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 |
|---|---|---|---|---|
3,930,466 | <p>I want to prove by induction that</p>
<p><span class="math-container">$$\left(\tfrac{1}{n}\sum_{k=1}^{n} x_k\right)^2\leq\tfrac{1}{n}\sum_{k=1}^{n} x_k^2$$</span></p>
<p>But I have no idea how. For n=1 it is trivial that
<span class="math-container">$$\left(\tfrac{1}{1}\sum_{k=1}^{1} x_k\right)^2\leq\tfrac{1}{1}\sum... | Macavity | 58,320 | <p>With a little rewriting, the Induction hypothesis is
<span class="math-container">$$\color{blue}{\left(\sum_{k=1}^n x_k\right)^2 \leqslant n \sum_{k=1}^n x_k^2}$$</span></p>
<p>For the inductive step, note
<span class="math-container">$$\left(\sum_{k=1}^n x_k +x_{n+1} \right)^2 = \color{blue}{\left(\sum_{k=1}^n x_k ... |
3,740,501 | <p>This question builds upon a previous question I asked <a href="https://math.stackexchange.com/questions/3548807/is-at-least-1-of-4-non-concyclic-points-contained-in-the-circle-through-the-othe">here</a></p>
<p>The answers helped me understand that for <span class="math-container">$4$</span> points that are not concy... | Axion004 | 258,202 | <p>Your answer is incorrect because you wrote <span class="math-container">$f(t+a)=f(t+5)$</span> when the step function had a right shift of <span class="math-container">$4$</span>. Applying</p>
<p><span class="math-container">$$\mathcal{L} \left(u(t-a)f(t)\right) = e^{-as} \mathcal {L} [f(t+a)],$$</span></p>
<p>we fi... |
63,796 | <p>I'm looking for a neater way to achieve this, which to me looks awkward and suggests that I am missing something...</p>
<p>Given a Dataset...</p>
<pre><code>ds = Dataset[{
<|"a"->1,"b"->1,"c"->3|>,
<|"a"->1,"b"->2,"c"->4|>,
<|"a"->2,"b"->3,"c"->5|>}];
</code></pre>
<p>... | Ymareth | 880 | <p>Typically as soon as I post this occurs to me...</p>
<pre><code>ds[GroupBy[#a &], MaximalBy[#b &]][Values, Merge[First]]
</code></pre>
<p>Since Values is a descending operator.</p>
|
51,732 | <p>A <em>Perron number</em> is a real algebraic integer $\lambda$ that is larger than the absolute value of any of its Galois conjugates. The Perron-Frobenius theorem says that any
non-negative integer matrix $M$ such that some power of $M$ is strictly positive has a
unique positive eigenvector whose eigenvalue is a Pe... | Misha Belolipetsky | 13,859 | <p>You can ask a similar question about Pisot and Salem numbers. Last year together with my project student Charlie Scarr we were looking, in particular, at a possible connection between distribution of the roots inside the unit circle and the Mahler measure of a polynomial. We did not progress too far but Charlie made... |
1,353,922 | <p>A function $f$ is continuous for all $x \geq 0$ and $f(x) \neq 0$ for all $x >0$. </p>
<p>If $\{f(x)\}^2 = 2\int_0^xf(t)dt $ then $f(x) = x$ for all $x \geq 0$.</p>
<p>But I am stuck with the sum. </p>
| Brian Cheung | 248,555 | <p>Differentiate both sides w.r.t. $x$ </p>
<p>Simplifying yields $2(f'(x)-1)f(x)=0$</p>
<p>i.e. $f'(x)=1$ or $f(x)=0$ (rej.) </p>
<p>Then integrate it back w.r.t. $x$ to get $f(x)=x+C$, use $f(0)=0$ and get $f(x)=x$.</p>
|
3,406,817 | <p>What intrinsic relationship is there between (riemann) integration and Euclidean geometry that enables one to get area under the curve as a integral? This can only be related to the definition of riemann integral as a limit of sums of areas of rectangles; or is it something else.</p>
| José Carlos Santos | 446,262 | <p>The Riemann integral is defined in such a way (through a limit of sums of areas of rectangles, as you wrote) that the area under the curve of a non-negative function is equal to the integral of that function.</p>
|
1,626,721 | <blockquote>
<p>In the <span class="math-container">$n$</span>-cell below, we are to tile it completely with cells of the form <span class="math-container">$\boxdot$</span> and <span class="math-container">$\boxtimes \hspace{-0.45 mm}\boxtimes$</span>. How many tilings are possible for a <span class="math-container">$1... | N. S. | 9,176 | <p><strong>Hint</strong> Let us figure out $H_{n+1}$. If the last tile is $\boxdot$, by erasing this we get a tiling of $n$ by the same tilings. There are $H_n$ such tilings, and by adding $\boxdot$ at the end we get a tiling of $n+1$.</p>
<p>If the last tiling is $\boxtimes \hspace{-0.45 mm}\boxtimes$ by erasing it w... |
479,851 | <p>I toss a coin a many times, each time noting down the result of the toss. If at any time I have tossed more heads than tails I stop.
I.e. if I get heads on the first toss I stop.
Or if toss T-T-H-H-T-H-H I will stop.
If I decide to only toss the coin at most 2n+1 times, what is the probability that I will get more ... | Brian M. Scott | 12,042 | <p>There is no harm in imagining that we toss the coin $2n+1$ times and simply look to see whether at some point we got a success (meaning more heads than tails at that point).</p>
<p>Say that a finite string of H’s and T’s is <em>balanced</em> if it contains an equal number of H’s and T’s, and that it is <em>complete... |
1,252,857 | <p>Let $f\colon \mathbb R \to \mathbb R$ be defined by $f(x)= 5x^3+3$. Is it onto?</p>
<p>According to me, if $y=5x^3+3$, then $x = \sqrt[3]{(y-3)/5}$ is not an element of $\mathbb R$ for all $y \in (-\infty,3)$ so all numbers in the codomain $(-\infty,3)$ wont have pre-images.</p>
<p>But many say $5x^3+7$ as an odd ... | Noiralef | 224,757 | <p>Why do you say that $x = \sqrt[3]{\frac{y-3}5}$ is not an element of $\mathbb R$ for all $y$?</p>
<p>For every $b \in \mathbb R$ there is exactly one $a \in \mathbb R$ such that $a^3 = b$, we call this $a = \sqrt[3]{b}$.
Maybe you are confused because for $b < 0$ there is no $a \in \mathbb R$ such that $a^2 = b$... |
2,751,300 | <blockquote>
<p>Figure out with distinct real numbers the system of equations.</p>
<p><span class="math-container">$$\frac{xy}{x-y} = \frac{1}{30}$$</span>
<span class="math-container">$$\frac{x^2y^2}{x^2+y^2} = \frac{1}{2018}$$</span></p>
</blockquote>
<p>I multiplied x-y both side on the first equation and square on ... | g.kov | 122,782 | <p>Since $x,y\ne0$,
we can use $u=1/x$, $v=1/y$:
\begin{align}
v-u &= 30
\tag{1}\label{1}
,\\
v^2+u^2 &= 2018
\tag{2}\label{2}
.
\end{align} </p>
<p>\begin{align}
\eqref{2}-\eqref{1}^2:\quad
2uv&=1118
,\\
v^2+u^2+2uv &= 2018+1118
,\\
(u+v)^2&=56^2
,\\
u+v&=\pm56
\tag{3}\label{3}
.
\end{align... |
18,136 | <p>I introduced the hypercube (to undergraduate students in the U.S.) in the
context of generalizations of the Platonic solids, explained its
structure, showed it rotating.
I mentioned <a href="https://matheducators.stackexchange.com/a/1824/511">Alicia Stott</a>, who discovered the <span class="math-container">$6$</spa... | Kafein | 13,579 | <p>I would posit it's impossible for most humans to truly grok higher dimensions. I've heard of extremely rare exceptions but I'm going to ignore them here.</p>
<p>Having an intuitive understanding of some fact about geometry requires having a <em>mental image</em> of what is going on. You can't have 4D mental images.... |
18,136 | <p>I introduced the hypercube (to undergraduate students in the U.S.) in the
context of generalizations of the Platonic solids, explained its
structure, showed it rotating.
I mentioned <a href="https://matheducators.stackexchange.com/a/1824/511">Alicia Stott</a>, who discovered the <span class="math-container">$6$</spa... | Aloizio Macedo | 4,817 | <p>My suggestion would be to demystify the concept and try to disassociate it from spatial interpretations at the first approach.</p>
<p>Having <span class="math-container">$n$</span> dimensions is just having <span class="math-container">$n$</span> variables. As one professor I had liked to repeat, a grocery store ow... |
3,886,643 | <p>I have doubts about how to solve this issue about bases in numbering systems:</p>
<p>"Determine the base b in <span class="math-container">$(104)_b = 8285$</span>"</p>
<p>can anybody help me?
Thanks.</p>
| Théophile | 26,091 | <p>Hint: In the number <span class="math-container">$104_b$</span>, the final digit represents the units, the middle digit represents the number of "<span class="math-container">$b$</span>"s, and the leftmost digit represents the number of "<span class="math-container">$b^2$</span>"s. (Compare to ba... |
3,140,052 | <p>I found a proof of this theorem in <a href="http://web.mit.edu/18.705/www/13Ed.pdf" rel="nofollow noreferrer">modern algebra</a> 2.30</p>
<blockquote>
<p>Proof: Set <span class="math-container">$S := \{ \text{ ideals } \mathfrak b \mid \mathfrak b ⊃ \mathfrak a \text{ and } \mathfrak b ∌ 1 \}$</span>. Then <span ... | Arthur | 15,500 | <p><span class="math-container">$S$</span> is a subset of the powerset of the elements in the ring. As such it can be easily constructed directly using the axioms of ZF set theory (the power set axiom and one of the axioms of comprehension / specification, specifically), starting with the set of all the elements of the... |
2,346,921 | <blockquote>
<p>Evaluate the following integral using Laplace transform
$$\int_0^\infty\frac{e^{-ax}\sin bx}{x}\,dx$$</p>
</blockquote>
<p>I obtained this partial result
$$=\int_0^\infty \frac{1}{p+1} \frac{b}{p^2+b^2}\,dp$$
and I am stuck here. I know that the final answer is $$\arctan\frac{b}{a}.$$ I would appr... | Robert Z | 299,698 | <p>Your partial result does not hold. According to <a href="https://en.wikipedia.org/wiki/Laplace_transform#Evaluating_improper_integrals" rel="noreferrer">Laplace transform properties</a>
$$\int_0^\infty\frac{e^{-ax}\sin bx}{x}\,dx=\cal{L}\left(\frac{\sin bx}{x}\right)(a)=\int_a^{+\infty}
\cal{L}\left(\sin bx\right)(... |
3,138,450 | <p>Let <span class="math-container">$A \subseteq \mathbb{R}$</span> be bounded above and let <span class="math-container">$c \in \mathbb
{R}$</span>. Define the set <span class="math-container">$c + A = \{c + a : a \in A\} $</span></p>
<p>Now since <span class="math-container">$a \leq \sup A , \forall a\in A$</span>. ... | fleablood | 280,126 | <blockquote>
<p>So now I have to prove that ∀ε>0,c+a>c+supA−ε.</p>
</blockquote>
<p>Well if that <em>is</em> what you have to prove, then you'd have <span class="math-container">$\forall \epsilon > 0, a > \sup A - \epsilon$</span> so <span class="math-container">$c+a > c +\sup A - \epsilon$</span>.</p>
<p>... |
3,249,391 | <p>By Mathematica, we find <span class="math-container">$$\sum_{n=1}^\infty \frac{4^n}{n^3\binom{2n}{n}}=\pi^2\log(2)-\frac{7}{2}\zeta(3).$$</span></p>
<blockquote>
<p>How to find the closed form for general series:
<span class="math-container">$$\sum_{n=1}^\infty \frac{4^n}{n^p\binom{2n}{n}}? \ \ (p\ge 3)$$</spa... | Robert Z | 299,698 | <p>Note that
<span class="math-container">$$\sum_{k=1}^\infty \frac{(4x)^n}{n^2{{2n}\choose n}}=2\arcsin^2(\sqrt{x}).$$</span>
Hence, for <span class="math-container">$p=3$</span> we have the integral form
<span class="math-container">$$\sum\limits\limits_{n=1}^\infty \frac{4^n}{n^3\binom{2n}{n}}=\int_{0}^1\frac{2\arcs... |
4,621,227 | <p>This is a very soft and potentially naive question, but I've always wondered about this seemingly common phenomenon where a theorem has some method of proof which makes the statement easy to prove, but where other methods of proof are incredibly difficult.</p>
<p>For example, proving that every vector space has a ba... | Thomas | 128,832 | <p>Questions like this always remind me of Hamiltons search for a multiplication in <span class="math-container">$\mathbb{R}^3$</span> which somehow extends the multiplication of real and complex numbers (*). He searched, in vain, for years. And he was, at his time, a really famous and renowned mathematician.</p>
<p>To... |
4,311,731 | <p>I was thinking about solutions of equation <span class="math-container">$x^2=i$</span> . First thought coming to my mind was <span class="math-container">$x=\pm \sqrt{i}$</span> . ( I know it's wrong ) .
Then I thought if we solve this equation like real problem then
<span class="math-container">$|x|=\sqrt i$</span>... | mathma | 270,091 | <p><span class="math-container">$\sqrt{i}$</span> can be defined, simply take and <span class="math-container">$8$</span>-th root of unity. You can also compute it explicitely in the following way, by Euler's formula <span class="math-container">$i=e^{i\pi/2}$</span> so a square root of <span class="math-container">$i$... |
3,042,308 | <p>Suppose a matrix <span class="math-container">$A$</span> has eigenvalues 0, 3, 7 and eigenvectors <span class="math-container">$\mathbf{u, v, w,}$</span> respectively. Find the least square minimum length solution for <span class="math-container">$A\mathbf{x} = \mathbf{u+v+w}$</span>.</p>
<p>This was on our enginee... | AmbretteOrrisey | 613,228 | <p><span class="math-container">$${1\over2}(2\operatorname{heavi}(x)-\operatorname{kron\delta}(x))(2\operatorname{heavi}(y)+\operatorname{kron\delta}(y)-1)+\operatorname{kron\delta}(x) ,$$</span>where <span class="math-container">$$\operatorname{heavi}$$</span>is the <em>Heaviside</em> function, &<span class="math-... |
2,699,536 | <blockquote>
<p>Given a continuous function $f:[0,1]\to\mathbb R$, prove that $$\forall t>0, \frac{1}{t}\cdot\ln\left(\int_{0}^{1}e^{-tf(x)}dx\right)\leq-\min f(x).$$</p>
</blockquote>
<p>I have no idea where to begin. Thought I could use the FTOC to come up with some form of antiderivative but I don't know if th... | Michael Hardy | 11,667 | <p>The inequality says
$$ \int_0^1 e^{-tf(x)} \, dx \leq e^{-t\min_x f(x)}. $$</p>
<p>Since $t>0,$ the quantity $e^{-t\min_x f(x)}$ is the largest value of the function $e^{-tf(x)}$ within $0\le x\le 1.$ So this just says that</p>
<p>$$
\int_0^1 g(x) \,dx \le (1-0)\times \max\{ g(x) : 0\le x\le 1 \}.
$$</p>
|
3,091,162 | <p>Let <span class="math-container">$G_1, G_2$</span> be two groups with at least one nontrivial proper subgroup each.</p>
<p>Let <span class="math-container">$S_1, S_2$</span> be the sets of proper subgroups of, respectively <span class="math-container">$G_1, G_2$</span>.</p>
<p>Suppose there exists a bijective func... | verret | 191,246 | <p>There are two pairs of examples of order <span class="math-container">$16$</span>. These are the smallest examples. One of these two pairs is <span class="math-container">$C_4\times C_4$</span> and <span class="math-container">$C_4\rtimes C_4$</span>. For both of these, the complete list of proper subgroups is:</p>
... |
3,783,186 | <p>I am trying to prove that <span class="math-container">$$2≤\int_{-1}^1 \sqrt{1+x^6} \,dx ≤ 2\sqrt{2} $$</span> I learned that the equation <span class="math-container">$${d\over dx}\int_{g(x)}^{h(x)} f(t)\,dt = f(h(x))h'(x) - f(g(x))g'(x) $$</span> is true due to Fundamental Theorem of Calculus and Chain Rule, a... | SarGe | 782,505 | <p><img src="https://i.stack.imgur.com/JhuEW.png" alt="enter image description here" /></p>
<p>[Edited after the <a href="https://math.meta.stackexchange.com/q/32326/782505">meta question</a> and <a href="https://chat.stackexchange.com/transcript/message/55188059#55188059">this</a> conversation.]</p>
<p>Let <span class... |
3,176,593 | <p>I'm having a hard time understanding how to find all solutions of the form <span class="math-container">$a_n = a^{(h)}_n+a_n^{(p)}$</span></p>
<p>I show that <span class="math-container">$a_n=n2^n \to a_n=2(n-1)2^{n-1} +2^n=2^n(n-1+1)=n2^n$</span>.</p>
<p>I can show that <span class="math-container">$a_n^{(h)}$</s... | Minus One-Twelfth | 643,882 | <p>Your homogeneous solution has <span class="math-container">$2^n$</span> in it already. When this happens, for the particular solution part, we cannot just use <span class="math-container">$C2^n$</span> (you have seen what happens if we do). Instead, <em>the rule in this scenario is to modify the guess by <strong>mul... |
632,850 | <blockquote>
<p>Suppose that $f(x)$ is differentiable on $[0,1]$ and $f(0) = f(1) = 0$. It is also
known that $|f''(x)| \le A$ for every $x \in (0,1)$. Prove that
$|f'(x)| \le A/2$ for every $x \in [0,1]$.</p>
</blockquote>
<p>I'll explain what I did so far. First using Rolle's theorem, there is some point $c \i... | Disintegrating By Parts | 112,478 | <p>If $f''$ exists on $[0,1]$ and $|f''(x)| \le A$, then $f'$ is Lipschitz continuous, which implies that $f'$ is absolutely continuous. So $f'(a)-f'(b)=\int_{a}^{b}f''(t)\,dt$ for any $0 \le a, b \le 1$. And, of course, the same is true of $f$ because $f$ is continuously differentiable.</p>
<p>Because $f(0)=0$, one h... |
4,401,028 | <p>Let <span class="math-container">$p$</span> be an odd prime and <span class="math-container">$ω=e^{2\pi i /p}$</span>. Determine if <span class="math-container">$1-ω$</span> is prime in <span class="math-container">$\mathbb{Z}[ω]=\mathbb{Z}+\mathbb{Z}ω+\mathbb{Z}ω^2+...+\mathbb{Z}ω^{p-2}$</span></p>
<p><strong>My at... | B. Goddard | 362,009 | <p>People sometimes work with "pure functions." So if you are considering <span class="math-container">$0$</span> to be a function, then you could write <span class="math-container">$\sin =0$</span>. It would be a false statement, because the two functions are not the same. The CAS Maple, for instance, will... |
123,712 | <p>How can I prove that the function
$$ f(x) = \left\{\begin{array}{l l} x &\text{if }x \in \mathbb{Q} \\ -x & \text{if } x \notin \mathbb{Q}
\end{array} \right. $$
is discontinuous for $x \neq 0$, using $\epsilon$'s and $\delta$'s?</p>
<p>I see that is truth. But I cannot prove using only $\eps... | David Mitra | 18,986 | <p>Let $x_0\ne 0$ be fixed. Informally, $f$ is continuous at $x_0$ would mean that $f(x)$ can be made as close to $f(x_0)$ as desired by taking $x$ sufficiently close to $x_0$. With the epsilon business, this means that $\color{maroon}{ \text{for every}}$ $\epsilon>0$, there is a $\delta>0$ such that $$\tag{1}
\... |
1,591,197 | <p>Let $ A \trianglelefteq G $ and $ B \trianglelefteq A $ a Sylow normal subgroup of $ A $. My textbook says then that $ B \trianglelefteq G $.</p>
<p>I don’t understand why that is.</p>
| Tsemo Aristide | 280,301 | <p>Since $B$ is a normal sylow $p$-subgroup there is one $p$-sylow subgroup in $A$ since all the sylow $p$-subgroups are conjugated. Let $g\in G, gBg^{-1}\subset A$ since $A$ is normal in $G$ and is a $p$-sylow subgroup since it has the same cardinal than $B$ thus is $B$ since $B$ is the unique $p$-sylow subgroup of $A... |
416,407 | <blockquote>
<p>What examples are there of habitual but unnecessary uses of the axiom of
choice, in any area of mathematics except topology?</p>
</blockquote>
<p>I'm interested in standard proofs that use the axiom of choice, but where
choice can be eliminated via some judicious and maybe not quite obvious
rephrasing. ... | Tom Leinster | 586 | <p>Turning ZM's <a href="https://mathoverflow.net/questions/416407/unnecessary-uses-of-the-axiom-of-choice#comment1068341_416407">comment</a> into an answer: Zorn's lemma is sometimes invoked to show that the maximal atlas (in the definition of differentiable manifolds) exists, but it is <a href="https://math.stackexch... |
416,407 | <blockquote>
<p>What examples are there of habitual but unnecessary uses of the axiom of
choice, in any area of mathematics except topology?</p>
</blockquote>
<p>I'm interested in standard proofs that use the axiom of choice, but where
choice can be eliminated via some judicious and maybe not quite obvious
rephrasing. ... | Tim Campion | 2,362 | <p>The existence of a Haar measure on any locally compact group was first proven by Weil using the axiom of choice. Cartan later supplied a choice-free proof.</p>
<p>Because the Haar measure is unique up to a scalar factor, this is an example where it seems "obvious" that choice really shouldn't be necessary.... |
416,407 | <blockquote>
<p>What examples are there of habitual but unnecessary uses of the axiom of
choice, in any area of mathematics except topology?</p>
</blockquote>
<p>I'm interested in standard proofs that use the axiom of choice, but where
choice can be eliminated via some judicious and maybe not quite obvious
rephrasing. ... | Kameryn Williams | 64,676 | <p>A good number of theorems in Ramsey theory and related areas are what logicians call <span class="math-container">$\Pi^1_2$</span> statements—those of the form "for every set of integers <span class="math-container">$X$</span> there is a set of integers <span class="math-container">$Y$</span> satisfying some pr... |
1,481,106 | <p>I am having a difficult time solving this problem. I have tried this several different ways, and I get a different result, none of which is correct, every time. I've derived an answer geometrically and cannot replicate it with a double integral.</p>
<p>Here's the problem: Use a double integral to find the area betw... | SchrodingersCat | 278,967 | <p>Surface area
$$=\int \int_S dS$$
$$=\int \int_S \frac{dx \,\ dy}{\hat n\cdot\hat k}$$
since the surface considered here is $z=0$
$$=\int^{y=2}_{y=-2} \int^{x=\sqrt{4-y^2}}_{x=1-\sqrt{4-y^2}} dx \,\ dy$$
$$=\int^{y=2}_{y=-2} (2\sqrt{4-y^2}-1)dy$$
$$= 4(\pi - 1)$$</p>
|
1,481,106 | <p>I am having a difficult time solving this problem. I have tried this several different ways, and I get a different result, none of which is correct, every time. I've derived an answer geometrically and cannot replicate it with a double integral.</p>
<p>Here's the problem: Use a double integral to find the area betw... | amd | 265,466 | <p>As a double integral, this is $I=\int_R dA$, where $R$ is the region given by the intersection of the two disks. This integral can be evaluated as an iterated integral in several ways. Picking up where you left off, we can integrate with respect to $y$ first, then with respect to $x$, i.e., $$I=2\int_\frac12^2 \int_... |
1,136,486 | <p>Can anyone help me find the first digit of $2015^{2015}$?</p>
<p>It is easy to find the last digit but I have no idea for the first digit.</p>
| Empy2 | 81,790 | <p>Take the base-ten logarithm.<br>
By the log laws, that will equal $2015\log2015$. Suppose that is $x+y$, where $x$ is a whole number and $0<y<1$.
The number has $x+1$ digits.<br>
$y$ tells you the first digit. Or rather, $10^y$ starts with the same digits $2015^{2015}$ does.</p>
|
1,136,486 | <p>Can anyone help me find the first digit of $2015^{2015}$?</p>
<p>It is easy to find the last digit but I have no idea for the first digit.</p>
| Mario Carneiro | 50,776 | <p>Now, you didn't say "by hand" as these calculations often do, but I'll assume that at least you're from before 1950 so that solutions like just <a href="http://www.wolframalpha.com/input/?i=2015%5E2015">asking Wolfram Alpha</a> about a mere 8000-digit number are beyond you.</p>
<p>We can calculate $\log_{10}(2015^{... |
674,621 | <p>I am trying to figure out what the three possibilities of $z$ are such that </p>
<p>$$
z^3=i
$$</p>
<p>but I am stuck on how to proceed. I tried algebraically but ran into rather tedious polynomials. Could you solve this geometrically? Any help would be greatly appreciated.</p>
| Dan | 1,374 | <p>Taking the absolute value of both sides: $|z^3| = |i|$, gives $|z| = 1$. So, $z = \cos (\theta) + i \sin (\theta)$ for some real $\theta$.</p>
<p>Using De Moivre's formula gives $z^3 = \cos(3\theta) + i \sin(3\theta)$. Given that $z^3 = i = 0 + 1i$, this means that $\cos(3\theta) = 0$ and $\sin(3\theta) = 1$. So... |
674,621 | <p>I am trying to figure out what the three possibilities of $z$ are such that </p>
<p>$$
z^3=i
$$</p>
<p>but I am stuck on how to proceed. I tried algebraically but ran into rather tedious polynomials. Could you solve this geometrically? Any help would be greatly appreciated.</p>
| IQ WANTER | 511,144 | <p>
$z^3=i$ $⟹ z=\sqrt [3] {i}$<br />
</p>
<p>We know,
$\sqrt [3] {i} =a+bi$</p>
<p>$⟹ i=(a+bi)^3 = a^3-3ab^2+3a^2bi-b^3i= a^3-3ab^2+(3a^2b-b^3)i$</p>
<p>From this we can say, </p>
<p>$a^3-3ab^2=0$________________(i)</p>
<p>$3a^2b-b^3=1$_______________(ii)</p>
<p>From (i) we get,</p>
<p>$a(... |
1,601,427 | <blockquote>
<p>Let $a,b,c$ be three nonnegative real numbers. Prove that $$a^2+b^2+c^2+3\sqrt[3]{a^2b^2c^2} \geq 2(ab+bc+ca).$$</p>
</blockquote>
<p>It seems that the inequality $a^2+b^2+c^2 \geq ab+bc+ca$ will be of use here. If I use that then I will get $a^2+b^2+c^2+3\sqrt[3]{a^2b^2c^2} \geq ab+bc+ca+3\sqrt[3]{a... | Mufasa | 49,003 | <p>I think I have got close to proving this but I'm not sure if this is a valid proof - but here goes...</p>
<p>Using AM-GM we can show that:
$$a^2+b^2+c^2\ge3\sqrt[3]{a^2b^2c^2}\tag{1}$$
$$ab+bc+ca\ge3\sqrt[3]{a^2b^2c^2}\tag{2}$$
We can also show that:
$$a^2+b^2+c^2\ge ab+bc+ca\tag{3}$$
We can therefore infer that:
$... |
615,396 | <p>What is considered a good format for writing problem sets in mathematics? Are there any good examples of problem sets that are well-written and formatted that you can show me?</p>
| Tyler Clark | 3,791 | <p>I like to write my proofs in LaTeX. I have gotten some nice templates offline for this and modified them to my preference. I have also created a commands file so I am not constantly writing out long commands (i.e. \mathbb{R} vs. \bb{R}). I can send you the .tex files for these if you are interested. </p>
<p>I also ... |
767,474 | <blockquote>
<p>If $a,b,c\in\mathbb R^+$ prove that:
$$\frac{a^2+1}{b+c}+\frac{b^2+1}{a+c}+\frac{c^2+1}{a+b}\ge 3$$</p>
</blockquote>
| evil999man | 102,285 | <p>By Cauchy Schwarz Lemma:
$$\frac{a^2}{b+c}+\frac{b^2}{a+c}+\frac{c^2}{a+b}\geqslant \frac{(a+b+c)^2}{2(a+b+c)}$$</p>
<p>Also By $HM\leqslant AM$</p>
<p>$$\frac 3 {\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{a+c}}\leqslant \frac{2(a+b+c)}{3}$$</p>
<p>$$\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{a+c}\geqslant\frac9 {2(a+b+c)}$$... |
189,218 | <p>I have raw images that I can import and view with ImageJ. They are the output of a program which I don't have the source code for, so I'm stuck with the output format.</p>
<p>They are imported as follows with ImageJ:</p>
<blockquote>
<p>Image type: 16 bit Unsigned</p>
<p>Width: 320</p>
<p>Height: 25600</p>
<p>Offset... | yode | 21,532 | <p>After some dig in version 13, I found a package that can do this indeed. And I have tried Canon and Sony, so far so good.</p>
<pre><code><< ImageFileTools`
</code></pre>
<p>Here is a function <code>ImageFileTools`Raw`RawGet</code>, and its usage is:</p>
<pre><code>? ImageFileTools`Raw`RawGet
</code></pre>
<p><... |
308,436 | <p>$P$ is probability. We have: $P(A) \ge \frac{2}{3}$, $P(B) \ge \frac{2}{3}$, $P(C) \ge \frac{2}{3}$ and $P(A \cap B \cap C)=0$. We have to find $P(A)$.
How to do it? Of course we have $P(A \cup B \cup C) = P(A) + P(B) + P(C) - P(A \cap B)- P(A \cap C)- P(B \cap C)$, but what next? Please, help me.</p>
| dtldarek | 26,306 | <p>First, observe that </p>
<p>$$P(X \cap Y) = 0 \text{ implies } P(X \cap Y^c) = P(X). \tag{1}$$</p>
<p>Then, bound $P(B \cap C)$ by
$$ P(B \cap C) = P(B) + P(C) - P(B \cup C) \geq 2\frac{2}{3}-1 = \frac{1}{3}.$$</p>
<p>Finally, apply $(1)$ to $P(A\cap (B \cap C)) = 0$ to get
$$P(A) = P(A \cap (B \cap C)^c) \leq \f... |
3,466,707 | <p>In that problem</p>
<p><span class="math-container">$$\lim\limits_{x \to \infty} \left(1-\frac{1}{x}\right)^\left(e^x\right)$$</span></p>
<p>I use <span class="math-container">$\ln$</span>, then it gave me <span class="math-container">$0\times\infty$</span> indeterminate, then I use L'Hospital Rule but I cannot re... | Bernard | 202,857 | <p><strong>Hint</strong>:</p>
<p>Determine first the limit of the log with an <em>asymptotic equivalent</em>:
<span class="math-container">$$\mathrm e^x\ln\Bigl(1-\frac 1x\Bigr)\sim_{x\to\infty} \mathrm e^x\cdot\Bigl(-\frac1x\Bigr).$$</span></p>
|
878,816 | <p>Solve inequality: $-5 < \frac{1}{x} < 0$ </p>
<p>I thought about how I can solve this. If I multiply all sides by $x$ I'm afraid I'm removing the answer, cause $\frac{x}{x}=1$. And when $x$ 'leaves' the inequality I'm left with no letter. </p>
<p>How do I get just $x$ in the middle without adding $x$ to ot... | J126 | 2,838 | <p>Since you know that $x\neq 0$ you can multiply by $x$. Just remember that $x<0$. So, multiplying by $x$ reverses the inequalities.</p>
<p>Then you get $-5x>1>0$. This gives $-5x>1$. Now divide by $-5$.</p>
|
1,360,658 | <p>prove: $$\frac{1-\cos 2\theta}{1-\cos\theta}=2\cos\theta-2$$</p>
<p>I'm thinking that there will be something to square in this? Because I notice that the $LHS$ looks like the half-angle identity....</p>
<p>Edit: I am so sorry guys, my grave mistake, the expression should have been equal to 2 instead like,</p>
<p... | Socre | 223,538 | <p>$$\frac{(1-(cos^{2}(\theta)-sin^2(\theta))}{(1-cos(\theta)}=\frac{1-cos(2\theta)}{(1-cos(\theta)}$$
Found By using property $cos(2\theta)=cos^{2}(\theta)-sin^{2}(\theta)$
$$\frac{2sin^{2}(\theta)}{1-cos(\theta)}=\frac{2(1-cos^{2}(\theta))}{(1-cos(\theta)}$$
Found By using the fact that $ 1- cos^{2}(\theta)=sin^{2}(\... |
1,330,362 | <p>I need to define in ZFC the following things:</p>
<ul>
<li>image and domain of a binary relation ($\{ x \mid (x,y)\in f \}$ would be a definition of domain, but it is a class for which is for me is not quite clear why it is a set)</li>
<li>f[X] for a binary relation $f$ and a set $X$</li>
</ul>
<p>Please describe ... | Ben Grossmann | 81,360 | <p>The best we can do (in terms of algebraic solutions) is as follows: rewrite the equation as
$$
a = \left(e^x\right)^{\ln(b)} + \left(e^x\right)^{\ln(c)}
$$
setting $y = e^x$, this becomes an equation on $y$:
$$
a = y^{\ln(b)} + y^{\ln(c)}
$$
Or, rearranging,
$$
y^{\ln(b)} + y^{\ln(c)} - a = 0
$$
In other words, we'r... |
1,524,109 | <p>Could anyone help me with this proof without using determinant? I tried two ways. </p>
<blockquote>
<p>Let $A$ be a matrix. If $A$ has the property that each row sums to zero, then there does not exist any matrix $X$ such that $AX=I$, where $I$ denotes the identity matrix. </p>
</blockquote>
<p>I then get stuck.... | Jimmy R. | 128,037 | <p><strong>Hint:</strong> You can sum the elements of a row by multiplying this row with a vector of $1$'s. Can you find now a matrix $X$ (with appropriate columns) such that $AX=Ο$?</p>
|
1,524,109 | <p>Could anyone help me with this proof without using determinant? I tried two ways. </p>
<blockquote>
<p>Let $A$ be a matrix. If $A$ has the property that each row sums to zero, then there does not exist any matrix $X$ such that $AX=I$, where $I$ denotes the identity matrix. </p>
</blockquote>
<p>I then get stuck.... | Pietro Paparella | 414,530 | <blockquote>
<p><strong>Theorem.</strong> If $A$ is an $n$-by-$n$ matrix, then $A$ is not invertible if and only if zero is an eigenvalue of $A$.</p>
</blockquote>
<p>If $e$ denotes the all-ones vector of appropriate size, then, by hypothesis, $Ae = 0 = 0e$, i.e., zero is an eigenvalue of $A$.</p>
|
2,053,017 | <p>The first thing I was trying to do is to substitute $a$ with something like $1 \over b$ or $1 + \frac{1}{b}$, but I got no results. Then I tried to use Squeeze theorem here, but I also got confused because, for example, $(a + 1)^{1/n}$ is not a decreasing function. </p>
| Mark Viola | 218,419 | <p>Without Loss of Generality, we assume $0<a<1$. Let $x_n=(1/a)^{1/n}-1$ so that $0\le x_n$. Then, we see using Bernoulli's Inequality that</p>
<p>$$a=\frac{1}{(1+x_n)^n}\le \frac{1}{1+nx_n}$$</p>
<p>whereupon isolating $x_n$ reveals</p>
<p>$$0 \le x_n \le \frac{1-a}{a\,n}\tag 1$$</p>
<p>Applying the squee... |
194,312 | <p>I'm preparing myself to a combinatorics test. A part of it will concentrate on the pigeonhole principle. Thus, I need some hard to very hard problems in the subject to solve.
I would be thankful if you can send me links\books\or just a lone problem.</p>
| Sungjin Kim | 67,070 | <p>IF $\alpha>0$ is irrational, then the fractional parts of $n\alpha$ forms a dense subset of $[0,1]$. </p>
<p>Given a positive integer $n>0$, Fibonacci numbers modulo $n$ is periodic. </p>
<p>These are very famous ones from number theory. There are proofs of these using pigeonhole principle. </p>
|
1,260,722 | <blockquote>
<p>Prove that <span class="math-container">$f=x^4-4x^2+16\in\mathbb{Q}[x]$</span> is irreducible.</p>
</blockquote>
<p>I am trying to prove it with Eisenstein's criterion but without success: for <strong>p=2</strong>, it divides <strong>-4</strong> and the constant coefficient 16, don't divide the leading ... | Jyrki Lahtonen | 11,619 | <p>The polynomial has no real roots, because it is equal to $(x^2-2)^2+12$. The remaining possibility is thus that it is a product of two quadratic factors. By Gauss' Lemma these need to have integer coefficients, so we are looking for a possibile factorization like
$$
p(x)=x^4-4x^2+16=(x^2+ax+b)(x^2+cx+d)
$$
with some... |
1,260,722 | <blockquote>
<p>Prove that <span class="math-container">$f=x^4-4x^2+16\in\mathbb{Q}[x]$</span> is irreducible.</p>
</blockquote>
<p>I am trying to prove it with Eisenstein's criterion but without success: for <strong>p=2</strong>, it divides <strong>-4</strong> and the constant coefficient 16, don't divide the leading ... | Sil | 290,240 | <p>After substitution
$$
g(x)=f(2x-1)=16x^4-32x^3+8x^2+8x+13
$$
it becomes irreducible by <a href="http://cms.dm.uba.ar/academico/materias/2docuat2011/teoria_de_numeros/Irreducible.pdf" rel="nofollow noreferrer">Murty's criterion</a> (see Theorem 1) for $n=4$, since $g(4)=2221$ is a prime. The criterion is not so well ... |
223,718 | <p>Suppose I have two lists to which I apply the <code>Line</code> command for example</p>
<pre><code>A = {{-4,0},{4,0}}
B = {{0,4},{0,-4}}
</code></pre>
<p>and I take <code>Line[A]</code> and <code>Line[B]</code>. Is there a way to get Mathematica to tell me the intersection points of the line? Of course this is a v... | kglr | 125 | <h3><code>Graphics`Mesh`FindIntersections</code></h3>
<pre><code>Graphics`Mesh`MeshInit[];
findIntersections = Complement[Graphics`Mesh`FindIntersections[Join[##]],
Join @@ Graphics`Mesh`FindIntersections /@ {##}] &;
</code></pre>
<p>Using <code>redLines</code> and <code>blueLines</code> from flinty's answer:... |
2,598 | <p>Should Wolfram Alpha Notebook questions be considered on-topic?</p>
<p>Here's an example: <a href="https://mathematica.stackexchange.com/questions/240780/calculating-double-integral-bounded-by-domain-in-wolfram-alpha-notebook">Calculating double integral bounded by domain in Wolfram Alpha Notebook</a></p>
<p>Here ar... | Penelope Benenati | 74,045 | <p><strong>Yes, most of the time (but not always). It seems to me that many users do not know well Notebook Edition and are not aware that Notebook Edition accepts almost all code syntax of Mathematica.</strong></p>
<p>To answer this question, one should first define "Wolfram Alpha Notebook questions." I use ... |
1,575,532 | <p>I'm having some difficulty with this problem out of my Calculus Book: </p>
<blockquote>
<p>Find the total area between region and the $x$-axis: $y=x^3-x^2-6x$, $-2 \le x \le 3.$</p>
</blockquote>
<p>I know I start with setting the function equal to $0$:</p>
<p>$$0=x^3-x^2-6x$$
$$0=x(x-3)(x-2)$$
Hence $x=3$ or ... | Vincenzo Zaccaro | 269,380 | <p>You use thr Rienmann integral. Then $A=\int_{-2 }^3(x^3-x^2-6x) dx=[x^4/4-x^3/3-3x^2]_{x-2}^{x=3}$. This is the area with sign. If you want the area without sign: $A= \int_{-2} ^0(x^3-x^2-6x) dx- \int_{0} ^3(x^3-x^2-6x) dx $. The polynomial is negative in $[0,3]$.</p>
|
1,575,532 | <p>I'm having some difficulty with this problem out of my Calculus Book: </p>
<blockquote>
<p>Find the total area between region and the $x$-axis: $y=x^3-x^2-6x$, $-2 \le x \le 3.$</p>
</blockquote>
<p>I know I start with setting the function equal to $0$:</p>
<p>$$0=x^3-x^2-6x$$
$$0=x(x-3)(x-2)$$
Hence $x=3$ or ... | gt6989b | 16,192 | <p>You factored incorrectly. Note that $$f(x) = x^3 - x^2 - 6x = x(x-3)(x+2)$$ so the roots lie at $-2,0,3$. The plot looks like</p>
<p><a href="https://i.stack.imgur.com/WoR2l.gif" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/WoR2l.gif" alt="enter image description here"></a></p>
<p>If you want the ... |
742,160 | <p>Answer true or false to each of the following questions. If a statement is true, prove it. If a statement is false, give a counterexample.</p>
<ol>
<li>For all sets $A$,$B$ and $C$: IF $A ⊆ B$ and $A ⊆ C$, Then $A ⊆ (B ∩ C)$</li>
<li>For all sets $A$ and $B$, if $|A| \le |B|$, then $A ⊆ B$</li>
</ol>
| Michal Boska | 140,880 | <p>Take a look at this picture (from <a href="https://en.wikipedia.org/wiki/File:Surjection.svg" rel="noreferrer">Wikipedia</a>):</p>
<p><a href="https://i.stack.imgur.com/XVMeJm.png" rel="noreferrer"><img src="https://i.stack.imgur.com/XVMeJm.png" alt="][2]][2]"></a></p>
<p>This function is NOT injection, because tw... |
6,090 | <p>I have been reading about Riemann Zeta function and have been thinking about it for some time.</p>
<p>Has anything been published regarding upper bound for the real part of zeta function zeros as the imaginary part of the zeros tend to infinity?</p>
<p>Thanks</p>
| timur | 2,473 | <p>De La Vallee-Pousin's theorem has been improved by Korobov and Vinogradov in the 50's and I believe their result is the strongest known asymptotic zero free region, cf. <em>The Riemann zeta-function: Theory and applications</em> by Alexandar Ivic. One can find more recent papers but my impression is that they don't ... |
4,506,196 | <p>If <span class="math-container">$R$</span> is a commutative Hilbert ring then each of its prime ideals is an intersection of maximal ideals. Is there a similar class of commutative rings for which every ideal is an intersection of prime ideals (that is, in which all ideals are radical)? Does this class have a name? ... | Eric Wofsey | 86,856 | <p>As Qiaochu Yuan's answer observed, such a ring must have the property that every finitely generated ideal is generated by an idempotent and therefore every ideal is generated by idempotents. Conversely, I claim that if every principal ideal in a commutative ring <span class="math-container">$R$</span> is generated ... |
1,027,707 | <p>In how many ways can 100 identical chairs be divided among 4 different rooms so that each room will have 10,20,30,40 or 50 chairs?</p>
<p>I'm having problems coming up with the generating function for this question. </p>
<p>The answer given is 68. </p>
| Community | -1 | <p>The question is incorrect, you need a tensor product in place of the product.
With that modification you can use/prove that</p>
<p>$\mathrm{Hom}_\mathrm{FDVect}(V,W) \cong W \otimes V^*$</p>
|
2,095,349 | <p>I test my skills in statistics and probabilities and I decided to work with distributions. So, I tried to solve the below problem </p>
<p><strong>Problem</strong></p>
<p>Suppose that a hospital serves in average $80$ citizens daily from a city with $11000$ citizens.
In a random day, what is the probability that t... | Sean Roberson | 171,839 | <p>Using the appropriate commands in R and Excel, the actual probability is $9.360672 \cdot 10^{-17}$. However, this is extremely small, so an answer of zero would also be acceptable.</p>
<p>When it comes to statistics and the normal distribution, don't expect "exact" answers. Why? Well, the probabilities are generate... |
2,008,653 | <p>According to Riemann (I think) the (exact) prime counting function is given by:
<span class="math-container">$$
\pi(x) = \operatorname{R}(x^1) - \sum_{\rho}\operatorname{R}(x^{\rho}) \tag{1}
$$</span>
with <span class="math-container">$ \operatorname{R}(z) = \sum_{n=1}^{\infty} \frac{ \mu (n)}{n} \operatorname{li... | Greg Hurst | 93,700 | <p>Borwein, Bradley, and Crandall state on page 249 <a href="https://cr.yp.to/bib/2000/borwein.pdf" rel="noreferrer">here</a> that</p>
<blockquote>
<p>... it should be the case that, in some appropriate sense
$$ \pi(x) \sim \text{Ri}(x) - \sum_{\rho} \text{Ri}(x^\rho) \tag{4} $$
with $\text{Ri}$ denoting the Rie... |
3,465,607 | <p>If I have a cubic where I know the turning points, can I find what its equation is?</p>
<p>I already know that the derivative is 0 at the turning points. Other than that, I'm not too sure how I can continue.</p>
<p>Suppose I have the turning points (-2,5) and (4,0). I have started doing the following:
<span class=... | Bernard | 202,857 | <p>Welcome! You simply forgot that having the turning points provides the derivative up to a nonzero constant factor, i.e.
<span class="math-container">$$y'(x)=K(x+2)(x-4),\quad K\in \mathbf R^*, \quad
\text{ whence }\;y(x)=K\biggl(\frac{x^3}3 -x^2-8x\biggr)+C.$$</span>
You now have two constants to adjust the ordinate... |
3,465,607 | <p>If I have a cubic where I know the turning points, can I find what its equation is?</p>
<p>I already know that the derivative is 0 at the turning points. Other than that, I'm not too sure how I can continue.</p>
<p>Suppose I have the turning points (-2,5) and (4,0). I have started doing the following:
<span class=... | marty cohen | 13,079 | <p>If the turning points
of a cubic polynomial
<span class="math-container">$f(x)$</span> are
<span class="math-container">$(a, b)$</span> and <span class="math-container">$(c, d)$</span>
then
<span class="math-container">$f(x)
=k(\dfrac{x^3}{3}-\dfrac{(a+c)x^2}{2}+acx)+h
$</span>
where
<span class="math-container">$k
... |
3,630,482 | <p>Show that, if 13 divides <span class="math-container">$n^2$</span> + <span class="math-container">$3n$</span> + <span class="math-container">$51$</span> then 169 divides <span class="math-container">$21n^2$</span> + <span class="math-container">$89n$</span> + <span class="math-container">$44$</span></p>
<p>We have ... | Joshua Wang | 773,061 | <p>First, turn the inequalities into lines by using an equals sign in place of the inequality sign. Then take every possible pair and find their point of intersection. Then, substitute each intersection point into the original system of inequalities to see if it lies on the solution region. Of all of the points that wo... |
4,260,645 | <p>I can't find any insights online on how useful the graph of function <span class="math-container">$f(x)$</span> (on the <span class="math-container">$y$</span> axis) versus it's derivative <span class="math-container">$f'(x)$</span>? (on the <span class="math-container">$x$</span> axis) does it provide some useful i... | Narasimham | 95,860 | <p>Such a graph treats <span class="math-container">$x$</span> as a <em>parameter</em>.</p>
<p>In dynamic systems ( vibrations/ oscillations etc. ) plots <span class="math-container">$ (f(t), f'(t))$</span> with time <span class="math-container">$t$</span> as parameter are called <em>phase portraits</em>; they are very... |
2,518,213 | <p>We work in the vector space $\mathbb{R}^k$ over $\mathbb{R}$. We define the (usual) infinite norm on $\mathbb{R}$:</p>
<p>$$\|x\|_{\infty} = \max\{|x_1|,|x_2|,\dots,|x_n|\}$$ where $x_1, x_2, \dots, x_n $ are the coordinates of of $x$ in $\mathbb{R}^k$. </p>
<p>We would like to prove the following fact:
$$\exists... | daw | 136,544 | <p>You can choose the sequence $(x_n)$ such that $\|x_n\|_\infty=1$. Then $\|x_n\|\to0$.</p>
<p>To arrive at a contradiction, I will use the Weierstrass theorem. First, we show that $x\mapsto \|x\|$ is continuous wrt the $\infty$-norm:
Write $x=\sum x_ie_i$ with the unit vectors $e_i$, then:
$$
\|x\|\le \sum_{i=1}^n |... |
97,329 | <p>This is inspired by <a href="https://mathoverflow.net/questions/97307/polynomials-all-of-whose-roots-are-rational">this </a> question. Let $f(x)=a_nx^n+...+a_0$ be a polynomial with rational coefficients. The sandard procedure of finding a rational root $p/q$ involves checking all $p$ that divide $a_0$ and all $q$ t... | Bruno | 16,178 | <p>You do not need the full power of LLL algorithm for finding rational roots. In particular, <a href="https://en.wikipedia.org/wiki/Berlekamp%E2%80%93Zassenhaus_algorithm" rel="nofollow noreferrer">Berlekamp-Zassenhaus algorithm</a> exponential-time algorithm to compute the irreducible factorization can be used to get... |
3,502,507 | <p>This is very similar to the <a href="https://math.stackexchange.com/questions/3501693/given-that-you-started-with-one-chip-what-is-the-probability-that-you-will-win">question</a> I've just asked, except now the requirement is to gain <span class="math-container">$4$</span> chips to win (instead of <span class="math-... | antkam | 546,005 | <p>This answer only addresses <strong>what's wrong with your recursion</strong>, since the other answers (both in this question and your earlier question) already gave many different ways to set up the right recursions (or use other methods).</p>
<p>The key mistake is what you highlighted. When you throw <span class=... |
1,251,334 | <blockquote>
<p>$||x-2|-3| >1$, then $x$ belongs to:<BR> (a) $(-\infty, -2) \cup (0,4) \cup(6,\infty)$<br> (b) $(-1,1)$<br> (c) $(-\infty, 1)\cup (1,
> \infty)$ <br> (d) $(-2,2)$</p>
</blockquote>
<p><strong><em>Answer:(a)</em></strong></p>
<p>My solution:</p>
<p>let $|x-2| = p,$<br>
$|p-3|>1$ and finally... | layman | 131,740 | <p>Ok, so $|x - 2| = p$, and so $|p - 3| > 1$.</p>
<p>Then that means $p - 3 > 1$ or $p - 3 < -1$.</p>
<p>Then that means $p > 4$ or $p < 2$.</p>
<p>But $p = |x - 2|$, so $|x - 2| > 4$ or $|x - 2| < 2$.</p>
<p>That means either $x - 2 > 4$ or $x - 2 < -4$ or $-2 < x - 2 < 2$.</p>
<... |
188,503 | <p>A Cayley table of an finite group has to have every element exactly once in every row and exactly once in every column.</p>
<p><strong>Proof</strong> that every element of a group has to be at most once in every row and at most once in every column:</p>
<p>Let $(G, \circ)$ be a group and $a, b, c, d \in G$ with:</... | William | 13,579 | <p>In the general setting, having at least one element in every row and column just means that given any $g$ and any $a$, there exists a $b$ and $b'$ such that $ab = g$ and $b'a = g$. This is true because you can let $b = a^{-1}g$ and $b' = ga^{-1}$. </p>
<p>Having exactly one element in each row or column is equivale... |
2,458,613 | <p>Say that we have $k$ red balls and $n-k$ blacks balls for $n$ balls total. Then, say we partition the balls into equal sized groups of size $m$. What is the expected number of groups with a red ball?</p>
<p>It seems clear that I should use linearity of expectation of some sort. I tried calculating the probability t... | true blue anil | 22,388 | <p>Let $X_i$ be an indicator random variable $=1$ if the $i^{th}$ group has a red ball, and $=0$ otherwise.</p>
<p>Then $P[i^{th}$ group has a red ball] $= \left[1 - \dfrac{\binom{n-k}{m}}{\binom{n}{m}}\right]$<br>
Now the expectation of an indicator r.v. is just the probability of the event it indicates, so $E[X_i] ... |
3,267,110 | <p>This integral
<span class="math-container">$$\int_{-\infty}^{\infty}\frac{ a^2x^2 dx}{(x^2-b^2)^2+a^2x^2}=a\pi, ~ a, b \in \Re$$</span>
looks suspiciously interesting as it is independent of the parameter <span class="math-container">$b$</span>. The question is: What is the best way of proving or disproving this?</... | Z Ahmed | 671,540 | <p><strong>One way:</strong></p>
<p>Denote the integral as <span class="math-container">$$I=\int_{-\infty}^{\infty}\frac{ a^2x^2 dx}{(x^2-b^2)^2+a^2x^2}.$$</span>
<span class="math-container">$$(x^2-b^2)^2+a^2x^2=(x^2+p^2)(x^2+q^2) \Rightarrow p=(a+c)/2,q=(a-c)/2, c=\sqrt{a^2-4b^2}.$$</span>
Then <span class="math-con... |
3,267,110 | <p>This integral
<span class="math-container">$$\int_{-\infty}^{\infty}\frac{ a^2x^2 dx}{(x^2-b^2)^2+a^2x^2}=a\pi, ~ a, b \in \Re$$</span>
looks suspiciously interesting as it is independent of the parameter <span class="math-container">$b$</span>. The question is: What is the best way of proving or disproving this?</... | Z Ahmed | 671,540 | <p><strong>How about this, please check it critically</strong></p>
<p>Denote the integral as <span class="math-container">$$I=\int_{-\infty}^{\infty}\frac{ a^2x^2 dx}{(x^2-b^2)^2+a^2x^2}.$$</span>
<span class="math-container">$$(x^2-b^2)^2+a^2x^2=(x^2-r^2)(x^2-s^2) \Rightarrow r=(d+ia)/2,s=(d-ia)/2, d=\sqrt{4b^2-a^2}.... |
132,957 | <p>Could you show me that how can I solve this as fast as possible, please? <br /></p>
<p>$$ ABCDEF \, \land \, DEFABC \; \large{ \in \, \mathbb{N^{+}} } $$ $$ ABCDEF \, = \, 6 \times (DEFABC) \\ $$ $$ (A+B+C+D+E+F)=\; ? $$</p>
<p>Thank you very much... :)</p>
| Wonder | 27,958 | <p>DEFABC = 142857 obviously here.</p>
<p>More systematically though, let the sum be X. The equation shows that X is divisible 3, so applying it again shows that X is divisible by 9. Assuming that ABCDEF are all distinct, we have to choose 4 digits to reject. Overall sum of 0-9 is 45, so it must be less than that. Sum... |
195,759 | <p>I have three wavelengths. Each of them has a different colour. I want to plot the resulting wave with colour that corresponds to the mixture of these colour intensities at every point. I tried this code:</p>
<pre><code>Plot[I1[y]+I2[y]+I3[y],{y,-0.0000001,0.0000001},ColorFunction ->colf]
colf[y_] := Blend[{{I1[y... | Roman | 26,598 | <p>This works:</p>
<pre><code>Plot[I1[y] + I2[y] + I3[y], {y, -0.0000001, 0.0000001},
ColorFunctionScaling -> False,
ColorFunction -> (Blend[{Blue, Yellow, Red}, {I1[#], I2[#], I3[#]}] &)]
</code></pre>
|
30,191 | <p>Does anybody know about software that exactly calculates the tree-width of a given graph and outputs a tree-decomposition? I am only interested in tree-decompositions of reasonbly small graphs, but need the exact solution and a tree-decomposition. Any comments would be great. Thanks!</p>
| supercooldave | 2,620 | <p>Some tree decomposition software is given here: <a href="http://hein.roehrig.name/dipl/" rel="nofollow">http://hein.roehrig.name/dipl/</a>. I haven't used it so can say nothing about it's quality.</p>
|
30,191 | <p>Does anybody know about software that exactly calculates the tree-width of a given graph and outputs a tree-decomposition? I am only interested in tree-decompositions of reasonbly small graphs, but need the exact solution and a tree-decomposition. Any comments would be great. Thanks!</p>
| Bart Jansen | 5,200 | <p>You could try the LibTW software, which is freely available from <a href="http://www.treewidth.com/" rel="nofollow">http://www.treewidth.com/</a> (also read their FAQ linked at the bottom of the page). It's written in Java so you can easily extend it with your own functionality.</p>
|
3,350,091 | <p>In 2d space I have two cameras with different field of view (<span class="math-container">$40$</span> and <span class="math-container">$46$</span> degrees in my example) and have some distance between them.<br>
Is would possible to describe this distance using just this two numbers? And using degrees as dimension of... | Matematleta | 138,929 | <p>I think what you want follows from the definition of the "partials" in an arbitrary manifold. If I misunderstood your question, please advise, and I will delete my answer. </p>
<p>Let <span class="math-container">$x\in \mathbb R^n$</span>, and <span class="math-container">$r_i$</span> the projections from <span cla... |
3,350,091 | <p>In 2d space I have two cameras with different field of view (<span class="math-container">$40$</span> and <span class="math-container">$46$</span> degrees in my example) and have some distance between them.<br>
Is would possible to describe this distance using just this two numbers? And using degrees as dimension of... | hal4math | 699,910 | <p><span class="math-container">$\newcommand{\R}{\mathbb{R}}$</span>
So because we are in <span class="math-container">$\mathbb{R}^n$</span> we can all imagine (or at least were told) how we should visualize what a tangent space is at a certain point <span class="math-container">$p$</span> in <span class="math-containe... |
2,243,900 | <p>I've researched this topic a lot, but couldn't find a proper answer to this, and I can't wait a year to learn it at school, so my question is:</p>
<blockquote>
<p>What exactly is calculus? </p>
</blockquote>
<p>I know who invented it, the Leibniz controversy, etc., but I'm not exactly sure what it is. I think I ... | erfink | 376,021 | <p>As other answers have broken down some of the applications and the categories of calculus, I'll try to give a more intuitive and motivating explanation.</p>
<p>At its core, calculus is about trying to do meaningful computations with quantities that are infinitely large or infinitely small ("infinitesimals"... |
292,639 | <p>Traditional Fourier analysis picks a period and then describes a function as: $$f(x) = \frac{1}{2} a_0 + \sum_{k=1}^\infty\, (a_k \cos{(\omega \cdot kx)} + b_n \sin{(\omega \cdot kx)})$$</p>
<p>I am wondering whether there is a way to Fourier-analyze a function in a way that the period is dependent on $x$. Let $g$ ... | copper.hat | 27,978 | <p>Note that you are solving $\int_{-a}^a f(x) dx$ where $f$ is odd. What does that say about $\int_{-a}^0 f(x) dx$ and $\int_{0}^a f(x) dx$?</p>
|
162,147 | <p>I have</p>
<pre><code>J = Table[{x10, y10, x10*y10}, {x10, 0, 1, 0.5}, {y10, 0, 1, 0.5}]
L = Table[{x10, y10, 2.0*x10*y10}, {x10, 0, 1, 0.5}, {y10, 0, 1, 0.5}]
</code></pre>
<p>I want the third elements of J and L to be added and the first and second elements are as they are (as they are the same in both cases) fo... | kglr | 125 | <pre><code>tJ = Table[{x10, y10, x10*y10}, {x10, 0, 1, 0.5}, {y10, 0, 1, 0.5}];
tL = Table[{x10, y10, 2.0*x10*y10}, {x10, 0, 1, 0.5}, {y10, 0, 1, 0.5}];
</code></pre>
<p>Several alternatives:</p>
<pre><code>tK1 = Map[{0, 0, 1} # &, tJ, {-2}] + tL;
tK2 = MapAt[2 # &, Mean[{tJ, tL}], {{All, All, -1}}];
tK3 = Me... |
177,091 | <p>What would $\int\limits_{-\infty}^\infty e^{ikx}dx$ be equal to where $i$ refers to imaginary unit? What steps should I go over to solve this integral? </p>
<p>I saw this in the Fourier transform, and am unsure how to solve this.</p>
| copper.hat | 27,978 | <p>Formally it is solved exactly as in the real case. The antiderivative is $\frac{1}{i k} e^{ikx}$. So, $\int e^{ikx} dx = \frac{1}{i k} e^{ikx} + C$.</p>
|
177,091 | <p>What would $\int\limits_{-\infty}^\infty e^{ikx}dx$ be equal to where $i$ refers to imaginary unit? What steps should I go over to solve this integral? </p>
<p>I saw this in the Fourier transform, and am unsure how to solve this.</p>
| jrand | 8,140 | <p>The definition of the Fourier inverse:</p>
<p>$$ f(t) = \mathfrak{F}^{-1}\{F(jw)\} = \frac{1}{2\pi} \int_{-\infty}^{+\infty} F(jw) e^{jwt} dw $$</p>
<p>The Fourier pair (in the angular frequency domain):</p>
<p>$$ \delta(t) \leftrightarrow 1 $$</p>
<p>The integral in the question: </p>
<p>$$ 2\pi \times \frac{1... |
3,986,785 | <p>On page 153 of <em>Linear Algebra Done Right</em> the second edition, it says:</p>
<blockquote>
<p>Define a linear map <span class="math-container">$S_1: \text{range}(\sqrt{T^*T} ) \to \text{range}(T)$</span> by:</p>
</blockquote>
<blockquote>
<p><strong>7.43:</strong> <span class="math-container">$S_1 (\sqrt{T^* ... | Will Harris | 851,396 | <p>A function <span class="math-container">$f: A \to B$</span> is well defined if for every <span class="math-container">$x \in A$</span>, <span class="math-container">$f(x)$</span> is equal to a single value in <span class="math-container">$B$</span>. So if <span class="math-container">$f(x)$</span> could equal two di... |
3,986,785 | <p>On page 153 of <em>Linear Algebra Done Right</em> the second edition, it says:</p>
<blockquote>
<p>Define a linear map <span class="math-container">$S_1: \text{range}(\sqrt{T^*T} ) \to \text{range}(T)$</span> by:</p>
</blockquote>
<blockquote>
<p><strong>7.43:</strong> <span class="math-container">$S_1 (\sqrt{T^* ... | Ruy | 728,080 | <p>When authors claim that something needs to be proven to be well defined that is because the definition given is under suspicion of having some flaw: it might not define anything at all. For example:</p>
<p>(Fake) Definition. Let <span class="math-container">$x$</span> be the smallest strictly positive real number.<... |
3,960,689 | <p>I study maths as a hobby and have come across this problem.
Two circles with centres A and B intersect at points P and Q, such that <span class="math-container">$\angle APB$</span> is a right angle. If AB = xcm and <span class="math-container">$\angle PAQ = \frac{1}{3}\pi$</span> radians, find in terms of x the leng... | A-Level Student | 764,058 | <p><strong>HINT</strong></p>
<p>We know that <span class="math-container">$\angle PAB=\frac{\pi}{6}$</span>, hence the radius of the larger circle is <span class="math-container">$x\cos{\frac{\pi}{6}}=\frac{x\sqrt3}{2}$</span>. Similarly, the radius of the smaller circle is
<span class="math-container">$x\sin{\frac{\pi... |
3,960,689 | <p>I study maths as a hobby and have come across this problem.
Two circles with centres A and B intersect at points P and Q, such that <span class="math-container">$\angle APB$</span> is a right angle. If AB = xcm and <span class="math-container">$\angle PAQ = \frac{1}{3}\pi$</span> radians, find in terms of x the leng... | Shubham Johri | 551,962 | <p><strong>Hint.</strong> For the common area,
<span class="math-container">$$\text{area of sector PBQ}+\text{area of sector PAQ}=\text{area of }\square\text{PBQA}+\text{common area of circles}$$</span></p>
<p>where <span class="math-container">$$\begin{align*}&\text{area of sector PAQ}=\frac16\pi r_1^2\\
&\tex... |
2,957,481 | <p>How many ways are there to roll a die six times such that there are more ones than twos?</p>
<p>I broke this up into six cases: </p>
<p><span class="math-container">$\textbf{EDITED!!!!!}$</span></p>
<p><span class="math-container">$\textbf{Case 1:}$</span> One 1 and NO 2s --> 1x4x4x4x4x4 = <span class="math-conta... | Ross Millikan | 1,827 | <p>Case 1 is wrong because there are <span class="math-container">$4$</span> numbers that are neither <span class="math-container">$1$</span> nor <span class="math-container">$2$</span>, so there are <span class="math-container">$6 \cdot 4^5$</span> ways to roll one <span class="math-container">$1$</span> and no <span ... |
4,305,593 | <blockquote>
<p>Prove that
<span class="math-container">$$ \int_{0}^{\infty} \frac{\sin^2 x-x\sin x}{x^3} \, dx = \frac{1}{2} - \ln 2 .$$</span></p>
</blockquote>
<p>Integration by parts gives</p>
<p><span class="math-container">\begin{align*}
&\lim_{R\to \infty} \int_{0}^{R} \frac{\sin^2 x-x\sin x}{x^3} \, dx \\
&... | Random Variable | 16,033 | <p>Alternatively, we could use the fact that <span class="math-container">$$\int_{0}^{\infty} t^{2} e^{-xt} \, \, \mathrm dt = \frac{2}{x^{3}}, \quad x >0,$$</span> and then apply Fubini's theorem.<span class="math-container">$$\begin{align} \int_{0}^{\infty} \frac{\sin^{2}x - x \sin x}{x^{3}} \, \mathrm dx &... |
52,364 | <p>A given user can interact in multiple ways with a website. Let's simplify a bit and say say a user can:</p>
<ul>
<li>Post a message</li>
<li>Comment a message</li>
<li>"like" something on the website via Facebook</li>
</ul>
<p><em>(after that we could add, following the site on twitter, buying something on the sit... | lucasdicioccio | 13,732 | <p>Just use fixed scores which adds up for each action. The data distribution will likely behave like a power-low.
If you want to prevent people from spamming the system, reward less and less as the number of actions grow (like XP in video games). You can use any $o(n)$ function (e.g. $\log$, $\mathrm{sqrt}$) with $n$ ... |
619,607 | <p>$\mathbf{(1)}$ Find $y^{\prime}$ of $y=8^{\sqrt x}$ </p>
<p>My try: </p>
<p>$\ln y=\ln(8)^{\sqrt x}$<br>
$\dfrac{1}{y}y^{\prime}=\sqrt{x}\ln8$<br>
I don't know how to proceed with right side. </p>
<p>$\mathbf{(2)}$ Find $y^{\prime}$ of $y=(t+4)(t+6)(t+7).$</p>
<p>This one I have no idea what to do so I don't ha... | Brian | 114,928 | <p><strong>(1)</strong> Note that $\ln\left(a^{b}\right) = b\ln\left(a\right)$ so, for $y = 8^{\sqrt{x}}$, $\ln y = \sqrt{x}\ln\left(8\right)$. Differentiating, we get $\frac{y'}{y} = \frac{\ln\left(8\right)}{2\sqrt{x}}$ and so $y'=\ln\left(8\right)\frac{8^{\sqrt{x}}}{2\sqrt{x}}$.</p>
<p><strong>(2)</strong> I assume ... |
4,218,513 | <p>This is an old Schwarz lemma problem from the August 2020 UMD qualifying exam for analysis, which is posted <a href="https://www-math.umd.edu/images/pdfs/quals/Analysis/Analysis-August-2020.pdf" rel="nofollow noreferrer">here</a>. The precise wording from the test is:</p>
<p>Suppose <span class="math-container">$f(z... | Geoffrey Sangston | 444,923 | <p>We know that the hyperbolic unit disk metric, <span class="math-container">$d(z, w) := |\frac{z - w}{1 - \bar{w}z}|$</span>, satisfies <span class="math-container">$d(g(z), g(w)) \le d(z, w)$</span> for analytic maps <span class="math-container">$g : \mathbb{D} \to \mathbb{D}$</span>. Note that <span class="math-con... |
81,887 | <p>The problem: Prove that for $n \in \mathbb N$:</p>
<p>$$ \left(1 + \frac{1}{n} \right)^n = 1 + \sum_{m=1}^{n} \frac{1}{m!} \left(1 - \frac{1}{n} \right) \left(1 - \frac{2}{n} \right) \cdots \left(1 - \frac{m-1}{n} \right). $$</p>
<p>The hint is to use the binomial theorem. So the left side can become:</p>
<p>$$ \... | Arturo Magidin | 742 | <p><strong>Hint.</strong> $\displaystyle 1 - \frac{k}{n} = \frac{n-k}{n}$. So
$$\left(1 - \frac{1}{n}\right)\left(1 - \frac{2}{n}\right)\cdots\left(1-\frac{m-1}{n}\right) = \frac{(n-1)(n-2)\cdots(n-(m-1))}{(n)(n)\cdots(n)}.$$</p>
|
1,794,673 | <p>Prove that there exist two real Gaussian distribution $X$ and $Y$ such that the vector $(X,Y)$ is not Gaussian. </p>
<p>The way that i tried to solve this problem was choosing $X=X$, and take before the linear combination $X-X$ or stuff like that; but in the definition of Gaussian distribution that i have the const... | Sergiy Maksymenko | 809,333 | <p>You can find the detailed proof in my joint paper with Olexandra Khokhliyuk for the case of weak Whitney topology:</p>
<p>[1] O. Khokhliuk, S. Maksymenko,
Smooth approximations and their applications to homotopy types.
Proceedings of the International Geometry Center, vol. 13, no. 2 (2020) 68-108
<a href="http://doi... |
3,464,383 | <p>I couldn't find any substantial list of 'strange infinite convergent series' so I wanted to ask the MSE community for some. By <em>strange</em>, I mean infinite series/limits that <strong>converge when you would not expect them to and/or converge to something you would not expect</strong>.</p>
<p>My favorite conver... | user97357329 | 630,243 | <p>You might find some interesting examples in the book, <strong>(Almost) Impossible Integrals, Sums, and Series</strong>. Here you have two examples:</p>
<p><strong>First example:</strong></p>
<blockquote>
<p><span class="math-container">$$\small\zeta(4)=\frac{4}{45}\sum_{i=1}^{\infty}\sum_{j=1}^{\infty}\sum_{k=1}... |
3,464,383 | <p>I couldn't find any substantial list of 'strange infinite convergent series' so I wanted to ask the MSE community for some. By <em>strange</em>, I mean infinite series/limits that <strong>converge when you would not expect them to and/or converge to something you would not expect</strong>.</p>
<p>My favorite conver... | Oscar Lanzi | 248,217 | <p>I would like to nominate an infinite product:</p>
<p><span class="math-container">$\prod_{n=2}^{\infty}\dfrac{n^3-1}{n^3+1}=\dfrac{2}{3}$</span></p>
<p>Proof: Factor thusly:</p>
<p><span class="math-container">$n^3-1=(n-1)(n^2+n+1)=((n-2)+1)(n^2+n+1)$</span></p>
<p><span class="math-container">$n^3+1=(n+1)(n^2-... |
294,972 | <p>I am seeking a particular integral of the differential equation:</p>
<p>$u^{(4)}(t) - 5u''(t) + 4u(t) - t^3 = 0$</p>
<p>I simply interested in the technique, not just an answer (Mathematica suffices for that). How would one apply undetermined coefficients in this case?</p>
| Mikasa | 8,581 | <p>We have $(D^4-5D+4)u=t^3$ where in $Du=u'$. So $$(D^2-1)(D^2-4)u=t^3$$ so $$D^4(D^2-1)(D^2-4)=0$$ So the general solution of the OE has a form of $$u=(At^3+Bt^2+Ct+E)+y_c$$ where in $y_c=C_1e^t+C_2e^{-t}+C_3e^{-2t}+C_4e^{2t}$.</p>
|
2,623,521 | <p>Prove that: $\tan (2\tan^{-1} (x))=2\tan (\tan^{-1} (x)+\tan^{-1} (x^3))$</p>
<p>My Attempt'
Let $\tan^{-1} (x)=A$
$$x=\tan (A)$$
Now,
L.H.S.$=\tan (2\tan^{-1} (x))$
$$=\tan (2A)$$
$$=\dfrac {2\tan (A)}{1-\tan^2 (A)}$$
$$=\dfrac {2x}{1-x^2}$$</p>
| mvw | 86,776 | <p>I still do no not understand the problem. But let us try to start anyway and discuss further.</p>
<p>With $m$ at rest at $A$ we have a force equilibrium
$$
0 = F_G + T_1 + T_{AC}
$$
In $x$-direction it is
$$
0 = 0 + \lVert T_1 \rVert \sin(30^\circ) - \lVert T_{AC} \rVert
$$
and in $y$-direction
$$
0 = -mg+ \lVert T... |
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