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 |
|---|---|---|---|---|
248,900 | <p>Let $\mathfrak{n}$ be a $2k$ dimensional $2$-step nilpotent Lie algebra and suppose that its center is $k$ dimensional. Does $\mathfrak{n}$ admit symplectic structure?</p>
<p>Let $\{f_1,\dots,f_k\}$ be a basis of the center of $\mathfrak{n}$ and complete it to a basis of $\mathfrak{n}$ $\{e_1,\dots,e_k,f_1,\dots,f_... | Tsemo Aristide | 80,891 | <p>Consider the following Lie algebra:</p>
<p>${\cal G}=Vect\{a,a',b,b',c,c',d,d'\}$ $[a,a']=[b,b']=c$, </p>
<p>$[a,b]=[a',b]=[a,b']=[a',b']=[c,a]=[c,b]=[c,a']=[c,b']=[c',a]=[c',a']=[c',b]=[c',b']=[c,c']=[d,a]=[d,a']=[d,b]=[d,b']=[d,c]=[d,c']=[d',a]=[d',a']=[d,b]=[d',b']=[d',c]=[d',c']=[d,d]=0.$</p>
<p>${\cal G}$ is... |
1,166,727 | <p>Show that $c|a$ and $c|b$ iff $c|gcd(a,b)$</p>
<p>I am only going to show that the if part is true and i have the solution to this proof just i found the if part of the proof dissatisfying.</p>
<p>since c|a, c|b and $c \le gcd(a,b)$ it follows that there exists an integer $f$ such that $gcd(a,b) = cf$ and hence $c... | Bernard | 202,857 | <p>L'Hospital rule is not the alpha and omega of limits computation! It works as far as linear approximations work.
$$\bigg(\frac{\ln x}{x}\bigg)^\frac{1}{x}=\mathrm e^{\tfrac{\ln\ln x -\ln x}x}.$$</p>
<p>We know that $\,\ln x =_{\infty}o(x),\enspace \ln\ln x=_{\infty}o(\ln x)\,$ hence $\,\ln\ln x=_{\infty}o(x)$, so
... |
2,382,897 | <p>Find all entire functions with $|f(z)|\geq e^{|z|}$ for all $z \in \mathbb{C}$. </p>
<p>I don't think there is any such entire function, and here is my thought: since $\Re(z) \leq |z|$, we know $|e^z|\leq e^{|z|}\leq |f(z)|$ for all $z$. Consider $g(z)=\frac{f(z)}{e^z}$. Since $e^z$ is never $0$, $g(z)$ is an enti... | Robert Israel | 8,508 | <p>It's a bit more complicated than necessary. $|f(z)| \ge e^{|z|} \ge 1$, so why not just use $f$ instead of $g$? </p>
<p>And, by the way, what are you contradicting?</p>
|
2,382,897 | <p>Find all entire functions with $|f(z)|\geq e^{|z|}$ for all $z \in \mathbb{C}$. </p>
<p>I don't think there is any such entire function, and here is my thought: since $\Re(z) \leq |z|$, we know $|e^z|\leq e^{|z|}\leq |f(z)|$ for all $z$. Consider $g(z)=\frac{f(z)}{e^z}$. Since $e^z$ is never $0$, $g(z)$ is an enti... | Jack D'Aurizio | 44,121 | <p>It looks fine to me. Assuming $|f(z)|\geq e^{|z|}$ we get that $g(z)\stackrel{\text{def}}{=}f(z) e^{-z}$ is an entire function whose modulus is always $\geq 1$, but that contradicts <a href="http://mathworld.wolfram.com/PicardsLittleTheorem.html" rel="nofollow noreferrer">Picard's little Theorem</a> or just Casorati... |
2,382,897 | <p>Find all entire functions with $|f(z)|\geq e^{|z|}$ for all $z \in \mathbb{C}$. </p>
<p>I don't think there is any such entire function, and here is my thought: since $\Re(z) \leq |z|$, we know $|e^z|\leq e^{|z|}\leq |f(z)|$ for all $z$. Consider $g(z)=\frac{f(z)}{e^z}$. Since $e^z$ is never $0$, $g(z)$ is an enti... | Marja | 469,632 | <p>From the inequality it follows that $f$ is never zero. </p>
<p>Therefore $1/f$ is entire and $|1/f|\leq e^{-|z|}\leq1$. Therefore $f$ ought to be constant.</p>
<p>But $e^{|z|}\rightarrow\infty$ as $z\rightarrow \infty$.</p>
<p>...</p>
<p>You probably don't want that, but in a sense $f(z)=\infty\in S^2$ is consta... |
17,423 | <p>In most of books on elementary algebra, intermediate algebra and college algebra, the degree of the non-zero polynomial <span class="math-container">$$f(x)=a_nx^n+\cdots a_1x+a_0$$</span> with <span class="math-container">$a_n\neq 0$</span> is defined to be <span class="math-container">$n$</span>. </p>
<p>But I am ... | Henry Towsner | 62 | <p>This: "On the other hand, without the proof the definition of the degree of polynomials is not even logically established." is not quite right. What is needed to establish the definition is the <em>fact</em> that the degree is well-defined, but this fact is stated (implicitly) by stating the definition.</p>
<p>The... |
258,598 | <pre><code>Solve[Sin[Tan[x]] - Tan[Sin[x]] == 0, x]
</code></pre>
<p>It says it runs out of methods. After a very long time. Plotting the expression shows there are periodic solutions and some very nasty bits, too. NSolve does no better.</p>
| Michael E2 | 4,999 | <p>Perhaps this:</p>
<pre><code>Block[{$MaxExtraPrecision = 200},
Solve[Sin[Tan[x]] - Tan[Sin[x]] == 0, x, Reals]
]
</code></pre>
<img src="https://i.stack.imgur.com/kNbny.png" width="450">
<p>If you want complex solutions, then you'll have to bound the domain probably — that is, more powerful methods exist on a boun... |
4,498,547 | <p>(Foreword: Sorry to bother this platform with this extremely simple confusion, but I've been stuck on this trivial question for a good 3.5 hours, leading to anguish/frustrated to the point that again (and in the past countless times) want to change another self-study textbook, or even give up (although it will not c... | B. Goddard | 362,009 | <p>There's a minus sign error here:</p>
<p><span class="math-container">$$\left(\int_{-h}^0 f(x) \; dx\right)^\prime = +f(-h).$$</span></p>
<p>I'm guessing you forgot the last step of the chain rule which would be to multiply by the derivative of <span class="math-container">$-h$</span>, which gives you another minus s... |
2,323,189 | <p>I've been asked to find the minimum and maximum of the following function:</p>
<p>$f(x,y) = x^2+y^2-x+1/4$ </p>
<p>On the region or restriction defined as:</p>
<p>$D$={${(x,y)\in\mathbb{R}^2:x^2+y^2\leq1; x+y\leq0}$}</p>
<p>First, I observed that $f$ is continuos, and after I did the graph of the region where I... | farruhota | 425,072 | <p>If the objective function is rearranged, the contour set will be a set of circles with radius $\sqrt{z-\frac12}$: $$\left(x-\frac12\right)^2+y^2=z-\frac12$$ </p>
<p>The objective is to find the maximum and minimum of $z$, that is the largest and smallest contour circle.</p>
<p>The feasible region and the extreme c... |
724,012 | <p>not sure how to approach the following $\frac{\sin(a)}{\sin(b)}<\frac{a}{b}<\frac{\tan(a)}{\tan(b)}$ for $0<b<a<\pi/2$. Hints would be appreciated!</p>
| Felix Marin | 85,343 | <p>$\newcommand{\+}{^{\dagger}}
\newcommand{\angles}[1]{\left\langle #1 \right\rangle}
\newcommand{\braces}[1]{\left\lbrace #1 \right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack #1 \right\rbrack}
\newcommand{\ceil}[1]{\,\left\lceil #1 \right\rceil\,}
\newcommand{\dd}{{\rm d}}
\newcommand{\down}{\downarrow}
\new... |
2,006,626 | <p>I was just wondering if the following proof makes logical sense and is set out in a manor which is easy to read and understand, to a mathematician.</p>
<p>Prove the $\lim_\limits{x\to2}\,\, 2x^2-6x+7=3$.</p>
<p>$\underline {Proof}:$</p>
<p>Let $\epsilon \gt 0$ be given.</p>
<p>$\underline{Observation}:$</p>
<p>... | hamam_Abdallah | 369,188 | <p>Let $f(x)=2x^2-6x+7$ and $\epsilon>0$.</p>
<p>$|f(x)-f(2)|=2|(x-1)(x-2)|$</p>
<p>as $x$ is near $2$, we can assume that</p>
<p>$|x-2|<\color{red}{1},\;$ for example. which gives</p>
<p>$1<x<3$ or $0<x-1<2$ and $|x-1|<2$.</p>
<p>thus</p>
<p>$|x-2|<inf(\color{red}{1},\frac{\epsilon}{4})$<... |
2,134,085 | <p>Can someone please explain where the <span class="math-container">$1$</span> goes in this expression?</p>
<blockquote>
<p>Find the measure of angle in radians by solving:
<span class="math-container">$$\sin^2 \theta +\cos \theta=1$$</span></p>
</blockquote>
<p>I got <span class="math-container">$$\sin^2 \thet... | B. Goddard | 362,009 | <p>There's quite a bit going on here. First, this is not an "identity", it's a conditional equation and you're asked to solve it. Second, replace $\sin^2 \theta$ by $1-\cos^2 \theta$ (you forgot to square the $\cos \theta.$) Third, the last time you write $\cos$, you leave it dis-embodied. You need to take the $\co... |
1,229,194 | <p>Problems with the following limits:</p>
<p>$$
1. \quad \quad \lim_{x\to0^+} e^{1/x} + \ln x \, .
$$</p>
<p>Substitutions such as $e^{1/x}=t$ and $1/x = t$ don't yield any useful results. </p>
<p>Pretty much the same with
$$
2. \quad \quad \lim_{x\to 0^+} e^{1/x} - 1/x \, ,
$$
Common denominator doesn't help much... | Barry Cipra | 86,747 | <p>If you accept that $\ln u\lt u$ for all $u$, it follows that $\ln(1/x)\lt1/x$, hence</p>
<p>$$e^{1/x}-1/x\lt e^{1/x}-\ln(1/x)=e^{1/x}+\ln x$$</p>
<p>Consequently, it suffices to show</p>
<p>$$\lim_{x\to0^+}(e^{1/x}-1/x)=\infty$$</p>
<p>Now we certainly have $\lim_{x\to0^+}e^{1/x}=\infty$. Let's write</p>
<p>$$... |
238,702 | <p>I am a highschool freshman, and I really like to have goals for my life, one of the big ones is my career of choice. Previously, I have always wanted to be a programmer, and I have written a lot of code. But it seems to me that programming can get slightly bland, whereas math never disappoints me. So my question ... | sacohe | 45,227 | <p>I'm a recent college grad, so I was in a very similar situation not too long ago. In high school, I realized that I liked math, I was good at math, and I was especially good at teaching/explaining/tutoring math, so I decided I was going to be a high school math teacher. When I was applying to college, I became con... |
1,876,639 | <blockquote>
<p><span class="math-container">$\tan^{-1}x, \tan^{-1}y, \tan^{-1}z $</span> are in arithmetic progression, as are <span class="math-container">$x$</span>, <span class="math-container">$y$</span>, <span class="math-container">$z$</span>. (We assume <span class="math-container">$y \ne 0,1,-1$</span>.) Show:... | Blue | 409 | <p>(Not an answer so much as an elaborate comment ...) </p>
<p>For fun, I made a "trigonograph" to see why we might expect the elements to be equal (item (3)). Here, the arithmetic progressions are $\alpha$, $\gamma$, $\beta$ and $\tan\alpha$, $\tan\delta$, $\tan\beta$. (Also, I take all values to be positive, and ass... |
940,410 | <p>Using the method showed <a href="https://math.stackexchange.com/questions/939051/what-is-the-closed-form-of-sum-n-1-infty-frac-it-j-0-left-n">here</a> proposed by Olivier Oloa with simplifications proposed by Anastasiya-Romanova, it is possible to prove that</p>
<p>$$\sum _{n=1}^{\infty }{\frac {{{\it J}_{0}\left(\... | Juan Ospina | 170,228 | <p>Using the method showed <a href="https://math.stackexchange.com/questions/939051/what-is-the-closed-form-of-sum-n-1-infty-frac-it-j-0-left-n">here</a> proposed by Olivier Oloa with simplifications proposed by Anastasiya-Romanova and which is essentially the method that David H is showing, I am obtaining</p>
<p>$$\s... |
1,562,082 | <p>Using the definition in Bartle's Introduction to Real Analysis, I am trying to gain an intuitive understanding of limits that tend to infinity.</p>
<p>Given Definition:</p>
<p>Let ($x_n$) be a sequence of real numbers.</p>
<p>(i) We say that ($x_n$) tends to $\infty$, and write $lim(x_n) = +\infty$ , if for every... | Yes | 155,328 | <p>Let me use divergence to plus infinity ($+\infty$) as an example. The formal definition for a real sequence to diverge to plus infinity says that, given any real number $\alpha$, we can find some term of the sequence such that every term of the sequence after this term is greater than $\alpha$.</p>
<p>If $n \geq 1$... |
4,042,039 | <p>Let <span class="math-container">$X$</span> be a finite set with <span class="math-container">$|X|=n$</span> equipped with the dicrete topology.Show that <span class="math-container">$C(X,Y)$</span> is homeomorphic to <span class="math-container">$Y^n$</span> where <span class="math-container">$Y^n=\overbrace{Y\time... | Henno Brandsma | 4,280 | <p>First note that if <span class="math-container">$X$</span> has the discrete topology then <span class="math-container">$C(X,Y)$</span> is just the set of <strong>all</strong> functions from <span class="math-container">$X$</span> to <span class="math-container">$Y$</span> (every function is continuous on a discrete ... |
1,862,782 | <p>I have the following series :</p>
<p>$$\sum_{k=1}^{\infty }\left ( \frac{k}{\sqrt{4k^{3}+1}} \right )$$</p>
<p>And I am trying to see if it's convergent or divergent. I first thought about the integral test, but it looks like it will not be easy to integrate it. I found a hint that it can be solved using the compa... | Malice Vidrine | 102,781 | <p>There is a sense in which they always coincide if $\prod_{j\in J}$ is understood to be an <em>internal</em> product--that is, a right adjoint to the functor $J^*:\mathbf{C}/1\to\mathbf{C}/J$ given by pullback along the unique morphism $J\to 1$. The intuition for thinking of such a right adjoint as a product is that ... |
2,241,326 | <p>I am not being able to understand the graphical method of solving this, any simple explanation will be appreciated.</p>
<p>A non-graphical calculation will be very helpful too.</p>
<p>Thank you so much in advance!</p>
| dantopa | 206,581 | <p>@DHMO works out the details. You don't want to integrate this function.</p>
<p>The plot of the function shows it is symmetrical, and strongly centered at the origin.</p>
<p>$$
f(x) = \frac{1}{1+x^{8}}
$$</p>
<p><a href="https://i.stack.imgur.com/4VZlz.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur... |
1,197,547 | <p>If $G=A * B$ is the free product of two groups $A$ and $B$ and $g \in G-A$, then prove that $gAg^{-1} \cap A=1$.</p>
<p>We know $A \cap B=1$, so if we write $g=a_1b_1a_2b_2 \ldots a_nb_n$, does not give me sufficient road to go? How should I approach it?</p>
| Joffan | 206,402 | <p>Hmm, those leprechauns are tricky...</p>
<p>Perhaps we need to rotate $16$ to get that multiple of 7, $91$. I can't get the rest of it to make sense, though - I tried working through other bases, no joy yet - I would have expected him to allude to going from $31_5$ to $331_5$, because they're very fond of dublin'.<... |
1,948,880 | <p>Suppose I have some $n$-dimensional vector space $V$ and a finite collection of $m$ distinct points $v_1,\dotsc, v_m\in V$. Is there a basis of $V$ such that the first coordinate of each $v_i$ is distinct?</p>
<p>This obviously fails when the base field is finite, but my intuition over $\mathbb{R}^n$ has convinced ... | Mike F | 6,608 | <p>You can think of this as being related to the following fun result. A short proof is given by Steve D <a href="https://mathoverflow.net/questions/26/can-a-vector-space-over-an-infinite-field-be-a-finite-union-of-proper-subspaces">here</a>.</p>
<blockquote>
<p><strong>Fact:</strong> If <span class="math-container">$V... |
4,373,862 | <p>The definition of interior-point says "point <span class="math-container">$p$</span> in a set <span class="math-container">$S$</span> is interior point of <span class="math-container">$S$</span> if <span class="math-container">$\exists \delta \gt 0 : \mathcal B \left( p,\delta \right) \subseteq S $</span> but ... | Adam Rubinson | 29,156 | <p>Yes.</p>
<p>Suppose <span class="math-container">$\ x \in \overline{B\left(p,\frac{\delta}{2}\right)}.\ $</span></p>
<p>Then by definition of <em>closure of a set in a metric space</em>, <span class="math-container">$\ \exists\ \{x_n\}_{n\in\mathbb{N}}\subseteq B\left(p,\frac{\delta}{2}\right)\ $</span> such that <s... |
4,445,869 | <p>Having been introduced to the <a href="https://en.wikipedia.org/wiki/Leibniz_integral_rule" rel="nofollow noreferrer">Feynman technique of integration</a>, it seemed natural to wonder if it could be done the other way:</p>
<ol>
<li><p>Introduce a new parameter <span class="math-container">$a$</span></p>
</li>
<li><p... | user619894 | 617,446 | <p>In fact, I just posted an answer with this kind on trick <a href="https://math.stackexchange.com/a/4445871/617446">enter link description here</a> where I first introduced a parameter, then differentiated, and finally reintegrated. I suppose there are more sophisticated examples around.</p>
|
3,809,546 | <blockquote>
<p>Using the change of variable evaluate <span class="math-container">$\iint_{R} x y\ dx\ dy,$</span> when the region <span class="math-container">$R$</span> is bounded by the curves <span class="math-container">$x y=1, x y=3, y=3 x, y=5 x$</span> in the <span class="math-container">$1^{\text {st }}$</span... | J.G. | 56,861 | <p>Since <span class="math-container">$u=xy$</span> while <span class="math-container">$v=y/x$</span>, the first two curves are the bound <span class="math-container">$1\le u\le 3$</span>, while the last two are <span class="math-container">$3\le v\le5$</span>. Now we just need to rewrite <span class="math-container">$... |
1,821,426 | <p>I would like to know if a homogeneous linear differential equation, with variable coefficients which are periodic, is stable.</p>
<p>So the differential equation can be written as,</p>
<p>$$
\dot{y}(t)=A(t)y(t), \tag{1}
$$</p>
<p>$$
A(t+T)=A(t). \tag{2}
$$</p>
<p>I would suspect that the solution could be of the... | Bilal Jafar Karaki | 312,149 | <p>Let $\lambda_{max}(t)$ is the maximum eigenvalue of [$A^T(t)+A(t)$].</p>
<p>If there is a finite constant $\gamma$ such that
$$ \int_\tau^t\lambda_{max}(\sigma)d\sigma\leq\gamma$$
$\forall t\geq\tau,$ then the system of variable coefficients ODE is uniformly stable.</p>
<p>The detailed proof of this claim can be... |
327,201 | <p>I'm reading Behnke's <em>Fundamentals of mathematics</em>:</p>
<blockquote>
<p>If the number of axioms is finite, we can reduce the concept of a consequence to that of a tautology.</p>
</blockquote>
<p>I got curious on this: Are there infinite sets of axioms? The only thing I could think about is the possible ex... | Thomas Andrews | 7,933 | <p>In first order Peano axioms the principal of mathematical induction is not one axiom, but a "template" called an <a href="http://en.wikipedia.org/wiki/Axiom_schema" rel="nofollow noreferrer">axiom scheme</a>. For every possible expression (or "predicate") with a free variable, <span class="math-c... |
4,351,449 | <p>Show that : <span class="math-container">$1 - |\phi(t)| \ge \frac{1-|\phi(2t)|}{4}$</span> where <span class="math-container">$\phi(t)$</span> is a characteristic function .</p>
<p>I am able to prove another part of the question <span class="math-container">$\Re(1-\phi(t)) \ge \Re(\frac{1-\phi(2t)}{4})$</span> . Ho... | JoeKyon | 957,600 | <p>Maybe you can consider
<span class="math-container">$1-\cos tx\geq4(1-\cos 2tx)$</span>
(relevant to estimation of the real part)</p>
|
2,235,258 | <p>$$
\int J_0(x)\sin x~{\rm d}x
$$</p>
<p>Where $J_0$ is Bessel function of first kind of order $0$</p>
<p>This what I tried </p>
<p>$$
\int J_0(x)\sin x~{\rm d}x= -J_0(x) \cos x - \int J_0'(x)\cos x~{\rm d}x
$$</p>
<p>$$
J_0'(x)=-J_1(x)
$$</p>
<p>$$
\int J_0(x)\sin x ~{\rm}x= -J_0(x) \cos x -(J_1(x)\sin x - \in... | mickep | 97,236 | <p>We integrate by parts,
$$
\begin{aligned}
\int J_0(x)\sin(x)\,dx
&=xJ_0(x)\sin(x)-\int x\bigl(-J_1(x)\sin(x)+J_0(x)\cos(x)\bigr)\,dx\\
&=xJ_0(x)\sin(x)-\int D\bigl(x J_1(x)\cos(x)\bigr)\,dx\\
&=xJ_0(x)\sin(x)-x J_1(x)\cos(x)+C.
\end{aligned}
$$</p>
<p><strong>Clarification</strong></p>
<p>In the second... |
999,389 | <p>In the "Make Money Game," the winning number is four digits, each selected at random from {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}, e.g. 0-3-9-6, 0-0-6-0, 9-4-7-9. A player may place any of the following types of bets. In each case tell the odds of the player winning.</p>
<p>I got, </p>
<pre><co... | Dániel G. | 81,032 | <ol>
<li><p>I'd say the probability is $6$ in $1000$ (since there can be
$10^3=1000$ possible combinations for the first three digits and
only the $3!=6$ permutations of our chosen numbers are "good"
outcomes).</p></li>
<li><p>Similarly that would be $3$ in $1000$ ($1000$ combinations but
this time only $3$ permutation... |
606,158 | <p>I'm really sorry for this basic, stupid question. I have been looking for answers online but I can't find any.</p>
<p>I don't understand the following summation:</p>
<p>$$\sum_{i=0}^{n-1}i=\frac{n(n-1)}2$$</p>
<p>I understand that the basic formula for summations of i is the following:</p>
<p>$$\sum_{i=1}^ni=\fr... | Ross Millikan | 1,827 | <p>You have $$\sum_{i=1}^n=\frac 12n(n+1)$$ If you don't like the substitution given by Brian M. Smith, you can just make the correction: $$\sum_{i=0}^{n-1}i=\sum_{i=1}^n=\frac 12n(n+1)-n=\frac 12n(n-1)$$ where the first equality just accounts for the deleted term and recognizes that the added term in the sum is $0$ ... |
1,505,908 | <p>Let $A$ and $B$ be two $n\times n$ matrices, and $x$ be a $n\times 1$ column vector. What is the derivative of $f=(x^TAx)Bx$ with respect to $x$?</p>
<p>I try to calculate it like </p>
<p>$\frac{\partial f}{\partial x}=\frac{\partial (x^TAx)}{\partial x}Bx+(x^TAx)\frac{\partial (Bx)}{\partial x}=[x^T(A+A^T)]Bx+(x^... | copper.hat | 27,978 | <p>I think direct expansion is the most straightforward here.
Look at $f(x+h)-f(x)$ and collect the terms that are linear in $h$.</p>
<p>Since $f(x+h) = (x^T+h^T) A(x+h) B (x+h)$, we get that
$Df(x)h = x^T AxBh + x^T A h Bx + h^T A x B x= (x^T AxB+Bx x^T A+Bx x^T A^T) h$.</p>
<p><strong>Aside</strong>: One needs to b... |
3,057,924 | <p>I was reading <a href="http://advancedintegrals.com/wp-content/uploads/2016/12/advanced-integration-techniques.pdf" rel="nofollow noreferrer">Advanced Integration Techniques</a>, and found that<span class="math-container">$$\int_{0}^{1}\sqrt{x}\sqrt{1-x}\,\mathrm dx =\frac{\pi}{8}$$</span></p>
<p>The book provides ... | Quanto | 686,284 | <p>Substitute <span class="math-container">$x=\frac12(1+t) $</span>
<span class="math-container">$$\int_{0}^{1}\sqrt{x}\sqrt{1-x}\ dx
=\frac14\int_{-1}^1\sqrt{1-t^2}\ dt=\frac{\pi}{8}$$</span></p>
|
1,636,807 | <p>Ok, here is what I think. Please correct me if I am wrong.
$$\sqrt{9} \neq 3$$
and also
$$\sqrt{9} \neq -3$$</p>
<p>Now let's assume, that above statements are false, then we have $-3 = \sqrt{9} = 3$ and since $3 \neq -3$ the assumption must be wrong.
Ok, square root must be equal to 3 and -3 at the same time. As o... | mweiss | 124,095 | <p>The equation $x^2 = 9$ does indeed have two solutions, and therefore it is not incorrect to say that "$9$ has two square roots", namely $x=3$ and $x=-3$. But the notation $\sqrt{9}$, by definition, refers to the <em>positive</em> square root of $9$. There is only one of those, and it is equal to $3$, not to $-3$.</... |
89,407 | <p>Can users of <em>Mathematica</em> on Windows 7 or Windows 8/8.1 upgrade their computers to Windows 10 – a process that began today? Does <em>Mathematica</em> work? Are all the settings, licenses, access to files and folders preserved? Is there anything one must be careful about?</p>
| RunnyKine | 5,709 | <p>I can confirm that upgrading to <strong>Windows 10</strong> from either <strong>Windows 7</strong> or <strong>Windows 8.1</strong> (you can't upgrade from <strong>Windows 8</strong> directly, you first have to upgrade to <strong>8.1</strong>) leaves all your settings and licenses intact and that includes your <em>Ma... |
320,589 | <p>The max flow-min cut theorem is one of the most famous theorems of discrete optimization, although it is very straightforward to prove using duality theory from linear programming. Are there any other examples of famous theorems that are also corollaries of LP duality, or duality of convex optimization? The Farkas... | Timothy Chow | 3,106 | <p>To elaborate on M. Winter's comment: Von Neumann's <a href="https://en.wikipedia.org/wiki/Minimax_theorem" rel="noreferrer">minimax theorem</a> for two-person zero-sum games can be thought of as a consequence of LP duality, although his first proof of the theorem <a href="https://www.jstor.org/stable/41134130" rel="... |
320,589 | <p>The max flow-min cut theorem is one of the most famous theorems of discrete optimization, although it is very straightforward to prove using duality theory from linear programming. Are there any other examples of famous theorems that are also corollaries of LP duality, or duality of convex optimization? The Farkas... | Fedor Petrov | 4,312 | <p><a href="https://en.wikipedia.org/wiki/Bondareva%E2%80%93Shapley_theorem" rel="nofollow noreferrer">Bondareva–Shapley_theorem</a> is quite famous in a game theory. In more modern terms, it unites the usual and dual description of the generalized permutohedron. </p>
|
1,261,819 | <p>I found lots of examples of ADMM formalization of equality constraint problems (all with single constraint). I am wondering how to generalize it for multiple constraints with mix of equality and inequality constraints.</p>
<p>I have a problem</p>
<blockquote>
<p>Minimize $||A_x||_1 + \lambda ||A_y||_2 $, such th... | Srdan | 235,443 | <p>The "constraint" $Ax+Bz=c$ is there to introduce variable splitting. The actual (problem) constraints are introduced through indicator functions (e.g. $g(z) = \mathbf{1}_{\|\cdot\| \leq \varepsilon} (z)$). To allow for multiple constraints, you can reformulate your problem such that $f$ or $g$ represents a sum of pe... |
3,254,331 | <blockquote>
<p>Prove <span class="math-container">$$\int_0^\infty\left(\arctan \frac1x\right)^2 \mathrm d x = \pi\ln 2$$</span></p>
</blockquote>
<p>Out of boredom, I decided to play with some integrals and Inverse Symbolic Calculator and accidentally found this to my surprise</p>
<p><span class="math-container">$... | user246336 | 246,336 | <p>First the substitution <span class="math-container">$x\mapsto 1/x$</span> and then integration by parts yield
<span class="math-container">$$\int_0^\infty\arctan^2x^{-1}\,dx=2\int_0^\infty\frac{\arctan x}{x(1+x^2)}\,dx$$</span>
so it suffices to evaluate the integral on the right. Define the function
<span class="ma... |
99,084 | <p>How can I prove directly that a locally compact topological group G is normal?
I have done this by showing that every locally compact topological group is strongly Paracompact. But I could not prove it directly.</p>
| Alex | 87,938 | <p>Hausdorfness is not required for the group. Let $H$ be a clopen $\sigma$-compact subgroup of $G$. $H$ exists and is normal. Thus $G$ is normal. This requires the knowledge that given $G$ a topological group and $H \subseteq G$ an open subgroup, $G$ is normal iff $H$ is normal. </p>
<p>Proof of my last statement is ... |
99,084 | <p>How can I prove directly that a locally compact topological group G is normal?
I have done this by showing that every locally compact topological group is strongly Paracompact. But I could not prove it directly.</p>
| kahen | 1,269 | <h2>Hints</h2>
<p>Start with a precompact symmetric open neighbourhood <span class="math-container">$U$</span> of the identity and form the subgroup <span class="math-container">$H = \cup_{n \in \mathbb N} U^n$</span>. Show that <span class="math-container">$H$</span> is clopen and <span class="math-container">$\sigma$... |
3,291,889 | <p>All rings are commutative ring with unity.</p>
<p>Let <span class="math-container">$A$</span> and <span class="math-container">$B$</span> are two <span class="math-container">$R$</span>-algebras and <span class="math-container">$I$</span> and <span class="math-container">$J$</span> are two ideals of <span class="ma... | Grey Fox | 668,075 | <p>We have that <span class="math-container">$\cos(3x)=4\cos^3(x)-3\cos(x)$</span> (by <a href="https://en.wikipedia.org/wiki/List_of_trigonometric_identities#Triple-angle_formulae" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/List_of_trigonometric_identities#Triple-angle_formulae</a>). </p>
<p>Then <span c... |
764,508 | <p>I can't solve this. Can any one show me detail solution
$$
\sqrt{x^2+x+1}+x^3=\sqrt{2x+2}+x^2+x
$$</p>
| Christian Blatter | 1,303 | <p>Let $f^{-1}=: g$ and denote by ${\rm rec}: z\mapsto {1/z}$ the reciprocal function. It seems you have available the formula
$$g'(y)={1\over f'\bigl(g(y)\bigr)}\qquad(c<y<d)\ .\tag{1}$$
This formula can be read as follows:
$$g'={\rm rec}\circ f'\circ g\ .\tag{2}$$
On the right side you have a composition of thr... |
802,293 | <p>Let $f: \mathbb{R}^{2} \rightarrow \mathbb{R}$ be a continuous function. Is $\overline{A} =\overline{\{x: f(x) < a\}} = \{x: f(x) \leq a\} = B$? </p>
<p>Since $B$ is closed and contains $A$, $\overline{A} \subset B$ as $\overline{A}$ is the smallest closed subset containing $A$. It remains to show $B \subset \ov... | clark | 33,325 | <p>Yes you are right, to see a counterexample take $ f(x,y)=a$.</p>
|
4,254,832 | <blockquote>
<p>Denote the distance between two sets <span class="math-container">$A,B \in \Bbb R^n$</span> as <span class="math-container">$d(A,B).$</span> If <span class="math-container">$d(A,B) > 0$</span> show that <span class="math-container">$m^*(A \cup B) = m^*(A) + m^*(B)$</span>.</p>
</blockquote>
<p>The p... | Mason | 752,243 | <p>An alternative approach: First show that if <span class="math-container">$O$</span> is open, then <span class="math-container">$O = \bigcup_{i = 1}^{\infty}Q_i$</span>, where the <span class="math-container">$Q_i$</span> are dyadic cubes disjoint except for possibly their boundaries. Then show <span class="math-cont... |
4,046,685 | <p>Clearly, the total number of subsets possible is <span class="math-container">$2^5$</span></p>
<p>For two elements to be common, both subsets need to have at least two elements, so we can form quite a lot of cases which satisfy both conditions.</p>
<p>Now there are far too many cases (IMO) for me to manually curate,... | V.G | 746,921 | <p>Let's say instead of choosing two subsets from a single set, we choose subsets from the two sets <span class="math-container">$A=\{1,2,3,4,5\}$</span> and <span class="math-container">$B=\{1,2,3,4,5\}$</span>, so number of subsets that can be chosen from <span class="math-container">$A$</span> are <span class="math-... |
419,091 | <blockquote>
<p><span class="math-container">$G$</span> is an infinite group.</p>
<ol>
<li><p>Is it necessary true that there exists a subgroup <span class="math-container">$H$</span> of <span class="math-container">$G$</span> and <span class="math-container">$H$</span> is maximal ?</p>
</li>
<li><p>Is it possible that... | Mikasa | 8,581 | <p>Rotman p. 324 problem 10.25:</p>
<blockquote>
<p>The following conditions on an abelian group are equivalent:</p>
<ul>
<li><p><span class="math-container">$G$</span> is divisible.</p>
</li>
<li><p>Every nonzero quotient of <span class="math-container">$G$</span> is infinite; and</p>
</li>
<li><p><span class="math-co... |
2,147,807 | <p>Essentially what the title asks. For an argument $x$, how can I analytically acquire values for the function:
$$
f(x)=\sum_{k=0}^{\infty}\frac{x^{2k+1}}{(2k+1)(k!)}
$$
Again, it is important that I know how to do this <strong>analytically</strong>, as there are other series comparable to this one that I also wish to... | Brevan Ellefsen | 269,764 | <p>You ask for specific values. By noting that this is simply
$$\sum_{k \ge 0} \int_0^x \frac{t^{2k}}{k!}dx= \int_0^x\sum_{k \ge 0} \frac{t^{2k}}{k!}dx= \int_0^xe^{t^2}dt = \frac{-i\sqrt{\pi}}{2}\operatorname{Erf}(ix)$$
All we have to do is look up known values of the <a href="https://en.wikipedia.org/wiki/Error_functi... |
558,156 | <p>Prove that $\binom{n}{r} + \binom{n}{r+1} = \binom{n+1}{r+1} $</p>
<p>Thanks in advance, my professor asked us to this a couple weeks ago, but I was enable to get to the right answer. </p>
<p>Good luck!</p>
<p>Here is what I got up to;</p>
<p>$\frac{(n+1)!}{(n-r)!(r+1)!} = \frac{(n)!}{(r)!(n-r)!} + \frac{(n)!}{... | Geoffrey | 105,204 | <p><strong>Hint</strong>: Recall that $(d+1)!=(d+1)*d!$ From here, try combining the fractions in the sum by giving them a common denominator.</p>
|
3,238,780 | <p>I am trying to work through a few problems, and one asks to sum over the Fibonacci numbers which are even-valued (it is the Euler Project problem #2). I realized that (if we index like <span class="math-container">$\langle 1, 2, 3, 5, 8, \ldots \rangle$</span>) that <span class="math-container">$F_n$</span> is even ... | Community | -1 | <p>There's a <em>mnemonic</em> to help you remember the quotient rule: "low d(high) minus high d(low), down below the square must go".</p>
<p>Alternatively, you could note that you have <span class="math-container">$\dfrac12x-\dfrac 32$</span>, so that the derivative is indeed <span class="math-container">$\dfrac 1... |
347,186 | <p>I have this question:</p>
<blockquote>
<p>Let $x, n$ be integers with $n \geq 2$ and $n$ not dividing $x$. Show that the order o($\bar{x}$) of $x \in Z_n$ is
$o(\bar{x})= \frac{n}{HCF(x, n)}$</p>
</blockquote>
<p>I've been thinking about it for ages but I still don't get why. A hint would be appreciated.</p>
| Did | 6,179 | <p><strong>Hint:</strong> Consider a series $\sum\limits_nx_n$ where $x_n=a_n-a_{n+1}$ for every $n\geqslant1$, for some sequence $(a_n)_{n\geqslant1}$. </p>
<ul>
<li>Express each partial sum $\sum\limits_{n=1}^Nx_n$ with $N\geqslant1$, as a simple function of $a_1$ and of another term of the sequence $(a_n)_{n\geqsla... |
760,098 | <blockquote>
<p>By drawing graph,or otherwise,find the <strong>number of roots</strong> of the equation
$x+2 \tan(x)= \pi/2$<br>
lying between $0$ and $2\pi$, and find the <strong>approximate value of the largest root</strong>.</p>
</blockquote>
<p>I found <strong>3</strong> roots by drawing a rough graph ,but... | Oria Gruber | 76,802 | <p>Approximate value - meaning we can use numerical methods.</p>
<p>To my knowledge there is no analytic solution to this equation in the domain mentioned.</p>
<p>We will attempt to find a root of $f(x)=x+2tan(x)-\frac{\pi}{2}$</p>
<p>notice that $f(0) = -\pi$ and that $f(2\pi)=\pi$ and it is continouos and so there... |
2,509,382 | <p>Two spheres of radii $r_1$ and $r_2$ intersect each other orthogonally. Prove that the circle formed by the intersection of the two spheres has a radius
$$\frac{r_1 r_2}{\sqrt{r_1^{2} + r_2^{2}}}.$$</p>
| Narasimham | 95,860 | <p>From Aretino's sketch considering similar right triangles</p>
<p><span class="math-container">$$CH= h ;\,CA= r_1;\,CB= r_2;\, \angle ACH= \angle HBC= \theta \,$$</span></p>
<p><span class="math-container">$$ \cos \theta = \frac{h}{r_1} ;\,\sin \theta=\frac{h}{r_2} ; $$</span></p>
<p>The result is arrived at by e... |
8,891 | <p>This is not a question. Just a request.</p>
<p>This Issue has been discussed previously and I think this can be implemented technically as well.</p>
<p>Writing title all in latex disables the options like "Open in New Tab" and other options in browsers. It may be user's own habit but I terribly hate it when the op... | Davide Cervone | 7,798 | <p>There is a way to get MathJax to not use its contextual menu (and use the browser's instead). On a page that has typeset equations, first enter</p>
<pre><code>javascript:MathJax.Menu.showContext(true);
</code></pre>
<p>in the URL location bar, or use your browser's developer console to type in the command (minus ... |
2,363,236 | <p>This may seem like a stupid question but</p>
<p>What method can one use to convert a decimal number, such as $0.672$ into a whole number?</p>
<p>This isn't rounding as $0.999$ should result in $0$, but saying "floor the number" isn't an answer as I'm looking for a mathematical method in which a number, $n$, can be... | Maxime Scott | 222,222 | <p>This can be a difficult concept to grasp, but as mentioned in the answers above, if you pick any two real numbers, there will be a rational number between them. Moreover, there will also be an irrational number between them.</p>
<p>This also implies that you can find a rational number as close to an irrational numb... |
1,894,699 | <p>$d(x,S) = \inf_{s \subset \mathbb{R}}\{|x-s|: s \in S\}, x\in \mathbb{R}$</p>
<p>I did notice this question was asked before, but most people were asking for tips or completely solve it for them. I want to get critique on my work as I seem to be the only to have attempted it.</p>
<p>How I went about it:</p>
<p>If... | arya | 309,614 | <p>You don't know if x,y and z are between -1 and +1.Take them as tan(a),tan(b),tan(c) and use tana.tanb + tanb.tanc + tana.tanc=1</p>
|
31,085 | <p>This was asked by my maths lecturer a couple of years ago and ive been wracking my brains ever since:</p>
<blockquote>
<p>Find a number that, when multiplied by
99 will give the original number but
with a 1 at the beginning and a 1 at
the end.</p>
<p>For example: 42546254 * 99 would equal
1425462541 ... | joriki | 6,622 | <p>$$112359550561797752809 \cdot 99 = 11123595505617977528091\;.$$</p>
<p>If you put what you say into an equation, it reads</p>
<p>$$10^n+1+10x=99x\;,$$
$$10^n+1=89x\;.$$</p>
<p>Then you just need to find an $n$ such that $10^n\equiv 88\pmod{89}$, and $n=22$ works.</p>
|
1,056,105 | <p>I was reading the section of Ravi Vakil's Algebraic Geometry notes where he discusses elliptic curves.</p>
<p>If we let an elliptic curve be $(E,p)$ (Where $p$ is the distinguished point), we have $\mathcal{O}(3p)$, which has $3$ sections, which gives us a closed embedding into $\mathbb{P}^2$. In the next sentence ... | Slade | 33,433 | <p>A nonsingular curve $C\subset\mathbb{P}^2$ of degree $d$ always has genus $g = \frac{(d-1)(d-2)}{2}$ (this can be computed explicitly), so $g=1$ implies $d=3$.</p>
|
4,361,279 | <p>In my book's table for antiderivatives of some functions, I came across the following,</p>
<p><span class="math-container">$$\int{e^{ax}dx=\frac{1}{a}e^{ax}} + C, \qquad a \neq0\tag{1}$$</span></p>
<p>I can't understand the reasoning behind the condition <span class="math-container">$a\neq0$</span>. Also,</p>
<p><sp... | QC_QAOA | 364,346 | <p>Alright, so the question is basically: what happens at <span class="math-container">$a=0$</span> in the integral</p>
<p><span class="math-container">$$\int e^{ax}dx$$</span></p>
<p>Well, as you noted, for <span class="math-container">$a\neq 0$</span> we have</p>
<p><span class="math-container">$$\int e^{ax}dx=\frac{... |
113,338 | <p>I am having a hard time understanding the meaning of the union operation in this equation.</p>
<p>$$C(A)=\bigcup_{x \in A}C(x)$$</p>
<p>For context, here is the sentence:</p>
<p>The candidate set for $x$ is $S \cap C(x)$. The candidate region for a set of points $A$ is $C(A)=\bigcup_{x \in A}C(x)$, with the candi... | Brett Frankel | 22,405 | <p>We say $y$ is in $C(A)$ if and only if $y$ is in $C(x)$ for some $x\in A$.
In other words, $\bigcup_{x \in A} C(x)$ is the set of points which are in at least one of the $C(x)$.
In general, the union of sets $\bigcup A_i$ is the set of elements contained in at least one of the $A_i$, and the subscript under the unio... |
789,420 | <p>If $M$ is a compact topological manifold WITH boundary does it follow that its homology groups are finitely generated and zero almost all of them? I know it is true in case it has no boundary (i.e. is properly a manifold).</p>
| bradhd | 5,116 | <p>Yes, it is still true: any compact topological manifold (with or without boundary) is homotopy-equivalent to a finite CW-complex, which has finitely-generated homology groups, only finitely many of which are nonzero.</p>
|
2,871,490 | <p>Let $F$ be a vector field such that $$\vec{F}=\langle x^2,y^2,z\rangle$$ Integration over the line segments which form the triangle with vertices $(0,0,0)$,$(0,2,0)$,$(0,0,2)$ can be achieved by parametrizing each segment and then evaluating $$\sum_{i=1}^3\int_{C_i}\vec{F}(r_i^{(x)}(t),r_i^{(y)}(t),r_i^{(z)}(t))\... | yoyo | 6,925 | <p>The orientation perserving isometry group of the plane is $SO_2(\mathbb{R})\rtimes\mathbb{R}^2$ (the orthogonal group $SO_2(\mathbb{R})$ are the rotations and $\mathbb{R}^2$ are the translations). This can be embedded in $SL_3(\mathbb{R})$ as follows:
$$
\left(
\begin{array}{ccc}
1&0&0\\
a&\cos\theta&am... |
1,665,714 | <p>I am looking for the following sets for all $z \in \mathbb{C}$</p>
<p>$$\{z: \cos(z)=0\} \text{ and } \{z: \sin(z)=0\}$$</p>
<p>I believe the best way to do this is consider the exponential form so</p>
<p>$$\cos(z)=\frac{e^{iz}+e^{-iz}}{2} \text{ and } \sin(z)=\frac{e^{iz}-e^{-iz}}{2i}$$</p>
<p>So for $\cos(z)=0... | Claude Leibovici | 82,404 | <p><em>This is not an answer since just based on numerical simulation.</em></p>
<p>Being stuck with any formal approach I tried, I just used numerical simulation and what I observed is that, if $$P_n=\prod_{k=1}^{\infty}\zeta (2kn)$$ then $\log(P_n-1)$ varies as a linear function of $n$. A basic linear regression $(1\... |
1,665,714 | <p>I am looking for the following sets for all $z \in \mathbb{C}$</p>
<p>$$\{z: \cos(z)=0\} \text{ and } \{z: \sin(z)=0\}$$</p>
<p>I believe the best way to do this is consider the exponential form so</p>
<p>$$\cos(z)=\frac{e^{iz}+e^{-iz}}{2} \text{ and } \sin(z)=\frac{e^{iz}-e^{-iz}}{2i}$$</p>
<p>So for $\cos(z)=0... | Igor Rivin | 109,865 | <p>For a reasonably large $n$ (that is, bigger than about $3$), $\zeta(n) \approx 1+ \frac{1}{2^n},$ which will give you the asymptotics you crave.</p>
|
4,142,894 | <p>Blitzstein, <em>Introduction to Probability</em> (2019 2 ed), p 58, Example 2.3.10 (Six-fingered man).</p>
<blockquote>
<p>A crime has been committed in a certain country. The perpetrator is one (and only one) of the <span class="math-container">$n$</span> men who live in the country. Initially, these n men are all ... | Tankut Dogrul | 928,949 | <p>Bayes rule states: <span class="math-container">$$P(A) = \frac{P(B|A)P(A)}{P(B|A)P(A)+P(B|A^c)P(A^c)} $$</span></p>
<p>We have:</p>
<ol>
<li>P(R) = Rugen is guilty = <span class="math-container">$\frac{1}{n}$</span> <em>since equally likely</em></li>
<li><span class="math-container">$P(R^c)$</span> = Someone else is... |
3,116,466 | <p>I integrate over the edge of a circle <span class="math-container">$K$</span> with radius 1/2</p>
<p><span class="math-container">$\int_{|z|=1/2}\frac{e^{1-z}}{z^{3}(1-z)}dz=\int_{|z|=1/2}-\frac{e^{1-z}}{z^{3}}\frac{1}{(z-1)}dz$</span> </p>
<p>By the Cauchy Integral form</p>
<p><span class="math-container">$f(w)=... | EditPiAf | 418,542 | <p>A graphical representation of the problem in the <span class="math-container">$x$</span>-<span class="math-container">$t$</span> plane may be relevant (see <a href="https://math.stackexchange.com/a/2941875">this post</a>). The method of characteristic provides the set of lines <span class="math-container">$x=t-t_0+... |
2,825,103 | <p>I'm having trouble coming up with a solution on the following question:</p>
<p><strong>Find a formula for the curvature of the cycloid given by</strong> $$ x = t−\sin (t)\ ,\ y = 1−\cos(t)$$.</p>
<p>I have the following:</p>
<p>$$r(t)=<t-\sin(t),1-\cos(t)>$$
$$r'(t)= <1-\cos(t), \sin(t)>$$</p>
<p>I'... | anomaly | 156,999 | <p>A f.g. group $G$ is quasi-isometric to the trivial group iff every $g\in G$ has bounded length (with respect to some generating set), i.e., iff $G$ is finite.</p>
|
2,469,690 | <blockquote>
<p>If $a+b+c=0$, for $a,b,c \in\mathbb R$, prove</p>
<p>$$ 3(a^2+b^2+c^2) \times (a^5+b^5+c^5) = 5(a^3+b^3+c^3) \times (a^4+b^4+c^4) $$</p>
</blockquote>
<p>I made this question as a more difficult (higher degree) version of <a href="https://math.stackexchange.com/questions/2469296/if-abc-0-prove-t... | Michael Rozenberg | 190,319 | <p>Let $a+b+c=3u$, $ab+ac+bc=3v^2$ and $abc=w^3$.</p>
<p>Hence, $u=0$ and
$$a^2+b^2+c^2=(a+b+c)^2-2(ab+ac+bc)=-6v^2,$$
$$a^3+b^3+c^3=a^3+b^3+c^3-3abc+3abc=$$
$$=(a+b+c)(a^2+b^2+c^2-ab-ac-bc)+3abc=3abc=3w^3,$$
$$a^4+b^4+c^4=(a^2+b^2+c^2)^2-2(a^2b^2+a^2c^2+b^2c^2)=$$
$$=36v^4-2((ab+ac+bc)^2-2abc(a+b+c))=36v^4-18v^4=18v^... |
3,455,552 | <p>In the <span class="math-container">$n$</span>-dimensional Euclidean space, given two vectors <span class="math-container">$\vec{OA}$</span> and <span class="math-container">$\vec{OB}$</span> (not collinear), their angle AOB is <span class="math-container">$\theta$</span>. Now there is a vector <span class="math-con... | Batominovski | 72,152 | <p>First note that <span class="math-container">$2$</span> is a primitive element modulo <span class="math-container">$19$</span>. To show this, observe that <span class="math-container">$\phi(19)=18$</span>. If <span class="math-container">$k$</span> is the smallest positive integer such that <span class="math-conta... |
1,793,594 | <p>Disclaimer: In the definition (Stewart Calculus, 7E): "Method of Lagrange Multipliers" part (b)- Evaluate $f$ at all extreme points $(x,y,z)$ from step a. The largest of these values is the maximum and the smallest of these values are the minimum. </p>
<p>It is impossible to infer whether an extreme point is a maxi... | amd | 265,466 | <p>In this case, simple geometric considerations tell you that you have a minimum: the objective function is the square of the distance from the origin and the constraint is the equation of a plane. </p>
<p>More generally, one can examine the <a href="https://en.wikipedia.org/wiki/Hessian_matrix#Bordered_Hessian" rel... |
595,864 | <blockquote>
<p><em>Theorem</em>: For every positive integer n greater than $2$, then $\phi(n)$ is an even integer.</p>
</blockquote>
<p>I know this theorem and the same is used much, but I was curious how it would be to demonstrate it, show me anyone know how or where to find it?</p>
| Community | -1 | <p>Note that if $x$ is relative prime to $n$, so is $n-x$ except for $n=2$, since you double count $x=1=2-1=n-x$.</p>
|
4,422,920 | <p>Original Question:</p>
<blockquote>
<p>Use the proof of Thm 7.2.4 given above to show that if <span class="math-container">$n \geq 2$</span>, then <span class="math-container">$\frac{e}{(n+1)^{\frac{1}{n}}} \frac{n}{n+1} < \frac{n}{n!^{\frac{1}{n}}} < \frac{e}{4^{\frac{1}{n}}}$</span>.</p>
</blockquote>
<block... | user99432 | 845,379 | <p>Take the <span class="math-container">$n$</span>-th root of <span class="math-container">$\frac{(n+1)^{n+1}}{n!}>e^n$</span> which gives</p>
<p><span class="math-container">$$\frac{(n+1)^\frac{n+1}{n}}{(n!)^{\frac{1}{n}}}>e.$$</span>
Now the left hand side is equal to <span class="math-container">$(n+1)\frac{(... |
1,441,932 | <p>I think I am making this problem far harder than it needs to be. Here is the statement: for each non-negative integer $n$, let $P_n$ be the space of real-valued polynomials of degree less than or equal to $n$. Find a Jordan Canonical basis for the map $T(f) = f' + f$.</p>
<p>My attempt: I let $\beta = \{1,x,x^2,...... | CopyPasteIt | 432,081 | <p>I want to address the question/comments with this answer, which contains some links/quotes and at the end I put in my two cents.</p>
<p>Mathematician Doron Zeilberger:</p>
<p>“For most people, the computer is only a tool, like a vacuum cleaner. For me, it’s like a colleague. Traditional mathematics is based on the... |
120,254 | <p>I want to use <code>Solve[]</code> inside a <code>Module</code>. If I make the variables solved for local
to the module, they are treated differently than if I leave them global.
For example,</p>
<pre><code>SolveIt[a_, b_] := Module[{x, soln},
soln = Solve[a x + b == 0, {x}];
Return[soln]
];
SolveIt[3, 4]
... | Szabolcs | 12 | <p>Do you really need to return <code>x</code> from your function (even as part of a larger expression)? There is simply no good way to do this if <code>x</code> has a global value, as <code>x = 1; x -> 2</code> immediately evaluates to <code>1 -> 2</code>.</p>
<p>You could return the solution value only (<code... |
4,243,792 | <p>Consider <span class="math-container">$f: \mathbb{R}^n \rightarrow \mathbb{R}$</span>. <span class="math-container">$f\left(\alpha x+\left(1-\alpha\right)x'\right)=\alpha f\left(x\right)+\left(1-\alpha\right)f\left(x'\right)$</span>, <span class="math-container">$\forall \alpha \in \left[0,1\right]$</span> and <span... | sadkangaroo | 332,200 | <p>First of all, this is a very good question, don't get discouraged by some people. The word you really want to use is "affine" instead of "linear", but many people use them interchangebly in informal discussion.</p>
<p>Note that adding a constant to <span class="math-container">$f$</span> preserve... |
3,954,579 | <p>I have two different questions, but they are related.</p>
<p>The first question is, Let <span class="math-container">$G$</span> be a finite abelian group. show that if <span class="math-container">$G$</span> contains (atleast) <span class="math-container">$2^n-1$</span> distinct elements of order 2, then there must ... | Bumblebee | 156,886 | <p>Hint: For your first question: show there is an injective group homomorphism <span class="math-container">$$\varphi: (\mathbb{Z}/2\mathbb{Z})^n\to G.$$</span> For the second question: show that the group has <span class="math-container">$2^n-1$</span> elements of order <span class="math-container">$2.$</span></p>
<p... |
2,000,268 | <p>We usually tend to say the "Average" is whether "Mean", "Median" or "Mode" and in colloquial usage "Average" is always equivalent to "Mean".</p>
<blockquote>
<p>But my <strong>question</strong> is: Is there any precise rigorous definition of "Average of a statistical population" in statistics (regardless of our k... | Abe Vallerian | 385,943 | <p>From a book (I forget the title) I read, the 'average' is basically a number which (we believe) can represent the whole population. Since Statistics usually deals with a huge number of data, we need a kind of 'summary' of the whole data, which is the 'average' itself. Thus, the choice of the type of average (mean, m... |
2,786,291 | <p>How to prove $l^1$ and $l^\infty$ are infinite dimensional spaces.</p>
<p>I know that a space is infinite dimensional if it has a subspace with infinite dimensions. But I don't know how to proceed. Any help will be appreciated.</p>
| Rgkpdx | 112,537 | <p>Suppose $l^1$ (or $l^\infty$) is finite dimensional, and let $\{e_n\}_1^N$ be a basis. Consider $\{0_n\}_1^{N+1}$ where $0_n$ is the vector zero everywhere and $1$ in the $n$-th position. You now get a contradiction as the set $\{0_n\}_1^{N+1}$ is linearly independent, implying that the cardinality of $\{e_n\}_1^N$ ... |
2,077,831 | <p>This is not a homework problem. It is meant as a challenge for people who really enjoy math and have time to spare.</p>
<p><strong>Background Info</strong></p>
<p>Suppose you have a 2D Cartesian coordinate system. There are three shapes: <em>R</em>, <em>C</em>, and <em>P</em>.</p>
<p><em>R</em> is a large rectang... | Joseph O'Rourke | 237 | <p><hr />
<a href="https://i.stack.imgur.com/YKogh.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/YKogh.jpg" alt="Star1"></a>
<hr />
<a href="https://i.stack.imgur.com/twsb... |
959,742 | <p>For a function $f(x) = x\sqrt{2+x^2}$ find out if it's bijective and if so, find its inverse.</p>
<p>The function is surjective because $x^2 > 0:\forall x\in\mathbb{R}$. I'm having difficulties proving that the function is injective.</p>
<p>I tried the following: $$f(x) = f(y) \iff x\sqrt{2+x^2} = y\sqrt{2+y^2}... | Some Math Student | 181,130 | <p>What do you know about monotonous functions? Is this function (strictly) monotonous, and if yes, does this tell you something about injectivity?</p>
|
98,913 | <p>I'm wondering to detect all the errors (i.e. their positions) in a codeword $(c_0, c_1, \cdots, c_{n-1})\in Q$ where $Q$ is an alphabet set with size $q$, i.e., to know whether $c_i$ is faulty or not, without asking for the exact initial value. Is it possible to achieve higher code rate than the ECC which corrects ... | Bruce Fang | 77,936 | <p>Do Error-locating(EL) codes proposed by Wolf and Elspas in 1963 satisfy your requirements?
This class of codes usually has higher rates. And tensor product parity codes(an extension of EL codes) also have higher rates.</p>
|
1,198,754 | <p>A rectangular prism has <strong>integer</strong> edge lengths. Find all dimensions such that its surface area equals its volume.</p>
<p>My Attempt at a Solution:</p>
<p>Let the edge lengths be represented by the variables $l, w, h$.</p>
<p>Then $$lwh = 2\,(lw +lh + wh) \implies lwh = 2lwh\left(\frac{1}{l} + \frac... | Tom | 217,400 | <p>There is $(4,5,20)$.</p>
<p>I can't think of a general approach that doesn't involve brute force, however.</p>
|
3,848,934 | <p>Let <span class="math-container">$T$</span> be some operator on an inner product space <span class="math-container">$(V, \langle\cdot,\cdot\rangle)$</span>, and <span class="math-container">$T^\dagger$</span> be its adjoint. I found too many questions about the proof of
<span class="math-container">$$Im(T^\dagger) =... | ΑΘΩ | 623,462 | <p>For the sake of completeness of approach, I would like to present here the general context of purely algebraic adjunction, where we consider no metric-inducing inner products and no topological vector spaces.</p>
<p>In the most general frame, consider an arbitrary ring <span class="math-container">$A$</span> togethe... |
307,370 | <p>Let $G$ be a finite group and $H$ a subgroup. Define a relation on $G$ by $$a\sim b\iff b^{-1}a \in H.$$</p>
<p>(0.) Show that this is an equivalence relation.</p>
<p>(i) Prove that for this relation $$[a] = \{ah: h\in H\}.$$</p>
<p>(ii) Prove that the cardinality $[a]$ equals the cardinality of $|H|$. </p>
<p>(... | Community | -1 | <p>$a \sim b$ is an equivalence relation because:</p>
<ul>
<li><em>reflexivity</em>: $a^{-1} a = 1 \in H$ because groups contain identity.</li>
<li><em>symmetry</em>: $b^{-1} a \in H$ implies $a^{-1} b = (b^{-1} a)^{-1} \in H$ because groups contain inverses.</li>
<li><em>transitivity</em>: $b^{-1} a, c^{-1} b \in H$ ... |
1,063 | <p><strong>Problem</strong></p>
<p>Using Mathematica's <code>Solve</code> operator can sometimes lead to an output involving a positive and negative root (say when solving for a variable such as <code>N^2</code>.</p>
<p><strong>What Is Happening</strong></p>
<p>Here I simply have a defined function <code>Radial</cod... | Vitaliy Kaurov | 13 | <p>You need to pass to <code>Solve</code> conditions for all three variables. <code>Solve</code> cannot decide whether <code>N</code> is positive or negative unless it knows the sign of <code>a</code> and <code>Z</code> , because their values define sign of <code>N</code>. Note, passing the assumptions to <code>Integra... |
868,613 | <p>When I say that $p_1=2$, I mean that the first prime in the standard ring of integers $(\mathbb{Z},*,+)$ is $2$. I was wondering whether the notion of ordering the primes like this can be generalized for any ring $R$. First of all, is it even possible to consistently equip a general ring (such as a finite one) with ... | Ivo Terek | 118,056 | <p>We have $\varnothing \subset A$ for <strong>any</strong> set $A$, including $ A = \{\varnothing \}$.</p>
<p>Suppose there is a set $A$ such that $\varnothing \not\subset A$. Then exists $x \in \varnothing$ such that $x \not\in A$. But this is a contradiction, because there is no element in $\varnothing$.</p>
|
1,052,512 | <p>Basically, the question started with a little argument I had with my friend. My friend said he thinks it's possible to draw only 2 lines on the letter "W" and make 6 triangles, and I played around with it, but I couldn't really do it, so I told him I don't think it's possible, and we need at least 3 lines. </p>
<p>... | Fmonkey2001 | 89,418 | <p>It only takes two lines to get six triangles? Let me try to add a picture real quick.<img src="https://i.stack.imgur.com/yr5FA.jpg" alt="j"></p>
<p>So far the max I can find with two lines is $5$. Ill keep trying!</p>
|
12,098 | <p>It's not really a typical math question. Today, while studying graphs, I suddenly got inquisitive about whether there exists a function that could possibly draw a heart-shaped graph. Out of sheer curiosity, I clicked on Google, which took me to <a href="http://mathworld.wolfram.com/HeartCurve.html">this page</a>.</p... | J. M. ain't a mathematician | 498 | <p>A somewhat late addition (I only found my yellowed notebooks containing these just now):</p>
<p>$$\left(2(1+\cos\,\varphi)\sin^3 t\qquad 2\cos\,\theta\;\sin^2 t \sin\,\varphi+\sin\,\theta\cos\,t\left(\cos\,2t-2\cos\,\varphi\;\sin^2 t-3\right)\right)^T$$</p>
<p>is a two-parameter family of curves that generate hear... |
3,318,909 | <p>I haven't learnt natural deduction yet so I'm completely stuck on how to proceed. One tip I was given was to use the properties of negation but again, that's not really helping.</p>
| Brian Moehring | 694,754 | <p>Write <span class="math-container">$\overline{\int_a^b} f(x)\,dx$</span> and <span class="math-container">$\underline{\int_a^b} f(x)\,dx$</span> for the upper and lower Riemann integrals, respectively. Since <span class="math-container">$f$</span> is bounded, there is some <span class="math-container">$M$</span> su... |
3,318,909 | <p>I haven't learnt natural deduction yet so I'm completely stuck on how to proceed. One tip I was given was to use the properties of negation but again, that's not really helping.</p>
| zhw. | 228,045 | <p>True, <span class="math-container">$f$</span> is Riemann integrable on <span class="math-container">$[0,1].$</span></p>
<p>Suppose <span class="math-container">$|f|\le M;$</span> we can assume <span class="math-container">$M>0.$</span></p>
<p>Let <span class="math-container">$\epsilon>0.$</span> Choose <span... |
167,934 | <p>Let $X$ be a proper algebraic variety. $X$ is said to have polynomial point count if there is a polynomial $P$ such that for all finite fields $\mathbb F_q$ with $q$ elements, $|X(_q)|=P(q)$.</p>
<p>If in addition $X$ is smooth, then by the Weil conjectures one can derive that $X$ has no odd cohomology.</p>
<p>My ... | Reladenine Vakalwe | 23,907 | <p>(I deleted my original answer since I realize I had misinterpreted the question a bit).</p>
<p>The cheekiest/simplest example that I know of a polynomial count variety with odd cohomology is $\mathbb{G}_m$. This example can be bootstrapped (standard `scissor formalism') to produce more complicated examples of such ... |
2,216,070 | <p>I'm having problems finding the primitive function to $\int \frac{1}{x \sqrt{8-x^2}} dx$. I've tried to use the substitution $t = x^2-8$, but then I just get stuck with $\int \frac{1}{(8-t)\sqrt{t}} dt$ instead. Using the substitution $t = \sqrt{x^2-8}$ doesn't get me much further either.</p>
<p>Any help is much ap... | hamam_Abdallah | 369,188 | <p><strong>hint</strong></p>
<p>multiply by $x $ and put $t=x^2$ to get</p>
<p>$$\int\frac {dt}{2t\sqrt {8-t}} $$</p>
<p>then</p>
<p>$8-t=u^2$</p>
<p>to obtain a rational fraction.</p>
|
537,250 | <p>$(X,K)$ a normed space
$E$ is a subspace of $X$.</p>
<p>if $\exists x_0 \in X$ that $||x_0||=d(x_0,E)=1$ then show</p>
<p>$$||e+\lambda x_0||\geq\frac{||e||}{2}\quad \forall e\in E \quad \forall\lambda\in K $$</p>
| copper.hat | 27,978 | <p>Note that $d(\lambda x_0,E) = |\lambda|$ and hence $\|e+\lambda x_0\| \ge |\lambda|$ (this is the key element here).</p>
<p>Also, $\|e+\lambda x_0\| \ge \|e\|-|\lambda| $ (always true if $\|x_0\| = 1$).</p>
<p>Combining gives:
$\|e+\lambda x_0\| \ge \max (\|e\|-|\lambda|, |\lambda|) = \frac{\|e\|}{2}+ \max (\frac{... |
2,471,838 | <p>If I have a a vector $\vec{v}=(x,y,z)$ in cartesian coordinates and I want to rotate my coordinate system by spherical angles $\theta$ and $\phi$ how would I find the resulting vector $\vec{v'}=(x',y',z')$ in cartesian coordinates?</p>
<p><a href="https://www.uwgb.edu/dutchs/MATHALGO/sphere0.htm" rel="nofollow nore... | Glorious Nathalie | 948,761 | <p>First express the axes of the rotated coordinate system in terms of the original one. The rotation is equivalent to two consecutive rotations: rotation about the <span class="math-container">$z$</span> axis by an angle <span class="math-container">$\phi$</span> followed by a rotation about the <span class="math-con... |
268,650 | <p>This may be a well known question:</p>
<p>Let $X$ be a complex analytic (finite intersection of zero locus of analytic functions) subset of $\mathbb{C}^n$. Is it true that the projection of $X$ to the first coordinate is a finite set or the complement of a finite set?</p>
<p>Edit: By the example in answers the con... | Ben McKay | 13,268 | <p>Let $X$ be the curve $(exp(2 \pi i z)-1)w=1$ in $\mathbb{C}^2_{z,w}$. Then $z$ cannot be an integer.</p>
|
906,318 | <p>How do you prove $f(x,y) = y - x$ is continuous? The domain is $\mathbb{R^{2}}$ and the codomain is $\mathbb{R}$. Is there an easy way to do it using the definition that the preimage of an open set is an open set? I don't have much experience proving multivariable functions are continuous.</p>
| M A Pelto | 171,159 | <p>Notice $|f(x,y)-f(x_o, y_o)| \leq |x-x_o|+|y-y_o| \leq 2 \sqrt{(x-x_o)^2+(y-y_o)^2}$. </p>
<p>You may workout the details of these weak inequalities if you would like.</p>
<p>Given $\epsilon>0$, define $\delta=\frac{\epsilon}{2}$. </p>
<p>This function is actually uniformly continuous on $\mathbb{R}^2$.</p>
|
906,318 | <p>How do you prove $f(x,y) = y - x$ is continuous? The domain is $\mathbb{R^{2}}$ and the codomain is $\mathbb{R}$. Is there an easy way to do it using the definition that the preimage of an open set is an open set? I don't have much experience proving multivariable functions are continuous.</p>
| user84413 | 84,413 | <p>Let $\epsilon>0$ be given, and take $\delta=\frac{\epsilon}{2}$.</p>
<p>If $||(x,y)-(a,b)||<\delta$, then $\sqrt{(x-a)^2+(y-b)^2}<\frac{\epsilon}{2}\implies|x-a|<\frac{\epsilon}{2}$ and $|y-b|<\frac{\epsilon}{2}$, so</p>
<p>$|f(x,y)-f(a,b)|=|(y-x)-(b-a)|=|(a-x)+(y-b)|\le|x-a|+|y-b|<\frac{\epsilon... |
4,539,750 | <p>I am having trouble understanding the validity of integrating both sides of an equation. I understand that an operation/manipulation can be performed to both sides of an equation, preserving the equality, eg. if two sides of an equation are equal, their derivatives are equal and hence it is valid to differentiate bo... | jdods | 212,426 | <p>More generally, if you have two derivatives of functions equal, say <span class="math-container">$f'(x)=g'(x)$</span>. It does not automatically follow that <span class="math-container">$f(x)=g(x)$</span>.</p>
<p>Although, assuming that <span class="math-container">$f'(x)=g'(x)$</span> on some interval, say <span cl... |
853,031 | <p>I was trying to show how to find $\pi$ value from formula $\pi R^2$, but I don't understand where is my mistake. </p>
<p>So I am calculating area using $n$ triangles <a href="https://i.stack.imgur.com/p5vzJ.jpg" rel="nofollow noreferrer">1</a></p>
<p>let $R=1$, then one triangle area is $1\cdot 1\cdot \dfrac{\sin\... | jdc | 7,112 | <p>Considering the bottom entry of your "rot," you want its dot product with each column to be zero. That can only happen if the entries are all the same, so the bottom row of rot is $\pm\big(\frac 1 2\ \frac 1 2\ \frac 1 2\ \frac 1 2\big)$. </p>
<p>Once you know that, you just need to find three more orthogonal unit ... |
5,563 | <p>This question applies to any package, but I encountered this problem while working with graphs.
There are symbols in the <code>Combinatorica</code> package (such as <code>Graph</code>, <code>IncidenceMatrix</code>, <code>EdgeStyle</code>, and others) that have the same name as analogous symbols in <code>System</code... | rm -rf | 5 | <p>Shadowing occurs only when there are two functions with the same name that are in <code>$ContextPath</code>. So right after you do <code><<Combinatorica`</code>, do the following:</p>
<pre><code>$ContextPath = Rest@$ContextPath;
</code></pre>
<p>What this does is that it removes <code>Combinatorica</code> (w... |
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