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 |
|---|---|---|---|---|
2,962,377 | <p>Rewrite <span class="math-container">$f(x,y) = 1-x^2y^2$</span> as a product <span class="math-container">$g(x) \cdot h(y)$</span> (both arbitrary functions)</p>
<p>To make more clear what I'm talking about I will give a example.</p>
<p>Rewrite <span class="math-container">$f(x,y) = 1+x-y-xy$</span> as <span class... | Jonathan | 606,065 | <p>Without loss of generality let <span class="math-container">$f(x) = ax^2 + b$</span> and <span class="math-container">$g(y) = cy^2 + d$</span>. Now
<span class="math-container">$$
1-x^2y^2 = f(x)g(y) = acx^2y^2 +adx^2 = cby^2 +bd
$$</span>
By comparing terms we obtain:
<span class="math-container">\begin{align}
ad&a... |
2,747,074 | <p>I've seen multiple ways on how to solve it online (and most likely the majority are wrong), but I don't know how to solve this fully so that I could get the full grade during my exam.</p>
<p>The question is as follows:</p>
<blockquote>
<p>Prove that if $\sum a_n$ is convergent with $a_n > 0$ for all $n$,
th... | J.G. | 56,861 | <p>For any $\epsilon >0$, there exists an $N$ such that all $n>N$ satisfy $|a_n|<\epsilon$. Hence $|a_n|^2<\epsilon |a_n|$.</p>
|
2,075,712 | <p>I want to show that if $A=[0,1)$ then its interior is $(0,1)$. I know that $int(A) \subset A$, and that $\forall x \in int(A)$ $\exists R>0 $ such that $B(x,R) \subset A$. Thus immediately we see that $0 \notin int(A)$ because $\not \exists R>0$ such that $B(0,R)\subset A$.</p>
<p>What I struggle to do is to ... | Fawad | 369,983 | <p>We have
\begin{align}
(r_1-r)(r_2-r)(r_3-r)&= \left(\frac{\Delta}{s-a} - \frac{\Delta}{s}\right)\left(\frac{\Delta}{s-b} - \frac{\Delta}{s}\right)\left(\frac{\Delta}{s-c}-\frac{\Delta}{s}\right)\\
&={\Delta}^3 \left(\frac{ abc}{s^3(s-a)(s-b)(s-c)}\right)\\
&=\frac{abc{\Delta}^2}{s^2\Delta}\\
&=4R{\l... |
3,996,790 | <p>I just realized that there may be a case where L'Hopital's rule fails, specifically</p>
<p><span class="math-container">$$ \lim_{x \to \infty} \frac{e^x}{e^x}$$</span></p>
<p>which evaluates to an indeterminate form, specifically <span class="math-container">$\frac{\infty}{\infty}$</span>. Sure, we can cancel the <s... | Community | -1 | <p>No. L'Hopital does not fail.</p>
<p>The L'Hopital rule says that (under some assumptions) if <span class="math-container">$\lim_{x\to\infty}\frac{f'(x)}{g'(x)}=L$</span> then
<span class="math-container">$$
\lim_{x\to\infty}\frac{f(x)}{g(x)}=L
$$</span></p>
<p>In your example,
<span class="math-container">$$
\lim_{x... |
3,996,790 | <p>I just realized that there may be a case where L'Hopital's rule fails, specifically</p>
<p><span class="math-container">$$ \lim_{x \to \infty} \frac{e^x}{e^x}$$</span></p>
<p>which evaluates to an indeterminate form, specifically <span class="math-container">$\frac{\infty}{\infty}$</span>. Sure, we can cancel the <s... | J.G. | 56,861 | <p>The rule lets us write <span class="math-container">$\lim_{x\to a}\frac{f}{g}=\lim_{x\to a}\frac{hf}{hg}=\lim_{x\to a}\frac{(hf)^\prime}{(hg)^\prime}$</span>, so sometimes you need a non-constant <span class="math-container">$h$</span> to make progress.</p>
<p>With examples of the form <span class="math-container">$... |
3,941,728 | <p>I know that every function from the empty set to any other set is the empty function.</p>
<p>I also know that there is no function to the empty set from any other set.</p>
<p>Now, what if both the domain and codomain are empty? Would such a function exist?</p>
| NirF | 782,124 | <p>You can take this approach: <br>
<span class="math-container">$$A_n=\begin{bmatrix}
0&1&2& .&.&. &n-1\\
1&0&1&2& .&.&n-2 \\
2&1&0&1&.&.&. \\
.&.&.&.&.&.&. \\
.&.&.&.&.&.&2 \\
.&.&.&.&am... |
1,013,776 | <p>I am looking at the following <strong>Theorem</strong>:</p>
<p>Let $\phi$ a type. We suppose that there is a set $Y$, such that $\forall x(\phi(x) \to x \in Y)$. Then, there is the set $\{ x: \phi(x) \}$.</p>
<p>and I try to understand its proof.</p>
<p>From the axiom shema of specification, there is the set $V=\... | ajotatxe | 132,456 | <p>$\{x:\phi(x)\}=V$, and you have proved that $V$ is a set.</p>
|
3,061,277 | <p>This concept I have asked a few people, but none of them are able to help me understand, so hope that there's a hero can save me from this problem!!!</p>
<p>My question occurs during substitution process, for example, sometimes we let <span class="math-container">$x = π - u$</span>. Then after some manipulation of... | Aniruddha Deshmukh | 438,367 | <p>This is called, not in the most humble manner, the "abuse" of notations. Let us go step by step in understanding what happened.</p>
<p>First, for the integral <span class="math-container">$\int\limits_{0}^{\pi} \dfrac{x \sin x}{1 + \cos^2 x} \ dx$</span>, we put <span class="math-container">$x = \pi - u$</span>, wh... |
3,831,310 | <p>I try to integrate</p>
<p><span class="math-container">$$\int \frac {dv}{\frac {-c}{m}v^2 - g \sin \theta}$$</span></p>
<p>I did substituted <span class="math-container">$u = \frac{c}{m}$</span> and <span class="math-container">$w = g \sin \theta$</span> to get</p>
<p><span class="math-container">$$-\int \frac {dv}{... | Alessio K | 702,692 | <p>You have <span class="math-container">$$-\int \frac{dv}{w+uv^2}=-\frac{1}{w}\int\frac{dv}{1+(v\sqrt{\frac{u}{w}})^2}$$</span></p>
<p>Now consider the substitution <span class="math-container">$v\sqrt{\frac{u}{w}}=\tan(x)$</span>.</p>
|
1,915,455 | <p>First of all they are all even... So it is at least $2$. But is it bigger than that? How do I find out?</p>
| SchrodingersCat | 278,967 | <p>Consider that all possible divisors of $n$ can be created by choosing from $p_1,p_2, \ldots, p_k$ in appropriate numbers.</p>
<p>So, for creating a particular divisor, we can choose $1$ $p_1$ or $2$ $p_1$'s or $3$ $p_1$'s and so on till a choice of all $a$ $p_1$'s. Then again we have the choice of not choosing any... |
3,696,265 | <p>This is from Vakil's FOAG: exercise 2.5 C, part b. I understand how objects in the extension sheaf from a sheaf on a base <span class="math-container">$\mathcal B$</span> of a topology are created, but I am having trouble understanding how to produce a morphism of sheaves given a morphism of sheaves on a base. </p>
... | Felix Marin | 85,343 | <p><span class="math-container">$\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\exp... |
77,642 | <p>$$\frac{1}{\sin(z)} = \cot (z) + \tan (\tfrac{z}{2})$$</p>
<p>I did this: </p>
<p><strong>First attempt</strong>: $$\displaystyle{\frac{1}{\sin (z)} = \frac{\cos (z)}{\sin (z)} + \frac{\sin (\frac{z}{2})}{ \cos (\frac{z}{2})} = \frac{\cos (z) }{\sin (z)} + \frac{2\sin(\frac{z}{4})\cos(\frac{z}{4})}{\cos^{2}(\frac{... | Sasha | 11,069 | <p>Let $w = \frac{z}{2}$. Then
$$
\cot(2w) + \tan(w) = \frac{\cos^2(w)-\sin^2(w)}{2 \sin(w) \cos(w)} + \frac{\sin(w)}{\cos(w)} = \frac{1}{\cos(w)} \left( \frac{\cos^2(w)-\sin^2(w) + 2 \sin^2(w)}{2 \sin(w)} \right)
$$
The numerator becomes 1, and we arrive at the result $\frac{1}{2 \sin(w) \cos(w)} = \frac{1}{\cos(2w)... |
77,642 | <p>$$\frac{1}{\sin(z)} = \cot (z) + \tan (\tfrac{z}{2})$$</p>
<p>I did this: </p>
<p><strong>First attempt</strong>: $$\displaystyle{\frac{1}{\sin (z)} = \frac{\cos (z)}{\sin (z)} + \frac{\sin (\frac{z}{2})}{ \cos (\frac{z}{2})} = \frac{\cos (z) }{\sin (z)} + \frac{2\sin(\frac{z}{4})\cos(\frac{z}{4})}{\cos^{2}(\frac{... | N. S. | 9,176 | <p>$$ \frac{\cos (z)}{\sin (z)} + \frac{\sin (\frac{z}{2})}{ \cos (\frac{z}{2})} =\frac{\cos (z)\cos (\frac{z}{2})+ \sin(z)\sin (\frac{z}{2}) }{\sin (z)\cos (\frac{z}{2})} =\frac{\cos (z-\frac{z}{2})}{\sin (z)\cos (\frac{z}{2})}$$</p>
|
255,230 | <p>A <em>complete linear hypergraph</em> is a <a href="https://en.wikipedia.org/wiki/Hypergraph" rel="nofollow noreferrer">hypergraph</a> $H=(V,E)$ such that </p>
<ol>
<li>$|e|\geq 2$ for all $e\in E$,</li>
<li>$|e_1\cap e_2|=1$ for all $e_1, e_2\in E$ with $e_1\neq e_2$, and</li>
<li>for all $v\in V$ we have $|\{e\in... | domotorp | 955 | <p>It is about $\sqrt n$.
If each edge has size at most $\sqrt n$, then you need at least $2n/\sqrt n=2\sqrt n$ edges to cover everything twice.
If there's an edge of size at least $\sqrt n$, then you need at least these many other edges to cover each of its points at least twice.
With some more involved argument of th... |
2,511,061 | <p>I will prove 0=1.
we know that, From Factorial definition
zero factorial is equal to one and
one factorial is equal to one.
so,0!=1!.
factorial get cancelled both sides,
we get 0=1.
Is this right..?</p>
| marty cohen | 13,079 | <p>Factorial ("!") is not an expression - it is an operator.</p>
<p>For example,
$\sin(0) = \sin(\pi) = 0$
but you can't cancel "sin"
to get
$0 = \pi$.</p>
|
638,906 | <p>Find the coefficient of $x$ in the expansion of $$\left(1-2x^3+3x^5\right)\left(1+\frac{1}{x}\right)^5.$$ Answer is $154$, but how?</p>
| lab bhattacharjee | 33,337 | <p>$$(1-2x^3+3x^5)\left(1+\frac1x\right)^5=\frac{(1-2x^3+3x^5)(1+x)^5}{x^5}$$</p>
<p>We need the coefficient of $x^6$ in $(1-2x^3+3x^5)(1+x)^5$ which is $-2\cdot\binom53+3\binom51$</p>
|
2,625,719 | <p>I'm getting an equation $$(\ln y - x)\frac{dy}{dx} - y\ln y = 0$$</p>
<p>Which I try to factorize and bring over to get:
$$\frac{dy}{dx} = \frac{y\ln y}{(\ln y - x)}$$</p>
<p>But this cannot be factorized further. I am intending to use either the the substitution $u=y/x$ or finding the integrating factor, but this... | David Quinn | 187,299 | <p>Hint...substitute $u=\ln y$ and you will arrive at a linear differential equation requiring an integrating factor.</p>
|
1,910,030 | <p>Let $V$ be a vector space and $U,W$ be its subspaces. Prove that if the union $U\cup W$ is a subspace of $V$, then $W \subseteq U$ or $U \subseteq W$.</p>
<p>I'm not sure where to begin at all really.</p>
| true blue anil | 22,388 | <ul>
<li><p>Re why multiply, you better go through an elementary presentation on <a href="https://people.richland.edu/james/lecture/m170/ch05-rul.html" rel="nofollow">probability rules.</a></p></li>
<li><p>Regarding $44$, do you understand if it is written as $\binom{11}1\binom41$ ?</p></li>
</ul>
<hr>
<p><em>Added e... |
86,277 | <p>I have a question: does the Heine-Borel theorem hold for the space $\mathbb{R}^\omega$ (where $\mathbb{R}^\omega$ is the space of countable sequences of real numbers with the product topology). That is, prove that a subspace of $\mathbb{R}^\omega$ is compact if and only if it is the product of closed and bounded su... | Cheerful Parsnip | 2,941 | <p>The statement that a subspace of $\mathbb R^\omega$ is compact if and only if it is the product of closed and bounded subspaces of $\mathbb R$ is false even for $\mathbb R^2$. Take the "plus sign" subset $(\{0\}\times [-1,1])\cup ([-1,1]\times\{0\})$. It is compact but not a product of subsets of $\mathbb R$. This... |
1,686,455 | <p>For $a_1, ... , a_2 > 0$, I need to prove that
$( \sum_{i=1}^{n} a_i)( \sum_{i=1}^{n} \frac{1}{a_i})\geq n^2 $.</p>
<p>When do we have inequality?</p>
<p>Should I do it by induction? any hints</p>
| Paolo Leonetti | 45,736 | <p>Equivalently, you have to prove that
$$
\sum_{1\le i ,j\le n: i\neq j}\frac{a_i}{a_j} \ge n^2-n.
$$
Now, $x+\frac{1}{x}\ge 2$ by am-gm, therefore
$$
\sum_{1\le i <j\le n}\frac{a_i}{a_j}+\frac{a_j}{a_i}\ge 2\binom{n}{2}=n^2-n.
$$</p>
|
1,686,455 | <p>For $a_1, ... , a_2 > 0$, I need to prove that
$( \sum_{i=1}^{n} a_i)( \sum_{i=1}^{n} \frac{1}{a_i})\geq n^2 $.</p>
<p>When do we have inequality?</p>
<p>Should I do it by induction? any hints</p>
| DeepSea | 101,504 | <p>Apply AM-GM inequality twice: $$LHS \geq n\sqrt[n]{a_1a_2\cdots a_n}\cdot n\sqrt[n]{\dfrac{1}{a_1a_2\cdots a_n}}=n^2=RHS$$, and equality occurs when $a_1=a_2=\cdots = a_n$.</p>
|
2,344,931 | <p>It is given that $a,b$ are roots of $3x^2+2x+1$ then find the value of:
$$\left(\dfrac{1-a}{1+a}\right)^3+\left(\dfrac{1-b}{1+b}\right)^3$$</p>
<p>I thought to proceed in this manner:</p>
<p>We know $a+b=\frac{-2}{3}$ and $ab=\frac{1}{3}$. Using this I tried to convert everything to <em>sum and product of roots</e... | Khosrotash | 104,171 | <p><span class="math-container">$$3a^2+2a+1=0 \to 3a^2+3a+1=a\\3b^2+2b+1=0\to 3b^2+3b+1=b\\a-1=3a(a+1)\\b-1=3b(b+1)$$</span>
so <span class="math-container">$$\left(\dfrac{1-a}{1+a}\right)^3+\left(\dfrac{1-b}{1+b}\right)^3=\\
\left(\dfrac{-3a(a+1)}{1+a}\right)^3+\left(\dfrac{-3b(b+1)}{1+b}\right)^3\\=-27(a^3+b^3)=-27(s... |
2,344,931 | <p>It is given that $a,b$ are roots of $3x^2+2x+1$ then find the value of:
$$\left(\dfrac{1-a}{1+a}\right)^3+\left(\dfrac{1-b}{1+b}\right)^3$$</p>
<p>I thought to proceed in this manner:</p>
<p>We know $a+b=\frac{-2}{3}$ and $ab=\frac{1}{3}$. Using this I tried to convert everything to <em>sum and product of roots</e... | Raffaele | 83,382 | <p>Plug $x=\frac{1-y}{1+y}$ in the given equation. We get:
$$\frac{3 (1-y)^2}{(y+1)^2}+\frac{2 (1-y)}{y+1}+1=0$$
Expanding and collecting, we have:
$$y^2-2 y+3=0$$
whose solutions are $$y_1=\frac{1-a}{1+a};\;y_2=\frac{1-b}{1+b}$$
We also know that sum of roots is $s=y_1+y_2=2$ and product is $p=y_1y_2=3$.</p>
<p>The s... |
2,130,823 | <blockquote>
<p>Is it possible for some integer $n>1$ that $2^n-1\mid 3^n-1$ ?</p>
</blockquote>
<p>I have tried many things, but nothing worked.</p>
| WafflesTasty | 70,877 | <p>I was looking for this as well, and eventually figured it out myself. So here's my solution for future reference. The short answer is, <span class="math-container">$2^n - 1$</span> never divides <span class="math-container">$3^n - 1$</span>. Here's the proof, making use of the Jacobi symbol.</p>
<p>Assume <span cla... |
197,421 | <p>Consider the quotient group $\mathbb{Q}/\mathbb{Z}$ of the additive group of rational numbers. Then how to find the order of the element $2/3 + \mathbb{Z} $ in $\mathbb{Q}/\mathbb{Z}$.</p>
| Makoto Kato | 28,422 | <p>Let $a = 2/3 + \mathbb{Z}$.
Since $3a = 0$ and $a \neq 0$, the order of $a$ is $3$.</p>
|
197,421 | <p>Consider the quotient group $\mathbb{Q}/\mathbb{Z}$ of the additive group of rational numbers. Then how to find the order of the element $2/3 + \mathbb{Z} $ in $\mathbb{Q}/\mathbb{Z}$.</p>
| Bill | 418,485 | <p>The order of a coset $a + \mathbb Z$ and $a$ has given by is the smallest positive integer $b$ such that $b(a+\mathbb Z)= \mathbb Z$.</p>
|
4,293,199 | <p>let me preface by saying this <em>is</em> from a homework question, but the question is not to plot the decision boundary, just to train the model and do some predictions. I have already done that and my predictions <em>seem</em> to be correct.</p>
<p>But I would like to verify my results by plotting the decision bo... | lab bhattacharjee | 33,337 | <p>Clearly the roots of <span class="math-container">$$4c^3-3c-\cos3x=0$$</span> are <span class="math-container">$c_r\cos(x+2r\pi/3), r=0,1,2$</span></p>
<p>Let <span class="math-container">$p=c_0,c_1=q,c_2=r$</span></p>
<p><span class="math-container">$p+q+r=0\implies p^3+q^3+r^3=3pqr$</span></p>
<p><span class="math... |
4,435,032 | <p>I have equations of two conic sections in general form. Is it possible to find minimal distance between them (if they are not intercross)?</p>
<p>I need it to calculate is two spacecrafts on two orbits (2d case) can collide or not. If minimal distance bigger than sum of radiuses of bounding circles I don't need to c... | Andreas Cap | 202,204 | <p>You are right, then tensor fields are indeed parallel and this is independent of torsion-freeness.
In fact, if you take any <span class="math-container">$G$</span>-invariant tensor <span class="math-container">$T_0\in\otimes^k\mathbb R^{n*}\otimes\otimes^\ell\mathbb R^n$</span>, this gives rise to a <span class="mat... |
1,132,187 | <p>$x^2 = y^2 + xy + 5$, where $x$ and $y$ are natural numbers.</p>
<p>Here is what I have so far:</p>
<ol>
<li><p>$x \neq y$ (from the equation).</p></li>
<li><p>$x$ is always odd (using the equation and assuming $2$ cases - $y$ is odd or $y$ is even).</p></li>
<li><p>Solving for the equation as a quadratic in $y$, ... | Pp.. | 203,995 | <p>Multiply by $4$.
$$4x^2=4y^2+4xy+20$$</p>
<p>$$5x^2=(2y+x)^2+20$$</p>
<p>So, try to solve $$5x^2=z^2+20$$</p>
<p>$z$ must be multiple of $5$. So put $z=5a$ to get $$x^2-5a^2=4$$</p>
<p>This is a <a href="http://en.wikipedia.org/wiki/Pell%27s_equation" rel="nofollow">Pell's equation</a> with a solution $x=3, a=1... |
2,391,505 | <p>Most ODE textbooks provide the following steps to the solution of a separable differential equation (here the exponential equation is used as an example):</p>
<p>$$\frac{dN}{dt}=-\lambda N(t) \Rightarrow \frac{dN}{N(t)}=-\lambda dt\Rightarrow \int\frac{1}{N}dN=-\lambda\int dt \Rightarrow ln\mid N\mid = -\lambda t+C... | Gribouillis | 398,505 | <p>I would say that</p>
<p>$$\frac{d(N(t)e^{\lambda t})}{dt} = \frac{d N(t)}{dt} e^{\lambda t} + N(t)\frac{d(e^{\lambda t})}{dt} = -\lambda N(t) e^{\lambda t}+ \lambda N(t) e^{\lambda t}=0$$
hence $N(t) e^{\lambda t}$ is a constant $A$ and $N(t) = A e^{-\lambda t}$.</p>
|
2,044,451 | <p>I'm trying to prove the following inequality:
$$- \ln (x) \leq (x)^{-\frac{1}{e}} $$</p>
<p>over $[0, 1]$. I'm not sure how to move forward. I know that the equality is at $x = e^{-e}$. Any help is greatly appreciated.</p>
| xpaul | 66,420 | <p>Let $f(x)=-\ln x-x^{-\frac1e}$. Then
$$ f'(x)=\frac1{ex}(-e+x^{-\frac1e})$$
and hence $f(x)$ is decreasing if $x>e^{-e}$ and increasing if $0<x<e^{-e}$. So if $x>e^{-e}$, then $f(x)<f(e^\frac1e)=0$. Thus the solution of $\ln x<x^{-\frac1e}$ is $x>e^{-e}$.</p>
|
4,219,689 | <p>I was trying to derive the perimeter of a circle using vector algebra and came across this.....</p>
<p>Taking a circle of radius <span class="math-container">$r$</span> with center at the origin, the position vector of a particular point on the circle making an angle <span class="math-container">$t$</span> with the ... | Narasimham | 95,860 | <p>Due to vector symmetry for any regular n- sided polygon average of sum of projections is from the center of the polygon:</p>
<p><span class="math-container">$$ \bar p =\dfrac{\Sigma p}{n} =\dfrac{18}{6} =3.$$</span></p>
<p>Proof : Vector sum of sides =0, the above is a general relation.</p>
|
375,910 | <p>Okay so basically I want to know if you can solve this log equation without the use of u substitution: </p>
<p>$${\log_4{\log_3{x}}} = 1$$</p>
<p>I believe that u substitution is the only way to solve this problem, but please prove me wrong if theres another way to do so.</p>
| André Nicolas | 6,312 | <p>Since I do not know what $u$-substitution means, it will have to be done another way. To say that $\log_4$ of something is $1$ means that the something is $4^1$. So $\log_3 x=4$. And this means that $x=3^4$.</p>
|
1,804,333 | <blockquote>
<p>How many divisors does $25^2+98^2$ have?</p>
</blockquote>
<hr>
<p>My Attempt:</p>
<p>Calculator is not allowed but using calculator I found $193\times53$ that means $8$ divisors and that $4$ of them are positive.</p>
| barak manos | 131,263 | <p>HINT:</p>
<p>$a^2+b^2=(a+b)^2-2ab$</p>
<hr>
<p>Hence:</p>
<p>$25^2+98^2=$</p>
<p>$(25+98)^2-2\cdot25\cdot98=$</p>
<p>$123^2-4900=$</p>
<p>$123^2-70^2=$</p>
<p>$(123-70)\cdot(127+70)=$</p>
<p>$53\cdot193$</p>
|
384,471 | <p>I'm attempting to evaluate the limit</p>
<p>$\lim_{x\rightarrow\infty}\frac{1}{\sqrt{x^{2}-4x+1}-x+2}$</p>
<p>I got it reduced to the following</p>
<p>$\lim_{x\rightarrow\infty}\frac{\sqrt{\frac{1}{\left(x-2\right)^{2}}-\frac{3}{\left(x-2\right)^{4}}}+1}{1-\frac{3}{\left(x-2\right)^{2}}-1}$</p>
<p>But putting in... | vadim123 | 73,324 | <p>Hint: multiply top and bottom by $\sqrt{x^2-4x+1}+x-2$.</p>
<p>Additional hint upon request: $\lim_{x\rightarrow \infty}\sqrt{x^2-4x+1}=\infty$ and $\lim_{x\rightarrow \infty} x-2=\infty$.</p>
|
384,471 | <p>I'm attempting to evaluate the limit</p>
<p>$\lim_{x\rightarrow\infty}\frac{1}{\sqrt{x^{2}-4x+1}-x+2}$</p>
<p>I got it reduced to the following</p>
<p>$\lim_{x\rightarrow\infty}\frac{\sqrt{\frac{1}{\left(x-2\right)^{2}}-\frac{3}{\left(x-2\right)^{4}}}+1}{1-\frac{3}{\left(x-2\right)^{2}}-1}$</p>
<p>But putting in... | DonAntonio | 31,254 | <p>Hint: multiply numerator and denominator by the denominator's conjugate:</p>
<p>$$\lim_{x\rightarrow\infty}\frac{1}{\sqrt{x^{2}-4x+1}-x+2}=\frac{\sqrt{x^2-4x+1}+x-2}{x^2-4x+1-x^2+4x-4}=\;\ldots$$</p>
<p><strong>Added under request :</strong> </p>
<p>$$\left(\sqrt{x^2-4x+1}+x-2\right)\frac{\frac1x}{\frac1x}=\frac{... |
6,661 | <p>Is there a simple explanation of what the Laplace transformations do exactly and how they work? Reading my math book has left me in a foggy haze of proofs that I don't completely understand. I'm looking for an explanation in layman's terms so that I understand what it is doing as I make these seemingly magical tran... | JP McCarthy | 19,352 | <p>I have been teaching the Laplace Transform to a night degree class (mature) of civil engineers. They are good students but not great mathematicians. They couldn't follow the method of how we use the Laplace Transform to solve differential equations until I told them this story:</p>
<blockquote>
<p>Suppose that yo... |
2,852,832 | <blockquote>
<p>How can we prove the following?</p>
<p>$-1+\frac1{12}{\pi }^{2}-\frac12\sum\limits_{n=2}^{\infty }\Gamma \left( n+1 \right) \sum\limits_{k=0}^{n+2}
\,{\frac {\zeta \left( k \right) \left( -
1 \right) ^{-2+k} \left( {2}^{2-k}-2 \right) }{\Gamma \left( -1+k
\right) \left( n+1-k \right) !}} =... | skbmoore | 321,120 | <p>In the following assume $|z|<1$ to begin with and I'll let $z \to 1$ to get the OP's left hand side formula:
$$ S(z):=\sum_{n=1}^\infty z^n\,n \sum_{j=2}^\infty\, \frac{(-1)^{j-1}}{j^2}\big(1-\frac{1}{j}\big)^{n-1} .$$
$S(z)$ is solvable in closed-form in the following manner: Interchange summations and to elimi... |
2,852,832 | <blockquote>
<p>How can we prove the following?</p>
<p>$-1+\frac1{12}{\pi }^{2}-\frac12\sum\limits_{n=2}^{\infty }\Gamma \left( n+1 \right) \sum\limits_{k=0}^{n+2}
\,{\frac {\zeta \left( k \right) \left( -
1 \right) ^{-2+k} \left( {2}^{2-k}-2 \right) }{\Gamma \left( -1+k
\right) \left( n+1-k \right) !}} =... | Sangchul Lee | 9,340 | <p>Write $\chi(n) = n \sum _{j=2}^{\infty} \frac{(-1)^{j-1}}{j^2} \left( 1-\frac{1}{j} \right)^{n-1}$ using OP's notation. Its partial sum up to the $(N+1)$-th term can be simplified by interchanging the order of summation:</p>
<p>\begin{align*}
\sum_{n=1}^{N+1} \chi(n)
&= \sum _{j=2}^{\infty} \frac{(-1)^{j-1}}{j^... |
142,734 | <p>What are some books that discuss elementary mathematical topics ('school mathematics'), like arithmetic, basic non-abstract algebra, plane & solid geometry, trigonometry, etc, in an insightful way? I'm thinking of books like Klein's <em>Elementary Mathematics from an Advanced Standpoint</em> and the books of the... | Terry Tao | 766 | <p>If first-order logic counts as "elementary mathematics", then I would like to suggest (the relevant chapters of) "<a href="http://en.wikipedia.org/wiki/G%C3%B6del,_Escher,_Bach">Godel, Escher, Bach</a>", by Douglas Hofstadter. (As an aside: Hofstadter's puzzle of encoding "n is a power of 10" as a predicate in Pean... |
142,734 | <p>What are some books that discuss elementary mathematical topics ('school mathematics'), like arithmetic, basic non-abstract algebra, plane & solid geometry, trigonometry, etc, in an insightful way? I'm thinking of books like Klein's <em>Elementary Mathematics from an Advanced Standpoint</em> and the books of the... | Renaud Dreyer | 35,306 | <p><a href="http://en.wikipedia.org/wiki/Mathematics_Made_Difficult" rel="noreferrer"><em>Mathematics Made Difficult</em></a> by Carl E. Linderholm.</p>
|
142,734 | <p>What are some books that discuss elementary mathematical topics ('school mathematics'), like arithmetic, basic non-abstract algebra, plane & solid geometry, trigonometry, etc, in an insightful way? I'm thinking of books like Klein's <em>Elementary Mathematics from an Advanced Standpoint</em> and the books of the... | UwF | 36,090 | <p>How about Lawvere&Schanuel's "Conceptual Mathematics: A First Introduction to Categories"? See, e.g., <a href="http://books.google.fr/books?id=h0zOGPlFmcQC&lpg=PP1&dq=lawvere&pg=PP1#v=onepage&q=lawvere&f=false" rel="nofollow">http://books.google.fr/books?id=h0zOGPlFmcQC&lpg=PP1&dq=law... |
142,734 | <p>What are some books that discuss elementary mathematical topics ('school mathematics'), like arithmetic, basic non-abstract algebra, plane & solid geometry, trigonometry, etc, in an insightful way? I'm thinking of books like Klein's <em>Elementary Mathematics from an Advanced Standpoint</em> and the books of the... | Michael Greinecker | 35,357 | <p>While it contains much beyond school mathematics, a lot of school mathematics is treated in a beautiful way in <a href="https://en.wikipedia.org/wiki/Mathematics,_Form_and_Function" rel="nofollow noreferrer">Mathematics, Form and Function</a> by Saunders Mac Lane.</p>
|
142,734 | <p>What are some books that discuss elementary mathematical topics ('school mathematics'), like arithmetic, basic non-abstract algebra, plane & solid geometry, trigonometry, etc, in an insightful way? I'm thinking of books like Klein's <em>Elementary Mathematics from an Advanced Standpoint</em> and the books of the... | Martin Peters | 60,435 | <p>Two great ones are:</p>
<ul>
<li>Fuchs, Tabachnikov: <em>Mathematical Omnibus</em> and </li>
<li>Arnold: <em>Lectures and Problems: A Gift to Young Mathematicians</em>.</li>
</ul>
|
375,016 | <p>I just started reading the ABC of category theory using the appendix of a text, the first chapter of a text that I have never read, and above all (I found out now that they handle well the theory) the wikipedia pages. I want to know only this: in all three sources are given the definition of the category, and in all... | Martin Brandenburg | 1,650 | <p>Both category theory and set theory can be seen as formal theories in the general sense of mathematical logic. It is well-known that category theory can be developed inside set theory (perhaps including Grothendieck's axiom of universes). But Lawvere has also shown the converse, via the <a href="http://ncatlab.org/n... |
3,247,982 | <p>If <span class="math-container">$f:X \to Y$</span> is a function and <span class="math-container">$U$</span> and <span class="math-container">$V$</span> are subsets of <span class="math-container">$X$</span>, then <span class="math-container">$f(U \cap V) = f(U) \cap f(V)$</span>.</p>
<p>I am a little lost on this ... | José Carlos Santos | 446,262 | <p>You can't prove it, since it is false. Take a set <span class="math-container">$X$</span> with more than one point, take <span class="math-container">$x,x'\in X$</span> with <span class="math-container">$x\neq x'$</span> and let <span class="math-container">$f$</span> be a constant function. Then<span class="math-co... |
1,275,810 | <p>Let <span class="math-container">$ c_0 = \{ x = (x_n)_{n \in \mathbb N} \in l^\infty : \lim_{n \to \infty} x_n = 0\}$</span>. Show that <span class="math-container">$c_0$</span> is a Banach space with the norm <span class="math-container">$\rVert \cdot \lVert_\infty$</span></p>
<p>I am capable of showing the space ... | Calvin Khor | 80,734 | <p>We note that
$$ |y_k| \leq |y_k - x^n_k| + |x^n_k| = \lim_{m→∞}|x^m_k - x^n_k| + |x^n_k| \leq\lim_{m→∞} ‖x^m - x^n‖ + |x^n_k|$$
which holds for arbitrary $n$. (the superscript is which sequence, the subscript is the index of the sequence) As $x^n∈ c_0$, choose $K_n$ large such that $|x^n_k |<ε$ for $k>K_n$.</p... |
696,145 | <blockquote>
<p>For each positive integer <span class="math-container">$n$</span>, define <span class="math-container">$f(n)$</span> such that <span class="math-container">$f(n+1) > f(n)$</span> and <span class="math-container">$f(f(n))=3n$</span>. What is the value of <span class="math-container">$f(10)?$</span><... | Zoltan Zimboras | 7,317 | <p>You already deduced that $f(1) = 2$. From the $f(f(n))=3n$ condition, we have that $f(f(1))=f(2)=3$, hence $f(2)=3$. Similarly, we have that $f(3)=f(f(2))=3\cdot 2=6$, and $f(6)=f(f(3))=3\cdot 3=9$, and $f(9)=f(f(6))=3 \cdot 6=18$. So until now we have that </p>
<p>$$f(1)=2\, , \; f(2)=3 \, , \; f(3)=6 \, ,\; f(6)... |
2,960,614 | <p>I am going through Tenenbaum and Pollard's book on differential equations and they define the differential <span class="math-container">$dy$</span> of a function <span class="math-container">$y = f(x)$</span> to be the function <span class="math-container">$$
(dy)(x,\Delta x) = f'(x) \cdot (d\hat{x})(x, \Delta x)
$$... | Santana Afton | 274,352 | <p>For <strong>every</strong> pair of disjoint sets <span class="math-container">$A$</span> and <span class="math-container">$B$</span>, assume the following holds:</p>
<blockquote>
<p><span class="math-container">$m^*(A\cup B) = m^*(A) + m^*(B).$</span></p>
</blockquote>
<p>Now, let <span class="math-container">$A... |
1,176,016 | <p>How do you show that if you have sets $B_1, B_2, \cdots ,B_n$ and a set $C$, then
$$(B_1\cap B_2, \cap \cdots B_n)\cup C= (B_1\cup C)\cap(B_2\cup C) \cap \cdots
\cap (B_n\cup C)\,?$$</p>
<p>Thanks</p>
| MonkeysUncle | 217,283 | <p>For the equations you write to be true, $x$ has to take on some discrete value. The equality is not true in general. For the equals sign to hold, both sides of the equation need to be constants because it's only an equality when x takes on specific values. This means you can't just take the derivative or integral of... |
4,335,896 | <blockquote>
<p><span class="math-container">$$\frac{1}{2\cdot4}+\frac{1\cdot3}{2\cdot4\cdot6}+\frac{1\cdot3\cdot5}{2\cdot4\cdot6\cdot8}+\frac{1\cdot3\cdot5\cdot7}{2\cdot4\cdot6\cdot8\cdot10}+\cdots$$</span>
is equal to?</p>
</blockquote>
<p><strong>My approach:</strong></p>
<p>We can see that the <span class="math-con... | Mr.Gandalf Sauron | 683,801 | <p>If you look at the Binomial expansion of</p>
<p><span class="math-container">$$(1-x)^{-\frac{1}{2}}$$</span> you get :-</p>
<p><span class="math-container">$$\sum_{r=0}^{\infty}\frac{\binom{2r}{r}x^{r}}{4^{r}}$$</span></p>
<p>So <span class="math-container">$$\int_{0}^{1}(1-x)^{-\frac{1}{2}}dx=\sum_{r=0}^{\infty}\fr... |
3,064,943 | <p>Consider the senquence of iid r.v. <span class="math-container">$(Y_k)_{k\geq1}$</span> such that <span class="math-container">$\mathbb{P}(Y_k=1)=\mathbb{P}(Y_k=-1)=\frac{1}{2}$</span> and then consider the process <span class="math-container">$X=(X_k)_{n\geq1}$</span> such that <span class="math-container">$X_n=\su... | Aphelli | 556,825 | <p>For a « martingale » proof, the convergence theorem for martingales states that if <span class="math-container">$(X_n)$</span> is a martingale wrt some filtration, and <span class="math-container">$\mathbb{E}[|X_n|]$</span> is bounded, then <span class="math-container">$X_n$</span> converges pointwise. </p>
<p>Now ... |
3,855,521 | <p>Suppose <span class="math-container">$f:A\bigoplus M \to A\bigoplus N$</span> and <span class="math-container">$g: A \to A$</span>, where <span class="math-container">$g = \pi_A\circ f\circ l_A$</span>, are isomorphisms of <span class="math-container">$R$</span>-module. Prove that <span class="math-container">$M\con... | Narasimham | 95,860 | <p>HINT</p>
<p><span class="math-container">$y=\pm 1$</span> is clearly a tendency around <span class="math-container">$x=0$</span> and the</p>
<p><span class="math-container">$y=\log[(4/\pi)^2 x^{m}] $</span> tends to pass through <span class="math-container">$(x=1, x=-1)$</span> as <span class="math-container">$y\... |
606,380 | <blockquote>
<p>If <span class="math-container">$abc=1$</span> and <span class="math-container">$a,b,c$</span> are positive real numbers, prove that <span class="math-container">$${1 \over a+b+1} + {1 \over b+c+1} + {1 \over c+a+1} \le 1\,.$$</span></p>
</blockquote>
<p>The whole problem is in the title. If you wanna h... | Michael Rozenberg | 190,319 | <p>Another way.</p>
<p>After full expanding we need to prove that:
<span class="math-container">$$\sum_{cyc}(a^2b+a^2c)\geq2(a+b+c)=2\sum_{cyc}a^{\frac{5}{3}}b^{\frac{2}{3}}c^{\frac{2}{3}},$$</span> which is true by Muirhead because <span class="math-container">$$(2,1,0)\succ\left(\frac{5}{3},\frac{2}{3},\frac{2}{3}\ri... |
1,678,687 | <p>Find the point $(x_0, y_0)$ on the line $ax + by = c$ that is closest to the origin. </p>
<p>According to this <a href="http://www.intmath.com/plane-analytic-geometry/perpendicular-distance-point-line.php" rel="nofollow noreferrer">source</a>, I thought that $\left( -\frac{ac}{a^{2}+b^{2}}, -\frac{bc}{a^{2}+b^{... | lab bhattacharjee | 33,337 | <p>$$(h-0)^2+\left(\dfrac{c-ah}b\right)^2=\dfrac{(a^2+b^2)h^2-2cah+c^2}{b^2}$$</p>
<p>Now $(a^2+b^2)h^2-2cah=\left(h\sqrt{a^2+b^2}-\dfrac{ca}{\sqrt{a^2+b^2}}\right)^2-\dfrac{(ca)^2}{a^2+b^2}\ge-\dfrac{(ca)^2}{a^2+b^2}$
the minimum value occurs iff
the equality occurs if $$h\sqrt{a^2+b^2}-\dfrac{ca}{\sqrt{a^2+b^2}}=0\i... |
1,678,687 | <p>Find the point $(x_0, y_0)$ on the line $ax + by = c$ that is closest to the origin. </p>
<p>According to this <a href="http://www.intmath.com/plane-analytic-geometry/perpendicular-distance-point-line.php" rel="nofollow noreferrer">source</a>, I thought that $\left( -\frac{ac}{a^{2}+b^{2}}, -\frac{bc}{a^{2}+b^{... | GoodDeeds | 307,825 | <p>$$ax_0+by_0=c$$
Thus,
$$y_0=\frac{c-ax_0}{b}$$
The distance of $(x_0,y_0)$ from $(0,0)$ is, $d=\sqrt{x_0^2+y_0^2}$</p>
<p>Thus,
$$d=\sqrt{x_0^2+\left(\frac{c-ax_0}{b}\right)^2}=\frac{\sqrt{b^2x_0^2+c^2+a^2x_0^2-2acx_0}}{b}$$
To minimize $d$, taking the derivative with respect to $x_0$,
$$\frac{d}{dx_0}(d)=\frac{2(a... |
3,827,422 | <p>Given a finite group <span class="math-container">$G$</span>, how do we know that there exists a map <span class="math-container">$\rho: G \rightarrow GL(V)$</span> such that <span class="math-container">$\rho(g_1\circ g_2) = \rho(g_1).\rho(g_2)$</span> for any <span class="math-container">$g_1, g_2\in G$</span>?</p... | CyclotomicField | 464,974 | <p>Permutation matrices fully capture the essential features of the symmetric groups. So for <span class="math-container">$S_n$</span> I would take all of the <span class="math-container">$n!$</span> permutation matrices of size <span class="math-container">$n \times n$</span> and this allows me to ensure any finite gr... |
904,074 | <p>Is this Convergent or Divergent</p>
<p>$$\int_0^1 \frac{1}{\sin(x)}\mathrm dx $$</p>
<p>So little background to see if I am solid on this topic otherwise correct me please :) </p>
<p>To check for convergence I can look for a "bigger" function and compare if that is convergent, If yes then for sure the one in ques... | Mikasa | 8,581 | <p>You could use the following facts to find proper values of <span class="math-container">$A$</span> as well. Maybe it helps you for other integrands:</p>
<blockquote>
<p>Let <span class="math-container">$\lim_{x\to a^+}~(x-a)^pf(x)=A$</span>. Then:</p>
<ul>
<li><p>If <span class="math-container">$p<1$</span> and <... |
2,845,451 | <p>I was reading a number theory book and it was stated that $d^k+(a-d)^k=a[d^{k-1}-d^{k-2}(a-d)+ . . .+(a-d)^{k-1}]$ for $k$ odd. How did they arrive at this factorization? Is there an easy way to see it?</p>
| Ethan Bolker | 72,858 | <p>Start by understanding how to factor
$$
x^n - y^n .
$$</p>
<p>Presumably you know how to do that when $n=2$. For $n=3$ you can check that
$$
x^3 - y^3 = (x-y)(x^2 + xy + y^2)
$$</p>
<p>Now guess for higher powers.</p>
<p>Then see what happens if $n$ is odd and you replace $y$ by $(-y)$.</p>
|
2,845,451 | <p>I was reading a number theory book and it was stated that $d^k+(a-d)^k=a[d^{k-1}-d^{k-2}(a-d)+ . . .+(a-d)^{k-1}]$ for $k$ odd. How did they arrive at this factorization? Is there an easy way to see it?</p>
| asdf | 436,163 | <p>They used that</p>
<p>$$x^k+y^k=(x+y)(x^{k-1}-x^{k-2}y- \dots +xy^{k-2}+y^{k-1})$$</p>
<p>holds for $k$ - odd and have plugged $x=d$, $y=a-d$</p>
|
3,772,193 | <p>If I put the two vectors 1,0,0 and 0,1,0 next to each other</p>
<p><span class="math-container">$$\begin{pmatrix} 1 & 0 \\ 0 & 1 \\ 0 & 0 \end{pmatrix}$$</span></p>
<p>I can see that they are independent since I cannot write <span class="math-container">$(1,0,0)^t$</span> as <span class="math-container">... | Arctic Char | 629,362 | <p>The correct statement is probably this one: If you have <span class="math-container">$n$</span> column vectors in the <span class="math-container">$n$</span>-dimensional space, so that when put one next to each other there is a row of <span class="math-container">$0$</span>'s in your matrix, then it's linearly depen... |
2,048,962 | <blockquote>
<p><strong>Question:</strong> How would you solve this sinusoidal equation:</p>
<blockquote>
<p>Solve $5\cos(6x)+6=9$. Assume $n$ is an integer and the answers are in degrees.</p>
<ul>
<li><p>$-8.86+n\cdot 60$</p></li>
<li><p>$-3.54+n\cdot 60$</p></li>
<li><p>$3.54+n\cdot 60... | Lalala | 1,022,719 | <p>Using the cosine law you need to set the values of b and c (we are just going to have 1 variable because b and c will be equal, so we will only use b.)(A would be 6x)<br>
<span class="math-container">$$b = (180 - 6x)/2$$</span>
and so the formula would be (if b and c are equal): <br>
<span class="math-container">$$(... |
2,439,069 | <p>$$\int_0^{\infty}\frac{dx}{2+\cosh (x)}$$</p>
<p>I'm not sure how to approach this problem. I tried substituting $\cosh (x) = \frac{e^x + e^{-x}}{2}$, and following the substitution $y = e^x$, I ended up with the integral </p>
<p>$$2 \cdot \int_1^{\infty} \frac{1}{(y+2)^2-3} dy$$</p>
<p>Which I wasn't able to ev... | George Coote | 445,167 | <p>An alternative approach for interest. With this integral it's relatively easy to rewrite $\cosh$ in terms of e, but in cases where it's less feasible, you can use the half-angle substitution for the hyperbolic tangent. </p>
<p>Using that, we have,
$$t = \tanh \frac x 2$$
$$\mathrm dx = \frac 2 {1-t^2} \mathrm dt$$... |
4,155,792 | <p>I have seen many math books, and some of them, very good books, that say that <span class="math-container">$0!=1$</span> 'by convention'. I think that <span class="math-container">$0!$</span> <em>must</em> be <span class="math-container">$1$</span> because it is the product of the empty set. That is, for <span class... | Shinrin-Yoku | 789,929 | <p>One way to define <span class="math-container">$n!$</span> is that it’s the number of bijections from a set with <span class="math-container">$n$</span> elements to itself ,If the set is empty then the number of elements is <span class="math-container">$0$</span> hence <span class="math-container">$0!=1$</span> ,si... |
282,139 | <p>Assume that $\gamma$ is an analytic simple closed curve in $\mathbb{R}^2$ which surrounds origin.</p>
<blockquote>
<p>Is there a polynomial vector field on the plane which is tangent to $\gamma$? In the other word, can an arbitrary analytic simple closed curve be realized as a closed orbit or a limit cycle of a p... | Will Sawin | 18,060 | <p>If every homology class is a winning loop for one of the two players, and each player has at least one winning loop, then the game cannot end in a draw.</p>
<p>Proof: If one player has a particular loop, then the other player cannot have any of the other loops, so they either have no loops or the same loop. If both... |
519,715 | <p>I feel it true that $xy=1$ implies $yx=1$ in the real Clifford Algebra $C_n$ (with respect to the quadratic form $Q=-x_1^2-……-x_n^2$), but I cannot prove it. Is it true?</p>
| Doug Spoonwood | 11,300 | <p>It oftentimes works out easiest to start by hypothesizing the antecedent or "left side" of the conditionals you want to prove. Then use the assumptions and the hypothesis to help you deduce the conclusion, and then use conditional introduction. So, if you want to prove (p→r), hypothesize p, and use p along with th... |
519,715 | <p>I feel it true that $xy=1$ implies $yx=1$ in the real Clifford Algebra $C_n$ (with respect to the quadratic form $Q=-x_1^2-……-x_n^2$), but I cannot prove it. Is it true?</p>
| Peter Smith | 35,151 | <p>Here's a <strong>hint</strong> for the second question.</p>
<p>You need to think strategically. You are aiming to prove a conditional. The default way of establishing a conditional $A \to C$ is to temporarily assume the antecedent $A$ and aim for the consequent $C$. If you can get there, you can then discharge the ... |
78,545 | <p>I know that ListLinePlot allow to plot data, but I would like to compare my data with some analytic function in the same plot. Is it possible to listlineplot data and plot a function at the same time?</p>
| Mark Adler | 94 | <p>Use <code>Show[]</code> to combine plots. E.g. <code>Show[plot1, plot2]</code>.</p>
|
78,545 | <p>I know that ListLinePlot allow to plot data, but I would like to compare my data with some analytic function in the same plot. Is it possible to listlineplot data and plot a function at the same time?</p>
| ciao | 11,467 | <p>A trivial way (many ways to do this...)</p>
<pre><code>data = Range@100;
myF[x_] := x^1/2 + Sin[x];
ListLinePlot[{data, myF /@ data}]
</code></pre>
<p><img src="https://i.stack.imgur.com/cfFEC.png" alt="enter image description here"></p>
|
949,853 | <p>If $\alpha: [a,b] \rightarrow \mathbb{R}$ is an increasing function, we can define the Riemann-Stieltjes integral $$\int_a^b f d \alpha$$ Does the function $\langle f,g\rangle = \int_a^b fg d\alpha$ define an inner product on the $\mathbb{R}$-space of all Riemann-Stieltjes integrable functions on $[a,b]$ (with respe... | bryanj | 54,886 | <p>If you're just trying to get the Schwarz inequality, I think you can look at it like this:</p>
<p>If both $\int |f|^2$ and $\int |g|^2$ are non-zero, the proof which involve dividing by $\langle f, f\rangle$ or $\langle g, g\rangle$ goes through.</p>
<p>So say we're worried about the situation where $\int |f|^2 = ... |
2,598,912 | <p>Let $f$ be some function in $L_{loc}^1(\mathbb{R})$ such that, for some $a \in \mathbb{R}$,</p>
<p>$$\int_{|x| \leq r} |f(x)|dx \leq (r+1)^a$$</p>
<p>for all $r \geq 0$. Show that $f(x)e^{-|tx|} \in L^1(\mathbb{R})$ for all $t \in \mathbb{R} \setminus \{0\}$. </p>
<p>I'm having a hard time finding use of the bou... | Did | 6,179 | <p>Assume without loss of generality that $f\geqslant0$ everywhere and that $t>0$, then note that, for every $x$, $$e^{-t|x|}=\int_{|x|}^\infty te^{-tr}dr$$ hence, applying <a href="https://en.wikipedia.org/wiki/Fubini%27s_theorem#Tonelli's_theorem_for_non-negative_functions" rel="nofollow noreferrer">Tonelli's ... |
972,385 | <p>Could someone please explain this question to me? I know that such integers do NOT exist, but I could not prove it. "Either solve it or give a brief explanation as to why it is impossible." Thank you!</p>
<p>Find integers $s$ and $t$ such that $15s + 11t = 1$.</p>
| GDumphart | 124,970 | <p><a href="http://en.wikipedia.org/wiki/Extended_Euclidean_algorithm" rel="nofollow">http://en.wikipedia.org/wiki/Extended_Euclidean_algorithm</a></p>
<p>$$3 \cdot 15 -4 \cdot 11 = 1$$</p>
|
1,809,500 | <p>Prove that $k!>(\frac{k}{e})^{k}$. </p>
<p>It is known that $e^{k}>(1+k)$. So if we multiply $k!$ on both sides, we get $k!e^{k}>(k+1)!$. Also $k^k>k!$. Now how to proceed ?</p>
| LiorGolan | 332,029 | <p>For some reason I cannot comment so I'll share my thought here: <a href="https://en.wikipedia.org/wiki/Stirling%27s_approximation" rel="nofollow">Stirling's approximation</a></p>
|
485,407 | <p>I have always used equations for the line (y=a + bx) in R2. Recently I came upon this thing called parametric equations. I cannot grasp the difference between them and the equations for lines that I used before. Of course, I can't understand the plane and the parametric equations for it. Any good example or referenc... | abiessu | 86,846 | <p>A set of parametric equations is called by that name because they have a parameter. Such equations are often used when a graph might not have certain "typical" and/or "nice" properties like being one-to-one or onto, e.g., a circle.</p>
<p>For a circle, a set of parametric equations might look like this:</p>
<p>$$... |
485,407 | <p>I have always used equations for the line (y=a + bx) in R2. Recently I came upon this thing called parametric equations. I cannot grasp the difference between them and the equations for lines that I used before. Of course, I can't understand the plane and the parametric equations for it. Any good example or referenc... | Blue | 409 | <p>A good way to look at it is this ...</p>
<blockquote>
<p><strong>Parametric equations let you draw curves with an Etch-a-Sketch.</strong></p>
</blockquote>
<p><img src="https://i.stack.imgur.com/N3gIs.jpg" alt="enter image description here"></p>
<p>(Image credit: <a href="http://pikajane.deviantart.com/art/Cont... |
402,310 | <p>Vector math is something I find very interesting. However, we have never been told the link between vectors in physics (usually represented as arrows, e.g. a force vector) and in algebra (e.g. represented like a column matrix). It was really never explained well in classes.</p>
<p>Here are the things I can't wrap m... | Max P | 79,458 | <p>You can represent a vector as an arrow in cartesian coordinates by drawing an arrow from (0,0) to the vector (row vector) for example <3,2> taken as a point on the plane (3,4). In other words, your first bullet point is correct.</p>
<p>Vector normalization is the process by which one takes an arbitrary vector (a... |
50,406 | <p>This is a question about support of modules under extension of scalars.</p>
<p>Let $f \colon A \to B$ be a homomorphism of commutative rings (with unity), and let $M$ be a finitely generated $A$-module.
Recall that the <em>support</em> of $M$ is the set of prime ideals $\mathfrak{p}$ of $A$ such that the localizat... | Angelo | 4,790 | <p>Take $A = \mathbb Z$, $M = \mathbb Q$, $B = \mathbb Z/p\mathbb Z$, were $p$ is a prime.</p>
|
85,651 | <p>Let $\Gamma$ be one of the classical congruence subgroups $\Gamma_0(N)$, $\Gamma_1(N)$ and $\Gamma(N)$ of $SL(2, \mathbb{Z})$.</p>
<p>How does the lower bound for the length of primitive geodesics on $\Gamma \backslash \mathbb{H}$ depending on $N \rightarrow \infty$?</p>
<p>Any suggestions?</p>
| Ian Agol | 1,345 | <p>For a hyperbolic element $A\in SL(2,\mathbb{Z})$, we have the length of the closed geodesic is given by $\ln[(tr^2(A)-2+\sqrt{tr^4(A)-4tr^2(A)})/4]$, and this is monotonic in $|tr(A)|$ for $|tr(A)|>2$. For $A\in \Gamma(N),\Gamma_1(N)$, we have $tr(A)\equiv 2 (\mod N)$, and $tr(A)\neq \pm 2$, so the smallest that ... |
2,903,110 | <p>How do I prove that the function $$f:(0,1)\rightarrow \mathbb R$$ defined by:</p>
<p>$$f(x) = \frac{-2x+1}{(2x-1)^2-1}$$</p>
<p>is onto?</p>
| user | 505,767 | <p>Note that </p>
<ul>
<li>$\lim_{x\to 0^+} f(x)=-\infty$</li>
<li>$\lim_{x\to 1^-} f(x)=\infty$</li>
</ul>
<p>therefore since $f(x)$ is continuous for $x\in(0,1)$ by IVT we have that $f(x)$ is onto.</p>
<p>Moreover note that for $x\in(0,1)$ we have that $f(x)$ </p>
<p>$$f'(x)=\frac{2x^2-2x+1}{x(x-1)^2x^2}\ge 0$$</... |
2,075 | <p>I've come across so many posts here and on the "main" math-SE site that voice complaints, frustrations, pet-peeves, grievances, or else are critical of another post/question, user, OP, etc. It is really an energy sapper! Certainly not a boost for morale.</p>
<p>Since I'm pretty new here, and feeling a bit ambival... | Mark | 7,667 | <p>Personally I am pretty new here as well and probably is not in a position to speak for the community, but from personal experience with other sites having civil debates and some doubts is a good thing and is a sign of maturity. </p>
|
2,075 | <p>I've come across so many posts here and on the "main" math-SE site that voice complaints, frustrations, pet-peeves, grievances, or else are critical of another post/question, user, OP, etc. It is really an energy sapper! Certainly not a boost for morale.</p>
<p>Since I'm pretty new here, and feeling a bit ambival... | Matt E | 221 | <p>Dear Amy,</p>
<p>I contribute on MathSE for the same reasons that I contribute on MO:</p>
<p>I enjoy thinking about math, and solving math problems. I also enjoy
talking about and explaining mathematical ideas. MO and MathSE provide the
opportunity to do all this.</p>
<p>Also, as a professional mathematician (... |
2,075 | <p>I've come across so many posts here and on the "main" math-SE site that voice complaints, frustrations, pet-peeves, grievances, or else are critical of another post/question, user, OP, etc. It is really an energy sapper! Certainly not a boost for morale.</p>
<p>Since I'm pretty new here, and feeling a bit ambival... | Andrew Stacey | 2,907 | <p>I've um'ed and ah'ed about contributing to this, but having just left my 9th answer, I'll pitch in.</p>
<p>The reason why I hesitate is that I don't consider myself to be a contributor to maths-SX, so in the strictest sense I can't answer this question. However, I'm a "conscientious objector" in that my non-partic... |
18,421 | <p>I am teaching 4th-grade kids. The topic is Fraction. Basic understanding of a fraction as a part of the whole and as part of the collection is clear to the kids. Several concrete ways exist to teach this basic concept. But when it comes to fraction addition/subtraction I could not find a way that teaches it concrete... | Amy B | 5,321 | <p>Use a piece of paper as your whole. To teach <span class="math-container">$3/2 + 4/3$</span> do the following.</p>
<ol>
<li>Give each child/group of children 6 pieces of paper.</li>
<li>One piece of paper should be left as a whole - the students can
write 1 whole on the paper.</li>
<li>Have the students fold 2 o... |
1,074,809 | <p>What is a non-decreasing sequence of sets and how come it can have a limit?</p>
<p>It appear in a probability theory book</p>
| A.S | 24,829 | <p>We want to define the domain of the function to be as large as possible. Note that $\sqrt x$ is defined over $x \ge 0$ only. The $\sin$ function is defined over all the reals. So the only restriction we have on the domain is that $2x-1 \ge 0$.</p>
|
325,860 | <p>Let $a,b \in \mathbb{R}$, $a<b$ and let $f$ be a differentiable real-valued function on an open subset of $\mathbb{R}$ that contains [a,b]. Show that if $\gamma$ is any real number between $f'(a)$ and $f'(b)$ then there exists a number $c\in(a,b)$ such that $\gamma=f'(c)$.</p>
<p>Hint: Combine mean value theore... | Dominic Michaelis | 62,278 | <p>Hint:<br>
Define $h(x)= f(x)- \gamma \cdot x$. Then search for a maximum (which must exist on a compact interval).</p>
<p>Wlog assume $h'(a)>0$ and $h'(b)<0$ (if that is not the case, observe $-h(x)$ which fulfills the properties.</p>
<p>We see that although the derivative is not continuous in general, it st... |
2,640,909 | <p>I encountered a problem with 4 variables and I was wondering if anyone knows how to solve this:
This is what is known:</p>
<p>$$ \left\lbrace
\begin{align}
a+b &= 1800 \\
c+d &= 12 \\
a/c &= 100 \\
b/d &= 250 \\
(a+b)/(c+d) &= 150
\end{align}
\right.$$</p>
<p>Below is a screenshot from a spread... | GNUSupporter 8964民主女神 地下教會 | 290,189 | <p>$$a/c = 100 \implies a = 100c \\ b/d = 250 \implies b = 250d$$</p>
<p>Substitute this into $a+b = 1800$ to get $100c+250d = 1800$. Divide by $50$ to get $2c+5d = 36$. From $c+d = 12$, we have $c = 12-d$, so $2(12-d)+5d = 36$.</p>
<p>$3d = 12 \iff d=4$, so $c = 12-4=8$. Hence $(a,b,c,d) = (800,1000,8,4)$</p>
|
342,491 | <p>How to prove the following:</p>
<p>$a_n = \left\{\left(1+\frac{1}{n}\right)^n\right\}$ is bounded sequence, $ n\in\mathbb{N}$</p>
| Siddhant Trivedi | 28,392 | <p>Let me denote $S_n=\{(1+\frac{1}{n})^n\}$
\begin{align*}
\bigg(1+\frac{1}{n}\bigg)^n &=1+ {_nC_1} \frac{1}{n}+ {_nC_2} \bigg(\frac{1}
{n}\bigg)^2+..........+{_nC_n} \bigg(\frac{1}{n}\bigg)^n\\
&=1+ n \frac{1}{n}+\frac{n(n-1)}{2!} \bigg(\frac{1}{n}\bigg)^2+ \frac{n(n-1)(n-2)}{2!} \bigg(\frac{1}{n}\bigg)^3+...... |
1,872,369 | <p>Prove that the equation $z^{3}e^{z}=1$ has infinitely many complex solutions.How many of them are real?</p>
<p>Use the argument principle,I choose a disk centered at $0$ with radius $R$ and get $\int_{\partial D}\dfrac {3z^{2}e^{z}+z^{3}e^{z}}{z^{3}e^{z}-1}dz$,and I don't how to do with this integral.Additionally,I... | Claude Leibovici | 82,404 | <p><em>Just for your curiosity</em>.</p>
<p>The equation $z^3\,e^z=1$ has three roots which express in terms of <a href="https://en.wikipedia.org/wiki/Lambert_W_function" rel="nofollow">Lambert function</a>. They are $$x_1=3 W\left(\frac{1}{3}\right)$$ $$x_2=3 W\left(-\frac{1}{3} (-1)^{1/3}\right)$$ $$x_3=3 W\left(\f... |
743,292 | <p>Three fair six-sided dice are thrown and the dice show three different numbers. Find the probability that at least one six is obtained. </p>
<p>Im unsure ofwhat type of question this is, I have tried combinations such as 6C1 over 18C3. </p>
<p>But im not sure if this is correct. Any guidance is much appreciate... | Oleg567 | 47,993 | <p>There are $N=\binom{6}{3}=6C3=C_6^3 = 20$ total triplets with different numbers.</p>
<p>There are $M=\binom{5}{2}=5C2=C_5^2 = 10$ these triplets with one six.</p>
<hr>
<p>Full list:</p>
<p>$1,2,3$<br>
$1,2,4$<br>
$1,2,5$<br>
$1,2,6$ *<br>
$1,3,4$<br>
$1,3,5$<br>
$1,3,6$ *<br>
$1,4,5$<br>
$1,4,6$ *<br>
$1,5,6$ *<... |
35,327 | <p>This is a bit of a follow-up to a previous question: <a href="https://mathematica.stackexchange.com/questions/35256/how-can-i-merge-multiple-sets-of-morphological-components-perhaps-selected-usin">How can I merge multiple sets of morphological components (perhaps selected using different metrics)?</a></p>
<p>I've r... | C. E. | 731 | <p>Here's a different solution I was working on. </p>
<pre><code> c = Import["http://i.stack.imgur.com/gSXIj.png"];
Manipulate[
Column[{
Show[
components // Colorize,
Graphics[Locator[x, Background -> Orange]]
],
Text["Color index: " <> ToString[components[[Seque... |
951,914 | <p>Here's my question. We know that the absolute value of X looks like:
<img src="https://i.stack.imgur.com/ePQo9.png" alt="Absolute Value of x"></p>
<p>Clearly, we can see, since the absolute value of x is always greater than or equal to 0, the area under the curve is always positive. Why then does it integrate to th... | mod0 | 171,143 | <p>I am answering my own stupid question for the sake of completeness. </p>
<p>I am confusing definite integral with the indefinite integral. I do notice that if I take any two points and take the difference between the values of the indefinite integral evaluated at these points, I get a positive value for the area. A... |
2,764,447 | <p>In the following question:</p>
<p><a href="https://i.stack.imgur.com/Rgzjr.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/Rgzjr.jpg" alt="enter image description here" /></a>
I'm getting the answer as LHL=-infinity while RHL= infinity</p>
<p>The answer given to this question is D.</p>
<p>However ... | user | 505,767 | <p>Indeed, I agree with you, by the usual definition we say that the two side limits exist and we have</p>
<ul>
<li>$\lim_{x\to1^+}f(x)=+\infty$</li>
<li>$\lim_{x\to1^-}f(x)=-\infty$</li>
</ul>
<p>since those are different we say that the limit at $x=1$ doesn't exist and then the function is discontinuos at $x=1$.</p... |
3,266,705 | <p>I know that if <span class="math-container">$f(x)$</span> is irreducible then <span class="math-container">$\langle f(x) \rangle$</span> is a prime ideal. Then I thought: is it maximal? Then I search about it, I find that it is not maximal ideal but cannot find any proof. </p>
<p>Any help is appreciated. Thanks.... | Ruben | 386,073 | <p>An ideal <span class="math-container">$I \subset \mathbb Z [X]$</span> is <em>maximal</em> if and only if the quotient <span class="math-container">$\mathbb Z [X] / I$</span> is a <em>field</em>.</p>
<p>Suppose <span class="math-container">$f \in \mathbb Z [X]$</span> is irreducible and let <span class="math-contai... |
1,741,229 | <p>As the title.
Or rather, for any integer $m$ which is not the characteristic, does such an 'integer division' exist?</p>
| Noah Schweber | 28,111 | <p>The crucial fact is the following: if $k$ is a field of characteristic zero, there is a natural embedding $e$ of $\mathbb{Q}$ into $k$. </p>
<p>To prove this, first note that subring of $k$ generated by $1_k$ is isomorphic to $\mathbb{Z}$ (since the characteristic is zero), yielding an embedding $d$ of $\mathbb{Z}$... |
102,280 | <p>What are the usual tricks is proving that a group is not simple? (Perhaps a link to a list?)</p>
<p>Also, I may well be being stupid, but why if the number of Sylow p groups $n_p=1$ then we have a normal subgroup?</p>
| Did | 6,179 | <p><strong>Summary:</strong> Like in several questions of the same ilk asked previously on the site, the recursion can only determine each $T(n)$ as a function of $T(2i+1)$ and $k\geqslant0$, where $n=(2i+1)2^k$. Furthermoree, the series considered by the OP are simply not relevant to the asymptotics of $T(n)$. The gro... |
56,804 | <p>I know the Galois group is $S_3$. And obviously we can swap the imaginary cube roots. I just can't figure out a convincing, "constructive" argument to show that I can swap the "real" cube root with one of the imaginary cube roots. </p>
<p>I know that if you have a 3-cycle and a 2-cycle operating on three elements, ... | Billy | 13,942 | <p>A brute-force way to see it is easy enough. The roots of $X^3 - 2$ are $\sqrt[3]{2},\; \omega\sqrt[3]{2},\; \omega^2\sqrt[3]{2}$, so the splitting field of your polynomial is $K = \mathbb{Q}(\sqrt[3]{2},\omega)$. You are asking why it's legitimate to send $\sqrt[3]{2}$ to $\omega\sqrt[3]{2}$. Well, let's try it. Let... |
56,804 | <p>I know the Galois group is $S_3$. And obviously we can swap the imaginary cube roots. I just can't figure out a convincing, "constructive" argument to show that I can swap the "real" cube root with one of the imaginary cube roots. </p>
<p>I know that if you have a 3-cycle and a 2-cycle operating on three elements, ... | Marty Green | 10,838 | <p>I recently found an answer to my own question that no one else has brought up, so I thought I'd post it. The thing that always bothered me about swapping the real and imaginary cube roots of two was the assymetry of it all. What makes the positive imaginary root distinguishable from the negative imaginary root?</p>
... |
56,804 | <p>I know the Galois group is $S_3$. And obviously we can swap the imaginary cube roots. I just can't figure out a convincing, "constructive" argument to show that I can swap the "real" cube root with one of the imaginary cube roots. </p>
<p>I know that if you have a 3-cycle and a 2-cycle operating on three elements, ... | dragoboy | 168,033 | <p>One can prove a more general result : Let $f(X)$ be an irreducible polynomial with rational coefficients and of degree $p$ prime. If $f$ has precisely two nonreal roots in the complex numbers, then the Galois group of $f$ is $S_p$</p>
<p>So taking here $p=3$ we're done</p>
|
2,481,463 | <p>I am self-learning elliptic functions and modular form but struggling with the very basics.
I have programmed the Eisenstein series
$$E_6 = \sum_{m,n \in \mathbb{Z}} \frac{1}{(m +n \tau)^6} $$
and produced the following image (color denotes the argument)</p>
<p><a href="https://i.stack.imgur.com/NZeZb.png" rel="no... | davidlowryduda | 9,754 | <p>First of all, I think you've chosen an excellent topic of self-study. It's very deep and beautiful, and there's still lots to be found.</p>
<p>I think the first thing you should ask yourself is <strong>what does it mean to be a <em>fundamental domain?</em></strong></p>
<p>You have a modular function $f(z)$. This m... |
2,845,001 | <p>If I know the length of a chord of a circle and the length of the corresponding arc, but do not know the circle's radius, is there a formula by which I can calculate the length of the sagitta?</p>
| Ted Shifrin | 71,348 | <p>I hadn't even known the term before you made me look it up.:) I don't see a way to compute it explicitly, but it is in fact uniquely determined. This is because knowing $r\theta=A$ and $r\sin\theta=B$, there is a unique $\theta$ ($0<\theta<\pi/2$) with $\dfrac{\sin\theta}{\theta} = \dfrac BA$, and then $r=\dfr... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.