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 |
|---|---|---|---|---|
788,245 | <p>$$\sum_{n=1}^{\infty}\frac{(n+2)!}{(3n-1)}$$ I know this series does not converge. Can someone show me how to prove that? Should i use criteria of Dalamber or any other criteria?</p>
| Hilario Fernandes | 142,528 | <p>If you prove that $(n+2)!$ grows faster than $(3n-1)$, the general term will grow, and the summatory will not converge. So, you can prove this inequality (by induction):</p>
<p>\begin{equation}
((n+1)-2)!- (n-2)! > (3(n+1)-1) - (3n-1)
\end{equation} </p>
<p>This proof method is easy because $a_{n+1} > a_n$.... |
3,260,776 | <blockquote>
<p>Suppose we are given any arbitrary collection of sets.</p>
<p>How do get a largest topology from the above arbitrary collection?</p>
</blockquote>
<p>How to construct a Topology from this collection and in which condition ?</p>
<p>I don't have any answer to me.</p>
<p>Any help or idea is appreciated.</p... | John Hughes | 114,036 | <p>There are a couple of possibilities. </p>
<p>I'm going to assume you have a set <span class="math-container">$U$</span> whose elements are all subsets of one "big" set <span class="math-container">$A$</span>. (For instance: you might have a set of intervals, all of which are subsets of <span class="math-container">... |
2,300,049 | <p>these are my toughs:</p>
<p>$$z^2 = 1 + 2i \Longrightarrow (x+yi)(x+yi) = 1 + 2i$$</p>
<p>so: $x^2-y^2 = 1$ and $2xy = 2$</p>
<p>then i got that $x = 1/y$ but i cant continue to find the real- and imaginary part of z anymore. Appriciated any help</p>
| Zain Patel | 161,779 | <p>You have $x^2 - y^2 = 1$ and $x = \frac{1}{y}$. Substitute the latter into the former to get $$\frac{1}{y^2} - y^2 = 1 \implies 1 - y^4 = y^2 \implies y^4 + y^2 - 1 =0.$$</p>
<p>You can solve the quadratic $u^2 + u -1 = 0$ giving solution $u = \frac{1}{2}(-1 \pm \sqrt{5})$. So you need to solve $y^2 = \frac{1}{2}(-... |
2,300,049 | <p>these are my toughs:</p>
<p>$$z^2 = 1 + 2i \Longrightarrow (x+yi)(x+yi) = 1 + 2i$$</p>
<p>so: $x^2-y^2 = 1$ and $2xy = 2$</p>
<p>then i got that $x = 1/y$ but i cant continue to find the real- and imaginary part of z anymore. Appriciated any help</p>
| Michael Hardy | 11,667 | <p>\begin{align}
z^2 = 1 + 2i & = |1+2i|(\cos\alpha+i\sin\alpha) \text{ where } \tan \alpha = \frac 2 1 \\[10pt]
& = \sqrt{1^2+2^2} (\cos\alpha+i\sin\alpha) = \sqrt 5 (\cos\alpha+i\sin\alpha)
\end{align}
Therefore
$$
z = \pm\left( \sqrt{\sqrt 5} \right)\left( \cos\frac\alpha2 + i \sin\frac\alpha 2 \right).
$$<... |
3,278,761 | <p>Suppose a student says : "if 17 is even, then 2 is not a divisor of 17". </p>
<p>Surely his teacher would tell him he is wrong, saying that when a number is even, this number has 2 as divisor. The teacher would correct with " if 17 were even, then 2 would be a divisor of 17". In other words, the student's claim con... | Maxime Ramzi | 408,637 | <p>The student would be right in claiming "if <span class="math-container">$17$</span> is even, then <span class="math-container">$2$</span> does not divide <span class="math-container">$17$</span>", as well as in claiming "if <span class="math-container">$17$</span> is even, then <span class="math-container">$2$</span... |
134,796 | <p>Example list below. All elements are in the form {1 or 0, 1 or 0, 1 or 0}, with a least one of the numbers 0 and 1 in the element (so excluding {1,1,1} and {0,0,0}) </p>
<pre><code>ListA = {{1, 1, 0}, {1, 1, 0}, {1, 1, 0}, **{0, 1, 1}**, {1, 0, 1}, {1, 0, 1}, {1,
0, 1}}
</code></pre>
<p>I want a command to rep... | Aisamu | 8,238 | <p>Just for kicks:</p>
<pre><code>f[{a__, a__, b__}] := a
f[{a__, b__, b__}] := b
f[{a__, b__, c__}] := c
{First@ListA} ~Join~ Map[f, Partition[ListA, 3, 1]] ~Join~ {Last@ListA}
</code></pre>
<blockquote>
<p>{{1, 1, 0}, {1, 1, 0}, {1, 1, 0}, {1, 0, 1}, {1, 0, 1}, {1, 0, 1}, {1,
0, 1}}</p>
</blockquote>
|
3,082,635 | <p>Prove that for a given prime <span class="math-container">$p$</span> and each <span class="math-container">$0 < r < p-1$</span>, there exists a <span class="math-container">$q$</span> such that </p>
<p><span class="math-container">$$rq \equiv 1 \bmod p$$</span></p>
<p>I've only taken one intro number theory ... | cqfd | 588,038 | <blockquote>
<p><strong>Theorem</strong>:If <span class="math-container">$g $</span> is the greatest common divisor of <span class="math-container">$r $</span> and <span class="math-container">$p $</span>, then
there exists integers <span class="math-container">$q $</span> and <span class="math-container">$k $</s... |
1,331,063 | <p>Let $\cal{H}$ be a Hilbert space, $T$ a bounded linear operator on $\cal{H}$, $S$ a trace class operator, then can one verify that
$$|Tr(TS)|\leq\|T\|\cdot|Tr(S)|?$$</p>
| R.N | 253,742 | <p>It is correct for positive operators, for more information see Gohberg, Krein, Introduction to the Theoryof Linear Non-Self-Adjoint
Operators in Hilbert Space page 27</p>
|
1,345,364 | <p>I am struggling with this question: </p>
<blockquote>
<p>Let $\{a_n\}$ be defined recursively by $a_1=\sqrt2$, $a_{n+1}=\sqrt{2+a_n}$. Find $\lim\limits_{n\to\infty}a_n$. HINT: Let $L=\lim\limits_{n\to\infty}a_n$. Note that $\lim\limits_{n\to\infty}a_{n+1}=\lim\limits_{n\to\infty}a_n$, so $\lim\limits_{n\to\infty... | marty cohen | 13,079 | <p>If you have a recursion
of the form
$a_{n+1}
=f(a_n)
$,
if $L = \lim_{n \to \infty} a_n
$,
then
we must have
$L = f(L)$.</p>
<p>In your case,
$f(x) = \sqrt{2+x}$.</p>
<p>Therefore,
for any limit $L$,
we must have
$L = \sqrt{2+L}$.</p>
<p>Squaring,
$L^2 = L+2$,
which is a standard quadratic equation.</p>
<p>Comp... |
4,112,771 | <p>This is <strong>Exercise 2.3.4</strong> of Robinson's <em>"A Course in the Theory of Groups (Second Edition)"</em>. My universal algebra is a little rusty, so <a href="https://math.stackexchange.com/q/3590724/104041">this question</a> is not what I'm looking for; besides, I ought to be able to use tools gi... | Arturo Magidin | 742 | <p>Let <span class="math-container">$W$</span> be a set of words, let <span class="math-container">$\mathfrak{W}$</span> be the corresponding variety.</p>
<p>The key observation is that for any word <span class="math-container">$w(x_1,\ldots,x_n)$</span>, any groups <span class="math-container">$G$</span> and <span cla... |
4,244,966 | <p>I have the ODE <span class="math-container">$y^2(1+y'^2)=4$</span> to solve this I used the substitution <span class="math-container">$y'=p$</span>
<span class="math-container">$$y^2(1+p^2)=4$$</span>
<span class="math-container">$$2y(1+p^2)dy+2py^2dp=0$$</span>
<span class="math-container">$$(p^2+1)dy+py\;dp=0$$</s... | Alessio K | 702,692 | <p>You didn't make a mistake, but you have arrived back to the original equation. Note that <span class="math-container">$c$</span> can also be <span class="math-container">$-2$</span>.</p>
<p><span class="math-container">$$y\sqrt{p^2+1}=\pm2\implies y^2(1+p^2)=4$$</span></p>
<p>Instead, note that the differential equa... |
567,204 | <p>I'm currently studying CS, and as i didn't do maths A level i'm finding the module particularly difficult. We've now changed topics and lecturer, going onto discrete maths; and i'm refusing to fall behind :P. So, i'm going to post regularly/daily questions, just to make sure i have an understanding.</p>
<p>Hopefull... | gt6989b | 16,192 | <p><strong>Domain</strong> is a set on which valid inputs to $f$ are defined. Since $f$ is defined on $a,b,c,d$, the domain is the entire $X$.</p>
<p><strong>Range</strong> is a set to which $f$ maps the input -- in other words, all possible outputs of $f$. Here, $\{1,2,5\}$. Sometimes range is called <strong>image</s... |
624,002 | <p>Determine whether $\mathbb{Z}\times \mathbb{Z}$ and $\mathbb{Z}\times \mathbb{Z}\times \mathbb{Z}$ are isomorphic groups or not.</p>
<p>pf) Suppose that these are isomorphic. Note that $\mathbb{Z}\times \mathbb{Z}$ is a subgroup of $\mathbb{Z}\times \mathbb{Z}$ and $\mathbb{Z}\times \mathbb{Z}\times\left \{ 0 \righ... | Mikasa | 8,581 | <p>We know that these two groups are <em>free abelian</em> in which for $\mathbb Z\oplus\mathbb Z$ and $\mathbb Z\oplus\mathbb Z\oplus\mathbb Z$ the basis sets don't have the <strong>same cardinal number</strong> , so according to this <a href="http://en.wikipedia.org/wiki/Free_group#Facts%20and%20theorems">Theorem 3.<... |
2,811,155 | <p>I learned recently that there are mathematical objects that can be proven to exist, but also that can be proven to be impossible to "construct". For example see this answer on MSE:<br>
<a href="https://math.stackexchange.com/questions/2808804/does-the-existence-of-a-mathematical-object-imply-that-it-is-possible-to-c... | Mikhail Katz | 72,694 | <p>Concerning your question:</p>
<blockquote>
<p>what does existence of an object really mean, if it is impossible to "find" it? What does it mean when we say that a mathematical object exists?</p>
</blockquote>
<p>I would note that the underlying issue indeed centers on a concern with <em>meaning</em> as your wor... |
1,652,297 | <p><strong>Thm</strong>
Let $V$ and $W$ be Vector spaces and let $T:V \to W$ be linear </p>
<p>If $\beta = \{ v_1,\dots ,v_n \}$ is a basis for $V$
then $$ R(T)=\text{span}(T(\beta))=\text{span}(\{ T(v_1),\dots,T(v_n) \} ) $$</p>
<hr>
<p><strong>Dimension Theorem</strong> </p>
<p>Let $V$ and $W$ be Vector spac... | Alex Mathers | 227,652 | <p>Do you feel you have intuition for what rank really <em>means</em>? The rank is the dimension of the <em>image</em> of the transformation, so if you lose any basis vectors in your transformation, the two will not be equal.</p>
<p>In particular here's a simple counter example: Let $V=\mathbb{R}^2$ and $T:V\to V$ be ... |
2,829,990 | <p>I want to calcurate</p>
<p><span class="math-container">$$ \lim_{n \to \infty} \int_{(0,1)^n} \frac{n}{x_1 + \cdots + x_n} \, dx_1 \cdots dx_n $$</span></p>
<p>I met this in studying Lebesgue integral. But, I don't know how to do at all. I would really appreciate if you could help me!</p>
<p>[Add]</p>
<p>Thanks to e... | Felix Marin | 85,343 | <p><span class="math-container">$\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\exp... |
289,708 | <p>The <a href="https://en.wikipedia.org/wiki/Catalan_number" rel="noreferrer">Catalan numbers</a> <span class="math-container">$C_n$</span> count both </p>
<ol>
<li>the Dyck paths of length <span class="math-container">$2n$</span>, and </li>
<li>the ways to associate <span class="math-container">$n$</span> repeated a... | Martin Rubey | 3,032 | <p>(This is what I have written just before my wife killed the internet connection 12 hours ago before she went to bed. I only show that $D\leq E \Rightarrow A\leq B$ where $D$ and $E$ are Dyck paths and $A$ and $B$ the corresponding binary trees. I didn't look at Timothy's answer yet, but I am guessing it's the same... |
172,617 | <p>I need to plot two datasets on the same plot. The datasets have the same x-range. However, I want to show only parts of the plot. </p>
<p>A minimal example would be</p>
<pre><code> h = π/100.;
i1 = ListLinePlot[Table[{i*h, Sin[i*h]}, {i, 0, 100}], PlotStyle -> Red];
i2 = ListLinePlot[Table[{i*h, Cos[... | kglr | 125 | <pre><code>h = π / 100;
{d1, d2} = Table[{i h, #[i h]}, {i, 0, 100}] & /@ {Sin, Cos};
{ms1, ms2} = {{Opacity[0], Red}, {Blue, Opacity[0]}};
{i1, i2} = ListLinePlot[#[[1]],
PlotStyle -> Thick,
Mesh -> {{π/2}},
MeshStyle -> None,
MeshShading -> #[[2]]] & /@
{{d1, ms1}, {d2, ms2}};
S... |
195,790 | <blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="https://math.stackexchange.com/questions/19796/name-of-this-identity-int-e-alpha-x-cos-beta-x-space-dx-frace-al">Name of this identity? $\int e^{\alpha x}\cos(\beta x) \space dx = \frac{e^{\alpha x} (\alpha \cos(\beta x)+\beta \sin(\beta x))}{\alpha... | davidlowryduda | 9,754 | <p>I tried to naively find common denominators at every step. Can you finish from here?</p>
<p>$$\dfrac{\frac{2}{x^2 + 2xh + h^2} - \frac{2}{x^2}}{h} = \dfrac{\frac{2x^2 - 2(x^2 + 2xh + h^2)}{x^2(x^2 + 2xh + h^2)}}{h} = \dfrac{2x^2 - 2(x^2 + 2xh + h^2)}{hx^2(x^2 + 2xh + h^2)}$$</p>
|
195,790 | <blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="https://math.stackexchange.com/questions/19796/name-of-this-identity-int-e-alpha-x-cos-beta-x-space-dx-frace-al">Name of this identity? $\int e^{\alpha x}\cos(\beta x) \space dx = \frac{e^{\alpha x} (\alpha \cos(\beta x)+\beta \sin(\beta x))}{\alpha... | GeoffDS | 8,671 | <p>The trick is simply to add the two fractions together.</p>
<p>$$\lim_{h \to 0} \frac{\frac{2}{(x + h)^2} - \frac{2}{x^2}}{h} = \lim_{h \to 0} \frac{\frac{2[x^2 - (x+h)^2]}{x^2(x + h)^2}}{h} = \lim_{h \to 0} \frac{\frac{2[-2hx - h^2]}{x^2(x + h)^2}}{h} = \lim_{h \to 0} \frac{2[-2x - h]}{x^2(x + h)^2} = \frac{2(-2x)}... |
3,577,021 | <p>Let inner product space V be defined over F or C and linear operators T on V, evaluate <span class="math-container">$T^{*}$</span> at the given vector in V.</p>
<p><span class="math-container">$V=R^2, T(a,b)=(2a+b,a-3b), x=(3,5)$</span></p>
<p>I know <span class="math-container">$T^{*}$</span> is the conjugate tra... | Graham Kemp | 135,106 | <p>By definition of conditioning over an event: <span class="math-container">$$\mathsf E(Y\mid Y>1-X)=\mathsf E(Y\mathbf 1_{Y>1-X})\div \mathsf E(\mathbf 1_{Y>1-X})$$</span></p>
<p><em>Always</em> use this before attempting to use the Law of Total Expectation. </p>
<p><span class="math-container">$$\begin{al... |
4,099,804 | <p>I need to characterize every finitely generated abelian group G that has the following property:
<span class="math-container">$$\frac{G}{S} \text{ is cyclic for every } \lbrace0\rbrace \lneq S\leq G$$</span>
Given the problems before this one, I believe I am supposed to use the structure theorem figure out the under... | Andrea Marino | 177,070 | <p>First question: how many factors can appear at most in the decomposition?</p>
<p>Second question: in every case you deduced above, which powers can appear? Can you have for example a factor <span class="math-container">$\mathbb{Z}/2^3\mathbb{Z}$</span>?</p>
<p>Good work!</p>
|
608,909 | <p>Is it possible to solve analytically the following equation?
$$\left(x+\frac{1}{x}\right)^{\frac{1}{x}}=A$$
with $A\gt 1$? I tried to transform it in the following:
$\frac{1}{x}\ln\left(x+\frac{1}{x}\right)=B$ with $B=\ln(A)$, but it seems to be still unsolvable. Is there some trick to solve it? Thanks.</p>
| Stefan Hamcke | 41,672 | <p>I just learned that the fact that both adjunction spaces are homotopy equivalent to each other can be seen as an immediate consequence of a general property:</p>
<blockquote>
<p>Let $h\mathbf{Top}^B$ denote the homotopy category under $B$, the quotient category of $(B\downarrow\mathbf{Top})$ where we identify $f\... |
608,909 | <p>Is it possible to solve analytically the following equation?
$$\left(x+\frac{1}{x}\right)^{\frac{1}{x}}=A$$
with $A\gt 1$? I tried to transform it in the following:
$\frac{1}{x}\ln\left(x+\frac{1}{x}\right)=B$ with $B=\ln(A)$, but it seems to be still unsolvable. Is there some trick to solve it? Thanks.</p>
| Ronnie Brown | 28,586 | <p>This is also proved in <a href="http://groupoids.org.uk/topgpds.html" rel="nofollow noreferrer">Topology and Groupoids</a> (as it was in the 1968 edition, "Elements of Modern Topology"); this has some pictures of the crucial mapping cylinder construction <span class="math-container">$M(f) \cup X$</span> which, if <s... |
2,144,140 | <p>We know that $(a,b)$ are open by definition. How do you prove that some arbitrary union of $(a,b)$ cannot give you $[c,d]$ ?</p>
| 5xum | 112,884 | <p>Because for every union of open intervals, let's call it $U$, we have the property:</p>
<blockquote>
<p>$$\forall x\in U \exists\epsilon>0: (x-\epsilon, x+\epsilon) \subseteq U$$</p>
</blockquote>
<p>Or, in english,</p>
<blockquote>
<p>Each element $x$ of $U$ has some neighborhood $(x-\epsilon, x+\epsilon)... |
1,885,492 | <p>Question.</p>
<blockquote>
<p>prove that if ${ a }_{ 1 },{ a }_{ 2 },...{ a }_{ n }>0$ then $$ \frac { { a }_{ 1 }+{ a }_{ 2 }+...+{ a }_{ n } }{ n } \ge \frac { n }{ \frac { 1 }{ { a }_{ 1 } } +\frac { 1 }{ { a }_{ 2 } } +...+\frac { 1 }{ { a }_{ n } } } $$ </p>
</blockquote>
<p><strong>Proof</strong>
$$... | Vincenzo Oliva | 170,489 | <p>As for the first equation, rearranging and squaring both sides we get $$72^2m^2-144m=48r^2,$$ which simplifies to $$108m^2-3m=3m(36m-1)=r^2.$$ Then $r^2=9k^2$ and $m=3n$ for some $k,n$, which yields $$n(108n-1)=k^2.$$ Since the factors of the LHS are coprime, both must be squares: say $n=y^2, 108n-1=x^2;$ therefore,... |
1,298,730 | <p>Find functions $f$ and $\alpha$ such that the improper Riemann-Stieltjes integral $\int_1^{\infty}|f|d\alpha$ converges, but $\int_1^{\infty}fd\alpha$ does not exist?</p>
<p>I'm really not sure how to start this problem, and I haven't been able to find another post on here that has considered this.</p>
<p>EDIT: I ... | zhw. | 228,045 | <p>I see this is the same basic idea as another answer, but here it is anyway: Set $\alpha (x) = -\cos x.$ Then $d\alpha (x) = \sin x \,dx.$ Define $f(t)= 1/t$ on $(0,\pi), (2\pi,3\pi), \dots, f(t) = -1/t$ on $ (\pi,2\pi),(3\pi,4\pi), \dots$ Then</p>
<p>$$\int_0^\infty |f(t)|\,d\alpha(t) = \int_0^\infty \frac{\sin t}{... |
1,438,999 | <p>If $(x-a)^2=(x+a)^2$ for all values of $x$, then what is the value of $a$?</p>
<p>At the end when you get $4ax=0$, can I divide by $4x$ to cancel out $4$ and $x$?</p>
| Brevan Ellefsen | 269,764 | <p>$$(x-a)^2 = (x+a)^2$$
$$x^2 - 2ax + a^2 = x^2 + 2ax + a^2$$
$$x^2 - 2ax + a^2 = x^2 + 2ax + a^2$$
$$-2ax = 2ax$$
$$-a = a$$
Note that this statement is only true when $a=0$, which is thus your solution.</p>
|
1,438,999 | <p>If $(x-a)^2=(x+a)^2$ for all values of $x$, then what is the value of $a$?</p>
<p>At the end when you get $4ax=0$, can I divide by $4x$ to cancel out $4$ and $x$?</p>
| E.H.E | 187,799 | <p>$$\frac{(x-a)^2}{(x+a)^2}=1$$
$$\left(\frac{x-a}{x+a}\right)^2=1$$</p>
<p>$$\left(1-\frac{2a}{x+a}\right)^2=1$$
it is clear that $\frac{2a}{x+a}$ should be equal to zero</p>
<p>so, the $$a=0$$</p>
|
970,409 | <p>Let matrices $A$, $B\in{M_2}(\mathbb{R})$, such that $A^2=B^2=I$, where $I$ is identity matrix. </p>
<p>Why can be numbers $3+2\sqrt2$ and $3-2\sqrt2$ eigenvalues for the Matrix $AB$? </p>
<p>Can be numbers $2,1/2$ the eigenvalues of matrix $AB$? </p>
| Yiorgos S. Smyrlis | 57,021 | <p>Set
$$
A=\left(\begin{matrix}0 & 3-2\sqrt{2} \\ 3+2\sqrt{2} & 0\end{matrix}\right),\quad
B=\left(\begin{matrix}0 & 1 \\ 1 & 0\end{matrix}\right).
$$
Then
$$
AB=\left(\begin{matrix} 3-2\sqrt{2} & 0 \\ 0 & 3+2\sqrt{2}\end{matrix}\right).
$$
The eigenvalues of $A,B$ are $\pm 1$, and hence $A^2=B... |
102,304 | <p>I have here a complex equation:</p>
<p>$$z^2 - (7+j)z + 24 +j7 = 0$$</p>
<p>How do we get the roots of this equation? I started using the quadratic formula $-b \pm \sqrt{ b^2-4ac}\over 2$, but it yielded too much complexity on it. Is there any way to directly attack this? Thanks.</p>
| Peđa | 15,660 | <p>Let's denote $z$ as : $z=a+jb$ , then we have :</p>
<p>$a^2-b^2+2abj-(7+j)(a+bj)+24+7j=0 \Rightarrow$</p>
<p>$\Rightarrow a^2-b^2+2abj - (7a+7bj+aj-b)+24+7j=0$</p>
<p>So , you have to solve following system of equations :</p>
<p>$\begin{cases}
a^2-b^2-7a+b+24=0 \\
2ab-7b-a+7=0\\
\end{cases}$</p>
|
102,304 | <p>I have here a complex equation:</p>
<p>$$z^2 - (7+j)z + 24 +j7 = 0$$</p>
<p>How do we get the roots of this equation? I started using the quadratic formula $-b \pm \sqrt{ b^2-4ac}\over 2$, but it yielded too much complexity on it. Is there any way to directly attack this? Thanks.</p>
| André Nicolas | 6,312 | <p>Although we do find the roots, the following is mainly a spoof of school algebra. </p>
<p>In school algebra, students are expected to solve very special equations of the form $ax^2+bx +c=0$, where $a$, $b$, and $c$ are integers, by <em>factoring</em>. The Quadratic Formula, and even the Rational Roots Theorem, ar... |
523,932 | <p>I've got a system of equations which is:<br></p>
<p>$\begin{cases} x=2y+1\\xy=10\end{cases}$</p>
<p>I have gone into this: $x=\dfrac {10}y$.
<br>
How can I find the $x$ and $y$?</p>
| Jay | 9,814 | <p>Notice that $10 = xy = (2y + 1)y = 2y^2 + y$. But then $$2y^2 + y - 10 = 0.$$
Can you solve this quadratic equation?</p>
<p>If you use the substitutions $x = \frac{10}{y}$ or $y = \frac{10}{x}$ then you are implicitly assuming either $y$ or $x$ is not $0$.</p>
|
1,514,628 | <p>I've been looking over some old assignments in my analysis course to get ready for my upcoming exam - I've just run into something that I have no idea how to solve, though, mainly because it looks nothing like anything I've done before. The assignment is as follows:</p>
<p>"Let $H$ be a Hilbert space, and let $(e_n... | Hamed | 191,425 | <p>You know that $\{e_n\}_{n\in \mathbb{N}}$ is orthonormal. So let $(a,b)$ be the notation for inner product of the space. Gram-Schmidt is the way to go actually,as you guessed. Convince yourself that with Gram-Schmidt you find
$$
f_1 = \frac{e_1+e_2}{\sqrt{2}}, \quad f_2=\frac{e_3+e_4}{\sqrt{2}},
\quad f_3=\frac{e_1... |
3,500,405 | <p>I have a similar question to what was asked already <a href="https://math.stackexchange.com/questions/2511111/prove-that-the-following-map-has-at-least-k-2-fixed-points">here</a></p>
<p>But I do not really understand the answer there.</p>
<p>The problem is: Let <span class="math-container">$x_0 \in S^1$</span> and... | Matematleta | 138,929 | <p>Let <span class="math-container">$p:t\mapsto e^{2\pi it}$</span> be the usual covering map of <span class="math-container">$S^1$</span>. Since the induced map <span class="math-container">$f_*$</span> takes <span class="math-container">$[g]$</span> to <span class="math-container">$k[g]$</span>, it is not hard to sho... |
1,419,784 | <p>When we consider calculations at tiniest of scales which number system would be more accurate, when we consider the binary number system( base 2) or the number system we generally use (base 10).</p>
<p>The other way to say it would be if we consider a number system with base 1,2,3,4.... , does it effects the accura... | Matt Samuel | 187,867 | <p>Without rounding, the base doesn't matter at all. With rounding, the base 10 number system is in a sense more general than binary because whenever something can be represented exactly in a finite number of digits in binary, the same is true in decimal, but there are numbers with a finite length base 10 representatio... |
1,419,784 | <p>When we consider calculations at tiniest of scales which number system would be more accurate, when we consider the binary number system( base 2) or the number system we generally use (base 10).</p>
<p>The other way to say it would be if we consider a number system with base 1,2,3,4.... , does it effects the accura... | Stefan Mesken | 217,623 | <p>The base in which we represent natural numbers does not effect natural numbers at all (the base $1$ however, isn't suitable - think about it). This is like asking "Is 'Times New Roman' or 'Comic Sans' (a better comparison is "Times New Roman" and a suitable variation of "Tengwar") more accurate when writing an essay... |
815,770 | <p>Compute $F_{1000} \bmod F_{11}$, where $F_n$ denote the Fibonacci numbers.</p>
<p>Progress:</p>
<p>$F_{11}=89$ . I believe you should find the period of $F_n \bmod 89$ and use that to solve it. But I'm not not getting anywhere from that.</p>
<p>Thanks!</p>
| heropup | 118,193 | <p>The correct relationship is $$F_{11k + n} \equiv F_n \cdot F_{10}^k \pmod {F_{11}},$$ and since $1000 = 11(90)+10$, we have $$F_{1000} \equiv F_{10} \cdot F_{10}^{90} = F_{10}^{91} \pmod {F_{11}}.$$ Now we must compute $$55^{91} \pmod {89},$$ and since $89$ is prime, by Fermat's little theorem, we have $$55^{89} \e... |
3,691,147 | <p>Consider the wave equation in one dimension <span class="math-container">$u_{tt}-u_{xx}=0$</span> together with a Fourier Transform along <span class="math-container">$t$</span>, ie <span class="math-container">$$\text{FT}[u](x,\omega)=\int_{-\infty}^{+\infty}u(x,t)\exp(-i\omega t)\mathrm{d}t.\tag{1}$$</span> The ab... | pluton | 30,598 | <p>If we consider for simplicity the left propagating wave, the solution reads <span class="math-container">$u(x,t)=T(x+t)$</span> where <span class="math-container">$T$</span> is a distribution. Its Fourier Transform in time is (because of translation)
<span class="math-container">$$
\text{FT}[u](x,\omega)=\int_{-\inf... |
50,113 | <p>What are some good books on field and Galois theory?</p>
| Jack Rousseau | 11,764 | <p><a href="https://projecteuclid.org/ebooks/notre-dame-mathematical-lectures/Galois-Theory/toc/ndml/1175197041" rel="nofollow noreferrer">Galois Theory</a> by Emil Artin is a nice treatment of the latter.</p>
|
119,810 | <p>My question today is about the minimization of an error function with two parameters. It is a function that measures the error of a set of points. The two parameters are the weights of a regressor. </p>
<p>$$\frac{1}{N}\sum_{t=1}^{N}[r^t-(w_1x^t+w_0)]^2$$ </p>
<p>The minimum should be calculated by taking partia... | Henry | 6,460 | <p>Let's start by ignoring the constant $\frac{1}{N}$. Then </p>
<p>$$\sum_t[r^t-(w_1x^t+w_0)]^2 $$ $$= \sum_t r^{2t} + w_1^2 \sum_t x^{2t} +N w_0^2 -2 w_1 \sum_t r^t x^t -2 w_0 \sum_t r^t+ 2 w_1 w_0 \sum_t x^t $$</p>
<p>Take the partial derivatives with respect to $w_0$ and $w_1$ and set them to zero and you get<... |
267,355 | <p>Let $H_i = (V_i, E_i)$ be <a href="https://en.wikipedia.org/wiki/Hypergraph" rel="nofollow noreferrer">hypergraphs</a> for $i=1,2$. Then we say that $H_1\cong H_2$ if there is a bijection $\varphi:V_1\to V_2$ such that $A\in E_1$ implies $\varphi(A) \in E_2$ and $B\in E_2$ implies $\varphi^{-1}(B)\in E_1$.</p>
<p>I... | Joel David Hamkins | 1,946 | <p>The answer is yes. </p>
<p>Consider the collection $\mathcal C$ of hypergraphs of the following form. They have underlying set $\omega$ as the vertices, the natural numbers. The finite edges in the hypergraph are all and only the sets of the form $\{0,1,\ldots,n\}$. And then the hypergraph can have any desired coll... |
267,355 | <p>Let $H_i = (V_i, E_i)$ be <a href="https://en.wikipedia.org/wiki/Hypergraph" rel="nofollow noreferrer">hypergraphs</a> for $i=1,2$. Then we say that $H_1\cong H_2$ if there is a bijection $\varphi:V_1\to V_2$ such that $A\in E_1$ implies $\varphi(A) \in E_2$ and $B\in E_2$ implies $\varphi^{-1}(B)\in E_1$.</p>
<p>I... | Peter Heinig | 108,556 | <p>Joel's answer is spot on, and makes full use of
your not requiring any further properties that your hypergraphs should have, but it might perhaps be nice to know that also with both </p>
<ul>
<li>less demands on the isomorphism</li>
<li>more demands on each of the hypergraphs </li>
</ul>
<p>the number of isomorph... |
2,946,379 | <p>The question posed is the following: Let <span class="math-container">$X$</span> be a Banach Space and let <span class="math-container">$T:X\to X$</span> be a Lipschitz-Continuous map. Show that, for <span class="math-container">$\mu$</span> sufficiently large, the equation
<span class="math-container">\begin{equat... | Henry Lee | 541,220 | <p>I think it should be:
<span class="math-container">$$I=\int_1^\infty\frac{e^x+e^{3x}}{e^x-e^{5x}}dx$$</span>
<span class="math-container">$u=e^x$</span> so <span class="math-container">$dx=\frac{du}{u}$</span> so:
<span class="math-container">$$I=\int_e^\infty\frac{u+u^3}{u-u^5}\frac{1}{u}du=\int_1^\infty\frac{1+u^2... |
2,264,791 | <p>I have a problem that I'm having trouble figuring out the distribution with given condition.</p>
<p>It is given that 1/(<span class="math-container">$X$</span>+1), where <span class="math-container">$X$</span> is an exponentially distributed random variable with parameter 1.</p>
<blockquote>
<p><strong>Original Prob... | Felix Marin | 85,343 | <p>$\newcommand{\bbx}[1]{\,\bbox[8px,border:1px groove navy]{\displaystyle{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
\n... |
936,525 | <p>I am following a proof in the text OPTIMIZATION THEORY AND METHODS a springer series by WENYU SUN and YA-XIANG YUAN. I come across what seems obvious that for a column vector $v$, with dimension $n\times 1$, $$\biggl\|I-\frac{vv^T}{v^Tv}\biggr\|=1,$$ where $I$ is an $n\times n$ matrix, and $\|.||$ is a matrix norm.... | leonbloy | 312 | <p>Letting $A=I-\frac{vv^T}{v^Tv}$ . Then $A$ is symmetric, and its eigenvalues are real. Besides $A x = x - v \frac{v^T x}{v^Tv}=\lambda x$ implies that either $x = \alpha v$, (which gives $\lambda=0$) or $v^T x=0$ which gives $\lambda=1$. Hence it's greater eigenvalue is 1 and, that's the spectral norm.</p>
|
267,707 | <p>The weight 2, level 1 Eisenstein series $E_2(z)$ is a non-holomorphic automorphic form. It is defined as the analytic continuation to $s = 0$ of the series
$$ E_2(z, s) = \sum_{\substack{m, n \in \mathbf{Z} \\ (m, n) \ne (0,0)}} \frac{\operatorname{Im}(z)^s}{(mz + n)^2 |mz + n|^{2s}} $$
which is convergent for $\ope... | paul garrett | 15,629 | <p>(I took @LSpice's inquiry as encouragement to add some remarks to @GH from MO's good answer... But/and one of the issues that helped me overcome my skepticism about the utility of representation theory long ago was its clarification of exactly these issues about "Hecke summation", Maass-Shimura operators, and such. ... |
2,661,443 | <p>For the equation $2^x = 7$</p>
<p>The textbook says to use log base ten to solve it like this $\log 2^x = \log 7$. </p>
<p>I then re-arrange it so that it reads $x \log 2 = \log 7$ then divide the RHS by $\log 2$ to isolate the $x$. I understand this part.</p>
<p>I can alternatively solve it in an easier way by s... | user | 505,767 | <p>The use of Log in base 10 is a common choice and we often find it in calculators since our standard system of counting is in base 10 (i.e. number of fingers).</p>
<p>To handle algebraic problems any other choice is fine depending upon the specific problem to solve by <a href="https://en.wikipedia.org/wiki/Logarithm... |
2,661,443 | <p>For the equation $2^x = 7$</p>
<p>The textbook says to use log base ten to solve it like this $\log 2^x = \log 7$. </p>
<p>I then re-arrange it so that it reads $x \log 2 = \log 7$ then divide the RHS by $\log 2$ to isolate the $x$. I understand this part.</p>
<p>I can alternatively solve it in an easier way by s... | Henry | 6,460 | <p>This works in any base because $\log_b (a^c) = c \log_b(a)$ </p>
<p>The practical reason for using base $10$ was a little old fashioned: it allowed the use of tables of logarithms instead of a calculator and reducing these calculations to addition and subtraction. Calculators then provided a $\log_{10}$ function a... |
2,661,443 | <p>For the equation $2^x = 7$</p>
<p>The textbook says to use log base ten to solve it like this $\log 2^x = \log 7$. </p>
<p>I then re-arrange it so that it reads $x \log 2 = \log 7$ then divide the RHS by $\log 2$ to isolate the $x$. I understand this part.</p>
<p>I can alternatively solve it in an easier way by s... | chux - Reinstate Monica | 83,175 | <blockquote>
<p>Why would the textbook suggest to use log base ten rather than simply using log base two?</p>
</blockquote>
<p>Hmmm, if we embrace binary, and use log<sub>2</sub>(), how about using base 2 for the constants and representation also? </p>
<p>Then $2^x = 7$ becomes $10^x = 111$ and is solved with $x =... |
224,226 | <p>I am trying to count the number of distinct colours in a <span class="math-container">$5\times5$</span> box, (a radius 2 filter) at all points over a quantized image. I cannot seem to get anything out of the following code except for a black square:</p>
<pre><code>img = ColorQuantize[ExampleData[{"TestImage&quo... | flinty | 72,682 | <p>I came up with another method. I can replace the colours in the original quantized image with single numbers to form a palletized grayscale image first. Then <code>ImageFilter</code> works and I get the same image as in <strong>@MarcoB's</strong> answer - seeing it with two different methods is enough confirmation f... |
1,712,457 | <blockquote>
<p>Assume $f$ is differentiable over an open interval $I$. Suppose $a<b$ are two numbers in $I$ with $f'(a) < f'(b)$. Show that if $f'(a) < 0 <f'(b)$, then neither $f(a)$ nor $f(b)$ can be the minimum value of $f$ over $[a,b]$.</p>
</blockquote>
<p>Intuitively this makes sense: $f$ must ch... | Paramanand Singh | 72,031 | <p>You need to understand the meaning of sign of derivative.</p>
<p><em>Let $f'(c) > 0$ then there is an interval $J$ with $c$ as an interior point such that if $x>c$ and $x \in J$ then $f(x)>f(c)$ and if $x<c$ and $x\in J$ then $f(x)<f(c)$.</em></p>
<p>Same way we have corresponding result for the cas... |
2,579,140 | <p>N. Elkies' page <a href="http://www.math.harvard.edu/~elkies/trinomial.html" rel="nofollow noreferrer">http://www.math.harvard.edu/~elkies/trinomial.html</a> ends with an information about octic trinomials "whose Galois group is contained in $G_{1344}$".</p>
<p>One of reported trinomials, $x^8+324x+567$, "has a sma... | Dietrich Burde | 83,966 | <p>The group $G_{168}$ <strong>is</strong> isomorphic to $PSL(2,7)$, and it is a subgroup of $G_{1344}$. Elkies writes "This Galois group is isomorphic with $G_{168}$, acting on $8$ letters via the other guise of that group, as $PSL_2(\mathbb{Z}/7\mathbb{Z})$." So you could understand the "contradiction" reviewing his ... |
2,184,593 | <p>By Cauchy's criterion of limit (not sequencial criterion), show that $$\lim_{x\to 0}(\sin{\frac{1}{x}}+x\cos{\frac{1}{x}})$$ does not exist.</p>
<p>Cauchy's criterion of limit </p>
<p>$\lim_{x\to c}f(x)=l$ iff for every $\epsilon>0$, there exists $\delta$ such that $$|f(x_2)-f(x_1)|<\epsilon$$ for $0<|x_1... | 5xum | 112,884 | <p><strong>Hint</strong>:</p>
<p>You can find, for any $\epsilon > 0$, a value of $x_1$ such that $0<x_1<\epsilon$ and $\sin\frac1{x_1}=1$, and a value of $x_2$ such that $0<x_2<\epsilon$ and $\sin\frac{1}{x_2}=-1$. </p>
|
2,184,593 | <p>By Cauchy's criterion of limit (not sequencial criterion), show that $$\lim_{x\to 0}(\sin{\frac{1}{x}}+x\cos{\frac{1}{x}})$$ does not exist.</p>
<p>Cauchy's criterion of limit </p>
<p>$\lim_{x\to c}f(x)=l$ iff for every $\epsilon>0$, there exists $\delta$ such that $$|f(x_2)-f(x_1)|<\epsilon$$ for $0<|x_1... | Math-fun | 195,344 | <p>Note first that $$\lim_{x \to 0}x \cos \frac 1x=0$$ so you can safely focus on $\sin \frac 1x$. Now consider $x_n=\frac 1{2n\pi+\frac{\pi}2}$ and $y_n=\frac1{n\pi}$ for $n\to \infty$ and see what happens.</p>
|
359,742 | <p>I have a mathematical problem that leads me to a particular necessity. I need to calculate the convolution of a function for itself for a certain amount of times. </p>
<p>So consider a generic function $f : \mathbb{R} \mapsto \mathbb{R}$ and consider these hypothesis:</p>
<ul>
<li>$f$ is continuos in $\mathbb{R}$.... | Daniel Naftalovich | 203,380 | <p>While I don't know about the general case of your $F^{(\infty)}(x)$ for any such generic function $f$, it seems that in some cases you can find explicit closed forms for $n$-fold convolutions for some functions $f$. Maybe then you can study the limit as $n\rightarrow\infty$ for your purposes?</p>
<p>For example, I... |
3,988,180 | <p>The task:
<span class="math-container">$$
\text{Calculate } \int_{C}{}f(z)dz\text{, where } f(z)=\frac{\bar{z}}{z+i}\text{, and } C \text{ is a circle } |z+i|=3\text{.}
$$</span>
Finding the circle's center and radius:
<span class="math-container">$$
|z+i|=|x+yi+i|=|x+(y+1)i|=3
\\
x^2+(y+1)^2=3^2
$$</span>
Parametri... | Mark Viola | 218,419 | <p>Note that on the circle <span class="math-container">$C$</span>, given by <span class="math-container">$|z+i|=3$</span>, we have <span class="math-container">$\overline{z+i}=\frac9{z+i}$</span>. Hence, on <span class="math-container">$C$</span></p>
<p><span class="math-container">$$f(z)=\frac9{(z+i)^2}+i\frac{1}{z+... |
3,988,180 | <p>The task:
<span class="math-container">$$
\text{Calculate } \int_{C}{}f(z)dz\text{, where } f(z)=\frac{\bar{z}}{z+i}\text{, and } C \text{ is a circle } |z+i|=3\text{.}
$$</span>
Finding the circle's center and radius:
<span class="math-container">$$
|z+i|=|x+yi+i|=|x+(y+1)i|=3
\\
x^2+(y+1)^2=3^2
$$</span>
Parametri... | Community | -1 | <p>Observe that
<span class="math-container">$$
\frac{\overline{z}}{z+i}
=\frac{\overline{z}\cdot \overline{(z+i)}}{(z+i)\cdot \overline{(z+i)}}
=\frac{\overline{z}\cdot \overline{(z+i)}}{|z+i|^2}
=\frac{1}{9}\cdot \overline{z}\cdot \overline{(z+i)}
$$</span></p>
<p>Your parametrization of the path should be
<span clas... |
2,804,129 | <p>To Evaluate the Limit $$L=\lim_{n \to \infty}\left(1+\sum_{k=1}^{n} \frac{1}{\binom{n}{k}}\right)^n \tag{1}$$</p>
<p>My try:</p>
<p>I tried to use $$\frac{1}{\binom{n}{k}}+\frac{1}{\binom{n}{k+1}}=\frac{n+1}{n} \frac{1}{\binom{n-1}{k}} $$</p>
<p>taking summation both sides from $k=1$ to $k=n$ we get</p>
<p>$$\su... | Kavi Rama Murthy | 142,385 | <p>Since teh detreminant of $M$ is 1 there can be only one solution. Equate $T(a+be^{2t}+ce^{4t})$ to $1+2e^{2}+3e^{4t}$ and you will get $a=13, b=-10,c=3$.</p>
|
2,595,247 | <p>What is equation of circle when two lines y=x and y=x-4 are tangent to a circle at (2,2) and (4,0) respectively.</p>
| Wouter | 89,671 | <p>Because $\lim_{n\rightarrow\infty} x^{n+1}=0$ whenever $x\in [0,1[$.</p>
<p>So your integrand approaches 0 everywhere except at $x=1$, which is just one point and thus doesn't influence the value of the integral.</p>
|
21,243 | <p>I know that this question is some sort of bridge between Informatics and Mathematics, not knowing the best place where to post this question, I opted for this place because of the type of answer I want (which concerns Math more than anything).</p>
<p>Consider to have a graph represented by a collection of nodes and... | Udara | 15,658 | <p>I'm also interested on the convergence of Markov chains on a graph and found your questions very interesting.
As per my knowledge, for adjacancy matrices, eigenvalues of AM implies many things. It implies the connectivity of the graph. If the difference between the largest eigenvalue and the second largest eigenval... |
1,515,478 | <p>Given a quadratic equation with one and only one root (for example $6-\sqrt{2}$ ). Does there exist integers $a,b$ and $c$ where $ax^2 + bx + c = 0$ for that root?</p>
| More water plz | 282,704 | <p>Yes. given $r_1 = 6-\sqrt{2}$, we know that $r_2 = 6+\sqrt{2}$</p>
<p>so the quadratic equation is given by </p>
<p>$(x-(6-\sqrt{2}))(x-(6+\sqrt{2}))=(x-6)^2-2 = x^2-12x+34$</p>
|
2,863,995 | <p><em>Problem:</em></p>
<p>Let $f_n: [0,1] \to \mathbb{R}$ be a sequence of measurable functions. </p>
<p>Suppose that $\int_{0}^{1}|f_n(x)|^2 ~ dx \le 1$ for $n \in \mathbb{N}$ and $f_n$ converges to $0$ a.e. </p>
<p>Show that $\lim_{n \to \infty} \int_{0}^{1} f_n(x) ~ dx = 0$.</p>
<p><em>Question:</em> <strong>I... | Mike Earnest | 177,399 | <p>Here is a way to solve the problem. Since $f_n\to 0$ a.e, and the measure space is finite, you have $f_n\to 0$ in probability. Given $\epsilon>0$, let $A_n=\{|f_n|>\epsilon\}$. Then
$$
\int_0^1 |f_n|\,dx=\int_{A_n} |f_n|\,dx + \int_{[0,1]\setminus A_n}|f_n|\,dx\le \int_0^1 f_n(x){\bf 1}_{A_n}(x)\,dx + \epsilon... |
295,517 | <p>My math is not incredibly strong and perhaps I have just not been searching for the right terms, but I have a summation that is part of an algorithm I've been working on and would really like to reduce it to just a formula, but am really struggling to find a solution (if one exists).</p>
<p>$\sum_{i=1}^{n}\frac{5}{... | Simply Beautiful Art | 272,831 | <p>In contrary to GEdgar's answer, we can use a different result:</p>
<p>$$\sum_{k=1}^n\frac1{k^{0.35}}\approx\zeta(0.35)+\frac1{0.65}n^{0.65}$$</p>
<p>where $\zeta(0.35)\approx-1.0105112244$</p>
<p>For example, with $n=100$,</p>
<p>$\zeta(0.35)+\frac1{0.65}(100)^{0.65}=29.685832082775940902507610997434927477716286... |
2,501,518 | <p>$\begin{pmatrix}
a \\
b
\end{pmatrix}
\begin{pmatrix}
a \\
b
\end{pmatrix}^T
\begin{pmatrix}
C & D \\
D^T & E
\end{pmatrix}=
\begin{pmatrix}
I_m & 0\\
0 & I_n
\end{pmatrix}
$</p>
<p>$a$ and $b$ are vectors with length $m$ and $n$ respectively. C has dimension $m$ by $m$ and E $n$ by $n$. </p>
<p>... | Community | -1 | <p>There is no solution because of the followingobvious reason. The rank of the right-hand side is $n+m$, while that of the left hand side is bounded by the rank of $\begin{pmatrix} a \\ b\end{pmatrix}$, that is $m$.</p>
|
490,064 | <p>Solve the Cauchy problem, $\forall t \in \mathbb{R}$,
$$ \begin{cases}
u''(t) + u(t) = |t|\\
u(0)=1, \quad u'(0) = -1
\end{cases} $$</p>
<p>The solution to the homogeneous equation is $A\cos(t) + B \sin(t)$. Empirically, $|t|$ is "more or less" a particular solution, however it is not differentiable in $0$... What ... | Amzoti | 38,839 | <p>As mentioned in the comments, you can solve the system for $t \ge 0$ and $t \lt 0$.</p>
<p>However, you can use Laplace Transforms to solve the problem, and will arrive at:</p>
<ul>
<li>$t \ge 0, u(t) = t - 2 \sin t + \cos t$</li>
<li>$t \lt 0, u(t) = \cos t - t$</li>
</ul>
<p>If we check the initial conditions, ... |
3,858,362 | <p>Solve <span class="math-container">$$\dfrac{x^3-4x^2-4x+16}{\sqrt{x^2-5x+4}}=0.$$</span>
We have <span class="math-container">$D_x:\begin{cases}x^2-5x+4\ge0\\x^2-5x+4\ne0\end{cases}\iff x^2-5x+4>0\iff x\in(-\infty;1)\cup(4;+\infty).$</span> Now I am trying to solve the equation <span class="math-container">$x^3-4... | Community | -1 | <p>Answer :
<span class="math-container">$\frac{x^3-4x^2-4x+16}{\sqrt{x^2 - 5x+4}}= \frac{x(x^2 - 4)-4(x^2 - 4)}{\sqrt{x^2 - 5x+4}}$</span>=<span class="math-container">$\frac{(x^2 - 4)(x-4)}{\sqrt{x^2 - 5x+4}}$</span></p>
<p><span class="math-container">$\sqrt{x^2 - 5x+4} = 0 $</span> if <span class="math-container">$... |
3,536,671 | <p>I have the following mathematical operations to use: Add, Divide, Minimum, Minus, Modulo, Multiply and Round.</p>
<p>With these I need to get a number, run it through a combination of these and return 0 if the number is negative or equal to 0 and the number itself if the number is greater than 0.</p>
<p>Is that po... | J. W. Tanner | 615,567 | <p>Take <span class="math-container">$x-\min(0,x)$</span>.</p>
<p>If <span class="math-container">$x>0$</span>, then <span class="math-container">$\min(0,x)=0$</span>, so <span class="math-container">$x-\min(0,x)=x$</span>.</p>
<p>If <span class="math-container">$x\le0$</span>, then <span class="math-container">$\... |
1,862,524 | <p>In the textbook I'm using to prepare the logic exam says that first order logic may be used to <strike>implement</strike> axiomatize data structures. There is an example of that:</p>
<p>"Stack": uses a language that contains the <strike>predicates</strike> functions <em>top</em>, <em>pop</em> and <em>push</em>, and... | George V. Williams | 54,806 | <p>The book is wrong, your derivation is also wrong.</p>
<p>Note that you substitute $u=8$ instead of $u=4$. You already changed the bounds to integrate from $u=0$ to $4$. The correct answer is $4(e^4 - 1)$.</p>
<p>$\int_0^4 \frac{1}{2} (8)e^u du$ $\to$ $\int_0^4 4e^u du \Big\vert_0^4=4e^\color{red}{4}-4=4(e^\color{r... |
4,280,426 | <blockquote>
<p>We have a bag with <span class="math-container">$3$</span> black balls and <span class="math-container">$5$</span> white balls. What is the probability of picking out two white balls if at least one of them is white?</p>
</blockquote>
<p>If <span class="math-container">$A$</span> is the event of first b... | Henno Brandsma | 4,280 | <p>The closed sets are all sets that are at most countable, or <span class="math-container">$\Bbb R$</span>. So the <em>only</em> closed superset of an uncountable set <span class="math-container">$A$</span> is <span class="math-container">$\Bbb R$</span> so</p>
<p><span class="math-container">$$A \text{ uncountable } ... |
3,102,905 | <p>I have the following sequence <span class="math-container">$$(x_{n})_{n\geq 1}, \ x_{n}=ac+(a+ab)c^{2}+...+(a+ab+...+ab^{n})c^{n+1}$$</span>
Also I know that <span class="math-container">$a,b,c\in \mathbb{R}$</span> and <span class="math-container">$|c|<1,\ b\neq 1, \ |bc|<1$</span>
I need to find the limit of... | exp ikx | 614,823 | <p>Hint: Take <span class="math-container">$ac$</span> common. For finding the limit of the sequence, consider <span class="math-container">$n \rightarrow \infty$</span> and apply the formula for summation of geometric series with infinite terms. Since your <span class="math-container">$r<1$</span>, <span class="mat... |
2,199,303 | <p>Consider the DE $$y''+\lambda y=0$$ where $\lambda$ is a constant </p>
<p>subject to the boundary conditions $$y(0)=0$$ and $$y(a)=0$$ where $a$ is a positive constant</p>
<p>I want to find the eigenvalues and eigenfunctions related to this problem</p>
<p>My attempt:</p>
<p>The auxiliary equation is $$m^2-\lambd... | Chappers | 221,811 | <p>Yes, your approach so far is correct. One should also check (preferably first, so you don't forget!) the $\lambda=0$ case: here, it is easy to check that the boundary conditions can't be satisfied.</p>
<p>One can actually extract the general solution from your analysis. In order for there to be a nonzero solution, ... |
2,292,324 | <p>I know what the answer to this question is, but I am not sure how the answer was reached and I would really like to understand it! I am omitting the base case because it is not relevant for my question.</p>
<p>Inductive hypothesis:</p>
<p>$$\frac{1}{1\cdot2} + \frac{1}{2\cdot3} + \frac{1}{3\cdot4} + \dotsb + \frac... | Mark Viola | 218,419 | <p>The base case $n=1$ is true. Assume that for some $n> 1$ </p>
<p>$$\sum_{k=1}^n\frac{1}{k(k+1)}=\frac{n}{n+1}$$</p>
<p>Then, we have</p>
<p>$$\begin{align}
\sum_{k=1}^{n+1}\frac{1}{k(k+1)}&=\sum_{k=1}^n\frac{1}{k(k+1)}+\frac{1}{(n+1)(n+2)}\\\\
&=\frac{n}{n+1}+\frac{1}{(n+1)(n+2)}\\\\
&=\frac{n(n+2... |
1,968,541 | <p>$$144x^5 − 121x^4 + 100x^3 − 81x^2 − 64x + 49 = 0 $$</p>
<p>I re-wrote it as </p>
<p>$$ 12^2x^5 - 11^2x^4 + 10^2x^3 - 9^2x^2 - 8^2x + 7^2 = 0 $$</p>
<p>And then as $$ \sum_{k=0}^{5} (k+7)^2(-1)^kr^k = 0 $$</p>
<p>But I don't know what to do with that. Thanks for any help!</p>
| Carl Schildkraut | 253,966 | <p>Generally, the best way to do these kinds of problems is by the rational root theorem. However, there is a nicer way here.</p>
<p>Consider this polynomial $\mod 2$. If it has a solution $n\in\mathbb{Z}$, then</p>
<p>$$-121n^4-81n^2+49\equiv 0\mod 2$$</p>
<p>$$n^4+n^2+1\equiv 0\mod 2$$</p>
<p>$$(n^2)(n^2+1)+1\equ... |
3,392,871 | <blockquote>
<p>Let <span class="math-container">$k>1$</span> and define a sequence <span class="math-container">$\left\{a_{n}\right\}$</span> by <span class="math-container">$a_{1}=1$</span> and <span class="math-container">$$a_{n+1}=\frac{k\left(1+a_{n}\right) }{\left(k+a_{n}\right)}$$</span>
(a) Show that <s... | Theo Bendit | 248,286 | <p>Turn the question of whether <span class="math-container">$(a_n)$</span> is monotone increasing into an inequality purely in terms of a single term <span class="math-container">$a_n$</span>. In particular,
<span class="math-container">$$a_n \le a_{n+1} = \frac{k(1 + a_n)}{k + a_n}.$$</span>
Simplifying, making the t... |
3,392,871 | <blockquote>
<p>Let <span class="math-container">$k>1$</span> and define a sequence <span class="math-container">$\left\{a_{n}\right\}$</span> by <span class="math-container">$a_{1}=1$</span> and <span class="math-container">$$a_{n+1}=\frac{k\left(1+a_{n}\right) }{\left(k+a_{n}\right)}$$</span>
(a) Show that <s... | trancelocation | 467,003 | <p>A possible way is to investigate the derivative of the function underlying the recursion and then use MVT together with induction:</p>
<ul>
<li><span class="math-container">$\boxed{f(x)} = k\frac{1+x}{k+x}= \frac{k+x + (k-1)x}{k+x}= 1 + (k-1)\frac{x}{k+x} = \boxed{1+ (k-1)\left(1 - \frac{k}{k+x} \right)}$</span></l... |
345,252 | <blockquote>
<p>Prove that $A \subset B$ if and only if $A \cap B = A$</p>
</blockquote>
<p>I'm having problems writing this proof using formal logic operators. I know the idea behind it: Since $A \subset B$, the only elements A and B have in common are those of A, so the intersection must be just A...</p>
| Berci | 41,488 | <p>$\Rightarrow:$ We always have $A\cap B\subseteq A$. For the reversed containment, if $a\in A$, then $a\in B$ as well by assumption ($A\subseteq B$), so $A\subseteq A\cap B$.</p>
<p>$\Leftarrow:$ We always have $A\cap B\subseteq B$.</p>
|
345,252 | <blockquote>
<p>Prove that $A \subset B$ if and only if $A \cap B = A$</p>
</blockquote>
<p>I'm having problems writing this proof using formal logic operators. I know the idea behind it: Since $A \subset B$, the only elements A and B have in common are those of A, so the intersection must be just A...</p>
| Code-Guru | 34,869 | <p>Since you are proving an "if and only if statement" you have two things to prove. Do you know what they are? If so, then you will see that in one direction you need to prove that two sets are equal? Do you know how to do this? What about proving that one set is a subset of another?</p>
|
426,974 | <p>Suppose the dynamical system <span class="math-container">$(X,T)$</span> has only proper factors (i.e. not <span class="math-container">$(X,T)$</span> itself) of zero topological entropy. Does the system <span class="math-container">$(X,T)$</span> also have zero entropy?</p>
| Ville Salo | 123,634 | <p>YES under strong additional assumptions.</p>
<blockquote>
<p>Theorem. Let <span class="math-container">$X$</span> be compact metrizable and <em>zero-dimensional</em>, and let <span class="math-container">$T : X \to X$</span> be an <em>aperiodic homeomorphism</em>. If <span class="math-container">$(X, T)$</span> has ... |
2,553,284 | <p>I know that
$$\ln e^2=2$$
But what about this?
$$(\ln e)^2$$
A calculator gave 1. I'm really confused.</p>
| BDSub | 504,256 | <p>Since $\ln e = 1 $
So $(\ln e)^2 = 1$</p>
|
2,553,284 | <p>I know that
$$\ln e^2=2$$
But what about this?
$$(\ln e)^2$$
A calculator gave 1. I'm really confused.</p>
| Fghj | 458,544 | <p>Consider the most important property of logarithm</p>
<p>$$ \log(m)^n = n \log (m) $$</p>
<p>So that, $$ \log_e(e)^2 = 2 \log_e(e) = 2 \ln(e) = 2 $$</p>
<p>And since, $$ [ln(e)^2] ≠ [ln(e)]^2 $$</p>
<p>As you might be thinking that $ ln(e)^2 $ is same as $ [ln(e)]^2 $ but that's not true.</p>
<p>Actually,
$$ l... |
529,260 | <p>Let $V$ be a complex vector space of dimension $n$ with a scalar product, and let $u$ be an unitary vector in $V$. Let $H_u: V \to V$ be defined as</p>
<p>$$H_u(v) = v - 2 \langle v,u \rangle u$$</p>
<p>for all $v \in V$. I need to find the minimal polynomial and the characteristic polynomial of this linear operat... | lhf | 589 | <p><em>Hint:</em> Consider an orthonormal basis containing $u$ and express $H$ in that basis.</p>
|
970,062 | <p>To show to quadratic forms are not equivalent, we can find rank, or discriminant or some element which is represented by either one only etc. But Is there a general criterion to show that two binary(right now I am only concerned for binary) quadratic forms are equivalent.
Like here is an example which uses variable ... | Bhaskar Vashishth | 101,661 | <p>Measure Theory and Probability Theory by Athreya, and Probability and Measure by Billingsley</p>
|
1,473,513 | <p>The motion of a pendulum is described by the differential equation</p>
<p><span class="math-container">$$ \ddot\theta +\frac gl \sin \theta = 0$$</span></p>
<p>if we integrate this equation with respect to <span class="math-container">$\theta$</span> we obtain</p>
<p><span class="math-container">$$ \frac 12 \dot... | Hosein Rahnama | 267,844 | <p>There is a tidy trick for that using <strong>chain-rule</strong>. Remember this once and for all. We have</p>
<p>$$\ddot \theta (t) + {g \over l}\sin \left( {\theta \left( t \right)} \right) = 0$$</p>
<p>where it is a <strong>nonlinear second order differential equation</strong>. Wow, it seems scary a little as we... |
141,484 | <p><strong>Bug introduced in 10.4.1 or earlier and fixed in 11.1.1</strong></p>
<hr>
<p>I recently installed MMA v11.1 and encountered an issue with the memory usage of the LinearModelFit[] command. It appears than when mixing numeric and nominal variables, the LinearModelFit[] command uses a very large block of memo... | yode | 21,532 | <p>This is what you after?</p>
<pre><code>string="[[1 4 5 6 2] [9 8 7 4 7] [10 3 1 0 5]],[[3 0 9 1 7] [5 6 11 1 7] [3 0 2 0 1]]";
ImportString[StringReplace[#,Whitespace->","],"RawJSON"]&/@StringSplit[string,","]
</code></pre>
<blockquote>
<p>{{{1,4,5,6,2},{9,8,7,4,7},{10,3,1,0,5}},{{3,0,9,1,7},{5,6,11,1,7},... |
1,993,693 | <blockquote>
<p>$$\lim_{x \rightarrow +\infty} \frac{2^x}{x}$$ $$\lim_{x \rightarrow
\infty} \frac{x^{50}}{e^x}$$</p>
</blockquote>
<p>I don't really know how to solve this.</p>
<p>As for the first one, I know that $\lim_{x \rightarrow \infty} a^x=0$ , I supposed that helps...?</p>
<p>How do I solve these (prefer... | egreg | 62,967 | <p>You can catch two birds with a stone. Note that $2^x=e^{x\log 2}$ so
$$
\frac{2^x}{x}=\frac{e^{x\log 2}}{x}=\frac{e^{x\log 2}}{x\log 2}\log 2
=\frac{e^t}{t}\log 2 \qquad(\text{for }t=x\log2)
$$
and
$$
\frac{x^{50}}{e^x}=\left(\frac{x}{e^{x/50}}\right)^{\!50}=
50^{50}\left(\frac{x/50}{e^{x/50}}\right)^{\!50}=
50^{50... |
2,384,538 | <p>I am studying Linear Algebra Done Right, chapter 2 problem 6 states:</p>
<blockquote>
<p>Prove that the real vector space consisting of all continuous real valued functions on the interval $[0,1]$ is infinite dimensional.</p>
</blockquote>
<p><strong>My solution:</strong></p>
<p>Consider the sequence of functio... | Jonas Meyer | 1,424 | <p>By definition, <span class="math-container">$x^n + \sum\limits_{k=0}^{n-1}a_k x^k$</span> is a nonzero polynomial of degree <span class="math-container">$n$</span>. A nonzero polynomial of degree <span class="math-container">$n$</span> can have at most <span class="math-container">$n$</span> zeros in any field, and... |
97,788 | <p>What exactly is the connection between knots and operator algebra? I heard that Jones established such a connection while discovering the celebrated Jones Polynomial. </p>
<p>Now Jones Polynomial is probably understood out of that context on its own, but what was it with Operator algebras in this space ? Can someon... | Daniel Moskovich | 2,051 | <p>The relationship between operator algebras and <em>braids</em> is fairly straightforward to explain, and is nicely written up in many places (<i>e.g.</i> in Kauffman's <a href="http://rads.stackoverflow.com/amzn/click/9810203446" rel="noreferrer">Knots and Physics</a>). Jones studied representations of the braid gr... |
97,788 | <p>What exactly is the connection between knots and operator algebra? I heard that Jones established such a connection while discovering the celebrated Jones Polynomial. </p>
<p>Now Jones Polynomial is probably understood out of that context on its own, but what was it with Operator algebras in this space ? Can someon... | Koushik | 30,081 | <p>you may looks into David Evans monumental book on quantum symmetries on operator algebras</p>
|
4,459,722 | <p>I have the vector field
<span class="math-container">\begin{align*}
X:\mathbb R^d&\to\mathbb R^d\\
x&\mapsto\frac{x}{\|x\|}
\end{align*}</span>
which is a differentiable vector field outside of the origin, and I am interested in its divergence. After some easy computation we get
<span class="math-container">... | robjohn | 13,854 | <p><strong>Function Away from the Origin</strong></p>
<p><span class="math-container">$\nabla\|x\|=\frac{x}{\|x\|}$</span>. Therefore,
<span class="math-container">$$
\begin{align}
\nabla\cdot\frac{x}{\|x\|}
&=\frac{\nabla\cdot x}{\|x\|}-x\cdot\frac{x}{\|x\|^3}\tag{1a}\\
&=\frac{d}{\|x\|}-\frac1{\|x\|}\tag{1b}\... |
4,459,722 | <p>I have the vector field
<span class="math-container">\begin{align*}
X:\mathbb R^d&\to\mathbb R^d\\
x&\mapsto\frac{x}{\|x\|}
\end{align*}</span>
which is a differentiable vector field outside of the origin, and I am interested in its divergence. After some easy computation we get
<span class="math-container">... | Calvin Khor | 80,734 | <p>(This is in addition to the <a href="https://math.stackexchange.com/a/4460505/80734">other</a> <a href="https://math.stackexchange.com/a/4459766/80734">two</a> answers: here, we give the 'correct' generalisation of <span class="math-container">$\operatorname{sgn}'=2\delta_0$</span> to higher dimensions. Put another ... |
2,280,052 | <p>Wolfram Alpha says:
$$i\lim_{x \to \infty} x = i\infty$$</p>
<p>I'm having a bit of trouble understanding what $i\infty$ means. In the long run, it seems that whatever gets multiplied by $\infty$ doesn't really matter. $\infty$ sort of takes over, and the magnitude of whatever is being multiplied is irrelevant. I.e... | Mark S. | 26,369 | <p>Briefly, Wolfram|Alpha preserves the $i$ because it's giving you a "direction" for the infinity. Just like $-7\displaystyle{\lim_{x\to\infty}}x=-\infty$ (the direction being "leftwards" on the real line/in the complex plane), $i\displaystyle{\lim_{x\to\infty}}x=i\infty$ (the direction being "upwards" in the complex ... |
2,280,052 | <p>Wolfram Alpha says:
$$i\lim_{x \to \infty} x = i\infty$$</p>
<p>I'm having a bit of trouble understanding what $i\infty$ means. In the long run, it seems that whatever gets multiplied by $\infty$ doesn't really matter. $\infty$ sort of takes over, and the magnitude of whatever is being multiplied is irrelevant. I.e... | fleablood | 280,126 | <p>In the reals all non-zero numbers have a parity. Either they are positive or they are negative. <span class="math-container">$\lim_{x\rightarrow \infty}|ax| =\infty$</span> (if <span class="math-container">$a \ne 0$</span>) because the magnitude of <span class="math-container">$ax$</span> gets infinitely large.</p... |
1,427,595 | <blockquote>
<p>The <a href="https://en.wikipedia.org/wiki/Cayley_table" rel="nofollow">Cayley table</a> tells us whether a group is abelian. Because the group operation of an abelian group is commutative, a group is abelian if and only if its Cayley table is symmetric along its diagonal axis.</p>
</blockquote>
<p>S... | Ben Sheller | 250,221 | <p>Number the elements of the group, and then think of the Cayley table as a matrix: in the $ij$-th entry, we have $g_ig_j$.</p>
<p>If the Cayley table is symmetric, then the $ij$-th entry is equal to the $ji$-th entry, so $g_ig_j=g_jg_i$, and so the group is abelian.</p>
<p>Conversely, if the group is abelian, then ... |
3,696,776 | <p>I was given this problem:</p>
<p><a href="https://i.stack.imgur.com/8ACor.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/8ACor.png" alt="Problem"></a></p>
<p>These are my calculations and I'm asking for verification:</p>
<p>Pointwise limit:</p>
<p><span class="math-container">$\lim_{n \to \in... | Saaqib Mahmood | 59,734 | <p>For all <span class="math-container">$x \in [R, +\infty)$</span>, we find that, since <span class="math-container">$R > 1$</span>, therefore we have
<span class="math-container">$$
\begin{align}
\lim_{n \to \infty} f_n(x) &= \lim_{n \to \infty} \frac{ x^{2n} }{ 1 + x^{2n} } \\
&= \lim_{n \to \infty} \frac... |
980,818 | <p>I'm working on a problem that involves the following summation:
$$y=\sum_{i=0}^{x}i2^i$$
I need to determine the largest value of $x$ such that $y$ is less than or equal to some integer K. Currently I'm using a lookup table approach which is fine, but I would really like to find and understand a solution that would ... | MPW | 113,214 | <p>You can compute this sum exactly to get $$y=x2^{x+1}+2$$.</p>
|
3,964,862 | <p>The arithmetic mean has the nice property of minimising the sum of squares, or in other words, minimising the sum of quadratic-euclidean distances. Formally, given a set of points <span class="math-container">$x_0, \dots, x_n \in \mathbb{R}^d$</span>, the arithmetic mean, <span class="math-container">$\mu = \frac{1}... | Bolito2 | 856,740 | <p>Let <span class="math-container">$\alpha = (x_1x_2...x_n)^\frac{1}{n} $</span>. Then, <span class="math-container">$\ln(\alpha) = \frac{1}{n} \sum{\ln(x_i)}$</span> so because of the property you stated, <span class="math-container">$$\ln(\alpha) = \underset{x \in \mathbb{R}^d}{\operatorname{argmin}} \sum_{i=1}^n d... |
123,054 | <blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="https://math.stackexchange.com/questions/33215/what-is-48293">What is 48÷2(9+3)?</a> </p>
</blockquote>
<p>In the field of real numbers, does the expression 10 / 2 * 5 make sense? Is it 25 or 1? Is it a bad question or the order of computati... | Ross Millikan | 1,827 | <p>The computer languages I have used explicitly say that 10/2*5=25. For writing, I would always include the parentheses to be clear.</p>
|
123,054 | <blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="https://math.stackexchange.com/questions/33215/what-is-48293">What is 48÷2(9+3)?</a> </p>
</blockquote>
<p>In the field of real numbers, does the expression 10 / 2 * 5 make sense? Is it 25 or 1? Is it a bad question or the order of computati... | Brian M. Scott | 12,042 | <p>The most widespread convention is that operations <strong>of equal precedence</strong> are performed from left to right in the absence of parentheses; by this convention $10/2\cdot 5=25$. However, it is violated often enough, intentionally or otherwise, that in such cases one should always supply enough parentheses ... |
1,255,334 | <p>A classmate and I are studying this following question from Stein-Shakarchi, Chapter 2, Exercise 12:</p>
<blockquote>
<p>Show that there are <span class="math-container">$f \in L^1(\mathbb{R}^d)$</span> and a sequence <span class="math-container">$\{f_n\}$</span> with <span class="math-container">$f_n \in L^1(\mathb... | shalop | 224,467 | <p>We will give an example in $\mathbb{R}^1$. The analagous construction in $\mathbb{R}^d$ is similar.</p>
<p>Fix $k \in \mathbb{Z}$. Take some enumeration $\{I^k_n\}_{n \in \mathbb{N}}$ of all subintervals of $[k,k+1]$ which have the form $[\frac{p}{q},\frac{p+1}{q}]$ for some integers $p,q$. Notice that $m(I^k_n) \t... |
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