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 |
|---|---|---|---|---|
529,861 | <p>If $m,n$ are coprime positive integers and $m-n$ is odd, then $(m-n),(m+n),m,n,$ are coprime each other?</p>
<p>How do I prove it?</p>
<p>Especially how do I prove $(m-n), (m+n)$ are coprime?</p>
| Real Hilbert | 54,006 | <p>Note that $x=\frac{1}{1+p} \text{for some}~ p>0 $</p>
<p>which implies $|x^k|=\frac{1}{(1+p)^k} \leq \frac{1}{kp}<\epsilon~ \forall~ k>\frac{1}{p\epsilon} $ </p>
|
2,316,286 | <blockquote>
<p><strong>Theorem</strong> <em>(Cauchy-Schwarz Inequality) : If $u$ and $v$ are vectors in an inner product space $V$, then</em>
$$\langle u,v\rangle ^2\leqslant \langle u,u\rangle \langle v,v\rangle .$$</p>
</blockquote>
<p><strong>Proof</strong> : If $u=0$, then $\langle u,v\rangle = \langle u,u\ra... | N. S. | 9,176 | <p>If $at^2+bt+c \geq 0$ then subbing $t=-\frac{b}{2a}$ you get
$$a(\frac{-b}{2a})^2+b(-\frac{b}{2a})+c \geq 0$$</p>
<p>Now, since $a>0$, multiplying by $4a$ you get
$$b^2-2b^2+4ac \geq 0$$</p>
|
732,121 | <p>I'm having trouble seeing why the bounds of integration used to calculate the marginal density of $X$ aren't $0 < y < \infty$.</p>
<p>Here's the problem:</p>
<p>$f(x,y) = \frac{1}{8}(y^2 + x^2)e^{-y}$ where $-y \leq x \leq y$, $0 < y < \infty$ </p>
<p>Find the marginal densities of $X$ and $Y$.</p>
... | Javier Gutierrez | 291,573 | <p>If you graph $x = y$ and $x = -y$ ;
and only look at $0 < y < \infty$ ;
you will only use quadrants I and IV.</p>
<p>Thus $x$ can only be positive.</p>
<p>$0 < y < \infty $ aren't x-bounds.</p>
|
170,014 | <p>Determine for which $a$ values $f = x^2+ax+2$ can be divided by $g= x-3$ in $\mathbb Z_5$. </p>
<p>I don't know if there are more effective (and certainly <strong>right</strong>) ways to solve this problem, I assume there definitely are, but as I am not aware of them, I thought I could proceed like this: I have div... | Did | 6,179 | <p>What is known is explained in C. Albanese, S. Lawi, <em>Laplace transform of an integrated gometric Brownian motion</em>, MPRF 11 (2005), 677-724, in particular in the paragraph of the Introduction beginning by <em>A separate class of models</em>...</p>
|
678,073 | <p>I am working with the multiplicative ring of integers modulo $2^{127}$.</p>
<p>Consider the set $E=\{(k,l) \mid 5^k \cdot 3^l \equiv 1\mod 2^{127}, k > 0, l> 0\}$.
I wonder if anybody knows or has an idea where to look for a result related to a lower bound for $M=\min\{k+l \mid (k,l)\in E \}$.</p>
<p>We hav... | benh | 115,596 | <p><strong>Claim:</strong></p>
<blockquote>
<p>Let $c=40647290924413185736448652556727923386$, then the <strong>set of solutions</strong> is
given by $$A = \{(k,l) \in (\Bbb Z/2^{126} \Bbb Z)^2 \mid 5^k3^l \equiv 1\bmod 2^{127}\}=
\{(cn, 2 n) \mid n \in \Bbb Z\}$$</p>
</blockquote>
<p>This gives an explicit form... |
2,883,625 | <p>Let $f:\Bbb R^2\to \Bbb R$ such that
$$(f_x)^2+(f_y)^2=4\Big(1-f(x,y)\Big)\Big(f(x,y)\Big)^2,\qquad 0<f(x,y)<1.$$
then which functions satisfy the above property?</p>
| Jaap Scherphuis | 362,967 | <p>It is not always possible for both of $a,b$ to satisfy $a,b<\sqrt{p}$.</p>
<p>A small counterexample is $p=5$, $x=1$, $y=4$. The only solutions are $(a,b) \in \{ (1,4), (2,3), (3,2), (4,1) \}$.</p>
<p>In fact, for every $p$ the values $(x,y)=(1,p-1)$ will lead to a counterexample because it forces $a+b=0 \mod p... |
2,883,625 | <p>Let $f:\Bbb R^2\to \Bbb R$ such that
$$(f_x)^2+(f_y)^2=4\Big(1-f(x,y)\Big)\Big(f(x,y)\Big)^2,\qquad 0<f(x,y)<1.$$
then which functions satisfy the above property?</p>
| Russ | 497,388 | <p>I'm somewhat new to this board, but if Python is permitted, here is an annotated function that implements Jaap's efficient algorithm (with some helpers up front):</p>
<p>(Again, sorry if posting code is inappropriate for this board - please let me know, and I will delete it. If appropriate, feel free to delete thi... |
1,439,850 | <p>So the problem states that the centre of the circle is in the first quadrant and that circle passes through $x$ axis, $y$ axis and the following line: $3x-4y=12$. I have only one question. The answer denotes $r$ as the radius of the circle and then assumes that centre is at $(r,r)$ because of the fact that the circl... | Nikolay Gromov | 269,576 | <p>Again substitute $y(x)=z(x)/x$ solution you find $x z'(x)=1-2 z(x)$ and then (by separating the variable)
$$
-\frac{1}{2}\log(1-2z)=c+\log(x)
$$
or
$$
z=\frac{1}{2}+\frac{c}{x^2}
$$
or
$$
y=\frac{c}{x^3}+\frac{1}{2x}
$$</p>
|
1,087,015 | <p>I'm looking for a polynomial $P(x)$ with the following properties:</p>
<ol>
<li>$P(0) = 0$.</li>
<li>$P\left(\frac13\right) = 1$</li>
<li>$P\left(\frac23\right) = 0$</li>
<li>$P'\left(\frac13\right) = 0$</li>
<li>$P'\left(\frac23\right) = 0$</li>
</ol>
<p>From 1 and 3 we know that $P(x) = x\left(x - \frac23\right)... | Ross Millikan | 1,827 | <p>Certainly higher orders could. Since you have five conditions, you could expect a quartic to work. The condition that $P(\frac 23)=P'(\frac 23)=0$ gives a factor $(x-\frac 23)^2$, so we expect $P(x)=x(x-\frac 23)^2(ax+b)$ Now apply the conditions at $x=\frac 13$ to get $a$ and $b$. We get $P(\frac 13)=\frac {a+... |
3,623,368 | <p>for example this equation: 505x-673y=1 . x=4 and y = 3. but how can I find them with mathematics. What would be the approach here?</p>
| Matteo | 686,644 | <p>This is a simply diophantine equation. To solve this, you have basically to express <span class="math-container">$673$</span> in terms of <span class="math-container">$505$</span> and <span class="math-container">$1$</span> using euclid algorithm. The result, as you have shown is <span class="math-container">$x=4$</... |
428,415 | <p>I tried using integration by parts twice, the same way we do for $\int \sin {(\sqrt{x})}$
but in the second integral, I'm not getting an expression that is equal to $\int x\sin {(\sqrt{x})}$.</p>
<p>I let $\sqrt x = t$ thus,
$$\int t^2 \cdot \sin({t})\cdot 2t dt = 2\int t^3\sin(t)dt = 2[(-\cos(t)\cdot t^3 + \int... | Nick | 83,568 | <p>Just continue your path of partial integration with the last integral ? The last integral is purely a cosine which is integrable and yields your sollution.</p>
|
1,275,848 | <p>Given two numbers $x$ and $y$, how to check whether $x$ is divisible by <strong>all</strong> prime factors of $y$ or not?, is there a way to do this without factoring $y$?.</p>
| 2'5 9'2 | 11,123 | <p>$x$ is divisible by all prime factors of $y$ if and only if for some $n$, $x^n\equiv0$ modulo $y$. You might compute $x^n$ modulo $y$ for $n=1$ up to say $\log_2(y)$ and see if $0$ arises as a result. For large numbers, where prime factorization is hard, but modular arithmetic is doable, this would be more efficient... |
1,275,848 | <p>Given two numbers $x$ and $y$, how to check whether $x$ is divisible by <strong>all</strong> prime factors of $y$ or not?, is there a way to do this without factoring $y$?.</p>
| Mark Bennet | 2,906 | <p>If $x$ is divisible by all the prime factors of $y$, then so is the highest common factor $h_1$ of $x$ and $y$.</p>
<p>To test whether $y$ has a prime factor $p$ which is not a factor of $x$ - well then $p$ is not a factor of $h_1$, but will be a factor of $y_1$ where $y=y_1h_1$. Let $h_2$ be the highest common fac... |
3,789,676 | <p>I am try to calculate the derivative of cross-entropy, when the softmax layer has the temperature T. That is:
<span class="math-container">\begin{equation}
p_j = \frac{e^{o_j/T}}{\sum_k e^{o_k/T}}
\end{equation}</span></p>
<p>This question here was answered at T=1: <a href="https://math.stackexchange.com/questions/9... | Alex | 38,873 | <p>It's called chain rule:<span class="math-container">$\frac{\partial L}{\partial s} = \frac{\partial L}{\partial y} \times\frac{\partial y}{\partial s}$</span>. For the first term, in case of Euclidean loss, it is <span class="math-container">$(y-L)$</span>. For the second, it is <span class="math-container">$\sigma(... |
802,877 | <blockquote>
<p>Find $\displaystyle\lim_{n\to\infty} n(e^{\frac 1 n}-1)$ </p>
</blockquote>
<p>This should be solved without LHR. I tried to substitute $n=1/k$ but still get indeterminant form like $\displaystyle\lim_{k\to 0} \frac {e^k-1} k$. Is there a way to solve it without LHR nor Taylor or integrals ?</p>
<p>... | Community | -1 | <p><strong>Hint:</strong> You have $\lim_{n\to\infty}\dfrac{e^\frac{1}{n}-e^0}{\frac{1}{n}}=\lim_{k\to0}\dfrac{e^k-e^0}{k}$. Use the definition of a differential.</p>
|
58,870 | <p>I am teaching a introductory course on differentiable manifolds next term. The course is aimed at fourth year US undergraduate students and first year US graduate students who have done basic coursework in
point-set topology and multivariable calculus, but may not know the definition of differentiable manifold. I ... | Georges Elencwajg | 450 | <p>I nominate Ehresmann's theorem according to which a proper submersion between manifolds is automatically a locally trivial bundle. It is incredibly useful, in deformation theory for example, but is sadly neglected in introductory courses and books on manifolds. It is completely elementary: witness <a href="http://ww... |
294,519 | <p>The problem I am working on is:</p>
<p>Translate these statements into English, where C(x) is “x is a comedian” and F(x) is “x is funny” and the domain consists of all people.</p>
<p>a) $∀x(C(x)→F(x))$ </p>
<p>b)$∀x(C(x)∧F(x))$</p>
<p>c) $∃x(C(x)→F(x))$ </p>
<p>d)$∃x(C(x)∧F(x))$</p>
<h2>-----------------------... | Barbara Osofsky | 59,437 | <p>You literally wrote down symbol by symbol what the statements were. But languages used for everyday communications only rarely uses quantifiers, and does not have absolute truth/falsehood. Pretend you are talking to a friend. If you gave your answer to say a), the friend would probably look at you as if you were cra... |
1,877,558 | <p>For instance, let $(\mathbb{R}, \mathfrak{T})$ be $\mathbb{R}$ with the usual topology. </p>
<p>Why is that $\mathfrak{T} \times \mathfrak{T}$ is a basis on $\mathbb{R} \times \mathbb{R}$ instead of topology?</p>
<p>It seems that people just take $\mathfrak{T} \times \mathfrak{T}$ as a basis by definition. There m... | 5xum | 112,884 | <p>The unit ball $$\{(x,y)| x^2+y^2<1\}$$ is not a cartesian product of two sets in $\mathbb R$, but it is an open set in $\mathbb R^2$.</p>
<p>This can easily be shown using basic set theory - you don't need any knowledge of topology. If you find it hard to show this fact, then it's good practice and I suggest you... |
2,221,033 | <p>My question is due to <a href="https://en.wikipedia.org/w/index.php?title=Imaginary_number&diff=prev&oldid=175488747" rel="noreferrer">an edit</a> to the Wikipedia article: <a href="https://en.m.wikipedia.org/wiki/Imaginary_number" rel="noreferrer">Imaginary number</a>.</p>
<p>The funny thing is, I couldn'... | Jatin | 809,617 | <p>A complex number z=a+ib where a and b are real numbers is called :
1- purely real , if b=0 ; e.g.- 56,78 ;
2- purely imaginary, if a=0 ,e.g.- 2i, (5/2)i ;
3- imaginary,if b≠ 0 ,e.g.- 2+3i,1-i,5i ;
0 is purely imaginary and purely real but not imaginary.</p>
|
4,374,391 | <blockquote>
<p>Find prime number <span class="math-container">$p$</span> such that <span class="math-container">$19p+1$</span> is a square number.</p>
</blockquote>
<p>Now, I have found out, what I think is the correct answer using this method.<br />
Square numbers can end with - <span class="math-container">$1, 4, 9,... | Khosrotash | 104,171 | <p>Another idea to show is to use a binary base like below
<span class="math-container">$$(0000000...000)_2=0\\(0000000...001)_2=1\\(0000000...010)_2=2\\
(0000000...011)_2=3\\(0000000...100)_2=4\\\vdots\\(0111111...111)_2=2^n-1\\(1000000...000)_2=2^n$$</span> and</p>
<p>there is <span class="math-container">$2^n$</span... |
4,288,460 | <blockquote>
<p>Suppose that <span class="math-container">$\{X_t : Ω → S := \mathbb{R}^d, t\in T\}$</span> is a stochastic
process with independent increments and let <span class="math-container">$\mathcal{B}_t :=\mathcal{B}_t^X$</span> (natural filtration) for all <span class="math-container">$t\in T$</span>. Show, fo... | Robert Shore | 640,080 | <p>I'll give you an alternative approach. (I'm assuming you don't know yet that the inverse image of a closed set under a continuous function is closed.) If <span class="math-container">$h(x)=0$</span> is always true, then <span class="math-container">$K= \Bbb R$</span> and is closed.</p>
<p>Otherwise, we will prove ... |
1,320,469 | <p>I am working on the following problem</p>
<blockquote>
<p>[R. Vakil] Exercise 19.8.B: Suppose $C$ is a curve of genus $g>1$ over
a field $k$ that is not algebraically closed. Show that $C$ has a
closed point of degree at most $2g-2$ over the base field.</p>
</blockquote>
<p>I have no idea how to do this q... | Phil Tosteson | 157,315 | <p>Okay, by hypothesis $\omega$ is degree $2g -2$ and it has a global section $s \in H^0(\omega)$. </p>
<p>Then $\omega$ is $\mathcal O(D)$ for the divisor $D = \sum_p \nu_p(s) \cdot p$, where the sum is taken over all points $p$. And $2g -2 =\deg \omega = \sum_p \nu_p(s) \cdot \deg p$, and all the $v_p(s)$ are posi... |
910,070 | <p>I am working on a weighted minimization problem. Without the weights, the error function can be expressed as $e^T e$. With weights, $e$ first need to element-wise multiple by $w$, then the same formula applies: $(w \circ e)^T (w \circ e)$. How do I express it in pure matrix form (without the $\circ$). The $\circ... | Memming | 24,717 | <p>Note that $$ (w \circ e)^\top (w \circ e) = e^\top W e$$
where $W = \mathrm{diag}^{-1}(w_1^2, \ldots, w_n^2)$ and $\circ$ denotes Hadamard (or Schur) product.</p>
|
3,366,569 | <p>I am trying to solve the following problem;</p>
<p>Write all elements of the following set: <span class="math-container">$ A=\left \{ x\in\mathbb{R}; \sqrt{8-t+\sqrt{2-t}}\in\mathbb{R}, t\in\mathbb{R} \right \}$</span> .</p>
<p>My assumption is that the solution is <span class="math-container">$\mathbb{R}$</span> ... | Lutz Lehmann | 115,115 | <p>You should have first tried the direct Euler-Cauchy approach by computing the characteristic polynomial <span class="math-container">$0=m(m-1)+m-1=m^2-1$</span> giving <span class="math-container">$x,x^{-1}$</span> as basis solutions. The right side is not of the form <span class="math-container">$x^r\ln(x)^p$</span... |
605,155 | <p>$\newcommand{\ker}{\operatorname{ker}}$</p>
<p>Proof that: $\ker AB\subseteq\ker A+\ker B$</p>
<p>my solution:</p>
<p>$x\in \ker AB\to ABx=0\to \begin{cases} Ax=0\to x\in \ker A\\Bx=0\to x\in \ker B\end{cases}$</p>
<p>$\to x\in \ker A+\ker B\to \ker AB\subseteq \ker A+\ker B$</p>
<p>Question: Do it right? if fa... | Mercy King | 23,304 | <p>$$
x\in \ker(AB) \iff ABx=0 \iff Bx\in \ker (A)\iff x \in B^{-1}(\ker(A)),
$$
i.e.
$$
\ker(AB)=B^{-1}(\ker(A)).
$$</p>
|
262,319 | <pre><code>ContourPlot[
EuclideanDistance[{-5, 0}, {x, y}]*
EuclideanDistance[{5, 0}, {x, y}], {x, -15, 15}, {y, -11, 11},
Contours -> Range[5, 150, 20], Frame -> False,
ContourLabels -> (Text[Style[#3, Directive[Blue, 15]], {#1, #2}] &),
AspectRatio -> Automatic,
ColorFunction -> (If[# &l... | Bob Hanlon | 9,362 | <p>Simplifying and evaluating the argument will also reduce the redundant labels.</p>
<pre><code>$Version
(* "13.0.0 for Mac OS X x86 (64-bit) (December 3, 2021)" *)
ContourPlot[
Evaluate[
Simplify[
EuclideanDistance[{-5, 0}, {x, y}]*
EuclideanDistance[{5, 0}, {x, y}]]],
{x, -15, 15}, {y, -11, 11... |
178,028 | <p>I am given $G = \{x + y \sqrt7 \mid x^2 - 7y^2 = 1; x,y \in \mathbb Q\}$ and the task is to determine the nature of $(G, \cdot)$, where $\cdot$ is multiplication. I'm having trouble finding the inverse element (I have found the neutral and proven the associative rule.</p>
| hmakholm left over Monica | 14,366 | <p><em>Hint</em>. Your set has a certain formal similarity with $\{x+\sqrt{-1}y \mid x^2+y^2=1\}$ (arising by writing $-1$ for both occurrences of $7$), which is the unit circle in the complex plane. In the complex unit circle, the multiplicative inverse is just the complex conjugate, i.e. what you get by negating the ... |
21,201 | <p>Next Monday, I'll have an interview at Siemens for an internship where I have to know about fluid dynamics/computational fluid dynamics. I'm not a physicist, so does somebody have a suggestion for a good book where I can read about some basics? Thank you very much.</p>
| anonymous | 5,334 | <hr>
<p>Let $G$ be a finite subgroup of $GL(V)$. I claim that the following are equivalent:</p>
<p>$(1)$ Any two elements of $G$ which are $GL(V)$ conjugate are also $G$ conjugate.</p>
<p>$(2)$ The representation ring $\mathbb{Q} \otimes \mathrm{Rep}(G)$ is spanned by representations $S^{\lambda}(V)$, where $S^{\lam... |
402,214 | <p>I recently obtained "What is Mathematics?" by Richard Courant and I am having trouble understanding what is happening with the Prime Number Unique Factor Composition Proof (found on Page 23).</p>
<p>The first part:</p>
<blockquote>
<p><img src="https://i.stack.imgur.com/h5rCh.png" alt="enter image description he... | Federica Maggioni | 49,358 | <p>You want to show that every positive integer can be expressed as a product of prime numbers, secondly you want that such a decomposition is unique (except for the order of factors). Call $\mathscr{F}$ the set of positive integers not satisfying your claim, i.e. the set of positive integers that can be written in mor... |
2,323,351 | <p>I thought we take $4$ vowels and find number of arrangements $4!$ and multiply it with arrangements that can be made with consonants that is $5!/2!$. However my approach seems to be wrong. </p>
| Evargalo | 443,536 | <p>Assume $n>=7$.</p>
<p>Starting from the set {1,3,5,7}, any solution set can be reached by increasing the gaps between the chosen integers : you have to place $n-7$ gaps in any of the 5 following spots: before the 1, between 1 and 3, between 3 and 5, between 5 and 7, or after 7.</p>
<p>For instance, if $n=20$, y... |
2,323,351 | <p>I thought we take $4$ vowels and find number of arrangements $4!$ and multiply it with arrangements that can be made with consonants that is $5!/2!$. However my approach seems to be wrong. </p>
| Christian Blatter | 1,303 | <p>Each admissible choice is a binary word of length $n$ containing exactly four ones, and satisfying the extra condition that after the first three ones there is at least one zero.</p>
<p>Given an admissible word $w$ remove the first zero after each of the first three ones, and you obtain a binary word $w'$ of lengt... |
2,565,802 | <p>Calculate the volume of the region bounded by $z=0, z=1,$, and $(z+1)\sqrt{x^2+y^2}=1$</p>
<p>The integral is $\int_{B}z\text{ dV}$</p>
<p>The area is like the thing between the top two green places. The first place is $z=1$, second is $z=0$</p>
<p>Clearly we have $0\leq z\leq 1$, but I'm not sure what to bound n... | Stephen Meskin | 465,208 | <p>Let the $n$ people around the table be labeled $1, 2, \ldots n $ and let $k$ be the # of those people to be selected.<br>
We will count using two cases. Case A: $1$ is selected. Case B: $1$ is not selected. </p>
<p>Case A: $ \binom{n-k-1}{k-1}$<br>
Case B: $ \binom{n-k}{k}$<br>
Their sum is $\frac{n}{k}\binom{n-k-... |
2,565,802 | <p>Calculate the volume of the region bounded by $z=0, z=1,$, and $(z+1)\sqrt{x^2+y^2}=1$</p>
<p>The integral is $\int_{B}z\text{ dV}$</p>
<p>The area is like the thing between the top two green places. The first place is $z=1$, second is $z=0$</p>
<p>Clearly we have $0\leq z\leq 1$, but I'm not sure what to bound n... | Christian Blatter | 1,303 | <p>Assume there are $N$ admissible quadruples. These involve $4N$ choices of an individual, and by symmetry each person is chosen the same number of times, hence ${N\over25}$ times. We now count the number of admissible quadruples having person $100=0$ as a member. The admissible choices of the remaining three persons ... |
1,177,493 | <p>If $p$ is a prime and $p \equiv 1 \bmod 4$, how many ways are there to write $p$ as a sum of two squares? Is there an explicit formulation for this?</p>
<p>There's a theorem that says that $p = 1 \bmod 4$ if and only if $p$ is a sum of two squares so this number must be at least 1. There's also the Sum of Two Squar... | mcmat23 | 180,748 | <p>If $p$ is prime $p \equiv 1 \bmod \ 4 $ then $\exists!(a, b), (b, a) \in \mathbb N^2 $ such that $p = a^2 + b^2$. In fact, suppose $p = a^2 + b^2 = c^2 + d^2$ then in $\mathbb Z[i]$ $(a + ib)(a - ib) = (c + id)(c - id)$ but $N(a + ib) = N(a - ib) = N(c + id) = N(c - id) = p$ where $N(a + ib) = a^2 + b^2$ is the norm... |
752,517 | <p>From Wikipedia</p>
<blockquote>
<p>...the free group $F_{S}$ over a given set $S$ consists of all expressions (a.k.a. words, or terms) that can be built from members of $S$, considering two expressions different unless their equality follows from the group axioms (e.g. $st = suu^{−1}t$, but $s ≠ t$ for $s,t,u \in... | Michael Hardy | 11,667 | <p><b>PS: I've written a third answer that shows the really simple way to do this, without trigonometric functions.</b></p>
<p>The usual topology on the circle $\{(x,y) : x^2+y^2=1\}$ can be characterized in at least the following two ways, and I think it's fairly easy to show the two ways yield the same topology:</p>... |
752,517 | <p>From Wikipedia</p>
<blockquote>
<p>...the free group $F_{S}$ over a given set $S$ consists of all expressions (a.k.a. words, or terms) that can be built from members of $S$, considering two expressions different unless their equality follows from the group axioms (e.g. $st = suu^{−1}t$, but $s ≠ t$ for $s,t,u \in... | The very fluffy Panda | 140,598 | <p>It is also useful to know that if X and Y are locally compact Hausdorff spaces and are homeomorphic, then so are their one point compactifications. Another way of looking at the problem would be by considering the steoreographic projection which extends to a map from $\mathbb{R}\cup \{\infty\}$ to $\displaystyle ... |
752,517 | <p>From Wikipedia</p>
<blockquote>
<p>...the free group $F_{S}$ over a given set $S$ consists of all expressions (a.k.a. words, or terms) that can be built from members of $S$, considering two expressions different unless their equality follows from the group axioms (e.g. $st = suu^{−1}t$, but $s ≠ t$ for $s,t,u \in... | Michael Hardy | 11,667 | <p><b>PS: I've written a third answer that shows the really simple way to do this, without trigonometric functions.</b></p>
<p>Since you're making a lot of the problem of proving surjectivity, I'm adding a new answer dealing with trigonometry and geometry rather than with point-set topology. You have
\begin{align}
x ... |
2,877,576 | <p>Is it true that in equation $Ax=b$,
$A$ is a square matrix of $n\times n$, is having rank $n$, then augmented matrix $[A|b]$ will always have rank $n$?</p>
<p>$b$ is a column vector with non-zero values.
$x$ is a column vector of $n$ variables.</p>
<p>If not then please provide an example.</p>
| PQR Theorist | 582,944 | <p>My algebra gives b=√(√7500) or 9.30604 cm, (same as your answer) leading to an (internal) angle of 111.47°--which agrees with @Bence's result. </p>
|
3,037,296 | <p>I'm confused of what <span class="math-container">$\sqrt {3 + 4i}$</span> would be after I used quadratic formula to simplify <span class="math-container">$z^2 + iz - (1 + i)$</span></p>
| timtfj | 619,670 | <p>Well. As a hint: a complex number can be represented by a real part and an imaginary part. Or, on the complex plane, it can be expressed as a distance from the origin (its magnitude) and an angle. Multiplying two complex numbers multiplies the magnitudes and adds the angles—and as with real numbers, there are two sq... |
3,124,158 | <p>So what I want to prove is
<span class="math-container">$$ |xy+xz+yz- 2(x+y+z) + 3| \leq |x^2+y^2+z^2-2(x+y+z)+3| $$</span>
for <span class="math-container">$x,y,z\in \mathbb{R}$</span>, and I'm aware that the RHS is just <span class="math-container">$|(x-1)^2+(y-1)^2+(z-1)^2|$</span>.</p>
<p>Now I'm able to prove ... | Macavity | 58,320 | <p><strong>Hint:</strong> You have shown <span class="math-container">$|xy+yz+zx|\leqslant |x^2+y^2+z^2|$</span>. Now replace <span class="math-container">$(x,y,z)$</span> with <span class="math-container">$(x-1,y-1,z-1)$</span>…</p>
|
1,913,689 | <blockquote>
<p>Let $f: X \rightarrow Y$ be a function. $A \subset X$ and $B \subset Y$.
Prove $A \subset f^{-1}(f(A))$.</p>
</blockquote>
<p>Here is my approach. </p>
<p>Let $x \in A$. Then there exists some $y \in f(A)$ such that $y = f(x)$. By the definition of inverse function, $f^{-1}(f(x)) = \{ x \in X$ suc... | Alberto Takase | 146,817 | <p><strong>Proposition.</strong> Let $X$ and $Y$ be sets. Let $f:X\to Y$. For each $A\in\mathscr{P}(X)$, $A\subseteq f^{-1}(f(A))$.</p>
<p><em>Proof.</em> Let $A\in\mathscr{P}(X)$ be arbitrary.
\begin{align}
f^{-1}(f(A))&=\{z\in X:f(z)\in f(A)\}\\
&=\{z\in X:f(z)\in\{y\in Y:(\exists x\in A)[f(x)=y]\}\}\\
&... |
2,007,373 | <p>At some point in your life you were explained how to understand the dimensions of a line, a point, a plane, and a n-dimensional object. </p>
<p>For me the first instance that comes to memory was in 7th grade in a inner city USA school district. </p>
<p>Getting to the point, my geometry teacher taught,</p>
<p>"a p... | ಠ_ಠ | 169,780 | <p>This is less an answer and more of an extended comment. You seem to be struggling with the idea of a point as contrasted with an <a href="https://ncatlab.org/nlab/show/infinitesimally+thickened+point" rel="noreferrer">infinitesimally thickened point</a>, and it sounds to me like you want to do geometry with with <a ... |
2,109,832 | <p>This is for beginners in probability!</p>
<p>Could someone give me a step by step on how to find the MGF of the binomial distribution?</p>
| y_prime | 112,175 | <p>Let's use a right triangle. The simple one, 3-4-5.</p>
<p>I can't draw a diagram since I suck at those, but let the angle $\theta$ be opposite of the side of length 3. So $\sin(\theta)=\frac{3}{5}$ and $\cos(\theta)=\frac{4}{5}.$ (This is actually derivable since $\cos^2\theta=1-\sin^2\theta=\frac{16}{25}.$ Note th... |
2,109,832 | <p>This is for beginners in probability!</p>
<p>Could someone give me a step by step on how to find the MGF of the binomial distribution?</p>
| User8128 | 307,205 | <p>Use $\sin(2x) =2\sin(x)\cos(x)$ and then $\cos(x)=\sqrt{1-\sin^2(x)}$.</p>
|
2,109,832 | <p>This is for beginners in probability!</p>
<p>Could someone give me a step by step on how to find the MGF of the binomial distribution?</p>
| Nosrati | 108,128 | <p>By formula $\sin x=2\sin x\cos x$ we write
$$\color{red}{\sin(2\arcsin\dfrac35)}=2\sin(\arcsin\dfrac35)\cos(\arcsin\dfrac35)$$
By formula $\cos x=\sqrt{1-\sin^2x}$
$$\color{red}{\sin(2\arcsin\dfrac35)}=2\sin(\arcsin\dfrac35)\sqrt{1-\sin^2(\arcsin\dfrac35)}$$
But $\sin(\arcsin x)=x$ then
$$\color{red}{\sin(2\arcsin\d... |
3,133,695 | <p>A spotlight on the ground shines on a wal 12 m away. If a man 2m tall walks from the spotlight toward the building at a speed of 1.6m/s, how fast is the length of his shadow on the building decreasing when he is 4m from the building.</p>
<p>How do you solve this word problem. I have drawn a picture to figure out th... | Parallelism Alert | 639,984 | <p>No, two non-congruent quadrilaterals with same sets of sides and angles don't exist.
Let's prove this:</p>
<p>Suppose that there exist such two quadrilaterals <span class="math-container">$ABCD$</span> and <span class="math-container">$A^{'}B^{'}C^{'}D^{'} $</span>.<br>
We have <span class="math-container">$AB=A^{... |
846,797 | <p>I encountered this calculation in a problem $\dfrac{\sin 150^o\times\sin 20^o}{\sin 80^o\times\sin 10^o}$ and calculated that it equals 1.</p>
<p>Is it just a coincidence or is there any identity that says $\sin 150^o\times\sin 20^o=\sin 80^o\times\sin 10^o$?</p>
<p>I am trying to use the addition formulae and<br... | Cookie | 111,793 | <p>We will need to utilize the following observations (the unit circle would help you see these visually):</p>
<ul>
<li>$\sin 150^\circ=\frac 12$ </li>
<li>$\cos 10^\circ = \sin 80^\circ$</li>
</ul>
<p>Also the double-angle identity is used:</p>
<ul>
<li>$\sin 20^\circ = 2 \cos 10^\circ \sin 10^\circ$ (double angle ... |
3,983,914 | <p><strong>Preliminary properties</strong>: Let the state vector <span class="math-container">$x(t)=[x_1(t),\dots,x_n(t)]^T\in\mathbb{R}^n$</span> be constrained to the dynamical system
<span class="math-container">$$
\dot{x} = Ax +
\begin{bmatrix}
\phi_1(x_1) \\
\vdots \\
\phi_n(x_1) \\
\end{bmatrix}, \ \ \ \ x(0) = ... | open problem | 876,065 | <p>Instead of the general case let us focus on the case where <span class="math-container">$\alpha_{i} = 1$</span> for all i. In this special case <span class="math-container">$sign(x_{1})|x_{1}|=x_{1}$</span> the <span class="math-container">$\phi(x_{1})$</span> term can then be absorbed into <span class="math-contain... |
1,791,146 | <p>I know that a set G with a binary operation $*$ is a group, if:</p>
<ol>
<li><p>$a*b\in G$, for all $a, b \in G$.</p></li>
<li><p>$*$ is associative:</p></li>
</ol>
<p>$$(a*b)*c=a*(b*c) \\ \text{for all }a, b, c\in G.$$</p>
<ol start="3">
<li>An identity element $e \in G$ exists, such that</li>
</ol>
<p>$$a*e = ... | AnotherPerson | 185,237 | <p>So we know that universal statements are true on empty domains, and existence statements are false. Being a group requires the existence of an identity element, and since the empty set cannot satisfy this (it has no elements) it is not a group. </p>
|
1,198,722 | <p>I am working with a standard linear program:</p>
<p>$$\text{min}\:\:f'x$$
$$s.t.\:\:Ax = b$$
$$x ≥ 0$$</p>
<p><strong>Goal:</strong> I want to enforce all nonzero solutions $x_i\in$ x to be greater than or equal to a certain threshold "k" if it's nonzero. In other words, I want to add a conditional bound to the LP... | Michael Grant | 52,878 | <p>This cannot be solved with linear programming, but it can be solved with mixed-integer linear programming (MILP). What you are looking for is something called a <em>semicontinuous variable</em> in the MILP community. A semicontinuous variable $x$ is constrained to the disjoint set
$$x\in\{0\}\cup[\ell,u] \qquad\text... |
2,393,525 | <p>I have two questions which I think both concern the same problem I am having. Is $...121212.0$ a rational number and is $....12121212....$ a rational number? The reason I was thinking it could be a number is when you take the number $x=0.9999...$, then $10x=9.999...$ . Therefore, we conclude $9x=9$ which means $x=1... | Eric Wofsey | 86,856 | <p>The "numbers" you have written are not real numbers at all, so they are not rational or irrational. The decimal expansion of a real number cannot continue infinitely to the left. Why? Well, intuitively, such a number would be "infinitely large" and there are no infinitely large real numbers. More precisely, an e... |
1,042,227 | <p>I want to verify that the solution to the difference equation</p>
<p>$m_x - 2pqm_{x-2} = p^2 + q^2$</p>
<p>with boundary conditions</p>
<p>$m_0 = 0$</p>
<p>$m_1 = 0$</p>
<p>is</p>
<p>$$m_x = -\frac{1}{2}(\frac{1}{\sqrt{2pq}} +1)(\sqrt{2pq})^x + \frac{1}{2}(\frac{1}{\sqrt{2pq}} - 1)(-\sqrt{2pq})^x + 1$$</p>
<p... | Did | 6,179 | <p>If, as I suspect, $p+q=1$, one might prefer the general formula, valid for every integer $x\geqslant0$, $$m_{2x}=m_{2x+1}=1-(2pq)^x.$$ In the general case, if $2pq\ne1$, try $$m_{2x}=m_{2x+1}=\left(1-(2pq)^x\right)\,\frac{p^2+q^2}{1-2pq}.$$ Finally, if $2pq=1$, $$m_{2x}=m_{2x+1}=x\,(p^2+q^2).$$
In each case, checkin... |
3,765,555 | <p>Let <span class="math-container">$\triangle ABC$</span> be an isosceles triangle with base <span class="math-container">$a$</span> and altitude to the base <span class="math-container">$b.$</span> I am trying to find the sides of the rectangle inscribed in <span class="math-container">$\triangle ABC$</span> if its d... | SarGe | 782,505 | <p><a href="https://i.stack.imgur.com/kGNSV.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/kGNSV.png" alt="enter image description here" /></a></p>
<p>Let <span class="math-container">$ABC$</span> be a an isosceles triangle as shown in the figure. <span class="math-container">$D(t,0)$</span> and <sp... |
3,820,465 | <p>I'm working on the following problem but I'm having a hard time figuring out how to do it:</p>
<p>Q: Let A and B be two arbitrary events in a sample space S. Prove or provide a counterexample:</p>
<p>If <span class="math-container">$P(A^c) = P(B) - P(A \cap B)$</span> then <span class="math-container">$P(B) = 1$</sp... | copper.hat | 27,978 | <p>Let <span class="math-container">$P$</span> be uniform on <span class="math-container">$S=\{1,2,3\}$</span> and let <span class="math-container">$A=\{1,2\}, B=\{2,3\}$</span>.</p>
<p><span class="math-container">$P(A \cap B) = P \{2\} = {1 \over 3}$</span>.
<span class="math-container">$P(B) = P \{2,3\} = {2 \over 3... |
385,789 | <p>Can anybody please help me this problem?</p>
<p>Let $K = \mathbb{F}_p$ be the field of integers module an odd prime $p$, and $G = \mathcal{M}^*_n(\mathbb{F}_p)$ the set of $n\times n$ invertible matrices with components in $\mathbb{F}_p$. Based on the linear (in)dependence of the columns of a matrix $M\in G$, get t... | rschwieb | 29,335 | <p>Hint: you are working with a pool of $p^n$ column vectors from $\Bbb F^n$. Of course, $n$ could be 1 or 2, but to get you going, what I say will venture up to 3.</p>
<p>When picking the first column, you'll have $p^n-1$ choices. (Anything except the zero vector.)</p>
<p>When you pick the second column, you'll have... |
3,913,732 | <p>The following was asked by a high school student which I could not answer. Please help</p>
<p>In the figure below, show that the bisector of <span class="math-container">$\angle AEB$</span> and <span class="math-container">$\angle AFD$</span> intersect at perpendicular <img src="https://i.stack.imgur.com/76tDO.jpg" ... | Claude Leibovici | 82,404 | <p>Using whole numbers, you want to solve
<span class="math-container">$$\cos \left(\frac{13}{10} \pi \cos (t)\right)=\cos \left(\frac{13\pi }{10}\right)=\sqrt{\frac{5}{8}-\frac{\sqrt{5}}{8}}$$</span> Taking the inverse
<span class="math-container">$$t=\cos ^{-1}\left(\frac{10 }{13 \pi }\cos ^{-1}\left(-\frac{1}{2} \... |
3,784,872 | <p><strong>Problem:</strong></p>
<p>Suppose <span class="math-container">$(X_n)_{n \geq 1}$</span> are indipendent random variables defined in <span class="math-container">$(\Omega, \mathscr{A},\mathbb{P})$</span>.
Define <span class="math-container">$Y=\limsup _{n \to \infty} \frac{1}{n} \sum_{1 \leq p \leq n}X_p$</sp... | Kavi Rama Murthy | 142,385 | <p>For any <span class="math-container">$k$</span> <span class="math-container">$\frac 1n \sum\limits_{p=1}^{k}X_p \to 0$</span>. So we can take the sum from <span class="math-container">$k+1$</span> to <span class="math-container">$\infty$</span>. This shows that <span class="math-container">$Y$</span> and <span clas... |
132,862 | <p>Is it true that given a matrix $A_{m\times n}$, $A$ is regular / invertible if and only if $m=n$ and $A$ is a basis in $\mathbb{R}^n$?</p>
<p>Seems so to me, but I haven't seen anything in my book yet that says it directly.</p>
| Aaron Meyerowitz | 84,560 | <p>As was noted in the question, the set $\mathbb{N}$ which we are trying to specify is infinite because we know $T \subseteq \mathbb{N}$ for the infinite set $T=\{ 0, S(0), S(S(0)), \ldots \}.$ I will rephrase your question as</p>
<blockquote>
<p><strong>How do we show that in fact $\mathbb{N}=\{ 0, S(0), S(S(0))... |
1,783,323 | <p>Given the transition matrix for a 2 state Markov Chain, how do I find the n-step transition matrix P^n? I also need to take n--> inf and find the invariant probability pi?</p>
| Carl | 339,093 | <p>The conditions that have to be fulfilled are for an stationary distribution on a finite markov chain to exist are:</p>
<ul>
<li>It is irreducible</li>
<li>Additionally if it is aperiodic then $P^n$ will converge against a projection matrix $ e \cdot \pi^T $ where $e = (1,\dots,1) $ and $ \pi $ is the stationary dis... |
3,987,470 | <p>I am reading about how a wrong formulation of the tower of Hanoi and the inductive hypothesis can lead to a dead-end.<br />
The example I am reading states the following:</p>
<blockquote>
<p>The task is to move N discs from a <em>specific</em> pole to another
<em>specific</em> pole. Assume there are poles <span clas... | Community | -1 | <p>You are right to be concerned about the second part.</p>
<p>Your statement of the only thing pending is correct. However, the second part is nonsense since the only sensible inductive hypothesis concerns 'moving n discs from pole A to pole B, assuming that 3 poles are available'.</p>
<p><strong>An added observation<... |
3,987,470 | <p>I am reading about how a wrong formulation of the tower of Hanoi and the inductive hypothesis can lead to a dead-end.<br />
The example I am reading states the following:</p>
<blockquote>
<p>The task is to move N discs from a <em>specific</em> pole to another
<em>specific</em> pole. Assume there are poles <span clas... | Ross Millikan | 1,827 | <p>The reason that the second leads to a dead end is because having moved the smallest disk to <span class="math-container">$C$</span> you can't move the bottom <span class="math-container">$n$</span> disks to <span class="math-container">$B$</span> because you can't use <span class="math-container">$C$</span> to trans... |
4,227,536 | <blockquote>
<p>Let <span class="math-container">$X$</span> be the product space <span class="math-container">$\Bbb R^{\Bbb R}$</span>. Let <span class="math-container">$A \subset X $</span> be the set of all characteristic functions of finite sets. Show that the constant map <span class="math-container">$g, g(x) = 1$<... | Henno Brandsma | 4,280 | <p>If <span class="math-container">$U$</span> is a basic open neighbourhood of <span class="math-container">$g$</span>, it is of the form <span class="math-container">$U = \bigcap_{x \in F} p_x^{-1}[U_x]$</span> for a finite subset <span class="math-container">$F \subseteq \Bbb R$</span> and for each <span class="math-... |
627,258 | <p>Helly everybody,<br>
I'm trying to find another approach to topology in order to justify the axiomatization of topology. My idea was as follows:</p>
<p>Given an <strong>arbitrary</strong> collection of subsets of some space: $\mathcal{C}\in\mathcal{P}^2(\Omega)$<br>
Define a closure operator by: $\overline{A}:=\big... | C-star-W-star | 79,762 | <p>Nearness Spaces</p>
<p>Such a space wouldn't constitute a topology in the open set definition, though it would give rise to some sort of space in which "being close" still makes sense for <strong>all points</strong> to subsets.</p>
<p>Imagine the following situation:<br>
$\Omega:=\mathbb{S}^2\cup\{\mathbb{B}^3\}$<... |
33,817 | <p>It is an open problem to prove that $\pi$ and $e$ are algebraically independent over $\mathbb{Q}$.</p>
<ul>
<li>What are some of the important results leading toward proving this?</li>
<li>What are the most promising theories and approaches for this problem?</li>
</ul>
| Evan Jenkins | 396 | <p><a href="http://en.wikipedia.org/wiki/Schanuel%27s_conjecture">Schanuel's conjecture</a> would imply this result. It states that if $z_1, \ldots, z_n$ are linearly independent over $\mathbb{Q}$, then $\mathbb{Q}(z_1, \ldots, z_n, e^{z_1}, \ldots, e^{z_n})$ has transcendence degree at least $n$ over $\mathbb{Q}$. In ... |
496,255 | <p>Let $u$ be an integer of the form $4n+3$, where $n$ is a positive integer. Can we find integers $a$ and $b$ such that $u = a^2 + b^2$? If not, how to establish this for a fact? </p>
| David Vaknin | 216,025 | <p>Let's assume x^2+y^2 = 4n+3, then either x or y has to be even. Let's assume x = 2z and write</p>
<p>(2z)^2+y^2 = 4n+3. This can also be written as follows: </p>
<p>(2z+y)^2-4zy = 4n+3 by rearrangement we can write </p>
<p>(2z+y)^2-1^2 =4n+2+4zy</p>
<p>(2z+y)^2-1^2=2(2n+2zy+1) and further </p>
<p>(2z+y-1)(2z+y+... |
1,012,895 | <p>I am stuck with my revision for the upcoming test.</p>
<p>The question asks"</p>
<p>An implementation of insertion sort spent 1 second to sort a list of ${10^6}$ records. How many seconds it will spend to sort ${10^7}$ records?</p>
<p>By using $\frac{T(x)}{T(1)}$ = $\frac{10^7}{10^6}$ I thought the answer was $10... | Aditya Hase | 190,645 | <p>Hint: Time complexity of Insertion sort is $O(n^2)$</p>
<p>Roughly speaking if $10^6$ records took $1$ second then </p>
<p>$10\times 10^6$ records will take $10^2\times1=100$ seconds</p>
|
3,620,375 | <p>I am asked to calculate the integral <span class="math-container">$$\int_C \frac{1}{z-a}dz$$</span> where <span class="math-container">$C$</span> is the circle centered at the origin with radius <span class="math-container">$r$</span> and <span class="math-container">$|a|\neq r$</span></p>
<p>I parametrized the cir... | Community | -1 | <p>By Cauchy's Integral Formula, we get <span class="math-container">$2\pi i$</span>, if <span class="math-container">$|a|\lt r$</span>. Otherwise we get <span class="math-container">$0$</span>, by Cauchy's theorem.</p>
|
132,003 | <p>I have a time consuming function that is going to be iterated in a <code>Nest</code> or <code>NestList</code> and I would like to know if there is a good way to monitor the progress. I have found a partial work-around, but it requires an extra global variable (n). </p>
<pre><code>fun[x_] := Module[{}, n++; Pause[1]... | Kuba | 5,478 | <p><code>fun</code> should not know about <code>n</code>:</p>
<pre><code>nestListWithMonitor[f_, init_, n_] := Module[{it = 0},
PrintTemporary[ProgressIndicator[Dynamic[it/n]]];
NestList[(it++; f[#]) &, init, n]
]
</code></pre>
<p><code>it</code> is highlighted <code>Red</code> but it doesn't matter if its pa... |
3,079,493 | <p>Let <span class="math-container">$$D_6=\langle a,b| a^6=b^2=1, ab=ba^{-1}\rangle$$</span> <span class="math-container">$$D_6=\{1,a,a^2,a^3,a^4,a^5,b,ab,a^2b,a^3b,a^4b,a^5b\}$$</span></p>
<p>I would like to compute its character table and its irreducible representations.</p>
<p>I will explain what I have done so fa... | Ben Dyer | 164,207 | <blockquote>
<ol>
<li>Is <span class="math-container">$D_6/\{a^2,a^4\}$</span> really abelian? I can not see it clearly.</li>
</ol>
</blockquote>
<p>In your notation it looks like <span class="math-container">$\{a^2,a^4\}$</span> is denoting a conjugacy class and not a (normal) subgroup, so presumably you mean <... |
3,565,015 | <p>I generated this polynomial after playing around with the golden ratio. I first observed that (using various properties of <span class="math-container">$\phi$</span>), <span class="math-container">$\phi^3+\phi^{-3}=4\phi-2$</span>. This equation has no significance at all, I just mention it because the whole problem... | Abhijeet Vats | 426,261 | <p>Here's a possible way to do it:</p>
<p><span class="math-container">$x^6-4x^4+2x^3+1 = (x^6+2x^3+1)-4x^4 = (x^3+1)^2 - 4x^4$</span></p>
<p><span class="math-container">$(x^3+1)^2-4x^4 = [x^3+1-2x^2][x^3+1+2x^2]$</span></p>
<p><span class="math-container">$x^6-4x^4+2x^3+1= [(x^3-x^2)+(1-x^2)][x^3+2x^2+1]$</span></... |
3,265,835 | <p>I came across some equation in physics which had a different kind of integration. Like it should have <span class="math-container">$dx2$</span> but had <span class="math-container">$d2x$</span> . And I did some substitution for solving it like putting <span class="math-container">$x= u^2$</span> and then double diff... | Joel Biffin | 332,245 | <p>This is simply an abuse of the notation for the infinitesimal <span class="math-container">$dx$</span>. Often scientists (most frequently Physicists) will use slightly misleading notation due to a few little "calculus tricks" which are not 100% mathematically sound but work 90% of the time. </p>
<p>In this case, I ... |
2,877,833 | <p><a href="https://i.stack.imgur.com/9wqM5.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/9wqM5.png" alt="enter image description here"></a></p>
<p>Look at this part:</p>
<blockquote>
<p>Define the vector $p = -\nabla f(x^*)$ and note that $p^T\nabla f(x^*)
= -||\nabla f(x^*)||^2 <0$. Becau... | user293794 | 293,794 | <p>I think what you're missing is the following fact: if $F:\mathbb{R}^n\rightarrow\mathbb{R}$ is continuous and satisfies $F(x_0)<0$ then there exists some $\delta>0$ such that $F(x)<0$ for all $x$ such that $|x-x_0|<\delta$. You should try to prove this from the limit definition of continuity. In your par... |
227,833 | <p>In the documentation article for <code>Polygon</code> in Mathematica 12, there is an example with the input:</p>
<pre><code>pol = Polygon[{{1, 0}, {0, Sqrt[3]}, {-1, 0}}]
</code></pre>
<p>In the documentation article the output is displayed as:</p>
<blockquote>
<pre><code>Polygon[{{1, 0}, {0, Sqrt[3]}, {-1, 0}}]
</c... | N0va | 42,436 | <p>Following the comments under m_goldberg Answer to this question (<a href="https://mathematica.stackexchange.com/a/227859">https://mathematica.stackexchange.com/a/227859</a>) the following code disables the SummaryBox for <code>Polygon</code> only without disabling all elided forms or modifying the protected symbol <... |
223,642 | <p>$z\cdot e^{1/z}\cdot e^{-1/z^2}$ at $z=0$.</p>
<p>My answer is removable singularity.
$$
\lim_{z\to0}\left|z\cdot e^{1/z}\cdot e^{-1/z^2}\right|=\lim_{z\to0}\left|z\cdot e^{\frac{z-1}{z^2}}\right|=\lim_{z\to0}\left|z\cdot e^{\frac{-1}{z^2}}\right|=0.
$$
But someone says it is an essential singularity. I don't know... | DonAntonio | 31,254 | <p>$$ze^{1/z}e^{-1/z^2}=z\left(1+\frac{1}{z}+\frac{1}{2!z^2}+...\right)\left(1-\frac{1}{z^2}+\frac{1}{2!z^4}-...\right)$$</p>
<p>So this looks like an essential singularity, uh? </p>
<p>I really don't understand how you made the following step:</p>
<p>$$\lim_{z\to 0}\left|z\cdot e^{\frac{z-1}{z^2}}\right|=\lim_{z\to... |
223,642 | <p>$z\cdot e^{1/z}\cdot e^{-1/z^2}$ at $z=0$.</p>
<p>My answer is removable singularity.
$$
\lim_{z\to0}\left|z\cdot e^{1/z}\cdot e^{-1/z^2}\right|=\lim_{z\to0}\left|z\cdot e^{\frac{z-1}{z^2}}\right|=\lim_{z\to0}\left|z\cdot e^{\frac{-1}{z^2}}\right|=0.
$$
But someone says it is an essential singularity. I don't know... | JavaMan | 6,491 | <p>First, notice that
$$\lim_{z \to 0} e^{1/z}$$ does not exist as you get different values when you approach $0$ along the real line $x + 0i$ from the right and from the left.</p>
<p>From there, it is not difficult to show that $\lim_{z \to 0} z e^{1/z} e^{-1/x^2}$ does not exist either. Finally, we need to show th... |
1,137,079 | <p>I'm new to the concept of complex plane. I found this exercise:</p>
<blockquote>
<p>Let $z,z_1,z_2\in\mathbb C$ such that $z=z_1/z_2$. Show that the length of $z$ is the quotient of the length of $z_1$ and $z_2$.</p>
</blockquote>
<p>If $z_1=x_1+iy_1$ and $z_2=x_2+iy_2$ then $|z_1|=\sqrt{x_1^2+y_1^2}$ and $|z_2|... | David K | 139,123 | <p>Given $z = z_1/z_2$, you can conclude that $z z_2 = z_1$,
so $|z_1| = |z z_2|$.
If you can show that $|z z_2| = |z||z_2|$
then you can divide both sides by $|z_2|$ to get the desired result.</p>
<p>So you just need to know that the modulus of a product of two numbers is the
product of the modulus of each number.</p... |
2,921,439 | <p>I got this summation from the book <a href="https://rads.stackoverflow.com/amzn/click/0201558025" rel="nofollow noreferrer">Concrete Mathematics</a> which I didn't exactly understand:</p>
<p>$$
\begin{align}
Sn &= \sum_{1 \leqslant k \leqslant n} \sum_{1 \leqslant j \lt k} {\frac{1}{k-j}} \\
&= \sum_{1 \leq... | Chinny84 | 92,628 | <p>$$
-x^3-x^2+4x + 4 = -x^2(x+1) + 4(x+1)
$$
Then we can see that
$$
\frac{x+1}{-x^3-x^2+4x + 4 } = \frac{x+1}{-x^2(x+1) + 4(x+1)} = \frac{1}{4-x^2}
$$</p>
|
1,141,632 | <p>I need to answer a question on fractals from the book <em>Fractals Everywhere</em> by M. Barsley and I have been struggling with it for a while:</p>
<p>Use collage theorem to help you find an IFS consisting of two affine maps in $\mathbb{R}^2$ whose attractor is close to this set:
<img src="https://i.stack.imgur.co... | heropup | 118,193 | <p>The basic idea is to think of two affine mappings that each map the entire picture to a proper subset of itself. Instead of trying to match up the entire set of points, all you need to do is think about the placement of the vertices of the bounding box such that certain points are identified under the mapping.</p>
... |
2,794,715 | <p>Is it right that</p>
<p><strong>$$\sqrt[a]{2^{2^n}+1}$$</strong></p>
<p>for every $$a>1,n \in \mathbb N $$ </p>
<p>is always irrational?</p>
| lhf | 589 | <p>$\sqrt[a]{m}$ is rational iff it is an integer iff $m$ is an $a$-th power.</p>
<p>The question deals with <a href="https://en.wikipedia.org/wiki/Fermat_number" rel="nofollow noreferrer">Fermat numbers</a>.</p>
<p>Apart from the easy counterexamples, <a href="https://en.wikipedia.org/wiki/Fermat_number#Primality_of... |
2,261,410 | <blockquote>
<p>The generating function for a Bessel equation is:</p>
<p>$$g(x,t) = e^{(x/2)(t-1/t))}$$</p>
<p>Using the product $g(x,t)\cdot g(x,-t)$ show that:</p>
<p>a) $$[J_0(x)]^2 + 2[J_1(x)]^2 + 2[J_2(x)]^2 + \cdots = 1$$</p>
<p>and consequently:</p>
<p>b)</p>
<p>$$|J_0(x)|\le 1, \... | Jack D'Aurizio | 44,121 | <p>By the <a href="http://mathworld.wolfram.com/Jacobi-AngerExpansion.html" rel="nofollow noreferrer">Jacobi-Anger expansion</a> we have:</p>
<p>$$e^{iz\sin\theta} = \sum_{n=-\infty}^{+\infty}J_n(z) e^{in\theta}\tag{1}$$
hence by <a href="https://en.wikipedia.org/wiki/Parseval%27s_theorem" rel="nofollow noreferrer">Pa... |
2,612,794 | <p>I have a very elementar question but I do not see where my mistake is. </p>
<p>Suppose we have a sequence $(x_n)$ with
$\lim_{n\to\infty}x_n=1$. Moreover, suppose that the sequence $({x_n}^c)$ for some constant $c>1$ has limit $\lim_{n\to\infty}{x_{n}}^c=c$.</p>
<p>Then
$$
\lim_{n\to\infty}\log({x_n}^c)=\log(c)... | user326210 | 326,210 | <p>If $x_n\rightarrow 1$, the only way to get $x_n^c\rightarrow c$ is if $c=1$. There is no other exponent $c$ which makes this work. </p>
<p>After all, if $x_n\rightarrow 1$, then $x_n^c \rightarrow 1^c = 1$ as well, by continuity (see below). But then if $1=\lim_{n\rightarrow\infty}x_n^c = c$, then necessarily $c=1$... |
1,428,143 | <p>Let $f:E\to F$ where $E$ and $F$ are metric space. We suppose $f$ continuous. I know that if $I\subset E$ is compact, then $f(I)$ is also compact. But if $J\subset F$ is compact, do we also have that $f^{-1}(J)$ is compact ?</p>
<p>If yes and if $E$ and $F$ are not necessarily compact, it still works ?</p>
| robjohn | 13,854 | <p>Clayton's answer based on non-injectivity is very good. However, we can also base a counterexample on non-surjectivity.</p>
<p>Consider $f:\mathbb{R}\mapsto[-1,1]$ given by
$$
f(x)=\frac{x}{\sqrt{x^2+1}}
$$
Then
$$
f^{-1}([-1,1])=\mathbb{R}
$$</p>
|
1,433,980 | <p>so the problem I m having deals with conditional probability. I am given so much information and don't know what to do with what. Here is the problem:</p>
<p>"A study investigated whether men and women place more importance on a mate's ability to express his/her feelings or on a mate's ability to make a good livin... | WW1 | 88,679 | <p>I disagree that </p>
<p>"35% of the participants were men that said feelings were more important" </p>
<p>means the same as </p>
<p>" Of the 71% that said feelings were important, 35% of them were men"</p>
<p>The original question does not contain any conditional probabilities.</p>
<p>I read the information pro... |
2,691,232 | <p>Let $E$ be a complex Hilbert space.</p>
<blockquote>
<p>I look for an example of $A,B\in \mathcal{L}(E)$ such that $A\neq 0$ and $B\neq 0$ but $AB=0$.</p>
</blockquote>
| user284331 | 284,331 | <p>Let $A=E_{1,1}$ and $B=E_{2,2}$, then $AB=0$, where $E_{1,1}$ is the matrix has scalar $1$ only at $(1,1)$ entry, $E_{2,2}$ is the matrix has scalar $1$ only at $(2,2)$ entry.</p>
|
8,997 | <p>I have a set of data points in two columns in a spreadsheet (OpenOffice Calc):</p>
<p><img src="https://i.stack.imgur.com/IPNz9.png" alt="enter image description here"></p>
<p>I would like to get these into <em>Mathematica</em> in this format:</p>
<pre><code>data = {{1, 3.3}, {2, 5.6}, {3, 7.1}, {4, 11.4}, {5, 14... | Peter | 28,135 | <p>Here is an Excel VBA function that will generate the list in an excel cell. Copy the cell and paste directly into mma:</p>
<p>Function ToMathematicaList(Y_Values, X_Values)</p>
<p>N1 = Y_Values.Count</p>
<p>stout = "{"</p>
<p>For J = 1 To (N1 - 1)</p>
<p>stout = stout & "{" & X_Values(J) & ", " &... |
481,167 | <p>Let $V$ be a $\mathbb{R}$-vector space.
Let $\Phi:V^n\to\mathbb{R}$ a multilinear symmetric operator.</p>
<p>Is it true and how do we show that for any $v_1,\ldots,v_n\in V$, we have:</p>
<p>$$\Phi[v_1,\ldots,v_n]=\frac{1}{n!} \sum_{k=1}^n \sum_{1\leq j_1<\cdots<j_k\leq n} (-1)^{n-k}\phi (v_{j_1}+\cdots+v_{j... | Anthony Carapetis | 28,513 | <p>This is true, here's a proof.</p>
<p>I'm going to use the polynomial notation $\Phi\left(v_{1},\ldots,v_{n}\right)=v_{1}\cdots v_{n}$
- note that the multilinearity and symmetry of $\Phi$ means that
manipulating these like polynomials (i.e. commuting elements, distributing
``multiplication'') is completely legitima... |
204,592 | <p>The matrix exponential is a well know thing but when I see online it is provided for matrices. Does it the same expansion for a linear operator? That is if $A$ is a linear operator then $$e^A=I+A+\frac{1}{2}A^2+\cdots+\frac{1}{k!}A^k+\cdots$$</p>
| Fly by Night | 38,495 | <p>As you have suggested, if $A$ is a linear operator then:</p>
<p>$$\exp A = I + A + \frac{1}{2}A^2 + \cdots + \frac{1}{k!}A^k + \cdots \, . $$</p>
<p>These are very common in physics. <a href="http://www.tcs.tifr.res.in/~pgdsen/pages/courses/2007/quantalgo/lectures/lec06.pdf" rel="nofollow">Here is a link</a> to a ... |
204,592 | <p>The matrix exponential is a well know thing but when I see online it is provided for matrices. Does it the same expansion for a linear operator? That is if $A$ is a linear operator then $$e^A=I+A+\frac{1}{2}A^2+\cdots+\frac{1}{k!}A^k+\cdots$$</p>
| Hagen von Eitzen | 39,174 | <p>The exponential series has a remarkably "ubiquitiuos" convergence. As soon as you have a $\mathbb Q$-algebra $M$ with a norm such that $||X Y||\le c\cdot ||X||\cdot ||Y||$ for some $c$, then $\exp(A)$ converges for all $A$ with respect to this norm. Hence if $M$ is complete, you indeed obtain an element of $M$. More... |
526,820 | <p>How do I integrate the inner integral on 2nd line? </p>
<p><img src="https://i.stack.imgur.com/uIxQX.png" alt="enter image description here"></p>
<hr>
<p>$$\int^\infty_{-\infty} x \exp\{ -\frac{1}{2(1-\rho^2)} (x-y\rho)^2 \} \, dx$$</p>
<p>I know I can use integration by substitution, let $u = \frac{x-y\rho}{\sq... | AstroSharp | 83,876 | <p>$$\ldots=\sqrt{1-\rho^2}\left[y\rho\sqrt{1-\rho^2}\underbrace{\int_{\mathbb{R}}ue^{-u^2/2}du}_{0\mbox{ odd integrant}}+y\rho{\int_{\mathbb{R}}e^{-u^2/2}du}\right]=y\rho\sqrt{1-\rho^2}\int_\mathbb{R}e^{-u^2/2}du=\ldots$$</p>
|
1,285,014 | <p>Let $R,S$ be commutative rings with identity.</p>
<p>Proving that $X \sqcup Y$ is an affine scheme is the same as proving that $Spec(R) \sqcup Spec(S) = Spec(R \times S)$.</p>
<p>I proved that if $R,S$ are rings, then the ideals of $R \times S$ are exactly of the form $P \times Q$, where $P$ is an ideal of $R$ and... | Rob Arthan | 23,171 | <p>I am assuming you mean truth table in the title. Consider a conjunction of literals such as $p \land \lnot q \land \lnot r$: this is true for the assignment given by the row with $(p, q, r) = (1, 0, 0)$ in the truth table and not for any other row. The DNF is the disjunction of the conjunctions corresponding to the ... |
2,409,183 | <p>Good evening all! I'm trying to find the eigenvalues and eigenvectors of the following problem</p>
<p>$$
\begin{bmatrix}
-10 & 8\\
-18 & 14\\
\end{bmatrix}*\begin{bmatrix}
x_{1}\\
x_{2}\\
\end{bmatrix}
$$</p>
<p>I've found that $λ_{1},_{2}=2$ where $... | Mark Fischler | 150,362 | <p>Clearly the surface integral vanishes if it has a factor of any component to an odd power. We can always write the surface integral in spherical coordinates, aligning the $z$ axis with one of the indices appearing to the $4$-th or $2$-nd power. </p>
<p>We can also make the substitution $u = \cos \theta$ where $du =... |
48,989 | <p>How to prove $\text{Rank}(AB)\leq \min(\text{Rank}(A), \text{Rank}(B))$?</p>
| ncmathsadist | 4,154 | <p>You know that a linear transformation cannot increase the dimension of its domain; i.e. If $T: V\rightarrow W$ is a linear transformation,
$$\dim(T(V))\le \dim(V).$$</p>
|
1,828,097 | <p>If we contruct two strainght lines as shown:<a href="https://i.stack.imgur.com/8K5Eo.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/8K5Eo.png" alt="enter image description here"></a></p>
<p>Then join them such that to complete a triangle. <a href="https://i.stack.imgur.com/Uvtnw.png" rel="nofoll... | avz2611 | 142,634 | <p>the problem is that when we define point we consider point to be dimensionless but line is comprised of points , so it should be dimensionless as well but that's not the case . To go around the problem consider an $\epsilon$ value that is the minimum distance between two points , now you will realize for that $\epsi... |
4,462,081 | <p>I actually already have the solution to the following expression, yet it takes a long time for me to decipher the first operation provided in the answer. I understand all of the following except how to convert <span class="math-container">$\left(1+e^{i\theta \ }\right)^n=\left(e^{\frac{i\theta }{2}}\left(e^{\frac{-i... | Matt E. | 948,077 | <p>The first operation is just a result of algebraic manipulation. See here:</p>
<p><span class="math-container">\begin{align}
1 + e^{i\theta} =
e^0 + e^{i\theta} =
e^{i\frac{\theta}{2} - i\frac{\theta}{2}} +
e^{i\frac{\theta}{2} + i\frac{\theta}{2}} =
e^{i\frac{\theta}{2}}e^{-i\frac{\theta}{2}} +
e^{i\frac{\theta... |
612,827 | <p>I'm self studying with Munkres's topology and he uses the uniform metric several times throughout the text. When I looked in Wikipedia I found that there's this concept of a <a href="http://en.wikipedia.org/wiki/Uniform_space" rel="nofollow">uniform space</a>.</p>
<p>I'd like to know what are it's uses (outside poi... | Willie Wong | 1,543 | <p>Let me quote from Warren Page's <em>Topological Uniform Structures</em>:</p>
<blockquote>
<p>This book aims to acquaint the reader with a slice of mathematics that is interesting, meaningful, and in the mainstream of contemporary <em>[Ed: book originally published 1978]</em> mathematical developments. Admittedly ... |
1,358,735 | <p>I'm sorry to sound like a dummy, but I've had trouble with Algebra all my life. I'm studying online with Khan Academy and one of the questions is: </p>
<p>Point $E$'s $y$-coordinate is $0$, but its $x$-coordinate is not $0$.
Where could point $E$ be located on the coordinate plane?</p>
<p>There is not graph or not... | Zain Patel | 161,779 | <p>If the point $E$ has a $y$-coordinate of $0$, then it lies on the $x$-axis. Can you plot the point $(5,0)$, $(100,0)$, $(-23, 0)$. Notice what they all have in common? They lie on the $x$-axis and have a $y$-coordinate of $0$.</p>
<p>The equation of the $x$-axis is $y=0$. This is why, when you want to find $x$-inte... |
735,470 | <p>I am having trouble with integrating the following:</p>
<p>$$\int \frac{\cos2x}{1-\cos4x}\mathrm{d}x$$</p>
<p>I have simplified it using the double angle: </p>
<p>$$\int \frac{1-2\sin^2x}{1-\cos4x}\mathrm{d}x$$</p>
<p>But i am stuck as I am not sure on how to continue on from here. Should i use the double angle ... | WimC | 25,313 | <p>Note that $1-\cos(4x)=1-(1-2\sin^2(2x))=2 \sin^2(2x)$.</p>
|
4,351,794 | <p>I am trying not exactly to solve equation, but just change it from what is on right side to what is on left side. But I didn't do any math for years and can't remember what to.</p>
<blockquote>
<p><span class="math-container">$$\frac{1}{2jw(1+jw)}=\frac{-j(1-jw)}{2w(1+w^2)}$$</span></p>
<p>Here, <span class="math-co... | Deepak | 151,732 | <p>Here, it is quite apparent <span class="math-container">$j$</span> represents the imaginary unit (<span class="math-container">$j^2 = -1$</span>). This is quite common usage in physics and electrical engineering. In mathematics, you should be using <span class="math-container">$i$</span> rather than <span class="mat... |
4,351,794 | <p>I am trying not exactly to solve equation, but just change it from what is on right side to what is on left side. But I didn't do any math for years and can't remember what to.</p>
<blockquote>
<p><span class="math-container">$$\frac{1}{2jw(1+jw)}=\frac{-j(1-jw)}{2w(1+w^2)}$$</span></p>
<p>Here, <span class="math-co... | HAD HAN | 1,010,307 | <blockquote>
<p>si <span class="math-container">$j^{2}=-1$</span><br />
<span class="math-container">$\dfrac{1}{2jw(1+jw)}=\dfrac{1}{2jw(1+jw)}\times\dfrac{j}{j}=\dfrac{1\times j}{2jw(1+jw)\times j}=\dfrac{j}{2j^{2}w(1+jw)}=\dfrac{j}{2(-1)w(1+jw)}=\dfrac{j}{-2w(1+jw)}=\dfrac{j}{-2w(1+jw)}\times \dfrac{1-jw}{1-jw)}=\dfr... |
2,916,158 | <p>I am trying to understand why we need Large deviation theory/principle.</p>
<p><strong>Here is what I understand so far</strong> based on the <a href="https://en.wikipedia.org/wiki/Large_deviations_theory#An_elementary_example" rel="noreferrer">Wikipedia</a>.
Let $S_n$ be a random variable which depends on $n$.
We... | saz | 36,150 | <p>Large deviation theory deals with the decay of probabilities of rare events on an exponential scale. If $(S_n)_{n \in \mathbb{N}}$ is a random walk, then "rare event" means that $\lim_{n \to \infty} \mathbb{P}(S_n \in A) =0$. Large deviation theory aims to determine the asymptotics of</p>
<p>$$\mathbb{P}(S_n \in A)... |
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