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 |
|---|---|---|---|---|
4,365,119 | <blockquote>
<p>If <span class="math-container">$\alpha,\beta,\gamma$</span> are the roots of the equation <span class="math-container">$x^3+x^2-x+1=0$</span>, find the value of <span class="math-container">$\prod\left(\frac1{\alpha^3}+\frac1{\beta^3}-\frac1{\gamma^3}\right)$</span></p>
</blockquote>
<p><strong>My Atte... | Sangchul Lee | 9,340 | <p><strong>1.</strong> Let <span class="math-container">$P(x) = x^3 + x^2 - x + 1$</span>. Since all the zeros of <span class="math-container">$P(x)$</span> are non-zero, <span class="math-container">$\gcd(P(x), x^3) = 1$</span>. Then the <a href="https://en.wikipedia.org/wiki/B%C3%A9zout%27s_identity#For_polynomials" ... |
1,764,921 | <p>I have to find the dual of the lp problem given below
Minimise $$z=-x_1+\frac43 x_2$$
subject to∶
$$\begin{array}[t]{l}
2x_1+4x_2\le16\\
-\frac{1}2 x_1-x_2\le4\\
-3x_1+4x_2\ge-24\\
x_1≥0,x_2≤0
\end{array}
$$</p>
<p>Also find the solution to the dual problem without solving it.</p>
| Math1000 | 38,584 | <p>By evaluating the objective function at each BFS of the primal, we see that $(x_1^\star,x_2^\star)=\left(\frac85,-\frac{24}5\right)$ is an optimal solution, with objective value $-8$. Consider the dual problem:
\begin{align}
\max & \quad 16y_1 +4y_2-24y_3\\
\mathrm{s.t.} &\quad 2y_1-\frac12 y_2-3y_3\geqslant... |
893,406 | <p>I have been given this example 2.0013−2.0005=0.0008. The destructive cancellation is the large common value, here the 2, that disappears. </p>
| Michael Hardy | 11,667 | <p>$$
\begin{align}
2.314890251 - 2.314890254 & = -0.000000003 \\[8pt]
2.314890251 - 2.314890253 & = -0.000000002
\end{align}
$$
One of these is $3\times\text{something}$ and the other is $2\times\text{something}$, where "something" is the same in both cases. For some purposes the ratio of $3$ to $2$ might be ... |
538,444 | <p><img src="https://i.stack.imgur.com/mBZOe.jpg" alt="enter image description here"></p>
<p>I'm studying and I'm not that sure how to answer this question. </p>
<p>Is $97.1%$ $=$ $650,000,000$?</p>
<p>I was going to find $2.9%$ of $650,000,000$ however this would be wrong as I would finding our the annum of the $20... | amWhy | 9,003 | <p>Determine the order of $G: \,|G|$, and the orders of $H$ in each case; then you can easily calculate the index of $H$ in $G$: $$[G:H] = \dfrac{|G|}{|H|}$$</p>
<hr>
<p>Now, since the order of a group is the number of elements it contains, and the order of the direct product of two sets is given by the <em>product o... |
1,534,322 | <p>I have a question that in short reads:</p>
<blockquote>
<p>If some element decays exponentially with a half life of 25,000 years, how long will it take for 99.9% of it to decay away?</p>
</blockquote>
<p>Using the exponential decay formula: $P(t)=P_0e^{-kt}$ where $k$ is the half life, $t$ is time, $P_0$ is the... | user247327 | 247,327 | <p>Since $\frac{\partial v}{\partial y}= 4x+ 2$, $v(x,y)= 4xy+ 2y+ \phi(x)$ where $\phi(x)$ is an arbitrary function of x (since partial differentiation with respect to y treats x as a constant, the derivative of any function of x only, with respect to y, is 0). From that $\frac{\partial v}{\partial x}= 4y+ \phi'(x)=... |
1,534,322 | <p>I have a question that in short reads:</p>
<blockquote>
<p>If some element decays exponentially with a half life of 25,000 years, how long will it take for 99.9% of it to decay away?</p>
</blockquote>
<p>Using the exponential decay formula: $P(t)=P_0e^{-kt}$ where $k$ is the half life, $t$ is time, $P_0$ is the... | user114403 | 376,623 | <p>There is a short cut method: Replace x by z=x+iy and y by 0 in u(x, y) and add a purely imaginary constant.The resulting function is the analytic function for which u(x, y) is the real part. Thus f(z) =u(z, 0)++iC</p>
|
3,286,415 | <p>Solve for the angle <span class="math-container">$x$</span> in the right triangle without trigonometry.</p>
<p><a href="https://i.stack.imgur.com/swWdP.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/swWdP.jpg" alt="enter image description here"></a></p>
<p>I don't know how to find the angle <sp... | Michael Rozenberg | 190,319 | <p>Let <span class="math-container">$QM$</span> be an altitude of <span class="math-container">$\Delta PQB$</span>.</p>
<p>Thus, <span class="math-container">$$MQ=QC=\frac{1}{2}PQ,$$</span> <span class="math-container">$$\measuredangle MPQ=30^{\circ},$$</span>
<span class="math-container">$$\measuredangle PAQ=15^{\cir... |
77,306 | <p>I'm not entirely sure what I want to ask here, so please bear with me!</p>
<p>I think the explanation they give us in school for how finding the antiderivative by substitution works is: $$\int f(g(t))g'(t)dt=(Fg)(t)=F(g(t))=F(u)=\int f(u)du$$</p>
<p>But I never really understood the equality $F(u)=\int f(u)du$... | Mariano Suárez-Álvarez | 274 | <p>The «substitution rule» works precisely because the chain rule is true for derivatives.</p>
|
607,403 | <p>Find,</p>
<p>$$x^2y^2-2xy+x=2$$</p>
<p>find the slope and the equation of the tangent line at $(2,0)$ by implicit differentiation.</p>
<hr>
<p>I have gotten this question wrong on a quiz and exam. I'm trying to understand it for my final. Part of the problem is that my text book says find $d/dx$ and then ignore ... | Cyriac Antony | 120,721 | <p>For your information, to ignore $dy/dx$ is a horrible advice.
Use a reliable textbook. You should be solving for $dy/dx$ instead.</p>
<p>[What you have to do is to bring terms containing $dy/dx$ on one side (say LHS) and other terms on the other side. Take $dy/dx$ as common factor on LHS. Divide the equation by wha... |
3,554,502 | <blockquote>
<p>Two players, A and B, play the following backgammon variant.</p>
<p>A series of steps is labelled <span class="math-container">$-2$</span>, <span class="math-container">$-1$</span>, <span class="math-container">$0$</span>, <span class="math-container">$1$</span> and <span class="math-container">$... | quarague | 169,704 | <p>One can analyse the expected values at each stage. First the base case without doubling, at stage 0, player A has an expected return of <span class="math-container">$0\$$</span> due to symmetry. At stage <span class="math-container">$1$</span>, he has a <span class="math-container">$50\%$</span> chance to win immedi... |
242,892 | <p>Let's take two independent, random variables $X$ and $Y$ from set $\{1,\ldots,n\}$
probability, that $X =i$ AND $Y = j$ is $\frac{1}{n^2}$.
I want to determine the expected value, so I started:
$$E[Z] = E[|X-Y|] = \sum_i (a_i) \cdot P(|X-Y|=a_i),$$ (sum is to $n$) but I don't now how to count $P(|X-Y|=a_i)$, cause ... | André Nicolas | 6,312 | <p>Draw the $n\times n$ grid of points with coordinates $(i,j)$, where $i$ and $j$ range from $1$ to $n$. The points $(x,y)$
such that $x-y=k$ are the points on the <strong>line</strong> $x-y=k$. For $k=0$ there are $n$ such points, the points on the diagonal. For $x-y=1$, there are $n-1$ points, for $x-y=2$ there ar... |
2,510,496 | <p>Let $\mathcal{H}$ be a complex Hilbert space. Let $T\in \mathcal{B}(\mathcal{H})$ and let $A\in \mathcal{B}(\mathcal{H})^+$ (i.e. $A^*=A$ and $\langle Ax\;| \;x\rangle \geq0,\;\forall x\in \mathcal{H}$).</p>
<blockquote>
<p>Assume that $AT=TA$. Why $A^{1/2}T=TA^{1/2}$?</p>
</blockquote>
<p>Thank you</p>
| C. Ding | 320,080 | <p>You can check that $p(A)T=Tp(A)$ for any polynomial $p$.</p>
<p>For any continuous function $f$ on $\sigma(A)$, there is a sequence $\{p_n\}$ of polynomials such that $p_n$ is convergent to $f$ uniformly. Hence $p_n(A)$ is convergent to $f(A)$ and thus $f(A)T=Tf(A)$. </p>
<p>Specially, $f(x)=\sqrt{x}, x\in \sigma... |
2,393,044 | <p>For two discrete random variables $X,Y$, it is known that their covariance is equal to $\text{Cov}(X,Y) = E(XY) - E(X)E(Y)$. Define a "variant" of the total variation distance in the following manner:
$$ \delta(X,Y) = \sup_{\alpha, \beta} \{| P(X = \alpha \text{ and } Y = \beta) - P(X = \alpha)P(Y = \beta)| .\} $$... | drhab | 75,923 | <p>Take $X_n=Y_n$ together with probability $P(X_n=\frac{k}{n})=\frac1{n}$ for $k=1,\dots,n$.</p>
<p>Then $\delta(X_n,Y_n)=|\frac1{n}-\frac1{n^2}|$ so it will approach $0$ if we let $n$ grow.</p>
<p>$\text{Cov}(X_n,Y_n)=\text{Cov}(X_n,X_n)=\text{Var}(X_n)$ and will approach the positive variance of standard uniform ... |
2,575,760 | <p>$g(x,y)=f(x^3+3y+2), f:R \rightarrow R$, directional derivative in the direction of the vector $(1,0)$ at the point $(0,0)$</p>
<p>My attempt:</p>
<p>I'm not sure how to prove this is zero?
Using the definition I got to $\lim_{h \rightarrow 0} \frac{f(h^3+2)-f(2)}{h}$ but I dont know how to compute that.
I'm aware... | Tsemo Aristide | 280,301 | <p>Consider $h(x,y)=x^3+3y+2$, $dh=3xdx+3dy$ implies that $d(f\circ h)_{(x,y)}(1,0)=f'(3x^2+2y+2)(3x)$ and if $(x,y)=(0,0)$ the result is $0$.</p>
|
3,138,158 | <p>How can I show that if the cube power of a matrix is the null matrix, then the matrix itself is singular?</p>
| pokemonfan | 274,554 | <p>A determinant free solution could be more enlightening. Note that if <span class="math-container">$A^3=0$</span> that then <span class="math-container">$im A^2 \subset ker A$</span> now there are two options. Either <span class="math-container">$im A^2=0$</span> and A is non-singular or not and then <span class="mat... |
508,522 | <p>Take a list of the first $n$ primes $P_n=\{2,3,5,7,11,\ldots\}$ and convert the sequence into a binary string</p>
<p>$$S_n = 101110111\ldots$$</p>
<p>Compress the string with your favorite compression algorithm (the details don't matter) and let the size of the string after compression be $C(S_n)$. Compare this to... | Ted | 15,012 | <p>Primes are always odd, so the last bit is always 1 (except for the initial prime 2, of course). So there is definitely some redundancy.</p>
<p>Another consideration is that your description scheme (listing all the bits of every number) is not optimal. To describe <em>any</em> subset of $\{1,2,\ldots,n\}$ takes at... |
1,324,884 | <p><strong>Def:</strong> A set $S$ in the Euclidean space $ \mathbb R^n $ is called a star shaped domain if there exists $x_o$ in $S$ such that for all $x$ in $S$ the line segment from $x_0$ to $x$ is in $S$.</p>
<p>From my last question asked on MSE it turned out that boundary of a star shaped domain may not be Path ... | hmakholm left over Monica | 14,366 | <p>Limits are about how functions behave <em>close to but NOT at</em> the chosen point. When you speak about $\lim_{x\to 0} f(x)$ the information you get is about what $f(x)$ does for <em>small nonzero values</em> of $x$. The limit doesn't care about what $f(0)$ or even whether $f(0)$ has a defined value.</p>
<p>It is... |
3,380,002 | <p>we need to find a mapping of those points onto <span class="math-container">$\mathbf{R}^2$</span> so that the two colors classes are can be separated by a line?</p>
<p>So I don't know the radius of those circles. But if I knew, I would take their norm. However that would be a mapping to <span class="math-container"... | Matteo Spadetto | 531,071 | <p>The algebraic closure <span class="math-container">$k$</span> of the field <span class="math-container">$\mathbb{Z}/2\mathbb{Z}$</span> can be regarded as a <span class="math-container">$\mathbb{Z}/2\mathbb{Z}$</span>-linear space. Then <span class="math-container">$k$</span> happens to be infinite dimensional: if <... |
2,107,874 | <p>What are the other angles in triangle and what is $\tan(\frac{\beta-\gamma}{2})$, if $\alpha = 60$ and $\frac{b}{c}=2+\sqrt3$?
and
I tried to solve it using law of sines and law of cosines but I couldn't find answer.</p>
| lab bhattacharjee | 33,337 | <p>Let $\dfrac{\beta-\alpha}2=x$ and we have $\beta+\alpha=120^\circ$</p>
<p>As $0<\beta,\alpha<\pi,-\dfrac\pi2<x<\dfrac\pi2$</p>
<p>$\beta=60^\circ+x,\alpha=60^\circ-x$</p>
<p>$$\dfrac{2+\sqrt3}1=\dfrac{\sin(60^\circ+x)}{\sin(60^\circ-x)}$$</p>
<p>Using <a href="https://brilliant.org/wiki/componendo-an... |
135,863 | <p>There are $3$ production lines in a factory. $2$ of them are new lines and $1$ of them is an old line. All production lines produce at the same rate. The old production line has $8\%$ defective rate while the new production lines have only $3\%$ defective rate. The components are shipped to customers in $100$-unit l... | Julián Aguirre | 4,791 | <p>On $\mathbb{R}^2$ $x\,y$ and $x^2-y^2$ are examples of harmonic functions that are neither concave nor convex.</p>
<p>If $u$ is harmonic on an open set $\Omega\subset\mathbb{R}^n$, then it verifies the mean value property:
$$
u(x)=\frac{1}{|B_R(x)|}\int_{B_R(x)}u(y)\,dy,\quad B_R(x)\subset\Omega
$$
where $B_R(x)$ i... |
19,124 | <p>To most people, a torus is a donut-like shape. Topologists like to describe the torus differently: you start with a square, and "identify opposite sides". We can imagine gluing together one pair of opposite sides to get a cylinder, and then gluing together opposite ends of the cylinder to get a torus. (Pro... | Eric Towers | 5,800 | <p>I have (once) used a square of felt with two strips of velcro glued to the felt. Then just "identify" one opposing pair of edges in an orientation preserving way via velcro. Demonstrate that one obtains a finite cylinder. Then "identify" the two circle edges in an orientation preserving way.</... |
4,443,170 | <p>Let <span class="math-container">$I$</span> be an uncountable set and let <span class="math-container">$2^I=\{0, 1\}^I$</span> be the set of all functions from <span class="math-container">$I$</span> into <span class="math-container">$\{0, 1\}$</span>. Consider the product measure <span class="math-container">$\mu$<... | eyeballfrog | 395,748 | <p>Suppose I have two solutions <span class="math-container">$(x_1,y_1)$</span> and <span class="math-container">$(x_2,y_2)$</span>. They must satisfy
<span class="math-container">$$
8x_1 = 29 + 3 y_1 \\
8x_2 = 29 + 3 y_2
$$</span>
Now subtract these equations to get
<span class="math-container">$$
8(x_1 - x_2) = 3(y_1... |
4,443,170 | <p>Let <span class="math-container">$I$</span> be an uncountable set and let <span class="math-container">$2^I=\{0, 1\}^I$</span> be the set of all functions from <span class="math-container">$I$</span> into <span class="math-container">$\{0, 1\}$</span>. Consider the product measure <span class="math-container">$\mu$<... | hpbhpb | 734,001 | <p><span class="math-container">$$x=8^{-1}\cdot 29=2^{-1}\cdot 2=2\cdot 2=4=1 (\mod 3).$$</span></p>
<p><span class="math-container">$$x=3n+1.$$</span></p>
<p><span class="math-container">$$8(3n+1)=29+3y \Leftrightarrow y=8n-7.$$</span></p>
|
3,289,226 | <p>I am unsure on how to prove this statement. I am relatively certain that it is true, any advice on how to start?</p>
<p>Edit: It has occurred to me that this will be significantly easier to show by restricting the simplex to be regular (and may be only true in this case).</p>
| JonathanZ supports MonicaC | 275,313 | <p>[Note: Some of my <span class="math-container">$n$</span>s are off by one because the simplexes I was thinking about weren't the standard simplexes. I don't think that changes the core idea though, and this is only a sketch of an idea.]</p>
<p>I haven't worked this all the way out, but it's got an <span class="math... |
81,079 | <p>Some time ago I met a forester. He told that there are only larches and spruces in his forest.
He also said that there are exactly $10$ spruces at the distance of exactly 1 km from each larch. Next, he argued that there are more larches than spruces in his forest. Can he be right?</p>
| Marc van Leeuwen | 18,880 | <p>Nice number theoretic question in disguise! Not being much of a botanist I'll
call larches X and spruces Y, or I won't be able to tell them apart. Also I
take "argued" in the question to mean "asserted", since one can obviously not
argue that there <em>must</em> be more larches than spruces, but there <em>can</em> b... |
1,757,914 | <p>My instructor proved the central limit theorem using the characteristic function. I think the proof is a standard one because I found basically the same proof in wikipedia.</p>
<p>So for i.i.d. ${X_1, X_2,\cdots, X_n}$ with $E[X_i]=\mu$ and $\text{Var}(X_i)=\sigma^2<+\infty$, he then defined
$$Y_i=\frac{X_i-\mu}... | Anthony Carapetis | 28,513 | <p>I would say your formula is correct.</p>
<p>Some might argue that the "correct" area of the Möbius strip is twice that value (i.e. with $\theta$ going all the way up to $4 \pi$) - the non-orientability makes things a little awkward. In particular if you use the language that a non-twisted strip has "two sides" whil... |
3,372,482 | <p>Looking at some of the more common pythagorean triples, I noticed a trend that there are usually two numbers which differ by only one or two, with more much smaller or larger number in comparison. For example:</p>
<p><span class="math-container">$$5,12,13$$</span>
<span class="math-container">$$8,15,17$$</span>
<sp... | poetasis | 546,655 | <p>The well-known Euclid's formula is <span class="math-container">$A=m^2-n^2\quad B=2mn\quad C=m^2+n^2$</span> and generates trivial triples, all primitives, doubles and square multiples of primitives but in a seemingly random pattern. A variation replaces <span class="math-container">$(m,n)$</span> with <span class="... |
287,516 | <p>I need to calculate value of $1000^{th}$ derivate of the following function at $0$:</p>
<p>$$
f(x) = \frac{x+1}{(x-1)(x-2)}
$$</p>
<p>I've done similar problems before (e.g. $f(x)= \dfrac{x}{e^{x}}$) but the approach I've used would not work in this case and I believe I should expand this function into a power ser... | Brian M. Scott | 12,042 | <p>Starting with the partial fraction decomposition $$f(x)=\frac{3}{x-2}-\frac{2}{x-1}\;,$$ use the geometric series sum to write </p>
<p>$$\begin{align*}
f(x)&=-\frac32\cdot\frac1{1-\frac{x}2}+2\frac1{1-x}\\\\
&=2\sum_{n\ge 0}x^n-\frac32\sum_{n\ge 0}\left(\frac{x}2\right)^n\\\\
&=\sum_{n\ge 0}\left(2-\fra... |
80,265 | <p>Let $\alpha$ be a root of the polynomial $X^3-X-1$. The polynomial has discriminant $-23$ which is not a square, so the splitting field must have Galois group $S_3$. I would like to figure out the splitting of primes in $\mathbb{Q}(\alpha)$. The ramified case is trivial, so assume that $p$ is not ramified. By lookin... | David E Speyer | 448 | <p>As Qiaochu says, it depends what you mean by explicit. Here is another sort-of-explicit answer.</p>
<p>Let $K$ be the splitting field of $x^3-x-1$. As you already realized, this is an $S_3$-extension of $\mathbb{Q}$, and the quadratic subfield is $\mathbb{Q}(\sqrt{-23})$. Let $p$ be a rational prime other than $23$... |
842,911 | <p>What is the following product in terms of $n$?</p>
<p>$$\frac{1}{2} \cdot \frac{3}{4} \cdot \frac{5}{6} \cdot \frac{7}{8} \cdot \cdots \cdot \frac{2n-1}{2n}$$</p>
<p>Thank you.</p>
| johannesvalks | 155,865 | <p>Given</p>
<p>$$
(1/2)(3/4)(5/6)(7/8)\cdots([2n-1]/2n)
$$</p>
<p>Thus</p>
<p>$$
\frac{1}{2} \frac{3}{4} \frac{5}{6} \cdots \frac{2n-1}{2n} = \frac{1 \cdot 2 \cdot 3 \cdot 4 \cdots [2n-1] \cdot 2n}{2^{2n} \cdot (1 \cdot 2 \cdot 3 \cdot 4 \cdots 2n)^2 } = \frac{[2n]!}{2^n (n!)^2}
$$</p>
|
1,229,488 | <p>I'm not sure where to begin on this problem. Do you begin by solving the integral of each one then applying the conditions to each respective integral? Any help would be appreciated.</p>
| user46372819 | 86,425 | <p>You need to integrate $f'''(x)$, then solve for the constant with the given value of $f''$. Then you repeat until you arrive at $f$.</p>
<p>$$f''(x)=\int f'''(x) \ dx =\int \sin(x) \ dx=-\cos(x) + c_1$$
$$f''(0)=-\cos(0)+c_1=-9$$
$$c_1=-8$$
$$f''(x)=-\cos(x)-8=-\big(\cos(x)+8\big)$$</p>
<p>Now repeat with $f''$.</... |
733,140 | <p>Suppose 26 athletes (Alice, Bob, Charlie, Derek.....Zak) run a race and each ordering of the athletes is equally likely. What's the probability that Alice finishes before Bob, and Bob finishes before Charlie. </p>
<p>For this I thought the answer was 1/4 because I thought that there are half of the possibilites whe... | robjohn | 13,854 | <p><strong>Hint:</strong> Show
$$
\mathrm{E}[X]\le a\mathrm{P}[X\le a]+\max(X)\mathrm{P}[X\gt a]
$$</p>
|
627,620 | <p>I am wondering if you can compute the SVD/eigenvectors of a rank r matrix of size n*n in time O(n^2 r)? My understanding is that standard eigenvector computations involve bringing matrix into Hesseberg form which costs O(n^3) operations.</p>
<p>Thanks.</p>
| Community | -1 | <p>In one sense, yes; the tails of all vectors in a vector space are at the same point.</p>
<p>There is a related notion of an <em>affine space</em>, which is more or less the same thing as a vector space, but vectors can have tails at any point. We get a vector space by picking a point (it can be any point) and call ... |
1,174,463 | <p><span class="math-container">$$\int x^3 \cos(2x^2-5)dx$$</span></p>
<p>I tried using the substitution method but to no avail.</p>
<p>I also tried integrating by parts but that didn't work either.</p>
| gt6989b | 16,192 | <p><strong>HINT</strong> Try by parts with $u = x^2$ and $dv = x \cos(\ldots) dx$, and then by parts again should take care of it.</p>
|
2,631,937 | <p>I understand the Divergence theorem is as follows, </p>
<p>$\iiint\limits_{v} (\nabla \bullet \textbf{F}) \;dV= \iint\limits_{dv} \textbf{F} \bullet d \textbf{S}$</p>
<p>My query is if i already know $(\nabla \bullet \textbf{F})= 0$ then can i assume the following,</p>
<p>$(\nabla \bullet \textbf{F})\iiint\limits... | A Slow Learner | 435,219 | <p>No.</p>
<p>Let's look at your equation:</p>
<p>$$
\nabla \cdot \mathbf F \iiint_vdV=\iint_{dv}\mathbf F\cdot dS
$$</p>
<p>Putting the meaning of the divergence theorem aside, let's just see the values of the LHS and RHS in your equation. The LHS is a function because $\nabla\cdot \mathbf F$ is a function of $(x,y... |
256,723 | <p>When are Noetherian conditions on a scheme genuinely essential in algebraic geometry? </p>
<p>I am under the impression that most of the time these conditions are imposed for expositional clarity and to simplify the commutative algebra involved (for example throughout Hartshorne and FGA), and that they can be remo... | Beren Sanders | 1,148 | <p>Just to give one example: There are certain questions about the derived category $D(R)$ of a commutative ring $R$ where noetherian assumptions can really result in different behaviour. For example, Neeman proved that if $R$ is noetherian then the Bousfield classes of $D(R)$ correspond to arbitrary subsets of the spe... |
406,255 | <p>$ \int_ {-\pi}^{\pi}\ f(x) \ dx $ is equivalent to integrating $f(x)$ over $-\pi < x < \pi$ (strictly less than) and also $-\pi \le x \le \pi$ (less than or equal). So I just need to reassure myself that the bounds of integration don't change when considering either less than or less than or equal. A quick co... | Jon Claus | 77,104 | <p>No, the bounds are not included which is crucial when evaluating improper integrals. For example, if $ f(x) = \sqrt{\left|\tan\left(\frac{x}{2}\right)\right|} $, then the integral converges despite the integrand not being defined at the bounds. </p>
|
76,832 | <p>I have a complex <code>Manipulate-</code>based demonstration containing a few sliders and a Setterbar. </p>
<p>I would like to make a movie out of it, avi, or whatever else. The movie will then be used for a journal publication as a Supporting material, so the quality should be good enough. </p>
<p>When the movie... | Kuba | 5,478 | <p>This is just an idea how to prepare frames to export, don't have time for more now:</p>
<pre><code>f = Interpolation[
{
{0, {1, 0, 0}},
{1, {2, 0, 0}},
{2, {2, 1, 0}},
{2.02, {1, 0, .5}},
{3, {1, 0, .5}},
{4, {2, 0, .5}},
{5, {2, 1, .5}},
{5.02, {1, 0, 1}},
{6, {1, 0, 1}},
{7, {2, 0,... |
76,832 | <p>I have a complex <code>Manipulate-</code>based demonstration containing a few sliders and a Setterbar. </p>
<p>I would like to make a movie out of it, avi, or whatever else. The movie will then be used for a journal publication as a Supporting material, so the quality should be good enough. </p>
<p>When the movie... | Michael E2 | 4,999 | <p>I wrote something to do something a little like what you wanted. Here I've adapted it so you can get a run through for each setting of the <code>SetterBar</code>. Some description of the functions.</p>
<p><code>autolist[control_pattern] :=</code> list of manipulate settings for the animation</p>
<p><code>specs</... |
222,114 | <p>It is known (via Kotschwar's uniqueness of backwards Ricci flows) that the isometry group of a Riemannian metric remains unchanged under the Ricci flow. But, one can easily observe that it can change at the limit. For example, one can perturb a sphere $S^2$ slightly so that it has no symmetry. As long as the Ricci f... | Igor Belegradek | 1,573 | <p>I think the phenomenon is much more general: If a sequence of metrics $d_i$ on a compact metric space $X$ converges (pointwise on $X\times X$) to a metric $d$, and if $h$ lies in the intersection of the isometry groups of $(X, d_i)$, then clearly $h$ is an isometry of $(X,d)$. </p>
|
4,076,273 | <p>Given a one dimensional random walk where I have <span class="math-container">$p$</span> probability to go forward and <span class="math-container">$q$</span> probability to go backwards.</p>
<p>I have to prove that, starting from the origin, the probability of reaching <span class="math-container">$a>0$</span> b... | Noether | 721,023 | <p>After what Ian's response.</p>
<p>Ian (<a href="https://math.stackexchange.com/users/83396/ian">https://math.stackexchange.com/users/83396/ian</a>), Probability to reach <span class="math-container">$a>0$</span> before returning to the origin in a one dimensional random walk., URL (version: 2021-03-25): <a href="... |
4,512,690 | <p>I am having problems in evaluating the following Cauchy principal value</p>
<p><span class="math-container">$$\int_{-\infty}^{+\infty} \frac{\cos(3x)}{x-1}d x$$</span></p>
<p>I know it is supposed to show my work, but I got barely nothing. I tried with using <span class="math-container">$\cos(x) = (z + z^{-1})/2$</s... | Enrico M. | 266,764 | <p>I am unable to draw the contour lines in the books-fancy way, so I will avoid it.</p>
<p>Let's proceed by considering the following integral instead:</p>
<p><span class="math-container">$$\int \frac{e^{3iz}}{z-1}\ \text{d}z$$</span></p>
<p>steppint into the complex plane, remembering that <span class="math-container... |
3,333,655 | <p>Mount Kea is a dormant volcano on the island of Hawaii, and also the highest seamount in the world as measured from its base on the oceanic plate. At an unknown distance along with the oceanic plate from the peak of Mauna Kea, the angle of elevation to the peak is approximately <span class="math-container">$45.29^\c... | Con | 682,304 | <p>The first thing you missed was that <span class="math-container">$h$</span> was in the denominator such that you will get the power <span class="math-container">$\frac{1}{h}$</span> in the logarithm (which was fixed in the first line by an edit now and still is not fixed later on). Afterwards you let <span class="ma... |
3,333,655 | <p>Mount Kea is a dormant volcano on the island of Hawaii, and also the highest seamount in the world as measured from its base on the oceanic plate. At an unknown distance along with the oceanic plate from the peak of Mauna Kea, the angle of elevation to the peak is approximately <span class="math-container">$45.29^\c... | J. W. Tanner | 615,567 | <p><span class="math-container">$\lim\limits_{h\to0}\dfrac{\ln(x+h)-\ln x}{h}=\lim\limits_{h\to0}\dfrac{\ln\left(1+\dfrac hx\right)}h=\lim\limits_{h\to0}\dfrac{\dfrac hx+o(h)}h=\dfrac 1x$</span></p>
|
2,551,410 | <p>Cardinals have a well-defined arithmetic, but I don't have enough background knowledge to check if this notion makes sense.</p>
| jgon | 90,543 | <p>I'm going to say no. $\newcommand{\RR}{\mathbb{R}}\RR(\alpha)$ has two possible meanings that I can think of depending on context. If $\alpha$ is an element of an (already existing) extension field $K$ of $\RR$, then $\RR(\alpha)$ is the smallest subfield of $K$ containing both $\RR$ and $\alpha$. If $\alpha$ is an ... |
1,745,023 | <p><img src="https://i.stack.imgur.com/qtk0E.jpg" alt="i want to know the way of solving (a) " /></p>
<p>If I found the sol of (a), I could find sol of (b) and (c) by using Hensel's lemma.</p>
<p>I want to know the way of solving (a) without testing all the cases (ex testing 1,2,3,4,...10)
I think that there will be ot... | aras | 277,932 | <p>You can't because that statement is not always true. Counterexample: If $B = \{ (1,0,0), (0,1,0), (0,0,1) \}$, then $A = \{ (1,1,0), (1,0,1), (0,1,1) \}$ is a set of linear combinations of the elements of $B$, but $A$ is linearly independent. </p>
|
183,031 | <p>I'm reading a book on analytic number theory. It asks me to prove:</p>
<blockquote>
<p>$$ \sum_p \frac{\log p}{p^s} = \frac{1}{s-1} + O(1) \tag{A}$$
and conclude, via integration, that:
$$ \sum_p \frac{1}{p^s} = \log \frac{1}{s-1} + O(1) \tag{B}$$</p>
</blockquote>
<p>Now, I know how to prove $(A)$ via Abe... | anon | 11,763 | <p>The first idea I had when you posted in chat was to use infinite Möbius inversion. We have</p>
<p>$$P(s)=\sum_p\frac{1}{p^s}=\sum_{n\ge1}\frac{\mu(n)}{n}\log\zeta(ns), $$</p>
<p>where $P(\cdot)$ is the prime zeta function and $\zeta(\cdot)$ is the Riemann zeta function. (For those who want to check this: write $\z... |
1,458,055 | <p>What i want to prove is following integral
\begin{align}
\int_{-\infty}^{\infty} e^{-x^2} \cos(x^2) dx=\frac{1}{2} \sqrt{\left(1+\sqrt{2}\right) \pi }
\end{align}
can you give some explicit method to obtain this result?</p>
| Jack D'Aurizio | 44,121 | <p>By using a substitution, $\cos\theta = \text{Re}(e^{i\theta})$ and the (direct and inverse) Laplace transform:
$$\int_{0}^{+\infty}e^{-x^2}\cos(x^2)\,dx = \frac{1}{2}\cdot\text{Re}\int_{0}^{+\infty}e^{(i-1)x}\frac{dx}{\sqrt{x}}=\frac{1}{2}\cdot\text{Re}\int_{0}^{+\infty}\frac{ds}{\sqrt{\pi s}\left((1-i)+s\right)}$$
... |
4,188,560 | <p><a href="https://i.stack.imgur.com/tlruT.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/tlruT.png" alt="Polynomials Problem" /></a></p>
<p><strong>OBSERVATIONS</strong></p>
<p>Let <span class="math-container">$$f(x)=1-x+x^2...+x^{16}-x^{17}$$</span>
If we try to replace <span class="math-containe... | Stefan Albrecht | 945,884 | <p>In another answer and the comments, a simpler approach has been pointed out. I want to show how to continue with your (correct and intuitive) approach.</p>
<p>Following your idea, we have that <span class="math-container">$$f(y)=\sum_{n=0}^{17}(-1)^n\sum_{k=0}^n\binom nky^k(-1)^{n-k}$$</span> We are only interested ... |
1,963,572 | <p>Is it possible that a function takes in an equation and pops out another equation by modifying the first one like f(2x+2=0)=3x+3=1. </p>
| Arthur | 15,500 | <p>In the broadest sense of the word "function", yes. A function $f:X\to Y$ (or just $f$ or just $X\to Y$ if the rest is understood) consists of three things:</p>
<ol>
<li>a <em>domain</em> $X$, which is the set of valid inputs</li>
<li>a <em>codomain</em> $Y$, which is the set of (potential) outputs</li>
<li>an <em>a... |
2,158,796 | <p>Let $K = \mathbb{Q}(\alpha)$ where $\alpha^3 - 50\alpha- 10= 0$.</p>
<p>Prove that $\{1, \alpha, \alpha^{2}\}$ is an integral basis of $\mathcal O_{K}$.</p>
<p>I know that the minimal polynomial is</p>
<blockquote>
<p>$$m_\alpha(x)=x^3 - 50x -10$$</p>
</blockquote>
<p>but I'm not sure where to even begin. I ha... | Mark | 143,872 | <p>First, compute the discriminant of $1,\alpha,\alpha^2$ (I assume you know how to do that?). That's $2^2 5^2 4973$. Next, the discriminant of ${\cal O}_K$ must divide that, and the quotient must be a square. However, the polynomial is 2-Eisenstein as well as 5-Eisenstein, which implies that $2$ and $5$ ramify, whi... |
1,465,138 | <p>I know how to prove that the derivative of $\ln(x)$ is ${1\over x}$ using the definition $f'(x) = {f(x+h) - f(x) \over h}$ but I have ran into trouble proving that the derivative of $\ln(f(x))$ is ${f'(x) \over f(x)}$. Is there a way to prove this strictly using the definition of the derivative?</p>
| Orca | 153,354 | <p>If $a.b=0$ , then either $a =0$ or $b=0$ or $a=b=0$. </p>
<p>Now from $f_x=0$, we have $x=0$ or $x^2+y^2 =1$.
From $f_y=0$, we have $y=0$ or $x^2+y^2 =-1$. </p>
<p>EDIT :
critical points would be, $(0,0)$, $(1,0)$, $(0, i)$ </p>
|
1,480,844 | <p>Let ${a_n}$ be a sequence of real numbers. The backwards differences of this
sequence are defined recursively:</p>
<p>The first difference $∇a_n$ is an new sequence defined by:$∇a_{n} = a_{n} - a_{n-1} $</p>
<p>The $(k + 1)$st difference $∇^{k+1}a_n $ is $∇^{k+1}a_n = ∇^{k}a_{n} - ∇^{k}a_{n-1} $ Find $∇a_{n}$ and... | Esoog | 165,061 | <p>Ok, for $a_n=n^2$ we have $∇a_n=a_n-a_{n-1}=n^2-(n-1)^2=n^2-n^2+2n-1=2n-1$. You had some signs wrong. As for the other, ${∇}^2a_n=∇(a_n-a_{n-1})=∇a_n-∇a_{n-1}=a_n-a_{n-1}-a_{n-1}+a_{n-2}=a_n-2a_{n-1}+a_{n-2}$. Applying this formula to $a_n=2n$, we get $∇^2a_n=2n-4(n-1)+2(n-2)=0$. This is the same as saying $∇ ∇a_n=∇... |
722,890 | <p>Ok, so I know that since the numerator has a higher power that long division is needed. So after doing that, the main fraction is $\frac{-6x-36}{x^2 + 12x + 36}$. I think that's right. But my problem is that after you factor the denominator they're both equal to $(x+6)$. So how would you use partial fraction decompo... | Martin Sleziak | 8,297 | <p>After the observation that the denominator is equal to $(x+6)^2$ (which was mentioned in other answer), this problem can be solved relatively easy using the substitution $y=x+6$.</p>
<p>$$
\begin{align*}
\frac{x^3}{(x+6)^2}
&=\frac{(y-6)^3}{y^2}=\\
&=\frac{y^3-3\cdot 6 y^2+3\cdot 6^2 y - 6^3}{y^2}=\\
&=... |
3,081,186 | <p>I am looking to figure out if the following integral converges or diverges.
<span class="math-container">$\int_{0}^\infty{\frac{dx}{\sqrt{x}*(1+x)}}$</span> = <span class="math-container">$\int_{0}^\infty{\frac{dx}{\mathrm{x}^{1/2}+\mathrm{x}^{3/2}}}$</span>.</p>
<p>I have set it less than or equal to <span class="... | DonAntonio | 31,254 | <p>If you care, observe that</p>
<p><span class="math-container">$$\int\frac{dx}{\sqrt x(1+x)}=2\int\frac{d\sqrt x}{1+\left(\sqrt x\right)^2}=2\arctan\sqrt x+C\;\ldots\ldots$$</span></p>
<p>and not only you solve what you want: you also <em>get a value</em> .</p>
|
3,088,503 | <p>For <span class="math-container">$r\in [0,1)$</span> define
<span class="math-container">$$
I(r) = \int_0^{2\pi}\frac{\cos(t) - r}{1 - 2r\cos t + r^2}\,dt.
$$</span>
Numerical experiments hint at <span class="math-container">$I(r) = 0$</span> for all <span class="math-container">$r\in [0,1)$</span> but I can't show ... | mechanodroid | 144,766 | <p>Not sure why none of the answers mention this approach, but you can just take the derivative with respect to <span class="math-container">$r$</span>:</p>
<p><span class="math-container">\begin{align}
I'(r) &= \int_0^{2\pi}\frac{d}{dr}\left(\frac{\cos t - r}{1 - 2r\cos t + r^2}\right)\,dt\\
&= \int_0^{2\pi} ... |
3,321,399 | <p><span class="math-container">$$
\lim\limits_{n \to \infty} \cos^2\left(\pi \sqrt[3]{n^3+n^2+2n}\right)
$$</span>
where <span class="math-container">$n \in \mathbb{N}$</span>.</p>
<p>In this question, what I thought was, since <span class="math-container">$n \to \infty$</span> and <span class="math-container">$\cos ... | rtybase | 22,583 | <p>Try with <span class="math-container">$$\cos(a+b)=\cos(a) \cos(b) - \sin(a) \sin(b) \tag{1}$$</span>
In this case
<span class="math-container">$$\cos^2\left(\pi \sqrt[3]{n^3+n^2+2n}\right)=\cos^2\left(\pi \sqrt[3]{n^3+n^2+2n}-\pi n+\pi n\right)=\\
\left(\cos\left(\pi \sqrt[3]{n^3+n^2+2n}-\pi n\right) \cdot \color{r... |
3,321,399 | <p><span class="math-container">$$
\lim\limits_{n \to \infty} \cos^2\left(\pi \sqrt[3]{n^3+n^2+2n}\right)
$$</span>
where <span class="math-container">$n \in \mathbb{N}$</span>.</p>
<p>In this question, what I thought was, since <span class="math-container">$n \to \infty$</span> and <span class="math-container">$\cos ... | marty cohen | 13,079 | <p><span class="math-container">$\lim _{n \to \infty} \cos^2(\pi \sqrt[3]{n^3+n^2+2n}) = ?
$</span></p>
<p><span class="math-container">$(n+1/3)^3
=n^3+3(1/3)n^2+3(1/9)n+1/27
=n^3+n^2+n/3+1/27
$</span>
so
<span class="math-container">$n^3+n^2+2n
\gt (n+1/3)^3
$</span>.</p>
<p><span class="math-container">$(n+1/3+a/n)... |
1,009,996 | <p>Show that if $a,b$ are rational numbers and $x$ is a positive real number then $x^a$$x^b$ $=$ $x^{(a+b)}$</p>
<p>I honestly have no idea how to even do this. Anyone have any hints or a good explanation?
Thanks in advance!</p>
| quadratische_hose | 190,547 | <p>The other answer took the calculus based approach, so I'll show you the algebraic approach to this proof.</p>
<p>First, take the log base x of both sides:</p>
<pre><code>log[(x^a)(x^b)]/log(x) = log[x^(a+b)]/log(x)
</code></pre>
<p>Then simplify using log properties:</p>
<pre><code>[log(x^a)+log(x^b)]/log(x) = l... |
1,080,833 | <p>There are several nontrivial powers containing every digit equally often, for example</p>
<p>$32043^2$ $2158479^3$ $69636^4$ $643905^5$ $3470187^6$</p>
<p>A necessary condition for a power with the desired property is that the base is
divisible by $3$ because the power must be a multiple of $9$.</p>
<p>My questi... | Michael Stocker | 118,272 | <p>Here is a $\frac{1}{3}$th of an answer.
There is an 10th power with the property. Here is a list of the smallest solutions for a given power:</p>
<p>Edit to keep the post small, see Jeppes comment for a list up to 24:</p>
<ul>
<li>10: 81785058</li>
</ul>
|
4,086,718 | <p>Prove <span class="math-container">$\lim\limits_{(x,y) \to (0,0)} \frac{xy(y^2-x^2)}{x^2+y^2}=0$</span></p>
<p>My attempt: For all <span class="math-container">$\epsilon > 0$</span> there is a <span class="math-container">$\delta > 0$</span> such that <span class="math-container">$\left | \frac{xy(y^2-x^2)}{x^... | Kavi Rama Murthy | 142,385 | <p>Your last line of inequalities should be changed to <span class="math-container">$|y^{2}-x^{2} |= |y^{2}|+|x^{2}|\leq y^{2}+x^{2}$</span> without the intermediate steps (which are wrong).</p>
<p>If <span class="math-container">$x=\sqrt 3$</span> and <span class="math-container">$y=1$</span> the inequality <span clas... |
4,086,718 | <p>Prove <span class="math-container">$\lim\limits_{(x,y) \to (0,0)} \frac{xy(y^2-x^2)}{x^2+y^2}=0$</span></p>
<p>My attempt: For all <span class="math-container">$\epsilon > 0$</span> there is a <span class="math-container">$\delta > 0$</span> such that <span class="math-container">$\left | \frac{xy(y^2-x^2)}{x^... | Vítězslav Štembera | 663,062 | <p>Otherwise it is ok. I would write it however a bit different:
<span class="math-container">\begin{align}
&(x-y)^2\ge 0~\implies~|xy|\le\frac{x^2+y^2}{2}\le x^2+y^2\\
&\left|\frac{xy(y^2-x^2)}{x^2+y^2}\right|\le|y^2-x^2|\le x^2+y^2\le\delta^2=\varepsilon
\end{align}</span>
when choosing <span class="math-cont... |
103,603 | <p>There is a lock with the password between 000-999. But we could use only two digits to open the lock. For example, if the password is 123, the lock is open when you try 1*3, or *23, or 12*.
questions:
a) compose a strategy that the try number of opening the lock is the smallest one under the worst situation
b) Prov... | MaxW | 23,782 | <p>Assuming:
(1) The lock requires a 3 digit key to open. </p>
<p>(2) Each of three positions can either be a digit (0-9) or an *</p>
<p>(3) The lock is cryptographically secure except for a crib for the 3 digit cipher in that one digit can be replaced by an "*". </p>
<p>(4) Each guess takes the same out effort/time... |
103,603 | <p>There is a lock with the password between 000-999. But we could use only two digits to open the lock. For example, if the password is 123, the lock is open when you try 1*3, or *23, or 12*.
questions:
a) compose a strategy that the try number of opening the lock is the smallest one under the worst situation
b) Prov... | TonyK | 1,508 | <p>Assuming Gerry Myerson's interpretation of the question (see comments to OP), the answer is somewhere between 36 and 64. For 36, see Steven Stadnicki's comment to the OP. For 64:</p>
<pre><code>000 (28) 111 (28) 222 (28) 333 (28) 444 (28)
555 (28) 666 (28) 777 (28) 888 (28) 999 (28)
012 (22) 021 (22) 102 (22) 120 (... |
1,990,274 | <p>The problem asks:</p>
<blockquote>
<p>A silo (base not included) is to be constructed in the form of a
cylinder surmounted by a hemisphere. The cost of construction per
square unit of surface area is $7$ times as great for the hemisphere
as it is for the cylindrical sidewall. Determine the dimensions to b... | Bharat | 79,100 | <ol>
<li><p><a href="http://videolectures.net/computationaltopology2013_ljubljana/?q=Topology" rel="nofollow"><strong>Summer School on Computational Topology and Topological Data Analysis, Ljubljana 2013</strong></a></p></li>
<li><p><a href="https://www.youtube.com/playlist?list=PLpG_ISEhQ6z0Q5MaIvdn5tBJFWfp9fZtQ" rel=... |
1,197,201 | <p>So I was given the task of proving the following variant of the triangle inequality using only the properties of the absolute value:</p>
<p>$\vert\lvert x\rvert -\lvert y \rvert \rvert \leq \lvert x+y\rvert$</p>
<p>Now my initial instinct was to divide it to four cases based on the possible values of x and y, and ... | graydad | 166,967 | <p>This is more difficult without the use of the triangle inequality, but I have a way of doing this proof with two cases instead of four. The only property of absolute value I use is: $|a|^2 = a^2$ for all $a \in \Bbb{R}$. So,
$$\begin{align}\vert\lvert x\rvert -\lvert y \rvert \rvert^2 = \left(\lvert x\rvert -\lvert... |
1,857,250 | <p>I am stuck on this problem - I really don't know how to set it up.</p>
<p>A soft drink can is $h$ centimeters tall and has a radius of $r$ cm. The cost of material in the can is $.0005$ cents per cm$^2$ and the soda itself costs $.001$ cents per cm$^3$. The cans are currently $9$ cm tall and have a radius of $4$ cm... | Vladimir Vargas | 187,578 | <p>Although you have a precise answer to this problem, you have a function $p(r,h)$ of price depending on the height and radius. You don't require calculus to answer the problem. However they are asking you to use calculus to estimate. This usually means the following:</p>
<p>Having a function $z = f(x,y)$, and knowin... |
761,578 | <p>I am looking at the Cantor Intersection Theorem from Apostol's Mathematical analysis.</p>
<p>Let {$Q_1, Q_2, $...} be a countable collection of nonempty sets in $R^n$ such that:</p>
<p>1) $Q_{k+1} \subset Q_k$</p>
<p>2) Each set $Q_k$ is closed and $Q_1$ is bounded.</p>
<p>Then the intersection $\bigcap_{k=1}^\i... | ah11950 | 136,851 | <p>These are getting a bit long for comments, so:</p>
<p>I'm not sure what you mean by 'all elements have been kept track'. For example, if $\Omega = \{1,2,3,4,5\}$, and $\sigma = (123)(45)$ acts on $\Omega$, then all elements of $\Omega$ are accounted for ('kept track'?), but $\sigma$ cannot be expressed as a single ... |
276,264 | <p>Say we are given a system of differential equations </p>
<p>$$
\left[ \begin{array}{c} x' \\ y' \end{array} \right] = A\begin{bmatrix} x \\ y \end{bmatrix}
$$ </p>
<p>Where $A$ is a $2\times 2$ matrix. </p>
<p>How can I in general solve the system, and secondly sketch a solution $\left(x(t), y(t) \right)$... | Wooster | 57,008 | <p>$\newcommand{\vect}[1]{\mathbb{#1}}$Try finding out about diagonalisation of matrices. (If you do not already know about this.)</p>
<p>The basic idea is that I can find a particular matrix P, and a diagonal matrix $\Lambda$; these combine in such a way that $A = P \Lambda P^{-1}$. (These matrices relate to the ei... |
3,318,166 | <p>This may be a silly question but I want to make sure my general intuition is correct. I don't think it is possible for an <span class="math-container">$n$</span>-dimensional manifold to live inside <span class="math-container">$\mathbb{R}^m$</span> for <span class="math-container">$m < n$</span> because in the "b... | Angina Seng | 436,618 | <p>Let <span class="math-container">$n>m$</span>.</p>
<p>If you could embed an <span class="math-container">$n$</span>-manifold in <span class="math-container">$\Bbb R^m$</span>, then you could
embed <span class="math-container">$\Bbb R^n$</span> in <span class="math-container">$\Bbb R^m$</span> (take a coordinate ... |
106,076 | <p>I am struggling to have Mathematica 10.3 solve a system of PDE's (with periodic boundary conditions and random initial conditions), but either it produces a set of very noisy InterpolatingFunction objects that are then nearly impossible to NIntegrate, or it fails to solve it altogether. My best try so far was this:<... | Alexei Boulbitch | 788 | <p>I only address your question about the integration. I tried your code for the integral for t=1. Indeed there was such a message: </p>
<p>NIntegrate::slwcon: Numerical integration converging too slowly; suspect one of the following: singularity, value of the integration is 0, highly oscillatory integrand, or Working... |
1,386,193 | <p>I am learning conditional expectation for fun. I cannot solve the following problem, I think it is just my vocabulary is limited. This problem came from a book on Stochastic Processes:</p>
<p>Let $\Omega = [0,1]\times [0,1]$ with the usual Lebesgue measure and $\sigma$-algebra of Borel sets. Let $X$ and $Y$ be rand... | Sungjin Kim | 67,070 | <p>This is to establish the following for Borel set $B\subseteq [0,1]$,
$$\begin{align}
\mathsf P(Y\in B)
& = \iint_{Y^{-1}(B)} f_{X,Y}(x,y) \mathrm dx\mathrm dy
\\
& = \iint_{[0;1]\times B} f_{X,Y}(x,y) \mathrm dx\mathrm dy
\\
& = \int_B \int_0^1 (x+y) \mathrm dx\mathrm dy
\\
& =\int_B (y + \frac 1 ... |
1,595,137 | <p>This is Singapore Mathematical Olympiad 2015 Grade 8/Secondary 2 Junior Round 1 Question 1.</p>
<blockquote>
<p>1.Among the five numbers, $\frac{5}{9},\frac{4}{7},\frac{3}{5},\frac{6}{11}$ and $\frac{13}{21}$, which one has the smallest value?</p>
</blockquote>
<p>So the obvious method is to either make the same... | Mark Bennet | 2,906 | <p>$35\lt 36$ so $\frac 59\lt \frac 47$; $25\lt 27$ so $\frac 59\lt \frac 35$; $55\gt 54$ so $\frac 59\gt \frac 6{11}$ and $126\lt 143$ so $\frac 6{11}\lt \frac {13}{21}$</p>
|
1,595,137 | <p>This is Singapore Mathematical Olympiad 2015 Grade 8/Secondary 2 Junior Round 1 Question 1.</p>
<blockquote>
<p>1.Among the five numbers, $\frac{5}{9},\frac{4}{7},\frac{3}{5},\frac{6}{11}$ and $\frac{13}{21}$, which one has the smallest value?</p>
</blockquote>
<p>So the obvious method is to either make the same... | Barry Cipra | 86,747 | <p>Order the numbers by increasing denominator,</p>
<p>$${3\over5},{4\over7},{5\over9},{6\over11},{13\over21}$$</p>
<p>and then think of them "free throw" statistics, i.e., at some point your (basketball) team has made $3$ of $5$ free throws, a little later it's made $4$ of $7$, and so forth. Note that each stat, by... |
2,030,264 | <p>I need help in proving that the function $f(A) =\operatorname{adj} (A)$ is continuous.</p>
<p>I'm not sure at all how to work with continuity and matrices..
Thanks.</p>
| Learnmore | 294,365 | <p>Let $A_n$ be a sequence such that $A_k\to A$ then there exists $m$ such that $n\ge m\implies $$||A_k-A||<\epsilon $ forall $\epsilon>0$</p>
<p>Now $||A||=\sum _{i,j=1}^n|a_{ij}|^2$</p>
<p>Hence $A_k\to A\implies ||A_k-A||=\sum_{i,j=1}^n|a_{i,j}^{(k)}-a|^2<\epsilon$</p>
<p>Hence $\text{Adjoint} (A_n)\to \... |
418,880 | <p>Helmut Wielandt <a href="https://books.google.com/books/about/Finite_Permutation_Groups.html?id=npviBQAAQBAJ" rel="nofollow noreferrer">discussed</a> an old question (Chap. 2, Section 15, which can be dated back to Camille Jordan):</p>
<p>Let <span class="math-container">$g\neq 1$</span> be a permutation in some fin... | Nick | 498,224 | <p>The idea of introducing K-functional is described in R. A. DeVore's article <em>Nonlinear approximation</em> p. 85.</p>
|
3,237,889 | <p>I have the following Maxwell's equations:</p>
<p><span class="math-container">$$\nabla \times \mathbf{h} = \mathbf{j} + \epsilon_0 \dfrac{\partial{\mathbf{e}}}{\partial{t}} + \dfrac{\partial{\mathbf{p}}}{\partial{t}},$$</span></p>
<p><span class="math-container">$$\nabla \times \mathbf{e} = - \mu_0 \dfrac{\partial... | J sx | 515,634 | <p>In general case, these equations don't apply and you need fourier sum for periodic signals and fourier integral for non periodic signals. However, in time harmonic case(sinusoidal time harmonic), the derivatives always operate on <span class="math-container">$e^{j \omega t}$</span>, so differentiating once means mul... |
3,237,889 | <p>I have the following Maxwell's equations:</p>
<p><span class="math-container">$$\nabla \times \mathbf{h} = \mathbf{j} + \epsilon_0 \dfrac{\partial{\mathbf{e}}}{\partial{t}} + \dfrac{\partial{\mathbf{p}}}{\partial{t}},$$</span></p>
<p><span class="math-container">$$\nabla \times \mathbf{e} = - \mu_0 \dfrac{\partial... | J_P | 676,091 | <p>The first thing to remember is that the Fourier transform <span class="math-container">$\mathcal{F}$</span> is linear: <span class="math-container">$$
\mathcal{F}(\alpha f+\beta g)=\alpha \mathcal{F}(f)+\beta \mathcal{F}(g)
$$</span>
It also changes differentiation into multiplication. Let <span class="math-containe... |
61,030 | <p><a href="http://reference.wolfram.com/language/ref/Dataset.html"><code>Dataset</code></a> rather conveniently truncates the display of a dataset that has many rows. However, occasionally, I would like to show more of a dataset without having to convert to e.g. a <a href="http://reference.wolfram.com/language/ref/Tab... | m_goldberg | 3,066 | <p>I have great sympathy for your frustration. I complained about this to Wolfram tech support back in August. This is a condensed version of my email.</p>
<blockquote>
<p>I really like the new structured datasets introduced in V10. However, I find one aspect of working with them extremely frustrating: the way they ... |
1,325,406 | <p>I'm having trouble grasping the proof of theorem E, section 5, chapter 1 in Halmos' <em>Measure Theory</em>.</p>
<p>Let $X$ be a nonempty set, and $\Omega$ a family of subsets of of $X$. Given $A\subset X$, denote by $\Omega \sqcap A$ the class of all sets of the form $E \cap A$ with $E \in \Omega$. Further, denote... | Brian M. Scott | 12,042 | <p>Let $F_0=B_0\cup(C_0\setminus A)$ and $F_1=B_1\cup(C_1\setminus A)$, where $B_i \in S(\Omega \sqcap A)$ and $C_i \in S(\Omega)$ for $i=0,1$. Then</p>
<p>$$\begin{align*}
F_0\setminus F_1&=\big(B_0\cup(C_0\setminus A)\big)\setminus\big(B_1\cup(C_1\setminus A)\big)\\
&=\Big(B_0\setminus\big(B_1\cup(C_1\setmin... |
1,386,549 | <p>What is the range of $x+\frac{1}{x}$ if $-2\leq x \leq 2$?<br></p>
<p>My try:<br></p>
<p>$-2\leq x \leq 2$<br></p>
<p>$\Rightarrow \frac{-1}{2}\leq x \leq \frac{1}{2}$<br></p>
<p>$\Rightarrow \frac{-5}{2}\leq x+\frac{1}{x} \leq \frac{5}{2}$<br></p>
<p>Am i correct or wrong?If not,please help me solve this.</p>
| Santeri | 256,364 | <p>In the first one you are saying: "For every faculty member, there exists a student...", whereas in the second one you are saying: "There is a student, and for all faculty members...". In this question we want the student to be fixed.</p>
<p>Also, the first one seems to actually read: "For all people y, if y is a fa... |
1,386,549 | <p>What is the range of $x+\frac{1}{x}$ if $-2\leq x \leq 2$?<br></p>
<p>My try:<br></p>
<p>$-2\leq x \leq 2$<br></p>
<p>$\Rightarrow \frac{-1}{2}\leq x \leq \frac{1}{2}$<br></p>
<p>$\Rightarrow \frac{-5}{2}\leq x+\frac{1}{x} \leq \frac{5}{2}$<br></p>
<p>Am i correct or wrong?If not,please help me solve this.</p>
| DanielWainfleet | 254,665 | <p>The first : Every faculty member was questioned by (at least) one student. The second : Some student asked every faculty member....</p>
|
27,968 | <p>What is the "standard" way to denote all positive (or non-negative) real numbers? I'd think</p>
<p>$$ \mathbb R^+ $$</p>
<p>but I believe that that is usually used to denote "all real numbers including infinity".</p>
<p>So is there a standard way to denote the set</p>
<p>$$ \{x \in \mathbb R : x \geq 0\} \; ?$$ ... | Hendrik Vogt | 2,575 | <p>I'd completely avoid using $\mathbb{R}^+$ since people won't know if $0$ is included or not. So $\mathbb{R}_0^+$ would be a possibility, but then how would you denote $\{x\in\mathbb{R}:x>0\}$? Again, with $\mathbb{R}^+$ people won't know that $0$ isn't included. Personally, I prefer writing $[0,\infty)$ and $(0,\... |
42,624 | <p>In the <a href="http://arxiv.org/abs/math/0606464v1">http://arxiv.org/abs/math/0606464v1</a> I read </p>
<p>"If you want to prove existence of exotic smooth structure on $\mathbb R^4$ you can do this if you are in
possession of a knot which is topologically slice but not smoothly slice (slice means zero slice genu... | Kelly Davis | 1,400 | <p>I wrote this up for my own personal edification when I was going through Gompf and Stipsicz. Please, excuse the pedantry. I had to make sure even I understood it.</p>
<p><strong>Definition:</strong> (<strong>Smoothly Slice Knot</strong>)
A knot $K$ in $\partial D^4$, where $D^n$ is the standard closed $n$--dimensio... |
801,155 | <p>Suppose $f(x)$ is some function with domain [0,1] and $\sum_{n\ge1}f\left(\frac 1n\right)$ converges,
than $a_n=f\left(\frac 1n\right)\to 0$ but does $b_n=f\left(\frac 1{n^2}\right)$ also converge to zero ?
I could not prove the last statement or find any counter-example,
I only guess that since nothing known about ... | Bruno Joyal | 12,507 | <p>Remark that $C$ is a curve with a node $N=(0,0)$, and, informally speaking, the map $\varphi$ realizes $C$ as the "quotient" of $\mathbf A^1$ obtained by identifying the points $\pm 1$ (which both map to $N$ on $C$). One can even make that precise : any morphism of $\mathbf A^1$ into an affine variety which maps $\p... |
2,950,011 | <p>The difference <span class="math-container">$$\log(2)-\sum_{n=1}^{100}\frac{1}{2^n n}$$</span> is </p>
<p>(1) less than <span class="math-container">$0$</span></p>
<p>(2) greater than <span class="math-container">$1$</span></p>
<p>(3) less than <span class="math-container">$\frac{1}{2^{100}101}$</span></p>
<p>(... | JV.Stalker | 416,274 | <p>You could also determine the value of the summa by using the polylogarithm functions:</p>
<p><span class="math-container">$\sum\limits_{n=1}^{\infty}\frac{1}{2^n\,n}=\sum\limits_{n=1}^{\infty}\frac{{\big(\frac{1}{2}}\big)^n}{n}=Li_1(\frac{1}{2})=\ln2$</span></p>
|
1,280,140 | <p>If $V$ is a $k$-vector space and $T : V \to k$ is a homomorphism (linear map) that is not the zero map, is it true that some element of $V$ is mapped to $1$ of $k$? Of course $k$ is a 1-dimensional $k$-vector space generated by $1$ but how can I see that there exists a $v \in V$ such that $T(v) = 1$?</p>
| Hagen von Eitzen | 39,174 | <p>Pick $v$ with $a:=Tv\ne0$ and consider $\frac 1a v$.</p>
|
4,465,858 | <p><strong>Problem Statement</strong></p>
<p>Let <span class="math-container">$j=0,...,N-1$</span>. I want to evaluate</p>
<p><span class="math-container">$$
S_N=\sum_{j=0}^{N-1}e^{i2\pi\frac{j}{N}}
$$</span></p>
<p>I think that <span class="math-container">$S_N=0$</span> for all <span class="math-container">$N\ge 2$</... | Bruno Aceves | 821,973 | <p>Your series method is valid. Observe that <span class="math-container">$\{e^{i2\pi \frac{j}{N}}\}_{j=0}^{N-1}$</span> are the <span class="math-container">$N$</span>-ths root of unity, so they are the roots of the polynomial <span class="math-container">$z^N-1$</span>. Then:</p>
<p><span class="math-container">$$z^N... |
4,465,858 | <p><strong>Problem Statement</strong></p>
<p>Let <span class="math-container">$j=0,...,N-1$</span>. I want to evaluate</p>
<p><span class="math-container">$$
S_N=\sum_{j=0}^{N-1}e^{i2\pi\frac{j}{N}}
$$</span></p>
<p>I think that <span class="math-container">$S_N=0$</span> for all <span class="math-container">$N\ge 2$</... | nwsteg | 1,003,350 | <p>I want to post another proof (similar to Bruno's) and also inspired by this <a href="https://www.kylem.net/math/roots_unity.html#:%7E:text=The%20n%20th%20roots%20of%20unity%20lie%20evenly%20on%20the,numbers%20as%20vectors%20is%20zero." rel="nofollow noreferrer">webpage</a>.</p>
<p>The Nth roots of unity are roots of... |
2,208,847 | <p>If I have a congruence like the following:</p>
<p>$$3x^2 + 5x + 1 \equiv 0 \pmod{q}$$</p>
<p>How would I use the law of quadratic reciprocity to determine if it has a solution? I really have no idea where to go with this. The law of quadratic reciprocity is that:</p>
<p>$$\left( \frac{p}{q} \right) = (-1)^{\frac{... | Bernard | 202,857 | <p>You don't have to use the law of quadratic reciprocity to solve this equation: multiply it by $9$ to obtain
$$9(3x2+5x+1)\equiv x^2+6x+9=(x+3)^2\equiv 0\mod 13.$$
As $\mathbf Z/13\mathbf Z$ is a field, the only solution is $\;x\equiv -3\equiv 10$.</p>
|
2,208,847 | <p>If I have a congruence like the following:</p>
<p>$$3x^2 + 5x + 1 \equiv 0 \pmod{q}$$</p>
<p>How would I use the law of quadratic reciprocity to determine if it has a solution? I really have no idea where to go with this. The law of quadratic reciprocity is that:</p>
<p>$$\left( \frac{p}{q} \right) = (-1)^{\frac{... | Peter | 82,961 | <p>An illustration of the use of the law</p>
<p>Let us calculate $(\frac{37}{73})$ :</p>
<p>We have two primes, both of the form $4k+1$, hence we have</p>
<p>$$(\frac{37}{73})=(\frac{73}{37})=(\frac{-1}{37})=1$$</p>
<p>Note that $73\equiv -1\mod 37$ and that $(\frac{-1}{p})=1$ holds for $p=2$ and for $p=4k+1$. Not ... |
2,053,714 | <p>what I tried was , I observed the cubes from 1-100.</p>
<p>$1^3=1$ </p>
<p>$2^3=8$ </p>
<p>$3^3=27$ </p>
<p>$4^3=64$ </p>
<p>$5^3=125$ </p>
<p>$6^3=216$ </p>
<p>$7^3=343$ </p>
<p>$8^3=512$ </p>
<p>$9^3=81$ </p>
<p>$10^3=1000$ </p>
<p>$11^3=1331$ </p>
<p>$12^3=1729$ </p>
<p>$13^3=2197$ </p>
<p>$14^3=274... | Jack M | 30,481 | <p>You need the following two facts:</p>
<ol>
<li>The last digit of $x^3$ depends only on the last digit of $x$.</li>
<li>The last two digits of $x^3$ depend only on the last two digits of $x$.</li>
</ol>
<p>More generally, if I multiply together two numbers $a$ and $b$, the last digit of $ab$ depends only on the las... |
4,508,923 | <p>I'm doing exercises on Bonnafe's beautiful book Representations of <span class="math-container">$SL_2(\mathbb{F}_q)$</span>. The first exercise is:</p>
<p>1.1. Let <span class="math-container">$k$</span> be the subfield of <span class="math-container">$\mathbb{F}_{q}$</span> generated by <span class="math-container"... | reuns | 276,986 | <p>I don't understand the hint.</p>
<ul>
<li><p>The point is that <span class="math-container">$$\Bbb{F}_{q^2}=\Bbb{F}_p(\zeta_{q+1})$$</span></p>
<p>(proof: write <span class="math-container">$q=p^n$</span>, then <span class="math-container">$\Bbb{F}_p(\zeta_{q+1})=\Bbb{F}_{p^m}$</span> where <span class="math-contain... |
1,221,795 | <blockquote>
<p>Suppose $f$ is continuous on $[a, b]$, if for every continuous function $g$ on $[a, b]$ with $g(a) = g(b) = 0, \int_{a}^{b}f(x)g(x) dx = 0$, Show $f(x) = 0, \forall x \in [a, b]$, </p>
</blockquote>
<p>I want to prove by contradiction, and then find a continuous $g$ such that $g(a) = g(b) = 0$ but $... | HK Lee | 37,116 | <p>Let $g(x)=f(x)(x-a)(b-x)$
Then $$\int_a^b f(x)^2 (x-a)(b-x) dx =0 $$
Note that $$f(x)^2 (x-a)(b-x)\geq 0,\ (x-a)(b-x) > 0\ (x\in
(a,b))$$</p>
<p>If for some $x\in [a,b]$, $f(x)\neq 0$ then since $f$ is continuous, there exists a closed
set $x\in [s,t]$ : $$ f(x)^2\geq c > 0\ {\rm on}\ [s,t] \subseteq [a,... |
322,365 | <p>Let $\phi:R[x]\rightarrow C$ defined for a fixed $z\in C$ as $\phi(p(x))=p(z)$ where $p\in R[x]$. Then this map is supposed to be surjective. Could someone explain me please why?</p>
| user1551 | 1,551 | <p>No. If $\chi_A(x)$ is the characteristic polynomial of $A$, then $\chi_A(x)+1$ is a degree-$n$ polynomial that is <em>necessarily</em> not equal to $\chi_A(x)$. Hence there always exists a degree-$n$ monic polynomial that is not the characteristic polynomial of $A$.</p>
<p>Alternatively, if every arbitrary degree-$... |
1,065,540 | <p>Which books cover the proof that Riemann Hypothesis is equivalent to the best error bound for the Prime Number Theorem?</p>
<p>My understanding is that Riemann Hypothesis is equivalent to the best bound of prime number theorem. Von Koch (1901) proved that the Riemann hypothesis is equivalent to the "best possible" ... | Marco Cantarini | 171,547 | <p>You can find a proof in -H. M. Edwards, "Riemann's Zeta Function", Dover publications, 1974.-</p>
|
1,065,540 | <p>Which books cover the proof that Riemann Hypothesis is equivalent to the best error bound for the Prime Number Theorem?</p>
<p>My understanding is that Riemann Hypothesis is equivalent to the best bound of prime number theorem. Von Koch (1901) proved that the Riemann hypothesis is equivalent to the "best possible" ... | davidlowryduda | 9,754 | <p>The method to prove the Prime Number Theorem can be quickly summarized, and further details can be found in any introductory book on analytic number theory.</p>
<p>The proof relies on understanding the analytic properties of the Riemann zeta function $\zeta(s)$. In particular, the classical approach to proving the ... |
2,377,728 | <p>Suppose $F$ is a family of sets. Then the intersection of $F$ is the set $\cap F$ defined as follows:</p>
<p>$$ \cap F = \{x \mid \forall A. A \in F \rightarrow x \in A)\}$$</p>
<p>I am having trouble understanding this. This is how I am currently reading it:</p>
<p>The intersection of $F$ (which is a family of s... | epi163sqrt | 132,007 | <p>Since
\begin{align*}
a_{n+1}&=c_na_n\\
a_n&=c_{n-1}a_{n-1}\\
&\ \ \,\vdots\\
a_1&=c_0a_0
\end{align*}</p>
<p>multiplication of LHS and RHS gives
\begin{align*}
\prod_{k=1}^{n+1}a_k&=\left(\prod_{k=0}^n c_k\right)\left(\prod_{k=0}^na_k\right)
\end{align*}</p>
<blockquote>
<p>Given that $a_n\ne... |
850,249 | <p>Express a complex number in modulus amplitude form</p>
<p>$\displaystyle 1+\sin \alpha +i\cos \alpha $</p>
<p>My Attempt: </p>
<p>$\displaystyle r\cos \theta= 1+\sin \alpha $</p>
<p>$\displaystyle r\sin \theta= \cos \alpha $</p>
<p>Squaring and adding.. $\displaystyle r^2= (1+\sin \alpha)^2+ \cos^2 \alpha$</p>
... | Claude Leibovici | 82,404 | <p><strong>Hint</strong> </p>
<p>Another way would consist in writing $$\displaystyle 1+\sin( \alpha) +i\cos( \alpha)=\Big(\sin(\frac{\pi}{2})+\sin( \alpha)\Big)+i \Big( \cos(\frac{\pi}{2})+\cos( \alpha)\Big)$$ Now, apply $$\sin(p)+\sin(q)=2 \sin \left(\frac{p+q}{2}\right) \cos \left(\frac{p-q}{2}\right)$$ $$\cos(p)+\... |
850,249 | <p>Express a complex number in modulus amplitude form</p>
<p>$\displaystyle 1+\sin \alpha +i\cos \alpha $</p>
<p>My Attempt: </p>
<p>$\displaystyle r\cos \theta= 1+\sin \alpha $</p>
<p>$\displaystyle r\sin \theta= \cos \alpha $</p>
<p>Squaring and adding.. $\displaystyle r^2= (1+\sin \alpha)^2+ \cos^2 \alpha$</p>
... | gniourf_gniourf | 51,488 | <p>A general fact is:
$$\forall z\in\mathbb{C},\ \lvert z\rvert^2+z^2=z(\overline z+z)=2\Re(z)z.$$
(a manifestation of the inscribed angle theorem?).
Hence:
$$\forall z\in\mathbb{C},\ \bigl\lvert\lvert z\rvert^2+z^2\bigr\rvert=2\bigl\lvert\Re(z)\bigr\rvert\lvert z\rvert,\ \text{and}\ \arg(z)=\begin{cases}\arg(z)&\t... |
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