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 |
|---|---|---|---|---|
222,093 | <p>For what value of m does equation <span class="math-container">$y^2 = x^3 + m$</span> has no integral solutions?</p>
| PAD | 27,304 | <p>Here is the solution in Ireland and Rosen (page 270). </p>
<p>Suppose the equation has a solution. Then $x$ is odd. For otherwise reduction modulo 4 would imply 3 is a square modulo 4. Write the equation as
$$y^2+1=(x+2)(x^2-2x+4)=(x+2) ((x-1)^2+3) \ . \ \ \ (*)$$
Now since $(x-1)^2 +3$ is of the form $4n+3$ th... |
688,742 | <p>Given $P\colon\mathbb{R} \to \mathbb{R}$ , $P$ is injective (one to one) polynomial function i need to formally prove that $P$ is onto $\mathbb{R}$</p>
<p>my strategy so far .......
polynomial function is continuous and since it one-to-one function it must be strictly monotonic and now i have no idea what to do ..... | Community | -1 | <p>If we <em>continuously</em> extend the real exponential to have values $e^{+\infty} = +\infty$ and $e^{-\infty} = 0$, then this can be seen as taking the limit of a continuous function, which can be done by simply plugging in the limiting value:</p>
<p>$$\lim_{x\to +\infty} \frac{11 - e^{-x}}{7} = \frac{11 - e^{-(+... |
688,742 | <p>Given $P\colon\mathbb{R} \to \mathbb{R}$ , $P$ is injective (one to one) polynomial function i need to formally prove that $P$ is onto $\mathbb{R}$</p>
<p>my strategy so far .......
polynomial function is continuous and since it one-to-one function it must be strictly monotonic and now i have no idea what to do ..... | Ilmari Karonen | 9,602 | <p><a href="https://math.stackexchange.com/a/688730">As EPAstor notes</a>, your reason (A) is closer to the truth, but it's not the <em>complete</em> answer.</p>
<p>The reason why $\displaystyle \lim_{x \to \infty} e^{-x} = 0$ implies $\displaystyle \lim_{x \to \infty} \frac{11 - e^{-x}}{7} = \frac{11}{7}$ is that</p>... |
1,946,438 | <p>I solved the equation $e^{e^z}=1$ and it seemed to easy so I suspect I must be missing something.</p>
<blockquote>
<p>Would someone please check my answer?</p>
</blockquote>
<p>My original answer:</p>
<p>$e^{e^z}=1$ if and only if $e^z = 2\pi i k$ for $k\in \mathbb Z$ if and only if $z=\ln(2\pi i k)$ for $k\in ... | BCLC | 140,308 | <p>I believe you are making too many jumps</p>
<p>Consider that in Approach 1 you said</p>
<blockquote>
<ol>
<li><p><span class="math-container">$e^{e^z}=1$</span> if and only if <span class="math-container">$e^z = 2\pi i k$</span></p>
</li>
<li><p><span class="math-container">$e^z = 2\pi i k$</span> for <span class="m... |
4,151 | <p>Since it's currently summer break, and I've a bit more time than normal, I've been organizing my old notes. I seem to have an almost unwieldy amount of old handouts and tests from classes previously taught. I'm hesitant to get rid of these, as they may provide useful for some future course. Because I adjunct at a fe... | SlyPuppy | 625 | <p>I use schoology.com to keep my notes and such.</p>
<p>It's great because it goes where I go plus it's free for both students and teachers.</p>
|
265,047 | <p>Let $X$ be a Banach space and let $T:X \rightarrow X$ be a bounded linear map. Show that: If $T$ is surrjective then its transpose $T':X' \rightarrow X'$ is bounded below.</p>
<p>My try: We know that $R^\perp_M = N_{M'}$ and since X is surrjective
$R_M = X$ hence $R_M^\perp = N_{M'} = 0$ so $M'$ is invertible and... | Community | -1 | <p>By expanding out the brackets you cans split it into two series:
$$\sum_{n=1}^\infty\frac{\left(\ln n+(-1)^n\right)n^{1/2}}{n\cdot n^{1/2}} = \sum_{n=1}^\infty{\frac{\ln n}{n}}+{\sum_{n=1}^\infty{\frac{(-1)^n}{n}}}$$
The second series is the alternating Harmonic, widely known to be convergent. This can be verified b... |
57,769 | <p>Consider a finitely axiomatized theory $T$ with axioms $\phi_1,...,\phi_n$ over a first-order language with relation symbols $R_1,...,R_k$ of arities $\alpha_1,...,\alpha_k$. Consider the atomic formulas written in the form $(x_1,...,x_{\alpha_j})\ \varepsilon R_j$.</p>
<p>Translate this theory into a (finite) set-... | bonnnnn2010 | 13,590 | <p>Fourier Analysis has many applications in Nonlinear PDEs, for example, Nonlinear Schrödinger Equation, a very often used method is Hardy-Littlewood decomposition, to get the well-posedness(existence, uniqueness and some kind of dependence on the initial data) of the solution. A good book is "Nonlinear Dispersive Equ... |
3,186,239 | <p>Two Independent variables have Bernoulli distribution:
<span class="math-container">$X_1$</span> with <span class="math-container">$b(n,p)$</span> and <span class="math-container">$X_2$</span> with <span class="math-container">$b(m,p)$</span>.
How can I find conditional distribution <span class="math-container">$\ma... | drhab | 75,923 | <p>Guide (preassuming that we are dealing with binomial distribution, see my comment on the question):</p>
<p><span class="math-container">$$P(X_1=k\mid X_1+X_2=t)P(X_1+X_2=t)=P(X_1=k, X_2=t-k)=P(X_1=k)P(X_2=t-k)$$</span>where the second equality rests on independence.</p>
<p>Further <span class="math-container">$X_1... |
663,435 | <p>Bob has an account with £1000 that pays 3.5% interest that is fixed for 5 years and he cannot withdraw that money over the 5 years</p>
<p>Sue has an account with £1000 that pays 2.25% for one year, and is also inaccessible for one year.</p>
<p>Sue wants to take advantage of better rates and so moves accounts each ... | Ross Millikan | 1,827 | <p>You should just be able to plug the values into your first equation to get the value of Bob's account at the end of 5 years. Note that A is the total value, not the amount earned. Also note that the interest rate needs to be expressed as a decimal. </p>
<p>For Sue, first calculate how much she has at the end of ... |
3,869,237 | <p>I know this is quite weird or it does not make much sense, but I was wondering, does <span class="math-container">$\int e^{dx}$</span> has any meaning or whether it makes sense at all? If it does means something, can it be integrated and what is the result?</p>
| alienare 4422 | 834,770 | <p>Well it makes sense when you think about it one way and it doesn't when you think about it another way...First of all <span class="math-container">$dx$</span> is supposed to be extremely close to <span class="math-container">$0$</span> so (as J.G. said)
that's going to equal <span class="math-container">$1+dx$</span... |
3,869,237 | <p>I know this is quite weird or it does not make much sense, but I was wondering, does <span class="math-container">$\int e^{dx}$</span> has any meaning or whether it makes sense at all? If it does means something, can it be integrated and what is the result?</p>
| Amaan | 814,546 | <p>Tool 1: Know, <span class="math-container">$\displaystyle \lim_{\Delta x\rightarrow 0} \frac{a^{\Delta x}-1}{\Delta x}=\ln(a) $</span>. Now, <span class="math-container">\begin{align*}
\int e^{\mathrm{d}x}&=\int e^{\mathrm{d}x}-1+1\\
&=\int \frac{e^{\mathrm{d}x}-1}{\mathrm{d}x}\mathrm{d}x+1\\
&=\int \ln... |
3,421,858 | <p><span class="math-container">$\sqrt{2}$</span> is irrational using proof by contradiction.</p>
<p>say <span class="math-container">$\sqrt{2}$</span> = <span class="math-container">$\frac{a}{b}$</span> where <span class="math-container">$a$</span> and <span class="math-container">$b$</span> are positive integers. </... | Deepak | 151,732 | <p>Your initial premise is that <span class="math-container">$b$</span> is <em>defined</em> to be the smallest positive integer such that <span class="math-container">$b\sqrt 2$</span> is a positive integer. This is equivalent to reducing <span class="math-container">$\frac ab$</span> to its lowest terms (that is, make... |
4,383,800 | <p>I can already see that the <span class="math-container">$\lim_\limits{n\to\infty}\frac{n^{n-1}}{n!e^n}$</span> converges by graphing it on Desmos, but I have no idea how to algebraically prove that with L’Hopital’s rule or induction. Where could I even start with something like this?</p>
<p>Edit: For context, I came... | trancelocation | 467,003 | <p>You can use the simple fact that</p>
<p><span class="math-container">$$e^n = \sum_{k=0}^{\infty}\frac{n^k}{k!} \stackrel{k=n}{>} \frac{n^n}{n!}$$</span></p>
<p>Hence, you get</p>
<p><span class="math-container">$$0<\frac{n^{n-1}}{n!e^n} < \frac{n^{n-1}}{n!\frac{n^n}{n!}} = \frac 1n$$</span></p>
|
703,031 | <p>In a sequence of integers, $A(n)=A(n-1)-A(n-2)$, where $A(n)$ is the $n$th term in the sequence, $n$ is an integer and $n\ge3$,$A(1)=1$,$A(2)=1$, calculate $S(1000)$, where $S(1000)$ is the sum of the first $1000$ terms.</p>
<p>How to approach these type of questions? Which topics should I study?</p>
| 5xum | 112,884 | <p>$$A(1) + A(2) + A(3) +\dots= \\=A(1) + A(2) + (A(2) - A(1)) + (A(3) - A(2)) + (A(4) - A(3)) + \dots$$</p>
<p>Can you see a pattern?</p>
|
178,823 | <p>How would I prove the following trig identity? </p>
<blockquote>
<p><span class="math-container">$$\frac{ \cos (A+B)}{ \cos A-\cos B}=-\cot \frac{A-B}{2} \cot \frac{A+B}{2} $$</span></p>
</blockquote>
<p>My work thus far has been:
<span class="math-container">$$\dfrac{2\cos\dfrac{A+B}{2} \cos\dfrac{A-B}{2}}{-2\s... | davidlowryduda | 9,754 | <p>First, note that $\dfrac{\cos(A + B)}{\cos A - \cos B} \neq -\cot \left( \dfrac{A-B}{2} \right) \cot \left(\dfrac{A+B}{2} \right)$</p>
<p>In particular, if we use something like $A = \pi/6, B = 2\pi/6$, then the left is $0$ as $\cos(\pi/2) = 0$ and the right side is a product of two nonzero things.</p>
<p>I suspec... |
225,866 | <p>If I define, for example,</p>
<pre><code>f[OptionsPattern[{}]] := OptionValue[a]
</code></pre>
<p>Then the output for <code>f[a -> 1]</code> is 1.</p>
<p>However, in my code, I have a function that must be called using the syntax <code>f[some parameters][some other parameters]</code>, and I want to add options to... | kglr | 125 | <p>If you <em>have to</em> use <code>Array</code>:</p>
<pre><code>Array[Through @ {x, y, z} @ vl[[#]] &, Length @ vl]
</code></pre>
<blockquote>
<pre><code> {{x[6], y[6], z[6]}, {x[9], y[9], z[9]}, {x[10], y[10], z[10]}}
</code></pre>
</blockquote>
<p>Also:</p>
<pre><code>f = Through /@ # /@ #2 &;
f[{x, y, z},... |
744,982 | <p>Could anyone let me know if the following linear programming problem can be solved in polynomial time or should be NP-hard?</p>
<p>$\min c^Tx$</p>
<p>s.t. $x^TQx\geq C^2, x\in [0,1]^n,c\in \mathbb{R}_+^n,Q\in\mathbb{R}_+^{n\times n}$ is positive semi-definite matrix.</p>
| AndreaCassioli | 130,183 | <p>I am afraid you can't.</p>
<p>This is a not convex problem due to the quadratic constraint. If your $Q$ matrix were negative semidefinite, then it would have been solvable in polynomial time. </p>
|
744,982 | <p>Could anyone let me know if the following linear programming problem can be solved in polynomial time or should be NP-hard?</p>
<p>$\min c^Tx$</p>
<p>s.t. $x^TQx\geq C^2, x\in [0,1]^n,c\in \mathbb{R}_+^n,Q\in\mathbb{R}_+^{n\times n}$ is positive semi-definite matrix.</p>
| RoyS | 774,804 | <p>My guess is that this problem can be solved in polynomial time.</p>
<p>It is well known that <span class="math-container">$\min_x c'x$</span>, such that <span class="math-container">$x'Qx \geq C$</span> can be solved in polynomial time (since SDP relaxation is tight). Your problem has extra bound constraints <span ... |
623,810 | <p>$\omega = y dx + dz$ is a differential form in $\mathbb{R}^3$, then what is ${\rm ker}(\omega)$? Is ${\rm ker}(\omega)$ integrable? Can you teach me about this question in details? Many thanks!</p>
| Branimir Ćaćić | 49,610 | <p>In this context, I presume that $\ker(\omega)$ denotes the subbundle of $T \mathbb{R}^3$ with fibre $\ker(\omega)_p := \ker(\omega_p)$ at each $p \in \mathbb{R}^3$, where $\ker(\omega_p)$ is the kernel of the functional $\omega_p : T_p \mathbb{R}^3 \to \mathbb{R}$. How does this functional look like? Well, setting $... |
1,597,891 | <p>Let $G$ be an abelian group of order $75=3\cdot 5^{2}$. Let $Aut(G)$ denote its group of automorphisms. Find all possible order of $Aut(G)$.</p>
<p>My approach is to first study its Sylow 5-subgroup. Since $n_{5}|3$ and $n_{5}\equiv 1\pmod{5}$, $n_{5}=1$. So $G$ has a unique Sylow 5-subgroup, denote $F$. By Sylow's... | p Groups | 301,282 | <p>$G$ is abelian, so Sylow subgroups are characteristic. Hence $G$ is product of characteristic Sylow subgroups, say $H_3$ and $H_5$. Then $Aut(G)\cong Aut(H_3)\times Aut(H_5)$. </p>
<p>$H_3$ is cyclic, whose automorphism group is well known. $H_5$ is either cyclic or $Z_5\times Z_5$. The automorphism groups in bot... |
4,224,417 | <p>I'm trying to learn a bit of Number Theory. And while I understand the definition of congruence relations modulo <span class="math-container">$n$</span> and that they are an equivalence relations, I fail to see the <em>motivation</em> for it. So what is congruence relation <span class="math-container">$\bmod n$</spa... | Roddy MacPhee | 903,195 | <p>Modulo weakens equality, helps divisibility, and allows dealing with huge numbers. It's relevant to Cryptography, and therefore cybersecurity. It can even help with divisor form:</p>
<p>Assume a number is of form <span class="math-container">$2^p-1$</span> , it's clear that any time <span class="math-container">$2^p... |
2,844,902 | <p>Does $$\int_{[1,z]}\frac{1}{u}du=\log(z)$$ where $z\in\mathbb C$ ? I know that on a closed circle that contain $0$ we have $$\int_C\frac{1}{z}dz=2i\pi=\log(1),$$</p>
<p>but for $$\int_{[1,z]}\frac{1}{u}du=\log(z)$$ I don't really know to compute the integral.</p>
| copper.hat | 27,978 | <p>Since $H$ is constant (and hence bounded) on a trajectory, you know that for any initial condition
that there is some $K$ such that ${1 \over 2} v^2(t) + {1 \over q^4(t)} \le K$
for all $t$.</p>
<p>In particular ${1 \over 2} v^2(t) \le K$ and
$ {1 \over q^4(t)} \le K$ for all $t$,
or
$|v(t)| \le \sqrt{2K}$ and
$|q(... |
4,073,757 | <p>Q: A coin is tossed untill k heads has appeared. If a mathematician knows how many heads appeared, can he figure out what is the probability that the coin was tossed <span class="math-container">$n$</span> times?</p>
<p>What I tried: The number of heads debends of the number of tosses. So I tried Bayes' theoream <sp... | X. Li | 417,726 | <p>The coin toss process can be described in the context of independent and identically distributed (i.i.d.) Bernoulli trials.</p>
<p>Suppose we define a random variable corresponding to the number of trials required to have <span class="math-container">$k$</span> successes (i.e. <span class="math-container">$k$</span>... |
1,088,338 | <p>There are at least a few things a person can do to contribute to the mathematics community without necessarily obtaining novel results, for example:</p>
<ul>
<li>Organizing known results into a coherent narrative in the form of lecture notes or a textbook</li>
<li>Contributing code to open-source mathematical softw... | Horst Grünbusch | 88,601 | <p>Find problems to solve. These can be open mathematical questions but (more important) new fields of application that may also need new mathematical concepts. Calculus e.g. was founded because it was needed for physics. So any scientist who dares to show his problems to mathematicians helps mathematics as well. </p>
|
2,628,220 | <p>Let $(a_{n})_{n \in \mathbb N_{0}}$ be a sequence in $\mathbb Z$, defined as follows:
$a_{0}:=0,
a_{1}:=2,
a_{n+1}:= 4(a_{n}-a_{n-1}) \forall n \in \mathbb N$. </p>
<p>Required to prove: $a_{n}=n2^{n} \forall n \in \mathbb N_{0}$</p>
<p>I have gone about it in the following: </p>
<p>Induction start: $n=0$ (condi... | Cuija Gaming | 524,563 | <p>Show that your premise holds for $n = 0$ and $n = 1$.</p>
<p>Then the induction step is: $A(n)~ ∧ ~ A(n+1) → A (n+2)$</p>
|
1,707,132 | <blockquote>
<p>Let $X$ be a contractible space (i.e., the identity map is homotopic to the constant map). Show that $X$ is simply connected.</p>
</blockquote>
<p>Let $F$ be the homotopy between $\mathrm{id}_X$ and $x_0$, that is $F:X\times [0,1]\to X$ is a continuous map such that
$$ F(x,0)=x,\quad F(x,1)=x_0$$
fo... | Ivin Babu | 704,464 | <p><img src="https://i.stack.imgur.com/JH7s6.jpg" alt="enter image description here" />
Using this lemma given in Munkres Topology ( which is not too difficult to understand once you've gone through the proof) can be used to show that any loop based at <span class="math-container">$x_0$</span> is path homotopic to the ... |
195,333 | <p>I'm looking to make a graph with the x-axis reversed and a frame and tick marks that are thicker than default. However, the x-axis tick marks do not maintain the specified thickness once I reverse the x-axis. I can't restore the thickness of the tick marks using TicksStyle or FrameTicksStyle. How can I get around th... | MassDefect | 42,264 | <p>If your tick marks are being overridden, it might be that you're using <code>Ticks</code> rather than <code>FrameTicks</code>.</p>
<p>This is how I usually go about making my own tick marks. Unfortunately, that seems to happen a lot more often than I would like.</p>
<pre><code>ticks[min_, max_, stepsz_, majorstep_... |
2,666,568 | <p>I have a dynamical system: $\dot{\mathbf x}$= A$\mathbf x$ with $\mathbf x$=
$\bigl( \begin{smallmatrix} x \\ y\end{smallmatrix} \bigr)$ and A =
$\bigl( \begin{smallmatrix} 3 & 0 \\ \beta & 3 \end{smallmatrix} \bigr). \beta$ real, time-independent.</p>
<p>I calculated the eigenvalue $\lambda$ = 3 with the ... | Carlos | 207,930 | <p>The eigenvectors for $\beta\ne0$ are $[0,1]^T$, linear dependent, therefore they don’t form a basis (not diagonalizable), thus defective.</p>
<p>The eigenvectors for $\beta=0$ are $[0,1]^T$, $[1,0]^T$, i.e. ,the standard basis. </p>
<p>The fixed points do not change for $\beta\ne0$, and they are clearly unstable ... |
1,093,396 | <p>I've been working on a problem from a foundation exam which seems totally straightforward but for some reason I've become stuck:</p>
<p>Let $f: \mathbb{ R } \rightarrow \mathbb{ R } ^n$ be a differentiable mapping with $f^\prime (t) \ne 0$ for all $t \in \mathbb{ R } $, and let $p \in \mathbb{ R } ^n$ be a point NO... | kremerd | 205,133 | <p>The first statement is actually wrong! Consider for example the function $f\colon\mathbb R\to\mathbb R^2,\; t\mapsto (0,e^t)$ with the point $p=(0,0)$.</p>
<p>The problem here is that the domain of $f$ is not compact. If you consider a function on a closed interval $f\colon[a,b]\to\mathbb R^n$ the statement becomes... |
4,231,509 | <p>I'm trying to prove that the group <span class="math-container">$(\mathbb{R}^*, \cdot)$</span> is not cyclic (similar to [1]). My efforts until now culminated into the following sentence:</p>
<blockquote>
<p>If <span class="math-container">$(\mathbb{R}^*,\cdot)$</span> is cyclic, then <span class="math-container">$\... | Robert Shore | 640,080 | <p>Note that for any <span class="math-container">$n \in \Bbb N, -1$</span> has no <span class="math-container">$n$</span>th root in <span class="math-container">$\Bbb R^*$</span> except (when <span class="math-container">$n$</span> is odd) <span class="math-container">$-1$</span> itself. Therefore, if <span class="ma... |
1,722,287 | <p>So far I know that when matrices A and B are multiplied, with B on the right, the result, AB, is a linear combination of the columns of A, but I'm not sure what to do with this. </p>
| xxxxxxxxx | 252,194 | <p>The image of $B$ is a subspace of dimension $\mathrm{rank}(B)$. Left multiplication by $A$ transforms this into a new subspace, which is the image of $AB$ having dimension $\mathrm{rank}(AB)$; this linear transformation cannot increase the dimension of the subspace.</p>
<p>Put in other words, $\mathrm{rank}(B)$ is ... |
1,722,287 | <p>So far I know that when matrices A and B are multiplied, with B on the right, the result, AB, is a linear combination of the columns of A, but I'm not sure what to do with this. </p>
| copper.hat | 27,978 | <p>If $AB$ has rank $r$ then there are vectors $v_1,...,v_r$ such that
$ABv_1,...,AB v_r$ are linearly independent.</p>
<p>Hence $Bv_1,...,B v_r$ must be linearly independent (or an immediate contradiction), and so $B$ must have rank $\ge r$.</p>
|
1,007,399 | <p>I came across following problem</p>
<blockquote>
<p>Evaluate $$\int\frac{1}{1+x^6} \,dx$$</p>
</blockquote>
<p>When I asked my teacher for hint he said first evaluate</p>
<blockquote>
<p>$$\int\frac{1}{1+x^4} \,dx$$</p>
</blockquote>
<p>I've tried to factorize $1+x^6$ as</p>
<p>$$1+x^6=(x^2 + 1)(x^4 - x^2 +... | Dylan | 135,643 | <p>Here's a nice "trick" my former professor taught me</p>
<p>$$ \int\frac{dx}{1+x^6} = \frac{1}{2} \int \frac{(1-x^2+x^4)+x^2+(1-x^4)}{(1+x^2)(1-x^2+x^4)} dx \\
= \frac{1}{2}\int \frac{dx}{1+x^2} + \frac{1}{2} \int \frac{x^2}{1+x^6} dx + \frac{1}{2} \int \frac{1-x^2}{1-x^2+x^4} dx \\
= \frac{1}{2}\int \frac{dx}{1+x^2... |
1,262,322 | <p>Suppose that virus transmision in 500 acts of intercourse are mutually independent events and that the probability of transmission in any one act is $\frac{1}{500}$. What is the probability of infection?</p>
<p>So I do know that one way to solve this is to find the probability of complement of the event we are tryi... | Karl | 203,893 | <p>You could compute the geometric series:
$$\frac{1}{500}+\left(\frac{499}{500}\right)\left(\frac{1}{500}\right)+\left(\frac{499}{500}\right)^2\left(\frac{1}{500}\right)+,...,+\left(\frac{499}{500}\right)^{499}\left(\frac{1}{500}\right)\\=\frac{\left(\frac{1}{500}\right)\left(1-\left(\frac{499}{500}\right)^{500}\right... |
85,470 | <p>We decided to do secret Santa in our office. And this brought up a whole heap of problems that nobody could think of solutions for - bear with me here.. this is an important problem.</p>
<p>We have 4 people in our office - each with a partner that will be at our Christmas meal.</p>
<p>Steve,
Christine,
Mark,
Mary,... | Daniel | 20,082 | <p>You take your own name out and your partners name out of the hat. You then draw a card which you keep. Hand the hat to your partner. They then get to draw a card, that they keep. You can now put your card and your partners card back in the hat. Hand the hat to someone else. Rinse and repteat.</p>
|
85,470 | <p>We decided to do secret Santa in our office. And this brought up a whole heap of problems that nobody could think of solutions for - bear with me here.. this is an important problem.</p>
<p>We have 4 people in our office - each with a partner that will be at our Christmas meal.</p>
<p>Steve,
Christine,
Mark,
Mary,... | Wolfgang Brehm | 223,307 | <p><strong>Edit:</strong> I just realized this method does not produce all possible derangements. Each pairing is random, but not as independent as it could be, because this method forces the derangement to be a single cycle.</p>
<p>Dr. Hannah Fry describes in her Book "The Indisputable Existence of Santa Claus&qu... |
1,798,261 | <p>what is multilinear coefficient? I heard it a couple of times and I tried to google it, all I am getting is multiple linear regression.
I am confused at this point. </p>
| Sri-Amirthan Theivendran | 302,692 | <p>The subspace is the null space of the matrix
\begin{bmatrix}
1&1&1\\
\end{bmatrix}
and hence is a $2$ dimensional subspace by the rank nullity theorem. One can check that $(1,-1,0)^T$ and $(0,1,-1)^T$ are two linearly independent vectors in the null space so that they form a basis for $W$.</p>
|
685,682 | <p>Should I worry about the following appeIarance of circularity in ZFC set theory? In constructing the universe of sets, you start with the empty set and then keep taking the power set over and over.</p>
<p>But this process continues through all the ordinals. The ordinals are there in the beginning to allow the con... | Asaf Karagila | 622 | <p>There is no circularity.</p>
<p>When a universe is given, all the sets there exists. Including the ordinals. However when we are given a universe of $\sf ZFC$ we can prove that the sets $V_\alpha$ form a strictly increasing and definable hierarchy, and every set is a member of some $V_\alpha$.</p>
<p>On the other ... |
685,682 | <p>Should I worry about the following appeIarance of circularity in ZFC set theory? In constructing the universe of sets, you start with the empty set and then keep taking the power set over and over.</p>
<p>But this process continues through all the ordinals. The ordinals are there in the beginning to allow the con... | mbsq | 20,990 | <p>Logically, you just prove things about the cumulative hierarchy from the ZFC axioms and it's not circular.</p>
<p>Conceptually, the way I like to think of it, the iterated powerset operation and the ordinals that you use to iterate build each other as you go. It's only by applying powerset to $V_\omega$ that we ge... |
3,534,254 | <blockquote>
<p><span class="math-container">$r\gt0$</span>, Compute
<span class="math-container">$$\int_0^{2\pi}\frac{\cos^2\theta }{ |re^{i\theta} -z|^2}d\theta$$</span>
when <span class="math-container">$|z|\ne r$</span></p>
</blockquote>
<p>The problem is related to Poisson kernel and harmonic function, but ... | Quanto | 686,284 | <p>Express <span class="math-container">$z$</span> also in polar form, <span class="math-container">$z = s^{i\alpha}$</span>. Then, </p>
<p><span class="math-container">$$|re^{i\theta} -z|^2=r^2+s^2-2rs\cos(\theta-\alpha)$$</span></p>
<p>and, with the variable change <span class="math-container">$t= \theta - \alpha$<... |
2,403,741 | <blockquote>
<p>Solve for $\theta$ the following equation.
$$\sqrt {3} \cos \theta - 3 \sin \theta = 4 \sin 2\theta \cos 3\theta.$$</p>
</blockquote>
<p>I tried writing sin and cos expansions but it is becoming too long.Please help me.</p>
| John Hughes | 114,036 | <p>Look at positive integer triples whose product is 36:</p>
<pre><code>1 1 36
1 2 18
1 3 12
1 4 9
1 6 6
2 2 9
2 3 6
3 3 4
</code></pre>
<p>For each, compute the sum:</p>
<pre><code>1 1 36 -> 38
1 2 18 -> 21
1 3 12 -> 16
1 4 9 -> 14
1 6 6 -> 13
2 2 9 -> 13
2 3 6 -> 11
3 3 4 -> 10... |
76,558 | <p>Reading books and papers on complexity theory, I am struck by the extreme degree to which proofs are stated in an intuitive, hand-wavy way. The alternative is to give a lot of details about the coding of complicated Turing machines
which simulate each other. Even proofs which basically take one line for a person to ... | Timothy Chow | 3,106 | <p>The same comment and question could be applied to just about any area of mathematics. To my knowledge, nobody has really worked on formalizing computational complexity theory. Formalization is still hard work; an experienced person will take about a week to formalize about a page of an undergraduate textbook. The... |
76,558 | <p>Reading books and papers on complexity theory, I am struck by the extreme degree to which proofs are stated in an intuitive, hand-wavy way. The alternative is to give a lot of details about the coding of complicated Turing machines
which simulate each other. Even proofs which basically take one line for a person to ... | Kaveh | 7,507 | <p>Generally complexity theorist prefer to use as little formalism as possible. $\mathsf{IP}=\mathsf{PSpace}$ is on the list <a href="http://www.cs.ru.nl/~freek/100/" rel="nofollow">here</a> but it doesn't seem that it has been verified with a proof assistant. </p>
<p>I doubt that complexity theorists would be interes... |
978,384 | <p>The following picture is constructed by connecting each corner of a square with the midpoint of a side from the square that is not adjacent to the corner. These lines create the following red octagon:</p>
<p><img src="https://i.stack.imgur.com/PZyGa.jpg" alt="enter image description here"></p>
<p>The question is, ... | Shadow | 256,351 | <p>Let us name all the points on the outer square like this:</p>
<p><a href="https://i.stack.imgur.com/PjeU9.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/PjeU9.png" alt="enter image description here"></a></p>
<p>So, AB, CD, EF, AD are side of the square of length say $\mathrm{a}$ and E, F, G and... |
754,888 | <p>The letters that can be used are A, I, L, S, T. </p>
<p>The word must start and end with a consonant. Exactly two vowels must be used. The vowels can't be adjacent.</p>
| MJD | 25,554 | <p><a href="http://cr.yp.to/papers/powers-ams.pdf" rel="nofollow">This paper</a> of Daniel J. Bernstein, “Detecting Perfect Powers in Essentially Linear Time” describes a fast algorithm which does exactly what you asked for: given a positive integer $n$, it finds positive integers $x$ and $k$ with $n = x^k$ and $k$ as ... |
2,797,902 | <p>AFAIK, every mathematical theory (by which I mean e.g. the theory of groups, topologies, or vector spaces), started out (historically speaking) by formulating a set of axioms that generalize a specific structure, or a specific set of structures. </p>
<p>For example, when people think of a “field” they AFAIK usually... | Ethan Bolker | 72,858 | <p>Several significant examples from mathematical physics:</p>
<ul>
<li><p>The usefulness of Hilbert space in the formalization of quantum
mechanics.</p></li>
<li><p>Riemannian manifolds as the appropriate language for general
relativity.</p></li>
<li><p><a href="https://en.wikipedia.org/wiki/Calabi%E2%80%93Yau_manifo... |
2,797,902 | <p>AFAIK, every mathematical theory (by which I mean e.g. the theory of groups, topologies, or vector spaces), started out (historically speaking) by formulating a set of axioms that generalize a specific structure, or a specific set of structures. </p>
<p>For example, when people think of a “field” they AFAIK usually... | Ethan Bolker | 72,858 | <p>Number theory (the "structure of the integers") had no applications for years - a fact that particularly pleased G. H. Hardy. </p>
<p>Now it's central to cryptography: prime factorization, discrete logarithms, elliptic curves.</p>
<p>See <a href="https://crypto.stackexchange.com/questions/59537/how-come-public-key... |
2,623,324 | <p>Assume that the measure space is finite for this to make sense. Also, we know that $L^p$ spaces satisfy log convexity, that is -
$$\|f\|_r \leq \|f\|_p^\theta \|f\|_q^{1-\theta}$$
where $\frac{1}{r}=\frac{\theta}{p} +\frac{1-\theta}{q}$.
The text which I am following says 'Indeed this is trivial when $q=\infty$, and... | Atmos | 516,446 | <p>Use that
$$
\left(n+1\right)^{1/\sqrt{n}}=e^{\left(1/\sqrt{n}\right)\ln\left(1+n\right)}
$$
And
$$
\ln\left(1+n\right)=\ln\left(n\right)+\ln\left(1+\frac{1}{n}\right)
$$
So it becomes
$$
\left(n+1\right)^{1/\sqrt{n}}=e^{\frac{\ln(n)}{\sqrt{n}}+\frac{\ln\left(1+\frac{1}{n}\right)}{\sqrt{n}}}
$$
Using exponential prop... |
2,449,581 | <p>There is a brick wall that forms a rough triangle shape and at each level, the amount of bricks used is two bricks less than the previous layer. Is there a formula we can use to calculate the amount of bricks used in the wall, given the amount of bricks at the bottom and top levels?</p>
| Alex F | 485,563 | <p>$S = (a_1 + a_n)/2*(a_n-a_1 + 2)/2$</p>
<p>Sum of the elements of an arithmetic progression is $S = (a_1+a_n)*n/2$. In case you do not now what $n$ is equal to, you can use the following formula:<br>
$S = (a_1 + a_n)/2*(a_n-a_1 + x)/x$, where $a_1$ is the first element, $a_n$ is the last element, $x$ is the "step",... |
4,218,729 | <p>You have a database of <span class="math-container">$25,000$</span> potential criminals. The probability that this database includes the art thief is <span class="math-container">$0.1$</span>. In a stroke of luck, you found a DNA sample of this thief from the crime scene. You compare this sample with a database of <... | lonza leggiera | 632,373 | <p>Given the wording of the question, I think your formulation of the problem is reasonable. However you're not really given enough information to evaluate <span class="math-container">$\ P(B)\ $</span> properly. Was the sample from the crime scene merely tested against random entries in the database <em>until</em> a ... |
2,867,479 | <p>From <a href="https://math.stackexchange.com/questions/2867457">ETS Major Field Test in Mathematics</a></p>
<blockquote>
<p>A student is given an exam consisting of
8 essay questions divided into 4 groups of
2 questions each. The student is required to
select a set of 6 questions to answer,
including at l... | BCLC | 140,308 | <p>$$\binom{8}{6}=28$$</p>
<p>Then let's take away the choices where we don't cover all 4 groups. There are only 4. Why? To not cover all 4 groups but to choose 6 means to pick all questions but 2 questions of the same group. That is, from 4 groups, pick 1 group to exclude or equivalently pick 3 groups to include $$\b... |
2,867,479 | <p>From <a href="https://math.stackexchange.com/questions/2867457">ETS Major Field Test in Mathematics</a></p>
<blockquote>
<p>A student is given an exam consisting of
8 essay questions divided into 4 groups of
2 questions each. The student is required to
select a set of 6 questions to answer,
including at l... | Xiaonan | 336,263 | <p>Because we only have four groups and you need to pick up 6 balls which means you need to pick up two groups of balls and one ball each from the other two groups so the final result would be</p>
<p>${4\choose2} \times 2 \times 2$ which is 24.</p>
|
2,844,060 | <p>How to find the points of discontinuity of the following function $$f(x) = \lim_{n\to \infty} \sum_{r=1}^n \frac{\lfloor2rx\rfloor}{n^2}$$ </p>
| Jakobian | 476,484 | <p>Using Stolz theorem:
$$ \lim_{n\to \infty} \sum_{r=1}^n \frac{\lfloor 2rx\rfloor}{n^2} = \lim_{n\to \infty} \frac{\lfloor2nx\rfloor}{n^2-(n-1)^2} = \lim_{n\to \infty} \frac{\lfloor2nx\rfloor}{2n-1} = x $$
Note that for the last limit you use squeeze theorem.</p>
<p>So the function is continuous</p>
|
1,183,185 | <p>i) Show that for a particle moving with velocity $v(t), $if $ v(t)·v′(t) = 0$ for all $t$ then the speed $v$ is constant.
</p>
<p>I did $(v(t))^2=|v(t)|^2=(v(t)\bullet(v(t)))$. </p>
<p>Therefore $\frac{d}{dt}(v(t))^2=2v(t)$</p>
<p>Also, $\frac{d}{dt}(v(t)\bullet(v(t))=2(v(t)·v′(t))$ </p>
<p>I'm stuck here.</p>... | heropup | 118,193 | <p>$$\begin{align*} \Pr[A] &= \Pr\left[\left(t_0 < \min_i X_i\right) \cap \left(\max_i X_i \le t_1\right)\right] \\ &= \Pr[t_0 < X_{(1)} \le X_{(N)} \le t_1] \\ &= \Pr[t_0 < X_1, X_2, \ldots, X_N \le t_1] \\ &= \Pr\left[\bigcap_{i=1}^N t_0 < X_i \le t_1 \right] \\ &\overset{\rm ind}{=} \... |
1,002,719 | <p>If we have</p>
<p>$f: \{1, 2, 3\} \to \{1, 2, 3\}$</p>
<p>and</p>
<p>$f \circ f = id_{\{1,2,3\}}$</p>
<p>is the following then always true for every function?</p>
<p>$f = id_{\{1,2,3\}}$</p>
| egreg | 62,967 | <p>Hint: consider $f$ defined by $f(1)=2$, $f(2)=1$ and $f(3)=3$.</p>
|
3,623,277 | <p>Show that the cycles <span class="math-container">$(1, 2, \ldots, n)$</span>, <span class="math-container">$(n, \ldots, 2, 1)$</span> are inverse permutations. </p>
| Bill Dubuque | 242 | <p><strong>Hint</strong> <span class="math-container">$ $</span> It's always true mod <span class="math-container">$3,\,$</span> so <a href="https://math.stackexchange.com/a/1864763/242">by CRT</a> we need only combine all roots <span class="math-container">$\{0,\pm1\}$</span> mod <span class="math-container">$5$</span... |
2,359,372 | <blockquote>
<p>Given that
$$\log_a(3x-4a)+\log_a(3x)=\frac2{\log_2a}+\log_a(1-2a)$$
where $0<a<\frac12$, find $x$.</p>
</blockquote>
<p>My question is how do we find the value of $x$ but we don't know the exact value of $a$? </p>
| Hypergeometricx | 168,053 | <p>Put $u=3x-2a$ to exploit the symmetry and note that $\frac 1{\log_2}a=\log_a 2$:
$$\begin{align}
\log_a(3x-4a)+\log_a(3x)&=\frac2{\log_2a}+\log_a(1-2a)\\
\log_a (\underbrace{3x-4a}_{u-2a})(\underbrace{3x}_{u+2a})&=\log_a 2^2(1-2a)\\
(u-2a)(u+2a)&=4(1-2a)\\
u^2-4a^2&=4(1-2a)\\
u^2&=4(a-1)^2\\
u=3x... |
1,855,650 | <p>Need to solve:</p>
<p>$$2^x+2^{-x} = 2$$</p>
<p>I can't use substitution in this case. Which is the best approach?</p>
<p>Event in this form I do not have any clue:</p>
<p>$$2^x+\frac{1}{2^x} = 2$$</p>
| Zain Patel | 161,779 | <p>Elucidate the problem by using the substitution $u = 2^x$, then you have $$u + \frac{1}{u} = 2$$</p>
<p>Multiply throughout by $u \neq 0$ to get $$u^2 +1 = 2u \iff u^2 - 2u + 1 = 0$$</p>
<p>This is an easy quadratic to solve, you should get $u = 1$ and hence you need only solve $2^x = 1 \iff x = 0$. </p>
|
1,855,650 | <p>Need to solve:</p>
<p>$$2^x+2^{-x} = 2$$</p>
<p>I can't use substitution in this case. Which is the best approach?</p>
<p>Event in this form I do not have any clue:</p>
<p>$$2^x+\frac{1}{2^x} = 2$$</p>
| H. Potter | 289,192 | <p>Substitute $y=2^x$. Hence, the equation is: $y+y^{-1}=2$ which is equivalent to: $y^2-2y+1=0$. Can you take it from here ? Once you find which $y$('s) satisfy the equation, try to find $x$ such that $2^x=y$.</p>
|
1,415,752 | <p>I test my answer using wolfram alpha pro but it gets a different result to what I am getting. This is homework.</p>
<p>My result is z= 2(y-1)</p>
<p>partial derivative with respect to y is </p>
<pre><code> x.y^x-1
</code></pre>
<p>partial derivative with respect to x is
ln(y).y^x</p>
<p>ln(1) is zero... | 5xum | 112,884 | <p>It is undefined.</p>
<p>The function $x\mapsto \sqrt{1-x}$ is only defined for $1-x>0$, so only for $x<1$.</p>
<p>In your case, the limit only takes undefined values, since it is a limit when $x$ is <strong>decreasing</strong> to $1$.</p>
<p>The limit $$\lim_{x\to 1^-}\sqrt{1-x}$$
on the other hand <em>does... |
161,029 | <p>I have not seen a problem like this so I have no idea what to do.</p>
<p>Find an equation of the tangent to the curve at the given point by two methods, without elimiating parameter and with.</p>
<p>$$x = 1 + \ln t,\;\; y = t^2 + 2;\;\; (1, 3)$$</p>
<p>I know that $$\dfrac{dy}{dx} = \dfrac{\; 2t\; }{\dfrac{1}{t}}... | Mohamed | 33,307 | <p>2nd method: eliminating parameter:</p>
<p>$x=1+ \ln t , y=t^2+2 \Leftrightarrow t=\exp(x-1), y=2+\exp(2x-2)$</p>
<p>Consider the function: $f(x)=2+ \exp(2x-2)$, then: $f'(x)=2 \exp(2x-2)$</p>
<p>The tangent at $x=1$ has equation: $Y=f'(1)(X-1)+ f(1)$, thus: $Y=2(X-1)+3$, thus:$$Y=2X+1$$</p>
|
206,712 | <p>Warning: very new to Mathematica
Currently I am trying to solve for <code>x</code>, but the variable <code>K</code> changes. So I'd like a value for <code>x</code>, for every <code>K</code>. However, this equation below gives me 2 complex numbers and 2 real, of which 1 real value is positive. I'm not sure how possi... | kglr | 125 | <p>try <a href="https://reference.wolfram.com/language/ref/Part.html" rel="nofollow noreferrer"><code>Part</code></a>:</p>
<pre><code>data[[All, All, {2, 3}]]
</code></pre>
<blockquote>
<p>{{{x1, y1}, {x1, y1}, {x1, y1}, {x1, y1}, {x1, y1}}, {{x2, y2}, {x2,
y2}, {x2, y2}}, {{x3, y3}, {x3, y3}, {x3, y3}, {x3, ... |
206,712 | <p>Warning: very new to Mathematica
Currently I am trying to solve for <code>x</code>, but the variable <code>K</code> changes. So I'd like a value for <code>x</code>, for every <code>K</code>. However, this equation below gives me 2 complex numbers and 2 real, of which 1 real value is positive. I'm not sure how possi... | Coolwater | 9,754 | <p>You can write <code>positions = data[[All, All, {2, 3}]]</code>, or by adjusting your code:</p>
<pre><code>positions = Table[{data[[j]][[i]][[2]], data[[j]][[i]][[3]]},
{j, 1, Length[data]}, {i, 1, Length[data[[j]]]}]
</code></pre>
|
192,883 | <p>Can anyone please give an example of why the following definition of $\displaystyle{\lim_{x \to a} f(x) =L}$ is NOT correct?:</p>
<p>$\forall$ $\delta >0$ $\exists$ $\epsilon>0$ such that if $0<|x-a|<\delta$ then $|f(x)-L|<\epsilon$</p>
<p>I've been trying to solve this for a while, and I think it w... | Ilmari Karonen | 9,602 | <p>There are two problems with your "backwards" definition, which I'll illustrate with examples:</p>
<ol>
<li><p>Let $f(x) = \sin x$ and let $a$, $L$ and $\delta$ be arbitrary real numbers. Then $\epsilon = |L| + 2$ satisfies your definition.</p></li>
<li><p>Let $f(x) = 1/x$ (for $x \ne 0$, and let $f(0) = 0$, just t... |
192,883 | <p>Can anyone please give an example of why the following definition of $\displaystyle{\lim_{x \to a} f(x) =L}$ is NOT correct?:</p>
<p>$\forall$ $\delta >0$ $\exists$ $\epsilon>0$ such that if $0<|x-a|<\delta$ then $|f(x)-L|<\epsilon$</p>
<p>I've been trying to solve this for a while, and I think it w... | TonyK | 1,508 | <p>Your condition can be translated into words as:</p>
<p>$f$ is bounded in every open ball around $a$</p>
<p>This is quite different from being continuous!</p>
<p>PS Not every function satisfies this condition, but the ones that don't are rather fierce. (In fact, in a strict sense, <em>almost no</em> function satis... |
440,242 | <p>I'm pretty sure almost all mathematicians have been in a situation where they found an interesting problem; they thought of many different ideas to tackle the problem, but in all of these ideas, there was something missing- either the "middle" part of the argument or the "end" part of the argumen... | Phil Harmsworth | 106,467 | <p>I think there's some good advice on how to conduct research in J.E. Littlewood's "The Mathematician's Art of Work", included in <em>Littlewood's miscellany</em>, CUP, 1986.</p>
<blockquote>
<p>"A <em>sine qua non</em> is an intense conscious curiosity about the subject, with a craving to exercise the ... |
3,424,259 | <p>The system in question is <span class="math-container">$$\begin{cases} x_1 -x_2 + x_3 = -1 \\ -3x_1 +5x_2 + 3x_3 = 7 \\ 2x_1 -x_2 + 5x_3 = 4 \end{cases}$$</span></p>
<p>After writing this in matrix-form and performing row-operations we can show that</p>
<p><span class="math-container">$$
\begin{matrix}
-1 ... | José Carlos Santos | 446,262 | <p>Yes, that is sufficient, since at that point you can assert that, no matter what the values of <span class="math-container">$x_2$</span>, and <span class="math-container">$x_3$</span> are, the number <span class="math-container">$x_2+3x_3$</span> cannot be equal to both <span class="math-container">$6$</span> and <s... |
3,424,259 | <p>The system in question is <span class="math-container">$$\begin{cases} x_1 -x_2 + x_3 = -1 \\ -3x_1 +5x_2 + 3x_3 = 7 \\ 2x_1 -x_2 + 5x_3 = 4 \end{cases}$$</span></p>
<p>After writing this in matrix-form and performing row-operations we can show that</p>
<p><span class="math-container">$$
\begin{matrix}
-1 ... | AlvinL | 229,673 | <p>Using the <a href="https://www.encyclopediaofmath.org/index.php/Kronecker-Capelli_theorem" rel="nofollow noreferrer">Kronecker-Capelli</a> criterion, the system of equations is solvable if and only if the rank of the system matrix is equal to the rank of the augmented matrix. The rank of the system matrix is clearly... |
3,306,341 | <p>Let <span class="math-container">$f\in Hom(R,R')$</span> be a surjective map and let <span class="math-container">$I$</span> be an ideal of <span class="math-container">$R$</span></p>
<p>Assume that <span class="math-container">$Ker(f)\subseteq I$</span> , prove that <span class="math-container">$f^{-1}(f(I))=I$</s... | k.stm | 42,242 | <p>More generally, show that <span class="math-container">$f^{-1}(f(I)) = \ker f + I$</span>. For <span class="math-container">$x ∈ R$</span>,
<span class="math-container">\begin{align*}
x ∈ f^{-1}(f(I)) &\iff f(x) ∈ f(I) \\
&\iff ∃y ∈ I\colon~f(x) = f(y) \\
&\iff ∃y ∈ I\colon~x - y ∈ \ker f\\
&\iff …?
... |
668,959 | <p>I don't know if this is an already existing conjecture, or has been proven: There is at least one prime number between <span class="math-container">$N$</span> and <span class="math-container">$N-\sqrt{N}$</span>.</p>
<p>Some examples:
<span class="math-container">$N=100$</span></p>
<p><span class="math-container">$... | Will Jagy | 10,400 | <p>the usual form of such things is to say that something is true for sufficiently large numbers. That is likely to be the case, but actual proofs, including the "sufficiently large" condition, have still not reached what you need. The best results are something like $x + x^{0.525};$ for large enough positive real numb... |
1,680,269 | <p>Here $\mathbb{Z}_{n}^{*}$ means $\mathbb{Z}_{n}-{[0]_{n}}$</p>
<p>My attempt:</p>
<p>$(\leftarrow )$</p>
<p>$p$ is a prime, then, for every $[x]_{n},[y]_{n},[z]_{n}$ $\in (\mathbb{Z}_{n}^{*},.)$ are verified the following:</p>
<p>1) $[x]_{n}.([y]_{n}.[z]_{n}) = ([x]_{n}.[y]_{n}).[z]_{n}$, since from the operatio... | GiantTortoise1729 | 219,849 | <p>Given a non-empty set $A$ of a metric space $X$, note the function $$\textrm{dist}_A(x) = \inf_{a\in A} |a-x|$$ is continuous and $\textrm{dist}_A(x) = 0$ iff $x \in \textrm{cl}(A)$. Also recall continuous functions on compact sets attain their infimums. Hopefully you can put the pieces together.</p>
|
2,166,917 | <p>$20$ questions in a test. The probability of getting correct first $10$ questions is $1$. The probability of getting correct next $5$ questions is $\frac 13$. The probability of getting correct last $5$ questions is $\frac 15$. What is the probability of getting exactly $11$ questions correctly?</p>
<p>This is the ... | user247327 | 247,327 | <p>The fact that "The probability of getting correct first 10 questions is 1" means that you need to get exactly one of the last 10 questions correct.</p>
<p>You can do that in either of two ways:
1) Get exactly one of the next 5 questions correct and get all of last 5 incorrect.
or
2) Get all of the next 5 questions ... |
2,231,949 | <p>To find the minimal polynomial of $i\sqrt{-1+2\sqrt{3}}$, I need to prove that
$x^4-2x^2-11$ is irreducible over $\Bbb Q$. And I am stuck. Could someone please help? Thanks so much!</p>
| ancient mathematician | 414,424 | <p>As there are no rational roots the only possible factorisation is into two quadratics whose coefficients we may assume to be integral by Gauss. As 11 is prime and as there is no $X^3$ term we must have
$$
(X^2 +\alpha X +\epsilon)(X^2 -\alpha X -11\epsilon)
$$
where $\epsilon=\pm1$.
Now look at the $X$ term and get ... |
3,328,387 | <p>Suppose I have two positive semi-definite <span class="math-container">$n$</span>-by-<span class="math-container">$n$</span> matrices <span class="math-container">$A$</span>, <span class="math-container">$B$</span> and an <span class="math-container">$n$</span>-by-<span class="math-container">$n$</span> identity mat... | Rodrigo de Azevedo | 339,790 | <p>Given symmetric matrices <span class="math-container">$\mathrm A \succ \mathrm O_n$</span> and <span class="math-container">$\mathrm B \succeq \mathrm O_n$</span>, let</p>
<p><span class="math-container">$$\mathrm X (t) := \int_0^t e^{-\tau \mathrm A} \mathrm B e^{-\tau \mathrm A} \,\mathrm d \tau$$</span></p>
<p>Us... |
720,969 | <p>While studying real analysis, I got confused on the following issue.</p>
<p>Suppose we construct real numbers as equivalence classes of cauchy sequences. Let $x = (a_n)$ and $y= (b_n)$ be two cauchy sequences, representing real numbers $x$ and $y$.</p>
<p>Addition operation $x+y$ is defined as $x+y = (a_n + b_n)$... | Jose Antonio | 84,164 | <p>You can certainly check that the operation is well-defined by checking $x+y=x'+y'$ where $x$ and $x'$ are equivalent and $y$ and $y'$ are equivalent. </p>
<p>Let $(x_n)$, $(x_n')$, $(y_n)$ and $(y'_n)$ where $x_n-x_n'\to 0$ and $y_n-y_n' \to 0$, i.e., are equivalents. </p>
<p>So in ordered to prove the claim we ne... |
1,550,841 | <p>$\int e^{2\theta}\ \sin 3\theta\ d\theta$</p>
<p>After Integrating by parts a second time, It seems that the problem will repeat for ever. Am I doing something wrong. I would love for someone to show me using the method I am using in a clean and clear fashion. Thanks. <a href="https://i.stack.imgur.com/FipiE.jpg" r... | Mirko | 188,367 | <p>You got $\displaystyle\int e^{2\theta}\sin(3\theta)d\theta=...=\sin(3\theta)\frac12e^{2\theta}-\frac32\Bigl(\frac12e^{2\theta}\cos(3\theta)+\frac32\int e^{2\theta}\sin(3\theta)d\theta\Bigr)$. </p>
<p>Now you could denote the integral you are looking for by $x$: </p>
<p>$\displaystyle x=\int e^{2\theta}\sin(3\the... |
34,557 | <p>I just came across a spam answer which is extremely vulgar (sexual). I flagged it for moderator attention, and then it occurred to me that I could edit it and erase its contents by blanking it till a moderator gets to look at it. Is this an acceptable thing to do?</p>
| peterh | 111,704 | <p>Rude/abuse/spam deletions are causing -100 rep loss to the OP and the post content is defaced. People having access to the deleted content (OP + mods + 10k+ users) can still see it, but even they need an extra click to see its edit history.</p>
<p><a href="https://i.stack.imgur.com/EMWKL.png" rel="nofollow noreferre... |
2,239,240 | <p>I'm looking to do some independent reading and I haven't been able to find rough prerequisites for Differential Topology at the level of Milnor or Guillemin and Pollack.</p>
<p>Is a semester of analysis (Pugh) and a semester of topology (Munkres) enough to make sense of most of it or should I take a second semester ... | Hagen von Eitzen | 39,174 | <p>By the rules of precedence, we compute
$$ (2+3)^2 = 5^2 = 25.$$
By the same rules, we compute
$$2^2+2\cdot 2\cdot 3+3^2=4+12+9=25. $$
The very fact that both computations produce the same result justifies us to write down the interesting fact
$$ (2+3)^2=2^2+2\cdot 2\cdot 3+3^2.$$</p>
|
107,399 | <p>Let's say we have a set a\of associations:</p>
<pre><code>dataset = {
<|"type" -> "a", "subtype" -> "I", "value" -> 1|>,
<|"type" -> "a", "subtype" -> "II", "value" -> 2|>,
<|"type" -> "b", "subtype" -> "I", "value" -> 1|>,
<|"type" -> "b", "subtype" -> ... | Edmund | 19,542 | <p>Instead of a nested association solution, would <code>Query</code> and <code>Select</code> be acceptable.</p>
<pre><code>Query[Select[#type == "a" && #subtype == "I" &], "value"]@dataset
(* {1} *)
</code></pre>
<p>This form is more descriptive on what is happening and does not require reshaping of the... |
107,399 | <p>Let's say we have a set a\of associations:</p>
<pre><code>dataset = {
<|"type" -> "a", "subtype" -> "I", "value" -> 1|>,
<|"type" -> "a", "subtype" -> "II", "value" -> 2|>,
<|"type" -> "b", "subtype" -> "I", "value" -> 1|>,
<|"type" -> "b", "subtype" -> ... | Mr.Wizard | 121 | <p>I believed this question to be a duplicate of <a href="https://mathematica.stackexchange.com/q/86578/121">Create Nested List from tabular data</a> and proposed, with minor variation, the same answer:</p>
<pre><code>fn[x_List] := GroupBy[x, First -> Rest, fn]
fn[{a_}] := a
nested = fn[dataset]
</code></pre>
<bl... |
1,613,185 | <p>There are five red balls and five green balls in a bag. Two balls are taken out at random. What is the probability that both the balls are of the same colour</p>
| drhab | 75,923 | <p>Hint:</p>
<p>If you have taken one ball out then in the bag there are $4$ balls that have the color of the drawn ball and $5$ that have <em>not</em> the color of the drawn ball.</p>
|
2,916,037 | <p>Some cute results have every digit doubled. </p>
<p>\begin{align}
99225500774400 = {} & \frac{40!}{31!} \\[8pt]
33554433 = {} & 2^{25} +1 \\[8pt]
222277 = {} & -22^{2^2}+77^3 \\[8pt]
8811551199 = {} & 95^5 + 64^5 \\[8pt]
7755660000 = {} & 95^5 + 65^4 \\[8pt]
334444448888 = {} & 6942^3 - 1... | joriki | 6,622 | <p>$$\frac15=0.001100110011\ldots_2\;,$$</p>
<p>$$\frac1{20}=0.001100110011\ldots_3\;,$$</p>
<p>and generally</p>
<p>$$\frac1{101(10-1)}=0.001100110011\ldots$$</p>
<p>in all bases.</p>
|
2,055,559 | <blockquote>
<p>Let <span class="math-container">$a,b,c$</span> be the length of sides of a triangle then prove that:</p>
<p><span class="math-container">$a^2b(a-b)+b^2c(b-c)+c^2a(c-a)\ge0$</span></p>
</blockquote>
<p>Please help me!!!</p>
| math110 | 58,742 | <p>$$a^2b(a-b)+b^2c(b-c)+c^2a(c-a)
=\dfrac{1}{2}[(a+b-c)(b+c-a)(a-b)^2+(b+c-a)(a+c-b)(b-c)^2+(a+c-b)(a+b-c)(c-a)^2]
≥0$$</p>
|
2,140,192 | <p>I want to show that $C^1[0,1]$ isn't a Banach Space with the norm:</p>
<p>$$||f||=\max\limits_{y\in[0,1]}|f(y)|$$</p>
<p>Therefore, I want to show that the sequence $\left \{ |x-\frac{1}{2}|^{1+\frac{1}{n}} \right \}$ converges to $|x-\frac{1}{2}|$, but I can't find $N$ in the definition of convergence. Could anyo... | Martin Argerami | 22,857 | <p>You have, using the Mean Value Theorem, the inequality
$$
1-e^t\leq |t|,\ \ \ \ t\leq0.
$$
Then, for $t\in[0,1]$ and $x>0$,
$$
|t^{1+x}-t|=|t|\,|t^x-1|\leq |t|\,|e^{x\log t}-1|\leq x\,|t\log t|\leq \frac xe.
$$
So
$$
\left|\,\left|x-\tfrac12\right|^{1+\tfrac1n}-\left| x-\tfrac12\right|\,\right |\leq\frac1{ne}
$... |
89,188 | <p>Norimatsu's lemma says that on a smooth projective complex variety $X$ of dimension $n$, then we have $H^i(X,\mathcal O_X(-A-E))=0$ for $i < n$ when $A$ is an ample divisor and $E$ is a simple normal crossings (SNC) divisor. Does this statement remain true if $E$ is an effective divisor with SNC support? In ot... | Sándor Kovács | 10,076 | <p>Let $X$ be an arbitrary smooth projective surface, $A$ an arbitrary ample divisor and $E\subset X$ a smooth proper curve. Consider the short exact sequence
$$
0\to \mathscr O_X(-A-(a+1)E)\to \mathscr O_X(-A-aE)\to \mathscr O_X(-A-aE)|_E\to 0
$$
If $H^0(X, \mathscr O_X(-A-aE))=0$, then
$H^1(X, \mathscr O_X(-A-(a+1)... |
2,484 | <p>It's been quite a while since I was tutoring a high school student and even longer since not a gifted one.</p>
<p>However, this time, something was amiss. I have asked him to show me how he does some exercise, and then another and the only thing I wanted to do was to shout:</p>
<blockquote>
<p><strong>You are do... | Toscho | 439 | <ol>
<li>JvR's answer is very good.</li>
<li>I'd like to point out a missing aspect:</li>
</ol>
<h1>Emotion</h1>
<p>Your student has developped highly negative emotions concerning math. He has lost motivation and is in despair concerning math. He feels, he'll never understand math also he knows, that it's important. He... |
3,141,510 | <p>Calculate sum
<span class="math-container">$$ \sum_{k=2}^{2^{2^n}} \frac{1}{2^{\lfloor \log_2k \rfloor} \cdot 4^{\lfloor \log_2(\log_2k )\rfloor}} $$</span></p>
<p>I hope to solve this in use of Iverson notation:</p>
<h2>my try</h2>
<p><span class="math-container">$$ \sum_{k=2}^{2^{2^n}} \frac{1}{2^{\lfloor \log... | TheSilverDoe | 594,484 | <p>Let's write
<span class="math-container">$$\sum_{k=2}^{2^{2^n}} \frac{1}{2^{\lfloor \log_2(k) \rfloor}4^{\lfloor \log_2(\log_2(k))\rfloor}} = \sum_{i=0}^{n-1} \sum_{k=2^{2^i}}^{2^{2^{i+1}}-1} \frac{1}{2^{\lfloor \log_2(k) \rfloor}4^{\lfloor \log_2(\log_2(k))\rfloor}} + \frac{1}{2^{2^n}4^{n}}$$</span> </p>
<p><span ... |
2,805,192 | <p><a href="https://i.stack.imgur.com/OhGi4.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/OhGi4.png" alt="enter image description here"></a></p>
<p>Definition:</p>
<p><a href="https://i.stack.imgur.com/mV4NU.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/mV4NU.png" alt="enter ... | Henno Brandsma | 4,280 | <p>The sketched argument is as follows: let $\delta >0$ be given.
Do the recursion: pick $x_0 \in X$ and having picked $x_0,\ldots,x_n$, pick $x_{n+1}$ such that $$\forall i\le n: d(x_i, x_{n+1}) \ge \delta \tag 1$$</p>
<p>If this process never stops, it's easy to verify that the set $\{x_n: n \ge 0\}$ is infinite... |
8,816 | <p>What is the result of multiplying several (or perhaps an infinite number) of binomial distributions together?</p>
<p>To clarify, an example.</p>
<p>Suppose that a bunch of people are playing a game with k (to start) weighted coins, such that heads appears with probability p < 1. When the players play a round, t... | BCnrd | 3,927 | <p>This is really a comment on Pete's comment for Mikhail's answer, but I am making it an answer because it raises a question which I think should be more widely known.</p>
<p>The construction of Aut-scheme uses the entire Hilbert scheme, which has countably many components (due to varying Hilbert polynomials), and it... |
1,459,334 | <p>How can we show that additive inverse of a real number equals the number multiplied by -1, i.e. how can we show that $(-1)*u = -u$ for all real numbers $u$?</p>
| Community | -1 | <p>$-u$ is the unique additive inverse of $u$. By distributivity we have
$$(-1)u+u=(-1)u+(1)u=(-1+1)u=0u=0$$
and hence $(-1)u=-u.$ To prove $0u=0$, observe $0u=(0+0)u=0u+0u$ and the additive identity element is unique.</p>
|
1,459,334 | <p>How can we show that additive inverse of a real number equals the number multiplied by -1, i.e. how can we show that $(-1)*u = -u$ for all real numbers $u$?</p>
| cr001 | 254,175 | <p>$(-1)*u = (-1)*u + u - u = (-1)*u + (1)*u - u = (-1+1)*u - u = -u$</p>
|
2,431,375 | <p>A continuous function $f$ on $[a,b]$, differentiable in $(a,b)$, has only 1 point where its derivative vanishes. What is true about this function?</p>
<p>A. $f$ cannot have an even number of extrema.</p>
<p>B. $f$ cannot have a maximum at one endpoint and minimum at the other.</p>
<p>C. $f$ might be monotonically... | gen-ℤ ready to perish | 347,062 | <p>I argue that the truthfulness of A depends on if “extrema” means “relative extrema,” “global extrema,” or both. After all, a function has to have a minimum and maximum value on an internal. For example, if $f(x)=x$ for $a\leq x\leq b$ only, then the extrema of $f$ are simply $a$ and $b$.</p>
<p>The prior example al... |
232,562 | <p>Ax-Grothendieck Theorem states that if $\mathbf K$ is an algebraically closed field, then any injective polynomial map $P:\mathbf K^n\longrightarrow \mathbf K^n$ is bijective.</p>
<blockquote><b>Question 1.</b> What does the inverse map of $P$ look like ? What kind of map is that ?</blockquote>
<p>$P^{-1}$ need no... | Tsemo Aristide | 80,891 | <p>$P$ has a polynomial inverse implies that the Jacobian of $P$ is a constant function. There is a conjecture known as the Jacobian conjecture which says that if the characteristic of $K$ is zero, $P$ has a polynomial inverse if and only if its Jacobian is a non zero constant.</p>
<p><a href="https://en.wikipedia.org... |
186,878 | <p>Is there a way to find the geoposition of a given distance from start in a <code>GeoPath</code>? I want to mark equidistant positions along a track, for example, a mark every 500 km along the path given by</p>
<pre><code>path=GeoGraphics[
GeoPath[{
Entity["City", {"Boston", "Massachusetts", "UnitedStates"}],
... | ZaMoC | 46,583 | <p>Here is a function for finding GeoPositions between 2 cities with certain step</p>
<pre><code>city1 = Entity["City", {"Boston", "Massachusetts", "UnitedStates"}];
city2 = Entity["City", {"Rochester", "NewYork", "UnitedStates"}];
city3 = Entity["City", {"Chicago", "Illinois", "UnitedStates"}];
geopath[c1_, c2_, ste... |
1,611,506 | <blockquote>
<p>$$\int (2x^2+1)e^{x^2} \, dx$$</p>
</blockquote>
<p>It's part of my homework, and I have tried a few things but it seems to lead to more difficult integrals. I'd appreciate a hint more than an answer but all help is valued.</p>
| Future | 299,525 | <p>Let $u = xe^{x^2}$. Then $du = (2x^2+1) e^{x^2}$.
Thus,</p>
<p>$$\int (2x^2 + 1)e^{x^2} = \int du = u + C = xe^{x^2} + C$$</p>
|
3,455,009 | <p>In the proof of the expectation of the binomial distribution,</p>
<p><span class="math-container">$$E[X]=\sum_{k=0}^{n}k \binom{n}{k}p^kq^{n-k}=p\frac{d}{dp}(p+q)^n=pn(p+q)^{n-1}=np$$</span></p>
<p>Why is <span class="math-container">$\sum_{k=0}^{n}k \binom{n}{k}p^kq^{n-k}= p\frac{d}{dp}(p+q)^n$</span>?</p>
<p>I ... | user | 505,767 | <p>We have for <span class="math-container">$z\neq 0$</span></p>
<p><span class="math-container">$$(z+1)^3=z^3 \iff \left(1+\frac1z\right)^3=1 \implies 1+\frac1z=1,e^{i\frac23 \pi},e^{-i\frac23 \pi}$$</span></p>
<p>therefore</p>
<p><span class="math-container">$$\frac1z =-1+e^{i\frac23 \pi},-1+e^{-i\frac23 \pi} \imp... |
19,880 | <p>I want to write down $\ln(\cos(x))$ Maclaurin polynomial of degree 6. I'm having trouble understanding what I need to do, let alone explain why it's true rigorously.</p>
<p>The known expansions of $\ln(1+x)$ and $\cos(x)$ gives:</p>
<p>$$\forall x \gt -1,\ \ln(1+x)=\sum_{n=1}^{k} (-1)^{n-1}\frac{x^n}{n} + R_{k}(x... | Ofir | 2,125 | <p>Since you know the polynomial of $\ln(1+t)$ and you know that $\cos(x)= 1 + (-\frac {x^2}{2!}+\frac{x^4}{4!}+x^4\epsilon(x))$ then you can "plug" it in the polynomial of $\ln(1+t)$. You now have that $\ln(\cos(x)) = P(x) + R(x)$ where P is a polynomial and R is the error such that $\lim _{x\rightarrow 0} \frac{R(x)}... |
19,880 | <p>I want to write down $\ln(\cos(x))$ Maclaurin polynomial of degree 6. I'm having trouble understanding what I need to do, let alone explain why it's true rigorously.</p>
<p>The known expansions of $\ln(1+x)$ and $\cos(x)$ gives:</p>
<p>$$\forall x \gt -1,\ \ln(1+x)=\sum_{n=1}^{k} (-1)^{n-1}\frac{x^n}{n} + R_{k}(x... | J126 | 2,838 | <p>Another way to do the problem is to realize that</p>
<p>$$
\frac{d}{dx}\ln(\cos x)=-\tan x
$$</p>
<p>And so the series will just be $\ln(\cos 0)=0$ plus a slightly adjusted series for $-\tan x$. This also might help you see that the series is valid, since you are doing it out the long way.</p>
|
1,973,686 | <p>I am stuck on two questions :</p>
<ol>
<li>If $f,g\in C[0,1]$ where $C[0,1]$ is the set of all continuous functions in $[0,1]$ then is the mapping $id:(C[0,1],d_2)\to (C[0,1],d_1)$ continuous ? where $id$ denotes the identity mapping.</li>
</ol>
<p>where $d_2(f,g)=(\int _0^1 |f(t)-g(t)|^2dt )^{\frac{1}{2}} $ and ... | Alexis Olson | 11,246 | <p>@marwalix got the first question.</p>
<p>For the second, consider the function $f(x) = \min\left\{1,\frac{1}{x}\right\}$.</p>
<p>$$\int_{\Bbb R} |f|^2 = \int_{-\infty}^{-1} \frac{dx}{x^2} + \int_{-1}^1 dx + \int_{1}^{\infty} \frac{dx}{x^2}= 1+2+1 = 4$$</p>
<p>but $\int_{\Bbb R}|f|$ doesn't converge.</p>
|
2,663,537 | <p>Suppose G is a group with x and y as elements. Show that $(xy)^2 = x^2 y^2$ if and only if x and y commute.</p>
<p>My very basic thought is that we expand such that $xxyy = xxyy$, then multiply each side by $x^{-1}$ and $y^{-1}$, such that $x^{-1} y^{-1} xxyy = xxyy x^{-1}$ , and therefore $xy=xy$.</p>
<p>I realiz... | operatorerror | 210,391 | <p>If $x$ and $y$ commute, clearly we have
$$
xyxy=xxyy=x^2y^2
$$
if instead
$$
xyxy=x^2y^2
$$
then hitting the left side with $x^{-1}$ and the right with $y^{-1}$ yields
$$
xy=yx
$$</p>
|
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