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 |
|---|---|---|---|---|
66,068 | <p>I have a list like this. </p>
<pre><code>cdatalist = {{1., 0.898785, Failed, Failed, 50., 25., "serial"}, {1., 1.31175,1., Failed, 50., 25., "serial"}, {1., 18.8025, Failed, 0.490235, 50., 25., "serial"}, {1., 19.6628, 0.990079, Failed, 50., 25., "serial"}, {1., 39.547, Failed, Failed, 50., 25., "serial"}, {1., 39.... | Gerli | 8,389 | <p>Try <a href="http://reference.wolfram.com/language/ref/Cases.html" rel="nofollow">Cases</a>.</p>
<p>If you want the third column to be Real:</p>
<pre><code>Cases[cdatalist, {_, _, _Real, __}][[All, 1 ;; 3]]
</code></pre>
<p>or</p>
<pre><code>Cases[cdatalist[[All, 1 ;; 3]], {_, _, _Real}]
</code></pre>
<p>depend... |
4,059,489 | <blockquote>
<p>Let <span class="math-container">$ A, B \in M_n (\mathbb{C})$</span> such that <span class="math-container">$(A-B)^2 = A -B$</span>. Then <span class="math-container">$\mathrm{rank}(A^2 - B^2) \geq \mathrm{rank}( AB -BA)$</span>.</p>
</blockquote>
<p>I tried to apply the basic inequalities without resul... | PTDS | 277,299 | <p><span class="math-container">$$\sqrt{\frac{x-8}{1388}}+\sqrt{\frac{x-7}{1389}}+\sqrt{\frac{x-6}{1390}}=\sqrt{\frac{x-1388}{8}}+\sqrt{\frac{x-1389}{7}}+\sqrt{\frac{x-1390}{6}} \tag {1}$$</span></p>
<p>Since <span class="math-container">$x \geq 1390$</span>, let us put <span class="math-container">$x = 1390 + k$</span... |
3,788,298 | <p>Let <span class="math-container">$f(x)$</span> be an integrable function on <span class="math-container">$[0,1]$</span> that obeys the property <span class="math-container">$f(x)=x, x=\frac{n}{2^m}$</span> where <span class="math-container">$n$</span> is an odd positive integer and m is a positive integer. Calculat... | Steven | 606,584 | <p>If we're using the Lebesgue integral, the value can be anything; simply define <span class="math-container">$f(x) = c$</span> outside the countable number of points you specify. If the Riemann integral is under consideration, the value must be <span class="math-container">$\frac12$</span>, since Riemann integrabilit... |
1,747,525 | <p>Given a number $N$, how can I write down a summation of all odd numbers divisible by 5 which are also less than $N$?</p>
<p>For instance, if $N = 27$ then I am looking for a series to generate $5+15+25$.</p>
<p>Its pretty clear the series looks like </p>
<p>$$\sum_{k=0}^{???} 5(2k+1)$$</p>
<p>but I am having tro... | Unit | 196,668 | <p>You want $5(2k+1) \le N < 5(2(k+1)+1)$, which means $k \le \frac{N/5-1}{2} < k+1$; now take floors.</p>
|
1,747,525 | <p>Given a number $N$, how can I write down a summation of all odd numbers divisible by 5 which are also less than $N$?</p>
<p>For instance, if $N = 27$ then I am looking for a series to generate $5+15+25$.</p>
<p>Its pretty clear the series looks like </p>
<p>$$\sum_{k=0}^{???} 5(2k+1)$$</p>
<p>but I am having tro... | Anurag A | 68,092 | <p>You want $5(2k+1) \leq N$. Thus
$$k \leq \frac{N}{10}-\frac{1}{2}.$$
Thus $k=\left\lfloor \frac{N}{10}-\frac{1}{2}\right\rfloor$.</p>
|
505,178 | <blockquote>
<p>Suppose $k$ is an algebraically closed field, and $f\in k[x, y]$ is
an irreducible polynomial in two variables. Furthermore, suppose that $f(u(x),
v(y))=f(x, y)$ for every $x, y\in k$, where $u\in k[x]$, $v\in k[y]$ are
polynomials of one variable. Can we conclude that either $u(x)=x$ or
$v(y)=y... | leshik | 15,215 | <p>The answer is no. Take <span class="math-container">$f(x,y)=x^2-1+y^2$</span> and <span class="math-container">$u(x)=-x,$</span> <span class="math-container">$\nu(y)=-y.$</span> If you want some more "nontrivial" examples, you can consider symmetries with respect to <span class="math-container">$x\to 1-x$</span> or ... |
3,736,580 | <p>Show that for <span class="math-container">$n>3$</span>, there is always a <span class="math-container">$2$</span>-regular graph on <span class="math-container">$n$</span> vertices. For what values of <span class="math-container">$n>4$</span> will there be a 3-regular graph on n vertices?</p>
<p>I think this q... | DanielV | 97,045 | <p>You could define the sequence recursively in terms of the average of the previous terms of the sequence:</p>
<p><span class="math-container">$$x_k = \begin{cases}
3 & \text{ if } & a_{k-1} > \pi \\
4 & \text{ if } & a_{k-1} < \pi \\
\end{cases}$$</span></p>
<p>where</p>
<p><span class="math-... |
3,736,580 | <p>Show that for <span class="math-container">$n>3$</span>, there is always a <span class="math-container">$2$</span>-regular graph on <span class="math-container">$n$</span> vertices. For what values of <span class="math-container">$n>4$</span> will there be a 3-regular graph on n vertices?</p>
<p>I think this q... | fleablood | 280,126 | <p>Yes. Consider <span class="math-container">$a \le \omega \le b$</span>. (In this specific case <span class="math-container">$a=3; b=4; \omega = \pi$</span>)</p>
<p>Define <span class="math-container">$x_1=\begin{cases}b &\omega \le \frac {a+b}2\\a &\omega >\frac{a+b}2\end{cases}$</span></p>
<p><span clas... |
2,714,450 | <p>Suppose $A$ and $B$ are two square matrices so that $e^{At}=e^{Bt}$ for infinite (countable or uncountable) values of $t$ where $t$ is positive.</p>
<p>Do you think that $A$ <strong>has to be equal to</strong> $B$?</p>
<p>Thanks,
Trung Dung.</p>
<hr>
<p>Maybe I do not state clearly or correctly.</p>
<p>I mean t... | Angina Seng | 436,618 | <p>Take $A=\pmatrix{0&1\\-1&0}$. Then $\exp(tA)=I$ for $t=2n\pi$ ($n$ integer). That is $\exp(tA)=\exp(tB)$ infinitely often for $B$ the zero matrix.</p>
|
2,714,450 | <p>Suppose $A$ and $B$ are two square matrices so that $e^{At}=e^{Bt}$ for infinite (countable or uncountable) values of $t$ where $t$ is positive.</p>
<p>Do you think that $A$ <strong>has to be equal to</strong> $B$?</p>
<p>Thanks,
Trung Dung.</p>
<hr>
<p>Maybe I do not state clearly or correctly.</p>
<p>I mean t... | Community | -1 | <p>$\textbf{Proposition 1.}$ Let $(t_k)$ be a positive sequence that converges to $l$ s.t., for every $k$, $t_k\not= l$. </p>
<p>If, (*) for every $k$, $e^{t_kA}=e^{t_kB}$, then $A=B$.</p>
<p>$\textbf{Proof.}$ There is $k$ s.t. $t_kA$ is $2i\pi$ congruence free (for every $\lambda,\mu\in spectrum(A)$, $t_k(\lambda-\m... |
927,188 | <p>This question has been on my mind for a very long time, and I thought I'd finally ask it here. </p>
<p>When I was 6, my dad pulled me out of school. The classes were too easy; the professors, too dull. My father had been man of philosophy his entire life (almost got a PhD in it) and regretted not having a more q... | paul garrett | 12,291 | <p>There is a lot of hype surrounding "math competitions", since "winning" and "competition" can easily be understood in broad cultural terms, whether or not we collectively think of these as either highest goals or as legitimate formative principles. Most professional mathematics and its practice does not resemble com... |
3,435,256 | <p>The following statement is given in my book under the topic <em>Tangents to an Ellipse</em>:</p>
<blockquote>
<p>The <a href="http://mathworld.wolfram.com/EccentricAngle.html" rel="nofollow noreferrer">eccentric angles</a> of the points of contact of two parallel tangents differ by <span class="math-container">$\... | Quanto | 686,284 | <p>According to the definition of the eccentric angle for the ellipse <span class="math-container">$\frac{x^2}{a^2}+ \frac{y^2}{b^2}=1$</span>,</p>
<p><span class="math-container">$$ t= \tan^{-1} \frac{ay}{bx}$$</span></p>
<p>evaluate </p>
<p><span class="math-container">$$t_2-t_1= \tan^{-1} \frac{ay_2}{bx_2} - \tan... |
2,339,707 | <p>Suppose I have five bins into which I want to place 15 balls. The bins have capacities $2$, $2$, $3$, $3$, and $7.$ I place the balls one at a time in the bins, randomly and uniformly amongst the bins that are not full (so for example, if after placing four balls, both of the bins with capacity $2$ are already full,... | Miguel | 259,671 | <p>Your solution is incorrect because you restrict to a particular value for $p_0$. It is true that if $v$ is an eigenvector of $A$, then you have:
$$A^n v=\lambda^n v$$
But you have to compute $A^n p_0$ instead, and you cannot choose $p_0$ because it is part of the problem statement. In other words, you cannot assume ... |
1,590,625 | <blockquote>
<p>If $f(x)=\log \left(\cfrac{1+x}{1-x}\right)$ for $-1 < x < 1$,then
find $f \left(\cfrac{3x+x^3}{1+3x^2}\right)$ in terms of $f(x)$.</p>
</blockquote>
<p><strong>My Attempt</strong>
$$f \left(\cfrac{3x+x^3}{1+3x^2}\right)=\log\left(\cfrac{1+\cfrac{3x+x^3}{1+3x^2}}{1-\cfrac{3x+x^3}{1+3x^2}}\rig... | Archis Welankar | 275,884 | <p>hint $$log(\frac{1+x}{1-x})=2(x+\frac{x^3}{3!}...\infty)$$ for $|x|<1$</p>
|
33,702 | <p>The code that I have written has an unintended consequence that I'm not sure how to get around. I want 3 rotation transforms to be applied simultaneously to 1 graphics object. Instead, I get 3 separate separate copies of the graphics object, one per transformation.</p>
<p>The documentation does state that this wi... | halirutan | 187 | <p>When tracing other plots, I often saw that plotting-related functions compile their arguments when possible. A quick look at the trace of your example suggests the same, because the <code>Exp</code> function is used only for the evaluation of <code>function</code>. This seems to indicate that your second argument <c... |
700,012 | <p>Ok, so the question is to prove by induction that:</p>
<p>$${n \choose k} \le n^k$$</p>
<p>Where $N$ and $k$ are integers, $k \le n$;</p>
<p>How do I approach this? Do i choose a $n$ and a $k$ to form my base case?</p>
| Stella Biderman | 123,230 | <p>You need to prove this for all $n$ and all $k$, right? Then, to do it by induction, you must take an arbitrary $n$ and then induct on $k$.</p>
|
298,791 | <blockquote>
<p>If a ring $R$ is commutative, I don't understand why if $A, B \in R^{n \times n}$, $AB=1$ means that $BA=1$, i.e., $R^{n \times n}$ is Dedekind finite.</p>
</blockquote>
<p>Arguing with determinant seems to be wrong, although $\det(AB)=\det(BA ) =1$ but it necessarily doesn't mean that $BA =1$.</p>
... | Martin Brandenburg | 1,650 | <p><strong>Lemma</strong>. Every surjective endomorphism $f : M \to M$ of a finitely generated $R$-module $M$ is an isomorphism.</p>
<p>Proof: $M$ becomes an $R[x]$-module, where $x$ acts by $f$. By assumption, $M=xM$. Nakayama's Lemma implies that there is some $p \in R[x]$ such that $(1-px)M=0$. This means $\mathrm{... |
27,455 | <p>Let $(f_n)_{n \geq 1}$ be disjointly supported sequence of functions in $L^\infty(0,1)$. Is the space $\overline{\mathrm{span}(f_n)}$ (the closure of linear span) complemented in $L^\infty(0,1)$? By complemented we mean that $L^\infty(0,1) = \overline{\mathrm{span}(f_n)} \oplus X$, where $X$ is a subspace of $L^\inf... | Philip Brooker | 11,532 | <p>The answer is <em>no</em>. The closed linear span of such a sequence is separable, and so if $\overline{span(f_n)}$ was complemented in $L^\infty (0, 1)$ then every Banach space isomorphic to $L^\infty (0, 1)$ would contain an infinite dimensional, separable complemented subspace. In particular, since $L^\infty (0, ... |
27,455 | <p>Let $(f_n)_{n \geq 1}$ be disjointly supported sequence of functions in $L^\infty(0,1)$. Is the space $\overline{\mathrm{span}(f_n)}$ (the closure of linear span) complemented in $L^\infty(0,1)$? By complemented we mean that $L^\infty(0,1) = \overline{\mathrm{span}(f_n)} \oplus X$, where $X$ is a subspace of $L^\inf... | Michael Causey | 52,539 | <p>If all but finitely many of the functions are the zero function, then the answer is yes, because any finite-dimensional subspace is complemented. But this is the trivial case. </p>
<p>In the nontrivial case, just note that if you normalize the nonzero functions in the sequence, they form a basic sequence $1$-equi... |
1,986,798 | <p>The way I solved the problem is to change the equation to $|x+2|=1-|y-3|$, and then square both sides. But I don't think it is the right way to solve the problem. I hope someone can either give me a hint or show me how to solve the problem.</p>
<blockquote>
<p>$|x+2|+|y-3|=1$ is an equation for a square. How many... | GFauxPas | 173,170 | <p>The Jacobian can be a row or column vector, which is to say a matrix with only one row or column. As CoffeeBliss says, the Jacobian of a function $\mathbb R^n \to \mathbb R^m$ is an $m \times n$ matrix.</p>
<p>There is an operation called the <strong>gradient</strong> of $f$ and it is defined by:</p>
<p>$$\boldsym... |
1,729,308 | <p>The sum of the first $n$ $(n>1)$ terms of the A.P. is $153$ and the common difference is $2$. If the first term is an integer , then number of possible values of $n$ is </p>
<p>$a)$ $3$</p>
<p>$b)$ $4$</p>
<p>$c)$ $5$</p>
<p>$d)$ $6$</p>
<p>My approach : I used the formula for the first $n$ terms of an A.P. ... | User | 311,480 | <p>one of the possible value of $n$ is $3$.</p>
<p><strong>REASON</strong></p>
<p>Let the first term of $AP$ be $a-2$.</p>
<p>$AP: a-2, a, a+2,a+4,a+6,a+8,a+10,\cdots$</p>
<p>Sum of first three terms of $AP= 3a$ which will give $a=51$(integer value).</p>
<p>Now generalising this pattern</p>
<p>we get other possib... |
331,654 | <p>After having received Brian M. Scott's permission (see comments in the selected answer) I am integrating his suggestions with my own solutions to form a complete answer to the questions apperaing below. </p>
<blockquote>
<p>Let $\mathscr{T}$ be the collection of subsets of $\Bbb R$ consisting
of $\emptyset, \Bb... | anon271828 | 54,360 | <p>Hint: For $(b)$, use the fact that $\Bbb Q$ is dense in $\Bbb R$. Maybe consider $\sqrt{2}+1/n\to\sqrt{2}$ as $n\to\infty$. For $(g)$, I'm not sure why you say there doesn't exist a finite cover; $\Bbb R$ itself is a perfectly good finite open cover. For $(j)$, I think you need more argument. It seems like you are j... |
2,913,921 | <p>As I know, the general complexity of matrix inversion is $O(n^3)$,
but it is a little bit high.
My matrix is $(I + A)$ , where $I$ is an $n \times n$ identity matrix and $A$ is a hermitian matrix and the norm of all its elements are very small (near $1/100$). Therefore, I think it's a particular matrix that close... | Pushpendre | 52,858 | <p>$(I+A)(I-A) = I - AA$. If $A$ is small enough, then you may be able to approximate $(I+A)^{-1}$ by $I-A$. </p>
<p>If you don't need to compute the inverse but only need to solve $(I+A)^{-1}v$ then you can improve accuracy by taking the rational series $I - A + AA/2 + ...$ upto the desired tolerance and then compute... |
3,089,493 | <p>Calculate the volume between <span class="math-container">$x^2+y^2+z^2=8$</span> and <span class="math-container">$x^2+y^2-2z=0$</span>. I don't know how to approach this but I still tried something:</p>
<p>I rewrote the second equation as: <span class="math-container">$x^2+y^2+(z-1)^2=z^2+1$</span> and then combin... | 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... |
1,516,925 | <p>Let $x,y,z$ be 3 non-zero integers defined as followed: </p>
<p>$$(x+y)(x^2-xy+y^2)=z^3$$</p>
<p>Let assume that $(x+y)$ and $(x^2-xy+y^2)$ are coprime
and set $x+y=r^3$ and $x^2-xy+y^3=s^3$</p>
<p>Can one write that $z=rs$ where $r,s$ are 2 integers?
I am not seeing why not but I want to be sure.</p>
| André Nicolas | 6,312 | <p>Yes, there exist such integers $r$ and $s$. It is simplest to use the Fundamental Theorem of Arithmetic (Unique Factorization Theorem). The result is easy to prove for negative $z$ if we know the result holds for positive $z$. Also, the result is clear for $z=1$. So we may assume that $z\gt 1$.</p>
<p>Let $z=p_1^{a... |
131,051 | <p>So we want to find an $u$ such that $\mathbb{Q}(u)=\mathbb{Q}(\sqrt{2},\sqrt[3]{5})$. I obtained that if $u$ is of the following form: $$u=\sqrt[6]{2^a5^b}$$Where $a\equiv 1\pmod{2}$, and $a\equiv 0\pmod{3}$, and $b\equiv 0\pmod{2}$ and $ b\equiv 1\pmod{3}$. This works since $$u^3=\sqrt{2^a5^b}=2^{\frac{a-1}{2}}5^{\... | Community | -1 | <p>If we take $u = \sqrt{2} + \sqrt[3]{5}$, such a $u$ <em>almost always</em> turns out to work. In fact let's try if a rational linear combination of $\sqrt{2}$ and $\sqrt[3]{5}$ will work. Let us now write $u$ as $u = a\sqrt{2} + b\sqrt[3]{5}$ for rationals $a$ and $b$.</p>
<p>Clearly we have that $\Bbb{Q}(u)\subse... |
2,242,846 | <p>I will try to prove the theorem in the title:</p>
<blockquote>
<p>Suppose S is closed, non-empty then if $b = \sup\{x: x \in S\}$ (least upper bound), $b \in S$.</p>
</blockquote>
<p>I also use the following <strong>Theorem</strong> which we proved in class: S is closed iff every Cauchy sequence in S converges t... | Bram28 | 256,001 | <p>Given your EE rule, you need to prove $\forall x (\neg p(x) \rightarrow \neg \forall x \ P(x))$, which you do by a universal introduction on $\neg p(x) \rightarrow \forall x \ p(x)$, which on its turn you prove by a conditional Introduction on a subproof that assumes $\neg p(x)$ and derives $\neg \forall x \ p(x)$ .... |
4,602,596 | <p>In an exercise, my teacher asked us to prove that <span class="math-container">$\ell^1$</span> is a Banach space. I was able to do so, but there are two steps in my proof that I'm not quite so sure they are correct. This is what I came up with:</p>
<hr />
<p>let <span class="math-container">$x^n=(x^n_1, x^n_2, ...)$... | Lucas Henrique | 274,595 | <p>You already figured out that the homogeneous equation is unbounded. Suppose that there are two bounded solutions <span class="math-container">$y_1, y_2$</span>. Thus <span class="math-container">$\tilde y:=y_1 - y_2$</span> is bounded and satisfies the associated homogeneous equation. But if <span class="math-contai... |
1,561,563 | <p>Two circles $\Gamma_1,\Gamma_2$ have centers $O_1,O_2$. Let $\Gamma_1\cap\Gamma_2=A,B$, with $A\neq B$. An arbitrary line through $B$ intersects $\Gamma_1$ at $C$ and $\Gamma_2$ at $D$. The tangents to $\Gamma_1$ at $C$ and to $\Gamma_2$ at $D$ intersect at $M$. Let $N=AM\cap CD$. Let $l$ be a line through $N$ paral... | Kay K. | 292,333 | <p>Another simple solution:</p>
<p>\begin{align}
&x\mapsto\sin u\\
&I=\int_0^{\pi/2} \frac{\cos u}{\sin u + \cos u}du\\
&u\mapsto \frac{\pi}2-v\\
&I=\int_0^{\pi/2} \frac{\sin v}{\sin v + \cos v}dv\\
&\therefore 2I=\int_0^{\pi/2} \frac{\sin u + \cos u}{\sin u + \cos u}du=\frac{\pi}{2}\\
&\theref... |
139,385 | <p>Can anyone help me prove if $n \in \mathbb{N}$ and is $p$ is prime such that $p|(n!)^2+1$ then $(p-1)/2$ is even?</p>
<p>I'm attempting to use Fermats little theorem, so far I have only shown $p$ is odd.</p>
<p>I want to show that $p \equiv 1 \pmod 4$</p>
| Ragib Zaman | 14,657 | <p><strong>Theorem</strong>: Let $p$ be an odd prime. Then $-1$ is a square modulo $p$ if and only if $ p = 1\mod 4.$</p>
<p>Proof: For any $x\neq 0$ in $\mathbb{Z}_p$ call the following the bundle generated by $x$:
$$ B(x) = \{ x, x^{-1}, -x, -x^{-1} \}. $$</p>
<p>Check $B(x) = B(x^{-1}) = B(-x) = B(-x^{-1}) .$ The ... |
1,499,949 | <p>Prove that for all event $A,B$</p>
<p>$P(A\cap B)+P(A\cap \bar B)=P(A)$</p>
<p><strong>My attempt:</strong></p>
<p>Formula: $\color{blue}{P(A\cap B)=P(A)+P(B)-P(A\cup B)}$</p>
<p>$=\overbrace {P(A)+P(B)-P(A\cup B)}^{=P(A\cap B)}+\overbrace {P(A)+P(\bar B)-P(A\cup \bar B}^{=P(A\cap \bar B)})$</p>
<p>$=2P(A)+\un... | Michael Medvinsky | 269,041 | <p>@Angelo Mark
In your comment to @Elekko you say</p>
<p>$$x^2=9 \Rightarrow x=9^{\frac{1}{2}}= \sqrt{9}=3$$
however
$x^2=9$ doesn't imply neigher $x=9^{\frac{1}{2}}$ or $x= \sqrt{9}$, although they usually mean exactly the same thing.</p>
<p>The main question is not whether $x^{1/n}=\sqrt[n]x$ or not but what is an... |
2,249,020 | <p><a href="https://i.stack.imgur.com/L7PXf.jpg" rel="nofollow noreferrer">The Math Problem</a></p>
<p>I have issues with finding the Local Max and Min, and Abs Max and Min, after I find the Critical Point. How do I do this problem in its entirety? </p>
| angryavian | 43,949 | <p>If you have two vectors $v=(v_1,v_2)$ and $w=(w_1,w_2)$, then their dot product is $$v \cdot w = v_1 w_1 + v_2 w_2.$$ Additionally, we also have
$$v \cdot w = \|v\| \|w\| \cos \theta$$ where $\|v\|=\sqrt{v_1^2+v_2^2}$, $\|w\|=\sqrt{w_1^2+w_2^2}$, and $\theta$ is the angle between $v$ and $w$. <a href="https://proofw... |
2,655,018 | <p>I have a quick question regarding a little issue.</p>
<p>So I'm given a problem that says "$\tan \left(\frac{9\pi}{8}\right)$" and I'm supposed to find the exact value using half angle identities. I know what these identities are $\sin, \cos, \tan$. So, I use the tangent half-angle identity and plug-in $\theta = \f... | Rohan Shinde | 463,895 | <p>Let $\tan \frac {9\pi}{8}= \tan \frac {\theta }{2}=a$</p>
<p>By half angle formulas $$\tan \theta=\frac {2\tan \frac {\theta }{2}}{1-\tan ^2\frac {\theta }{2}}$$
Hence we get $$1=\frac {2a}{1-a^2}$$
Hence we get $a^2+2a-1=0$</p>
<p>Solve the quadratic to get the answer.</p>
<p>Note : By using quadratic formula we... |
1,473,862 | <p>Let $X$ be a random variable with mean $\mu$ and variance $\sigma^2$. Let $a < \mu$ and consider the probability
$$
F_X(a) = \mathbb{P}(X \leq a) = \mathbb{P}(X - \mu \leq a - \mu).
$$
If $a > \mu$, Cantelli's inequality (see here <a href="https://en.wikipedia.org/wiki/Cantelli%27s_inequality" rel="nofollow"... | Clement C. | 75,808 | <p>If you do not have any further assumption on the values $X$ can take (e.g., is it lower bounded a.s.?), then you cannot get any meaningful lower bound. For any $\varepsilon\in[0,1]$ (and wlog the case $\mu=0$), consider the random variable defined by
$$
X = \begin{cases}
-x\frac{1-\varepsilon}{\varepsilon} & \te... |
1,473,862 | <p>Let $X$ be a random variable with mean $\mu$ and variance $\sigma^2$. Let $a < \mu$ and consider the probability
$$
F_X(a) = \mathbb{P}(X \leq a) = \mathbb{P}(X - \mu \leq a - \mu).
$$
If $a > \mu$, Cantelli's inequality (see here <a href="https://en.wikipedia.org/wiki/Cantelli%27s_inequality" rel="nofollow"... | soakley | 84,631 | <p>There is a one-sided Chebyshev's inequality. For $k \gt 0,$ $$P \ [ \ X \leq \mu - k \sigma \ ] \leq { {1} \over {k^2 + 1} } $$</p>
|
3,371,638 | <p>Measure space <span class="math-container">$(X, \mathcal{A}, ν)$</span> has <span class="math-container">$ν(X) = 1$</span>. Let <span class="math-container">$A_n \in \mathcal{A} $</span> and denote </p>
<p><span class="math-container">$B := \{x : x ∈ A_n$</span> for infinitly many n }.</p>
<p>I want to prove that... | copper.hat | 27,978 | <p>Presumably you are using the Euclidean norm. (The unit ball with the <span class="math-container">$l_1, l_\infty$</span> norms <strong>are</strong> polyhedral.)</p>
<p>The Euclidean norm is strictly convex, so if <span class="math-container">$a \neq b$</span> then <span class="math-container">$\|ta+(1-t)b\|^2 < t... |
2,203,988 | <p>I'm reading the book <i>Heat Transfer</i> by J.P. Holman. On the chapter of unsteady-state conduction, page 140, the author remarks:</p>
<blockquote>
<p>The final series solution is therefore:
$${\theta(x,t) \over \theta_i} =
{4\over \pi} \sum^{\infty}_{n=1} {1\over n} e^{-\left({n\pi/L}\right)^2\alpha \,t}\si... | JMP | 210,189 | <p>For order doesn't matter, each ring can be placed on one of $3$ fingers. This results in a unique string, for example $12323113$, which results in finger $1$ having rings $1,6,7$, finger $2$ has rings $2,4$ and finger $3$ has rings $3,5,8$.</p>
<p>This clearly groups each finger's rings as distinct, and gives $3^8$... |
3,125,093 | <p>Let us remember, the conditions to apply L'Hôpital's Rule:</p>
<p>Let suppose:</p>
<p><span class="math-container">$f(x)$</span> and <span class="math-container">$g(x)$</span> are real and differentiable for all <span class="math-container">$x\in (a,b)$</span> </p>
<p>1-) <span class="math-container">$ \lim_{x\t... | J.G. | 56,861 | <p>Repeatedly using the existence of inverses in groups gives <span class="math-container">$$ijk=k^2\implies ij=k,\,ijk=i^2\implies jk=i,\,kijk=k^3\implies kij=k^2=j^2\implies ki=j.$$</span>Define <span class="math-container">$m:=k^{-1}ji=i^2$</span>; we would normally call this <span class="math-container">$-1$</span>... |
209,842 | <p>I was looking for a general way of formulating solutions for work and time problems.</p>
<p>For example,</p>
<p>30 soldiers can dig 10 trenches of size 8*3*3 ft in half a day working 8 hours per day. How many hours will 20 soldiers take to dig 18 trenches of size 6*2*2 ft working 10 hours per day?</p>
<p>Now i kn... | lab bhattacharjee | 33,337 | <p>$30$ soldiers can dig $10$ trenches of size $8\cdot 3\cdot 3$ cube fit in $4$ hours.</p>
<p>$1$ soldier can dig $10$ trenches of size $8\cdot 3\cdot 3$ cube fit in $4\cdot 30$ hours.</p>
<p>$1$ soldier can dig $1$ trench of size $8\cdot 3\cdot 3$ cube fit in $\frac{4\cdot 30}{10}$ hours.</p>
<p>$1$ soldier can di... |
293,026 | <p>The question is to show that $A\sin(x + B)$ can be written as $a\sin x + b\cos x$ for suitable a and b.</p>
<p>Also, could somebody please show me how $f(x)=A\sin(x+B)$ satisfies $f + f ''=0$?</p>
| Michael Hardy | 11,667 | <p>If
$$
f(x) = A\sin(x+B)
$$
then
$$
f'(x) = A\cos(x+B)\cdot\frac{d}{dx}(x+B) = A\cos(x+B)\cdot1,
$$
and
$$
f''(x) = -A\sin(x+B)\cdot\frac{d}{dx}(x+B) = -A\sin(x+B).
$$
So
$$
f''(x)+f(x) = -A\sin(x+B)+A\sin(x+B) = 0.
$$</p>
<p>For the initial question, the standard trigonometric identity
$$
\sin(x+B) = \sin x\cos B+ ... |
1,575,397 | <p>I need help calculating
$$\lim_{n\to\infty}\left(\frac{1}{n^{2}}+\frac{2}{n^{2}}+...+\frac{n}{n^{2}}\right) = ?$$</p>
| Adhvaitha | 228,265 | <p>We have
$$1+2+\cdots+n = \dfrac{n(n+1)}2$$
Hence, we need
$$\lim_{n \to \infty}\left(\dfrac1{n^2} + \dfrac2{n^2} + \cdots + \dfrac{n}{n^2}\right) = \lim_{n \to \infty}\left(\dfrac{n(n+1)}{2n^2}\right) = \dfrac12$$</p>
|
1,991,238 | <p>How can I integrate this? $\int_{0}^{1}\frac{\ln(x)}{x+1} dx $</p>
<p>I've seen <a href="https://math.stackexchange.com/questions/108248/prove-int-01-frac-ln-x-x-1-d-x-sum-1-infty-frac1n2">this</a> but I failed to apply it on my problem.</p>
<p>Could you give some hint?</p>
<p>EDIT : From hint of @H.H.Rugh, I'v... | grand_chat | 215,011 | <p>Way late to the party, but here's a general result, and an elementary derivation:</p>
<hr />
<p><strong>Claim:</strong> Let <span class="math-container">$(a_n)$</span> and <span class="math-container">$(b_n)$</span> be sequences of positive integers with <span class="math-container">$a_n\to\infty$</span> and <span ... |
4,292,618 | <p>I have the following function <span class="math-container">$$\frac{1}{1+2x}-\frac{1-x}{1+x} $$</span>
How to find equivalent way to compute it but when <span class="math-container">$x$</span> is much smaller than 1? I assume the problem here is with <span class="math-container">$1+x$</span> since it probably would b... | njuffa | 114,200 | <p>The formula as written has numerically acceptable behavior in the vicinity of the singularities. Outside this region it can be transformed algebraically into the rational function <span class="math-container">$\frac{2x^{2}}{2x^{2}+3x+1}$</span>. But in this form, there is an issue with premature overflow in intermed... |
3,352,834 | <p><span class="math-container">$A^2 + A - 6I = 0$</span></p>
<p>A= <span class="math-container">$\begin{bmatrix}a & b\\c & d\end{bmatrix}$</span></p>
<p>I was asked to find
<span class="math-container">$a + d$</span>, and <span class="math-container">$ad - bc$</span></p>
<p><span class="math-container">$a+d... | Peter Foreman | 631,494 | <p>You have that
<span class="math-container">$$(A+3I)(A-2I)=0$$</span>
so the matrix
<span class="math-container">$$A=2I=\pmatrix{2&0\\0&2}$$</span>
is a valid such <span class="math-container">$A$</span>.</p>
|
3,352,834 | <p><span class="math-container">$A^2 + A - 6I = 0$</span></p>
<p>A= <span class="math-container">$\begin{bmatrix}a & b\\c & d\end{bmatrix}$</span></p>
<p>I was asked to find
<span class="math-container">$a + d$</span>, and <span class="math-container">$ad - bc$</span></p>
<p><span class="math-container">$a+d... | Mostafa Ayaz | 518,023 | <p>Actually, in the characteristic equation of a matrix <span class="math-container">$A$</span> like<span class="math-container">$$|\lambda I-A|=\lambda^n+a_{n-1}\lambda^{n-1}+\cdots +a_1\lambda+a_0=0$$</span>we have <span class="math-container">$$a_0=|-A|=(-1)^n |A|\\a_{n-1}=\text{tr}(A)$$</span>so here we have three ... |
11,994 | <p>Now that we get to see the SE-network wide list of "hot" questions, I am just shaking my head in disbelief. At the time I am writing this, the two hot questions from Math.SE are titled (get a barf-bag, quick)</p>
<ul>
<li><a href="https://math.stackexchange.com/q/599520/8348">https://math.stackexchange.com/q/599520... | Post No Bulls | 111,742 | <p>There is at least something that can be done to reduce the degree of embarrassment that "hot" questions bring to Math.SE: <strong>improve their titles</strong>. The titles are what ~3 million daily visitors to SE actually see; relatively few will click through to the question (although the absolute numbe... |
3,091,353 | <p>There are 2 definitions of <strong><em>Connected Space</em></strong> in my lecture notes, I understand the first one but not the second. The first one is:</p>
<blockquote>
<p>A topological space <span class="math-container">$(X,\mathcal{T})$</span> is connected if there does not exist
<span class="math-conta... | Hagen von Eitzen | 39,174 | <p>If <span class="math-container">$U$</span> is closed and open, then so is <span class="math-container">$V:=U^\complement$</span>. So if such <span class="math-container">$U$</span> exists that is neither empty nor the whole space, we have <span class="math-container">$X=U\cup V$</span> with <span class="math-contain... |
749,714 | <p>Does anyone know how to show this preferable <strong>without</strong> using modular</p>
<p>For any prime $p>3$ show that 3 divides $2p^2+1$ </p>
| lab bhattacharjee | 33,337 | <p><strong>Generalization</strong>:</p>
<p>If $q$ is prime not dividing integer $\displaystyle m,(m,q)=1\implies m^{q-1}\equiv1\pmod q$ (by Fermat's Little Theorem)</p>
<p>$\displaystyle(q-1)m^{q-1}\equiv (-1)\cdot1\pmod q\equiv-1\iff (q-1)m^{q-1}+1\equiv0\pmod q$</p>
<p>Here $q=3$</p>
|
3,717,506 | <p>I am reading some text about even functions and found this snippet:</p>
<blockquote>
<p>Let <span class="math-container">$f(x)$</span> be an integrable even function. Then,</p>
<p><span class="math-container">$$\int_{-a}^0f(x)dx = \int_0^af(x)dx, \forall a \in \mathbb{R}$$</span></p>
<p>and therefore,</p>
<p><span c... | Travis Willse | 155,629 | <p>This answer expands on A. Kriegman's and folds in some of my comments thereunder.</p>
<p>Let <span class="math-container">$P_n(k)$</span> denote the fraction of values of <span class="math-container">$n$</span>-term sequences with value <span class="math-container">$k$</span>, which we can interpret as the probabili... |
2,247,968 | <blockquote>
<p>$a,b$ are elements in a group $G$. Let $o(a)=m$ which means that $a^m=e$, $\gcd(m,n)=1$ and $(a^n)*b=b*(a^n)$. Prove that $a*b=b*a$.</p>
</blockquote>
<p><em>Hint: try to solve for $m=5,n=3$.</em></p>
<p>I am stuck in this question and can't find an answer to it, can anyone give me some hints?</p>... | egreg | 62,967 | <p>The constant function $f(x)=1$ is a counterexample for (a) and (d).</p>
<p>Since $f(0)=1$, (c) is obviously false.</p>
<p>It remains to show (b) is true. Suppose $f(x)<0$ for some $x\in[0,2]$. Then the minimum of $f$ is negative. What's the derivative at a point of minimum, if this point is in $(0,2)$? Can you ... |
2,258,697 | <p>I recently encountered this question and have been stuck for a while. Any help would be appreciated!</p>
<p>Q: Given that
$$\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = \frac{1}{5} \tag{1} \label{eq:1}$$
$$abc = 5 \tag{2} \label{eq:2}$$
Find $a^3 + b^3 + c^3$. It wasn't specified in the question but I think it can be... | amd | 265,466 | <p>The first two columns are obviously linearly independent, while the last two columns are duplicates of the first, so the nullity of this matrix is 2, which means that it has $0$ as an eigenvalue of multiplicity two. The row sums all equal $2$, so that’s another eigenvalue with associated eigenvector $(1,1,1,1)^T$ (r... |
1,262,036 | <p>In complex analysis, this seems to be a really helpful way to avoid having to expand out Laurent series. I am unclear, however, when it is appropriate to use this property.</p>
<p>In specific, I'm worried I CAN'T use this method on the following:</p>
<p>$$\frac{e^z}{z^3 \sin(z)}$$ at the origin. This looks really ... | zhw. | 228,045 | <p>Expanding the series is not so bad really. Rewrite the thing as</p>
<p>$$\frac{1}{z^4}\cdot \frac{e^z}{(\sin z)/z}.$$</p>
<p>We want the coefficient of $z^3$ in the expansion of the second quotient. Now $(\sin z)/z = 1 - (z^2/6 + O(z^4)),$ so its reciprocal is $1+(z^2 + O(z^4)).$ So we are looking at</p>
<p>$$(1+... |
15,205 | <p>I'm a young math student. And I live with the effort of always wanting to understand everything I study, in mathematics. This means that for every thing I face I must always understand every single demonstration, studying the basics every time if I don't remember them. And this makes it impossible for me to prepare ... | kcrisman | 1,608 | <p>I don't have any specific resources, but I suggest that you might find a little success by finding some physics applications where there really is a difference between the "algebra-based" and "calc-based" physics that use e.g. multivariate calc or integration by parts or something.</p>
<p>As an example, I think tha... |
1,549,138 | <p>I have a problem with this exercise:</p>
<p>Proove that if $R$ is a reflexive and transitive relation then $R^n=R$ for each $n \ge 1$ (where $R^n \equiv \underbrace {R \times R \times R \times \cdots \times R} _{n \ \text{times}}$).</p>
<p>This exercise comes from my logic excercise book. The problem is that I've ... | Leox | 97,339 | <p>It is enought to prove that if a relation $R$ is transitive and reflexive then $R^2=R.$</p>
<p>By definition of transitive realtion we have that $R^2 \subseteq R.$ Let us prove that $R \subseteq R^2.$ Let $(a,b) \in R$. Since $R$ is reflexive then $(b,b) \in R$. Then by definition of composition we get that $... |
53,188 | <p>Recently I read the chapter "Doctrines in Categorical Logic" by Kock, and Reyes in the Handbook of Mathematical Logic. And I was quite impressed with the entire chapter. However it is very short, and considering that this copy was published in 1977, possibly a bit out of date. </p>
<p>My curiosity has been sparked ... | Buschi Sergio | 6,262 | <p>For a introduction:</p>
<p>1) Notes on Logic and Set theory (cap. 1, cap 3)
P.T. Johnstone. </p>
<p>2) Locally Presentable And Accessible Categories by J Adamek J Rosicky (Cap. 3 & cap. 5)</p>
<p>FOr a comprehensive view:</p>
<p>1) Sketches of an Elephant: A Topos Theory Compendium (VOl 2, cap D1)</p>
<p>3)... |
984,915 | <blockquote>
<p>If $A=\{a_1,...,a_n\}$ and $B=\{b_1,...,b_n\}$ are two bases of a vector space
$V$, there exists a unique matrix $M$ such that for any $f\in V$,
$[f]_A=M[f]_B$.</p>
</blockquote>
<p>My textbook uses this theorem without a proof, so I'm trying to show that it's true myself. Consider $[f]_A = (c_1,... | hickslebummbumm | 168,882 | <p>Note that $\frac{-1}{n} < \frac{-n}{n^2+1} \leq \frac{(-1)^n n}{n^2 +1} \leq \frac{n}{n^2+1} < \frac{1}{n}$. The outer two sequences both converge to $0$, it follows from the sandwich theorem that $\lim_{n \rightarrow \infty}\frac{(-1)^n n}{n^2 + 1} = 0$, too.</p>
|
984,915 | <blockquote>
<p>If $A=\{a_1,...,a_n\}$ and $B=\{b_1,...,b_n\}$ are two bases of a vector space
$V$, there exists a unique matrix $M$ such that for any $f\in V$,
$[f]_A=M[f]_B$.</p>
</blockquote>
<p>My textbook uses this theorem without a proof, so I'm trying to show that it's true myself. Consider $[f]_A = (c_1,... | Bumblebee | 156,886 | <p>HINT:$$\dfrac{n}{n^2+1}=\dfrac{1}{n+\dfrac1n}\to 0\,\,\,\,\, \text{as}\,\,\,\,\,n\to\infty$$</p>
|
3,050,497 | <p>The operator is given by
<span class="math-container">$$A=\begin{pmatrix}
1 & 0 & 0\\
1 & 1 & 0\\
0 & 0 & 4
\end{pmatrix}$$</span>
I have to write down the operator <span class="math-container">$$B=\tan(\frac{\pi} {4}A)$$</span>
I calculate <span class="math-container">$$\mathcal{R} (z) =\fra... | Ankit Kumar | 595,608 | <p>Note that <span class="math-container">$$1+a+b+c+ab+bc+ca+abc=(1+a)(1+b)(1+c)$$</span>
<span class="math-container">$$(1+a)(1+b)(1+c)=1623=1\cdot3\cdot541$$</span>
<span class="math-container">$$\implies a=0,\ b=2,\ c=540.$$</span>
Note that <span class="math-container">$0\leq a<b<c$</span>. Those three number... |
977,446 | <p>Prove that $A\cap B = \emptyset$ iff $A\subset B^C$. I figured I could start by letting $x$ be an element of the universe and that $x$ is an element of $A$ and not an element of $B$. </p>
| ajotatxe | 132,456 | <p>You must prove both implications, that is: if $A\cap B=\emptyset$ then $A\subset B^c$ and conversely: if $A\subset B^c$ then $A\cap B=\emptyset$.</p>
<p><strong>For the first:</strong> A good way to prove that some set is a subset of another one is supposing that $x$ is in the subset and proving that $x$ is in the ... |
614,749 | <p><strong>The game:</strong></p>
<p>Given $S = \{ a_1,..., a_n \}$ of positive integers ($n \ge 2$). The game is played by two people. At each of their turns, the player chooses two <strong>different</strong> non-zero numbers and subtracts $1$ from each of them. The winner is the one, for the last time, being able to... | Greg Martin | 16,078 | <p>I'm more interested in the competitive part of the question, so my answer deals with the following modification to (d): The starting position can be any sequence of $n\ge3$ positive integers, regardless of whether the total sum is even or odd (although that overall parity cannot change during the game, of course). T... |
614,749 | <p><strong>The game:</strong></p>
<p>Given $S = \{ a_1,..., a_n \}$ of positive integers ($n \ge 2$). The game is played by two people. At each of their turns, the player chooses two <strong>different</strong> non-zero numbers and subtracts $1$ from each of them. The winner is the one, for the last time, being able to... | Barry Cipra | 86,747 | <p>This answers parts a)-c).</p>
<p>The set $\mathbb{W}$ is fairly small: It consists of sets $S=\{a,a\}$ and $S=\{1,\ldots,1\}$ with an even number of $1$'s. Every other set can, by (in)judicious play, lead to a losing position of the form $S=\{a\}$. This can be proved by induction on the total of the integers in ... |
253,152 | <p>So I was given $f(x)$ continuous and positive on $[0,\infty)$, and need to show that $g(x)$ increasing on $(0,\infty)$</p>
<p>And $g(x)={\int_0^xtf(t)dt\over \int_0^xf(t)dt} $</p>
<p>So my approach is I want to show that $g'(x)>0$, so I used FTC and quotient rule to take the derivative of $g'(x)$, but then I go... | Nameless | 28,087 | <p>You want to show that for $x>0$, \begin{equation}g^{\prime}(x)=\frac{xf(x)\int_0^xf(t)dt-f(x)\int_0^xtf(t)dt}{(\int_0^xf(t)dt)^2}
>0\end{equation}
which implies since $f$ is positive,
$0<x\int_0^xf(t)dt-\int_0^xtf(t)dt=\int_0^x(x-t)f(t)dt$ which is true since $x>t>0$ and $f$ is positive. </p>
|
3,893,440 | <p>Suppose we have <span class="math-container">$4$</span> books on Math, <span class="math-container">$5$</span> books on English and <span class="math-container">$6$</span> books on History. In how many ways you can put them on your bookshelf if you want :- <br/>
<span class="math-container">$1)$</span> The first boo... | fleablood | 280,126 | <ol>
<li>The first book must be a math book:</li>
</ol>
<p>There are <span class="math-container">$4$</span> choices for that book.</p>
<p>One you choice that math book, there are <span class="math-container">$14$</span> books remaining. They can go in any order. SO there are <span class="math-container">$14!$</span>... |
1,067,051 | <p>How can I find the point of intersection of <span class="math-container">$y=e^{-x}$</span> and <span class="math-container">$y=x$</span> ?</p>
<p><a href="https://i.stack.imgur.com/VoX32.png" rel="nofollow noreferrer">Here's the graph</a></p>
| George V. Williams | 54,806 | <p>The solution to this equation can be expressed in terms of the Lambert-W function.</p>
<p>$$ e^{-x} = x $$
$$ 1 = xe^x $$
$$ x = W(1) \approx 0.567$$</p>
<p>Note that the last step is by definition.</p>
|
3,409,598 | <p>Given three equation</p>
<p><span class="math-container">$$\log{(2xy)} = (\log{(x)})(\log{(y)})$$</span>
<span class="math-container">$$\log{(yz)} = (\log{(y)})(\log{(z)})$$</span>
<span class="math-container">$$\log{(2zx)} = (\log{(z)})(\log{(x)})$$</span></p>
<p>Find the real solution of (x, y, z)</p>
<p>What s... | Milten | 620,957 | <p>You are sort of close. But you treat all the <span class="math-container">$x_i$</span> as if they were <span class="math-container">$x$</span>. Also, the answer should be an <span class="math-container">$n$</span>-dimensional vector.</p>
<p>The gradient is <span class="math-container">$\nabla f = (\frac{\partial f}... |
1,365,489 | <p>What is the value of the following expression?</p>
<p>$$\sqrt[3]{\ 17\sqrt{5}+38} - \sqrt[3]{17\sqrt{5}-38}$$</p>
| g.kov | 122,782 | <p>\begin{align}
x&=\sqrt[3]{17\sqrt{5}+38} - \sqrt[3]{17\sqrt{5}-38}
\end{align}</p>
<p>Note that
\begin{align}
17\sqrt{5}-38&=\frac{1}{17\sqrt{5}+38}.
\end{align}</p>
<p>Let $a=\sqrt[3]{17\sqrt{5}+38}$.
Then we have
\begin{align}
x^3&=\left(a-\frac1a\right)^3
\\
x^3&= a^3-3a+\frac3a-\frac{1}{a^3... |
126,739 | <p><strong>I changed the title and added revisions and left the original untouched</strong> </p>
<p>For this post, $k$ is defined to be the square root of some $n\geq k^{2}$. Out of curiousity, I took the sum of one of the factorials in the denominator of the binomial theorem;
$$\sum _{k=1}^{\infty } \frac{1}{k!} \... | Tibor Pogany | 113,270 | <p>\begin{align*}
S_m &= \sum_{k \geq 0} \frac1{(k+m)k!}
= \sum_{k \geq 0} \frac{\Gamma(m+k)}{\Gamma(m+k+1) k!} \\
&= \frac1m \sum_{k \geq 0} \frac{(m)_k}{(m+1)_k} \frac1{k!}
= \frac1m {}_1F_1(m; m+1; 1)\,.
\end{align*}
The desired value $S_2 = 1$, since the confluent hypergoemetric ... |
1,231,772 | <p>Motivated by Baby Rudin Exercise 6.9</p>
<p>I need to show that $\int_0^\infty \frac{|\cos x|}{1+x} \, dx$ diverges.</p>
<p>My attempt: </p>
<p>$\frac{|\cos x|}{1+x} \geq \frac{\cos^2 x}{1+x}$, and then $\int_0^\infty \frac{\cos^2 x}{1+x} \, dx + \int_0^\infty \frac{\sin^2 x}{1+x} \, dx = \int_0^\infty \frac{1}{1... | wlad | 228,274 | <p>$\int_{\pi/2}^{\infty} \frac{|\cos x|}{1+x} =
\sum_{r=0}^{\infty} \int_{\pi/2 + \pi r}^{\pi/2 + \pi r+1} \frac{|\cos x|}{1+x}dx \geq
\sum_{r=0}^{\infty} \int_{\pi/2 + \pi r}^{\pi/2 + \pi r+1} \frac{|\cos x|}{1+(\pi/2 + \pi r+1)}dx =
\sum_{r=0}^{\infty} \frac{1}{1+(\pi/2 + \pi r+1)} \int_{\pi/2 + \pi r}^{\pi/2 + \pi ... |
1,231,772 | <p>Motivated by Baby Rudin Exercise 6.9</p>
<p>I need to show that $\int_0^\infty \frac{|\cos x|}{1+x} \, dx$ diverges.</p>
<p>My attempt: </p>
<p>$\frac{|\cos x|}{1+x} \geq \frac{\cos^2 x}{1+x}$, and then $\int_0^\infty \frac{\cos^2 x}{1+x} \, dx + \int_0^\infty \frac{\sin^2 x}{1+x} \, dx = \int_0^\infty \frac{1}{1... | Adhvaitha | 228,265 | <p>We have
\begin{align}
I_n & = \int_{n\pi}^{(n+1)\pi} \dfrac{\vert \cos(x) \vert}{1+x}dx = 2\int_{n\pi}^{n\pi+\pi/2} \dfrac{\vert \cos(x) \vert}{1+x}dx \geq 2\int_{n\pi}^{n\pi+\pi/6} \dfrac{\vert \cos(x) \vert}{1+x}dx\\
& \geq 2 \int_{n\pi}^{n\pi+\pi/6} \dfrac{1/2}{1+x}dx =\int_{n\pi}^{n\pi+\pi/6} \dfrac{dx}{... |
755,571 | <p>$$a_n=3a_{n-1}+1; a_0=1$$</p>
<p>The book has the answer as: $$\frac{3^{n+1}-1}{2}$$</p>
<p>However, I have the answer as: $$\frac{3^{n}-1}{2}$$</p>
<p>Based on:</p>
<p><img src="https://i.stack.imgur.com/4vJrQ.png" alt="enter image description here"></p>
<p>Which one is correct?</p>
<p>Using backwards substit... | lhf | 589 | <p>Write $b_n= a_n +\alpha$ and find $\alpha$ such that the recurrence reduces to $b_{n+1}=3b_n$. You'll find that $\alpha=1/2$ works. Then of course $b_n=3^nb_0=3^n(a_0+\alpha)$ and $a_n=b_n-\alpha=3^na_0+(3^n-1)\alpha$.</p>
|
2,887,880 | <p>I read this <a href="https://www.reddit.com/r/math/comments/8frbe2/what_is_a_natural_way_to_represent_nonlinear/" rel="nofollow noreferrer">reddit</a> post and this <a href="https://math.stackexchange.com/q/1388566/553404">SE thread</a> discussing how to represent nonlinear/linear transforms in matrix notations but ... | Bananach | 70,687 | <p>In some sense everything you can possibly define can be encoded in numbers and you are free to write those numbers in matrix form to get a linear operator. Thus the answer to your question is trivially yes.</p>
<p>However, the answer is also nontrovially yes. Such a yes is dependent on what connections between the ... |
1,190,759 | <p>I was trying to show the following
$\int_{-\infty}^{\infty} x^{2n}e^{-x^2}dx = (2n)!{\sqrt{\pi}}/4^nn!$ by using $\int_{-\infty}^{\infty} e^{-tx^2}dx = \sqrt{\pi/t}$
thus</p>
<p>I differentiated this exponential integral n times to get the following. </p>
<p>$\int_{-\infty}^{\infty} \frac{d^ne^{-tx^2}}{dt^n}dx ... | Alijah Ahmed | 124,032 | <p>The first thing is your answer should be
$$\int_{-\infty}^{\infty} \frac{d^ne^{-tx^2}}{dt^n}dx=\frac{1\times 3\times 5 \times ... \times (2n-1)\times \sqrt{\pi}t^{-\frac{2n-1}{2}}}{2^{n}} $$
based on the repeated differentiation of $t^{-1/2}\sqrt{\pi}$</p>
<p>Then, if you multiply both numerator and denominator by ... |
949,664 | <p>Consider the Laplace transform $\int_{0}^{\infty} e^{-px}f(x)\,dx$ <br/>
Assume $f(x)=1$ , then the Laplace transform is $\frac {1}{p}$. <br/>
Assume $f(x)=x$ , then the Laplace transform is $\frac {1}{p^2}$.<br/>
The question is, what will happen to the $f(x)$ after getting transformed?<br/>
Why should the function... | Christian Blatter | 1,303 | <p>The Laplace transform ${\cal L}$ is applied solely to known or unknown functions which are defined explicitly or implicitly in terms of <em>"analytic formulas"</em>. What makes ${\cal L}$ useful are alone its <em>formal</em> algebraic properties, encoded in certain rules of term manipulation. A central ingredient of... |
1,955,393 | <p>I have been trying to evaluate this limit:</p>
<p>$$\lim_{n\to\infty}{\sqrt[n]{4^n + 5^n}}$$</p>
<p>What methods should I try in order to proceed?</p>
<p>I was advised to use "Limit Chain Rule", but I believe there is a different approach.</p>
| E.H.E | 187,799 | <p>Hint:
$$\lim_{n\to\infty}{\sqrt[n]{4^n + 5^n}}=\lim_{n\to\infty}5{\sqrt[n]{\frac{4}{5^n}^n + 1}}$$</p>
|
990,512 | <p>Suppose that the probability of $x=0$ is $p$, and the probability of $x=1$ is $1-p=q$. Consider the random sequence $X=\{X_i\}_{i=1}^{\infty}$. We map this sequence by $C$ to a point in the interval $[0,1]$ as below:</p>
<p>$1)$ we look at the first random variable. If it is $0$, then we update the interval to $I_1... | Did | 6,179 | <p>At the $n$th step, $I_n=J$, where $J$ is one of $2^n$ disjoint intervals whose union is $(0,1)$. Each of these intervals $J$ has length $p^kq^{n-k}$ for some $0\leqslant k\leqslant n$ and the probability that $I_n=J$ is $p^kq^{n-k}$. This holds for every $n$ hence $C(X)$ is uniformly distributed in $(0,1)$.</p>
|
898,543 | <p>I have the random vector $(X,Y)$ with density function $8x^{2}y$ for $0 < x < 1$, $0 < y < \sqrt{x}$ I am trying to find the marginal distributions of $X$ and $Y$. For $X$ this seems to be simply the integral $\int_{0}^{\sqrt{x}}8x^{2}y = 4x^{3}$, which is also the given solution, and follows the general... | Hypergeometricx | 168,053 | <p>Draw a Venn diagram!</p>
<p>Let probabilities be as follows:</p>
<p>$a$=Blue only</p>
<p>$b$=Blue and Left</p>
<p>$c$=Left only</p>
<p>$d$=None</p>
<p>From probabilities given, </p>
<p>$$\begin{align}
\frac b{a+b}=\frac 17 \quad \Rightarrow a&=6b\\
\frac b{b+c}=\frac 13 \quad \Rightarrow c&=2b\\
d&... |
1,687,336 | <p>I've been searching through the internet and through SE to find something to help me understand generating functions, but I haven't found anything that would solve my problem with them.</p>
<p>I understand that </p>
<p>$$\frac1{1-x}=\sum_{n\ge 0}x^n\;,\tag{1}$$</p>
<p>gives the sequence $(1, 1, 1, 1,...) $ becaus... | Community | -1 | <p>Take $\dfrac1{1-x}=1+x+x^2+x^3+x^4+...$, multiply by $4$ and replace $x$ by $x^3$ to get $\dfrac4{1-x^3}=4+4x^3+4x^6+4x^9+4x^{12}+...$. Is this unclear ?</p>
|
2,762,953 | <p>I've studied Markov Process with 2x2 matrices.
Using the linear algebra and calculus procedures is clear to me how a Markov chain works.</p>
<p>However, i'm still not able to grasp the intuitive and immediate meaning of a Markov chain.
Why intuitively, for $n\rightarrow +\infty $, the state of the system is indepen... | amd | 265,466 | <p>Suppose that your process with matrix $P$ has a unique stationary distribution $\mathbf\pi_\infty$. This vector is an eigenvector of $1$, since $\mathbf\pi_\infty P=\mathbf\pi_\infty$, and the other eigenvalue $\lambda$ of $P$ has absolute value less than one. Every state vector $\mathbf\pi$ can be decomposed into t... |
1,448,363 | <p>I have gotten to the next stage where you write it as $\frac{1}{\left(\frac 34\right)}$ to the power of $3$, now I am stuck</p>
<p>I've got it now, thanks everyone.</p>
| jameselmore | 86,570 | <p>Hint:
$$\left(\frac{b}{a}\right)^{-n} = \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$$</p>
|
2,617,235 | <p>Given a triangle $\Delta$ABC, how to draw any inscribed equilateral triangle whose vertices lie on different sides of $\Delta$ABC?</p>
| g.kov | 122,782 | <p>Another way is to draw a bisector $AD$ of $\angle CAB$,
$D\in BC$,
find the intersection of the sides $AC$ and $AB$
with the line $AD$, rotated $\pm30^\circ$ around $D$:</p>
<p><a href="https://i.stack.imgur.com/Nq1A2.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/Nq1A2.png" alt="enter image de... |
1,708,996 | <p>If $x = a( \theta +\sin \theta)$ and $y = a(1-\cos \theta)$ then $\frac{dy}{dx}$ will be equal to : </p>
<p>$a) \sin \frac{\theta}{2}$</p>
<p>$b) \cos \frac{\theta}{2}$</p>
<p>$c) \tan \frac{\theta}{2}$</p>
<p>$d) \cot \frac{\theta}{2}$</p>
<p>I have solved till : $\frac{dy}{dx} = \frac{\sin \theta}{1 + \cos \t... | Archis Welankar | 275,884 | <p>Hint $$\sin(x)=2\sin(x/2)\cos(x/2)$$ and $$1+\cos(x)=2\cos^2(x/2)$$</p>
|
3,581,724 | <p>I suspect a simple wooden toy "lead screw" was made by advancing a cylindrical rotary cutting tool ( <em>Cylindrical End Mill Cutter</em>) along the surface of the rotating wooden dowel (base cylinder), resulting in a helical cut (the axes of the cylinders are orthogonal (<em>skew</em>).</p>
<p><a href="https://i.s... | Community | -1 | <p>If I am right, the cutting surface is described by a revolving vertical circle, the center of which describes an helix of vertical axis.</p>
<p>Parametrically:</p>
<p><span class="math-container">$$\begin{cases}x=(R+r\cos u)\cos\ t,\\y=(R+r\cos u)\sin t,\\z=r\sin u+a^2t.\end{cases}$$</span></p>
<p>At time <span c... |
3,581,724 | <p>I suspect a simple wooden toy "lead screw" was made by advancing a cylindrical rotary cutting tool ( <em>Cylindrical End Mill Cutter</em>) along the surface of the rotating wooden dowel (base cylinder), resulting in a helical cut (the axes of the cylinders are orthogonal (<em>skew</em>).</p>
<p><a href="https://i.s... | Narasimham | 95,860 | <p>With videos the question is more clear. We can consider the simplest router section to be a rectangular groove like a channel in a flat plate.</p>
<p>When wrapped around a cylinder isometrically we have a helical strip as channel bottom flanked by two developable helicoids on either side.</p>
<p>The cross section ... |
2,553,175 | <p>How can I verify that
$$1-2\sin^2x=2\cos^2x-1$$
Is true for all $x$?</p>
<p>It can be proved through a couple of messy steps using the fact that $\sin^2x+\cos^2x=1$, solving for one of the trigonemtric functions and then substituting, but the way I did it gets very messy very quickly and you end up with a bunch of ... | A Piercing Arrow | 502,334 | <p>First, get the trigonometric functions on one side and integers on the other. Then, divide by $2$ to get $\sin{x^2}+\cos{x}^2 = 1.$ Since we know this to be true (check out proof of unit circle), we have hereby finished our proof!</p>
|
2,067,794 | <p>Let $A$ be a ring and $u,v \in A^\times$. When do we have that $u + v \in A^\times$?</p>
<p>I think that A is needed to be an integral domain. For example consider $\mathbb{Z/6}$. Both $1$ and $5$ is a unit but their sum $1+5=0$ is not a unit.</p>
| B. Goddard | 362,009 | <p>In any field, the sum of two units is a unit, unless they're additive inverses. For a non-field example, let $\zeta$ be a primitive sixth-root of unity and let $A$ be the ring of integers in $\mathbb{Q}[\zeta]$. Then note that $\zeta^2+1 = \zeta$. (Because $0 = \zeta^6 - 1 = (\zeta^3-1)(\zeta^3+1) = (\zeta-1)(\zet... |
2,067,794 | <p>Let $A$ be a ring and $u,v \in A^\times$. When do we have that $u + v \in A^\times$?</p>
<p>I think that A is needed to be an integral domain. For example consider $\mathbb{Z/6}$. Both $1$ and $5$ is a unit but their sum $1+5=0$ is not a unit.</p>
| ToThichToan | 389,952 | <p>I don't sure about an answer for your question. However, there is a nice result of F. Beukers and Schlickewei about the number of solutions of the unit equation in a finitely generated subgroup. </p>
|
67,985 | <p>Consider $X_1,X_2$ i.i.d. standard normal random variables(mean 0, variance 1). Are the random variables $Y=X_1+X_2$ and $Z=X_1-X_2$ dependent? I am not sure how to prove this one way or the other.</p>
| Sasha | 11,069 | <p>If you are familiar with the concept of characteristic function, it is easiest to compute characteristic function for $(Y, Z)$. For independent variables, the characteristic function would factor into a product:</p>
<p>$$
\begin{eqnarray}
\mathbb{E}\left( \exp( i t_1 Y + i t_2 Z ) \right) &=&
\math... |
67,985 | <p>Consider $X_1,X_2$ i.i.d. standard normal random variables(mean 0, variance 1). Are the random variables $Y=X_1+X_2$ and $Z=X_1-X_2$ dependent? I am not sure how to prove this one way or the other.</p>
| Michael Lugo | 173 | <p>$X_1$ and $X_2$ are independent standard normals, so $(X_1, X_2)$ has rotationally symmetric density, namely
$$ {1 \over 2\pi} \exp(-(x_1^2 + x_2^2)/2). $$
If you change coordinates with $u = (x_1 + x_2)/\sqrt{2}, v = (x_1 - x_2)/\sqrt{2}$ (so the change from $(x_1, x_2)$ to $(u,v)$ is area-preserving) then this bec... |
362,716 | <p>Let <span class="math-container">$E$</span> be a separable <span class="math-container">$\mathbb R$</span>-Banach space, <span class="math-container">$\rho_r$</span> be a metric on <span class="math-container">$E$</span> for <span class="math-container">$r\in(0,1]$</span> with <span class="math-container">$\rho_r\le... | Benoît Kloeckner | 4,961 | <p>I can answer assuming some regularity on the Markov semigroup, which I would expect to be satisfied in most cases. Specifically, assume local (in time) Lipschitz continuity on your Markov semigroup, i.e.
<span class="math-container">$$\forall s_0>0, \exists C>0, \forall s\in[0,s_0], \forall \mu_1,\mu_2 : \mat... |
206,227 | <p>I was given the following problem:</p>
<p>Let $V_1, V_2, \dots$ be an infinite sequence of Boolean variables. For each natural number $n$, define a proposition $F_n$ according to the following rules: </p>
<p>$$\begin{align*}
F_0 &= \text{False}\\
F_n &= (F_{n-1} \ne V_n)\;.
\end{align*}$$</p>
<p>Use induc... | Berci | 41,488 | <p>So, given $V_i$ Boolean variables, i.e. all of them are either true or false, and you define 'propositions' $F_0,F_1,\dots$, they will be also <strong>evaluated</strong> as <em>true</em> or <em>false</em> (we can call it just another set of Boolean variables, built up using $V_i$), such that
$$F_0:=\text{false}, \ F... |
2,764,073 | <p>I recently was working on a question posted in an AP calculus BC multiple choice sheet which asked:</p>
<p>Let f(x) be a positive, continuous deceasing function. If $\int_1^∞ f(x)dx$ = 5, then which of the following statements must be true about the series $\sum_1^∞f(n)$?</p>
<p>(a) $\sum_1^∞f(n)$ = 0</p>
<p>(b) ... | José Carlos Santos | 446,262 | <p>Since:</p>
<ul>
<li>$\displaystyle\int_1^2f(x)\,\mathrm dx<f(1)$</li>
<li>$\displaystyle\int_2^3f(x)\,\mathrm dx<f(2)$</li>
<li>$\displaystyle\int_3^4f(x)\,\mathrm dx<f(3)$</li>
</ul>
<p>and so on, you have$$\int_1^{+\infty}f(x)\,\mathrm dx<\sum_{n=1}^\infty f(n).$$</p>
|
3,091,090 | <p>I came across this question the other day and have been trying to solve it by using some simple algebraic manipulation without really delving into L'Hospital's Rule or the Power Series as I have just started learning limit calculations.
We needed to find :
<span class="math-container">$$\lim_{x \to 0} \frac {x\cos x... | Mefitico | 534,516 | <p>1) If I understand your question, you are conjecturing that:</p>
<p><span class="math-container">$$
\lim_{x \to a} \frac{f(x)}{g(x)}= \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)}
$$</span></p>
<p>This is wrong. You may try to prove it, but as a simple counterexample should suffice:</p>
<p>Let <span class="math... |
2,785,993 | <p>Let</p>
<ul>
<li><p><span class="math-container">$k(0)=11$</span></p>
</li>
<li><p><span class="math-container">$k(1)=1101$</span></p>
</li>
<li><p><span class="math-container">$k(2)=1101001$</span></p>
</li>
<li><p><span class="math-container">$k(3)=11010010001$</span></p>
</li>
<li><p><span class="math-container">... | Cesareo | 397,348 | <p>The formation law is clearly</p>
<p>$$
n_k = 2^k n_{k-1}+1
$$</p>
<p>with $n_1=3$</p>
<p><code>n0 = 3;
For[i = 2, i < 50, i++, n1 = 2^i n0 + 1;
If[PrimeQ[n1], Print[n1, " ", IntegerString[n1, 2]]]; n0 = n1]</code></p>
<p>obtaining</p>
<p>n = 13 -- 1101</p>
<p>n = 271302750695377321080849818469209754627603... |
54,486 | <p>Many colour schemes and colour functions can be accessed using <a href="http://reference.wolfram.com/mathematica/ref/ColorData.html"><code>ColorData</code></a>.</p>
<p>Version 10 introduced new default colour schemes, and a new customization option using <a href="http://reference.wolfram.com/mathematica/ref/PlotThe... | Mr.Wizard | 121 | <p>For ease of direct access I have found through digging the following relationships for indexed colors:</p>
<pre><code>map = {"Default" -> 97, "Earth" -> 98, "Garnet" -> 99, "Opal" -> 100, "Sapphire" -> 101,
"Steel" -> 102, "Sunrise" -> 103, "Textbook" -> 104, "Water" -> 105,
"Bold... |
2,452,084 | <ul>
<li><em>I'm having trouble understanding why the arbitrariness of $\epsilon$ allows us to conclude that $d(p,p')<0$. It seems we could likely
conclude a value such as $\frac {\epsilon}{100}$ couldn't we?? The
other idea that would normally work is the limit (as $n$ approaches
$\infty$, $p$ approaches $p'$) but ... | Paramanand Singh | 72,031 | <p>The crucial thing to note here is that $p, p'$ are fixed points in $X$ (ie they don't depend on $n$) and hence $d(p, p') $ is non-negative specific real number and let's denote this specific non-negative number by $A$. It should now be clear that in the inequality $A<\epsilon$ the number $A$ is fixed and $\epsilo... |
2,452,172 | <p>Let $(a_n)_{n \geq 1}$ be a decreasing sequence of positive reals. Let $s_n = a_1 + a_2 + ... + a_n$ and </p>
<p>\begin{align}
b_n = \frac{1}{a_{n+1}} - \frac{1}{a_n}, n \geq 1
\end{align}</p>
<p>Prove that if $(s_n)_{n \geq 1}$ is convergent, then $(b_n)_{n \geq 1}$ is unbounded.</p>
<p>My attempt: If $\lim_{n \... | alphacapture | 334,625 | <p>Hint: Suppose for the sake of contradiction that the $b_n$ were bounded. Then show that $\sum{a_n}$ is divergent.</p>
|
646,779 | <p>Prove that if $p$ and $q$ are polynomials over the field $F$, then the degree of their sum is less than or equal to whichever polynomial's degree is larger</p>
<p>$$\deg(p+q)\leq \max \left\{\deg(p),\deg(q) \right\}$$</p>
<p>Currently, I am taking it case by case, but I was curious if there was a way to do a proof... | Bill Dubuque | 242 | <p>Here's one way. Let $\, \rm d := \deg.\,$ Suppose for contradiction $\, {\rm d}(f+g) > {\rm d}(f),\, {\rm d}(g).\,$ Choose such a counterexample of minimal degree $\,d = {\rm d}(f+g).\,$ Necessarily $\,d > 0\,$ since it is true for constants. Since $(f+g)(0) = f(0)+g(0)\,$ subtracting the constant terms from ... |
346,198 | <p>Recently I was playing around with some numbers and I stumbled across the following formal power series:</p>
<p><span class="math-container">$$\sum_{k=0}^\infty\frac{x^{ak}}{(ak)!}\biggl(\sum_{l=0}^k\binom{ak}{al}\biggr)$$</span></p>
<p>I was able to "simplify" the above expression for <span class="math-container"... | RobPratt | 141,766 | <p>You might be able to use the fact that
<span class="math-container">$$\sum_{k=0}^\infty b_{ak}=\sum_{k=0}^\infty \left(\frac{1}{a}\sum_{j=0}^{a-1} \exp\left(2\pi ijk/a\right)\right)b_k.$$</span>
For example, when <span class="math-container">$a=1$</span>, taking <span class="math-container">$b_k = \frac{x^k}{k!}\sum... |
346,198 | <p>Recently I was playing around with some numbers and I stumbled across the following formal power series:</p>
<p><span class="math-container">$$\sum_{k=0}^\infty\frac{x^{ak}}{(ak)!}\biggl(\sum_{l=0}^k\binom{ak}{al}\biggr)$$</span></p>
<p>I was able to "simplify" the above expression for <span class="math-container"... | AccidentalFourierTransform | 106,114 | <p>The explicit formula is as follows:
<span class="math-container">$$
S_a=\frac{1}{a^2}\left(\sum_{z^a=2^a}+2\sum_{p_a(z)=0}\right)e^{az}
$$</span>
where the polynomials <span class="math-container">$p_a$</span> are given by <a href="http://oeis.org/A244608" rel="noreferrer">A244608</a>. For example,
<span class="math... |
217,429 | <blockquote>
<p>Let $A\subset\mathbb R$. Show for each of the following statements that it is either true or false.</p>
<ol>
<li>If $\min A$ and $\max A$ exist then $A$ is finite.</li>
<li>If $\max A$ exists then $A$ is infinite.</li>
<li>If $A$ is finite then $\min A$ and $\max A$ do exist.</li>
<li>If ... | Brian M. Scott | 12,042 | <p>HINTS: You’re given that $P$ is the point $\left(x,-\frac7{25}\right)$ on the unit circle in the fourth quadrant. The fact that $P$ is on the unit circle tells you that $$x^2+\left(-\frac7{25}\right)^2=1\;;$$ why? And what are the values of $x$ making this true?</p>
<p>The fact that $P$ is in the fourth quadrant te... |
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