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 |
|---|---|---|---|---|
1,714 | <p>I know of two good mathematics videos available online, namely:</p>
<ol>
<li>Sphere inside out (<a href="https://www.youtube.com/watch?v=BVVfs4zKrgk" rel="nofollow noreferrer">part I</a> and <a href="https://www.youtube.com/watch?v=x7d13SgqUXg" rel="nofollow noreferrer">part II</a>)</li>
<li><a href="https://www.yo... | Leandro Vendramin | 17,845 | <p>For a course on cluster algebras (by S. Fomin): <a href="http://qgm.au.dk/video/mc/cluster/" rel="nofollow">http://qgm.au.dk/video/mc/cluster/</a></p>
<p>EDIT: Some graduate short-courses in FCEyN, UBA, Buenos Aires, Argentina:</p>
<ul>
<li>J. Harris, Intersection Theory</li>
<li>R. Hartshorne, Introduction to Def... |
1,714 | <p>I know of two good mathematics videos available online, namely:</p>
<ol>
<li>Sphere inside out (<a href="https://www.youtube.com/watch?v=BVVfs4zKrgk" rel="nofollow noreferrer">part I</a> and <a href="https://www.youtube.com/watch?v=x7d13SgqUXg" rel="nofollow noreferrer">part II</a>)</li>
<li><a href="https://www.yo... | Community | -1 | <p>I feel that I have something new to add. I sometimes make mathematical videos, and I should make more. Here is one about a 3D diagram that I made of the happy family, in group theory. There are a lot of other videos on my channel, however, not pertaining to mathematics such as singing and art, but I hope that in ... |
1,714 | <p>I know of two good mathematics videos available online, namely:</p>
<ol>
<li>Sphere inside out (<a href="https://www.youtube.com/watch?v=BVVfs4zKrgk" rel="nofollow noreferrer">part I</a> and <a href="https://www.youtube.com/watch?v=x7d13SgqUXg" rel="nofollow noreferrer">part II</a>)</li>
<li><a href="https://www.yo... | debapriyay | 9,586 | <p>The video that I am adding here is a new upload in Youtube.
Here is the link: <a href="https://www.youtube.com/watch?v=aUl28Pjz89M" rel="nofollow noreferrer">https://www.youtube.com/watch?v=aUl28Pjz89M</a></p>
<p>I have seen many videos on 21-card trick and also read many blogs on it. A very few of them has attempte... |
736,624 | <blockquote>
<p>Heaviside's function $H(x)$ is defined as follows:</p>
<p>$H(x) = 1$ if $x > 0$, </p>
<p>$H(x) = \frac 1 2$ if $x= 0$</p>
<p>$H(x) = 0$ if $x < 0$ </p>
</blockquote>
<p>Define $F(x) := \int^x_0 H(t) \ dt, \ x \in \mathbb R$. </p>
<p>$H(x)$ is continuous except at $x=0$ so the in... | Siminore | 29,672 | <p>The left derivative of $F$ at zero is zero, the right derivative is one. Hence $F$ cannot be differentiable at zero. This is your mistake. Be careful, since the FTC requires that the function to be integrated be continuous in the whole interval. </p>
|
361,755 | <p>Let $S$ be a multiplicatively closed subset of a commutative noetherian ring $A$. Let $M$ and $N$ be finitely generated $A$-modules. If $M_S$ is isomorphic to $N_S$, show that $M_t$ is isomorphic to $N_t$ for some $t \in S.$</p>
| Community | -1 | <p>Since $M$ is finitely presented, we have that $$\text{Hom}_A(M,N)_S \cong \text{Hom}_{A_S}(M_S, N_S) \text{ (*)}$$</p>
<p>via the map which takes $f/s$ to the product of the constant map $1/s$ and the map from $M_S$ to $N_S$ induced by $f$, by THM 7.11 in Matsumura's "Commutative Ring Theory". Take $f \in \text{Ho... |
1,452,425 | <p>From what I have been told, everything in mathematics has a definition and everything is based on the rules of logic. For example, whether or not <a href="https://math.stackexchange.com/a/11155/171192">$0^0$ is $1$ is a simple matter of definition</a>.</p>
<p><strong>My question is what the definition of a set is?<... | Mauro ALLEGRANZA | 108,274 | <p><em>Long Comment</em> to Asaf's answer, trying to add some more background.</p>
<p>We can compare the issue regarding the "definition" of <em>set</em> with Geometry.</p>
<p>Euclid's <a href="https://en.wikipedia.org/wiki/Euclid%27s_Elements" rel="noreferrer">Elements</a> opens with five <a href="http://ale... |
87,538 | <p>My problem is the following:</p>
<p>I have a finite surjective morphism $f: X\rightarrow Y$ of noetherian schemes and know that $Y$ is a regular scheme.
(Indeed, in my situation, the two schemes are topologically the same and the arrow is topologically the identity.)</p>
<p>I don't know if $f$ is étale or smooth. ... | Mahdi Majidi-Zolbanin | 16,046 | <p>Here is a possible affirmative answer. At any point $P\in X$, write $f_P^{\#}$ for the induced local homomorphism of stalks $\mathcal{O}_{Y,f(P)}\rightarrow\mathcal{O}_{X,P}$. Assume: </p>
<ol>
<li>$X$ is Cohen-Macaulay;</li>
<li>At any point $P\in X$, we have: $\dim \mathcal{O}_{X,P}=\dim {\mathcal{O}}_{{Y,f(P)}}+... |
1,722,948 | <blockquote>
<p>$$\frac{1}{x}-1>0$$</p>
</blockquote>
<p>$$\therefore \frac{1}{x} > 1$$</p>
<p>$$\therefore 1 > x$$</p>
<p>However, as evident from the graph (as well as common sense), the right answer should be $1>x>0$. Typically, I wouldn't multiple the x on both sides as I don't know its sign, bu... | MathematicsStudent1122 | 238,417 | <p>Your answer would be correct were we assuming $x$ is positive. You must keep in mind, though, that the inequality fails for $x<0$ (recall that the inequality sign switches when we multiply by a negative number). </p>
<p>For $x>0$, we have a solution $1>x$. In other words, $0<x<1$ is a solution. </p>
... |
1,893,280 | <p>How to show $\frac{c}{n} \leq \log(1+\frac{c}{n-c})$ for any positive constant $c$ such that $0 < c < n$?</p>
<p>I'm considering the Taylor expansion, but it does not work...</p>
| RRL | 148,510 | <p>Hint: With $x = (c/n)/(1 - c/n)$</p>
<p>$$\log(1 + x) = \int_1^{1+x} \frac{dt}{t} \geqslant \frac{x}{1+x}$$</p>
|
1,893,280 | <p>How to show $\frac{c}{n} \leq \log(1+\frac{c}{n-c})$ for any positive constant $c$ such that $0 < c < n$?</p>
<p>I'm considering the Taylor expansion, but it does not work...</p>
| Hagen von Eitzen | 39,174 | <p>Starting with
$$ e^t\ge 1+t\qquad\text{for all }t\in\Bbb R$$
(possibly the most useful inequality about the exponential)
we find by plugging in $-\frac cn$ for $t$
$$ e^{-c/n}\ge 1-\frac cn= \frac{n-c}{n}=$$
and then after taking reciprocals (both sides are positive!)
$$e^{c/n} \le \frac n{n-c}=1+\frac c{n-c}$$
Fi... |
2,948,118 | <p>I understand that for a function or a set to be considered a vector space, there are the 10 axioms or rules that it must be able to pass. My problem is that I am unable to discern how exactly we prove these things given that my book lists some weird general examples.</p>
<p>For instance: the set of all third- degre... | Miles Zhou | 596,572 | <p>With any Riemann sum with maximum partition size <span class="math-container">$\delta \lt \epsilon/2$</span>, we have <span class="math-container">$|U-L|<(f(1^+)-f(1^-))*\delta=2\delta<\epsilon$</span>.</p>
|
1,464,747 | <p>I am trying to solve this question:</p>
<blockquote>
<p>How many ways are there to pack eight identical DVDs into five indistinguishable boxes so that each box contains at least one DVD?</p>
</blockquote>
<p>I am very lost at trying to solve this one. My attempt to start this problem involved drawing 5 boxes, an... | General Thiha | 344,304 | <p>It is called stirling numbers.
S(8,5)=1050 ways</p>
|
1,464,747 | <p>I am trying to solve this question:</p>
<blockquote>
<p>How many ways are there to pack eight identical DVDs into five indistinguishable boxes so that each box contains at least one DVD?</p>
</blockquote>
<p>I am very lost at trying to solve this one. My attempt to start this problem involved drawing 5 boxes, an... | Ranendra Bose | 354,326 | <p>Identical objects in indistinguishable(identical) boxes.
Use partition of numbers but $x_1 \geq x_2 \geq x_3 \geq x_4 \geq x_5 \geq 1$ since $(3,0,0,0,0)$ is the same as $(0,3,0,0,0)$ //identical offices.
Now $$x_1+x_2+x_3+x_4+x_5=8$$ but $x_i' = x_i-1$ so all $x_i'\geq 0$.
Hence, $$x_1'+x_2'+x_3'+x_4'+x_5'=3 =8-(... |
194,954 | <p>Is there a reason to use <code>Hold*</code> attributes for functional code (e.g. no intention to mutate input)? I'd expect performance gains as in pass by value vs pass by reference. </p>
<p>E.g. </p>
<pre><code>data = RandomReal[1, 10^8];
data // Function[x, x[[1]]] // RepeatedTiming
</code></pre>
<blockquote>
... | Carl Woll | 45,431 | <p>I don't understand why you think there will be a difference in timing for your example. In both cases the head is evaluated, the data variable is evaluated, the <a href="http://reference.wolfram.com/language/ref/Function" rel="noreferrer"><code>Function</code></a> is evaluated, and then the <a href="http://reference... |
1,130,302 | <p>I am struggling with following challenge in my free time programming project $-$ how is it possible to make reflection vector that reflects along normal with angle that is not larger than some $\alpha$?</p>
<p><img src="https://i.stack.imgur.com/6bmgv.png" alt="Example of normal and limited angle reflection"></p>
... | mesel | 106,102 | <p>Let <span class="math-container">$G$</span> act on the left coset of <span class="math-container">$H$</span> by left multiplication. Then we have an homomorphism <span class="math-container">$\phi$</span> from <span class="math-container">$G$</span> to <span class="math-container">$S_3$</span>. As <span class="math-... |
3,399,195 | <p>So I've seen various questions with the limit 'equal' to <span class="math-container">$\infty$</span> or that the limit doesn't exist in a case where the function tends to <span class="math-container">$\infty$</span>.</p>
<p>For example, the limit of <span class="math-container">$\sqrt{x}$</span> as <span class="mat... | Allawonder | 145,126 | <p>Depends on the context in which you're working.</p>
<p>If it's with the reals, for example, then such limits simply fail to exist. However, we do sometimes work with the extended reals <span class="math-container">$[-\infty,+\infty],$</span> and clearly in that case we can say something like <span class="math-conta... |
3,009,362 | <p>I need to find
<span class="math-container">$$\lim_{x\rightarrow -5} \frac{2x^2-50}{2x^2+3x-35}$$</span></p>
<p>Looking at the graph, I know the answer should be <span class="math-container">$\frac{20}{17}$</span>, but when I tried solving it, I reached <span class="math-container">$0$</span>.</p>
<p>Here are the... | Siong Thye Goh | 306,553 | <p>For your way <span class="math-container">$1$</span>, check the computation of your denominator, it should give you <span class="math-container">$0$</span> again.</p>
<p>For your way <span class="math-container">$2$</span>, check your factorization in your denominator as well.</p>
<p>Use L'hopital's rule:</p>
<p>... |
2,628,149 | <p>I am having trouble finding the general solution of the following second order ODE for $y = y(x)$ without constant coefficients: </p>
<p>$3x^2y'' = 6y$<br>
$x>0$</p>
<p>I realise that it may be possible to simply guess the form of the solution and substitute it back into the the equation but i do not wish to us... | nonuser | 463,553 | <p>You can try to use this approximation $$f(x+h)\doteq f(x)+hf'(x)$$ if $h$ is suficently small. Take $f(x)=\sqrt[4]{x}$, $x=16$ and $h=-1$. So we get
$$ \sqrt[4]{x+h} \doteq \sqrt[4]{x} +{h\over 4\sqrt[4]{x^3}}$$</p>
<p>So $$ \sqrt[4]{15} \doteq \sqrt[4]{16} -{1\over 4\sqrt[4]{16}^3}= 2-{1\over 32}$$</p>
|
912,217 | <p>Let $X$ be a R.V whose pdf is given by
$$f(x)=p\frac{1}{\sqrt{2\pi\sigma_1^2}}\exp\left(-\frac{(x-\mu_1)^2}{2\sigma_1^2}\right)+
(1-p)\frac{1}{\sqrt{2\pi\sigma_2^2}}\exp\left(-\frac{(x-\mu_2)^2}{2\sigma_2^2}\right)$$</p>
<p>clearly $X\sim pN(\mu_1,\sigma_1^2)+(1-p)N(\mu_2,\sigma_2^2)=N(p\mu_1+(1-p)\mu_2,p^2\sigma_... | Alex | 38,873 | <p>Let's assume $\Delta x>0$. Then it immediately follows from the 'increasing' part that $ f(x + \Delta x)- f(x) \geq 0$, and, since it is differentiable, the ratio is also positive. </p>
|
1,335,878 | <p>Let $N\unlhd K$ be a normal subgroup of a given group $K$ and let $$q:K\to K/N$$ be the natural quotient map. Let $A\subseteq K$ be a subset of $K$ and let the <a href="https://en.wikipedia.org/wiki/Conjugate_closure" rel="nofollow">conjugate closure</a> of $A$ in $K$ be denoted by $\langle A^K\rangle$.</p>
<p><str... | DanielWainfleet | 254,665 | <p>Let $B$ be a real or complex Banach space, and let $B^*$ be its dual space.That is, $B^*$ is the space of continuous linear functionals $g:B\to S$ (where $S$ is $ R$ for real space $X$ ,and $ S$ is $ C$ for complex space $X$)......For $g\in B^*$ we define $||g||=\sup \{|g(x)|/||x|| :x \ne 0\}$ . Now let $ M_g=\{x\in... |
84,034 | <p>Vieta's theorem states that given a polynomial
$$ a_nx^n + \cdots + a_1x+a_0$$
the quantities
$$\begin{align*}s_1&=r_1+r_2+\cdots\\
s_2&=r_1 r_2 +r_1 r_3 + \cdots \end{align*}$$
etc., where $r_1,\dots, r_n$ are the roots of the given polynomial, are given by
$$s_i = (-1)^i \frac{a_{n-i}}{a_n} .$$</p>
<p>S... | B. Decoster | 9,143 | <p>Quick answer: you're not going to find the roots in any quicker way with this method. Remember that in general, for polynomials of degree 5 or more, you cannot find explicit formulas for the roots. You simply cannot. With this method or another.</p>
<p>Now, what is Vieta's theorem? It is in fact just expanding the ... |
84,034 | <p>Vieta's theorem states that given a polynomial
$$ a_nx^n + \cdots + a_1x+a_0$$
the quantities
$$\begin{align*}s_1&=r_1+r_2+\cdots\\
s_2&=r_1 r_2 +r_1 r_3 + \cdots \end{align*}$$
etc., where $r_1,\dots, r_n$ are the roots of the given polynomial, are given by
$$s_i = (-1)^i \frac{a_{n-i}}{a_n} .$$</p>
<p>S... | J. M. ain't a mathematician | 498 | <p><em>Numerically</em>, however, for a <em>monic</em> polynomial $p(x)=x^n+c_{n-1} x^{n-1}+\cdots+c_1 x+c_0$, one can treat the Vieta equations relating the $n$ roots $x_k$ and the $n$ remaining coefficients $c_j$ as a system of simultaneous nonlinear equations, and then apply the multivariate version of Newton-Raphso... |
2,972,957 | <p><strong>Artin's Theorem-</strong> Let <span class="math-container">$E$</span> be a field and <span class="math-container">$G$</span> be a group of automorphisms of <span class="math-container">$E$</span> and <span class="math-container">$k$</span> be the set of elements of <span class="math-container">$E$</span> fix... | R.C.Cowsik | 293,582 | <p>The group <span class="math-container">$G$</span> generated by <span class="math-container">$n$</span> automorphisms has atleast <span class="math-container">$n$</span> elements, and the fixed field of <span class="math-container">$G$</span> is the same as the fixed field
of the set of <span class="math-container">$... |
2,252,579 | <p>$$ \lim_{n\to\infty}\left (\frac n {n+1} \right )^{2n} = \lim_{n\to\infty}\left (\frac{n+1}{n} \right )^{-2n} =\lim_{n\to\infty} \left (1 + \frac 1n \right )^{-2n}= \left (\lim_{n\to\infty}\left (1 + \frac 1n \right )^{n} \right )^{-2} = e^{-2}$$</p>
<p>What I don't understand is why is it a -2 and not +2? Also, ... | Mark Viola | 218,419 | <p>Note that $x=\left(\frac{1}{x}\right)^{-1}$. But, perhaps a better way to avoid any confusion is to write</p>
<p>$$\lim_{n\to \infty}\left(\frac{n}{n+1}\right)^{2n}=\lim_{n\to \infty}\left(\frac{1}{\left(1+\frac1n\right)^n}\right)^2=\left(\frac{1}{e}\right)^2=\frac1{e^2}=e^{-2}$$</p>
|
221,017 | <p>If I have the following list:</p>
<pre><code>https://pastebin.com/nqyf4yY5
</code></pre>
<p>How can I find the closest value to "89" in the "T[C]" column and its corresponding value in the "DH,aged-DH,unaged (J/g)" column?.</p>
<p>Thank you in advanced,</p>
| Rohit Namjoshi | 58,370 | <pre><code>data = Import["~/Downloads/data.txt"] // ToExpression // Part[#, 6 ;;] &;
nf = Nearest[data[[All, 2]] -> {"Index", "Element"}];
data[[nf[89][[1, 1]]]]
(* {3.87*10^-6, 89.2592, 5.13099, 0.0107504, 0.0102723, 0.0123268, 0.0000417117} *)
</code></pre>
|
221,017 | <p>If I have the following list:</p>
<pre><code>https://pastebin.com/nqyf4yY5
</code></pre>
<p>How can I find the closest value to "89" in the "T[C]" column and its corresponding value in the "DH,aged-DH,unaged (J/g)" column?.</p>
<p>Thank you in advanced,</p>
| creidhne | 41,569 | <p>With your data, assuming the column headings are in row 5:</p>
<pre><code>data[[5]]
(* {"Time(s)","T[C]","K(T)=k^(1/n)","dx/dT","x(t)","DH,aged-DH,unaged (J/g)","Check dx"} *)
</code></pre>
<p>... find the column numbers for columns "T[C]" and "DH,aged-DH,unaged (J/g)":</p>
<pre><code>{c1, c2} = Flatten@{
Posi... |
3,314,561 | <p>Consider the triangle <span class="math-container">$PAT$</span>, with angle <span class="math-container">$P = 36$</span> degres, angle <span class="math-container">$A = 56$</span> degrees and <span class="math-container">$PA=10$</span>. The points <span class="math-container">$U$</span> and <span class="math-contain... | Philip Roe | 430,997 | <p>It is a fact that not all physical problems have smooth solutions. Often this situation arises from a set of conservation laws that are expressed mathematically by applying such laws to a finite control volume to obtain an integral equation. Then we let the size of the control volume go to zero and arrive at some PD... |
122,468 | <p>I know how to find the number of solutions to the equation:</p>
<p><span class="math-container">$$a_1 + a_2 + \dots + a_k = n$$</span></p>
<p>where <span class="math-container">$n$</span> is a given positive integer and <span class="math-container">$a_1$</span>, <span class="math-container">$a_2$</span>, <span class... | Minh Hien | 821,566 | <p>Number of solutions is <span class="math-container">$S(n, k)$</span>.</p>
<p>If <span class="math-container">$a_1=1$</span>, we need the number of solutions of: <span class="math-container">$a_2+⋯+a_k=n-1$</span>, which <span class="math-container">$S(n-1, k-1)$</span></p>
<p>If <span class="math-container">$a_1>... |
2,398,215 | <p>If $f$ is continuous on $\mathbb{R}$ any of the following conditions are satisfied then $f$ must be a constant.</p>
<p>(1).$f(x)=f(mx),\forall x\in \mathbb{R},|m|≠1,m\in \mathbb{R}$</p>
<p>(2).$f(x)=f(2x+1),\forall x\in \mathbb{R}$</p>
<p>(3).$f(x)=f(x^2),\forall x\in \mathbb{R}$.</p>
<p>Suppose $f$ satisfy (1).... | Gabriel Romon | 66,096 | <ol start="2">
<li>Let $y=x+1$, so that $\forall y \in \mathbb R, f(y-1)=f(2y-1)$. Then define $g(y)=f(y-1)$, so that $\forall y, g(y)=g(2y)$. </li>
</ol>
<p>Using the result of 1., $g$ is constant and equal to $g(0)=f(-1)$, so $f$ is also constant and equal to $f(-1)$.</p>
<ol start="3">
<li>Note that $f(-x)=f((-x)... |
809,336 | <p>If G is abelian then factor group G/H is abelian.</p>
<p>How about the converse of this statement? </p>
<p>Is it true?</p>
| Brian Fitzpatrick | 56,960 | <p>If $G$ is your favorite nonabelian group, then $G/G$ is the trivial group!</p>
<p>Also, if $H$ has prime index $p$, then $G/H\simeq\Bbb Z_p$.</p>
|
1,232,532 | <p>First, I'm not looking for an answer here, I'm just looking to understand the problem so that I can prove it. I'm trying to analyzing the worst case running time of an algorithm, and it must has summation notation. What keeping me back is that I don't understand how to express <code>doSomething(n-j)</code> in summat... | Michael Hoppe | 93,935 | <p>Lhs is defined for $x\geq1/2$; for those $x$ the lhs is non-negative whereas the rhs is negative.</p>
<p>Some explanation: squaring is “if” but not “iff”: if $x=5$ then $ x^2=25$, but if $x^2=25$ then $x$ may be $-5$ as well.</p>
|
680,319 | <p>Let's restate this question in using mathematical notation. Let $n,k \in \mathbb{N}$. Let $f(n)=\left\lfloor{\frac{n}{k}}\right\rfloor$. Is it possible to rewrite this using the addition, multiplication, and exponentiation operators? I know it's possible for the case where $k=1$. Quite simply, note that $f(n)=\left\... | user44197 | 117,158 | <p>Here is something to get started.
Suppose you have the function
$$
f(n,k) = \left\{\begin{array}{ll}1 & \text{if $k | n$}\\\\0 & \text{if $k \not | n$}\end{array}\right.
$$
Then you should be able to cobble together the floor function (havent't thought this through).</p>
<p>Let
$$
\theta = e^{i 2 \pi/k}... |
2,170,382 | <p>I'm working on a question that asks to:</p>
<p>Find the area in the first quadrant bounded by the curves;
$\ xy = 1, xy=5, y=e^2x, y=e^5x $. </p>
<p>I would very much appreciate help solving this question (including the method of how to find the transformation expressions for $\ u$ and $v$ to use in the Jacobian... | DonAntonio | 31,254 | <p>An idea: </p>
<p>Try to write everything as function of $\;x\;$ to find intersection points. After all we have two hyperbolas and two straight lines:</p>
<p>$$\begin{cases}y=\frac1x\\{}\\y=\frac5x\\{}\\y=e^2x\\{}\\y=e^5x\end{cases}\;\;\implies\begin{cases}e^2x=\frac1x\implies x=\frac1e\\{}\\e^2x=\frac5x\implies x=... |
4,488,991 | <p>Let <span class="math-container">$M,M'$</span> be oriented connected compact smooth manifolds of the same dimension, let <span class="math-container">$S$</span> be a smooth manifold,
and let <span class="math-container">$\nu : S\times M\rightarrow M'$</span> be some smooth map.
Let <span class="math-container">$\nu_... | poetasis | 546,655 | <p>In support of the answer by @heropup , all triples where
<span class="math-container">$\space B =\dfrac{A^2-1}{2}\space$</span> and
<span class="math-container">$\space C-B=1\space$</span> can be generated by</p>
<p><span class="math-container">\begin{align}
A &=&&2k+1\\
B &= 2 k^2 + &&2 k\\
... |
2,657,301 | <p>On the set of natural numbers$\mathbb { N} $, define the operations $a \oplus b := \max(a,b)$
and $a\otimes b := a+b$ Is $(\mathbb {N},\oplus,\otimes)$ is ring? commutative ring with unity? Field?</p>
<p>My solution :</p>
<p>1- $(\mathbb {N},\oplus) $ is abelian groub because : </p>
<p>a. It is comutative
$a \... | Stefan4024 | 67,746 | <p>The identity can't be dependent on $a$. Anyway you want to have $a = a \oplus e = \max(a,e)$. So for each $a$ we must have $a \ge e$, which in natural numbers is $0$ (or $1$ if you consider $\mathbb{N}$ as the set of positive integers). But then there aren't any inverses, as for $a \not = 0$ we have that $a \oplus b... |
2,657,301 | <p>On the set of natural numbers$\mathbb { N} $, define the operations $a \oplus b := \max(a,b)$
and $a\otimes b := a+b$ Is $(\mathbb {N},\oplus,\otimes)$ is ring? commutative ring with unity? Field?</p>
<p>My solution :</p>
<p>1- $(\mathbb {N},\oplus) $ is abelian groub because : </p>
<p>a. It is comutative
$a \... | rschwieb | 29,335 | <p>As soon as you saw $a\oplus a=a$ for every $a$, you should have immediately seen that if this were a ring, $a=a\oplus a\oplus -a=a\oplus -a=0$, a contradiction if there exists $a\neq 0$. So it clearly can't be a ring.</p>
<p>But it is easy to see that it is a sub-semiring of the <a href="https://en.wikipedia.org/w... |
3,888,766 | <p>I need to prove this identity:</p>
<p><span class="math-container">$$2\cos\left(2\pi ft + \phi\right)\cos(2\pi ft) = \cos(4\pi ft + \phi) + \cos(\phi)$$</span></p>
<p>I know I have to use some identity or property but I can't find any to do it.</p>
| Fourier_T | 575,542 | <p>Multiply the two matrices as shown and you will have following equations:</p>
<ol>
<li><span class="math-container">$$a+3b=1$$</span></li>
<li><span class="math-container">$$4a-b=0$$</span>
From (2), we have: <span class="math-container">$$b=4a$$</span>
Putting in (1), we get: <span class="math-container">$$a+3(4a)=... |
19,596 | <p>I am trying to rearrange and manipulate some vector differential equations in <em>Mathematica</em>. As far as I understand you have to tell <em>Mathematica</em> that a variable is a vector by specifying the components of the vector. For example</p>
<pre><code>r = {x, y, z};
</code></pre>
<p>If I want to define vec... | jonsq | 5,506 | <p>Our package VEST (Vector Einstein Summation Tools) is designed to do exactly this. It's described <a href="http://arxiv.org/abs/1309.2561">here</a> and can be downloaded from <a href="https://github.com/jonosquire/VEST">github</a> along with a tutorial. </p>
<p>I realize this question is now very old, but thought t... |
66,671 | <p>$$\text{ABC- triangle:} A(4,2); B(-2,1);C(3,-2)$$<br>
Find a D point so this equality is true:</p>
<p>$$5\vec{AD}=2\vec{AB}-3\vec{AC}$$</p>
| Américo Tavares | 752 | <p>$$\text{The given vectors } \overrightarrow{AB}=B-A\text{ and }\overrightarrow{AC}=C-A \text{ and the solution }D=A+\overrightarrow{AD}$$</p>
<p><img src="https://i.stack.imgur.com/Iunwu.jpg" alt="enter image description here"></p>
<p>Let $(x,y)$ be the coordinates of $D$. The equation</p>
<p>$$5\overrightarrow{A... |
478,517 | <blockquote>
<p>Construct a topological mapping of the open disk $|z|<1$ onto the whole plane.</p>
</blockquote>
<p>I represent $z=re^{i\theta}$. I thought about the bijection from $(0,1)$ to $(0,\infty)$, which is given by $x\rightarrow \dfrac1x-1$. Applying this to the norm, we will get the mapping $re^{i\theta... | miracle173 | 11,206 | <p>Put a sphere on a plane. Draw a line from the center $M$ of the sphere to a point $P$ in the plane. This line intersects the surface of lower hemisphere of the sphere in a point $Q$. so you get a 1-1 mapping from the surface of the lower hemisphere to the plan. Draw Line through $Q$ vertical to the plane. This line... |
454,040 | <p>I need to know whether There exists any continuous onto map from $(0,1)\to (0,1]$</p>
<p>could any one give me any hint?</p>
| Mathronaut | 53,265 | <p>$f(x)=|\sin (\pi x)|$ will work</p>
|
3,128,862 | <p>I'm really stuck in this fairly simple example of conditional probability, I don't understand the book reasoning:</p>
<blockquote>
<p>An ordinary deck of 52 playing cards is randomly divided into 4 piles of 13 cards each.
Compute the probability that each pile has exactly 1 ace. </p>
<p><strong>Solution.</... | Floris Claassens | 638,208 | <p>An automorphism is a permutation which is by definition a bijection from <span class="math-container">$\mathcal{P}$</span> to <span class="math-container">$\mathcal{P}$</span>. Let <span class="math-container">$\phi$</span> be an automorphism and <span class="math-container">$L,L'$</span> parallel lines. Suppose <sp... |
2,867,042 | <blockquote>
<p>Find the value of
$$\tan\theta \tan(\theta+60^\circ)+\tan\theta \tan(\theta-60^\circ)+\tan(\theta + 60^\circ) \tan(\theta-60^\circ) + 3$$
(The answer is $0$.)</p>
</blockquote>
<p>My try: Let $\theta$ be $A$, $60^\circ -\theta$ be $B$, and $60^\circ + \theta$ be $C$. I simplified the result and ... | Ross Millikan | 1,827 | <p>The number of symmetric chains is much higher. You have $n \choose i$ ways to choose $C_i$. Then you choose the $n-2i$ elements that will get added, but you choose them in order, so the total number of chains is $${n \choose i}{n-i \choose n-2i}(n-2i)!$$
For example, if $n=6, i=2$ we have ${6 \choose 2}=15$ ways t... |
1,654,545 | <p>My teacher explained this problem to us - "There are $3$ mailboxes. $3$ people put letters in at random. There is no preference for any of the $3$ mailboxes. Compute the probability that each mailbox contains $1$ letter."</p>
<p>I tried this problem on my own and got the wrong answer. I understand the teacher's so... | Archis Welankar | 275,884 | <p>Please note that each letter is different so $111$ can have $3!$ ways same for $1,2$ so there are more than $10$ ways . thus this is the flaw of your solution put a,b,c as letters and A,B,C as mailboxes and then count ways but still personally i think teachers solution is fast and easy as always writing sample space... |
1,654,545 | <p>My teacher explained this problem to us - "There are $3$ mailboxes. $3$ people put letters in at random. There is no preference for any of the $3$ mailboxes. Compute the probability that each mailbox contains $1$ letter."</p>
<p>I tried this problem on my own and got the wrong answer. I understand the teacher's so... | N. F. Taussig | 173,070 | <p>By counting how many envelopes were placed in each mailbox, you treated the three letters as if they were identical. However, they are distinct. Consequently, the outcomes you listed are not equally likely.</p>
<p>Suppose the letters are $a$, $b$, and $c$ and the mailboxes are $A$, $B$, $C$. Since there are thre... |
2,114,619 | <p>An intruder has a cluster of 64 machines, each of which can try 10^6 passwords per second. How long does it take him to try all legal passwords if the requirements for the password are as follows:</p>
<ul>
<li>passwords can be 6, 7, or 8 characters long</li>
<li>each character is either a lower-case letter or a dig... | BranchedOut | 364,830 | <p>The number of possible passwords is $\sum_{k=0}^{2}10\cdot(10+26)^{5+k}=806,014,126,080=8.06\cdot10^{11}$.</p>
<p>The rate of password input is $64\cdot10^6$ passwords per second. Thus the time to try all of them is </p>
<p>$$\frac{8.06\cdot10^{11}\,\text{passwords}}{64\cdot10^6\,\text{passwords/second}}=12,593.97... |
259,308 | <p>The output of <code>ListPointPlot3D</code> is shown below:
<a href="https://i.stack.imgur.com/ypt73.png" rel="noreferrer"><img src="https://i.stack.imgur.com/ypt73.png" alt="enter image description here" /></a>
I only want to connect the dots in such a way that it forms a ring-like mesh. However, when I use <code>Li... | kglr | 125 | <pre><code>bdr = BoundaryDiscretizeRegion @ DelaunayMesh[pts]
</code></pre>
<p><a href="https://i.stack.imgur.com/dRX8U.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/dRX8U.png" alt="enter image description here" /></a></p>
<p>With some manual tweaking of the threshold <code>t</code>, we can filter ... |
2,397,564 | <p>Question:</p>
<p>Prove that if $ \ A\cup B \subseteq C \cup D,\ A \cap B =$ ∅ $\land \ C \subseteq A \implies B \subseteq D$.</p>
<p>My attempt:</p>
<p>Let $ \ x\in B \implies x \in A \cup B \implies x \in C \cup D \because A\cup B \subseteq C \cup D$.</p>
<p>Now, $ x \in C \lor x\in D$. If $\ x \in C \implies... | Eric | 309,041 | <blockquote>
<p><strong>Theorem.</strong> Let $A,~B,~C,~D$ be (any) sets. If $A\cup B\subseteq C\cup D$ and $A\cap B=\emptyset$ and $C\subseteq A$, then $B\subseteq D$.</p>
</blockquote>
<p><em>Proof.</em>(in much detailed, giving you the instruction that what should you do in every step)</p>
<p>Let $A,~B,~C,~D$ be... |
3,715,715 | <p>Let’s say I have a set <span class="math-container">$X$</span> and a set <span class="math-container">$Y$</span>, and <span class="math-container">$X \subseteq Y$</span>. Is it possible to state that <span class="math-container">$|X| \leq |Y|$</span> (<span class="math-container">$|X|$</span> cardinality of <span cl... | Arturo Magidin | 742 | <p>Not only can you say it, you can prove it!</p>
<p>Recall the definitions: for sets <span class="math-container">$A$</span> and <span class="math-container">$B$</span>, we say that</p>
<ul>
<li><span class="math-container">$|A|\leq |B|$</span> if and only if there exists a one-to-one function <span class="math-cont... |
2,539,693 | <p>A number theory textbook asked us to compare $\tan^{-1}(\frac{1}{2})$ and $\sqrt{5}$. In fact, these are rather close:</p>
<p>\begin{eqnarray*}
\tan^{-1} \frac{1}{2} &=& 0.46364 \\ \\
\frac{1}{\sqrt{5}} &=& 0.44721
\end{eqnarray*}</p>
<p>So at least numerically I think we have the answer that the ... | Robert Z | 299,698 | <p>Hint. Note that for $x\in (0,1)$, if $d$ is an odd positive integer then
$$\arctan(x)> \sum_{k=0}^d\frac{(-1)^kx^{2k+1}}{2k+1}.$$</p>
|
2,539,693 | <p>A number theory textbook asked us to compare $\tan^{-1}(\frac{1}{2})$ and $\sqrt{5}$. In fact, these are rather close:</p>
<p>\begin{eqnarray*}
\tan^{-1} \frac{1}{2} &=& 0.46364 \\ \\
\frac{1}{\sqrt{5}} &=& 0.44721
\end{eqnarray*}</p>
<p>So at least numerically I think we have the answer that the ... | Ross Millikan | 1,827 | <p>You can use the Taylor series for $\arctan x$
$$\arctan (\frac 12) \gt \frac 12-\frac 1{2^3\cdot 3}\gt 0.45833$$ where we have an alternating series so the error is of the sign of the first neglected term and $$11^2=121 \lt 5\cdot 5^2\\\frac 1{\sqrt 5} \lt \frac 5{11}\lt 0.45455$$</p>
|
1,163,033 | <p>I want to calculate $ 8^{-1} \bmod 77 $ </p>
<p>I can deduce $ 8^{-1} \bmod 77$ to $ 8^{59} \bmod 77 $ using Euler's Theorem.</p>
<p>But how to move further now. Should i calculate $ 8^{59} $ and then divide it by $ 77 $ or is there any other theorem i can use ? </p>
| paw88789 | 147,810 | <p>For small to intermediate size moduli, you can use the method of adding the modulus, and cancelling when possible. (For larger moduli, you may want to use the extended Euclidean Algorithm.)</p>
<p>Here's adding the modulus on your problem:</p>
<p>$8x\equiv 1\equiv 78 \pmod{77}$. So $4x\equiv 39 \pmod{77}$.</p>
... |
1,163,033 | <p>I want to calculate $ 8^{-1} \bmod 77 $ </p>
<p>I can deduce $ 8^{-1} \bmod 77$ to $ 8^{59} \bmod 77 $ using Euler's Theorem.</p>
<p>But how to move further now. Should i calculate $ 8^{59} $ and then divide it by $ 77 $ or is there any other theorem i can use ? </p>
| Bill Dubuque | 242 | <p>Below are a few methods.</p>
<hr>
<p>${\rm mod}\ 77\!:\ \ \dfrac{1}8\equiv \dfrac{-76}8\equiv \dfrac{-19}{2}\equiv \dfrac{58}2\equiv 29\,\ $ by fraction fiddling</p>
<hr>
<p>${\rm mod}\ 77\!:\ \ \dfrac{1}8 \equiv \dfrac{10}{80}\equiv \dfrac{10}{3}\equiv\dfrac{87}3\equiv 29\,\ $ by <a href="https://math.stackexc... |
623,428 | <blockquote>
<p>Suppose $$
Y = X^TAX,
$$ where $Y$ and $A$ are both known $n\times n$, real, symmetric matrices. The unknown matrix $X$ is restricted to $n\times n$.</p>
</blockquote>
<p>I think there should be at least one real valued solution for $X$. How do I solve for $X$? </p>
| Disintegrating By Parts | 112,478 | <p>Suppose $Y$ is of full rank, but $A$ is not. You can't do what you want then.</p>
|
4,336,659 | <p>For a beta distribution with parameters <span class="math-container">$a$</span> and <span class="math-container">$b$</span>, we can interpret it as the distribution of the probability of heads for a coin we tossed <span class="math-container">$a+b$</span> times and saw <span class="math-container">$a$</span> heads a... | Shannon Starr | 1,003,218 | <p>You answered your own question completely and entirely, using the observation by <a href="https://math.stackexchange.com/users/356647/lee-david-chung-lin">lee-david-chung-lin</a>. Actually a Beta(a,b) distribution for <span class="math-container">$a,b \in \{1,2,\dots\}$</span> is the same as the Bayesian posterior f... |
679,904 | <p>The question is let $a \in \mathbb{R} $ does not contain 0. Prove that $|a+\frac{1}{a}| \ge 2$. I have no idea how to start this problem and any help on it would be greatly appreciated.</p>
| triwer23 | 129,160 | <p>Note that it is enough to consider the case $a>0$. Also note that for $x,y$ non negative, $x+y\geq 2\sqrt{xy}$. </p>
|
239,136 | <p>I was given this question and I'm not really sure how to approach this...</p>
<p>Assume $(r,s) = 1$. Prove that If $G = \langle x\rangle$ has order $rs$, then $x = yz$, where $y$ has order $r$, $z$ has order $s$, and $y$ and $z$ commute; also prove that the factors $y$ and $z$ are unique.</p>
| Dan Shved | 47,560 | <p>Here is a hint: you can set $y=x^{sn}$ and $z=x^{rm}$. To find the appropriate $n$ and $m$ you can use <a href="http://en.wikipedia.org/wiki/B%C3%A9zout%27s_identity" rel="nofollow">Bezout's identity</a>.</p>
|
621,742 | <p>How do you get from$$\int^\infty_0\int^\infty_0e^{-(x+y)^2} dx\ dy$$to
$$\frac{1}{2}\int^\infty_0\int^u_{-u}e^{-u^2} dv\ du?$$ I have tried using a change of variables formula but to no avail.<br>
Edit: Ok as suggested I set $u=x+y$ and $v=x-y$, so I can see this gives $dx dy=\frac{1}{2}dudv$ but I still can't see h... | Yiorgos S. Smyrlis | 57,021 | <p>Hint. Set $u=x+y$, $v=x-y$.</p>
<p>Then
$$
\{(x,y): x,y\ge 0\}=\{(u,v) : u>0, v\in(-u,u)\},
$$
and
$$
dx\,dy=\frac{1}{2}du\,dv,
$$
as
$$
\frac{\partial (x,y)}{\partial(u,v)}=\frac{1}{2}
$$</p>
|
621,742 | <p>How do you get from$$\int^\infty_0\int^\infty_0e^{-(x+y)^2} dx\ dy$$to
$$\frac{1}{2}\int^\infty_0\int^u_{-u}e^{-u^2} dv\ du?$$ I have tried using a change of variables formula but to no avail.<br>
Edit: Ok as suggested I set $u=x+y$ and $v=x-y$, so I can see this gives $dx dy=\frac{1}{2}dudv$ but I still can't see h... | Felix Marin | 85,343 | <p>$\newcommand{\+}{^{\dagger}}%
\newcommand{\angles}[1]{\left\langle #1 \right\rangle}%
\newcommand{\braces}[1]{\left\lbrace #1 \right\rbrace}%
\newcommand{\bracks}[1]{\left\lbrack #1 \right\rbrack}%
\newcommand{\ceil}[1]{\,\left\lceil #1 \right\rceil\,}%
\newcommand{\dd}{{\rm d}}%
\newcommand{\ds}[1]{\displayst... |
964,387 | <p>I'm attempting to teach myself topology for graduate school this summer, but I'm having a tough time.
I'm trying to prove that the Euclidean topology on $\mathbb{R}^{m+n}$ is equivalent to the product topology on $\mathbb{R}^m \times \mathbb{R}^n$. I realize to do this that I should make a homeomorphism between the... | Pedro | 23,350 | <p><em>Hint</em> Suppose $\lVert \cdot \rVert_1$ and $\lVert \cdot \rVert_2$ are the Euclidean norms in $ \Bbb R^n,\Bbb R^m$ respectively. Then $\lVert (x,y)\rVert = (\lVert x\rVert_1^2+\lVert y\rVert_2^2)^{1/2}$ is the Euclidean norm in $\Bbb R^n\times\Bbb R^m$. </p>
|
964,387 | <p>I'm attempting to teach myself topology for graduate school this summer, but I'm having a tough time.
I'm trying to prove that the Euclidean topology on $\mathbb{R}^{m+n}$ is equivalent to the product topology on $\mathbb{R}^m \times \mathbb{R}^n$. I realize to do this that I should make a homeomorphism between the... | Ri-Li | 152,715 | <p>Just use $f:\Bbb R^m \times \Bbb R^n \to \Bbb R^{m+n}$. $((x_1,...,x_m),(x_{m+1},...,x_{m+n}))\to (x_1,...,x_{m+n})$ Now see that it is bijective and continuous. Actually it some kind of identity function. [As you are having problem, some things you need to clarify,
1) Eucledean topology is same as the metric topolo... |
582,478 | <p>Please simplify this logic expression for me with helping boolean algebra :</p>
<p>A'C'D + A'BD + BCD + ABC + ACD'</p>
<p>I know that must use consensus theorem .</p>
<p>my solve :</p>
<p>STEP 1 : Terms 1 & 3 ---eliminate---> Term 2</p>
<p>STEP 2 : Terms 3 & 5 ---eliminate---> Term 4</p>
<p>STEP 3 : Te... | Suraj M S | 85,213 | <p>given
$$ A'C'D+A'BD+BCD+ABC+ACD'$$
use distributive law $xyz+pqr=(xyz+p).(xyz+qr)$
$$\to (A'C'D+A'D).(A'C'D+B) +(BCD+BC).(BCD+A)+ACD'$$
$$ \to A'C'D+B+BCD+A+ACD'$$
$$ \to A'C'D+B(1+CD)+A(1+CD')$$
$$ \to A'C'D+A+B$$
$$ \to (A+A').(A+C'D) +B$$
$$ \to A+B+C'D$$</p>
|
582,478 | <p>Please simplify this logic expression for me with helping boolean algebra :</p>
<p>A'C'D + A'BD + BCD + ABC + ACD'</p>
<p>I know that must use consensus theorem .</p>
<p>my solve :</p>
<p>STEP 1 : Terms 1 & 3 ---eliminate---> Term 2</p>
<p>STEP 2 : Terms 3 & 5 ---eliminate---> Term 4</p>
<p>STEP 3 : Te... | Donald Splutterwit | 404,247 | <p>Using a truth table Let $X=A'C'D + A'BD + BCD + ABC + ACD'$. (We shall leave entries blank where their value is zero)
\begin{eqnarray*}
\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline
A & B & C & D & & A'C'D & A'BD & BCD & ABC & ACD' & & X \\ \hline
0&0&0&0&... |
3,428,668 | <p>How to prove this</p>
<p><span class="math-container">$$S = \{(x, y) | Ax + By ≥ c, x ≥ 0, y ≥ 0\}$$</span>
where <span class="math-container">$A$</span> is an <span class="math-container">$m \times n$</span> matrix, <span class="math-container">$B$</span> is a positive semi-definite <span class="math-container">$m... | user2661923 | 464,411 | <p><span class="math-container">$27 \times 24 = 648 \;\equiv 18 \mod(63).$</span></p>
<p>Therefore, the 2nd equation should be </p>
<p><span class="math-container">$18y + 48 \;\equiv 24 \mod(63).$</span></p>
|
3,498,199 | <p>Suppose if a matrix is given as</p>
<p><span class="math-container">$$ \begin{bmatrix}
4 & 6\\
2 & 9
\end{bmatrix}$$</span></p>
<p>We have to find its eigenvalues and eigenvectors.</p>
<p>Can we first apply elementary row operation . Then find eigenvalues.</p>
<p>Is their any relation on the matrix if ... | Simply Beautiful Art | 272,831 | <p>As noted, you cannot apply elementary row operations to <span class="math-container">$A$</span> and expect the eigenvalues/vectors be preserved. However, you can apply elementary row operations to <span class="math-container">$|A-\lambda I|=0$</span> to solve for <span class="math-container">$\lambda$</span>. In you... |
221,351 | <p>I asked the following question (<a href="https://math.stackexchange.com/questions/1487961/reference-for-every-finite-subgroup-of-operatornamegl-n-mathbbq-is-con">https://math.stackexchange.com/questions/1487961/reference-for-every-finite-subgroup-of-operatornamegl-n-mathbbq-is-con</a>) on math.stackexchange.com and ... | p Groups | 84,772 | <p>This is not exact answer to your question, but long back, I had seen this result in an article in Monthly. I hope it will be useful.</p>
<p><a href="http://www.jstor.org/stable/pdf/2695329.pdf?acceptTC=true" rel="nofollow">http://www.jstor.org/stable/pdf/2695329.pdf?acceptTC=true</a></p>
|
465,255 | <p>Does there exists any form of Algebra where operators can be assumed as variables?</p>
<p>For example:
$$
1+2\times3=7
$$
can be considered as:
$$
1\:(\mathrm{\,X})\:2\:(\mathrm{Y})\:3=7
$$
?</p>
| qaphla | 85,568 | <p>The total number of points scored on the exam would be $200 * 25 + 304 * 25 + 350 * 25 + 250 * 25 = 27,600$, out of a maximum possible $400 * (25 * 4) = 40,000$.</p>
<p>Taking $\frac{27,600}{40,000}$ and simplifying it gives $\frac{276}{400} = \frac{69}{100}$.<br>
As there are $100$ total points on the exam, each s... |
10,427 | <p>I like Mathematica, but it's syntax baffles me.</p>
<p>I am trying to figure out how to minimize the whitespace around a graphic.</p>
<p>For example,</p>
<pre><code>ParametricPlot3D[{r Cos[t], r Sin[t], r^2}, {r, 0, 1}, {t, 0, 2 \[Pi]},
Boxed -> True, Axes -> False]
</code></pre>
<p><img src="https://i.s... | Dr. belisarius | 193 | <p>Actually, there isn't white space at all:</p>
<pre><code>Show[RegionPlot3D[True, {x, -1, 1}, {y, -1, 1}, {z, 0, 1},
PlotStyle -> Directive[Yellow, Opacity[0.5]], Mesh -> None,
Boxed -> False, Axes -> False, PlotRangePadding -> 0],
ParametricPlot3D[{r Cos[t]... |
10,427 | <p>I like Mathematica, but it's syntax baffles me.</p>
<p>I am trying to figure out how to minimize the whitespace around a graphic.</p>
<p>For example,</p>
<pre><code>ParametricPlot3D[{r Cos[t], r Sin[t], r^2}, {r, 0, 1}, {t, 0, 2 \[Pi]},
Boxed -> True, Axes -> False]
</code></pre>
<p><img src="https://i.s... | Vitaliy Kaurov | 13 | <p>I think what you are looking for is <a href="http://reference.wolfram.com/mathematica/ref/ViewAngle.html" rel="noreferrer"><strong>ViewAngle</strong></a> option. The graph below compares default <code>Automatic</code> versus custom setting for <code>ViewAngle</code>. The image are framed intentionally to see clearly... |
2,416,424 | <p>It is known that the collection of finite mixtures of Gaussian Distributions over $\mathbb{R}$ is dense in $\mathcal{P}(\mathbb{R})$ (the space of probability distributions) under convergence in distribution metric.</p>
<p>I'm interested to know the following:</p>
<p>Let $P_X$ be a random variable with finite $p$ ... | Jaroslaw Matlak | 389,592 | <p>By denoting $t=2^x$ you obtain the following inequality:</p>
<p>$$\frac{16-t}{t-8}>0$$</p>
<p>You can now multiplicate both sides of inequality by $-1$:
$$\frac{t-16}{t-8}<0$$</p>
<p>Ranges of $t$ satisfying this inequality are the same, as ranges of $t$ satisfying this one:
$$(t-16)(t-8)<0$$
with condit... |
1,524,349 | <p>This is Problem 45 in Chapter 19 in Michael Spivak's book "Calculus".</p>
<ol start="45">
<li>(a) Suppose that $\frac {f(x)} x$ is integrable on every interval [a, b] for $0$ < a < b, and that $\lim_{x\to0}f(x)=A$ and $\lim_{x\to\infty}f(x)=B$. Prove that for all $\alpha$, $\beta$ > $0$ we have</li>
</ol>
<p... | John Dawkins | 189,130 | <p>Another approach: Observe that
$$
\int_a^b {f(\alpha x)-f(\beta x)\over x}\,dx =\int_{\alpha a}^{\beta a}{f(t)\over t}\,dt - \int_{\alpha b}^{\beta b}{f(t)\over t}\,dt.
$$
The first integral on the right side of the above display can be written as
$$
\int_{\alpha a}^{\beta a}{f(t)-A\over t}\,dt+A\log(\beta/\alpha),... |
1,747,696 | <p>First of all: beginner here, sorry if this is trivial.</p>
<p>We know that $ 1+2+3+4+\ldots+n = \dfrac{n\times(n+1)}2 $ .</p>
<p>My question is: what if instead of moving by 1, we moved by an arbitrary number, say 3 or 11? $ 11+22+33+44+\ldots+11n = $ ?
The way I've understood the usual formula is that the first n... | Christopher Carl Heckman | 261,187 | <p>You get what is called an arithmetic series. More details are at <a href="https://en.wikipedia.org/wiki/Arithmetic_progression" rel="nofollow">https://en.wikipedia.org/wiki/Arithmetic_progression</a> .</p>
|
3,554,646 | <p>Let <span class="math-container">$f:(a,b)\to \mathbb{R}$</span> be injective and continuous. Prove that</p>
<ul>
<li><span class="math-container">$f$</span> is monotonic.</li>
<li>The image of <span class="math-container">$f$</span> is <span class="math-container">$(c,d)$</span> or maybe (<span class="math-containe... | Ansar | 719,973 | <p>Put <span class="math-container">$c = \mbox{inf Im} f$</span> and <span class="math-container">$d = \mbox{sup Im} f$</span>. If <span class="math-container">$c<y<d$</span>, then <span class="math-container">$c\leq c'<y<d'\leq d$</span> for some <span class="math-container">$c',\ d'\in\mbox{Im} f$</span>.... |
4,226,455 | <blockquote>
<p>Let <span class="math-container">$X$</span> be a set and <span class="math-container">$Y$</span> a topological space. What is the topology on <span class="math-container">$X$</span> induced by constant maps <span class="math-container">$f:X \to Y$</span>?</p>
</blockquote>
<p>The induced topology is <sp... | Bram28 | 256,001 | <p>You typically can't remove quantifiers and still have a proposition.</p>
<p>For example, take <span class="math-container">$\forall x \ P(x)$</span>. If you take away the quantifier, you are left with <span class="math-container">$P(x)$</span>. But <span class="math-container">$P(x)$</span> is not a proposition. Wit... |
4,226,455 | <blockquote>
<p>Let <span class="math-container">$X$</span> be a set and <span class="math-container">$Y$</span> a topological space. What is the topology on <span class="math-container">$X$</span> induced by constant maps <span class="math-container">$f:X \to Y$</span>?</p>
</blockquote>
<p>The induced topology is <sp... | ryang | 21,813 | <ol>
<li><p>The formula <span class="math-container">$$\forall x{\in} D\:\:P(x)$$</span> is logically equivalent to
<span class="math-container">$$\forall x\,\big(x{\in} D\to P(x)\big).$$</span> These formulae are vacuously true when <span class="math-container">$∀x\;x\notin D,$</span> since a <strong>vacuously true</s... |
2,965,717 | <p>How would you prove that <span class="math-container">$$\displaystyle \prod_{k=1}^\infty \left(1+\dfrac{1}{2^k}\right) \lt e ?$$</span></p>
<p>Wolfram|Alpha shows that the product evaluates to <span class="math-container">$2.384231 \dots$</span> but is there a nice way to write this number? </p>
<p>A hint about so... | Awe Kumar Jha | 605,905 | <p>Proof of the Lemma by Induction.
Let <span class="math-container">$P(n): (1+a_1)(1+a_2)...(1+a_n) ≤ 1 + s_n + ... + \frac {s_n^n}{n!}$</span>
For<span class="math-container">$n=1$</span> the result is obvious.
Let <span class="math-container">$P(m): (1+a_1)(1+a_2)...(1+a_m) ≤ 1 + s_m + ... + \frac {s_m^m}{m!}$</span... |
1,768,700 | <p>According to my knowledge, to prove that $24^{31}$ is congruent to $23^{32}$ mod 19, we must show that both numbers are divisible by 19 i.e. their remainders must be equal with mod 19. Please correct me if I'm wrong.</p>
<p>So, I was able to reduce $23^{32}$ and find its mod 19, which is 17 but I am having a bit of... | Umberto P. | 67,536 | <p>After some calculation you have
\begin{align*} 24 &\equiv 5 \bmod 19 \\
24^2 &\equiv 25 \equiv 6 \bmod 19 \\
24^4 &\equiv 36 \equiv -2 \bmod 19 \\
24^8 & \equiv 4 \bmod 19 \\
24^{16} & \equiv 16 \equiv -3 \bmod 19 \end{align*}</p>
<p>Now multiply:
$$24^{31} \equiv 5\cdot 6 \cdot (-2) \cdot 4 \cd... |
4,255,587 | <p>There was a quiz posted on F*cebook by someone. Here's the problem.</p>
<p><a href="https://i.stack.imgur.com/fGu2W.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/fGu2W.jpg" alt="enter image description here" /></a></p>
<p>And here's my attempt:</p>
<p><a href="https://i.stack.imgur.com/EMP6g.jpg... | Vasili | 469,083 | <p>Let <span class="math-container">$\alpha$</span> is the base angle. Then the area of one (bigger) blue triangle is <span class="math-container">$\frac{16^2}{2} \tan (2\pi-2\alpha)$</span>, the area of the second blue triangle is <span class="math-container">$\frac{7^2}{2} \tan \alpha$</span>. On the other hand, <spa... |
3,068,197 | <p>I'm working through this problem and I haven't been able to make any progress. The textbook provides the answer of <span class="math-container">$ {9 \choose 4}$</span> but I'm not sure as to how they got this result. </p>
| trancelocation | 467,003 | <ul>
<li>If you write the <span class="math-container">$8$</span> <span class="math-container">$1's$</span> horizontally beside each other, the number of possible places to put zeros between them or at the left-most or right-most position is <span class="math-container">$\color{blue}{9}$</span></li>
<li>The number of w... |
3,068,197 | <p>I'm working through this problem and I haven't been able to make any progress. The textbook provides the answer of <span class="math-container">$ {9 \choose 4}$</span> but I'm not sure as to how they got this result. </p>
| robjohn | 13,854 | <p>Consider the two atoms: <span class="math-container">$x\to01$</span> and <span class="math-container">$y\to1$</span>. If we arrange <span class="math-container">$4$</span> <span class="math-container">$x$</span>s and <span class="math-container">$5$</span> <span class="math-container">$y$</span>s, we get all the all... |
75,791 | <p>When will a probabilistic process obtained by an "abstraction" from a deterministic discrete process satisfy the Markov property?</p>
<p>Example #1) Suppose we have some recurrence, e.g., $a_t=a^2_{t-1}$, $t>0$. It's a deterministic process. However, if we make an "abstraction" by just considering the one partic... | Community | -1 | <p>As @Did points out, you want to know when "lumping" preserves the Markov property. See Lemma 2.5 in <em>Markov Chains and Mixing Times</em> by Levin, Peres, and Wilmer. The book is available online <a href="http://pages.uoregon.edu/dlevin/MARKOV/markovmixing.pdf" rel="nofollow">here</a>.</p>
|
3,449,274 | <p>I have equation <span class="math-container">$2b^2 - 72b - 406=0$</span>. I divided it with 2 and I got <span class="math-container">$b^2 - 36b - 203=0$</span>. My teacher then wrote <span class="math-container">$(b-29)(b-7)=0$</span> but I don’t understand how he got that. When I try to solve that equation I get <s... | Matt Samuel | 187,867 | <p>A horizontal asymptote only tells you that it tends to a certain value as <span class="math-container">$x$</span> goes to infinity in some direction. It can certainly cross the asymptote any number of times without contradicting this, including infinitely many times. For example,
<span class="math-container">$$f(x)... |
2,245,631 | <blockquote>
<p>$x+x\sqrt{(2x+2)}=3$</p>
</blockquote>
<p>I must solve this, but I always get to a point where I don't know what to do. The answer is 1.</p>
<p>Here is what I did: </p>
<p>$$\begin{align}
3&=x(1+\sqrt{2(x+1)}) \\
\frac{3}{x}&=1+\sqrt{2(x+1)} \\
\frac{3}{x}-1&=\sqrt{2(x+1)} \\
\frac{(3-x... | zwim | 399,263 | <p>$f(x)=x+x\sqrt{2x+2}$</p>
<p><a href="https://i.stack.imgur.com/m0hnj.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/m0hnj.png" alt="enter image description here"></a></p>
<p>We are trying to find $f(x)=3$ but notice that for $-1\le x\le 0$ then $f(x)\le 0$.</p>
<p>$f'(x)=1+\sqrt{2x+2}+\frac {... |
4,215,724 | <p><span class="math-container">$f\colon \mathbb{R}^2\to \mathbb{R}$</span> such that <span class="math-container">$f_x(2,-1)=1$</span> and <span class="math-container">$f_y(2,-1)=1$</span> and <span class="math-container">$g(x,y)=\langle x^2y,x-y\rangle$</span> and <span class="math-container">$h = f\circ g$</span> th... | Sebathon | 482,453 | <p>If <span class="math-container">$\langle \cdot,\cdot \rangle$</span> defines a vector, then you did it wrong. Using the <a href="https://en.wikipedia.org/wiki/Chain_rule#General_rule" rel="nofollow noreferrer">chain rule</a> (the case when <span class="math-container">$k=1$</span>) we have</p>
<p><span class="math-c... |
233,169 | <p>I had to redo the problem because there was a mistake. With the given function from a previous problem, I was solving <a href="https://mathematica.stackexchange.com/questions/231664/adding-a-point-in-a-manipulate-command">link</a>, I found that the parabola created a trajectory on the graph, ie another parabola.</p>... | cvgmt | 72,111 | <p>To get the trajectory equation about the minimum, we can do as below.</p>
<pre><code>f[x_, a_] := x^2 - 2*(a - 2)*x + a - 2;
min = Minimize[f[x, a], x];
Eliminate[{x, y} == ({x, y} /. Last@min /. y -> First@min), a]
</code></pre>
<blockquote>
<p>-y == -x + x^2</p>
</blockquote>
<p>To plot the minimum points, her... |
108,060 | <p>Suppose:
$$\sum_{n=2}^{\infty} \left( \frac{1}{n(\ln(n))^{k}} \right) =\frac{1}{ 2(\ln(2))^{k} } +\frac{1}{ 3(\ln(3))^{k} }+...,
$$
by which $k$ does it converge?</p>
<p>When I use comparison test I get inconclusive result:</p>
<p>$\lim_{n\rightarrow\infty} \frac{u_{n+1}}{u_{n}}=\frac{n\ln(n)^{k}}{(n+1)\ln(n+1)^{... | Community | -1 | <p>I think Cauchy's Condensation test will do the deal.</p>
<p>$$\sum a_n \text{converges} \iff \sum2^na_{2^n} \text{converges}$$</p>
<p>And, on applying this test, and simplifying the test, you'll need to compare it with the series $\sum n^{-p}$. For what values of $p$ does this series converge?</p>
|
327,860 | <p>Let <span class="math-container">$A$</span> be a symmetric <span class="math-container">$d\times d$</span> matrix with integer entries such that the quadratic form <span class="math-container">$Q(x)=\langle Ax,x\rangle, x\in \mathbb{R}^d$</span>, is non-negative definite. For which <span class="math-container">$d$</... | Zhi-Wei Sun | 124,654 | <p>Such problems were investigated by L. J. Mordell in the 1930s. Two related papers of Mordell are as follows:</p>
<ol>
<li><p>L. J. Mordell, A new Warings problem with squares of linear forms, Quarterly J. (Oxford series) 1 (1930), 276–288. </p></li>
<li><p>L. J. Mordell, On binary quadratic forms expressable as a s... |
78,725 | <p>The general theorem is: for all odd, distinct primes $p, q$, the following holds:
$$\left( \frac{p}{q} \right) \left( \frac{q}{p} \right) = (-1)^{\frac{p-1}{2}\frac{q-1}{2}}$$</p>
<p>I've discovered the following proof for the case $q=3$:
Consider the Möbius transformation $f(x) = \frac{1}{1-x}$, defined on $F_{p}... | Grigory M | 152 | <p>Well, at least it can be extended to a proof of the case $q=5$.
$\def\lf#1#2{\left(\dfrac{#1}{#2}\right)}$</p>
<p><strong>0.</strong> $\lf5p=1\iff\exists\phi\in\mathbb F_p:\phi^2+\phi-1=0$.</p>
<p>// Note that in $\mathbb R$ one can take $\phi=2\cos(2\pi/5)$. So in the next step we'll use something like 'rotation ... |
62,000 | <p>Let $I,J,K$ be three non-void sets, and let $\gamma$:$I\times J\times K\rightarrow\mathbb{N}$.
Is there some nonempty set $X$, together with some functions {$\{ f_{i}:X\rightarrow X;i\in I\} $},
some subsets {$\{ \Omega_{j}\subset X;j\in J\} $}, and some
points {$\{p_{k}\in X;k\in K} $} s.t. $\mid f_{i}^{-1}\left(p_... | Zev Chonoles | 1,916 | <p>Here is a second attempt (see edit history for previous version).</p>
<p>For each $t\in\mathbb{N}$, let
$$P_{i,j,k,t}=\{1_{i,j,k,t},\ldots,n_{i,j,k,t},\ldots,\gamma(i,j,k)_{i,j,k,t}\}$$
(so that for each choice of $i\in I$, $j\in J$, $k\in K$, and $t\in\mathbb{N}$, we have a disjoint set of size $\gamma(i,j,k)$).... |
4,513,678 | <p>Suppose <span class="math-container">$f(x) = ax^3 + bx^2 + cx + d$</span> is a cubic equation with roots <span class="math-container">$\alpha, \beta, \gamma.$</span> Then we have:</p>
<p><span class="math-container">$\alpha + \beta + \gamma= -\frac{b}{a}\quad (1)$</span></p>
<p><span class="math-container">$\alpha\b... | Sourav Ghosh | 977,780 | <p>A function of two or more variables is said to be symmetric function if <span class="math-container">$f$</span> remains unaltered by an interchange of any two of it's variables.</p>
<p>If <span class="math-container">$\alpha, \beta, \gamma$</span> are roots of a cubic equation and the function <span class="math-cont... |
2,525,573 | <p>Find the domain and range of $y=\sqrt {x-2}$</p>
<p>My Attempt:
$$y=\sqrt {x-2}$$
For $y$ to be defined,
$$(x-2)\geq 0$$
$$x\geq 2$$
So $dom(f)=[2,\infty)$.</p>
| Michael Rybkin | 350,247 | <p>Here, I'm using the property: $a^b=a^c \Leftrightarrow b=c $ where $a > 0$ and $a\ne1$. Here's how it goes:</p>
<p>$$
e^{3x-1} = 5e^{2x}\implies\\
e^{3x-1} = e^{\ln{5}}\cdot e^{2x}\implies\\
e^{3x-1} = e^{\ln{5}+2x}\implies\\
3x-1 = \ln{5} + 2x\implies\\
3x-2x= \ln{5}+1\implies\\
x= \ln{5}+1\implies\\
x= \ln{5}+... |
416,940 | <p><span class="math-container">$\DeclareMathOperator\Spec{Spec}\newcommand{\perf}{\mathrm{perf}}\DeclareMathOperator\SHC{SHC}$</span>I have just finished reading the paper "The spectrum of prime ideals in tensor triangulated categories" in which Balmer proposes his notion of spectrum which nowadays is consid... | Maxime Ramzi | 102,343 | <p>I am not an expert in tt-geometry, but let me try to answer some of your questions.</p>
<p>(1) You are correct, the Balmer spectrum is typically not well-suited to study the "big" categories - this is because all definitions that appear only use "finitary" things : tensor products, cones/extensio... |
4,492,566 | <blockquote>
<p>To which degree must I rotate a parabola for it to be no longer the graph of a function?</p>
</blockquote>
<p>I have no problem with narrowing the question down by only concerning the standard parabola: <span class="math-container">$$f(x)=x^2.$$</span></p>
<p>I am looking for a specific angle measure. O... | Taladris | 70,123 | <p>Let <span class="math-container">$P_\theta$</span> be the parabola obtained from the parabola <span class="math-container">$P$</span> of equation <span class="math-container">$y=x^2$</span> by a rotation of angle <span class="math-container">$\theta\in(0,2\pi)$</span>.</p>
<p>A rotation of <span class="math-containe... |
4,492,566 | <blockquote>
<p>To which degree must I rotate a parabola for it to be no longer the graph of a function?</p>
</blockquote>
<p>I have no problem with narrowing the question down by only concerning the standard parabola: <span class="math-container">$$f(x)=x^2.$$</span></p>
<p>I am looking for a specific angle measure. O... | Brian Drake | 843,797 | <p>You write (emphasis added):</p>
<blockquote>
<p>I am looking for a specific angle measure. <strong>One</strong> such measure must exist as
the reflection of <span class="math-container">$f$</span> over the line <span class="math-container">$y=x$</span> is certainly no longer
well-defined.</p>
</blockquote>
<p>It sou... |
2,880,566 | <p>In an optimization problem I finally get to the point where I have to solve</p>
<p>$$x +\sec(x)(\tan(x)\cos(2x)+\tan(x)-2\sin(2x)) =0$$</p>
<p>which obviously leads to</p>
<p>$$x=-\sec(x)(\tan(x)\cos(2x)+\tan(x)-2\sin(2x))$$</p>
<p>Nevertheless, this couldn't in any case help knowing the optimal size of the angl... | Multigrid | 541,516 | <p>Hint: Use $\cos(2x) = 2\cos^2(x) - 1$ and $\sin(2x) = 2\sin(x) \cos(x)$</p>
<p>by the way do you get $x = 0$ and 2 other solutions (symmetric)?</p>
|
340,264 | <p>Given that</p>
<p>$L\{J_0(t)\}=1/(s^2+1)$</p>
<p>where $J_0(t)=\sum\limits^{∞}_{n=0}(−1)n(n!)2(t2)2n$,</p>
<p>find the Laplace transform of $tJ_0(t)$. </p>
<p>$L\{tJ_0(t)\}=$_<strong><em>_</em>__<em>_</em>__<em>_</em>___<em></strong>---</em>___?</p>
| Alex Youcis | 16,497 | <p>Now, I want to say something more than just the run-of-the-mill answer, if that's ok with you.</p>
<p>There is a general principle which says that a polynomial $F(T_1,\ldots,T_n)\in\mathbb{Z}[T_1,\ldots,T_n]$ has a zero in $\mathbb{Z}_p$ if and only if $F(T_1,\ldots,T_n)$ has a zero in $\mathbb{Z}/p^n\mathbb{Z}$ fo... |
3,365,112 | <p>Pretty simple, for <span class="math-container">$a,b \in \mathbb R$</span>, show that <span class="math-container">$|a-b|<\frac{|b|}{2}$</span> implies <span class="math-container">$\frac{|b|}{2}<|a|$</span>. I can see this graphically on the number line, but I can't seem to show it algebraically.</p>
<p>I'm ... | azif00 | 680,927 | <p>First, rewrite <span class="math-container">$|a-b|<\cfrac{|b|}{2}$</span> as <span class="math-container">$|b-a|<\cfrac{|b|}{2}$</span>. </p>
<p>Now, I'll use your second version of the triangle inequality (without the extra bars) and we obtain
<span class="math-container">$$|b|-|a| \leq |b-a|<\frac{|b|}{2... |
3,365,112 | <p>Pretty simple, for <span class="math-container">$a,b \in \mathbb R$</span>, show that <span class="math-container">$|a-b|<\frac{|b|}{2}$</span> implies <span class="math-container">$\frac{|b|}{2}<|a|$</span>. I can see this graphically on the number line, but I can't seem to show it algebraically.</p>
<p>I'm ... | Mohammad Riazi-Kermani | 514,496 | <p><span class="math-container">$$ |a-b|<\frac {|b|}{2} \implies -\frac {|b|}{2} < a-b<\frac {|b|}{2}$$</span></p>
<p><span class="math-container">$$\implies b-\frac {|b|}{2} <a< b+\frac {|b|}{2}$$</span></p>
<p>For <span class="math-container">$b>0$</span> we get <span class="math-container">$$\fra... |
2,713,391 | <p>I can find the answers to similar questions online, but what I'm trying to do is develop my own intuition so I can find the answers. I am quite sure I am wrong, so could you look over my reasoning?</p>
<p>If $X = (1,2,3,4,5,6,7,8)$,</p>
<blockquote>
<ol>
<li>How many strings over X of length 5?</li>
</ol>
</... | drhab | 75,923 | <p>What you call simplest solution is a correct one (and by far the most simple and elegant one).</p>
<p>If you still insist to go the other direction then your answer must be:$$\binom51\times7^4+\binom52\times7^3+\binom53\times7^2+\binom54\times7^1+\binom55\times7^0=15961$$</p>
<p>Here $\binom5{k}$ is the number of ... |
69,948 | <p>Has anyone ever created a "pairing function" (possibly non-injective)
with the property to be nondecreasing wrt to product of arguments, integers n>=2, m>=2. (We can also assume that n and m are bounded by an integer K, if useful) :</p>
<p>n m > n' m' => p(n,m) > p(n',m') </p>
<p>If yes what does it look lik... | Joel David Hamkins | 1,946 | <p>Your second property is simply inconsistent with the nature
of a pairing function, since we want $p(n,m)=p(n',m')\iff
n=n'$ and $m=m'$. That is, a pairing function must be
one-to-one on pairs, but multiplication is not, since
$2\cdot 6=3\cdot 4$.</p>
<p>The first property, however, is easy to arrange as follows
(an... |
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