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 |
|---|---|---|---|---|
606,431 | <p>Can someone explain to me how to solve this using inverse trig and trig sub?</p>
<p>$$\int\frac{x^3}{\sqrt{1+x^2}}\, dx$$</p>
<p>Thank you. </p>
| Farshad Nahangi | 50,728 | <p>You can use hyperbolic substitution, i.e. let $x=\sinh t$ then
\begin{align*}
\int\frac{x^3}{\sqrt{1+x^2}}\, dx&=\int \sinh^3 t\, dt\\
&=\int (\cosh^2t -1)\sinh t\, dt\\
&=\frac{\cosh^3 t}{3}-\cosh t+C\\
&=\frac{(\sqrt{1+x^2})^3}{3}-\sqrt{1+x^2}+C
\end{align*}
Also you can use triangle substitution :... |
2,839,945 | <blockquote>
<p>Let $p$ be a prime in $\mathbb{Z}$ of the form $4n + 1, n \in \mathbb{N}$. Show that $\left(\frac{-1}{p}\right) = 1$ (here $\left(\frac{\#}{p}\right)$ is the Legendre symbol). Hence prove that $p$ is not a prime in the ring $\mathbb{Z}[i]$.</p>
</blockquote>
<p>Here is my solution:</p>
<p>Since $p &... | Robert Soupe | 149,436 | <p>Yes, it's correct, though I'd like to unpack it a little, and work it out with a specific prime.</p>
<blockquote>
<p>Since $p > 2$, we have $$\left(\frac{-1}{p}\right) = 1$$ if and only if $$(-1)^{\frac{p - 1}{2}} \equiv 1 \pmod p$$ if and only $(-1)^{2n} \equiv 1 \pmod p$, which is true.</p>
</blockquote>
<p... |
3,075,979 | <p>Prove that <span class="math-container">$$\frac{k^7}{7}+\frac{k^5}{5}+\frac{2k^3}{3}-\frac{k}{105}$$</span> is an integer using mathematical induction.</p>
<p>I tried using mathematical induction but using binomial formula also it becomes little bit complicated.</p>
<p>Please show me your proof.</p>
<p>Sorry if t... | Angina Seng | 436,618 | <p>Call the expression <span class="math-container">$f(k)$</span>. As it's a degree <span class="math-container">$7$</span> polynomial, it obeys the recurrence
<span class="math-container">$$\sum_{j=0}^8(-1)^j\binom8jf(k-j)=0.$$</span>
Thus
<span class="math-container">$$f(k)=8f(k-1)-28f(k-2)+56f(k-3)-70f(k-4)+56f(k-5)... |
4,301,632 | <blockquote>
<p><span class="math-container">$$X^2 = \begin{bmatrix}1&a\\0&1\\\end{bmatrix}$$</span> where <span class="math-container">$a \in \Bbb R \setminus \{0\}$</span>. Solve for matrix <span class="math-container">$X$</span>.</p>
</blockquote>
<hr />
<p>I was practicing for matrix equations and this is t... | Erdel von Mises | 981,313 | <p>HINT: Let <span class="math-container">$X = (I + N)$</span> where <span class="math-container">$I$</span> is the identy matrix and <span class="math-container">$N$</span> is the nilpotent matrix with all the diagonal entries and the lower left coner being zero. Take the second power of <span class="math-container">$... |
1,050,382 | <p>In $\mathbb{R}^5$ there is given vector space $V$. Its dimension is 3. In $\mathbb{R}^{6,5}$ consider the subset $X = \{A \in \mathbb{R}^{6,5} : V \subset \ker A\}$. I have to show that $X$ is a vector space in $\mathbb{R}^{6,5}$ and find its dimension. To show that $X$ is vector space consider $x_1, x_2 \in X$ and ... | Ivo Terek | 118,056 | <p>Let's take a quick look. I'll try a sketch/give a hint. It seems that so far, so good. Indeed, take $A,B \in X$, $\lambda \in \Bbb R$. To show that $A+\lambda B \in X$, we have to show that $V \subset \ker(A+\lambda B)$, assuming that $V \subset \ker A \ \cap \ker B $. But that's true, since given ${\bf v} \in V$ we... |
1,050,382 | <p>In $\mathbb{R}^5$ there is given vector space $V$. Its dimension is 3. In $\mathbb{R}^{6,5}$ consider the subset $X = \{A \in \mathbb{R}^{6,5} : V \subset \ker A\}$. I have to show that $X$ is a vector space in $\mathbb{R}^{6,5}$ and find its dimension. To show that $X$ is vector space consider $x_1, x_2 \in X$ and ... | 2'5 9'2 | 11,123 | <p>For the dimension of $X$, if $A\in X$, $A$ has to nullify $V$ and can do anything on the $2$-dimensional orthogonal complement of $V$. So the dimension of $X$ is the same as the dimension of the space of linear transformations from a $2$-dimensional space to a $6$-dimensional space, which is $12$.</p>
<p>EDIT: A co... |
1,425,935 | <p>How would I solve this trigonometric equation?</p>
<p>$$3\cos x \cot x + \sin 2x = 0$$</p>
<p>I got to this stage: $$3 \cos x = -2 \sin^2x$$</p>
<p>Is is a dead end or is there a easier way solve this equation? </p>
| mathlove | 78,967 | <p>You might have missed $\cos x=0$.</p>
<p>$$3\cos x\cot x+\sin 2x=0$$
$$3\cos x\cdot\frac{\cos x}{\sin x}+2\sin x\cos x=0$$
$$3\cos^2x+2\sin^2x\cos x=0$$
$$\cos x(3\cos x+2\sin^2x)=0$$
$$\cos x(3\cos x+2-2\cos^2x)=0$$</p>
|
1,368,899 | <p>I am fairly new to statistics and just recently encountered queueing theory.</p>
<p>I have programmed a simulation for a $M/M/1$ queue in which I specify the inter-arrival times and service times. I input say, an exponential distribution with a mean of $1$ for both the inter-arrival and service time.</p>
<p>I also... | lab bhattacharjee | 33,337 | <p>$$\dfrac{9\alpha^2}{2\alpha^2}=\dfrac{a^2}{a-b}$$</p>
<p>$$\implies9b=9a-2a^2=-2\left(a-\dfrac94\right)^2+2\left(\dfrac94\right)^2\le2\left(\dfrac94\right)^2$$ as $a$ is real</p>
|
2,184,776 | <p>So there's an almost exact question like this here: </p>
<p><a href="https://math.stackexchange.com/questions/576268/use-a-factorial-argument-to-show-that-c2n-n1c2n-n-frac12c2n2-n1#576280">Use a factorial argument to show that $C(2n,n+1)+C(2n,n)=\frac{1}{2}C(2n+2,n+1)$</a></p>
<p>However, I'm getting stuck in just... | Lærne | 252,762 | <p>I think your solution is correct. Here is the visualization. Take $2n+1$ marbles. You want to pick $n+1$ among them. However you notice one very specific tiny red marble among them. It captures your attention and your decision to pick that marble or not to pick it comes before picking any other marbles.</p>
<p... |
2,184,776 | <p>So there's an almost exact question like this here: </p>
<p><a href="https://math.stackexchange.com/questions/576268/use-a-factorial-argument-to-show-that-c2n-n1c2n-n-frac12c2n2-n1#576280">Use a factorial argument to show that $C(2n,n+1)+C(2n,n)=\frac{1}{2}C(2n+2,n+1)$</a></p>
<p>However, I'm getting stuck in just... | Joffan | 206,402 | <p>Another route is through the recurrence relations.</p>
<p>Note that $C(2n,n+1) = C(2n,n-1)$</p>
<p>Then</p>
<p>$\begin{align}
2\cdot \left( C(2n,n+1)+C(2n,n) \right) &= C(2n,n-1)+2C(2n,n) + C(2n,n+1)\\[1ex]
&= C(2n+1,n)+C(2n+1,n+1) \\[1ex]
&= C(2n+2,n+1) \\
\end{align}$</p>
|
119,584 | <p>It is known that there are multiplicative version concentration inequalities for
sums of independent random variables. For example, the following
multiplicative version <strong>Chernoff</strong> bound.</p>
<hr>
<p><strong>Chernoff bound:</strong></p>
<p>Let $X_1,\ldots,X_n$ be independent random variables and $X_... | Iosif Pinelis | 36,721 | <p><span class="math-container">$\newcommand{\de}{\delta}$</span>
The "dependent" version of the multiplicative Chernoff bound can be proved quite similarly to the "independent" case. Indeed, let <span class="math-container">$E_{i-1}$</span> denote the conditional expectation given <span class="math-container">$X_1,\do... |
18,048 | <p>When taking a MOOC in calculus the exercises contain 5 options to select from. I then solve the question and select the option that matches my answer. Obviously only one of the options is correct. But there are (quite a few) times where my solution is wrong even though it is one of the available options. </p>
<p>My... | Tom Au | 1,333 | <p>If the answer requires a complicated formula, I would present several "permutations" of the formula, only one of which is right. Example:</p>
<p>If <span class="math-container">$f(x) = \frac{u(x)}{v(x)}$</span>, what is <span class="math-container">$f'(x)$</span>? (Quotient Rule.</p>
<blockquote>
<p>Answer... |
2,792,770 | <p>I found the following question in a test paper:</p>
<blockquote>
<p>Suppose $G$ is a monoid or a semigroup. $a\in G$ and $a^2=a$. What can we say
about $a$?</p>
</blockquote>
<p>Monoids are associative and have an identity element. Semigroups are just associative. </p>
<p>I'm not sure what we can say about $a... | Mees de Vries | 75,429 | <p>Small note: we only call such a function a random variable when $X$ <em>is</em> measurable.</p>
<p>A very simple example would be $\Omega = \Omega' = \{0, 1\}$, with $\mathcal F = \{\emptyset, \Omega\}$ and $\mathcal F' = \mathcal P(\Omega)$, and $X = \mathrm{id}_\Omega$. Then $\{1\}$ is measurable in $(\Omega', \m... |
2,459,123 | <p>My attempt:</p>
<p>Step 1 $n=4 \quad LHS = 4! = 24 \quad RHS=4^2 = 16$</p>
<p>Therefore $P(1)$ is true.</p>
<p>Step 2 Assuming $P(n)$ is true for $n=k, \quad k!>k^2, k>3$</p>
<p>Step 3 $n=k+1$</p>
<p>$LHS = (k+1)! = (k+1)k! > (k+1)k^2$ (follows from Step 2)</p>
<p>I am getting stuck at this s... | David Bowman | 366,588 | <p>Another construction: Just look at the interval $[0,1]$ and iterate this process in each interval. Let $A_n$ be the fat cantor set of measure $\displaystyle \frac{n}{n+1}$. Then $A_n^c$ is open and dense. By Baire Category, $\displaystyle \bigcap_{n \in \mathbb{N}} A_n^c$ is dense. By construction it is $G_\delta$, ... |
3,745,159 | <p>Let <span class="math-container">$A,B,C$</span> be <span class="math-container">$n\times n$</span> matrices with real entries such that their product is pairwise commutative. Also <span class="math-container">$ABC=O_{n}$</span>. If
<span class="math-container">$$k=\det\left(A^3+B^3+C^3\right).\det\left(A+B+C\right)$... | Aditya Narayan Sharma | 335,483 | <p>Assume that <span class="math-container">$d=(m,n)$</span> and <span class="math-container">$p=(2^n-1,2^m-1)$</span> so that <span class="math-container">$p | (2^n-1)$</span> & <span class="math-container">$p | (2^m-1)$</span> since <span class="math-container">$p$</span> is their GCD.
Now since p divides both of... |
1,831,134 | <p><a href="https://i.stack.imgur.com/cQEeH.jpg" rel="nofollow noreferrer">Worked examples</a></p>
<p>Can somebody please explain to me how the generator matrix is obtained when we are given the codewords of the binary code in the examples attached.</p>
<p>I tried arranging the codes in a matrix with each row being a... | Ashwin Ganesan | 157,927 | <p>Given a set $\mathcal{C}$ of codewords, before we can construct a generator matrix, we need to verify that $\mathcal{C}$ is a linear subspace - ie, the sum (and also scalar multiples in the non-binary case) of any two codewords must be a codeword. In the link given, the subsets $\mathcal{C}$ given are all subspaces... |
665,596 | <p>Let $b_n$ be the number of lists of length $100$ from the set $\{0,1,2\}$ such that the sum of their entries is $n$. How does $b_{198}$ equal ${100\choose 2}+100$?</p>
| Felix Marin | 85,343 | <p>$\newcommand{\+}{^{\dagger}}
\newcommand{\angles}[1]{\left\langle #1 \right\rangle}
\newcommand{\braces}[1]{\left\lbrace #1 \right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack #1 \right\rbrack}
\newcommand{\ceil}[1]{\,\left\lceil #1 \right\rceil\,}
\newcommand{\dd}{{\rm d}}
\newcommand{\down}{\downarrow}
\new... |
1,134,177 | <p>Consider this primality test: Fix an initial segment of primes (e.g. 2,3,5,7), and combine a $b$-pseudoprime test for each b in that list. For several such initial segments, find the first $n$ for which the test gives an incorrect answer.</p>
<p>Hey all! I'm not quite understanding what the aforementioned question ... | Joffan | 206,402 | <p>My interpretation is as follows:</p>
<p>Taking some group of primes $p_i =\{2,3,\ldots\}$, then for numbers $n$ in turn not divisible by any of these, apply the Fermat pseudoprime test using each $p_i$ to each $n$:
$$ p_i^{n-1} \equiv1 \bmod n \implies n \text{ is prime} $$</p>
<p>Find the first number that gets... |
2,369,081 | <blockquote>
<p>Evaluate the integral $$\int_0^1\frac{x^7-1}{\log (x)}\,dx $$</p>
</blockquote>
<p>[1]: <a href="https://i.stack.imgur.com/lcK2p.jpgplz" rel="nofollow noreferrer">https://i.stack.imgur.com/lcK2p.jpgplz</a> I'm trying to solve this definite integral since 2 hours. Please, I need help on this.</p>
| sn24 | 199,128 | <p>You can use feynman approach
Consider \begin{align}
f(n)=\int_0^1 \frac{x^n-1}{\log(x)}\,dx&
\end{align}
Then
\begin{align}
f'(n)=\int_0^1 \frac{x^n\log(x)}{\log(x)}\,dx&
\end{align}
So \begin{align}
f'(n)= \frac{1}{n+1}
\end{align}
\begin{align}
f(n)=\log(n+1)+c
\end{align}
For c \begin{align}
f(0)=0 , c=... |
3,336,311 | <p><strong>Help me factor these polynomials</strong> </p>
<ul>
<li><span class="math-container">$(x+\sqrt{2})^2$</span> - 8</li>
<li>14a - 49<span class="math-container">$a^2$</span> + 100<span class="math-container">$b^2$</span> - 1</li>
</ul>
| David G. Stork | 210,401 | <p><span class="math-container">$$(x + 3 \sqrt{2})(x - \sqrt{2})$$</span></p>
<p><span class="math-container">$$7 a (2 - 7a) + (10 b + 1)(10 b - 1)$$</span></p>
|
1,823,736 | <p><a href="http://www.math.drexel.edu/~dmitryk/Teaching/MATH221-SPRING'12/Sample_Exam_solutions.pdf" rel="nofollow">Problem 10c from here</a>.</p>
<blockquote>
<p>Thirteen people on a softball team show up for a game. Of the $13$ people who show up, $3$ are women. How many ways are there to choose $10$ players ... | Charles | 306,633 | <p>This looks like a homework problem, and I think there might even be a tag for such problems.</p>
<p>Regardless, from the graph you can see that you can construct the area with two integrals (vertical <em>or</em> horizontal). I'll illustrate vertical.</p>
<p>The total area between functions $f(x)$ and $g(x)$ where ... |
1,703,120 | <p>So I have a vector <span class="math-container">$a =( 2 ,2 )$</span> and a vector <span class="math-container">$b =( 0, 1 )$</span>.<br />
As my teacher told me, <span class="math-container">$ab = (-2, -1 )$</span>.</p>
<p><span class="math-container">$ab = b-a = ( 0, 1 ) - ( 2, 2 ) = ( 0-2, 1-2 ) = ( -2, -1 )$</sp... | MPW | 113,214 | <p>They are related by the fact that $$\mathbf a- \mathbf b = -(\mathbf b- \mathbf a)$$</p>
<p>The difference is the direction. Generally, the vector from a starting point to an ending point is $$(\textrm{terminal point})-(\textrm{initial point})$$</p>
|
2,187,509 | <p>We're currently implementing the IBM Model 1 in my course on statistical machine translation and I'm struggling with the following appplication of the chain rule.
When applying the model to the data, we need to compute the probabilities of different alignments given a sentence pair in the data. In other words to com... | Barry Cipra | 86,747 | <p>Rewrite the inequality to be proved as</p>
<p>$$\left(1+u\over2\right)^2e^{(1+u)/2}-\ln\left(1+u\over2\right)-1\gt0$$</p>
<p>for $-1\lt u\lt1$. This can be rewritten as</p>
<p>$$(1+u)^2\sqrt ee^{u/2}-4\ln(1+u)+4\ln2-4\gt0$$</p>
<p>Now $e^{u/2}\ge1+{u\over2}$, so </p>
<p>$$(1+u)^2\sqrt ee^{u/2}\ge(1+2u)\sqrt e\... |
66,009 | <p>Hi I have a very simple question but I haven't been able to find a set answer. How would I draw a bunch of polygons on one graph. The following does not work:</p>
<pre><code>Graphics[{Polygon[{{989, 1080}, {568, 1080}, {834, 711}}],
Polygon[{{1184, 1080}, {989, 1080}, {834, 711}, {958, 541}}],
Polygon[{{1379,... | Szabolcs | 12 | <p>This is not a full answer, just a start towards a solution.</p>
<p>The culprit is <code>Dispatch</code>, which became <a href="http://reference.wolfram.com/mathematica/ref/AtomQ.html" rel="noreferrer">atomic</a> in version 10, and comparison wasn't implemented for it.</p>
<p>Here's a small test in version 9:</p>
<pr... |
71,166 | <p>This question have been driving me crazy for months now. This comes from work on multiple integrals and convolutions but is phrased in terms of formal power series.</p>
<p>We start with a formal power series</p>
<p>$P(C) = \sum_{n=0}^\infty a_n C^{n+1}$</p>
<p>where $a_n = (-1)^n n!$</p>
<p>With these coefficien... | Gerald Edgar | 454 | <p>Of course your series $P(C)$ diverges. But it is a transseries. Or an asymptotic series. In fact, one of the best known. The series
$$
\sum_{n=0}^\infty (-1)^n n! C^{n+1}
$$
is the asymptotic series (as $C \downarrow 0$) for the function
$$
p(C) = -e^{1/C} \mathrm{Ei}(-1/C) .
$$
So, of course, your series $P_T(C... |
2,263,230 | <p>Let's say I wanted to express sqrt(4i) in a + bi form. A cursory glance at WolframAlpha tells me it has not just a solution of 2e^(i<em>Pi/4), which I found, but also 2e^(i</em>(-3Pi/4))</p>
<p>Why do roots of unity exist, and why do they exist in this case? How could I find the second solution? </p>
| symplectomorphic | 23,611 | <p>You're solving the equation $z^2=4i$. According to the Fundamental Theorem of Algebra, this equation has two complex roots. You can find them in many ways. </p>
<p>The most elementary approach: assume $z=a+bi$, where $a$ and $b$ are real. Then $(a+bi)^2=(a^2-b^2)+2abi=4i$. Equating real and imaginary parts, you nee... |
119,904 | <p>It is possible to do simple math between TemporalSeries objects. For example</p>
<pre><code>es=EventSeries[{{{2016, 1, 1}, 2}, {{2016, 1, 3}, 2.1}}];
td=TemporalData[{{{2016, 1, 1}, 2}, {{2016, 1, 3}, 3.1}}];
es*td (* works fine *)
</code></pre>
<p>This returns a TemporalData object with the path <code>{{366059520... | Michael Stern | 86 | <p>Given two <code>TimeSeries</code> (<code>ts1</code> and <code>ts2</code>), I was able to speed up my results about 60x with the following:</p>
<ol>
<li><code>paths=Map[#["Path"] &, {ts1,ts2}];</code></li>
<li><code>commonDates=Intersection[paths[[1, All, 1]], paths[[2, All, 1]]];</code></li>
<li><code>fakeDatep... |
119,904 | <p>It is possible to do simple math between TemporalSeries objects. For example</p>
<pre><code>es=EventSeries[{{{2016, 1, 1}, 2}, {{2016, 1, 3}, 2.1}}];
td=TemporalData[{{{2016, 1, 1}, 2}, {{2016, 1, 3}, 3.1}}];
es*td (* works fine *)
</code></pre>
<p>This returns a TemporalData object with the path <code>{{366059520... | Jonathan Kinlay | 36,994 | <p>One of the regular tasks in statistical arbitrage is to compute correlations between a large universe of stocks, such as the S&P500 index members, for example. Mathematica/WL has some very nice features for obtaining financial data and manipulating time series. And of course it offers all the commonly required s... |
119,904 | <p>It is possible to do simple math between TemporalSeries objects. For example</p>
<pre><code>es=EventSeries[{{{2016, 1, 1}, 2}, {{2016, 1, 3}, 2.1}}];
td=TemporalData[{{{2016, 1, 1}, 2}, {{2016, 1, 3}, 3.1}}];
es*td (* works fine *)
</code></pre>
<p>This returns a TemporalData object with the path <code>{{366059520... | sakra | 68 | <p>For some applications, an alternative to using <code>TimeSeries</code> expressions is using associations which support both a fast <code>KeyIntersection</code> and <code>KeyUnion</code> operation.</p>
<p>The example by Jonathan Kinlay from another answer to this question can be rewritten using associations in the fo... |
85,374 | <p>I'm currently using <code>WolframLibraryData::Message</code> to generate messages from a library function, like this:</p>
<pre><code>Needs["CCompilerDriver`"]
src = "
#include \"WolframLibrary.h\"
DLLEXPORT mint WolframLibrary_getVersion() {return WolframLibraryVersion;}
DLLEXPORT int Wolfra... | Simon Woods | 862 | <p>You have not defined any message text in <em>Mathematica</em>.</p>
<p>The text you supply in the C code is the message tag, e.g</p>
<pre><code>libData->Message("myerror");
</code></pre>
<p>Then you need to define the actual message content in <em>Mathematica</em>:</p>
<pre><code>LibraryFunction::myerror = "He... |
693,640 | <p>Assmue $f(x_{1},x_{2},\cdots,x_{n})$ is a second degree real polynomial with $n(n\ge 2)$ variables.
Let $S$ be such that $f(x_{1},x_{2},\cdots,x_{n})$ is the set of minimum and maximum points.
In other words：
$$S=\{(b_{1},b_{2},\cdots,b_{n})\in R^n| f(x_{1},x_{2},\cdots,x_{n})\le f(b_{1},b_{2},\cdots,b_{n}),\forall... | Einar Rødland | 37,974 | <p>There is actually a simple reason why this is true. The set $S$ of extremal points of a second degree polynomial must be an affine subspace. An affine subspace which is invariant under permutations of the coordinates must contain a point on the form $(p,\ldots,p)$.</p>
<p>To see this, assume we have two distrinct p... |
2,756,139 | <p>Let $I=[0,1]$ and $$X = \prod_{i \in I}^{} \mathbb{R}$$
That is, an element of $X$ is a function $f:I→\mathbb{R}$.</p>
<p>Prove that a sequence $\{f_n\}_n ⊆ X$ of real functions converges to some $f ∈ X$ in the product topology on $X$, if and only if it converges pointwise, i.e. for every $x ∈ I$, $f_n(x) → f(x)$ i... | Henno Brandsma | 4,280 | <p>An element of $X = \prod_{i \in [0,1]} \mathbb{R}$ is just a function from $[0,1]$ to $\mathbb{R}$, i.e. a "choice" of a point in $\mathbb{R}$ for each $x \in [0,1]$. It's the same whether we write $(x_i)_{i \in [0,1]}$ or just the function $f$ that sends $i$ to $x_i$. Just writing $f$ is often easier in notations.<... |
1,493,965 | <p>We're given the power series $$ \sum_1^\inf \frac{j!}{j^j}z^j$$</p>
<p>and are asked to find radius of convergence R. I know the formula $R=1/\limsup(a_n ^{1/n})$, which leads me to compute $\lim \frac{j!^{1/j}}{j}$, and then I'm stuck.</p>
<p>The solution manual calculates R by $1/\lim|\frac{a_{j+1}}{a_j}|$, but ... | Julián Aguirre | 4,791 | <p>$Y$ is a normed space with norm defined as $\|x\|=\sup_{|s-t_0|\le\delta}|x(s)|$. From here we get a distance $d(x,y)=\|x-y\|$. What you have shown is
$$
\|\Delta T(x)\|=_{\text{def}}\|T(x+\Delta x)-T( x)\|\le q\|\Delta x\|=_{\text{def}}q\|(x+\Delta x)-x\|.
$$
Call $y=x+\Delta x$. The above inequality is now
$$
\|... |
3,693,196 | <p>This is known as the factor formula. It is used for the addition of sin functions. I don't understand how the two are equal though. How would you get to the right side of the equation using the left?</p>
<p><span class="math-container">$$\sin(s) + \sin(t) = 2 \sin\left(\frac{s+t}{2}\right) \cos \left(\frac{s-t}{2}\... | Anas A. Ibrahim | 650,028 | <p><span class="math-container">$$P(A \text{ given that } B) = \frac{P(A \text{ and } B)}{P(B)}$$</span>
where <span class="math-container">$A,B$</span> are events, this is another way of writing <strong>Bayes' theorem</strong>. </p>
<p>Just take <span class="math-container">$A$</span> as the event that the train will... |
2,664,370 | <p>I don't know how to start, i've noticed that it can be written as $$\lim_{x\to 0} \frac{2^x+5^x-4^x-3^x}{5^x+4^x-3^x-2^x}=\lim_{x\to 0} \frac{(5^x-3^x)+(2^x-4^x)}{(5^x-3^x)-(2^x-4^x)}$$</p>
| Michael Rozenberg | 190,319 | <p>$$\lim_{x\rightarrow0}\frac{2^x+5^x-4^x-3^x}{5^x+4^x-3^x-2^x}=\lim_{x\rightarrow0}\frac{\frac{2^x-1}{x}+\frac{5^x-1}{x}-\frac{4^x-1}{x}-\frac{3^x-1}{x}}{\frac{5^x-1}{x}+\frac{4^x-1}{x}-\frac{3^x-1}{x}-\frac{2^x-1}{x}}=$$
$$=\frac{\ln2+\ln5-\ln4-\ln3}{\ln5+\ln4-\ln3-\ln2}.$$
I used $$\lim_{x\rightarrow0}\frac{a^x-1}{... |
377,354 | <p>I'm referencing this page: <a href="http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/cfINTRO.html#sqrtalgsalg" rel="nofollow">An Introduction to the Continued Fraction</a>, where they explain the algebraic method of solving the square root of $14$.</p>
<p>$$\sqrt{14} = 3 + \frac1x$$</p>
<p>So, $x_0 = 3... | TonyK | 1,508 | <p>It's really easy to compute $\left\lfloor \frac{\sqrt a + b}{c}\right\rfloor$ for integers $a,b,c$. Just use the fact that</p>
<p>$$\left\lfloor \frac{r + b}{c}\right\rfloor = \left\lfloor \frac{\lfloor r \rfloor + b}{c}\right\rfloor$$</p>
<p>for real $r$ and integers $b,c$.</p>
<p>Here, $\lfloor\sqrt{14}\rfloor ... |
1,253,475 | <p>I'm trying to prove the following statement: if E is a subspace of V, then dim E + dim $E^{\perp}$ = dim V. I know this is true because when these two subspaces are added, they are equal to V, but I'm not sure how to rigorously say this, could I get a little help?</p>
| mookid | 131,738 | <p>You need to prove that $$
V = E + E^\perp
$$ and $$
E \cap E^\perp = \{0\}
$$
then it follows from an elementary theorem.</p>
|
302,179 | <p>The question I am working on is:</p>
<blockquote>
<p>"Use a direct proof to show that every odd integer is the difference of two squares."</p>
</blockquote>
<p>Proof:</p>
<p>Let n be an odd integer: $n = 2k + 1$, where $k \in Z$</p>
<p>Let the difference of two different squares be, $a^2-b^2$, where $a,b \in Z... | AndreasT | 53,739 | <p>Here you have another approach...</p>
<p>Note that
$$
\sum_{i=1}^n
{
i
}
=
\frac{n(n+1)}2
$$
so that the sum of the first $n$ odd naturals is
$$
\sum_{i=1}^n
{
(2i-1)
}
=
2\left(
\sum_{i=1}^n
{
i
}
\right)
- n
=
n(n+1)-n
=
n^2
$$
We have showed that the sum of the fir... |
2,260,051 | <p>Let $M$ and $N$ be two square matrices of same order, and $M^2 = N^4$.</p>
<p>Can any such $M,N$ exist when the following relations do not hold?</p>
<ol>
<li><p>$M = N^2$, and </p></li>
<li><p>$M = -N^2$ ?</p></li>
</ol>
| badjohn | 332,763 | <p>There are plenty of nilpotent matrices. So, pick $M$ of degree 2 and $N$ of degree 4.</p>
<p>Or look for roots of the identity matrix. $M$ could be any reflection and $N$ a rotation by 90 degrees.</p>
|
4,326,547 | <p>I've solved linear ODEs before. This however is something completely new to me. I want to solve it without using approximations or anything.</p>
<p><span class="math-container">$s''( t) s( t) =( s'( t))^{2} +B( s( t))^{2} s'( t) -g\cdot s( t) s'( t)$</span></p>
<p>These are the equations I started with</p>
<p><spa... | Highvoltagemath | 664,767 | <p>Me and my friend got it! (This was also exactly what Sal in the comments suggested doing)
<span class="math-container">$ \begin{array}{l}
v( s) =s'( t) \Longrightarrow \\
\frac{dv( s)}{dt} =\frac{dv( s)}{ds} \cdot \frac{ds}{dt} =v'( s) \cdot v( s) \ \text{literally the chain rule}\\
\Longrightarrow \\
v'( s) \cdot v... |
3,853,509 | <blockquote>
<p>prove <span class="math-container">$$\sum_{cyc}\frac{a^2}{a+2b^2}\ge 1$$</span> holds for all positives <span class="math-container">$a,b,c$</span> when <span class="math-container">$\sqrt{a}+\sqrt{b}+\sqrt{c}=3$</span> or <span class="math-container">$ab+bc+ca=3$</span></p>
</blockquote>
<hr />
<p><str... | José Carlos Santos | 446,262 | <p>Note that<span class="math-container">\begin{align}f(x)&=\frac1{2+3x^2}\\&=\frac12\times\frac1{1+\frac32x^2}\\&=\frac12\left(1-\frac32x^2+\frac{3^2}{2^2}x^4-\frac{3^3}{2^3}x^6+\cdots\right)\text{ if }\left|\frac32x^2\right|<1\\&=\frac12-\frac3{2^2}x^2+\frac{3^2}{2^3}x^4-\frac{3^3}{2^4}x^6+\cdots\e... |
3,540,613 | <p>The integral to solve:</p>
<p><span class="math-container">$$
\int{5^{sin(x)}cos(x)dx}
$$</span></p>
<p>I used long computations using integration by parts, but I don't could finalize:</p>
<p><span class="math-container">$$
\int{5^{sin(x)}cos(x)dx} = cos(x)\frac{5^{sin(x)}}{ln(5)}+\frac{1}{ln(5)}\Bigg[ \frac{5^{s... | user577215664 | 475,762 | <p><span class="math-container">$$\int{5^{sin(x)}cos(x)dx} = \cos(x)\frac{5^{sin(x)}}{ln(5)}+.........$$</span></p>
<p>You made a confusion with:</p>
<p><span class="math-container">$$\int a^xdx =\int e^{x \ln |a|}dx=\dfrac {a^x}{\ln |a|}$$</span>
You don't have <span class="math-container">$x$</span> here but a func... |
4,550,991 | <p>This is question is taken from an early round of a Norwegian national math competition where you have on average 5 minutes to solve each question.</p>
<p>I tried to solve the question by writing every number with four digits and with introductory zeros where it was needed. For example 0001 and 0101 would be the numb... | Lourrran | 1,104,122 | <p>Consider an alphabet with 9 letters (0,1,2,3,5,6,7,8,9)</p>
<p>Consider all words that you can create with 4 letters in this alphabet : <span class="math-container">$9^4$</span> words.</p>
<p>So <span class="math-container">$4 \times 9^4$</span> letters will be used.</p>
<p>Each letter is used equally, so each lette... |
302 | <p>I know that the Fibonacci numbers converge to a ratio of .618, and that this ratio is found all throughout nature, etc. I suppose the best way to ask my question is: where was this .618 value first found? And what is the...significance?</p>
| John Stillwell | 1,587 | <p>As previous answers have pointed out, both the golden ratio and the Fibonacci numbers go back thousands of years. However, I believe the connection
between the two was discovered around 1730. At that time, Daniel Bernoulli and
Abraham de Moivre independently came up with the generating function
for the Fibonacci num... |
3,060,250 | <p>Let <span class="math-container">$f : \mathbb{R} \to \mathbb{R}$</span> be differentiable. Assume that <span class="math-container">$1 \le f(x) \le 2$</span> for <span class="math-container">$x \in \mathbb{R}$</span> and <span class="math-container">$f(0) = 0$</span>. Prove that <span class="math-container">$x \le f... | EuxhenH | 281,807 | <p>I am assuming that you mean <span class="math-container">$1\leq f'(x) \leq 2$</span>, otherwise, as stated by John Omielan, it is not possible for <span class="math-container">$f(0)$</span> to be <span class="math-container">$0$</span>.</p>
<p>Since <span class="math-container">$f$</span> is differentiable on any i... |
58,926 | <p>Is it well known what happens if one blows-up $\mathbb{P}^2$ at points in non-general position (ie. 3 points on a line, 6 on a conic etc)? Are these objects isomorphic to something nice? </p>
| Francesco Polizzi | 7,460 | <p>In both examples you are considering, the anticanonical model is a <em>singular</em> del Pezzo surface.</p>
<p>In fact, let $X$ be the blow-up of $\mathbb{P}^2$ at three points lying on a line $L$. By Bezout's theorem, the birational map associated with the linear system of cubics through the three points contract... |
8,567 | <p>When highlighting text using <code>Style</code> and <code>Background</code>, as in <code>Style["Test ", White, Background -> Lighter@Blue]</code> is there a way to pad (ie, enlarge) the bounding box? </p>
<p>The bottom of the background seems coincident with the base of the text: <img src="https://i.stack.imgu... | Szabolcs | 12 | <p>If I understand it correctly, the gist of your question is going from</p>
<pre><code>HoldComplete[x, y, z, up]
</code></pre>
<p>to</p>
<pre><code>HoldComplete[1, y^2, z, up]
</code></pre>
<p>assuming the following definitions:</p>
<pre><code>_[___, up, ___] ^= "UpValueEvaluated"
x = 1
f[x_] := x^2
</code></pre>... |
3,335,060 | <blockquote>
<p>The numbers of possible continuous <span class="math-container">$f(x)$</span> defiend on <span class="math-container">$[0,1]$</span> for which <span class="math-container">$I_1=\int_0^1 f(x)dx = 1,~I_2=\int_0^1 xf(x)dx = a,~I_3=\int_0^1 x^2f(x)dx = a^2 $</span> is/are</p>
<p><span class="math-container"... | Yuriy S | 269,624 | <p>An extended comment.</p>
<p>Not really a proof, but an interesting consequence:</p>
<p><span class="math-container">$$\log 20=4 \log 2+\log \left(1+\frac{1}{4}\right)<3$$</span></p>
<p><span class="math-container">$$\log 2< \frac34 -\frac14 \log \left(1+\frac{1}{4}\right) $$</span></p>
<p><span class="math... |
1,803,416 | <p>Does the function $d: \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$ given by:
$$d(x,y)= \frac{\lvert x-y\rvert} {1+{\lvert x-y\rvert}}$$ define a metric on $\mathbb{R}^n?$</p>
<p>How do you go about proving this? Do I need to just show that it satisfies the three conditions to be a metric? If so how do I show t... | Math1000 | 38,584 | <p>In general, if $(E,d)$ is a metric space, then $d':=\frac d{1+d}$ is a metric. The triangle inequality is the only nontrivial property here. If $x,y,z\in E$, then $d(x,z)\leqslant d(x,y)+d(y,z)$, so so monotonicity of the map $t\mapsto\frac t{1+t}$ on $[0,\infty)$ yields
\begin{align}
d'(x,z) &= \frac{d(x,z)}{1+... |
127,225 | <p>I got stuck solving the following problem:</p>
<pre><code>Table[Table[
Table[
g1Size = x; g2Size = y;
vals =
FindInstance[(a1 - a2) - (b1 - b2) == z && a1 + b1 == g1Size &&
a2 + b2 == g2Size && a1 + a2 == g1Size && b1 + b2 == g2Size &&
a1 > 0 &am... | mikado | 36,788 | <p>Here is an implementation based on <code>ListConvolve</code></p>
<pre><code>lcimplementation[mat_, n_] :=
With[{c = ConstantArray[1, Length[First[mat]]]},
Last[Fold[
Take[ListConvolve[{c, #2}, #1, {1, -1}, 0, Times, Plus, 1],
UpTo[n + 1]] &, {c}, mat]]]
</code></pre>
<p>Compare this with the ref... |
2,612,416 | <p>Can you please help me with this limit? I can´t use L'Hopital rule.</p>
<p>$$\lim_{x\to \infty} \frac{\sqrt{4x^2+5}-3}{\sqrt[3]{x^4}-1} $$</p>
| marwalix | 441 | <p>Rewrite the fraction as follows</p>
<p>$${\sqrt{4+{5\over x^2}}-{3\over x}\over \sqrt[3]{x}-{1\over x}}$$</p>
<p>And this has $0$ as limit at $+\infty$</p>
<p>At $1$ we have a $0/0$ indetermination. Many ways to solve, the most elementary being the conjugate radicals.</p>
<p>Multiply both numerator and denominat... |
90,459 | <p>I want to find the degree of $\mathbb{Q}(\sqrt{3+2\sqrt{2}})$ over $\mathbb{Q}$. I observe that
$3+2\sqrt{2}=2+2\sqrt{2}+1=(\sqrt{2}+1)^2$ so
$$
\mathbb{Q}(\sqrt{3+2\sqrt{2}})=\mathbb{Q}(\sqrt{2}+1)=\mathbb{Q}(\sqrt{2})
$$
so the degree is 2.</p>
<p>Is there a more mechanical way to show this without noticing the... | Bill Dubuque | 242 | <p>This reduces to checking if the radical <span class="math-container">$\:\sqrt{3+2\sqrt{2}}\:$</span> denests. While there are <a href="https://math.stackexchange.com/a/4697/242">general algorithms</a>, simple cases like this can be tackled by employing an easy formula that I discovered as a teenager.</p>
<p><strong>... |
1,831,191 | <p>I am confused about the following Theorem:</p>
<p>Let <span class="math-container">$f: I \to \mathbb{R}^n$</span>, <span class="math-container">$a \in I$</span>. Then the function <span class="math-container">$f$</span> is differentiable at <span class="math-container">$a$</span> if and only if there exists a functi... | Eric Wofsey | 86,856 | <p>The theorem states that $\varphi(a)=f'(a)$ for this particular value of $a$. It doesn't say that $\varphi(x)=f'(x)$ for <em>all</em> $x$, or indeed for any value of $x$ besides the single value $x=a$. So the fact that $\varphi$ is continuous at $a$ doesn't tell you that $f'$ is continuous at $a$, since continuity ... |
4,080,234 | <p>This may be a stupid question but I was looking over a proof and one of the steps simplifies <span class="math-container">$|x|/x^2=1/|x|$</span> and I was wondering what the rigorous justification of that is. Is it because <span class="math-container">$x^2$</span> is essentially the same as <span class="math-contai... | Aditya_math | 798,141 | <p>Yes!</p>
<p>For intuition, split into <span class="math-container">$2$</span> cases, <span class="math-container">$x$</span> positive and negative</p>
|
4,080,234 | <p>This may be a stupid question but I was looking over a proof and one of the steps simplifies <span class="math-container">$|x|/x^2=1/|x|$</span> and I was wondering what the rigorous justification of that is. Is it because <span class="math-container">$x^2$</span> is essentially the same as <span class="math-contai... | lone student | 460,967 | <p>If <span class="math-container">$x\in\mathbb R$</span>, then <strong>by definition</strong> of absolute value:</p>
<ul>
<li><p>If <span class="math-container">$x>0$</span>, then <span class="math-container">$|x|=x$</span>, which implies <span class="math-container">$$|x|^2=x^2$$</span></p>
</li>
<li><p>If <span c... |
1,902,138 | <p>It's common to see a plus-minus ($\pm$), for example in describing error
$$
t=72 \pm 3
$$
or in the quadratic formula
$$
x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}
$$
or identities like
$$
\sin(A \pm B) = \sin(A) \cos(B) \pm \cos(A) \sin(B)
$$</p>
<p>I've never seen an analogous version combining multiplication with div... | guestDiego | 338,527 | <p>Perhaps because
$$
a\frac{\times}{\div}b
$$
(typographically quite horrible) is written as
$$
a\cdot b^{\pm1}
$$</p>
|
1,902,138 | <p>It's common to see a plus-minus ($\pm$), for example in describing error
$$
t=72 \pm 3
$$
or in the quadratic formula
$$
x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}
$$
or identities like
$$
\sin(A \pm B) = \sin(A) \cos(B) \pm \cos(A) \sin(B)
$$</p>
<p>I've never seen an analogous version combining multiplication with div... | gowrath | 255,605 | <p>There are many function (square roots for ex.) where $f(x) = f(-x)$ and for $f^{-1}$, the $\pm$ is useful. If there were common functions where $f(x) = f(\frac{1}{x})$ it might be a thing. Can anyone think of example functions like this? </p>
|
4,121,607 | <p>I want to find a function which satisfies certain following limits.</p>
<p>The question is:
Find a function which satisfies</p>
<p><span class="math-container">$$
\lim_{x\to5} f(x)=3, \text{ and } f(5) \text{ does not exist}
$$</span></p>
<p>I would think that because it says <span class="math-container">$f(5)$</spa... | jjagmath | 571,433 | <p>Take the following function:</p>
<p><span class="math-container">$f:\Bbb R \setminus \{5\} \to \Bbb R$</span> given by <span class="math-container">$f(x) =3$</span></p>
|
4,017,554 | <p>Simple question - how to prove that:</p>
<p><span class="math-container">$\sqrt {x\times y} = \sqrt x \times \sqrt y$</span> ?</p>
<p>If I use the exponentation the answer seems easy, because</p>
<p><span class="math-container">$(x\times y)^n = x^n \times y^n$</span> because I get</p>
<p><span class="math-containe... | J.G. | 56,861 | <p>If <span class="math-container">$z:=\sqrt{x}\sqrt{y}\ge0$</span> then <span class="math-container">$z^2=\sqrt{x}\sqrt{y}\sqrt{x}\sqrt{y}=\sqrt{x}\sqrt{x}\sqrt{y}\sqrt{y}=xy$</span> so <span class="math-container">$z=\sqrt{xy}$</span>.</p>
|
97,672 | <p>Given that I have a set of equations about varible $x_0,x_1,\cdots,x_n$, which own the following style:</p>
<p>$
\left(
\begin{array}{cccccccc}
\frac{1}{6} & \frac{2}{3} & \frac{1}{6} & 0 & 0 & 0 & 0 & 0 \\
0 & \frac{1}{6} & \frac{2}{3} & \frac{1}{6} & 0 & 0 & 0... | xzczd | 1,871 | <pre><code>l = 5; s = 3;
(* Solution 1 *)
# + SparseArray[#2, {l, l}] & @@ Internal`PartitionRagged[mat\[Transpose], {l, s}];
LinearSolve[%\[Transpose], yValues]
(* Solution 2 *)
Module[{m = #[[;; l]]}, m[[;; s]] += #[[-s ;;]]; m] &[mat\[Transpose]];
LinearSolve[%\[Transpose], yValues]
</code></pre>
|
126,553 | <p>In <a href="http://www.icpr2010.org/pdfs/icpr2010_MoAT5.1.pdf" rel="nofollow">this paper</a>, in the Formula at the beginning of 2.2, we have</p>
<p>$B=\{b_i(O_t)\}$</p>
<p>where </p>
<p>$i=0,1$ - the number of probability formula</p>
<p>$O_t$ - the state at moment $t$</p>
<p>$b_i(O_t)$ - two probabilities or e... | yoyostein | 28,012 | <p>My guess is that he meant it as $B_i:=b_i(O_t)$</p>
|
2,352,313 | <p>If $f_n$ is the number of permutations of numbers $1$ to $n$ that no number is in it's place(I think same as $D_n$)and $g_n$ is the number of the same permutations with exactly one number in it's place Prove that $\mid f_n-g_n \mid =1$.</p>
<p>I need a proof using mosly combinatorics not mostly algebra.I think we s... | Paul Aljabar | 435,819 | <p>I think this might be part of the way to a solution</p>
<p>Assuming that
$$
g_{n} = n f_{n-1}
$$</p>
<p>Use one of the recurrence relations for derangements:
$$
f_{n} = n f_{n-1} + (-1)^{n}
$$</p>
<p>$$
\begin{aligned}
f_{n} - g_{n}
&=
f_{n} - n f_{n-1}
\\
&=
f_{n} - n (n-1) f_{n-2} - n (-1)^{n-1}
\\
&... |
2,159,915 | <p>Consider the following system of ODE:</p>
<p>$$\begin{array}{ll}\ddot y + y + \ddot x + x = 0 \\
y+\dot x - x = 0 \end{array}$$</p>
<p><strong>Question</strong>: How many initial conditions are required to determine a unique solution?</p>
<p>A naive reasoning leads to four: $y(0),\dot y(0), x(0)$ and $\dot x(0)$.... | the_candyman | 51,370 | <p>I guess that the first answer is $4$, even if the dynamical matrix is rank deficient. Consider this example:
$$\left[\begin{array}{c}\dot{x}\\\dot{y}
\end{array}\right] = \left[\begin{array}{cc}1 & 0\\ 1 & 0
\end{array}\right] \cdot \left[\begin{array}{c}x\\y
\end{array}\right].$$
Following you thoughts, eve... |
182,346 | <p>Let's call a polygon $P$ <em>shrinkable</em> if any down-scaled (dilated) version of $P$ can be translated into $P$. For example, the following triangle is shrinkable (the original polygon is green, the dilated polygon is blue):</p>
<p><img src="https://i.stack.imgur.com/M0LOu.png" alt="enter image description here... | Włodzimierz Holsztyński | 8,385 | <p>Let $\ L\ $ be a Hilbert space. Let $\ P\subseteq L\ $ be a non-empty compact subset. Then $\ P\ $ is called $\ \mu$-shrinkable $\ \Leftarrow:\Rightarrow$</p>
<p>$$\exists_{q\in L}\ \ \mu\cdot P\ +\ q\ \subseteq\ P$$</p>
<p>for arbitrary $\ \mu\ge 0\ $ (thus $\ \mu \le 1\ $ when $\ |P|>1$).</p>
<p>Let $\ m(P)... |
1,647,327 | <p>suppose S is a metric space and $B(S)$ is the set of bounded functions and $C_b(S)$ is the set consisting of bounded continuous functions.</p>
<p>Prove that $C_b(S)$ is a closed subspace of $B(S)$.</p>
<p>I thought of looking at the complement $B(S) \backslash C_b(S) = \{f| \text{ f is bounded and not continuous}\... | s.harp | 152,424 | <p>The norm on the function space is given by the sup norm $\|f\|:=\sup\{|f(x)|x\in S\}$. Convergence of functions in sup-norm is called uniform convergence. It is a general statement that a uniform limit of continuous functions is again continuous.</p>
<p>To see that let $x_n \to x$, let $f_m \in C_b(S)$ and $f_m \to... |
3,062,701 | <p>I want to solve this system by Least Squares method:<span class="math-container">$$\begin{pmatrix}1 & 2 & 3\\\ 2 & 3 & 4 \\\ 3 & 4 & 5 \end{pmatrix}\begin{pmatrix}x\\y\\z\end{pmatrix} =\begin{pmatrix}1\\5\\-2\end{pmatrix} $$</span> This symmetric matrix is singular with one eigenvalue <span ... | Mostafa Ayaz | 518,023 | <p>Note that <span class="math-container">$$\begin{pmatrix}3&4&5\end{pmatrix}\begin{pmatrix}x\\y\\z\end{pmatrix}=-2$$</span>and <span class="math-container">$$\Big[2\cdot\begin{pmatrix}2&3&4\end{pmatrix}-\begin{pmatrix}1&2&3\end{pmatrix}\Big]\begin{pmatrix}x\\y\\z\end{pmatrix}=\begin{pmatrix}3&a... |
3,360,694 | <p><span class="math-container">$U(n)$</span> is the collection of positive integers which are coprime to n forms a group under multiplication modulo n.</p>
<p>What is the order of the element 250 in <span class="math-container">$U(641)$</span>?</p>
<p>My attempt:
Here 641 is a prime number.
So <span class="math-cont... | Hagen von Eitzen | 39,174 | <p>A priori, the order of <span class="math-container">$250$</span> could be any divisor of <span class="math-container">$640$</span>.
Since you found the inverse and it is an obvious square (<span class="math-container">$100=10^2$</span>), it is clear that the order of <span class="math-container">$250$</span> is at ... |
1,323,317 | <p>I have to compute the following quantity:</p>
<p>$$
1) \sum\limits_{k=0}^{n} \binom{n}{k}k2^{n-k}
$$</p>
<p>Moreover, I have to give an upper bound for the following quantity:</p>
<p>$$
2) \sum\limits_{k=1}^{n-2} \binom{n}{k}\frac{k}{n-k}
$$</p>
<p>As regards 1), I see that $\binom{n}{k}k2^{n-k}=\frac{n! ... | Brian M. Scott | 12,042 | <p>HINT for the first problem: $\binom{n}kk2^{n-k}$ is the number of ways to start with $n$ people, choose a team of size $k$, choose one of the team members to be captain, and then choose some subset of the remaining $n-k$ people to be substitutes. Thus, $\sum_{k=0}^n\binom{n}kk2^{n-k}$ is the total number of ways to ... |
1,323,317 | <p>I have to compute the following quantity:</p>
<p>$$
1) \sum\limits_{k=0}^{n} \binom{n}{k}k2^{n-k}
$$</p>
<p>Moreover, I have to give an upper bound for the following quantity:</p>
<p>$$
2) \sum\limits_{k=1}^{n-2} \binom{n}{k}\frac{k}{n-k}
$$</p>
<p>As regards 1), I see that $\binom{n}{k}k2^{n-k}=\frac{n! ... | Alex R. | 22,064 | <p>For the first one, $(x+2)^n=\sum_{k=0}^n \binom{n}{k}x^k2^{n-k}$. Now differentiate in $x$:</p>
<p>$n(x+2)^{n-1}=\sum_{k=1}^n \binom{n}{k}kx^{k-1}2^{n-k}.$ </p>
<p>Now plug in $x=1$. </p>
<p>For the second one, $(x+y)^n=\sum_{k=0}^n \binom{n}{k}x^ky^{n-k}$. Notice that:</p>
<p>$$\frac{(x+y)^n}{y}=\sum_{k=0}^n \b... |
716,859 | <p>Define the mean of order $p$ of $a$ and $b$ as $s_p(a,b)$ $=$ $({a^p + b^p\over 2})^{1/p}$.</p>
<p>I have to find the limit of the sequence $s_n(a,b)$. I already know this sequence is bounded above by $b$ (from a previous question) and if I assume the limit exists I can show it is $b$. What I cannot show is that th... | Pi89 | 130,217 | <p>Remark that for two positive numbers $a$ and $b$, $a^p+ b^p \underset{+ \infty}{\sim} max(a,b)^p$.
It should be helpful to answer the question</p>
|
4,436,210 | <p>I have been given this exercise: Calculate the double integral:</p>
<blockquote>
<p><span class="math-container">$$\iint_D\frac{\sin(y)}{y}dxdy$$</span>
Where <span class="math-container">$D$</span> is the area enclosed by the lines: <span class="math-container">$y=2$</span>, <span class="math-container">$y=1$</span... | Arturo Magidin | 742 | <p>You cannot just look at the final product if you did not carefully note steps in which you were assuming facts about the value of <span class="math-container">$a$</span>. So let us take a careful look at the Gaussian elimination process.</p>
<p>Starting from
<span class="math-container">$$\left(\begin{array}{rrr|r}
... |
79,726 | <p>Let $R$ be a commutative ring with unity. Let $M$ be a free (unital) $R$-module.</p>
<p>Define a <em>basis</em> of $M$ as a generating, linearly independent set.</p>
<p>Define the <em>rank</em> of $M$ as the cardinality of a basis of $M$ (as we know commutative rings have IBN, so this is well defined).</p>
<p>A <... | Arturo Magidin | 742 | <p>In <em>Some remarks on the invariant basis property</em>, Topology <strong>5</strong> (1966), pp. 215-228, MR <strong>33</strong> #5676, P.M. Cohn proved that there are rings in which the notion of rank is well defined, but which admit free modules of rank $t$ that can be generated by fewer than $t$ elements. Howev... |
152,405 | <p>This question complement a previous MO question: <a href="https://mathoverflow.net/questions/95837/examples-of-theorems-with-proofs-that-have-dramatically-improved-over-time">Examples of theorems with proofs that have dramatically improved over time</a>.</p>
<p>I am looking for a list of</p>
<h3>Major theorems in ma... | Victor Protsak | 5,740 | <p><a href="http://www.math.harvard.edu/~yihang/GZSeminarnotes/talk%201.pdf"><strong>Gross-Zagier formula</strong></a> (1986) relating the heights of Heegner points on elliptic curves and the derivatives of $L$-series was a major source of progress in number theory in the last 25 years (cutting pretty close here!). Onc... |
152,405 | <p>This question complement a previous MO question: <a href="https://mathoverflow.net/questions/95837/examples-of-theorems-with-proofs-that-have-dramatically-improved-over-time">Examples of theorems with proofs that have dramatically improved over time</a>.</p>
<p>I am looking for a list of</p>
<h3>Major theorems in ma... | Wlodek Kuperberg | 36,904 | <p>The <a href="http://en.wikipedia.org/wiki/Kepler_conjecture">Sphere packing problem in $\mathbb{R}^3$, <em>a.k.a.</em> the Kepler Conjecture.</a> Although the first accepted proof was published just about 10 years ago, the conjecture is very old, and there were several unsuccessful attempts at it for quite a long ti... |
152,405 | <p>This question complement a previous MO question: <a href="https://mathoverflow.net/questions/95837/examples-of-theorems-with-proofs-that-have-dramatically-improved-over-time">Examples of theorems with proofs that have dramatically improved over time</a>.</p>
<p>I am looking for a list of</p>
<h3>Major theorems in ma... | Sasha Patotski | 32,741 | <p>As far as I know, the Quillen equivalence between simplicial sets and topological spaces is one of such <a href="http://ncatlab.org/nlab/show/model+structure+on+simplicial+sets#classical_model_structure" rel="nofollow">theorems</a>.</p>
|
152,405 | <p>This question complement a previous MO question: <a href="https://mathoverflow.net/questions/95837/examples-of-theorems-with-proofs-that-have-dramatically-improved-over-time">Examples of theorems with proofs that have dramatically improved over time</a>.</p>
<p>I am looking for a list of</p>
<h3>Major theorems in ma... | Abdelmalek Abdesselam | 7,410 | <p>The construction of the $\Phi^4_3$ quantum field theory model.
This was done in the early seventies by <a href="http://isites.harvard.edu/fs/docs/icb.topic1494462.files/Phi%5E4_3_Online_1973%20Fort%20der%20Physik.pdf" rel="noreferrer">Glimm-Jaffe</a>, <a href="http://projecteuclid.org/euclid.cmp/1103859849" rel="nor... |
28,347 | <p>I often end up with a function that contains the term $1/(1 + x^2/y^2)$, and I need to evaluate this in the limit $y\rightarrow 0$. By hand, I can rewrite this as $y^2/(y^2 + x^2)$, but how can I tell <em>Mathematica</em> to make such a simplification?</p>
<p>I have tried using <code>1/(1 + x^2/y^2) // Simplify</co... | Alexei Boulbitch | 788 | <pre><code> expr = 1/(1 + x^2/y^2)
(* 1/(1 + x^2/y^2) *)
Simplify[expr, ComplexityFunction -> (Count[#, _Power[_, -2]] &)]
(* y^2/(x^2 + y^2) *)
</code></pre>
|
2,069,507 | <p><a href="https://i.stack.imgur.com/B4b88.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/B4b88.png" alt="The image of parallelogram for help"></a></p>
<p>Let's say we have a parallelogram $\text{ABCD}$.</p>
<p>$\triangle \text{ADC}$ and $\triangle \text{BCD}$ are on the same base and between two... | David K | 139,123 | <p>As already hinted in comments and answers, you appear to have
applied Heron's formula
to the areas of the triangles $\triangle ADC$ and $\triangle BCD$.
Then observing that the only variables in Heron's formula are
the three sides of the triangle and its area,
and that three of these quantities in the formula for $\... |
1,325,563 | <p>Can someone show me:</p>
<blockquote>
<p>If $x$ is a real number, then $\cos^2(x)+\sin^2(x)= 1$.</p>
<p>Is it true that $\cos^2(z)+\sin^2(z)=1$, where $z$ is a complex variable?</p>
</blockquote>
<p>Note :look [this ] in wolfram alpha showed that's true !!!!</p>
<p>Thank you for your help</p>
| Mercy King | 23,304 | <p>$$
\cos^2z+\sin^2z=(\cos z+i\sin z)(\cos z-i\sin z)=e^{iz}e^{-iz}=e^{iz-iz}=e^0=1.
$$</p>
|
1,325,563 | <p>Can someone show me:</p>
<blockquote>
<p>If $x$ is a real number, then $\cos^2(x)+\sin^2(x)= 1$.</p>
<p>Is it true that $\cos^2(z)+\sin^2(z)=1$, where $z$ is a complex variable?</p>
</blockquote>
<p>Note :look [this ] in wolfram alpha showed that's true !!!!</p>
<p>Thank you for your help</p>
| Ruvi Lecamwasam | 77,395 | <p>You can use the <a href="https://www.wikiwand.com/en/Identity_theorem#/An_improvement">identity theorem</a>. As they are just sums of exponentials, $\sin(z)$ and $\cos(z)$ are holomorphic, and on the real axis $\sin^2(x)+\cos^2(x)=1$. As $\mathbb{R}$ is a set with an accumulation point (namely any point in $\mathbb{... |
3,618,791 | <p>Given <span class="math-container">$p\in[0,1]$</span>, prove or disprove that the sum
<span class="math-container">$$\sum_{n=k}^\infty\sum_{j=0}^k\left(\matrix{n\\j}\right)p^j(1-p)^{n-j}$$</span>
is bounded by a constant that does not depend on <span class="math-container">$k$</span>.</p>
<p>The terms <span class="... | Anton Vrdoljak | 744,799 | <p>(a) Given sum is nothing else than: </p>
<p><span class="math-container">$\sum_{r=0}^{10}u_r=u_0+u_1+u_2+u_3+u_4+u_5+u_6+u_7++u_8+u_9+u_{10}$</span>.</p>
<p>Next, we have <span class="math-container">$u_n=0.5^n$</span>, so we have to find:</p>
<p><span class="math-container">$0.5^0 + 0.5^1 + 0.5^2 + 0.5^3 + 0.5^4... |
384,318 | <p>Let $X$ be a topological space and let $A,B\subseteq X$ be closed in $X$ such that $A\cap B$ and $A\cup B$ are connected (in subspace topology) show that $A,B$ are connected (in subspace topology).</p>
<p>I would appreciate a hint towards the solution :)</p>
| Abel | 71,157 | <p>The general idea of a proof could go as follows. I'll leave the details to you:</p>
<p>Let $\mathbf{2}$ be the two point space with discrete topology and let $f\colon A\to \mathbf{2}$ be a continuous function. $f|_{A\cap B}\colon A\cap B\to\mathbf{2}$ is a continuous function and $A\cap B$ is connected, thus $f|_{A... |
3,712,699 | <p>Let <span class="math-container">$K$</span> be a number field with ring of integers <span class="math-container">$\mathcal{O}_K$</span> and let <span class="math-container">$p$</span> be a rational prime. Let <span class="math-container">$(p) = \mathfrak{p}_1^{e_1}\ldots\mathfrak{p}_r^{e_r}$</span> be the prime fact... | Wojowu | 127,263 | <p>One can prove the general case by essentially reducing to the Galois case (though see a remark at the end). Let <span class="math-container">$L/\mathbb Q$</span> be a Galois extension containing <span class="math-container">$K$</span>. Any prime <span class="math-container">$\mathfrak q$</span> lying above <span cla... |
4,646,715 | <blockquote>
<p>Let <span class="math-container">$A\in \operatorname{Mat}_{2\times 2}(\Bbb{R})$</span> with eigenvalues <span class="math-container">$\lambda\in (1,\infty)$</span> and <span class="math-container">$\mu\in (0,1)$</span>. Define <span class="math-container">$$T:S^1\rightarrow S^1;~~x\mapsto \frac{Ax}{\|Ax... | Anne Bauval | 386,889 | <p>Your two eigenvectors <span class="math-container">$x,y$</span> are only assumed to be non-zero, so they are not in <span class="math-container">$S^1.$</span></p>
<p>A point <span class="math-container">$z\in S^1$</span> is fixed by <span class="math-container">$T$</span> iff <span class="math-container">$Az=\|Az\|z... |
3,001,700 | <p>I am trying to find an <span class="math-container">$x$</span> and <span class="math-container">$y$</span> that solve the equation <span class="math-container">$15x - 16y = 10$</span>, usually in this type of question I would use Euclidean Algorithm to find an <span class="math-container">$x$</span> and <span class=... | Mohammad Riazi-Kermani | 514,496 | <p>You have <span class="math-container">$16-15=1$</span></p>
<p>What about <span class="math-container">$$ x=-10+16k, y= -10+15k ?$$</span></p>
<p>That implies</p>
<p><span class="math-container">$$ 15 x-16y=10$$</span>
Which is a solution </p>
|
188,900 | <p><strong>Bug introduced in 10.0 and persisting through 11.3 or later</strong></p>
<hr>
<p>In <code>11.3.0 for Microsoft Windows (64-bit) (March 7, 2018)</code> writing:</p>
<pre><code>f[w_, x_, y_, z_] := w*x^2*y^3 - z*(w^2 + x^2 + y^2 - 1)
eqn = {D[f[w, x, y, z], w] == 0,
D[f[w, x, y, z], x] == 0,
... | Somos | 61,616 | <p>Thanks for asking! In version 9.0 only 16 solutions are returned and they are all valid. In version 10.2 there are 20 solutions, with the extra 4 all being invalid. Contragulations! I think you found a bug. You may want to click "Help", then "Give Feedback...", and then fill out the form in your browser to report.</... |
4,032,969 | <p>I have an integral that depends on two parameters <span class="math-container">$a\pm\delta a$</span> and <span class="math-container">$b\pm \delta b$</span>. I am doing this integral numerically and no python function can calculate the integral with uncertainties.</p>
<p>So I have calculated the integral for each mi... | seVenVo1d | 551,567 | <pre><code>from numpy import sqrt
from scipy import integrate
import uncertainties as u
from uncertainties.umath import *
#Important Parameters
C = 2997.92458 # speed of light in [km/s]
eta = 4.177 * 10**(-5)
a = u.ufloat(0.1430, 0.0011)
b = u.ufloat(1089.92, 0.25)
gama = u.ufloat(0.6736, 0.0054)
@u.wrap
def D_zrec_... |
154,893 | <p>I am having trouble figuring this out.</p>
<p>$$\sqrt {1+\left(\frac{x}{2}- \frac{1}{2x}\right)^2}$$</p>
<p>I know that $$\left(\frac{x}{2} - \frac{1}{2x}\right)^2=\frac{x^2}{4} - \frac{1}{2} + \frac{1}{4x^2}$$ but I have no idea how to factor this since I have two x terms with vastly different degrees, 2 and -2.<... | Américo Tavares | 752 | <p>Since
$$\begin{equation*}
\left( \frac{x}{2}-\frac{1}{2x}\right) ^{2}=\frac{x^{2}}{4}-\frac{1}{2}+
\frac{1}{4x^{2}},
\end{equation*}$$</p>
<p>we have</p>
<p>$$\begin{eqnarray*}
1+\left( \frac{x}{2}-\frac{1}{2x}\right) ^{2} &=&1+\left( \frac{x^{2}}{4}-
\frac{1}{2}+\frac{1}{4x^{2}}\right) \\
&=&1+\f... |
2,669,942 | <p>I am very confused about "integration by substitution". For example:</p>
<p>We know that $\int\ x^2 dx$ = $x^3/3 +C$.</p>
<p>Just for doing it, maybe for the sake of practicing, or for testing with integration by substitution, we could have made the substitution $x^2=u$. Then $x=u^{1/2}$ and $dx= (u^{-1/2}/2) du$.... | ncmathsadist | 4,154 | <p>The purpose of the $du$ is to suck up the extra factor of the inner function spilt out by the chain rule.</p>
|
2,669,942 | <p>I am very confused about "integration by substitution". For example:</p>
<p>We know that $\int\ x^2 dx$ = $x^3/3 +C$.</p>
<p>Just for doing it, maybe for the sake of practicing, or for testing with integration by substitution, we could have made the substitution $x^2=u$. Then $x=u^{1/2}$ and $dx= (u^{-1/2}/2) du$.... | Rene Schipperus | 149,912 | <p>The chain rule is
<span class="math-container">$$\frac{d}{dx}f(g(x))=f^{\prime}(g(x))g^{\prime}(x)$$</span> now we can integrate both sides of this equation to get</p>
<p><span class="math-container">$$f(g(x))=\int f^{\prime}(g(x))g^{\prime}(x)dx$$</span></p>
<p>If we were to write <span class="math-container">$y=g(... |
631,214 | <p>Two kids starts to run from the same point and the same direction of circled running area with perimeter 400m. The velocity of each kid is constant. The first kid run each circle in 20 sec less than his friend. They met in the first time after 400 sec from the start. Q: Find their velocity.</p>
<p>I came with one e... | mathlove | 78,967 | <p>If we can use the fact that
$$\frac{\ln(u)}{u}\to 0\ \ \text{($u\to\infty$)},$$
consider $x=1/u.$</p>
|
631,214 | <p>Two kids starts to run from the same point and the same direction of circled running area with perimeter 400m. The velocity of each kid is constant. The first kid run each circle in 20 sec less than his friend. They met in the first time after 400 sec from the start. Q: Find their velocity.</p>
<p>I came with one e... | Ulrik | 53,012 | <p>One can also use the following power series, which converge for $-1 < x \leq 1$:
$$\ln(1+x) = \sum_{n=1}^\infty \frac{(-1)^{n+1}}{n} x^n$$
Then $x \ln x$ becomes
$$x \ln x = (x-1) \ln x + \ln x = \sum_{n=1}^\infty \frac{(-1)^{n+1}}{n}(x-1)^{n+1} + \sum_{n=1}^\infty \frac{(-1)^{n+1}}{n}(x-1)^n = (x-1) + \sum_{n=1}... |
186,890 | <p>Working with other software called SolidWorks I was able to get a plot with a curve very close to my data points:</p>
<p><a href="https://i.stack.imgur.com/DooKo.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/DooKo.png" alt="enter image description here"></a></p>
<p>I tried to get a plot as acc... | MassDefect | 42,264 | <p>If you want to reproduce that graph exactly, add in the derivatives at each point when using the Interpolate function.</p>
<pre><code>Clear["Global`*"]
dados = {{{0}, 0, 0}, {{1}, 1000, 0}, {{2}, -750, 0}, {{3}, 250,
0}, {{4}, -1000, 0}, {{5}, 0, 0}};
Plot[
Interpolation[dados][x],
{x, 0, 5},
ImageSize ->... |
3,671,223 | <p>First and foremost, I have already gone through the following posts:</p>
<p><a href="https://math.stackexchange.com/questions/2463561/prove-that-for-all-positive-integers-x-and-y-sqrt-xy-leq-fracx-y">Prove that, for all positive integers $x$ and $y$, $\sqrt{ xy} \leq \frac{x + y}{2}$</a></p>
<p><a href="https://ma... | Gary | 83,800 | <p>We would like to prove
<span class="math-container">$$
\frac{{x + y}}{2} \ge \sqrt {xy}
$$</span>
for all non-negative <span class="math-container">$x$</span>, <span class="math-container">$y$</span>. If this was true, then we would also have
<span class="math-container">$$
x + y \ge 2\sqrt {xy},
$$</span>
<span cl... |
2,716,585 | <p>Evaluate $\sum_{k=1}^{n-3}\frac{(k+3)!}{(k-1)!}$. </p>
<p>My strategy is defining a generating function,
$$g(x) = \frac{1}{1-x} = 1 + x + x^2...$$
then shifting it so that we get,
$$f(x)=x^4g(x) = \frac{x^4}{1-x}= x^4+x^5+...$$
and then taking the 4th derivative of f(x).
Calculating the fourth derivative is going t... | epi163sqrt | 132,007 | <p>Here is a variation using generating functions <em>without</em> differentiation. It is convenient to use the <em>coefficient of</em> operator $[x^n]$ to denote the coefficient of $x^n$ of a series. This way we can write for instance
\begin{align*}
[x^k](1+x)^n=\binom{n}{k}\tag{1}
\end{align*}</p>
<blockquote>
<p>... |
2,397,874 | <p>I am new to modulus and inequalities , I came across this problem:</p>
<p>$ 2^{\vert x + 1 \vert} - 2^x = \vert 2^x - 1\vert + 1 $ for $ x $</p>
<p>How to find $ x $ ?</p>
| lab bhattacharjee | 33,337 | <p>Hint:</p>
<p>As $(5+2\sqrt6)(5-2\sqrt6)=1$</p>
<p><strong>Method</strong>$\#1$:</p>
<p>set $(5+2\sqrt6)^{x^2-3}=a$ to find $$a+\dfrac1a=10$$</p>
<p>So, we have $$(5+2\sqrt6)^{x^2-3}=a=(5+2\sqrt6)^{\pm1}$$</p>
<p><strong>Method</strong>$\#2$:</p>
<p>$$a+\dfrac1a=5+2\sqrt6+\dfrac1{5+2\sqrt6}$$</p>
|
2,397,874 | <p>I am new to modulus and inequalities , I came across this problem:</p>
<p>$ 2^{\vert x + 1 \vert} - 2^x = \vert 2^x - 1\vert + 1 $ for $ x $</p>
<p>How to find $ x $ ?</p>
| nonuser | 463,553 | <p>If you write $a = (5+2\sqrt6)^{x^2-3}$ then you have $a+1/a =10$ and thus $a_{1,2} = {5\pm 2\sqrt6}$. </p>
<p>So $x^2-3 = \pm 1$ and thus $x = \pm 2, \pm \sqrt{2}$.</p>
|
249,074 | <p>I am trying to solve the following problem:</p>
<blockquote>
<p>Show that a unit-speed curve $\gamma$ with nowhere vanishing curvature is a geodesic on the ruled surface $\sigma(u,v)=\gamma(u)+v\delta(u)$, where $\gamma$ is a smooth function of $u$, if and only if $\delta$ is perpendicular to the principal normal... | Kevin | 51,522 | <p>are you enrolled in UofCalgary PMAT 423? I have the same question as you.</p>
<p>(=>) this is what I was thinking as well. Just remember that we're supposed to show that γ is perpendicular to the principal normal of γ, not to γ". Use γ" = kn (not N which is the standard unit normal of the surface).</p>
<p>(<=) ... |
2,512,424 | <p>It is an easy exercise to show that all finite groups with at least three elements have at least one non-trivial automorphism; in other words, there are - up to isomorphism - only finitely many finite groups $G$ such that $Aut(G)=1$ (to be exact, just two: $1$ and $C_2$).</p>
<p>Is an analogous statement true for a... | YCor | 35,400 | <p>Mikko's nice answer concerns finite groups $G$. Let me here answer for infinite groups $G$ (but still finite automorphism groups, as in the question).</p>
<p>The picture is indeed very different:</p>
<blockquote>
<p>For $A=C_2$ cyclic, there exists uncountably many non-isomorphic (abelian countable) groups $G$ w... |
1,049,841 | <p>Out of interest </p>
<p>If i have the map $\phi: R \longrightarrow R/I $ where $R$ is a ring and $I$ is a nilpotent ideal ?</p>
<p>then would i be right in saying that if i were to apply this map to the jacobson radical of $R$ it would take me to the jacobson radical of $R/I$ </p>
<p>i.e. is the following true:
$... | rschwieb | 29,335 | <p>Firstly, any maximal right ideal contains any nilpotent ideal. We want to prove this because it match the maximal right ideals of both rings exactly.</p>
<p>If $M$ were a maximal right ideal not containing $I$, then $M+I=R$, and in particular $m+i=1$ for some $m\in M$, $i\in I$. Pick $n$ such that $i^n=0$. Then $1=... |
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