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 |
|---|---|---|---|---|
4,450,722 | <blockquote>
<p>Q: <span class="math-container">$f\in L^1(\mathbb{R}^n),\alpha>0$</span> and
<span class="math-container">$$\int_{\mathbb{R}^n} |f(x+h)-f(x)|\,dx\leq |h|^{1+\alpha},\ h\in\mathbb{R}^n.$$</span>
Show that <span class="math-container">$f=0$</span> a.e.</p>
</blockquote>
<p>My attempt:
Take <span class=... | greg | 357,854 | <p><span class="math-container">$
\def\op#1{\operatorname{#1}}
\def\bbR#1{{\mathbb R}^{#1}}
\def\e{\varepsilon}
\def\o{{\tt1}}
\def\f{\frac{\tt1}{f}}
\def\p{\partial}
\def\A{{\cal A}}\def\L{{\cal L}}
\def\LR#1{\left(#1\right)}
\def\BR#1{\Big(#1\Big)}
\def\trace#1{\op{Tr}\LR{#1}}
\def\Diag#1{\op{Diag}\LR{#1}}
\def\diag#... |
1,609,758 | <p>Is the answer of $\ \ 2415^n \ mod \ \ 2556 \ \ $ is only $711$ and $1989$ for $n\in \mathbb{Z}^{+} >1 \ \ ?$ </p>
<p>If it's true, how to prove that ?</p>
<p>I try to expand $(2556 - 141)^n $ for any $n\in \mathbb{Z}^{+} >1$ , but it doesn't make sense.</p>
<p>According to <a href="http://www.wolfr... | user127022 | 127,022 | <p>Uniform convergence is established in Theorem 3.a.1 in König's <em>Eigenvalue Distribution of Compact Operators</em> (DOI: <a href="https://doi.org/10.1007/978-3-0348-6278-3" rel="nofollow noreferrer">10.1007/978-3-0348-6278-3</a>)</p>
<p><strong>Theorem</strong>: Let $(\Omega, \mu)$ be a finite measure space and $... |
183,123 | <p>By using Cantor's Theorem: if We have a Set A, with n members the Cardinality of the Power Set is $P(A)=2^n$.
There is Formula to calculate n Power Sets $P(...P(P(A))...)$ of a set A with m elements without doing the simple iteration $(((2^2)^2)..)^2^m$ n times?
Many thanks</p>
| Mark Bennet | 2,906 | <p>Once you have corrected the formulation of your question as noted in the comments, you have an expression for the cardinality of the power set iterate.</p>
<p>You can express comparable expressions more economically using <a href="http://en.wikipedia.org/wiki/Knuth%27s_up-arrow_notation" rel="nofollow">Knuth's up a... |
340,701 | <p>I'm thinking about the following problem. Assume that one is given a sequence of binary data $(x_i)_i\in\{0,1\}^n$ and that wants to determine whether or not this sequence has been generated by a Bernoulli-like random mechanism, such as coin tosses.</p>
<p>My first idea was to use classical statistical test theory,... | Michael Hardy | 11,667 | <p>The Bernoulli distribution is any probability distribution supported on the set $\{0,1\}$. That means there is some number $p\in[0,1]$ such that the probability assigned to the set $\{1\}$ is $p$ and that assigned to $\{0\}$ is $1-p$.</p>
<p>One may speak of <b>independent</b> Bernoulli trials or of dependent Bern... |
1,110,831 | <p>I have $\pi:4y-3z-4=0 \quad A=(2,4,4) \quad B=(2,-2,-4)$.
I have to calculate the equation of the plane $\sigma$ throw $A$ and $B$ and orthogonal to $\pi$.
What is the solution?
Thanks in advice!</p>
| André Nicolas | 6,312 | <p>In accordance with the usual convention, we avoid caps, which make equations harder to solve. </p>
<p>We are looking for positive integer solutions $(x,y)$ of
$$\frac{ax+b}{c-x}=y,$$
where $a$, $b$, $c$ are non-negative and $x\lt c$.</p>
<p>Equivalently, we want, together with the bound on $x$, $ax+b=cy-xy$, that... |
1,056,194 | <p>With the regular trig functions, if I ever end up with something like $\operatorname{trig}_1(\operatorname{arctrig}_2(f(x))$, where $\text{trig}_1$ and $\text{trig}_2$ are two arbitrary trigonometric functions, I can draw a right triangle to find a formula for this that doesn't involve any trigonmetric functions.</p... | Teoc | 190,244 | <p>Hint: Use $\sinh^{-1}(x)=\log(x+\sqrt{1+x^2})$ and use $\cosh(x)=\dfrac{e^{x}+e^{-x}}{2}$</p>
<p>When plugging in $3x$, you should get $\cosh(\sinh^{-1}(3x))=\dfrac{3x+\sqrt{1+9x^2}+\frac{1}{3x+\sqrt{1+9x^2}}}{2}$.</p>
<p>That simplifies to $\dfrac{2\sqrt{9x^2+1}}{2}=\sqrt{9x^2+1}$</p>
|
3,221,884 | <p><strong>Exercise :</strong></p>
<blockquote>
<p>Let <span class="math-container">$H$</span> be a Hilbert space and <span class="math-container">$A,B \in \mathcal{L}(H)$</span> be self-adjoint operators with <span class="math-container">$0 \leq A \leq B$</span> and <span class="math-container">$B \in \mathcal{L}_c... | Martin Argerami | 22,857 | <p>I can think of two arguments. None of them is entirely elementary, though, in the sense that they depend on having square roots available. </p>
<ul>
<li><p>Let <span class="math-container">$\{x_n\}$</span> be a sequence in the unit ball of <span class="math-container">$H$</span>. Since <span class="math-container">... |
3,221,884 | <p><strong>Exercise :</strong></p>
<blockquote>
<p>Let <span class="math-container">$H$</span> be a Hilbert space and <span class="math-container">$A,B \in \mathcal{L}(H)$</span> be self-adjoint operators with <span class="math-container">$0 \leq A \leq B$</span> and <span class="math-container">$B \in \mathcal{L}_c... | Disintegrating By Parts | 112,478 | <p>Because <span class="math-container">$B-A \ge 0$</span>, then <span class="math-container">$[x,y]=\langle (B-A)x,y\rangle$</span> is a pseudo inner product, lacking only positive definiteness. As such, the Cauchy-Schwarz inequality holds
<span class="math-container">$$
|[x,y]|^2 \le [x,x][y,y] \\
|\l... |
2,887,176 | <p>Can transfinite induction be used to demonstrate that 0.9...=1?</p>
<p>More generally, can it be used to prove limits of sequences?</p>
| David K | 139,123 | <p>Sequences and series such as the one we use for $0.999\ldots=1$ all have the same infinite length, so there isn't anywhere to "transfinitely induct" <em>to.</em> </p>
<p>Moreover, we already use ordinary finite induction to prove limits of sequences.</p>
<p>So I really don't see how transfinite induction applies h... |
1,590,619 | <p>I must find the sum $$\sum_\limits{n=1}^{\infty}\frac1{n(2n+1)}$$ I have already tried partial fractions, but obtained an indeterminate form of infinity minus infinity. Perhaps completing the square would help?</p>
| Community | -1 | <p>For $|x|<1$, it holds $$\sum_{k=1}^\infty \frac{x^k}{k}=-\ln(1-x)$$</p>
<p>Hence, for $|x|<1$, it holds $$\sum_{k=1}^\infty \frac{x^{2k}}{k}=-\ln(1-x^2)=-\ln(1-x)-\ln(1+x)$$</p>
<p>Hence, for $|x|<1$ it holds $$\sum_{k=1}^\infty \frac{x^{2k+1}}{k(2k+1)}=-\int_0^x\ln(1-t^2)\,dt=(1-x)\ln(1-x)-(1+x)\ln(1+x)+... |
3,339,979 | <p>The problem is as follows:</p>
<p>"Prove that if the limit <span class="math-container">$f(x)$</span> as <span class="math-container">$x$</span> tends to <span class="math-container">$a$</span> is <span class="math-container">$L$</span>, then there exists an <span class="math-container">$m\gt0$</span> and <span cla... | Kavi Rama Murthy | 142,385 | <p>What you have done is fine. You can end the proof by observing that if <span class="math-container">$M >|L|+1$</span> then we have both <span class="math-container">$M >L+1$</span> and <span class="math-container">$-M <1-L$</span>. </p>
|
3,372,482 | <p>Looking at some of the more common pythagorean triples, I noticed a trend that there are usually two numbers which differ by only one or two, with more much smaller or larger number in comparison. For example:</p>
<p><span class="math-container">$$5,12,13$$</span>
<span class="math-container">$$8,15,17$$</span>
<sp... | user | 505,767 | <p>All pythagorean triples are given by</p>
<p><span class="math-container">$$a = m^2 - n^2 ,\ \, b = 2mn ,\ \, c = m^2 + n^2 $$</span></p>
<p>for any pair <span class="math-container">$m>n>0$</span>.</p>
<p>In order to have a small <span class="math-container">$a$</span> we can set <span class="math-container... |
668,580 | <p><strong>Theorem</strong> <em>Suppose</em> $$\begin{align}(a)&\sum_{n=0}^\infty a_n\quad\text{converges absolutely}\\(b)&\sum_{n=0}^\infty a_n=A\\(c)&\sum_{n=0}^\infty b_n=B\\(d)&c_n=\sum_{k=0}^n a_k b_{n-k}\quad(n=0,1,2,\dots)\end{align}$$ Then $$\sum_{n=0}^\infty c_n=AB$$</p>
<p><strong>Proof.</str... | meta | 126,987 | <p>You're considering integers modulo $p$ under multiplication - so they're refering to an integer being equivalent to 0 mod $p$ if it is a multiple of $p$. </p>
<p>Then since $(x+1)(x-1) = 0$ in $\mathbb{Z}_p^*$, have that (in $\mathbb{Z}$) $p$ divides $(x+1)(x-1)$. Then either $p$ divides $(x+1)$ or $p$ divides $(x-... |
3,148,902 | <p>I'm trying to find an explicit formula for the following recurrence relation:</p>
<blockquote>
<p><span class="math-container">$a_0 = 1$</span>, <span class="math-container">$\forall n \ge 1: a_n = n^3a_{n-1} + (n!)^3$</span></p>
</blockquote>
<p>So far though, my attempts have been unsuccessful.</p>
<hr>
<p>... | Myunghyun Song | 609,441 | <p><strong>Hint</strong>: Divide by <span class="math-container">$(n!)^3$</span> and define <span class="math-container">$\displaystyle b_n=\frac{a_n}{(n!)^3}$</span> to get
<span class="math-container">$$
b_n =b_{n-1}+1.
$$</span></p>
|
2,669,054 | <p>Question: What is the area of the region enclosed by the curves:</p>
<p>$$2y = 4\sqrt{x},\quad y = 3,\quad \text{and} \quad 2y + 2x = 6.
$$
I have tried calculate all the definite integrals but I am not sure which curve I am supposed to subtract and which one is supposed to come first. And also, I am a little confu... | johnnyb | 298,360 | <p>The answers given are more specific, but I wanted to give you a more general answer. When you have multiple lines like this, what they are asking you to do is to think about the shape that it makes. For any particular part of the graph, there will only be two curves that matter. However, to find out <em>which</em... |
2,669,054 | <p>Question: What is the area of the region enclosed by the curves:</p>
<p>$$2y = 4\sqrt{x},\quad y = 3,\quad \text{and} \quad 2y + 2x = 6.
$$
I have tried calculate all the definite integrals but I am not sure which curve I am supposed to subtract and which one is supposed to come first. And also, I am a little confu... | g.kov | 122,782 | <p><a href="https://i.stack.imgur.com/Qdb3z.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/Qdb3z.png" alt="enter image description here"></a></p>
<p>As far as the bounding curves
can be easily expressed either as $y=f(x)$ or $x=f(y)$,
there are more options to calculate the area.
For example, we ... |
138,857 | <p>At an ambulance stations there are 5 nurses and 4 medics. Find in how many ways can we create a team with three people, where at least one of them is a medic.</p>
<p>I solved the problem this way: there are $4\cdot C_5^2$ ways to form a team with one medic and two nurses and $C_4^2 \cdot 5$ ways to form a team with... | Peter Taylor | 5,676 | <p>The quicker way to look at it is that there are $\binom{9}{3}$ ways of selecting a team of 3 people, of which $\binom{5}{3}$ don't have a medic. So the answer is $\binom{9}{3} - \binom{5}{3} = 84 - 10 = 74$, which agrees with your calculation.</p>
|
3,923,706 | <p>Take two spaces <span class="math-container">$X_1$</span> <span class="math-container">$X_2$</span> and consider <span class="math-container">$\mu\in\mathcal{P}(X_1\times X_2)$</span> i.e the space of Borel probability measures. Let <span class="math-container">$\mu_1,\mu_2$</span> be Borel probability measures on <... | Euleroid Photography | 1,153,775 | <p>Writing this up for completeness, even though the question is very old:</p>
<p>Let <span class="math-container">$\mu_1 = \delta_x$</span>. Let <span class="math-container">$A \in \mathcal{B}(X_1)$</span>, <span class="math-container">$B \in \mathcal{B}(X_2)$</span>. We want to show that
<span class="math-container">... |
1,229,488 | <p>I'm not sure where to begin on this problem. Do you begin by solving the integral of each one then applying the conditions to each respective integral? Any help would be appreciated.</p>
| MPW | 113,214 | <p><strong>Hint:</strong> The key is to realize that the function $$G(x)=b + \int_a^x g(t)\;dt$$ satisfies both $$G'(x)=g(x)$$ and $$G(a)= b.$$</p>
|
1,967,460 | <p>Given for $\alpha>1$
$$\int^{\pi}_0 \frac{dx}{\alpha-\cos x}=\frac{\pi}{\sqrt{\alpha^2-1}}$$</p>
<p>Prove that $$\int^{\pi}_0 \ln \left(\frac{b-\cos x}{a- \cos x}\right)dx=\pi \left(\frac{b+\sqrt{b^2-1}}{a+\sqrt{a^2-1}}\right)$$
if $a, b>1$</p>
| Nan | 372,333 | <p>First, notice the integrand can be written as
$$
\begin{align}
\ln (\frac{b-\cos x}{a- \cos x}) &=\ln(b-\cos x)-\ln(a-\cos x)\\
& =\ln(y-\cos x)|^{y=b}_{y=a}\\
& =\int^b_a\frac{1}{y-\cos x}dy
\end{align}
$$
So the integration equals
$$
\begin{align}
\int^{\pi}_0 \ln (\frac{b-\cos x}{a- \cos x})dx &=... |
2,809,012 | <p>I'm struggling with the concept of constants in the context of derivatives.
For example;</p>
<p>$$f(\theta)=r(\cos\theta-1)$$ where $r$ is a constant.
$$f'(\theta)=-r\sin\theta$$</p>
<p>Why is the product rule not used in the above?</p>
| tinlyx | 164,036 | <p>You can use the product rule (if you like). Just need to note that the derivative of a constant ($r$ here) is $0$, and you will get the same result.</p>
|
2,809,012 | <p>I'm struggling with the concept of constants in the context of derivatives.
For example;</p>
<p>$$f(\theta)=r(\cos\theta-1)$$ where $r$ is a constant.
$$f'(\theta)=-r\sin\theta$$</p>
<p>Why is the product rule not used in the above?</p>
| Neo | 563,385 | <p>You can still apply the product rule in the following way:
$$ f'(x)= \frac {dr}{dx}(cos\theta-1)+r \frac{d(cos\theta-1)}{dx} = 0(cos\theta-1)+r(-sin\theta+0)=-rsin\theta $$ </p>
<p>Since we have the following differentiation rule (which is a direct corollary of the product rule):</p>
<p>$$ \frac{d(kf(x))}{dx} = k... |
495,812 | <p>Let $A$ be an $m \times n$ matrix, and let $A'$ be the result of a sequence of elementary row operations on $A$. Prove that the rows of $A$ span the same space as the rows of $A'$.</p>
<p>I know how to prove that A and A' have both the same rank (rank (A)= dim(Vect(A)) ), but I am no sure how to prove that they bot... | Pedro | 23,350 | <p>Suppose that $v_1,\ldots, v_m$ are the rows of $A$, in $\Bbb R^n$. You have to show that $$\langle v_1,\ldots,v_i,\ldots,v_i+v_j,\ldots,v_m\rangle=\langle v_1,\ldots,v_i,\ldots,v_j ,\ldots,v_m\rangle$$</p>
<p>$$\langle v_1,\ldots,\lambda v_i, \ldots,v_m\rangle=\langle v_1,\ldots,v_i,\ldots,v_m\rangle \;\;,\;\lambda... |
233,483 | <p>Now that <em>AlphaGo</em> <a href="https://en.wikipedia.org/wiki/AlphaGo_versus_Lee_Sedol" rel="noreferrer">has just beaten</a> Lee Sedol in Go and <em>Deep Blue</em> <a href="https://en.wikipedia.org/wiki/Deep_Blue_versus_Garry_Kasparov" rel="noreferrer">has beaten</a> Garry Kasparov in chess in 1997, I wonder what... | Alexandre Eremenko | 25,510 | <p>There is no doubt that computer will HELP doing mathematics, and they are already helping. Computers are better in doing some jobs, humans are better in doing other jobs. I believe that there will always remain something in mathematics that only humans can do.</p>
<p>It is difficult to predict, of course what exactl... |
233,483 | <p>Now that <em>AlphaGo</em> <a href="https://en.wikipedia.org/wiki/AlphaGo_versus_Lee_Sedol" rel="noreferrer">has just beaten</a> Lee Sedol in Go and <em>Deep Blue</em> <a href="https://en.wikipedia.org/wiki/Deep_Blue_versus_Garry_Kasparov" rel="noreferrer">has beaten</a> Garry Kasparov in chess in 1997, I wonder what... | abo | 20,716 | <p>The advantage that humans will have over computers in mathematics is that humans will be the judges of the value of whatever mathematics is done. That is, the judgment of what mathematics is worth doing - what is an interesting question, what has beauty - will be done by humans, and this will give humans an intrins... |
1,867,078 | <p>The primitive element theorem is a basic result about field extensions. I was wondering whether there are nice geometric ways to visualize it or think about it. Since field spectra are singletons, it has to be about the non-trivial automorphisms of points (I think), and I don't know how to think about it.</p>
| MooS | 211,913 | <p>Reformulate the primitive element theorem as follows:</p>
<blockquote>
<p>Let $\mathfrak m \subset k[X_1, \dotsc, X_n]$ be a maximal ideal, such that the obtained field extension is separable. For
general $(a_1, \dotsc, a_n) \in k^n$, the map $$k[T] \to k[X_1, \dotsc, X_n]/\mathfrak m, T \mapsto a_1X_1 + \dotsb... |
76,832 | <p>I have a complex <code>Manipulate-</code>based demonstration containing a few sliders and a Setterbar. </p>
<p>I would like to make a movie out of it, avi, or whatever else. The movie will then be used for a journal publication as a Supporting material, so the quality should be good enough. </p>
<p>When the movie... | Nasser | 70 | <p>The export to movie is easy. (just write <code>p=Manipulate[..]</code> then export <code>p</code> to movie.</p>
<p>Controlling the sequences as you want, I think have to be programmed in. <code>Autorunsquences</code> does not give one full control of all the scenario needed.</p>
<p>Here is the Manipulate you have... |
485,755 | <p>Let's call a point $P$ which satisfies the following condition 'a rational point'.</p>
<p><strong>Condition</strong>: Each distance $PA, PB, PC$ from a point $P$ to three vertices $A, B, C$ of an equilateral triangle $ABC$ which has edge-length $1$ is <strong>rational number</strong>.</p>
<p>I'm interested in this... | mathlove | 78,967 | <p>I've just got the following:</p>
<p><strong>The answer for Question 1</strong> : There exist an infinite number of rational points in the equilateral triangle ABC.</p>
<p>$\left(PA, PB, PC\right)=$
$$\left(\frac{n^4+10n^2+9}{n^4+4n^3+10n^2-12n+9}, \frac{n^4-4n^3+10n^2+12n+9}{n^4+4n^3+10n^2-12n+9}, \frac{8n^3+24n}{... |
1,745,023 | <p><img src="https://i.stack.imgur.com/qtk0E.jpg" alt="i want to know the way of solving (a) " /></p>
<p>If I found the sol of (a), I could find sol of (b) and (c) by using Hensel's lemma.</p>
<p>I want to know the way of solving (a) without testing all the cases (ex testing 1,2,3,4,...10)
I think that there will be ot... | Kamil | 331,974 | <p>As it was said before, <strong>A</strong> must not be lineary dependent. The easy way to define it is to make a matrix, where coefficients are the coefficients from linear combinations of vectors from <strong>A</strong> through <strong>B</strong>, and then calculate its determinant. If det = 0 $ \Rightarrow$ <strong... |
183,031 | <p>I'm reading a book on analytic number theory. It asks me to prove:</p>
<blockquote>
<p>$$ \sum_p \frac{\log p}{p^s} = \frac{1}{s-1} + O(1) \tag{A}$$
and conclude, via integration, that:
$$ \sum_p \frac{1}{p^s} = \log \frac{1}{s-1} + O(1) \tag{B}$$</p>
</blockquote>
<p>Now, I know how to prove $(A)$ via Abe... | anon | 11,763 | <p>What if you don't integrate over all of $(s,\infty)$? Say, fix $S$ large enough, and...</p>
<p>$$\int_s^S\sum_{p}\frac{\log p}{p^\sigma}d\sigma=\int_s^S\frac{1}{\sigma-1}+O(1)d\sigma=\log\frac{1}{s-1}+O(1) $$</p>
<p>whereas</p>
<p>$$\int_s^S\sum_{p}\frac{\log p}{p^\sigma}d\sigma=\sum_p\int_s^S\frac{\log p}{p^\sig... |
183,031 | <p>I'm reading a book on analytic number theory. It asks me to prove:</p>
<blockquote>
<p>$$ \sum_p \frac{\log p}{p^s} = \frac{1}{s-1} + O(1) \tag{A}$$
and conclude, via integration, that:
$$ \sum_p \frac{1}{p^s} = \log \frac{1}{s-1} + O(1) \tag{B}$$</p>
</blockquote>
<p>Now, I know how to prove $(A)$ via Abe... | saradi | 34,619 | <p>Well we know that </p>
<p>$$\zeta(\sigma)= \frac{1}{\sigma-1}$$ when $$ \sigma \to1^+$$ since zeta has a simple pole in $\sigma=1$ with residual one.
Now your function $$\sum_p\frac{\log p}{p^s}$$ can be expressed in terms of the zeta function. </p>
<p>see <a href="http://people.math.jussieu.fr/~demarche/enseignem... |
2,051,843 | <p>I am analyzing the below puzzle presented <a href="https://books.google.ie/books?id=8EDCAgAAQBAJ&pg=PA37&lpg=PA37&dq=in%20an%20election,%20two%20candidates,%20Albert%20and%20Benjamin&source=bl&ots=GGih2A4OTC&sig=had3_VHldWIeszXN1HtMpG31YnE&hl=en&sa=X&ved=0ahUKEwjghIXlk-jQAhVYF8AKH... | JeremyDouglass | 396,141 | <p>Each letter sequence gives the votes in the order they are drawn from the box.</p>
<p>Here are the tallies for the first vote sequence:</p>
<blockquote>
<p><strong>AAABB</strong></p>
</blockquote>
<pre><code>1. A | A=1 | B=0
2. A | A=2 | B=0
3. A | A=3 | B=0
4. B | A=3 | B=1
5. B | A=3 | B=2
</code></pre>
<p>T... |
180,504 | <p>I want to plot a curve given by $$f(x,y)\equiv y-y^3+3yx^2=0,$$ that passes through $(0,0)$ and $f(x,y) \to + \infty$ as $|x|,|y| \to \infty$. I thought of using ContourPlot to do so but when I use ContourPlot[y-y^3+3yx^2==0,{x,x_min,x_max},{y,y_min,y_max}], there are three other curves in addition to the one I want... | Giancarlo | 43,659 | <p>Thanks to Kuba's comment now I see:
<code>HoldPattern</code> prevents the evaluation of its argument, so <code>Cases</code> don't match the element in the list <code>a->b/2</code> that is</p>
<pre><code>Times[Rational[1, 2], b]
</code></pre>
<p>with the unevaluated pattern</p>
<pre><code>Times[b, Power[2, -1]]... |
1,458,055 | <p>What i want to prove is following integral
\begin{align}
\int_{-\infty}^{\infty} e^{-x^2} \cos(x^2) dx=\frac{1}{2} \sqrt{\left(1+\sqrt{2}\right) \pi }
\end{align}
can you give some explicit method to obtain this result?</p>
| omegadot | 128,913 | <p>So let us follow through on Eclipse Sun's suggestion. In doing so we will find that while the double integral can indeed be separated into two parts (two double integrals), neither of them turns out to be equal to zero.</p>
<p>Let
$$I = \int^\infty_{-\infty} e^{-x^2} \cos (x^2) \, dx = 2 \int^\infty_0 e^{-x^2} \cos... |
3,551,183 | <p>Let <span class="math-container">$f$</span> be a function from <span class="math-container">$\mathbb{R}^n$</span> to <span class="math-container">$\mathbb{R}^m$</span>. </p>
<p>In general, mathematicians like to generalize a mathematical concept very very much. </p>
<p>I think they like generalization more than ... | Peter Melech | 264,821 | <p>Note that if <span class="math-container">$f:X\rightarrow Y$</span> is a differentiable map between normed spaces, then its derivative is a mapping <span class="math-container">$f':X\rightarrow \mathcal{L}(X,Y),x\mapsto f'(x)$</span>.
Indeed for <span class="math-container">$x_0\in X$</span> the derivative is the li... |
79,012 | <p>Let
$$f(z)=\frac{1}{(z-1)(z-2)}$$
and let
$$R_1=\Bigl\{z\Bigm| 1<|z|<2\Bigr\}\quad\text{ and }\quad R_2=\Bigl\{z\Bigm| |z|>2\Bigr\}.$$</p>
<p>How do you find the Laurent series convergent on $R_1$? Also how do you do it for $R_2$?</p>
<p>I'm having serious trouble with this as I can't see how to expand ... | leslie townes | 18,076 | <p>I assume you want the Laurent series expansion for this function at $0$ (since this is the point those annuli are centered at). The standard way to do this is not to use the integral formula at all--- just the uniqueness of the Laurent series expansion (which I guess is often proved using the integral formula). Yo... |
4,470,820 | <p>I shall find the integral by Feynman’s Technique Integration on a particular integral</p>
<p><span class="math-container">$\displaystyle I(a)=\int_{0}^{\pi} \ln (a \cos x+1) d x,\tag*{} $</span>
where <span class="math-container">$-1\leq a \leq 1.$</span>
<span class="math-container">$\displaystyle \begin{aligned}I^... | Claude Leibovici | 82,404 | <p>Concerning the antiderivative
<span class="math-container">$$I=\int \log (b \cos (x)+c)\,dx=2\int \frac 1{t^2+1}\Big[\log[ (c-b)t^2+(b+c)]-\log(t^2+1)\Big]\,dt$$</span> But
<span class="math-container">$$\log[ (c-b)t^2+(b+c)]=\log(c-b)+\log \left(t-\sqrt{\frac{b+c}{b-c}}\right)+\log \left(t+\sqrt{\frac{b+c}{b-c}}\ri... |
4,470,820 | <p>I shall find the integral by Feynman’s Technique Integration on a particular integral</p>
<p><span class="math-container">$\displaystyle I(a)=\int_{0}^{\pi} \ln (a \cos x+1) d x,\tag*{} $</span>
where <span class="math-container">$-1\leq a \leq 1.$</span>
<span class="math-container">$\displaystyle \begin{aligned}I^... | Jack D'Aurizio | 44,121 | <p>You can use Chebyshev polynomials of the second kind. The identity</p>
<p><span class="math-container">$$ U_n(x) = 2^n \prod_{k=1}^{n}\left(x-\cos\frac{\pi k}{n+1}\right) $$</span></p>
<p>leads to</p>
<p><span class="math-container">$$\log U_n(A) = n\log 2+\sum_{k=1}^{n}\log\left(A-\cos\frac{\pi k}{n+1}\right) $$</s... |
2,961,323 | <p>In transport theory in plasma physics, there's an important integral called the Coulomb logarithm, which relates to the scattering cross section off the Yukawa potential. It can be written as
<span class="math-container">$$
\ell(\Lambda) = \int_0^\infty \cos^2\left(\int_0^{u^*}\left[1 - 2\frac{u}{\xi}\exp\left(-\fra... | Yuriy S | 269,624 | <p>I have tried to expand the integrand up to first order in <span class="math-container">$1/ \Lambda$</span>, however I still obtained a divergent integral.</p>
<p>I will still provide the attempt, as it might be useful.</p>
<p>First, I denote:</p>
<p><span class="math-container">$$t=\frac{1}{\Lambda}$$</span></p>
... |
98,605 | <p>The <a href="http://en.wikipedia.org/wiki/Characteristic_%28algebra%29"><em>characteristic</em> of a ring</a> (with unity, say) is the smallest positive number $n$ such that $$\underbrace{1 + 1 + \cdots + 1}_{n \text{ times}} = 0,$$ provided such an $n$ exists. <em>Otherwise, we define it to be $0$.</em> </p>
<p>Bu... | Bill Dubuque | 242 | <p>Recall that an R-algebra is a ring A containing a central image of the ring R. This image is $\,\cong$ R/I so it is <em>characterized</em> by the kernel I. For example, if R = $\mathbb Z$ then an R-algebra is simply a ring A, and the kernel $\rm\ I = (n)\ $ characterizes the canonical image of $\mathbb Z$ in A, vi... |
98,605 | <p>The <a href="http://en.wikipedia.org/wiki/Characteristic_%28algebra%29"><em>characteristic</em> of a ring</a> (with unity, say) is the smallest positive number $n$ such that $$\underbrace{1 + 1 + \cdots + 1}_{n \text{ times}} = 0,$$ provided such an $n$ exists. <em>Otherwise, we define it to be $0$.</em> </p>
<p>Bu... | dezign | 32,937 | <p>Not only does adding 1 infinitely times in a field not make sense, but also if adding 1 a positive number of times never yields zero, then certainly adding it zero times yields zero, thus it only seems natural to refer to such a field as "characteristic zero". </p>
|
624,826 | <p>It is known that in $p\%$ of cases, Joe helps Bill in his game of chess; In all the rest of cases Bill plays alone. When he does receive help, his probability of victory is $0.95$, and without help, $0.75$. It is known that if Bill lost a game , the probability Joe helped him is $3/13$. Calculate $p$. </p>
<p>Im pr... | heropup | 118,193 | <p>This question is problematic because in chess, there is a nonzero probability of a game resulting in a draw. <strong>In order to answer the question, it is necessary to assume no draws occur.</strong> We will first operate under this assumption, and later explain why this is necessary.</p>
<p>Let $H$ be the event... |
1,503,256 | <p>There are 8 employees at a small company. The company have a Christmas tradition where each employee has to bring a small gift. The gifts are then put in a bag and later distributed by a random draw by each of the employees. </p>
<p>a) What is the probability that at least one of the employees don't not get their ... | Balloon | 280,308 | <ul>
<li><p>The 6 is true, by writing $(A^6B^6)^{-1}=(B^{-1})^6(A^{-1})^6.$</p></li>
<li><p>The 4 is true by writing $(ABA^{-1})^5=AB\underset{=I_n}{\underbrace{A^{-1}A}}BA^{-1}ABA^{-1}ABA^{-1}ABA^{-1}=AB^5A^{-1}$</p></li>
<li><p>For the 1-2-5, you can
see that they hold if $AB=BA$ but find a counter-example with $AB\... |
1,338,050 | <p>I can visualize a first-order tensor as a segment (a vector), but I'm not sure how to visualize a second-order tensor. The book that I'm trying to study is "Vector and tensor analysis with applications" (A. Borisenko and I. Tarapov)</p>
| Zach466920 | 219,489 | <p>Apparently rotation and scaling matrices can be considered tensors of rank two. This is what the matrix in a affine transformation does. Look at this answer <a href="https://math.stackexchange.com/questions/1333965/are-affine-transformation-matrices-tensors">here</a></p>
|
3,008,942 | <p>A fair 6-sided die is rolled 5 times and the result is recorded for each roll.</p>
<p>a) How many different sequences of results are possible?</p>
<p>b) Of the possible sequences of results, how many of them contain exactly 3 rolls of a 4?
<br/>
<br/>
<br/>
I'm pretty sure part (a) is <span class="math-container">... | 5xum | 112,884 | <p>For (a), you are correct.</p>
<p>For (b), first, try to solve a simpler question:</p>
<blockquote>
<p>How many results contain <span class="math-container">$4$</span> in their first three rolls, and no <span class="math-container">$4$</span> after that?</p>
</blockquote>
<p>Now, answer another similar question:... |
1,171,733 | <p>Let $\{x_n\}$ be a sequence in $\mathbb{R}$ where $\forall n\in\mathbb{N}:x_n\neq 0$ and it converges to some $x\neq 0$. If the sequence is NOT monotone is it ever true that $\frac{1}{x_n}\rightarrow\frac{1}{x}$? If so what other conditions (if any) are needed and how would you show it. </p>
<p>Thanks in advance a... | EmarJ | 45,731 | <p>Consider the mapping: $g: \mathbb{R}^+\to \mathbb{R}^+, \quad x \mapsto \frac{1}{x}$. $g$ is continuous in its domain. This implies that if $x_n \to x$ then $g(x_n) \to g(x)$, that is your thesis.</p>
<p>Same argument can be used in the case of negative values.</p>
<p>EDIT:
As MPW wrote it is sufficient to take $g... |
3,088,503 | <p>For <span class="math-container">$r\in [0,1)$</span> define
<span class="math-container">$$
I(r) = \int_0^{2\pi}\frac{\cos(t) - r}{1 - 2r\cos t + r^2}\,dt.
$$</span>
Numerical experiments hint at <span class="math-container">$I(r) = 0$</span> for all <span class="math-container">$r\in [0,1)$</span> but I can't show ... | Dr. Sonnhard Graubner | 175,066 | <p>Hint: <span class="math-container">$$\cos(t)=\frac{1-a^2}{1+a^2}$$</span> and <span class="math-container">$$dt=\frac{2da}{1+a^2}$$</span>
Now your indefinite integral is given by <span class="math-container">$$\int-\frac{2 \left(a^2 r+a^2+r-1\right)}{\left(a^2+1\right)
\left(a^2 r^2+2 a^2 r+a^2+r^2-2 r+1\right)}... |
3,321,399 | <p><span class="math-container">$$
\lim\limits_{n \to \infty} \cos^2\left(\pi \sqrt[3]{n^3+n^2+2n}\right)
$$</span>
where <span class="math-container">$n \in \mathbb{N}$</span>.</p>
<p>In this question, what I thought was, since <span class="math-container">$n \to \infty$</span> and <span class="math-container">$\cos ... | Eric Towers | 123,905 | <p><span class="math-container">$\sqrt[3]{n^3 + n^2 + 2n} = n \sqrt[3]{1 + \frac{1}{n} + \frac{2}{n^2}}$</span></p>
<p>In the limit, this cube root <span class="math-container">$\rightarrow 1 + \frac{1}{3n}$</span>, so this subexpression <span class="math-container">$\rightarrow n\pi + \pi/3$</span>, so the argument to... |
2,270,401 | <p>I have heard about Euler bricks, and i have noticed that there are an <strong>strong text</strong>infinite number of euler bricks(<a href="https://math.stackexchange.com/questions/2264401/euler-bricks-and-the-4th-dimension">as for 4D bricks we are still pending even one</a>!) in existence. I get there may be infinit... | Old Peter | 340,536 | <p><strong>You ask, “could someone actually use a formula to calculate a perfect Euler brick?”</strong></p>
<p>IMHO, it seems unlikely, for this is such an obvious way. As far as I recall, some parametric solutions have been proven to not lead to a perfect cuboid.
Massive brute force has also failed, so unless you hav... |
2,270,401 | <p>I have heard about Euler bricks, and i have noticed that there are an <strong>strong text</strong>infinite number of euler bricks(<a href="https://math.stackexchange.com/questions/2264401/euler-bricks-and-the-4th-dimension">as for 4D bricks we are still pending even one</a>!) in existence. I get there may be infinit... | Jackson Vliet | 969,141 | <p>Generating Euler Bricks is hard with just a formula, so I made a Python code to generate them for you. Of course, this will generate already known bricks, and will take a massive amount of time. For with each for loop, you have to cube the range and that is the number of sets it must test. Some improvement could be ... |
1,529,885 | <p>I have $24$ items randomly divided into $4$ groups (it can happen that group has $0$ items). How can I prove that when I remove a maximum of $12$ items (of my choice) three boxes will have the same amount?</p>
| Pieter21 | 170,149 | <p>W.l.o.g., any counter example could be written as $a,b,c,d$ where:</p>
<ol>
<li>$a \leq b \leq c \leq d$,</li>
<li>$a + b + c + d = 24$</li>
<li>$a = 0$, because for any counter example $a,b,c,d$, we have another counter example $0, b, c, d+a$</li>
<li>$b + c \geq 13$, at least 13 items must be removed for case $0,... |
1,197,201 | <p>So I was given the task of proving the following variant of the triangle inequality using only the properties of the absolute value:</p>
<p>$\vert\lvert x\rvert -\lvert y \rvert \rvert \leq \lvert x+y\rvert$</p>
<p>Now my initial instinct was to divide it to four cases based on the possible values of x and y, and ... | Fermat | 83,272 | <p><strong>The proof of triangle inequality</strong></p>
<p>As you know $xy\le |xy|=|x||y|$ $$(x+y)^2 = |x+y|^2=x^2+2xy+y^2\le |x|^2+2|x||y|+|y|^2$$ taking square root of both sides gives $|x+y|\le |x|+|y|$</p>
<p><strong>The proof of the given inequality</strong>
$$|x|=|x+y-y|\le |x+y|+|y|\implies |x|-|y|\le|x+y|$$
... |
1,857,250 | <p>I am stuck on this problem - I really don't know how to set it up.</p>
<p>A soft drink can is $h$ centimeters tall and has a radius of $r$ cm. The cost of material in the can is $.0005$ cents per cm$^2$ and the soda itself costs $.001$ cents per cm$^3$. The cans are currently $9$ cm tall and have a radius of $4$ cm... | Jan Eerland | 226,665 | <blockquote>
<p>Notice, for a cylinder:</p>
<ul>
<li><span class="math-container">$$\text{Volume cylinder}=\pi r^2h$$</span></li>
<li><span class="math-container">$$\text{Surface area cylinder}=2\pi r(r+h)$$</span></li>
</ul>
<p>You can prove that using the function <span class="math-container">$f(x)=r$</span> where <s... |
276,264 | <p>Say we are given a system of differential equations </p>
<p>$$
\left[ \begin{array}{c} x' \\ y' \end{array} \right] = A\begin{bmatrix} x \\ y \end{bmatrix}
$$ </p>
<p>Where $A$ is a $2\times 2$ matrix. </p>
<p>How can I in general solve the system, and secondly sketch a solution $\left(x(t), y(t) \right)$... | Community | -1 | <p>If you don't want change variables then,</p>
<p>there is a simple way for calculate $e^A$(all cases).</p>
<p>Let me explain about. Let A be a matrix and $p(\lambda)=\lambda^2-traço(A)\lambda+det(A)$ the characteristic polynomial. We have 2 cases:</p>
<p>$1$) $p$ has two distinct roots</p>
<p>$2$) $p$ has one roo... |
3,602,660 | <p>More clearly the question is:</p>
<p>Prove that R is a field by showing that <span class="math-container">$\mathbb{R}$</span> forms an isomorphisim to <span class="math-container">$\mathbb{C}$</span> as rings (explicitly find an isomorphism between them and prove it is an isomorphism).</p>
<p>Let <span class="math... | YiFan | 496,634 | <p>The claim that a complex polynomial of degree <span class="math-container">$n$</span> has <span class="math-container">$n$</span> roots is only true when the roots are counted with <em>multiplicity</em>. In your case, the polynomial <span class="math-container">$x^3$</span> has the root <span class="math-container">... |
179,442 | <p>Recently I've encountered the following equation in a book (don't remember the source, I just took some notes down on my scrap paper) </p>
<p>$$\dot x=t-x^2,\quad x(1)=1.$$ Then I've recognized that this is a Riccati type equation, but not explicitly solvable. The exercise proposed there was however to make a quali... | uforoboa | 15,453 | <p><strong>First point.</strong> </p>
<ul>
<li><p>Set $f(t)=-\sqrt t$. Then $f(t)$ is strictly decreasing and convex on $[1,+\infty)$. Moreover, it goes to $-\infty$ in an infinite time.</p></li>
<li><p>$\forall t=\bar t$ such that $x(t)=-\sqrt t$, $x(t)$ has a local minimum. Indeed $x'(\bar t)=0$ and $x''(\bar t)=1-2... |
106,076 | <p>I am struggling to have Mathematica 10.3 solve a system of PDE's (with periodic boundary conditions and random initial conditions), but either it produces a set of very noisy InterpolatingFunction objects that are then nearly impossible to NIntegrate, or it fails to solve it altogether. My best try so far was this:<... | Anton Antonov | 34,008 | <p>I computed the PDE system solutions without the <code>MaxStep</code> specification. (As suggested by user21.) </p>
<p>The numerical integration seems to be fast enough with the default <code>NIntegrate</code> options. If I use <code>AccuracyGoal -> 6</code> the integration becomes 3 times faster. If I remove the... |
2,030,264 | <p>I need help in proving that the function $f(A) =\operatorname{adj} (A)$ is continuous.</p>
<p>I'm not sure at all how to work with continuity and matrices..
Thanks.</p>
| Rodrigo de Azevedo | 339,790 | <p>Let $\mathrm F : \mathbb R^{n \times n} \to \mathbb R^{n \times n}$ be defined by </p>
<p>$$\mathrm F (\mathrm X) = \mbox{adj} (\mathrm X)$$</p>
<p>Note that each entry of matrix $\mathrm F (\mathrm X)$ is a <strong>cofactor</strong> of $\mathrm X$, i.e., $\pm 1$ multiplied by the determinant of a $(n-1) \times (n... |
61,030 | <p><a href="http://reference.wolfram.com/language/ref/Dataset.html"><code>Dataset</code></a> rather conveniently truncates the display of a dataset that has many rows. However, occasionally, I would like to show more of a dataset without having to convert to e.g. a <a href="http://reference.wolfram.com/language/ref/Tab... | gbrill | 27,849 | <p>That functionality is, actually, included. But it is not resident in the Dataset itself; as is always the case in Mathematica, you have to explicitly apply it. </p>
<p>The expectation of how you graphically manipulate data in Mathematica notebook UI revolves around functional approaches that you apply yourself wit... |
61,030 | <p><a href="http://reference.wolfram.com/language/ref/Dataset.html"><code>Dataset</code></a> rather conveniently truncates the display of a dataset that has many rows. However, occasionally, I would like to show more of a dataset without having to convert to e.g. a <a href="http://reference.wolfram.com/language/ref/Tab... | Trad Dog | 19,069 | <p>Helllo there, </p>
<p>I have encountered the same issues with Dataset. Resurecting one of my old thread (I have a tag for being a Necromancer;
([Previous post on generic display] [1]: <a href="https://mathematica.stackexchange.com/questions/21117/how-can-i-create-an-advanced-grid-interface-on-mathematica/57053#5705... |
3,527,470 | <p>Compute <span class="math-container">$$\lim\limits_{n\to \infty} \int\limits_0^1 x^{2019} \{nx\} dx,$$</span> where <span class="math-container">$\{a\}$</span> denotes the fractional part of the real number <span class="math-container">$a$</span>.<br>
I firstly tried to apply the substitution <span class="math-conta... | PythonSage | 596,065 | <p>I have a slightly different approach. Might not be the best. We have <span class="math-container">$\int\limits_{0}^{1}x^{2019}\{nx\}dx$</span>.</p>
<p>Say <span class="math-container">$x\in[\frac{r-1}{n},\frac{r}{n})$</span>, then <span class="math-container">$nx\in[0,1)$</span>. Hence, <span class="math-container"... |
3,144,178 | <blockquote>
<p>Let <span class="math-container">$z=a+ib,w=c+id\in\mathbb{C}. $</span>Prove <span class="math-container">$|z|-|w|\leq|z-w|$</span></p>
</blockquote>
<p>Solution: If <span class="math-container">$|z|-|w|\leq 0$</span> then we are done since <span class="math-container">$|z-w|\geq 0$</span>. Suppose <s... | J. Doe | 613,013 | <p>From triangle inequality,
<span class="math-container">$$ \\ |z|=|z-w+w|\leq |z-w|+|w|\Rightarrow
\\ |z|-|w|\leq |z-w| $$</span></p>
|
3,357,166 | <p>It is well-known that the image of a compact set is compact and the image of a connected set is connected under a continuous map. Here <span class="math-container">$A$</span> is perfect if <span class="math-container">$A$</span> it is equal to the derived set of itself.</p>
<p>I would like to ask if this property h... | Henno Brandsma | 4,280 | <p>No, any compact metric space <span class="math-container">$X$</span> is the continuous image of <span class="math-container">$\{0,1\}^\mathbb{N}$</span>, the Cantor cube, which is perfect. So also for the non-perfect <span class="math-container">$X=\{0,1\}$</span> (projection ) or the non-perfect <span class="math-c... |
1,386,549 | <p>What is the range of $x+\frac{1}{x}$ if $-2\leq x \leq 2$?<br></p>
<p>My try:<br></p>
<p>$-2\leq x \leq 2$<br></p>
<p>$\Rightarrow \frac{-1}{2}\leq x \leq \frac{1}{2}$<br></p>
<p>$\Rightarrow \frac{-5}{2}\leq x+\frac{1}{x} \leq \frac{5}{2}$<br></p>
<p>Am i correct or wrong?If not,please help me solve this.</p>
| lodrik | 230,397 | <p><strong>Hint:</strong> Just read it out loud. The first says (simplified): "For every faculty member $y$, there is a student that asked $y$ a question". The second formula reads "There is a student that asked every faculty member a question".</p>
|
27,968 | <p>What is the "standard" way to denote all positive (or non-negative) real numbers? I'd think</p>
<p>$$ \mathbb R^+ $$</p>
<p>but I believe that that is usually used to denote "all real numbers including infinity".</p>
<p>So is there a standard way to denote the set</p>
<p>$$ \{x \in \mathbb R : x \geq 0\} \; ?$$ ... | user02138 | 2,720 | <p>The following is also pretty common notation for the non-negative reals: $\mathbb{R}_{\geq 0}$ or $\mathbb{R}_{+}$.</p>
|
1,014,453 | <p>If two chords of a circle intersect and are $\perp$ to each other, is it possible to find the distance from the intersection point of the chords to the center?</p>
<p>I was trying to use the power of a point argument.</p>
| Edward Jiang | 179,276 | <p>The short answer is yes, it is possible. </p>
<p><img src="https://i.stack.imgur.com/fmbaz.png" alt="enter image description here"></p>
<p>The general outline of finding the distance $KO$ is this: </p>
<p>Given the lengths $AK, BK, CK$, and $DK$, use the Pythagorean Theorem to find the lengths of the sides of cyc... |
1,014,453 | <p>If two chords of a circle intersect and are $\perp$ to each other, is it possible to find the distance from the intersection point of the chords to the center?</p>
<p>I was trying to use the power of a point argument.</p>
| Steve Kass | 60,500 | <p>Draw congruent chords parallel to those you have, and get this symmetric picture.</p>
<p><img src="https://i.stack.imgur.com/4ELLv.png" alt="enter image description here"></p>
<p>The central rectangle has width $AB-2BK$ and height $CD-2CK$. The center of the circle is at the intersection of the rectangle's diagona... |
135,783 | <p>Assume that $ B \hookrightarrow A $ is an injective ring homomorphism. </p>
<blockquote>
<blockquote>
<p><b> Question: </b> If $\mathfrak{p}$ is an associated prime of $B$, is there an associated prime $ \mathfrak{q}$ of $A$ such that $ \mathfrak{q} \cap B = \mathfrak{p}$?</p>
</blockquote>
</blockquote>
<... | Olivier Benoist | 2,868 | <p>Let me explain why the statement is true. Let $i:B\to A$ be an injection of noetherian rings and $\mathfrak{p}$ be an associated prime of $B$ : it is the annihilator of a nonzero element $f\in B$.</p>
<p>Let us first localize at $\mathfrak{p}$. More precisely, replacing $B\to A$ by $B_{\mathfrak{p}}\to A\otimes_B B... |
2,157,250 | <p>Exponential Blowup of Subset Construction
Definition: $L_n = \{w ∈ \{0, 1\}^\star\,|\,w_{|w|-n}=1\}$. In other words,
$L_n$ is the set of binary words having a $1$ in the nth position from the end. Here $|w|$ is the length of the word $w$.</p>
<p>Note: an NFA with only $n+1$ states can be designed to recognized $L_... | dxiv | 291,201 | <blockquote>
<p>Is there a more logical way to solve this problem in limited time?</p>
</blockquote>
<p>Hint: if you take the premise to mean that there do in fact exist $a,b$ such as $f=f^{-1}$ then, for a shortcut, you can simply plug in $x=0$ and determine that $f\big(f(0)\big)=f(\frac{-1}{b})=0 \implies a=b\,$. ... |
1,280,140 | <p>If $V$ is a $k$-vector space and $T : V \to k$ is a homomorphism (linear map) that is not the zero map, is it true that some element of $V$ is mapped to $1$ of $k$? Of course $k$ is a 1-dimensional $k$-vector space generated by $1$ but how can I see that there exists a $v \in V$ such that $T(v) = 1$?</p>
| Michael Albanese | 39,599 | <p>As $T$ is not the zero map, there is $v \in V$ such that $T(v) = a \neq 0$. Then $\frac{1}{a}T(v) = 1$, so by the linearity of $T$, $T(\frac{1}{a}v) = 1$.</p>
|
4,089,740 | <p>From a complex number, <span class="math-container">$z=x+iy$</span> and its conjugate there is a simple way to evaluate the real part and imaginary part,
<span class="math-container">\begin{equation}
\text{Re}(z)=\frac{z+z^*}{2}
\end{equation}</span>
and
<span class="math-container">\begin{equation}
\text{Im}(z)=\fr... | runway44 | 681,431 | <p>The imaginary axis in <span class="math-container">$\mathbb{C}$</span> is spanned by <span class="math-container">$\{i\}$</span>, so it makes sense for <span class="math-container">$\mathrm{Im}()$</span> to extract the coefficient of <span class="math-container">$i$</span> from the expression <span class="math-conta... |
4,089,740 | <p>From a complex number, <span class="math-container">$z=x+iy$</span> and its conjugate there is a simple way to evaluate the real part and imaginary part,
<span class="math-container">\begin{equation}
\text{Re}(z)=\frac{z+z^*}{2}
\end{equation}</span>
and
<span class="math-container">\begin{equation}
\text{Im}(z)=\fr... | Nullius in Verba | 910,867 | <p>The conjugate can be defined purely in terms of quaternion addition and multiplication.</p>
<p><span class="math-container">$q^*=-{1\over 2}(q+iqi+jqj+kqk)$</span></p>
<p>Then as mentioned in other answers:</p>
<p><span class="math-container">$Re(q)={1\over 2}(q+q^*)$</span></p>
<p><span class="math-container">$Im(q... |
1,221,795 | <blockquote>
<p>Suppose $f$ is continuous on $[a, b]$, if for every continuous function $g$ on $[a, b]$ with $g(a) = g(b) = 0, \int_{a}^{b}f(x)g(x) dx = 0$, Show $f(x) = 0, \forall x \in [a, b]$, </p>
</blockquote>
<p>I want to prove by contradiction, and then find a continuous $g$ such that $g(a) = g(b) = 0$ but $... | Krishan Bhalla | 213,622 | <p>Elaborating on N. S.'s comment, if $f$ is non zero at some $x_0 \in [a,b]$, then it is non-zero in some region $[x_0- \delta , x_0 + \delta ]$. WLOG take $f > 0$ in this region.</p>
<p>Then find a $g$ s.t. $g(a) = g(b) = 0$ but $g(x) > 0$ $ \forall x \in [x_0-\delta,x_0+\delta]$ (think of defining $g$ piecewi... |
795,991 | <p>What I'm really wondering is, does Lagrange polynomial interpolation have an answer for every question of "what's the next integer in this sequence"? Does it define an infinite integer sequence to extend every finite integer sequence?</p>
<p>Let $S$ be a finite set of points, at least two of them, all having consec... | GuestODE123 | 151,435 | <p>This is simple ODE with total differential.</p>
<p>In this case we have:</p>
<p>$$P(x,y) = 3x^2y^2 - y^3 + 2x$$</p>
<p>$$Q(x,y) = 2x^3y - 3xy^2 + 1$$</p>
<p>and partial derivate of each function is </p>
<p>$$\frac{dP}{dy} = 6x^2y - 3y^2$$</p>
<p>$$\frac{dQ}{dx} = 6*x*x*y - 3*y*y$$</p>
<p>So we obtain $$\frac{... |
2,905,129 | <blockquote>
<p>If we have a matrix $A \in \mathbb{R}^{n \times n}$ and $A=T\Lambda T^{-1}$ for $T\in \mathbb{R}^{n \times n}$, how to prove that each column vector $t^i \in \mathbb{R}^n$ that $At^i = \lambda(t^i)$?</p>
</blockquote>
<p>I'm having trouble understanding how to prove that each element $t^i$ of $T$ is ... | gt6989b | 16,192 | <p>So $A$ and $T$ are both $n \times n$ matrices. Note that any eigenvalue/eigenvector pair $\vec{x}, \lambda$ must satisfy
$$
A \vec{x} = \lambda \vec{x} \quad (*)
$$
and so $\vec{x} \in \mathbb{R}^n$. If you take $k$ such pairs $\left(\vec{t}_i, \lambda_i\right)_{i=1}^k$, each one satisfying an equation like $(*)$, y... |
2,295,515 | <p>I was doing a problem and I ran into this triangle:</p>
<p><a href="https://i.stack.imgur.com/sFSxM.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/sFSxM.png" alt="enter image description here"></a></p>
<p>The problem itself is not that important for this question, but rather the explanation of ... | Pablo | 381,844 | <p>The answer to what I was looking for is called the t<strong>riangle midsegment theorem</strong>. The theorem states:</p>
<blockquote>
<p>In a triangle, the segment joining the midpoints of any two sides will be parallel to the third side and half its length</p>
</blockquote>
<p>For instance:</p>
<p><a href="htt... |
901,703 | <p>Let $G$ be a $p$-group and $H$ be a $p$-subgroup of $G$; $H$ is not normal in $G$. Then prove that the number of distinct conjugates of $H$ in $G$ divides $|G|$.
Notice that the conjugates may have nontrivial intersection.</p>
| Nicky Hekster | 9,605 | <p><strong>Hint:</strong> </p>
<p>The number of distinct conjugates of subgroup $H\leq G$ equals index $$[G:N_G(H)]$$ Here $N_G(H)=\{g \in G : g^{-1}Hg=H\}$ is the normalizer of $H$ in $G$.</p>
|
18,231 | <p>I'm stuck on the following problem :</p>
<p>Does there exist a real borel measure $\mu$ on $[0,1]$ such that</p>
<p>$$\int_{0}^{1}x^n d\mu = e^{-n^2}$$
for all $n \geq 1$ ?</p>
<p>Does anyone have any hint?</p>
<p>Thank you</p>
| Pete L. Clark | 299 | <p>I find it telling that you do not list any intermediate level courses among those you have taken: especially, real analysis, topology or algebra. These are the courses which provide the theoretical foundation for all later study of mathematics (and are, as in Roy Smith's answer, arbitrarily challenging: one cannot ... |
4,457,260 | <p>I need to translate the distance between two points into an angle from 0 to 359. To do this I use the new position coordinates which is defined by a vector and subtract the original position. This generates a difference which I can then use to define the slopes absolute value (I know it can be done with out the abso... | OGFandEGF | 1,046,673 | <p>I have a stupid question :</p>
<p>Can I replace <span class="math-container">$\binom{1}{1}\binom{24-9}{6}$</span> by <span class="math-container">$\binom{18-9+5}{5}$</span>? (It means in case <span class="math-container">$x_1=9$</span>). Thus:</p>
<p><span class="math-container">$\binom{24}{6}-\binom{14}{5}-6\binom... |
240,470 | <p>I usually produce graphics with Mathematica and then use Keynote (I am on a Mac running the latest Big Sur OS) for post processing. Until version 12.0.0.0 I would directly highlight the graphics in Mathematica, copy it (Cmd+C) and then paste it (Cmd+V) directly in Keynote; this would produce a perfect vector copy of... | ihojnicki | 18,638 | <p>It is a known bug. Unfortunately, there is no way to change the current behavior.</p>
|
2,615,176 | <p>The nature of the curve $f(x) = x (\log_{2} x)^6$ is baffling me. I expect the curve looks rather like plot of $f(x) = x\log_{2} x$; however probing on Wolfram Alpha says it does not. The shape of the graph looks more like a squashed $f(x) = x^2$ which is not all that surprising but just doesn't seem right.</p>
<p>... | Ross Millikan | 1,827 | <p>The vertical asymptote in your first graph comes because you have $x^6$ in the denominator, so overall you have $x^{-4},$ which goes to $+\infty$ as $ \to 0$. As $x$ increases $x \log^6 x$ will increase faster than $x$, but eventually slower than any $x^{1+\epsilon}$</p>
|
1,676,491 | <p>$\lim_{(x,y)\rightarrow 0}\frac{xy^2}{x^2+y^4}$</p>
<p>if we consider the path $x=y^2$ and get
$\lim_{(y^2,y)\rightarrow 0}\frac{y^2(y^2)}{(y^2)^2+y^4}=\lim_{y\rightarrow 0}\frac{y^4}{2y^4}=\frac{1}{2}$</p>
<p>but if I used Mathematrica to draw a graph of this function
<a href="https://i.stack.imgur.com/D6VWA.png"... | David K | 139,123 | <p>For
$
\lim_{(x,y)\rightarrow (0,0)}\frac{xy^2}{x^2+y^4},
$
if you set $x = -y^2$, the limit is
$$
\lim_{y\rightarrow 0}\frac{(-y^2)y^2}{(y^2)^2+y^4} = -\frac 12,
$$
and if you set either $x = 0$ or $y = 0$, the limit is
$$
\lim_{y\rightarrow 0}\frac{0(y^2)}{0^2+y^4}
= \lim_{x\rightarrow 0}\frac{x(0^2)}{x^2+0^4} = 0.... |
2,828,070 | <p>If $$\log_{3x+5} (ax^2+8x+3)>2$$ find the interval in which x lies.</p>
<p>Answer is $\left(-\dfrac43 , -\dfrac{23}{22}\right)$. Now hint me how to solve the problem please.</p>
| TRUSKI | 91,784 | <p>Let us consider an experiment:
Pick a number uniformly from the interval $[0,1]$.</p>
<p>Let the event </p>
<p>$A =$ The picked number is $\leq 0.5$, and</p>
<p>$B =$ The picked number is $\geq 0.5$.</p>
<p>Here $P(A \cap B) = P$(the picked number is .5) = 0 (property of continious uniform random variable). Hope... |
1,471,174 | <p>Let $\psi: \mathbb R \to \Bbb R$ be a given analytic function.</p>
<p>How can I construct a smooth (that is, infinitely differentiable) function $\Phi:\Bbb R\to \Bbb R$ such that $\psi|_D=\Phi|_D$ for some interval $D$, and $\Phi$ is bounded above and below?</p>
<p>E: I've been reminded in the comments that such ... | Robert Israel | 8,508 | <p>Hint: look up "bump function".</p>
|
2,059,406 | <p>I know that any index 2 subgroup is normal.</p>
<p>But is it the biggest?</p>
| Robert Israel | 8,508 | <p>I presume we're talking about finite groups here. Since the order of a subgroup must divide the order of the group, the only subgroup bigger than an index $2$ subgroup (if that exists) is the group itself.</p>
|
2,321,285 | <p>For my abstract algebra course I have to decide whether
$$(*)\qquad X^4 + a^2, \; a \in \mathbb{Z}\;\text{odd}$$ is irreducible over $\mathbb{Q}[X]$ and $\mathbb{Z}[X]$. </p>
<p>Since the degree of the polynomial is 4 and thus even I should be able to split it into two polynomials of degree two $if$ the coefficie... | José Carlos Santos | 446,262 | <p>Why did you not finish the computations? You already knew that $c+e=d^2$, that $d(c-e)=0$, and that $ce=a^2$. From the second equality, you can deduce that $d=0$ or $c=e$. But $d=0$ is not really a possibility, because then $c+e=0$, that is $c=-e$. This is not compatible with $ce=a^2$.</p>
<p>So, we know that $c+e=... |
1,479,177 | <p>If there exists $r \in \mathbb{R}$ with $r < 1, K \in \mathbb{N}$, such that $|X_n|^{1/n} < r$ for all $n > K$, then the series $X_n$ (Summation of $X_n$ from $n$ to $\infty$) is absolutely convergent.</p>
<p>We know that $|X_n| < r^n$ for all $n > K$ so sum of all terms would be bounded by sum of f... | user5216 | 250,930 | <p>$17\mathbb{Z}$ is prime ideal in $\mathbb{Z}$ and since $\mathbb{Z}$ is principle ideal domain, which kind of ideal is $17\mathbb{Z}$?</p>
|
1,590,518 | <p>Define $$p(x)= \sum_{i=1}^{n+1}2^{i}\cdot\frac{\prod_{j=1,j\neq i}^{n+1}(x-j)}{\prod_{j=1,j\neq i}^{n+1}(i-j)}.$$</p>
<p>Find $p(n+2)$.</p>
<p>The polynomial is rather complicated. I tried but can not succeed.</p>
<p>According to my teacher, the result is $2^{n+2}-1$.</p>
<p>But how we can prove it.</p>
| Dr. Sonnhard Graubner | 175,066 | <p>HINT: for 1) prove by induction that $$\sum_{i=1}^n 4i-3=2n(n+3)$$
for 2) show that $$\frac{x^4+2x^3+4x^2+6x+3}{x^3-3x^2-9x-5}={\frac {{x}^{2}+3}{-5+x}}$$
for 3) the limit doesn't exist<br>
for 4) the limit doesn't exist<br>
for 5) prove by L'Hospital that the limit is given by $$-\frac{1}{4}$$</p>
|
1,590,518 | <p>Define $$p(x)= \sum_{i=1}^{n+1}2^{i}\cdot\frac{\prod_{j=1,j\neq i}^{n+1}(x-j)}{\prod_{j=1,j\neq i}^{n+1}(i-j)}.$$</p>
<p>Find $p(n+2)$.</p>
<p>The polynomial is rather complicated. I tried but can not succeed.</p>
<p>According to my teacher, the result is $2^{n+2}-1$.</p>
<p>But how we can prove it.</p>
| Jan Eerland | 226,665 | <p>HINTS:</p>
<ul>
<li>1)</li>
</ul>
<p>$$\lim_{n\to\infty}\left(\frac{2n^2+5}{1+5+\dots+(4n-3)}+\frac{5}{n}\right)=$$
$$\lim_{n\to\infty}\left(\frac{2n^2+5}{\sum_{k=1}^{n}(4k-3)}+\frac{5}{n}\right)=$$
$$\lim_{n\to\infty}\left(\frac{2n^2+5}{2n^2-n}+\frac{5}{n}\right)=$$
$$\lim_{n\to\infty}\frac{2n^2+5}{2n^2-n}+\lim_{... |
2,468,458 | <p>Given 80 people, how do you arrange so that each person:</p>
<ul>
<li>listens to exactly 12 other people</li>
<li>is listened to by 12 other people</li>
</ul>
<p>This is an interesting question my friend posed and neither of us could find eh solution to. I feel like the answer is easy but still can't manage to fig... | Bram28 | 256,001 | <p>Split the $80$ people into $5$ clusters of $16$.</p>
<p>Then, split each cluster of $16$ into $4$ groups of $4$, and have each person within any group of $4$ listen to all the other persons in all the other $3$ groups of $4$ within that cluster of $16$.</p>
<p>If you don't want to allow two people listening to eac... |
1,384,455 | <p>Let $X$ be a topological space of infinite cardinality. Is it possible for any $X$ to be homeomorphic to $X\times X$ $?$</p>
<p>For example, $\mathbb R$ is not homeomorphic to $\mathbb R^{2}$, and $S^{1}$ is not homeomorphic to $S^{1} \times S^{1}$ . What other topological spaces might... | Tomasz Kania | 17,929 | <p>Many Banach spaces are <em>linearly</em> homeomorphic to their Cartesian squares. For instance all classical spaces including $c_0$, $\ell_\infty$, $C(K)$ for $K$ compact metric, $L_p(\mu)$ for $p\in [1,\infty]$ etc. </p>
|
2,779,458 | <p>I was working on the following definite integral:</p>
<p>$\displaystyle\int^{\infty}_{-\infty}{xe^{-x^{2}}}dx$</p>
<p>I decided to use $u$-substitution to make the integral simpler, by letting $u = -x^{2} \Rightarrow du = - 2x\hspace{1 mm}dx$.</p>
<p>The lower bound then becomes $-(-\infty)^{2}=-\infty$, while th... | Atmos | 516,446 | <p>In fact you can simply use that for $N>0$
$$
\int_{-N}^{N}xe^{-x^2}\text{d}x=-\frac{1}{2}\left(e^{-N^2}-e^{-N^2}\right)=0
$$
Then taking $\displaystyle N \rightarrow +\infty$ leads you to the result.</p>
<p>I think it is a better approach as $e^{\pm \infty}$ has no sense ( for me )</p>
<p>Note that $\displaysty... |
2,676,479 | <p>Say you have $f(x) $ and $g(x)$ and $g(x) = f^{-1}(x) $.</p>
<p>I observed that these two curves need not intersect, for example with $f(x) = e^x$ and $g(x) = \ln x $ never intersecting each other.</p>
<p>I also observed that a function can either have one, two, or three intersections with its inverse, but I was u... | flawr | 109,451 | <p>You can in fact have arbitrary many intersections with an inverse. Consider for instance the function $f(x) = \sin(x) + x$ which has an inverse on $\mathbb R$ (because it is strictly monotonic almost everywhere as $f'(x) = \cos(x)+1 \geq 0 \forall x$ and $f'(x) = \cos(x)+1 > 0 \forall x \in \pi(\mathbb Z +1/2)$)... |
2,676,479 | <p>Say you have $f(x) $ and $g(x)$ and $g(x) = f^{-1}(x) $.</p>
<p>I observed that these two curves need not intersect, for example with $f(x) = e^x$ and $g(x) = \ln x $ never intersecting each other.</p>
<p>I also observed that a function can either have one, two, or three intersections with its inverse, but I was u... | Jan Bohr | 317,480 | <p>I assume that $f\colon \mathbb{R}\rightarrow \mathbb{R}$ is a bijection, such that $g$ is defined on all of $\mathbb{R}$. This can be adjusted to other situations (say $\mathbb{R} \rightarrow (0,\infty)$ in the case of the exponential function, then $g$ is only defined on $(0,\infty)$.)</p>
<p>Assume $f(x_0) =x_0$,... |
1,078,493 | <p>How can I prove that for each $n\in\Bbb Z^+$ $$\frac{f_{2n}}{f_{2n-1}}\leq\frac{1+\sqrt{5}}{2}$$ where each $f_i$ is a term of the Fibonacci sequence. </p>
<p>Any help is really appreciated</p>
| 5xum | 112,884 | <p>Use induction. For $n=1$, the statement is obviously true. For $n+1$, use these two facts:</p>
<ul>
<li>You already know that $\frac{f_{2n}}{f_{2n-1}}\leq \frac{1+\sqrt5}{2}$</li>
<li>You know that $f_{2(n+1)} = f_{2n+2} = f_{2n+1} + f_{2n}$ and that $f_{2n+1} = f_{2n} + f_{2n-1}$.</li>
</ul>
<p>Using this, you ca... |
23,704 | <p>I hit a snag whilst revising some log rules, could anyone confirm my suspicion:</p>
<p>$$\log _b \left( x \right) = \log _b \left( y \right) \rightarrow x = y ?$$</p>
| NebulousReveal | 2,548 | <p>Yes because $b^{\log_{b} y} = b^{\log_{b} x} \Longleftrightarrow y = x$. </p>
|
23,704 | <p>I hit a snag whilst revising some log rules, could anyone confirm my suspicion:</p>
<p>$$\log _b \left( x \right) = \log _b \left( y \right) \rightarrow x = y ?$$</p>
| Mohammad Akhtar | 7,493 | <p>Yes it does, by the following argument: Suppose that $\mathrm{log}_b(x) = \mathrm{log}_b(y)$. Then $b^{\mathrm{log}_b(x)} = b^{\mathrm{log}_b(y)}$. But now (by the definition of $\mathrm{log_b(\cdot)}$) we know that $b^{\mathrm{log}_b(x)} = x$, so we conclude that $x = y$ as required.</p>
|
24,835 | <p>If we accept the axiom of choice, we take the responsibility of living in a world in which, e.g., a ball in Euclidean 3-space is equiscindable to two isometric copies of itself (Banach-Tarski). So we have a very natural and seemingly harmless axiom (that most matehmaticians are probably willing to accept and use) on... | Simon Thomas | 4,706 | <p>In fact, the opposite is true. If suitable large cardinals exist, then all projective sets of reals (i.e. the definable sets that you are likely to come across in real life) are non-pathological. More precisely, they are measurable, have the Baire property, are either countable or have a perfect subset, etc.</p>
|
24,835 | <p>If we accept the axiom of choice, we take the responsibility of living in a world in which, e.g., a ball in Euclidean 3-space is equiscindable to two isometric copies of itself (Banach-Tarski). So we have a very natural and seemingly harmless axiom (that most matehmaticians are probably willing to accept and use) on... | Andrés E. Caicedo | 6,085 | <p>(This was a reply to John that grew beyond the alloted limit.)</p>
<blockquote>
<p>large cardinals have consequences that were initially surprising to set theorists.</p>
</blockquote>
<p>But, John, this is simply because our intuitions about large cardinals took a bit to form, just as one would expect with any r... |
423,172 | <p>Let us suppose you have a certain function $f(x)$ and you want to find out in which intervals this function is decreasing, constant or increasing. I know you need to follow these steps:</p>
<ol>
<li>Find out $f'(x)$.</li>
<li>Find out the values for which $f'(x)=0$. In other words, we need to find out the zeroes of... | Berci | 41,488 | <p>There are also other kind of <em>separation points</em>, in the given example it is $x=0$, where the function is not defined.</p>
<p>So, choose arbitrary values $r$ from the intervals $(-\infty,0)$, $(0,1)$ and $(1,+\infty)$ now.</p>
<p>In case we didn't have more separation points --assuming that the given functi... |
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