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 |
|---|---|---|---|---|
1,012,123 | <p><em><strong>The differentiable fuction $z=z(x,y)$ is given implicitly by equation $f(\frac{x}{y},z)=0$, where $f(u,v)$ is supposed to be differentiable and $\frac{\partial f}{\partial v}(u,v)\neq0$. Verify that
$$x\frac{\partial z}{\partial x}+y\frac{\partial z}{\partial y}=0.$$</strong></em>
This is the exercise 6... | JJacquelin | 108,514 | <p>$f(u,v)=0$ where $u=\frac{x}{y}$ and $v=z(x,y)$
$$du=\frac{1}{y}dx-\frac{x}{y^2}dy$$
$$dv=\frac{\partial z}{\partial x}dx+\frac{\partial z}{\partial y}dy$$
Since $f(u,v)=0$, we have :
$$\frac{\partial f}{\partial u}du+\frac{\partial f}{\partial v}dv=0$$
$$\frac{\partial f}{\partial u}\big(\frac{1}{y}dx-\frac{x}{y^2}... |
1,012,123 | <p><em><strong>The differentiable fuction $z=z(x,y)$ is given implicitly by equation $f(\frac{x}{y},z)=0$, where $f(u,v)$ is supposed to be differentiable and $\frac{\partial f}{\partial v}(u,v)\neq0$. Verify that
$$x\frac{\partial z}{\partial x}+y\frac{\partial z}{\partial y}=0.$$</strong></em>
This is the exercise 6... | Derso | 177,211 | <p>We need the partial derivative of $f$ relative to $x$. Thus, consider $y$ and $z$ to be constants and derivate $f(u(x),v(x))$ relative to $x$, with $u(x)=\frac{x}{y}$ and $v(x)=z$. We have, by the Chain Rule
$$\frac{\partial f}{\partial x}(u(x),v(x))=\frac{\partial f}{\partial u}(u(x),v(x))\cdot\frac{\partial u}{\pa... |
1,697,472 | <p>I was in a roll playing game last night. In combat we throw 3 six sided dice when we attack and roll some amount of dice if we hit our target by rolling high enough on the attack (for example the dice sum to at least 10, or 13, or something like that). </p>
<p>The number the dice sum to is called the target number.... | DonAntonio | 31,254 | <p>Define</p>
<p>$$f(x)=\frac{\log x}{\sqrt x}\implies f'(x)=\frac{\frac1{\sqrt x}-\frac{\log x}{2\sqrt x}}{x}=\frac{2-\log x}{2x\sqrt x}<0\iff \log x>2\iff x>e^2$$</p>
<p>and thus the function is monotonic descending for $\;x>e^2\;$ , and the sequence $\;\frac{\log n}{\sqrt n}\;$ is monotonic descending ... |
1,697,472 | <p>I was in a roll playing game last night. In combat we throw 3 six sided dice when we attack and roll some amount of dice if we hit our target by rolling high enough on the attack (for example the dice sum to at least 10, or 13, or something like that). </p>
<p>The number the dice sum to is called the target number.... | zhw. | 228,045 | <p>Hints:</p>
<p>1) Take the derivative of $\ln x/\sqrt x.$</p>
<p>2) The expression equals $[(1-1/(n+1))^{n+1}]^{n^2/(n+1)}.$</p>
<p>3) No, that quotient of factorials is not accurate. Instead, note that $(2k-1)/(3k-1) < 2/3, k = 1,2,\dots $</p>
<p>4) Yes, of course you can write it that way.</p>
|
3,203,090 | <p><a href="https://i.stack.imgur.com/RdEok.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/RdEok.png" alt="enter image description here"></a></p>
<p>Could someone help me? I'm not sure how to start</p>
<p>My professor also gave me this hint: </p>
<p><a href="https://i.stack.imgur.com/91Oy2.png" r... | trancelocation | 467,003 | <p>It might be useful to depict the situation by a tree diagram:</p>
<p><span class="math-container">\begin{array}{cccccccc}
A & \stackrel{\frac{5}{6}}{\rightarrow} & B & \stackrel{\frac{2}{3}}{\rightarrow} & C & \stackrel{\frac{1}{2}}{\rightarrow} & A & \stackrel{\frac{5}{6}}{\rightarrow... |
234,722 | <p>I got this questions on one of my example sheets in a first year algebra course:</p>
<p>"Let $B:W×V→K$ be a bilinear map, where $V,W$ are vector spaces over a a field $K$. Let $U$ be a subspace of $V$. If we denote by $U^{\perp}= \{w \in W|B(w,u)=0,\forall u∈U \},$ prove that $dim(U)+dim(U^{\perp})≥dim(W)$.</p>
<p... | Elchanan Solomon | 647 | <p>The dimension of $V$ appears to be irrelevant, as we can replace it with $U$. So the infinite-dimensional case is really when $\dim \, (W) = \infty$, and so we need to show that one of the quantities on the left-hand side is infinite. Suppose that $\dim \, U$ is finite. What can you conclude about $\dim \, U^\perp$... |
753,997 | <p>Someone told me that math has a lot of contradictions. </p>
<p>He said that a lot of things are not well defined.</p>
<p>He told me two things that I do not know.</p>
<ul>
<li>$1+2+3+4+...=-1/12$</li>
<li>what is infinity $\infty$?</li>
</ul>
<p>Since I am not a math specialist and little. How to disprove the pr... | Mathias | 141,599 | <p>In mathematics contradiction is a useful tool to prove some theorems. If you want prove some statement, sometimes its easier to disprove that the statement is not true. So you assume its not true and derive a contradiction, which proves that the statement is in fact true.</p>
<p>Apart from that, mathematics is buil... |
3,791,848 | <p>I'm trying to solve the following question from the <a href="https://www.math.ucla.edu/%7Echparkin/gre/GREProb.pdf" rel="nofollow noreferrer">real analysis</a> section:</p>
<blockquote>
<ol>
<li>Let <span class="math-container">$K$</span> be a nonempty subset of <span class="math-container">$\mathbb R^n$</span> wher... | Umesh Shankar | 816,291 | <p>I would just like that to add that if the range was the reals endowed with the bounded metric, <span class="math-container">$d(x,y)=\frac{|x-y|}{1+|x-y|}$</span>, then the statement is not true for metric spaces even if the <span class="math-container">$Dom(f)$</span> satisfied the Heine-Borel property.</p>
|
137,483 | <p>Let $M$ be a closed Riemannian manifold and $\omega$ and $\eta$ two differential forms of the same degree. Then one can consider $\int_M \omega \wedge *\eta$, where $*$ denotes the Hodge star operator. Can you tell me, why this defines a scalar product or at least where I can find a proof of this fact? In particular... | Will Jagy | 10,400 | <p>Low level tidbits from Frank Warner, <em>Foundations of Differentiable Manifolds and Lie Groups</em>, exercise 13 on page 79. One may also talk about the inner product on forms this way: homogeneous forms of different degrees have inner product zero. Then let
$$ \langle w^1 \wedge \cdots \wedge w^p, v^1 \wedge \... |
1,336,692 | <p>Why $y=e^x$ is not an algebraic curve over $\mathbb R$? I can say that is not a algebraic curve over $\mathbb C$ because $e^x$ is a periodic function, but what about $\mathbb R$?</p>
<p><strong>EDIT:</strong></p>
<p>I don't want to use trascendence of $e$. Or, I can ask this question for $y=2^x$.</p>
<p><strong>U... | lhf | 589 | <p>Suppose $x$ and $e^x$ satisfy a polynomial equation $f(x,e^x)=0$ where $f(x,y)$ has minimal degree in $y$.</p>
<p>Write $f(x,y)=p(x)y^n+g(x,y)$, where $p(x)y^n$ is the leading term in $y$.</p>
<p>Differentiate $p(x)e^{nx}+g(x,e^x)=0$ and get $np(x)e^{nx}+p'(x)e^{nx}+h(x,e^x)=0$, for some $h$.</p>
<p>Subtract $n$ ... |
2,709,696 | <p>stuck on a question and can't seem to make any progress:</p>
<p>We have an insurance company who expects the number of accidents their policy holders will have is Poisson distributed. The Poisson mean $\Delta$ follows a Gamma distribution with the $\Gamma$(2,1) density function being $f_{\Delta}(\lambda) = \lambda ... | Valborg | 546,072 | <p>Without loss of generality, we may assume that $m$ and $n$ are non-negative. We begin by rearranging the equation to read as follows:
$$
2^n(2^{n+1}+1)=(m-1)(m+1)
$$
This implies that $2^{n-1}$ exactly divides either $m-1$ or $m+1$. If it is the case that $2^{n-1}$ exactly divides $m-1$, we can write $m-1=k2^{n-1}$ ... |
2,863,179 | <p>I'll state the Cantor's theorem proof as is it is in my study texts:</p>
<p>Theorem (Cantor): Let $X$ be any set. Then $|X|<|\mathcal{P}(X)|$</p>
<p>Proof: Define map $\varphi:X\rightarrow\mathcal{P}(X)$ by $\varphi:x\mapsto\{x\}$. $\varphi$ is injective, thus $|X|\leq|\mathcal{P}(X)|$. Now suppose there is a b... | Mees de Vries | 75,429 | <blockquote>
<p>In the proof $A$ is supposed to be the set of all members of $X$ (thus numbers) that are not in the range of $\psi$</p>
</blockquote>
<p>No, this is not quite true. The construction is more subtle than that. For any $x \in X$, the value $\psi(x) \in \mathcal P(X)$ is some set -- a subset of $X$ to be... |
321,794 | <p>Assume $$G_1=\mathbb Z_5 \times \mathbb Z_{5^2}\times \mathbb Z_{5^3}\times \mathbb Z_{5^4} \times\ldots$$ $$G_2= \mathbb Z_{5^2}\times \mathbb Z_{5^3}\times \mathbb Z_{5^4} \times \ldots$$ How do I prove $G_1$ and $G_2$ aren't isomorphic? I asked this question here <a href="https://math.stackexchange.com/question... | Community | -1 | <p>obviously the reason they are not isomorphic is that $Z_5$ only occurs in one.</p>
<p>so how do we turn that into a proof? $Z_5$ is generated by an element of order 5, but we certainly have elements of order 5 in both groups so we will have to be able to say something different about the element of order $5$ in $Z_... |
321,794 | <p>Assume $$G_1=\mathbb Z_5 \times \mathbb Z_{5^2}\times \mathbb Z_{5^3}\times \mathbb Z_{5^4} \times\ldots$$ $$G_2= \mathbb Z_{5^2}\times \mathbb Z_{5^3}\times \mathbb Z_{5^4} \times \ldots$$ How do I prove $G_1$ and $G_2$ aren't isomorphic? I asked this question here <a href="https://math.stackexchange.com/question... | Andreas Caranti | 58,401 | <p>I may easily be wrong, but it seems to me that while all elements of order $5$ in $G_2$ are fifth multiples, not all elements of order $5$ in $G_1$ are.</p>
<p><em>PS</em> I had written <em>powers</em> instead of <em>multiples</em>, because I had automatically translated in my head all involved factors as "a cyclic... |
3,256,475 | <p>I'm sorry that I don't know how to upload the context, I'll just write the example:</p>
<blockquote>
<p>Example 7. The product of two quotient maps need not be a quotient map.</p>
<p>Proof: Let <span class="math-container">$X=\mathbb{R}$</span> and <span class="math-container">$\mathbb Q$</span> is set of rational n... | nonuser | 463,553 | <p>Write <span class="math-container">$$x^2(y-2)^2 = -y^3+3y^2-1$$</span></p>
<p>Since <span class="math-container">$y-2\mid -y^3+3y^2-1$</span> and <span class="math-container">$y\equiv 2\pmod{y-2}$</span> we have</p>
<p><span class="math-container">$$0\equiv -y^3+3y^2-1 \equiv -8+12-1 \equiv 3 \pmod{y-2}$$</span> ... |
1,914 | <p><a href="https://matheducators.stackexchange.com/a/1528/42">This answer</a> describes an analogy between finite state machines and mazes. This allows for some playful exercises, like </p>
<blockquote>
<p>Draw a representation of a word accepted by the following automaton...</p>
</blockquote>
<p>which sums up to ... | Benjamin Dickman | 262 | <p>There are many examples of games that children play (in the same way that kids might enjoy mazes) which have deep mathematics underlying them. An oft-cited example of this is the game of <a href="http://en.wikipedia.org/wiki/Hex_%28board_game%29" rel="nofollow noreferrer"><strong>Hex</strong></a>:</p>
<p><img src="... |
1,249,726 | <p>I am wondering whether it is possible to construct an experiment, where the probability of occurrence of an event comes out to be an irrational number. </p>
| demitau | 221,294 | <p>It depends on how do you define experiment. Theoretically you can take an unfair coin where the probability of the tail is $\sqrt{2}/2$. Or one can construct a dice with the area of one of the faces being irrational (but this example is more complex to settle correctly).</p>
|
1,249,726 | <p>I am wondering whether it is possible to construct an experiment, where the probability of occurrence of an event comes out to be an irrational number. </p>
| John Bentin | 875 | <p>A famous example is <a href="https://en.wikipedia.org/wiki/Buffon%27s_needle" rel="noreferrer">Buffon's needle</a>. A needle is tossed randomly onto a horizontal plane ruled with parallel lines whose distance apart is the same as the length of the needle. The probability that the needle crosses a line is $2/\pi$.</p... |
2,981,450 | <blockquote>
<p>Prove that there do not exist natural <span class="math-container">$n$</span> such that <span class="math-container">$(1+i)^n=1$</span>.</p>
</blockquote>
<p>I try to prove with the binomial and proving by induction but it isn't working</p>
<p><img src="https://i.stack.imgur.com/erGAb.png" alt="try ... | Robert Z | 299,698 | <p>Note that for any <span class="math-container">$n\in\mathbb{Z}$</span>,
<span class="math-container">$$|(1+i)^n|=|(1+i)|^n=(\sqrt{2})^{n}.$$</span>
What may we conclude about the equation <span class="math-container">$(1+i)^n=1$</span>?</p>
|
1,631 | <p>Using <code>MouseAppearance</code> one can change the cursor image when passing over an expression.</p>
<p>Is it possible to change the cursor image for the entire notebook front end (not just one expression within it)?</p>
| Mr.Wizard | 121 | <p>Have you looked through the included documentation?</p>
<p><a href="http://reference.wolfram.com/mathematica/guide/ImageProcessing.html" rel="nofollow">guide/ImageProcessing</a></p>
<p><a href="http://reference.wolfram.com/mathematica/tutorial/ImageProcessing.html" rel="nofollow">tutorial/ImageProcessing</a></p>
... |
302,519 | <p>My teacher gave us an homework. I solved it, but I don't think I have the right answer.</p>
<p><strong>PROBLEM</strong></p>
<p>We have three coins identical in appearance.</p>
<ul>
<li>Coin A falls on tails and heads with equal probability</li>
<li>Coin B falls twice as much on tails as heads</li>
<li>Coin C alwa... | Alfonso Fernandez | 54,227 | <p>What you have to do in this case is update your priors according to your observation. Your first prior distribution for selecting a coin was (I assume) $\left(\frac{1}{3},\frac{1}{3},\frac{1}{3}\right)$.</p>
<p>Now that you've seen tails, the probabilities for the coin you're holding change; using Bayes' theorem:
$... |
1,781,957 | <p>How would you go about expressing the following as a limit?
$$\int_0^1 \ln(x) dx$$
I know how to express limits on simple equations, but have no clue how to go about expressing an integral as a limit.</p>
<p>Do I need to expand it as if I was performing a normal integration?</p>
| JKnecht | 298,619 | <p>Integrate by parts</p>
<p>$$= \lim_{\epsilon \to 0^+} x \ln(x) |_\epsilon^1 - \int_0^1 1 \: \text{dx}$$</p>
<p>$$= -\left ( \lim_{\epsilon \to 0^+} \epsilon \ln(\epsilon) \right) - \int_0^1 1 \: \text{dx}$$</p>
<p>$$= -1$$</p>
|
83,882 | <p>Suppose I have a quadratic polynomial in two variables <code>x</code> and <code>y</code> in which the squares with respect to <code>x</code> and <code>y</code> have already been completed:</p>
<pre><code> q = -72 + 9 (-2 + x)^2 + 4 (3 + y)^2 ;
</code></pre>
<p>How might I extract the "constant" part <code>-72</cod... | kglr | 125 | <pre><code>♯ = # & @@ # &
♯ @ q
</code></pre>
<blockquote>
<p>-72</p>
</blockquote>
<p><strong>tl;dr</strong></p>
<pre><code>♯♯ = # & @ ## & @@ # &
♯♯ @ q
</code></pre>
<blockquote>
<p>-72</p>
</blockquote>
|
1,019,833 | <p>Let $x \in A \cap B$. Suppose we have $U \subset A,V \subset B$ such that $ x \in U$ and $x \in V$. So, $x \in U \cap V$. Further, suppose there exists a set $G \subset U \cap V $ so that $x \in G$. Does it follow that </p>
<p>$$ x \in G \subset U \cap V \subset A \cap B $$</p>
<p>?? </p>
| Martín-Blas Pérez Pinilla | 98,199 | <p>You are saying that
$$x\in G$$
$$G\subset U\cap V$$
and $U \subset A,V \subset B$ implies $ U \cap V \subset A \cap B$.
Nothing new in your last line!</p>
|
1,478,460 | <p>For a linear autonomous system in the plane
$$ \mathbf{\dot{x}} = \begin{pmatrix} a & b\\ c & d \end{pmatrix}\mathbf{x} \qquad (a,b,c,d \in \mathbb{R})$$
with determinant $D = ad - bc$ and trace $T = a + d$ we have the characteristic polynomial
$$ \chi(\lambda) = \lambda^2 - T\lambda + D$$
and the eigenvalue... | arney | 84,299 | <p>Inspired by <a href="https://math.stackexchange.com/users/265466/amd">amd</a>'s help I <em>did the math</em>:</p>
<p>The original system
$$ \mathbf{\dot{x}} = \begin{pmatrix} a & b\\ c & d \end{pmatrix}\mathbf{x}, \qquad D = ad - bc, \qquad T = a + d $$
has the eigenvalues
$$ \lambda_{\pm} = \frac{T \pm i\s... |
2,112,161 | <p>This is a question out of "Precalculus: A Prelude to Calculus" second edition by Sheldon Axler. on page 19 problem number 54.</p>
<p>The problem is Explain why $(a−b)^2 = a^2 −b^2 $ if and only if $b = 0$ or $b = a$.</p>
<p>So I started by expanding $(a−b)^2$ to $(a−b)^2 = (a-b)(a-b) = a^2 -2ab +b^2$. To Prove tha... | Community | -1 | <p>This is a geometrical approach.</p>
<p>Let $a-b=d$. Then $d^2 + b^2 = a^2$ and the using <a href="https://en.wikipedia.org/wiki/Pythagorean_theorem#Converse" rel="nofollow noreferrer">Converse of Pythagoras theorem</a> there must be a triangle having $a, b, a-b$ as sides, which is impossible because $b + (a - b) = ... |
115,325 | <p>I need a simple proof that a line cannot intersect a circle at three distinct points.</p>
| user 1591719 | 32,016 | <p>Take any 3 distinct points on a circle and notice that each angle of the triangle formed by those 3 points is higher than 0 and smaller than 180 degrees. Any of the angles formed by 3 distinct points on a line (degenerate triangle) takes a value of either 0 degrees or 180 degrees.</p>
<p>The proof is complete.</p>
|
2,910,860 | <p>Question: [For which positive real numbers $a$ and $b$ is</p>
<p>$u(x,y) = \cosh(ax)\sin(by)$</p>
<p>harmonic? When $a$ and $b$ satisfy this condition find a holomorphic function $f(z)$ such that $\Re f = u$]</p>
<hr>
<p>I got $a=b$ for the condition so that</p>
<p>$u(x,y) = \cosh(ax)\sin(ay)$ and $v(x,y) = -\s... | Phil H | 554,494 | <p>The angle at $B$ (DBC) made by the $6, 8, 10$ triangle is $90$ degrees. The angle at $B$ (ABC) made by the $5, 6, 7$ triangle is determined from the law of cosines $7^2 = 5^2 + 6^2 -2(5)(6)\cos(B)$. So $B = 78.4630$ degrees.</p>
<p>Looking end on along the edge of length 6, we see a foreshortened AB as a length of:... |
3,744,077 | <p>When a group of people need to decide a winner or leader between them, one approach would be that a random hidden integer is chosen with uniform distribution on <span class="math-container">$\{0, 1, ..., n\}$</span> and all <span class="math-container">$p$</span> participants publicly choose a number.</p>
<p>Then, t... | afreelunch | 604,896 | <p>An interesting problem!</p>
<p>The case of <span class="math-container">$p = 2$</span> is fairly straightforward. For simplicity, suppose that each player must choose a real number in <span class="math-container">$[0, 1]$</span> (ignoring integer issues) and let <span class="math-container">$x_1 \in [0, 1]$</span> a... |
3,629,734 | <p>Suppose <span class="math-container">$\{x_n\}$</span> is a sequence of real numbers such that <span class="math-container">$\lim_{n\to +\infty}x_n=0$</span>. If the product sequence <span class="math-container">$\{nx_n\}$</span> is bounded (<span class="math-container">$\exists M>0$</span> such that <span class="... | Michael Rozenberg | 190,319 | <p>Let <span class="math-container">$A$</span> be a given point and <span class="math-container">$a$</span> and <span class="math-container">$b$</span> be given lines.</p>
<p>Also, let <span class="math-container">$R_{A}^{\alpha}$</span> be a rotation around <span class="math-container">$A$</span> by <span class="math... |
2,011,511 | <p>Let $f(x)=\sum_{n=1}^\infty \frac{(-1)^n}{\sqrt{n}}e^{\frac{x}{n}}$.</p>
<p>Is f differentiable on $\mathbb{R}$?</p>
<p>I can prove the series does not uniformly convergent on $\mathbb{R}$. But this does not imply f is not differentiable. </p>
| Robert Israel | 8,508 | <p>Write
$$ f(x) - f(0) = \sum_{n=1}^\infty \frac{(-1)^n}{\sqrt{n}} \left( e^{x/n} - 1 \right) $$
Now $e^{x/n} - 1 = x/n + O((x/n)^2)$, and since $\sum_n 1/n^{3/2}$ converges we find that this series converges uniformly on compact subsets of $\mathbb C$. Thus $f(x) - f(0)$ is an entire function, and then so is $f(x)$... |
2,011,511 | <p>Let $f(x)=\sum_{n=1}^\infty \frac{(-1)^n}{\sqrt{n}}e^{\frac{x}{n}}$.</p>
<p>Is f differentiable on $\mathbb{R}$?</p>
<p>I can prove the series does not uniformly convergent on $\mathbb{R}$. But this does not imply f is not differentiable. </p>
| Mark Viola | 218,419 | <blockquote>
<p><strong>I thought it might be instructive to present a "brute-force" way forward that relies on straightforward analysis and an inequality for the exponential function. To that end, we now proceed.</strong></p>
</blockquote>
<hr>
<p>$$\begin{align}
\left|\frac{f(x+h)-f(x)}h -\sum_{n=1}^\infty\frac{... |
1,873,702 | <p>My question is very simple. Suppose we have a polynomial defined as follows:
$$p(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots+a_0 $$
where all of the $a_n$'s are all real and positive. Is there something that we can say about the roots of $p(x)$? Can we say the roots of $p(x)$ all contain negative real parts?</p>
<p>Thank... | Bernard | 202,857 | <p>The simplest counter-example is $\;x^5+x^4+x^3+x^2+x+1$: its roots are
$$\mathrm e^{\tfrac{\mathrm ik\pi}3},\quad k=1,\dots,5$$
and two of them have positive real parts.</p>
<p>Of course, this polynomial is not irreducible in $\mathbf Q[x]$.</p>
|
3,145,550 | <blockquote>
<p>Prove if <span class="math-container">$x$</span> and <span class="math-container">$y$</span> are real numbers with <span class="math-container">$x \lt y$</span>, then there are infinitely many rational numbers in the interval <span class="math-container">$[x,y]$</span>.</p>
</blockquote>
<p>What I go... | NicNic8 | 24,205 | <p>Consider the interval <span class="math-container">$[x,y]$</span>. Find two rational numbers <span class="math-container">$q_{min}$</span> and <span class="math-container">$q_{max}$</span> such that <span class="math-container">$x < q_{min} < q_{max} < y$</span>. Let <span class="math-container">$r=q_{min... |
2,197,241 | <p>Let $R$ be a commutative ring with identity, let $N = \{x|x \in R, x^k=0$ for some natural number k$\}$, and let $L = \{x|x \in R, 1+xr$ invertible for all $ r \in R\}$. I am trying to find an example of a ring R such that $N$ is strictly smaller than $L$. </p>
<p>It is not hard to show that $N \subseteq L$ and s... | Ben West | 37,097 | <p>Any local domain should work. If $R$ is a local domain with unique maximal ideal $\mathfrak{m}$, then $L=\mathfrak{m}$. But as a domain, $\{0\}$ is a prime ideal, so $N=\{0\}$. </p>
<p>For some concrete examples, you could look at <a href="https://en.wikipedia.org/wiki/Valuation_ring" rel="nofollow noreferrer">valu... |
437,596 | <blockquote>
<p>Show that $\mathbb{R}^n\setminus \{0\}$ is simply connected for $n\geq 3$.</p>
</blockquote>
<p>To my knowledge I have to show two things:</p>
<ol>
<li><p>$\mathbb{R}^n\setminus \{0\}$ is path connected for $n\geq 3$.</p></li>
<li><p>Every closed curve in $\mathbb{R}^n\setminus \{0\}, n\geq 3$ is nu... | Ronnie Brown | 28,586 | <p>The easy part of the Seifert-van Kampen theorem and which does not require an algebraic context (e.g. groups or groupoids) is that if <span class="math-container">$X$</span> is the union of a family <span class="math-container">$\mathcal U$</span> of path connected and simply connected open sets <span class="math-co... |
48,378 | <p>Simplified real world problem that came up yesterday.</p>
<p>A new course on my university can be opened if all the attendants can be split into (one or more) groups of n±2 people, where n≥5. So for example if n is 5 and there are 11 attendants, they can be split into groups of 5+0 and 5+1 people (or 5-1 and 5+2). ... | Arturo Magidin | 742 | <p>If all you care about is the valid numbers of attendants, and not the many possible ways to realize the splitting, you can do it as follows: for each $k$, the number of attendants can be anywhere from $kn-2k=k(n-2)$ through $kn+2k=k(n+2)$. To see this, note that you can achieve $k(n-2)$ by dividing them into $k$ gro... |
48,378 | <p>Simplified real world problem that came up yesterday.</p>
<p>A new course on my university can be opened if all the attendants can be split into (one or more) groups of n±2 people, where n≥5. So for example if n is 5 and there are 11 attendants, they can be split into groups of 5+0 and 5+1 people (or 5-1 and 5+2). ... | Gerry Myerson | 8,269 | <p>Find the smallest integer $k$ such that $k(n-2)\le (k-1)(n+2)$. Then you can open any course with $k(n-2)$ attendants or more. That leaves a finite problem to solve, looking at the classes with fewer than $k(n-2)$ attendants. The value of $k$ is the smallest integer not exceeding $(n+2)/4$. </p>
<p>For example, if ... |
48,378 | <p>Simplified real world problem that came up yesterday.</p>
<p>A new course on my university can be opened if all the attendants can be split into (one or more) groups of n±2 people, where n≥5. So for example if n is 5 and there are 11 attendants, they can be split into groups of 5+0 and 5+1 people (or 5-1 and 5+2). ... | Henry | 6,460 | <p>If $n=15$ then with one group you can make any number in $[13,17]$, with two groups any in $[26,34]$, in three any in $[39,51]$, in four any in $[52,68]$, in five any in $[65,85]$ and now the sets are overlapping you can clearly make any higher number. So you can make any number of attendants from $39$ upwards, plu... |
2,467,244 | <p>I have a proof that I know has a fundamental error in it, but I have no idea where the error is. Here is the proof:</p>
<blockquote>
<p>Proof: Suppose that $x \in A^{\complement} \cup B^{\complement}$. This implies that $x \in A^{\complement}$ or $x \in B^{\complement}$. This disjunction will be true if $x \in A^... | drhab | 75,923 | <p>There is no fundamental error in it. </p>
<p>You proved legally that $x\in A^{\complement}\implies x\in(A\cap B)^{\complement}$ so that $A^{\complement}\subseteq(A\cap B)^{\complement}$. </p>
<p>What lacks somehow is a statement as: </p>
<p>"...likewise we prove that $B^{\complement}\subseteq(A\cap B)^{\complemen... |
832,814 | <blockquote>
<p>Is the Krull dimension of any commutative semilocal Hilbert ring equal to zero? </p>
</blockquote>
<p>I appreciate any help from anyone! </p>
| Asaf Karagila | 622 | <p><strong>HINT:</strong> What happens if you remove a single point from $A$? Or any finite number of points from $A$? </p>
|
1,561,316 | <p>A polynomial in x has m nonzero terms. Another polynomial in x has n nonzero terms, where m is less than n. These polynomials are multiplied and all like terms are combined. The resulting polynomial has a maximum of how many nonzero terms? How would you prove that the answer is mn?</p>
| Henry | 6,460 | <p>Each non-zero coefficient in the product must be the result of at least one pair from the original polynomials so the number cannot exceed $mn$.</p>
<p>$mn$ is a possible result, for example from $(x^{(m-1)n}+x^{(m-2)n}+\cdots+x^{2n}+x^{n}+1)(x^{n-1}+x^{n-2}+\cdots+x^2+x^1+1)$ </p>
|
1,683,977 | <p>How should I compute the derivative of $e^{x\sin x}$ ?
I am a student of class 11, so can you explain me how to do this without high level mathematics ( I know first principles )
I know that derivative of $e^x$ is $e^x$, but I cannot understand what to do with that $\sin x$?</p>
| Itakura | 229,346 | <p>$y=e^{x\cdot sinx}$, let $u=x\cdot sinx$</p>
<p>$\frac{dy}{dx}=\frac{dy}{du}\cdot \frac{du}{dx}$ (chain rule)</p>
<p>$\frac{dy}{du}=e^u$</p>
<p>$\frac{du}{dx}=x\cdot cosx + 1\cdot sinx$ (product rule)</p>
<p>$y'=e^u\cdot (x\cdot cosx+sinx)$</p>
<p>therefore: $y'=e^{xsinx}(x\cdot cosx+sinx)$</p>
|
89,654 | <p>I am new to <em>Mathematica</em>, and I need help with integration of the following Reliability expression (from a well-known Reliability Model). </p>
<p>$$R(t)=e^{-Ne^{-bt_i}(1-e^{-bt})}, t \geq 0$$</p>
<p>I have simplified this by saying $m = b*t_i$. </p>
<pre><code>Integrate[Exp[-N*Exp[-m]*(1 - Exp[-b*t])], {t... | Michael E2 | 4,999 | <p>Since your integrand does not approach zero but a finite positive number,</p>
<pre><code>Limit[Exp[-16.136 (1 - Exp[-0.012*t])], t -> Infinity]
(* 9.82255*10^-8 *)
</code></pre>
<p>the integral over <code>{t, 0, Infinity}</code> does not converge.</p>
<hr>
<p>By the way, the error in the <code>NIntegrate[in... |
4,320,453 | <p>In the context of Lebesgue Integrals, I have came across <span class="math-container">$L^2$</span> as the set of measurable functions <span class="math-container">$f:[a,b] \rightarrow \mathbb{C}$</span> that have Lebesgue integrable squares - that is <span class="math-container">$x \to |f(x)|^2$</span> is Lebesgue ... | Wuestenfux | 417,848 | <p>Hint: <span class="math-container">$\mathbb{R}^n$</span> is an <span class="math-container">$n$</span>-dim. <span class="math-container">$\mathbb{R}$</span>-vector space and <span class="math-container">$\mathbb{C}^n$</span> is a <span class="math-container">$2n$</span>-dim, <span class="math-container">$\mathbb{R}$... |
1,463,457 | <p>Given a function </p>
<p>$$F(x)= \begin{cases} x^2 & \text{when }x \in \mathbb Q \\3x & \text{when }x \in\mathbb Q^c \end{cases}$$</p>
<p>Show that $F$ is continuous or not on $x=3$ with $\epsilon-\delta$.</p>
<p>I tried to deal with problems just like doing on Dirichlet functions. Mistakenly or not, I co... | Yes | 155,328 | <p><em>Hint:</em> Note that $x^{2} \geq 3x$ for all $x \geq 3$ and that $x^{2} \leq 3x$ for all $0 \leq x \leq 3$;
now
$
|F(3+h) - F(3)| \leq (3+h)^{2} - 9 = 6h+h^{2} \to 0
$
as $h \to 0+$
and
$
|F(3+h) - F(3)| \leq |(3+h)^{2} - 9| = |6h+h^{2}| \to 0
$
as $h \to 0-$,
so $|F(3+h) - F(3)| \to 0$ as $h \to 0$.</p>
|
4,145,772 | <p>I asked a question before, and was directed <a href="https://math.stackexchange.com/questions/3447575/why-is-the-tangent-of-an-angle-called-that/3447586#3447586">here</a> instead. :^( Hopefully, someone reading on this thread might still be able to help answer my question.</p>
<p>I am trying to understand how the w... | Michael Hardy | 11,667 | <blockquote>
<p>"a straight line or plane that touches a curve or curved surface at a point, but if extended does not cross it at that point"</p>
</blockquote>
<p>That is wrong. The tangent line to the graph of <span class="math-container">$y = \sin x$</span> at <span class="math-container">$x=0$</span> cross... |
355,095 | <p>For arbitrary formulas $A,B,C$ it holds that:</p>
<ol>
<li>$\{A,B\} \vDash C $ if $A \vDash (B \Rightarrow C)$</li>
<li>$(A \Rightarrow B) \vDash C$ if $A \vDash (B \Rightarrow C)$</li>
<li>$A \vDash C$ if $A \vDash (B \Rightarrow C)$</li>
</ol>
<p>I know that only first one holds, can someone explain me why?</p>
| hmakholm left over Monica | 14,366 | <p>The canonical way to show that the two last implications <em>don't</em> hold would be to find formulas you can plug in for $A$, $B$ and $C$, such that the entailment to the right of the "if" is logically valid, but the one to the left isn't.</p>
<p>For example, try setting $B\equiv P$, $C\equiv Q$ and $A\equiv (P\R... |
397,830 | <blockquote>
<p>If $a,b \in \mathbb{Z}$ and odd, show $8 \mid (a^2-b^2)$.</p>
</blockquote>
<p>Let $a=2k+1$ and $b=2j+1$. I tried to get $8\mid (a^2-b^2)$ in to some equivalent form involving congruences and I started with
$$a^2\equiv b^2 \mod{8} \Rightarrow 4k^2+4k \equiv 4j^2+4j \mod{8}$$
$$\Rightarrow k^2+k-j^2-j... | Vidyanshu Mishra | 363,566 | <p>The given condition says that $a,b$ both are odd. So let, $a=2k+1$ and $b=2l+1$ where $k,l$ are some integers.</p>
<p>Now, $a^2-b^2=4k^2+4k+1-(4l^2+4l^2+1)=4(k^2-l^2+k-l)$ </p>
<p>$$=4((k^2-k)(l^2-l))=4(k(k-1)+l(l-1)$$.</p>
<p>Notice that $k(k-1),l(l-1)$ are products of two consecutive integers and hence divisib... |
397,830 | <blockquote>
<p>If $a,b \in \mathbb{Z}$ and odd, show $8 \mid (a^2-b^2)$.</p>
</blockquote>
<p>Let $a=2k+1$ and $b=2j+1$. I tried to get $8\mid (a^2-b^2)$ in to some equivalent form involving congruences and I started with
$$a^2\equiv b^2 \mod{8} \Rightarrow 4k^2+4k \equiv 4j^2+4j \mod{8}$$
$$\Rightarrow k^2+k-j^2-j... | Blaise Thunderstorm | 396,960 | <p>observe that if $x=2k+1$</p>
<p>$x^2=4k(k+1)+1$</p>
<p>as $k(k+1)$ is a product of two numbers hence is even</p>
<p>so $x^2$ is of form $8k+1$</p>
<p>so $a^2-b^2=8p+1-8q-1=8(p-q)$</p>
<p>a multiple of $8$!!!</p>
|
3,725,762 | <p>Prove that <span class="math-container">$ \sum_{i=1}^{N} a_i \leq \sqrt{N \sum_{i=1}^{N}a_i^2} $</span>. Well i choose <span class="math-container">$u=(1,\ldots,1)$</span> and <span class="math-container">$v=(a_1,\ldots,a_N)$</span> whit <span class="math-container">$a_i$</span> positive and. Apply <span class="mat... | PrincessEev | 597,568 | <p>Recall the Cauchy-Schwarz inequality for sums:</p>
<p><span class="math-container">$$\left( \sum_1^N a_n b_n \right)^2 \le \left( \sum_1^N a_n^2 \right) \left( \sum_1^N b_n^2 \right)$$</span></p>
<p>Let <span class="math-container">$b_n = 1$</span>. Then this simplification results:</p>
<p><span class="math-containe... |
82,734 | <p>Can anyone please recommend some good reading on the geometry of linear groups and their actions?</p>
<p>An example of the kind of question I am interested in: Explicitly describe a fundamental domain for the action of $GL_2(\mathbb{Z})$ on $GL_2(\mathbb{R})$, and compute the volume of the quotient. </p>
<p>I'm fa... | paul garrett | 15,629 | <p>In addition to Siegel's "Geometry of Numbers", Godement's Seminaire Bourbaki from 1967-8 does "reduction theory" including a very nice adelic version of Minkowski-Siegel-Borel.</p>
<p>If the goal is obtaining a Siegel-set approximation to a fundamental domain, rather than a precise fundamental domain, the GL(n,Q) c... |
149,601 | <p>I want to use NSolve to find the root of a function <code>myFun</code> that is only meaningful when evaluated on a numeric argument:</p>
<pre><code>Clear[myFun];
myFun[x_?NumericQ] := N[Sin[x]]
myFun[1]
(* 0.841471 *)
NSolve[{myFun[x] == 0.5, x > 0, x < 2}, x]
(* NSolve[{myFun[x] == 0.5, x > 0, x <... | SPPearce | 23,105 | <p>m_goldberg has explained why <code>NSolve</code> doesn't work here, if your function does need <code>?Numeric</code> you probably want to use <code>FindRoot</code> to solve it:</p>
<pre><code>FindRoot[myFun[x] - 0.5, {x, 0, 2}]
(* {x -> 0.523599} *)
</code></pre>
|
528,068 | <p>This problem has two parts:
$a)$ Let $k>0$, find the minimum of the function $f(x,y)=x+y$ over the set S=$\{(x,y) \in \mathbb R^2_{> 0}:xy=k\}$. $b)$ Prove that for every $(x,y) \in \mathbb R^2_{> 0}$ the inequality $\frac{x+y} {2}\geq \sqrt{xy}$ holds.$$$$
I want to find the minimum of $f$ restricted to t... | boojum | 882,145 | <p>You already <em>had</em> proven that the point <span class="math-container">$ \ (\sqrt{k} \ , \sqrt{k}) \ $</span> is at the minimum when you made your gradient calculation. You found that this point lies on the "level surface" <span class="math-container">$ \ x \ + \ y \ = \ 2\sqrt{k} \ $</span> , which... |
909,690 | <p><img src="https://i.stack.imgur.com/Z8Hq1.png" alt="enter image description here"></p>
<p>Part (a) I am familiar with:</p>
<p>(a) P(batch is rejected) = P(X greater than or equal to 3)</p>
<p>and n = 15 and p(defective) = 0.1 </p>
<p>This gives me the correct answer of 0.1841</p>
<p>I am stuck at part 2! I have... | TonyK | 1,508 | <p>You want the point where the perpendicular bisector of the two points cuts the $y$-axis. </p>
<p><img src="https://i.stack.imgur.com/qIHrm.png" alt="enter image description here"></p>
<p>The slope of the line between $(5,-5)$ and $(1,1)$ is $-\frac{3}{2}$, so the slope of the normal is $\frac23$. Hence the normal... |
1,588,361 | <p>I know if a matrix has a left and right inverse then the inverses are the same and are (is) unique and the original matrix is a square matrix, thus if I have a matrix which has multiple left inverses for example then it has no right inverse and is a non-square matrix. But if a matrix has a unique left inverse then ... | Jendrik Stelzner | 300,783 | <p>It has to be a square matrix. This can be shown by using basic properties about linear equation systems:</p>
<p>Let $A$ be a $m \times n$-matrix. An $n \times m$-matrix $B$ is a left inverse of $A$ if and only if
$$
\sum_{k=1}^m B_{ik} A_{kj}
= (B \cdot A)_{ij}
= I_{ij}
= \delta_{ij}
\quad
\text{for every $1 ... |
2,947,806 | <p>Clearly it isn't, a quick sketch would show it, but I need an analytical proof. The obvious suggestion would be to view the limit on both sides and try to find a inconsistency but how does that work exactly in this case.</p>
<p>Evaluating from both sides the definition of differentiability yields already an intuiti... | user2662833 | 89,483 | <p>What you've done is totally fine. Remember, something is only differentiable if it is continuous and these two limits are the same:</p>
<p><span class="math-container">$$ L_r = \lim_{\Delta x \rightarrow 0^+} {f(x+\Delta x) - f(x) \over \Delta x} $$</span>
<span class="math-container">$$ L_l = \lim_{\Delta x \right... |
415,573 | <p>Show that there holds $$\sqrt{1+x} < 1+(x/2)$$ for all $x > 0$ .</p>
<p>I need guidance in doing this question. Can anyone help please? I'll be thankful!</p>
| Patrick | 50,809 | <p>My guess is that this problem is intended to give you practice with Rolle'e Theorem but why not just square both sides:</p>
<p>$$\sqrt{1+x}<1+\frac{x}{2}$$</p>
<p>$$1+x<1+x+\frac{x^2}{4}$$ whenever $x>0$. </p>
|
2,245,916 | <p>Find by the method of characteristic, the integral surface of $$pq=xy$$ which passes through the curve $$z=x,y=0$$</p>
<p>By strip condition, there is a unique initial strip $$x_{0}=s,y_{0}=0,z_{0}=s,p_{0}=1,q_{0}=0$$</p>
<p>And the characteristic equations are </p>
<p>$$\frac{dx}{dt}=f_p,\\ \frac{dy}{dt}=f_q,
\... | doraemonpaul | 30,938 | <p>Hint:</p>
<p>Let <span class="math-container">$\begin{cases}p=x^2\\q=y^2\end{cases}$</span> ,</p>
<p>Then <span class="math-container">$z_x=z_pp_x+z_qq_x=2xz_p$</span></p>
<p><span class="math-container">$z_y=z_pp_y+z_qq_y=2yz_q$</span></p>
<p><span class="math-container">$\therefore2xz_p2yz_q=xy$</span> with <s... |
2,756,762 | <p>I am trying to solve the following equation:
$$
z^3 + z +1=0
$$</p>
<p>Attempt: I tried to factor out this equation to get a polynomial term, but none of the roots of the equation is trivial.</p>
| José Carlos Santos | 446,262 | <p>Indeed, none of the roots is trivial. In this case, your best option is probably to apply <a href="https://en.wikipedia.org/wiki/Cubic_function#Cardano's_method" rel="nofollow noreferrer">Cardano's formula</a>. It gives a real root and two complex non-real roots.</p>
|
2,756,762 | <p>I am trying to solve the following equation:
$$
z^3 + z +1=0
$$</p>
<p>Attempt: I tried to factor out this equation to get a polynomial term, but none of the roots of the equation is trivial.</p>
| Quanto | 686,284 | <p>Substitute <span class="math-container">$x=\frac2{\sqrt3}\sinh t$</span> to rewrite the equation <span class="math-container">$x^3+x+1=0 $</span> as</p>
<p><span class="math-container">$$4\sinh^3t+3\sinh t + \frac{3\sqrt3}2=0$$</span></p>
<p>Comparing with the identity <span class="math-container">$4\sinh^3t+3\sinh... |
283,245 | <p>Draks gave the identity, <a href="https://math.stackexchange.com/questions/102413/higher-order-trigonometric-function">Higher Order Trigonometric Function</a> $$\sum_{k=0}^\infty \frac{(-1)^k x^{km}}{(km)!}=\frac{1}{m}\sum_{k=0}^{m-1} \exp( e^{i\frac{2k+1}{m}\pi}x )$$ How can this be proven?</p>
| anon | 11,763 | <p>Writing $\zeta_{2m}$ for a $2m$th root of unity, $e^{\large i\pi \frac{2k+1}{m}n}=e^{\large2\pi i(\frac{k}{m}n+\frac{1}{2m}n)}=\zeta_m^{kn}\zeta_{2m}^n$. Thus</p>
<p>$$\frac{1}{m}\sum_{k=0}^{m-1}\exp\left(e^{i\pi \frac{2k+1}{m}}x\right)=\frac{1}{m}\sum_{k=0}^{m-1}\sum_{n=0}^\infty\frac{e^{\pi i\frac{2k+1}{m}n}x^n}{... |
116,727 | <p>I was thinking a bit about isometric embeddings into Hilbert spaces and got the following idea.</p>
<p>First, as we recall, many vector spaces over the reals are isomorphic to $\mathbb{R}^{\alpha}$ for some cardinal number $\alpha$. [EDIT: As Carl Mummert remarked below, not every vector space is of this form, as I... | ncmathsadist | 4,154 | <p>Neither. Their interior is empty and their closure is the entire line.</p>
|
116,727 | <p>I was thinking a bit about isometric embeddings into Hilbert spaces and got the following idea.</p>
<p>First, as we recall, many vector spaces over the reals are isomorphic to $\mathbb{R}^{\alpha}$ for some cardinal number $\alpha$. [EDIT: As Carl Mummert remarked below, not every vector space is of this form, as I... | Rafa | 25,524 | <p>Remember that in metric spaces (those with a notion of distance), we can check if a set <span class="math-container">$A$</span> is closed by checking that the limits of elements in <span class="math-container">$A$</span> rest in <span class="math-container">$A$</span> (i.e., for any convergent sequence <span class="... |
3,163,005 | <blockquote>
<p>If <span class="math-container">$a,b,c\in\mathbb{R}$</span> and <span class="math-container">$2a+b+3c=20.$</span> Then minimum value of <span class="math-container">$a^2+4b^2+c^2$</span> is </p>
</blockquote>
<p>what i try</p>
<p>Cauchy schwarz inequality</p>
<p><span class="math-container">$$(a^2+... | Word Shallow | 466,835 | <p>I think this solution is more simple.</p>
<p>From <span class="math-container">$$2a+b+3c=20\Leftrightarrow b=20-2a-3c$$</span></p>
<p>Then <span class="math-container">$$A=a^2+4\left(20-2a-3c\right)^2+c^2$$</span></p>
<p><span class="math-container">$$=17a^2+48ac-320a+37c^2-480c+1600$$</span></p>
<p><span class=... |
3,341,577 | <p>Consider the matrix <span class="math-container">$B = \begin{bmatrix} 2 & 2 \\ 1 & 3 \end{bmatrix}$</span>. Find projection matrices <span class="math-container">$P_1, P_2$</span> such that (1) <span class="math-container">$B = \lambda_1 P_1 + \lambda_2 P_2$</span> where <span class="math-container">$\lambda... | Gerry Myerson | 8,269 | <p><span class="math-container">$$B=\lambda_1P_1+\lambda_2P_2=P_1+4P_2=P_1+4(I-P_1)=4I-3P_1$$</span> so <span class="math-container">$P_1=(1/3)(4I-B)$</span>. </p>
|
172,871 | <p>I have another problem that I can't figure out.</p>
<p>A triangle har 2 of its summits in points (7.788,0,1.95), (0,7.788,1.95), and the last one on the curve with all the points (7.788,7.788,a^2+1.95), a is a real number. Calculate the area f(a) of the triangle as a function of a and calculate where it takes its m... | Dovendyr | 58,101 | <p>Thank you to everyone.
The last line that @Coolwater named worked fine after I fixed the missing parenthesis, so I will go for it (until test on Monday, then I will explore how it works!)</p>
<p>I don't understand what happens in the last line though!</p>
<pre><code>({x, y, z} - d).n/n[[1]] == 0 // ExpandAll
</cod... |
4,529,181 | <p>I let my base step be <span class="math-container">$n=1$</span>,
<span class="math-container">$5^1 \geq 4(1) + 1$</span> is true.
We can now assume that it is true for some number <span class="math-container">$n = k$</span> where <span class="math-container">$k$</span> is a natural number.
We wish to show the claim ... | eMathHelp | 166,193 | <p><span class="math-container">$5^{k+1}=5\cdot 5^k\ge 5(4k+1)=16k+4k+5\ge4k+5=4(k+1)+1$</span></p>
|
3,375,280 | <p>Let A and B two bonded sets from <span class="math-container">$R$</span> i want to prove that the set <span class="math-container">$AB=\{ab,a\in A,b\in B\}$</span> is bounded </p>
<p>Let <span class="math-container">$c\in AB$</span> then there exists <span class="math-container">$a\in A$</span> and <span class="mat... | drhab | 75,923 | <p>For your flaw see the answer of Calvin Lin.</p>
<p>Hint:</p>
<p>Find a positive constant <span class="math-container">$c$</span> such that <span class="math-container">$|a|\leq c$</span> and <span class="math-container">$|b|\leq c$</span> for every <span class="math-container">$a\in A$</span> and <span class="math... |
690,209 | <p>I have a question that says this:</p>
<blockquote>
<p>Classify $\mathbb{Z}_5 \times \mathbb{Z}_4 \times \mathbb{Z}_8 / \langle(1,1,1)\rangle$ according to the fundamental theorem of finitely generated abelian groups.</p>
</blockquote>
<p>I would like to see how it is correctly answered. This is not homework; I'd... | hmakholm left over Monica | 14,366 | <p>I'm assuming that $\langle(1,1,1)\rangle$ means the subgroup generated by $(1,1,1)$, or in other words $\{(k\bmod 5,k\bmod 4,k\bmod 8)\mid k\in\mathbb Z\}$.</p>
<p>In that case we can see that each of the cosets that make up the quotient must contain an element of the form $(0,x,0)$. Namely, assume that $(a,b,c)$ i... |
690,209 | <p>I have a question that says this:</p>
<blockquote>
<p>Classify $\mathbb{Z}_5 \times \mathbb{Z}_4 \times \mathbb{Z}_8 / \langle(1,1,1)\rangle$ according to the fundamental theorem of finitely generated abelian groups.</p>
</blockquote>
<p>I would like to see how it is correctly answered. This is not homework; I'd... | Steven Alexis Gregory | 75,410 | <p>For convenience, let $\mathcal G = \mathbb{Z}_5 \times \mathbb{Z}_4 \times \mathbb{Z}_8 / \langle(1,1,1)\rangle$</p>
<p>Since $\text{ord}\langle(1,1,1)\rangle = 40$, then $\text{ord}
(\mathcal G) = \dfrac{5 \times 4 \times 8}{40} = 4$</p>
<p>Using <a href="https://math.stackexchange.com/questions/1320861/characte... |
931 | <p>What do moderators do? Is it just a matter of dealing with posts flagged for moderator attention, or are there other things too?</p>
| Scott Morrison | 3 | <p>Dealing with flags for moderator attention is the main day-to-day obligation. Currently (October 2013) we do pretty well at this, averaging under 3 hours to process a flag. For various reasons, there are actually more flags for moderator attention now than there was before the transition to 2.0.</p>
<p>The software... |
308,102 | <p>Let $f:M \to N$ be a smooth weakly conformal map between connected $d$-dimensional Riemannian manifolds, i.e. $f$ satisfies $df^Tdf =(\det df)^{\frac{2}{d}} \, \text{Id}_{TM}$. </p>
<blockquote>
<p>Assume $d \ge 3$ and that $df=0$ at some point. Is it true that $f$ is constant?</p>
</blockquote>
<p>A proof for ... | Vít Tuček | 6,818 | <p>I am no expert in this field, but I just looked into <strong>Harmonic Morphisms Between Riemannian Manifolds</strong> by <em>Paul Baird</em> and <em>John C. Wood</em>, specifically into chapter 14. Here they give more general definition of weakly conformal mapping (which it seems is the standard definition). Namely,... |
308,102 | <p>Let $f:M \to N$ be a smooth weakly conformal map between connected $d$-dimensional Riemannian manifolds, i.e. $f$ satisfies $df^Tdf =(\det df)^{\frac{2}{d}} \, \text{Id}_{TM}$. </p>
<blockquote>
<p>Assume $d \ge 3$ and that $df=0$ at some point. Is it true that $f$ is constant?</p>
</blockquote>
<p>A proof for ... | David E Speyer | 297 | <p>I am not at all an expert in this field. However, I believe I can show that there are no examples where $Df$ vanishes to finite order. Thus, in particular there are no analytic solutions and, if the OP is right about two-jet-determination, there are no solutions.</p>
<p>Let $f: M \to N$ be a weakly conformal map be... |
3,212,371 | <p>Let <span class="math-container">$(X,d)$</span> be a metric space and let <span class="math-container">$x_n$</span> be a sequence in <span class="math-container">$X$</span>. Let <span class="math-container">$x\in X$</span>. Define a sequence <span class="math-container">$\{y_n\}$</span> by <span class="math-contain... | Cornman | 439,383 | <p>It is <span class="math-container">$\lim_{n\to\infty} \frac{n^n}{(n+1)^n}=\lim_{n\to\infty}\left(\frac{n}{1+n}\right)^n=\lim_{n\to\infty}\left(\frac{n+1-1}{n+1}\right)^n=\lim_{n\to\infty}\left(1-\frac{1}{n+1}\right)^n=\frac1e$</span></p>
|
1,708,090 | <p>Suppose we have the original Riemann sum with no removed partitions, where $f(x)$ is continuous and reimmen integratable on the closed interval $[a,b]$.
$$\lim_{n\to\infty}\sum_{i=1}^{n}f\left(a+\left(\frac{b-a}{n}\right)i\right)\left(\frac{b-a}{n}\right)$$</p>
<p>If we remove $s$ partitions for every $d$ partition... | zhw. | 228,045 | <p>Let's simplify and assume $f$ is Riemann integrable on $[0,1].$ Fix $d,s \in \mathbb N,0<s<d.$ Consider the uniform partition of $[0,1]$ into subintervals of length $1/n,$ thinking of $n$ here as being larger than $d.$ Choose $c_i \in [(i-1)/n,i/n].$ </p>
<p>The key to this is to look at the indices $i$ in gr... |
4,377,170 | <p>I have the following system:</p>
<p><span class="math-container">$$
(a + tb)\cos \theta = x \\
(b + ta) \sin \theta = y \\
$$</span></p>
<p>with the constraints</p>
<p><span class="math-container">$$a > b$$</span> <span class="math-container">$$\theta \in (-\pi, \pi]$$</span> <span class="math-container">$$t >... | TurlocTheRed | 397,318 | <p><span class="math-container">$x=(a+tb)\cos\theta, y=(b+at)\sin\theta$</span></p>
<p>So <span class="math-container">$t=\frac{1}{b}(\frac{x}{\cos(\theta)}-a)$</span></p>
<p><span class="math-container">$y=\sin\theta[b+\frac{a}{b}(\frac{x}{\cos \theta}-a)]=\sqrt{1-\cos^2\theta}[b+\frac{a}{b}(\frac{x}{\cos \theta}-a)]$... |
3,517,528 | <p>Let <span class="math-container">$X$</span> be the the space of polynomials on <span class="math-container">$[0, 1]$</span> and let <span class="math-container">$\|p\| = \max|p'(x)|$</span>, where <span class="math-container">$p'$</span> is the derivative of p. Is <span class="math-container">$\| . \|$</span> a norm... | José Carlos Santos | 446,262 | <p>No, because <span class="math-container">$1\neq0$</span>, but <span class="math-container">$\lVert1\rVert=0$</span>.</p>
|
189,385 | <p>I have some questions about the definition of tangent space that arose after reading the book Differential Geometry of Curves and Surfaces of Manfredo do Carmo.</p>
<p>First, what's the best way to define tangent space? Using tangent vector to curves or derivations? </p>
<p>I've researched about the concept of der... | James S. Cook | 36,530 | <p>For smooth manifolds the tangent space given by equivalence classes of smooth curves and the set of derivations on smooth functions is equivalent. You can give an isomorphism between these constructions. Each has its advantages. </p>
<ul>
<li><p>For example, in the equivalence class of curves set-up the differentia... |
189,385 | <p>I have some questions about the definition of tangent space that arose after reading the book Differential Geometry of Curves and Surfaces of Manfredo do Carmo.</p>
<p>First, what's the best way to define tangent space? Using tangent vector to curves or derivations? </p>
<p>I've researched about the concept of der... | Vishesh | 27,826 | <p>I am just trying to add my understanding to the mix. As in one of the books I used, the author remarks that the definition of tangent vectors and ergo the tangent space is much more intuitive and "pleasant" when curves are employed.</p>
<p>However as the answers already given so neatly tell, it is much more benefic... |
3,788,418 | <p>If you take a bus to work in the morning there is a 20% chance you’ll arrive late. When you go
by bicycle there is a 10% chance you’ll be late. 70% of the time you go by bike, and 30% by
bus. Given that you arrive late, what is the probability you took the bus?</p>
<p>P(took the bus | being late) = <strong>P(took th... | Simon Terrington | 302,396 | <p>Good question well first you gotta take the bus <span class="math-container">$(0.3)$</span> then you need to be late on the bus <span class="math-container">$(0.2)$</span> so your bolded probability is <span class="math-container">$0.3 \times 0.2$</span> otherwise known as <span class="math-container">$0.06$</span> ... |
246,278 | <p>Let $0 < a < b < \infty$. Define $x_1=a$, $x_2=b$, and $x_{n+2} = \frac{x_n + x_{n+1}}{2}$ for $n \geq 1$. Does $\{x_n\}$ converge? If so, to what limit?</p>
| Brian M. Scott | 12,042 | <p>HINT: Note that <span class="math-container">$\dfrac{x_n+x_{n+1}}2$</span> is just the arithmetic mean of <span class="math-container">$x_n$</span> and <span class="math-container">$x_{n+1}$</span>, so geometrically it’s their midpoint: it’s halfway between <span class="math-container">$x_n$</span> and <span class="... |
682,156 | <p>I have come across two contradicting definitions of characteristics function (CHF). In wikipedia CHF is defined as the inverse Fourier transform (FT) of probability density function (PDF) and at some places (e.g. <a href="http://www.math.nus.edu.sg/~matsr/ProbI/Lecture6.pdf" rel="nofollow">http://www.math.nus.edu.sg... | Jason | 195,308 | <p>If the density exists, then the characteristic function will be a scaled form of the Fourier transform. There are multiple conventions for the Fourier transform - I use
$$\hat f(t)=\int_{-\infty}^\infty f(x)e^{-2\pi ixt}\,\mathrm d x$$
in which case the characteristic function of a random variable with density $f$ i... |
16,719 | <p>What are your favourite <em>scholarly</em> books? My favourite is definitely G.N. Watson's <a href="http://books.google.com/books?id=Mlk3FrNoEVoC&printsec=frontcover&dq=watson+bessel&source=bl&ots=SOToCOngC4&sig=3jym5VIo2ESQR4IlZREpEdy6oXE&hl=en&ei=WeWKS9yGNYWWtgfhtN21Dw&sa=X&oi=... | Mariano Suárez-Álvarez | 1,409 | <p>Nelson Dunford and Jacob T. Schwartz's <em>Linear Operators</em>. One of the most awe-inspiring books ever written.</p>
|
16,719 | <p>What are your favourite <em>scholarly</em> books? My favourite is definitely G.N. Watson's <a href="http://books.google.com/books?id=Mlk3FrNoEVoC&printsec=frontcover&dq=watson+bessel&source=bl&ots=SOToCOngC4&sig=3jym5VIo2ESQR4IlZREpEdy6oXE&hl=en&ei=WeWKS9yGNYWWtgfhtN21Dw&sa=X&oi=... | Mariano Suárez-Álvarez | 1,409 | <p>A rather amazing book, in the spirit of Dickson's on the history of number theory, is Th. Muir's <em>The Theory of Determinants in the Historical Order of Development</em> which, in four volumes, gives more information and references about determinants than most humans can survive.</p>
|
357,417 | <p>Say I have a matrix $A$ and its row reduced echelon form looks like this:
$$
\begin{bmatrix}
1 &-3 &2 &-7\\
0 &0 &0 &0\\
0 &0 &0 &0\\
0 &0 &0 &0
\end{bmatrix}
$$
I see it has $3$ free variables.... does this mean that the $\text{Null}(A) = \mathbb{R}^3$?
.... Or ... | user1337 | 62,839 | <p>No. The Nullspace of $A$ lies within $\mathbb R^4$, 4 as in the number of columns (and not $\mathbb R^3$). It is given by the solutions to first equation that you found, $x_1-3x_2+2x_3-7x_4=0$.</p>
|
357,417 | <p>Say I have a matrix $A$ and its row reduced echelon form looks like this:
$$
\begin{bmatrix}
1 &-3 &2 &-7\\
0 &0 &0 &0\\
0 &0 &0 &0\\
0 &0 &0 &0
\end{bmatrix}
$$
I see it has $3$ free variables.... does this mean that the $\text{Null}(A) = \mathbb{R}^3$?
.... Or ... | Clive Newstead | 19,542 | <p>You can certainly say $\operatorname{null}(A) \cong \mathbb{R}^3$, but all this is saying is that the null space is $3$-dimensional. If you want to say what the null space is <em>equal</em> to then you need to do more work; in particular, it needs to be expressed as a subspace of $\mathbb{R}^4$.</p>
|
3,618,029 | <p>In <em>Cormen's Introduction to Algorithm's book</em>, I'm attempting to work the following problem:</p>
<blockquote>
<p>Show that the solution to the recurrence relation <span class="math-container">$T(n) = T(n-1) + n$</span> is <span class="math-container">$O(n_2)$</span> using substitution
<span class="math-... | CHAMSI | 758,100 | <p>Let <span class="math-container">$ n \in\mathbb{N}^{*} $</span>, and <span class="math-container">$ k $</span> be an integer less than <span class="math-container">$ n $</span>, we have that : <span class="math-container">\begin{aligned} T\left(k\right)&=T\left(k-1\right)+k\\ \Longrightarrow\sum_{k=2}^{n}{\left(... |
1,370,561 | <p>Consider a function $f \in C([-1,1],\mathbb{R})$ and suppose that </p>
<p>$$\int_{-1}^{1} f(t)t^{2n}dt=0$$</p>
<p>for all $n \in \mathbb{N}_0$. I want to show that under this assumption the function $f$ has to be odd, that is $f(x)=-f(-x)$ for all $x \in [-1,1]$.</p>
<p>Let $A$ denote the set $span\{t^{2n}:n \in ... | zhw. | 228,045 | <p>You can write $f= f_e + f_o,$ where $f_e(x) = (f(x) + f(-x))/2, f_o(x) = (f(x) - f(-x))/2.$ Note that $f_e$ is even, $f_o$ is odd. Since $f_o$ is odd, we have</p>
<p>$$\int_{-1}^1 f_o(t)t^{2n}\, dt = 0$$</p>
<p>for all $n.$ Our hypothesis then implies</p>
<p>$$\int_{-1}^1 f_e(t)t^{2n}\, dt = 0$$</p>
<p>for all $... |
2,948,984 | <p>We have five different pairs of gloves. Three people choose at random one left and one right glove. What is probability, that each person doesn't have a pair.
My attempt:
<span class="math-container">$|\Omega| = {{10}\choose{6}}$</span>
First person can choose the left glove in five ways, right glove-four ways.
Seco... | N. F. Taussig | 173,070 | <p>Since the gloves are distinct, there are <span class="math-container">$5 \cdot 4 \cdot 3 = 60$</span> ways for the three people to choose three of the five left gloves and the same number of ways for them to choose three of the five right gloves. Hence, there are <span class="math-container">$60^2 = 3600$</span> se... |
2,948,984 | <p>We have five different pairs of gloves. Three people choose at random one left and one right glove. What is probability, that each person doesn't have a pair.
My attempt:
<span class="math-container">$|\Omega| = {{10}\choose{6}}$</span>
First person can choose the left glove in five ways, right glove-four ways.
Seco... | ArsenBerk | 505,611 | <p>Let <span class="math-container">$G_{ir}$</span> and <span class="math-container">$G_{il}$</span> be the right and left gloves of pair <span class="math-container">$i$</span>. Then we can seperate right and left pairs as two sets such as <span class="math-container">$L = \{G_{1l}, G_{2l}, G_{3l}, G_{4l}, G_{5l}\}$</... |
2,948,984 | <p>We have five different pairs of gloves. Three people choose at random one left and one right glove. What is probability, that each person doesn't have a pair.
My attempt:
<span class="math-container">$|\Omega| = {{10}\choose{6}}$</span>
First person can choose the left glove in five ways, right glove-four ways.
Seco... | drhab | 75,923 | <p>Give the three people numbers <span class="math-container">$1,2,3$</span> and let them choose gloves one by one.</p>
<p>For <span class="math-container">$i=1,2,3$</span> let <span class="math-container">$E_i$</span> denote the event that person <span class="math-container">$i$</span> chooses a pair.</p>
<p>To be f... |
4,009,431 | <p>Let <span class="math-container">$V$</span> and <span class="math-container">$W$</span> be finite dimensional complex vector spaces and let <span class="math-container">$f: V\xrightarrow[]{}W$</span> and <span class="math-container">$g: W\xrightarrow[]{}V$</span> be linear maps. Suppose that <span class="math-contai... | Fred | 380,717 | <p>We have that <span class="math-container">$fg= id_W$</span>. Now show that <span class="math-container">$(gf)^2=gf.$</span> Hence <span class="math-container">$gf:V \to V$</span> is a projection.</p>
<p>Show that <span class="math-container">$V_0:=ker(gf)$</span> and <span class="math-container">$V_1:= im(gf)$</spa... |
242,177 | <p>Show that exactly one of:
\begin{cases} B^Tv = 0\\ d^Tv = 1 \end{cases}
or
$$Bu=d$$
has a solution.
I tried with Farkas lemma, but I run into trouble. </p>
| user1551 | 1,551 | <p>I know very little optimization theory, but if we perform <a href="http://en.wikipedia.org/wiki/Singular_value_decomposition" rel="nofollow">SVD</a> to get $B=U\Sigma V^T$ and absorb the orthogonal matrices $U,V$ into $u,v,d$, the two systems can be rewritten as
\begin{align}
&\begin{cases} \Sigma^T \tilde{v} = ... |
242,177 | <p>Show that exactly one of:
\begin{cases} B^Tv = 0\\ d^Tv = 1 \end{cases}
or
$$Bu=d$$
has a solution.
I tried with Farkas lemma, but I run into trouble. </p>
| saz | 36,150 | <p>Let $u = u_1 - u_2$ such that $u_1, u_2 \geq 0$. Then the second equation is equivalent to</p>
<p>$$B \cdot (u_1-u_2) = d, u_1, u_2 \geq 0 \quad \Leftrightarrow \quad \begin{pmatrix} B & -B \end{pmatrix} \cdot \begin{pmatrix} u_1 \\ u_2 \end{pmatrix} = d, u_1,u_2 \geq 0$$</p>
<p>Moreover the first equation is... |
4,467,187 | <p>In <a href="https://math.stackexchange.com/questions/4564289/lemniscate-numbers-and-others-what-would-be-the-properties">this post</a> user William Ryman asked what would happen if we try to build "complex numbers" with shapes other than circle or hyperbola in the role of a "unit circle".</p>
<p>... | geetha290krm | 1,064,504 | <p>Counter-example for <span class="math-container">$h$</span>: Let <span class="math-container">$A$</span> be the open unit disk so that <span class="math-container">$\overline A=A$</span>. If <span class="math-container">$f(z)=z$</span> then <span class="math-container">$h(z)=\overline z ^{2}$</span> which is not ana... |
4,154,210 | <p>Does <span class="math-container">$$\lim_{\epsilon \to 0^+} \left(\int_{1/2}^{1-ε}＋\int_{1＋ε}^{3/2}\right)\frac{\log x}{(x-1)^2} dx$$</span> exist?</p>
<p>Hint says <span class="math-container">$$\lim_{\epsilon \to 0^+} \left( \frac{\log x-a-b(x-1)-c(x-1)^2}{(x-1)^3}\right)$$</span> exists only if <span class="mat... | Diger | 427,553 | <p>I'd introduce <span class="math-container">$x=1/u$</span> in the first integral <span class="math-container">$$\int_{1/2}^{1-\epsilon} \frac{\log x}{(x-1)^2} \, {\rm d}x = -\int_{1/(1-\epsilon)}^2 \frac{\log u}{(u-1)^2} \, {\rm d}u $$</span> so that
<span class="math-container">$$ - \int_{1/(1-\epsilon)}^2 \frac{\lo... |
3,892,668 | <p>Find the extreme values of <span class="math-container">$f(x,y)=x^2y$</span> in <span class="math-container">$D=\{x^2+8y^2\leq24\}$</span></p>
<p>It was easy to find using Lagrange multipliers the local extreme values on <span class="math-container">$\partial{D}$</span> since we have the condition <span class="math-... | J.G. | 56,861 | <p>Rescaling any point with <span class="math-container">$x^2+8y^2<24$</span> so that <span class="math-container">$x^2+8y^2=24$</span> increases <span class="math-container">$|x^2y|$</span>, so extrema of <span class="math-container">$x^2y$</span>, which can change its sign under <span class="math-container">$(x,\,... |
652,960 | <p>I am lost on this one. I'm still new to ring theory, as we're only a couple weeks into the course, but it's already well over my head. I know that $R$ is an integral domain, so the additive and multiplicative identities are not equal, and if both $a$ and $b$ are nonzero, then their product will be nonzero. So $u$ ca... | Hagen von Eitzen | 39,174 | <p>We may assume wlog. that $a\ne 0$.
From $b\in(a)$, there exists $u\in R$ with $b=ua$. From $a\in(b)$, there exists $v\in R$ with $a=vb$. Then $a=vua$, hence $(vu-1)a=0$. Since $r$ is a domain and $a\ne0$, we conclude $vu=1$.</p>
|
75,777 | <p>I'm looking for references to (as many as possible) elementary proofs of the Weyl's equidistribution theorem, i.e., the statement that the sequence $\alpha, 2\alpha, 3\alpha, \ldots \mod 1$ is uniformly distributed on the unit interval. With "elementary" I mean that it does not make use of complex analysis in partic... | Tony Huynh | 2,233 | <p>Not sure if this qualifies, but there is a short proof using Fourier Analysis. No hardcore stuff, just Fejér's Theorem. See <a href="https://books.google.com/books?id=OcZ5iKsGrmoC&pg=PA11" rel="nofollow noreferrer">Chapter 3</a> of Körner's <em>Fourier Analysis</em> book (there are 110 chapters in the book, s... |
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