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 |
|---|---|---|---|---|
3,689,051 | <p>Let <span class="math-container">$(X,d_{X})$</span> be a metric space, and let <span class="math-container">$(Y,d_{Y})$</span> be another metric space. Let <span class="math-container">$f:X\to Y$</span> be a function. Then the following two statements are logically equivalent:</p>
<p>(a) <span class="math-container... | Noob mathematician | 779,382 | <p>Let us assume b) prove a) first.</p>
<p>Let take <span class="math-container">$\epsilon\gt 0$</span> and <span class="math-container">$x\in X$</span>. We have the ball <span class="math-container">$B(f(x),\epsilon)\subset Y$</span> is an open subset. Its inverse image <span class="math-container">$f^{-1}\left(B(f(x... |
2,400,110 | <p>Let's say that we have a matrix of transfer functions:
$$G(s) = C(sI-A)^{-1}B + D$$</p>
<p>And we create the sensitivity matrix transfer function:</p>
<p>$$S(s) = (I+GK)^{-1}$$</p>
<p>Where $K$ is our controller gain matrix.</p>
<p>We also create the complementary sensitivity transfer function matrix:</p>
<p>$$... | euraad | 443,999 | <p>I found the answer after hours of resarsch. I was looking in Essentials of Robust Control by Doyle and Zhou.</p>
<p>How to check if the system is stabilizable:</p>
<p>The matrix
$$rank\begin{bmatrix}
(\lambda_i I-A) & B
\end{bmatrix} = n$$</p>
<p>has full-row rank $n$ for all eigenvalues $ \lambda_i \in \Re ... |
492,392 | <p>I found this math test very interesting. I would like to know how the answer is being calculate?</p>
| mrf | 19,440 | <p>Divide both sides by $98{}{}{}{}$.</p>
|
67,434 | <p>Let $f: \mathbb{R}^n \to L^2(\mathbb{R}^d) $ be a Bochner-integrable function (all measures are the Lebesgue measure). Does then $ \int_{\mathbb{R}^n} f(x) d\lambda^n (y) = \int_{\mathbb{R}^n} f(x)(y) d\lambda^n $ hold for $\lambda^d$-almost all $y \in \mathbb{R}^d$? I.e. can one compute such Bochner integrals just... | Martin Väth | 165,275 | <p>To make Gerald Edgar's answer complete: There always <strong>does</strong> exist a product-measurable choice.</p>
<p>More precisely, if <span class="math-container">$f\colon\mathbb{R}^n\to L_2(\mathbb{R}^d)$</span> is measurable then there exists a product measurable function <span class="math-container">$g\colon\ma... |
3,008,564 | <p>Let <span class="math-container">$f:X\longrightarrow{Y}$</span>, <span class="math-container">$s:Y\longrightarrow{X}$</span> be continuous functions where <span class="math-container">$fs=id$</span> how can I prove that Y has the quotient topology for <span class="math-container">$f$</span>?</p>
| Dante Grevino | 616,680 | <p>You can use the universal property. The function <span class="math-container">$s:Y\to X$</span> is a quotient if and only if it satisfies the following two properties:
1) <span class="math-container">$s$</span> is suryective;
2) for every topological space <span class="math-container">$Z$</span> and every function ... |
3,008,564 | <p>Let <span class="math-container">$f:X\longrightarrow{Y}$</span>, <span class="math-container">$s:Y\longrightarrow{X}$</span> be continuous functions where <span class="math-container">$fs=id$</span> how can I prove that Y has the quotient topology for <span class="math-container">$f$</span>?</p>
| Matematleta | 138,929 | <p>If you don't want to use the universal property of the quotient, you can argue directly:</p>
<p><span class="math-container">$f$</span> is surjective because <span class="math-container">$f\circ s=1_Y:$</span> let <span class="math-container">$y\in Y.$</span> Then, <span class="math-container">$f(s(y))=y$</span> so... |
33,311 | <p>I'm (very!) new to <em>Mathematica</em>, and trying to use it plot a set in $\mathbb{R}^3$. In particular, I want to plot the set</p>
<p>$$ \big\{ (x,y,z) : (x,y,z) = \frac{1}{u^2+v^2+w^2}(vw,uw,uv) \text{ for some } (u,v,w) \in \mathbb{R}^3\setminus\{0\} \big\}. $$
This is just the image of function from $\mathbb{... | Nasser | 70 | <p>This is good reason to use the second argument of dynamics.</p>
<p><img src="https://i.stack.imgur.com/ezhK3.gif" alt="enter image description here"></p>
<pre><code>Manipulate[
function; (*just to allow tracking, since not explicity in the command*)
ListLinePlot[data, PlotRange -> {{start, stop}, Au... |
730,253 | <p>let $x,y$ such
$$2^x+5^y=2^y+5^x=\dfrac{7}{10}$$</p>
<p>prove or disprove $x=y=-1$ is the only solution for the system.</p>
<p>My try: since
$$2^x-2^y=5^x-5^y$$</p>
<p>But How can prove or disprove $x=y$?</p>
| Barry Cipra | 86,747 | <p>Here's $99\%$ of a proof. Unfortunately I'm stuck on the final, crucial $1\%$. </p>
<p><strong>Edit</strong>: Having finally understood Greg Martin's answer, I now see how to get unstuck on the crucial $1\%$. I'm <strong>inserting</strong> the rest of the proof at the pertinent point, but leaving the rest alone... |
2,884,784 | <p>Since there there several <a href="https://en.wikipedia.org/wiki/Locally_compact_space" rel="nofollow noreferrer">inequivalent ways</a> to define the local compactness, I first give the defintion in Fred H. Croom's Principles of Topology.</p>
<p><em>Definition</em>: A space $X$ is locally compact at a point $x$ if ... | tmaths | 584,390 | <p>The usual term for those "functions" is the term "operator". For example, the operator $T$ that "swap" coordinates is defined by $T(f) = g$ with $g(x,y) = f(y,x)$.</p>
|
2,884,784 | <p>Since there there several <a href="https://en.wikipedia.org/wiki/Locally_compact_space" rel="nofollow noreferrer">inequivalent ways</a> to define the local compactness, I first give the defintion in Fred H. Croom's Principles of Topology.</p>
<p><em>Definition</em>: A space $X$ is locally compact at a point $x$ if ... | Ignat Insarov | 278,244 | <p>What you are describing are transformations of expressions or syntax trees. Computer scientists have a lot to do with those things.</p>
<p>While mathematicians are ordinarily interested in <em>"extrinsic"</em> equality, where <span class="math-container">$f = g$</span> when <span class="math-container">$\forall x. ... |
2,795,652 | <blockquote>
<p>Given a set of decimal digits. And given a set of primes $\mathbb{P}$, find some $p \in \mathbb{P}$ such that $p^n, n \in \mathbb{N} $ contained in itself all the digites from a given set, and it does not matter in what order.</p>
</blockquote>
| Ross Millikan | 1,827 | <p>$2^{100}=1267650600228229401496703205376$ includes all the digits so it will answer your request for any set. </p>
<p>For any given prime it is believed that there is a highest power that does not contain all the digits, so if your list of primes doesn't include $2$ I would just raise one of them to the $1000$ pow... |
661,542 | <p>Permutation without consecutive numbers, such as 1,2 or 5,4?</p>
| Rebecca J. Stones | 91,818 | <p>There's not that many cases, we could draw a <a href="http://en.wikipedia.org/wiki/Decision_tree" rel="nofollow noreferrer">decision tree</a>:</p>
<p><img src="https://i.stack.imgur.com/tniz8.png" alt="Decision tree"></p>
<p>By symmetry, starting with either $1$ or $5$ results in the same number of sequences. Sim... |
2,197,065 | <p>Baire category theorem is usually proved in the setting of a complete metric space or a locally compact Hausdorff space.</p>
<p>Is there a version of Baire category Theorem for complete topological vector spaces? What other hypotheses might be required?</p>
| Emilio Novati | 187,568 | <p>There are many physical problems that concern some mean distance especially in statistical mechanics. <a href="https://en.wikipedia.org/wiki/Mean_inter-particle_distance" rel="nofollow noreferrer">This</a>, about the mean distance between the molecules of a gas, is the more classical that I know.</p>
|
391,020 | <p>Does there exist a non-zero homomorphism from $\mathbb{Z}/n\mathbb{Z}$ to $\mathbb{Z}$ ? If yes, state the mapping. How is this map exactly? </p>
| Seirios | 36,434 | <p>Let $\varphi : \mathbb{Z}_n \to \mathbb{Z}$ be a homomorphism. Using isomorphism theorem, $\mathbb{Z}_n / \text{ker}(\varphi)$ is isomorphic to $\text{Im}(\varphi)$. Because $\text{Im}(\varphi) \leq\mathbb{Z}$, either $\text{Im}(\varphi)= \{0\}$ or $\text{Im}(\varphi) \simeq \mathbb{Z}$; on the other hand, $\mathbb... |
3,837,745 | <p>The two equations are</p>
<p><span class="math-container">$$\begin{aligned}3x^2-12y=& \ 0\\
3y^2-12x=& \ 0
\end{aligned}$$</span></p>
<p>Using system of equations find all of the solutions?</p>
<p>I found the first one to be <span class="math-container">$(0,0).$</span> The answer key says <span class="math-c... | Math Lover | 801,574 | <p><span class="math-container">$3x^2 - 12y = 0 => x^2 = 4y$</span></p>
<p><span class="math-container">$3y^2 - 12x = 0 => 4x = y^2$</span></p>
<p>Multiplying both, <span class="math-container">$4x^3 = 4y^3$</span> or <span class="math-container">$x = y$</span></p>
<p>Substituting in one of the equations,</p>
<p>... |
3,273,830 | <p>Okay so I've started to study derivatives and there is this idea of continuity. The book says <em>"a real valued function is considered continuous at a point iff the graph of a function has no break at the point of consideration, which is so iff the values of the function at the neighbouring points are close enough ... | Ruben | 386,073 | <p>Continuity isn't a property of a point of the graph, but a property of neighborhoods of points of the graph. You need information about a neighborhood of a point to say anything about continuity at that point.</p>
<p>When we say <em>continuous at a point</em> we are really saying something about neighborhoods of th... |
106,708 | <p><strong>My fragile attempt:</strong> Note that if $1987^k-1$ ends with 1987 zeros, that means $1987^k$ has last digit 1 (and 1986 "next" ones are zeros). For this to be satisfied, $k$ has to be in form $k=4n$, where $n\in N$. This means out number can be written in form </p>
<p>$$ [(1987^n)^2+1][1987^n+1][1987^n-1]... | probably_right | 120,381 | <p>I don't think it's possible, here is why:
We have established that in 1987^k k is of the form 4n or 2m
and also that 1987^k ends in 1, so
1987^K = 1987^2m = (1987^m -1)(1987^m +1)
Since 1987^k ends in 1, i.e. it is an odd number
Both (1987^m -1) and (1987^m +1) are odd (as multiplication of only 2 odd numbers is odd... |
738,435 | <p>I am having a problem understanding how to determine if a function is one to one.</p>
<p>The problem is: Show that the function f(x) = 3x+4 is one-to-one.</p>
<p>Also, I'm being thrown off by the notation x[subset 1] = x[subset 2], what does that mean, loosely speaking? "In my eyes" that would mean two different v... | ojblass | 65,909 | <p>Draw a graph and if there are any values of x for which there are two values of y then the function is not one to one.</p>
|
522,714 |
<p>$$
dz_t \sim O\left(\sqrt{dt}\,\right)
$$</p>
<p>$z$ is a Brownian motion random variable, for reference. I just don't understand what the $\sim O$ part means. I've looked up the page for Big O notation on wikipedia because I thought it might be related, but I can't see the link.</p>
| ziang chen | 38,195 | <p>Oh, mistake $(x^2)^2+(\frac1{x^2})^2=(x^2+\frac1{x^2})^2-2$ !!
en
$$x+\frac1x=2\sqrt3$$
and
$$x-\frac1x=2\sqrt2$$
so
$$x^2+\frac1{x^2}=(x+\frac1x)^2-2=10$$
and
$$x^2-\frac1{x^2}=(x+\frac1x)(x-\frac1x)=4\sqrt6$$
It follows that
$$x^4-\frac1{x^4}=(x^2+\frac1{x^2})(x^2-\frac1{x^2})=40\sqrt6$$</p>
|
1,758,194 | <p>Consider the function f: {-1, +1} -> R defined by</p>
<p>$f(x)= \arcsin (\frac{1+x^2}{2x})$.</p>
<p>Due to the following two inequalities :</p>
<p>(i) $1+x^2 \geq 2x$</p>
<p>(ii)$1+x^2 \geq -2x$ , </p>
<p>the function can only be defined at $x=1$ and $x=-1$. I have learnt that the epsilon delta definition only ... | Lord_Farin | 43,351 | <p>Recall the (most commonly used) definition of $\lim\limits_{x\to 1} f(x) = \frac\pi2$:</p>
<p>$$\forall\epsilon > 0~\exists \delta>0: \forall x \in \operatorname{dom}(f): 0<|x-1|<\delta \implies |f(x)-\frac\pi2|<\epsilon$$</p>
<p>(Note that we need to exclude the situation $x=1$ to distinguish betwe... |
2,708,142 | <p>It is known that a stochastic matrix is a square matrix $\boldsymbol{T}$ that satisfies</p>
<ol>
<li>$\boldsymbol{T_{ij}} \geq 0$ for all $i,j$.</li>
<li>$\sum_{j}^{ }\boldsymbol{T_{ij}}=1.$ </li>
</ol>
<p>Assume $\boldsymbol{P}$ is the transition matrix of a Markov chain on $k$ states. $\boldsymbol{I}$ is the $k ... | Pietro Paparella | 414,530 | <p>If $Q= [q_{ij}]$ and $P=[p_{ij}]$, then the $(i,j)$-entry of $(1-\alpha)I_n + \alpha P$, $\alpha \in (0,1)$, is given by
$$
[(1-\alpha)I_n + \alpha P]_{ij} =
\begin{cases}
1-\alpha+\alpha p_{ii}, & i=j \\
\alpha p_{ij}, & i \ne j.
\end{cases}
$$
Since the sum of nonnegative matrices is nonnegative and since ... |
2,708,142 | <p>It is known that a stochastic matrix is a square matrix $\boldsymbol{T}$ that satisfies</p>
<ol>
<li>$\boldsymbol{T_{ij}} \geq 0$ for all $i,j$.</li>
<li>$\sum_{j}^{ }\boldsymbol{T_{ij}}=1.$ </li>
</ol>
<p>Assume $\boldsymbol{P}$ is the transition matrix of a Markov chain on $k$ states. $\boldsymbol{I}$ is the $k ... | Pietro Paparella | 414,530 | <p>Here's another proof, which is far more elegant.</p>
<blockquote>
<p><strong>Exercise:</strong> If $A$ is a square complex matrix, then every row of $A$ sums to one if and only if $Ae = e,$ i.e., $(1,e)$ is a right eigenpair of $A$ ($e$ denotes the all-ones vector). </p>
</blockquote>
<p>Thus, if $P$ is row sto... |
673,928 | <p>How could I solve, apart from brute force, the congruence</p>
<p>$$ \frac{x(x-1)}{2} \equiv 0\ ( \text{mod} \ 2) \ ? $$</p>
<p>I mean, for what values of $y$ has the polynomial $ x(x-1)=2y $ only integer solutions and even for example $ x=f(y) $ is an EVEN number ?
Thanks.</p>
| gt6989b | 16,192 | <p>Well, at $t=0$, the exponent becomes $1$, and at $t \to \infty$, the exponent tends to $e^{-kt} \to 0$.</p>
|
527,328 | <p>$$\lim_{x\rightarrow\infty}\left(\frac{x+1}{x-2}\right)^{2x-1}$$</p>
<p>What are the steps to solve it? Probably the division should be multiplied by some expression.</p>
| lab bhattacharjee | 33,337 | <p>As $\displaystyle \frac{x+1}{x-2}=1+\frac3{x-2},$</p>
<p>$$\lim_{x\to\infty}\left(\frac{1+x}{-2+x}\right)^{-1+2x}$$</p>
<p>$$=\left(\lim_{x\to\infty}\left(1+\frac3{-2+x}\right)^{\frac{x-2}3}\right)^{\lim_{x\to\infty}\frac{3(2x-1)}{x-2}}$$</p>
<p>$$=e^6$$ as $\lim_{x\to\infty}\frac{3(2x-1)}{x-2}=3\lim_{x\to\infty}... |
484,281 | <p>I recently got interested in mathematics after having slacked through it in highschool. Therefore I picked up the book "Algebra" by I.M. Gelfand and A.Shen</p>
<p>At problem 113, the reader is asked to factor $a^3-b^3.$</p>
<p>The given solution is:
$$a^3-b^3 = a^3 - a^2b + a^2b -ab^2 + ab^2 -b^3 = a^2(a-b) + ab(... | Umberto | 92,940 | <p>I am not sure I get your question. The last formula is just obtained factoring a and b. For example, the first part</p>
<p>$a^2(a-b) = a^3 -a^2b$</p>
<p>So once you get the last formula you can take $(a-b)$ and obtain</p>
<p>$(a-b)(a^2+ab+b^2)$</p>
<p>and to obtain the second formula you simply add and subtract ... |
484,281 | <p>I recently got interested in mathematics after having slacked through it in highschool. Therefore I picked up the book "Algebra" by I.M. Gelfand and A.Shen</p>
<p>At problem 113, the reader is asked to factor $a^3-b^3.$</p>
<p>The given solution is:
$$a^3-b^3 = a^3 - a^2b + a^2b -ab^2 + ab^2 -b^3 = a^2(a-b) + ab(... | J. W. Perry | 93,144 | <p>Being able to quickly derive the factorization for a difference of cubes (and most other things) prevented me from having to memorize many things in school. I cannot help but share this with you as it was something I must have done a hundred times in a pinch. I will answer your question by delivering a proper full d... |
384,090 | <p>Find all real numbers $x$ for which $$\frac{8^x+27^x}{12^x+18^x}=\frac76$$</p>
<p>I have tried to fiddle with it as follows: </p>
<p>$$2^{3x} \cdot 6 +3^{3x} \cdot 6=12^x \cdot 7+18^x \cdot 7$$
$$ 3 \cdot 2^{3x+1}+ 2 \cdot 3^{3x+1}=7 \cdot 6^x(2^x+3^x)$$
Dividing both sides by $6$ gives us
$$2^{3x}+3^{3x}=7 \cdot ... | lab bhattacharjee | 33,337 | <p>Let us put $2^x=a,3^x=b$ to remove the indices to improve clarity</p>
<p>So, $8^x=(2^3)^x=(2^x)^3=a^3$ and similarly, $27^x=b^3$</p>
<p>$12^x=(2^2\cdot3)^x=(2^x)^2\cdot3^x=a^2b$ and similarly, $18^x=ab^2$</p>
<p>So, the problem reduces to $$\frac{a^3+b^3}{ab(a+b)}=\frac76$$</p>
<p>$\displaystyle\implies 6(a^2-a... |
384,090 | <p>Find all real numbers $x$ for which $$\frac{8^x+27^x}{12^x+18^x}=\frac76$$</p>
<p>I have tried to fiddle with it as follows: </p>
<p>$$2^{3x} \cdot 6 +3^{3x} \cdot 6=12^x \cdot 7+18^x \cdot 7$$
$$ 3 \cdot 2^{3x+1}+ 2 \cdot 3^{3x+1}=7 \cdot 6^x(2^x+3^x)$$
Dividing both sides by $6$ gives us
$$2^{3x}+3^{3x}=7 \cdot ... | hkBattousai | 21,550 | <p>$$
\begin{array}{rcl}
6(8^x + 27^x) &=& 7(12^x + 18^x) \\
6(2^{3x} + 3^{3x}) &=& 7(3^x2^{2x} + 3^{2x}2^x) \\
\end{array}
$$</p>
<p>Substitute $a\!=\!2^x$ and $b\!=\!3^x$ for simplicity:</p>
<p>$$
\begin{array}{rcl}
6(a^3 + b^3) &=& 7(a^2b + ab^2) \\
6(a+b)(a^2 - ab + b^2) &=& 7ab(a+... |
3,372,952 | <blockquote>
<p>I know that all the subspaces of <span class="math-container">$\mathbb{R}^{3}$</span> over <span class="math-container">$\mathbb{R}$</span> are:</p>
<ul>
<li><p>Zero subspace.</p></li>
<li><p>Lines passing through origin.</p></li>
<li><p>Planes passing through origin.</p></li>
<li><p><spa... | mathcounterexamples.net | 187,663 | <p>The dimension of a linear subspace <span class="math-container">$F$</span> of a vectorial space is lower or equal to the dimension of the space itself <span class="math-container">$E$</span>. </p>
<p>Here, <span class="math-container">$\dim E = \dim \mathbb R^3=3$</span>. Hence the dimensions of the subspaces belon... |
222,701 | <p>Below in code are two versions. I believe plot1 might be correct but not sure why its shifted to the left and how to fix it. plot2 is copied directly from the wolfman online documentation but using my own data. It doesn't look normalized and the continuous curve is almost flat. I know something is likely wrong with ... | MarcoB | 27,951 | <p>To summarize <a href="https://mathematica.stackexchange.com/users/19758/jimb">@JimB</a>'s comment, you should plot the PDF of the normal distribution that most closely resembles your data, i.e. the one with the same mean and standard deviation:</p>
<pre><code>Show[
Histogram[raw, Automatic, "PDF"],
Plot[
PD... |
2,028,881 | <p>So, I have been told that for every $x\in\mathbb{N}$, $\ln{x}$ is irrational by using the fact that $e$ is transcendental number.</p>
<p>But, how to prove that $\log_{2}{3}$ is irrational number? My idea is to rewrite the form</p>
<p>$$\log_2 3 = \frac{\ln 3}{\ln 2}$$</p>
<p>But, if both $x, y$ irrational, it isn... | Michael Hardy | 11,667 | <p>Suppose $\log_2 3 = \dfrac m n$ for some $m,n\in\{1,2,3,\ldots\}.$</p>
<p>Then $2^{m/n} = 3.$</p>
<p>Therefore $2^m = 3^n.$</p>
<p>Therefore an even number equals an odd number.</p>
|
2,028,881 | <p>So, I have been told that for every $x\in\mathbb{N}$, $\ln{x}$ is irrational by using the fact that $e$ is transcendental number.</p>
<p>But, how to prove that $\log_{2}{3}$ is irrational number? My idea is to rewrite the form</p>
<p>$$\log_2 3 = \frac{\ln 3}{\ln 2}$$</p>
<p>But, if both $x, y$ irrational, it isn... | Bill Dubuque | 242 | <p>More generally the logs of primes $\rm\,p_i\,$ are linearly independent over $\,\Bbb Q\,$ since if</p>
<p>$$\rm\ \: c_1 \log\,2 \, +\, c_2 \log\, 3\, + \cdots +\, c_n\log\,p_n =\ 0,\ \: c_i\in\mathbb Q\:,\: $$</p>
<p>then multiplying by a common denominator we can assume all $\rm\ c_i \in \mathbb Z,\:$ so, expo... |
1,142,530 | <p>Find an equation of the plane.
The plane that passes through the point
(−3, 2, 1) and contains the line of intersection of the planes
x + y − z = 4
4x − y + 5z = 2</p>
<p>I know the normal to plane 1 is <1,1,-1> and the normal to plane 2 is <4,-1,5>. The cross product of these 2 would give a vector that is ... | Trucker | 153,069 | <p>If <span class="math-container">$\vec{r}=(x,y,z)$</span> is the position of a point on the intersection, then <span class="math-container">$<\vec{r},\vec{n}_1>=4$</span> and <span class="math-container">$<\vec{r},\vec{n}_2>=2$</span>. Where <span class="math-container">$\vec{n}_1=(1,1,-1)$</span> and <sp... |
507,494 | <p>$\begin{align*}\frac{n!}{e}-!n&=n!\sum_{k=n+1}^\infty \frac{(-1)^k}{k!}\\&=\frac{(-1)^{n+1}}{n+1}\left(1-\frac1{n+2}+\frac1{(n+2)(n+3)}-\cdots\right)\end{align*}$</p>
<p>and hence</p>
<p>$$\frac1{n+2} \leq \left|\frac{n!}{e}-!n\right| \leq \frac1{n+1}$$</p>
<p>I can understand how$$ \left|\frac{n!}{e}-!n\... | abiessu | 86,846 | <p>Your derivation and your answer are correct, and only one minor piece of detail is helpful in guaranteeing the result:</p>
<p>$${1\over n+2}\le |{n!\over e}-!n|\le {1\over n+1}\tag{1}$$</p>
<p>The far LHS of $(1)$ is shown by the following condition:</p>
<p>$${1\over n+1}-{1\over (n+1)(n+2)}+{1\over (n+1)(n+2)(n+... |
4,614,269 | <p>Given the curve <span class="math-container">$y=\frac{5x}{x-3}$</span>. To examine its asymptotes if any.</p>
<p><strong>We are only taking rectilinear asymptotes in our consideration</strong></p>
<p>My solution goes like this:</p>
<blockquote>
<p>We know that, a straight line <span class="math-container">$x=a$</spa... | epi163sqrt | 132,007 | <p>We show the following is valid for <span class="math-container">$0\leq m<n$</span>
<span class="math-container">\begin{align*}
\color{blue}{\sum_{i=m+1}^n\binom{n+i}{m}\binom{2n}{n+i}=\sum_{i=1}^n\binom{n-i}{m}\binom{2n}{n-i}}\tag{1}
\end{align*}</span></p>
<blockquote>
<p>We start with the right-hand side of (1)... |
266,981 | <p>Wikipedia presents a recursive definition of the Pfaffian of a skew-symmetric matrix as $$ \operatorname{pf}(A)=\sum_{{j=1}\atop{j\neq i}}^{2n}(-1)^{i+j+1+\theta(i-j)}a_{ij}\operatorname{pf}(A_{\hat{\imath}\hat{\jmath}}),$$ where $\theta$ is the Heaviside step function. I have failed to find a proper reference for t... | მამუკა ჯიბლაძე | 41,291 | <p>And now for something entirely different :D</p>
<p>The category $\mathbf{Set}^{\mathbb C^\circ}$ of presheaves of sets on a small category $\mathbb C$ is monoidal closed wrt cartesian products (i. e. cartesian closed) - in fact it is a topos. </p>
<p>Now it happens that the category $\mathbf{Set}^{\mathbb C}$ of p... |
72,876 | <p>A quiver in representation theory is what is called in most other areas a directed graph. Does anybody know why Gabriel felt that a new name was needed for this object? I am more interested in why he might have felt graph or digraph was not a good choice of terminology than why he thought quiver is a good name. (I r... | user46121 | 46,121 | <p>When doing my thesis it seemed it was simply because multiple edges and loops are allowed when working with quivers and therefore it is easier to use this well defined term rather than referring to it as a directed graph which seems to have a much broader meaning.</p>
|
3,741,859 | <p>I am trying to prove that, for all non-negative integers <span class="math-container">$x$</span> and all non-negative real numbers <span class="math-container">$p$</span>,
<span class="math-container">$$
\left(p-x\right)\left(x+1\right)^p+x^{p+1}\geq0.
$$</span>
I've been at this for a while and I'm stuck. I've trie... | Polygon | 341,812 | <p>I got it!</p>
<p>The proof is rather long and there is probably an easier way, but I believe this is all correct.</p>
<p>Consider the function <span class="math-container">$f(x)=(1+x)^{-p}+px$</span>. I start by showing that this function is never less than <span class="math-container">$1$</span> when <span class="m... |
189,781 | <p>I want to add custom Mesh lines onto a Plot3D, with different MeshStyle than Mesh lines which already exist. As a MWE, starting with a meshed plot</p>
<pre><code>Plot3D[Exp[-(x^2+y^2)],{x,-3,3},{y,-3,3},PlotRange->All,Mesh->Automatic]
</code></pre>
<p>I want to go back and add extra mesh (keeping the existi... | Michael E2 | 4,999 | <p>One difficulty in generating mesh lines is that lines that are judged to pass through a polygon in the surface graphics will not be rendered as a line on the surface. Presumably the rendering software/hardware is programmed with several possibilities in mind, that lines might lie in the plane of a polygon and that ... |
184,487 | <p>For any domain $A$ let $A^\times$ be its group of units.</p>
<p>Let $A$ be a noetherian domain with only finitely many prime ideals, and field of fractions $K$. </p>
<p>Is the group $K^\times/A^\times$ finitely generated?</p>
| Agustí Roig | 664 | <p>Today I don't have the time to think about your proof, but I would like to share some heuristics about roofs. Maybe they could be helpful.</p>
<p>These roofs $(s,f)$ (or its classes modulo the equivalence relation you have recalled) are the morphisms of the <em>localised category</em> $S^{-1}B$, or $B[S^{-1}]$. The... |
224,670 | <p>If I have a data that I fit with <code>NonlinearModelfit</code> that fits a data based on two fitting parameters, <code>c1</code> and <code>c2</code>.</p>
<p>When I used <code>nlm["ParameterTable"] // Quiet</code> I get the following table:</p>
<p><a href="https://i.stack.imgur.com/ZmQQO.png" rel="nofollow... | flinty | 72,682 | <p>Assuming the c1/c2 estimates came from a large sample, the <code>eq</code> has a non-central ratio of Gaussians distribution. You'd have to simulate or use <code>NExpectation</code> here:</p>
<pre><code>dc1 = NormalDistribution[8.08318, 0.692171];
dc2 = NormalDistribution[21.1577, 3.13379];
td = TransformedDistribut... |
3,288,651 | <p>I have a problem understanding a proof about ideals, which states that every ideal in the integers can be generated by a single integer. And with that I realized that I also don't really understand ideals in general and the intuition behind them. </p>
<p>So let me start by the definition of an ideal. For <span clas... | José Carlos Santos | 446,262 | <ul>
<li>Yes, <span class="math-container">$u$</span> and <span class="math-container">$v$</span> are integers.</li>
<li>No, the assertion “if <span class="math-container">$a,b\in\mathbb Z$</span>, then there is a <span class="math-container">$d\in\mathbb Z$</span> such that <span class="math-container">$(a,b)=(d)$</sp... |
3,537,297 | <p>I am asked the following question (<strong>non-calculator</strong>):</p>
<blockquote>
<p>Solve for <span class="math-container">$0 \leq x \leq 2 \pi$</span></p>
<p><span class="math-container">$$4 \sin x =\sqrt{3} \csc x + 2 - 2\sqrt{3} $$</span>
<strong>Answer:</strong> <span class="math-container">$x = \... | Andrew Chin | 693,161 | <p><span class="math-container">\begin{align}
4\sin x &=\sqrt3\csc x+2-2\sqrt3\\
4\sin^2x&=\sqrt3+2\sin x-2\sqrt3\sin x\\
4\sin^2x-2\sin x+2\sqrt3\sin x-\sqrt3&=0\\
2\sin x(2\sin x-1)+\sqrt3(2\sin x-1)&=0\\
(2\sin x-1)(2\sin x+\sqrt3)&=0
\end{align}</span></p>
<p>Can you take it from here?</p>
|
1,008,610 | <blockquote>
<p>If <span class="math-container">$a$</span> is a group element, prove that <span class="math-container">$a$</span> and <span class="math-container">$a^{-1}$</span> have the same order.</p>
</blockquote>
<p>I tried doing this by contradiction.</p>
<p>Assume <span class="math-container">$|a|\neq|a^{-1}|$</... | user496040 | 496,040 | <p>Let's say
a is an element and n is its order ,then
<span class="math-container">$$a^n=e$$</span>
Repeatedly multiplication by <span class="math-container">$a^{-1}$</span> n times
<span class="math-container">$$(a^{-1})^{n}•(a)^{n}=e•(a^{-1})^n$$</span>
<span class="math-container">$$(a^{-1}•a)^{n}=(a^{-1})^n$$</span... |
1,633,722 | <p>Consider the family of $n \times n$ real matrices $A$, for which there is a $n \times n$ real matrix $B$ with $AB-BA=A$. How large can the rank of a matrix in this family be?</p>
<h2>Motivation</h2>
<p>Prasolov's book contains an exercise about proving that if $A,B$ are matrices with $AB-BA=A$ then $A$ cannot be i... | adjan | 219,722 | <p>If $c \in span\{x,y\}$, then $c$ can be expressed as a linear combination of $x$ and $y$:</p>
<p>$$c = \alpha x + \beta y$$</p>
<p>Thus you only have to solve the linear equation system</p>
<p>$$\begin{bmatrix}
0 \\
-2 \\
3 \\
\end{bmatrix} = \alpha \begin{bmatrix}
2 \\
... |
1,633,722 | <p>Consider the family of $n \times n$ real matrices $A$, for which there is a $n \times n$ real matrix $B$ with $AB-BA=A$. How large can the rank of a matrix in this family be?</p>
<h2>Motivation</h2>
<p>Prasolov's book contains an exercise about proving that if $A,B$ are matrices with $AB-BA=A$ then $A$ cannot be i... | Bernard | 202,857 | <p>You don't even have to do backwards substitutions. Just proceed further to put the matrix in row echelon form: you get
$$\begin{bmatrix}
1 & 1 & 3 \\
0 & 1 & -2 \\
0 & 0 & 0 \\
\end{bmatrix}$$
Thus the augmented matrix has rank $2$, like $\begin{bmatrix}
... |
228,389 | <p>How and where is it proved that WKL$_0$ proves the compactness theorem for countable models? (This is a follow-up to a comment of F. Dorais.)</p>
| Noah Schweber | 8,133 | <p>The proof is basically the usual Henkinization proof of completeness (and hence compactness as a corollary), using your favorite proof system. Starting with a consistent theory, first add constants and complete the theory, then build the term model. </p>
<p>The only noneffective step here is completing the theory, ... |
1,515,667 | <blockquote>
<p>Show that the limit $\lim_{(x,y)\to (0,0)}\frac{2e^x y^2}{x^2+y^2}$
does not exist</p>
</blockquote>
<p>$$\lim_{(x,y)\to (0,0)}\frac{2e^x y^2}{x^2+y^2}$$</p>
<p>Divide by $y^2$:</p>
<p>$$\lim_{(x,y)\to (0,0)}\frac{2e^x}{\frac{x^2}{y^2}+1}$$</p>
<p>$$=\frac{2(1)}{\frac{0}{0}+1}$$</p>
<p>Since $\... | zoli | 203,663 | <p>The limit exists if for any $\varepsilon>0$ there exists a $\delta>0$ such that
$$\frac{2e^x y^2}{x^2+y^2}<\varepsilon\tag 1$$
if the distance of $(x,y)$ from the origin is less than $\delta$. That is, the limit exists only if it is independent of the path of $(x,y)$ when approaching $(0,0)$.</p>
<p>In the... |
1,515,667 | <blockquote>
<p>Show that the limit $\lim_{(x,y)\to (0,0)}\frac{2e^x y^2}{x^2+y^2}$
does not exist</p>
</blockquote>
<p>$$\lim_{(x,y)\to (0,0)}\frac{2e^x y^2}{x^2+y^2}$$</p>
<p>Divide by $y^2$:</p>
<p>$$\lim_{(x,y)\to (0,0)}\frac{2e^x}{\frac{x^2}{y^2}+1}$$</p>
<p>$$=\frac{2(1)}{\frac{0}{0}+1}$$</p>
<p>Since $\... | Hirshy | 247,843 | <p>In addition to the answers showing you a way: when dealing with limits you can't just put in the values for (in this case) $x$ and $y$. With the same argument you would say, that $$\lim\limits_{x\to 0} \frac{\sin(x)}{x}=\frac{0}{0}=\text{undefined}$$ whereas it is easy to show (and something one should know) that $$... |
1,067,762 | <p>Is what I am doing below correct, please assist. </p>
<p>$$\sum_{k=-\infty}^{-1}\frac{e^{kt}}{1-kt} = \sum_{k=1}^{\infty}\frac{e^{-{kt}}}{1-kt}$$ </p>
<p>Is this the rule on how to "invert" the limits, and does it matter if there are imaginary numbers in the sum; or not or is it all the same with both pure real an... | lab bhattacharjee | 33,337 | <p>HINT:</p>
<p>$$\tan(B+C)=\frac{\tan B+\tan C}{1-\tan B\tan C}=\infty$$</p>
<p>Alternatively,</p>
<p>$\tan B\cdot\tan C=\cdots=1\iff\sin B\sin C=\cos B\cos C\iff\cos(B+C)=0$</p>
|
3,429,514 | <p>so I am having a problem understanding finding solution to that equation:
<span class="math-container">$$x'' +w^2 x =f sin ω t$$</span>
with initial conditions x(0)=0 and x'(0)=0. I know that is an explanation of how to solve that:
<br>
<br></p>
<p><em>I had to make the euler formula substitution and then multip... | Gribouillis | 398,505 | <p>What they mean is that when <span class="math-container">$\omega\neq 0$</span> and <span class="math-container">$f\neq 0$</span>, the equation <span class="math-container">$x'' + \omega^2 x = f \sin (\omega t)$</span> has a particular solution of the form
<span class="math-container">\begin{equation}
x(t) = A t \sin... |
2,425,916 | <p>What is this set describing?</p>
<p>{$n∈\mathbb{N}|n\ne 1$ and for all $a∈\mathbb{N}$ and $b∈\mathbb{N},ab=n$ implies $a= 1$ or $b= 1$}</p>
<p>Is it describing a subset of natural numbers, excluding 1, that is the product of two other natural numbers, of which one must be 1? Isn't that just every natural number ex... | gen-ℤ ready to perish | 347,062 | <p>Might it help if I translate to ordinary English?</p>
<ol>
<li>$n$ must be a natural number other than $1$</li>
<li>one must be able to write $n$ as the product of two natural numbers $a$ and $b$</li>
<li>one of the aforementioned $a$ or $b$ must equal $1$</li>
<li>the prior statements must hold for all natural $a$... |
2,982,978 | <p><strong>Exercise :</strong></p>
<blockquote>
<p>Let <span class="math-container">$S: l^1 \to l^1$</span> be the right-shift operator :
<span class="math-container">$$S(x_1,x_2,\dots) = (0,x_1,x_2,\dots)$$</span>
Prove that <span class="math-container">$S$</span> is bounded and find its norm.</p>
</blockquote>... | Tito Eliatron | 84,972 | <p><span class="math-container">$$\|S(x_1,x_2,\dots)\|_1=\|(0,x_1,x_2,\cdots)\|_1=0+|x_1|+|x_2|+\dots = |x_1|+|x_2|+\dots = \|(x_1,x_2,\dots)\|_1.$$</span></p>
<p>In particular, <span class="math-container">$$\|S(x_1,x_2,\cdots)\|_1\le 1\cdot \|(x_1,x_2,\cdots)\|_1.$$</span></p>
<p>This inequality says that <span cla... |
3,921,252 | <p>Let <span class="math-container">$M$</span> be a connected topological <span class="math-container">$n$</span>-manifold. Note this implies <span class="math-container">$M$</span> is path-connected.</p>
<p>Let <span class="math-container">$a,b \in M$</span>. Must there always exist a continuous <span class="math-cont... | Tyrone | 258,571 | <p>If <span class="math-container">$M$</span> is a topological manifold and <span class="math-container">$a\in M$</span> is any point, then the inclusion <span class="math-container">$\{a\}\hookrightarrow M$</span> is a cofibration (i.e. has the <a href="https://en.wikipedia.org/wiki/Homotopy_extension_property" rel="n... |
3,921,252 | <p>Let <span class="math-container">$M$</span> be a connected topological <span class="math-container">$n$</span>-manifold. Note this implies <span class="math-container">$M$</span> is path-connected.</p>
<p>Let <span class="math-container">$a,b \in M$</span>. Must there always exist a continuous <span class="math-cont... | Aphelli | 556,825 | <p>Let’s prove something stronger: for each such manifold <span class="math-container">$M$</span> (without boundary), the path-connected component of the identity in the group of homeomorphisms of <span class="math-container">$M$</span> acts transitively on <span class="math-container">$M$</span>.</p>
<p>Since it’s a c... |
621,757 | <p>I was wondering what the definition of a partial derivative of such a map is? </p>
<p>Is it $\partial_if_j$ or is it anything else? The reason why I have doubts is that I found the definition that a partial derivative is the total derivative of a function $f(\,\cdot\,,x_2): x_1 \mapsto f(x_1,x_2)$, where $(x_1,x_2)... | Ulrik | 53,012 | <p>The most natural definition would be this, I think: A map $f: \mathbb{R}^n \to \mathbb{R}^m$ can always be decomposed into $m$ maps $f_1, f_2, \ldots, f_m : \mathbb{R}^n \to \mathbb{R}$ that satisfy
$$f(x) = (f_1(x), f_2(x), \ldots, f_m(x))$$
Then one can define the partial derivative of $f$ with respect to the $i$t... |
621,757 | <p>I was wondering what the definition of a partial derivative of such a map is? </p>
<p>Is it $\partial_if_j$ or is it anything else? The reason why I have doubts is that I found the definition that a partial derivative is the total derivative of a function $f(\,\cdot\,,x_2): x_1 \mapsto f(x_1,x_2)$, where $(x_1,x_2)... | Stefan Smith | 55,689 | <p>You already accepted an answer, but there is an even simpler and more general way to look at it, if you know what a topological vector space is (for example, any vector space with a norm is a topological vector space). If $S$ is a (real) topological vector space, and $f:\mathbb{R}^n \to S$, then define</p>
<p>$$ \... |
621,757 | <p>I was wondering what the definition of a partial derivative of such a map is? </p>
<p>Is it $\partial_if_j$ or is it anything else? The reason why I have doubts is that I found the definition that a partial derivative is the total derivative of a function $f(\,\cdot\,,x_2): x_1 \mapsto f(x_1,x_2)$, where $(x_1,x_2)... | janmarqz | 74,166 | <p>It helps a lot to think the derivative in matrix terms as $$Jf=\left[\begin{array}{cccc}
{\partial f_1\over\partial x_1}&{\partial f_1\over\partial x_2}&\cdots&{\partial f_1\over\partial x_n}\\
{\partial f_2\over\partial x_1}&{\partial f_2\over\partial x_2}&\cdots&{\partial f_2\over\par... |
3,077,808 | <p>I'm trying to find for which values <span class="math-container">$r$</span> the following improper integral converges.
<span class="math-container">$$\int_0^\infty x^re^{-x}\, dx$$</span>
What I have so far is that <span class="math-container">$x^r < e^{\frac{1}{2}x}$</span> for <span class="math-container">$x \g... | Michael Hardy | 11,667 | <p><span class="math-container">\begin{align}
& \text{For small enough }a>0, \\[8pt]
& x^r > x^r e^{-x} > 0.9 x^r \text{ when } 0<x<a.
\end{align}</span>
Therefore <span class="math-container">$\displaystyle \int_0^a x^r e^{-x} \, dx$</span> converges for precisely those values of <span class="ma... |
395,604 | <p>Lets say you have a sequence $S = (0, 1, 2, 3, 4, 5, 6, 7, 8)$</p>
<p>And another sequence $T = (0, 1, 2, 3)$</p>
<p>Is there any specific mathematical term that defines the relationship between $S$ and $T$, that specifically says that $S$ starts with $T$?</p>
<p>I thought $T$ would be called an <em>initial sub-... | Asaf Karagila | 622 | <p>I usually use "initial segment", and "tail segment" or "end segment" or "final segment" to denote the last part of a sequence.</p>
<p>If you want to indicate the segment is not everything, then "proper initial segment" should suffice.</p>
|
2,581,735 | <p>I'm studying Neural Networks for machine learning (by Geoffrey Hinton's course) and I have a question about learning rule for linear neuron (lecture 3).</p>
<p>Linear neuron's output is defined as: $y=\sum_{i=0}^n w_ix_i$</p>
<p>Where $w_i$ is weight connected to input #i and $x_i$ is input #i.</p>
<p>Learning pr... | AnonymousCoward | 565 | <p>tl;dr if you want to understand what you are doing beyond following "the delta rule", you should review some basic vector calc and look up "gradient-descent" which is what you are doing.</p>
<p>We want to find $w$'s that minimize $E$ (think of $E(w)$ as a function of $w = (w_1, ...,w_n)$ since the inputs $x$ are h... |
234,466 | <p>This is the second problem in Neukirch's Algebraic Number Theory. I did the proof but it feels a bit too slick and I feel I may be missing some subtlety, can someone check it over real quick?</p>
<p>Show that, in the ring $\mathbb{Z}[i]$, the relation $\alpha\beta =\varepsilon\gamma ^n$, for $\alpha,\beta$ relativ... | joriki | 6,622 | <p>The proof looks OK, but you might want to make the role of the fact that $\alpha$ and $\beta$ are relatively prime more apparent. You probably mean the right thing, but the current wording sounds a bit like you're using that to get the prime decompositions, not for concluding that $p_1$ through $p_r$ and $p_s$ throu... |
145,046 | <p>I'm a first year graduate student of mathematics and I have an important question.
I like studying math and when I attend, a course I try to study in the best way possible, with different textbooks and moreover I try to understand the concepts rather than worry about the exams. Despite this, months after such an in... | Michael Hardy | 11,667 | <p>Let's say you can't remember what Theorem 42 says, from a course you took nine semesters ago. Then 20 years later someone mentions a term that sounds vaguely familiar to you---you're not sure why. It's in a discussion of a topic you're curious about. You look in book indexes for terms sounding vaguely related, an... |
145,046 | <p>I'm a first year graduate student of mathematics and I have an important question.
I like studying math and when I attend, a course I try to study in the best way possible, with different textbooks and moreover I try to understand the concepts rather than worry about the exams. Despite this, months after such an in... | A.Magnus | 80,219 | <p>I am a late-comer to this question but I hope I have a good answer to this question.</p>
<p>I am a HS math teacher, inevitably after some times into a semester, an overwhelmed student would come up to me and asks if he has to remember all of these formulas when going to college. Generally, this is my answer: No, yo... |
325,337 | <p>There's a continuous function $f: \mathbb{Q} \rightarrow \mathbb{R}$</p>
<p>Prove that $ \exists t \in \mathbb{R} \setminus \mathbb{Q} \ \ \exists g \in \mathbb{R} :\ \ \lim _{q \rightarrow t, \ q\in \mathbb{Q}}f(q)=g$.</p>
<p>Prove that there are $2^{\aleph _0}$ such numbers $t$.</p>
<p>I know that if a functio... | Paul | 17,980 | <p><strong>HINT:</strong> </p>
<blockquote>
<p>Question 2, because $R$ has $2^{\aleph_0}$ numbers, and $Q$ has countable numbers, so $R\setminus Q$ still has $2^{\aleph_0}$ numbers...</p>
</blockquote>
<hr>
<p><strong>ADDED:</strong> Suppose that $|R\setminus Q|=\kappa<2^{\aleph_0}$, then $|R|=|R\setminus Q \cu... |
325,337 | <p>There's a continuous function $f: \mathbb{Q} \rightarrow \mathbb{R}$</p>
<p>Prove that $ \exists t \in \mathbb{R} \setminus \mathbb{Q} \ \ \exists g \in \mathbb{R} :\ \ \lim _{q \rightarrow t, \ q\in \mathbb{Q}}f(q)=g$.</p>
<p>Prove that there are $2^{\aleph _0}$ such numbers $t$.</p>
<p>I know that if a functio... | Martin Sleziak | 8,297 | <p>Here is my attempt; I hope that there is a simpler solution. (But if you have already seen the facts I mention below, it is not that complicated.)</p>
<p>Let us define
$$\omega(f,x)=\limsup\limits_{\substack{t\to x\\t\in\mathbb Q}} f(t)-\liminf\limits_{\substack{t\to x\\t\in\mathbb Q}} f(t).$$
(This make sense for ... |
1,453,330 | <p>I want to prove that this function is one-to-one</p>
<p>$ f: \mathbb R \rightarrow \mathbb R$ ,
$$f(x) = \left\{\begin{array}{ll}
2x+1 & : x \ge 0\\
x & : x \lt 0
\end{array}
\right.$$
And for this function</p>
<p>$ f: \mathbb R \rightarrow \mathbb R$,
$$f(x) = \left\{\begin{array}{ll}
x^2 & : x \gt... | Community | -1 | <p>For Function 1:</p>
<p>case 1: $f(x) > 0 \implies x \geq 0$</p>
<p>$f(x) = f(y) \implies2x + 1 = 2y + 1 \implies x=y$</p>
<p>case 2: $f(x) < 0 \implies x<0 $</p>
<p>$f(x) = f(y) \implies x=y$</p>
<p>case 3: $x\geq0$ and $y<0$</p>
<p>$x\geq0 \implies f(x) \geq 1$</p>
<p>$y<0 \implies f(y) < 0... |
3,444,463 | <p>Everyone knows that the shortest distance between two points is a straight line. How can we prove it mathematically? </p>
| IamWill | 389,232 | <p>This follows from the <a href="https://en.wikipedia.org/wiki/Euler%E2%80%93Lagrange_equation" rel="nofollow noreferrer">Euler-Lagrange equation</a> from the calculus of variations. Let <span class="math-container">$(a,y(a)),(b,y(b))$</span> be two points in the plane, where <span class="math-container">$y$</span> is... |
639,665 | <p>How can I calculate the inverse of $M$ such that:</p>
<p>$M \in M_{2n}(\mathbb{C})$ and $M = \begin{pmatrix} I_n&iI_n \\iI_n&I_n \end{pmatrix}$, and I find that $\det M = 2^n$. I tried to find the $comM$ and apply $M^{-1} = \frac{1}{2^n} (comM)^T$ but I think it's too complicated.</p>
| gerw | 58,577 | <p>Hint: Compute the inverse of
$$\begin{pmatrix}1 & i \\ i & 1\end{pmatrix}$$
and try the same pattern.</p>
|
639,665 | <p>How can I calculate the inverse of $M$ such that:</p>
<p>$M \in M_{2n}(\mathbb{C})$ and $M = \begin{pmatrix} I_n&iI_n \\iI_n&I_n \end{pmatrix}$, and I find that $\det M = 2^n$. I tried to find the $comM$ and apply $M^{-1} = \frac{1}{2^n} (comM)^T$ but I think it's too complicated.</p>
| Ittay Weiss | 30,953 | <p>A general strategy in such cases is to first compute the result for some small values of $n$, then try to discern a general pattern for the answer, formulate your guess, and attempt to prove it. </p>
|
639,665 | <p>How can I calculate the inverse of $M$ such that:</p>
<p>$M \in M_{2n}(\mathbb{C})$ and $M = \begin{pmatrix} I_n&iI_n \\iI_n&I_n \end{pmatrix}$, and I find that $\det M = 2^n$. I tried to find the $comM$ and apply $M^{-1} = \frac{1}{2^n} (comM)^T$ but I think it's too complicated.</p>
| Woria | 119,309 | <p>I like the @gerw's idea. But if you want a direct way, you can use the method below:</p>
<p>If
$$M^{-1}=\begin{pmatrix}A & B \\ C & D\end{pmatrix}$$
then,
$$\begin{pmatrix}I_{n} & O \\ O & I_{n}\end{pmatrix}=I_{2n}=MM^{-1}=\begin{pmatrix} I_n&iI_n \\iI_n&I_n \end{pmatrix} \begin{pmatrix}A... |
201,358 | <p>I would like to be able to extract elements from a list where indices are specified by a binary mask.</p>
<p>To provide an example, I would like to have a function <code>BinaryIndex</code> doing the following:</p>
<pre><code>foo = {a, b, c}
mask = {True, False, True}
BinaryIndex[mask, foo]
(* expected output *) {a... | Ulrich Neumann | 53,677 | <p>Try </p>
<pre><code>Pick[foo, mask]
(* {a, c}*)
</code></pre>
|
3,403,364 | <p>Here's a little number puzzle question with strange answer:</p>
<blockquote>
<p>In an apartment complex, there is an even number of rooms. Half of the rooms have one occupant, and half have two occupants. How many roommates does the average person in the apartment have? </p>
</blockquote>
<p>My gut instinct was ... | badjohn | 332,763 | <p>Try a very simple example first: 2 rooms. So, 1 room has 1 person and the other has 2. There are 3 people. 1 of those people has 1 room mate. The other 2 people have 1 roommate. So, 2 out of 3 people have a roommate. </p>
|
1,790,311 | <p>Show the following equalities $5 \mathbb{Z} +8= 5\mathbb{Z} +3= 5\mathbb{Z} +(-2)$.</p>
<p>$5 \mathbb{Z} +8=\{5z_{1}+8: z_{1} \in \mathbb{Z}\}$,</p>
<p>$5 \mathbb{Z} +3=\{5z_{2}+3: z_{2} \in \mathbb{Z}\}$,</p>
<p>$5 \mathbb{Z} +(-2)=\{5z_{3}+(-2): z_{3} \in \mathbb{Z}\}$.</p>
<p>So, how can prove to use these de... | Community | -1 | <p>We have $5 \mathbb{Z}+8=5\mathbb{Z}+5+3=5(\mathbb{Z}+1)+3$. So, $\mathbb{Z}+1$ =$\mathbb{Z}$. Thus, $5 \mathbb{Z}+8= 5 \mathbb{Z}+3$.</p>
<p>Now, we will show $5 \mathbb{Z}+3=5 \mathbb{Z}+(-2)$.
$5 \mathbb{Z}+3=$5 $\mathbb{Z}+5-2=5$ $(\mathbb{Z}+1)+(-2)$. Thus,$5 \mathbb{Z}+3=5 \mathbb{Z}+(-2)$.</p>
<p>Therefore, ... |
23,312 | <p>What is the importance of eigenvalues/eigenvectors? </p>
| Arturo Magidin | 742 | <h3>Short Answer</h3>
<p><em>Eigenvectors make understanding linear transformations easy</em>. They are the "axes" (directions) along which a linear transformation acts simply by "stretching/compressing" and/or "flipping"; eigenvalues give you the factors by which this compression occurs. </p>
<p>The more directions ... |
1,693,630 | <p><a href="https://i.stack.imgur.com/TVeGv.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/TVeGv.jpg" alt="enter image description here"></a></p>
<p>This is my attempt at finding $\frac{d^2y}{dx^2}$. Can some one point out where I'm going wrong here?</p>
| Doug M | 317,162 | <p>Update...
I see 2 different mistakes. One in the middle and one at the end.</p>
<p>$\frac{dy}{dx}=\frac{dy}{dt}\frac{dt}{dx} = \frac {2t}{3t^2-12}$</p>
<p>$\frac{d^2y}{dx^2} = \frac{d}{dt}(\frac{dy}{dx})\frac{dx}{dt}$. Looks like you left out the $\frac{dx}{dt}$.</p>
<p>Toward the end, you dropped the exponent.... |
3,739,144 | <p>I have the following proposition proved in my lectures notes, but I think there are a couple of errors and there is one think I don't get:</p>
<p>If <span class="math-container">$p_n$</span> is a Cauchy sequence in a metric space <span class="math-container">$(X,d)$</span>, the set <span class="math-container">$\{p_... | Brian M. Scott | 12,042 | <p>The wording ‘if <span class="math-container">$p_n$</span> has <span class="math-container">$p_0$</span> as a limit point’ is very sloppy, but not for the reason that you suggest at (1). It’s sloppy because <span class="math-container">$p_n$</span> is a single point, not a sequence; what the author means is ‘if <span... |
4,079,711 | <p>Calculate <span class="math-container">$$\iint_D (x^2+y)\mathrm dx\mathrm dy$$</span> where <span class="math-container">$D = \{(x,y)\mid -2 \le x \le 4,\ 5x-1 \le y \le 5x+3\}$</span> by definition.</p>
<hr />
<p>Plotting the set <span class="math-container">$D$</span>, we notice that it is a parallelogram. I tried... | Henno Brandsma | 4,280 | <p>It follows from the universal property for mappings for initial topologies, which I showed and explained <a href="https://math.stackexchange.com/a/1548818/4280">here</a>. The map <span class="math-container">$t \to (f(t), g(t))$</span> has compositions with the two projections that are precisely <span class="math-co... |
3,071,751 | <p>Is there a parametrization for the figure '8' curve, which is self-intersected?</p>
| timtfj | 619,670 | <p>This is an example of a <a href="https://en.wikipedia.org/wiki/Lissajous_curve" rel="nofollow noreferrer">Lissajous figure</a>. (If you've got an oscilloscope with separate <span class="math-container">$x$</span> and <span class="math-container">$y$</span> inputs and a couple of signal generators you can have hours ... |
2,592,282 | <p>If $f$and $g$ is one to one then ${f×g}$ and ${f+g}$ is one to one </p>
<p>Is this true? If not, I need some clarification to understand </p>
<p>Thanks</p>
| fleablood | 280,126 | <p>One to one means that if $f(a) = k$ and $f(b) = k$ then $a=b$.</p>
<p>So if $f(a) + g(a) = k + j = m$ and $f(b) + g(b) = l + h = m$ does it follow that $a = b$ and that $k = l$ and $j = h$? </p>
<p>What if $f(a) = k; g(a) = j$ but $f(b) = j$ and $g(b) = k$. Is that ever possible? Why wouldn't it be?</p>
<p>And... |
1,823,487 | <p>There are $m$ different people and a <strong>circle</strong> that has $m+r$ identical seats. How many ways can we put those people in the circle?</p>
<p>If the seats were not identical then the solution was: $ \frac{1*(m+r-1)!}{r!} $</p>
<p>I can't understand how the fact that the seats are identical affects my s... | Aster rovels | 335,675 | <p>To understand the identical seats consider this</p>
<p>Example: for combination it would be that first select any one of the persons among $m$ then chose a seat for him and because all seats are identical hence there is no restriction so you have to choose from $n$ seats i.e. $^mC_1.^{n}C_1$, then pickup a seat aga... |
3,117,459 | <p>I am interested in approximating the natural logarithm for implementation in an embedded system. I am aware of the Maclaurin series, but it has the issue of only covering numbers in the range (0; 2).</p>
<p>For my application, however, I need to be able to calculate relatively precise results for numbers in the ran... | Arthur | 15,500 | <p>That's basically how a computer might do it. We have
<span class="math-container">$$
\ln (a\cdot 2^b) = \ln(a)+ b\cdot\ln(2)
$$</span>
So <span class="math-container">$\ln(2)$</span> is just a constant that the computer can remember, and <span class="math-container">$b$</span> is an integer, and <span class="math-co... |
3,117,459 | <p>I am interested in approximating the natural logarithm for implementation in an embedded system. I am aware of the Maclaurin series, but it has the issue of only covering numbers in the range (0; 2).</p>
<p>For my application, however, I need to be able to calculate relatively precise results for numbers in the ran... | jmerry | 619,637 | <p>Here's an <a href="https://en.wikipedia.org/wiki/Binary_logarithm#Iterative_approximation" rel="nofollow noreferrer">algorithm</a> for calculating base <span class="math-container">$2$</span> logarithms to whatever floating-point precision you're using. It takes one squaring and some bit operations per bit of precis... |
3,857,071 | <p>I'm doing work with wind direction data, and will be coding a function that checks whether the a given wind direction is bewteen lower and upper limit or bound</p>
<p>e.g.:</p>
<ol>
<li>Is 5 degrees is between 315 degrees and 45 degrees? True</li>
<li>Is 310 degrees between 315 degrees and 45 degrees? False</li>
<li... | Hagen von Eitzen | 39,174 | <p>Is <span class="math-container">$\beta$</span> between <span class="math-container">$\alpha$</span> and <span class="math-container">$\gamma$</span> ?</p>
<p>First, what you decide when <span class="math-container">$\beta=\alpha$</span> or <span class="math-container">$\beta=\gamma$</span> is perhaps up to interpret... |
2,780,216 | <p>How many integer solutions are there to the equation $a + b + c = 21$ if $a \geq3$,
$b \geq 1$, and $c \geq 2b$?</p>
| Robert Israel | 8,508 | <p>If $a = 3+A$, $b =1+B$, $c = 2b + d=2B+d+2$, the equation is $A + 3 B + d = 15$ with $A, B, d \ge 0$. Now for each $B$ from $0$ to $5$,
the number of $(A,d)$ pairs is $16 - 3 B$. So...</p>
|
216,082 | <ol>
<li><p>How to prove that $\mathbb{R}^k$ is connected?</p></li>
<li><p>Let $C$ be an infinite connected set in $\mathbb{R}^k$. How can I show that $C\bigcap \mathbb{Q}^k$ is nonempty?</p></li>
</ol>
| Brian M. Scott | 12,042 | <p>(1) I’ll suggest one of many approaches to proving that $\Bbb R^2$ is connected; it generalizes easily to $\Bbb R^k$. Note that $\Bbb R^2$ is the union of the $x$-axis, which I’ll call $A$, and the straight lines $L_a$ whose equations are $x=a$ for real numbers $a$. Each of these sets is homeomorphic to $\Bbb R$, so... |
1,722,692 | <p>I am asked to find</p>
<p>$$\lim_{x \to 0} \frac{\sqrt{1+x \sin(5x)}-\cos(x)}{\sin^2(x)}$$</p>
<p>and I tried not to use L'Hôpital but it didn't seem to work. After using it, same thing: the fractions just gets bigger and bigger.</p>
<p>Am I missing something here?</p>
<p>The answer is $3$</p>
| Paramanand Singh | 72,031 | <p>The usual / simplest way is to proceed as follows
\begin{align}
L &= \lim_{x \to 0}\frac{\sqrt{1 + x\sin 5x} - \cos x}{\sin^{2}x}\notag\\
&= \lim_{x \to 0}\frac{\sqrt{1 + x\sin 5x} - \cos x}{\sin^{2}x}\cdot\frac{\sqrt{1 + x\sin 5x} + \cos x}{\sqrt{1 + x\sin 5x} + \cos x}\notag\\
&= \lim_{x \to 0}\frac{(1... |
30,305 | <p>I want to call <code>Range[]</code> with its arguments depending on a condition. Say we have </p>
<pre><code>checklength = {5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6}
</code></pre>
<p>I then want to call <code>Range[]</code> 13 times (the length of <code>checklength</code>) and do <code>Range[5]</code> when <code>... | Mr.Wizard | 121 | <p>For your "XXX" expression you need <a href="http://reference.wolfram.com/mathematica/ref/Sequence.html" rel="nofollow noreferrer"><code>Sequence</code></a> or an equivalent. But since <code>If</code> does not have the attribute <a href="http://reference.wolfram.com/mathematica/ref/SequenceHold.html" rel="nofollow n... |
2,922,077 | <p>An urn contains $29$ red, $18$ green, and $12$ yellow balls. Draw two balls without replacement What is the probability that the number of red balls in the sample is exactly $1$ or the number of yellow balls in the sample is exactly $1$ (or both)? What about with replacement? </p>
<p>I can't seem to figure this out... | dxiv | 291,201 | <p>Hint: using the <a href="https://en.wikipedia.org/wiki/Generalized_mean" rel="nofollow noreferrer">generalized mean inequality</a>, for $\,p \ge 1\,$:</p>
<p>$$
\sqrt[p]{\dfrac{|a+b|^p+|a-b|^p}{2}} \ge \dfrac{|a+b|+|a-b|}{2}
$$</p>
<p>What's left to note is that $\,|a+b|+|a-b| \ge 2|b|\,$, which follows fro... |
1,842,989 | <p>This question might be silly, but while teaching a tutorial as a TA, I suddenly had the need to bring up a tautological statement in first order logic that involved the quantifiers $\forall\exists\forall\exists$ in this order. That is, a tautology of the form $\forall x\exists y\forall z\exists w $ $\phi(x,y,z,w)$ f... | user21820 | 21,820 | <p>Try a stronger version of the infinitude of primes!</p>
<blockquote>
<p>Given any natural $m$, there is a least prime bigger than $m$.</p>
</blockquote>
<p>Which can be stated over first-order arithmetic (with $+,\cdot,<,0,1$) as:
$\def\imp{\rightarrow}$</p>
<blockquote>
<p>$φ \overset{def}= \forall m\ \ex... |
1,691,981 | <p>There is a symbolic notation for the set of all eigenvalues
$$\operatorname{spec} \varphi = \lbrace \lambda \in K \mid \lambda \textrm{ is an eigenvalue} \rbrace$$
There is also a notation for the eigenspace
$$V_\lambda = \lbrace \alpha \in V \mid \varphi(\alpha) = \lambda \alpha \rbrace$$
Is there any standard nota... | Ethan Bolker | 72,858 | <p>Not sure this is an answer, too long for a comment.</p>
<p>I'd be surprised if there were a standard notation for the set of eigenvectors, since that set doesn't have nice geometry. (Think about it for the eigenvectors of the matrix with 1, 2, 2, on the diagonal.)</p>
<p>Arguments about eigenvectors usually begin<... |
4,490,664 | <p>I want to examine the number of non-trivial zeros of <span class="math-container">$\zeta(s)$</span> in the small substrip given by <span class="math-container">$a \leq \Re s \leq 1$</span>, where <span class="math-container">$a>0$</span> is an absolute constant, and <span class="math-container">$U \leq \Im s \leq... | LurchiDerLurch | 744,506 | <p>Firstly, it should come as no surprise that results on <span class="math-container">$N(\sigma, T)$</span> only give bounds. After all, any other kind of approximation would disprove RH.</p>
<p>I would not consider myself an expert in the field, but it seems to me as if any kind of formula for <span class="math-conta... |
2,336,744 | <p>Any two co-prime number $a,b$ with $a>b+2$ we have $a^2+b^2$ is not divisible by $a-b$, $a,b \in \mathbb{N}$.
But how to prove this?</p>
| ajotatxe | 132,456 | <p>Since $a-b$ divides $a^2-b^2$, if it also divides $a^2+b^2$ then $a-b$ divides $2b^2$.</p>
<p>Now use that $a$ and $b$ are coprime to see that $a-b$ and $b^2$ are coprime. So by Euclides' lemma $a-b$ divides $2$. This contradicts the condition $a-b>2$.</p>
|
4,642,388 | <p>As an enthusiast in Mathematics, yet aware of this site policy, I am sure enough that this question has the fate to remain closed in the future, yet I have made my mind emotionally that before it occurs, someone will be able to answer or even comment on this work.</p>
<p>In 2022, I published a book named <a href="ht... | Primo Petri | 137,248 | <p>To expand a first-order theory to a second-order theory conservatively (i.e. without adding new first-order definable sets) you have to add a sort for every "definition-scheme". I.e. a sort for every partitioned formula <span class="math-container">$\varphi(x;y)$</span>.</p>
<p>The canonical expansion of a... |
3,072,995 | <p>The only thing I know with this equation is <span class="math-container">$y=\frac{x^2+1}{x+1}=x+1-\frac{2x}{x+1}$</span>.</p>
<p>Maybe it can be solved by using inequality.</p>
| Mostafa Ayaz | 518,023 | <p>The equation <span class="math-container">$$y={x^2+1\over x+1}=a\to x^2-ax-(a-1)=0$$</span>for <span class="math-container">$x\ne -1$</span> has a real root if and only if <span class="math-container">$$\Delta\ge0 \iff a^2+4a-4\ge 0\iff a\ge 2\sqrt 2-2\text{ or }a\le -2\sqrt 2-2$$</span>in that case, the root is <sp... |
3,319,122 | <p>This is from Tao's Analysis I: </p>
<p><a href="https://i.stack.imgur.com/DYQxE.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/DYQxE.png" alt="enter image description here"></a></p>
<p>So far I managed to show (inductively) that these sets do exist for for every <span class="math-container">$\m... | Kavi Rama Murthy | 142,385 | <p>Interpret the function as <span class="math-container">$1$</span> when <span class="math-container">$x=0$</span>. Claim: <span class="math-container">$(1-x)e^{x} \leq 1$</span> for all <span class="math-container">$x$</span>. Use the fact that <span class="math-container">$e^{-x} \geq 1-x$</span> for all real <span... |
3,398,645 | <p>I have a doubt about value of <span class="math-container">$e^{z}$</span> at <span class="math-container">$\infty$</span> in one of my book they are mentioning that as <span class="math-container">$\lim_{z \to \infty} e^z \to \infty $</span></p>
<p>But in another book they are saying it doesn't exist.I am confused... | edm | 356,114 | <p><span class="math-container">$e^z$</span> has an essential singularity at <span class="math-container">$z=\infty$</span> so we would not say <span class="math-container">$\lim\limits_{z\to\infty}e^z=\infty$</span>, but instead we say the limit does not exist.</p>
|
2,701,582 | <p>I thought I was doing this right until I checked my answer online and got a different one. I worked through the problem again and got my original answer a second time so this one is bothering me since the other similar ones I have done checked out okay. Please let me know if I'm doing something wrong, thanks!</p>
<... | anon | 11,763 | <p>Generating sets can be very small. So small, in fact, that the number of subsets of $A$ can be dwarfed by the number of subgroups of $G$. As others have pointed out, the symmetric group $S_n$ is generated by $2$ elements.</p>
<p>Here's a family of abelian examples inspired by geometry. As a vector space, we know $\... |
3,450,908 | <p>I know how to solve problems, but I don't understand the core theory that sticks these two functions with this special relation. </p>
<p>Could you help me find the core idea that I need to understand in order to be satisfied with my understanding? </p>
| GEdgar | 442 | <p>The values of <span class="math-container">$\exp x$</span> are positive. The domain of <span class="math-container">$\log x$</span> is the positive numbers.<br>
If <span class="math-container">$x$</span> is a real number, then <span class="math-container">$\log(\exp(x)) = x$</span>.<br>
If <span class="math-contain... |
623,796 | <p>What's the domain of the function $f(x) = \sqrt{x^2 - 4x - 5}$ ?</p>
<p>Thanks in advance.</p>
| Community | -1 | <p>The function $x\mapsto\sqrt x$ is defined on the interval $[0,+\infty)$ so the given function $f$ is defined for $x$ such that $x^2-4x-5\ge0$.</p>
<p>The reduced discriminant $\Delta'=9$ hence the roots are: $x_1=-1$ and $x_2=5$ and then $f$ is defined on $(-\infty,-1]\cup[5,+\infty)$.</p>
|
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