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 |
|---|---|---|---|---|
2,152,872 | <p>I am working on a problem in Artin's Algebra related to the algebraic geometry talked in Chapter 11. The problem number is 9.2., F.Y.I.</p>
<p>Here goes the problem:</p>
<blockquote>
<p>Let $f_1, \dots, f_r$ be complex polynomials in the variables $x_1, \dots, x_n$, let $V$ be the variety of their common zeros, ... | A.G. | 115,996 | <p>Note that if you allow loops and parallel edges then the problem is very simple: just draw 5 nodes and $1+1+2+3+3=10$ edge ends, then join the edge ends whichever way you like. This is possible as $10$ is even (if the total were odd it would be impossible).
<a href="https://i.stack.imgur.com/rZ9hV.png" rel="nofollow... |
2,122,389 | <p>The problem goes so : you have a parking lot with 8 parking spaces and 8 cars, of which 4 are red and 4 are white. What is the probability of :</p>
<p>a) 4 white cars being parked next to each other ?</p>
<p>b) 4 white cars and 4 red cars being parked next to each other ?</p>
<p>c) red and white cars being parked... | Simply Beautiful Art | 272,831 | <p>I'm not so sure of indefinite integration, but if we add bounds, it comes out beautifully, as shown <a href="https://en.wikipedia.org/wiki/Leibniz_integral_rule#Examples_for_evaluating_a_definite_integral" rel="noreferrer">here</a>.</p>
<p>Take the derivative with respect to $m$ to get</p>
<p>$$\begin{align}-I'(-m... |
2,122,389 | <p>The problem goes so : you have a parking lot with 8 parking spaces and 8 cars, of which 4 are red and 4 are white. What is the probability of :</p>
<p>a) 4 white cars being parked next to each other ?</p>
<p>b) 4 white cars and 4 red cars being parked next to each other ?</p>
<p>c) red and white cars being parked... | Jack D'Aurizio | 44,121 | <p>This is a well-known problem about the <a href="https://en.wikipedia.org/wiki/Poisson_kernel" rel="nofollow noreferrer">Poisson kernel</a>.<br>
Since $\log\|z\|=\text{Re}\log(z)$ and for every $n\in\mathbb{Z}$ we have $\int_{0}^{2\pi}e^{ni\theta}\,d\theta=2\pi\,\delta(n)$,</p>
<p>$$\forall r\in\mathbb{R},\quad \int... |
23,566 | <p>I love math, and I used to be very good at it. The correct answers came fast and intuitively. I never studied, and redid the demonstration live for the tests (sometimes inventing new ones). I was the one who answered the tricky questions in class (8 hours of math/week in high school)... You get the idea.</p>
<p>As ... | phren0logy | 9,279 | <p>Linear Algebra: Try MIT's OpenCourseware with the inimitable Gilbert Strang.</p>
<p>Calculus: Try Calculus Made Easy, available as a free and nicely typeset PDF.</p>
<p>I also second Khan Academy.</p>
|
23,566 | <p>I love math, and I used to be very good at it. The correct answers came fast and intuitively. I never studied, and redid the demonstration live for the tests (sometimes inventing new ones). I was the one who answered the tricky questions in class (8 hours of math/week in high school)... You get the idea.</p>
<p>As ... | Community | -1 | <p>Perhaps of interest/relevance:</p>
<p><a href="http://johnlawrenceaspden.blogspot.com/2011/01/effortless-superiority.html" rel="nofollow">http://johnlawrenceaspden.blogspot.com/2011/01/effortless-superiority.html</a></p>
|
1,022,950 | <p>I was reading linear dependence between vectors, where I come across below explanation:</p>
<hr>
<p>In a rectangular xy-coordinate system every vector in the plane can be expressed in
exactly one way as a linear combination of the standard unit vectors. For example, the
only way to express the vector (3, 2) as a l... | Timbuc | 118,527 | <p>$$\left|\;\sqrt\frac{n+1}n-1\;\right|=\frac{\sqrt{n+1}-\sqrt n}{\sqrt n}=\frac1{\sqrt n\left(\sqrt{n+1}+\sqrt n\right)}$$</p>
<p>Take now <strong>any</strong> $\;\epsilon>0\;$ . We want to check for which $\;n$' s we have</p>
<p>$$\frac1{\sqrt n\left(\sqrt{n+1}+\sqrt n\right)}<\epsilon$$</p>
<p>But now we c... |
184,682 | <p>I have difficulties with a rather trivial topological question: </p>
<p>A is a discrete subset of $\mathbb{C}$ (complex numbers) and B a compact subset of $\mathbb{C}$. Why is $A \cap B$ finite? I can see that it's true if $A \cap B$ is compact, i.e. closed and bounded, but is it obvious that $A \cap B$ is closed?<... | André Nicolas | 6,312 | <p>The result is not correct, for the set $A$ of all $\frac{1}{n}$, where $n$ ranges over the positive integers, is discrete. Let $B$ be the unit disk.</p>
<p>As you observed, everything would be fine if $A\cap B$ were closed. But in this case it isn't.</p>
|
2,664,286 | <p>I am confused between the usage of two words <em>for all</em> and <em>any</em>. Let us consider the example of the definition normal subgroups, A subgroup $H$ is said to be normal if $\forall g \in G, g^{-1}Hg = H$ but if I rephrase the definition of normal subgroup to $H$ is normal in $G$ if for any $g \in G, g^{-1... | Michael Rozenberg | 190,319 | <p>Let $Q'$ be symmetrical to $Q$ with respect to the line $l:y=-2x+7.$</p>
<p>Also, let $PQ'\cap l=\{R\}$.</p>
<p>Thus, $R$ is a needed point.</p>
|
674,310 | <p>I am having trouble with a proof for linear algebra. Could somebody explain to me how to prove that if $A$ and $B$ are both $n\times n$ non singular matrices, that their product $AB$ is also non singular. </p>
<p>A place to start would be helpful. Thank you for your time. </p>
| zipirovich | 127,842 | <p>Depends how far into linear algebra you are and what you can use. One possible and very short solution: a square matrix is nonsingular iff its determinant is nonzero. Now use the property for $\det(AB)$.</p>
|
157,992 | <p>Please, help me</p>
<p>Prove that $(1, i);(1,-i)$ are characteristic vectors of $\begin{bmatrix} a & b \\ -b & a \end{bmatrix}$</p>
<p>I've found the polynomial characteristic: $\lambda^2-2a\lambda+a^2+b^2$ and the roots are:</p>
<p>$\lambda_{1} = \frac{a+ib}{\lambda} \\
\lambda_{2} = \frac{a-ib}{\lambda... | DonAntonio | 31,254 | <p>So you have to check whether $$\begin{pmatrix}a&b\\\!\!\!-b&a\end{pmatrix}\binom{1}{i}=\lambda\binom{1}{i}\,\,,\,\lambda=\lambda_1\,\,or\,\,\lambda_2$$and the same for the other given vector. Well, now just verify that you indeed get the above matrix equation right when you use the values you got for $\,\lam... |
99,572 | <p>One of the most useful tools in the study of convex polytopes is to move from polytopes (through their fans) to toric varieties and see how properties of the associated toric variety reflects back on the combinatorics of the polytopes. This construction requires that the polytope is rational which is a real restrict... | Neil Strickland | 10,366 | <p>The DJ construction works with a simplicial complex $K$ and a subtorus $W\leq\prod_{v\in V}S^1$ (where $V$ is the set of vertices of $K$). People tend to be interested in the case where $|K|$ is homeomorphic to a sphere, but that isn't really central to the theory. However, it is important that we have a simplicia... |
2,601,380 | <p>I'm trying to figure out the impedance of a capacitor. My textbook tells me the answer is $\frac{-i}{\omega C}$ and plugging that into the equation does work but I wanted to come up with that answer myself. So I wrote out the equation with what I know:</p>
<p>$$-V_0\omega C\sin\omega t = Re\left( \frac{V_0(\cos\ome... | Jan Eerland | 226,665 | <p>For a capacitor, there is the relation:</p>
<p>$$\text{I}_\text{C}\left(t\right)=\text{C}\cdot\frac{\text{d}\text{V}_\text{C}\left(t\right)}{\text{d}t}\tag1$$</p>
<p>Considering the voltage signal to be:</p>
<p>$$\text{V}_\text{C}\left(t\right)=\text{V}_\text{p}\sin\left(\omega t\right)\tag2$$</p>
<p>It follows ... |
463,650 | <p>Consider the sequence $\left \{ x_{n} \right \}$ that satisfies the condition:
$$\left | x_{n+1}-x_{n} \right |< \frac{1}{2^{n}}
\ \ \ for\ all\ n=1,2,3,...$$
Part (1): Prove that the sequence $\left \{ x_{n} \right \}$ is convergent.</p>
<p>Part (2): Does the result in part (1) hold if we only assume that $\l... | N. S. | 9,176 | <p>For part (2) you can also use $x_n = \ln(n)$.</p>
<p>Note that by the MVT we have for some $c_n \in (n,n+1)$:</p>
<p>$$ \frac{\ln(n+1)-\ln(n)}{n+1-n}= \frac{1}{c_n} < \frac{1}{n} \,.$$</p>
|
160,491 | <pre><code>Histogram[RandomVariate[NormalDistribution[0, 1], 200]]
</code></pre>
<p>How to calculate the area under the histogram.</p>
| Jack LaVigne | 10,917 | <p>Here is one sample of the random generated data</p>
<pre><code>data = RandomVariate[NormalDistribution[0, 1], 200];
</code></pre>
<p>and the associated historgram</p>
<pre><code>Histogram[data]
</code></pre>
<p><img src="https://i.stack.imgur.com/q31Yg.png" alt="Mathematica graphics"></p>
<p>The coordinates of ... |
160,491 | <pre><code>Histogram[RandomVariate[NormalDistribution[0, 1], 200]]
</code></pre>
<p>How to calculate the area under the histogram.</p>
| Szabolcs | 12 | <p>Why would you want to calculate that?</p>
<p>The third argument of <code>Histogram</code> controls the meaning of the bin height. The default is <code>"Count"</code>. Thus your result for <code>Histogram[data]</code> would simply be <code>Length[data]</code>.</p>
|
2,633,720 | <blockquote>
<p>Prove by induction that $$ (k + 2)^{k + 1} \leq (k+1)^{k +2}$$ for $ k > 3 .$</p>
</blockquote>
<p>I have been trying to solve this, but I am not getting the sufficient insight. </p>
<p>For example, $(k + 2)^{k + 1} = (k +2)^k (k +2) , (k+1)^{k +2}= (k+1)^k(k +1)^2.$</p>
<p>$(k +2) < (k +1)^... | Akababa | 87,988 | <p>Try taking log of both sides and prove $\frac{\log x}x$ is decreasing.</p>
<p>Or by induction try to show $(\frac{k+1}k)^k\leq k$:</p>
<p>$$(1+1/k)^k\leq \sum_{i=0}^k \binom ki k^{-i}<\sum_{i=0}^k 1=k+1$$</p>
|
1,714,654 | <p>Show that a box (rectangular parallelopiped) of maximum volume V with prescribed surface area is a cube.
Let $$V=xyz$$
$$S=2xy + 2yz + 2zx$$
$S$ is constant.</p>
<p>Using Lagrange method, I am stuck at $V_x$$_x$=$0$=$V_y$$_y$=$V_z$$_z$ at the (only) critical point. How to approach this. </p>
| Mark Viola | 218,419 | <p>Recall from elementary geometry that the sine function satisfies the inequalities </p>
<p>$$|\theta\cos(\theta)|\le |\sin(\theta)|\le |\theta|$$</p>
<p>for $|\theta|\le \pi/2$. </p>
<p>Letting $\theta =x^2-y^2$ we can write</p>
<p>$$|\cos(x^2-y^2)|\le\left|\frac{\sin(x^2-y^2)}{x^2-y^2}\right|\le|1|$$</p>
<p>whe... |
106,464 | <p>I'd like to prove the following:</p>
<blockquote>
<p>If $\mathfrak{a} \subseteq k[x_0, \ldots, x_n]$ is a homogeneous ideal, and if $f \in k[x_0,\ldots,x_n]$ is a homogeneous polynomial with $\mathrm{deg} \ f > 0$, such that $f(P) = 0 $ for all $P \in Z(\mathfrak{a})$ in $\mathbb P^n$, then $f^q \in \mathfrak{... | azarel | 20,998 | <p>$\bf Hint:$ Let $\overline Z(\mathfrak a)=\{p\in \mathbb A^{n+1}:\forall g\in\mathfrak a \ (g(p)=0)\}$ (the affine variety determined by $\mathfrak a$). Note that $f\in \overline Z(\mathfrak a)$ and aplly the standard Nullstellensatz. </p>
|
322,302 | <p>Conjectures play important role in development of mathematics.
Mathoverflow gives an interaction platform for mathematicians from various fields, while in general it is not always easy to get in touch with what happens in the other fields.</p>
<p><strong>Question</strong> What are the conjectures in your field prove... | Mare | 61,949 | <p>The strong no loop conjecture for quiver algebras <span class="math-container">$A$</span> states that a simple module <span class="math-container">$S$</span> with <span class="math-container">$Ext_A^1(S,S) \neq 0$</span> has infinite projective dimension. It was proven here <a href="https://www.sciencedirect.com/sci... |
322,302 | <p>Conjectures play important role in development of mathematics.
Mathoverflow gives an interaction platform for mathematicians from various fields, while in general it is not always easy to get in touch with what happens in the other fields.</p>
<p><strong>Question</strong> What are the conjectures in your field prove... | efs | 109,085 | <p>Recently, Dasgupta, Kakde and Ventullo proved Gross's conjecture on the value at zero of the <span class="math-container">$p$</span>-adic <span class="math-container">$L$</span>-function constructed by Cassou-Noguès, and Deligne and Ribet. The article, <em>On the Gross-Stark Conjecture</em>, was published in Annals ... |
322,302 | <p>Conjectures play important role in development of mathematics.
Mathoverflow gives an interaction platform for mathematicians from various fields, while in general it is not always easy to get in touch with what happens in the other fields.</p>
<p><strong>Question</strong> What are the conjectures in your field prove... | user27976 | 27,976 | <p>Kiran Kedlaya finished the proof of Deligne's conjecture (1.2.10) made in La conjecture de Weil, II, which is definitely "noteworthy", and perhaps "not so famous" compared to the original Weil conjectures.</p>
<p>Colloquium talk: <em><a href="https://www.math.rutgers.edu/news-events/list-all-events/icalrepeat.detai... |
322,302 | <p>Conjectures play important role in development of mathematics.
Mathoverflow gives an interaction platform for mathematicians from various fields, while in general it is not always easy to get in touch with what happens in the other fields.</p>
<p><strong>Question</strong> What are the conjectures in your field prove... | Denis Nardin | 43,054 | <p>Tyler Lawson's <a href="https://arxiv.org/abs/1703.00935" rel="noreferrer">recent proof</a> that the Brown-Peterson spectrum <span class="math-container">$BP$</span> at the prime 2 has no <span class="math-container">$E_∞$</span>-ring structure. This was later generalized at odd primes, using similar methods, by <a ... |
322,302 | <p>Conjectures play important role in development of mathematics.
Mathoverflow gives an interaction platform for mathematicians from various fields, while in general it is not always easy to get in touch with what happens in the other fields.</p>
<p><strong>Question</strong> What are the conjectures in your field prove... | Stopple | 6,756 | <p>This MO question <a href="https://mathoverflow.net/questions/115447/the-riemann-zeros-and-the-heat-equation">The Riemann zeros and the heat equation</a> describes the Newman conjecture. Very briefly, a deformation parameter is introduced into an integral representation of the Riemann zeta function, creating a funct... |
322,302 | <p>Conjectures play important role in development of mathematics.
Mathoverflow gives an interaction platform for mathematicians from various fields, while in general it is not always easy to get in touch with what happens in the other fields.</p>
<p><strong>Question</strong> What are the conjectures in your field prove... | Peter Shor | 2,294 | <p><a href="https://en.wikipedia.org/wiki/Connes_embedding_problem" rel="noreferrer">Connes' embedding conjecture</a> (from 1976) about the structure of infinite-dimensional von Neumann algebras was shown to be false in the paper</p>
<ul>
<li>Zhengfeng Ji, Anand Natarajan, Thomas Vidick, John Wright, Henry Yuen, <span ... |
322,302 | <p>Conjectures play important role in development of mathematics.
Mathoverflow gives an interaction platform for mathematicians from various fields, while in general it is not always easy to get in touch with what happens in the other fields.</p>
<p><strong>Question</strong> What are the conjectures in your field prove... | David White | 11,540 | <p>The <a href="https://en.wikipedia.org/wiki/Stabilization_hypothesis" rel="nofollow noreferrer">Baez-Dolan Stabilization Hypothesis</a> was posed in 1995. It involves the relationship between weak <span class="math-container">$n$</span>-categories as <span class="math-container">$n$</span> varies. Specifically, if on... |
3,686,921 | <blockquote>
<p>Prove that for all triangles with angles <span class="math-container">$\alpha, \beta, \gamma$</span>, <span class="math-container">$$\frac{\sin\alpha}{\cos\alpha + 1} + \frac{\sin\beta}{\cos\beta + 1} + \frac{\sin\gamma}{\cos\gamma + 1} = \frac{\cos\alpha + \cos\beta + \cos\gamma + 3}{\sin\alpha + \si... | CryoDrakon | 193,538 | <p>Using your transformation I a get sligtly different equation:
<span class="math-container">$$a+b+c=\frac{\sum_{cyc}\frac{1}{1+a^2}}{\sum_{cyc}\frac{a}{1+a^2}}$$</span>
<span class="math-container">$$\sum_{cyc}\frac{a(a+b+c)}{1+a^2}=\sum_{cyc}\frac{1}{1+a^2}$$</span>
<span class="math-container">$$\sum_{cyc}\frac{a^2... |
3,680,536 | <p>In Eric Lengyel's book, <em>Mathematics for 3D Game Programming and Computer Graphics, 3<sup>rd</sup> Edition</em>, there is a theorem that states</p>
<blockquote>
<p>An <span class="math-container">$n \times n$</span> matrix M is invertible if and only if the rows of M form a linearly independent set of vectors.... | user790072 | 790,072 | <p>The proof appears to be using the fact that linearly independent subsets of <span class="math-container">$\Bbb{R}^n$</span> have at most <span class="math-container">$n$</span> vectors in them. For the sake of illustration, let's say we're at step C with <span class="math-container">$j = 3$</span> in an <span class=... |
1,327,944 | <p>I'm a hobbyist working on a mechanical sorting machine to sort magic the gathering cards. I'm by no means a mathematician though, and I was wondering if you all wouldn't mind helping me out with a math puzzle to determine the best route to go with my machine.</p>
<p>The average Magic: the Gathering set of cards con... | Kingrames | 256,266 | <p>Well unless you're using exclusively cards from way back in ye oldene dayse (tm), there's a collector number on the card itself, so they're technically already sorted.</p>
<p>Before I begin, let me say: Alphabetically is the absolute WORST way to sort M:tG cards. I personally sort by more effective category: white ... |
1,849,758 | <p>Does there exist a name for the class of commutative rings with identity that satisfy the following:</p>
<p>For any 2 ideals $I_1,I_2$ of R,we have : $I_1 I_2= (I_1\cap I_2)(I_1+I_2) $</p>
<p>I would also like to see an example of a ring not satisfying the above property.</p>
<p>Thank you</p>
| E.R | 325,912 | <p>If $R$ is a noetherian domain with that property $R$ is called Dedekind domain, see Larsen and McCarthy' <em>Multiplicative Theory of Ideals</em>, theorem 6.20. However, see D. D. Anderson's homepage, he has a paper with a similar title for arbitrary commutative rings. </p>
|
4,062,242 | <p>Is <span class="math-container">$$\int_1^\infty \frac{\log x}{x^2}dx$$</span>
finite? How to solve this?</p>
| BCLC | 140,308 | <p>Use integration by parts. Keep trying different parts until it works.</p>
<ol>
<li>This doesn't work:</li>
</ol>
<p><span class="math-container">$u= \frac1x$</span>, <span class="math-container">$dv=\frac{\ln(x)dx}{x}$</span></p>
<p><span class="math-container">$du= -\frac{dx}{x^2}$</span>, <span class="math-contain... |
542,454 | <p>There are two tangent lines on $f(x) = \sqrt{x}$ each with the $x$-value $a$ and $b$ respectively. </p>
<p>I need to prove that $c$, the $x$ value of the point at which the two lines intersect each other, is equal to $\sqrt{ab}$, the geometric mean of $a$ and $b$. </p>
<p>I have been trying many different ways of ... | TBrendle | 97,380 | <p>The two points of tangency are $(a, \sqrt{a})$ and $(b, \sqrt{b})$. If either of $a$ or $b$ is $0$ then the result holds trivially, so assume $ab \neq 0$.</p>
<p>Because the curve is a parabola, the intersection of the tangents, $(x_0, y_0)$, has $y_0=(\sqrt{a}+\sqrt{b})/2$. That's because projecting perpendicularl... |
2,747,896 | <p>I am a bit confused with cardinality at the moment. I know that the cardinality of $\mathbb{R}$ is equal to $\lvert(0,1)\rvert$, but does that mean they are equal to infinity, if not what are they equal to ?</p>
| saulspatz | 235,128 | <p>Their cardinality is referred to as "the cardinality of the continuum." There are infinitely many different infinite cardinal numbers. In many contexts, though, it enough to say that the cardinality is "uncountably infinite" or that the set is "uncountable." An example of a countable infinite set is the integers.... |
3,768,198 | <p>Show that <span class="math-container">$\|uv^T-wz^T\|_F^2\le \|u-w\|_2^2+\|v-z\|_2^2$</span>, assuming <span class="math-container">$u,v,w,z$</span> are all unit vectors.</p>
| Ben Grossmann | 81,360 | <p>Another approach: note that for orthogonal matrices <span class="math-container">$U,V,$</span> we have
<span class="math-container">$$
\|uv^T - wz^T\|_F^2 = \|U(uv^T - wz^T)V\|_F^2 = \|(Uu)(Vv)^T - (Uw)(Vz)^T\|_F^2.
$$</span>
So without loss of generality, we can assume that <span class="math-container">$u = v = (1,... |
2,059,192 | <p>I was reading about Sobolev spaces and came across the notation $\dot{H}^1, \dot{H}^{-1}, \dot{H}^t$. I'm familiar with $H^1, H^{-1}, H^t$, but not the dot, and I can't find these spaces defined anywhere. Is this notation common, and could you explain it to me or point me to a reference?</p>
<p>I have more or less ... | Michał Miśkiewicz | 350,803 | <p>I have seen another meaning of $\dot{H}^s(\mathbb{R}^d)$ in <em>Fourier analysis and nonlinear PDEs</em> by Bahouri-Chemin-Danchin. In this book, this is called a <em>homogenous Sobolev space</em> defined as the set of all tempered distributions such that $\hat{u} \in L^1_{loc}$ and
$$ ||u||_{\dot{H}^s}^2 := \int_{... |
102,738 | <p>I imported two sets data
one: </p>
<pre><code>data1={{0., 5.02512*10^-10}, {0.06668, 2.99284*10^-8}, {0.13336,
3.22116*10^-8}, {0.20004, 2.58191*10^-8}, {0.26672,
1.99125*10^-7}, {0.3334, 1.21646*10^-8}, {0.40008,
3.35916*10^-7}, {0.46676, 3.79768*10^-7}, {0.53344,
1.02102*10^-7}, {0.60012, 1.17535*10^-6}, {0.666... | WReach | 142 | <pre><code>SetOptions[EvaluationNotebook[]
, StyleDefinitions -> Notebook @
{ Cell[StyleData[StyleDefinitions -> "Default.nb"]]
, Cell[StyleData["ItemNumbered"], CounterBoxOptions -> {CounterFunction -> (#*2-1&)}]
, Cell[StyleData["SubitemNumbered"], CounterBoxOptions -> {CounterFunction -> ... |
3,605,636 | <p>Let <span class="math-container">$m_{a},m_{b},m_{c}$</span> be the lengths of the medians and <span class="math-container">$a,b,c$</span> be the lengths of the sides of a given triangle , Prove the inequality : </p>
<p><span class="math-container">$$m_{a}m_{b}m_{c}\leq\frac{Rs^{2}}{2}$$</span></p>
<p>Where : </p>
... | Quanto | 686,284 | <p><a href="https://i.stack.imgur.com/6VkTE.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/6VkTE.png" alt="enter image description here"></a></p>
<p>Note that the triangles ABD and EDC are similar. Then,</p>
<p><span class="math-container">$$\frac{AD}{BD}=\frac{CD}{ED}\implies \frac{m_a}{\frac a2}... |
344,725 | <p>in $\Delta ABC$,and </p>
<p>$$\dfrac{\sin{(\dfrac{B}{2}+C)}}{\sin^2{B}}=\dfrac{\sin{(\dfrac{C}{2}+B)}}{\sin^2{C}}$$</p>
<p>prove that $B=C$</p>
<p>I think $\sin{(\dfrac{B}{2}+C)}\sin^2{C}=\sin{(\dfrac{C}{2}+B)}\sin^2{B}$</p>
<p>then
$$\sin{(\dfrac{B}{2})}\cos{C}\sin^2{C}+\cos{\dfrac{B}{2}}\sin^3C=\sin{\dfrac{C}{... | coffeemath | 30,316 | <p>No manipulations I tried worked, so I began to think the statement was false. So I set up the function
$$f(b,c)=\sin(b/2+c)\sin^2(c),$$
noting the sides of your equation are equal iff $f(b,c)=f(c,b).$ So if $b=c$ followed from that it should be true that for example the function
$$g(x)=f(x,x+.5)-f(x+.5,x)$$
should n... |
2,471,680 | <p>I am working with a theorem and i need the reference of the above limit.
Kindly guide.</p>
| user387832 | 387,832 | <p>Make use of the monotone convergence theorem.</p>
<p>From Wikipedia:</p>
<p><em>If a sequence of real numbers is decreasing and bounded below, then its infimum is the limit.</em></p>
<p>So you'll need to show that $0$ is the infimum of the sequence and that this sequence is decreasing.</p>
<p><em>Hint</em>: Show... |
2,860,156 | <p>Let $A\in \mathbb{M}_3(\mathbb{R})$ be a symmetric matrix whose eigen-values are $1,1$ and $3$. Express $A^{-1}$ in the form $\alpha I +\beta A$, where $\alpha, \beta \in \mathbb{R}$.</p>
| InsideOut | 235,392 | <p>Let $\alpha,\beta\in \Bbb R$ such that $A^{-1}=\alpha I+\beta A$. Then $$A(\alpha I+\beta A)=\alpha A+ \beta A^2=I$$</p>
<p>This is equivalent to say that $\beta A^2 + \alpha A-I=0$, that is $A$ is a solution of the polynomial $$p(x)=\beta x^2+\alpha x-1=0.$$ This is also equivalent to say that the eigenvalues $1,3... |
1,081,417 | <p>This is exercise number $57$ in Hugh Gordon's <em>Discrete Probability</em>. </p>
<hr>
<p>For $n \in \mathbb{N}$, show that</p>
<p>$$\binom{\binom{n}{2}}{2}=3\binom{n+1}{4}$$</p>
<hr>
<p>My algebraic solution:</p>
<p>$$\binom{\binom{n}{2}}{2}=3\binom{n+1}{4}$$</p>
<p>$$\binom{\frac{n(n-1)}{2}}{2}=\frac{3n(n+1... | Grigory M | 152 | <p>$\binom{\binom n2}2$ counts pairs of (distinct) 2-element subsets of $n$-element set. Union of such pair is either 4-element set (and each 4-element set is counted 3 times: there are 3 ways to divide 4-set into 2 pairs) or 3-element set (and each 3-element set is also counted 3 times). That gives $3\binom n4+3\binom... |
3,325,114 | <blockquote>
<p>You are on an island inhabited only by knights, who always tell the truth, and knaves, who always lie. You meet two women who live there and ask the older one,</p>
<blockquote>
<p>"Is at least one of you a knave?"</p>
</blockquote>
<p>She responds yes or no, but! you do not yet have enough inf... | Floris Claassens | 638,208 | <p>Just write a truth table: You've got 4 possibilities:</p>
<ul>
<li>Woman 1 is a knight and woman 2 is a knight, answers are no and yes.</li>
<li>Woman 1 is a knight and woman 2 is a knave, answers are yes and yes. </li>
<li>Woman 1 is a knave and woman 2 is a knight, answers are no and no. </li>
<li>Woman 1 is a kn... |
3,325,114 | <blockquote>
<p>You are on an island inhabited only by knights, who always tell the truth, and knaves, who always lie. You meet two women who live there and ask the older one,</p>
<blockquote>
<p>"Is at least one of you a knave?"</p>
</blockquote>
<p>She responds yes or no, but! you do not yet have enough inf... | AgentS | 168,854 | <p><span class="math-container">$a$</span>: older woman is knight<br>
<span class="math-container">$b$</span>: younger woman is knight</p>
<p>Behavior of <code>older woman</code> can be represented with the boolean function
<span class="math-container">$$f(a,b)=a(a'+ b') + a'(ab) = a b'$$</span>
Behavior of <code>you... |
1,557,165 | <p>Prove that
$$\int_1^\infty\frac{e^x}{x (e^x+1)}dx$$
does not converge.</p>
<p>How can I do that? I thought about turning it into the form of $\int_b^\infty\frac{dx}{x^a}$, but I find no easy way to get rid of the $e^x$.</p>
| Enrico M. | 266,764 | <p>Collect $e^x$ and have</p>
<p>$$\int\frac{e^x}{e^x(x + e^{-x})} = \int \frac{dx}{x + xe^{-x}}$$</p>
<p>Now substitute $e^{-x} = y$ so that $dy = -y dx$ and extrema changes from $1/e$ to $0$:</p>
<p>$$\int_0^{1/e} \frac{dy}{y(- \log(y))(1+y)}$$</p>
<p>which has a pole along the path of integration, so then the in... |
1,557,165 | <p>Prove that
$$\int_1^\infty\frac{e^x}{x (e^x+1)}dx$$
does not converge.</p>
<p>How can I do that? I thought about turning it into the form of $\int_b^\infty\frac{dx}{x^a}$, but I find no easy way to get rid of the $e^x$.</p>
| Lucian | 93,448 | <p>Since $~\dfrac{dx}x=d~\big(\ln x\big),~$ we'll just let $x=e^t.~$ This yields $\displaystyle\int_0^\infty\frac{e^{e^t}}{e^{e^t}+1}~dt.~$ Simplifying </p>
<p>both sides by $e^{e^t}$ leads to $\displaystyle\int_0^\infty\frac1{1+e^{-e^t}}~dt,~$ which, as far as I'm able to see, diverges </p>
<p>as shamelessly as $\di... |
187,545 | <p><span class="math-container">$\DeclareMathOperator\GL{GL}\DeclareMathOperator\L{\mathfrak{L}}$</span>The free Lie algebra <span class="math-container">$\L(V)$</span> generated by an <span class="math-container">$r$</span>-dimensional vector space <span class="math-container">$V$</span> is, in the
language of <a href... | Daniel Barter | 4,002 | <p>What you are describing is the algebraic operad Lie. More details can be found <a href="https://mathoverflow.net/questions/87309/an-n-dimensional-representation-of-the-symmetric-group-s-n2"> here</a>. The Whitehouse modules are exactly what you get when you take Lie onto the other side of Schur Weyl duality. The seq... |
3,451,374 | <p>Given that I have a random variable <span class="math-container">$\max\{K-X, 0\}$</span> where <span class="math-container">$k>0$</span> is a constant and <span class="math-container">$x$</span> is uniformly distributed on <span class="math-container">$[-K, K]$</span> or I guess more generally with any distributi... | Siddhant | 687,664 | <p>Hint :- Consider <span class="math-container">$Y = max(X_1,X_2)$</span>, then <span class="math-container">$P(Y \leq y) = P(X_1 \leq y,X_2 \leq y)$</span>. If <span class="math-container">$X_1$</span> and <span class="math-container">$X_2$</span> are independent then <span class="math-container">$P(Y \leq y) = P(X_1... |
3,638,028 | <p>find <span class="math-container">$f\circ f$</span> for the function <span class="math-container">$f\colon \mathbb R^2\to \mathbb R^2$</span> (,)=(−,)
I know that if (,)=(−,), then () is its inverse reflected about the -axis. If this is the case then <span class="math-container">$f\circ f$</span> = f^-1(−f^-1(−)). I... | Henry | 6,460 | <p>Just factor out the <span class="math-container">$(k+1)$</span> term</p>
<p>or if you prefer expand and then factor:</p>
<p><span class="math-container">$$\frac {k(k+1)+2(k+1)}2=\frac {k^2+3k+2}2=\frac {(k+1)(k+2)}2$$</span></p>
|
2,281,510 | <p>Why do we replace y by x and then calculate y for calculating the inverse of a function?</p>
<p>So, my teacher said that in order to find the inverse of any function, we need to replace y by x and x by y and then calculate y. The reason being inverse takes y as input and produces x as output.</p>
<p>My question is... | StuartMN | 439,545 | <p>Good question , If <span class="math-container">$y=f(x)$</span> then for <span class="math-container">$x$</span> the function <span class="math-container">$f$</span> determines a unique <span class="math-container">$y$</span> .If there is an inverse function then for each <span class="math-container">$y$</span> the ... |
1,840,778 | <p>In rectangle $ABCD$, we have $AD = 3$ and $AB = 4$. Let $M$ be the midpoint of $\overline{AB}$, and let $X$ be the point such that $MD = MX$, $\angle MDX = 77^\circ$, and $A$ and $X$ lie on opposite sides of $\overline{DM}$. Find $\angle XCD$, in degrees. </p>
<p><img src="https://i.stack.imgur.com/3TsZm.png" alt="... | Stefan4024 | 67,746 | <p>Find the $\angle ADM$ from the right-angled triangle. This will help you find $\angle XDC$, as $\angle XDC = 77^{\circ} - \angle CDM = 77^{\circ} - 90^{\circ} + \angle ADM$. Then use Sine Theorem on $\triangle DMX$ to find the length of $\overline{DX}$ and then finally use Sine Theorem on $\triangle DXC$ to find $\a... |
340,886 | <p>Suppose $x=(x_1,x_2),y = (y_1,y_2) \in \mathbb{R}^2$. I noticed that
\begin{align*}
\|x\|^2 \|y\|^2 - \langle x,y \rangle^2 &=
x_1^2y_1^2 + x_1^2 y_2^2 + x_2^2 y_1^2 + x_2^2 y_2 ^2 - (x_1^2 y_1^2 + 2 x_1 y_1 x_2 y_2 + x_2^2 y_2^2) \\
&=(x_1 y_2)^2 - 2x_1 y_2 x_2 y_1 + (x_2 y_2)^2 \\
&=(x_1 y_2 - x_2 y_1)... | user1551 | 1,551 | <p>When $n\ge3$, $\|x\|^2\|y\|^2 - \langle x,y\rangle^2$ is not the square of any polynomial $p(x,y)$. Keep all entries other than $x_1$ fixed and let
\begin{align*}
q(x_1) &= \|x\|^2\|y\|^2 - \langle x,y\rangle^2,\\
\Rightarrow q\,'(x_1) &= 2(x_1 \|y\|^2 - y_1\langle x,y\rangle)
\end{align*}
If $q$ is a square... |
75,900 | <p>I have got a following equation: </p>
<pre><code>-(c - x)/Sqrt[b^2 + (c - x)^2] + x/Sqrt[a^2 + x^2] == 0.
</code></pre>
<p>Trying to solve it for x, so I evaluate</p>
<pre><code>Solve[-(c-x)/Sqrt[b^2+(c-x)^2]+x/Sqrt[a^2+x^2] == 0,x].
</code></pre>
<p>This produces</p>
<blockquote>
<pre><code>{{x -> (a*c)/(a ... | chuy | 237 | <p>You can try using the option <a href="http://reference.wolfram.com/language/ref/MaxExtraConditions.html" rel="nofollow"><code>MaxExtraConditions</code></a> </p>
<blockquote>
<p>Solve gives generic solutions only. Solutions that are valid only when
continuous parameters satisfy equations are removed. Additional
... |
3,132,380 | <p>To compute this I used the fact that <span class="math-container">$S(n,2) = 2^{n-1}-1$</span> and used the recurrence relation <span class="math-container">$S(n,k) = kS(n-1,k) + S(n-1,k-1)$</span>, and used induction to get that <span class="math-container">$S(n,3)=\dfrac{3^{n-1}+1}{2}-2^{n-1}$</span>.</p>
<p>But i... | Mike Earnest | 177,399 | <p><span class="math-container">$3^{n-1}$</span> counts ternary sequences of length <span class="math-container">$n-1$</span>, symbols in <span class="math-container">$\{0,1,2\}. $</span> </p>
<p><span class="math-container">$(3^{n-1}+1)/2$</span> counts ternary sequences whose first nonzero symbol is a <span class="... |
1,534,675 | <p>Since differential quantities are defined as any variable /function tending to zero ($\lim_{x\to0} x= dx$). This is basically the smallest value that we can imagine. Doesn't this mean that there is only one smallest value that ew can imagine? Doesn't the existence of another differential quantity, say $dl$ also mean... | Community | -1 | <p>A <a href="https://en.wikipedia.org/wiki/Differential_of_a_function" rel="nofollow">differential</a> is not a quantity, it is not "tending to zero" and the equation $\lim_{x\to0} x= dx$ is meaningless (by the way, $\lim_{x\to0} x= 0$). An expression like $dl>dh$ also has no meaning.</p>
|
1,358,270 | <p>If we have a function $f=f(r, \theta, \phi)$, where $(r, \theta, \phi)$ are spherical coordinates on $\mathbb{R}^3$, how do we compute the gradient $\nabla f$ by using the formula
$$\nabla f \cdot d\vec{r} = df ?$$
Here $\vec{r}$ is the position vector and $df=\frac{\partial f}{\partial r}dr +\frac{\partial f}{\par... | Alex M. | 164,025 | <p>Let us split the integral into $\int \limits _{- \infty} ^{-R} + \int \limits _{-R} ^R + \int \limits _R ^\infty$, where $R>1$ is large enough in order for the possible roots of the denominator to be included in $[-R, R]$. Let us first analyze the third integral, for $y > R$, which is a type 1 improper integra... |
4,218 | <p>I could imagine a system of categorizing the questions that would work alongside the current tagging system. If you select the "homework" tag (or some special tag or option), it would give you the option to specify which textbook problem your question pertains to in terms of title/chapter/section/problem number. M... | Martin Sleziak | 8,297 | <p>This might be closer to a longer comment than to an answer - but I hope it's ok to post it here anyway.</p>
<hr>
<p>I have been more than once in the situation that I stumbled upon a problem when reading a mathematical book. Usually I thought that I am either missing an important point in some proof or that the au... |
288,974 | <p>Alright this maybe really funny but I want to know why is this wrong. We often come across identities which we prove by multiplying both the sides of the identity by a certain entity but why don't we multiply it by $0$. That way every identity will be proved in one single line. That is so stupid. I mean, by that way... | Todd Wilcox | 41,686 | <p>The first proof (unlike the second) immediately leads the reader to the following, more definite style of writing essentially the same thing:
$$
\sin^2(x)=\sin^2(x)\frac{\cos^2(x)}{\cos^2(x)}=\frac{\sin^2(x)}{\cos^2(x)}\cos^2(x)=\tan^2(x)\cos^2(x)\text{.}
$$
This is a more explicit demonstration of what I believe An... |
288,974 | <p>Alright this maybe really funny but I want to know why is this wrong. We often come across identities which we prove by multiplying both the sides of the identity by a certain entity but why don't we multiply it by $0$. That way every identity will be proved in one single line. That is so stupid. I mean, by that way... | dhpratik | 60,095 | <p>the reason behind this is is a basic primary school rule.
to prove <code>x=y</code></p>
<p>this condition should also hold true.</p>
<pre><code>x/y=1
</code></pre>
<p>so if u divide both sides by <code>'0'</code></p>
<pre><code>x*0=y*0
0=0
then lhs=rhs
but
how do u answer this
lhs/rhs = 1
0/0=?
</code></pre>
|
1,303,577 | <p>I have started to learn about the properties of the <a href="http://en.wikipedia.org/wiki/Quadratic_residue" rel="nofollow">quadratic residues modulo n (link)</a> and reviewing the list of quadratic residues modulo $n$ $\in [1,n-1]$ I found the following possible property:</p>
<blockquote>
<p>(1) $\forall\ p \gt ... | Asvin | 68,188 | <p>More generally, Let $N(n) = \{\textrm{the number of quadratic residues in Z/nZ}\}$.</p>
<p>Then for $n,m$ coprime, $N(nm) = N(n)N(m)$. This will follow from the chinese remainder theorem($Z/mnZ = Z/mZ\times Z/nZ$) and basic group theory. Try working through the details.</p>
<p>Example: For $p$ prime: $N(p) = p+1/... |
1,612,353 | <blockquote>
<p>In how many ways out of $20$ students you can select $1$ treasurer, $1$ secretary and $3$ more representatives?</p>
</blockquote>
<p>I understand that for single selections I can multiply with the availability of the persons. Like for treasurer I can have $20$ options, for secretary then I have $19$ ... | N. F. Taussig | 173,070 | <p>You are correct that since the treasurer can be selected in $20$ ways and the secretary can be selected in $19$ ways, the number of ways of selecting the treasurer and secretary is $20 \cdot 19$. </p>
<p>It remains to select three representatives. There are $18$ ways to pick the first representative, $17$ ways to... |
801,562 | <p>We consider that $R$ is a commutative ring with $1_R$.</p>
<p>Each $c \in R^*$(if we see it as a constant polynomial), divides each polynomial of $R[X]$.</p>
<p>($c \in R^*$ means that $c$ is invertible.)</p>
<p>I haven't undersotod it..Could you explain it to me?</p>
<p>Does it mean that if we have a polynomial... | Bill Dubuque | 242 | <p><strong>Hint</strong> $\ $ An invertible element remains invertible in every extension ring (having the same $\,\color{#c00}1)$. Therefore $\,c c' = \color{#c00}1\,$ for $\,c'\in R\,$ yields $\, f = \color{#C00}1\cdot f = (cc')f = c(c'f),\,$ so $\,c\mid f\,$ in $\,R[x].$</p>
|
4,491,251 | <p>Per the question title, what's the easiest way to evaluate the following?
<span class="math-container">$$\int_0^{\pi/6}\sec x\,dx$$</span></p>
<p>You can do something like computing the derivatives of <span class="math-container">$\sec x$</span> and <span class="math-container">$\tan x$</span>, adding them up, compu... | egreg | 62,967 | <p>It's not difficult (and a standard exercise) to compute
<span class="math-container">$$
\int\frac{2}{\sin2t}\,dt=\int\frac{\cos^2t+\sin^2t}{\sin t\cos t}\,dt=
\int\Bigl(\frac{\cos t}{\sin t}+\frac{\sin t}{\cos t}\Bigr)\,dt=\log\lvert\tan t\rvert+c
$$</span>
How do you transform a cosine into sine? Easy, with the com... |
1,249,707 | <blockquote>
<p>Assume <span class="math-container">$V$</span> to be a finite dimensional vector space. Define the algebraic multiplicity <span class="math-container">$am(\lambda)$</span> of an eigenvalue <span class="math-container">$\lambda$</span> of a linear operator <span class="math-container">$T:V\to V$</span> a... | Marc van Leeuwen | 18,880 | <p>Don't use the Cayley-Hamilton theorem; it is less elementary than what you need. And in any case in Axler's book it (8.20) <em>follows</em> results to the effect you are asking about (8.10, 8.18). In fact Axler <em>defines</em> the (algebraic) multiplicity of <span class="math-container">$\lambda$</span> as <span cl... |
4,574,692 | <p>The theorem goes: Let <span class="math-container">$A_{1}, A_{2} ... \in \mathcal{A}$</span> with <span class="math-container">$A_{N}$</span> increasing to <span class="math-container">$\Omega$</span> and <span class="math-container">$\mu (A_{N}) < \infty$</span> for all <span class="math-container">$N \in \mathb... | geetha290krm | 1,064,504 | <p>Hint: <span class="math-container">$$\tilde{d}(f, g) $$</span> <span class="math-container">$$ = \sum_{N = m+1}^{\infty} \frac{2^{-N}}{1 + \mu(A_{N})} \int_{A_{N}} \text{min}\{1, d(f(\omega), g(\omega))\} d\mu $$</span> <span class="math-container">$$+\sum_{N = 1}^{m} \frac{2^{-N}}{1 + \mu(A_{N})} \int_{A_{N}} \text... |
3,055,324 | <p>I need some help with constructing a proof for the following statement,<span class="math-container">$ \frac{P_1 P_2}{hcf(m,n)} = lcm(P_1,P_2)$</span> where <span class="math-container">$P_1$</span> and <span class="math-container">$P_2$</span> are polynomials with real coefficients.</p>
<p>I know how to do the sam... | Bill Dubuque | 242 | <p>This proof works in any gcd domain. We use the <span class="math-container">$\,\overbrace{{\rm involution}\,\ x' :=\, ab/x}^{\rm\large cofactor\ duality\ \ }\ $</span> on the divisors of <span class="math-container">$\rm\:ab,\,$</span> which exposes <span class="math-container">$\rm\color{#c00}{cofactor\ reflecti... |
188,947 | <p>I have a rather complex looking plot which is a combination of graphics objects, generated by </p>
<pre><code>data = Import["o2ld.csv"];
data2 = Import["stemld.csv"];
a1 = ListContourPlot[data, Contours -> 25, Axes -> False,
PlotRangePadding -> 0, Frame -> False,
ColorFunction -> "DarkRainbow", Pl... | Robert Jacobson | 27,662 | <p>This is one of those "I'm not even sure how to ask this" kind of questions. You could be asking any (or all) of the following questions.</p>
<h2>How do I add a legend to appear on a <code>ListContourPlot</code>?</h2>
<p>All you need to do is add <code>PlotLegends->Automatic</code> to your <code>ListContourPlot<... |
1,375,365 | <p>Find all polynomials for which </p>
<p>What I have done so far:
for $x=8$ we get $p(8)=0$
for $x=1$ we get $p(2)=0$</p>
<p>So there exists a polynomial $p(x) = (x-2)(x-8)q(x)$</p>
<p>This is where I get stuck. How do I continue?</p>
<p><strong>UPDATE</strong></p>
<p>After substituting and simplifying I get
$(x-... | Valentin | 31,877 | <p>Following the method outlined in <a href="https://math.stackexchange.com/questions/3888/find-polynomials-such-that-x-16p2x-16x-1px/4150#4150">this answer</a> we can write the original equation in the form
$$\frac{\sigma p}{p} = \frac{\sigma^3 r}{r}$$
where $\sigma p(x) = p(2x)$ and $r(x)=8-x$. Using the "additive no... |
2,946,384 | <p>How to prove that any integer n which is not divisible by 2 or 3 is not divisible by 6?</p>
<p>The point was to prove separately inverse, converse and contrapositive statements of the given statement: "for all integers n, if n is divisible by 6, then n is divisible by 3 and n is divisible by 2".
I have the proof f... | Community | -1 | <p>There are a ton of mistakes here, unfortunately. The key issue is that you've got something like</p>
<p><span class="math-container">$$\frac{t + t^3}{t - t^5} = 1 + t^2 - \frac{1}{t^4} - \frac{1}{t^2}$$</span></p>
<p>where you've just mixed-and-matched all four terms. This is a (very) incorrect manipulation of the... |
444,486 | <p>I am teaching myself real analysis, and in this particular set of lecture notes, the <a href="http://www.math.louisville.edu/~lee/RealAnalysis/IntroRealAnal-ch01.pdf" rel="nofollow">introductory chapter on set theory</a> when explaining that not all sets are countable, states as follows:</p>
<blockquote>
<p>If $S... | Steven Clontz | 86,887 | <p>One might consider <span class="math-container">$\mathbb C$</span> and <span class="math-container">$\mathbb R^2$</span> to be isomorphic in some sense, since <span class="math-container">$f(a+bi)=(a,b)$</span> defines a bijection. If we define addition in <span class="math-container">$\mathbb R^2$</span> coordinate... |
1,054,595 | <p>I have been thinking about this problem for a while and I still can't come up with a solution. Could you please point me in a direction? Here's the problem.</p>
<pre><code>Let A, B be two 2x2 matrices, A = a b . A and B belong to M2(C). A*B - B*A = A.
c d
Prove that for every n... | DumpsterDoofus | 93,655 | <p>Assuming <span class="math-container">$\|\|_2$</span> refers to the Frobenius norm, we have the obvious statement
<span class="math-container">$$\text{MaxElement}(A-B)<\sqrt{\epsilon}$$</span>
where <span class="math-container">$\text{MaxElement}$</span> returns the element with largest absolute value, but I'm no... |
1,671,357 | <p>I'm trying to solve a minimization problem whose purpose is to optimize a matrix whose square is close to another given matrix. But I can't find an effective tool to solve it.</p>
<p>Here is my problem:</p>
<blockquote>
<p>Assume we have an unknown Q with parameter $q11, q12,q14,q21,q22,q23,q32,q33,q34,q41,q43... | Jean Marie | 305,862 | <p>Nice answer, @Samrat Mukhopadhyay</p>
<p>I would like to add something that can bring a supplementary light to the issue.</p>
<p>It is based on a property that has been overlooked, i.e., that $rank(P)=1$.</p>
<p>$P=\begin{pmatrix}
1 \\ 1 \\ 1 \\ 1
\end{pmatrix}
\begin{pmatrix}
0.48 & 0.24 & 0.16 & 0.... |
1,671,357 | <p>I'm trying to solve a minimization problem whose purpose is to optimize a matrix whose square is close to another given matrix. But I can't find an effective tool to solve it.</p>
<p>Here is my problem:</p>
<blockquote>
<p>Assume we have an unknown Q with parameter $q11, q12,q14,q21,q22,q23,q32,q33,q34,q41,q43... | Johan Löfberg | 37,404 | <p>As you ask about MATLAB, here is an implementation and solution using YALMIP (disclaimer, developed by me).</p>
<p>First, set up data and model</p>
<pre><code>G = repmat([.48 .24 .16 .12],4,1);
Q = sdpvar(4);
Q(1,3)=0;
Q(3,1)=0;
Q(2,4)=0;
Q(4,2)=0;
residual = Q*Q-G;
Objective = residual(:)'*residual(:);
Model = [... |
3,334,031 | <p>I was doing some practice problems that my professor had sent us and I have not been able to figure out one of them. The given equation is:</p>
<p><span class="math-container">$-y^2dx +x^2dy = 0$</span></p>
<p>He then asks us to verify that:</p>
<p><span class="math-container">$ u(x, y) = \frac{1}{(x-y)^2}$</span... | MafPrivate | 695,001 | <p>Actually, we can set a function and try to find a formula. Though I <strong>haven't finished</strong>, I hope it will inspire you guys.</p>
<p>Let a function <span class="math-container">$f_{n} \left(x\right)$</span> mean that the number of ways to take <span class="math-container">$n$</span> steps from <span class... |
786,827 | <blockquote>
<p>Two balls are chosen at random from a box containing 12 balls, numbered 1;2; : : : ;12. Let X be the
larger of the two numbers obtained. Compute the PMF of X, if the sampling is done</p>
<p>(a) without replacement;</p>
<p>(b) with replacement</p>
</blockquote>
<p>I understand the numerato... | Mario Carneiro | 50,776 | <p>You can use a trick to make this a regular least-squares optimization, in an extension of the method for <a href="https://math.stackexchange.com/questions/213658/get-the-equation-of-a-circle-when-given-3-points">finding the center of a circle through three points</a>. Draw a perpendicular bisector for every pair of ... |
786,827 | <blockquote>
<p>Two balls are chosen at random from a box containing 12 balls, numbered 1;2; : : : ;12. Let X be the
larger of the two numbers obtained. Compute the PMF of X, if the sampling is done</p>
<p>(a) without replacement;</p>
<p>(b) with replacement</p>
</blockquote>
<p>I understand the numerato... | AnonSubmitter85 | 33,383 | <p>For each point you are given, we assume that it satisfies the following relation:</p>
<p>$$
( x_i - x_0 )^2 + ( y_i - y_0)^2 = r^2.
$$</p>
<p>This can be written as</p>
<p>$$
\left[
\begin{array}{ccc}
1 & -2x_i & -2y_i
\end{array}
\right]
\left[
\begin{array}{c}
x_0^2 + y_0^2 -r^2 \\
x_0 \\
y_0
\end{array... |
4,107,920 | <p>the question tells me that <span class="math-container">$P(A|B)>P(A)$</span> and needs me to prove: <Br></p>
<ol>
<li><span class="math-container">$P(B|A)>P(B)$</span> <br></li>
<li><span class="math-container">$P(B^c|A)<P(B^c)$</span></li>
</ol>
<p>In general all I want to ask is do I need to care that <sp... | tommik | 791,458 | <blockquote>
<p>I think that getting an answer about (1) will be enough for me to keep going.</p>
</blockquote>
<p>What you are asking is immediate. In fact,</p>
<p>If</p>
<p><span class="math-container">$$\mathbb{P}[A|B]>\mathbb{P}[A]$$</span></p>
<p>this means that</p>
<p><span class="math-container">$$\mathbb{P}[... |
1,488,388 | <p><strong>The Statement of the Problem:</strong></p>
<p>Let $G$ be a finite abelian group. Let $w$ be the product of all the elements in $G$. Prove that $w^2 = 1$.</p>
<p><strong>Where I Am:</strong></p>
<p>Well, I know that the commutator subgroup of $G$, call it $G'$, is simply the identity element, i.e. $1$. But... | Community | -1 | <p>$w = g_1...g_r$, where $g_1,..., g_r$ are the elements of $G$ of order $2$ (all other elements can be paired with their inverses, or in the case of $e$, can be removed), then $w^2= g_1^2...g_r^2= e^r = e$.</p>
|
1,488,388 | <p><strong>The Statement of the Problem:</strong></p>
<p>Let $G$ be a finite abelian group. Let $w$ be the product of all the elements in $G$. Prove that $w^2 = 1$.</p>
<p><strong>Where I Am:</strong></p>
<p>Well, I know that the commutator subgroup of $G$, call it $G'$, is simply the identity element, i.e. $1$. But... | egreg | 62,967 | <p>Yes, your argument is essentially the way to go and the derived subgroup is indeed irrelevant. To make the proof more formal, you can do as follows.</p>
<p>The map $g\mapsto g^{-1}$ is bijective on $G$. Writing $G=\{g_1,g_2,\dots,g_n\}$ and
$$
w=g_1g_2\dots g_n
$$
we have
$$
w^{-1}=(g_1g_2\dots g_n)^{-1}=\color{red... |
3,710,377 | <p>In <em>Postmodern Analysis</em> by Jurgen Jost, the Lebesgue integral of a step function is defined as follows:</p>
<p>Suppose we have a step function <span class="math-container">$t:W\subset\mathbb{R}^d\to \mathbb{R}$</span> defined on a cube <span class="math-container">$W\subset\mathbb{R}^d$</span> given by
<sp... | Sam | 496,121 | <p>Suppose either f or g (WLOG f) is a bijection, and thus has an inverse <span class="math-container">$f^{-1}$</span>.</p>
<p>Then <span class="math-container">$f(g(x))=x \Rightarrow g(x)=f^{-1}(x)$</span>.</p>
<p>And so <span class="math-container">$g(f(x))=f^{-1}(f(x))=x$</span>.</p>
<p>So if such a pair f,g were... |
1,773,375 | <p>I am having a problem with solving this equation.</p>
<p>I've tried different ways but nothing works.</p>
<p>$$y^2\frac{dy}{dx}+2xy=e^y$$</p>
| MathCurious314 | 201,890 | <p>I am not sure what you are asking for, but I am assuming that you want to find $\frac{dy}{dx}$, so</p>
<p>$$y^2 \cdot \frac{dy}{dx} + 2xy=e^y$$</p>
<p>$$y^2 \cdot \frac{dy}{dx}=e^y - 2xy$$</p>
<p>$$\frac{dy}{dx}=\frac{e^y - 2xy}{y^2}$$</p>
|
1,773,375 | <p>I am having a problem with solving this equation.</p>
<p>I've tried different ways but nothing works.</p>
<p>$$y^2\frac{dy}{dx}+2xy=e^y$$</p>
| Nikunj | 287,774 | <p>Notice that L.H.S can be written as an exact differential, i.e:
$$\frac{d}{dy}(xy^2)=e^y$$
$$\implies d(xy^2)=e^ydy$$
Integrating both sides, we get:
$$xy^2=e^y+C$$</p>
|
51,509 | <p>Here is a problem due to Feynman. If you take 1 divided by 243 you get 0.004115226337 .... It goes a little cockeyed after 559 when you're carrying out the decimal expansion, but it soon straightens itself out and repreats itself nicely. Now I want to see how many times it repeats itself. Does it do this indefinitel... | eldo | 14,254 | <p>For this particular case (could be easily extended):</p>
<pre><code>Count[#, Max@#] &[ StringLength /@ Rest@StringSplit[ToString@N[1/243, 10^6], "00"]]
37037
</code></pre>
<p>With <code>10^6</code> digits after the decimal point there are <code>37037</code> repetitions.</p>
|
3,888,365 | <p>I have been trying to understand this limit:</p>
<p><span class="math-container">$$\lim_{x \to 0}\frac{tan(x)-sin(x)}{x^2}$$</span></p>
<p>When aplying the l'Hopital rule I arrive to the limit being <span class="math-container">$0$</span> but when doing things organically I get an indetermination:</p>
<p><span class... | RRL | 148,510 | <p>Rearrange this as <span class="math-container">$\frac{\sin x}{x} \frac{1}{\cos x}\frac{1- \cos x}{x}$</span> and use the standard limit <span class="math-container">$\frac{1- \cos x}{x} \to 0$</span> as <span class="math-container">$x \to 0$</span>.</p>
|
2,384,422 | <p>I'm really stuck on how to go about solving the following first order ODE; I've got little idea on how to approach it, and I'd really appreciate if someone could give me some hints and/or working for a solution so I can have a reference point on how to approach these sorts of problems.</p>
<p>The following is one o... | velut luna | 139,981 | <p>The particular solution is
$$y=\frac{1}{2}x^2 e^{-\sin x}$$</p>
<p>Then solve for the homogeneous solution.</p>
<p>Can you take it from here?</p>
|
1,630,480 | <p>Assume $U\subset\mathbb{R}\times\mathbb{R}^{n}=\mathbb{R}^{n+1}$, $U$ is open and $(t_0, \bf{x}$$_0)\in U$. Assume ${\bf f} (= {\bf f}(t,{\bf x})) : U \to \mathbb{R}$ is <em>continuous</em>. Then the following is called an <em>initial value problem</em>, with <em>initial condition</em>:</p>
<p>\begin{align*}
\frac{... | Alind Shukla | 600,280 | <p>If number of matrices are let us assume <span class="math-container">$M$</span>.
Then number of ways to Multiply <span class="math-container">$M$</span> Matrices are =
<span class="math-container">$$[(2N)!/(N+1)!N! ]$$</span>
Where <span class="math-container">$N=M-1$</sp... |
536,362 | <p>Let $\Sigma = \sigma(\mathcal C)$ be the $\sigma$-algebra generated by the countable collection of sets $\mathcal C \subset \mathcal{P}(X)$. How can I prove that if $\mu$ is a $\sigma$-finite measure on $(X,\Sigma)$ then $L^p(X)$ is separable for $1 \le p < \infty$?</p>
<p>I know that simple functions are dense ... | ncmathsadist | 4,154 | <p>Begin as follows. You have a countable collection of sets generating the algebra. Now take the finite intersections of all elements of this collection; it is countable. Now take the finite unions of the countable family of finite intersections. This is an algebra of sets; the smallest $\sigma$-algebra generating ... |
1,631,589 | <p>Consider the sequence $\{\frac{x^n}{n!}\}_n$ for any number $x$.</p>
<p>By choosing $m>x$ and letting $n>m$ , show that:</p>
<p>$\frac{x^n}{n!} < \frac{x^n}{m^n} < \frac{m^m}{(m-1)!}$</p>
<p>Am using the squeeze theorem , but unable to start third inequality.</p>
| Marcin Malogrosz | 140,932 | <p>Consider a graph $G=(V,E)$ such that $|V|=n$ and $|E|={n \choose 2}$ (i.e. any two vertices are connected). Than the LHS of your identity is the number of unordered pairs of edges. Any such pair is either determined by four vertices or by three. In the first case we first choose 4 vertices and then connect them in a... |
1,457,063 | <p>I am utterly confused on how to solve this problem. I found a lemma that says $|A\cup B|=|A|+|B|$ is true if the two sets are disjoint which makes sense, but how do I prove the entire statement. </p>
| Adam Hughes | 58,831 | <p>Write the disjoint unions and use your original result. That is:</p>
<blockquote>
<p>$$\begin{cases}A\cup B=A\setminus B\cup B\setminus A \cup A\cap B \\ A = A\setminus B \cup A\cap B\\ B=B\setminus A\cup A\cap B\end{cases}.$$</p>
</blockquote>
<p>Since you know that these are all disjoint, you can use the origi... |
1,457,063 | <p>I am utterly confused on how to solve this problem. I found a lemma that says $|A\cup B|=|A|+|B|$ is true if the two sets are disjoint which makes sense, but how do I prove the entire statement. </p>
| zoli | 203,663 | <p>$|A|+|B|$ contains twice those elements that are contained in both sets. So, if you want to calculate the true number of elements of $A\cup B$, $|A\cup B|$ then you have to subtract the number of elements that are taken into account twice, that is, you have to subtract $|A\cap B|$ form $|A|+|B|$. As a result</p>
<p... |
56,394 | <p>Hi!</p>
<p>While studying C*-algebras I found 2 different definitions for non degenerate representations (<em>-homomorphisms $\pi:\mathcal{A} \rightarrow B(\mathcal{h})$ where $\mathcal{A}$ is a C</em>-algebra and $B(\mathcal{h})$ is the space of bounded linear operators on some Hilbert space $\mathcal{h}$):</p>
<... | Jan Jitse Venselaar | 3,897 | <p>Yes they are. This is Proposition I.9.2 in Theory of Operator Algebras I by Takesaki.</p>
<p>Short proof:</p>
<p>2) => 1): suppose $\pi(a)\xi = 0$ for all $a$. Then $(\pi(a)\eta|\xi) = 0$ for all $\eta\in h$
and $a\in\mathcal{A}$ hence $\xi=0$.</p>
<p>1) => 2): Take $\xi \in h$ orthogonal to all $\pi(a) \eta$. Th... |
1,765,538 | <p>If $N$ is the set of all natural numbers, $R$ is a relation on $N \times N$, defined by $(a,b) \simeq (c,d)$ iff $ad=bc$, how can I prove that $R$ is an equivalence relation ?</p>
| JKnecht | 298,619 | <p>Hint:</p>
<p>You need to prove that $R$ is reflexive, symmetric and transitive.</p>
<p>I leave the first two to you. They are very straight forward.</p>
<p>Now suppose $(a,b) \simeq (c,d)$ and $(c,d) \simeq (e,f)$</p>
<p>Then $ad = bc$ and $cf=de$</p>
<p>Thus</p>
<p>$(ad)(cf)=(bc)(de)$</p>
<p>and by cancellin... |
2,439,111 | <p>By definition, a function $ f: \mathbb{R} \rightarrow\mathbb{R}$ is linear iff</p>
<ol>
<li>$f(x+y)=f(x)+f(y)$ $ \forall x,y \in \mathbb{R}$ </li>
<li>$ f(bx) = bf(x)$ $ \forall b,x \in \mathbb{R}$</li>
</ol>
<p>I am trying to prove the following statement: </p>
<p>If $ f $ is a linear map defined above then $f$... | M. Van | 337,283 | <p>It is difficult to give a first step without giving away the whole solution, since the solution consists of just one step:
$$f(x)=x \cdot f(1)=f(1) \cdot x$$</p>
|
3,856,180 | <p><span class="math-container">$X = \{0 ,1\}^{\mathbb N}$</span> be the metric space. Can anyone please tell me how to define a continuous injective function from <span class="math-container">$X = \{0 ,1\}^{\mathbb N}$</span> to the cantor set ?</p>
<p>Can anyone please give an idea ?</p>
| Community | -1 | <p>For each <span class="math-container">$x\in X$</span>, we have a sequence of zeros and ones. Meanwhile the Cantor set is the set of all real numbers in the unit interval whose ternary expansion contains no <span class="math-container">$1$</span>'s. So the natural map would be to send a given sequence <span class="... |
2,040,293 | <p>I am trying to follow this tutorial: <a href="http://ctms.engin.umich.edu/CTMS/index.php?example=InvertedPendulum&section=SystemModeling" rel="nofollow noreferrer">http://ctms.engin.umich.edu/CTMS/index.php?example=InvertedPendulum&section=SystemModeling</a></p>
<p>I am stuck to understand how to make a sta... | martini | 15,379 | <p>Yes it's true. For defining the power $x^m$ with general $m \in \mathbf R$ and $x \in (0,\infty)$, there are two possibilities: </p>
<ol>
<li><p>The direct way (using some definition of $\exp$ and $\log$ that does not use non-integral powers, e.g. the power series): We define $$ x^m := \exp(m\log x) $$
then the pro... |
2,788,498 | <p>Suppose $T([a,-b])=[−x,y]$ and $T([a,b])=[x,y]$. Find a matrix $A$ such that $T(x)=Ax$ for all $x\in\mathbb{R}^2$.</p>
| Lutz Lehmann | 115,115 | <p>As $1$ is not a root of the characteristic polynomial (of the (homogeneous) differential operator on the left side), the degree of the polynomial factor stays the same and you have to try to fit the parameters in
$$
y_p(x)=(ax+b)e^x
$$
to the equation.</p>
|
2,788,498 | <p>Suppose $T([a,-b])=[−x,y]$ and $T([a,b])=[x,y]$. Find a matrix $A$ such that $T(x)=Ax$ for all $x\in\mathbb{R}^2$.</p>
| User1234 | 235,058 | <p>Hint: $$$$Here is an alternate solution using the Method of Annihilators:$$$$
$y_1=4xe^x$ is clearly annihilated by $D^2-2D+1=(D-1)^2$ where $D$ denotes the derivative operator. Hence, using this annihilator on both sides of the original ODE, the ODE can be "rewritten" as $$(D^2+1)(D-1)^2y=(D+i)(D-i)(D-1)^2y=0$$ $$$... |
3,575,334 | <p>I am trying to show that <span class="math-container">$\int_{-b}^{b} \frac{f(N+\frac{1}{2} + it)}{e^{2\pi i(N+\frac{1}{2} + it)}-1} dt \to 0$</span> as <span class="math-container">$N \to \infty$</span> where <span class="math-container">$|f(N+1/2+it)| \le A/(1+(N+1/2)^2)$</span> for some constant <span class="math-... | J. W. Tanner | 615,567 | <p>The intersection <span class="math-container">$(A-C)\cap(C-B)$</span> is empty, </p>
<p>because any element of <span class="math-container">$A-C$</span> is not in <span class="math-container">$C$</span>, and any element of <span class="math-container">$C-B$</span> is in <span class="math-container">$C$</span>, </p>... |
1,425,433 | <p>I have problems proceeding in solving the following differential equation $$xy' + y + (y')^2 = 0.$$</p>
<p>After solving for $\frac{dy}{dx}$ in the quadratic and using the substitution $u^2 = x^2 - 4y$ in the discriminant, I obtain $\frac{du}{dx} = 2 \frac{x}{u} -1$. Please how can I proceed? </p>
| najayaz | 169,139 | <p>First make the substitution $p=y'$ to get $$px+y+p^2=0$$
Now differentiate wrt $x$ $$p'x+2p+2pp'=0$$
$$p'=\frac{-2p}{x+2p}$$
Put $p=vx\implies p'=v+xv'$
$$v+xv'=\frac{-2v}{1+2v}$$
$$xv'=\frac{-3v-2v^2}{1+2v}$$
$$\frac{1+2v}{3v+2v^2}dv=-\frac{dx}x$$
Now integrating,
$$\ln v+2\ln(2v+3)+3\ln x=c$$
Now put back $v=\fra... |
1,425,433 | <p>I have problems proceeding in solving the following differential equation $$xy' + y + (y')^2 = 0.$$</p>
<p>After solving for $\frac{dy}{dx}$ in the quadratic and using the substitution $u^2 = x^2 - 4y$ in the discriminant, I obtain $\frac{du}{dx} = 2 \frac{x}{u} -1$. Please how can I proceed? </p>
| JJacquelin | 108,514 | <p>$$xy+y+y’^2=0$$
Let : $y’=p$
$$y=-xy’-y’^2=-xp-p^2$$
$$ \frac{dy}{dp} =-p\frac{dx}{dp}-x-2p$$
$$p=y’=\frac{dy}{dx}=\frac{dy}{dp} \frac{dp}{dx} =-p-(x+2p) \frac{dp}{dx}$$
$$2p=-(x+2p) \frac{dp}{dx}$$
$$2p\frac{dx}{dp}+x=-2p$$
The solution of this first order linear ODE is :
$$x=-\frac{2}{3}p+\frac{C}{2\sqrt{p}}$$
$$... |
1,017,411 | <blockquote>
<p>Let <span class="math-container">$R$</span> be a commutative Ring with <span class="math-container">$1$</span> and <span class="math-container">$M$</span> a <span class="math-container">$R$</span>-Module. <span class="math-container">$$\varphi: \begin{cases}R & \longrightarrow \text{end}_R(M) \\ a &... | egreg | 62,967 | <p>My impression is that you're overcomplicating things.</p>
<h3>For $a\in R$, $\lambda_a\colon M\to M$ is an $R$-endomorphism of $M$</h3>
<p>This is just a simple verification.</p>
<h3>$\varphi$ is a ring homomorphism</h3>
<p>Indeed, $\varphi(ab)=\lambda_{ab}$ and this is the map
$$
x\mapsto \lambda_{ab}(x)=(ab)x=... |
216,421 | <p>How do I go about proving this? Do I have to show total boundedness (I don't know how to use the finiteness of the residue field, and this seems like something that it might pertain to).</p>
| Makoto Kato | 28,422 | <p>Let $A$ be a DVR.
Let $P$ be its maximal ideal.</p>
<p><strong>Lemma 1</strong>
$P^n/P^{n+1}$ is, as an $A$-module, isomorphic to $A/P$ for every integer $n > 0$.</p>
<p>Proof:
Let $\pi$ be a generator of $P$.
Let $\phi\colon P^n \rightarrow P^n/P^{n+1}$ be the canonical $A$-homomorphism.
Let $g\colon A \righta... |
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