qid int64 1 4.65M | question large_stringlengths 27 36.3k | author large_stringlengths 3 36 | author_id int64 -1 1.16M | answer large_stringlengths 18 63k |
|---|---|---|---|---|
2,469,720 | <p>Math problem:</p>
<blockquote>
<p>Find $x$, given that $ \, 2^2 \times 2^4 \times 2^6 \times 2^8 \times \ldots \times 2^{2x} = \left( 0.25 \right)^{-36}$</p>
</blockquote>
<p>To solve this question, I changed the left side of the equation to $2^{2+4+6+ \ldots + 2x}$ and the right side to: $\frac{2^{74}}{3^{36}}$... | Community | -1 | <p>Take the base-2 logarithm of both members, and you get</p>
<p>$$2+4+6+\cdots+2x=(-2)(-36)$$</p>
<p>or</p>
<p>$$1+2+3+\cdots+x=36.$$</p>
<p>$36$ is the eighth triangular number.</p>
<hr>
<p>Even though this is irrelevant to the given problem, you convert a power of $2$ to a power of $3$ by writing</p>
<p>$$2^a... |
370,007 | <p>A river boat can travel a 20km per hour in still water. The boat travels 30km upstream against the current then turns around and travels the same distance back with the current. IF the total trip took 7.5 hours, what is the speed of the current? Solve this question algebraically as well as graphically..</p>
<p>I st... | Alexander Gruber | 12,952 | <p><em>Hint/roadmap:</em></p>
<p>If $|a|$ and $|b|$ are coprime <strong>and $a$ and $b$ commute</strong>, then $|ab|=|a||b|$. In particular, this holds for all pairs of elements in abelian groups.</p>
<p>This follows from these facts:</p>
<ol>
<li><p><em>if $ab=ba$</em>, then $(ab)^x=a^xb^x$, $\hspace{17pt}$(<em>no... |
1,356,367 | <p>Is it true that projection is a normal matrix? It's clear that orthogonal projection is, but what about non-orthogonal projection?</p>
<p>By normal matrix, I mean matrix A such that $AA' = A'A$.</p>
| James Pak | 187,056 | <p>$$\frac{n!}{n_1!n_2!},\;\; n_1+ n_2=n,$$</p>
<p>counts the permutations of the sequence
$$\underbrace{p_1...p_1}_{n_1}\underbrace{p_2...p_2}_{n_2}.$$</p>
<p>Coincidentally, $$\frac{n!}{n_1!n_2!}=\frac{n!}{k!(n-k)!}={n \choose k}, \;\; n_1=k=\text{number of successes}.$$</p>
<p>Therefore, ${n \choose k}$, disguise... |
2,572,302 | <p>I want to calculate the limit: $$ \lim_{x\to +\infty}(1+e^{-x})^{2^x \log x}$$
The limit shows itself in an $1^\infty$ Indeterminate Form. I tried to elevate $e$ at the logarithm of the function:</p>
<p>$$\lim_{x\to +\infty} \log(e^{(1+e^{-x})^{2^x \log x}}) = e^{\lim_{x\to +\infty} \log((1+e^{-x})^{2^x \log x})} =... | user | 505,767 | <p>We have:</p>
<p>$$(1+e^{-x})^{2^x logx}=e^{2^x \log x\log(1+e^{-x})}\to e^0 = 1$$</p>
<p>indeed</p>
<p>$$2^x \log x\log(1+e^{-x})=\frac{2^x\log x}{e^{x}}\log\left[\left(1+\frac{1}{e^x}\right)^{e^x}\right]\to 0\cdot \log e=0$$</p>
<p>$$\frac{2^x\log x}{e^{x}}=\frac{\log x}{\left(\frac{e}{2}\right)^{x}}\stackrel{\... |
353,087 | <p>Solve the interior Dirichlet Problem</p>
<p>$$(r^2u_r)_r+\dfrac{1}{\sin\phi}(\sin\phi~u_\phi)_\phi+\dfrac{1}{\sin^2\phi}u_{\theta\theta}=0\,, \,\,\,\,\,\,\, 0<r<1 $$</p>
<p>where $u(1,\phi)=\cos3\phi$</p>
| doraemonpaul | 30,938 | <p>$(r^2u_r)_r+\dfrac{1}{\sin\phi}(\sin\phi~u_\phi)_\phi+\dfrac{1}{\sin^2\phi}u_{\theta\theta}=0$</p>
<p>$r^2u_{rr}+2ru_r+u_{\phi\phi}+\cot\phi~u_\phi+\csc^2\phi~u_{\theta\theta}=0$</p>
<p>Note that this PDE is separable.</p>
<p>Let $u(r,\phi,\theta)=f(r)g(\phi)h(\theta)$ ,</p>
<p>Then $r^2f''(r)g(\phi)h(\theta)+2r... |
2,189,832 | <p>Take the matrix
$$
\begin{matrix}
1 & 1 & 1 & 1 \\
1 & 1 & 1 & 1 \\
1 & 1 & 1 & 1 \\
1 & 1 & 1 & 1 \\
\end{matrix}
$$</p>
<p>I tried to calculalte the eigenvalues of this matrix and got to a point where I found that the eigenva... | Scientifica | 164,983 | <p>This means that you made a mistake while computing the eigenvalues, so you should check your work and find what you did wrong.</p>
<p>Actually, you can solve this problem pretty easly. We're working on a four-dimensional vector space, and I guess that's $\mathbb{R}^4$ (or $\mathbb{C}^4$, not much difference in this... |
3,553,644 | <p>I am taking a Introduction to Calculus course and am struggling to understand how derivatives can represent tangent lines.</p>
<p>I learned that derivatives are the rate of change of a function but they can also represent the slope of the tangent to a point. I also learned that a derivative will always be an order... | José Carlos Santos | 446,262 | <p>What happens is that, for each <span class="math-container">$a$</span> in the domain of <span class="math-container">$f$</span>, <span class="math-container">$f'(a)$</span> is the <em>slope</em> of the the tangent to the graph of <span class="math-container">$f$</span> at the point <span class="math-container">$\big... |
3,553,644 | <p>I am taking a Introduction to Calculus course and am struggling to understand how derivatives can represent tangent lines.</p>
<p>I learned that derivatives are the rate of change of a function but they can also represent the slope of the tangent to a point. I also learned that a derivative will always be an order... | Community | -1 | <ul>
<li>The formula defining the derivative function is not itself the equation of the tangent; this formula gives you , for each tangent ( one tangent for each point <span class="math-container">$(x, f(x))$</span> of the graph of <span class="math-container">$f$</span> ), the slope of this line. And a slope is a <em>... |
2,062,706 | <p>I have the following function:</p>
<p>\begin{equation}
f(q,p) = q \sqrt{p} + (1-q) \sqrt{1 - p}
\end{equation}</p>
<p>Here, $q \in [0,1]$ and $p \in [0,1]$.</p>
<p>Now, given some value $q \in [0,1]$ what value should I select for $p$ in order to maximize $f(q,p)$? That is, I need to define some function $g(q)$ s... | mfl | 148,513 | <p>We have that </p>
<p>$$\nabla f=\left(\sqrt{p}-\sqrt{1-p},\frac{q}{2\sqrt p}-\frac{1-q}{2\sqrt{1-p}}\right)$$ doesn't vanish on $(0,1)\times (0,1).$ Thus the maximum is achieved on the boundary. Now we have</p>
<p>\begin{cases}f(0,p)&=1-p\\f(1,p)&=p\\f(q,0)&=\sqrt{1-q}\\ f(q,1)&=\sqrt{q}\end{cases}... |
2,048,054 | <p>I need to find signed distance from the point to the intersection of 2 hyperplanes. I was quite sure that this is something that every mathematician do twice a week :) But not found any good solution or explanation for same problem.</p>
<p>In my case the hyperplanes is defined as $y = w'*x + x_0$, but it is ok to d... | N. S. | 9,176 | <p>$$[0,1] \cup [2,3]$$ is metric, disconnected and has infinitely many elements.</p>
<p>Note: if there are infinitely many <strong>components</strong> take each component as an open set and use this open cover to show that the space is not compact.</p>
|
1,704,410 | <p>If we have two groups <span class="math-container">$G,H$</span> the construction of the direct product is quite natural. If we think about the most natural way to make the Cartesian product <span class="math-container">$G\times H$</span> into a group it is certainly by defining the multiplication</p>
<p><span class... | Cameron Williams | 22,551 | <p>I take a different point of view than most people on this, I think, which is due to the way I first encountered it (along the lines of mathematical physics). To me the direct product is lacking in much structure. You effectively just slap two groups together and call it a day - much like when you combine two subspac... |
1,013,484 | <p>I've this function : $f(x,y)= \dfrac{(1+x^2)x^2y^4}{x^4+2x^2y^4+y^8}$ for $(x,y)\ne (0,0)$ and $0$ for $(x,y)=(0,0)$</p>
<p>It's admits directional derivatives at the origin?</p>
| 2'5 9'2 | 11,123 | <p>A directional derivative needs a direction. Maybe you travel vertically straight through $(0,0)$, along the line $x=0$. Then your function is constantly $0$, so of course that particular directional derivative exists. </p>
<p>Otherwise, travel in the direction of the line $y=kx$. Then away from $(0,0)$ your functio... |
4,192,869 | <p>What is the difference between a set being an element of a <span class="math-container">$\sigma$</span>-algebra compared to being a subset of a <span class="math-container">$\sigma$</span>-algebra?</p>
| user6247850 | 472,694 | <p>A subset of a <span class="math-container">$\sigma$</span>-algebra is a set of sets, but an element of a <span class="math-container">$\sigma$</span>-algebra is a set of elements of the underlying space. For example, if we take <span class="math-container">$\Omega = \{1,2,3\}$</span> and the <span class="math-conta... |
1,034,697 | <p>The exercise goes like this:</p>
<p>-Let $W= {(x,y,z)|2x+3y-z=0}$ Then $W\subseteq\mathbb{R}^3$, find the dimension of $W$.</p>
<p>-Find the dimension $[\mathbb{R}^3|W]$</p>
<p>This was a problem from my algebra exam, it was a team exam and this problem was solved by another member of the team (we...he had it rig... | JohnD | 52,893 | <p>To show $W$ is a subspace, use the Subspace Theorem: $0\in W$, $W$ is closed under addition and scalar multiplication.</p>
<p>To compute the dimension of $W$, just note that $2x+3y-z=0$ implies $x=-3y+z$. So you have two free variables: $y$ and $z$. Thus, $W$ is two dimensional. </p>
<p>Since $\mathbb{R}^3$ is thr... |
426,114 | <p>What is the terminology of two point support in this lemma?</p>
<p><img src="https://i.stack.imgur.com/fecF5.png" alt="enter image description here"></p>
| coffeemath | 30,316 | <p><strong>Hint:</strong> If $c(x)=c_0 \in (0,1)$ then it may be that $f$ has a value exceeding $f(c_0)$ which occurs at some $x<c$. In this case a small movement of $x$ will only move $c(x)$ near $c_0$ so the max will stay the same. </p>
<p>On the other hand it may be that $f$ has its maximum value on the interval... |
426,114 | <p>What is the terminology of two point support in this lemma?</p>
<p><img src="https://i.stack.imgur.com/fecF5.png" alt="enter image description here"></p>
| Stefan Hamcke | 41,672 | <p>Note that the set $C:=\{(x,t)\mid x\in\Bbb R,\ t\in[0,c(x)]\}$ is closed and contains the graph of $c$, the set $G(c) = \{(x,c(x))\mid x\in\Bbb R\}$, which is closed in $\Bbb R \times [0,1]$</p>
<p>Show that the preimage of an open subbase set $(-\infty,a)$ under $F$ is open: This is the set $\Bbb R-\{x\mid \exists... |
244,433 | <p>I have a list:</p>
<pre><code>data = {{2.*10^-9, 0.0025}, {4.*10^-9, 0.0025}, {6.*10^-9, 0.0025}, {8.*10^-9, 0.0025}, {1.*10^-8, 0.0025}, {7.*10^-9, 0.0023}, {3.*10^-9, 0.0025},...}
</code></pre>
<p>And I wanted to remove every third pair and get</p>
<pre><code> newdata = {{2.*10^-9, 0.0025}, {4.*10^-9, 0.0025}, {8.... | Nasser | 70 | <p>I am sure there are many ways to do this. One direct way could be to build the index and use it to select the entries.</p>
<pre><code>data = {{2.*10^-9, 0.0025}, {4.*10^-9, 0.0025}, {6.*10^-9,
0.0025}, {8.*10^-9, 0.0025}, {1.*10^-8, 0.0025}, {7.*10^-9,
0.0023}, {3.*10^-9, 0.0025}};
</code></pre>
<p>And now... |
244,433 | <p>I have a list:</p>
<pre><code>data = {{2.*10^-9, 0.0025}, {4.*10^-9, 0.0025}, {6.*10^-9, 0.0025}, {8.*10^-9, 0.0025}, {1.*10^-8, 0.0025}, {7.*10^-9, 0.0023}, {3.*10^-9, 0.0025},...}
</code></pre>
<p>And I wanted to remove every third pair and get</p>
<pre><code> newdata = {{2.*10^-9, 0.0025}, {4.*10^-9, 0.0025}, {8.... | Pillsy | 531 | <p>There are many ways to do this, but my favorite is to use the the stride and end arguments in <a href="http://reference.wolfram.com/language/ref/Drop.html" rel="noreferrer"><code>Drop</code></a>:</p>
<pre><code>Drop[data, {3, -1, 3}] === newdata
(* True *)
</code></pre>
|
6,431 | <p>I hate to sound like a broken record, but closing <a href="https://math.stackexchange.com/q/219906/12042">this question</a> as <em>not constructive</em> makes no sense to me. The canned explanation reads in relevant part:</p>
<blockquote>
<p>We expect answers to be supported by facts, references, or specific expe... | Emily | 31,475 | <p>I personally voted close for the reason that the "question is not a good fit for our Q&A format." I feel as though a user, especially one who has asked previous questions, should understand that MSE is not a dumping site for homework problems on which that they wish to avoid actual work. In particular, because t... |
1,876,133 | <p>Let x be a object that is not a set</p>
<p>Let S be a set</p>
<p>Would the following statement:</p>
<p>x ⊆ S</p>
<p>evaluate to False, or considered not a well formed statement (as x is not even a set).</p>
| Hagen von Eitzen | 39,174 | <p>A more general claim is true:</p>
<p><strong>Claim.</strong> Let $f\colon \Bbb R\to\Bbb R$ be differentiable such that $f'(x)\ge 0$ for all $x\in \Bbb R$. Also assume that between any two points, there is at least one point with non-zero derivative. Then $f$ is strictly increasing. <em>(So in principle many more t... |
1,876,133 | <p>Let x be a object that is not a set</p>
<p>Let S be a set</p>
<p>Would the following statement:</p>
<p>x ⊆ S</p>
<p>evaluate to False, or considered not a well formed statement (as x is not even a set).</p>
| Community | -1 | <p>Because $f'(x) > \forall x \gt x_0$ then $f$ is strictly increasing 0n $(x_0, \infty)$. Now, because $f$ continuous it follows $f$ is strictly increasing on $[x_0, \infty)$. Similar, $f$ is strictly increasing on $(-\infty, x_0]$ therefore $f$ is strictly increasing on $(-\infty, \infty)$</p>
|
3,526,586 | <p>1) Let <span class="math-container">$A \in \mathbb{R}^{n \times n}$</span> be a matrix with nonzero determinant. Show that there exists <span class="math-container">$c>0$</span> so that for every <span class="math-container">$v \in \mathbb{R}^{n},\|A v\| \geq c\|v\|$</span></p>
<p>My attempt:
Since <span class="... | Akshya Kumar | 1,009,890 | <p>U=span{(1,1,1),(-1,2,1)}
Observe that (0,1,0)=(1\3)(1,1,1)+(1/3)(-1,2,-1)
Since (0,1,0)belongs to U
So orthogonal projection of (0,1,0) on U is (0,1,0).
Hence, option 1 is correct.</p>
|
3,202,797 | <p>Why is solving the system of equations
<span class="math-container">$$1+x-y^2=0$$</span>
<span class="math-container">$$y-x^2=0$$</span>
the same as minimizing
<span class="math-container">$$f(x,y)=(1+x-y^2)^2 + (y-x^2)^2$$</span></p>
<p>Originally I thought it was because if you take the partial derivatives of <sp... | Mohammad Riazi-Kermani | 514,496 | <p>Note that <span class="math-container">$$ f(x,y)=(1+x-y^2)^2 + (y-x^2)^2$$</span> is sum of two squares which is always non-negative.</p>
<p>The minimum value of <span class="math-container">$f(x,y)$</span> is zero which is attained when both squares are zero.</p>
<p>Your system of equations are simply making the ... |
39,684 | <p>In order to have a good view of the whole mathematical landscape one might want to know a deep theorem from the main subjects (I think my view is too narrow so I want to extend it).</p>
<p>For example in <em>natural number theory</em> it is good to know quadratic reciprocity and in <em>linear algebra</em> it's good... | Gadi A | 1,818 | <p>It might be <strong>too</strong> obvious, but <a href="http://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems" rel="nofollow">Gödel's incompleteness theorems</a> are certainly landmarks, not only because of their historical significance and "popularity" outside mathematics (where usually they are quoted... |
39,684 | <p>In order to have a good view of the whole mathematical landscape one might want to know a deep theorem from the main subjects (I think my view is too narrow so I want to extend it).</p>
<p>For example in <em>natural number theory</em> it is good to know quadratic reciprocity and in <em>linear algebra</em> it's good... | Arturo Magidin | 742 | <p>For <strong>Field Theory</strong>, we have <strong>The Fundamental Theorem of Galois Theory</strong>. It is a basic tool not only for fields, but for algebraic number theory among other fields, and it establishes a beautiful connection between group theory and field theory.</p>
|
410,905 | <p>If A is real, symmetric, regular, positive definite matrix in $R^{n.n}$ and $x,h\in R^n$, why is it $\langle Ah,x\rangle = \langle h,A^T x\rangle =\langle Ax,h\rangle$?
Is there some rule or theorem for this?</p>
| DonAntonio | 31,254 | <p>I think this follows straightforward from definitions:</p>
<p>$\,A\,$ is symmetric and real$\;\implies A^*=A^t=A\;$ , so since the inner product is real we get</p>
<p>$$\langle x,y\rangle =\langle y,x\rangle \;\;\text{ and}\;\;\langle y,A^tx\rangle=\langle A^tx,y\rangle=\langle Ax,y\rangle\;\ldots $$</p>
|
58,209 | <p>Question: Of the following, which is the best approximation of
$$\sqrt{1.5}(266)^{3/2}$$</p>
<p>$$(A)~1,000~~~~(B)~2,700~~~~(C)~3,200~~~~(D)~4,100~~~~(E)~5,300$$</p>
<p>I used $1.5\approx1.44=1.2^2$ and $266\approx256=16^2$. Therefore the approximation by me is $4096$, so I chose $(D)$ which is wrong. The correct ... | Jacob | 825 | <p>$$\sqrt{1.5}(266)^{3/2} \approx \sqrt{\frac{16}{9}}(256)^{3/2} \approx \frac{4}{3} \times 4096 \approx 5460 $$</p>
<p>Hence, $(E)$</p>
<p><strong>N.B.</strong> <a href="https://math.stackexchange.com/questions/58209/how-to-do-this-approximation/58212#58212">kuch nahi's answer</a> is probably the "right" one ; this... |
1,416,041 | <blockquote>
<p>How many ways are there to plane n indistinguishable balls into n urns so that exactly one urn is empty?</p>
</blockquote>
<p>Why is the answer for this question n(n-1)?</p>
| Bamboo | 155,045 | <p>It does not matter which urn is empty, and there are $n$ urns, so we have $n$ choices of which urn to leave empty.</p>
<p>Next, every other urn must have at least one ball in it (since we can only have one empty urn), so must distribute $n-1$ of the balls into these $n-1$ urns. We have one ball left and we can cho... |
1,416,041 | <blockquote>
<p>How many ways are there to plane n indistinguishable balls into n urns so that exactly one urn is empty?</p>
</blockquote>
<p>Why is the answer for this question n(n-1)?</p>
| NeitherNor | 262,655 | <p>Hint: what is the maximal number of balls in a urn when the condition is satisfied, and how many urns will have the maximal number? Say you choose the urn having zero balls first. How many different choices do you have? Then, given that you have already chosen the urn with zero balls: how many choices do you have fo... |
1,840,352 | <blockquote>
<p>For every $A \subset \mathbb R^3$ we define $\mathcal{P}(A)\subset \mathbb R^2$ by
$$ \mathcal{P}(A) := \{ (x,y) \mid \exists_{z \in \mathbb R}:(x,y,z) \in A \} \,. $$
Prove or disprove that $A$ closed $\implies$ $\mathcal{P}(A)$ closed and $A$ compact $\implies$ $\mathcal{P}(A)$ compact.</p>
</bl... | DanielWainfleet | 254,665 | <p>Let $A=\{(x,0,z): xz=1\},$ which is closed in $R^3.$ Then $P(A)=\{(x,0):x\ne 0\},$ which is not closed in $R^2.$</p>
<p>Observe that $P$ is Lipschitz-continuous, as $\|P(u)-P(v)\|=\|P(u-v)\|\leq \|u-v\|.$ The continuous image of a compact space is compact. So if $A\subset R^3$ is compact then $P(A)$ is compact.</p>... |
4,112,958 | <p>This is a Number Theory problem about the extended Euclidean Algorithm I found:</p>
<p>Use the extended Euclidean Algorithm to find all numbers smaller than <span class="math-container">$2040$</span> so that <span class="math-container">$51 | 71n-24$</span>.</p>
<p>As the eEA always involves two variables so that <s... | hamam_Abdallah | 369,188 | <p><strong>Other way</strong></p>
<p><span class="math-container">$$71n\equiv 24\pmod {51}\iff$$</span>
<span class="math-container">$$20n\equiv 24\pmod {51}\iff$$</span>
<span class="math-container">$$5n\equiv 6\pmod {51}\iff$$</span>
<span class="math-container">$$5n\equiv -45\pmod {51}\iff$$</span>
<span class="math... |
644,935 | <p>I'm having trouble integrating $3^x$ using the $px + q$ rule. Can some please walk me through this?</p>
<p>Thanks</p>
| JPi | 120,310 | <p>If I'm correct in what you think the $px+q$ rule is then take $f(x)=e^x$, $p=\log 3$, and $q=0$, with $\log$ the natural logarithm (of course!).</p>
|
12,878 | <p>I wish the comments didn't have a lower bound for characters.Many times all I want to say is "yes". Can someone explain me what is the purpose of this l.b.?</p>
| Alexander Gruber | 12,952 | <p>Situations warranting a simple "yes" or "no" exist, but they should be in the extreme minority. Discussions here should be more in depth than that. Comments are not a chat room. The limit exists to remind users to write good content, and the inconvenience of being unable to leave short comments pales, in my view, ... |
7,647 | <p>Given a polyhedron consists of a list of vertices (<code>v</code>), a list of edges (<code>e</code>), and a list of surfaces connecting those edges (<code>s</code>), how to break the polyhedron into a list of tetrahedron?</p>
<p>I have a convex polyhedron.</p>
| Hugh Thomas | 468 | <p>If I understand your question correctly, you're saying that the given information is the face structure of a 3-dimensional convex polytope, and you would like a subdivision of the polytope into tetrahedra. </p>
<p>Here is one way to proceed. First, subdivide all the faces into triangles. Then pick your favourite... |
4,059,420 | <p>If f is holomorphic at every point on the open disc <span class="math-container">$$D=\{z:|z|\lt1\}$$</span>
I want to show that f is constant</p>
| johnnyb | 298,360 | <p>The symbol <span class="math-container">$\infty$</span>, as is currently used, simply means that the value is beyond the range supported by the real numbers. It is not itself a number. For instance, let's say that we only knew single-digit numbers. We might have values, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, and <span cla... |
398,857 | <p>Please help me solve this and please tell me how to do it..</p>
<p>$12345234 \times 23123345 \pmod {31} = $?</p>
<p>edit: please show me how to do it on a calculator not a computer thanks:)</p>
| André Nicolas | 6,312 | <p>We want to replace these big numbers by much smaller ones that have the same remainder on division by $31$.</p>
<p>Take your first big number $12345234$. Divide by $31$ with the calculator. My display reads $398233.3548$. Subtract $398233$. My calculator shows $0.354838$. Multiply by $31$. The calculator gives $10.... |
679,544 | <p>How to prove this for positive real numbers?
$$a+b+c\leq\frac{a^3}{bc}+\frac{b^3}{ca}+\frac{c^3}{ab}$$</p>
<p>I tried AM-GM, CS inequality but all failed.</p>
| Michael Rozenberg | 190,319 | <p>Another way.</p>
<p>By Rearrangement
<span class="math-container">$$\sum_{cyc}\left(\frac{a^3}{bc}-a\right)=\sum_{cyc}\frac{a}{bc}(a^2-bc)\geq\frac{1}{3}\sum_{cyc}\frac{a}{bc}\sum_{cyc}(a^2-bc)=\frac{1}{6}\sum_{cyc}\frac{a}{bc}\sum_{cyc}(a-b)^2\geq0.$$</span></p>
|
552,238 | <p>Reasons that $(\mathbb R^n, +, \mathcal Z)$ is not a topological group: Given any two distinct points $\vec{p},\vec q \in \mathbb R ^n$ let $P$ be the unique hyperplane through $\vec p$ which is perpendicular to the vector $\vec p- \vec q$. $P$ is closed with respect to $\mathcal Z$ and hence its complement is an op... | Alex Youcis | 16,497 | <p>The problem is that the product topology is not the same as the Zariski topology. So, while maps $\mathbb{R}^{2n}\to\mathbb{R}$ are continuous, polynomial ones, this is in the Zariski topology, not the product topology.</p>
<p>In fact, every $T_0$ topological group is already Hausdorff. Every variety over $\mathbb{... |
552,238 | <p>Reasons that $(\mathbb R^n, +, \mathcal Z)$ is not a topological group: Given any two distinct points $\vec{p},\vec q \in \mathbb R ^n$ let $P$ be the unique hyperplane through $\vec p$ which is perpendicular to the vector $\vec p- \vec q$. $P$ is closed with respect to $\mathcal Z$ and hence its complement is an op... | Martin Brandenburg | 1,650 | <p>Let me mention that the correct "replacement" for the usual euclidean space in algebraic geometry is the affine space. In particular, this means that you don't have the product topology of the underlying topological spaces (which wouldn't make much sense anyway since this doesn't incorporate the algebraic structure)... |
2,031,964 | <p>I am required to prove that the following series
$$a_1=0, a_{n+1}=(a_n+1)/3, n \in N$$
is bounded from above and is monotonously increasing through induction and calculate its limit. Proving that it's monotonously increasing was simple enough, but I don't quite understand how I can prove that it's bounded from above... | hamam_Abdallah | 369,188 | <p><strong>Hint</strong></p>
<p>Let $f(x)=\frac{x+1}{3}$</p>
<p>$f$ has one fixed point $L=\frac{1}{2}=f(L)$</p>
<p>Now, you can prove by induction that</p>
<p>$$\forall n\geq 0\;\;a_n\leq L$$</p>
<p>using the fact that $f$ is increasing at $\; \mathbb R$ and $a_{n+1}=f(a_n)$.</p>
|
2,031,964 | <p>I am required to prove that the following series
$$a_1=0, a_{n+1}=(a_n+1)/3, n \in N$$
is bounded from above and is monotonously increasing through induction and calculate its limit. Proving that it's monotonously increasing was simple enough, but I don't quite understand how I can prove that it's bounded from above... | Simply Beautiful Art | 272,831 | <p>See that for any $a_n<\frac12$, we have</p>
<p>$$a_{n+1}=\frac{a_n+1}3<\frac{\frac12+1}3=\frac12$$</p>
<p>Thus, it is proven that since $a_0<\frac12$, then $a_1<\frac12$, etc. with induction.</p>
<hr>
<p>We choose $\frac12$ since, when solving $a_{n+1}=a_n$, we result with $a_n=\frac12$, the limit of... |
399,804 | <p>The Question was:</p>
<blockquote>
<p>Express $2\cos{X} = \sin{X}$ in terms of $\sin{X}$ only.</p>
</blockquote>
<p>I have had dealings with similar problems but for some reason, due to I believe a minor oversight, I am terribly vexed.</p>
| Double AA | 22,307 | <p>Try using $\cos{x}^2 + \sin{x}^2 =1$ or some other known identity, but solving for $\cos{x}$ in terms of $\sin{x}$. Then just substitute.</p>
|
1,057,675 | <p>I was asked to prove that $\lim\limits_{x\to\infty}\frac{x^n}{a^x}=0$ when $n$ is some natural number and $a>1$. However, taking second and third derivatives according to L'Hôpital's rule didn't bring any fresh insights nor did it clarify anything. How can this be proven? </p>
| Luiz Cordeiro | 58,818 | <p>In some cases, it is interesting (or simply makes notation simpler) to consider the extended real numbers: $\overline{\mathbb{R}}=[-\infty,\infty]$.</p>
<p>Formally, pick your favorite objects which are not real numbers, and let's conveniently call them $-\infty$ and $\infty$. Define the set $\overline{\mathbb{R}}=... |
58,772 | <p>Is there any general way to find out the coefficients of a polynomial.</p>
<p>Say for e.g.<br>
$(x-a)(x-b)$ the constant term is $ab$, coefficient of $x$ is $-(a+b)$ and coefficient of $x^2$ is $1$.</p>
<p>I have a polynomial $(x-a)(x-b)(x-c)$.
What if the number is extended to n terms.?</p>
| Geoff Robinson | 13,147 | <p>This question leads naturally to the elementary symmetric polynomials. If we have a polynomial such as $p(x) = (x - t_1)(x-t_2) \ldots (x-t_n) = \prod_{j=1}^{n}(x-t_i)$, we may write it as $x^{n} + \sum_{j=1}^{n} (-1)^{j}x^{n-j}s_{j}(t_1,\ldots ,t_n).$ Here $t_1,t_2,\ldots ,t_n$ can be elements of any (commutative) ... |
2,208,113 | <p>Let $x$ and $y \in \mathbb{R}^{n}$ be non-zero column vectors, from the matrix $A=xy^{T}$, where $y^{T}$ is the transpose of $y$. Then the rank of $A$ is ?</p>
<hr>
<p>I am getting $1$, but need confirmation .</p>
| John Wayland Bales | 246,513 | <p>If there is a solution, the symmetry of the equations suggests we investigate the possibility that $x=y=z$</p>
<p>So assume that $x=y=z=a$.</p>
<p>$$2a=\sqrt{4a-1}$$</p>
<p>Thus $4a^2-4a+1=0$, or $(2a-1)^2=0$. Thus $x=y=z=\frac{1}{2}$.</p>
<p>This is a solution. But, as per the comment of dxiv below, this does n... |
1,556,298 | <p>If we have $p\implies q$, then the only case the logical value of this implication is false is when $p$ is true, but $q$ false.</p>
<p>So suppose I have a broken soda machine - it will never give me any can of coke, no matter if I insert some coins in it or not.</p>
<p>Let $p$ be 'I insert a coin', and $q$ - 'I ge... | skyking | 265,767 | <p>You have to distinguish between cases where the truth values for $p$ and/or $q$ are given and cases when they're not. If the truth values for $p$ and $q$ are given then only that row of the truth table is relevant. If on the other hand they are given completely or partially all the relevant rows are relevant. </p>
... |
1,413,145 | <p>I would like to find a way to show that the sequence $a_n=\big(1+\frac{1}{n}\big)^n+\frac{1}{n}$ is eventually increasing.</p>
<p>$\hspace{.3 in}$(Numerical evidence suggests that $a_n<a_{n+1}$ for $n\ge6$.)</p>
<p>I was led to this problem by trying to prove by induction that $\big(1+\frac{1}{n}\big)^n\le3-\f... | Arin Chaudhuri | 404 | <p>Let $a_n = (1 + 1/n)^n.$ </p>
<p>We want to show $a_{n+1} - a_{n} \geq \dfrac{1}{n(n+1)}$ for large $n$. </p>
<p>$\dfrac{a_{n+1}}{a_n} = \left(1 + \dfrac{1}{n}\right) \left(1 - \dfrac{1}{(n+1)^2}\right)^{n+1}.$</p>
<p>The RHS can be expanded as</p>
<p>$\left(1 + \dfrac{1}{n}\right) \left(1 - \dfrac{1}{(n+1)^2}\r... |
1,734,419 | <p>I have tried to show that : $2730 |$ $n^{13}-n$ using fermat little theorem but i can't succeed or at a least to write $2730$ as $n^p-n$ .</p>
<p><strong>My question here</strong> : How do I show that $2730$ divides $n^{13}-n$ for $n$ is integer ?</p>
<p>Thank you for any help </p>
| André Nicolas | 6,312 | <p>Hint: Use Fermat's Theorem, working separately modulo $2$, $3$, $5$, $7$, and $13$. (Note that $2730=2\cdot 3\cdot 5\cdot 7\cdot 13$.)</p>
|
8,695 | <p>I have a parametric plot showing a path of an object in x and y (position), where each is a function of t (time), on which I would like to put a time tick, every second let's say. This would be to indicate where the object is moving fast (widely spaced ticks) or slow (closely spaced ticks). Each tick would just be... | jVincent | 1,194 | <p>One way to do it would be to find analytically the parallel line yourself, and then creating a <code>Graphics[]</code> with all the lines. Here is a rough working example that's specific to this one expression and scales the lines according to the differential of the function, but it outlines the gist of the idea. <... |
28,532 | <p><code>MapIndexed</code> is a very handy built-in function. Suppose that I have the following list, called <code>list</code>:</p>
<pre><code>list = {10, 20, 30, 40};
</code></pre>
<p>I can use <code>MapIndexed</code> to map an arbitrary function <code>f</code> across <code>list</code>:</p>
<pre><code>{f[10, {1}],... | Kuba | 5,478 | <p>I'm sure one can improve following solution</p>
<pre><code>SetAttributes[mapIndexedAt, HoldRest];
mapIndexedAt[f_, list_, pos_] := Do[list = MapAt[f[#, pos[[i]]] &, list, pos[[i]]]
, {i, Length@pos}]
l = {1, 1, 1, 1};
mapIndexedAt[(#1 + #2) &, l, {2, 3}]
</code></pre>
<... |
113,797 | <p>I'm trying to extract every 21st character from this text, s (given below), to create new strings of all 1st characters, 2nd characters, etc.</p>
<p>I have already separated the long string into substrings of 21 characters each using</p>
<pre><code> splitstring[String : str_, n_] :=
StringJoin @@@ Partitio... | JungHwan Min | 35,945 | <p><code>StringTake</code> and <code>Span</code> would be useful.</p>
<p>For example, to get the second characters:</p>
<pre><code>StringTake[s, 2 ;; ;; 21]
(* "LLFOWJZSPLJJHNHQOQLYSOPWQOSZNTTLTOHETNJOJOODOCJFQOJ" *)
</code></pre>
<p>To get all of the strings:</p>
<pre><code>StringTake[s, Array[# ;; ;; 21&, 21]... |
81,715 | <p>I am a graduate student in physics trying to learn differential geometry on my own, out of a book written by Fecko.</p>
<p>He defines the gradient of a function as:</p>
<p>$
\nabla f = \sharp_g df = g^{-1}(df, \cdot )
$</p>
<p>This makes enough sense to me. However, when I try to calculate the gradient of a fu... | Hans Lundmark | 1,242 | <p>The components that the formula $g^{ij} \partial_j f$ refers to are taken with respect to the natural tangent space basis induced by the coordinate system; these vectors are often denoted by $(\partial/\partial r, \partial/\partial \theta, \partial/\partial \phi)$, and they differ from the orthonormal frame $(\hat{r... |
3,710,710 | <p>Suppose <span class="math-container">$A(t,x)$</span> is a <span class="math-container">$n\times n$</span> matrix that depends on a parameter <span class="math-container">$t$</span> and a variable <span class="math-container">$x$</span>, and let <span class="math-container">$f(t,x)$</span> be such that <span class="m... | peek-a-boo | 568,204 | <p>Yes, there is a chain rule for such functions. Before getting to that, let's just briefly discuss partial derivatives for multivariable functions.</p>
<blockquote>
<p>Let <span class="math-container">$V_1, \dots, V_n, W$</span> be Banach spaces (think finite-dimensional if you wish, such as <span class="math-cont... |
78,423 | <p>How far can one go in proving facts about projective space using just its universal property?</p>
<p>Can one prove Serre's theorem on generation by global sections, calculate cohomology, classify all invertible line bundles on projective space?</p>
<p>I don't like many proofs of some basic technical facts very aes... | Anton Geraschenko | 1 | <p>I think the answer is "probably not." The reason is that projective space has <em>two</em> universal properties which are used to prove different kinds of things about it. One of these is the slick universal property you like, and the other is the clunky one which results in unpleasantries.</p>
<p>Though each unive... |
4,383,557 | <p>This question came up in an oral exam. During the course we studied a bit of the theory of lie algebras and some representation theory.</p>
<p>The question: show that the lie algebra <span class="math-container">$\mathfrak{g_2}$</span> has a dimension <span class="math-container">$14$</span> representation, where di... | Torsten Schoeneberg | 96,384 | <p>Hint 1: What is that Lie algebra's dimension?</p>
<p>Hint 2: Surely in your class the concept of the <em>adjoint representation</em> was mentioned?</p>
|
464,426 | <p>Find the limit of $$\lim_{x\to 1}\frac{x^{1/5}-1}{x^{1/3}-1}$$</p>
<p>How should I approach it? I tried to use L'Hopital's Rule but it's just keep giving me 0/0.</p>
| Pedro | 23,350 | <p><strong>Hint</strong> Take $x=t^{15}$. Then use long division, or $$\frac{t^a-1}{t-1}=1+t+t^2+\cdots+t^{a-1}$$</p>
<p>You can also think about derivatives of $x^{1/3},x^{1/5}$ at $x=1$.</p>
|
405,866 | <p>Original question:
For compact metric spaces, plenty of subtly different definitions converge to the same concept. Overtness can be viewed as a property dual to compactness. So is there a similar story for overt metric spaces?</p>
<p>Edit: Since overtness is trivially true assuming the Law of the Excluded Middle, cl... | Paul Taylor | 2,733 | <p>David Roberts has rubbed the magic lamp and the genie appears!</p>
<p>Even though the notion of overtness does depend on the strength of the ambient logic,
I believe the question here is with the notion of metric space, rather than the choice of a model of mathematics.</p>
<p>The natural answer is that any metric sp... |
147,661 | <p>Let $R$ be a (noncommutative) ring and $a \in R$ such that $a(1-a)$ is nilpotent. Why is $1+a(t-1)$ a unit in $R[t,t^{-1}]$? Probably one just has to write down an inverse element, but I could not find it. Perhaps there is a trick related to the geometric series which motivates the choice of the inverse element, whi... | wxu | 4,396 | <p><strong>Should be read first:</strong> If $R$ is commutative, then both side is easy. Pick any prime ideal $\mathfrak{p}$ of $R$, $1+a(t-1)=1-a+at$ is a unit in $R/\mathfrak{p}[t,t^{-1}]$ if and only if either $1-a=0$ or $a=0$ in $R/\mathfrak{p}$. So if and only if $(1-a)a\in \mathfrak{p}$.</p>
<hr>
<p>Since $(1+... |
4,463,373 | <p>If <span class="math-container">$\frac{\partial f}{\partial x}(0,0) = \frac{\partial f}{\partial y}(0,0) = 0$</span>, then <span class="math-container">$f'((0,0);d)=0$</span> (directional derivative) for every direction <span class="math-container">$d \in \mathbb{R}^n$</span>.</p>
<p>Is this true? I'm trying to find... | sadman-ncc | 942,091 | <p>Your derivative calculation is correct but can be simplified in a different way for making the differentiability of <span class="math-container">$f$</span> much more obvious. By definition, <span class="math-container">$$\frac{d}{dx}(|x|)=\frac{x}{|x|}.$$</span>
Now, if <span class="math-container">$f(x)=x|x^3|$</sp... |
435,079 | <p>This is exercise from my lecturer, for IMC preparation. I haven't found any idea.</p>
<p>Find the value of</p>
<p>$$\lim_{n\rightarrow\infty}n^2\left(\int_0^1 \left(1+x^n\right)^\frac{1}{n} \, dx-1\right)$$</p>
<p>Thank you</p>
| marty cohen | 13,079 | <p>I get an answer that differs from
that of user17762.
This is because the error term
is not one term of order
$\frac{x^{2n}}{n^2}$
but a number of such terms.</p>
<p>I get that the limit
is between
3/4 and 7/8,
but only have an infinite series
for the value.</p>
<p>My complete analysis follows.</p>
<p>$(1+x^n)^{1... |
435,079 | <p>This is exercise from my lecturer, for IMC preparation. I haven't found any idea.</p>
<p>Find the value of</p>
<p>$$\lim_{n\rightarrow\infty}n^2\left(\int_0^1 \left(1+x^n\right)^\frac{1}{n} \, dx-1\right)$$</p>
<p>Thank you</p>
| Sangchul Lee | 9,340 | <p>By integration by parts,</p>
<p>\begin{align*}
\int_{0}^{1} (1 + x^{n})^{\frac{1}{n}} \, dx
&= \left[ -(1-x)(1+x^{n})^{\frac{1}{n}} \right]_{0}^{1} + \int_{0}^{1} (1-x)(1 + x^{n})^{\frac{1}{n}-1}x^{n-1} \, dx \\
&= 1 + \int_{0}^{1} (1-x) (1 + x^{n})^{\frac{1}{n}-1} x^{n-1} \, dx
\end{align*}</p>
<p>so that... |
335,116 | <p>As a PhD student, if I want to do something algebraic / linear-algebraic such as representation theory as well as do PDEs, in both the theoretical and numerical aspects of PDEs, would this combination be compatible and / or useful? Is it feasible?</p>
<p>I'd be grateful for an online resource to look into.</p>
<p>... | Robert Furber | 61,785 | <p>This goes back to the beginning of the subject of unitary representations of locally compact noncompact groups. Wigner was looking for all possible generalizations of the Dirac equation to higher spin, and developing the representation theory of the Poincaré group is how he obtained his results (Bargmann did this in... |
335,116 | <p>As a PhD student, if I want to do something algebraic / linear-algebraic such as representation theory as well as do PDEs, in both the theoretical and numerical aspects of PDEs, would this combination be compatible and / or useful? Is it feasible?</p>
<p>I'd be grateful for an online resource to look into.</p>
<p>... | RBega2 | 127,803 | <p>Peter Olver has an interesting book on <a href="https://www.springer.com/gp/book/9781468402742" rel="noreferrer">Symmetry and PDEs</a>. Another area to consider (that is particularly important for geometric PDEs) are exterior differential systems. <a href="https://services.math.duke.edu/~bryant/MSRI_Lectures.pdf" ... |
1,275,461 | <p>I am trying to proof that $-1$ is a square in $\mathbb{F}_p$ for $p = 1 \mod{4}$. Of course, this is really easy if one uses the Legendre Symbol and Euler's criterion. However, I do not want to use those. In fact, I want to prove this using as little assumption as possible.</p>
<p>What I tried so far is not really... | Thomas Andrews | 7,933 | <p>You can use Wilson's theorem: $(p-1)!\equiv-1\pmod p$ and then show that $$(p-1)!\equiv 1\cdot 2\cdots \frac{p-1}{2} \left(-\frac{p-1}{2}\right)\cdots(-2)(-1) = \left(\left(\frac{p-1}{2}\right)!\right)^2(-1)^{\frac{p-1}{2}}\pmod p$$</p>
<p>This gives an exactly formula for a solution of $a^2=-1$, although not an ef... |
745,674 | <p>Let $E$ be a complex vector space of dimension 3. Let $f$ be a non zero endomorphism such that $f^2=0$. I want to show that there is a basis $B=\{b_1,b_2,b_3\}$ of $E$ such that
$$f(b_1)=0, f(b_2)=b_1,f(b_3)=0$$</p>
<p><strong>Edit</strong> Here is how i see the answer now: </p>
<p>$f$ being non zero there exists ... | Valentin Waeselynck | 141,752 | <p>You're not doing it in the right order. Choose $b_2$ so $f(b_2) \neq 0$. Then $b_1 = f(b_2)$, and choose any other $b_3 \in Ker(f)$ that independent from $b_1$.</p>
|
3,037,793 | <p>I have to write the following sentence "If professors are unhappy all students fail their exams" in logic and my answer is:</p>
<p>∀x [Prof(x) ∧ Unhappy(x)] ⇒ [∀y stud(y) ⇒ fail_exam(x,y)]</p>
<p>However, the answer of my teacher is:</p>
<p>∀x ∀y( prof(x) ∧ unhappy(x) ∧ stud(y) ) ⇒ fail exam(x, y))</p>
<p>Can s... | Bram28 | 256,001 | <p>They are equivalent ... although to show that, I will first insist on adding a few parentheses so as to indicate the proper scope of the quantifiers, giving us:</p>
<p><span class="math-container">$\forall x ((Prof(x) \land Unhappy(x)) \rightarrow \forall y (Stud(y) \rightarrow FailExam(x,y)))$</span></p>
<p>and</... |
3,037,793 | <p>I have to write the following sentence "If professors are unhappy all students fail their exams" in logic and my answer is:</p>
<p>∀x [Prof(x) ∧ Unhappy(x)] ⇒ [∀y stud(y) ⇒ fail_exam(x,y)]</p>
<p>However, the answer of my teacher is:</p>
<p>∀x ∀y( prof(x) ∧ unhappy(x) ∧ stud(y) ) ⇒ fail exam(x, y))</p>
<p>Can s... | Jorge Adriano Branco Aires | 74,765 | <p>You've been answered that they are equivalent. It's also worth knowing the transformation from <span class="math-container">$(P\land Q) \rightarrow R$</span> to <span class="math-container">$P\rightarrow Q \rightarrow R$</span> is known by the name of <em><a href="https://en.wikipedia.org/wiki/Currying" rel="nofollo... |
3,999,996 | <p>I know how you can show this geometrically, but is there any way to prove this algebraically?</p>
| jjuma1992 | 715,368 | <p>You can use the following two known inequalities
<span class="math-container">$$x \leq |x|$$</span> and <span class="math-container">$$\left|\int_a^b f(x)dx\right|\leq \int_a^b|f(x)|dx.$$</span>
We have
<span class="math-container">\begin{align*}
\int_0^1x\sin^{-1}xdx &\leq \left|\int_0^1x\sin^{-1}xdx\right|\\
&... |
1,246,705 | <p>I was doing some linear algebra exercises and came across the following tough problem :</p>
<blockquote>
<p>Let $M_{n\times n}(\mathbf{R})$ denote the set of all the matrices whose entries are real numbers. Suppose $\phi:M_{n\times n}(\mathbf{R})\to M_{n\times n}(\mathbf{R})$ is a nonzero linear transform (i.e. t... | Gabriel Romon | 66,096 | <p>Since $\varphi$ is linear, it's only natural to investigate how it acts on the canonical basis of $M_n(k)$. Let $E_{ij}$ denote the matrix with a single one for the entry $(i,j)$ and $0$'s elsewhere. It is standard that $E_{ij}E_{kl} = \delta_{jk}E_{il}$, hence $\varphi(E_{ij})\varphi(E_{kl}) = \delta_{jk}\varphi(E... |
2,764,381 | <p>one thing I don't understand is what is sin(0) and sin(1) exactly? I am alright with the concept of radian (pi) but don't understand 0 and 1. What does it mean?</p>
| fhorrobin | 558,456 | <p>I am not completely sure what you are asking but I will try to answer. In the context of the question in the title, $[0,1] $ means that they are talking about $y=\sin x $ for values on $x $ on the closed interval from 0 to 1 (rather than say all of the real numbers. </p>
<p>In terms of what this means, the sin func... |
3,505,397 | <p>well, I have to find the Taylor polynomial of <span class="math-container">$f(x,y)=\sin(x)\sin(y)$</span> at <span class="math-container">$(0,\pi/4)$</span>. I found:</p>
<p>Is <span class="math-container">$T_3(x,y)=-\frac{1}{12}\sqrt{2}x(16x^2+48y^2-24\pi+3\pi^2)$</span> correct?</p>
| Quanto | 686,284 | <p>Use the identity <span class="math-container">$\cos^{-1}x+\sin^{-1}x=\frac\pi2 $</span>,</p>
<p><span class="math-container">$$\cos^{-1}\left(\sin\frac{16\pi}{7}\right)
=\frac\pi2 - \sin^{-1}\left(\sin\frac{16\pi}{7}\right)
=\frac\pi2 - \sin^{-1}\left(\sin\frac{2\pi}{7}\right)
=\frac\pi2-\frac{2\pi}{7}=\frac{3\pi}{... |
578,337 | <p>For $n=1,2,3,\dots,$ and $|x| < 1$ I need to prove that $\frac{x}{1+nx^2}$ converges uniformly to zero function. How ?. For $|x| > 1$ it is easy. </p>
| Sugata Adhya | 36,242 | <p>$f_n(0)=0\to0$ and for $x\ne0,$ $|f_n(x)|=\left|\dfrac{x}{1+nx^2}\right|\le\left|\dfrac{1}{nx}\right|\to0$ as $n\to\infty$</p>
<p>So, $f(x)=\displaystyle\lim_{n\to\infty}f(x)\equiv0$ on $(-1,1).$ Let $\forall~n\ge1,$$$M_n=\displaystyle\sup_{|x|<1}\left|\dfrac{x}{1+nx^2}-0\right|=\displaystyle\sup_{|x|<1}\left... |
489,109 | <p>I've been stumped on this problem for hours and cannot figure out how to do it from tons of tutorials.</p>
<p>Please note: This is an intro to calculus, so we haven't learned derivatives or anything too complex.</p>
<p>Here's the question: </p>
<p>Let $f(x) = x^5 + x + 7$. Find the value of the inverse function a... | Henry Swanson | 55,540 | <p>In general, polynomials won't have an inverse. This one happens to have one, but it's not fun to express, as far as I know.</p>
<p>Since you only need to find the inverse at a particular number, not any $y$, just plug it in and rearrange until something looks nice: $x^5 + x + 7 = 1035$ means $x(x^4 + 1) = 1028$. Th... |
1,407,683 | <p>I am new to differential geometry. It is surprising to find that the linear connection is not a tensor, namely, not coordinate-independent. </p>
<p>Can we bypass this ugly object? Only intrinsic quantities should appear in a textbook. </p>
| Amitai Yuval | 166,201 | <p>In contrast to what is written in the question and some of the comments, a connection is <em>independent</em> of coordinates.</p>
<p>The intrinsic way to define connections is the following. For simplicity, we treat only connections on the tangent bundle, even though a similar definition can be applied for any vect... |
1,407,683 | <p>I am new to differential geometry. It is surprising to find that the linear connection is not a tensor, namely, not coordinate-independent. </p>
<p>Can we bypass this ugly object? Only intrinsic quantities should appear in a textbook. </p>
| jxnh | 132,834 | <p>As mentioned in a previous answer, connections are quite intrinsic. I will take the slightly more pedestrian view that a connection is a collection of maps from type $(k,l)$ tensor fields to type $(k,l+1)$ fields that is linear, satisfies a Leibniz rule, commutes with contraction, agrees with the differential for sm... |
2,104,984 | <p>For every $x\in \mathbb{R}$, $f(x+6)+f(x-6)=f(x)$ is satisfied. What may be the period of $f(x)$?
I tried writing several $f$ values but I couldn't get something like $f(x+T)=f(x)$.</p>
| Simply Beautiful Art | 272,831 | <p>It follows then that</p>
<p>$$\begin{align}f(x)&=f(x+6)+f(x-6)\\&=f(x+12)+f(x)+f(x-6)\\\implies0&=f(x+12)+f(x-6)\\\implies f(x+12)&=-f(x-6)\\\implies f(x+18)&=-f(x)\\\implies f(x+36)&=-f(x+18)=f(x)\end{align}$$</p>
<blockquote>
<p>$$f(x+36)=f(x)$$</p>
</blockquote>
|
1,443,812 | <p>Suppose that $X$ and $Y$ have joint p.d.f.</p>
<p>$$ f(x, y) = 3x, \; 0 < y < x < 1.$$ </p>
<p>Find $f_X(x)$,the marginal p.d.f. of $X$.</p>
<p>this is what i got</p>
<p>$$f_X(x) = \int_0^x f(x, y)dy = \int_0^x 3x dy = 3x^2$$
for $0 < x < 1$.</p>
<p>however, if want to know whether $X$ and $Y$ ar... | drhab | 75,923 | <p><strong>Hint</strong>:</p>
<p>The pdf takes value $0$ on $\{\langle x,y\rangle\mid x<y\}$ so that $P(X<Y)=0$.</p>
<p>Then $P(X<c\wedge Y>c)=0$ for each $c$. </p>
<p>If $X$ and $Y$ are indeed independent then: $$P(X<c\wedge Y>c)=P(X<c)P(Y>c)$$
for each $c$.</p>
<p>So to prove that $X$ and... |
4,041,758 | <p>If you have a line passing through the middle of a circle, does it create a right angle at the intersection of the line and curve?</p>
<p>More generally, is it valid to define an angle created between a line and a curve? Is the tangent to the curve at the point of intersection a valid interpretation (I.e a semi circ... | Directions In Physics | 888,931 | <blockquote>
<p>More generally, is it valid to define an angle created between a line and a curve?</p>
</blockquote>
<p>Yes. It can be done with a little calculus. Calculus gives us a workable definition of a unique tangent vector at each point on a curve. So, we can use calculus to translate questions about curves int... |
1,912,570 | <p>There is a proof from stein for this assertion,</p>
<p><a href="https://i.stack.imgur.com/fqRKU.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/fqRKU.jpg" alt="enter image description here"></a>
<a href="https://i.stack.imgur.com/fhZmE.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgu... | Reveillark | 122,262 | <p>They are not equal because the cubes may not intersect neither $E_1$ nor $E_2$, so you might be counting 'excessive' cubes in the large sum.</p>
<p>Here's a hint to see that the inequality holds, using the following:</p>
<blockquote>
<p><strong>Proposition</strong>: Let $\{a_n\}$ be a sequence of non-negative te... |
215,983 | <p>I was expecting to get the answer to the above by doing either of the following:</p>
<pre><code>Options[Graph[{1 -> 2, 2 -> 1}], GraphLayout]
AbsoluteOptions[Graph[{1 -> 2, 2 -> 1}], GraphLayout]
</code></pre>
<p>However the result to both is</p>
<blockquote>
<p>GraphLayout->Automatic</p>
</blockquo... | Bob Hanlon | 9,362 | <p>Using <a href="https://reference.wolfram.com/language/ref/Rasterize.html" rel="nofollow noreferrer"><code>Rasterize</code></a> to do a "visual" comparison with some suspects</p>
<pre><code>grR = Rasterize@Graph[{1 -> 2, 2 -> 1}];
Select[{#,
Rasterize@
Graph[{1 -> 2, 2 -> 1}, GraphLayout ->... |
4,417,896 | <p>I have only found information regarding doing this by integration by parts. By differentiating under the integral sign, I let
<span class="math-container">$$I_n = \int_0^\infty x^n e^{-\lambda x} dx $$</span>
and get <span class="math-container">$\frac{dI_n}{d\lambda} = -I_{n+1} $</span> and therefore <span class="m... | Quanto | 686,284 | <p>Differentiate under the integral sign <span class="math-container">$n$</span> times as follows
<span class="math-container">$$\int_0^\infty x^n e^{-\lambda x} dx=(-1)^n\frac{d^n}{d\lambda^n} \int_0^\infty e^{-\lambda x} dx=
(-1)^n\frac{d^n}{d\lambda^n}\frac1\lambda=\frac{n!}{\lambda^{n+1}}
$$</span></p>
|
1,017,965 | <p>Just as the title, my question is what is the matrix representation of Radon transform (Radon projection matrix)? I want to have an exact matrix for the Radon transformation. </p>
<p>(I want to implement some electron tomography algorithms by myself so I need to use the matrix representation of the Radon transforma... | Guillermo González Sánchez | 739,077 | <p>Radon transform consists in a rotation plus a projection in X axis.</p>
<p>This code will throw True for any squared matrix where there is an inscribed circle containing the target values.</p>
<pre><code>from skimage.transform import radon, rotate
import numpy as np
def get_image_center(img):
center = (im... |
2,307,785 | <p>Things I understand:</p>
<p>Shannon entropy- </p>
<ul>
<li>is the expected amount of information in an event from that
distribution. </li>
<li>In game of 20 questions to guess an item, it is the
lower bound on the number questions one could ask.</li>
</ul>
<p>Doubt:</p>
<p>It gives the lower bound on the number ... | leonbloy | 312 | <blockquote>
<p>In game of 20 questions to guess an item, it is the lower bound on the number questions one could ask.</p>
</blockquote>
<p>That's incorrect. One could be lucky and get the right answer on the first question. The entropy gives a lower bound on the <em>average</em> number questions one needs to ask.</... |
2,961,971 | <blockquote>
<p><span class="math-container">$$\sum_{n=1}^\infty\frac{(2n)!}{2^{2n}(n!)^2}$$</span></p>
</blockquote>
<p>Can I have a hint for whether this series converges or diverges using the comparison tests (direct and limit) or the integral test or the ratio test?</p>
<p>I tried using the ratio test but it fa... | Nosrati | 108,128 | <p>Every term of the series is
<span class="math-container">$$\frac{(2n)!}{2^{2n}(n!)^2}=\dfrac{\Gamma(2n+1)}{2^{2n}\Gamma^2(n)}=\dfrac{\Gamma(n+\frac12)}{n\Gamma(n)\sqrt{\pi}}>\dfrac{1}{2n}$$</span>
by the formula
<span class="math-container">$$\dfrac{\Gamma(n)}{\Gamma(2n)}=\dfrac{\sqrt{\pi}}{2^{2n-1}\Gamma(n+\fra... |
2,961,971 | <blockquote>
<p><span class="math-container">$$\sum_{n=1}^\infty\frac{(2n)!}{2^{2n}(n!)^2}$$</span></p>
</blockquote>
<p>Can I have a hint for whether this series converges or diverges using the comparison tests (direct and limit) or the integral test or the ratio test?</p>
<p>I tried using the ratio test but it fa... | user | 505,767 | <p><strong>HINT</strong></p>
<p>We have that</p>
<p><span class="math-container">$$\frac{(2n)!}{2^{2n}(n!)^2} =\frac1{4^n}\binom{2n}{n} \sim \frac{1}{\sqrt{2\pi n}}$$</span></p>
<p>indeed recall that by <a href="https://en.wikipedia.org/wiki/Binomial_coefficient#Bounds_and_asymptotic_formulas" rel="nofollow noreferr... |
16,795 | <p>Consider a finite simple graph $G$ with $n$ vertices, presented in two different but equivalent ways:</p>
<ol>
<li>as a logical formula $\Phi= \bigwedge_{i,j\in[n]} \neg_{ij}\ Rx_ix_j$ with $\neg_{ij} = \neg$ or $ \neg\neg$ </li>
<li>as an (unordered) set $\Gamma = \lbrace [n],R \subseteq [n]^2\rbrace$ </li>
</ol>
... | Greg Kuperberg | 1,450 | <p>It is a strange question, but maybe a useful answer can make it a bit better.</p>
<p>Certainly for many purposes a graph will look totally different from its complement. For instance, a graph and its complement have completely different spectra, diameter, perfect matchings, etc. So that side of the question is ki... |
16,795 | <p>Consider a finite simple graph $G$ with $n$ vertices, presented in two different but equivalent ways:</p>
<ol>
<li>as a logical formula $\Phi= \bigwedge_{i,j\in[n]} \neg_{ij}\ Rx_ix_j$ with $\neg_{ij} = \neg$ or $ \neg\neg$ </li>
<li>as an (unordered) set $\Gamma = \lbrace [n],R \subseteq [n]^2\rbrace$ </li>
</ol>
... | Theo Johnson-Freyd | 78 | <p>OK, I'll bite.</p>
<p>I've never liked the word "graph". Some people use it to mean "set $V$ of 'vertices' a collection $E$ of size-two subsets of $V$". Some mean "set $V$ and a symmetric $V\times V$ matrix valued in nonnegative integers, i.e. a map $V\times V \to \mathbb N$, telling you how often a vertex is con... |
3,434,242 | <p>If I need to get some variables values from a vector in Matlab, I could do, for instance, </p>
<pre><code>x = A(1); y = A(2); z = A(3);
</code></pre>
<p>or I think I remember I could do something like</p>
<blockquote>
<p>[x, y, z] = A;</p>
</blockquote>
<p>However Matlab is not recognizing this format. what w... | Thales | 382,892 | <p>The right way to do it is:</p>
<pre><code>x = A(1); y = A(2); z = A(3);
</code></pre>
<p>This <code>[x, y, z] = A</code> makes no sense because A is not a function. You can use more variables in the left-hand side of an assignment as the outputs of a function. If <code>A</code> is a variable, then you just can't ... |
435,936 | <p>Does anyone know when
$x^2-dy^2=k$ is resoluble in $\mathbb{Z}_n$ with $(n,k)=1$ and $(n,d)=1$ ?
I'm interested in the case $n=p^t$</p>
| Brian M. Scott | 12,042 | <p>Yes. Let $A$ be the set (still to be constructed), and for $n\in\Bbb Z^+$ let $A_n=\{k\in A:k\le n\}$. Suppose that you’ve constructed $A_n$. For any $\epsilon>0$ choose $m\in\Bbb Z^+$ large enough so that $\frac{n}m<\epsilon$, and let $A_m=A_n$. (In other words, omit from $A$ every integer $k$ such that $n<... |
2,032,241 | <p>In Euler's (number theory) theorem one line reads: since $d|ai$ and $d|n$ and $gcd(a,n)=1$ then $d|i$. I've been staring at this for over an hour and I am not convinced why this is true could anyone explain why? I have tried all sorts of lemma's I've seen before but I honestly just can't see it and I feel I'm going ... | Bill Dubuque | 242 | <p>Clearer: since $\,a\,$ and $\,i\,$ are coprime to $\,n\,$ so too is their product $\,ai\,$ (by Euclid's Lemma).
By the Euclidean algorithm $\ 1 = \gcd(ai,n) = \gcd(\color{#c00}{ai\bmod n},\,n) = \gcd(\color{#c00}{s_i},n).\,$ So $\,s_i\,$ is coprime to $\,n\, $ and $\,0\le s_i < n,\,$ hence $\,s_i\in S,\,$ by th... |
239,653 | <p>It is usually told that Birch and Swinnerton-Dyer developped their famous conjecture after studying the growth of the function
$$
f_E(x) = \prod_{p \le x}\frac{|E(\mathbb{F}_p)|}{p}
$$
as $x$ tends to $+\infty$ for elliptic curves $E$ defined over $\mathbb{Q}$, where the product is defined for the primes $p$ where $... | Joe Silverman | 11,926 | <p>I believe that the original experimental observation was that the product seems to converge to $\infty$ if $E(\mathbb Q)$ is infinite, and to a finite value if $E(\mathbb Q)$ is finite. Also, I may be mis-remembering, but I thought they looked at $$\prod_p \frac{p}{\#E(\mathbb F_p)},$$ with the limit conjecturally b... |
24,593 | <p>Traditionally, I have always taught evaluating expressions before teaching linear equations. But, I was recently given a remedial class of students that have to cover the bare minimums (and we have until mid-December to finish). Luckily, I have great flexibility with what I can do to the syllabus, so for the first t... | Tommi | 2,083 | <p>Your context is students who have not learned much from the mathematics education they have been exposed to thus far. I guess, but can not be sure, that they have already been exposed to someone showing them algorithms and asking them to repeat. Because they are performing badly now, likely the teachers have tried t... |
801,668 | <p>Sources: <a href="https://rads.stackoverflow.com/amzn/click/0495011665" rel="nofollow noreferrer"><em>Calculus: Early Transcendentals</em> (6 edn 2007)</a>. p. 206, Section 3.4. Question 95.<br>
<a href="https://rads.stackoverflow.com/amzn/click/0470383348" rel="nofollow noreferrer"><em>Elementary Differential Equat... | Nick Peterson | 81,839 | <p>Think of $\frac{dy}{dx}$ as a function of $x$; say it is $f(x)$. Then $x$ is also a function of $t$. So, by the <em>exact same</em> chain rule argument that you gave,
$$
\frac{d}{dt}\left[\frac{dy}{dx}\right]=\frac{d}{dt}[f(x)]=\frac{df}{dx}\cdot\frac{dx}{dt}.
$$
But
$$
\frac{df}{dx}=\frac{d}{dx}\left[\frac{dy}{dx}\... |
3,040,110 | <p>What is the Range of <span class="math-container">$5|\sin x|+12|\cos x|$</span> ?</p>
<p>I entered the value in desmos.com and getting the range as <span class="math-container">$[5,13]$</span>.</p>
<p>Using <span class="math-container">$\sqrt{5^2+12^2} =13$</span>, i am able to get maximum value but not able to fi... | arctic tern | 296,782 | <p>Without loss of generality we may assume <span class="math-container">$0\le \theta\le\pi/2$</span> so <span class="math-container">$\cos \theta$</span> and <span class="math-container">$\sin \theta$</span> are positive.</p>
<p>You can solve this with calculus (set the derivative equal to zero and solve for <span cl... |
2,279,120 | <p>How do we prove that for any complex number $z$ the minimum value of $|z|+|z-1|$ is $1$ ?
$$
|z|+|z-1|=|z|+|-(z-1)|\geq|z-(z-1)|=|z-z+1|=|1|=1\\\implies|z|+|z-1|\geq1
$$</p>
<p>But, when I do as follows
$$
|z|+|z-1|\geq|z+z-1|=|2z-1|\geq2|z|-|1|\geq-|1|=-1
$$
Since LHS can never be less than 0, $|z|+|z-1|\geq0$</p>... | egreg | 62,967 | <p>The triangle inequality tells you that
$$
|z|+|z-1|=|z|+|1-z|\ge|z+1-z|=1
$$
which is what you did. Taking into account that for $z=1/2$ you have equality, the minimum is $1$.</p>
<p>The other approach simply gives you a different lower bound, which is not very informative:
$$
|z|+|z-1|\ge|z+z-1|=|2z-1|\ge0
$$
is a... |
25,485 | <p>Not sure if this is more appropriate for here or for Math.SE, but here goes:</p>
<p>How does one who is self-studying mathematics determine if a textbook is too hard for you?</p>
<p>Math is hard in general, but when does a textbook cross that line from being challenging to being nearly intractable?</p>
<p>Sometimes ... | Rusi | 17,672 | <p>Lovely question; I await better answers than mine; meanwhile here's my provisional one(s)...</p>
<h1>tl;dr</h1>
<p>(a) tempo (b) flow (c) practice-to-success</p>
<p>Note 1: These are related but different enough to merit separate discussions.<br />
Note 2: The commonality and differenc s are clearer in music/arts th... |
4,631,463 | <p>I came cross the following equation:<br />
<span class="math-container">$$\lim_{x\to+\infty}\frac{1}{x}\int_0^x|\sin t|\mathrm{d}t=\frac{2}{\pi}$$</span>
I wonder how to prove it.</p>
<p>Using the Mathematica I got the following result:
<a href="https://i.stack.imgur.com/CVWgR.png" rel="nofollow noreferrer"><img src... | Prem | 464,087 | <p>Let <span class="math-container">$x=A+\pi C$</span> where <span class="math-container">$A$</span> is between <span class="math-container">$0$</span> & <span class="math-container">$\pi$</span> , with Integer <span class="math-container">$C$</span>.</p>
<p><span class="math-container">$D(A,C) = \frac{1}{A+\pi C}\... |
254,126 | <p>If 0 < a < b, where a, b $\in\mathbb{R}$, determine $\lim \bigg(\dfrac{a^{n+1} + b^{n+1}}{a^{n} + b^{n}}\bigg)$</p>
<p>The answer (from the back of the text) is $\lim \bigg(\dfrac{a^{n+1} + b^{n+1}}{a^{n} + b^{n}}\bigg) = b$ but I have no idea how to get there. The course is Real Analysis 1, so its a course o... | AnonymousCoward | 565 | <p>Lets pursue this induction: Suppose that in what follows, at each step, $M_{d-1}\cap M_1$ is $\mathbb{R}^2$ less $d$ distinct points. You have a short exact sequence: </p>
<p>$$ 0 \rightarrow \Omega^*(M_1\cup M_{d-1}) \rightarrow \Omega^*(M_1)\oplus\Omega^*(M_{d-1}) \rightarrow \Omega^*(M_d) \rightarrow 0$$</p>
<... |
4,359,019 | <p>Let <span class="math-container">$P \in \mathbb{R}^{N\times N}$</span> be an orthogonal matrix and <span class="math-container">$f: \mathbb{R}^{N \times N} \to \mathbb{R}^{N \times N}$</span> be given by <span class="math-container">$f(M) := P^T M P$</span>. I am reading about random matrix theory and an exercise i... | QuantumSpace | 661,543 | <p>Since <span class="math-container">$F$</span> is a dense subspace of <span class="math-container">$E$</span>, it is natural to define
<span class="math-container">$$\varphi: E' \to F': f \mapsto f\vert_F.$$</span>
Clearly this map is linear. To see that it is isometric, you will have to invoke density of <span class... |
3,095,710 | <p>Factor <span class="math-container">$x^8-x$</span> in <span class="math-container">$\Bbb Z[x]$</span> and in <span class="math-container">$\Bbb Z_2[x]$</span></p>
<p>Here what I get is <span class="math-container">$x^8-x=x(x^7-1)=x(x-1)(1+x+x^2+\cdots+x^6)$</span> now what next? Help in both the cases in <span clas... | Ri-Li | 152,715 | <p>After my edit I finally got the answer it was under my nose the polynomial <span class="math-container">$1+\cdots +x^6$</span> does not have zeros at <span class="math-container">$0$</span> and <span class="math-container">$1$</span> so it can't be factored as a multiple of a one degree polynomial and one other as a... |
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