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 |
|---|---|---|---|---|
338,480 | <blockquote>
<p>Find the point where equations $x=t^2-t$ and $y= t^3 -3t-1$ cross itself.</p>
</blockquote>
<p>This's the first time I meet this kind of problem, can someone give me some idea? Thank you.</p>
| Yimin | 30,330 | <p>There are some $p,q$ such that</p>
<p>$t^2-t = p$</p>
<p>$t^3-3t-1=q$</p>
<p>has two different solutions $t_1,t_2$,</p>
<p>consider the third solution to the second equation as $t_3$, then $t_1+t_2+t_3 = 0$[why?]. And from the first equation $t_1+t_2 = 1$, thus $t_3 =-1$, plug into the second equation,</p>
<p>$... |
91,252 | <p>Let $v_1, \ldots, v_n$ be a set of vectors in a vector space $V$. Show that $v_1, \ldots, v_n$ is a basis of $V$ if and only if for any non-zero linear function $f$ on $V$ there is a vector $v$ in $\operatorname{span}(v_1, \ldots, v_n)$ such that $f(v) \neq 0$. </p>
<p>Suppose that $v_1, \ldots, v_n$ is not a basis... | hmakholm left over Monica | 14,366 | <p>Um, unless I'm misreading the question, the thing to be proved is false. Counterexample: Let $V=\mathbb R^1$ (a real vector space) and let $n=2$, $v_1=5$, and $v_2=42$. Then $\mathrm{span}(5,42)$ is $\mathbb R^1$ itself, and it is certainly the case that for any nonzero linear $f$, there is a vector $v$ in $\mathbb ... |
2,287,312 | <p>So the 12 marbles are distinct with each marble being a different color. Therefore I have to take into account, the different colors and different amount each person have. For example, person 1 can have 2 marbles, person 2 can have 7 marbles, and person 3 can have 3 marbles with each marble being a different color. ... | Daniel Akech Thiong | 169,316 | <p>Let $x$ be a real number. Then by the density of rational numbers there exists a sequence $\{x_{n} \}$ of rational numbers converging to $x$. But if $f$ is continuous then we have $f(x_{n}) \to f(x)$. But then $f$ would be identically zero, which is a contradiction. </p>
|
2,287,312 | <p>So the 12 marbles are distinct with each marble being a different color. Therefore I have to take into account, the different colors and different amount each person have. For example, person 1 can have 2 marbles, person 2 can have 7 marbles, and person 3 can have 3 marbles with each marble being a different color. ... | aeyalcinoglu | 87,855 | <p>I'd like to add an answer which uses directly epsilon-delta definition.</p>
<p>Let $x \in \mathbb{R}$ and $x \neq 0$. We want to show that $f$ is not continuous at $x$. So we want to find an $\epsilon > 0$ such that for all $\delta > 0$ the following condition is not satisfied:
$$\forall x' \in \mathbb{R} \te... |
2,287,312 | <p>So the 12 marbles are distinct with each marble being a different color. Therefore I have to take into account, the different colors and different amount each person have. For example, person 1 can have 2 marbles, person 2 can have 7 marbles, and person 3 can have 3 marbles with each marble being a different color. ... | Ovi | 64,460 | <p>Let $a \in \mathbb{Q}$, $a \not = 0$ and select $\epsilon = \dfrac {|a|}{2}$. In order for the limit to exist, there must exist a $\delta$ such that $|f(x) - f(a)| < \dfrac {|a|}{2}$ whenever $x \in (a- \delta, a+\delta) - \{a\}$. Because of the density of irrationals, for any $\delta$ there exists an irrationa... |
1,536,632 | <p>I have the following system of equations </p>
<p>\begin{align}
\frac{dx}{d \tau} &= x \left(1-x-\frac{y}{x+b} \right) \\
\frac{dy}{d \tau} &= cy \left(-1+a\frac{x}{x+b} \right)
\end{align}</p>
<p>I am asked to show that if $a<1$, the only nonnegative equilibria are $(0,0), (1,0)$.</p>
<p>So first it is... | Bernard | 202,857 | <p>Reducion modulo $61$ is a ring homomorphism, hence
$$\bigl(25^{1202}+3\bigr)^2=\bigl(5^{2404}+3\bigr)^2\equiv\bigl(5^{2404\bmod60}+3\bigr)^2=628^2\equiv 18^2=324\equiv19\mod61.$$</p>
|
2,496,211 | <p>I was wondering if you could help me with this proof I have for metric spaces.
I understand the theorem and the idea I just don't understand how to proceed.
I am attaching the question.</p>
<p>Here is how I think I should proceed: So, Let U = Q(i.e. rational numbers), Y = [0,1], and X = R(i.e. real numbers).</p>
<... | angryavian | 43,949 | <p>Assuming you meant $U = \mathbb{Q} \cap Y$, your example does not work, since $U$ is both not open in $Y$ and not open in $X$.</p>
<p>A counterexample can be made with $U=Y = \{0\}$ in $X=\mathbb{R}$.</p>
|
2,496,211 | <p>I was wondering if you could help me with this proof I have for metric spaces.
I understand the theorem and the idea I just don't understand how to proceed.
I am attaching the question.</p>
<p>Here is how I think I should proceed: So, Let U = Q(i.e. rational numbers), Y = [0,1], and X = R(i.e. real numbers).</p>
<... | bradipolpo | 329,956 | <p>Consider the metric space $X=\mathbb{R}$ and the closed subset $Y=[0,1]$. What are the open sets in $Y$? Are they all open in $X$?</p>
|
236,308 | <p>I solved the following equation the hard way:
$$\sqrt{x+1} +\sqrt{x+33}=\sqrt{x+6} +\sqrt{x+22}$$
The only solution is $x=3$.
I am wondering if there is some easy observation that solves the equation without squaring both sides?</p>
| John Bentin | 875 | <p><strong>Edit:</strong> My answer was wrong. Please refer to Matthew Conroy's correct answer instead. My version of his solution is $$x=\dfrac{(4s_2-s_1^2)^2-64s_4}{64s_3-8s_1(4s_2-s_1^2)},$$where $s_1, s_2, s_3, s_4$ are respectively the cubic, quadratic, linear, and constant coefficients of the polynomial expansion... |
3,012,130 | <p><span class="math-container">$4, 15, 13, 7, 22, -1, 31, -9, 40, -17, 49$</span>.</p>
<p>What comes next? The answer is <span class="math-container">$-25$</span>, but why?</p>
| TurlocTheRed | 397,318 | <p>Break up the sequence into the even ordered terms and odd ordered.</p>
|
86,459 | <p>I have problems with the following logarimthic equation:</p>
<p>$$\log _a \left(\frac{x+\sqrt{x^2+5}}{5}\right) = b$$</p>
<p>How can I compute $ \log _a (x-\sqrt{x^2-5})$ in terms of $b$?</p>
| anon | 11,763 | <p>I assume you have a typo that is preventing others from answering. Let</p>
<p>$$b=\log_a\left(\frac{x+\sqrt{x^2-5}}{5}\right),\qquad \tilde{b}=\log\left(x-\sqrt{x^2-5}\right).$$</p>
<p>Now use the rules </p>
<ul>
<li>$\log(u)+\log(v)=\log(uv)$ </li>
<li>$(z-w)(z+w)=z^2-w^2$</li>
<li>$\log_a(1)=0$</li>
</ul>
<p>i... |
129,693 | <p><img src="https://i.stack.imgur.com/tFha8.png" alt="enter image description here"></p>
<p>I'm lost on what's happening here. This is regarding MinML( "an idealized programming language" ) . More pics below: Thank You Very Much</p>
<p><img src="https://i.stack.imgur.com/BYjTF.png" alt="enter image description here"... | Harry Peter | 83,346 | <p>$\int\dfrac{\sin x}{1+x^2}dx$</p>
<p>$=\int\sum\limits_{n=0}^\infty\dfrac{(-1)^nx^{2n+1}}{(2n+1)!(x^2+1)}dx$</p>
<p>$=\int\sum\limits_{n=0}^\infty\dfrac{(-1)^nx^{2n}}{2(2n+1)!(x^2+1)}d(x^2+1)$</p>
<p>$=\int\sum\limits_{n=0}^\infty\dfrac{(-1)^n(x^2+1-1)^n}{2(2n+1)!(x^2+1)}d(x^2+1)$</p>
<p>$=\int\sum\limits_{n=0}^... |
3,963,229 | <p>I tried to find this limit using polar representation <span class="math-container">$x=r \cos\theta $</span> and <span class="math-container">$y=r \sin \theta$</span>
i got something like <span class="math-container">$\ln (\frac{1}{\cos2 \theta})$</span> which makes me feel im in wrong way !</p>
<p>is my approach cor... | Piotr Piękos | 382,084 | <p>Before calculating you should get some intuition about how the function behaves in the neighbourhood of 0 and whether the limit will depend on the trajectory you choose to converge to zero.
What will be the limit of <span class="math-container">$(x_n, y_n)$</span> where <span class="math-container">$x_n = 0$</span> ... |
329,067 | <p>I have a question that is as follows:</p>
<blockquote>
<p>For each integer $n \geq 3$, construct a 3-regular graph on $2n$ vertices such that $G_n$ does not have any 3-cycles.</p>
</blockquote>
<p>Here is what I have:</p>
<p>I have $2n$ vertices numbered $1, 2, \ldots, 2n$, and a vertex $k$ connected to $k-1$, ... | Douglas S. Stones | 139 | <p>What you have described is an example of a <a href="http://en.wikipedia.org/wiki/Circulant_graph" rel="nofollow noreferrer">circulant graph</a>, and your method will pan out (as per Ross Millikan's answer).</p>
<p>I'd also like to add that there's examples that are not only $3$-cycle free, but have no odd length cy... |
4,625,921 | <p><a href="https://i.stack.imgur.com/mIKMW.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/mIKMW.png" alt="Two polar curves." /></a></p>
<p>I am looking for the area outside the circle <span class="math-container">$r = 3\cos\Theta$</span> and inside the limaçon <span class="math-container">$r = 1 + ... | David Quinn | 187,299 | <p>Hint…consider the region you require to find as two equal parts and work out the upper region.</p>
<p>Draw a straight line from the pole to the point of intersection in the first quadrant.</p>
<p>The upper limits are not the same for both integrals.</p>
<p>The region inside the circle has upper limit <span class="ma... |
33,907 | <p><strong>Note:</strong> Cross-posted at <a href="http://community.wolfram.com/groups/-/m/t/137895?p_p_auth=8QnKtT9I" rel="nofollow">http://community.wolfram.com/groups/-/m/t/137895?p_p_auth=8QnKtT9I</a>.</p>
<p>I am to design a two step gearbox. The first step is to choose the number of teeth in each cog wheel in o... | Daniel Lichtblau | 51 | <p>I think in general you might have to enforce the gcd (divisibility) constraint after the fact. For this particular example I get a result on the first try that has those numbers relatively prime, so we can accept that solution.</p>
<p>Anyway, here is the code. For objective function I sum the teeth and add a penalt... |
3,240,240 | <blockquote>
<p>If a right circular cone has three mutually perpendicular generators
then find its semi-vertical angle.</p>
</blockquote>
<p>We see that if <span class="math-container">$ax^2+by^2+cz^2+2fyx+2gzx+2hxy=0$</span> has three mutually perpendicular generators, then <span class="math-container">$a+b+c=0$<... | Chandrasenachary | 731,992 | <p>Assume 3 mutually perpendicular generators as
coordinate axes
The axis of circular cone makes equal angles with the axes
if <span class="math-container">$(l,m,n )$</span> are DC's of axes then
<span class="math-container">$$l=m=n=\cos\theta=\left(\frac13\right)^\frac12$$</span>
<span class="math-container">$$... |
143,173 | <p>I have a small question that I think is very basic but I am unsure how to tackle since my background in computing inequalities is embarrassingly weak - </p>
<p>I would like to show that, for a real number <span class="math-container">$p \geq 1$</span> and complex numbers <span class="math-container">$\alpha, \beta$... | Community | -1 | <p>Without loss of generality, we can assume that that $\lvert \alpha \rvert \geq \lvert \beta \rvert$. Now essentially you want to prove that $$\displaystyle \left \lVert \frac{z + 1}{2} \right \Vert^p \leq \displaystyle \frac{\lVert z \rVert^p + 1}{2},$$ where $\displaystyle z = \frac{\alpha}{\beta}$ and $\lVert z \r... |
57,057 | <p>Let $a,b,c,d,e$ be positive real numbers which satisfy $abcde=1$. How can one prove that:
$$\frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d} +\frac{1}{e}+ \frac{33}{2(a + b + c + d+e)} \ge{\frac{{83}}{10}}\ \ ?$$</p>
| user782220 | 21,556 | <p>Replace with reciprocals so the problem becomes</p>
<p>$a + b + c + d + e + {33 \over 2}{1 \over {1 \over a} + {1 \over b} + {1 \over c} + {1 \over d} + {1 \over e}} \geq {83 \over 10}$</p>
<p>Sort the numbers so that $a\le b\le c\le d\le e$ and assume that $a,b,c,d,e$ are not all equal to $1$. Since $a<1$ and ... |
2,848,251 | <p>From: Philip Johnson-Laird <a href="https://dof.princeton.edu/about/clerk-faculty/emeritus/philip-nicholas-johnson-laird" rel="nofollow noreferrer">BA PhD Psychology (UCL)</a>, Stuart Professor of Psychology Emeritus at Princeton. (Author isn't a logician.) <a href="https://www.amazon.ca/How-We-Reason-Philip-... | Eric Wofsey | 86,856 | <p>A vs C and A vs D are impossible because A is in the game and B is not, violating [1].</p>
<p>If B is not in the game, then A cannot be in the game either without violating [1]. But then the players must be C and D, which violates [2]. So B must be in the game. All combinations with B are easily seen to be possi... |
3,295,973 | <p>Let <span class="math-container">$(\Omega,\mathcal A,\mu)$</span> be a measure space, <span class="math-container">$p,q\ge1$</span> with <span class="math-container">$p^{-1}+q^{-1}=1$</span> and <span class="math-container">$f:\Omega\to\mathbb R$</span> be <span class="math-container">$\mathcal A$</span>-measurable ... | Cornman | 439,383 | <p>Keep in mind, that 'squaring' is not an equivalent term transformation.
It manipulates the domain of your equation and can therefor create 'fake solutions'.</p>
<p>So either you have to be more careful, when squaring, or you have to check every solution if it is fake or not. (Which means that the original equation... |
3,887,526 | <p>I need to prove <span class="math-container">$\frac{|x+y+z|}{1+|x+y+z|} \le \frac{|x|}{1+|y|+|z|}+\frac{|y|}{1+|x|+|z|}+\frac{|z|}{1+|x|+|y|}$</span>. I've tried to use triangle inequality or to explore the form of <span class="math-container">$(a+b+c)^2$</span> but it won't get me anywhere. I would be grateful for ... | Calvin Lin | 54,563 | <p>We will prove a stronger statement.</p>
<p>Lemma: <span class="math-container">$f(x) = \frac{x}{1+x}$</span> is an increasing function on <span class="math-container">$ x \geq 0$</span>.<br />
This is obvious by setting <span class="math-container">$ f(x) = 1 - \frac{1}{1+x}$</span>, which is increasing on <span cla... |
351,815 | <p>Having trouble understanding this. Is there anyway to prove it?</p>
| Sijo Joseph | 72,833 | <p>Why does the function $exp(x)$ converge?</p>
<p>Since </p>
<p>$$\exp(x)=\sum_{i=0}^{\infty} \frac{x^{n}}{n!}$$ for large $n$, $x^n$ grows slowly compared with $n!$.</p>
|
351,815 | <p>Having trouble understanding this. Is there anyway to prove it?</p>
| ramo | 113,068 | <p>$n!> k^n$ if $n\ge ke$</p>
<p>try it for yourself with various combinations of $n$ and $k$</p>
<p>use $\log(n!)$ and $k \log(n)$ for large $n$</p>
|
2,028,594 | <p>Given a simple undirected connected graph. What is the minimum number of edge-disjoint paths needed to cover all edges of the graph? I have only found information about the vertex-covering one.</p>
<p>My friend suggests that such value equals <em>u/2</em>, with <em>u</em> being number of vertices with odd degree, o... | hmakholm left over Monica | 14,366 | <p>Assuming that the "paths" we're talking about are actually trails, that is, walks that may repeat vertices but may <em>not</em> repeat edges (which seems to be the most natural context for requiring them to be edge-disjoint), that sounds right to me.</p>
<p>Clearly <em>at least</em> that many paths will be needed, ... |
786,271 | <p>Suppose $a, b$ and $n$ are positive integers, all greater than one. If $a^n+b^n$
is prime, what can you relate $n$ with 2?</p>
<p>My approach: for $a^n+b^n$ to be prime $\forall n>1$, $a$ and $b$ has to be coprimes.
But how do I ascertain anything about $n?$</p>
| JEM | 47,652 | <p>Since this problem is not amenable to the standard repertoire of 1st order techniques, we might use some asymptotic analysis to understand the qualitative features of this ODE. Firstly, the derivative is always positive and bounded as $x$ or $y$ or both approach $\pm\infty$. The limiting differential equation in all... |
389,269 | <p>Let <span class="math-container">$G$</span> be a nonabelian finite simple group all of whose Sylow subgroups of odd order are cyclic.</p>
<p>If we further assume that its Sylow <span class="math-container">$2$</span>-subgroup is dihedral, then due to Suzuki, we know that <span class="math-container">$G\cong \operato... | Noam D. Elkies | 14,830 | <p>Partial answer:</p>
<p>Any group whose order is a power of <span class="math-container">$2$</span> times a squarefree number
must satisfy this condition. This condition holds for some of
Suzuki's groups <span class="math-container">${\rm Sz}(2^k)$</span> (with <span class="math-container">$k \geq 3$</span> odd),
wh... |
910,020 | <p>Show that if $G$ is a group of order $168$ that has a normal subgroup of order $4$ , then $G$ has a normal subgroup of order $28$.</p>
<p><strong>Attempt:</strong> $|G|=168=2^3.3.7$</p>
<p>Then number of sylow $7$ subgroups in $G = n_7 = 1$ or $8$.</p>
<p>Given that $H$ is a normal subgroup of order $4$ in $G$. <... | Morten Krogh | 185,971 | <p>You can show it by using the quotient group as Andrea does, but you can also directly show that $n_7=1$.</p>
<p>A 7-group acts on the normal 4-group by conjugation. This action is an
automorphism. The automorphism group of 4-groups have order 2 or 6 and hence a
7-seven group must act trivially. This means that the ... |
338,383 | <p>Multivariate polynomial indexed by <span class="math-container">${1, \ldots, n}$</span> are acted on by <span class="math-container">$S_n$</span>: for <span class="math-container">$\sigma \in S_n$</span>, define <span class="math-container">$\sigma(x_i) = x_{\sigma(x_i)}$</span>, etc. Symmetric polynomials are those... | Sam Hopkins | 25,028 | <p>If you're interested specifically in the cyclic group you may be interested in the "cyclic quasi-symmetric functions" studied recently in this paper of Adin, Gessel, Reiner, and Roichman: <a href="https://arxiv.org/abs/1811.05440" rel="nofollow noreferrer">Cyclic quasi-symmetric functions</a>. These "... |
2,819,950 | <p>I am a university graduate with a B.S. in mathematics who has been developing software for the past 8 years. I recently discussed a mutual interest in topology with a friend who is just about to complete his degree. Due to time constraints both of us missed our chance to take a topology course during our undergradua... | Adrian Keister | 30,813 | <p>I'd recommend Crossley's <a href="https://smile.amazon.com/Essential-Topology-Springer-Undergraduate-Mathematics/dp/1852337826/ref=sr_1_1?ie=UTF8&qid=1529007801&sr=8-1&keywords=crossley%20essential%20topology" rel="nofollow noreferrer"><em>Essential Topology</em></a>. It's pitched at a great level, and v... |
2,066,765 | <p>Are there any examples of subspaces of $\ell^{2}$ and $\ell^{\infty}$ which are not closed?</p>
| Community | -1 | <p>Yes. For instance, the sequences which are eventually $0$.</p>
|
981,181 | <p>I'm stuck with a logic problem like this</p>
<blockquote>
<p>I eat ice cream if I am sad.</p>
<p>I am not sad.</p>
<p>Therefore I am not eating ice cream.</p>
</blockquote>
<p>Is this conclusion logical? The first sentence can be understood both like "ice cream <span class="math-container">$\implies$</span> sad... | Sebastian Negraszus | 103,176 | <p>In logic, there is a difference between <em><a href="http://en.wikipedia.org/wiki/Logical_consequence" rel="nofollow">implication</a></em> ($a \implies b$) and <em><a href="http://en.wikipedia.org/wiki/Logical_equivalence" rel="nofollow">equivalence</a></em> ($a \iff b$). The word "if" (without "only") usually means... |
73,219 | <p>For Riemann surfaces there are at least to possible notions of hyperbolicity. The classical one given by the Uniformization Theorem, or equivalently the type problem, which essentially says that a simply connected Riemann surfaces is conformally equivalent to one of the following:</p>
<ul>
<li>Riemann Sphere $\math... | Alain Valette | 14,497 | <p>About the related question: it is a result of Babai that a (connected, locally finite) vertex-transitive, planar graph is isomorphic to the 1-skeleton of an Archimedean tiling of the sphere, or the Euclidean plane, or the hyperbolic plane. So assuming transience singles out the hyperbolic plane, and implies Gromov-h... |
180,495 | <blockquote>
<p>Suppose $T$ is an everywhere defined linear map from a Hilbert space $\mathcal{H}$ to itself. Suppose $T$ is also symmetric so that $\langle Tx,y\rangle=\langle x,Ty\rangle$ for all $x,y\in\mathcal{H}$. Prove that $T$ is a bounded directly from the uniform boundedness principle and not the closed grap... | user66081 | 66,081 | <p>Both proofs are valid for a linear operator $T : X \to X'$ on a Banach space $X$, that is symmetric in the sense that $\langle T x, y \rangle = \langle T y, x \rangle$, $x, y \in X$, where $\langle \cdot, \cdot \rangle$ is the duality pairing.</p>
|
4,278,202 | <p>Suppose, I am working on <span class="math-container">$Eq(X)$</span> = the set of all equivalence relations on <span class="math-container">$X$</span>. Let <span class="math-container">$\theta_1, \theta_2$</span> denote arbitrary equivalence relations.</p>
<p>Then how does <span class="math-container">$\theta_1 \lor... | Asinomás | 33,907 | <p>Recall that an equivalence relation over <span class="math-container">$X$</span> is a subset of <span class="math-container">$X\times X$</span>. In this case the order is containment.</p>
<p>The meet <span class="math-container">$\land$</span> is going to be the intersection as it usually works out. Note that the in... |
3,468,537 | <p>I tried to proof this but I'm not sure if it's fine or if I'm missing something. Any help or hints are appreciated.
This is what I got:</p>
<p>By hypothesis we know that <span class="math-container">$p||G|$</span> then <span class="math-container">$|G|=pm$</span>. Supposing that <span class="math-container">$p|m$</... | Alex Mathers | 227,652 | <p>You don't have to make this distinction about whether <span class="math-container">$p$</span> divides <span class="math-container">$m$</span> or not; the whole point here is that you can define</p>
<p><span class="math-container">$$H:=\{g\in G\mid\text{ord$(g)$ is a power of $p$}\}$$</span></p>
<p>and under the as... |
766,765 | <blockquote>
<p>Let $f:[0,\infty]\rightarrow \mathbb{R}$ a bounded function in each bounded interval. If $\lim_{x\rightarrow \infty} [f(x+1)-f(x)]=L$ then prove that $\lim_{x\rightarrow \infty}\displaystyle\frac{f(x)}{x}=L$</p>
</blockquote>
<p>I appreciate any hint to solve this problem. </p>
<p>Thanks a lot!</p>
| YTS | 126,222 | <p>This is another proof:</p>
<p>Given $\epsilon>0$ there exist $y>0$ such that if $x>y$ then:</p>
<p>$$L-\epsilon<f(x+1)-f(x)<L+\epsilon$$</p>
<p>we can conclude the following inequalytis valid for every $n$</p>
<p>$$L-\epsilon<f(x+2)-f(x+1)<L+\epsilon$$</p>
<p>$$L-\epsilon<f(x+2)-f(x+1)&l... |
184,241 | <p>I am having issues with the code:</p>
<pre><code>CC = p0^2/(p0 - L)
Dbuy = (1 - CC)*(L/p0) + (CC - p0)*Log[CC/p0]
Dwait = (1 - p1)*(L/(p1 - p0))
Dlock = (1 - p1)*(1 - L/p0 - L/(p1 - p0)) + (p1 - CC)*(1 - L/p0) - L*Log[(p1 - p0)/(CC - p0)]
RevLocking1 = Table[NMaximize[{x*Dbuy*p0 + x*Dwait*0.4*p1 + x*Dlock*(p0 + 0.4... | Sjoerd Smit | 43,522 | <p>A function that is of great use here, is <code>AssociationTranspose</code> in the <code>GeneralUtilities</code> package. It does the same thing as regular transpose, but works with <code>List</code> and <code>Associations</code> alike (and mixtures of the two):</p>
<pre><code>listData = {
<|"Input" -> 1, "d... |
64,491 | <p>I need to change the ticks in the xaxis for representation pH values, but the ticks are not dispersed along the range of xaxis. Another question: why the vertical line defined with Epilog is not visible?</p>
<pre><code>Kb1 := Kw/Ka1
Kb2 := Kw/Ka2
oh := Kw/x
f[x_] := Cs Kb2 oh/(Kb2 oh + oh^2 + Kb1 Kb2)
ff[x_] = Simp... | Junho Lee | 16,245 | <p><strong>Version 9</strong></p>
<p>I made <code>findD</code> for that.</p>
<pre><code>findD[{x1_, x2_}, n_] := FindDivisions[-Log[10, #] & /@ {x1, x2}, n]
myTicks[xmin_, xmax_] := {10^-#, #} & /@ findD[{xmin, xmax}, 10]
</code></pre>
<p>Have try this code.</p>
<pre><code>Kb1 := Kw/Ka1; Kb2 := Kw/Ka2; oh :... |
3,202,530 | <p>This seems easy. But it isn't. The diameter is given as <span class="math-container">$16$</span> and it asks you to find the coordinates of point <span class="math-container">$9$</span>. It's tempting to say that it is <span class="math-container">$(4, 4\sqrt{3})$</span>, but that isn't the answer. What the heck am ... | Dylan | 135,643 | <p>Let <span class="math-container">$u = \dot y$</span>. Then <span class="math-container">$\ddot y = \dfrac{du}{dt} = \dfrac{du}{dy}\dfrac{dy}{dt} = u \dfrac{du}{dy}$</span></p>
<p>So you have</p>
<p><span class="math-container">$$ u \frac{du}{dy} = \frac{C}{y} $$</span></p>
<p>Integrating both sides gives</p>
<p>... |
3,202,530 | <p>This seems easy. But it isn't. The diameter is given as <span class="math-container">$16$</span> and it asks you to find the coordinates of point <span class="math-container">$9$</span>. It's tempting to say that it is <span class="math-container">$(4, 4\sqrt{3})$</span>, but that isn't the answer. What the heck am ... | Jean Marie | 305,862 | <p>In the case of differential equation <span class="math-container">$y(t)y''(t)=\alpha t$</span> you have asked in your recent question <a href="https://math.stackexchange.com/q/3239474">Anyone know how to solve this differential equation $yy''=\alpha t$</a>, I am afraid that there are no closed form formula.<... |
987,962 | <p>Find an $x$ in $\Bbb R$ for which rank of the matrix $$A=\begin{bmatrix}1 & 1&1&1 \\1 & -1&-1&1\\1 &-3 &-3 &x \end{bmatrix}$$
is as minimal/maximal as possible.</p>
<p>I was thinking of row reducing the matrix and then count the pivot.</p>
| Martingalo | 127,445 | <p>Let us compute $F(x) = \int_{-\infty}^x f(y)dy$ the distribution function:
If $x<0$ then clearly $F(x)=0$.</p>
<p>If $0<x<1$ then
$$F(x) = \int_{-\infty}^x f(y)dy = \int_{0}^x f(y)dy = \int_0^x ydy = \frac{x^2}{2}.$$</p>
<p>If $1<x<2$ then
$$F(x) = \int_{-\infty}^{x} f(y) dy = \int_0^1 f(y)dy+ \int_... |
53,596 | <p>It's "well known" that, for any weight $k$ and level $N$, the space $S_k(\Gamma_1(N))$ of cusp forms of that weight and level has a basis in which all the Hecke operators act by matrices with entries in $\mathbb{Z}$; consequently all the Hecke eigenvalues are algebraic numbers (indeed algebraic integers).</p>
<p>I ... | Marc Palm | 10,400 | <p>Here is another proof: Deligne and Serre have proven that the corresponding L-function equal the Artin L-function of a Galois representation. Deligne had proven similar facts known for weight $k \geq 2$ modular forms before that, and their proof essentially relies on the former results. This implies algebraicity and... |
3,794,148 | <p><span class="math-container">$n$</span> is fixed.
<br />
<span class="math-container">$ x \in ]0,1[$</span>
<br />
Coud you help me prove, without theorem of dominated convergence :
<span class="math-container">$$ \int_{1}^{n} \frac{x^t}{t} dt \underset{x \to 1^{-}}{\rightarrow} \int_{1}^{n} \frac{1}{t} dt$$</span><... | RRL | 148,510 | <p>Hint:</p>
<p><span class="math-container">$$\left|\int_1^ n \frac{x^t}{t} \, dt - \int_1^ n \frac{1}{t} \, dt\right| \leqslant \int_1^n\left| \frac{x^t}{t} - \frac{1}{t}\right| \, dt,$$</span></p>
<p>and <span class="math-container">$(x,t) \mapsto x^t/t $</span> is uniformly continuous on <span class="math-contai... |
3,794,148 | <p><span class="math-container">$n$</span> is fixed.
<br />
<span class="math-container">$ x \in ]0,1[$</span>
<br />
Coud you help me prove, without theorem of dominated convergence :
<span class="math-container">$$ \int_{1}^{n} \frac{x^t}{t} dt \underset{x \to 1^{-}}{\rightarrow} \int_{1}^{n} \frac{1}{t} dt$$</span><... | zhw. | 228,045 | <p>By the MVT, for any fixed <span class="math-container">$t\ge 1$</span></p>
<p><span class="math-container">$$\tag 1 x^t-1 = x^t-1^t = tc_x^{t-1}(x-1)$$</span></p>
<p>for some <span class="math-container">$c_x\in (x,1).$</span> The absolute value of the last expression is therefore bounded above by <span class="math-... |
1,341,231 | <p>Let $f$ be continuous on $[0,1]$, and let $\alpha>0$. Find: $\lim\limits_{x\to 0}{x^{\alpha}\int_{x}^{1}{f(t)\over t^{\alpha +1}}dt}$. I tried integration by parts, but I am not sure if $f$ is integrable and to what extent. It also gets really messy. Besides, I am not sure if I am to express the limit using $f$, ... | vasmous | 249,266 | <p>I will do exactly the same thing. I just finished my degree in mathematics but in our department there is not a single course of Number Theory, and since I will start my graduate courses in October I thought it will be a great idea to study Number Theory on my own. So, I asked one of my professors, who is interested... |
4,308,155 | <p>Given an operator <span class="math-container">$T \in B(H)$</span>, denote its spectrum by <span class="math-container">$\sigma(T)$</span>. We write <span class="math-container">$\Omega := \sigma(T)\setminus \{0\}$</span>. The following result is well-known and <em><strong>can be assumed without proof</strong></em>:... | Martin Argerami | 22,857 | <p>That's precisely where you get to use that your <span class="math-container">$T$</span> is compact.</p>
<p>Fix <span class="math-container">$\delta>0$</span>, and consider <span class="math-container">$\Omega_\delta=\{\lambda\in\Omega:\ \lambda>\delta\}$</span>. Suppose <span class="math-container">$\Omega_\de... |
2,682,008 | <p>I’ve been studying general form Ricatti Differential Equations recently, and I’m confused as to why a general form solution is not possible. What would it mean if a general form solution were possible? </p>
| Fred | 380,717 | <p>If a special solution $y_1$ of the Ricatti Differential Equation is known, then the general solution is obtained as $y=y_1+u$, where $u$ satisfies a Bernoulli equation.</p>
<p>If a special solution of the Ricatti Differential Equation is not known, then, in general, the general form of the solutions is not possible... |
43,690 | <p>I have to apologize because this is not the normal sort of question for this site, but there have been times in the past where MO was remarkably helpful and kind to undergrads with similar types of question and since it is worrying me increasingly as of late I feel that I must ask it.</p>
<p>My question is: what ca... | ohai | 9,501 | <p>I suspect that most interesting mathematical results raise more questions than they settle, so in this case you would not have anything to worry about.</p>
|
43,690 | <p>I have to apologize because this is not the normal sort of question for this site, but there have been times in the past where MO was remarkably helpful and kind to undergrads with similar types of question and since it is worrying me increasingly as of late I feel that I must ask it.</p>
<p>My question is: what ca... | Gerry Myerson | 3,684 | <p>If you are an undergraduate, you don't yet know what you can contribute to mathematics, and in particular you don't yet know whether, or at what level, you can create original mathematics. Fortunately, you don't have to work this out entirely for yourself; if you go on to graduate work in mathematics, you will have ... |
43,690 | <p>I have to apologize because this is not the normal sort of question for this site, but there have been times in the past where MO was remarkably helpful and kind to undergrads with similar types of question and since it is worrying me increasingly as of late I feel that I must ask it.</p>
<p>My question is: what ca... | Alok Gandhi | 10,429 | <p>Muad you raised not only a very important question but also a fairly enigmatic one. I completed my Maters in Mathematics 13 years ago and left mathematics then and there, what I realized during these years was that I could not sleep without thinking something new to discover or invent in pure mathematics. The number... |
43,690 | <p>I have to apologize because this is not the normal sort of question for this site, but there have been times in the past where MO was remarkably helpful and kind to undergrads with similar types of question and since it is worrying me increasingly as of late I feel that I must ask it.</p>
<p>My question is: what ca... | Bill Thurston | 9,062 | <p>It's not <em>mathematics</em> that you need to contribute to. It's deeper than that: how might you contribute to humanity, and even deeper, to the well-being of the world, by pursuing mathematics? Such a question is not possible to answer in a purely intellectual way, because the effects of our actions go far beyo... |
275,517 | <p>Some mathematical concepts are ended with "-oid", such as Matroid, greedoid, groupoid. What does that mean? Do these concepts share something in common? Thanks!</p>
| Tom Au | 12,506 | <p>If you have a "center" of an object, it means "equidistant" from the edges (spacially).</p>
<p>If you have a <a href="http://en.wikipedia.org/wiki/Centroid" rel="nofollow"><strong>centroid</strong></a>, it means the <strong>weighted</strong> "center" of an object, where the weightS are something like the density of... |
328,868 | <p>It is easily shown that the function $$\begin{cases} \exp \left(\frac{1}{x^2-1} \right) & |x| < 1 \\ 0 & \text{otherwise} \\ \end{cases}$$
is smooth and has compact support in $\mathbb R$. I tried playing with it to find a function with the following properties:</p>
<p>a. $f(x)=0$ for $x \le 0$</p>
<p>b... | muzzlator | 60,855 | <p>Let $$h(x) = \left\{\begin{array}{c} 0 & x \leq 0 \\ e^{-1/x} & x > 0 \end{array} \right. $$</p>
<p>Then consider $$ g(x) = \frac{h(x)}{ h(x) + h(1-x)} $$</p>
<p>It is smooth because it is the ratio of smooth functions with the denominator never $0$.</p>
<p>To verify it is increasing, we can calculat... |
3,380,479 | <p>i'm doing a seminar about Galois Theory but I have a problem with the definition of purely inseparable element and the one of purely inseparable extension, I read the definitions but i would like to see an example for each one of them. If somebody can help me please. </p>
| user | 505,767 | <p>We need to specify for what number we need to calculate the "least residue of mod 13".</p>
<p>For example</p>
<p><span class="math-container">$$20 \equiv 7 \mod 13$$</span></p>
<p>then to find the <a href="https://en.wikipedia.org/wiki/Modular_multiplicative_inverse" rel="nofollow noreferrer"><strong>inverse</str... |
37,977 | <p>I have two lists, say <code>a</code> and <code>b</code>, both of length <code>n</code>. I'd like to compute the following:</p>
<ul>
<li>minimum of $a[i]/b[i]$ where $i=1, 2, ...n$ and $b[i]>0$</li>
</ul>
<p>I'd also like to know the index of the element where the min occurs.</p>
| Gabriel | 744 | <p>I would do this by first combining the lists using Transpose, and then filtering the division pairs using either a pattern with <code>Cases</code> or a predicate using <code>Select</code></p>
<pre><code>Cases[Transpose[{a, b}], {_, _?Positive}]
</code></pre>
<p>or</p>
<pre><code>Select[Transpose[{a, b}], #[[2]] &... |
37,977 | <p>I have two lists, say <code>a</code> and <code>b</code>, both of length <code>n</code>. I'd like to compute the following:</p>
<ul>
<li>minimum of $a[i]/b[i]$ where $i=1, 2, ...n$ and $b[i]>0$</li>
</ul>
<p>I'd also like to know the index of the element where the min occurs.</p>
| Ray Koopman | 8,159 | <p>If the minimum can be assumed to be unique then use <code>f1</code>, which returns the position of only the first occurrence of the minimum value. Otherwise use <code>f2</code>, which is slower but returns the positions of all occurrences.</p>
<pre><code>f1[a_,b_] := With[{i = Ordering[#1,1][[1]]}, {#1[[i]],#2[[i]]... |
3,310,599 | <p>Let <span class="math-container">$Y$</span> be a random variable on <span class="math-container">$[0,1]$</span> and let <span class="math-container">$X\sim U[0,1]$</span> be a uniformly distributed random variable that is independent of <span class="math-container">$Y$</span>.</p>
<p>Prove that <span class="math-co... | IljaKlebanov | 692,074 | <p>Start with conditioning on the event <span class="math-container">$Y=y$</span>, which is simpler to think through:
<span class="math-container">$$
P(X<Y|Y=y) = P(X<y|Y=y) \stackrel{\text{indep.}}{=}P(X<y)
= \int_{0}^{y}1\, \mathrm dx
=y.
$$</span>
Conditioning an event <span class="math-container">$A$</span... |
2,530,221 | <p>Let $X$ be a topological space, and equip its homeomorphism group $\text{Homeo}(X)$ with the compact-open topology. I probably need to restrict $X$ to be locally compact for it to play off with $\text{Homeo}(X)$ nicely.</p>
<p>The usual definition of $X$ being <b>homogeneous</b> is that given $x,y \in X$ there exis... | Moishe Kohan | 84,907 | <p>You are asking the following: Pick $x\in X$ and for $G=Homeo(X)$ consider the orbit map $o_x: G\to X$, $o_x(g)=g(x)$. Assuming that $o_x$ is surjective, when does it have a (continuous) section $s: X\to G$?</p>
<p>A general observation is that if $X$ is not contractible, then no section $s: X\to G$ is homotopic to... |
1,123,647 | <p>I am looking into starting an amateur project for online tutoring. I need a math/geometry program that I can use to create shapes, graph functions, and create animation. I know of many types of software (such as geogebra) but they have restrictions for commercial use. In other words, I can't use the material or feat... | Mark Fischler | 150,362 | <p>1) Contact the geogebra support people; there may be some provision to pay a fee for the right to distribute-for-profit screenshots or animations produced. </p>
<p>2) See if Mathematica meets your needs -- but this is not a cheap solution, the commercial cost is about $2500. However, Mathematica is a truly professi... |
3,115,566 | <p>Convergence of the series for <span class="math-container">$a \in \mathbb R$</span> <span class="math-container">$$\sum_{n=1}^\infty\sin\left(\pi \sqrt{n^2+a^2} \right)$$</span> </p>
<p>I saw this problem in a calculus book and it gave a hint that says </p>
<p><strong>HINT</strong> First show that <span class="mat... | robjohn | 13,854 | <p><strong>Hint:</strong>
<span class="math-container">$$
\begin{align}
\sum_{n=1}^\infty\sin\left(\pi\sqrt{n^2+a^2}\right)
&=\sum_{n=1}^\infty(-1)^n\sin\left(\pi\sqrt{n^2+a^2}-\pi n\right)\\
&=\sum_{n=1}^\infty(-1)^n\sin\left(\frac{\pi a^2}{\sqrt{n^2+a^2}+n}\right)
\end{align}
$$</span>
where <span class="math... |
3,528,162 | <p>I'm encountering the following interesting math problem:</p>
<p>A device consists of 5 independently working blocks. </p>
<p>Each of them has a damage probability of 1/4. Upon damage of 1, 2 or 3 blocks, the probabilities for shutting the device down are respectively 1/5 , 2/5, 4/5. </p>
<p>If more blocks are da... | jvdhooft | 437,988 | <p>Since there are <span class="math-container">$n = 5$</span> blocks, and each block has a probability <span class="math-container">$p =\frac{1}{4}$</span> of being damaged, the expected number of damaged blocks <span class="math-container">$d$</span> equals:</p>
<p><span class="math-container">$$d = n \cdot p = 5 \c... |
3,250,593 | <p>Revising the modal logic principles, I have encountered an negative introspection axiom:
<span class="math-container">$$ \vDash \neg \square \alpha \longrightarrow \square \neg \square \alpha $$</span>
with additional information, that it is a direct effect from Euclidean property of accessibility relation <span cla... | mattrym | 679,574 | <p>By the time, I have found a possible solution, I would appreciate some comments:</p>
<p>Let's get any possible world <span class="math-container">$w \in \mathbb{W}$</span>, such as <span class="math-container">$\kappa, w \vDash \neg \square \alpha $</span>. Then,from a definition of <span class="math-container">$\s... |
2,326,072 | <p>We define 3 sequences $(a_n),(b_n),(c_n)$ with positive terms so that
$$ a_{n+1}\leq\frac{b_n+c_n}{3}\ ,\ b_{n+1}\leq\dfrac{a_n+c_n}{3}\ ,\ c_{n+1}\leq\dfrac{a_n+b_n}{3} $$
Check if any of $(a_n),(b_n),(c_n)$ converge, and if they do find their limit.</p>
<p>PROOF</p>
<p>My part of the proof is this: By adding the... | Andre | 303,581 | <p>Yes because you have the "sandwiched" inequality $0 < a_{n} < x_{n} \rightarrow 0$, so $a_{n} \rightarrow 0$.</p>
<p>EDIT: Of course, you have to make sure that $x_n \rightarrow 0$ in the first place, but you have that it is bounded by a geometric sequence which converges to zero which implies the stated conv... |
1,012,652 | <p>I have a problem with the following question.</p>
<p>For which $n$ does the following equation have solutions in complex numbers</p>
<p>$$|z-(1+i)^n|=z $$</p>
<p>Progress so far.</p>
<ol>
<li><p>Let $z=a+bi$.</p></li>
<li><p>Since modulus represents a distance, the imaginary part of RHS has to be 0. This immedia... | Matt Samuel | 187,867 | <p>Note that $1+i=\sqrt{2}(\cos(\pi/4)+i\sin(\pi/4))$. We have </p>
<p>$|z-(1+i)^n|=|z-(\sqrt{2})^n(\cos(n\pi/4)+i\sin(n\pi/4))|=z$</p>
<p>$\sqrt{(z-(\sqrt{2})^n\cos(n\pi/4))^2+(\sqrt{2})^{2n}\sin^2(n\pi/4)}=z$</p>
<p>so</p>
<p>$(z-2^{n/2}\cos(n\pi/4))^2+2^n\sin^2(n\pi/4)=z^2$,</p>
<p>$-2^{n/2+1}z\cos(n\pi/4)+2^n\... |
1,012,652 | <p>I have a problem with the following question.</p>
<p>For which $n$ does the following equation have solutions in complex numbers</p>
<p>$$|z-(1+i)^n|=z $$</p>
<p>Progress so far.</p>
<ol>
<li><p>Let $z=a+bi$.</p></li>
<li><p>Since modulus represents a distance, the imaginary part of RHS has to be 0. This immedia... | Lucian | 93,448 | <ul>
<li>Obviously, <em>z</em> is a point on the positive semi-axis, since absolute values are always positive reals.</li>
<li>$|z_1-z_2|$ represents the distance between two points on the complex plane.</li>
<li>Therefore, <em>z</em> is equally distanced from the origin <em>O</em>, and from $A_n=(1+i)^n$. This implies... |
3,340,374 | <p>I'm writing up several proofs for myself, all of which have a particular sticking point.</p>
<p>Essentially, I want to prove that for a function <span class="math-container">$f$</span> of two real variables, we have</p>
<p><span class="math-container">$\lim_{h \to 0} \frac{f(x \;+\; h, \; y \;+\; h) \; - \; f(x, \... | Allawonder | 145,126 | <p>There's no way anything could be messed up since <em>the second coordinate is always fixed.</em> It's only the first that's varying. That is, we're considering the change from the point <span class="math-container">$(x,y+h)$</span> to the point <span class="math-container">$(x+h,y+h).$</span> Throughout this change,... |
2,508,103 | <p>Could someone, please, define a cycle in a $3-$Uniform Hypergraph? I looked at different resources, but they have different definitions and I am so confused!!</p>
| Donald Splutterwit | 404,247 | <p>An example of a ("pentagon") $5-$cycle in a $3-$ uniform hypergraph.</p>
<p><a href="https://i.stack.imgur.com/lCAAF.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/lCAAF.png" alt="enter image description here"></a></p>
<p>The definition of a cycle is very similar to that for a graph it is a seq... |
2,508,103 | <p>Could someone, please, define a cycle in a $3-$Uniform Hypergraph? I looked at different resources, but they have different definitions and I am so confused!!</p>
| Misha Lavrov | 383,078 | <p>The problem is that there <em>are</em> different notions of what cycles are in hypergraphs. You always have to specify which you mean.</p>
<p>In a $3$-uniform hypergraph:</p>
<ul>
<li>A <em>loose</em> cycle consists of edges $\{v_1,v_2,v_3\}, \{v_3,v_4,v_5\}, \dots, \{v_{2k-1}, v_{2k}, v_1\}$. From each edge of th... |
3,402,800 | <p>Let,<span class="math-container">$E$</span> be a zero dimensional sheaf on an algebraic surface <span class="math-container">$S$</span> over the field of complex numbers,i.e <span class="math-container">$dim(Supp(E)) =0$</span></p>
<p>Then my question is the following : Is it true that <span class="math-container">... | Slup | 348,401 | <p>This can be also viewed as a consequence of a general fact due to Grothendieck (which can be proved by noetherian induction).</p>
<p><em>Let <span class="math-container">$X$</span> be a noetherian topological space of dimension <span class="math-container">$n$</span>. Then for all <span class="math-container">$i>... |
893,259 | <p>I have tried this problem multiple times, I have the solution but not the steps. I keep getting the wrong answer. I believe it may be in the algebra after I have taken the Laplace on both sides.</p>
<p>$y''+y = \sin(t) ;\:\: y(0) = 1, \:\: y'(0) = -1 $</p>
| Dmoreno | 121,008 | <p>Follow the next steps I describe for you below:</p>
<ul>
<li>Take Laplace transform on both sides of the ODE to have:</li>
</ul>
<p>$$ s^2 Y(s) - s y_0 - y'_0 + Y(s) = \frac{1}{s^2+1}, \quad Y(s) = \mathcal{L}_s [y(t)]$$</p>
<ul>
<li>Substitute data and solve for $Y(s)$:</li>
</ul>
<p>$$Y(s) = \frac{1}{s^2+1} \... |
3,990,717 | <p>I'm currently a self-learner on Abstract Algebra and Group theory. The book that I'm currently reading is <em>A First Course In Abstract Algebra</em> by <em>John B. Fraleigh</em>. So I know this question might truly be a basic question which every student would be taught in class and there is confusion about it. How... | Community | -1 | <p>Conjugation by any element of the group is an automorphism, and thus preserves order. All that is left is to note that <span class="math-container">$ab$</span> and <span class="math-container">$ba$</span> are conjugates: <span class="math-container">$b(ab)b^{-1}=ba$</span>.</p>
|
137,253 | <p>We have the following: $A,B,C$ are sets. </p>
<p>$C = \{ab: a \in A, b \in B\}$. </p>
<p>What is the relationship between $\sup(C),\sup(A)$, and $\sup(B)$?. </p>
<p>Is it: $$\sup(C) \le \sup(A) \sup(B)\;,$$ and why?.</p>
| Yuval Filmus | 1,277 | <p>Suppose for simplicity that $A$ and $B$ are finite. For all $a \in A$, we have $a \leq \max A$. For all $b \in B$, we have $b \leq \max B$. What can you say about $ab$ (assuming $A,B\subseteq \mathbb{R}_{\geq 0}$)? What can you deduce about $\max C$?</p>
<p>You can also think about the extreme case $A = \{a\}$, $B ... |
546,083 | <p>Thanks to internet, I found and understand how to solve diophantine $x^2 - Dy^2 = 1$. Now I would like to solve the following diophantine equation :
$$x^2 - 2y^2 = x - 2y$$
but I don't know how to do it, even if I could read articles explaining how to solve the diophantine $x^2 - Dy^2 = c$.</p>
| Shaun | 104,041 | <p>That's not the only answer in fact.</p>
<p>For any $\alpha\in\mathbb{C}$, we define, for any complex $z=a e^{i\theta }\neq 0$, the multifunction
$$[z^{\alpha }]:=\{e^{\alpha (\log |z|+i\theta )} : \theta\in\{\arg (z)+2k\pi i: k\in\mathbb{Z}\}\},$$
taking $z^{\alpha }$ as $w=be^{i\phi }\in [z^{\alpha }]$ such that $... |
2,115,451 | <p>$$\lim_{n\to \infty}\frac{\frac{1}{2\ln2}+\frac{1}{3\ln3}+\ldots+\frac{1}{n\,\ln\,n}}{\ln(\ln\,n)}$$</p>
<p>The result is $1$ (according to the book, though it does not show the steps, which I'm interested in).</p>
<p>I've applied the theorem and it led me to an equally unhelpful limit.</p>
| Rene Schipperus | 149,912 | <p>If you apply Stolz you get</p>
<p>$$\frac{1}{n\ln n(\ln \ln n-\ln\ln(n-1))}$$ Now take $e$ to the power of the denominator you get
$$\left(\frac{\ln n}{\ln(n-1)}\right)^{n\ln n}$$
Write this as
$$\left(1+\frac{a_n}{n\ln n}\right)^{n\ln n}$$</p>
<p>Where $$a_n=n\ln n \frac{\ln (1+\frac{1}{n-1})}{\ln (n-1)}$$ its e... |
134,424 | <p>I would like to create a function where I can define which case I want to use to create a path.</p>
<pre><code>p1 = {40, 48}; p2 = {50, 116}; p3 = {63, 160};
listPurple = Symbol["p" <> ToString[#]] & /@ Range[3];
disksPurple = {Purple, Disk[#, 2] & /@ listPurple};
Graphics[{disksPurple}, ImageSize -&g... | glS | 27,539 | <p>Here is an attempt at automatically finding the best paths to use to join the points.
This avoids having to explicitly give the horizontal and vertical specification for every point.</p>
<pre><code>ClearAll[generateDirections, hvPath]
generateDirections[pts_, initialDirection_] :=
Module[{d = initialDirection /. ... |
134,424 | <p>I would like to create a function where I can define which case I want to use to create a path.</p>
<pre><code>p1 = {40, 48}; p2 = {50, 116}; p3 = {63, 160};
listPurple = Symbol["p" <> ToString[#]] & /@ Range[3];
disksPurple = {Purple, Disk[#, 2] & /@ listPurple};
Graphics[{disksPurple}, ImageSize -&g... | Simon Woods | 862 | <p>You can simply provide two definitions for <code>f</code></p>
<pre><code>f[p1_, p2_, lVertical] := {p1, {p1[[1]], p2[[2]]}}
f[p1_, p2_, lHorizontal] := {p1, {p2[[1]], p1[[2]]}}
</code></pre>
|
14,858 | <p>Let $V$ be a complex vector space of dimension 6 and let $G\subset {\mathbb P}^{14}\simeq {\mathbb P}(\Lambda^2V)$ be the image of the Plucker embedding of the Grassmannian $Gr(2, V)$.</p>
<ol>
<li>Why the degree of $G$ is 14? or in general, how to calculate the degree of a Plucker embedding?</li>
</ol>
<p>Let ${\... | Andrea Ferretti | 828 | <p>Guess you are reading Beauville-Donagi :-)</p>
<p>To compute the degree of the Grassmannian you can note that the hyperplane divisor under the Plucker embedding has class $\sigma_1$, so the degree of the Grassmannian is just $\sigma_1^8$, which you can compute by Schubert calculus. See the paragraph about Grassmann... |
807,795 | <p>Task is to find an eigenvector of the following linear operator:<br>
$f \to \int^{x}_{-x} f(t)dt$ in the linear span $\langle cos(x), sin(x), ...,cos(mx),sin(mx)\rangle$.</p>
<p>I know how to find eigenvectors and have the general idea about what linear span and linear operators are, but still I feel like I'm a bi... | Just_a_fool | 114,428 | <p>Note: </p>
<p>$\sum_{k=1}^n\sqrt{k^2+1}$ > $a_k=\sum_{k=1}^n\sqrt{k^2}$</p>
<p>for any positive n</p>
<p>As the definition of the RHS series (lets call it $c_n$) suggests, it is equal to the sum of the positive integers up to n.
This sum is given by the formula:</p>
<p>$$c_n=(n/2)(n+1)$$</p>
<p>Plugging this in... |
807,795 | <p>Task is to find an eigenvector of the following linear operator:<br>
$f \to \int^{x}_{-x} f(t)dt$ in the linear span $\langle cos(x), sin(x), ...,cos(mx),sin(mx)\rangle$.</p>
<p>I know how to find eigenvectors and have the general idea about what linear span and linear operators are, but still I feel like I'm a bi... | Mick A | 153,109 | <p>Try to show $b_{n+1} - b_n < 0$:</p>
<p>$b_{n+1} - b_n = \left[\dfrac{2a_{n+1}}{n+1} - (n+1)\right] - \left[\dfrac{2a_n}n - n\right]$</p>
<p>$ = \dfrac1{n(n+1)}\left[2n \sum_{k=1}^{n+1}\sqrt{k^2+1} - 2(n+1) \sum_{k=1}^n\sqrt{k^2+1} - n(n+1)\right]$</p>
<p>$ = \dfrac1{n(n+1)} \left[2n \sqrt{(n+1)^2 + 1} - 2 \s... |
3,213,253 | <p>We're given dynamical system:</p>
<p><span class="math-container">$$
\dot x = -x + y + x (x^2 + y^2)\\
\dot y = -y -2x + y (x^2 + y^2)
$$</span></p>
<p>Question is what's the largest constant <span class="math-container">$r_0$</span> s.t. circle <span class="math-container">$x^2+y^2 < r_0^2$</span> lies in the ... | AVK | 362,247 | <p>(This is only a "numerical" answer to the question)</p>
<p>A basin of attraction can be found numerically using reverse-time (or backward) integration. If we choose some initial point and integrate backwards for a sufficient time, we can obtain the set of the system states that it had before it reached the selected... |
2,615,427 | <p>I saw the following question in my linear algbra book, and found it rather strange:</p>
<p>"Let $V$ be a real vector space of dimension $n$. Let $L,K \colon V \rightarrow \mathbb{R}$ be linear transformations, so that $\ker(L) \subset \ker(K)$.
Prove that $K=\lambda L$ for a $\lambda \in \mathbb{R}$, using the foll... | The Phenotype | 514,183 | <p>Assume $0\leq a<b$.</p>
<p>Suppose $a^2\geq b^2$, then $a^2\geq b^2=b\cdot b>ab>a\cdot a=a^2$, which is a contradiction in itself ($a^2<a^2$ is not possible), so $a^2<b^2$.</p>
<p>Suppose $\sqrt{a}\geq \sqrt{b}$, then $a=\sqrt{a}\cdot \sqrt{a}\geq \sqrt{b}\cdot\sqrt{a}\geq\sqrt{b}\cdot \sqrt{b}=b$, ... |
301,905 | <p>I'm having troubles to solve this integration: $\int_0^1 \frac{x^{2012}}{1+e^x}dx$</p>
<p>I've tried a lot using so many techniques without success. I found $\int_{-1}^1 \frac{x^{2012}}{1+e^x}dx=1/2013$, but I couldn't solve from 0 to 1.</p>
<p>Thanks a lot.</p>
| Community | -1 | <p>You have $$\int_{-1}^{1} \frac{x^{2012}}{1+e^{x}} \ dx =\underbrace{\int_{-1}^{0}\frac{x^{2012}}{1+e^{x}}}_{I_{1}} \ dx + \int_{0}^{1}\frac{x^{2012}}{1+e^{x}} \ dx \qquad \cdots (1)$$</p>
<p>In $I_{1}$ put $x=-t$, then you have $dx = -dt$, and so the limits range from $t=0$ to $t=1$. So you have $$I_{1}= -\int_{1}^... |
46,505 | <p>What do I do if I have to solve the usual quadratic equation $X^2+bX+c=0$ where $b,c$ are in a field of characteristic 2? As pointed in the comments, it can be reduced to $X^2+X+c=0$ with $c\neq 0$.</p>
<p>Usual completion of square breaks. For a finite field there is <a href="http://www.ams.org/mathscinet-getitem?... | Tim Dokchitser | 3,132 | <p>I think this solves $X^2+X+c=0$ over $F((t))$:</p>
<p>I want to assume that $c\in F[[t]]$. If not, say $c=at^{-m}+...$, then the quadratic has no solutions when $m$ is odd or $a$ is not a square, and otherwise the substitution $X\mapsto X+\sqrt{a}t^{-m/2}$ gives a new equation with smaller $m$. So, after finitely m... |
1,075,983 | <p>Consider the sum $$S = \sum_{k=1}^n k$$</p>
<p>As I was computing the first triangle number with over 500 divisors (Project Euler), I came across the hypothesis that most triangle numbers have an even number of divisors (if $n=8$, then $S = 36$ has $9$ divisors).</p>
<p>Example:</p>
<pre><code>The number 3 has 2 ... | Mark Bennet | 2,906 | <p>Here is a way of making substantial progress.</p>
<p>Divisors come in pairs except when a number is square, so you are looking for triangle numbers which are also squares i.e. $$\frac {n(n+1)}2=m^2$$</p>
<p>Next multiply both sides by $8$ and add $1$ to obtain $$4n^2+4n+1=8m^2+1$$</p>
<p>If you then put $p=2n+1, ... |
1,075,983 | <p>Consider the sum $$S = \sum_{k=1}^n k$$</p>
<p>As I was computing the first triangle number with over 500 divisors (Project Euler), I came across the hypothesis that most triangle numbers have an even number of divisors (if $n=8$, then $S = 36$ has $9$ divisors).</p>
<p>Example:</p>
<pre><code>The number 3 has 2 ... | Travis Willse | 155,629 | <p>First, note that for any positive integer $S$, its factors come in pairs $(n, \frac{S}{n})$; if $n$ and $\frac{S}{n}$ are the same, then $S = n^2$, that is, $S$ is a square number and $n$ is its square root. So, a positive integer has an odd number of factors iff it is square, and hence we may rephrase the question ... |
2,520,669 | <p>How are these equal? </p>
<blockquote>
<p>$$|\sqrt{x} - \sqrt{c}| = \frac{|x-c|}{|\sqrt{x} + \sqrt{c}|},$$ </p>
</blockquote>
| symplectomorphic | 23,611 | <p>Hint:</p>
<p>You're supposed to spot derivatives here. Write your quotient as </p>
<p>$$\frac{\cos(x+h)-\cos x}{h}\cdot\frac{h}{(x+h)^{1/2}-x^{1/2}}$$</p>
|
2,520,669 | <p>How are these equal? </p>
<blockquote>
<p>$$|\sqrt{x} - \sqrt{c}| = \frac{|x-c|}{|\sqrt{x} + \sqrt{c}|},$$ </p>
</blockquote>
| Michael Hardy | 11,667 | <p>\begin{align}
& \lim_{h\to0} \frac{\cos(x+h) - \cos(x)}{(x+h)^{1/2} - x^{1/2}} = \frac{\lim_{h\to0} \Big(\cos(x+h) - \cos x\Big) / h}{\lim_{h\to0} \Big( (x+h)^{1/2} - x^{1/2} \Big) / h} = \frac{\dfrac d {dx} \cos x }{\dfrac d {dx} x^{1/2}} = \cdots\cdots
\end{align}</p>
|
1,750,998 | <p>It is given that a and b belong to non negative real numbers and $a^3+b^3=2$. Then prove that $3(a^4+b^4)+2a^4b^4\leq8$</p>
| Macavity | 58,320 | <p>By some quick AM-GM we have the following:</p>
<ol>
<li>$3=a^3+b^3+1 \ge 3ab \implies ab \le 1$. </li>
<li>Also $a^3+1+1 \ge 3a \implies 3a \le 4-b^3$ and similarly $3b \le 4-a^3$. </li>
</ol>
<p>Now we have all the ingredients to do:
$$LHS = 3a (a^3)+3b (b^3)+2ab(ab)^3 \le (4-b^3)a^3+(4-a^3)b^3+ 2\cdot 1(ab)^3 ... |
1,750,998 | <p>It is given that a and b belong to non negative real numbers and $a^3+b^3=2$. Then prove that $3(a^4+b^4)+2a^4b^4\leq8$</p>
| Jean Marie | 305,862 | <p>(This is not a new solution but a comment with a graphical display)</p>
<p>It might be of interest to have an idea of the tightness of the condition. This can be put in evidence by displaying the graphical representation of </p>
<p>$$y=3((2 - b^3)^{4/3} + b^4) + 2(2 - b^3)^{4/3} b^4 - 8$$</p>
<p>obtained by plugg... |
2,939,808 | <p><img src="https://i.stack.imgur.com/5DeV4.jpg" alt="enter image description here"></p>
<p>q 23</p>
<p>my solution<img src="https://i.stack.imgur.com/urLMH.jpg" alt="enter image description here"></p>
<p>I attempted like above.Then to check i plotted graph
on desmos. so it showed me three roots.One before zero al... | Avinash N | 253,506 | <p>Number of roots=3.
<a href="https://i.stack.imgur.com/gNP8F.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/gNP8F.jpg" alt="enter image description here"></a></p>
<p><a href="https://i.stack.imgur.com/zRB3c.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/zRB3c.jpg" alt="enter i... |
427,596 | <p>Just a quick question. I'm trying to understand the answer to <a href="https://math.stackexchange.com/questions/424063/group-action-on-a-manifold-with-finitely-many-orbits">one of my previous questions</a>. The precise problem I want to show is as follows. </p>
<blockquote>
<p>Let $G$ be a group acting faithful... | Jyrki Lahtonen | 11,619 | <p>This is not true. $G=GL_2(\mathbb{R})$ acts faithfully on $\mathbb{R}^2$, no?</p>
|
3,206,838 | <blockquote>
<p>The number of 5-digit numbers of the form abcde where a,b,c,d,e belong to <span class="math-container">${0,1,2,...9}$</span> and <span class="math-container">$b = a + c$</span>, <span class="math-container">$d = c + e $</span> are?</p>
</blockquote>
<p>I tried to reason out that out of the 5 digits w... | Community | -1 | <p>For each choice of <span class="math-container">$c$</span>, there are <span class="math-container">$10-c$</span> choices for <span class="math-container">$d$</span> (any of <span class="math-container">$c,\dots,9$</span>), and <span class="math-container">$9-c$</span> for <span class="math-container">$a$</span> (<sp... |
2,878,073 | <p><strong>Problem:</strong> $\{a_n\}_{n\in \mathbb N}, \quad a_{n+1}=\sqrt{2+a_n}, \quad \forall n\geq 1, \quad a_1=\sqrt{2}$</p>
<p><strong>Solution:</strong></p>
<p>We assume $a_n$ converges. Then is $\lim_{n\to\infty}a_n=\lim_{n\to\infty}a_{n+1}$</p>
<p>So we get</p>
<p>$$$a_n=\sqrt{2+a_n} \Rightarrow a_n^2-a_n... | Fred | 380,717 | <p>If $(a_n)$ is convergent, then denote the limit of the sequence by $a$. Then we get</p>
<p>$$a=\sqrt{2+a}.$$</p>
<p>Hence $a=-1$ or $a=2$ (and not $a_n=-1$ or $a_n=2$ as you wrote). Since all $a_n \ge 0$, we see that $a_n \to 2$.</p>
<p>All this considerations are made under the assumption that $(a_n)$ is conver... |
3,243,906 | <p>I am doing a problem where I am differentiating from first principles, but I can't simplify the final expression:</p>
<p><span class="math-container">$\frac{-2xh - h^2}{x^4h + 2x^3h^2+x^2h^3}$</span></p>
<p>Could someone explain it in steps?</p>
| learningstudent | 677,133 | <p>Factorising h from both N&D to get<span class="math-container">$$=-\frac{h+2x}{x^2(x^2+2xh+h^2)}$$</span>
<span class="math-container">$$=-\frac{h+2x}{x^2(x+h)^2}$$</span></p>
|
749,731 | <p>if $G' <H < G$ then $H$ is normal in $G$. ($G'$ is the commutator subgroup of $G$.)</p>
<p>This is what I do:</p>
<p>because $G' < H$ we have $\frac{H}{G'} \triangleleft \frac{G}{G'}$. because $\frac{G}{G'}$ is abelian then $\frac{\frac{G}{G'}}{\frac{H}{G'}} \approx \frac{G}{H} $ is abelian and it means ... | tchappy ha | 384,082 | <p>I am reading "Topics in Algebra 2nd Edition" by I. N. Herstein.<br />
This problem is the same problem as Problem 5(d) on p.65 in Herstein's book.<br />
I could not solve this problem, so I referred to "An Introduction to Algebraic Systems" (in Japanese) by Kazuo Matsuzaka.</p>
<blockquote>
<p>Le... |
79,317 | <p>This concerns one of those "well known" facts, referred to in a recent preprint I've been looking at. In principle it's elementary, but I can't pin down an explicit textbook reference for it. Start with two finite groups $A,B$ and their product $G:=A \times B$, working over a splitting field $K$ for the groups i... | Geoff Robinson | 14,450 | <p>I'm not sure what Fein does, but can't you do everything with Brauer characters, at least in the splitting field case. Use the fact that the Brauer characters of the PIMs are the unique class funcstion $\theta_i$ such that $\langle \theta_i, \phi_j \rangle = \delta_{ij}$where the $\phi_j$ are the Brauer characters o... |
1,419,105 | <p>I had a course in the construction of numbers last semester. I understand the potencial of most of the proofs, for example: I guess I can answer decently why commutativity is important.</p>
<p>But when it comes to the proof of uniqueness of $0,1$, I have no idea why that is important. For $\Bbb{N}$, I guess it's im... | Zubin Mukerjee | 111,946 | <p>It's nice to make sure we aren't working in the <a href="http://mathworld.wolfram.com/TrivialRing.html" rel="nofollow">trivial ring</a>. Ideally the axioms we assume for $\mathbb{N}$ should eliminate the possibility that $\mathbb{N}$ is the trivial ring; a proof that $0 \neq 1$ is one way to do that.</p>
|
9,302 | <p>Say I have a symmetric matrix. I have the concept of 2-norm as defined on wikipedia. Now I want to prove (disprove?) that the norm of a symmetric matrix is maximum absolute value of its eigenvalue. I would really appreciate if this can be done only using simple concepts of linear algebra.</p>
<p>I am quite new to m... | Ross Millikan | 1,827 | <p>Given a symmetric matrix, you have a complete set of eigenvalues and orthogonal eigenvectors. Any vector can be represented as a linear combination of the eigenvectors. Multiply your matrix by an arbitrary unit vector decomposed into the eigenvectors. Then note that the maximum length of the resultant vector is a... |
1,934,268 | <p>Let's say I have an arbitrary matrix $A$. </p>
<p>An eigenvector of $A$ would be a vector $\vec{x}$, such that </p>
<ol>
<li>$A\vec{x} = \lambda \vec{x}$</li>
<li>$\vec{x} \in N(A-\lambda I)$ i.e. $(A - \lambda I)\vec{x} = \vec{0}$</li>
</ol>
<p>where $\lambda$ is an eigenvalue of $\vec{x}$</p>
<hr>
<p>Now are ... | Josu Etxezarreta Martinez | 360,329 | <p>The first statement is correct by the definition of nullspace, as $x\in N(A-\lambda I)$, then all those vectors should fulfill the relationship $(A-\lambda I)x=0$, from where we can derive easily the relation $Ax=\lambda x$.</p>
<p>For the statement you make that if $x\in N(a-\lambda I)$ then $x\in N(A)$, it is fal... |
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