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 |
|---|---|---|---|---|
618,823 | <p>Reading a book I encountered the following claim, which I don't understand. Let $X$ be a smooth projective curve over $\mathbb{C}$, and $q\in X$ a rational point.
Denote by $\pi_i: X^n\to X$ the $i$-th projection of the cartesian $n$-product of the curve onto $X$ itself. The claim is that</p>
<blockquote>
<p>The ... | Community | -1 | <p>The result you want is the following.</p>
<blockquote>
<blockquote>
<p><strong>Hartshorne II Ex. 7.5 (c):</strong> If $\mathcal{L}, \mathcal{M}$ are two ample line bundles on a Noetherian scheme $X$, then $\mathcal{L}\otimes \mathcal{M}$ is ample. </p>
</blockquote>
</blockquote>
<p><strong>Proof:</strong... |
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... | joriki | 6,622 | <p>Your proof is OK, but there are some things you could improve:</p>
<p>It’s perhaps worth explaining how it is that the two sums count the same thing in much the same way, and thus should have the same number of terms, but don’t. This is because the terms in the second sum with <span class="math-container">$n-i\lt m$... |
314,236 | <p>The integral $\int_{-\infty}^{\infty} e^{ix} dx$ diverges. I have read (in a wikipedia article) that the principal value of this integral vanishes: $P \int_{-\infty}^{\infty} e^{ix} dx = 0$. How can one see that?</p>
<p>Thank you for your effort!</p>
| Diógenes | 184,463 | <p>You might want to consider the this as a the distribution:</p>
<p>$$P.V.\left(\int_{-\infty}^{\infty}e^{ixt}dx\right)_{t=1}$$</p>
<p>Now the distribution needs to be tested against test functions as:</p>
<p>$$\int_{-\infty}^{\infty}P.V.\left(\int_{-\infty}^{\infty}e^{ixt}dx\right)\phi(t)dt$$</p>
<p>Which is, bec... |
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... | john | 8,751 | <p>Given a group $G$ the set of functions $Set(X,G)$ admits a pointwise group structure -- in this way, the category of groups is cotensored as a $Set$-category. Likewise any of the standard algebraic categories.</p>
<p>Similarly when you encounter a functor category $[C,D]$ equipped with some ``pointwise" structure ... |
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... | ಠ_ಠ | 56,938 | <p>The category of sheaves of abelian groups on a space $X$ is powered and copowered over abelian groups as follows:</p>
<p>let $t: X \to \bullet$ denote the terminal map to the 1-point space. Then the direct image is the global sections functor $t_* = \Gamma(X, -)$ and the inverse image $t^{-1}$ constructs the consta... |
1,331,850 | <p>I haven't done something like this in a long time. How do I set something like this up? Can someone help me with the beginning or give me some direction?</p>
<p><img src="https://i.stack.imgur.com/jluer.png" alt="enter image description here">
<img src="https://i.stack.imgur.com/dv5yu.png" alt="enter image descript... | Dmoreno | 121,008 | <p>Your vector field is irrotational, i.e., $\nabla \wedge \mathbf{F} = 0 $, and there exists $G$ such that $\nabla G = \mathbf{F}$. We can see that $G = x^2/ 2 + 2 x y + y^2/2$ and <a href="https://en.wikipedia.org/wiki/Gradient_theorem" rel="nofollow"><strong><em>therefore</em></strong></a>:</p>
<p>$$ \int_C \mathbf... |
992,487 | <p>Consider the following problem.</p>
<p>A collection of $n$ countries $C_1, \dots, C_n$ sit on an EU commission. Each country $C_i$ is assigned a voting weight $c_i$. A resolution passes if it has the support of a proportion of the panel of at least $A$, taking into account voting weights. Each country $C_i$ has a p... | Nameless | 68,482 | <p>This is a very interesting problem! On your theorem: I don't understand why you maximize $z$, which is just a density (up to that point everything seems fine). You want to maximize $P(V\ge A)$. You assume for sufficiently many countries that this can be approximated by a normal distribution, invoking some version of... |
992,487 | <p>Consider the following problem.</p>
<p>A collection of $n$ countries $C_1, \dots, C_n$ sit on an EU commission. Each country $C_i$ is assigned a voting weight $c_i$. A resolution passes if it has the support of a proportion of the panel of at least $A$, taking into account voting weights. Each country $C_i$ has a p... | Matt B. | 164,029 | <p><a href="http://en.m.wikipedia.org/wiki/Roy" rel="nofollow">http://en.m.wikipedia.org/wiki/Roy</a>'s_safety-first_criterion</p>
<p>(About related finance problems).</p>
<p>trying to chose a portfolio that maximizes the probability of meeting a threshold is probably close to the problem you're mentioning here.</p>
... |
607,044 | <p>I'm looking at some work with Combinatorial Game Theory and I have currently got:
(P-Position is previous player win, N-Position is next player win)</p>
<p>Every Terminal Position is a P-Position,</p>
<p>For every P-Position, any move will result in a N-Position,</p>
<p>For every N-Position, there exists a move t... | Henry | 6,460 | <p>$\gamma$ is the limit of the sum of the slightly bigger than triangle pieces of this diagram (from Wikipedia)</p>
<p><img src="https://i.stack.imgur.com/LKfZK.png" alt="enter image description here"></p>
<p>As $n$ increases, the sum increases, but clearly has an upper bound of $1$ and therefore converges to a limi... |
607,044 | <p>I'm looking at some work with Combinatorial Game Theory and I have currently got:
(P-Position is previous player win, N-Position is next player win)</p>
<p>Every Terminal Position is a P-Position,</p>
<p>For every P-Position, any move will result in a N-Position,</p>
<p>For every N-Position, there exists a move t... | Community | -1 | <p>Let $$u_n=\sum_{k=1}^n\frac 1 k-\log n$$
then
$$u_{n}-u_{n-1}=\frac{1}{n}+\log\left(1-\frac 1 n\right)\sim_\infty-\frac{1}{2n^2}$$
so the series $\displaystyle\sum_{n\ge2}u_{n}-u_{n-1}$ is convergent by asymptotic comparison and then the sequence $(u_n)_n$ is convergent by telescoping.</p>
|
2,662,717 | <p>Let $(f_k)_{k=m}^\infty$ be a sequence of differentiable functions $f_k:[a,b]\rightarrow R$ whose derivatives are continuous. Suppose there exists a sequence $(M_k)_{k=m}^\infty$ in $R$ with $|f_k'|\le M_k$ for all $x\in X, k\geq m,$ and such that $\sum_{k=m}^\infty M_k$ converges. Assume also that there is some $x_... | Donald Splutterwit | 404,247 | <p>The neat way to show this is to use matricies
\begin{eqnarray*}
\begin{bmatrix}
0 &1 \\1 &1 \\
\end{bmatrix}^m =
\begin{bmatrix}
F_{m-1} &F_m \\F_m &F_{m+1} \\
\end{bmatrix} .
\end{eqnarray*}
\begin{eqnarray*}
\begin{bmatrix}
0 &1 \\1 &1 \\
\end{bmatrix}^n =
\begin{bmatrix}
F_{n-1} ... |
2,027,556 | <p>The definition I have of a tensor product of vector finite dimensional vector spaces $V,W$ over a field $F$ is as follows: Let $v_1, ..., v_m$ be a basis for $V$ and let $w_1,...,w_n$ be a basis for $W$. We define $V \otimes W$ to be the set of <strong>formal linear combinations</strong> of the mn symbols $v_i \otim... | Eric Wofsey | 86,856 | <p>The term "linear combination" has no meaning unless you are talking about elements of a vector space. The symbols $v_i\otimes w_j$ are no more than symbols, and there is not (yet) any vector space that they are elements of, so it is meaningless to talk about linear combinations of them. However, we can talk about... |
4,528,823 | <p>I'm trying to understand the proof that every bounded sequence has a convergent subsequence. The proof goes as follows:</p>
<blockquote>
<p>Let <span class="math-container">$\{a_{n}\}$</span> be a bounded sequence of real numbers. Choose <span class="math-container">$M\ge 0$</span>
such that <span class="math-contai... | Theo Bendit | 248,286 | <p>The point <span class="math-container">$a$</span> is in <span class="math-container">$\bigcap_n E_n$</span>, which means it is in every <span class="math-container">$E_n$</span>. The set <span class="math-container">$E_n$</span> is the result of throwing away the first <span class="math-container">$n - 1$</span> (I'... |
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... | egwene sedai | 655 | <p>You can plot another one with the desired mesh only, e.g.</p>
<pre><code>f[x_, y_] := Exp[-(x^2 + y^2)]
p0 = Plot3D[f[x, y], {x, -3, 3}, {y, -3, 3}, PlotRange -> All,
Mesh -> Automatic]
p2 = Plot3D[f[x, y], {x, -3, 3}, {y, -3, 3},
Mesh -> {{{0, {Red, Thickness[.01]}}}, {{0, {Red,
Thickness[.0... |
3,896,327 | <p>For the first part of this question, I was asked to find the either/or version and the contrapositive of this statement, which I found as follows:</p>
<p>i) either <span class="math-container">$n \leq 7$</span>, or <span class="math-container">$n^2-8n+12$</span> is composite</p>
<p>ii) if <span class="math-container... | Stinking Bishop | 700,480 | <p><span class="math-container">$$n^2-8n+12=(n-4)^2-2^2=(n-2)(n-6)$$</span></p>
<p>Thus, if <span class="math-container">$n>7$</span>, then both of the factors <span class="math-container">$n-2$</span> and <span class="math-container">$n-6$</span> are greater than <span class="math-container">$1$</span> and <span cl... |
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... | Wuestenfux | 417,848 | <p>Well, the proof is correct. What are the basic steps?</p>
<ol>
<li><p>If <span class="math-container">$I\ne \{0\}$</span> is an ideal of <span class="math-container">$\Bbb Z$</span> and <span class="math-container">$0\ne a\in I$</span>, then <span class="math-container">$-a=(-1)a\in I$</span> and so <span class="ma... |
2,974,839 | <p>I solved this question this way.</p>
<p>First, there are two ways that the cards will not be alternating:</p>
<p>B - R - B - R - B - R</p>
<p>R - B - R - B - R - B</p>
<p>Second, there are 6! (720) possible orders in which the cards can be dealt.</p>
<p>So, the answer is 2/720. Is this correct?</p>
| Community | -1 | <p>There are actually <span class="math-container">$6!/(3!3!)=720/36=20$</span> ways, so the probability is <span class="math-container">$2/20=1/10$</span>. </p>
|
2,974,839 | <p>I solved this question this way.</p>
<p>First, there are two ways that the cards will not be alternating:</p>
<p>B - R - B - R - B - R</p>
<p>R - B - R - B - R - B</p>
<p>Second, there are 6! (720) possible orders in which the cards can be dealt.</p>
<p>So, the answer is 2/720. Is this correct?</p>
| J.G. | 56,861 | <p>Without loss of generality, the first card is blue. Then choose a red one with probability <span class="math-container">$\frac{3}{5}$</span>, a blue one with probability <span class="math-container">$\frac{2}{4}=\frac{1}{2}$</span>, a red one with probability <span class="math-container">$\frac{2}{3}$</span>, and th... |
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>
| Carl Mummert | 5,442 | <p>The statement for the completeness theorem is due to Harvey Friedman, 1976, "Systems of second order arithmetic with restricted induction II", p. 558 of: Meeting of the Association for Symbolic Logic, John Baldwin, D. A. Martin, Robert I. Soare and W. W. Tait, <em>The Journal of Symbolic Logic</em>, Vol. 41, No. 2 ... |
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}$$</p>
<p>Since $\... | cr001 | 254,175 | <p>Indeed your proof is not valid.</p>
<p>Let $y=x$, $\lim_{(x,y)\to (0,0)}\frac{2e^x y^2}{x^2+y^2}=1$</p>
<p>Let $y=2x$, $\lim_{(x,y)\to (0,0)}\frac{2e^x y^2}{x^2+y^2}=\large{8\over5}$</p>
<p>So it cannot exist.</p>
|
3,429,514 | <p>so I am having a problem understanding finding solution to that equation:
<span class="math-container">$$x'' +w^2 x =f sin ω t$$</span>
with initial conditions x(0)=0 and x'(0)=0. I know that is an explanation of how to solve that:
<br>
<br></p>
<p><em>I had to make the euler formula substitution and then multip... | Community | -1 | <p>A solution of the homogeneous equation will <em>never</em> work as a solution of the non-homogeneous equation for a very simple reason: when plugged in the LHS, it yields <span class="math-container">$0$</span> !</p>
<p>So when the RHS belongs to the homogeneous equation, the ansatz must be different. In the case o... |
4,295,459 | <blockquote>
<p>Find the Taylor series of <span class="math-container">$$\frac{1}{(i+z)^2}$$</span> centered at <span class="math-container">$z_0 = i$</span>.</p>
</blockquote>
<p>Im thinking if I could find the Taylor series for <span class="math-container">$$\frac{1}{i+z}$$</span> I could use that <span class="math-c... | GEdgar | 442 | <p>To find the Taylor series of <span class="math-container">$1/(i+z)$</span> in powers of <span class="math-container">$z-i$</span>. I could write <span class="math-container">$w = z-i$</span> and find the Taylor series of that in powers of <span class="math-container">$w$</span>.</p>
<p><span class="math-container">... |
395,604 | <p>Lets say you have a sequence $S = (0, 1, 2, 3, 4, 5, 6, 7, 8)$</p>
<p>And another sequence $T = (0, 1, 2, 3)$</p>
<p>Is there any specific mathematical term that defines the relationship between $S$ and $T$, that specifically says that $S$ starts with $T$?</p>
<p>I thought $T$ would be called an <em>initial sub-... | xavierm02 | 10,385 | <p>In computer science (which is, at least at the beginning, maths), you call them words or strings over an alphabet $X$ which contains your letters.</p>
<p>For this alphabet $X$, you define $X^n$ by the usual Cartesian product and $X^*=\bigcup\limits_{n\in \Bbb N}X^n$. $X^*$ is the set of all words written with lette... |
1,521,739 | <p>The following is the Meyers-Serrin theorem and its proof in Evans's <em>Partial Differential Equations</em>:</p>
<blockquote>
<p><a href="https://i.stack.imgur.com/XnzXY.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/XnzXY.png" alt="enter image description here"></a>
<a href="https://i.stack... | benleis | 148,578 | <p>I'm going to assume the OP wanted the vertex and focus of the original tilted parabola since they already had rotated it to a standard form where those values are easier to find.</p>
<p>Rather than rotating, its convenient to do everything in place since we're looking for the focus and by definition: for any point... |
3,597,829 | <p>Edit: I am using the natural logarithm in what follows.</p>
<p>I want to figure out how to show by hand that the maximum of
<span class="math-container">$$\log(4)c+\log(3)a+\log(2)x$$</span>
when
<span class="math-container">$$a\geq 0, c\geq 0, x \geq 0, y \geq 0,$$</span>
<span class="math-container">$$a+c+x+y=1,... | Yuri Negometyanov | 297,350 | <p><span class="math-container">$\color{brown}{\textbf{Preliminary transformations.}}$</span></p>
<p>The task is to maximize the function
<span class="math-container">$$f(x,a,c) = (x+2c)\log 2 + a\log 3\tag1$$</span>
over non-negative <span class="math-container">$a,c,x,y,$</span> under the conditions
<span class="mat... |
3,683,739 | <p><strong>Question</strong>:</p>
<p>I am trying to solve a principal value integral involving a square root. Using Mathematica I can get an answer but I would like to know a general approach to obtain them by hand. To be clear I am interested in a clear explanation of the method not just the solution.</p>
<p>The pri... | Green Fish | 548,931 | <p><strong>Preliminary definitions and equivalences</strong>:</p>
<p>Firstly we define (as per the question):
<span class="math-container">\begin{equation}
\begin{split}
I_{B}\left(y\right)=&\frac{1}{\pi} \int_{0}^{B} x\sqrt{B^2-x^2}\left(\frac{\mathcal{P}_{x+y}}{x+y}+\frac{\mathcal{P}_{x-y}}{x-y}\right)\mathrm{d}... |
3,683,739 | <p><strong>Question</strong>:</p>
<p>I am trying to solve a principal value integral involving a square root. Using Mathematica I can get an answer but I would like to know a general approach to obtain them by hand. To be clear I am interested in a clear explanation of the method not just the solution.</p>
<p>The pri... | Ron Gordon | 53,268 | <p>The evaluation of the Cauchy principal value integral via contour integration is relatively straightforward. To begin, consider the contour integral</p>
<p><span class="math-container">$$\oint_C dz \, \frac{z \sqrt{z^2-B^2}}{z-y} $$</span></p>
<p>where <span class="math-container">$C$</span> is the following cont... |
194,373 | <p>Let $\Omega$ be a bounded smooth domain and define $\mathcal{C} = \Omega \times (0,\infty)$. Below, $x$ refers to the variable in $\Omega$ and $y$ to the variable in $(0,\infty)$. The map $\operatorname{tr}_\Omega:H^1(\mathcal C) \to L^2(\Omega)$ refers to the trace operator ($\operatorname{tr}_\Omega u = u(\cdot,0)... | Joonas Ilmavirta | 55,893 | <p>The trace only depends on values near the boundary.
That is, if $\phi\in C^\infty([0,\infty))$ is one in a neighborhood of zero, then $\operatorname{tr}_\Omega(\phi u)=\operatorname{tr}_\Omega(u)$ for every $u\in H^1(\mathcal C)$.
With this in mind, you can formally apply a cut-off to your constant function and trea... |
1,047,489 | <p>let $f(x)$ be a function such that </p>
<p>$$f(0) = 0$$</p>
<p>$$f(1) =1$$ $$f(2) = 2$$ $$f(3) = 4$$ $$f'(x) \text{is differentiable on } \mathbb{R}$$ Prove that there is a number in the interval $(0,3)$ such
that $0 < f''(x)<1$</p>
<p>I'm really stuck.
thanks</p>
| user193702 | 193,702 | <p>In the interval $(2,3)$ we can get $f'(\xi)=2$, and in $(0,1)$ we can get $f'(\delta)=1$, so in $[\delta,\xi]$ we can get $0<f"(\theta)<1$, considering $\xi-\delta>1$.</p>
|
243,210 | <p>I have difficulty computing the $\rm mod$ for $a ={1,2,3\ldots50}$. Is there a quick way of doing this?</p>
| Hans Engler | 9,787 | <p>Why don't you try separation of variables? This leads to a closed form solution that exists for all $x < \sqrt{2 \log \frac{3}{2}}$.</p>
|
25,413 | <p>Background - I am tutoring a second year college sophomore for a class titled Single Variable Calculus, and whose curriculum looks to be similar to the AB calculus I tutor in my High School.</p>
<p>We are on limits and L’Hôpital’s Rule, and I see this among the questions (note, all the worksheet questions are meant ... | user52817 | 1,680 | <p>The squeeze theorem merely introduces a basic framework for future analytic thinking. For example, the inequality</p>
<p><span class="math-container">$$\sum_{k=1}^n\frac1k>\int_1^{n+1}\frac{1}{x}\, dx$$</span></p>
<p>can be established easily by drawing boxes that fit above the curve <span class="math-container">... |
25,413 | <p>Background - I am tutoring a second year college sophomore for a class titled Single Variable Calculus, and whose curriculum looks to be similar to the AB calculus I tutor in my High School.</p>
<p>We are on limits and L’Hôpital’s Rule, and I see this among the questions (note, all the worksheet questions are meant ... | Xander Henderson | 8,571 | <p>Most of the other answers in this thread focus on the mathematics. This is appropriate, as this is a Q&A site for mathematics educators. However, I suspect that the question being answered ("Should I teach the squeeze theorem?") has already been addressed here many times. The distinguishing question... |
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>
| karakfa | 14,900 | <p>General block form inverse with $A$ and $D$ invertible is</p>
<p>$$\begin{pmatrix}A & B \\ C & D\end{pmatrix}^{-1}= \begin{pmatrix}X & -A^{-1}BY \\ -D^{-1}CX & Y\end{pmatrix}$$</p>
<p>where $X=(A-BD^{-1}C)^{-1}$ and $Y = (D-CA^{-1}B)^{-1}$</p>
<p>substituting values will give you $X=Y=\frac12I$ an... |
3,985,917 | <p>I am trying to show algebraically that <span class="math-container">$8^3>9^{8/3}$</span>. This came from trying to complete the base case of an induction proof.</p>
<p>I have struggled because <span class="math-container">$8$</span> and <span class="math-container">$9$</span> cannot be manipulated to be the same ... | David Lui | 445,002 | <p>We want to compare <span class="math-container">$x^y $</span> vs. <span class="math-container">$y^x$</span>, for <span class="math-container">$x, y > e$</span>. Take log base <span class="math-container">$y$</span> of both sides, we get <span class="math-container">$y \log_y(x) = y \ln(x) / \ln(y)$</span> vs. <sp... |
23,312 | <p>What is the importance of eigenvalues/eigenvectors? </p>
| 911 | 5,312 | <h3>A short explanation</h3>
<p>Consider a matrix <span class="math-container">$A$</span>, for an example one representing a physical transformation (e.g rotation). When this matrix is used to transform a given vector <span class="math-container">$x$</span> the result is <span class="math-container">$y = A x$</span>.<... |
507,827 | <p>Let $a_n$ be a positive sequence. Prove that
$$\limsup_{n\to \infty} \left(\frac{a_1+a_{n+1}}{a_n}\right)^n\geqslant e.$$</p>
| njguliyev | 90,209 | <p>Since I solved this problem several years ago, I didn't write my solution immediately, so that others could think on this problem. Now I am writing my own solution:</p>
<p>It starts as the solution by Ju'x, i.e. we can safely assume that $a_1 = 1$ and suppose the converse inequality. Then there exists $N \in \mathb... |
142,677 | <p>Consider the following list of equations:</p>
<p>$$\begin{align*}
x \bmod 2 &= 1\\
x \bmod 3 &= 1\\
x \bmod 5 &= 3
\end{align*}$$</p>
<p>How many equations like this do you need to write in order to uniquely determine $x$?</p>
<p>Once you have the necessary number of equations, how would you actually ... | jmc | 30,839 | <p>Note that even if you specify the residue of $x$ modulo all integers $n$, this need not determine an integer.</p>
<p>This is where $\hat{\mathbb{Z}}$ comes into the picture. I think you should know this as a fact, just to be complete (no pun intended). But I warn you that the theory going into $\hat{\mathbb{Z}}$ is... |
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>
| John_dydx | 82,134 | <p>You can approach it this way:</p>
<p>Find $\large\frac{dx}{dt}$ and $\large\frac{dy}{dt}$. Note that $\large\frac{dt}{dx}= \frac{1}{\large\frac{dx}{dt}}$
$$\frac{dy}{dx}= \frac{dy}{dt} \cdot \frac{dt}{dx}$$</p>
<p>And then differentiate again to find $\frac{d^2y}{dx^2}$</p>
<p>$$ \frac{dx}{dt} = 3t^2 - 12$$</p>
... |
400,749 | <p>The Extreme Value Theorem says that if $f(x)$ is continuous on the interval $[a,b]$ then there are two numbers, $a≤c$ and $d≤b$, so that $f(c)$ is an absolute maximum for the function and $f(d)$ is an absolute minimum for the function. </p>
<p>So, if we have a continuous function on $[a,b]$ we're guaranteed to have... | user79202 | 79,202 | <p>First of all, $f(x) = x^{-2}$ isn't well defined on the domain $[-1,1]$, specifically where $x = 0$, so you can't really say that it is discontinuous. But if you assign it any value at $x = 0$, so $f(0) := c$ for $c \in \mathbb{R}$, it is discontinuous.
Then, as shown by yourself, continuity isn't needed to find a f... |
703,125 | <blockquote>
<p>Let $\{A_\alpha\}$ be a collection of connected subspaces of $X$; let
$A$ be connectted subspace of $X$. Show that if $A\cap A_\alpha \neq
\emptyset$ for all $\alpha$, then $A\cup(\cup A_\alpha)$ is connected.</p>
</blockquote>
<p>I know this theorem:<img src="https://i.stack.imgur.com/6tILz.png"... | Kaladin | 133,789 | <p>If you assume that $A\cup(\cup A_{\alpha})$ is disconnected. Therefor it can be written as the union of two disjoint non-empty closed subsets $U_{1}$ and $U_{2}$. But because $A\cap A_{\alpha}\neq \emptyset$ for all $\alpha$ you get $A\cap U_{1}\neq \emptyset$ and $A\cap U_{2}\neq \emptyset$. But this would mean th... |
3,071,751 | <p>Is there a parametrization for the figure '8' curve, which is self-intersected?</p>
| BadAtAlgebra | 611,990 | <blockquote>
<p><span class="math-container">$x = \frac{a\sqrt{2}\cos(t)}{\sin^2(t) + 1}; \qquad y = \frac{a\sqrt{2}\cos(t)\sin(t)}{\sin^2(t) + 1}$</span></p>
</blockquote>
<p>Check out this source: </p>
<blockquote>
<p><a href="https://en.wikipedia.org/wiki/Lemniscate_of_Bernoulli" rel="nofollow noreferrer">htt... |
3,071,751 | <p>Is there a parametrization for the figure '8' curve, which is self-intersected?</p>
| md2perpe | 168,433 | <p>An easier parameterization of an <a href="https://www.wolframalpha.com/input/?i=plot%20%7B%20x%3Dsin(2t),%20y%3Dcos(t)%20%7D" rel="nofollow noreferrer">8-like figure</a> is <span class="math-container">$(x,y) = (\sin 2t, \cos t),$</span>
where <span class="math-container">$0 \leq t \leq 2\pi.$</span> It can easily b... |
625,975 | <p>I'm just starting to learn computability. Some treatments of the subject use a relation they call $T$, which I <em>think</em> is called the universal recursive relation. It's defined something like this (<a href="http://www.its.caltech.edu/~jclemens/courses/02ma117a/handouts/handout6.pdf" rel="nofollow">http://www.i... | MJD | 25,554 | <p>You have misunderstood $c$ here. It is not an output. It is a <em>complete description</em> of a computation, including all the state transitions, tape head movements, and tape modifications that the machine makes on its way to the final state. </p>
<p>It is computable to check whether a single step of this long ... |
4,084,486 | <p>I am studying different integral transform methods, and I am confused on why saying things such as
<span class="math-container">$$
\mathcal{F}^{-1}[1] = \delta(x)
$$</span>
is valid? If you actually plug this in,
<span class="math-container">$$
\mathcal{F}^{-1}[1] = \frac{1}{2\pi}\int_{-\infty}^{\infty}e^{ikx}dx
$$<... | Oliver Díaz | 121,671 | <p>Some initial remarks:</p>
<p>@Mark Viola considers the Fourier transform defined on the space of tempered distributions which contains not only copies of nice functions (integrable functions for example) but also finite measures, and many other things.</p>
<p>The <span class="math-container">$\delta$</span> "fu... |
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>... | Michael E2 | 4,999 | <p>Even simpler than mapping or <code>Table</code>, since <a href="http://reference.wolfram.com/mathematica/ref/Range.html" rel="nofollow noreferrer"><code>Range</code></a> is <a href="http://reference.wolfram.com/mathematica/ref/Listable.html" rel="nofollow noreferrer"><code>Listable</code></a>:</p>
<pre><code>Range[... |
926,069 | <p>Say that two $m\times n$ matrices, where $m,n\ge 2$, are <em>related</em> if one can be obtained from the other after a finite number of steps, where at each step we add any real number to all elements of any one row or column. For example, $\left(\begin{array}{cc}
0 & 0\\0 & 0
\end{array}\right)$
and
$\left... | frog | 84,997 | <p>Your series is the Fourier-Series of the $2\pi$-peridodic extension of the function
$$
f(x):=\frac{\pi-x}{2}
$$
defined on $[0,2\pi]$, hence the sign will depend on $x$…
EDIT:
Thus your statement
$$\operatorname{sgn}\left(\sum\limits_{k=1}^\infty\frac{\sin (kx)}{k}\right)=\operatorname{sgn}(\sin x)$$
is correct.</p>... |
128,015 | <p>For the function $\frac{1}{x}$ on the real line, one can use a modified principal value integral to consider it as a distribution p.f.$(\frac{1}{x}),$ and one can do a similar construction to make $\frac{1}{x^m}$ into a distribution for $m>1.$ In the complex plane, the function $\frac{1}{z^m}$ is locally integrab... | user23078 | 23,078 | <p>It's interesting to think distributiions as the boundary of analytic functions, which I believe is originated from Saito. You may find the following result helpful</p>
<p>Let I be an open interval on $\mathbb{R}$, and
$$
Z={z\in \mathbb{C};\Re z\in I,0<\Im z <\gamma}
$$
is a one sided complex neighborhood. ... |
733,101 | <p>I've been stuck for a while on this question and haven't found applicable resources.</p>
<p>I have 10 choices and can select 3 at a time. I am allowed to repeat choices (combination), but the challenge is that ABA and AAB are not unique.</p>
<p>10 choose 3 is the question.</p>
<p>I have been working on a smaller ... | André Nicolas | 6,312 | <p>Let us suppose that we have $m$ distinct object "types," but many objects of each type. You want to select $k$ objects, where selections may be repeated. Consider the equation
$$x_1+x_2+\cdots+x_{m}=k.\tag{1}$$
Any selection of $k$ objects, not necessarily distinct, corresponds to a solution of Equation (1) in non-... |
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... | hasManyStupidQuestions | 606,791 | <p><strong>Attempt</strong> (<em>community wiki</em>):</p>
<p>Note that for 2. I've been sloppy in declaring that <span class="math-container">$\mathfrak{F} \subseteq \mathcal{P}(\mathfrak{M})$</span> because the Tarskian semantics for multi-sorted logic don't require that. In general <span class="math-container">$\mat... |
3,072,995 | <p>The only thing I know with this equation is <span class="math-container">$y=\frac{x^2+1}{x+1}=x+1-\frac{2x}{x+1}$</span>.</p>
<p>Maybe it can be solved by using inequality.</p>
| Community | -1 | <p>Out of the given answers, one simple way is to get the <span class="math-container">$x$</span> in terms of <span class="math-container">$y$</span> and solve it :)</p>
<p><span class="math-container">$y=\frac{x^2+1}{x+1}$</span>
<span class="math-container">$$\Rightarrow yx + y = x^2 +1 \Rightarrow -x^2 + yx +(y-1)=... |
3,319,122 | <p>This is from Tao's Analysis I: </p>
<p><a href="https://i.stack.imgur.com/DYQxE.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/DYQxE.png" alt="enter image description here"></a></p>
<p>So far I managed to show (inductively) that these sets do exist for for every <span class="math-container">$\m... | Mishikumo2019 | 631,353 | <p>No, the function is really decreasing ! Indeed, if you differentiate the function, you have : <span class="math-container">$$f'(x)=-\left(\frac{e^x(x-1)+1}{(e^x-1)^2} \right)\le 0. $$</span>
And the function is continuous when <span class="math-container">$x=0$</span>. Indeed : <span class="math-container">$$e^x-1\s... |
2,370,716 | <p>Give the postive integer $n\ge 2$,and $x_{i}\ge 0$,such $$x_{1}+x_{2}+\cdots+x_{n}=1$$
Find the maximum of the value
$$x^2_{1}+x^2_{2}+\cdots+x^2_{n}+\sqrt{x_{1}x_{2}\cdots x_{n}}$$</p>
<p>I try
$$x_{1}x_{2}\cdots x_{n}\le\left(\dfrac{x_{1}+x_{2}+\cdots+x_{n}}{n}\right)^n=\dfrac{1}{n^n}$$</p>
| Michael Rozenberg | 190,319 | <p>Also we can make the following.</p>
<p>Let $x_1=x_2=...=x_{n-1}=0$ and $x_n=1$.</p>
<p>Hence, we get a value $1$ and it's a maximal value because for this we need to probe that
$$\sum_{i=1}^nx_i^2+\sqrt{\prod_{i=1}^nx_i}\leq\left(\sum_{i=1}^nx_i\right)^2,$$
which is true by AM-GM.</p>
<p>Indeed, let $\prod\limits... |
2,586,625 | <p>Consider the set $$S=\{x\mid x\in S\}.$$</p>
<p>For every element $x$, either $x\in S$ or $x\not\in S.$</p>
<p>If we know that $x$ is in fact element of $S$, then, by definition, $x\in S$ so it is true that $x\in S$.</p>
<p>If we know that $x\not\in S$, then, by definition, $x\not\in S$ so it is true that $x\not\... | CopyPasteIt | 432,081 | <p>There is the concept in $\text{Set Theory}$ of a 'collectivizing relation', in other words, a relation that can be used to define sets.</p>
<p>Starting with any set $S$, </p>
<p>$\tag 1 \text{There exists a set } X \text{ such that } X = \{x\mid x\in S \text{ AND } P(x) \}$</p>
<p>where P(x) is a statement about ... |
2,586,625 | <p>Consider the set $$S=\{x\mid x\in S\}.$$</p>
<p>For every element $x$, either $x\in S$ or $x\not\in S.$</p>
<p>If we know that $x$ is in fact element of $S$, then, by definition, $x\in S$ so it is true that $x\in S$.</p>
<p>If we know that $x\not\in S$, then, by definition, $x\not\in S$ so it is true that $x\not\... | JDH | 413 | <p>To define an object, such as a set, means to provide a property that that object and only that object has. </p>
<p>In your case, however, it turns out that <em>every</em> set $S$ has the property that $$S=\{x\mid x\in S\}.$$
This is just because every set is the set consisting of its elements. </p>
<p>So you haven... |
1,583,887 | <p>This problem is from an an Introduction to Abstract Algebra by Derek John that I am solving.</p>
<p>I am trying to prove that any group of order 1960 aren't simple, so I am doing it by contradiction, but I got stuck in the middle.</p>
<p>Suppose that $|G| = 1960 = 2^3 * 5 * 7^2$, by Sylow theory we have 2,5,7 sub... | Jyrki Lahtonen | 11,619 | <p>Posting my chatroom solution. It is more complicated than Alex Jordan's argument, but does give a bit more information about the groups of size $1960$.
More precisely, it shows that either a Sylow 5-subgroup, denoted $P_5$, or a Sylow 7-subgroup, denoted $P_7$, must be a normal subgroup. Also, we will see that $G$ a... |
3,561,807 | <p>There is this question regarding constrained optimisation. It says, a rectangular parallelepiped has all eight vertices on the ellipsoid <span class="math-container">$x^{2}+3y^{2}+3z^{2}=1$</span>. Using the symmetry about each of the planes, write down the surface area of the parallelepiped and therefore find the m... | Robert Lewis | 67,071 | <p>With <span class="math-container">$n$</span> odd, we have</p>
<p><span class="math-container">$n = 2k + 1. \; k \in \Bbb Z; \tag 1$</span></p>
<p>then</p>
<p><span class="math-container">$n^2 = 4k^2 + 4k + 1, \tag 2$</span></p>
<p>whence</p>
<p><span class="math-container">$n^2 - 1 = 4k^2 + 4k = 4(k^2 + k) \Lon... |
3,398,645 | <p>I have a doubt about value of <span class="math-container">$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 $</span></p>
<p>But in another book they are saying it doesn't exist.I am confused... | Kavi Rama Murthy | 142,385 | <p><span class="math-container">$|e^{i2\pi n}|=1$</span> for all <span class="math-container">$n$</span> and <span class="math-container">$|i2\pi n| \to \infty$</span>. So it is not true that <span class="math-container">$|e^{z}| \to \infty$</span> as <span class="math-container">$|z| \to \infty$</span>. Of course th... |
3,398,645 | <p>I have a doubt about value of <span class="math-container">$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... | robjohn | 13,854 | <p>In real analysis,
<span class="math-container">$$
\lim\limits_{x\to\infty}e^x=\infty
$$</span>
because the limit is taken along the positive real axis. Similarly,
<span class="math-container">$$
\lim\limits_{x\to-\infty}e^x=0
$$</span>
because the limit is taken along the negative real axis.</p>
<p>However, in comp... |
623,796 | <p>What's the domain of the function $f(x) = \sqrt{x^2 - 4x - 5}$ ?</p>
<p>Thanks in advance.</p>
| mathematics2x2life | 79,043 | <p>For the function $f$ you gave to be undefined, the input to the square root must be negative. First, notice that
$$
x^2-4x-5=(x+1)(x-5)
$$
So for this to be negative one of the terms must be negative and the other must be positive. So what numbers satisfy $x+1>0$ and $x-5<0$ or $x+1<0$ and $x-5>0$?</p>
|
623,796 | <p>What's the domain of the function $f(x) = \sqrt{x^2 - 4x - 5}$ ?</p>
<p>Thanks in advance.</p>
| Emi Matro | 88,965 | <p>The domain is $\text{domain}(f)\geq 0$, because since you are dealing with the reals, it cannot be negative inside the square root.
To determine domain, solve: $x^2-4x-5=0$: </p>
<p>$(x-5)(x+1)=0 \implies x=5,-1$ These are the roots, so the domain is: $(-\infty,-1] \cup [5,\infty)$. </p>
|
2,183,809 | <p>The problem:</p>
<blockquote>
<p>If $p$ is prime and $4p^4+1$ is also prime, what can the value of $p$ be?</p>
</blockquote>
<p>I am sure this is a pretty simple question, but I just can't tackle it. I don't even known how I should begin...</p>
| Will Jagy | 10,400 | <p>$$ 4 p^4 + 1 = (2 p^2 + 2p+1)(2 p^2 - 2p +1) = (2p(p+1)+1) (2p(p-1)+1)$$</p>
|
182,091 | <p>3D graphics can be easily rotated interactively by clicking and dragging with the mouse.</p>
<p>Is there a simple way to achieve the same for animated 3D graphics? I would like to rotate them interactively (in real time) <em>while</em> the animation is running.</p>
<hr>
<p>Here's an example animation, mostly take... | Kuba | 5,478 | <p>Here's another approach. There could be a problem in case plot's options change during animation, PlotRange/Ticks etc, currently only initial ones are preserved. Will try to come up with something more general later.</p>
<pre><code>DynamicModule[{viewPoint, viewVertical, plot}
,
plot[t_] :=
Plot3D[sol[t, x, y]... |
128,122 | <p>Original Question: Suppose that $X$ and $Y$ are metric spaces and that $f:X \rightarrow Y$. If $X$ is compact and connected, and if to every $x\in X$ there corresponds an open ball $B_{x}$ such that $x\in B_{x}$ and $f(y)=f(x)$ for all $y\in B_{x}$, prove that f is constant on $X$. </p>
<p>Here's my attempt:
Cover ... | Mike | 28,355 | <p>If you assume that U_0 is not equal to X initially, then the boundary points of U_0 must be in X, which implies by hypothesis that U_0 is closed, which is a contradiction unless U_0 equals X. Does this argument tacitly rely on the connectedness of X?</p>
|
2,185,072 | <p>Let A be the matrix: $$\begin{pmatrix} 1&2&3&2&1&0\\2&4&5&3&3&1\\1&2&2&1&2&1 \end{pmatrix}$$.</p>
<p>Show that {$\bigl( \begin{smallmatrix} 1 \\ 4\\3\end{smallmatrix} \bigr)$, $\bigl( \begin{smallmatrix} 3\\4\\1 \end{smallmatrix} \bigr)$} is a basis for th... | amd | 265,466 | <p>By finding the rref of $A$ you’ve determined that the column space is two-dimensional and the the first and third columns of $A$ for a basis for this space. The two given vectors, $(1,4,3)^T$ and $(3,4,1)^T$ are obviously linearly independent, so all that remains is to show that they also span the column space. You ... |
3,836,059 | <p>The following question is a last year's Statistics exam question I tried to solve (without any luck). Any help would be grateful. Thanks in advance.</p>
<p>An Atomic Energy Agency is worried that a particular nuclear plant has leaked radio-active material. They do <span class="math-container">$5$</span> independent ... | BruceET | 221,800 | <p>I will show the test and its result, leaving it to you to justify that it is a
LR test based on the sufficient statistic.</p>
<p>The sum <span class="math-container">$T$</span> of five readings is 18. Under <span class="math-container">$H_0: \lambda_T = 5(2) = 10,$</span> one has <span class="math-container">$P(T \g... |
287,043 | <p>Consider the problem of finding the limit of the following diagram:</p>
<p>$$ \require{AMScd} \begin{CD}
& & & & E
\\ & & & & @VVV
\\ && C @>>> D
\\ & & @VVV
\\A @>>> B
\end{CD} $$</p>
<p>The abstract definition of the limit involves an adjunction... | Mike Shulman | 49 | <p>This sort of calculus is central to the abstract study of homotopy limits via <a href="https://ncatlab.org/nlab/show/derivator" rel="noreferrer">derivators</a>. See <a href="https://arxiv.org/abs/1112.3840" rel="noreferrer">this paper</a> and <a href="https://arxiv.org/abs/1306.2072" rel="noreferrer">this one</a> f... |
85,814 | <p>how to solve $\pm y \equiv 2x+1 \pmod {13}$ with Chinese remainder theorem or iterative method?</p>
<p>It comes from solving $x^2+x+1 \equiv 0 \pmod {13}$ (* ) and background is following:</p>
<blockquote>
<p>13 is prime. (* ) holds under Euclidean lemma if and only if $4(x^2+x+1) \equiv \pmod {13}$
or if and ... | Phira | 9,325 | <p>Either, $\pm y-1$ is divisible by 2, so you divide by 2, or $\pm y+12$ is divisible by 2, so divide this by 2.</p>
|
3,840,643 | <p>Assume that given three predicates are presented below:</p>
<p><span class="math-container">$H(x)$</span>: <span class="math-container">$x$</span> is a horse</p>
<p><span class="math-container">$A(x)$</span>: <span class="math-container">$x$</span> is an animal</p>
<p><span class="math-container">$T(x,y)$</span>: <s... | lemontree | 344,246 | <p>Hints:</p>
<p>"<span class="math-container">$x$</span> is a <span class="math-container">$P$</span>'s tail" means that <span class="math-container">$x$</span> is a tail of <span class="math-container">$y$</span> and <span class="math-container">$y$</span> is a <span class="math-container">$P$</span>.</p>
<... |
1,785,414 | <p>I am trying to find a closed form for the integral $$I=\int_{\frac{\pi}{4}}^{\frac{3\pi}{4}}\frac{\lfloor|\tan x|\rfloor}{|\tan x|}dx$$ So far, my reasoning is thus: write, by symmetry through $x=\pi/2$, $$I=2\sum_{n=1}^{\infty}n\int_{\arctan n}^{\arctan (n+1)}\frac{dx}{|\tan x|}=2\sum_{n=1}^{\infty}n\ln\frac{\sin\a... | Jack D'Aurizio | 44,121 | <p>Maybe we are lucky. We may notice that:
$$ 1+\frac{2n+1}{n^2(n+1)^2} = 1+\frac{1}{n^2}-\frac{1}{(n+1)^2} $$
and the roots of the polynomial $x^2(x+1)^2+2x+1$ are given by
$$ \alpha = \frac{1}{2}\left(-1-\sqrt{2}-\sqrt{2\sqrt{2}-1}\right), $$
$$ \beta = \frac{1}{2}\left(-1-\sqrt{2}+\sqrt{2\sqrt{2}-1}\right), $$
$$ \g... |
66,801 | <p>In short, I am interested to know of the various approaches one could take to learn modern harmonic analysis in depth. However, the question deserves additional details. Currently, I am reading Loukas Grafakos' "Classical Fourier Analysis" (I have progressed to chapter 3). My intention is to read this book and then ... | Anil P | 15,700 | <p>you can also look as a primer lecture notes on topological groups Higgins, london Mathematical society very easy to read</p>
|
2,704,955 | <p>In my test on complex analysis I encountered following problem:</p>
<blockquote>
<p>Find $\oint\limits_{|z-\frac{1}{3}|=3} z \text{Im}(z)\text{d}z$</p>
</blockquote>
<p>So first I observed that function $z\text{Im}(z)$ is not holomorphic at least on real axis. Therefore we have to intgrate using parametrization.... | Mark Viola | 218,419 | <p>Note that since $\text{Im}(z)=\frac1{2i}(z-\bar z)$, that </p>
<p>$$z\text{Im}(z)=\frac1{2i}(z^2-|z|^2)$$</p>
<p>Since $z^2$ is analytic, we have</p>
<p>$$\begin{align}
\oint_{|z-\frac13 |=3}z\text{Im}(z)\,dz&=\frac i2\oint_{|z-\frac13 |=3}|z|^2\,dz\\\\
&=-\frac {3}2 \int_0^{2\pi} \left|\frac13 +3e^{i\phi... |
1,470,819 | <p>Let $f$ be defined (and real-valued) on $[a,b]$. For any $x\in [a,b]$ form a quotient $$\phi(t)=\dfrac{f(t)-f(x)}{t-x} \quad (a<t<b, t\neq x),$$ and define $$f'(x)=\lim \limits_{t\to x}\phi(t),$$ provided this limit exists in accordance with Defintion 4.1. </p>
<p>I have one question. Why Rudin considers $t\i... | Fabrice NEYRET | 277,841 | <p>In the x-semilog plot, the area under the curve is $A = \int exp(x)*f(exp(x)) $. The variable change $y = exp(x)$ yields $dx = dy/y$, so $ A = \int y/y*f(y) = \int f(y)$</p>
|
2,505,171 | <p>How many numbers are there if you do not allow leading $0$'s? </p>
<p>In how many of the numbers in each case is no digit $j$ in the $j$th place?</p>
<p>If leading $0$'s are allowed? </p>
<p>If they are not allowed? </p>
<p>I know how to answer this if the numbers $0$ through $9$ can be repeated, but I am gett... | Leonhard Euler | 481,442 | <p>10! (10 factorial)</p>
<p>Choose any of the 10 numbers (maybe a 6).
Then you have 9 numbers to choose for the next digit, and so on.</p>
<p>When you get rid of the leading 0, simply take away the 9! numbers that had a leading 0. </p>
<p>i.e. 10! - 9!</p>
|
1,593,679 | <p>While proving that $$\int^{\infty}_0 \frac{\sin x}xdx$$
I saw the Laplace Transform proof. <br>
It used that $$\cal L\left\{\frac{\sin t}{t}\right\}=\int^\infty_0 \cal L\left\{\sin(t)\right\}d\sigma$$
So for understanding it, I tried:
$$\cal L\left\{\frac{\sin t}{t}\right\}=\int^\infty_0e^{-st}\frac{\sin t}{t}dt=\i... | Fabian | 7,266 | <p>Your formula is a (too) short notation, suppressing the variable of the Laplace transform. It should read
$$\mathcal{L}\left\{\frac{\sin t}{t} \right\} (0) = \int_0^\infty \mathcal{L}\{ \sin t\}(\sigma) \,d\sigma.$$</p>
<p>This follows from the rule `<a href="https://en.wikipedia.org/wiki/Laplace_transform#Properti... |
1,652,747 | <p>Ok, so I think I'm getting the hang of this. Is this more or less on the right track?</p>
<p>$$e^{z-2}=-ie^2$$
$$e^ze^{-2}=-ie^2$$
$$e^z=-ie^4$$
$$\ln(e^z)=\ln(-ie^4)$$
$$z=\ln|-i|+iarg(-i)+2\pi ik+4$$
$$z=\frac{i\pi}{2}-\frac{i\pi}{2}+4+2\pi ik$$
$$z=4+2\pi ik$$</p>
| Kerr | 275,679 | <p>Note that $e^{z-2}=-ie^2=e^{2-\frac{i \pi}{2}+2\pi ki}$, so $$z-2=2-\frac{i \pi}{2}+2\pi ki$$
$$z=4+i(2\pi k-\frac{ \pi}{2}).$$
In your solution the last two rows are seems to be wrong, since $arg(-i)=-\pi/2$.</p>
|
3,477,795 | <p>How to use the absolute value function to translate each of the following statements into a single inequality.</p>
<p>(a) <span class="math-container">$\ x ∈ (-4,10) $</span> </p>
<p>(b) <span class="math-container">$\ x ∈ (-\infty,2] \cup[9,\infty) $</span></p>
<p>I think in the first one the absolute value of ... | hamam_Abdallah | 369,188 | <p><strong>hint</strong></p>
<p>Let <span class="math-container">$\epsilon>0$</span> given and consider the partition <span class="math-container">$\sigma$</span> definef by
<span class="math-container">$$\Bigl(1,2-\frac{\epsilon}{7},2+\frac{\epsilon}{7} ,4-\frac{\epsilon}{7}, 4+\frac{\epsilon}{7},7\Bigr)$$</span>
... |
3,477,795 | <p>How to use the absolute value function to translate each of the following statements into a single inequality.</p>
<p>(a) <span class="math-container">$\ x ∈ (-4,10) $</span> </p>
<p>(b) <span class="math-container">$\ x ∈ (-\infty,2] \cup[9,\infty) $</span></p>
<p>I think in the first one the absolute value of ... | CyclotomicField | 464,974 | <p>Since <span class="math-container">$\int_a^b f(x) dx = \int_a^cf(x) dx + \int_c^b f(x)dx$</span> you can split this integral into three parts, as <span class="math-container">$\int_1^7 f(x) dx = \int_1^2 f(x) dx + \int_2^4 f(x)dx + \int_4^7 f(x) dx$</span>. Since <span class="math-container">$f(x)$</span> is constan... |
192,784 | <p>First let me try to describe in more details below the approach of
"reordering" digits of Pi, which is used in OEIS A096566</p>
<p><a href="https://oeis.org/A096566" rel="nofollow noreferrer">https://oeis.org/A096566</a></p>
<p>and what I have done analyzing it so far.</p>
<p>I am looking at first 620 "reordered"... | binn | 39,264 | <p>"some sort of transition from initial order (within first 620 digits) to total randomness ... Does such concept of transition from order to randomness exist ?"</p>
<p>If the phenomenon is that pi digit summaries are different from your expectation when you use a small sample size but are close to your expectation w... |
40,500 | <blockquote>
<p>What are the most
fundamental/useful/interesting ways in
which the concepts of Brownian motion,
martingales and markov chains are
related?</p>
</blockquote>
<p>I'm a graduate student doing a crash course in probability and stochastic analysis. At the moment, the world of probability is a conf... | Simon Lyons | 9,564 | <p>Levy's characterisation of Brownian motion:</p>
<p>If $X$ is a continuous martingale and $X$ has quadratic variation process $[ X ]_t = t$ then $X$ is a standard Brownian motion.</p>
|
59,486 | <p>Chern-Weil theory tells us that the integral Chern classes of a flat bundle over a compact manifold (i.e. a bundle admitting a flat connection) are all torsion. Given a compact manifold $M$ whose integral cohomology contains torsion, one can then ask which (even-dimensional) torsion classes appear as the Chern clas... | Ben Wieland | 4,639 | <p>The answer to Tom's formulation is no. It's possible if you restrict to finitely generated groups that my argument falls apart, but I doubt this is essential.</p>
<p>Take a group $\Gamma$ so that $B\Gamma^+=K(Q/Z,2n-1)$, ie, $H^k(\Gamma;Z)=H^k(K(Q/Z,2n-1);Z)$. Since $Ext(Q/Z,Z)=\hat Z$, there lots of interesting cl... |
2,676,200 | <blockquote>
<p>A hyperbola has equation $\frac{x^2}{4}-\frac{y^2}{16}=1$. Show that every other line parallel to this asymptote, $y=2x$, intersects the hyperbola exactly once.</p>
</blockquote>
<p><a href="https://i.stack.imgur.com/gFB6v.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/gFB6v.png" ... | cansomeonehelpmeout | 413,677 | <p>You are actually done at $$c^2+4cx+16=0$$</p>
<p>Remember that you're solving for $x$, and you get only one such $x$, unless $c=0$.</p>
|
11,629 | <p>The basic logic course in school gives the impression that logic has both the syntax and the semantics aspects. Recently, I wonder whether the syntax part still plays an essential role in the current studies. Below are some of my observations, I hope the idea from the community can make them more complete.</p>
<p>M... | Neel Krishnaswami | 1,610 | <p>I think your observation is a very good one, but this phenomenon is limited to classical logic and does not continue to hold when we move to intuitionistic or substructural logics.</p>
<p>One way of understanding the role of syntax is to take the connectives of logic as explaining what counts as a legitimate proof ... |
46 | <p>I have solved a couple of questions myself in the past, and I think some of them are interesting to the public and will most likely appear in the future. One example for this is the question how to enable antialiasing in the Linux frontend, for which there is no native support right now. My question would now be whe... | Brett Champion | 69 | <p>If it's an interesting question, ask it. But give someone else a chance to answer it first -- you may learn something new.</p>
|
2,975,665 | <p>I am asked to find the sum of the series <span class="math-container">$$\sum_{n=0}^\infty\frac{(x+1)^{n+2}}{(n+2)!}$$</span></p>
<p>For some reason (that I don't understand) I can't apply the techniques for finding the sum of the series that I usually would to this one? I think the others that I have done have been... | Claude Leibovici | 82,404 | <p>What your teacher was meaning is that
<span class="math-container">$$\frac d{dx} \sum_{n=0}^\infty\frac{(x+1)^{n+2}}{(n+2)!}= \sum_{n=0}^\infty\frac{(x+1)^{n+1}}{(n+1)!}\tag 1$$</span>
<span class="math-container">$$\frac {d^2}{dx^2} \sum_{n=0}^\infty\frac{(x+1)^{n+2}}{(n+2)!}= \sum_{n=0}^\infty\frac{(x+1)^{n}}{(n)!... |
628,236 | <p>I am trying to give a name to this axiom in a definition: </p>
<p>$(X \bullet R) \sqcup (Y \bullet S) \equiv (X \sqcup Y) \bullet (R \sqcup S)$</p>
<p>(for all $X, Y, R, S$) where $\sqcup$ is the join of a lattice and $\bullet$ is some binary operation. It feels related to monotonicity/distributivity but I don't k... | Community | -1 | <p>You can see it is a cone. It is irreducible because its smooth part is connected and dense. Compute where it is smooth using the Jacobian.</p>
|
31,261 | <p>Given a 3-manifold $M$, one can define the Kauffman bracket skein module $K_t(M)$ as the $C$-vector space with basis "links (including the empty link) in $M$ up to ambient isotopy," modulo the skein relations, which can be found in the second paragraph of section two of <a href="http://arxiv.org/abs/math/0402102" re... | Charlie Frohman | 4,304 | <p>There is a gap in the proof that the peripheral ideal is nontrivial in that paper. </p>
<p>Thang Le and Stavros came up with a more algebraic way of definining a closely related
ideal that they could prove was nontrivial.</p>
<p>I think its a great problem. A good starting point might be to prove it for Torus k... |
2,164,946 | <p>I'm working on arc length calculation and area of surface of revolution in calculus and I'm really quite stuck on the process of how to do this. Here is a particular problem that I'm struggling with:</p>
<blockquote>
<p>Find the surface area of the surface of revolution generated by revolving the graph $$y=x^3; \... | Ahmed Al Dahmani | 443,872 | <p>You should place the X^3 inside the integral as the formula. Review the formula and you should be able to see your mistake :)</p>
|
1,688,184 | <blockquote>
<p>Prove that $2\sqrt 5$ is irrational</p>
</blockquote>
<p><strong>My attempt:</strong></p>
<p>Suppose $$2\sqrt 5=\frac p q\quad\bigg/()^2$$ </p>
<p>$$\Longrightarrow 4\cdot 5=\frac{p^2}{q^2}$$</p>
<p>$$\Longrightarrow 20\cdot q^2=p^2$$</p>
<p>$$\Longrightarrow q\mid p^2$$</p>
<p>$$\text{gcd}(p,q)... | Roman83 | 309,360 | <p>$$2\sqrt5 = \frac pq, \gcd (p,q=1)$$
$$20q^2=p^2 \Rightarrow 5|p^2 \Rightarrow 5|p \Rightarrow 25|p^2$$
Let $p=5p_1$
$$20q^2=25p_1^2 \Rightarrow 5|q $$
$$\gcd (p,q)\geq 5$$
Сontradiction.</p>
|
540,217 | <p><strong>Question:</strong></p>
<p>$\int ^1_0 \frac {\ln x}{1-x^2}dx$ - converges or diverges?</p>
<p><strong>What we did:</strong></p>
<p>We tried to compare with $-\frac 1x$ and $-\frac 1{x-1}$ but ended up finding that these convergence tests fail. Our book says this integral diverges, but Wolfram on the other ... | Riemann1337 | 98,640 | <p>It converges and its value is $-\pi^2/8$.</p>
|
556,855 | <p>Given a $\triangle ABC$ with sides $AB=BC$ and $\angle B=100^\circ $,
prove that $$a^3 + b^3 = 3a^2b$$
where $a=AB=BC$ and $b=AC$,</p>
<p>I have tried to use simultaneously the sine and cosine rules as well as the Pythagorean Theorem with all my attempts failing to prove that $LHS =RHS$. I would greatly appreciate... | Priyatham | 106,406 | <p>A straight forward application of cosine rule should tell you that
$$
b = 2a\sin(50)
$$</p>
<p>Consider </p>
<p>$$
\begin{equation}
\begin{split}
a^3 + b^3 - 3a^2b & = a^3(1+8\sin^350-6\sin50) \\
& = a^3(1+8\frac{(3\sin50 - \sin 30)}{4}-6\sin50) \\
& = a^3(1+6\sin50-2\sin30-6\sin50) \\
& = 0
\end{s... |
556,855 | <p>Given a $\triangle ABC$ with sides $AB=BC$ and $\angle B=100^\circ $,
prove that $$a^3 + b^3 = 3a^2b$$
where $a=AB=BC$ and $b=AC$,</p>
<p>I have tried to use simultaneously the sine and cosine rules as well as the Pythagorean Theorem with all my attempts failing to prove that $LHS =RHS$. I would greatly appreciate... | Jonas Kgomo | 45,379 | <p>$a^3+b^3=3a^2b$
using cosine</p>
<p>$b^2=a^2+a^2-2aa\cos \beta,\beta<100$ </p>
<p>$b^2=2a^2(1-\cos \beta)\\let \\(\frac{b}{a})^2=t^2=2(1-\cos \beta)$</p>
<p>$a^3+b^3=3a^2b\implies 1+t^3=3t\implies t^3-3t+1=0$</p>
<p>$t(t^2-3)=-1\implies t^2(t^2-3)^2=2(1-\cos \beta)(1-\cos\beta-3)^2\\=2(1-\cos \beta)(1+4\cos \... |
2,284,178 | <p>Let the roots of the equation:
$2x^3-5x^2+4x+6$ be $\alpha,\beta,\gamma$</p>
<ol>
<li>State the values of $\alpha+\beta+\gamma,\alpha\gamma+\alpha\beta+\beta\gamma,\alpha\beta\gamma$</li>
<li>Hence, or otherwise, determine an equation with integer coefficients which has $\frac{1}{\alpha^2}\frac{1}{\beta^2}\frac{1}{... | Donald Splutterwit | 404,247 | <p>We have
\begin{eqnarray*}
\alpha+\beta+\gamma =\frac{5}{2} \\
\alpha\beta+\beta\gamma+\gamma\alpha =2 \\
\alpha\beta\gamma =-3
\end{eqnarray*}
Now calculate
\begin{eqnarray*}
(\alpha+\beta+\gamma)^2 =\alpha^2+\beta^2+\gamma^2+2(\alpha\beta+\beta\gamma+\gamma\alpha) \\
(\alpha\beta+\beta\gamma+\gamma\alpha )^2=\alp... |
1,590,262 | <p>Let $ABC$ be of triangle with $\angle BAC = 60^\circ$
. Let $P$ be a point in its interior so that $PA=1, PB=2$ and
$PC=3$. Find the maximum area of triangle $ABC$.</p>
<p>I took reflection of point $P$ about the three sides of triangle and joined them to vertices of triangle. Thus I got a hexagon having area doubl... | Rory Daulton | 161,807 | <p>Let $\theta=\measuredangle PAB$ in the triangle you specify. Then
$\measuredangle PAC=60°-\theta$.</p>
<p><a href="https://i.stack.imgur.com/8gVGc.png" rel="noreferrer"><img src="https://i.stack.imgur.com/8gVGc.png" alt="enter image description here"></a></p>
<p>By the law of cosines,</p>
<p>$$2^2=1^2+c^2-2\cdot... |
1,590,262 | <p>Let $ABC$ be of triangle with $\angle BAC = 60^\circ$
. Let $P$ be a point in its interior so that $PA=1, PB=2$ and
$PC=3$. Find the maximum area of triangle $ABC$.</p>
<p>I took reflection of point $P$ about the three sides of triangle and joined them to vertices of triangle. Thus I got a hexagon having area doubl... | achille hui | 59,379 | <p>Let $\mathcal{A}$ be the area of $\triangle ABC$.
Let $\theta$ and $\phi$ be the angles $\angle PAC$ and $\angle BAP$ respectively.<br>
We have $\theta + \phi = \angle BAC = \frac{\pi}{3}$.
As functions of $\theta$ and $\phi$, the side lengths $b$, $c$ and area $\mathcal{A}$ are:</p>
<p>$$
\begin{cases}
c(\theta) &... |
1,977,577 | <p>if a function is lebesgue integrable, does it imply that it is measurable?
(without any other assumption)</p>
<p>The reason why I ask this is because royden, in his book, kind of imply about a measurable function when assuming the function to be lebesgue integrable</p>
| True_False | 608,343 | <p>Actually, this is the converse of the following theorem which you can start from its end to answer your question:</p>
<p>Let <span class="math-container">$f$</span> be a bounded measurable function on a set of finite measure <span class="math-container">$E$</span>. Then, <span class="math-container">$f$</span> is i... |
1,303,183 | <p>If I have a vector space $V$ ( of dimension $n$ ) over real numbers such that $\{v_1,v_2...v_n\}$ is the basis for the space ( not orthogonal ). Then I can write any vector $l$ in this space as $l=\sum_i\alpha_iv_i$. Here $\alpha_1,\alpha_2...\alpha_n$ are the coefficients that define the vector $l$ according to th... | Nick Alger | 3,060 | <p>Here is an alternative approach based on the existence and uniqueness of the matrix inverse for square matrices with full column rank. It is slightly circular in the sense that proofs of the existence and uniqueness of this matrix inverse usually use arguments similar to the other answers here. However, hopefully t... |
258,215 | <p>How can we show that if $f:V\to V$
Then for each $m\in \mathbb {N}$ $$\operatorname{im}(f^{m+1})\subset \operatorname{im}(f^m)$$
Please help,I am stuck on this.</p>
| Philip Benj | 52,337 | <p>The inverse image of a function exists if it is bijective.</p>
|
3,259,193 | <p>I have a simple question about notation regarding limits, specifically, <span class="math-container">$$\lim_{\|x\| \rightarrow \infty}f(x).$$</span> </p>
<p><strong>Question:</strong> </p>
<p><span class="math-container">$\lim_{\|x\| \rightarrow \infty}f(x)$</span>:</p>
<p>In words what we are doing is taking the... | KcH | 36,116 | <p>You use the analogy of <span class="math-container">$\lim_{x \rightarrow \infty} f(x)$</span>, saying</p>
<blockquote>
<p>I would visualize our value of just tending towards "infinity" on a graph.</p>
</blockquote>
<p>This says to me that you are thinking of functions with domain <span class="math-container">$\... |
312,649 | <p>Hello how to show the following:</p>
<p>Let $(X,\tau)$ be a topological space then a single point is compact but not
necessarily closed.</p>
<p>Thank you!</p>
| Robert Israel | 8,508 | <p>The topological space $X$ is $T_1$ (i.e. for any two distinct points $x$, $y$, $x$ has an open neighbourhood that does not contain $y$) if and only if all single-point sets are closed.</p>
|
1,549,490 | <p>$f(x) = 3x - \frac{1}{x^2}$</p>
<p>I am finding this problem to be very tricky:</p>
<p><a href="https://i.stack.imgur.com/7RjYG.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/7RjYG.jpg" alt="enter image description here"></a></p>
<p><a href="https://i.stack.imgur.com/jwUHp.jpg" rel="nofollow n... | Mikasa | 8,581 | <p>Just a hint:</p>
<p>Use anothe version of differentiation: </p>
<p>$f'(a)=\lim_{x\to a}\frac{f(x)-f(a)}{x-a}= \lim_{x\to a}\frac{(3x-1/x^2)-(3a-1/a^2)}{x-a}=\cdots=\lim_{x\to a}\frac{3a^2x^2(x-a)-(x-a)(x+a)}{x^2a^2(x-a)}=??$</p>
|
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