can just solve
<span class="math-container">$$x\triangle e=2(x+e)=x\implies e=-x/2$$</span>
But the identity cannot depend on <span class="math-container">$x$</span>, so there is no identity in the first place.</p>
|
1,315,450 | <blockquote>
<p>Prove that for all $n\in\mathbb{N}$ and $x>0$,
$$2^{-1+\frac{1}{n}}\left(x+1\right)\leq\left(x^{n}+1\right)^{\frac{1}{n}}$$</p>
</blockquote>
<p>The last class was about Taylor polynomial of functions, so I thought this might give me a solutions, but looking at the derivatives the only think I ... | Ewan Delanoy | 15,381 | <p>If you put $t=x-1$ and $d=x^n+1-2\bigg(\frac{x+1}{2}\bigg)^n$, you have</p>
<p>$$
\begin{array}{lcl}
d &=& 1+(1+t)^n-2\bigg(1+\frac{t}{2}\bigg)^n \\
&=& 1+\sum_{j=0}^n \binom{n}{j} t^j -\sum_{j=0}^n \binom{n}{j} \frac{t^j}{2^{j-1}} \\
&=& 1+\sum_{j=0}^n \binom{n}{j} \frac{2^{j-1}-1}{2^{j-1}}... |
2,339,521 | <p>In a game, each trial consists of two possible outcomes, success or failure. Two trials $H$ and $K$ are carried out. The probability of success for trial $H$ is $x$, and the probability of success for trial $K$ is $2/5$ if trial $H$ is a success, and $x/2$ if trial $H$ is a failure. Given that the probability of tri... | Henno Brandsma | 4,280 | <p>So $K$ is done after $H$,so we condition on the result of $H$</p>
<p>To get 1 success: $$P(H \land \lnot K) + P(\lnot H \land K) = P(\lnot K|H)P(H) + P(K| \lnot H)P(\lnot H) = \frac{3}{5} x + \frac{x}{2}(1-x)$$</p>
<p>and this should equal $\frac{1}{5}$</p>
|
269,398 | <p>I have a python code for data analysis, that uses the <a href="https://matplotlib.org/3.5.0/tutorials/colors/colormaps.html" rel="nofollow noreferrer">"seismic" color scale</a> for 2D density plots. However, I also need to do some other plots with Mathematica (because of packages etc), for which I would li... | J. M.'s persistent exhaustion | 50 | <p>There's no need to guess if you look at <a href="https://github.com/matplotlib/matplotlib/blob/main/lib/matplotlib/_cm.py#L190-L193" rel="noreferrer">the Matplotlib source</a>:</p>
<pre><code>seismicColors[x_?NumericQ] /; 0 <= x <= 1 :=
Blend[{RGBColor[0., 0., 0.3], RGBColor[0., 0., 1.], RGBColor[1., 1., 1.],... |
3,052,746 | <p>I want to solve <span class="math-container">$2x = \sqrt{x+3}$</span>, which I have tried as below:</p>
<p><span class="math-container">$$\begin{equation}
4x^2 - x -3 = 0 \\
x^2 - \frac14 x - \frac34 = 0 \\
x^2 - \frac14x = \frac34 \\
\left(x - \frac12 \right)^2 = 1 \\
x = \frac32 , -\frac12
\end{equation}$$</span>... | KM101 | 596,598 | <p>You made a mistake when completing the square.</p>
<p><span class="math-container">$$x^2-\frac{1}{4}x = \frac{3}{4} \color{red}{\implies\left(x-\frac{1}{2}\right)^2 = 1}$$</span></p>
<p>This is easy to spot since <span class="math-container">$(a\pm b)^2 = a^2\pm2ab+b^2$</span>, which means the coefficient of the l... |
3,052,746 | <p>I want to solve <span class="math-container">$2x = \sqrt{x+3}$</span>, which I have tried as below:</p>
<p><span class="math-container">$$\begin{equation}
4x^2 - x -3 = 0 \\
x^2 - \frac14 x - \frac34 = 0 \\
x^2 - \frac14x = \frac34 \\
\left(x - \frac12 \right)^2 = 1 \\
x = \frac32 , -\frac12
\end{equation}$$</span>... | Leonhard Euler | 481,442 | <p><span class="math-container">$$\begin{equation}
4x^2 - x -3 = 0 \\
(4x + 3)(x - 1) = 0 \\
x = -\frac34 , 1
\end{equation}$$</span></p>
|
2,343,027 | <p>I am having a problem in proving this map to be one-one. It is not said anything about the relationship about $K$ and $R$. Or is it not necessary that they be related somehow. Please help.</p>
| José Carlos Santos | 446,262 | <p>Suppose that $f(x)=f(y)$ with $x,y\in K$ and $x\neq y$. Then, if $z=x-y$, $f(z)=0$ with $z\neq 0$. But$$1=f(1)=f\left(z\times z^{-1}\right)=f(z)\times f(z^{-1})\text,$$which is impossible, since $f(z)=0$.</p>
|
3,956,828 | <p>I can find the nth integral of <span class="math-container">$\ln(z)$</span> as follows:
<span class="math-container">\begin{aligned}
\left(\frac d{dz}\right)^{-n}\ln(z)&=\frac1{\Gamma(n)}\int\limits_0^z(z-t)^{n-1}\ln(t)dt\\
&=\frac1{n!}\left[\int\limits_0^z\frac1t(z-t)^ndt-z^n\ln(0)\right]\\
&=\frac1{n!}... | Varun Vejalla | 595,055 | <p><em>This is not a derivation of the integral, but rather a proof of your conjecture.</em></p>
<p>Your expression is <span class="math-container">$$f_n(z)=\frac1{n!}\ln\left( 1+\frac{z}{k}\right)( k+z)^{n} -\sum ^{n}_{i=1}\frac{H_{n} -H_{n-i}}{i!( n-i) !} k^{n-i} z^{i}$$</span></p>
<p>I reversed the order of summatio... |
4,422,512 | <p>I do have a matrix of following form</p>
<p><span class="math-container">$$M:=\left(\begin{array}{c|ccc}
A & & * &\\
\hline 0 & & &\\
0 & & B &\\
0 & & &\\
\end{array}\right)$$</span></p>
<p>Here the <span class="math-container">$0$</span>'s represent matrices of ... | AnCar | 447,933 | <p>If <span class="math-container">$B$</span> is not invertible, then its rows are linearly dependent and this is preserved if you pad the rows with the same number of <span class="math-container">$0$</span>s at the beginning, implying that <span class="math-container">$M$</span> is not invertible either.
Now if <span ... |
2,615,626 | <p>The problem I have is:</p>
<blockquote>
<p>$\lim \limits_{x \to \infty} \sin{(x)}\ e^{-x}$</p>
</blockquote>
<p>Things I've tried:</p>
<ol>
<li><p>Researching how to do this problem, I've come across kind of similar examples that use either Taylors Rule, L'Hopitals Rule, or the Squeeze Theorem. Not sure which o... | user284331 | 284,331 | <p>$|(\sin x)e^{-x}|\leq e^{-x}\rightarrow 0$ as $x\rightarrow\infty$.</p>
|
542,808 | <p>I had a little back and forth with my logic professor earlier today about proving a number is irrational. I proposed that 1 + an irrational number is always irrational, thus if I could prove that 1 + irrational number is irrational, then it stood to reason that was also proving that the number in question was irrati... | Casteels | 92,730 | <p>Look at the contrapositive: If $x$ is rational, then $x+n$ is rational. Clearly this is a true statement.</p>
|
542,808 | <p>I had a little back and forth with my logic professor earlier today about proving a number is irrational. I proposed that 1 + an irrational number is always irrational, thus if I could prove that 1 + irrational number is irrational, then it stood to reason that was also proving that the number in question was irrati... | Community | -1 | <p>Suppose <span class="math-container">$x$</span> is irrational and <span class="math-container">$x+\dfrac pq=\dfrac mn$</span> then, <span class="math-container">$x=\dfrac mn-\dfrac pq=\dfrac{mq-np}{nq}$</span> so, <span class="math-container">$x$</span> would then be rational. :) </p>
|
542,808 | <p>I had a little back and forth with my logic professor earlier today about proving a number is irrational. I proposed that 1 + an irrational number is always irrational, thus if I could prove that 1 + irrational number is irrational, then it stood to reason that was also proving that the number in question was irrati... | Fredrik Meyer | 4,284 | <p>A proof in the style of "mathematics made difficult": Note that a number $r$ is rational if and only if $\mathbb Q(r) = \mathbb Q$. Now it is easy to see that $\mathbb Q(\gamma) = \mathbb Q(r+\gamma)$ for all rational $r$ and arbitrary $\gamma$. So if $r+\gamma$ is irrational, then $\gamma$ is also. </p>
|
542,808 | <p>I had a little back and forth with my logic professor earlier today about proving a number is irrational. I proposed that 1 + an irrational number is always irrational, thus if I could prove that 1 + irrational number is irrational, then it stood to reason that was also proving that the number in question was irrati... | user88377 | 88,377 | <p>This follows essentially from the fact that as additive abelian groups $\mathbb Q$ is a normal subgroup of $\mathbb R$, which is true since both groups are abelian and the latter contains the former. Hence we have a quotient homomorphism $\varphi: \mathbb R \rightarrow \mathbb R / \mathbb Q$. $r \in \mathbb R$ is ra... |
542,808 | <p>I had a little back and forth with my logic professor earlier today about proving a number is irrational. I proposed that 1 + an irrational number is always irrational, thus if I could prove that 1 + irrational number is irrational, then it stood to reason that was also proving that the number in question was irrati... | Joe | 623,665 | <p>Lemma:</p>
<blockquote>
<p>If <span class="math-container">$a$</span> is rational, and <span class="math-container">$b$</span> is rational, then <span class="math-container">$a+b$</span> is rational.</p>
</blockquote>
<p>Proof:</p>
<blockquote>
<p>Let <span class="math-container">$a=p/q$</span>, and <span class="mat... |
2,530,788 | <p>$x + y + z = 0$;</p>
<p>$x^2 + y^2 + z^2 = 1$;</p>
<p>$x^3 + y^3 + z^3 = 0$;</p>
<p>I understand that there are multiple solutions which are the permutations of $(\sqrt{ 2 }/2, 0, -\sqrt{2}/2).$</p>
<p>How do i go about solving for it? I have tried the normal brute force gaussian elimination method, Cramer's rul... | Dietrich Burde | 83,966 | <p>This is easy to solve by a direct calculation. Not even Vieta is needed. Substituting $z=-x-y$ the last equation gives
$$
xy(x+y)=0
$$
Obviously $x=y=0$ gives a contradiction, and also we may assume $xy\neq 0$; for $y=-x$ then the second equation is now very easy. </p>
|
228,651 | <p>When testing to determine the convergence or divergence of series with positive terms, there's a common way by comparing them with other series which we already know converge or diverge.</p>
<p>My question is, how do we choose the proper to-be-compared series? I hope to get some detailed <strong>methodology</strong... | Community | -1 | <p>In your case, the convergence of $\displaystyle \sum_{n=2}^{\infty} \dfrac1{n \log n}$ can be checked by using the following convergence test. If we have a monotone decreasing sequence, then $\displaystyle \sum_{n=2}^{\infty} a_n$ converges iff $\displaystyle \sum_{n=2}^{\infty} 2^na_{2^n}$ converges.</p>
<p>Note t... |
2,932,305 | <p>What are the intercepts of the planes <span class="math-container">$x = 0$</span> and <span class="math-container">$2y + 3z = 12$</span>? The word intercept is confusing me because I don't understand if I should say they intersect at point <span class="math-container">$(0,6,0)$</span> or the intercept is at <span cl... | Winter Soldier | 562,471 | <p>As @DavidPeterson as pointed out, the question alludes to the intersection of the two planes.</p>
<p>Now, you can use the fact that if the two planes intersect, then the intersection will be given by a straight line in space. Let the equation of the line be given by:
<span class="math-container">$$ \mathbf{r}(t) = ... |
272,144 | <p>Consider a multi-value function <span class="math-container">$f(z)=\sqrt{(z-a)(z+\bar a)}, \Im{a}>0,\Re{a}>0$</span>. To make the function be single-valued, one needs to make a cut. Suppose <span class="math-container">$a=e^{i\theta}$</span>, my choice of the branch cut is <span class="math-container">$e^{it},... | Michael E2 | 4,999 | <p><em>Update</em></p>
<p>Here's a simplified numerical way, by integrating an ODE without branch points from the origin to <span class="math-container">$z$</span>, determining the sign by whether the path crosses the branch cut (<em>update 2:</em> originally, I used <code>WhenEvent</code> to keep track of the sign, an... |
80,056 | <p>I am toying with the idea of using slides (Beamer package) in a third year math course I will teach next semester. As this would be my first attempt at this, I would like to gather ideas about the possible pros and cons of doing this. </p>
<p>Obviously, slides make it possible to produce and show clear graphs/pictu... | Thierry Zell | 8,212 | <p>I think you already touched on the two main points: pretty pictures are so much better than anything done on a chalkboard is the pro, but you cannot decently unwind any argument on slides. </p>
<p>I've used them intensively, I do it a lot less now. (Here's a con you did forget about: they take a <strong>lot</strong... |
80,056 | <p>I am toying with the idea of using slides (Beamer package) in a third year math course I will teach next semester. As this would be my first attempt at this, I would like to gather ideas about the possible pros and cons of doing this. </p>
<p>Obviously, slides make it possible to produce and show clear graphs/pictu... | BSteinhurst | 11,332 | <p>If you intend to post your slides online after class then you run the risk of students not even taking notes/digesting the material on their own (I've had this feeling myself) or feeling that they don't have to attend class. This is obviously a con but the other side is that the students then have a good outline of ... |
80,056 | <p>I am toying with the idea of using slides (Beamer package) in a third year math course I will teach next semester. As this would be my first attempt at this, I would like to gather ideas about the possible pros and cons of doing this. </p>
<p>Obviously, slides make it possible to produce and show clear graphs/pictu... | André Henriques | 5,690 | <blockquote>
<p>I would never, never use slides for a course.</p>
</blockquote>
<p><i>That said:</i><br>
I do sometimes show my student pictures taken from the web.<br> For example, I recently showed <a href="http://upload.wikimedia.org/wikipedia/en/0/03/Compound_of_five_cubes.png">this picture</a> to the students i... |
80,056 | <p>I am toying with the idea of using slides (Beamer package) in a third year math course I will teach next semester. As this would be my first attempt at this, I would like to gather ideas about the possible pros and cons of doing this. </p>
<p>Obviously, slides make it possible to produce and show clear graphs/pictu... | matthias beck | 3,193 | <p>I use a hybrid version for some of my classes which take place in a room that allows this: I use computer slides (and animations, computations, etc.) <b>and</b> the board. I learned this from my colleague Serkan Hosten, and it works really well in some classes. E.g., I use slides for definitions and theorems (includ... |
80,056 | <p>I am toying with the idea of using slides (Beamer package) in a third year math course I will teach next semester. As this would be my first attempt at this, I would like to gather ideas about the possible pros and cons of doing this. </p>
<p>Obviously, slides make it possible to produce and show clear graphs/pictu... | Jim Conant | 9,417 | <p>Full disclosure: I stole the following idea from my wife. </p>
<p>For some courses, like calculus, I will create slides with beamer, leaving blank spots to fill in during class. I then print the slides out on paper and present them with the document camera during lecture. When I get to an example, I will work it ou... |
80,056 | <p>I am toying with the idea of using slides (Beamer package) in a third year math course I will teach next semester. As this would be my first attempt at this, I would like to gather ideas about the possible pros and cons of doing this. </p>
<p>Obviously, slides make it possible to produce and show clear graphs/pictu... | Terry Tao | 766 | <p>Slides can, in principle, enhance a lecture, but there is one important difference between slides and blackboard that definitely needs to be kept in mind, and that is that slides are much more transient than a blackboard. Once one moves on from one slide to the next, the old slide is completely gone from view (unle... |
466,576 | <p>Some conventional math notations seem arbitrary to English speakers but were mnemonic to non-English speakers who started them. To give a simple example, Z is the symbol for integers because of the German word <em>Zahl(en)</em> 'number(s)'. What are some more examples?</p>
| MJD | 25,554 | <p>The Klein $V$-group is the four-element group with generators $a$ and $b$ and $a^2 = b^2 = (ab)^2 = 1$. The $V$ is for <em>vierergruppe</em> = "four-group".</p>
|
466,576 | <p>Some conventional math notations seem arbitrary to English speakers but were mnemonic to non-English speakers who started them. To give a simple example, Z is the symbol for integers because of the German word <em>Zahl(en)</em> 'number(s)'. What are some more examples?</p>
| MJD | 25,554 | <p>The logical-or symbol $\lor$ is a stylized letter ‘V’, the first letter of the Latin word <em>vel</em>.</p>
<p>(The $\land$ symbol arose later, derived by analogy from $\lor$.)</p>
|
466,576 | <p>Some conventional math notations seem arbitrary to English speakers but were mnemonic to non-English speakers who started them. To give a simple example, Z is the symbol for integers because of the German word <em>Zahl(en)</em> 'number(s)'. What are some more examples?</p>
| ratchet freak | 17,442 | <p>Eigen (as in the eigen vectors of a matrix) is Dutch/German for "own".</p>
|
466,576 | <p>Some conventional math notations seem arbitrary to English speakers but were mnemonic to non-English speakers who started them. To give a simple example, Z is the symbol for integers because of the German word <em>Zahl(en)</em> 'number(s)'. What are some more examples?</p>
| Marko Riedel | 44,883 | <p>In Polya's enumeration theorem the letter $Z(G)$ which is used for the cycle index of the permutation group $G$ originates with the German word <em>Zyklenzeiger</em>, I think.</p>
|
751,670 | <p>I've just started to learn about Cohen-Macaulay rings. I want to show that the following rings are Cohen-Macaulay:</p>
<p>$k[X,Y,Z]/(XY-Z)$ and $k[X,Y,Z,W]/(XY-ZW)$.</p>
<p>Also I am looking for a ring which is not Cohen-Macaulay.</p>
<p>Can anyone help me?</p>
| Community | -1 | <p><em>bruns herzog</em> book <em>Cohen Macaulay rings</em><br>
<img src="https://i.stack.imgur.com/7f7lw.png" alt="enter image description here"></p>
<p><img src="https://i.stack.imgur.com/UE9vc.png" alt="enter image description here"></p>
<p>note that the rings $k[X,Y,Z]$ and $k[X,Y,Z,W]$, are integral domains and ... |
272,114 | <p>Yesterday, my uncle asked me this question:</p>
<blockquote>
<p>Prove that $x = 2$ is the unique solution to $3^x + 4^x = 5^x$ where $x \in \mathbb{R}$.</p>
</blockquote>
<p>How can we do this? Note that this is not a diophantine equation since $x \in \mathbb{R}$ if you are thinking about Fermat's Last Theorem.<... | Community | -1 | <p>For all $x_j>x_i$ and $0<a<1$, $a^{x_i}>a^{x_j}$ . </p>
<p>Hence
\begin{align}
\left(\frac{3}{5}\right)^{x} + \left(\frac{4}{5}\right)^{x} - 1 < \left(\frac{3}{5}\right)^{2} + \left(\frac{4}{5}\right)^{2} - 1 = 0
\end{align}</p>
<p>for all $x>2$. Hence, there is no solution for $x>2$.</p>
<p... |
632,043 | <p>tl;dr: why is raising by $(p-1)/2$ not always equal to $1$ in $\mathbb{Z}^*_p$?</p>
<p>I was studying the proof of why generators do not have quadratic residues and I stumbled in one step on the proof that I thought might be a good question that might help other people in the future when raising powers modulo $p$.<... | Bill Dubuque | 242 | <p>You've attempted to apply a method of computing $k$-th roots outside its domain of applicability.</p>
<p>It <em>is</em> true that if $\,(k,p\!-\!1) = 1\,$ then $\,{\rm mod}\ p\!-\!1\!:\ \,k^{-1}\! = \color{#c00}{1/k \equiv i}\,$ exists, so $\ g^{\Large{j/k}} \equiv (g^{\Large j})^{\large{ \color{#c00}{1/k}}} \equiv... |
179,583 | <p>I have a fairly large array, a billion or so by 500,000 array. I need to calculate the singular value decomposition of this array. The problem is that my computer RAM will not be able to handle the whole matrix at once. I need an incremental approach of calculating the SVD. This would mean that I could take one or a... | Squirtle | 29,507 | <p>So is there any other property that the matrix has? For example, is it sparse or is it symmetric, real/complex, etc... As there are optimized algorithms for various situations, SVD may not be the best option -- it would be helpful to know what you are trying to get from the SVD.</p>
|
17,143 | <p>My next project I'd like to start working on is Domain Coloring. I am aware of the beautiful discussion at:</p>
<p><a href="https://mathematica.stackexchange.com/questions/7275/how-can-i-generate-this-domain-coloring-plot">How can I generate this "domain coloring" plot?</a></p>
<p>And I am studying it. H... | murray | 148 | <p>There are many approaches to domain coloring. The approach in @cormullion's answer is to use <code>Re</code> and <code>Im</code> as the mesh functions. Another way is to color points just by $\text{Arg}(f(z))$ and then superimpose contours for $\text{Abs}(f(z))$.</p>
<p>While that may readily be done in "pure <em>M... |
2,250,733 | <p>I have a general idea to solve the problem, which is to pair up 2s and 5s in the numerator and denominator, cancel those that are common, and the remaining pairs of 2s and 5s are the number of 0s left. Since 130 choose 70 is so large, how do I do this?</p>
| Angina Seng | 436,618 | <p><a href="https://en.wikipedia.org/wiki/Kummer%27s_theorem" rel="nofollow noreferrer">Kummer's theorem</a>
states that the power of a prime $p$ dividing a binomial coefficient
$\binom nk$ is the number of carries needed when adding $k$ to $n-k$ in
base $p$ notation.</p>
<p>Here $k=70$ and $n-k=60$. In base $2$, $k=(... |
3,287,710 | <p>I want to calculate the length of a clothoid segment from the following available information.</p>
<ol>
<li>initial radius of clothoid segment </li>
<li>final radius of clothoid segment</li>
<li>angle (i am not really sure which angle is this, and its not
documented anywhere)</li>
</ol>
<p>As a test case: I need t... | Empy2 | 81,790 | <p>I would introduce <span class="math-container">$z=-x-y$</span> so it is a shape on the plane <span class="math-container">$x+y+z=0$</span>. Then there are <span class="math-container">$(r,\theta)$</span> with
<span class="math-container">$$x=r\cos\theta\\y=r\cos(\theta+2\pi/3)\\
z=r\cos(\theta-2\pi/3)$$</span>
Then... |
1,413,150 | <p>So for a periodic function <span class="math-container">$f$</span> (of period <span class="math-container">$1$</span>, say), I know the Riemann-Lebesgue Lemma which states that if <span class="math-container">$f$</span> is <span class="math-container">$L^1$</span> then the Fourier coefficients <span class="math-cont... | Arin Chaudhuri | 404 | <p>Let $a_n = (1 + 1/n)^n.$ </p>
<p>We want to show $a_{n+1} - a_{n} \geq \dfrac{1}{n(n+1)}$ for large $n$. </p>
<p>$\dfrac{a_{n+1}}{a_n} = \left(1 + \dfrac{1}{n}\right) \left(1 - \dfrac{1}{(n+1)^2}\right)^{n+1}.$</p>
<p>The RHS can be expanded as</p>
<p>$\left(1 + \dfrac{1}{n}\right) \left(1 - \dfrac{1}{(n+1)^2}\r... |
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