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 |
|---|---|---|---|---|
3,012,130 | <p><span class="math-container">$4, 15, 13, 7, 22, -1, 31, -9, 40, -17, 49$</span>.</p>
<p>What comes next? The answer is <span class="math-container">$-25$</span>, but why?</p>
| Maged Saeed | 492,661 | <ul>
<li>Firs observation, first term and second term add up to 19, third and fourth add to 20, fifth and sixth add to 21 and so on..</li>
</ul>
<p>According to that, the the next number is <span class="math-container">$49+x = 24 \implies x = -25$</span></p>
<ul>
<li>Second observation, second and third terms add to ... |
3,012,130 | <p><span class="math-container">$4, 15, 13, 7, 22, -1, 31, -9, 40, -17, 49$</span>.</p>
<p>What comes next? The answer is <span class="math-container">$-25$</span>, but why?</p>
| Somos | 438,089 | <p>As a general rule, the simplest kind of sequences of numbers are linear recurrence sequences. It is a matter of finding the recurrence relation. Using the first <span class="math-container">$10$</span> terms of the sequence, and linear algebra, the generating function appears to be</p>
<p><span class="math-containe... |
86,459 | <p>I have problems with the following logarimthic equation:</p>
<p>$$\log _a \left(\frac{x+\sqrt{x^2+5}}{5}\right) = b$$</p>
<p>How can I compute $ \log _a (x-\sqrt{x^2-5})$ in terms of $b$?</p>
| André Nicolas | 6,312 | <p>There is presumably a typo in the question. But for fun we show that the version with presumed typo is not as awful as it looks. </p>
<p>Let $w$ be the second logarithm. Then
$$5a^b=x+\sqrt{x^2+5}\qquad\text{and}\qquad a^w=x-\sqrt{x^2-5}.$$</p>
<p>Take the first equation, bring $x$ to the left-hand side, square. ... |
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>... | Gigaboggie | 469,528 | <p>As a commenter mentions, the solution is:</p>
<p>$$y(x)=c_1 e^{\frac{x^2}{2}}+\sqrt{\frac{\pi }{2}} e^{\frac{x^2}{2}} \text{erf}\left(\frac{x}{\sqrt{2}}\right)$$</p>
<p>One thing to note here is that the erf(x) function is sometimes not all that well known. According to wolfram alpha, <a href="https://www.wolfr... |
732,791 | <p>The proof given for the inequality in my lecture notes seems confusing to me. The proof is as follows:</p>
<blockquote>
<p>Let $\lambda$ be any number and consider the vector $\lambda\textbf{v}-\textbf{w}$. Since $||\lambda\textbf{v}-\textbf{w}||\geqslant 0$ we have $$||\lambda\textbf{v}-\textbf{w}||^2\geqslant 0... | Mathias711 | 103,851 | <p><strong>P(0)</strong>=0.5*0.7*0.7 = 0.245 (all fail)</p>
<p><strong>P(>1)</strong> = 1 - <strong>P(0)</strong> = 1 - 0.245 = <strong>0.755</strong></p>
|
732,791 | <p>The proof given for the inequality in my lecture notes seems confusing to me. The proof is as follows:</p>
<blockquote>
<p>Let $\lambda$ be any number and consider the vector $\lambda\textbf{v}-\textbf{w}$. Since $||\lambda\textbf{v}-\textbf{w}||\geqslant 0$ we have $$||\lambda\textbf{v}-\textbf{w}||^2\geqslant 0... | kmbrgandhi | 132,855 | <p>The probability that you have no successes is:
$$
0.5 \cdot 0.7 \cdot 0.7 = \frac{49}{200}
$$
Therefore the probability of at least one success is
$
1 - \frac{49}{200} = \boxed{\frac{151}{200}}
$</p>
<p>Note that this assumes independence. </p>
|
1,471,415 | <p>I am failing to understand how to compute the derivative of a few exponential functions. Let's start with this one:</p>
<p>$$
v = 1 - e^{-t/\tau}
$$</p>
<p>The derivative is</p>
<p>$$
\frac{dv}{dt} = \frac{1-v}{\tau}
$$</p>
<p>Can someone walk me through this? If this is explained somewhere else, I'd love to k... | Henricus V. | 239,207 | <p>$$\begin{align*}
\frac{d}{dt} (1 - e^{-t/\tau}) &= -\frac{d}{dt}e^{-t/\tau} \\
&= - (-1/\tau) e^{-t/\tau} \\
&= \frac{e^{-t/\tau}}{\tau} \\
&= \frac{1 - (1 - e^{-t/\tau})}{\tau} \\
&= \frac{1 - v}{\tau}
\end{align*}$$</p>
|
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>
| robjohn | 13,854 | <p>Using the identity
$$
\sin(A)-\sin(B)=2\sin\left(\frac{A-B}{2}\right)\cos\left(\frac{A+B}{2}\right)
$$
we get
$$
\begin{align}
\lim_{x\to0}\frac{\sin\left(x^2+\frac1x\right)-\sin\left(\frac1x\right)}{x}
&=\lim_{x\to0}\frac{2\sin\left(\frac{x^2}{2}\right)\cos\left(\frac{x^2}{2}+\frac1x\right)}{x}\\
&=\lim_{x\... |
976,392 | <p>I would need a proof that <span class="math-container">$n \left(1-p^{\frac{1}{n}}\right)$</span> is increasing in <span class="math-container">$n \in \mathbf{N}$</span> for any <span class="math-container">$p \in (0,1)$</span>.</p>
<h3>Context</h3>
<p>I am working on a larger question and this is the last missing pi... | konewka | 174,857 | <p>Take the derivative to $n$:</p>
<p>$$\frac{d}{dn}n(1-p^{\frac{1}{n}})=1-p^{\frac{1}{n}}+\frac{np^{\frac{1}{n}}\log p}{n^2}=1+\left(\frac{1}{n}\log p-1\right)p^{\frac{1}{n}}$$</p>
<p>This will have to be greater than $0$ for every $n\in\mathbb{N},p\in(0,1)$, so
$$1+\left(\frac{1}{n}\log p-1\right)p^{\frac{1}{n}}>... |
976,392 | <p>I would need a proof that <span class="math-container">$n \left(1-p^{\frac{1}{n}}\right)$</span> is increasing in <span class="math-container">$n \in \mathbf{N}$</span> for any <span class="math-container">$p \in (0,1)$</span>.</p>
<h3>Context</h3>
<p>I am working on a larger question and this is the last missing pi... | gtrrebel | 169,563 | <p>One could also use elementary estimates.</p>
<p>Fix $n$, and set $p = x^{n(n+1)}$. Now the proving the statement is equivalent to
$$
n(1-x^{n+1}) \leq (n+1)(1-x^{n})
$$
which can be written as
$$
nx^{n}(1-x) \leq (1-x^{n}).
$$
Since $x \in (0, 1)$, we may divide by $1-x$ to get
$$
nx^n \leq 1+x+\ldots + x^{n-1}.
$$... |
329,067 | <p>I have a question that is as follows:</p>
<blockquote>
<p>For each integer $n \geq 3$, construct a 3-regular graph on $2n$ vertices such that $G_n$ does not have any 3-cycles.</p>
</blockquote>
<p>Here is what I have:</p>
<p>I have $2n$ vertices numbered $1, 2, \ldots, 2n$, and a vertex $k$ connected to $k-1$, ... | Ross Millikan | 1,827 | <p>If $k$ is part of a 3-cycle you must have an edge between two vertices that $k$ is connected to. So there would have to be and edge between two of $k-1, k+1, k+n$. Clearly there is no edge between $k-1, k+1$. Can there be an edge between $k+n$ and $k \pm 1$?.</p>
|
3,130,749 | <p>I am trying to differentiate <span class="math-container">$\frac{v^4-10v^2\sqrt{v}}{4v^2}$</span>.</p>
<p>I have tried splitting the fraction and doing the division before finding the differential, but I am still not getting the right answer.</p>
<p><span class="math-container">$$
\frac{v^4-10v^2\sqrt{v}}{4v^2}
=\... | bjcolby15 | 122,251 | <p>Hint: To make your work a little easier...</p>
<p>Let <span class="math-container">$w = \sqrt {v}$</span>. Then <span class="math-container">$w^2 = v$</span>, <span class="math-container">$w^4 = v^2$</span>, and <span class="math-container">$w^8 = v^4.$</span> You have <span class="math-container">$$\dfrac {w^8 -... |
3,130,749 | <p>I am trying to differentiate <span class="math-container">$\frac{v^4-10v^2\sqrt{v}}{4v^2}$</span>.</p>
<p>I have tried splitting the fraction and doing the division before finding the differential, but I am still not getting the right answer.</p>
<p><span class="math-container">$$
\frac{v^4-10v^2\sqrt{v}}{4v^2}
=\... | Michael Lee | 621,024 | <p>After simplifying the expression from...</p>
<p><span class="math-container">$$ \frac{v^4-10v^2\sqrt{v}}{4v^2} $$</span>.</p>
<p>To...</p>
<p><span class="math-container">$$\frac{v^2-10\sqrt{v}}{4}$$</span>.</p>
<p>I evaluated and had gotten...</p>
<p><span class="math-container">$$\dfrac {v}{2} - \dfrac {5}{\s... |
143,173 | <p>I have a small question that I think is very basic but I am unsure how to tackle since my background in computing inequalities is embarrassingly weak - </p>
<p>I would like to show that, for a real number <span class="math-container">$p \geq 1$</span> and complex numbers <span class="math-container">$\alpha, \beta$... | user65536 | 65,536 | <p>In fact, one advantage of the argument in john w.'s answer is that it can be extended further to prove <span class="math-container">$$(a+b)^n\leq p^na^n+q^nb^n,$$</span> where <span class="math-container">$\frac{1}{p}+\frac{1}{q}=1$</span>. In particular, one can choose the coefficient of <span class="math-containe... |
3,295,973 | <p>Let <span class="math-container">$(\Omega,\mathcal A,\mu)$</span> be a measure space, <span class="math-container">$p,q\ge1$</span> with <span class="math-container">$p^{-1}+q^{-1}=1$</span> and <span class="math-container">$f:\Omega\to\mathbb R$</span> be <span class="math-container">$\mathcal A$</span>-measurable ... | Michael Rozenberg | 190,319 | <p>Yes, <span class="math-container">$x=-2$</span> is invalid because <span class="math-container">$$x=\sqrt{2-x}\geq0.$$</span></p>
<p>I like the following reasoning.</p>
<p><span class="math-container">$x$</span> increases, <span class="math-container">$\sqrt{2-x}$</span> decreases, which says that our equation has... |
3,295,973 | <p>Let <span class="math-container">$(\Omega,\mathcal A,\mu)$</span> be a measure space, <span class="math-container">$p,q\ge1$</span> with <span class="math-container">$p^{-1}+q^{-1}=1$</span> and <span class="math-container">$f:\Omega\to\mathbb R$</span> be <span class="math-container">$\mathcal A$</span>-measurable ... | Michael Rybkin | 350,247 | <p><span class="math-container">$$
\sqrt{2-x}=x.
$$</span></p>
<p>There are two things that we can glean from that statement above: <span class="math-container">$2-x\ge0$</span> and <span class="math-container">$x\ge0$</span>. The expression under the square root must be greater or equal to zero and if something equal... |
351,815 | <p>Having trouble understanding this. Is there anyway to prove it?</p>
| Macavity | 58,320 | <p>Let me give a
Hint: Let <span class="math-container">$f(n) = \dfrac{n! }{ a^n}$</span>, for <span class="math-container">$ a > 1$</span>. What is <span class="math-container">$\dfrac{f(n+1)}{f(n)}$</span>??</p>
|
351,815 | <p>Having trouble understanding this. Is there anyway to prove it?</p>
| Robert Mastragostino | 28,869 | <p>If you're not quite in the market for a full proof:</p>
<p>$$a^n=a\times a\times a\times a...\times a$$
$$n!=1\times 2\times 3\times 4...\times n$$</p>
<p>Now what happens as $n$ gets much bigger than $a$? In this case, when $n$ is huge, $a$ will have been near some number pretty early in the factorial sequence. T... |
351,815 | <p>Having trouble understanding this. Is there anyway to prove it?</p>
| Sijo Joseph | 72,833 | <p>Use the striling's approximation to $n!$ for large numbers we get,<br>
$$ \log(n!)=n \log n -n. $$
also we have
$$\log(a^n)=n\log a.$$
Now divide the equations we get,
$$ \frac{\log(n!)}{\log(a^n)}=(n \log n -n)/n\log a. $$
$$ \frac{\log(n!)}{\log(a^n)}=\log n/\log a-1/\log(a). $$
for large a (a>1) we can neglect ... |
351,815 | <p>Having trouble understanding this. Is there anyway to prove it?</p>
| hat180 | 346,969 | <p>Assume that $x>a>0$. Then:
$$\frac{x!}{a^x}=\frac{a!\Pi^x_{i=a+1}i}{a^x}>a!\frac{(a+1)^{x-a}}{a^x}=\frac{a!}{(a+1)^a}\frac{(a+1)^x}{a^x}=\frac{a!}{(a+1)^a}(1+\frac{1}{a})^x\to_{x\to\infty}\infty$$</p>
|
4,209,055 | <p>Let <span class="math-container">$S=\sum_{k=1}^{m}e^{2\pi ik^2/m}$</span>,if <span class="math-container">$m$</span> is odd,how to directly calculate the absolute value of <span class="math-container">$S=\sqrt{m}$</span>.Don't use Gauss sum since here it says "it's easily shown"<a href="https://i.stack.img... | Sangchul Lee | 9,340 | <p>We have</p>
<p><span class="math-container">\begin{align*}
|S|^2
&= \sum_{k=0}^{m-1}\sum_{l=0}^{m-1} \exp\left(\frac{2\pi i}{m}(l^2-k^2)\right) \\
&= \sum_{k=0}^{m-1}\sum_{d=0}^{m-1} \exp\left(\frac{2\pi i}{m}(2kd+d^2)\right) \tag{$l\equiv k+d \pmod{m}$} \\
&= \sum_{d=0}^{m-1} \Biggl[ \sum_{k=0}^{m-1} \e... |
890,200 | <p>Let $x_1:=a>0$ and $x_{n+1}:=x_n+1/x_n$ for $n\in\mathbb{N}$. Determine whether $(x_n)$ converges or diverges.</p>
<p>My answer: $(x_n)$ is divergent.</p>
<p>Proof: Assume that $(x_n)$ converges to $x$. Then $\lim (x_{n+1})=\lim (x_n)$. That is, $x=x+1/x$. This equation has no solution. Hence, $(x_n)$ is... | lhf | 589 | <p>Your proof is fine.</p>
<p>Note that <em>divergent</em> simply means that it is <em>not convergent</em>, not that it goes to $\infty$.</p>
|
2,845,049 | <p>Let $\beta_m\searrow 0$ such that $\alpha_m:=\beta_m-\beta_{m+1}\searrow 0$.</p>
<p>Define $b_n:=\inf\{m:\alpha_m<2^{-n}\}$. Is it true that
$$
\sum_{n=1}^\infty \frac{b_n}{2^n}<\infty?
$$</p>
<p>For example, if $\beta_m=\frac 1 m$, then $b_n\sim 2^{n/2}$, so that the above series converges.</p>
<p>A critic... | zaphodxvii | 557,340 | <p>Your critical case $ \alpha_{m} = \frac{1}{\lg(m)} $ is a counter-example.</p>
<p>Notice that $ b_{n} = 2^{(2^{n+1})} $, since we have</p>
<p>$$ \frac{1}{\lg(m)} < 2^{-n} \implies m > 2^{(2^{n})} $$</p>
<p>and $\alpha_{m}$ is monotonous. </p>
<p>Therefore, </p>
<p>$$ \sum_{n=1}^{\infty} \frac{2^{(2^{n+1})... |
1,842,779 | <p>This is Q28 from Australian Maths Competition 2014.</p>
<p>A circle is surrounded by 6 other circles,in a hexagonal formation.The leftmost circle is 0,which the rightmost circle is 1000.Each of the five missing numbers is the average of its neighbours. What is the largest of the 5 missing numbers?</p>
<p>What I tr... | heropup | 118,193 | <p>Let the numbers be arranged as follows: $$\begin{array}{ccccc} & a & & b & \\ 0 & & c & & 1000 \\ & d & & e & \end{array}$$ Then we have the relationships $$\begin{align*} 3a &= b + c \\ 3d &= c + e \\ 3b &= a + c + 1000 \\ 3e &= c + d + 1000 \\ 6c &am... |
338,383 | <p>Multivariate polynomial indexed by <span class="math-container">${1, \ldots, n}$</span> are acted on by <span class="math-container">$S_n$</span>: for <span class="math-container">$\sigma \in S_n$</span>, define <span class="math-container">$\sigma(x_i) = x_{\sigma(x_i)}$</span>, etc. Symmetric polynomials are those... | benblumsmith | 12,419 | <p>The name of the body of theory you are asking for is "invariant theory of permutation groups." You will also find relevant papers by searching for "polynomial permutation invariants."</p>
<p>It falls under the broader rubric of "invariant theory of finite groups", which is a developed f... |
4,020,412 | <p>I was reading about ODE variable separation solving and the book says that assuming that a function can be expressed as the product of two single-variable functions loses generality, which I understand. I cannot, however, prove that it does. For example: the function <span class="math-container">$\sqrt{x+t}$</span> ... | Hagen von Eitzen | 39,174 | <p>Note that <span class="math-container">$f(x,y)=X(x)T(t)$</span> implies things like
<span class="math-container">$$\tag1f(x_1,t_1)f(x_2,t_2)=f(x_1,t_2)f(x_2,t_1)$$</span>
for all <span class="math-container">$x_1,x_2,t_1,t_2$</span>.</p>
<p>In the concrete example <span class="math-container">$f(x,t)=\sqrt{x+t}$</sp... |
4,020,412 | <p>I was reading about ODE variable separation solving and the book says that assuming that a function can be expressed as the product of two single-variable functions loses generality, which I understand. I cannot, however, prove that it does. For example: the function <span class="math-container">$\sqrt{x+t}$</span> ... | JJacquelin | 108,514 | <p>To prove that <span class="math-container">$\sqrt{x+t}=X(x)T(t)$</span> is not valid in general it is suffisant to show a conter-example.</p>
<p><span class="math-container">$$\text{Suppose that}\quad \sqrt{x+t}=X(x)T(t)\quad
\text{is true any }(x,t) \tag 1$$</span>
<span class="math-container">$$\text{then}\quad ... |
156,321 | <p>I have data that represents a cyclist’s power output in 1-second intervals, sampled while the cyclist was working to a prescribed training regime.</p>
<p>I want to find sequences within that data where the cyclist sustained a power output (plus or minus a certain percentage threshold) for fixed lengths of time. So:... | Lukas Lang | 36,508 | <p>As noted in the comments, use <a href="http://reference.wolfram.com/language/ref/MovingMap.html" rel="nofollow noreferrer"><code>MovingMap</code></a> and <a href="http://reference.wolfram.com/language/ref/MinMax.html" rel="nofollow noreferrer"><code>MinMax</code></a>:</p>
<pre><code>SeedRandom[99]
data = RandomFunc... |
918,509 | <p>Let $X$ be a random variable with pdf $f$. I would like to know why:</p>
<p>$$\operatorname{E} [X] = \int_\Omega X \, \mathrm{d}P = \int_\Omega X(\omega) P(\mathrm{d}\omega)= \int_{-\infty}^\infty x f(x)\, \mathrm{d}x . $$</p>
<p>I mean I don't get it why it is all equal and what the notation in third term mean... | Ncat | 36,880 | <p>Returned to the problem and things became clear. Thank you to <a href="https://math.stackexchange.com/users/172006/almagest">almagest</a> for the inspiration to consider his recursive function.</p>
<p>To keep everything in one spot, I'll start from the beginning defintions:</p>
<p>Let $S \colon \mathbb{N}^{2} \to ... |
482,793 | <p>As title says, how does one show that a function is continuous over some interval (let us say over some interval of real numbers?)</p>
<p>Would(Can) this involve derivative?</p>
| amWhy | 9,003 | <p>If you know that a given function $f$ is <em>differentiable</em> on some interval $(a, b) \subseteq \mathbb R$, then it certainly follows that $f$ is continuous on that interval (since differentiability requires continuity and more). </p>
<p>But a function $f$ <em>can be</em> continuous on an interval, even though ... |
2,066,765 | <p>Are there any examples of subspaces of $\ell^{2}$ and $\ell^{\infty}$ which are not closed?</p>
| DanielWainfleet | 254,665 | <p>For $1< x<2$ the space $l^x$ is a non-closed vector subspace of $l^2.$ </p>
<p>Let $(A_j)_{j\in \mathbb N}$ be a strictly increasing positive real sequence converging to $1/x.$ </p>
<p>For $i,j, \in \mathbb N$ let $B_{i,j}=0$ if $j>i,$ and $B_{i,j}=j^{-A_j}$ if $j\leq i.$ Let $v_i=(B_{i,j})_{j\in \math... |
14,828 | <p>I am reading the book by Goldrei on Classic Set Theory. My question is more of a clarification. It is on if we are overloading symbols in some cases. For instance, when we define $2$ as a natural number, we define $$2_{\mathbb{N}} = \{\emptyset,\{\emptyset\} \}$$ When we define $2$ as an integer, $2_{\mathbb{Z}}$ is... | Qiaochu Yuan | 232 | <p>Yes, in the sense that in ZF, each of the above definitions defines a different collection of sets. Also yes, in the sense that one should always be aware of context when discussing a mathematical object.</p>
<p>The mathematical fact that allows us to overload the symbol $2$ here is the existence of a sequence of ... |
14,828 | <p>I am reading the book by Goldrei on Classic Set Theory. My question is more of a clarification. It is on if we are overloading symbols in some cases. For instance, when we define $2$ as a natural number, we define $$2_{\mathbb{N}} = \{\emptyset,\{\emptyset\} \}$$ When we define $2$ as an integer, $2_{\mathbb{Z}}$ is... | Arturo Magidin | 742 | <p>Yes. And no. </p>
<p>You start with $\mathbb{N}$, and define $+$ and $\times$ (and $\lt$ and so on) appropriately.</p>
<p>Then you define an equivalence relation on $\mathbb{N}\times\mathbb{N}$ given by
$$(a,b)\sim(c,d) \Longleftrightarrow a+d=b+c,$$
and call the quotient set $(\mathbb{N}\times\mathbb{N})/\sim$ b... |
192,537 | <p>Can you give me an example that there is a $f \in C_0^{\infty}(\mathbb C)$, such that the equation $\bar \partial u=f$ has no $C_0^{\infty}(\mathbb C)$ solution?</p>
| Summer | 13,534 | <p>We take a $f\in C^\infty_0(\mathbb{C})$ such that supp$f$ is contained in the unit ball and $f \geq 0$ and $f=1$ on some smaller ball. </p>
<p>If we have a compactly-supported solution for the equation, it has to be $\displaystyle u:\zeta\mapsto \frac{1}{2i\pi}\int_{\mathbb{C}}\frac{f(z)}{z-\zeta}dz\wedge d\bar{z}$... |
192,537 | <p>Can you give me an example that there is a $f \in C_0^{\infty}(\mathbb C)$, such that the equation $\bar \partial u=f$ has no $C_0^{\infty}(\mathbb C)$ solution?</p>
| Moya | 192,336 | <p>With respect to the new bounty:</p>
<p>Here's a way to see this. Suppose $u$ is a compactly supported solution to $\bar{\partial}u=f$. Then we have, for large enough $R>0$</p>
<p>$$0=\int_{\partial D(0,R)} u(z)dz = 2i\int_{D(0,R)} \bar{\partial}u(z)\: d\bar{z}\wedge dz=2i\int_{D(0,R)} f(z)\:d\bar{z}\wedge dz$$
... |
2,571,629 | <blockquote>
<p>Let $\mathbb Z$ denote the set of integers and $\mathbb Z_{\ge 0}$
denote the set $\{0,1,2,3,...\}$. Consider the map $f:\mathbb Z_{\ge
0}\times \mathbb Z \to \mathbb Z$ given by $f(m,n)=2^m\cdot(2n+1)$. Then
the map $f$ is</p>
<p>(A)injective but not surjective.</p>
<p>(B)surjective b... | Community | -1 | <p><strong>Hint:</strong></p>
<p>Note that $0$ is not mapped by any pair of $(m,n)$ (as $2^m \geq 1$ for $m \in \mathbb{Z}_{>0}$). So, $\implies \, ?$</p>
|
3,468,537 | <p>I tried to proof this but I'm not sure if it's fine or if I'm missing something. Any help or hints are appreciated.
This is what I got:</p>
<p>By hypothesis we know that <span class="math-container">$p||G|$</span> then <span class="math-container">$|G|=pm$</span>. Supposing that <span class="math-container">$p|m$</... | HallaSurvivor | 655,547 | <p>I read your question as asking "if <span class="math-container">$p \mid |G|$</span> and <span class="math-container">$G$</span> is Abelian, then there is only one Sylow <span class="math-container">$p$</span>-subgroup". If this is not what you meant, please let me know and I'll update my answer.</p>
<p>The proof th... |
179,094 | <p>I have a very long expression in terms of the three functions u1, u2, u3. I am writing below only a small number of terms. I would like to define a <strong><em>rule</em></strong> that keeps terms of order three or less of any multiplication of u1, u2,u3 and their derivatives. </p>
<pre><code> ss= u2[x, y, z, t]^3 ... | David G. Stork | 9,735 | <pre><code>Series[Sin[x + y + z], {x, 0, 3}, {y, 0, 3}, {z, 0, 3}]
</code></pre>
|
179,094 | <p>I have a very long expression in terms of the three functions u1, u2, u3. I am writing below only a small number of terms. I would like to define a <strong><em>rule</em></strong> that keeps terms of order three or less of any multiplication of u1, u2,u3 and their derivatives. </p>
<pre><code> ss= u2[x, y, z, t]^3 ... | Carl Woll | 45,431 | <p>You can use a variation of the idea I gave <a href="https://mathematica.stackexchange.com/a/174472/45431">here</a>:</p>
<pre><code>Normal @ Series[
ss /. {f:u1|u2 -> (s f[#1,#2,#3,#4]&)},
{s, 0, 3}
] /. s->1
</code></pre>
<blockquote>
<p>u2[x, y, z, t]^3 + Derivative[0, 0, 0, 2][u1][x, y, z, t]... |
2,602,438 | <p>For an embedded software implementation I would like to compute,</p>
<p>$S(b) = \sum_{i=1}^{N}log( x_i - b )$,</p>
<p>for various values of $b$. Here $x_i$ is an array of fixed numbers.</p>
<p>Is there a fast way to do this without having to recompute the sum?</p>
<p>--</p>
<p>I tried looking at the Taylor expa... | Community | -1 | <p>If $x^2$ is not larger or equal to $0$, then $x$ is not a real number.</p>
|
2,602,438 | <p>For an embedded software implementation I would like to compute,</p>
<p>$S(b) = \sum_{i=1}^{N}log( x_i - b )$,</p>
<p>for various values of $b$. Here $x_i$ is an array of fixed numbers.</p>
<p>Is there a fast way to do this without having to recompute the sum?</p>
<p>--</p>
<p>I tried looking at the Taylor expa... | Konstantin | 509,087 | <p>You stated:
$$x\in \mathbb{R} \Rightarrow x^2\geq 0$$</p>
<p>Thus the counterpositive is:
$$x^2 \not\geq 0 \Rightarrow x\not\in \mathbb{R}$$</p>
|
2,793,144 | <p>Consider the following
Proposition: </p>
<ul>
<li>Let $A\subseteq B$. Also, let $B\subseteq C$. Thus, $A\subseteq C$.</li>
</ul>
<p>Proof: </p>
<ul>
<li>Let $A\subseteq B$. Also, let $B\subseteq C$.
<blockquote>
<p><strong>What goes here?</strong> Assume $x\in A$. As $x\in A$ and $A\subseteq B$, $x\in B$.
A... | José Carlos Santos | 446,262 | <p>Yes, that works, since all those series that you mentioned converge.</p>
|
2,258,720 | <p>Define a set X of integers recursively as follows:</p>
<p>Base: 5 is in X</p>
<p>Rule 1: If x is in X and x>0, then x+3 is in X</p>
<p>Rule2: If x is in X and x>0, then x+5 is in X</p>
<p>Show that every integer n>7 is in X</p>
<p>I am pretty new to this stuff and I am very lost on this. Please help!</p>
| drhab | 75,923 | <p>It cannot be shown.</p>
<p>$X=\{5,8\}\cup\{n\in\mathbb Z\mid n\geq10\}$ satisfies base and rules but does not contain $9$.</p>
|
256,806 | <p>How can I prove or disprove that $\lim\limits_{n\to \infty} (n+1)^{1/3}−n^{1/3}=\infty$?</p>
<p>My guess is that it is false but I can't prove it.</p>
| Dan Brumleve | 1,284 | <p>The difference of the cube roots of two consecutive positive numbers is always less than the difference of the numbers themselves which is $1$. So, the limit is not $\infty$.</p>
|
2,314,744 | <p>Prove that there is no integer $n \geq 2$ for which $$\frac{3^n - 2^n}{n}$$ is an integer</p>
<p>I really don't know how to start with it except with the parity of n (n being even clearly doesn't make the fraction an integer)</p>
<p>Edit:
If I take n=pk for a prime p then all i get is to disprove the congruence 3... | Bill Dubuque | 242 | <p>Suppose prime $\,\color{#0a0}{p\mid n\mid 3^n\!-2^n}.\,$ Then $p$ is odd so $\bmod p\!:\ 2$ is invertible so $\,3/2\,$ exists so</p>
<p>$$ \bmod p\!:\ \ (3/2)^{\large p-1}\!\equiv \color{#0a0}{1\equiv (3/2)^{\large n}}\Rightarrow\, (3/2)^{\large \color{#c00}{(p-1,n)}}\equiv 1$$ </p>
<p>However if $\,p\,$ is the <e... |
895,325 | <p>I am dealing with some nice rings that are always isomorphic to some fairly nice quotient ring of a polynomial ring. A typical example is:</p>
<p>$$ \mathbb{C}[X,XY,XY^2] \cong \frac{\mathbb{C}[U,V,W]}{\langle V^2 - UW \rangle}. $$</p>
<p>I would like a nice way to write the Kahler differentials of such rings. Fo... | Adam Hughes | 58,831 | <p>First note $f(1)=1$, then let compute $f(2^k)= 1$.</p>
<p>Note that the total number of divisors function is $\tau(n)$ which is multiplicative, so considering if $n=2^km$ with $m$ odd, we can see that $f(n)=f(m)$, so</p>
<p>$$f(2^km)={\tau(n)\over \tau(2^k)}=\tau(m)$$</p>
<p>and we know $\tau$ is multiplicative.<... |
2,416,848 | <p>Unfortunately,I've no idea of dealing with this problem.The term <em>isomorphism class</em> leads me to think of fundamental theorem of finitely generated abelian groups but i think i'm going in wrong direction.</p>
<p>Please give me some hint?suggestions?Anything?</p>
| Kenny Lau | 328,173 | <p>$\Bbb Z_5[i] \cong \Bbb Z_5[x]/\langle x^2+1 \rangle = \Bbb Z_5[x]/\langle(x+2)(x+3)\rangle \cong \Bbb Z_5[x]/\langle x+2 \rangle \times \Bbb Z_5[x]/\langle x+3 \rangle$ since $\langle x+2\rangle + \langle x+3 \rangle = \langle 1 \rangle$.</p>
<p>So it is isomorphic to $\Bbb Z_5 \times \Bbb Z_5$.</p>
<p>So its uni... |
3,202,530 | <p>This seems easy. But it isn't. The diameter is given as <span class="math-container">$16$</span> and it asks you to find the coordinates of point <span class="math-container">$9$</span>. It's tempting to say that it is <span class="math-container">$(4, 4\sqrt{3})$</span>, but that isn't the answer. What the heck am ... | Claude Leibovici | 82,404 | <p>Considering the original equation
<span class="math-container">$$y''=\frac C {y}$$</span> you need to use first the fact that
<span class="math-container">$$\frac{d^2t}{dy^2}=-\frac{\dfrac{d^2y}{dt^2}}{\left(\dfrac{dy}{dt}\right)^3}\implies \dfrac{d^2y}{dt^2}=-\frac{\dfrac{d^2t}{dy^2}}{\left(\dfrac{dt}{dy}\right)^3}... |
296,259 | <p>My friends argue this $d_t( \partial_{\dot{x}} g)=1+2\dot{\dot{x}} \not = \partial_t (\partial_{\dot{x} }g)$ where $g=t\dot x + x^2 + \dot{x}^2$. Why?</p>
| hhh | 5,902 | <p>$g=t\dot x+x^2+\dot x^2$</p>
<p><strong>Total derivative</strong></p>
<blockquote>
<p>$\frac{dg}{dt}=\dot x+2x\frac{dx}{dt}+2\dot x \frac{d\dot x}{dt}$</p>
</blockquote>
<p><strong>Partial derivative</strong> where we consider other variables as constants</p>
<blockquote>
<p>$\frac{\partial g}{\partial t}=\d... |
3,072,198 | <p>How can one explain that <span class="math-container">$$\frac{d}{dx}\left(\int_0^x{\cos(t^2+t)dt}\right) = \cos(x^2+x)$$</span>
Without solving the integral?</p>
<p>I know it's related to the fundamental theorem of calculus, but here we have a derivative with respect to <span class="math-container">$x$</span>, whil... | Lorenzo B. | 392,037 | <p>By the fundamental theorem of calculus, <span class="math-container">$\int_a^b{f(t)dt}=F(b)-F(a)$</span>, where <span class="math-container">$F'(x)=f(x)$</span>. In this case:
<span class="math-container">$$\int_0^x{\cos(t^2+t)dt}=F(x)-F(0)$$</span>
<span class="math-container">$$\frac{d}{dx}(F(x)-F(0))=F'(x)=f(x)=\... |
3,690,260 | <blockquote>
<p>Find two vectors <span class="math-container">$v_1$</span> and <span class="math-container">$v_2$</span>, whose sum is <span class="math-container">$\langle 0,-2\rangle$</span> where <span class="math-container">$v_1$</span> is parallel to <span class="math-container">$\langle 4,-3\rangle $</span>, wh... | Arturo Magidin | 742 | <p>It is not true that being perpendicular to <span class="math-container">$\langle 4,-3\rangle$</span> means it must equal zero; the dot product with <span class="math-container">$\langle 4,-3\rangle$</span> must equal zero, which tells you that <span class="math-container">$v_2=\langle x_2,y_2\rangle$</span> must be ... |
3,179,874 | <p>This question came up on a recent linear algebra exam of mine, and it's been bothering me ever since. The group is defined such that every element plus the identity matrix is invertible:</p>
<p><span class="math-container">$$(G,\ast):=\{\textbf{A}\in G | \textbf{A}+\textbf{E}_{n} \text{ is invertible}\}$$</span></p... | DanielWainfleet | 254,665 | <p>Hint.</p>
<p>Let the identity matrix be <span class="math-container">$I.$</span> Let the "zero matrix" be <span class="math-container">$0.$</span> We have <span class="math-container">$$A*B=(B+I)(A+I)-I.$$</span> And <span class="math-container">$A*0=0*A=A$</span> so <span class="math-container">$0$</span> is the ... |
987,962 | <p>Find an $x$ in $\Bbb R$ for which rank of the matrix $$A=\begin{bmatrix}1 & 1&1&1 \\1 & -1&-1&1\\1 &-3 &-3 &x \end{bmatrix}$$
is as minimal/maximal as possible.</p>
<p>I was thinking of row reducing the matrix and then count the pivot.</p>
| John | 105,625 | <p>For $y\in[1,2]$ $F(y)=F(1)+\int_{1}^y 2-t \, dt=\frac{1}{2}+2(y-1)-\frac{1}{2}(y^2-1)=-\frac{1}{2}y^2+2y-1$</p>
|
2,975,505 | <p>Assume that g : <strong>R</strong> → <strong>R</strong> is a bijection and define f : <strong>R</strong> → <strong>R</strong> by</p>
<p>f(x) = 2g(x) + 1.</p>
<p>Determine, with proof, whether f is a bijection.</p>
<p>My opinion:</p>
<p><em>I know that a function should either be increasing or decreasing in inter... | trancelocation | 467,003 | <p>You may proceed as follow.</p>
<ul>
<li><span class="math-container">$f([0,2\pi]) \subset [0,2\pi ] \Rightarrow f$</span> has a fixpoint according to the <a href="http://mathworld.wolfram.com/FixedPointTheorem.html" rel="nofollow noreferrer">fixed point theorem</a>.</li>
<li><span class="math-container">$|f'(x)| =|... |
970,872 | <p>There is a common brain teaser that goes like this:</p>
<p>You are given two ropes and a lighter. This is the only equipment you can use. You are told that each of the two ropes has the following property: if you light one end of the rope, it will take exactly one hour to burn all the way to the other end. But it d... | Ranajoy | 281,213 | <p>Let's say yes half burns in 5 mins and half burns in 55 mins. We now lit fire in both the ends. Now whatever rate might be of burning, the total rope get burned out at 30 mins. </p>
<p>Now take the 2nd rope , lit it's two ends, and lit any two more portions of the rope. That makes the rope burn down in 15 mins. So ... |
2,682,008 | <p>I’ve been studying general form Ricatti Differential Equations recently, and I’m confused as to why a general form solution is not possible. What would it mean if a general form solution were possible? </p>
| JJacquelin | 108,514 | <p>The well known way to solve the Riccati ODE is first to find a particular solution. Then it is easy to transform it to a first order linear ODE. See Eqs.[7-8} in :
<a href="http://mathworld.wolfram.com/RiccatiDifferentialEquation.html" rel="nofollow noreferrer">http://mathworld.wolfram.com/RiccatiDifferentialEquatio... |
43,690 | <p>I have to apologize because this is not the normal sort of question for this site, but there have been times in the past where MO was remarkably helpful and kind to undergrads with similar types of question and since it is worrying me increasingly as of late I feel that I must ask it.</p>
<p>My question is: what ca... | Todd Trimble | 2,926 | <p>I appreciate the OP's question, but it seems to me that </p>
<blockquote>
<p>One can rewrite their books in modern language and notation or guide others to learn it too but I never believed this was the significant part of a mathematician work; which would be the creation of original mathematics</p>
</blockquote>... |
43,690 | <p>I have to apologize because this is not the normal sort of question for this site, but there have been times in the past where MO was remarkably helpful and kind to undergrads with similar types of question and since it is worrying me increasingly as of late I feel that I must ask it.</p>
<p>My question is: what ca... | Bob Pego | 9,057 | <p>On this issue I find a deal of comfort in the concluding paragraph of G. H. Hardy's <i>A Mathematician's Apology</i>:</p>
<p>The case for my life, then, or for that of any one else who has
been a mathematician in the same sense which I have been one, is
this: that I have added something to knowledge, and helped ot... |
3,146,447 | <p>I have a function <span class="math-container">$y=1/(1+e^{-x})$</span>. I have been asked to use the first derivative to find any stationary points and then use the second derivative to classify them and provide points of inflection.</p>
<p>When I derive the function, I get the result <span class="math-container">$... | Community | -1 | <p>The first derivative is</p>
<p><span class="math-container">$$\frac{e^{-x}}{(1+e^{-x})^2},$$</span> which has no roots. Hence there are no stationary points.</p>
<p>The second derivative is</p>
<p><span class="math-container">$$\frac{e^{-2x}-e^{-x}}{(1+e^{-x})^3},$$</span> which has a single root at <span class="... |
4,567,552 | <p>Calculate the residues of <span class="math-container">$\frac{1}{z^4+1}$</span> in terms of it's poles. What I've done is reduce <span class="math-container">$(x-x^3)(x-x^5)(x-x^7)$</span> mod <span class="math-container">$(x^4+1)$</span> to get <span class="math-container">$4x^3$</span>. So we get <span class="math... | Mark Viola | 218,419 | <p>Using L'Hospital's Ruel, the residues are</p>
<p><span class="math-container">$$\lim_{z\to z_n}\frac{z-z_n}{z^4+1}=\lim_{z\to z_n}\frac1{4z^3}=\frac1{4z_n^3}=-\frac14 z_n$$</span></p>
<p>where <span class="math-container">$z_n=e^{i(2n-1)\pi/4}$</span>, for <span class="math-container">$n=1,2,3,4$</span>, are the loc... |
544,763 | <p>Let $A\subseteq[a,b]$ be Lebesgue measurable, such that: $m(A)>\frac{2n-1}{2n}(b-a)$. I need to show that $A$ contains an arithmetic sequence with <em>n</em> numbers ($a_1,a_1+d,...,a_1+(n-1)*d$ for some d).</p>
<p>I thought about dividing [a,b] into n equal parts, and show that if I put one part on top of the o... | Niels J. Diepeveen | 3,457 | <p><strong>Hint:</strong> You are on the right track. Have you noticed that the length of each of
your sub-intervals is $\frac{b-a}{n}$, while the total length of all the missing pieces is only $\frac{b-a}{2n}$?</p>
|
888,486 | <p>Q: Solve the following limits using a table of values</p>
<p>$$\lim_{x\to0}\frac {4^x-1}{8^x-1}$$</p>
<p>I tried rewriting it as</p>
<p>$$\lim_{x\to0}\frac{2^{2x}-1}{2^{3x}-1}$$</p>
<p>but I do not know where to go from this, I got this question from <a href="http://math.bard.edu/~mbelk/math141/LimitsExercises.p... | Adam Hughes | 58,831 | <p>$\require{cancel}$</p>
<p>Yes it can be solved algebraically if you write it as</p>
<p>$$\lim_{x\to 0}{\cancel{(2^x-1)}(2^x+1)\over \cancel{(2^x-1)}(2^{2x}+2^x+1)}$$</p>
<p>then cancel the factors in common and be left with</p>
<p>$$\lim_{x\to 0}{(2^x+1)\over (2^{2x}+2^x+1)}$$</p>
<p>into which you can just plu... |
2,021,324 | <blockquote>
<p>Showing $\frac12 3^{2-p}(x+y+z)^{p-1}\le\frac{x^p}{y+z}+\frac{y^p}{x+z}+\frac{z^p}{y+x}$ with $p>1$, and $x,y,z$ positive </p>
</blockquote>
<p>By Jensen I got taking $p_1=p_2=p_3=\frac13$, (say $x=x_1,y=x_2,z=x_3$)</p>
<p>$\left(\sum p_kx_k\right)^p\le\sum p_kx^p$ which means;</p>
<p>$3^{1-p}(x... | user257 | 337,759 | <p>WLOG, $x+y+z=1\tag1$ then by power mean inequality;</p>
<p>$\sum\limits_{cyc}\left(\frac13(x+y)^{-1}\right)^{-1}\le(x+y)^{\frac 13}(y+z)^{\frac13}(x+z)^{\frac1 3}$</p>
<p>and AM-GM</p>
<p>$(x+y)^{\frac 13}(y+z)^{\frac13}(x+z)^{\frac13}\le\frac13(x+y)+\frac13(y+z)+\frac13(x+z)\overset{(1)}=\frac23$</p>
<p>$\displ... |
1,312,783 | <p>I have come across the following formula:</p>
<p>$$u(n)=\sum_{m=-\infty}^{n}\delta(m)$$</p>
<p>where $u(n)$ is the Unit Step and $\delta(m)$ is the Delta Function:</p>
<p>What I can't understand is how this formula "works".</p>
<p>Expanding the formula we have:</p>
<p>$$u(n)=...+\delta(0)+\delta(1)+\delta(2)+..... | JimmyK4542 | 155,509 | <p>The function $f(x) = \sqrt{x+1}-\sqrt{x}$ is defined for $x \ge 0$, and differentiable for $x > 0$. As it turns out, there is no point $x > 0$ at which $f'(x) = 0$ or $f'(x)$ is undefined, which is why you got no solution. </p>
<p>Since $f(x)$ is continuous for $x \ge 0$ and differentiable for $x > 0$, thi... |
18,861 | <p>I hope I am asking my question in the right forum.</p>
<p>I am trying to introduce some mathematical problems (Better to be famous in the math community) to a group of senior high school students with a typical background in high school mathematics like (differentiation and applications - basic probability- basic pl... | ruferd | 8,527 | <p>Some problems that are easy to understand (but not understand the solution to):</p>
<ul>
<li><a href="https://en.wikipedia.org/wiki/Fermat%27s_Last_Theorem" rel="noreferrer">Fermat's Last Theorem</a></li>
<li><a href="https://en.wikipedia.org/wiki/Four_color_theorem" rel="noreferrer">Four Color Theorem</a></li>
<li>... |
1,072,750 | <p>I'm looking at this solution to this problem:</p>
<p><img src="https://i.stack.imgur.com/qQTTU.png" alt="enter image description here"></p>
<p>I'm getting thrown off by the special case where $n = 2$. If $n = 2$, why must it be that $x = 1$? All that we then know is that $x^2 = 1$ or that $x = x^{-1}$. However, I ... | Slade | 33,433 | <p>You're plugging in wrong: $n=2$ is not the case $x^2 = 1$, but rather the case $x^1 = 1$.</p>
<p>But the special case isn't part of the proof, just an illustration. In general, the point is that if some power—any power—of $x$ equals $1$, then $x$ is invertible.</p>
|
3,697,011 | <blockquote>
<p>For what value of <span class="math-container">$a$</span> would the following function have exactly one solution?
<span class="math-container">$$a^2x^2+3x-5\frac{1}{a}=0$$</span></p>
</blockquote>
<p>I know that it needs to become
<span class="math-container">$$\frac{3}{2}x^2+3x+\frac{3}{2}=0$$</... | Harish Chandra Rajpoot | 210,295 | <p>The equation of parabola: <span class="math-container">$y=a^2x^2+3x-\frac{5}{a}$</span> can be rewritten as follows
<span class="math-container">$$\left(x+\frac{3}{2a^2}\right)^2=\frac{1}{a^2}\left(y+\frac{5}{a}+\frac{9}{4a^2}\right)\equiv X^2=4AY$$</span>
The given quadratic equation: <span class="math-container">$... |
1,201,717 | <p>Inn$(G)=\{\varphi_g \in \text{Aut}(G) \mid g \in G\}$</p>
<p>If $\varphi_g, \varphi_h \in \text{Inn}(G)$, then
$$\varphi_g \varphi_h (x) =\varphi_g(hxh^{-1})=ghxh^{-1}g^{-1}=ghx(gh)^{-1}=\varphi_{gh} \in \text{Inn}(G)$$
Also, since $\varphi_g\varphi_g^{-1}=x$, and $\varphi_g\varphi_{g^{-1}}=x$,
$$\varphi_g^{-1}=\va... | slider | 156,607 | <p>It's well-defined because you defined it.</p>
<p>The times when you need to check something is well-defined is when you give a "definition" that involves making some extra choices; you then have to check that your construction is independent of these choices. You didn't make any choices so your question doesn't see... |
3,296,638 | <p>Are two planes parallel if the magnitude of the cross product of their normal vectors is equal to <span class="math-container">$0$</span>?</p>
<p><span class="math-container">$|| \vec n_1 \times \vec n_2|| = 0$</span></p>
| David | 651,991 | <p>Indeed:</p>
<ul>
<li>Two planes are parallel if, and only if, their normal vectors are parallel</li>
<li>Two vectors are parallel if, and only if, their cross product is the null vector.</li>
</ul>
|
3,528,162 | <p>I'm encountering the following interesting math problem:</p>
<p>A device consists of 5 independently working blocks. </p>
<p>Each of them has a damage probability of 1/4. Upon damage of 1, 2 or 3 blocks, the probabilities for shutting the device down are respectively 1/5 , 2/5, 4/5. </p>
<p>If more blocks are da... | David Quinn | 187,299 | <p>Hint 1. The number of damaged blocks follows a Binomial distribution with <span class="math-container">$n=5$</span> and <span class="math-container">$p=\frac14$</span></p>
<p>Hint 2. You need to calculate the conditional probability of 2 blocks damaged given that the device breaks down</p>
|
255,215 | <p>I am very new to Mathematica and already spent a lot of time trying to do this but failed.</p>
<p>I am trying to solve an ODE:</p>
<pre><code>solution = DSolve[{-((m (1 + m) + 4/(9 (-2/3 + t) t)) y[t]) +
2 (-1/3 + t) y'[t] + (-2/3 + t) t y''[
t] == (-4 (1 + C/2))/(9 (-2/3 + t) t), y[1] == 1, y'[1] == C}... | user64494 | 7,152 | <p>Here is an answer for <code>m==2</code> in version 13.1 on Windows 10 (<code>C</code> is reserved in WL.)</p>
<pre><code>ClearAll[m, c]; $Assumptions = {m \[Element] Integers, c \[Element] Reals}; m = 2;
solution = AsymptoticDSolveValue[{-((m (m + 1) + 4/(9 (-2/3 + t) t)) y[t]) +
2 (-1/3 + t) y'[t] + (-2/3 + t) t... |
2,246,522 | <p>I found a form of the lagrange theorem that I don't know,</p>
<p>I didn't find something similar on the internet.</p>
<blockquote>
<p>Suppose that $f:[a,b]\times[c,d] \subset \mathbb{R}^2 \longrightarrow \mathbb{R}$ is a continuous function. Consider the function</p>
<p>\begin{align}
I:[c,d] &\longright... | Community | -1 | <p>Note that the following is equivalent to MVT:</p>
<blockquote>
<p>Let <span class="math-container">$a\in \Bbb R$</span> and <span class="math-container">$k>0$</span>. If <span class="math-container">$f: [a,a+k] \to \Bbb R$</span> is continuous on <span class="math-container">$[a,a+k]$</span> and differentiable on... |
1,690,248 | <p>Hello I am hoping to find some direction on solving this try it yourself problem in my textbook.</p>
<p>Let S be an arbitrary set of symbols and let $\Phi = \{v_0 \equiv t | t \in T^S\} \cup \{\exists v_0 \exists v_1 \neg v_0 \equiv v_1\}$. </p>
<p><em>Note: $T^S$ is the set of all S-terms.</em></p>
<p>Show that ... | Ross Millikan | 1,827 | <p>If you want the average speed over the last $5$ minutes you can just add up the last $300,000$ speed values and divide by $300,000$. Once you get the sum started you can just update it by adding the new value and subtracting the old one. Storing the speed data in a <a href="https://en.wikipedia.org/wiki/Circular_b... |
1,690,248 | <p>Hello I am hoping to find some direction on solving this try it yourself problem in my textbook.</p>
<p>Let S be an arbitrary set of symbols and let $\Phi = \{v_0 \equiv t | t \in T^S\} \cup \{\exists v_0 \exists v_1 \neg v_0 \equiv v_1\}$. </p>
<p><em>Note: $T^S$ is the set of all S-terms.</em></p>
<p>Show that ... | mapexmbirch | 321,339 | <p>Cheers for your help guys.</p>
<p>I did it on my own using a spreadsheet and patience. The formula is below, Avmph is the current average speed and inc is incremented after each time the speed is measured. </p>
<p>Avmph =((Avmph*(inc-1))+mph)/inc </p>
|
2,956,460 | <p>Consider the "formal definition" here <a href="https://en.wikipedia.org/wiki/Limit_of_a_sequence" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Limit_of_a_sequence</a>. I checked some references and this is often precisely the definition in all words and terms used in this article. I claim that this defini... | Community | -1 | <p><span class="math-container">$N$</span> need not be a function of <span class="math-container">$\epsilon$</span>. For a given <span class="math-container">$\epsilon$</span>, many <span class="math-container">$N$</span> are possible (in particular all values larger than <span class="math-container">$N$</span> also wo... |
2,956,460 | <p>Consider the "formal definition" here <a href="https://en.wikipedia.org/wiki/Limit_of_a_sequence" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Limit_of_a_sequence</a>. I checked some references and this is often precisely the definition in all words and terms used in this article. I claim that this defini... | user21820 | 21,820 | <p>The wikipedia article states that we call <span class="math-container">$x$</span> the limit of the (real) sequence <span class="math-container">$(x_n)$</span> if the following condition holds:</p>
<blockquote>
<p><span class="math-container">$∀ ε > 0 ( ∃ N ∈ \mathbb N ( ∀ n ∈ \mathbb N ( n ≥ N ⇒ | x_n − x | &l... |
1,253,382 | <p>I am almost sure that this would have been asked before, but how can one find
$$
\int \frac{x^2}{1+x^2} dx?
$$
If I had a $x^2 - 1$ in the denominator, then I could factor into $(x-1)(x+1)$ and use partial fractions. But here I have an irreducible $1+x^2$.</p>
<p>I am sure that this question has already been answer... | user222031 | 222,031 | <p>Hint: $$x^2 = x^2 + 1 - 1 $$
or you could try this substitution:
$$x = \tan(\theta)$$</p>
|
3,340,374 | <p>I'm writing up several proofs for myself, all of which have a particular sticking point.</p>
<p>Essentially, I want to prove that for a function <span class="math-container">$f$</span> of two real variables, we have</p>
<p><span class="math-container">$\lim_{h \to 0} \frac{f(x \;+\; h, \; y \;+\; h) \; - \; f(x, \... | Ninad Munshi | 698,724 | <p><span class="math-container">$$\lim_{h\to 0} \frac{f(x+h,y+h)-f(x,y+h)}{h} = \lim_{h\to 0} \frac{f(x+h,y+h)-f(x,y)+f(x,y)-f(x,y+h)}{h}$$</span> <span class="math-container">$$ = \lim_{h\to 0} \frac{f(x+h,y+h)-f(x,y)}{h} - \lim_{h\to 0} \frac{f(x,y+h)-f(x,y)}{h}$$</span> <span class="math-container">$$ = \sqrt{2}\lim... |
2,260,172 | <p>I know that if two topological spaces $X$ and $Y$ are homeomorphic then so are their one point compactifications $X^*$ and $Y^*$. If $X$ and $Y$ (say both are smooth manifolds) are diffeomorphic what do we know about $X^*$ and $Y^*$? Are they diffeomorphic too or only homeomorphic? If the latter is true, are there ... | freakish | 340,986 | <p>One point compactification of a manifold does not always have to be a manifold. For more details read this:</p>
<p><a href="https://math.stackexchange.com/questions/240339/one-point-compactification-of-manifold">One-point compactification of manifold</a></p>
<p>But even when it is, I don't think it has to be uniqu... |
3,402,800 | <p>Let,<span class="math-container">$E$</span> be a zero dimensional sheaf on an algebraic surface <span class="math-container">$S$</span> over the field of complex numbers,i.e <span class="math-container">$dim(Supp(E)) =0$</span></p>
<p>Then my question is the following : Is it true that <span class="math-container">... | Dominique Mattei | 549,131 | <p>I develop my comment here. I assume your sheaf is coherent as you are interested in algebraic surfaces.</p>
<p>Thus let <span class="math-container">$E$</span> be a coherent sheaf on an algebraic variety. Assume <span class="math-container">$Supp(E)=\{x_1, \dots, x_n\}$</span> is finite. Write <span class="math-con... |
893,259 | <p>I have tried this problem multiple times, I have the solution but not the steps. I keep getting the wrong answer. I believe it may be in the algebra after I have taken the Laplace on both sides.</p>
<p>$y''+y = \sin(t) ;\:\: y(0) = 1, \:\: y'(0) = -1 $</p>
| IAmNoOne | 117,818 | <p>You know that the LHS is,</p>
<p>$$\mathcal{L}(y'') + \mathcal{L}(y)= [s^2 \mathcal{L}(y) - sy(0) - y'(0)] + \mathcal{L}(y) = \mathcal{L}(y)(s^2+1) - s + 1, $$</p>
<p>and for the RHS we have, $$\mathcal{L}(\sin t) = \dfrac{1}{s^2 + 1}.$$</p>
<p>Equate both sides and isolate for $\mathcal{L}(y)$ to get, $$\mathcal... |
1,768,188 | <p>I used to compute complexity of an algorithm which reaches to constant value after x level because of $a^\frac{1}{x}=1$. Now I need to find $x$ to reach answer.</p>
<blockquote>
<p>To describe more : my recursive algorithm is $T(n)=2T(\sqrt{n})+log_{2}^{n}$ ; ans $T(1)=1$. I tried to solve it using binary tree wh... | Riccardo Orlando | 335,442 | <p>If $a =1$, any $x\neq 0$ will do. Otherwise, you would need $1/x=0$ which of course never happens (in the usual $\Bbb{R}$)</p>
|
1,768,188 | <p>I used to compute complexity of an algorithm which reaches to constant value after x level because of $a^\frac{1}{x}=1$. Now I need to find $x$ to reach answer.</p>
<blockquote>
<p>To describe more : my recursive algorithm is $T(n)=2T(\sqrt{n})+log_{2}^{n}$ ; ans $T(1)=1$. I tried to solve it using binary tree wh... | Ethan Hunt | 292,757 | <p>You could say $\lim_{x\to\infty}a^{1/x}=1$, but has been already noted there is no specific value for $x$ that results in $a^{1/x}$ equaling $1$. </p>
|
4,490,258 | <p>I've recently started learning hyperbolic functions and inverse hyperbolic functions, and I came across this equation involving inverse hyperbolic functions. I tried to solve it numerically (I got x=-0.747), but how would you solve it analytically? I don't know how to type in latex so please forgive me.</p>
<p><span... | jjagmath | 571,433 | <p>We have</p>
<p><span class="math-container">$$\sinh^{-1}(x)=-\cosh^{-1}(x+2)$$</span></p>
<p>Apply <span class="math-container">$\cosh^2(x)$</span> on both sides</p>
<p><span class="math-container">$$\cosh^2(\sinh^{-1}(x))=\cosh^2(-\cosh^{-1}(x+2))=(x+2)^2$$</span></p>
<p>since <span class="math-container">$\cosh$</... |
877,850 | <p>I'm trying to calculate the expected area of a random triangle with a fixed perimeter of 1. </p>
<p>My initial plan was to create an ellipse where one point on the ellipse is moved around and the triangle that is formed with the foci as the two other vertices (which would have a fixed perimeter) would have all the ... | bonsoon | 48,280 | <p>Note $\pi/2$ is $90^\circ$. Consider the rectangle in the following diagram:
<img src="https://i.stack.imgur.com/E5tRg.png" alt="enter image description here"></p>
|
1,879,509 | <p>Please forgive the crudeness of this diagram.</p>
<p><a href="https://i.stack.imgur.com/AoS2Q.png" rel="noreferrer"><img src="https://i.stack.imgur.com/AoS2Q.png" alt="enter image description here"></a></p>
<p>(I took an image from some psychobabble website and tried to delete the larger circle that's not relevant... | Robert Israel | 8,508 | <p>Let's say the circles each have radius $1$. Rotating a bit for convenience, their centres could be at $[1,0]$, $[0,1]$, $[-1,0]$, $[0,-1]$. The area of your black region is the area of a circle minus twice the area common to two adjacent circles. See e.g. <a href="http://mathworld.wolfram.com/Circle-CircleIntersec... |
710,374 | <p>Let $A$ be an $m\times n$ matrix. If the rank of $A$ is $m$, then prove there exists a matrix $B$, wich is $n \times m$, such that $AB=\text{I}_m$</p>
| Brad S. | 133,841 | <p>Suppose $A$ is an $(mxn)$ matrix with $rank(A) = m$</p>
<p>then $n \ge m $ and the column space of $A$ spans all of $\mathbb{R}^m$.</p>
<p>So, $\forall b\in \mathbb{R}^m, \exists x\in \mathbb{R}^n$ s.t. $Ax = b$.</p>
<p>Choose $b_i$ to be the $i^{th}$ column of $I_m$. The the columns of $B (nxm)$ are the vectors ... |
1,619,464 | <p>I tried using the universal property of tensor products to show that there are mutually inverse maps from $M \times N$ to $M' \times N$, and use this to show that $M \cong M'$, but I didn't get far. I know that this is true if $N = R$, but I couldn't think of a counterexample.</p>
| Michael Albanese | 39,599 | <p>This is not true in general. For example, $\mathbb{Q}\otimes_{\mathbb{Z}}(\mathbb{Z}/p\mathbb{Z}) \cong 0 \cong 0\otimes_{\mathbb{Z}}(\mathbb{Z}/p\mathbb{Z})$ but $\mathbb{Q}$ and $0$ are not isomorphic as $\mathbb{Z}$-modules.</p>
|
216,055 | <p>Does anyone have a suggestion for the best computer program to perform calculations in the 2nd Weyl algebra? </p>
| unknown | 16,739 | <p>I use GAP (<a href="http://www.gap-system.org/" rel="nofollow">http://www.gap-system.org/</a>) with the GBNP package (<a href="http://www.win.tue.nl/~amc/pub/grobner/chap0.html" rel="nofollow">http://www.win.tue.nl/~amc/pub/grobner/chap0.html</a>).
You define Weyl algebra as a quotient of free associative algebra by... |
3,213,253 | <p>We're given dynamical system:</p>
<p><span class="math-container">$$
\dot x = -x + y + x (x^2 + y^2)\\
\dot y = -y -2x + y (x^2 + y^2)
$$</span></p>
<p>Question is what's the largest constant <span class="math-container">$r_0$</span> s.t. circle <span class="math-container">$x^2+y^2 < r_0^2$</span> lies in the ... | Cesareo | 397,348 | <p>After solving the system</p>
<p><span class="math-container">$$
\dot r = \frac{r}{2}(-2-\sin(2 \phi)+2r^2) \\
\dot \phi = -(1+\cos^2(\phi))
$$</span></p>
<p>we have</p>
<p><span class="math-container">$$
\left\{
\begin{array}{rcl}
r & = & \frac{\sqrt{3} \sqrt{3-\cos \left(2 \sqrt{2} \left(t-2 c_1\right... |
2,794,228 | <p>If we define a sequence $a_n$ where $n\in\mathbb{N}$ by $$a_n=\frac{1}{({2n\pi})^{\frac{1}{3}}}$$
Then how could one show the inequality $$a_n-a_{n+1}\leq Cn^{-\frac{4}{3}}$$</p>
<p>I have tried got to a point where $$a_n-a_{n+1}\leq(\frac{2\pi}{(2n\pi)(2(n+1)\pi)})^{1/3}$$</p>
| user | 505,767 | <p>We have that</p>
<p>$$a_n-a_{n+1}=\frac{1}{({2n\pi})^{\frac{1}{3}}}-\frac{1}{( {2(n+1)\pi})^{\frac{1}{3}} }=\frac{1}{{(2(n+1)\pi})^{\frac{1}{3}}}\left(\left(1+\frac1n\right)^\frac13-1\right)\le$$</p>
<p>$$\le\frac{1}{(2\pi)^\frac13n^\frac13}\left(1+\frac1{3n}-1\right)=\frac{1}{3(2\pi)^\frac13}\frac1{n^\frac43}$$</... |
46,505 | <p>What do I do if I have to solve the usual quadratic equation $X^2+bX+c=0$ where $b,c$ are in a field of characteristic 2? As pointed in the comments, it can be reduced to $X^2+X+c=0$ with $c\neq 0$.</p>
<p>Usual completion of square breaks. For a finite field there is <a href="http://www.ams.org/mathscinet-getitem?... | Timothy Wagner | 10,775 | <p>Here is a paper that might help</p>
<p><a href="http://www.raco.cat/index.php/PublicacionsMatematiques/article/viewFile/37927/40412">http://www.raco.cat/index.php/PublicacionsMatematiques/article/viewFile/37927/40412</a></p>
|
3,523,491 | <p>Consider a <span class="math-container">$m \times n$</span> matrix <span class="math-container">$M$</span> such that each cell in <span class="math-container">$M$</span> is equal to the sum of its adjacent cells (sharing either an edge or a corner with this cell). What are the values in this matrix. </p>
<p>I am tr... | David E Speyer | 448 | <p>The point of this answer is to show that the only solutions are Jaroslaw Matlak's solution (when <span class="math-container">$m \equiv 1 \bmod 2$</span> and <span class="math-container">$n \equiv 2 \bmod 3$</span>), the transpose of his solution, and linear combinations of the above (when <span class="math-containe... |
691,927 | <p>Now before I begin, I know this question has been asked multiple times but all the answers but I had so many questions of my own that I figured I should make a new question as my thoughts are different than previous answers. </p>
<p>Now I will ask the question first then explain my thoughts and troubles :)</p>
<p>... | JPi | 120,310 | <p>Don't try to take derivatives. Note that the density of the uniform distribution is</p>
<p>$$ \frac{1}{b-a} I(a<X<b),$$</p>
<p>where $I$ is the <em>indicator function</em>. So your likelihood function should be something like</p>
<p>$$\frac{1}{(b-a)^n} \prod_{i=1}^n I(a<X_i<b).$$</p>
<p>Now eyeball... |
1,416,661 | <p>Consider a "spinner": an object like an unmagnetized compass needle that can pivots freely around an axis, and is stable pointing in any direction. You give it a spin and see where it comes to rest, measuring the resulting angle (divided by 2π) as a number from 0 to 1.</p>
<p>I am bit confused, when i look into the... | zoli | 203,663 | <p>You can depict this uniform distribution in many different ways:</p>
<p>(a) Angle measured in radian and divided by $2\pi$ results in a uniform distribution over the interval $[0,1)$. In this case the pdf is</p>
<p>$$f_1(x)=\begin{cases}
1,& \text{ if } 0\le x <1\\
0,& \text{ otherwise. }
\end{cases}$$<... |
4,605,607 | <blockquote>
<p><strong>4.3.</strong> Identify the following rings:</p>
<ol>
<li><p><span class="math-container">$\mathbb{Z}[x] / (x^2 - 3, 2x + 4)$</span>,</p>
</li>
<li><p><span class="math-container">$\mathbb{Z}[i] / (2 + i)$</span>,</p>
</li>
<li><p><span class="math-container">$\mathbb{Z}[x] / (6, 2x - 1)$</span>,... | Jendrik Stelzner | 300,783 | <p>The ideal <span class="math-container">$I = (6, 2x - 1)$</span> contains the element <span class="math-container">$3(2x - 1) = 6x - 3$</span>, but also the element <span class="math-container">$6x$</span>, and therefore the element <span class="math-container">$3 = 6x - (6x - 3)$</span>.
But <span class="math-contai... |
2,939,808 | <p><img src="https://i.stack.imgur.com/5DeV4.jpg" alt="enter image description here"></p>
<p>q 23</p>
<p>my solution<img src="https://i.stack.imgur.com/urLMH.jpg" alt="enter image description here"></p>
<p>I attempted like above.Then to check i plotted graph
on desmos. so it showed me three roots.One before zero al... | John Wayland Bales | 246,513 | <p>Let us define the continuous function</p>
<p><span class="math-container">$$ U(x)=f(x)-h(x)-2x^2+2=2^x+3^x-2(x^2+x+1) $$</span></p>
<p>We know that as <span class="math-container">$x$</span> increases the exponential portion will eventually overpower the polynomial portion and the function will be positive from th... |
3,859,737 | <p>I saw this problem on a math board / insta:</p>
<p><span class="math-container">$$\lim_{x\rightarrow3}\frac{\sqrt{3x}-3}{\sqrt{2x-4}-\sqrt{2}}$$</span></p>
<p>My first step would be take a derivative of the numerator and denominator to see if the limit exists or not, since just plugging in gets me 0/0 which is undef... | user | 505,767 | <p>As noticed there is a mistake with a negative exponent, as an alternative by rationalization</p>
<p><span class="math-container">$$\frac{\sqrt{3x}-3}{\sqrt{2x-4}-\sqrt{2}}=\frac{\sqrt{3x}-3}{\sqrt{2x-4}-\sqrt{2}}\frac{\sqrt{3x}+3}{\sqrt{3x}+3}\frac{\sqrt{2x-4}+\sqrt{2}}{\sqrt{2x-4}+\sqrt{2}}=$$</span></p>
<p><span c... |
2,861,406 | <p>I want to calculate the answer of the integral $$\int\frac{dx}{x\sqrt{x^2-1}}$$ I use the substitution $x=\cosh(t)$ ($t \ge 0$) which yields $dx=\sinh(t)\,dt$. By using the fact that $\cosh^2(t)-\sinh^2(t)=1$ we can write $x^2-1=\cosh^2(t)-1=\sinh^2(t)$. Since $t\ge 0$, $\sinh(t)\ge 0$, and we have $\sqrt{x^2-1}=\si... | packetpacket | 578,206 | <p>The constant! You can't forget the constant!</p>
<p>$$2\arctan(x+\sqrt{x^2-1}) = \arctan(\sqrt{x^2-1}) + \frac{\pi}{2}$$ for $x \ge1$ and
$$2\arctan(x+\sqrt{x^2-1}) = \arctan(\sqrt{x^2-1}) - \frac{\pi}{2}$$ for $x\le1$ (Have a go at proving these statements) </p>
<p>As the solutions differ by a constant, there is ... |
3,206,838 | <blockquote>
<p>The number of 5-digit numbers of the form abcde where a,b,c,d,e belong to <span class="math-container">${0,1,2,...9}$</span> and <span class="math-container">$b = a + c$</span>, <span class="math-container">$d = c + e $</span> are?</p>
</blockquote>
<p>I tried to reason out that out of the 5 digits w... | user614287 | 614,287 | <p>If <span class="math-container">$c=0$</span>, then depending of <span class="math-container">$b \neq 9$</span>(because <span class="math-container">$a$</span> cannot be <span class="math-container">$0$</span>), <span class="math-container">$a$</span> could be from <span class="math-container">$1$</span> to <span cla... |
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