qid int64 1 4.65M | question large_stringlengths 27 36.3k | author large_stringlengths 3 36 | author_id int64 -1 1.16M | answer large_stringlengths 18 63k |
|---|---|---|---|---|
1,676,347 | <p>Suppose that $V$ and $W$ are vector spaces, $g:V\rightarrow W$ a linear map. Show that g is surjective $\iff$ For any vector space $U$ the map $g^{\ast}:Hom(W,U) \rightarrow Hom(V,U)$ defined by $g^{\ast}(f) = f\circ g$ is injective.</p>
<p>I've failed too much trying to solve this problem. Any hint could be useful... | Bernard | 202,857 | <p>One way is easy: if $g$ is surjective, $g*$ is injective – in other words, for any $v\colon W\to U$ , if $v\circ g=0$, then $v=0$.</p>
<p>Conversly, suppose $g*$ is injective, and consider the linear map $p\colon W \longrightarrow \operatorname{coker} g=W/g(V)g$.</p>
<p>By definition, $g*(p)=p\circ g=0$, hence$p=0... |
1,883,459 | <p>I am trying to make a block diagonal matrix from a given matrix by multiplying the given matrix to some other matrices.
Say $A$ is an $N \times N$ matrix, I want to make an $A^\prime$ matrix with size $kN \times kN$ such that $A^\prime$ has $A$ as its diagonal element $k$ times. In fact $A^\prime$ is the direct sum... | Med | 261,160 | <p>It is not possible to find matrices $B$ and $C$ to get what you want. To give a simple example, the matrix A' is a general form of the identity matrix "I". </p>
<p>$B_{kN×N}.1_{N×N}.C_{N×kN}=I_{kN×kN}$</p>
<p>Having a column vector "B" and a row vector "C", you cannot get the identity matrix. Because the identity ... |
2,463,565 | <p>I want to use the fact that for a $(n \times n)$ nilpotent matrix $A$, we have that $A^n=0$, but we haven't yet introduced the minimal polynomials -if we had, I know how to prove this.</p>
<p>The definition for a nilpotent matrix is that there exists some $k\in \mathbb{N}$ such that $A^k=0$.</p>
<p>Any ideas?</p>
| Ben Grossmann | 81,360 | <p>Note that if $\operatorname{rank}(A^k) = \operatorname{rank}(A^{k+1})$, then $\operatorname{rank}(A^j) = \operatorname{rank}(A^k)$ for all $j \geq k$. To see that this is the case, note if $\operatorname{rank}(A^k) = \operatorname{rank}(A^{k+1})$, then the restriction of $A$ to the image of $A^{k}$ is an invertible... |
1,555,429 | <p>Hi I am trying to solve the sum of the series of this problem:</p>
<p>$$
11 + 2 + \frac 4 {11} + \frac 8 {121} + \cdots
$$</p>
<p>I know its a geometric series, but I cannot find the pattern around this. </p>
| Olivier Oloa | 118,798 | <ul>
<li>If your series is</li>
</ul>
<p>$$
S_1=11+11\times\sum_{n=1}^{\infty}\frac{2^n}{11^{n}}
$$
then you may use </p>
<blockquote>
<p>$$
x+x^2+\cdots+x^n+\cdots=\frac{x}{1-x}, \quad |x|<1, \tag1
$$ </p>
</blockquote>
<p>with $x=\dfrac2{11}$.</p>
<ul>
<li>If your series is</li>
</ul>
<p>$$
S_2=11+2+\sum_... |
3,361,153 | <p>I am given two boolean expression<br>
1) <span class="math-container">$x_1 \wedge x_2 \wedge x_3$</span><br>
2) <span class="math-container">$(x_1 \wedge x_2) \vee (x_3 \wedge x_4)$</span> </p>
<p>Now I need to know which expression is trivial and which is non-trivial. I wanted to know what is the procedure of doi... | Dr. Sonnhard Graubner | 175,066 | <p>Note that <span class="math-container">$$n\equiv 0,1,2\mod 3$$</span> so <span class="math-container">$$n^2\equiv 0,1 \mod 3$$</span></p>
|
3,361,153 | <p>I am given two boolean expression<br>
1) <span class="math-container">$x_1 \wedge x_2 \wedge x_3$</span><br>
2) <span class="math-container">$(x_1 \wedge x_2) \vee (x_3 \wedge x_4)$</span> </p>
<p>Now I need to know which expression is trivial and which is non-trivial. I wanted to know what is the procedure of doi... | Anurag A | 68,092 | <p>By division algorithm, <span class="math-container">$n=3q+r$</span>, where <span class="math-container">$r \in \{0,1,2\}$</span>. Thus
<span class="math-container">$$n^2=9q^2+6qr+r^2=3(3q^2+2qr)+r^2.$$</span>
Since <span class="math-container">$3 \mid n^2$</span>, this implies <span class="math-container">$3 \mid r... |
3,989,591 | <p>The following question I read in a book, but the book does not give proof. I doubt the correctness of the result</p>
<p>let <span class="math-container">$p>3$</span> be prime number. prove or disprove
<span class="math-container">$$(x+1)^{2p^2}\equiv x^{2p^2}+\binom{2p^2}{p^2}x^{p^2}+1\pmod {p^2}\tag{1}$$</span><... | Abhijeet Vats | 426,261 | <p>Yea, so we are partitioning <span class="math-container">$[0,2]$</span> into <span class="math-container">$n$</span> equal pieces each of length <span class="math-container">$\frac{2}{n}$</span>. In particular:</p>
<p><span class="math-container">$$x_i = 0+\frac{2i}{n} = \frac{2i}{n}$$</span></p>
<p>Then, we pick ou... |
3,715,987 | <p>The domain is: <span class="math-container">$\forall x \in \mathbb{R}\smallsetminus\{-1\}$</span></p>
<p>The range is: first we find the inverse of <span class="math-container">$f$</span>:
<span class="math-container">$$x=\frac{y+2}{y^2+2y+1} $$</span>
<span class="math-container">$$x\cdot(y+1)^2-1=y+2$$</span>
<sp... | Vishu | 751,311 | <p>How about using calculus?</p>
<p><span class="math-container">$$f(x)=\frac{x+2}{(x+1)^2}=\frac{1}{x+1}+\frac{1}{(x+1)^2} \\ f’(x) =\frac{-1}{(x+1)^2}-\frac{2}{(x+1)^3} =0 \\ \implies x=-3$$</span> which you can see clearly corresponds to a minimum. We see <span class="math-container">$f(-3) =-\frac 14$</span>. Now,... |
3,715,987 | <p>The domain is: <span class="math-container">$\forall x \in \mathbb{R}\smallsetminus\{-1\}$</span></p>
<p>The range is: first we find the inverse of <span class="math-container">$f$</span>:
<span class="math-container">$$x=\frac{y+2}{y^2+2y+1} $$</span>
<span class="math-container">$$x\cdot(y+1)^2-1=y+2$$</span>
<sp... | Michael Hardy | 11,667 | <p><span class="math-container">$$\text{You wrote: } x=\frac{y+2}{y^2+2y+1} $$</span>
<span class="math-container">$$x(y^2+2y+1)=y+2$$</span>
<span class="math-container">$$
xy^2 + ( 2x-1 )y +(x-2)=0 \tag 1
$$</span>
<span class="math-container">$$
ay^2 + by + c = 0, \quad\text{where } a=x,\, b=(2x-1), \, c = x-2 \tag ... |
688,168 | <p>$\int\limits_{y=0}^{3}\int\limits_{x=y}^{\sqrt{18-y^2}} 7x + 3y$ $dxdy$</p>
<p>Okay so I converted this into polar form because I was told to do so
I got the integral of $(7r\cos\theta + 3r\sin\theta)rdrd\theta$ where $0\le \theta \le \pi/4$ and $0\le r \le \sqrt{18}$</p>
<p>I think I'm making a mistake solving th... | walkar | 98,077 | <p>I'm not sure about polar form, but you can evaluate the integral as such:</p>
<p>$\int\limits_{y=0}^{3}\int\limits_{x=y}^{\sqrt{18-y^2}} 7x + 3y$ $dxdy$</p>
<p>$\int\limits_{0}^{3}(\frac{7}{2}x^2 + 3y\cdot x)|_{\sqrt{18-y^2}}^y$ $dy$</p>
<p>$\int\limits_{0}^{3}(\frac{7}{2}y^2 + 3y^2))-(\frac{7}{2}(18-y^2) + 3y\cd... |
688,168 | <p>$\int\limits_{y=0}^{3}\int\limits_{x=y}^{\sqrt{18-y^2}} 7x + 3y$ $dxdy$</p>
<p>Okay so I converted this into polar form because I was told to do so
I got the integral of $(7r\cos\theta + 3r\sin\theta)rdrd\theta$ where $0\le \theta \le \pi/4$ and $0\le r \le \sqrt{18}$</p>
<p>I think I'm making a mistake solving th... | Lion | 125,746 | <p>Let $I$ denote the integral value. By calculating the integral by the polar form, we have:
\begin{equation}
I=\int_0^{\frac{\pi}{4}}\int_0^\sqrt{18}7r^2\cos(\theta)+3r^2\sin(\theta)drd\theta\\
=\int_0^{\frac{\pi}{4}}\int_0^\sqrt{18}7r^2\cos(\theta)drd\theta+\int_0^{\frac{\pi}{4}}\int_0^\sqrt{18}3r^2\sin(\theta)drd\t... |
3,629,186 | <p>Assume that <span class="math-container">$x=x(t)$</span> and <span class="math-container">$y=y(t)$</span>. Find <span class="math-container">$dx/dt$</span> given the other information.</p>
<p><span class="math-container">$x^2−2xy−y^2=7$</span>; <span class="math-container">$\frac{dy}{dt} = -1$</span> when <span cla... | TravorLZH | 748,964 | <p><span class="math-container">$$x^2-2xy-y^2=7$$</span>
Now, differentiate the both side of the equation:</p>
<p><span class="math-container">$$2x{\mathrm{d}x\over\mathrm{d}t}-2y{\mathrm{d}x\over\mathrm{d}t}-2x{\mathrm{d}y\over\mathrm{d}t}-2y{\mathrm{d}y\over\mathrm{d}t}=0$$</span></p>
<p>Rearranging this equation w... |
101,526 | <p>Is there a notion of <i>"smooth bundle of Hilbert spaces"</i> (the base is a smooth finite dimensional manifold, and the fibers are Hilbert spaces) such that:</p>
<blockquote>
<p><b>1•</b> A smooth bundle of Hilbert spaces over a point is the same thing as a Hilbert space.</p>
<p><b>2•</b> If <span class="... | TaQ | 12,643 | <p><strong>This is not an answer</strong> but rather a comment to Peter Michor's answer. Anyway, I post it as an answer to get more flexibility in text formulation and to get more visibility. Namely, I think there is a crucial error which completely breaks down the argument so that generally <em>it is not possible</em>... |
615,067 | <p>I heard that Weil proved the Riemann hypothesis for finite fields. Where can I found the details of the proof? I found the following sketch but I was unable to fill the details: </p>
<p>Motivation: I try to understand the elementary theory of finite fields but I'm not an expert of algebraic geometry, it would be ni... | Dietrich Burde | 83,966 | <p>Very nice articles with many historal details are the survey articles $1,2,3,4$ by Peter Roquete, <a href="http://www.rzuser.uni-heidelberg.de/~ci3/rv4.pdf" rel="nofollow"><em>On the Riemann hypothesis in characteristic $p$, its origin and development</em></a>. The link is for the last part, part $1,2,3$ are also av... |
3,414,208 | <blockquote>
<p>In the beginning A=0. Every time you toss a coin, if you get head, you increase A by 1, otherwise decrease A by 1. Once you tossed the coin 7 times or A=3, you stop. How many different sequences of coin tosses are there?</p>
</blockquote>
<p>The tricky part of this problem is the combination of the r... | JMoravitz | 179,297 | <p>Continue flipping even after having reached <span class="math-container">$A=3$</span>.</p>
<p>There are <span class="math-container">$2^7$</span> different sequences of seven flips.</p>
<p>All sequences of the form <code>HHHxxxx</code> should have counted the same. There are <span class="math-container">$2^4$</sp... |
4,569,458 | <p>I wonder if there is any trick in this problem. The following graph is a regular hexagon with its center <span class="math-container">$C$</span> and one of the vertices <span class="math-container">$A$</span>. There are <span class="math-container">$6$</span> vertices and a center on the graph, and now assume we per... | HackR | 1,088,726 | <p>We can look at this graph as an electric network where every edge has conductance <span class="math-container">$c(xy)=1$</span> (and hence resistance <span class="math-container">$1$</span> as well) where <span class="math-container">$xy$</span> represents an edge. We label the hexagon with <span class="math-contain... |
2,405,505 | <p>How to prove that the infinite product $\prod_{n=1}^{+\infty} \left(1-\frac{1}{2n^2}\right)$ is positive ?</p>
<p>Thanks</p>
| Raffaele | 83,382 | <p>As you can see <a href="http://functions.wolfram.com/ElementaryFunctions/Sin/08/0001/" rel="nofollow noreferrer">here</a> <span class="math-container">$\sin z$</span> can be expressed by an infinite product, namely</p>
<p><span class="math-container">$$\sin z=z \prod _{n=1}^{\infty } \left(1-\frac{z^2}{\pi ^2 n^2}\r... |
4,008,987 | <p>I am reading a math book and in it, it says, "Let <span class="math-container">$V$</span> be the set of all functions <span class="math-container">$f: \mathbb{Z^n_2} \rightarrow \mathbb{R}.$</span> I know that <span class="math-container">$\mathbb{Z^n_2}$</span> is just the cyclic group of order <span class="ma... | Asker | 201,024 | <blockquote>
<p>[...] I don't get what the actual function means.</p>
</blockquote>
<p><span class="math-container">$f:\bf{Z}_2^n \to R$</span> simply assigns to each element of <span class="math-container">$\bf{Z}_2^n$</span> an element of <span class="math-container">$\bf{R}$</span>.</p>
<blockquote>
<p>How would suc... |
403,184 | <p>A (non-mathematical) friend recently asked me the following question:</p>
<blockquote>
<p>Does the golden ratio play any role in contemporary mathematics?</p>
</blockquote>
<p>I immediately replied that I never come across any mention of the golden ratio in my daily work, and would guess that this is the same for al... | Wojowu | 30,186 | <p>Golden ratio is relevant in most places where you consider Fibonacci numbers. One occurrence where it is particularly visible is the proof by Bugeaud, Mignotte and Siksek of the fact that the largest perfect power in the Fibonacci series is 144 (arXiv version <a href="https://arxiv.org/abs/math/0403046" rel="norefer... |
403,184 | <p>A (non-mathematical) friend recently asked me the following question:</p>
<blockquote>
<p>Does the golden ratio play any role in contemporary mathematics?</p>
</blockquote>
<p>I immediately replied that I never come across any mention of the golden ratio in my daily work, and would guess that this is the same for al... | Jakub Konieczny | 14,988 | <p>The golden ratio is closely related to the <a href="https://en.wikipedia.org/wiki/Zeckendorf%27s_theorem" rel="noreferrer">Zeckendorf representation</a> which is one of the simplest examples of a numeration system (other than the usual base-<span class="math-container">$b$</span> represenations). As such, it's an im... |
403,184 | <p>A (non-mathematical) friend recently asked me the following question:</p>
<blockquote>
<p>Does the golden ratio play any role in contemporary mathematics?</p>
</blockquote>
<p>I immediately replied that I never come across any mention of the golden ratio in my daily work, and would guess that this is the same for al... | Denis Serre | 8,799 | <p>In numerical analysis, a basic problem consists in approaching a solution <span class="math-container">$x^*$</span> of <span class="math-container">$f(x)=0$</span>, where <span class="math-container">$f:{\mathbb R}\rightarrow{\mathbb R}$</span> is at least continuous and more often <span class="math-container">$C^2$... |
3,667,798 | <p>Find values of <span class="math-container">$a$</span> for which the integral <span class="math-container">$$\int^{\infty}_{0}e^{-at}\sin(7t)dt$$</span> converges</p>
<p>What i try</p>
<p><span class="math-container">$$\int^{\infty}_{0}e^{-at}\sin(7t)dt$$</span></p>
<p><span class="math-container">$$=\frac{1}{a^2... | J.G. | 56,861 | <p>For <span class="math-container">$z\in\Bbb C$</span>,<span class="math-container">$$\int_0^\infty e^{-zt}dt=\frac1z(1-\lim_{t\to\infty}e^{-zt}),$$</span> which converges iff<span class="math-container">$$0=\lim_{t\to\infty}|e^{-zt}|=\lim_{t\to\infty}e^{-(\Re z)t},$$</span>i.e. <span class="math-container">$\Re z>... |
1,336,424 | <p>Find the minimum distance between point $M(0,-2)$ and points $(x,y)$ such that: $y=\frac{16}{\sqrt{3}\,x^{3}}-2$ for $x>0$ .</p>
<p>I used the formula for distance between two points in a plane to get: $$d=\sqrt{x^{2}+\frac{256}{3x^{6}}}$$ And this is where I cannot come up with how to proceed. I tried calculus ... | georg | 144,937 | <p>HINT:</p>
<p>Searching a minimum value for $d^2$. I would say - in point minimum for $d^2$ will be minimized too 'd'.</p>
|
101,972 | <p>I have a grammatically computable function $f$, which means that a grammar $G = (V,\Sigma,P,S)$ exists, so that</p>
<p>$SwS \rightarrow v \iff v = f(w)$.</p>
<p>Now I have to show that, given a grammatically computable function $f$, a Turing Machine $M$ can be constructed, so that $M$ computes $f$.</p>
<p>Here is... | Dilip Sarwate | 15,941 | <p>I am not sure what <em>Euclid's algorithm</em> in the title of the question is referring to, but as Marc van Leeuwen says, polynomial long division is the way to go. Since doing it <em>by hand</em> is asked about, I will suggest the following based on long experience of doing such divisions by hand and making many ... |
426,306 | <p>If <span class="math-container">$K = \mathbb{Q}(\sqrt{d})$</span> is a real quadratic field, then any unit <span class="math-container">$u \in \mathcal{O}_K^\times$</span> with <span class="math-container">$u > 1$</span> must not be too small: indeed, such a <span class="math-container">$u = u_1 + u_2 \sqrt{d}$</... | Wojowu | 30,186 | <p>There will not be a smallest one. As you say, <span class="math-container">$O_K^\times$</span> has rank <span class="math-container">$2$</span>, meaning it has two multiplicatively independent generators <span class="math-container">$u_1,u_2$</span>, which we may take to be positive. This is equivalent to saying <sp... |
3,803,360 | <p>Convergence of <span class="math-container">$\sum\sum_{k, n=1}^\infty\frac{1}{(n+3)^{2k}}$</span>.</p>
<p>What I tried:</p>
<p>For the iterated summation, <span class="math-container">$\sum_{n=1}^\infty(\sum_{k=1}^\infty\frac{1}{(n+3)^{2k}})=\sum_{n=1}^\infty\lim_{k\to\infty}\frac{1-(\frac{1}{n+3})^{2k}}{1-(\frac{1}... | Eric Towers | 123,905 | <p>The <a href="https://en.wikipedia.org/wiki/Geometric_series" rel="nofollow noreferrer">geometric series</a> doesn't start at <span class="math-container">$k = 0$</span>, so the numerator is not <span class="math-container">$1$</span>. (That is, the numerator is the first term in the series, which is not <span class... |
684,755 | <p>In section 10 of <em>Topology</em> by Munkres, the minimal uncountable well-ordered set $S_{\Omega}$ is introduced. Furthermore, it is remarked that,</p>
<blockquote>
<p>Note that $S_{\Omega}$ is an uncountable well-ordered set every section of which is countable. Its order type is in fact uniquely determined by ... | user642796 | 8,348 | <p>This stems from the fact that given two well-ordered sets $X, Y$ exactly one of the following holds (<a href="https://math.stackexchange.com/q/332793/8348">see this previous question</a>):</p>
<ol>
<li>$X$ is order-isomorphic to $Y$;</li>
<li>There is a (unique) $a \in X$ such that $\{ x \in X : x < a \}$ is ord... |
2,985,172 | <p>Let <span class="math-container">$f(x) = \dfrac{1}{3}x^3 - x^2 - 3x.$</span> Part of the graph <span class="math-container">$f$</span> is shown below. There is a maximum point at <span class="math-container">$A$</span> and a minimum point at <span class="math-container">$B(3,-9)$</span>. </p>
<p><a href="https://i.... | Parcly Taxel | 357,390 | <p>All your stated answers (to the problems before (b)(iii)) are correct. For (b)(iii), do the calculation in steps:</p>
<ul>
<li>Reflect <span class="math-container">$B$</span> in the <span class="math-container">$x$</span>-axis. This sends <span class="math-container">$(3,-9)$</span> to the point <span class="math-c... |
3,143,649 | <p>I am reading the book <em>Random perturbation of dynamical sustem</em> of Freidlin and Wantzell (2nd edition). On page 20, they define a Markov process as follow:</p>
<blockquote>
<p>Let <span class="math-container">$(\Omega ,\mathcal F,\mathbb P)$</span> a probability space and <span class="math-container">$(X,\... | Gautam Shenoy | 35,983 | <p>To put it in simple words,
Q1: You begin in state <span class="math-container">$x$</span> with probability 1.</p>
<p>Q2: (The Markov property: ) Given the entire history upto time <span class="math-container">$t$</span> (that's what <span class="math-container">$|\mathcal{F_t}$</span> means), the present sample <s... |
4,563,707 | <p>Sequence given : 6, 66, 666, 6666. Find <span class="math-container">$S_n$</span> in terms of n</p>
<p>The common ratio of a geometric progression can be solved is <span class="math-container">$\frac{T_n}{T_{n-1}} = r$</span>, where r is the common ratio and n is the</p>
<p>When plugging in 66 as <span class="math-c... | insipidintegrator | 1,062,486 | <p>Let’s look at it another way:
<span class="math-container">$$6666=6\times 1111=6\times \dfrac{9999}{9}$$</span><span class="math-container">$$=\dfrac69\cdot(10^4-1)=\dfrac23(10^4-1).$$</span>
Thus, the general term is <span class="math-container">$$T_n=\underbrace{6666…6}_{n \ \text{times}\ 6}=6\times\dfrac{\overbra... |
1,903,263 | <p>I am developing an algorithm that approximates a curve using a series of linear* segments. The plot below is an example, with the blue being the original curve, and the red and yellow being an example of a two segment approximation. The x-axis is time and the y-axis is attenuation in dB. The ultimate goal is to use ... | Kuifje | 273,220 | <p>Your problem reminds me of a similar one developed <a href="http://dspace.mit.edu/bitstream/handle/1721.1/5097/OR-300-94-26970351.pdf" rel="nofollow">here</a>, in Chapter 4. </p>
<p>It is solved with a graph theory approach, more specifically with a shortest path algorithm that gives you the optimal set of knots.</... |
7,080 | <p>What is the right definition of the symmetric algebra over a graded vector space V over a field k?</p>
<p>More generally: What is the right definition of the symmetric algebra over an object in a symmetric monoidal category (which is suitably (co-)complete)?</p>
<p>Two possible definitions come to my mind:</p>
<p... | Qiaochu Yuan | 290 | <p>My understanding is that the "right" way to define the symmetric algebra comes from a braiding that tells you how the symmetric group acts on tensor products. And an easy and general way to get such a braiding is to consider the category of representations of a <a href="https://mathoverflow.net/questions/4640/are-s... |
417,181 | <p>We have to prove that if the difference between two prime numbers greater than two is another prime,the prime is $2$.
It can be proved in the following way.</p>
<p>1)$Odd -odd =even$. </p>
<p>Therefore the difference will always even.</p>
<p>2)The only even prime number is $2$.Therefore the difference will be $2$... | A.S | 24,829 | <p>The series $$\sum_{n=4}^\infty \frac{1}{n\log n \log\log n}$$ does not converge. Use the integral test.</p>
|
69,658 | <p>Given a fiber bundle $f: E\rightarrow M$ with connected fibers we call the image $f^*(\Omega^k(M))\subset \Omega^k(E)$ the subspace of basic forms. Clearly, for any vertical vector field $X$ on $E$ we have that the interior product $i_X(f^*\omega)$ and the Lie derivative $L_X(f^*\omega)$ vanish for all $\omega \in \... | Willie Wong | 1,543 | <p>If you added a connectedness assumption (which is essential, as I mentioned in my comment to your main question), what you desire follows from </p>
<p><strong>Prop</strong> Let $f:M\to N$ be a submersion such that for every $y\in N$, $f^{-1}(y)$ is a connected submanifold of $M$. Then if $\mu\in\Omega^k(M)$ is a di... |
2,601,601 | <p>Consider the complex matrix $$A=\begin{pmatrix}i+1&2\\2&1\end{pmatrix}$$ and the linear map $$f:M(2,\mathbb{C})\to M(2,\mathbb{C}),\qquad X\mapsto XA-AX.$$</p>
<p>I want to find a basis of $\ker f$.</p>
<p>I already know the canonical basis $\{E_{11},E_{12},E_{21},E_{22}\}$ and computed $$f(E_{11})=\begin{... | GNUSupporter 8964民主女神 地下教會 | 290,189 | <p>From your computations, we know that the range of $f$ has dimension two, so we just need two linearly independent matrices $K_1$ and $K_2$ so that $f(K_1)=f(K_2)=0$.</p>
<p>Set $K_1=(E_{11}+E_{22})/2$ and $K_2=(E_{12}+E_{21})/2$. From the question's computation results, it's easy to see that $f(K_1)=f(K_2)=0$. Fi... |
2,601,601 | <p>Consider the complex matrix $$A=\begin{pmatrix}i+1&2\\2&1\end{pmatrix}$$ and the linear map $$f:M(2,\mathbb{C})\to M(2,\mathbb{C}),\qquad X\mapsto XA-AX.$$</p>
<p>I want to find a basis of $\ker f$.</p>
<p>I already know the canonical basis $\{E_{11},E_{12},E_{21},E_{22}\}$ and computed $$f(E_{11})=\begin{... | Mohammad Riazi-Kermani | 514,496 | <p>You have already found$$f(E_{11})=\begin{pmatrix}0&2\\-2&0\end{pmatrix},f(E_{12})=\begin{pmatrix}2&0\\0&-2\end{pmatrix},f(E_{21})=\begin{pmatrix}-2&0\\0&2\end{pmatrix},f(E_{22})=\begin{pmatrix}0&-2\\2&0\end{pmatrix}$$Note that $$f(E_{11})+f(E_{22})=\begin{pmatrix}0&0\\0&0\end{... |
2,886,973 | <p>The Wikipedia article gives an <a href="https://en.wikipedia.org/wiki/Gauss%E2%80%93Bonnet_theorem#Interpretation_and_significance" rel="nofollow noreferrer">interesting example</a> of the Gauss-Bonnet theorem:</p>
<blockquote>
<p>As an application, a torus has Euler characteristic 0, so its total curvature must ... | Julian Rosen | 28,372 | <p>There is a version of the Gauss-Bonnet formula that works for manifolds that are not smooth.</p>
<p>Gaussian curvature is a curvature density, and its integral over a region gives the total curvature in that region. There is also a notion of point curvature: if you have a bunch of angles summing to $\theta$ that me... |
253,271 | <p>I recently found that <code>\[Gradient]</code> and <code>\[InlinePart]</code> both expand (contract) to special symbols in MMA.</p>
<p>So far as I can tell (see <a href="https://mathematica.stackexchange.com/questions/134506/inlinepart-what-is-it-and-what-happened-to-it">InlinePart. What is it and what happened to i... | Adam | 74,641 | <p>I copy pasted the input strings from the Listing of (not all) Named Characters, excluding the unicode characters and dashes. There are 1009 elements in that <code>list</code>.</p>
<p>I went grepping for the text <code>\[Gradient]</code>.</p>
<ul>
<li>Mathematica/SystemFiles/Libraries/Linux-x86-64/libWolframEngine.s... |
384,450 | <p>I don't know what this double-arrow $\twoheadrightarrow$ means!</p>
| FiveLemon | 76,591 | <p>This is usually used in category theory to denote an <a href="http://en.wikipedia.org/wiki/Epimorphism" rel="nofollow noreferrer">epimorphism</a>. </p>
<p>Related question:
<a href="https://math.stackexchange.com/questions/20015/special-arrows-for-notation-of-morphisms">Special arrows for notation of morphisms</a>... |
29,703 | <p>For an <a href="http://www.bekirdizdaroglu.com/ceng/Downloads/ISCE10.pdf">image denoising problem</a>, the author has a functional $E$ defined </p>
<p>$$E(u) = \iint_\Omega F \;\mathrm d\Omega$$</p>
<p>which he wants to minimize. $F$ is defined as </p>
<p>$$F = \|\nabla u \|^2 = u_x^2 + u_y^2$$</p>
<p>Then, the ... | Glen Wheeler | 4,322 | <p>This is essentially a matter of definitions. The steepest descent gradient flow of a functional $F$ in an inner product space $S(M,N)$ (for example) is a family $u:M\times [0,T)\rightarrow N$ which satisfies
$$
\partial_t F = - \lVert u_t \rVert^2.
$$
For example, suppose $\Sigma$ is a surface immersed in $\mathb... |
3,276,877 | <blockquote>
<p>Insert <span class="math-container">$13$</span> real numbers between the roots of the equation: <span class="math-container">$x^2 +x−12 = 0$</span> in a few ways that these <span class="math-container">$13$</span> numbers together with the roots of the equation will form the first <span class="math-co... | PM. | 416,252 | <p>Exterior angle of triangle = sum of 2 interior opposite angles
<span class="math-container">$$
\angle ACB = 2 \angle EAC \\
\angle ABC = 2 \angle DAB \\
$$</span>
<span class="math-container">$$
\Rightarrow \angle EAC + \angle DAB = 0.5(\angle ACB + \angle ABC) = 0.5(180-\alpha)
$$</span>
<span class="math-container... |
37,052 | <p>This is my first question with mathOverflow so I hope my etiquette is up to par here.</p>
<p>My question is regarding a <span class="math-container">$3\times3$</span> magic square constructed using the la Loubère method (see <a href="http://en.wikipedia.org/wiki/Magic_square#Method_for_constructing_a_magic_square_of... | Marcel Bischoff | 10,718 | <p>One should be able to obtain the formula from the appendix of:</p>
<ul>
<li>Hellmut Baumgärtel, Matthias Jurke, Fernando Lledó, <em>Twisted duality of the CAR-Algebra</em>, J.Math.Phys. 43 (2002) 4158-4179, <a href="https://doi.org/10.1063/1.1483376" rel="nofollow noreferrer">https://doi.org/10.1063/1.1483376</a>, <... |
2,445,023 | <p>I have a trouble in calculating which function grows faster. </p>
<p>$f(n) = 3\log_4 n + \sqrt{n} + 3 \\
g(n) = 4\log_3 n + \log n + 200$</p>
<p>Can someone let me know how to solve this? </p>
| Penguino | 90,137 | <p>You can ignore the constants at the end because they don't effect the growth rate of the functions. If you then compare e to the power of f and g, you will see that e^g is of the form cn^2, while e^f is of the form cne^sqrt(n). Can you tell which of these grows faster?</p>
|
2,107,837 | <p>Let $f:\mathbb{R}^2\to\mathbb{R}^2$ be a continuous map such that $\forall \ x : |x-f(x)|<5777$. </p>
<p>Show that $f$ is surjective.</p>
| John | 253,426 | <p>The notation is not very precise, but $X\cap Y$ is the event "both $X$ and $Y$ happen". This means that "I do not get number 3 on the first thrown" AND "I do not get number 3 on the first four thrown", which is equivalent to say "I do not get number 3 on the first four thrown", which is $Y$. </p>
|
2,107,837 | <p>Let $f:\mathbb{R}^2\to\mathbb{R}^2$ be a continuous map such that $\forall \ x : |x-f(x)|<5777$. </p>
<p>Show that $f$ is surjective.</p>
| Community | -1 | <p>Note that $X\cap Y=Y$ if and only if $Y\subset X$. </p>
|
3,669,937 | <p>I had this problem where i had the application <span class="math-container">$\varphi: \mathbb Z[i] \Rightarrow \mathbb Z/(2)$</span> where <span class="math-container">$\varphi(a+bi)=\bar{a}+\bar{b}$</span>. I had to find the kernel and prove that is a factor ideal. I proofed that the kernel is formed by all the com... | Pedro Juan Soto | 601,282 | <p>Since <span class="math-container">$\mathbb{Z}[i]$</span> is a <a href="https://en.wikipedia.org/wiki/Gaussian_integer#Euclidean_division" rel="nofollow noreferrer">Euclidean domain</a> and this implies that it is a <a href="https://en.wikipedia.org/wiki/Gaussian_integer#Principal_ideals" rel="nofollow noreferrer">P... |
167,262 | <p>I make a circle with radius as below</p>
<pre><code>Ctest = Table[{0.05*Cos[Theta*Degree], 0.05*Sin[Theta*Degree]}, {Theta, 1, 360}] // N;
</code></pre>
<p>And herewith is my list of data points</p>
<pre><code>pts = {{0., 0.}, {0.00493604, -0.00994539}, {0.00987001, -0.0198918}, {0.0148019, -0.0298392}, {0.019731... | jkuczm | 14,303 | <p>You could use symmetry of your equation to "factorize" problem of finding solutions. Since some variables have same coefficients, your equation is invariant with respect to permutations of those variables. You could group them, and replace with lower number of variables, representing sums of original variables.</p>
... |
1,743,482 | <p>I was doing this question on convergence of improper integrals where in our book they have used the fact that $2+ \cos(t) \ge1$. Can somebody prove this?</p>
| MooS | 211,913 | <p>On $(\mathbb Z/p\mathbb Z)^*=\{1,2, \dotsc, p-1\}$ consider the equivalence relation, defined by $x \sim y$ iff $x=-y$ or $x=y^{-1}$ or $x=-y^{-1}$.</p>
<p>Since $p$ is odd, we have $x \neq -x$, i.e. any equivalence class has $2$ or $4$ elements:</p>
<ul>
<li>If $x^2=1$, we have $[x] = \{x,-x\}$</li>
<li>If $x^2=-... |
423,718 | <p>I have a general question, that deals with the question how I am able to find out whether a particular curve(in $\mathbb{C}$) is positively oriented? Take e.g. $ y(t)=a+re^{it}$. Obviously this one is positively oriented, but is there a fast general method to proof this? </p>
| Fly by Night | 38,495 | <p>Orientation only really applies to simple, closed curves. </p>
<p>For a general curve, you might find the winding number to be useful.</p>
<p>See: <a href="http://mathworld.wolfram.com/ContourWindingNumber.html" rel="nofollow">http://mathworld.wolfram.com/ContourWindingNumber.html</a></p>
|
1,443,335 | <p>The rank of a linear transformation from V into W is defined:</p>
<blockquote>
<p>If V is finite-dimensional, the <em>rank</em> of T is the dimension of the range of T and ...</p>
</blockquote>
<p>However, there is no guarantee the range of T is finite-dimensional, in which case the dimension of it cannot be def... | user133281 | 133,281 | <p>Note that $m$ is an integer iff $15-2n$ is a divisor of $17+n$. Now, $15 -2n \mid 17+n$ if and only if $$15-2n \mid 2(17+n)+(15-2n) = 49$$ (note that $15-2n$ is odd). So we find an integral value of $m$ if and only if $15-2n$ is one of the divisors $-49$, $-7$, $-1$, $1$, $7$ or $49$ of $49$. This corresponds to $n$... |
361,862 | <p>I would like you to expose and explain briefly some examples of theorems having some hypothesis that are (as far as we know) actually necessary in their proofs but whose uses in the arguments are extremely subtle and difficult to note at a first sight. I am looking for hypothesis or conditions that appear to be almo... | Timothy Chow | 3,106 | <p>This is not a perfect example because the subtle hypotheses in question were not "unnoticed"; nevertheless I think it fulfills several of your other criteria. Let us define the "Strong Fubini theorem" to be the following statement:</p>
<blockquote>
<p>If <span class="math-container">$f:\mathbb{R}^2 \to \mathbb{R... |
1,231,365 | <p>I am from a non-English speaking country. Should we say monotonous function or monotonic function?</p>
| Unit | 196,668 | <p>"Monotonic" or "monotone", but not "monotonous" (boring).</p>
|
1,231,365 | <p>I am from a non-English speaking country. Should we say monotonous function or monotonic function?</p>
| dilpreet | 230,441 | <p>monotonic for sure is to be used. Monotonic functions are those functions which are either increasing or decreasing. They are such that for each specific value of x there is a unique y(value of function) which does not repeat for any other x.</p>
|
381,011 | <p>I should prove this claim:</p>
<blockquote>
<p>Every undirected graph with n vertices and $2n$ edges is connected.</p>
</blockquote>
<p>If it is false I should find a counterexample.
I was thinking to consider the complete graph with $n$ vertices. Such a graph is connected and contains $\frac{n(n-1)}{2}$ nodes. ... | ccorn | 75,794 | <p>Consider two copies of graphs with $m$ vertices amd $2m$ edges, e.g. of $K_5$, the complete graph with $m=5$ vertices. With two copies, you have $n=2m$ vertices and $2n$ edges, but the graph is not connected. Therefore the claim is false.</p>
|
381,011 | <p>I should prove this claim:</p>
<blockquote>
<p>Every undirected graph with n vertices and $2n$ edges is connected.</p>
</blockquote>
<p>If it is false I should find a counterexample.
I was thinking to consider the complete graph with $n$ vertices. Such a graph is connected and contains $\frac{n(n-1)}{2}$ nodes. ... | Douglas S. Stones | 139 | <p>The claim is true for up to $6$ vertices. The smallest counterexample (unique up to isomorphism) is this graph on $7$ vertices:</p>
<p><img src="https://i.stack.imgur.com/nxUhB.png" alt="Smallest counterexample"></p>
<p>The vertices are ascribed their degree, so we can easily verify there are $14$ edges via the <... |
3,628,358 | <p>As stated, I need to prove that, up to isomorphism, the only simple group of order <span class="math-container">$p^2 q r$</span>, where <span class="math-container">$p, q, r$</span> are distinct primes, is <span class="math-container">$A_5$</span> (the alternating group of degree 5).</p>
<p>Now I know the following... | Derek Holt | 2,820 | <p>Here is a sketch solution. I can give more detail, but it depends on which results you are familiar with.</p>
<p>Let <span class="math-container">$G$</span> be simple of order <span class="math-container">$p^2qr$</span>.
By Burnside's Transfer Theorem, <span class="math-container">$p$</span> must be the smallest of... |
54,878 | <p>Consider the 2 parameter family of linear systems </p>
<p>$$\frac{DY(t)}{Dt} = \begin{pmatrix}
a & 1 \\
b & 1 \end{pmatrix} Y(t)
$$</p>
<p>In the ab plane, identify all regions where this system posseses a saddle, a sink, a spiral sink, and so on. </p>
<p>I was able... | 40 votes | 85,506 | <p>Summarizing the comments: the best way to begin is to look at determinant $a-b$ and trace $a+1$: </p>
<ul>
<li>$a-b<0$: saddle</li>
<li>$a-b> 0$ and $a+1=0$: stable center</li>
<li>$a-b> 0$ and $a+1<0$: stable node or spiral, depending on $(a+1)^2-4(a-b)$ being positive or negative</li>
<li>$a-b> 0$ ... |
697,984 | <p>I want to check whether the position operator $A$, where $Af(x)=xf(x)$ , is self-adjoint. For this to be true it has to be Hermitian and also the domains of it and its adjoint must be equal. The Hilbert space I'm working with is of course $L^2(\mathbb{R}) $ with the natural inner product. The problem I'm having is w... | Jimmy R. | 128,037 | <p>The event on the LHS in english can be described as "The event that either A but not B or B but not A occurs". </p>
<p>This event is the union of two disjoint sets $A \cap B^c$ and $A^c \cap B$. They are disjoint because $$(A\cap B^c) \cap(A^c \cap B)=(A\cap A^c)\cap (B\cap B^c)=\emptyset\cap\emptyset=\emptyset$$Th... |
1,531,291 | <p>I want to find the radius of convergence of </p>
<p><span class="math-container">$$\sum_{k = 0}^{\infty}\frac{ k^{2 k + 5} \ln^{10} k \ln \ln k}{\left(k!\right)^2} \,x^k$$</span></p>
<p>I know formulae
<span class="math-container">$$R=\dfrac{1}{\displaystyle\limsup_{k\to\infty} \sqrt[k]{\left\lvert a_k\right\rver... | xpaul | 66,420 | <p>You can use
<span class="math-container">$$ R=\lim_{k\to\infty}\frac{a_k}{a_{k+1}}. $$</span>
In fact,
<span class="math-container">\begin{eqnarray}
R&=&\lim_{k\to\infty}\frac{ k^{2 k + 5} \ln^{10} k \ln \ln k}{\left(k!\right)^2}\frac{\left((k+1)!\right)^2}{ (k+1)^{2 (k+1) + 5} \ln^{10} (k+1) \ln \ln (k+1)}... |
189,380 | <p>How can I solve this ODE:</p>
<p>$y(x)+Ay'(x)+Bxy'(x)+Cy''(x)+Dx^{2}y''(x)=0$</p>
<p>Can you please also show the derivation.</p>
| Tunococ | 12,594 | <p><a href="http://en.wikipedia.org/wiki/Frobenius_method" rel="nofollow">Frobenius method</a> is the most general method I know for this case. Assume your solution is of the form $y = x^r\sum_{n=0}^\infty a_n x^n$, plug it in, and solve for $r$ and $a_n$. You should get two $r$, say $r_1$ and $r_2$. If $r_1 - r_2$ is ... |
3,541,897 | <p>While searching for non-isomorph subgroups of order <span class="math-container">$2002$</span> I just encountered something, which I want to understand. Obviously I looked for abelian subgroups first and found <span class="math-container">$2002=2^2*503$</span> so we have the groups
<span class="math-container">$$
\m... | Olivier Roche | 649,615 | <p>Here's where the hypothesis <span class="math-container">$\gcd (n, m) = 1$</span> plays a role : if <span class="math-container">$d := \gcd(n, m) \neq 1$</span>, then <span class="math-container">$\mathbb{Z} /nm \mathbb{Z}$</span> has an element of order <span class="math-container">$d^2$</span> but <span class="mat... |
2,878,508 | <p>How can I determine $ f(x)$ if $f(1-f(x))=x$ for all real $x$?
I have already recognized one problem caused from this: it
follows that $ f(f(x))=1-x $, which is discontinuous. So how can I construct a function $f(x)$?</p>
<p>Best regards and thanks,
John</p>
| Adrian Keister | 30,813 | <p>This is a partial answer.</p>
<p>We know that $f(x)$ is invertible, because $f^{-1}(x)=1-f(x),$ from the original; from here we get the very interesting relationship of $f(x)+f^{-1}(x)=1.$ Suppose we try to find out what $f(0)$ is (set it equal to $a$). By repeated alternating applications of $f$ and the equation $... |
184,266 | <p>Let $a,b,c$ and $d$ be positive real numbers such that $a+b+c+d=4.$ </p>
<p>Prove the inequality </p>
<blockquote>
<p>$$a^2bc+b^2cd+c^2da+d^2ab \leq 4 .$$ </p>
</blockquote>
<p>Thanks :) </p>
| Michael Rozenberg | 190,319 | <p>Let $\{a,b,c,d\}=\{x,y,z,t\}$, where $x\geq y\geq z\geq t$.</p>
<p>Hence, since $(x,y,z,t)$ and $(xyz,xyt,xzt,yzt)$ are the same ordered,</p>
<p>by Rearrangement and AM-GM we obtain:
$$a^2bc+b^2cd+c^2da+d^2ab=a\cdot abc+b\cdot bcd+c\cdot cda+d\cdot dab\leq$$
$$\leq x\cdot xyz+y\cdot xyt+z\cdot xzt+t\cdot yzt=xy(xz... |
510,151 | <p>Prove by induction that $2k(k+1) + 1 < 2^{k+1} - 1$ for $ k > 4$.
Can some one pls help me with this?</p>
<p>I reformulated like this</p>
<p>$ 2k(k+1) + 1 < 2^{k+1} - 1 $</p>
<p>$ 2k^2+2k+2<2^{k+1}$</p>
<p>and I tried like this
Take $k=k+1$</p>
<p>$ 2^{k+2} -1 > 2(k+1)(k+2) + 1 $</p>
<p>$... | kedrigern | 97,299 | <p>Sequence $1/n$ converges to $0$ iff for every $\varepsilon>0$ there is $n\in\mathbb{N}$ so that for every $k > n$ $$d(\frac1k,0) < \varepsilon.$$ Since $d$ is the discrete metric and $\frac1k > 0$, $d(\frac1k,0) = 1$. So if you take for example $\varepsilon=\frac12$ there is no $n$ as required above.</p>... |
2,878,814 | <ol>
<li>If a function $f(x)$ is continuous and increasing at point $x=a,$ then there is a nbhd $(x-\delta,x+\delta),\delta>0$ where the function is also increasing.</li>
<li>if $f' (x_0)$ is positive, then for $x$ nearby but smaller than
$x_0$ the values $f(x)$ will be less than $f(x_0)$, but for $x$
nearby but la... | WW1 | 88,679 | <p>Let $AB = AB'\equiv x$
then
$$ BC = x\cos\theta
\\AC = x\sin\theta $$</p>
<p>And
$$ AC'=x \sin(\theta-\alpha)
\\ \implies AC'=x\sin\theta \cos\alpha -x\cos\theta \sin \alpha
\\AC'= AC \cos\alpha - BC\sin\alpha
$$</p>
|
3,084,934 | <p>I want to prove or disprove that the Fourier transform <span class="math-container">$\mathcal F \colon (\mathcal S(\mathbb R^d), \lVert \cdot \rVert_1) \to L^1(\mathbb R^d)$</span> is unbounded, where <span class="math-container">$\lVert\cdot \rVert_1$</span> denotes the <span class="math-container">$L^1(\mathbb R^d... | Locally unskillful | 494,915 | <p>We say that <span class="math-container">$\lim_{x\to\ a} f(x) = \infty$</span>, if <span class="math-container">$\forall M>0, \exists δ>0$</span> s.t. whenever <span class="math-container">$\vert x-a \vert < δ$</span>, then <span class="math-container">$f(x) > M $</span>.</p>
<p>We say that <span class=... |
34,487 | <p>A few years ago Lance Fortnow listed his favorite theorems in complexity theory:
<a href="http://blog.computationalcomplexity.org/2005/12/favorite-theorems-first-decade-recap.html" rel="nofollow">(1965-1974)</a>
<a href="http://blog.computationalcomplexity.org/2006/12/favorite-theorems-second-decade-recap.html" rel=... | Suresh Venkat | 972 | <p>There's the <a href="http://lucatrevisan.wordpress.com/2008/08/06/bounded-independence-and-dnfs/" rel="nofollow">Bazzi/Razborov</a>/<a href="http://www.cs.toronto.edu/~mbraverm/Papers/FoolAC0v7.pdf" rel="nofollow">Braverman</a> sequence on fooling AC0 circuits. </p>
|
704,921 | <p>This is the question:
$$
\frac{(2^{3n+4})(8^{2n})(4^{n+1})}{(2^{n+5})(4^{8+n})} = 2
$$
I've tried several times but I can't get the answer by working out.I know $n =2$, can someone please give me some guidance? Usually I turn all the bases to 2, and then work with the powers, but I probaby make the same mistake ever... | Cookie | 111,793 | <p>We have
\begin{align}
\frac{(2^{3n+4})(8^{2n})(4^{n+1})}{(2^{n+5})(4^{8+n})} &= \frac{(2^{3n+4})(2^{3})^{2n}(2^2)^{n+1}}{(2^{n+5})(2^2)^{8+n}} \\
&=\frac{2^{3n+4+6n+2n+2}}{2^{n+5+16+2n}} \\
&=\frac{2^{11n+6}}{2^{3n+21}} \\
&=2^{(11n+6)-(3n+21)} \\
&=2^{8n-15}
\end{align}</p>
<p>Also from the ori... |
1,114,502 | <p>I attempted the following solution to the birthday "paradox" problem. It is not correct, but I'd like to know where I went wrong.</p>
<p>Where $P(N)$ is the probability of any two people in a group of $N$ people having the same birthday, I consider the first few values.</p>
<p>For two people, the probability that ... | David | 119,775 | <p>$\def\hor{\ \hbox{or}\ }$Continuing your notation, write $P(ABC)$ for the probability that $A,B$ and $C$ share a birthday. We can also write something like $P(ABBC)$, but it is the same as $P(ABC)$. By the principle of inclusion/exclusion we have
$$\eqalign{P(3)
&=P(AB\hor AC\hor BC)\cr
&=P(AB)+P(AC)+P... |
428,843 | <p>Consider the lines in the image below:</p>
<p><img src="https://i.stack.imgur.com/AWmrd.png" alt="enter image description here"></p>
<p>Given a set of arbitrary points $p1$ and $p2$ where the direction of travel is from the former to the latter, I want to be able to directional arrow marks as in the image above.</... | Hagen von Eitzen | 39,174 | <p>a) Note that $0<u<v$ implies $0<\sqrt u<\sqrt v$. This allows you to show the claim by starting from $0<n<n+\sqrt {n+1}$ and walking your way to the outer $\sqrt{}$.</p>
<p>b) Follow the hint</p>
<p>c) By induction: $0<x_1<2$ and $0<x_n<2$ implies $1+\sqrt 2 x_n<1+2\sqrt 2<4$</p... |
1,116,009 | <p>Suppose $\alpha$,a,b are integers and $b\neq-1$. Show that if $\alpha$ satisfies the equation $x^2+ax+b+1=0$,then prove $a^2+b^2$ is composite.</p>
<p>I am starting with this study course of polynomials and finding it very difficult to understand. Please help me with the question. Thanks in advance ! </p>
| Muphrid | 45,296 | <p>I know of no algebra in which 3d arrays of numbers can be manipulated with the same ease as matrices. Part of that has to do with how any linear map from one space to another space can be represented with a matrix. You can chain such maps together sensibly (and really, only in one way up to the order of how you co... |
41,155 | <p>Lauren has 20 coins in her piggy bank, all dimes and quarters. The total amount of money is $3.05. How many of each coin does she have?</p>
| Murta | 2,266 | <p>Here is a start point. First, let's import your data</p>
<pre><code>SetDirectory@NotebookDirectory[]
data=Import["data.csv"]
</code></pre>
<p>Now <code>data</code> should be something like this:</p>
<pre><code>{{ ,ABC,DIA,ACE,SJ,KARMA,NOVICE},{ABC,0,2,3,1,2,1},{DIA,1,0,3,1,2,1},{ACE,2,1,0,3,1,2},{SJ,2,3,1,0,1,3},... |
69,476 | <p>Hello everybody !</p>
<p>I was reading a book on geometry which taught me that one could compute the volume of a simplex through the determinant of a matrix, and I thought (I'm becoming a worse computer scientist each day) that if the result is exact this may not be the computationally fastest way possible to do it... | Emil Jeřábek | 12,705 | <p>If the polynomial is given as $\alpha_0x^0+\dots+\alpha_nx^n$ and you do not know a priori anything about the $\alpha_i$’s, then you can’t do better than <a href="http://en.wikipedia.org/wiki/Horner_scheme">Horner’s scheme</a> (which takes $n$ additions and multiplications). If you know that the polynomial is sparse... |
43,886 | <p>Has work been done on looking at what happens to the exponents of the prime factorization of a number $n$ as compared to $n+1$? I am looking for published material or otherwise. For example, let $n=9=2^0\cdot{}3^2$, then,</p>
<p>$$
9 \;\xrightarrow{+1}\; 10
$$</p>
<p>$$
2^0\cdot{}3^2 \;\xrightarrow{+1}\; 2^1\cd... | Edison | 11,857 | <p>If we could deduce the prime factorization of a number $n+1$ from the prime factorization of the number $n$, then by induction we could in particular arrive at the prime factorization of all numbers. By the way, this would allow us to find new prime numbers. Since prime numbers seem to appear at random in the set of... |
1,336,937 | <p>I think: <em>A function $f$, as long as it is measurable, though Lebesgue integrable or not, always has Lebesgue integral on any domain $E$.</em></p>
<p>However Royden & Fitzpatrick’s book "Real Analysis" (4th ed) seems to say implicitly that “a function could be integrable without being Lebesgue measurable”. I... | srnoren | 99,255 | <p>On pg. 73 of Royden & Fitzpatrick, Lebesgue integrability is defined for bounded functions on domains of finite measure, without the assumption of measurability. However, this theorem that you have reveals that functions of this type can't be integrable unless they are measurable. Hence, the definition of integr... |
3,898,818 | <p>A (UK sixth form; final year of high school) student of mine raised the interesting question of how to prove that the total angle in the Spiral of Theodorus (formed by constructing successive right-angled triangles with hypotenuses of <span class="math-container">$\sqrt{n}$</span>), diverges.</p>
<p>He identified th... | PM 2Ring | 207,316 | <p>Rather than concentrating on the angle, consider the arc length of the spiral.</p>
<p>At the <span class="math-container">$n$</span>th step, we add a right triangle of base <span class="math-container">$\sqrt n$</span> and hypotenuse <span class="math-container">$\sqrt{n+1}$</span>, with the outer side (of course) o... |
3,226,028 | <h2>Problem</h2>
<p>I want to know how to solve the differential equation
<span class="math-container">$$ \dot{x} + a\cdot x - b\cdot \sqrt{x} = 0 $$</span> for <span class="math-container">$a>0$</span> and both situations: for <span class="math-container">$b > 0$</span> and <span class="math-container">$b < ... | D.B. | 530,972 | <p>One way of working this out is to make the substitution <span class="math-container">$y = \sqrt{x}$</span>. Then,
<span class="math-container">$$\frac{dx}{b\sqrt{x}-ax} \rightarrow \frac{2ydy}{by-ay^2} = \frac{2ydy}{y(b-ay)}.$$</span>
You can treat the integral in <span class="math-container">$y$</span> with partia... |
1,639,568 | <p>The above applies $\forall x,y \in \mathbb{R}$</p>
<p>I've tried: $x + y \ge 0$</p>
<p>$$x + y \ge x$$</p>
<p>$$ (x + y)^2 \ge 2xy$$</p>
<p>$$\frac{(x + y)^2}{2} \ge xy$$</p>
<p>But the closest I get is $\dfrac{x+y}{\sqrt{2}} \ge \sqrt{xy}$</p>
<p>Any ideas?</p>
| AsdrubalBeltran | 62,547 | <p>Note that:
$$(x-y)^2\ge 0\implies x^2+y^2\ge2xy\implies x^2+2xy+y^2\ge4xy\implies(x+y)^2\ge4xy$$</p>
|
1,014,303 | <blockquote>
<p>Is
$$\sum^\infty_{n=4}\frac{3^{2n}}{(-10)^n}$$
Convergent or Divergent? Explain why.</p>
</blockquote>
<p>I know I can do:
$$\sum^\infty_{n=4}\frac{9^{n}}{(-10)^n} \Rightarrow \sum^\infty_{n=4}\bigg(\frac{9}{-10}\bigg)^n$$
But I'm not sure where to go from here. The negative denominator is really... | user26857 | 121,097 | <p>$I$ not prime (because it is maximal among the <em>non-f.g.</em> ideals, and all prime ideals are f.g. by hypothesis) implies that there are two ideals $I_1,I_2$ such that $I_1I_2\subseteq I$, but $I_1\nsubseteq I$, and $I_2\nsubseteq I$. Now take $J_i=I_i+I$ and observe that $J_1J_2\subseteq I$, and $I\subsetneq J... |
1,014,303 | <blockquote>
<p>Is
$$\sum^\infty_{n=4}\frac{3^{2n}}{(-10)^n}$$
Convergent or Divergent? Explain why.</p>
</blockquote>
<p>I know I can do:
$$\sum^\infty_{n=4}\frac{9^{n}}{(-10)^n} \Rightarrow \sum^\infty_{n=4}\bigg(\frac{9}{-10}\bigg)^n$$
But I'm not sure where to go from here. The negative denominator is really... | Patrick Da Silva | 10,704 | <p>The ideal $I$ cannot be prime, because by (a) $I$ is not finitely generated (the ideal $I$ is maximal in the collection of non-finitely generated ideals, but it is not necessarily a maximal ideal!). If $I$ were prime, $I$ would be finitely generated by assumption on $R$, a contradiction. Therefore $I$ is not prime ;... |
88,122 | <p>For the easiest case, assume that $L/E$ is Galois and $E/K$ is Galois. Under what conditions can we conclude that $L/K$ is Galois? I guess the general case can be a bit tricky, but are there some "sufficiently general" cases that are interesting and for which the question can be answered?</p>
<p>EDIT: Since Jyrki's... | Keenan Kidwell | 628 | <p>This isn't an answer, nor is it very general, just an illustration with an example. Let $K$ be a number field, $E/K$ finite Galois, and $L/E$ the Hilbert class field of $E$ (the maximal unramified abelian extension of $E$, which I'll take to be inside some algebraic closure $\overline{K}$ of $K$). Then $L/E$ is Galo... |
3,780,575 | <p>We know that if <span class="math-container">$f$</span> is continuous on [a,b] and <span class="math-container">$f:[a,b] \to \mathbb{R}$</span>, then there exists <span class="math-container">$c \in [a,b]$</span> with <span class="math-container">$f(c)(a-b) = \int_a^bf(x)dx$</span></p>
<p>If we change ''f is continu... | user10354138 | 592,552 | <p>First of all, it is consider very bad style to just throw random equations around. Write down in English what exactly you are doing --- is it an assumption you make? A given condition? Some logical deduction from earlier? And every sentence should start with an English word not an equation, unless you absolutely ... |
2,077,958 | <p>Or more abstractly, let $T \in \mathcal{L}(U,V)$ be a linear map over finite dimensional vector spaces, I need to prove that $T^*$ and $T^* T$ have the same range.</p>
<p>The direction $v \in range(T^*T) \rightarrow v \in range(T^*)$ is obvious. I'm stuck on the other direction.
Suppose $u\in range(T^*)$, then the... | Noble Mushtak | 307,483 | <p>$A$ and $A^TA$ have the same null space. Therefore, the orthogonal complements of their null spaces are the same. It is well-known that the orthogonal complement of any matrix $M$ is the column space of $M^T$, so $A^T$ and $(A^TA)^T=A^TA$ have the same column space.</p>
|
344,345 | <p>Are there any relationship between the scalar curvature and the simplicial volume? </p>
<p>The simplicial volume is zero (positive) on Torus (Hyperbolic manifold) and those manifolds does not admit a Riemannian metric with positive scalar curvature. What do we know about the simplicial volume of a Riemannian mani... | Paul Siegel | 4,362 | <p>In a <a href="https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/101-problemsOct1-2017.pdf" rel="noreferrer">preliminary version</a> of what would become Gromov's "A Dozen Problems, Questions and Conjectures about Positive Scalar Curvature", he writes on page 88:</p>
<blockquote>
<p>Neither is one able to pro... |
194,218 | <blockquote>
<p>Let A, B be two sets. Prove that <span class="math-container">$A \subset B \iff A \cup B = B$</span></p>
</blockquote>
<p>I'm thinking of using disjunctive syllogism by showing that <span class="math-container">$\neg \forall Y(Y \in A).$</span> However, I'm not sure how the proving steps should proceed ... | rschwieb | 29,335 | <p>Here are the (very straightforward) first steps you should have thought of beginning with:</p>
<p>In one direction, suppose $A\subseteq B$: then $A\cup B\subseteq B\cup B\dots$</p>
<p>In the other direction, suppose $A\cup B=B$: then $A\subseteq A\cup B\subseteq\dots$</p>
|
3,599,893 | <p>I had this idea to build a model of Earth in Minecraft. In this game, everything is built on a 2D plane of infinite length and width. But, I wanted to make a world such that someone exploring it could think that they could possibly be walking on a very large sphere. (Stretching or shrinking of different places is OK... | Captain Lama | 318,467 | <p>What you want to do is not possible because there is no flat sphere. That is, there is no way to put a metric on a topological sphere such that the curvature is everywhere zero. This can be shown using the <a href="https://en.wikipedia.org/wiki/Gauss%E2%80%93Bonnet_theorem" rel="noreferrer">Gauss-Bonnet theorem</a>:... |
3,223,618 | <p>I have this system of linear equations with parameter:</p>
<p><span class="math-container">$ ax + 4y + z =0 $</span></p>
<p><span class="math-container">$2y + 3z = 1$</span> </p>
<p><span class="math-container">$3x -cz=-2$</span></p>
<p>What I did was to put those equations into a matrix and transform that matr... | Brian Fitzpatrick | 56,960 | <p>This is a situation where <a href="https://en.wikipedia.org/wiki/Cramer%27s_rule" rel="nofollow noreferrer">Cramer's rule</a> works quite well. </p>
<p>Our system is of the form <span class="math-container">$A\vec{x}=\vec{b}$</span> where
<span class="math-container">\begin{align*}
A &= \left[\begin{array}{rrr}... |
3,881,390 | <p>I tried multiplying both sided by 4a
which leads to <span class="math-container">$(6x+4)^2=40 \pmod{372}$</span>
now I'm stuck with how to find the square root of a modulo.</p>
| fleablood | 280,126 | <p>Might be easier to factor or use the quadratic formula.</p>
<p><span class="math-container">$3x^2 + 4x - 2\equiv 0\pmod {31}$</span> so abusing notation where <span class="math-container">$\sqrt {k}$</span> will mean the congruence <span class="math-container">$a$</span> where <span class="math-container">$a^2 \equi... |
3,891,749 | <p>I got a pretty good idea for the proof but it feels like it's missing some details.</p>
<p>Proof:
<span class="math-container">$$X \times Y \Leftrightarrow a\in X \land b \in Y \Leftrightarrow a\in X \land b\in Z$$</span></p>
<p>Since <span class="math-container">$b\in Y \Leftrightarrow b\in Z$</span>, then <span cl... | Derek Luna | 567,882 | <p>Let <span class="math-container">$y \in Y$</span>. Since <span class="math-container">$X \neq \emptyset $</span> , let <span class="math-container">$x \in X$</span>. Consider <span class="math-container">$(x,y) \in X \times Y = X \times Z$</span> which implies <span class="math-container">$y \in Z$</span>. The other... |
2,828,487 | <p>If $\mathcal{R}$ is a von Neumann algebra acting on Hilbert space $H$, and $v \in H$ is a cyclical and separating vector for $\mathcal{R}$ (hence also for its commutant $\mathcal{R}'$), and $P \in \mathcal{R}, Q \in \mathcal{R}'$ are nonzero projections, can we have $PQv = 0$?</p>
<p>[note i had briefly edited this... | Community | -1 | <p>In my undergrad, I took two one-semester courses following Royden's <em>Real Analysis</em>. I liked this book because Royden (generally) has a sufficient amount of detail. I found it a good grounding in the fundamentals of measure theory and metric spaces.</p>
<p>After this, I took a course in Fourier analysis that... |
231,479 | <p>Is there a function that can create hexagonal grid?</p>
<p>We have square grid graph, where we can specify <code>m*n</code> dimensions:</p>
<pre><code>GridGraph[{m, n}]
</code></pre>
<p>We have triangular grid graph (which works only for argument <code>n</code> up to 10 - for unknown reason):</p>
<pre><code>GraphDat... | Szabolcs | 12 | <p>With <a href="http://szhorvat.net/mathematica/IGraphM" rel="noreferrer">IGraph/M</a>:</p>
<pre><code>IGMeshGraph@IGLatticeMesh["Hexagonal", {6, 4}]
</code></pre>
<p><a href="https://i.stack.imgur.com/mWWa5.png" rel="noreferrer"><img src="https://i.stack.imgur.com/mWWa5.png" alt="enter image description her... |
231,479 | <p>Is there a function that can create hexagonal grid?</p>
<p>We have square grid graph, where we can specify <code>m*n</code> dimensions:</p>
<pre><code>GridGraph[{m, n}]
</code></pre>
<p>We have triangular grid graph (which works only for argument <code>n</code> up to 10 - for unknown reason):</p>
<pre><code>GraphDat... | cvgmt | 72,111 | <p><strong>Edit-4</strong></p>
<p>Besides of type <code>{m,n,o}</code>,here we want to find the type <code>{n[1],n[2],n[3],n[4],n[5],n[6]}</code>. Simple calculate, for example</p>
<pre><code>Solve[Array[n, 6] . CirclePoints[{0, 0}, {1, 0}, 6] == 0,
Array[n, 6], PositiveIntegers]
</code></pre>
<p>We can find that it ... |
117,285 | <p>Let $R \subseteq A \times A$ and $S \subseteq A \times A$ be two arbitary equivalence relations.
Prove or disprove that $R \cup S$ is an equivalence relation.</p>
<p>Reflexivity: Let $(x,x) \in R$ or $(x,x) \cup S \rightarrow (x,x) \in R \cup S$</p>
<p>Now I still have to prove or disprove that $R \cup S$ is symme... | Ashish kakran | 541,198 | <p>I don't know if i m right or wrong. But here is how i can think of it.</p>
<p>Lets say R and S are two equivalence relations on nonempty set A.</p>
<p>To answer whether R union S is equivalence relation? consider the fact that R forms partitions on A and S also forms some partitions</p>
<p>My intuition: taking un... |
1,936,043 | <p>I would like to prove that the sequence $n^{(-1)^{n}}$ is divergent. </p>
<p>My thoughts: I know $(-1)^n$ is divergent, so $n$ to the power of a divergent sequence is still divergent? I am not sure how to give a proper proof, pls help!</p>
| user159517 | 159,517 | <p>The argument that "n to the power of a divergent sequence is divergent" does not make sense (consider $n^{-n}$ for example.). Regarding your sequence: if this sequence were convergent against some limit value $a$, then every subsequence would have to converge against the same value $a$. Now look at the subsequences ... |
1,441,905 | <blockquote>
<p>Find the range of values of $p$ for which the line $ y=-4-px$ does not intersect the curve $y=x^{2}+2x+2p$</p>
</blockquote>
<p>I think I probably have to find the discriminant of the curve but I don't get how that would help.</p>
| D.L. | 95,150 | <p>The two equations give $x^2+(2+p)x+2p+4=0$, so you have to calculate the discriminant of this. You get : $-7p^2-4p-12$ which is always negative. </p>
|
4,327,729 | <p>In this question, I would like to investigate the location of the absolute value in the arcsecant integral.</p>
<p>Following <a href="https://math.stackexchange.com/questions/3735966/why-the-derivative-of-inverse-secant-has-an-absolute-value">this answer</a> and <a href="https://math.stackexchange.com/questions/1449... | John Wayland Bales | 246,513 | <p>Given that</p>
<p><span class="math-container">$$ \sec^{-1}(u)+C=\int \frac{1}{|u|\sqrt{u^2-1}} du $$</span></p>
<p>let <span class="math-container">$u=|x|$</span>. Then <span class="math-container">$du=\dfrac{|x|}{x}dx$</span> and</p>
<p><span class="math-container">\begin{eqnarray}
\sec^{-1}(|x|) +C&=&\int... |
2,011,181 | <blockquote>
<p><strong>Question:</strong> Find the area of the shaded region given $EB=2,CD=3,BC=10$ and $\angle EBC=\angle BCD=90^{\circ}$.</p>
</blockquote>
<p><a href="https://i.stack.imgur.com/BFf2h.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/BFf2h.jpg" alt="Diagram"></a></p>
<p>I first ... | David Quinn | 187,299 | <p>You are almost there. Just add the right hand pairs of equations you have so you get $$\frac{AX}{CD}+\frac{AX}{EB}=1$$ </p>
<p>Substituting the values, you get $AX=\frac 65$ so the required area is...?</p>
|
19,356 | <p>So I was wondering: are there any general differences in the nature of "what every mathematician should know" over the last 50-60 years? I'm not just talking of small changes where new results are added on to old ones, but fundamental shifts in the nature of the knowledge and skills that people are expected to acqui... | Timothy Chow | 3,106 | <p>I believe that one shift is that "what every mathematician should know" is nowadays much less a specific body of mathematical facts and much more a facility with navigating the ocean of mathematical knowledge.</p>
<p>For example, I might not need to have advanced computer programming skills, but I do need to have s... |
1,995,663 | <p>My brother in law and I were discussing the four color theorem; neither of us are huge math geeks, but we both like a challenge, and tonight we were discussing the four color theorem and if there were a way to disprove it.</p>
<p>After some time scribbling on the back of an envelope and about an hour of trial-and-e... | dxiv | 291,201 | <p>One possible 4-coloring of such.</p>
<p><a href="https://i.stack.imgur.com/z23kq.png" rel="noreferrer"><img src="https://i.stack.imgur.com/z23kq.png" alt="enter image description here"></a></p>
|
79,041 | <p>Let <span class="math-container">$\mathfrak{g}$</span> be the Lie algebra of a Lie group <span class="math-container">$G$</span> which acts on a manifold <span class="math-container">$M$</span>.
It is quite standard that the basic forms in <span class="math-container">$\Omega^*(M) \otimes W(\mathfrak{g}^*)$</span> f... | Praphulla Koushik | 118,688 | <p>As mentioned by the user SGP, the book <a href="https://www.springer.com/gp/book/9783540647973" rel="nofollow noreferrer">Supersymmetry and Equivariant de Rham Theory</a> by Victor W Guillemin and Shlomo Sternberg discuss about Cartan model.
One of the intentions is to prepare the reader to understand Cartan's pape... |
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