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 |
|---|---|---|---|---|
4,189,417 | <p>The book I am currently reading, is written in a local language, uses the symbol <span class="math-container">$$\gtreqqless$$</span></p>
<p>Noted usage in the book (translated):</p>
<blockquote>
<p>Let <span class="math-container">$f(x) = x^3e^{-3x}$</span> for x > 0. Then what is the maximum value of <span class... | ultralegend5385 | 818,304 | <p>It is similar to <span class="math-container">$\pm$</span>, so for example, <span class="math-container">$(a\pm b)^2=a^2\pm 2ab+b^2$</span>, if you take a plus you take it everywhere, if you take a minus, you take it everywhere.</p>
<p>Likewise, if you consider <span class="math-container">$1-x>0$</span>, you hav... |
284,507 | <p>In the textbook "Topology without tears" I found the definition.</p>
<p>$(X, \tau)$ is diconnected iff there exists open sets $A,B$ with $X = A \cup B$ and $A \cap B = \emptyset$.</p>
<p>In Walter Rudin: Principles of Analysis, I found.</p>
<p>$E \subseteq X$ is connected iff it is not the union of two nonempty s... | k.stm | 42,242 | <p>They are equivalent.</p>
<p>Let <span class="math-container">$X$</span> be disconnected as in the first sense.
Then there are (nonempty) open sets <span class="math-container">$A$</span>, <span class="math-container">$B$</span> in <span class="math-container">$X$</span> such that <span class="math-container">$X = A ... |
4,200,434 | <p>I am having difficulty with what should be a routine question, Exercise 2.2.2 (b) of <em>Understanding Analysis</em> by Stephen Abbott (2015).</p>
<blockquote>
<p><strong>Exercise 2.2.2.</strong> Verify, using the definition of convergence of a sequence, that the following sequences converge to the proposed limit.</... | Community | -1 | <p>After the edit to the OP which changes the <span class="math-container">$n^2$</span> in the denominator to <span class="math-container">$n^3$</span>:</p>
<p>Observe that
<span class="math-container">$$2n^3 \leq 2n^3 + 6 = 2(n^3 + 3)$$</span>
so
<span class="math-container">$$2n^2 = \frac{2n^3}{n} \leq \frac{2(n^3+3)... |
449,457 | <p>A desk has three drawers. The first contains two gold coins, the second has two silver coins and the third has one gold and one silver coin. A drawer is selected at random and a coin is drawn at random from the drawer. Suppose that the coin selected was silver. Use Bayes's Theorem to find the probability that the ot... | angryavian | 43,949 | <p>You select a drawer and select a coin from that drawer.</p>
<p>Let $A$ denote the event that the coin that you select is silver.</p>
<p>Let $B$ denote the event that the other coin in the drawer is gold.</p>
<p>You are equally likely to pick any coin, so</p>
<p>$$P(A) = \frac{3\text{ silver coins}}{6 \text{ coin... |
2,139,951 | <p>A set S of integers $\{s_1, s_2, . . . , s_k\}$ such that 0 ∈ S is called a set of digits
modulo m if and only if any integer b can be written as
$$b = f_nm^n + f_{n−1}m^{n−1} + · · · + f_1m + f_0$$
for some $n \geq 0$ and some $f_i$ ∈ S, moreover such expression is unique assuming
that $f_n \neq 0$.
Show that:</p>
... | fleablood | 280,126 | <p>1) $S$ must contain a complete representative of modulo $m$ and $k \ge n$.</p>
<p>Suppose not: Let $a\in \mathbb Z$ be such that $a \not \equiv s_i \mod m$ for any $s_i$ in $S$. Then $a = \sum_{i=0}^n s_i m^i; s_i \in S$ is impossible as $\sum_{i=0}^n s_i m^i \equiv s_0 \mod m$ but $a \not \equiv s_0 \mod m$. So... |
1,791,815 | <p>I'm trying to find an elementary way to see that the 1st de Rham cohomology of the n-sphere is zero for $n>1$, $H^1(S^n) = 0$.</p>
<p>This is part of an attempt to find the de Rham cohomology of the n sphere generally. I have shown that $H^k(S^n) = H^{k-1}(S^{n-1})$ (for $k>1$) but to find the de Rham cohomol... | R Mary | 309,615 | <p>I think the function you are looking for is $f(x)=\int_{\gamma_x} \alpha$ where $\gamma_x$ is a path from some chosen base point $x_0$ to $x$. But then for this to be well defined you have to say that all paths between $x$ and $x_0$ are homotopic, i.e. that your sphere is simply connected, so the whole argument gets... |
1,791,815 | <p>I'm trying to find an elementary way to see that the 1st de Rham cohomology of the n-sphere is zero for $n>1$, $H^1(S^n) = 0$.</p>
<p>This is part of an attempt to find the de Rham cohomology of the n sphere generally. I have shown that $H^k(S^n) = H^{k-1}(S^{n-1})$ (for $k>1$) but to find the de Rham cohomol... | Thomas | 284,057 | <p>Write $S^n$ as the union of two open sets $U= S^n-N$, $V=S^n-S $where $N,S$ are the north and south pole. Note that the intersection $U\cap V$ is connected if $n\geq 2$. $U,V$ are contracile (diffeomorphic to $R^,$, and on each of these sets the form $\alpha$ is exact and andmits primitive $f_U, f_V$. On the inter... |
3,738,719 | <p>We got</p>
<p><span class="math-container">$$x'=\frac{t^2x^2+1}{2t^2}$$</span></p>
<p><span class="math-container">$$\frac{dx}{dt}=\frac{t^2x^2+1}{2t^2}$$</span></p>
<p><span class="math-container">$$2t^2dx=\left(t^2x^2+1\right)dt$$</span></p>
<p><span class="math-container">$$0=\left(t^2x^2+1\right)dt+\left(-2t^2dx... | user577215664 | 475,762 | <p><span class="math-container">$$x'=\frac{t^2x^2+1}{2t^2}$$</span>
<span class="math-container">$$x'=\frac{x^2}{2}+\frac{1}{2t^2}$$</span>
Change the variable
<span class="math-container">$$x=u-\dfrac 1t$$</span>
<span class="math-container">$$u'=\frac{u^2}{2}-\dfrac ut$$</span>
This is Bernoulli 's differential equat... |
302,599 | <p><strong>Problem:</strong></p>
<p>For every $n$ in $\mathbb{N}$, we consider the sets $A_{n}:=\left \{ (2n+1)\lambda :\ \lambda \in \mathbb{N}\right \}$. The question is to find $\bigcap_{n=1}^{\infty }A_{n}$, and $\bigcup_{n=1}^{\infty }A_{n}$.</p>
<p>For the intersection, I think it is the empty set, because for... | André Nicolas | 6,312 | <p>These are all the positive integers that have a factor of the shape $2n+1$. That's the collection of all positive integers that have an odd factor $\ge 3$. Every positive integer qualifies, <strong>except</strong> for the powers $1,2,4,8,\dots$ of $2$.</p>
<p>So our set is the complement of the set of powers of $... |
524,387 | <p>Having watched an <a href="http://www.youtube.com/watch?v=NircIRdNUrE" rel="nofollow">integralCALC video lesson</a>, given</p>
<p>$$w=xe^{y/z}, x = t^2, y = 1-t, z=1+2t$$
which could be rewritten as
$$w=t^2 e^\frac{1-t}{1+2t}$$</p>
<p>How does $dw/dt$ differ from $\partial w/\partial t$? Is there some intuition ... | dezign | 32,937 | <p>There's no difference. It's just that we use straight $d$s to denote derivatives of functions with respect to one variable. It may seem like the function $w$ depends on more than one variable, but the other variables all depend on $t$, so at the end of the day all the values of the function $w$ are determined by the... |
3,660,825 | <p>I know this question has been asked many times and there is good information out there which has clarified a lot for me but I still do not understand how the addition and multiplication tables for <span class="math-container">$GF(4)$</span> is constructed?</p>
<p>I'm just starting to learn about fields in general, ... | Wuestenfux | 417,848 | <p>Well, take an irreducible polynomial of degree 2 over <span class="math-container">$GF(2)$</span>. There is exactly one namely <span class="math-container">$f(x)=x^2+x+1$</span>.
Then the quotient ring <span class="math-container">$GF(2)[x]/\langle f(x)\rangle$</span> is a field, <span class="math-container">$GF(4)$... |
992,125 | <p>If I rolled $3$ dice how many combinations are there that result in sum of dots appeared on those dice be $13$?</p>
| G Cab | 317,234 | <p>the formula introduced by Masacroso applies to a bulk of different schemes in combinatorics and diophantine geometry, all stemming out from finding
<span class="math-container">$$N_{\,b} (s,r,m) = \text{No}\text{. of solutions to}\;\left\{ \begin{gathered}
0 \leqslant \text{integer }x_{\,j} \leqslant r \hfill \... |
2,918,497 | <p>From <em>Linear Algebra</em> by Friedberg, Insel, and Spence:</p>
<blockquote>
<p>Given <span class="math-container">$M_1=\begin{pmatrix} 1&0\\0 &1\end{pmatrix}$</span>, <span class="math-container">$M_2=\begin{pmatrix} 0&0\\0 &1\end{pmatrix}$</span> and <span class="math-container">$M_3=\begin{pmatr... | gt6989b | 16,192 | <p><strong>HINT</strong></p>
<p>Span is the set of all linear combinations. Note that
$$
aM_1 + bM_2 + cM_3 =
\begin{pmatrix}
a & c \\
c & a+b
\end{pmatrix}
$$</p>
<p>Can you finish finding the span?</p>
<hr>
<p><strong>HINT 2</strong></p>
<p>The way you phrased it, the span does not <em>include</em> all s... |
1,357,488 | <p>Let $E$ together with $g$ be a inner product space(over field $\mathbb R$) , $\text{dim}E=n<\infty$ and $\{e_1,\cdots,e_n\}$ is orthonormal basis of $E$ that $\{e^1,\cdots,e^n\}$ is its dual basis. Now we define $\omega:=e^1\wedge\cdots\wedge e^n$ as element of volume of $E$.</p>
<p>How can I prove that $\omega... | Joonas Ilmavirta | 166,535 | <p>The equation you wish to prove is linear in each of the vectors $u_j$ and $v_j$.
Therefore it suffices to show the identity when these vectors are basis vectors.
There are $n$ basis vectors from which we now want to choose the $n$ vectors $u_1,\dots,u_n$.
If we choose any two to be the same, then both sides of the i... |
3,822,822 | <p>Four cards are face down on a table. You are told that two are red and two are black, and you need to guess which two are red and which two are black. You do this by pointing to the two cards you’re guessing are red (and then implicitly you’re guessing that the other two are black). Assume that all configurations ar... | Anatoly | 90,997 | <p>HINT: with <span class="math-container">$4$</span> cards, of which two are red and two are black, there are <span class="math-container">$\frac{4!}{2!2!}=6$</span> possible sequences. When you choose two cards, you automatically define the other two cards and then choose a sequence.</p>
<p>Now consider the two card... |
4,634,263 | <p>I have the following problem that asks to find the shock curve for the following IVP
<span class="math-container">$$u_t + (u^2)_x = 0, \quad u(x,0) = \frac{1}{2}e^{-x},$$</span>
to then obtain a weak solution from this curve using Rankine Hugoniot condition. I'm pretty lost on how to raise the problem. In Evans ther... | Ethan Bolker | 72,858 | <p>@FD_bfa offers a <a href="https://math.stackexchange.com/questions/4634210/why-are-two-disjoint-events-defined-to-be-independent-if-one-has-zero-probabilit/4643623#4643623">good answer</a>: a definition should cover edge and degenerate cases with no modification.</p>
<p>Here's another way to look at it. Two events a... |
3,081,076 | <p>In do Carmo, one exercise gives a plane in <span class="math-container">$\mathbb R^3$</span>, <span class="math-container">$ax +by +cz+d = 0$</span>, and tells us to show that <span class="math-container">$|d|/\sqrt{a^2 + b^2 + c^2}$</span> measures the distance from the plane to the origin.</p>
<p>However, this se... | Alex J Best | 31,917 | <p>Something similar was already asked at <a href="https://math.stackexchange.com/questions/3021423/independence-of-points-on-elliptic-curve">Independence of points on Elliptic curve</a> the answers there are the same as above but maybe a little more extensive.</p>
<p>Edit: I just noticed the question asker is the sam... |
1,023,013 | <blockquote>
<p>Prove:$\forall a_1,b_1,a_2,b_2: \left|\max(a_1,b_1) - \max(a_2,b_2)\right| \le \max(\left|a_1-a_2\right|, \left| b_1-b_2 \right|) $</p>
</blockquote>
<p>How can I prove it easily? Or should I just go over each case?
All those cases make me confused!</p>
| Empy2 | 81,790 | <p>$$\frac12(a_n^2-a_n-a_{n+1}^2+a_{n+1})=\frac12(a_n-a_{n+1})(a_n+a_{n+1}-1)$$<br>
Can you show that $|\frac12(a_n+a_{n+1}-1)|<1$?</p>
|
554,869 | <p>I want to show that the following conditions are equivalent for a nonzero element $a$ in a Boolean algebra $\mathcal{B}$:</p>
<p>1) for all $x\in\mathcal{B},a\leq x$ or $a\leq x'$</p>
<p>2) for all $x,y\in\mathcal{B},a\leq x\sqcup y\Rightarrow a\leq x$ or $a\leq y$</p>
<p>3) $a$ is minimal among nonzero elements ... | egreg | 62,967 | <p>(1)$\implies$(2) If $a\le x\sqcup y$ and $a\not\le x$, then $a\le x'$. Thus $a\le(x\sqcup y)\sqcap x'=\ldots\le\ldots$</p>
<p>(2)$\implies$(3) Let $b\le a$; then $a\le b\sqcup b'$, so $a\le b$ or $a\le b'$. In the first case $a=b$; in the second case $b\le b'$, so …</p>
<p>(3)$\implies$(1) …</p>
|
90,342 | <p>Here $\mathbb{T}^n := (\mathbb{R} / \mathbb{Z})^n$ is the topological group of the n-dimensional torus and $k \in \mathbb{Z}^n$ is a non-null vector, I'm working about the subgroup</p>
<p>$S = \{x \in \mathbb{T}^n : k \cdot x = 0_{\mathbb{T}^n}\}$</p>
<p>where $\cdot$ is the scalar product. I think that $S$ is iso... | James Cranch | 14,901 | <p>It is known, in fact, that every compact connected abelian Lie group is isomorphic to a torus.</p>
<p>Your construction has a danger of producing disconnected subgroups, however: this occurs when the coordinates of $k$ are all multiples of some fixed $i>1$. (You can see this even in small dimensions).</p>
|
1,990,033 | <p>Suppose I have a point $P(x_1, y_1$) and a line $ax + by + c = 0$. I draw a perpendicular from the point $P$ to the line. The perpendicular meets the line at point $Q(x_2, y_2)$. I want to find the coordinates of the point $Q$, i.e., $x_2$ and $y_2$.</p>
<p>I searched up for similar questions where the coordinates ... | 655321 | 197,645 | <p>Since
$$ \sin a \cdot \sin b = \frac{1}{2} \left[ \cos (a - b) - \cos (a+b) \right] $$
$$ \sin a \cdot \cos b = \frac{1}{2} \left[ \sin (a + b) + \sin (a - b) \right] $$
and
$$ \sin a - \sin b = 2 \sin\left[ \frac{1}{2} (a-b) \right] \cos \left[ \frac{1}{2} (a+b) \right] $$
we can use these identities to solve
$$ 4 ... |
1,143,893 | <p>Struggling with basic AS Statistical maths, Any help would be much appreciated.</p>
<p>The two events $A,B$ are such that $P(A)= 0.65, P(A\cup B)= 0.93$</p>
<p>Evaluate $P(B)$ given that $A$ and $B$ are independent. (4 Marks) </p>
<p>Thank you.</p>
| user 1 | 133,030 | <p>$P(AUB) = P(A) + P(B) - P(A\cap B) $.</p>
|
3,342,761 | <p>I'm trying to get an equation for a solid angle of a segment of octahedron in the same vein as described in this article <a href="http://www.rorydriscoll.com/2012/01/15/cubemap-texel-solid-angle/" rel="nofollow noreferrer">cubemap-texel-solid-angle</a>. I ended up having to integrate
<span class="math-container">$$\... | Jean Marie | 305,862 | <p>All 8 facets of an octaedron are alike and interchangeable. </p>
<p>Then, the looked for solid angle is </p>
<blockquote>
<p><span class="math-container">$$\frac18(4\pi)=\frac{\pi}{2}.$$</span></p>
</blockquote>
|
1,880,150 | <p>I work in a warehouse where we take components and put them together to make a finished good product.</p>
<p>We have these values for each component:</p>
<ol>
<li><p>Quantity Required</p></li>
<li><p>Quantity Used</p></li>
<li><p>Variance (Quantity Used- Quantity Required)</p></li>
<li><p>Overage ((Variance/Quanti... | Justthisguy | 81,821 | <p>It's a good rule of thumb to never add quantities that have different units. When you add, for example, the entries in the first column, the first entry has units "number of component A", whereas the second entry has units "number of component B". So what are the units of the sum?</p>
<p>A good way to aggregate thi... |
961,304 | <p>I am studying a book and I am stagnating on a what should be a straightfoward proof:</p>
<p>Show that if $X$ is compact, $V\subset X$ is open and $x\in V$, then there exists an open set $U$ in $X$ with $x\in U\subset \bar U\subset V$</p>
<p>I don't know how to find the appropriate set $U$. I am guessing you need t... | Ilmari Karonen | 9,602 | <p>Let $X$ be a <a href="//en.wikipedia.org/wiki/Random_variable" rel="nofollow">random variable</a> uniformly distributed over the interval $[a,b]$. If $f : [a,b] \to \mathbb R$ is a (Borel) <a href="//en.wikipedia.org/wiki/Measurable_function" rel="nofollow">measurable function</a>, then $Y = f(X)$ is also a random ... |
3,267,153 | <p>I want to understand an equation I saw in a paper. If <span class="math-container">$\theta$</span> is some angle, and <span class="math-container">$\sigma_y = \begin{bmatrix} 0 & -i \\ i & 0 \end{bmatrix}$</span>, then it is stated that <span class="math-container">$e^{-i \theta \sigma_y } = R_{\theta}$</sp... | Marc Bogaerts | 118,955 | <p>A simple method (based on the chinese remainder theorem) goes as follows:</p>
<p>To solve <span class="math-container">$f \operatorname{mod} p = u$</span> and <span class="math-container">$f \operatorname{mod} q = v$</span> (where p and q are coprime polynomials):</p>
<p>put <span class="math-container">$\ \bar{p... |
1,010,820 | <p>I tried the following :</p>
<p>\begin{align}\sec\theta + \tan\theta&=4\\
\frac1{\cos\theta} + \frac{\sin\theta}{\cos\theta}&=4\\
\frac{1+\sin\theta}{\cos\theta}&=4\\
\frac{1+\sin\theta}4&=\cos\theta\end{align}</p>
<p>now don't know how to evaluate further ?</p>
| user72272 | 72,272 | <p>one way is to use the half angle formule like @Mark Bennet suggested. then you have
$$\sec(\theta)=\frac{1+t^2}{1-t^2},\qquad\tan(\theta)=\frac{2t}{1-t^2}$$ where $t=\tan\left(\frac{\theta}{2}\right)$. This results in the following equationt
$$\frac{1+2t+t^2}{1-t^2}=\frac{(1+t)^2}{(1-t)(1+t)}=\frac{1+t}{1-t}=4\\t=\f... |
2,717,842 | <p>I want to show that if $X$ is a locally connected topological space, $A\subseteq X$ is a subspace and $f:X \rightarrow A$ is continuous such that $f|_{A} = Id_{A}$, then $A$ must be locally connected as well.</p>
<p>My progress so far:</p>
<p>Take $U\subseteq A$ $A-$open and $x\in U$. Since $f^{-1} (U)$ is $X-$ope... | William Elliot | 426,203 | <p>Let K be the nonnegative x and y axis.<br>
Project every point of the 1st quadrant at a 45 degree angle o K.<br>
Thus K is a retract of the first quadrant.<br>
In general, every V is retract of the space between the two lines. </p>
<p>Let C be the comb space of all lines from (0,0) to (1,1/n)<br>
for all n in N a... |
1,976,001 | <p>The statement goes as following: if $3 \mid 2a$, then $3 \mid a$ and $a$ is an integer. In my approach, I used prime factorization, but is this actually valid? This was my approach:</p>
<p>$$3 \mid 2a \implies 2a = 2 \cdot 3 \cdot k, k \in \mathbb{N}$$</p>
<p>$$\frac{2a}{2} = \frac{2\cdot 3 \cdot k}{2}$$</p>
<p>$... | Hagen von Eitzen | 39,174 | <p>Note that $2a=3k$ implies
$$ a=3a-2a=3(a-k)$$</p>
|
2,347,995 | <p>Not sure what I'm doing wrong.</p>
<p>Here's my work:</p>
<p>Expressing the first part $A \setminus (B\setminus C)$ using logical symbols:</p>
<p>$A \land \neg(B \land \neg C)$ becomes</p>
<p>$A \land \neg B\lor C$ (De Morgan's law)</p>
<p>While the second expression $(A \setminus B) \cup (A \cap C)$ is </p>
<... | Michael Rozenberg | 190,319 | <p>Since $\cos$ is an even function, we can assume $0<x<1$.
we need to prove that $$\frac{2\sin^2\frac{x}{2}}{x^2}\geq\frac{1}{4}$$ or
$$\sin\frac{x}{2}\geq\frac{1}{\sqrt2}\cdot\frac{x}{2}.$$
Let $f(x)=\sin{x}-\frac{1}{\sqrt2}x$, where $0<x<0.5$.</p>
<p>Thus, $$f'(x)=\cos{x}-\frac{1}{\sqrt2}>\cos\frac{1... |
311,950 | <p>I'm trying to show that any subset bounded of $\Bbb{R}^k$ is totally bounded.</p>
<p>Here is what I did: </p>
<p>(1)A subset of a totally bounded Set is bounded:</p>
<p>Proof: Let $X$ be a totally bounded subset and $Y\subset X$ then there exists an $\epsilon /2$-net $\{x_1,x_2,..,x_n\}$ and $X\subset \displaysty... | Emanuele Paolini | 59,304 | <p>If $B$ is bounded in $\mathbb R^n$ it is contained in some cube $[-R,R]^n$. Then for any $\epsilon>0$ just consider all balls of radius $\epsilon$ centered in the set
$$
(\frac \epsilon 2\mathbb Z)^n \cap [-R,R]^n
$$
these balls are a finite number (less then $(4R/\epsilon+1)^n$) and cover the whole cube $[-R,R... |
1,508,753 | <p>Show that for $x,y\in\mathbb{R}$ with $x,y\geq 0$, the arithmetic mean-quadratic mean inequality
$$\frac{x+y}{2}\leq \sqrt{\frac{x^2+y^2}{2}}$$
holds.</p>
<p>After my calculations I'll get: </p>
<p>$$-x^2+2xy-y^2$$
which can't be $\leq 0$.</p>
| Yes | 155,328 | <p>Initially we may guess that
$$
\frac{1}{1-x} \to -1
$$
as $x \to 2$. To prove this, note that we have $x\neq 1$ only if
$$
\bigg| \frac{1}{1-x} - (-1) \bigg| = \bigg|\frac{2-x}{1-x} \bigg| = \bigg|\frac{x-2}{x-1} \bigg|;
$$
we have $0 < |x-2| < 1/2$ only if $||x-1|-1| \leq |x-2| < 1/2$, only if $1/2 < |... |
1,508,753 | <p>Show that for $x,y\in\mathbb{R}$ with $x,y\geq 0$, the arithmetic mean-quadratic mean inequality
$$\frac{x+y}{2}\leq \sqrt{\frac{x^2+y^2}{2}}$$
holds.</p>
<p>After my calculations I'll get: </p>
<p>$$-x^2+2xy-y^2$$
which can't be $\leq 0$.</p>
| Ian Miller | 278,461 | <p>The definition of convergence says a sequence $\{a_n\}$ converges to a limit $L$ if given an $\epsilon$ however small there exists $N(\epsilon)$ such that $|a_n-L|<\epsilon$ for all $n>N(\epsilon)$.</p>
<p>So let $x_n$ be values getting progressively closer to 2. I.e. For any $\epsilon_2$ there exists $N_2(\e... |
48,679 | <p>I've been going through Fermats proof that a rational square is never congruent. And I've stumbled upon something I can't see why is. Fermat says: ''If a square is made up of a square and the double of another square, its side is also made up of a square and the double of another square'' Im having difficulties unde... | Phil Wild | 7,842 | <p>Another example from theoretical high-energy physics I've encountered: sometimes when physicists have some equation of motion for an arbitrary number $N$ of particles with positions $x_i$, e.g. something of the form $\frac{1}{N}\sum_i f(x_i) + \frac{1}{N^2}\sum_{ij} g(x_i, x_j) = 0$, they wish to know what the solut... |
48,679 | <p>I've been going through Fermats proof that a rational square is never congruent. And I've stumbled upon something I can't see why is. Fermat says: ''If a square is made up of a square and the double of another square, its side is also made up of a square and the double of another square'' Im having difficulties unde... | Peter Shor | 2,294 | <p>The <a href="http://en.wikipedia.org/wiki/Replica_trick" rel="noreferrer">replica method</a> and the <a href="http://en.wikipedia.org/wiki/Cavity_method" rel="noreferrer">cavity method</a> have been used by physicists to calculate thermodynamic quantities in various statistical mechanics settings (including quite a ... |
3,975,162 | <p>Let <span class="math-container">$(X, A, µ)$</span> be a positive metric space. If <span class="math-container">$\mu(X) < \infty$</span> and <span class="math-container">$(A_n)_{(n \in N^*)},A \in X$</span> <br />
show that if <span class="math-container">$\mu(A\bigtriangleup A_n)\rightarrow 0$</span> then <span ... | Infinity_hunter | 826,797 | <p>Just observe that <span class="math-container">$y$</span> does not divide any number in the form of <span class="math-container">$ky+ 1$</span> since it divides <span class="math-container">$ky$</span>. Now note that <span class="math-container">$(y+1)^2 = 1 + ky$</span> where <span class="math-container">$k = y + ... |
1,606,709 | <p>I am studing Kähler differentials and I tried to understand the geometric motivation behind these settings. What I do not understand is the role which plays the diagonal in all these theory. The cotangent sheaf is later defined in terms of the diagonal map. Why is this geometrically interesting? I tried to write a s... | DeepSea | 101,504 | <p><strong>Hint</strong>: Let $a =\dfrac{1}{2}$ and $b$ be the expression you are trying to calculate, then $b^2 = a + b$. You can use quadratic equation to finish.</p>
|
1,022,184 | <p>A game is played as follows: A random number $X$ is chosen uniformly from $[0,
1]$. Then a sequence $Y_1, Y_2,\ldots$ of random numbers is chosen independently
and uniformly from $[0, 1]$. The game ends the first time that $Y_i > X$. You are
then paid $(i-1)$ dollars. What is a fair entrance fee for this game?</p... | alexjo | 103,399 | <p>Let $Z$ represent the payment. Then
$$\Bbb{P}(Z = k|X = x) = \Bbb{P}(Y_1 \le x, Y_2 \le x, \ldots , Y_k \le x, Y_{k+1} > x)
= x^k(1 − x) .$$
Therefore,
$$\Bbb{P}(Z = k) = \int_0^1 x^k(1 − x)\operatorname{d}x=\left[\frac{x^{k+1}}{k+1}-\frac{x^{k+2}}{k+2}\right]_0^1=\frac{1}{k+1}-\frac{1}{k+2}=\frac{1}{(k+1)(k+2)}.... |
3,565,632 | <blockquote>
<p>Given <span class="math-container">$x_1 := a > 0$</span> and <span class="math-container">$x_{n+1} := x_n + \frac{1}{x_n}$</span> for <span class="math-container">$n \in \mathbb{N}$</span>, determine whether <span class="math-container">$(x_n)$</span> converges or diverges.</p>
</blockquote>
<p>Si... | Alex Ortiz | 305,215 | <p>Here is a suggestion:</p>
<p>Try proving the Liebniz rule for differentiating the product of two functions on <span class="math-container">$\mathbb R$</span>:</p>
<p><span class="math-container">$$
(fg)^{(n)}(x) = \sum_{k=0}^n{n\choose k}f^{(k)}(x)g^{(n-k)}(x),
$$</span>
and apply this to <span class="math-contain... |
2,164,333 | <p>Let $T$ be linear transformation from $V$ to $W$. I know how to prove the result that nullity$(T) = 0$ if and only if $T$ is an injective linear transformation. But I still don't intuitively understand why the kernel only containing the zero vector means that $T$ is injective, and vice versa. In contrast, the relati... | Maczinga | 411,133 | <p>Because if $\ker(T)\ne\{0\}$ then you have several manners to produce the null vector. Hence you cannot be injective.</p>
|
2,164,333 | <p>Let $T$ be linear transformation from $V$ to $W$. I know how to prove the result that nullity$(T) = 0$ if and only if $T$ is an injective linear transformation. But I still don't intuitively understand why the kernel only containing the zero vector means that $T$ is injective, and vice versa. In contrast, the relati... | Leif | 413,652 | <p>If you have for example some linear mapping given by the matrix $A$ such that $f(X) = AX$, you can look at the Kernel. If this mapping is injective, the only vector from $X$ that would give me zero would be a trivial vector, otherwise $X$ (it's actually a null space of matrix $A$) contains more vectors that could gi... |
395,850 | <p>Is Dirac delta a function? What is its contribution to analysis? </p>
<p>What I know about it:
It is infinite at 0 and 0 everywhere else. Its integration is 1 and I know how does it come.</p>
| fgp | 42,986 | <p>The delta <em>distribution</em> is not a function from <span class="math-container">$\mathbb{R} \to \mathbb{R}$</span>. For <em>any</em> such function <span class="math-container">$f$</span> that is zero except at finitly many points, as the delta distribution is, <span class="math-container">$\int f= 0$</span> (thi... |
338,638 | <p>There is a wonderful series of articles by Flajolet et. al. about Mellin Transforms and the asymptotic analysis of generating functions. In particular, on page 45 of the article <a href="http://algo.inria.fr/flajolet/Publications/mellin-harm.pdf" rel="nofollow noreferrer">Mellin Transforms and Asymptotics: Harmonic ... | reuns | 84,768 | <blockquote>
<p><span class="math-container">$$F(s)=\sum_{k}\lambda_{k}\mu_{k}^{-s}$$</span> extends meromorphically with finitely many poles in each strip because <span class="math-container">$$\Gamma(s)F(s)= \int_0^\infty f(t)t^{s-1}dt, \qquad f(t)=\sum_{k}\lambda_{k} e^{-t \mu_{k}}$$</span> where <span class="math... |
121,431 | <p>I need a good reference for the basic definitions of the dual of locally compact group (not necessarily abelian), its natural topology, $\sigma$-algebra, and the Plancherel measure on it (when they are defined). This topic seems pretty standard to me, but when I needed a basic reference on this (both to check my mem... | Uwe Franz | 30,364 | <p>Hartmut Fuehr's book (Abstract harmonic analysis of continuous wavelet transforms,
Springer Lecture Notes in Mathematics, Nr. 1863, 2005, X, 193 p., Softcover
ISBN: 3-540-24259-7), contains a "-reasonably self-contained- exposition of Plancherel theory", see also <a href="http://www.matha.rwth-aachen.de/~fuehr/book... |
2,069,849 | <p>For a given $x\in\left\{ 0,1\right\} ^{n}$, define $x_{\setminus k}=\left(x_{1},\dots,x_{k-1},x_{k+1},\dots,x_{n}\right)$. [Is there standard notation for this?]</p>
<p><strong>Prove or disprove the following claim:</strong></p>
<p>Let $X$ be some random variable with values in $\left\{ 0,1\right\} ^{n}$. There ex... | Will Sherwood | 124,497 | <p>Throughout the tune, we receive 12 partridges in pear trees, 22 turtle doves, 30 french hens, etc. So to total everything up, I think this sum is more compact:
$$\sum_{n=1}^{12}(13-n)f(n)$$</p>
|
1,161,183 | <p>Find the number of ways of choosing three numbers from the set ${1,2,3,…,20}$, so that the sum of the three numbers is divisible by $3$. I wrote $a+b+c=3k$, where $k=1,2,...20$, till $k=6$, there were no problems, but then restrictions started($a<21$). Even if I come to know of the correct method from here, the s... | Théophile | 26,091 | <p>The three numbers might all be multiples of 3, in which case there are $6 \choose 3$ ways to pick them.</p>
<p>There are several other possibilities: they might all be congruent to 1 modulo 3, for example. Find all such possibilities and count accordingly.</p>
|
1,161,183 | <p>Find the number of ways of choosing three numbers from the set ${1,2,3,…,20}$, so that the sum of the three numbers is divisible by $3$. I wrote $a+b+c=3k$, where $k=1,2,...20$, till $k=6$, there were no problems, but then restrictions started($a<21$). Even if I come to know of the correct method from here, the s... | Asinomás | 33,907 | <p>We only need concern ourselves with congruence $\bmod 3$.</p>
<p>There are $7$ numbers that are $1\bmod 3$</p>
<p>There are $7$ numbers that are $2\bmod 3$</p>
<p>There are $6$ numbers that are $0\bmod 3$</p>
<p>We now count the ways to add to $0\bmod 3$:</p>
<hr>
<p>Three zeros:</p>
<p>$0+0+0$ (There are $\b... |
4,577,925 | <p>Which expression is larger,
<span class="math-container">$$
99^{50}+100^{50}\quad\textrm{ or }\quad 101^{50}?
$$</span></p>
<p>Idea is to use the Binomial Theorem:</p>
<p>The right hand side then becomes
<span class="math-container">$$
101^{50}=(100+1)^{50}=\sum_{k=0}^{50}\binom{50}{k}1^{50-k}100^k=100^{50}+\sum_{k=... | robjohn | 13,854 | <p>The binomial theorem says
<span class="math-container">$$
101^{50}=\sum_{k=0}^{50}(+1)^k\binom{50}{k}100^{50-k}
$$</span>
and
<span class="math-container">$$
99^{50}=\sum_{k=0}^{50}(-1)^k\binom{50}{k}100^{50-k}
$$</span>
Furthermore, <span class="math-container">$100^{50}=2\binom{50}{1}100^{49}$</span>, which is the... |
4,577,925 | <p>Which expression is larger,
<span class="math-container">$$
99^{50}+100^{50}\quad\textrm{ or }\quad 101^{50}?
$$</span></p>
<p>Idea is to use the Binomial Theorem:</p>
<p>The right hand side then becomes
<span class="math-container">$$
101^{50}=(100+1)^{50}=\sum_{k=0}^{50}\binom{50}{k}1^{50-k}100^k=100^{50}+\sum_{k=... | cr001 | 254,175 | <p><span class="math-container">$$\sum_{k=0}^{49}\binom{50}{k}100^k$$</span><span class="math-container">$$=50\cdot100^{49}+1225\cdot100^{48}+ ...$$</span><span class="math-container">$$>62\cdot100^{49}$$</span><span class="math-container">$$>99^{50}$$</span></p>
<p>The last inequality comes from <span class="ma... |
19,294 | <p>Instead of taking a one to one correspondence meaning each set has the same number of elements. why not use the concept of coverings of topology? The irrational numbers covers the whole numbers but not vice versa?</p>
<p>A hierarchy of coverings instead of infinities. Wouldn't that make those infinities more mana... | Konrad Waldorf | 3,473 | <p>A nice point of view is to consider principal bundles with structure group $B\mathbb{C}^\times$. One can probably take any abelian Lie group instead of $\mathbb{C}^\times$. Principal $B\mathbb{C}^\times$-bundles are one way to give a precise meaning to what you are calling "patching together classifying spaces". </p... |
2,456,561 | <p>A bowl contains 16 chips, of which 6 are red, 7 are white and 3 are blue. If four chips are taken at random and without replacement, find the probability that there is at least 1 chip of each colour.</p>
<p>Why is the answer not the following:</p>
<p><a href="https://i.stack.imgur.com/25ImM.png" rel="nofollow nore... | JMoravitz | 179,297 | <p>Your thought process seems to be the following:</p>
<ul>
<li>Pick the red chip</li>
<li>Pick the white chip</li>
<li>Pick the blue chip</li>
<li>Pick one more chip</li>
</ul>
<p>When you picked your one more chip however, the final chip will be an additional chip of an earlier color. Supposing that the chips were... |
3,056,031 | <p>From the first chapter of Arfken's Mathematical Methods for physicists (rotation of the coordinate axis):</p>
<blockquote>
<p>To go on to three and, later, four dimensions, we find it convenient to use a more compact notation. Let
<span class="math-container">\begin{equation}
x → x_1,
\textbf{ }
y → x_2
\end{e... | Manraj | 584,828 | <p>You can verify-</p>
<p><span class="math-container">$\frac{\partial x_1'}{\partial x_1}=a_{11},\frac{\partial x_1'}{\partial x_2}=a_{12},\frac{\partial x_2'}{\partial x_1}=a_{21}\ and\ \frac{\partial x_2'}{\partial x_2}=a_{22}$</span></p>
|
1,856,171 | <p>Functional</p>
<p>$F(x)=(x^2-x+1)Q(x)+x-1$</p>
<p>$G(x)=(x^2-x+1)T(x)+x+1$</p>
<p>$F(x).G(x)=(x^2-x+1)H(x)+ax+b$</p>
<p>find a and b</p>
| G Cab | 317,234 | <p>Suppose you mean a vector ($\mathbf{v}$) "specular" to the given one ($\mathbf{u}$) with respect to the line $L$.
Since the line is passing through the origin, let's fix our mind on taking the vectors as orientated segments from the origin.<br>
If you do the cross product $\mathbf{w} = \mathbf{u} \times \mathbf{L}
$... |
3,123,681 | <p>let us consider following problem taken from book</p>
<p><em>An appliance store purchases electric ranges from
two companies. From company A, 500 ranges are
purchased and 2% are defective. From company B,
850 ranges are purchased and 2% are defective.
Given that a range is defective, find the probability
that it... | DXT | 372,201 | <p>We have <span class="math-container">$$x-\frac{x^3}{3}<\tan^{-1}(x)<x,x>0$$</span></p>
<p>So <span class="math-container">$$\sum^{n}_{k=1}\bigg[\frac{1}{n+k}-\frac{1}{3(n+k)^3}\bigg]<\sum^{n}_{k=1}\tan^{-1}\bigg(\frac{1}{n+k}\bigg)<\sum^{n}_{k=1}\frac{1}{n+k}$$</span></p>
<p>Now Using Limit <span cl... |
629,340 | <p>Is it true that if for each partition of a graph G's vertices into two non empty sets there is an edge with end points in both sides then G is connected? Intuitively this seems true to me. But I cannot prove this. I would very much appreciate some assistance. Thanks </p>
| Nick Peterson | 81,839 | <p>Yes, it is!</p>
<p>A graph is connected precisely when there is a path between any two vertices; however, another characterization is this:</p>
<p>Let $R$ be the relation on the vertex set $V$ defined by $vRw$ iff there is a path between $v$ and $w$. It is not too hard to show that this is an equivalence relation... |
14,568 | <p>The Poisson summation says, roughly, that summing a smooth $L^1$-function of a real variable at integral points is the same as summing its Fourier transform at integral points(after suitable normalization). <a href="http://en.wikipedia.org/wiki/Poisson_summation_formula">Here</a> is the wikipedia link.</p>
<p>For m... | Darsh Ranjan | 302 | <p>In what follows, I'll use the convention $$ \hat{f}(\xi) = \int_{-\infty}^{\infty} f(x)e^{-2\pi i x \xi}dx,$$ so that $$ f(x) = \int_{-\infty}^{\infty} \hat{f}(\xi)e^{2\pi i x \xi}d\xi.$$</p>
<p>I like the following interpretation of Poisson summation, which also gives a generalization: Consider the Dirac comb dist... |
14,568 | <p>The Poisson summation says, roughly, that summing a smooth $L^1$-function of a real variable at integral points is the same as summing its Fourier transform at integral points(after suitable normalization). <a href="http://en.wikipedia.org/wiki/Poisson_summation_formula">Here</a> is the wikipedia link.</p>
<p>For m... | paul garrett | 15,629 | <p>A variant on @Darsh Ranjan's argument: since $u=\sum_{n\in\mathbb Z}\delta(x-n)$ is invariant under translation by $\mathbb Z$, and is annihilated by multiplication by $e^{2\pi inx}-1$ for all $n\in \mathbb Z$ (this captures the order-zero aspect!) it suffices to show that there is a unique (up to scalar multiples) ... |
4,509,098 | <p><span class="math-container">$△ABC$</span> - triangle with <span class="math-container">$45°, 105°$</span> and <span class="math-container">$30°.$</span></p>
<p>Perimeter of triangle is <span class="math-container">$\sqrt6 + 2\sqrt3 - \sqrt2.$</span></p>
<p>Find the longest side?</p>
<p>Do you have any ideas? Seems ... | abcdefu | 1,080,275 | <p>The longest side will be the side that is opposite to the largest angle of the triangle.</p>
<p>You haven't defined the angles clearly so I can't tell which side is largest but the approach remains same.</p>
<p>No need of any law of sines or cosines.</p>
<p><strong>EDIT</strong></p>
<p>Let the sided opposite to <spa... |
3,651,173 | <p>I am trying to use generating function to find closed form formula for this expression:
<span class="math-container">$$\sum_{n \geq 0} \frac{n^2+4n+5}{n!}$$</span>
but I don't how to start this. any suggestion or hints. thank you</p>
| failedentertainment | 583,480 | <p>Recall that <span class="math-container">$e^x = \sum_{n \geq 0} \frac{x^n}{n!}$</span> so that <span class="math-container">$$xe^x = \sum_{n\geq 0} \frac{x^{n+1}}{n!} = \sum_{n \geq 0} (n+1) \frac{x^{n+1}}{(n+1)!} = \sum_{n \geq 1} n \frac{x^n}{n!}$$</span> where the latter equality follows by reindexing. Similarly,... |
2,440,785 | <p>if $f: X \to X$ is continuous where $X$ is a topological space with a cofinite topology, then:</p>
<p>$$(i) \ f^{-1}(x) \text{ is finite for all $x$} \\ \text{or} \\ (ii) \ f \text{ is constant}$$</p>
<p>My approach:</p>
<p>I couldn't build up a proper approach here to be honest. I believe we need to use the fac... | Community | -1 | <p>If the pre-image of every point is finite, you are done.</p>
<p>Otherwise there is a point with infinite pre-image. But single points are closed, so the pre-image must be closed. The only closed set that is NOT finite is all of $X$. So now you have a point whose pre-image is all of $X$, which means $f$ is constant.... |
177,915 | <p>I have two inequalities that I can't prove:</p>
<ol>
<li>$\displaystyle{n\choose i+k}\le {n\choose i}{n-i\choose k}$</li>
<li>$\displaystyle{n\choose k} \le \frac{n^n}{k^k(n-k)^{n-k}}$</li>
</ol>
<p>What is the best way to prove them? Induction (it associates with simple problems, but sometimes I find it difficult... | Davide Giraudo | 9,849 | <ol>
<li><p>Let $S_j^n:=\{A\subset [n],|A|=j\}$. Let $S'\subset S_k^{n-i}\times S_i^n$ which consists of pairs of disjoint subsets. The map
$$\varphi\colon S'\to S_{i+k}^n,\varphi(A,B)=A\cup B$$
is onto, hence $|S'|\geq |S_{i+k}^n|=\binom n{i+k}$. Since $S'\subset S_k^{n-i}\times S_i^n$, its cardinal is smaller than t... |
3,422,505 | <p>The problem is as follows:</p>
<blockquote>
<p>In a certain shopping mall which is many stories high there is a glass
elevator in the middle plaza. One shopper ridding the elevator notices
a kid drops a spheric toy from the top of the building where is
located the toy store. The shopper riding the elevator ... | Doug M | 317,162 | <p>If the elevator is moving at a constant velocity, the acceleration due to gravity is the same inside the elevator as outside.</p>
<p><span class="math-container">$a = -10\frac {m}{s^2}$</span></p>
<p>Velocities will be different due to the different frames of refference.</p>
<p>For the person in the elevator the ... |
176,033 | <p>What is the highest dimension for which the space of reduced positive definite quadratic forms (or the fundamental domain of $SL_n(\mathbb{R})/SL_n(\mathbb{Z})$) has been explicitly calculated? I know it's been done for $n \leq 7$ in 1970's. Has there been any progress since then? If not, is it because there is some... | JHM | 20,516 | <p>I very much enjoy Douglas Grenier's "Fundamental Domains for the General linear group", Pacific J. of Math., Vol 132, 2, 1988, wherein fundamental domains $F_n$ for the symmetric spaces $S_n \simeq SL(n,R) / SO(n)$ of $n \times n$ positive definite unimodular symmetric matrices are constructed inductively. That is, ... |
2,016,781 | <p>Premise:</p>
<ol>
<li>Let P(n) return the real part of the nth nontrivial root of the zeta function</li>
<li>The first several roots of the zeta function are already known </li>
</ol>
<p>Proof:</p>
<ol>
<li>Pick any integer n. For example, 1.</li>
<li>Solve for P(n)</li>
<li>Solve for P(n+1)</li>
<li>Both (2) and... | 5xum | 112,884 | <p>The problem with this proof is that it isn't a proof. Step number 4 claims something that was not proven in previous steps. We don't know what $f$ is, that's the whole point.</p>
<p>Also, even if this "proof" was correct, this is not how a proof by induction. A proof of induction would look like:</p>
<ol>
<li>Prov... |
1,752,106 | <p>You roll $n$ fair dice. Let X be the number of pairs of dice that sum to $7$.
Write $X$ as a sum of indicator variables and find $E[X]$</p>
<p>This is how I approached the problem</p>
<p>Let $S$ be all the strings length n on [6].</p>
<p>Let $X(S)$ be the number of ways to add up the numbers on the n dice that ad... | Graham Kemp | 135,106 | <p>You wish to find the expected count of pairs of dice whose results add up to seven.</p>
<p>Eg: If the results are <code>1,2,3,4,3,6,2,5,4</code> then there are $7$ ways to select pairs which add up to seven, because there is one <code>1</code> and one <code>6</code>, two <code>2</code> and one <code>5</code>, two <... |
789,043 | <p>Given a finite sequence of decimal digits $a_1,a_2,...,a_n$ prove that there exists a natural number $m$ such that decimal representation of $2^m$ starts with that sequence of digits.</p>
<p>Thanks for your help :)</p>
| Community | -1 | <p>It is not a proof, just a try to find by approximations the first solutions, without the guarantee that they exist.</p>
<p>I tried a small script to select candidates which were submited next to a <a href="http://www.wolframalpha.com/input/?i=2%5E2621" rel="nofollow">large digital computer</a> for example manually.... |
13,937 | <p>Prove that $$\sum_{\substack{x,y \in \mathbb N \\ ax-by \ne 0 }}\frac1{|ax-by|xy}$$ converges, where $a, b \in \mathbb N$
</p>
<p>This 2-D problem can be proved by an integral test. I'm looking for some other proofs that can be easily generalized to higher dimensional cases like below,</p>
<p>Prove that $$\sum_{\s... | Jack Schmidt | 583 | <p>For abelian groups, the ring End(A) is very important. As far as non-abelian groups A go, End(A) is not even (usually considered) a group.</p>
<p>"Adding" homomorphisms doesn't work in the non-abelian case.</p>
<p>If you define (f+g)(x) = f(x) + g(x), then (f+g)(x+y) = f(x+y) + g(x+y) = f(x) + f(y) + g(x) + g(y),... |
1,922,417 | <p>This is not for the Maths part of the General GRE. This is for the GRE <a href="https://www.ets.org/gre/subject/about" rel="noreferrer">Subject Test</a> in <a href="https://www.ets.org/gre/subject/about/content/mathematics" rel="noreferrer">Maths</a>. Feel free to add or comment.</p>
<ul>
<li><p><a href="https://ma... | BCLC | 140,308 | <h2>Top web resource:</h2>
<p><a href="https://en.wikipedia.org/wiki/DM_Ashura" rel="noreferrer">DM Ashura</a> aka <a href="https://www.mathsub.com/" rel="noreferrer">Bill Shillito</a>: <a href="https://www.mathsub.com/" rel="noreferrer">https://www.mathsub.com/</a></p>
<hr />
<h2>Official resources:</h2>
<p>(Official)... |
1,903,762 | <blockquote>
<p>Largest number that leaves same remainder while dividing 5958, 5430 and 5814 ?</p>
</blockquote>
<hr>
<p>$$5958 \equiv 5430 \equiv 5814 \pmod x$$
$$3\times 17\times 19 \equiv 5\times 181\equiv 3\times 331\pmod x$$
$$969 \equiv 905\equiv 993\pmod x$$</p>
<p>After a bit of playing with the calculato... | B. Goddard | 362,009 | <p>Since $5958 \equiv 5430 \mod{x}$, then $x | 5958 -5430 = 528$. Likewise $x |
5958 - 5814 = 144$ and $x | 5814 - 5430 = 384$. So $x$ is a common divisor of
$528,$ $144,$ and $384$. So you want $x=\gcd(528, 144, 384) = 48$.</p>
|
6,534 | <p>I apologize if this isn't the right place to ask this question.</p>
<p>Two features of stackexchange would be very useful for a personal math blog -- Latex works great, and comments and replies can be voted upon.</p>
<p>Is there any way to use the stackexchange functionality in a personal math blog?</p>
| GNUSupporter 8964民主女神 地下教會 | 290,189 | <h1>Short answer</h1>
<p>View my profile for a quick start, and <a href="https://vincenttam.gitlab.io/bhdemo" rel="nofollow noreferrer">my minimal demo site</a> (<a href="https://gitlab.com/vincenttam/bhdemo" rel="nofollow noreferrer">source</a>).</p>
<h1>Complete answer</h1>
<p>I recommend a <em>static blog</em> power... |
346,855 | <p>Let's start with the text of the assignment:</p>
<p><em>Let $V$ be a (finite-dimensional) vector space, and let $L<V$ be a subspace of $V$. Prove that $[a] = a + L$ holds for every vector $a\in{V}$.</em></p>
<p>(where $[S]$ in general is the set of all possible linear combinations of vectors from $S$, a.k.a. th... | lel | 60,013 | <p>To answer my own question... it was a misunderstanding after all. My reasoning is correct but the $[]$ brackets were actually representing simply the equivalence class for a relation mentioned earlier, but I missed it!</p>
|
2,960,054 | <p>We are told to proof the Minkowski inequality specialized to sequences:</p>
<p><span class="math-container">$$
\left(\sum_{i=1}^n|a_i+b_i|^p\right)^{1/p}\leq
\left(\sum_{i=1}^n|a_i|^p\right)^{1/p}
+\left(\sum_{i=1}^n|b_i|^p\right)^{1/p}
$$</span></p>
<p>when <span class="math-container">$p\geq1$</span> </p>
<p>We... | River Li | 584,414 | <p>Denote
<span class="math-container">$$S := \sum_{i=1}^n |a_i|^p, \quad \mathrm{and}\quad T := \sum_{i=1}^n |b_i|^p.$$</span></p>
<p>If <span class="math-container">$S = 0$</span> or <span class="math-container">$T = 0$</span>, the desired inequality is true.
In the following, assume that <span class="math-container"... |
9,304 | <p>I have the following question: I have a file that has structure:</p>
<pre><code>x1 y1 z1 f1
x2 y2 z2 f2
...
xn yn zn fn
</code></pre>
<p>I can easily visualize it with <em>Mathematica</em> using <code>ListContourPlot3D</code>. But could you please tell me how I can plot contour plot for this surface? I mean with t... | halirutan | 187 | <p>Ok, lets give this a try. @Mr.Wizard already showed you, how you can use <code>Interpolate</code> to make a function from your discrete data and since you didn't provide some test-data, I just assume we are speaking of an isosurface of a function $f(x,y,z)=c$ which is defined in some box in $\mathbb{R}^3$. </p>
<p>... |
2,909,036 | <p>I'm reading a book and there is some strange proof (strange for me) of the theorem that within each interval, no matter how small, there are rational points. Proof: we need only take a denominator $n$ large enough so that the interval $[0, \frac{1}{n}]$ is smaller than the interval $[A, B]$ in question; then at leas... | hamam_Abdallah | 369,188 | <p>$\Bbb R$ is Archimedian thus</p>
<p>$$\exists n>0 \; : \; n(B-A)>1$$</p>
<p>therefore</p>
<p>$$nA<nA+1<nB$$</p>
<p>Let $m=\lfloor nA+1\rfloor$
then</p>
<p>$$nA<m\le nA+1<nB$$</p>
<p>finally</p>
<p>$$A<\frac{m}{n}<B$$</p>
|
235,145 | <p>I have the following 3D-model:</p>
<pre><code>RevolutionPlot3D[Sqrt[E^-x (1 + E^x)^2], {x, 0, 4},
RevolutionAxis -> {1, 0, 0}]
</code></pre>
<p>Is there a way to make this model into a solid so that I can export it to let it be printed by a 3D printer. Now it is kind of hollow from the inside but I want it to b... | cvgmt | 72,111 | <p>Revolve three parametric curves. <code>{x, f[x]}</code> , <code>{0, y}</code> and <code>{4, y}</code></p>
<pre><code>SetOptions[RevolutionPlot3D, Mesh -> False];
f[x_] := Sqrt[E^-x (1 + E^x)^2];
a = RevolutionPlot3D[{x, f[x]}, {x, 0, 4},
RevolutionAxis -> {1, 0, 0}, PlotStyle -> Red];
b = RevolutionPlot... |
105,158 | <p>Wigner's D-matrices is defined as $D_{m'm}^j(\phi,\theta,\psi)=\langle jm'|R(\phi,\theta,\psi)|jm\rangle$; it produces a square matrix (indices $m$ and $m'$) of dimension $2j+1$. It is also the case that these matrices (for all positive $j$ multiple of $1/2$) are a representation of the rotation group $SU(2)... | J. M. ain't a mathematician | 498 | <p>For $n=-1,0$, finding the roots of the spherical Bessel functions $j_n(x)$ and $y_n(x)$ is somewhat easy, since:</p>
<p>$$\begin{array}{ll}
j_{-1}(x)&=&\frac{\cos\,x}{x}&\quad&y_{-1}(x)&=&\mathrm{sinc}(x)\\
j_0(x)&=&\mathrm{sinc}(x)&\quad&y_0(x)&=&-\frac{\cos\,x}{x}... |
219,165 | <p>I have the following equation which I want to solve it for r:</p>
<pre><code>TGB5 = (-((2 q^2)/r^5) + 2/r + (16 P \[Pi] r)/3)/(4 \[Pi]);
rlarge5 = Last[r /. Solve[TGB5 == T, r, Reals]] // Normal
</code></pre>
<p>The answer is a root object:</p>
<blockquote>
<p>Root[-3 q^2 + 3 #1^4 - 6 [Pi] T #1^5 + 8 P [Pi] #1... | Kheeyal | 1,894 | <p>In Mathematica 12.1, the following code gives the right answer:</p>
<pre><code>r /. Last[AsymptoticSolve[TGB5 == T, r, {T, Infinity, 8}]] // Expand
</code></pre>
<p>which is:</p>
<pre><code> -((4 P^2)/(9 \[Pi]^3 T^5)) + (128 P^4 q^2)/(81 \[Pi] T^5) - P/(
3 \[Pi]^2 T^3) - 1/(2 \[Pi] T) + (3 T)/(4 P)
</code></pre>... |
252,049 | <p>I have seen on the nLab that we can <a href="https://ncatlab.org/nlab/show/loop+space+object" rel="nofollow">view the loop space as a particular homotopy pullback</a>, and that it is even the way a "loop space object" is defined in general (when it exists). </p>
<p>Can someone give me some intuition of why it is so... | Emily | 130,058 | <p>Here are some asides on my other answer.</p>
<p>On general loop space objects: See the discussion around (specially above) [Remark 1.1.2.6, <a href="http://www.math.harvard.edu/~lurie/papers/HA.pdf" rel="nofollow noreferrer">HA</a>] and [Section 2.3.2, <a href="https://drive.google.com/file/d/0ByJtAd7Hyh5XNGxCdVdiR... |
3,334,910 | <p>Question. <span class="math-container">$\square ABCD$</span> is a square with <span class="math-container">$AB = 10$</span>. Circle <span class="math-container">$O$</span> inscribes the <span class="math-container">$\square ABCD$</span>. The center of the arc is <span class="math-container">$A$</span>. What is the a... | Quanto | 686,284 | <p><a href="https://i.stack.imgur.com/Wxgzb.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/Wxgzb.png" alt="enter image description here"></a></p>
<p>Evaluate the areas of the circular sectors ABP and OPE, as well as the quadrilateral APOF and then subtract them from the half square to get the shade... |
114,278 | <p>If I wanted to show that a group of order $66$ has an element of order $33$, could I just say that it has an element of order $3$ (by Cauchy's theorem, since $3$ is a prime and $3 \mid 66$), and similarly that there must be an element of order $11$, and then multiply these to get an element of order $33$? I'm pretty... | Pete L. Clark | 299 | <p>Step 1: Show that any finite group order $4k+2$ has an index two subgroup. (<a href="https://math.stackexchange.com/questions/108796/group-of-order-4k2-prove-that-the-following-permutation-is-odd">Hint</a>.)</p>
<p>Step 2: Show that any group of order $33$ is cyclic. </p>
|
38,746 | <p>Divide $\mathbb{R}^2$ into disjoint unit squares, let Q be a tile (a polyomino, a finite set of connected unit squares). If, for every finite set S of unit squares in $\mathbb{R}^2$, - I can find a finite set of disjoint Q-tiles (a tiling) such that S is a subset of this tiling, how to prove that I can tile entire $... | Alon Amit | 308 | <p>Consider a sequence $S_1, S_2, S_3, \ldots$ of finite regions that exhaust the plane, namely $\cup S_i = \mathbb{R}^2$. For instance, you may take $S_i$ to be the square of edge side $2i+1$ centered at the origin.</p>
<p>Each $S_i$ is coverable with copies of your tile $Q$. Define a tree as follows: The vertices at... |
38,746 | <p>Divide $\mathbb{R}^2$ into disjoint unit squares, let Q be a tile (a polyomino, a finite set of connected unit squares). If, for every finite set S of unit squares in $\mathbb{R}^2$, - I can find a finite set of disjoint Q-tiles (a tiling) such that S is a subset of this tiling, how to prove that I can tile entire $... | JDH | 413 | <p>I know three different proofs of this.</p>
<ul>
<li><p>First, you can appeal to König's lemma. View the set
of tilings of increasingly large squares based at the
origin (that is, they minimally cover such a square) as a
tree under the subtiling relation. This is a finitely
branching tree, since any tiling of a... |
849,486 | <p>Find all <em>distinct</em> integers $x$ and $y$ that satisfy the following equation.</p>
<p>$$
x\log y=y\log x.
$$</p>
<p>Obviously, if $x=y$, the equation is satisfied.
I found $x=2$, and $y=4$. I think we cannot find all solutions (if there are many).</p>
<p>P.S. The base of the $\log(\cdot)$ is not important.<... | Answer | 160,449 | <p>Here using some of the $\log$ laws we can find all solution classes.</p>
<p>$$a \log (b)=\log(b^a)$$ </p>
<p>and</p>
<p>$$e^{\log(x)}=x$$ shall be all that is required.</p>
<p>$$x\log y = y \log x$$</p>
<p>$$\log y^x = \log x^y$$
$$e^{\log (y^x)} = e^{\log (x^y)}$$
$$y^x = x^y$$</p>
<p>Once we have reached thi... |
849,486 | <p>Find all <em>distinct</em> integers $x$ and $y$ that satisfy the following equation.</p>
<p>$$
x\log y=y\log x.
$$</p>
<p>Obviously, if $x=y$, the equation is satisfied.
I found $x=2$, and $y=4$. I think we cannot find all solutions (if there are many).</p>
<p>P.S. The base of the $\log(\cdot)$ is not important.<... | puru | 151,916 | <p>We have $x^y = y^x$, with $0 < x < y$ (WLOG).</p>
<p>Let $y= rx$, where $r > 1$.
Then $x^{rx} = (rx)^x$.
Taking the natural logarithm of both sides, $rx \ln x = x (\ln r + \ln x)$.
Dividing by $x$, and rearranging, $(r−1) \ln x = \ln r.$
(Notice that $r = 1$ would be a solution, from which $y = x$.)
With $... |
4,127,468 | <p>Suppose we have two functions <span class="math-container">$f,g:\Bbb R\rightarrow \Bbb R$</span>. The chain rule states the following about the derivative of the composition of these functions, namely that
<span class="math-container">$$
(f \circ g)'(x) = f′(g(x))\cdot g′(x).
$$</span>
However, the equivalent expre... | littleO | 40,119 | <p>In my opinion the usual way of writing the chain rule in Leibniz notation is confusing and, I would say, bad. It's a frequent source of confusion on this website.</p>
<p>The function that is called <span class="math-container">$z$</span> on the left is not the same as the function that is called <span class="math-co... |
91,252 | <p>Let $v_1, \ldots, v_n$ be a set of vectors in a vector space $V$. Show that $v_1, \ldots, v_n$ is a basis of $V$ if and only if for any non-zero linear function $f$ on $V$ there is a vector $v$ in $\operatorname{span}(v_1, \ldots, v_n)$ such that $f(v) \neq 0$. </p>
<p>Suppose that $v_1, \ldots, v_n$ is not a basis... | Arturo Magidin | 742 | <p>As Henning points out, the statement is false as written; but if you add the assumption that $v_1,\ldots,v_n$ are linearly independent, then the result is true. </p>
<p>As to your idea, it's either completely wrong or <em>almost</em> right, depending on what you mean by "complement."</p>
<p>If by "complement" you ... |
1,188,928 | <p>I know that on a smooth projective variety any coherent sheaf has a finite locally free resolution. I read somewhere that this implies that any object in $D^b(X)$ for $X$ smooth projective is then isomorphic in $D^b(X)$ to a complex of locally free objects. It seems to me it should be a proof by induction on the len... | Jeremy Rickard | 88,262 | <p>Let
$$\mathcal{F}_\bullet=\dots\stackrel{d^\mathcal{F}_3}{\to}\mathcal{F}_2
\stackrel{d^\mathcal{F}_2}{\to}\mathcal{F}_1
\stackrel{d^\mathcal{F}_1}{\to}\mathcal{F}_0
\stackrel{d^\mathcal{F}_0}{\to}0\to\dots$$
be a bounded complex of coherent sheaves (assuming without loss of generality that $\mathcal{F}_i=0$ for $i&... |
635,893 | <p>I am trying to prove following statement:</p>
<blockquote>
<p>$[m,n]$ is a set of functions defined as $f \in [m,n] \leftrightarrow f: \{1,...,m\} \rightarrow \{1,...,n\}$. The size of $[m,n]$ is $n^m$ for $m,n \in \mathbb{N}_{\gt0}$.</p>
</blockquote>
<p>I have tried to prove it but I am not entirely sure about... | Carsten S | 90,962 | <p>HINT: If $\int_a^b f(x)\,dx\ne0$, then $\int_0^a f(x)\,dx\ne\int_0^b f(x)\,dx$.</p>
|
3,444,500 | <p>I need to prove that <span class="math-container">$3x^6+12x^5+9x^4-24x^3+9x^2-4x+3=0$</span> does not have any real root. I tried analyzing the derivatives to see the maxima and minima but I can't compute them exactly, so I couldn't proceed further. Hints are appreciated.</p>
| Axion004 | 258,202 | <p>One approach would be applying <a href="https://en.wikipedia.org/wiki/Sturm%27s_theorem" rel="nofollow noreferrer">Sturm's Theorem</a> to find the total number of real roots of the polynomial</p>
<p><span class="math-container">$$p(x)=3x^6+12x^5+9x^4-24x^3+9x^2-4x+3$$</span></p>
<p>Sturm's theorem expresses the nu... |
3,237,234 | <p>Find the maximum and minimum
value of <span class="math-container">$$\cos 2x + 3\sin x.$$</span></p>
| auscrypt | 675,509 | <p>I think you mean
<span class="math-container">$$\frac{\mathrm d}{\mathrm dx}\int_2^{x^4}\tan(t^2)\,\mathrm dt$$</span>
as the integral is usually done with respect to a "dummy variable"; reusing variables like that is bad practice.</p>
<p>In this case, using <a href="https://en.wikipedia.org/wiki/Leibniz_integral_r... |
3,237,234 | <p>Find the maximum and minimum
value of <span class="math-container">$$\cos 2x + 3\sin x.$$</span></p>
| HK Lee | 37,116 | <p>When <span class="math-container">$F(x)=\int^x_0\ \tan\ t^2\ dt$</span>, then <span class="math-container">$$ \frac{d}{dx}\ F =_{{\rm Fundamental \ Theorem\ of\ Calculus}} \tan\
x^2$$</span> so that <span class="math-container">$$ \frac{d}{dx}\int_2^{x^4}\ \tan\ t^2\ dt
=\frac{d}{dx}\{F(x^4) +C\} =_{{\rm Chain\ Rule... |
1,605,100 | <p>$$\lim_{n\to\infty}\sum_{k=1}^n \frac{n}{n^2+k^2}$$</p>
<p>I tried this using powerseries just putting as $x=1$ there ,even tried thinking subtracting $s_{n+1} - s_{n}$ would be of some help, also I thought of writing the denominator as product of two complex numbers and then doing the partial fractions but it did ... | Tsemo Aristide | 280,301 | <p>Since $H$ is a subgroup of finite index, $G/H$ has a finite numbers of elements. Let $S_H=\{ aHa^{-1},a\in H\}$, consider the map $f:S_H\rightarrow G/H$ such that $f(a)$ is the class of $a$. We show that $f$ is injective. Suppose that $f(a)=f(b)$, this implies that $b\in aH$, thus $b=ah, h\in H$, $bHb^{-1}=ahH{(ah)... |
2,287,312 | <p>So the 12 marbles are distinct with each marble being a different color. Therefore I have to take into account, the different colors and different amount each person have. For example, person 1 can have 2 marbles, person 2 can have 7 marbles, and person 3 can have 3 marbles with each marble being a different color. ... | RRL | 148,510 | <p>Hint:</p>
<p>If $x$ is irrational then by density of rationals there is a sequence of rationals $(x_n)$ converging to $x$ such that</p>
<p>$$x = \lim_n x_n =\lim_n f(x_n) \neq f(x) = 0.$$</p>
<p>Now consider rational $x \neq 0$.</p>
|
2,287,312 | <p>So the 12 marbles are distinct with each marble being a different color. Therefore I have to take into account, the different colors and different amount each person have. For example, person 1 can have 2 marbles, person 2 can have 7 marbles, and person 3 can have 3 marbles with each marble being a different color. ... | Alex Ortiz | 305,215 | <p>For continuity at the origin, let $\epsilon>0$ be given. Then we claim that if $|x| < \epsilon$, then $|f(x) - f(0)| =|f(x)| < \epsilon$. Indeed, if $x$ is rational, then $|f(x)| = 0 < \epsilon$ and if $x$ is irrational, then $|f(x)| = |x| < \epsilon$.</p>
<p>If $a$ is irrational, then by density of ... |
1,224,692 | <p>What is the most computationally efficient way to find the layer on which a ball (i) belongs when arranged in a tetrahedron or 3 dimensional triangle with a triangular base. The ball on the top layer is numbered one. The balls on the second layer are numbered 2 - 4. The fifth layer 4-10 and so on. </p>
| Mastrel | 152,513 | <p>Suppose there exists a non-zero solution to $(A-\lambda M)x=0$. Then that means that $A-\lambda M$ is not one-to-one (as $(A-\lambda M)0 =0 = (A-\lambda M)x$ but $x \not = 0$), thus $A - \lambda M$ is not invertible and hence $A- \lambda M$ has determinant $0$. </p>
|
1,890,496 | <p>I am so confused on how to find the domain of this function $3^\sqrt{x^2-3x}$ without graphing it. I have no idea what to do in this situation.</p>
| SquirtleSquad | 339,227 | <p>Hint: Think of this as the composition of $3^x$, $\sqrt{x}$, and $x^2-3x$. Which of these has a domain that isn't all real numbers? And what is the domain of a composition?</p>
|
2,382,304 | <p>Find the general solution of the equation $$ \frac{dy}{dx}=1+xy $$</p>
<p>My attempt: Arranging it in standard linear equation form $\frac{dy}{dx}+Py=Q$, we get
$$\frac{dy}{dx}+(-x)y=1$$
Hence, the integrating factor(I.F.) = $e^{\int-xdx}=e^{-x^2/2}$
Hence, the solution is $$y(e^{-x^2/2})=\int{e^{-x^2/2}dx}+c$$</p>... | hamam_Abdallah | 369,188 | <p>The incomplete equation is
$$\frac {y'}{y}=x $$</p>
<p>thus
$$\ln (\frac {y}{\lambda})=\frac {x^2}{2} $$
and
$$y_h=\lambda e^{\frac {x^2}{2}} $$</p>
<p>the constante variation method gives</p>
<p>$$\lambda'(x)=e^{\frac {-x^2}{2}} $$</p>
<p>the general solution is</p>
<p>$$y_g=\Bigl(\lambda+\int e^{\frac {-x^2}{... |
423,938 | <p>I don't know if I apply for this case sin (a-b), or if it is the case of another type of resolution, someone with some idea without using derivation or L'Hôpital's rule? Thank you.</p>
<p>$$\lim_{x\to0}\frac{\sin(x^2+\frac{1}{x})-\sin\frac{1}{x}}{x}$$</p>
| Community | -1 | <p>We have
$$\frac{\sin(x^2+\frac{1}{x})-\sin\frac{1}{x}}{x}=\frac{\sin x^2\cos\frac{1}{x}+\sin\frac{1}{x}\cos x^2-\sin\frac{1}{x}}{x}$$
and since $\sin x^2\sim_0 x^2$ and $\left|\cos \frac{1}{x}\right|\leq 1$ and since $\cos x^2\sim_0 1-\frac{x^4}{2}$ we can see easily that the given limit is $0$.</p>
|
104,368 | <p>I can show that there infinitely many solutions to this equation. Is it possible that the set
of rational solutions is dense?</p>
| Ramsey | 12,107 | <p>Over $\mathbb{C}$, elliptic curves with, say, a point of order $N$ can be identified with the quotient of the upper half plane $\mathbb{H}$ by $\Gamma_1(N)$ just by associating to the class of $\tau\in \mathbb{H}$ the isomorphism class of the air $(\mathbb{C}/(\mathbb{Z} \oplus \mathbb{Z}\tau), 1/N)$. Under this id... |
3,587,990 | <p>I am facing the following combination problem : </p>
<p>I do have <span class="math-container">$B$</span> colored boxes, each one containing <span class="math-container">$N_b$</span> balls of the given color.
The total number of balls is thus <span class="math-container">$\sum_{b=1}^{B} N_b = N$</span>.</p>
<p>I w... | RobPratt | 683,666 | <p>You can solve the problem via integer linear programming as follows. Let nonnegative integer decision variable <span class="math-container">$x_{b,p}$</span> be the number of balls of color <span class="math-container">$b$</span> distributed to player <span class="math-container">$p$</span>. Let binary decision var... |
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