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 |
|---|---|---|---|---|
1,121,354 | <p>I need help understanding the following solution for the given problem. </p>
<p>The problem is as follows: Given a field $F$, the set of all formal power series $p(t)=a_0+a_1 t+a_2 t^2 + \ldots$ with $a_i \in F$ forms a ring $F[[t]]$. Determine the ideals of the ring.</p>
<p>The solution: Let $I$ be an ideal and $... | Brian M. Scott | 12,042 | <p>We’re supposing that $I$ is an ideal in $F[[t]]$. For each $p(t)=\sum_{k\ge 0}a_kt^k\in I$, let $$a_p=\min\{k:a_k\ne 0\}\;,$$ so that</p>
<p>$$p(t)=a_{a_p}t^{a_p}+a_{a_p+1}t^{a_p+1}+\ldots=t^{a_p}\left(\underbrace{a_{a_p}+a_{a_p+1}t+a_{a_p+2}t^2+\ldots}_{q(t)\in F[[t]]}\right)\;.\tag{1}$$</p>
<p>Among all elements... |
1,612,220 | <p>This is an exercise page 7 from Sutherland's book Introduction to Metric and Topological Spaces.</p>
<blockquote>
<p>Suppose that <span class="math-container">$V,X,Y$</span> are sets with <span class="math-container">$V\subseteq X\subseteq Y$</span> and suppose that <span class="math-container">$U$</span> is a subse... | kccu | 255,727 | <p>No, your proof is not valid. The assumption that $x\in X$ does <em>not</em> imply that $x \in V$ and $x \in X \cap V$. For instance, take the sets $V=\{1,2,3\}$ and $X=\{1,2,3,4\}$. Then $4 \in X$, but $4 \notin V$ and $4 \notin V\cap X$. It might help to argue by contradiction for this direction (though it can also... |
1,465,490 | <p>I have a set of target coordinates and a set of actually clicked coordinates which should be approximately the same, but not identical. The y coordinates are equal, however, the x-coordinates differ, such that negative coordinates are closer together than larger/positive coordinates.
e.g.:</p>
<pre><code>target_y ... | MathAdam | 266,049 | <p>I hope this sketch will help you solve the problem. </p>
<p>You will need to identify which portion of the diagram contributes to the total length of the belt. You will also need to know how to calculate the circumference of a circle. </p>
<p>How much of each circle is covered by a belt? How much more belt lengt... |
40,493 | <p>I am trying to use Mathematica to solve a relatively simple ODE involving parameter(s). I would like to use a set of conditions to solve for the particular solution of the ODE. I understand how to make Mathematica find values for the constants that arise during the process of solving the ODE, but what about solving ... | Chris Degnen | 363 | <pre><code>DSolve[D[T[t], t] == -k*(T[t] - Ta), T[t], t]
</code></pre>
<blockquote>
<p>{{T[t] -> Ta + E^(-k t) C[1]}}</p>
</blockquote>
<p>At t = 0, (substituting t = 0 above), <code>T[0] = Ta + C[1]</code>,</p>
<p>therefore <code>C[1] = T[0] - Ta</code>.</p>
<p>Substituting C[1] gives:</p>
<pre><code>T[t] ->... |
562,802 | <p>I have been recently investigating the sequence 1,11,111,... I found, contrary to my initial preconception, that the elements of the sequence can have a very interesting multiplicative structure. There are for example elements of the sequence that are divisible by primes like 7 or 2003.</p>
<p>What I am interested ... | Cameron Buie | 28,900 | <p>In fact, every number coprime with $10$ (that is, those that aren't integer multiples of $2$ and/or $5$) divides some element of that sequence. See <a href="https://math.stackexchange.com/questions/226895/help-on-proving-that-every-natural-number-co-prime-with-10-is-a-factor-of-a-repu">this question</a>.</p>
<p>On ... |
1,203,922 | <p>Show that it is possible to divide the set of the first twelve cubes $\left(1^3,2^3,\ldots,12^3\right)$ into two sets of size six with equal sums.</p>
<p>Any suggestions on what techniques should be used to start the problem?</p>
<p>Also, when the question is phrased like that, are you to find a general case that ... | Rosie F | 344,044 | <p>Work modulo 7. $x^3=1$ if $x\in A=\{1, 2, 4, 8, 9, 11\}$; $x^3=-1$ if $x\in B=\{3, 5, 6, 10, 12\}$; $x^3=0$ if $x=7$. $3042=4=-3$ mod 7. As barak manos has pointed out, $11$ and $12$ are in different groups. Each group must have either three more elements of $B$ than of $A$, or four more elements of $A$ than of $B$.... |
231,773 | <p>Let $G=(V,E)$ be an undirected graph. We form a graph $H=(V',E')$ from $G$ such that </p>
<ul>
<li>$V' = V \cup \{ w_e \mid e \in E \}$, and </li>
<li>$E' = \{ aw_e, bw_e \mid ab = e \in E \} \cup \{ w_e w_f \mid e,f \text{ are adjacent edges in }G \}$. </li>
</ul>
<p>Informally, $H$ is built from $G$ by subdividi... | few_reps | 39,552 | <p>Let $X_p$ be the set of isometry classes of 4-dimensional positive definite lattices satisfying the property $L^\sharp/L\simeq (\mathbf Z/p)^2$. The set $X_n$ is stable under the involution $\tau : L\mapsto pL^\sharp$. </p>
<p>The question seems to ask whether or not $X_p^\tau$ contains an even lattice. Will Jagy ... |
3,583,330 | <p>I've approached the problem the following way : </p>
<p>Out of the 7 dice, I select any 6 which will have distinct numbers : 7C6.</p>
<p>In the 6 dice, there can be 6! ways in which distinct numbers appear.</p>
<p>And lastly, the last dice will have 6 possible ways in which it can show a number.</p>
<p>So the re... | rogerl | 27,542 | <p>As noted in my comment above, you are double counting. Here's a different way of approaching the problem: let's compute the probability of throwing two 1's and one each of the other five numbers. So first choose the two dice that come up 1, in <span class="math-container">$\binom{7}{2}$</span> ways. The remaining fi... |
3,583,330 | <p>I've approached the problem the following way : </p>
<p>Out of the 7 dice, I select any 6 which will have distinct numbers : 7C6.</p>
<p>In the 6 dice, there can be 6! ways in which distinct numbers appear.</p>
<p>And lastly, the last dice will have 6 possible ways in which it can show a number.</p>
<p>So the re... | AlanD | 356,933 | <p>Here's a very simple, structured way of doing it. Consider a multinomial distribution with 6 outcomes. In <span class="math-container">$n=7$</span> trials, you want exactly two of one face and exactly one of all other faces. There are 6 equally likely situations, since it is equally likely that a face will be the on... |
33,387 | <p>I was told the following "Theorem": Let $y^{2} =x^{3} + Ax^{2} +Bx$ be a nonsingular cubic curve with $A,B \in \mathbb{Z}$. Then the rank $r$ of this curve satisfies</p>
<p>$r \leq \nu (A^{2} -4B) +\nu(B) -1$</p>
<p>where $\nu(n)$ is the number of distinct positive prime divisors of $n$.</p>
<p>I can not find a n... | Álvaro Lozano-Robledo | 14,699 | <p>The bound is also proved in Knapp's "Elliptic Curves", p. 107, <a href="https://books.google.com/books?id=-e_qVoKF8H8C&pg=PA107&lpg=PA107&dq=upper%20bound%20rank%20elliptic%20curve&source=bl&ots=lmME7KN0rg&sig=oZRogKQbpic1p5Qkb1ZJKeMZvWM&hl=en&sa=X&ved=0CDQQ6AEwA2oVChMIsK2HpsOpxwI... |
2,009,557 | <p>I am pretty sure this question has something to do with the Least Common Multiple. </p>
<ul>
<li>I was thinking that the proof was that every number either is or isn't a multiple of $3, 5$, and $8\left(3 + 5\right)$.</li>
<li>If it isn't a multiple of $3,5$, or $8$, great. You have nothing to prove.</li>
<li>But if... | JMP | 210,189 | <p>If we have $a,b$ let $\gcd(a,b)=d$ and $a=da',b=db'$. Then $\operatorname{lcm}(a,b,a+b)=d\cdot\operatorname{lcm}(a',b',a'+b')$.</p>
<p>And $\gcd(a,a+b)=\gcd(b,a+b)=d$.</p>
<p>So $a,b,a+b|n$ if and only if $\operatorname{lcm}(a,b,a+b)|n$.</p>
<p>The LCM of $3,5,8$ is $120$, so $n$ must be a multiple of $120$ in yo... |
4,613,471 | <p>I would like to figure out the power series expansion of <span class="math-container">$f(z)=\frac{1}{(z+1)^2}$</span> around <span class="math-container">$z_0=1$</span>. Somehow expanding this into a geometric series would be the way to go I suppose, however, I fail to see how this can be rearranged in terms of (z-1... | CHAMSI | 758,100 | <p><span class="math-container">\begin{aligned}\int{\frac{\mathrm{d}u}{u\sqrt{u^{2}+u+1}}}&=\int{\frac{\mathrm{d}u}{u\sqrt{\left(u+\frac{1}{2}\right)^{2}+\frac{3}{4}}}}\\ &=\int{\frac{\frac{2}{\sqrt{3}}\,\mathrm{d}u}{u\sqrt{1+\left(\frac{2u+1}{\sqrt{3}}\right)^{2}}}}\\ &=2\int{\frac{\mathrm{d}y}{\left(\sqrt... |
4,613,471 | <p>I would like to figure out the power series expansion of <span class="math-container">$f(z)=\frac{1}{(z+1)^2}$</span> around <span class="math-container">$z_0=1$</span>. Somehow expanding this into a geometric series would be the way to go I suppose, however, I fail to see how this can be rearranged in terms of (z-1... | user170231 | 170,231 | <p>Continuing from your <span class="math-container">$u$</span>-integral, <a href="https://en.wikipedia.org/wiki/Euler_substitution" rel="nofollow noreferrer">substitute</a></p>
<p><span class="math-container">$$t = \frac{\sqrt{u^2+u+1}-1}u \implies u = \frac{2t-1}{1-t^2} \implies du = \frac{2(t^2-t+1)}{(1-t^2)^2} \, d... |
3,608,097 | <p>Three strings totaling a length <span class="math-container">$U= 3 a + 4b + 2 \pi r$</span> cut into three parts together enclose minimum total area</p>
<p><span class="math-container">$$ A= \frac{\sqrt3 a^2}{4} + b^2+\pi r^2,$$</span></p>
<p>when they are made into shapes of an equilateral triangle, square and ci... | alepopoulo110 | 351,240 | <p>Hint: this is a power series around <span class="math-container">$0$</span> so it converges to an interval around <span class="math-container">$0$</span> and there is a well known formula for the radius of convergence, namely </p>
<p><span class="math-container">$$R=\frac{1}{\limsup_{n\to\infty}|a_n|^{1/n}}$$</span... |
1,353,015 | <p>Given a positive singular measure $\mu$ on $[-\pi,\pi]$, we define a singular inner function by</p>
<p>$$S(z)=\exp\left(-\int_{-\pi}^{\pi}\frac{e^{i\theta}+z}{e^{i\theta}-z}\,d\mu(\theta)\right).$$</p>
<p>It is stated in many different sources that the radial limits $\lim_{r\rightarrow1^{-}}S(re^{2\pi it})$ equal ... | David C. Ullrich | 248,223 | <p>If $\mu$ is a positive singular measure on $[0,2\pi)$ then $$\lim_{h\to0}\frac{\mu((x-h,x+h))}{2h}=+\infty$$for $\mu$-almost every $t$. That much I know is in Rudin <em>Real and Complex Analysis</em>, for example.</p>
<p>It follows that if $\mu$ is a singular measure on the circle then $$\lim_{r\to1}P[\mu](re^{it})... |
1,514,388 | <p>$$ \lim_{x\to \infty} \left(\frac{1}{(x^2+x)\left(\ln\frac{x+1}{x}\right)^2}\right) $$</p>
<p>I know the answer is 1, but why does it tend to 1?
Can you manipulate the function and the "$\ln$" to make it obvious? </p>
<p>Much appreciated. </p>
| Zelos Malum | 197,853 | <p><strong>Hint:</strong>
$$\log (\frac{x+1}{x})=\log (1+\frac{1}{x})$$
and with $\frac{1}{x}\to 0$
we get
$$\log (1+\frac{1}{x})\to\frac{1}{x}$$</p>
|
9,696 | <p>I am tutoring a Grade 2 girl in arithmetic. She has demonstrated an ability to add two-digit numbers with carrying. For example: </p>
<p>$$\;\;14\\
+27\\
=41$$ </p>
<p>I asked her to write this out horizontally, and this is what she produced. </p>
<p>$$12+47=41$$ </p>
<p>She evidently is failing to see the n... | mweiss | 29 | <blockquote>
<p>She evidently is failing to see the numbers and is confounding the vertical addition of the digits with the horizontal reading of the numbers.</p>
<p>With practice, and prompting, she is able to get this right, but it seems as though she sees the sum as a matrix of four digits, and is missing the... |
362,250 | <p>Let <span class="math-container">$\nu$</span> be a <em>finite</em> Borel measure on <span class="math-container">$\mathbb{R}^n$</span> and define the shift operator <span class="math-container">$T_a$</span> on <span class="math-container">$L^p_{\nu}(\mathbb{R}^n)$</span> by <span class="math-container">$f\to f(x+a)$... | Jochen Wengenroth | 21,051 | <p>You get a rather obvious bound for <span class="math-container">$\|T_a\|_{op}$</span> from
<span class="math-container">$$ \int|f(x+a)|^p h(x)dx =\int |f(y)^p|h(y-a)dy = \int |f(y)|^ph(y) \left|h(y-a)/h(y)\right|dy \le c\int|f(y)|^ph(y)dy$$</span> with <span class="math-container">$c=\|h(y-a)/h(y)\|_\infty$</span>.<... |
3,009,543 | <p>I am having great problems in solving this:</p>
<p><span class="math-container">$$\lim\limits_{n\to\infty}\sqrt[3]{n+\sqrt{n}}-\sqrt[3]{n}$$</span></p>
<p>I am trying to solve this for hours, no solution in sight. I tried so many ways on my paper here, which all lead to nonsense or to nowhere. I concluded that I h... | MSDG | 447,520 | <p>As you suggested,
<span class="math-container">\begin{align}
\sqrt[3]{n+\sqrt n} - \sqrt[3]{n} &= \frac{n+\sqrt n - n}{(n+\sqrt n)^{2/3}+\sqrt[3]{n(n+\sqrt n)} + n^{2/3}}\\
&= \frac{\sqrt n}{(n+\sqrt n)^{2/3}+\sqrt[3]{n(n+\sqrt n)} + n^{2/3}}.
\end{align}</span>
The fastest growing term is evidently <span cl... |
1,989,950 | <p>I was doing proof of open mapping theorem from the book Walter Rudin real and complex analysis book and struck at one point. Given if $X$ and $Y$ are Banach spaces and $T$ is a bounded linear operator between them which is $\textbf{onto}$. Then to prove $$T(U) \supset \delta V$$ where $U$ is open unit ball in $X$ an... | 111 | 382,126 | <p>$$ \sec^2 \theta - \tan^2 \theta= 1 $$
$$ (\sec\theta - \tan\theta)(\sec\theta + \tan\theta) = 1 $$
I hope you can take it from here. </p>
|
860,247 | <p>Simplify $$\frac{3x}{x+2} - \frac{4x}{2-x} - \frac{2x-1}{x^2-4}$$</p>
<ol>
<li><p>First I expanded $x²-4$ into $(x+2)(x-2)$. There are 3 denominators. </p></li>
<li><p>So I multiplied the numerators into: $$\frac{3x(x+2)(2-x)}{(x+2)(x-2)(2-x)} - \frac{4x(x+2)(x-2)}{(x+2)(x-2)(2-x)} - \frac{2x-1(2-x)}{(x+2)(x-2)(2-x... | Semiclassical | 137,524 | <p>Let $x=\sinh^2 u$. (This is the same transformation as CountIblis used, but I'll employ it slightly differently.) Observe that $dx=2\cosh u \sinh u \, du$ and $$x+x^2=\sinh^2 u+\sinh^4 u=\sinh^2 u(1+\sinh^2 u)=\sinh^2 \cosh^2 u$$ since $\cosh^2 u-\sinh^2 u=1$.</p>
<p>Therefore \begin{align}
\int_0^1 x^2 \sqrt{x+x^2... |
316,878 | <p>Why is the function not analytic in the complex plane? I believe it is analytic on real plane.</p>
<p>$e^{(-\frac{1}{z^2})}$ where $z\in\mathbb{C}$. </p>
<p>Well a complex function should be infinitely differentiable and should converge. this happens on real plane. But what happens in complex plane?</p>
| Elmar Zander | 10,076 | <p>The function looks pretty well behaved on the real line (not plane!), however, already here you see some pecularities. E.g. it's infinitely differentiable at $x=0$, however, the Taylor series at that point has zero radius of convergence. The reason becomes obvious if you look at the function in the complex plane. Tr... |
1,923,149 | <p>Preimage is defined as $X = \{x \in \mathbb{R}^n:Ax \in S\}$, where $A$ is a linear mapping $A \in \mathbb{R}^{m,n}$, and $S \subseteq \mathbb{R}^m$ is a sequentially-closed set.</p>
<p>The definition of sequential-closedness is that $S$ is sequentially-closed iff every convergent sequence in $S$ has its limit in $... | dxiv | 291,201 | <p>Let $P(z) = n! f(z) = z(z-1)...(z-n+1)$ with $n \ge 1$. Since $n!$ is a strictly positive constant factor, $P(z)$ will have the same convexity properties as $f(z)$.</p>
<p>$P$ is a polynomial of degree $n$ and has $n$ distinct real roots $\{0,1,..,n-1\}$. It follows that its derivative $P'$ will have $n-1$ distinct... |
319,058 | <p>Denote <span class="math-container">$\square_m=\{\pmb{x}=(x_1,\dots,x_m)\in\mathbb{R}^m: 0\leq x_i\leq1,\,\,\forall i\}$</span> be an <span class="math-container">$m$</span>-dimensional cube.</p>
<p>It is all too familiar that <span class="math-container">$\int_{\square_1}\frac{dx}{1+x^2}=\frac{\pi}4$</span>.</p>
... | Iosif Pinelis | 36,721 | <p>Here is an elementary proof. Using the formula
<span class="math-container">\begin{equation}
\frac1{a^n}=\frac1{\Gamma(n)}\,\int_0^\infty u^{n-1} e^{-u\,a}\,du \tag{1}
\end{equation}</span>
with <span class="math-container">$a=1+\|x\|^2$</span>,
denoting your integral by <span class="math-container">$J_n$</span>,... |
2,540,007 | <p>I have the following question. Find the matrix representation of the transormation $T:\mathbb{R}^3\to\mathbb{R}^3$ that rotates any vector by $\theta=\frac{\pi}{6}$ along the vector $v=(1,1,1)$.</p>
<p>A hint is given to find the rotation matrix about the $z-axis$ by $\frac{\pi}{6}$which is $$
\begin{bmatrix}
\f... | Anders Beta | 464,504 | <p>There is a way to do the rotation of an arbitrary vector $\vec A$ about an arbitrary non-zero vector $\vec V$ by an angle $\theta$ directly without changing basis (at least in an overt way). You only need to use scalar and vector products. First you define the unit vector $\hat u$ along the direction of $\vec V$ by ... |
2,551,683 | <blockquote>
<p>A jet has a $5\%$ chance of crashing on any given test flight. Once it crashes the program will be halted. Find the probability that the program lasts less than three flights.</p>
</blockquote>
<p>The correct answer to this question is $0.1426$, but I can't figure out how to get it.</p>
<p>Here's my... | Vedvart1 | 354,933 | <p>Consider this.</p>
<p>The probability of the jet crashing on the first is simply $5\%$, or $.05$.</p>
<p>The probability of the jet crashing on the second run isn't as immediate. First, the jet has to <em>not</em> crash on the first run, which has probability $.95$. Then it actually has to crash, with probability ... |
2,252,317 | <p>I have a rather challenging question on my assignment and I have put in my best effort for now. I think I just need a tiny nudge to set me in the right direction to finish this proof. If you could have a look, that would be great!</p>
<hr>
<p><strong>Background on Cosets and Operations Defined on $V/W$</strong></p... | sunspots | 110,953 | <p>Suppose <span class="math-container">$v_{1} + W = v_{1}^{'} + W$</span> and <span class="math-container">$v_{2} + W = v_{2}^{'} + W.$</span> Then, as you note, we have <span class="math-container">$v_{1}-v_{1}^{'} \in W$</span> and <span class="math-container">$v_{2}-v_{2}^{'} \in W.$</span> As <span class="math-con... |
3,011,862 | <p>Test the convergence <span class="math-container">$$\int_0^1 \frac{x^n}{1+x}dx$$</span></p>
<p>I have used comparison test for improper integrals..by comparing with <span class="math-container">$1/(1+x)$</span>...
so I found it convergent ..
But the solution set says that it is convergent if <span class="math-conta... | user3482749 | 226,174 | <blockquote>
<p>But having 5 numbers to choose from, why is the first pick only 4 different possible ways ? </p>
</blockquote>
<p>Because there aren't five possible numbers to choose from. One of them you have already used in the last position. </p>
|
152,467 | <p>Can you please explain to me how to get from a nonparametric equation of a plane like this:</p>
<p>$$ x_1−2x_2+3x_3=6$$</p>
<p>to a parametric one. In this case the result is supposed to be </p>
<p>$$ x_1 = 6-6t-6s$$
$$ x_2 = -3t$$
$$ x_3 = 2s$$</p>
<p>Many thanks.</p>
| Sean | 23,682 | <p>One way to do it is to let $x_1 = t$ and $x_2=s$ and then solve for $x_3$.</p>
|
2,956,744 | <p><a href="https://i.stack.imgur.com/lhDyB.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/lhDyB.png" alt="https://i.imgur.com/vvQDuVa.png" /></a></p>
<p>I'll provide a quickly-drawn representation of what the problem is.
Basically, there is a line <span class="math-container">$$l: y=-x+b$$</span> a... | Community | -1 | <p>In general, you should draw the graph of the one variable function. Then take the right and left limit at the "corner" point of that graph, which gives you different values. In your function <span class="math-container">$f$</span>, it happens at <span class="math-container">$x=2$</span>.</p>
|
2,561,125 | <p>Hey having trouble finishing this question.</p>
<p>Prove by induction that $n^3 \le 2^n$ for all natural numbers $n\ge 10$.</p>
<p>This is what I have so far:</p>
<p>Base step: For $n = 10$ </p>
<p>$1000 \le 1024$</p>
<p>Assumption Step: For $n = k$</p>
<p>Assume $k^3 \le 2^k$</p>
<p>Induction step: For $n =... | jgon | 90,543 | <p>Note that $(k+1)^3+(k-1)^3 = 2k^3 + 6k$, so as long as $(k-1)^3 \ge 6k$, we'll have that $(k+1)^3 \le 2k^3$. This is equivalent to showing that $(k-1)^3 \ge 6(k-1)+6$.
Now $$(k-1)^3=(k-1)^2(k-1)\ge 81(k-1) = 6(k-1)+75(k-1)\ge 6(k-1)+6.$$
Hence the inequality holds, which implies that $(k+1)^3\le 2k^3$. </p>
<p>Now ... |
2,561,125 | <p>Hey having trouble finishing this question.</p>
<p>Prove by induction that $n^3 \le 2^n$ for all natural numbers $n\ge 10$.</p>
<p>This is what I have so far:</p>
<p>Base step: For $n = 10$ </p>
<p>$1000 \le 1024$</p>
<p>Assumption Step: For $n = k$</p>
<p>Assume $k^3 \le 2^k$</p>
<p>Induction step: For $n =... | ℋolo | 471,959 | <p>When you have $$k^3<2^k$$ you can do 2 things, one multiply by 2 and get $2k^3<2^{k+1}$ and prove that $(k+1)^3<2k^3$ for $k\ge 10$ or you can multiply both sides by $\frac{(k+1)^3}{k^3}$ and get $(k+1)^3<\frac{(k+1)^3}{k^3}2^k$ then you prove that $\frac{(k+1)^3}{k^3}<2$ for $k\ge 10$</p>
<hr>
<p>I... |
4,132,998 | <p>I figure it has something to do with minimal polynomials and the Jordan canonical form, I just can't piece it together. I would really appreciate a hint on how to start it.</p>
| Asinomás | 33,907 | <p>If the range of <span class="math-container">$X^k$</span> is equal to the range of <span class="math-container">$X^{k+1}$</span> then the range of <span class="math-container">$X^{k+i}$</span> is equal to the range of <span class="math-container">$X^k$</span> for all positive <span class="math-container">$k$</span>.... |
25,414 | <p>I'm running in to some problems with generating a persistent HSQLDB and during some troubleshooting I came upon the following behavior.</p>
<pre><code>Needs["DatabaseLink`"]
tc = OpenSQLConnection[
JDBC["hsqldb", ToFileName[Directory[], "temp"]], Username -> "sa"]
CloseSQLConnection[tc]
</code></pre>
<p>The ab... | fredt | 68,530 | <p>There shouldn't be a need to close the JRE in order to delete the <strong>.lck</strong> or <strong>.log</strong> file.</p>
<p>The <strong>.lck</strong> file exists for the duration of access to database, which starts with the JDBC connection. The purpose of the <strong>.lck</strong> file is to prevent another proce... |
274,249 | <p>I'd like to write the current data, but the Monitor and PutAppend are being written in such a way that the function <strong>pot</strong> does not match the current variables <strong>(a and b)</strong>. How can I fix this?</p>
<pre><code>f[a_, b_] = Exp[-Sqrt[b*x + Cos[c]]] + a*y;
PP[a_, b_] := NIntegrate[f[a, b], {... | Bob Hanlon | 9,362 | <pre><code>$Version
(* "13.1.0 for Mac OS X x86 (64-bit) (June 16, 2022)" *)
Clear["Global`*"]
data = {{1985, 1.133431849`}, {1986, 1.160159779`}, {1987,
1.258499165`}, {1988, 1.621471213`}, {1989, 1.680828727`}, {1990,
1.915023226`}, {1991, 1.970753299`}, {1992, 1.995476342`}, {1993,
... |
2,043,457 | <p>Does anybody know of a succint way to compute the residue of $f(z)=z^m/(1-e^{-z})^{n+1}$ at $z=0$? I am only interested in the nontrivial case $m<n$.
Induction seems complicated/inefficient, so I am looking for a "trick", perhaps with Lagrange inversion?</p>
| Bernard | 202,857 | <p>It is isomorphic, <em>as an abelian group</em>, to $\mathbf C^4$. However it cannot be a $\mathbf C$-vector space for the very simple following reason:</p>
<p>Matrices $\begin{pmatrix}a&b\\c&d\end{pmatrix}$ are characterised by the fact that its diagonal elements $a$ and $d$ are conjugate.</p>
<p>Now if $d... |
4,463 | <p>It seems that most authors use the phrase "elementary number theory" to mean "number theory that doesn't use complex variable techniques in proofs." </p>
<p>I have two closely related questions.</p>
<ol>
<li>Is my understanding of the usage of "elementary" correct?</li>
<li>It appears that advanced techniques fro... | Qiaochu Yuan | 290 | <p>I more or less agree with Kevin; "elementary" to me means "from first principles." Another way I would put this is that if Gauss didn't know it, it's not elementary.</p>
|
4,463 | <p>It seems that most authors use the phrase "elementary number theory" to mean "number theory that doesn't use complex variable techniques in proofs." </p>
<p>I have two closely related questions.</p>
<ol>
<li>Is my understanding of the usage of "elementary" correct?</li>
<li>It appears that advanced techniques fro... | Zach Conn | 1,104 | <p>To me elementary number theory encompasses those results that can be completely derived starting with the definition of prime and composite within a book of reasonable size (say, a few hundred pages at most) without using any material not found in the first three years of a traditional undergraduate mathematics educ... |
744,377 | <p>I just don't understand how to complete $\epsilon - N$ proofs. I don't know what my goal is or why they prove what they do. I have asked two questions on here in the past, but I simply don't 'get it'.</p>
<p>So first we set $\epsilon \gt 0$ and we want to find $N \in \mathbb{N}$ such that $n \geq N$, we then take t... | Flatfoot | 142,723 | <p>Suppose you and your friend examine the sequence $a_n=3+1/n$. You look at $a_1, a_2, ...a_{100}$ and notice that $a_n$ approaches the value 3. </p>
<p><strong>So you tell your friend</strong>: "I think that $a_n$ approaches the value 3."</p>
<p><strong>Friend</strong>: "What do you mean?"</p>
<p><strong>You</stro... |
459,579 | <blockquote>
<p>Find the value of $3^9\cdot 3^3\cdot 3\cdot 3^{1/3}\cdot\cdots$</p>
</blockquote>
<p>Doesn't this thing approaches 0 at the end? why does it approaches 1?</p>
| mnsh | 58,529 | <p>$$3^9 * 3^3 * 3 * 3^{\frac{1}{3}} * ...=$$</p>
<p>$$3^{9\sum_{n=0}^{\infty}3^{-n}}=$$
$$3^{9*1.5}=$$
$$3^{13.5}$$</p>
|
192,394 | <p>I'm re-reading some material from Apostol's Calculus. He asks to prove that, if $f$ is such that, for any $x,y\in[a,b]$ we have</p>
<p>$$|f(x)-f(y)|\leq|x-y|$$</p>
<p>then:</p>
<p>$(i)$ $f$ is continuous in $[a,b]$</p>
<p>$(ii)$ For any $c$ in the interval,</p>
<p>$$\left|\int_a^b f(x)dx-(b-a)f(c)\right|\leq\fr... | Michael Hardy | 11,667 | <p>You need to prove
$$
(b-c)^2 + (a-c)^2 \le (b-a)^2.
$$
A bit of algebra reduces this to
$$
c^2-bc-ac+ab \le 0.
$$
Factor by grouping:
$$
(c-a)(c-b)\le 0.
$$
This just says $c$ is between $a$ and $b$. It says that regardless of whether $a\le b$ or $b\le a$.</p>
|
2,619,907 | <p>Let $\lambda$ be a partition of length $n$ and suppose its largest diagonal block, the Durfee square of $\lambda$, has size $r$. By this I mean that $\lambda = (\lambda_1,\ldots,\lambda_n)$ is a non-increasing sequence of numbers, which I depict by the following diagram</p>
<p>\begin{align*}
&\square \cdots \sq... | frame95 | 212,233 | <p>We will prove that $\lambda_j'-j+1 \neq k-\lambda_k$ (*) for every $j,k$, by induction on $n+\lambda_1$ (the sum of lengths of $\lambda, \lambda'$). As we mentioned in comments, this proves the lemma. Note that (*) rewrites as $(j-\lambda_j') +(k-\lambda_k) \neq 1$.</p>
<p>$n+\lambda_1 = 1$: $\lambda= \lambda'=(1)$... |
2,889,075 | <p>let $A$ be an infinite subset of $\mathbb R$ that is bounded above and let $u=\sup A$. Show that there exists an increasing sequence $ (x_n) $ with $x_n \in A $ for all $n\in \mathbb N$ such that $u = \lim_{n\rightarrow\infty} x_n$.</p>
<p>If $u$ is in $A$ then the proof is trivial. If $u$ does not belong to $A$... | salvarico | 379,130 | <p>Why is the proof when $u \in A$ trivial? How can we find a strictly increasing sequence that converges to $\sup A$ or to prove that such a sequence exists?</p>
|
189,266 | <p><strong>Q1:</strong> If a Morse function on a smooth closed $n$-manifold $X$ has critical points of only index $0$ and $n$, does it follow that $X\approx \mathbb{S}^n\coprod\ldots\coprod\mathbb{S}^n$?</p>
<p>I think the following question is essential in regard to the one above:</p>
<p><strong>Q2:</strong> If $f$ ... | MJD | 25,554 | <p>What if you used $$d(a,b) = \sum |a_i - b_i|$$ It is continuous, and has the usual metric properties such as the triangle inequality, and for $a,b\in \{0, 1\}^n$ it is identical to the Hamming distance.</p>
|
2,961,023 | <p>Is it allowed to solve this inequality <span class="math-container">$x|x-1|>-3$</span> by dividing each member with <span class="math-container">$x$</span>? What if <span class="math-container">$x$</span> is negative?</p>
<p>My textbook provides the following solution:</p>
<blockquote>
<p>Divide both sides b... | Vasili | 469,083 | <p>Because, the inequality holds for <span class="math-container">$x \ge 0$</span>, let's consider the case <span class="math-container">$x<0$</span>. <br>In this case, <span class="math-container">$|x-1|=-x+1$</span> and we have <span class="math-container">$-x^2+x+3>0$</span>. The roots of the quadratic are <sp... |
2,202,724 | <p><strong>Method 1:</strong></p>
<p><img src="https://i.stack.imgur.com/vRVgX.png" alt="Method 1 image hyperlink"></p>
<p><strong>Method 2:</strong></p>
<p><img src="https://i.stack.imgur.com/pwww8.png" alt="Method 2 image hyperlink"></p>
<p>In these two images, you will see that I have integrated $\sin^3 x$ using... | creative | 166,713 | <p>Consider $f(x)=x^2+3$. Then $f'(x)=2x$. Isn't it ? Now if integrate this $2x$ again what do I get ? I will get $x^2$. So what happened in this entire process ? I lost the "+3" in the original function. So to compensate for these losses that we add a "+c" at the end. This is just an elementary motivation. When we add... |
2,202,724 | <p><strong>Method 1:</strong></p>
<p><img src="https://i.stack.imgur.com/vRVgX.png" alt="Method 1 image hyperlink"></p>
<p><strong>Method 2:</strong></p>
<p><img src="https://i.stack.imgur.com/pwww8.png" alt="Method 2 image hyperlink"></p>
<p>In these two images, you will see that I have integrated $\sin^3 x$ using... | fleablood | 280,126 | <p>$\int f(x) dx$ is a function, $F(x)$ so that $F'(x) = f(x)$. If $F(x)$ is such a function, then if $G(x) = F(x) +k$ for some constant $k$ then $G'(x) = F'(x)$ and ... well, then $\int f(x) dx$ could logically just as likely be $G(x)$ as well as $F(x)$ and there is no way to say one is correct and the other is wrong... |
365,287 | <p>Let $([0,1],\mathcal{B},m)$ be the Borel sigma algebra with lebesgue measure and $([0,1],\mathcal{P},\mu)$ be the power set with counting measure. Consider the product $\sigma$-algebra on $[0,1]^2$ and product measure $m \times \mu$.</p>
<p>(1) Is $D=\{(x,x)\in[0,1]^2\}$ measurable?</p>
<p>(2) If so, what is $m \t... | mathcounterexamples.net | 187,663 | <p>I'm using</p>
<blockquote>
<p>Franco Vivaldi <em>Mathematical Writing</em> (Springer Undergraduate Mathematics Series)</p>
</blockquote>
<p>which I find very useful. It contains many definitions of basic mathematical objects (sets, maps...) as well as examples of proves.</p>
|
1,151,726 | <p>The following question is from Fred H. Croom's book "Principles of Topology"</p>
<blockquote>
<blockquote>
<p>In <span class="math-container">$\mathbb{R}^n$</span>, let <span class="math-container">$R$</span> denote the set of points having only rational coordinates and <span class="math-container">$I$</sp... | Eugene Zhang | 215,082 | <p>Part 2: Every irrational point is a limit point of some rational points since in any neighborhood of an irrational point, there is at least one rational point. Contrary of it that every rational point is a limit point of some irrational points also holds for the same reason. So</p>
<p>$R'= R^n$ and $I'=R^n$</p>
<p... |
157,074 | <blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="https://math.stackexchange.com/questions/155685/prove-binomp-1k-equiv-1k-pmod-p">Prove $\binom{p-1}{k} \equiv (-1)^k\pmod p$</a> </p>
</blockquote>
<p>The question is as follows:</p>
<blockquote>
<p>Let $p$ be prime. Show that ${p \choose k}\b... | DonAntonio | 31,254 | <p>Remember Wilson's Theorem for a prime $\,p\,$: $$(p-1)!=-1\pmod p$$and from what you already proved we get $$\binom {p-1}{k}=\binom{p}{k}-\binom{p-1}{k-1}=\binom{p-1}{k-1}\pmod p$$Now just observe $$\frac{(p-1)!}{(p-k)!}=(p-k+1)(p-k+2)\cdot ...\cdot (p-1)\equiv(1-k)(2-k)\cdot ...\cdot (-1)(\text{mod } p)\Longrightar... |
2,313,060 | <p>$f(\bigcap_{\alpha \in A} U_{\alpha}) \subseteq \bigcap_{\alpha \in A}f(U_{\alpha})$</p>
<p>Suppose $y \in f(\bigcap_{\alpha \in A} U_{\alpha})$
$\implies f^{-1}(y) \in \bigcap_{\alpha \in A} U_{\alpha} \implies f^{-1}(y) \in U_{\alpha}$ for all $\alpha \in A$</p>
<p>$\implies y \in f (U_{\alpha})$ for all $\alph... | Tucker | 256,305 | <p>Let $y\in f(\cap_{\alpha\in A}U_{\alpha})$, be arbitrary.
Then there exists an $x\in\cap_{\alpha\in A}U_{\alpha}$, for which $y=f(x)$. Since $x\in\cap_{\alpha\in A}U_{\alpha}$, we know that $x\in U_{\alpha}$ for each $\alpha\in A$. $y=f(x)\in f(U_{\alpha})$ for each $\alpha\in A$, implies</p>
<p>$$
y\in \cap_{\alph... |
3,206,138 | <p>I have the matrix <span class="math-container">$A=\begin{pmatrix}
1 & -1\\
1 &1
\end{pmatrix}$</span> and <span class="math-container">$A^{n}=\begin{pmatrix}
x_{n} & -y_{n}\\
y_{n} &x_{n}
\end{pmatrix}$</span></p>
<p>At first exercise I found that <span class="math-container">$2A-A^{2}=2I_{2}$<... | DINEDINE | 506,164 | <p>Here’s what is wrong in your method. When you square both side by writing <span class="math-container">$\frac{x}{1−x}=i\Longrightarrow \frac{x²}{(1−x)²}=−1$</span> you have admitted others solutions which are not necessarily solution of the first equation. With this method you may verify which ,of the solution you h... |
3,206,138 | <p>I have the matrix <span class="math-container">$A=\begin{pmatrix}
1 & -1\\
1 &1
\end{pmatrix}$</span> and <span class="math-container">$A^{n}=\begin{pmatrix}
x_{n} & -y_{n}\\
y_{n} &x_{n}
\end{pmatrix}$</span></p>
<p>At first exercise I found that <span class="math-container">$2A-A^{2}=2I_{2}$<... | TonyK | 1,508 | <p>Let's keep things simple. Suppose you are given <span class="math-container">$$x=1$$</span>
Squaring both sides gives <span class="math-container">$$x^2=1$$</span>
And the solutions of the quadratic equation <span class="math-container">$x^2=1$</span> are <span class="math-container">$x=1$</span> and <span class="ma... |
1,375,958 | <p>I am looking for a bounded funtion $f$ on $\mathbb{R}_+$ satisfying $f(0)=0$, $f'(0)=0$ and with bounded first and second derivatives. My intitial idea has been to consider trigonometric functions or compositions of them, but I still haven't found an adequate one. Any ideas would be greatly appreciated.</p>
| Ben Grossmann | 81,360 | <p>A good candidate is
$$
f(x) = \cos(x) - 1
$$</p>
|
939,725 | <p>Given that $a_0=2$ and $a_n = \frac{6}{a_{n-1}-1}$, find a closed form for $a_n$.</p>
<p>I tried listing out the first few values of $a_n: 2, 6, 6/5, 30, 6/29$, but no pattern came out. </p>
| Robert Israel | 8,508 | <p>The function $f(x) = \dfrac{6}{x-1}$ is a fractional linear transformation.<br>
Composition of these corresponds to matrix multiplication:
$$\text{if}\ T(x) = \dfrac{a_{11} x + a_{12 }}{a_{21} x + a_{22}} \ \text{and} \ S(x) = \dfrac{b_{11} x + b_{12}}{b_{21} x + b_{22}}$$
$$\text{then}\ T(S(x)) = \dfrac{c_{11} x ... |
1,075,879 | <p>I have to prove or disprove the following statement:</p>
<blockquote>
<p>If a group $G$ acts on a set $X$, then every subgroup $H$ of $G$ acts on the set $X$ as well, and every orbit of the action $G$ on $X$ is an union of orbits of the action $H$ on $X$.'</p>
</blockquote>
<p>But I have absolutely no clue what ... | Tryst with Freedom | 688,539 | <p>It is a two step proof:</p>
<ol>
<li><p>Show that an element belonging to a certain orbit is an equivalence relation. <a href="https://math.stackexchange.com/questions/2430073/prove-that-is-in-the-same-g-orbit-as-is-an-equivalence-relation">Refer</a></p>
</li>
<li><p>Show that equivalence relation partitions set int... |
779,987 | <p>I want to understand intuitively why it is that the gradient gives the direction of steepest ascent. (I will consider the case of $f:\mathbb{R}^2\to\mathbb{R}$)</p>
<p>The standard proof is to note that the directional derivative is $$D_vf=v\cdot \nabla f=|\nabla f|\,\cos\theta$$ which is maximized at $\theta=0$. T... | Johannes Hahn | 62,443 | <p>Maybe the following helps to understand the intuition behind the object $\langle \nabla f,v\rangle$ occuring in the standard proof: $\nabla f(x)$ is the vector composed of the directional derivatives of $f$ in the directions of the $n$ standard basis vectors $e_1,\ldots e_n$. Now consider a unit vector $v$ in the 1-... |
189,068 | <p>I am trying to derive a meaningful statistic from a survey where I have asked the person taking the survey to put objects in a certain order. The order the person puts the objects is compared to a correct order and I want to calculate the error.</p>
<p>For example:</p>
<p>Users order: 1, 3, 4, 5, 2</p>
<p>Corre... | Robert Israel | 8,508 | <p>You have to read the "fine print". For a substitution $g(x) = t$ in an integral $\int_a^b f(x)\ dx$ to be valid you need $g$ to be one-to-one on the interval $[a,b]$. $\sin x + \cos x$ is not one-to-one on $[0,\pi/2]$. </p>
|
189,068 | <p>I am trying to derive a meaningful statistic from a survey where I have asked the person taking the survey to put objects in a certain order. The order the person puts the objects is compared to a correct order and I want to calculate the error.</p>
<p>For example:</p>
<p>Users order: 1, 3, 4, 5, 2</p>
<p>Corre... | Tunococ | 12,594 | <p>When you do a substitution, make sure that your variables are mapped $1$-to-$1$ in the range that you are interested in. In this case, the mapping $x \mapsto \sin x + \cos x$ is not $1$-to-$1$ for $x \in [0, \pi/2]$. When $x = 0$, $t = 1$, and as $x$ increases $t$ increases until $x$ reaches $\pi/4$, which is when $... |
2,094,657 | <p>I found this interesting problem on AoPS forum but no one has posted an answer. I have no idea how to solve it.</p>
<blockquote>
<p>$$
\int_0^\infty \sin(x^n)\,dx
$$
For all positive rationals $n>1$, $I_n$ denotes the integral as above.</p>
<p>If $P_n$ denotes the product
$$
P_n=\prod_{r=1}^{n-1}I_{\b... | math110 | 58,742 | <p>since
$$T_{\frac{r}{n}}=\int_{0}^{+\infty}\sin{(x^{\frac{r}{n}})}dx=\Gamma\left(\dfrac{r}{n}+1\right)\sin{\dfrac{\pi r}{2n}}=\dfrac{r}{n}\Gamma\left(\dfrac{r}{n}\right)\sin{\dfrac{\pi r}{2n}},r=1,2,,\cdots,n-1$$
proof see:<a href="https://de.wikibooks.org/wiki/Formelsammlung_Mathematik:_Bestimmte_Integrale:_Form_R(x... |
3,156,359 | <p>I am currently attempting to solve a system of quadratic (and linear) systems that I have run into while trying to triangulate sound.</p>
<p>My hypothetical setup includes 3 sensors on a perfectly equilateral triangle, with one sensor located at <span class="math-container">$(0,0)$</span> and the other two located ... | Claude Leibovici | 82,404 | <p>After comments, let us work with <span class="math-container">$4$</span> sonsors. So we have
<span class="math-container">$$(x - a_1)^2 + (y - b_1)^2 = c^2(t_1-\tau)^2\tag 1$$</span>
<span class="math-container">$$(x - a_2)^2 + (y - b_2)^2 = c^2(t_2-\tau)^2\tag 2$$</span>
<span class="math-container">$$(x - a_3)^2 +... |
2,550,568 | <p>Suppose we have an alphabet of $a$ letters and a word $w$ of length $r$. What is the probablity that $w$ will appear in a sequence of $n$ letters drawn at random from the given alphabet?</p>
<p>I have posted a general question since there seem to be a few of these questions appearing, and this is intended as a gene... | Rus May | 17,853 | <p>The <a href="http://sites.math.rutgers.edu/~zeilberg/gj.html" rel="nofollow noreferrer">Goulden-Jackson cluster method</a> gives a simple algorithm to compute the generating function for the number of words that avoid a list of "bad words". (Besides being relevant to your problem, the referenced paper is written in ... |
4,765 | <p>I have a grid made up of overlapping <span class="math-container">$3\times 3$</span> squares like so:</p>
<p><img src="https://i.stack.imgur.com/BaY9s.png" alt="Grid"></p>
<p>The numbers on the grid indicate the number of overlapping squares. Given that we know the maximum number of overlapping squares (<span clas... | Jyotirmoy Bhattacharya | 1,195 | <p>The hard part is to show that for any $x$ such that $0 \le x \le 2\pi$, and any $\epsilon>0$ there exists a real number $y$ and two integers $m$ and $n$ such that $|y-x|<\epsilon$ and $n=2\pi m+y$. Hint: break up $[0,2\pi]$ into small subintervals, remember that $\pi$ is irrational and apply the pigeonhole princi... |
4,765 | <p>I have a grid made up of overlapping <span class="math-container">$3\times 3$</span> squares like so:</p>
<p><img src="https://i.stack.imgur.com/BaY9s.png" alt="Grid"></p>
<p>The numbers on the grid indicate the number of overlapping squares. Given that we know the maximum number of overlapping squares (<span clas... | Robin Chapman | 226 | <p>Jmoy's is a good hint, but I think an even nicer approach is to
prove that the points $(\cos n,\sin n)$ are dense on the unit circle,
and then deduce the results for the individual terms.</p>
|
815,868 | <blockquote>
<p>Consider the following system describing pendulum</p>
<p><span class="math-container">$$\begin{align} & \frac{dx}{dt} = y, \\ & \frac{dy}{dt} = − \sin x. \end{align}$$</span></p>
<p>I need to classify all critical points of the system.</p>
</blockquote>
<p>All critical points are of the form <sp... | Yuan Gao | 154,382 | <p>For stability of stationary points you only need to look at the Jacobian matrix</p>
<p>$\begin{pmatrix} f_{x} && f_{y}\\
g_{x} && g_{y}
\end{pmatrix}$, where $f = y$ and $g = -sin(x)$.</p>
<p>For two dimensional dynamical system the behavior of stationary points are well-studied. This is summarized... |
3,017,602 | <p>I am trying to solve the following differential equation:
<span class="math-container">$$\frac{dy}{dx}=\frac{y^{1/2}}{2}, \quad y>0$$</span></p>
<p>Here is what I tried:
<span class="math-container">$$
\begin{split}
\frac{dy}{dx} &= \frac{y^{1/2}}{2} \\
2y^{1/2}dy &= dx\\
6y^{3/2} &= x+c\\
y &= \... | gt6989b | 16,192 | <p>Relevant equation to integrate should be
<span class="math-container">$$
2y^{-1/2}dy = dx
$$</span>
This integrates to
<span class="math-container">$$
x+C=4y^{1/2}
$$</span></p>
|
3,840,253 | <blockquote>
<p>How to show that <span class="math-container">$\csc x - \csc\left(\frac{\pi}{3} + x \right) + \csc\left(\frac{\pi}{3} - x\right) = 3 \csc 3x$</span>?</p>
</blockquote>
<p>My attempt:<br />
<span class="math-container">\begin{align}
LHS &= \csc x - \csc\left(\frac{\pi}{3} + x\right) + \csc\left(\frac... | Alessio K | 702,692 | <p>We have <span class="math-container">$$\sin x \sin\left(\frac{\pi}{3} + x\right) + \sin\left(\frac{\pi}{3} - x\right) \cos\left(x + \frac{\pi}{6}\right) \\$$</span>
<span class="math-container">$$=\sin(x)[\sin(\frac{\pi}{3})\cos(x)+\cos(\frac{\pi}{3})\sin(x)]$$</span>
<span class="math-container">$$+(\sin(\frac{\pi... |
4,249,281 | <p>In a game, 6 balls are chosen from a set of 40 balls numbered from 1 to 40. Find the probability that the number 30 is drawn and it is the highest number drawn in at least one of the next five games.</p>
<p>I have <span class="math-container">$X\sim \operatorname{Bin}(5,6/29)$</span> and <span class="math-container"... | Henry | 6,460 | <p>Hints:</p>
<ul>
<li>For a particular game, you want the probability that <span class="math-container">$30$</span> is drawn and the other five numbers drawn are smaller.
<ul>
<li>one way is to look at the probability all six are <span class="math-container">$30$</span> or less, and the probability all six are <span c... |
4,249,281 | <p>In a game, 6 balls are chosen from a set of 40 balls numbered from 1 to 40. Find the probability that the number 30 is drawn and it is the highest number drawn in at least one of the next five games.</p>
<p>I have <span class="math-container">$X\sim \operatorname{Bin}(5,6/29)$</span> and <span class="math-container"... | Math Lover | 801,574 | <p>Based on your comment, a simpler approach is to see that the probability of choosing number <span class="math-container">$30$</span> in a game and it being the highest number is,</p>
<p><span class="math-container">$ \displaystyle p = {29 \choose 5} / {40 \choose 6} = \frac{87}{2812}$</span></p>
<p>So, we have <span... |
2,594,829 | <p>I'm having trouble knowing when my ansatz is wrong. For example if my ansatz to this is $y_p=a\cos{x}+b\sin{x},$ I get nowhere. How can I make a correct ansatz and are there any general rules to determine proper ansatz?</p>
<p><strong>Note:</strong> I know one can sovle this using eulers formula and all that but th... | The Phenotype | 514,183 | <p>It's all okay, your $y_p$ is correct.</p>
<p>$2\cos x=y_p''−2y_p'+y_p=-a\cos{x}-b\sin{x}-2(-a\sin{x}+b\cos{x})+a\cos{x}+b\sin{x}=2a\sin x-2b\cos x$</p>
<p>so $a=0$ and $b=-1$.</p>
|
3,805,089 | <p>I have directional vectors <span class="math-container">$a, b, c, d$</span> in vector 2 space as seen in the images below. Unfortunately I don't have the sufficient vocabulary to explain this in more mathematical terms. In rough terms I need to check if vector <span class="math-container">$c$</span> and <span class=... | Gribouillis | 398,505 | <p>The coordinates of <span class="math-container">$c$</span> in the basis <span class="math-container">$(a, b)$</span> are <span class="math-container">$\frac{\det(c,b)}{\det(a, b)}$</span>
and <span class="math-container">$\frac{\det(a, c)}{\det(a, b)}$</span> (these are Cramer's formulas). If both are positive, the ... |
2,949,789 | <p>Suppose I have some function <span class="math-container">$V(x)=x+log(c)$</span>, where <span class="math-container">$x$</span> is a continuous random variable and <span class="math-container">$c$</span> a constant bounded on <span class="math-container">$[0,1]$</span>. I have some queries regarding the following:</... | drhab | 75,923 | <p>Let <span class="math-container">$X$</span> be a random variable on probability space <span class="math-container">$(\Omega,\mathcal A,P)$</span>.</p>
<p>Then <span class="math-container">$X$</span> is a measurable function <span class="math-container">$\Omega\to\mathbb R$</span>, so it takes values in <span class=... |
2,632,273 | <p>so basically I want to know why when we have something like:</p>
<p>$$v(x) = x - y + 1$$
If we take the derivative with respect to x, it yields:</p>
<p>$$v'(x) = 1 - \frac{dy}{dx}$$</p>
<p>Now I still don't understand why when it comes to implicit differentiation, we need to tag a $y'$ or $\frac{dy}{dx}$ after ev... | Doug M | 317,162 | <p>$y$ is a function of $x$ so when $x$ varies, it is going to cause $y$ to vary along with it.</p>
<p>If you want to treat $y$ as a variable that is independent from $x$ that is called partial differentiation, and it would give the same result as setting $\frac {dy}{dx} = 0$</p>
|
5,612 | <p>This is driving me nuts: I'm trying to control the parameters for a relatively large system of ODEs using Manipulate.</p>
<pre><code>With[{todo =
Module[
{sol, ode, timedur = 40},
ode = Evaluate[odes /. removeboundaries /. moieties];
sol = NDSolve[Join[ode, init], vars, {t, 0, timedur}];
Plot[Evaluate... | celtschk | 129 | <p>Your problem probably is that the right hand side of the <code>{todo = ...}</code> is evaluated before inserting into the Manipulate. Use <code>{todo := ...}</code> instead to disable premature evaluation. However if calculation of <code>ode</code> and <code>sol</code> does not involve <code>params</code> in any way... |
1,824,280 | <p>The question is from one of the past exams in a course I am doing. I have gotten halfway through it but cannot figure out how to finish it off.</p>
<p>So the first part was to prove that $4 \mid n^2 - 5 $ if $n$ is an odd integer. </p>
<p>Here is a brief proof (without intricate details):<br>
Consider $n = 2k+1$<b... | barak manos | 131,263 | <p>Alternatively, you can simply consider the following cases:</p>
<ul>
<li>$n\equiv0\pmod8 \implies n^2-5\equiv0^2-5\equiv- 5\equiv3\not\equiv0\pmod8$</li>
<li>$n\equiv1\pmod8 \implies n^2-5\equiv1^2-5\equiv- 4\equiv4\not\equiv0\pmod8$</li>
<li>$n\equiv2\pmod8 \implies n^2-5\equiv2^2-5\equiv- 1\equiv7\not\equiv0\pmod... |
1,824,280 | <p>The question is from one of the past exams in a course I am doing. I have gotten halfway through it but cannot figure out how to finish it off.</p>
<p>So the first part was to prove that $4 \mid n^2 - 5 $ if $n$ is an odd integer. </p>
<p>Here is a brief proof (without intricate details):<br>
Consider $n = 2k+1$<b... | lab bhattacharjee | 33,337 | <p>$$n^2\equiv5\pmod8\implies n^2\equiv5\pmod2\equiv1\implies n\text{ must be odd}$$</p>
<p>Now $(2a+1)^2=8\cdot\dfrac{a(a+1)}2+1\equiv1\pmod8\not\equiv5$</p>
|
1,824,280 | <p>The question is from one of the past exams in a course I am doing. I have gotten halfway through it but cannot figure out how to finish it off.</p>
<p>So the first part was to prove that $4 \mid n^2 - 5 $ if $n$ is an odd integer. </p>
<p>Here is a brief proof (without intricate details):<br>
Consider $n = 2k+1$<b... | zxcvber | 329,909 | <p>For any integer $n$, one of these has to be true.
$$n \equiv 0, \pm1, \pm2, \pm3, \pm4 \ (\rm mod 8)$$</p>
<p>Square both sides, to get</p>
<p>$$n^2 \equiv 0,1, 4, 9, 16 \ (\rm mod 8)$$
Then,
$$n^2 \equiv 0,1, 4 \ (\rm mod 8)$$
(9 is congruent to 1, 16 is congruent to 0)</p>
<p>Therefore, $n^2-5$ can never be div... |
1,478,038 | <p>Polyhedrons or three dimensional analogues of polygons were studied by Euler who observed that if one lets $f$ to be the number of faces of a polyhedron, $n$ to be the number of solid angles and $e$ to be the number of joints where two faces come together side by side $n-e+f=2$.</p>
<p>It was later seen that a seri... | NeitherNor | 262,655 | <p>Something like $\mathcal{N}(0,\sigma^2)$? That's the symbol for a normal distribution: <a href="https://en.wikipedia.org/wiki/Normal_distribution" rel="nofollow">https://en.wikipedia.org/wiki/Normal_distribution</a></p>
<p>The formula basically says that b is normally distributed with zero mean and variance $\sigma... |
74,188 | <blockquote>
<p>Let <span class="math-container">$a,c \in \mathbb R$</span> with <span class="math-container">$a \neq 0$</span>, and let <span class="math-container">$b \in \mathbb C$</span>. Define <span class="math-container">$$S=\{z\in \mathbb C: az\bar{z}+b\bar{z}+\bar{b}z+c=0\}.$$</span></p>
<p>a. Show that <span ... | Zarrax | 3,035 | <p>Where $z = x + iy$, your equation can be rewritten as
$$a(x^2 + y^2) + 2Re(b\bar{z}) + c = 0$$
Writing $b = b_1 + b_2i$ this is the same as
$$a(x^2 + y^2) + 2b_1x + 2b_2y + c = 0$$
Complete the square now....</p>
|
4,046,532 | <p><strong>QUESTION 1:</strong> Let <span class="math-container">$f, g: S\rightarrow \mathbb{R}^m$</span> be differentiable vector-valued functions and let <span class="math-container">$\lambda\in \mathbb{R}$</span>. Prove that the function <span class="math-container">$(f+g):S\rightarrow \mathbb{R}^m$</span> is also d... | Rafael | 894,475 | <p>Question 2. You do not need to extend <span class="math-container">$f$</span> on <span class="math-container">$S$</span> (it is an answer to your comment to another answer) because <span class="math-container">$f$</span> is already defined on <span class="math-container">$S$</span>, it is already a normal function. ... |
1,828,042 | <p>This is my first question on this site, and this question may sound disturbing. My apologies, but I truly need some advice on this.</p>
<p>I am a sophomore math major at a fairly good math department (top 20 in the U.S.), and after taking some upper-level math courses (second courses in abstract algebra and real an... | Rustyn | 53,783 | <p>At some point, math has less to do with intelligence and more to do with patience and methodology in learning. <BR><BR></p>
<p>If you really want to continue studying math, then why not "try" grad school. For me, grad school was less about getting a master's degree and more about learning more mathematics.<BR><BR><... |
2,886,460 | <blockquote>
<p>Let $\omega$ be a complex number such that $\omega^5 = 1$ and $\omega \neq 1$. Find
$$\frac{\omega}{1 + \omega^2} + \frac{\omega^2}{1 + \omega^4} + \frac{\omega^3}{1 + \omega} + \frac{\omega^4}{1 + \omega^3}.$$</p>
</blockquote>
<p>I have tried combining the first and third terms & first and la... | dxiv | 291,201 | <p>Alt. hint: let $\,z=\omega+\dfrac{1}{\omega}\,$ so that $\,z^2=\omega^2+\dfrac{1}{\omega^2}+2\,$, then use that $\,\omega^4=\bar\omega\,$ and $\,\omega^3=\bar\omega^2\,$ so the sum is:</p>
<p>$$\frac{\omega}{1 + \omega^2} + \frac{\omega^2}{1 + \omega^4} + \frac{\bar\omega^2}{1 + \bar\omega^4} + \frac{\bar\om... |
3,442,173 | <p>Give a counter example to each of the following:<br>
(a) G is a connected graph with a cut-vertex, then G contains a bridge.
(b) G is a tree if and only if it contains no cycle.</p>
| SlipEternal | 156,808 | <p>Suppose that <span class="math-container">$F(x)$</span> is an antiderivative of <span class="math-container">$f'(x)$</span>. We know that <span class="math-container">$F(x)$</span> and <span class="math-container">$f(x)$</span> differ by only a constant. Therefore:</p>
<p><span class="math-container">$$f(x) = F(x) ... |
4,255,430 | <p>I know that <strong>if</strong> both of the limits
<span class="math-container">$$
\lim_{x\to a} f(x) \quad\text{and}\quad \lim_{x\to a} g(x)
$$</span>
exist (so they are both equal to real numbers), then
<span class="math-container">$$
\lim_{x\to a} f(x) + g(x) = \lim_{x\to a} f(x) + \lim_{x\to a} g(x)
$$</span>
... | gamma Integrator | 908,678 | <p>you can write the sum as <span class="math-container">$$ x^2 + 4 * 2x + 2* \dfrac{32}{x^3} $$</span></p>
|
93,274 | <p>Wielandt wrote a paper titled "Remarks on diagonable matrices".</p>
<p>According to Mathematische Werke - Mathematical Works : Linear Algebra and Analysis
by Helmut Wielandt, Hans Schneider, Bertram Huppert (Editor) page 260 this paper from Wielandt remained unpublished (at least from the 1950s to the 1980s).</p>
... | cimo | 22,899 | <p>The Mutzin Taussky theorem states in fact an equivalence between the two propositions you gave. If you understand french, here is a paper presenting a problem ( starting from page 16 till 22) whose purpose is to prove that theorem.
<a href="http://agreg.org/Rapports/rapport2009.pdf" rel="nofollow">http://agreg.org/R... |
2,174,413 | <p>Proof by induction, that </p>
<p>$$x_n=10^{(3n+2)} + 4(-1)^n\text{ is divisible by 52, when n}\in N $$</p>
<p>for now I did it like that:</p>
<p>$$\text{for } n=0:$$
$$10^2+4=104$$
$$104/2=52$$
<br>
$$\text{Let's assume that:}$$
$$x_n=10^{(3n+2)} + 4(-1)^n=52k$$
$$\text {so else}$$
$$4(-1)^n=52k-10^{3n+2}$$</p>
... | Juniven Acapulco | 44,376 | <p>Assume that for $n\in\Bbb N$, $$10^{(3n+2)} + 4(-1)^n\text{ is divisible by 52.} $$ This means that $$\frac{10^{(3n+2)} + 4(-1)^n}{52}\in\Bbb Z.$$ Now,
$$\begin{align}
\frac{10^{[3(n+1)+2]} + 4(-1)^{n+1}}{52}&=\frac{10^{(3n+2)}10^3 + 4(-1)^n(-1)}{52}\\
&=\frac{10^{(3n+2)}10^3 + 4(-1)^n(10^3-1001)}{52}\\
&am... |
2,004 | <h1>Background:</h1>
<p>It is troublesome to upload some files to Stack exchange. A possibility is to give some Google Drive or Dropbox URL but it maybe unaccessible to some people. Using the SE Uploader gives some error information like the following:</p>
<p><img src="http://o8aucf9ny.bkt.clouddn.com/2016-10-15-17-0... | yode | 21,532 | <p>Thank for the J.M.'s and SqRoots' help in <a href="https://mathematica.stackexchange.com/questions/128688/how-to-call-picture-bed-api">this post</a>,I make a function to do this,one can use it encrypt any express in a URL.Now you just share your URL to another:</p>
<pre><code>ShareAny[expr_] :=
Module[{list, imag... |
2,004 | <h1>Background:</h1>
<p>It is troublesome to upload some files to Stack exchange. A possibility is to give some Google Drive or Dropbox URL but it maybe unaccessible to some people. Using the SE Uploader gives some error information like the following:</p>
<p><img src="http://o8aucf9ny.bkt.clouddn.com/2016-10-15-17-0... | Chris Degnen | 363 | <p>You can store limited amounts of data as QR images as I used here:</p>
<p><a href="https://mathematica.stackexchange.com/a/73088/363">https://mathematica.stackexchange.com/a/73088/363</a></p>
<p>You don't need to scan them; the data can be retrieved from the image URL.</p>
|
3,479,765 | <p>I am attempting to do this using Cauchy's integral theorem and formula. However I am unable to conclude if a singularity exists at all for me to apply any of those two techniques or any other theorem. </p>
| lab bhattacharjee | 33,337 | <p>If <span class="math-container">$$f(a,b)=\dfrac1{x^a(x+c)^b},$$</span></p>
<p><span class="math-container">$$f(a,b)=\dfrac1c\cdot\dfrac{x+c-x}{\cdots}=\dfrac{f(a,b-1)}c-\dfrac{f(a-1,b)}c$$</span></p>
<p>We can use this reduction formula repeatedly until at least one of <span class="math-container">$a,b$</span> bec... |
1,893,168 | <p>$$\lim_{x\to 0} {\ln(\cos x)\over \sin^2x} = ?$$</p>
<p>I can solve this by using L'Hopital's rule but how would I do this without this?</p>
| Aakash Kumar | 346,279 | <p>$$\frac{\log\left(\cos\left(x\right)\right)}{\sin^{2}\left(x\right)}=\frac{\log\left(1+\cos\left(x\right) -1 \right)}{\sin^{2}\left(x\right)}$$
$$\lim_{x\to 0} \frac{\log\left(1+\cos\left(x\right) -1 \right)}{\sin^{2}\left(x\right)}.\frac{x^2}{x^2}.\frac{\left(\cos\left(x\right) -1 \right)}{\left(\cos\left(x\right) ... |
1,893,168 | <p>$$\lim_{x\to 0} {\ln(\cos x)\over \sin^2x} = ?$$</p>
<p>I can solve this by using L'Hopital's rule but how would I do this without this?</p>
| Mark Viola | 218,419 | <blockquote>
<p>In <a href="https://math.stackexchange.com/questions/1589429/how-to-prove-that-logxx-when-x1/1590263#1590263">THIS ANSWER</a>, I showed using only the limit definition of the exponential function and Bernoulli's Inequality that the logarithm function satisfies the inequalities </p>
<p>$$\frac{x-1... |
2,668,839 | <blockquote>
<p>Finding range of $$f(x)=\frac{\sin^2 x+4\sin x+5}{2\sin^2 x+8\sin x+8}$$</p>
</blockquote>
<p>Try: put $\sin x=t$ and $-1\leq t\leq 1$</p>
<p>So $$y=\frac{t^2+4t+5}{2t^2+8t+8}$$</p>
<p>$$2yt^2+8yt+8y=t^2+4t+5$$</p>
<p>$$(2y-1)t^2+4(2y-1)t+(8y-5)=0$$</p>
<p>For real roots $D\geq 0$</p>
<p>So $$16... | choco_addicted | 310,026 | <p>Let
$$g(t)=\frac{t^2+4t+5}{2t^2+8t+8}=\frac{1}{2}+\frac{1}{2t^2+8t+8},$$
then
$$
g'(t)=-\frac{2(2t^2+8t+8)(4t+8)}{(2t^2+8t+8)^2}=-\frac{1}{(t+2)^3}.
$$
Thus $g$ strictly decreases in the interval $[-1,1]$, and $g(-1)$ and $g(1)$ are maximum and minimum respectively.</p>
|
2,668,839 | <blockquote>
<p>Finding range of $$f(x)=\frac{\sin^2 x+4\sin x+5}{2\sin^2 x+8\sin x+8}$$</p>
</blockquote>
<p>Try: put $\sin x=t$ and $-1\leq t\leq 1$</p>
<p>So $$y=\frac{t^2+4t+5}{2t^2+8t+8}$$</p>
<p>$$2yt^2+8yt+8y=t^2+4t+5$$</p>
<p>$$(2y-1)t^2+4(2y-1)t+(8y-5)=0$$</p>
<p>For real roots $D\geq 0$</p>
<p>So $$16... | Mohammad Riazi-Kermani | 514,496 | <p>Let $t=sin(x)$, $$y=\frac{t^2+4t+5}{2t^2+8t+8}=\frac {1}{2}[1+ \frac {1}{(t+2)^2}]$$</p>
<p>Note that $$-1\le t \le 1$$ Thus $$ 1\le y \le5/9$$ </p>
|
1,959,080 | <p>A book claims that $9(9_9) = 9^{387420489}$.</p>
<p>I've never seen such an expression, and I've been unable to find anything about it on Google...</p>
<p>How is it supposed to be evaulated?</p>
<p>For reference, the name of the book is <code>Pasatiempos curiosos e instructivos</code> and this is the page where t... | adjan | 219,722 | <p>$9_9$ seems to be a strange notation of $9^9$, since $9^9 = 387,420,489$.</p>
|
1,315,265 | <p>Let $X=\mathcal{L}_2 [-1,1]$ and for any scalar $\alpha$ we define $E_\alpha=\{f\in \mathcal{L}: f \text{ continuous in } [-1,1] \text{ and } f(0)=\alpha \}$.</p>
<ol>
<li>Prove $E_\alpha$ is convex for any $\alpha$.</li>
<li>Prove $E_\alpha$ is dense in $\mathcal{L}_2$</li>
<li>Prove there is no $f\in X^*$ that se... | Callus - Reinstate Monica | 94,624 | <p>This was getting too long for a comment. Analysis is not my thing, but I think you've reduced the problem to showing that for a continuous function $f\in \mathcal{C}^0(\left[0,1\right])$ and $\epsilon > 0$, you can find $g\in E_\alpha$ such that $\left|f-g\right|_2<\epsilon$. So, given a continuous $f$, let ... |
3,894,437 | <p>Let <span class="math-container">$\mathcal{A}$</span> be a finite set and consider the set of all sequences <span class="math-container">$\mathcal{A}^{\mathbb{Z}}$</span> on <span class="math-container">$\mathbb{Z}$</span> with values in <span class="math-container">$\mathcal{A}$</span>. This set has a cardinality o... | Rob Arthan | 23,171 | <p>The set of sequences <span class="math-container">$a_i$</span> that are constant for all but finitely many <span class="math-container">$i$</span> is countable and dense in <span class="math-container">$A^\Bbb{Z}$</span>.</p>
|
214,219 | <p>I have defined a function p(V,T):</p>
<pre><code>p[V_, T_] = (R*T/(V - b)) - (a/(V^2))
</code></pre>
<p>And then I used: </p>
<pre><code>Integrate[p[V, T], {V, V1, V2}]
</code></pre>
<p>And there is no output. What I have done wrong?
Obs.: I have also tried <code>dintt</code></p>
| Nasser | 70 | <blockquote>
<p>And there is no output</p>
</blockquote>
<p>sometimes when Integrate seems to hang or take long time, use the option <code>GenerateConditions -> False</code></p>
<pre><code>ClearAll[p, V, T, V1, V2, R, a, b];
p[V_, T_] := (R*T/(V - b)) - (a/(V^2));
Integrate[p[V, T], {V, V1, V2}, GenerateConditio... |
214,219 | <p>I have defined a function p(V,T):</p>
<pre><code>p[V_, T_] = (R*T/(V - b)) - (a/(V^2))
</code></pre>
<p>And then I used: </p>
<pre><code>Integrate[p[V, T], {V, V1, V2}]
</code></pre>
<p>And there is no output. What I have done wrong?
Obs.: I have also tried <code>dintt</code></p>
| Bob Hanlon | 9,362 | <pre><code>$Version
(* "12.0.0 for Mac OS X x86 (64-bit) (April 7, 2019)" *)
Clear["Global`*"]
p[V_, T_] = (R*T/(V - b)) - (a/(V^2));
</code></pre>
<p>Include an assumption</p>
<pre><code>Assuming[V2 > V1, Integrate[p[V, T], {V, V1, V2}]]
(* ConditionalExpression[
a (-(1/V1) + 1/V2) +
R T (-Log[-b + V1] + ... |
3,212,499 | <p>I'm struggling to find a solution to this exercise:</p>
<blockquote>
<p>Consider a set of 65 girls and a set of 5 boys. Prove that there are 3
girls and 3 boys such that either every girl knows every boy or no
girl knows any of the boys.</p>
</blockquote>
<p>I know I should use the Ramsey Theorem but I have ... | Μάρκος Καραμέρης | 563,059 | <p>Name the boys <span class="math-container">$A_1,...,A_5$</span>. Now for every girl let <span class="math-container">$(x_1,x_2,...,x_5), x_i=0$</span> or <span class="math-container">$1$</span> represent if she knows or doesn't know the corresponding boy. You have at most <span class="math-container">$2^5=32$</span>... |
163,468 | <p>I have a <code>Graph</code>, and I want to group some of its vertices into communities. <code>CommunityGraphPlot</code> uses force directed layout and its doesn't look like the original graph after I apply <code>CommunityGraphPlot</code>. I don't want the vertices of same community to come close so that the communit... | halmir | 590 | <p>If you want to draw blob like CommunityGraphPlot:</p>
<pre><code>iBlobs[style_, pts_, size_] :=
Block[{epts},
epts = Flatten[Tuples[CoordinateBounds[#, size]] & /@ pts, 1];
{style,
FilledCurve@
BSplineCurve[MeshPrimitives[ConvexHullMesh[epts], 1][[All, 1, 1]],
SplineClosed -> True]}
]
{... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.