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 |
|---|---|---|---|---|
4,048,102 | <p>I came up with the following ODE:</p>
<p><span class="math-container">$\frac{dx}{dt}+\frac{dy}{dt}=k_1(a-x-y)(b-x)+k_2(a-x-y)(c-y)$</span></p>
<p>where <span class="math-container">$k_1,k_2,a,b,c$</span> are constants. I know under changes of variable, the equation can be changed into the following form:</p>
<p><spa... | JLinsta | 895,235 | <p>First start with the base case, i.e. show it is true for <span class="math-container">$n=4$</span>. For the inductive case, assume <span class="math-container">$n! \geq 2^n$</span> and show <span class="math-container">$(n+1)! \geq 2^{n+1}$</span>. Hint: <span class="math-container">$(n+1)! = (n+1)n!$</span></p>
<p>... |
2,128,825 | <p>$$\int^{x}_{0}tf(t)dt = x\sin(x)+\cos(x)-1$$
find $f(\pi), f'(x)$</p>
<p>This question confuses me because usually, the way I have seen questions like these, they have been in the form:</p>
<p>$$H(x) = \int^{x}_{a} f(t)dt$$</p>
<p>This form is kind of different so I am not sure how I would solve it.</p>
<p>My t... | S.C.B. | 310,930 | <p>I'm afraid your answer is incorrect. </p>
<p>If you differentiate both sides, like normal, then using the product rule, we get that $$ xf(x)=x\cos x \iff f(x)=\cos x$$ So $f(\pi)=-1, f'(x)=-\sin x$. </p>
<p><strong>EDIT</strong></p>
<p>As pointed out by @celtschk, we require that $f(x)$ is continous in order for ... |
2,128,825 | <p>$$\int^{x}_{0}tf(t)dt = x\sin(x)+\cos(x)-1$$
find $f(\pi), f'(x)$</p>
<p>This question confuses me because usually, the way I have seen questions like these, they have been in the form:</p>
<p>$$H(x) = \int^{x}_{a} f(t)dt$$</p>
<p>This form is kind of different so I am not sure how I would solve it.</p>
<p>My t... | Community | -1 | <blockquote>
<p>Why is my method wrong?</p>
</blockquote>
<p>To do correctly, we have to use the concept of <a href="https://en.m.wikipedia.org/wiki/Leibniz_integral_rule#General_form:_Differentiation_under_the_integral_sign" rel="nofollow noreferrer">differentiation under the integral sign</a>. We thus have, $$\fra... |
241,683 | <p>EDIT: I adjusted the vertices to have labels that are integers (like the weights). Can the answer be adapted to this case?</p>
<p>I use simultaneous display of vertices and weights (this topic is related to another question posted <a href="https://mathematica.stackexchange.com/questions/154513/vertex-labels-of-graph... | mikado | 36,788 | <p>I think you would like to show the result in terms of <code>1-h</code> rather than <code>-(-1+h)</code>. To achieve this, I would do the following</p>
<pre><code>expr = FullSimplify[Integrate[t1^2*E^((1 - h)*s0*t1), {t1, 0, T}]];
rule = h -> 1 - HoldForm[1 - h];
expr /. rule;
</code></pre>
<p>You can use the r... |
99,936 | <p>I am new to this and I've plotted this:Plot[3 ArcSin[x + 4] - 16, but I don't know what/how to specify the range?</p>
| eldo | 14,254 | <pre><code>fun = 3 ArcSin[x + 4] - 16;
FunctionDomain[fun, x]
</code></pre>
<blockquote>
<p>-5 <= x <= -3</p>
</blockquote>
<pre><code>Plot[fun, {x, -5, -3}]
</code></pre>
<p><a href="https://i.stack.imgur.com/UtRkv.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/UtRkv.jpg" alt="enter ima... |
241,586 | <p>It is perhaps well known that the sign function is discontinuous, if defined for $f:\mathbb{R}\rightarrow \mathbb{R}$. However, if we were to define the sign function for $f:\mathbb{R} \setminus \left \{ 0 \right \}\rightarrow \mathbb{R}$, would the sign function still remain discontinuous? </p>
<p>My belief is yes... | Martin Argerami | 22,857 | <p>Any function becomes continuous if you remove its points of discontinuity from the domain. </p>
<p>In your case, the only discontinuity is at $0$, so by removing $0$ from the domain you make the function continuous. </p>
|
1,877,632 | <p>My integral calculus is rusty.
How do I calculate the interior area (blue region) of four bounding circles?<br><br>
<a href="https://i.stack.imgur.com/VtQIy.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/VtQIy.png" alt="enter image description here"></a></p>
| preferred_anon | 27,150 | <p>Consider the square joining the centres of the circles:</p>
<p><a href="https://i.stack.imgur.com/YBzUD.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/YBzUD.png" alt="enter image description here"></a></p>
|
2,560,602 | <p>Part (a)
Suppose $f'(z)$ is a complex derivation of $f(z)$. Since $f'(z)$ takes complex values, $f'(z)$ is a $2 \times 1$ column vector in $\mathbb{R}^2$. </p>
<p>Part (b)
If I interpret $f$ as function between $\mathbb{R}^2 \rightarrow \mathbb{R}^2$, then the derivation $f'(z)$ becomes a $2 \times 2$ linear tranf... | zhw. | 228,045 | <p>They don't match up because they are different objects. This should be familiar: for functions $f:\mathbb R\to \mathbb R,$ $f'(x)$ is a number, while $Df(x)$ is the unique linear transformation from $\mathbb R$ to $\mathbb R$ such that $f(x+h)-f(x) = Df(x)(h) + o(h).$ We do have $Df(x)(h) = f'(x)h$ for all $h,$ so t... |
3,919,925 | <p>If we randomly pick a real number from the number line, the probability of picking a number (say x) is 0. This is true for all real numbers x and it makes sense to me why this must be true. But seemingly there is a paradox lurking here. Suppose we pick the number y. This number also had probability 0 but was still c... | Arthur | 15,500 | <p>Probability zero does not mean impossible. Just as probability 1 doesn't mean guaranteed to happen.</p>
<p>A different example of the same phenomenon is this: Flip a coin until you get a heads. The probability that you get a heads at some point and therefore stop is 1. But it clearly isn't completely, absolutely gua... |
410,763 | <p>I am having hard time recalling some of the theorems of vector calculus. I want to calculate the volume integral of the curl of a vector field, which would give a vector as the answer. Is there any formula? As far as I can recall, maybe I can write</p>
<p><span class="math-container">$$\int \nabla \times \vec{A} \ \... | Muphrid | 45,296 | <p>This is one of the many guises of the fundamental theorem of calculus (aka generalized Stokes' theorem). The fundamental theorem can be written like so, in geometric calculus, on a region $\Omega$ that is $n$-dimensional in $\mathbb R^n$:</p>
<p>$$\int_\Omega |dV| \, \nabla A = \oint_{\partial \Omega} |dS| \, \hat... |
2,411,890 | <blockquote>
<p>If $x$ is a real number, then $|x+1| \leq 3$ implies that $-4 \leq x \leq 2$.</p>
</blockquote>
<p>I've tried to prove this by exhaustion, is that the right way to prove it? </p>
| basket | 294,706 | <p>A good thing to think about expressions like $| x+ a| \leq b $ is that they are equivalent to saying $|x - (-a)| \leq b$, or that the 'distance' between $x$ and $-a$ is less than $b$. Then in the context of your problem it is clear that $x$ lies between the bounds given.</p>
|
2,411,890 | <blockquote>
<p>If $x$ is a real number, then $|x+1| \leq 3$ implies that $-4 \leq x \leq 2$.</p>
</blockquote>
<p>I've tried to prove this by exhaustion, is that the right way to prove it? </p>
| Etched | 404,854 | <p>I'd say that the easiest way to prove this is to simply solve the inequality for x.</p>
<p>We know that:</p>
<p>|x+1| = 3, simplifies to:</p>
<p>x+1 = +3 AND x+1=-3 </p>
<p>So by this logic, we can say that:</p>
<p>x+1 ≤ 3 AND x+1 ≥ -3 (switching the ≤ to ≥ when we change the sign on the 3)</p>
<p>x ≤ ... |
1,815,975 | <p>Someone is planning a round-the-world trip that involves visiting $2n$
cities, with two cities from each of $n$ different countries. He can choose a city to start and end the journey in, with the other $2n-1$ cities being visited exactly once. However, he has the restriction that the two cities from each c... | Henry | 6,460 | <p>Apart from being a count, rather than a probability, this is essentially the same as <a href="https://math.stackexchange.com/questions/465318/showing-probability-no-husband-next-to-wife-converges-to-e-1">Showing probability no husband next to wife converges to $e^{-1}$</a> so you can rewrite your expression as $$\di... |
695,948 | <p>I don't have complex analysis at my beck and call, and I only have a low level of knowledge in topology, but I need to prove that this metric space (for any real $r$ and $R$ with $r < R$)$$ X = \{ (x, y) \in \mathbb{R}^2 \ | \ r \leq x^2 + y^2 \leq R \}$$</p>
<p>with the Manhattan metric $d((x_1, y_1), (x_2, y_2... | Disintegrating By Parts | 112,478 | <p>Assume you have a continuous $g$ as stated by you.</p>
<p>Show that, for each $t \in [0,1]$, there exists a continuous function $\theta_{t} : [0,1]\rightarrow\mathbb{R}$ such that $\theta_{t}(0)=-\pi/2$ and such that $g(s,t)=|g(s,t)|(\cos\theta_{t}(s),\sin\theta_{t}(s))$. (Connectedness of $[0,1]$ can be helpful.) ... |
242,199 | <p>I was reading the solution to this problem and noticed it used $X = \sum_{i=1}^n X_i$ and $Y = \sum_{i=1}^n Y_i$. I think I understand all other parts except this one. Would you please explain why $X = \sum_{i=1}^n X_i$ and $Y = \sum_{i=1}^n Y_i$?</p>
<p><img src="https://i.stack.imgur.com/hHP8n.png" alt="enter ima... | Brian M. Scott | 12,042 | <p>$X_k$ and $Y_k$ are always either $0$ or $1$. $X_k=1$ precisely when the $k$-th die shows a $1$, and $Y_k=1$ precisely when the $k$-th die shows a $2$. When you sum the $X_k$’s, you add $1$ for each die that shows a $1$ and $0$ for each die that shows something else, so the total is simply the number of dice showing... |
666,103 | <p>I am studying elementary number theory, and just started learning about divisors. I always, try to read several other sources mostly because it helps me understand ideas better, also the textbook I am using- is not always clear for me. </p>
<p>Some sources state that 0|0 is not possible, while others allow 0|0. <... | Marc van Leeuwen | 18,880 | <p>There is no mathematical reason to deny $0\mid 0$ of truth (and I don't think that even those who would not write that, would explicitly write $0\nmid0$ either), but there might be a linguistic one: the symbol is pronounced "divides", and I think most people would avoid saying "$0$ divides $0$" because we all know o... |
2,633,246 | <p>I need to find how many ways you can arrange the characters in Permutation with the $N$ appearing before all of the vowels but after the $P$. I understand that you can place the $N$ and $P$ $C(6,2)$ ways, but I'm not sure how to accommodate for the vowels. I could place the $N$, the $P$, and the vowels $(6,3)$ ways ... | Joffan | 206,402 | <p>Group 1: Permute the vowels $AEIOU$ $5! $ ways, then add $PN$ on front</p>
<p>Group 2: Separately permute the remaining consonants $RMTT$, $\frac {4!}{2!}$ ways</p>
<p>Interleave the two groups choosing $4$ from $11$, $\binom {11}{4}$</p>
<p>Multiply accumulated options</p>
|
169,998 | <p>If I have a point that is considered the origin and two lines that extend outwards infinitely to two other points, what is the best way to determine whether or not a fourth point resides on or within the angle created by those points?</p>
<p>The process I'm currently using is to get the angle of all three lines tha... | erlking | 31,089 | <p>If I understand you correctly you want to know whether the fourth point lies in the cone generated by the three other points with the origin as its apex. In that case, you can do this: Let $O,A,B$ denote the given vectors where $O$ is the origin and $C$ the fourth one. Unless $A$ and $B$ are linearly dependent you c... |
124,955 | <p>Euler and Lamé are said to have proven FLT for $n=3$ that is, they are believed to have shown that $x^3 + y^3 = z^3$ has no nonzero integer solutions. According to Kleiner they approached this by decomposing $x^3 + y^3$ into $(x + y)(x + y\omega)(x + y\omega^2)$ where $\omega$ is the primitive cube root of unity or ... | Antonio Vargas | 5,531 | <p>Ribenboim's <em>Fermat's Last Theorem for Amateurs</em> is an excellent resource for this.</p>
<p>You can also check out <a href="http://fermatslasttheorem.blogspot.ca/" rel="noreferrer">this blog</a>, but I always found it hard to navigate.</p>
|
4,128,041 | <p>Assume <span class="math-container">$Z,B \in C^{2 \times 2}$</span> and that <span class="math-container">$c \in C$</span> is an eigenvalue of <span class="math-container">$Z$</span> and <span class="math-container">$u \in C$</span> is an eigenvalue of <span class="math-container">$B$</span>. Then <span class="math... | Siong Thye Goh | 306,553 | <p>Let <span class="math-container">$Z=diag(1,-1)$</span> and <span class="math-container">$B=diag(0,1)$</span>, it is possible to form <span class="math-container">$4$</span> distinct sums but <span class="math-container">$Z+B$</span> only have two eigenvalues.</p>
<p>Hence the statement is false.</p>
|
2,496,309 | <p>i am wondering if the unity elements of a ring form a ring ? In other words do they form an abelian group under addition ? I have tryied but i have not reached to a conclusive answer. Thanks for any comment.</p>
| Andrew Tindall | 497,231 | <p>The method you'll want to use is multiplying the denominator and numerator both by $\sqrt{h+4} + 2$. Remember to switch the minus to a plus ( or vice versa), otherwise you are just squaring the denominator.</p>
|
84,036 | <p>I was doing an optimization but facing a problem getting what exactly Minimize function do. I run the following code:</p>
<pre><code> Log1[x_] := If[x == 0, 0, Log2[Abs[x]]];
VEntropy[x_] := -(x Log1[x] + (1 - x) Log1[1 - x]);
Prob[a_, b_, x_, y_] := 1 - 1/((a/x)^2 + (b/y)^2);
Cost[a_, b_, x_, y_] :=... | dantopa | 29,452 | <p>This sidesteps the question. To find the minimum on the constrained surface try <code>FindMinimum[Cost[0.27, 0.96286, x, Sqrt[1 - x^2]], {x, 0.1}]</code>
with the result
<code>{0.873162, {x -> 0.244627}}</code>.</p>
<p>The curve takes this form <code>Plot[Cost[0.27, 0.96286, x, Sqrt[1 - x^2]], {x, -1, 1}]</cod... |
468,291 | <p>I want to evaluate
$$\lim_{n \to \infty} n^{3/2}\int_0^1 \frac{x^2}{(1+x^2)^n}\ dx$$</p>
<p>All that I needed is an intergrable control function $g(\cdot)$ independent of $n \in \mathbb{N}$ such that $n^{3/2} \frac{x^2}{(1+x^2)^n}\leq g(x)$, but I do not find direct control function anyway....</p>
| Ron Gordon | 53,268 | <p>I would attack this a different way. Rewrite the integral as</p>
<p>$$I(n) = \int_0^1 dx \, x^2 \, e^{-n \log{(1+x^2)}}$$</p>
<p>Note that the maximum value of the integrand is at $x_0=1/\sqrt{n-1}$. Thus, as $n\to\infty$, the integral value is dominated by $x\in [x_0-\epsilon,x_0+\epsilon]$, for small $\epsilon... |
4,052,739 | <p>Given fundamental groupoid <span class="math-container">$\Pi_1(S^1)$</span> of the circle, how can one define a topology on it? The information on <a href="https://ncatlab.org/nlab/show/fundamental+groupoid#topologizing_the_fundamental_groupoid" rel="nofollow noreferrer">nlab</a> did little help other than the fact ... | Community | -1 | <p>Grupoid of <span class="math-container">$Y$</span> is the factor of the set of all map <span class="math-container">$[0,1]\to Y$</span> modulo homotpies. On the set of all maps between two toplogical spaces <span class="math-container">$Map(X,Y)$</span> there is so-called compact-open toplogy:<a href="https://en.wik... |
4,052,739 | <p>Given fundamental groupoid <span class="math-container">$\Pi_1(S^1)$</span> of the circle, how can one define a topology on it? The information on <a href="https://ncatlab.org/nlab/show/fundamental+groupoid#topologizing_the_fundamental_groupoid" rel="nofollow noreferrer">nlab</a> did little help other than the fact ... | Noel Lundström | 520,038 | <p>In order to make the groupoid <span class="math-container">$\Pi_1(X)$</span> into a topological groupoid we need to put a topology on the set of objects of <span class="math-container">$\Pi_1(X)$</span> and a topology on the set of morphisms of <span class="math-container">$\Pi_1(X)$</span>. This is an example of a ... |
1,461,919 | <p>I am asked to calculate $\displaystyle \lim_{n\rightarrow\infty} \int_n^{n+1}\frac{\sin x} x \, dx$.</p>
<p>Before letting the $\lim$ to confuse me I used integration by parts, but it didn't get me far.</p>
<p>Any hint?</p>
| DeepSea | 101,504 | <p>Let $y = x-n \Rightarrow dx = dy \Rightarrow \displaystyle \int_{n}^{n+1} \dfrac{\sin x}{x} dx = \displaystyle \int_{0}^1 \dfrac{\sin(y+n)}{y+n}dy$. We treat this as Lebesgue integral, and the function $\dfrac{\sin(y+n)}{y+n}$ is dominated by $1$ which is integrable over $[0,1]$ because $\sin(y+n) \leq 1 \leq y+n, ... |
9,022 | <p>Sorry if this question is below the level of this site: I've read that the quotient of a Hausdorff topological group by a closed subgroup is again Hausdorff. I've thought about it but can't seem to figure out why. Is it obvious? A simple yes or no (with reference is possible) is all I need.</p>
| Jorge Vitório Pereira | 605 | <p>Edit: Below I expand my crude original answer "<a href="http://en.wikipedia.org/wiki/Hausdorff_space" rel="nofollow">Yes</a>" as requested
by the community.</p>
<hr>
<p>Yes. Let $G$ be the group and $H$ be the closed subgroup. The <a href="http://en.wikipedia.org/wiki/Kernel_of_a_function" rel="nofollow">kernel</... |
492,408 | <p>solve recurrence relation $a_n = 6 a_{n–1} – 9 a_{n–2}$,
where $a_0 = 1$ and $a_1 = 6$ and Verify, using Principle of Mathematical
Induction, that $a_n = 3^n + n 3^n$.</p>
<p>ans: i have done so far...<br>
put $a_n=b_n$<br>
$a_n-6a_{n-1}+9a_{n-2}=0$<br>
$b_n-6b_{n-2}+9a_{n-2}=0$<br>
$b^2-6b+9=0$, $b=3,3$</p>
<p>ge... | Brian M. Scott | 12,042 | <p>It makes no sense to ‘put $a_n=b_n$’ when you have no $b_n$ in the problem. What you wanted is the auxiliary equation $b^2-6b+9=0$, which you correctly solved to find the general solution</p>
<p>$$a_n=(c_1+c_2n)3^n\tag{1}$$</p>
<p>for some constants $c_1$ and $c_2$. You determine those by using the known values of... |
328,197 | <p>Let $R$ be a commutative ring with unity, and let $S\subset R$ be any finite set. Then
$$ \sum_{L \subset S} \prod_{x \in L} (x-1) = \prod_{x \in S} x,$$
which is easy enough to show by induction.</p>
<p>Does this follow from any sort of general principle? Perhaps inclusion-exclusion, in some form?</p>
| anon | 11,763 | <p>$$\begin{array}{cl} \sum_{L \subseteq S} \prod_{x \in L} (x-1) & = \sum_{L\subset S}\sum_{M\subseteq L}(-1)^{|L\setminus M|}\prod_{x\in M}x \\
& =\sum_{M\subseteq S}\left[\sum_{M\subseteq L\subseteq S}(-1)^{|L\setminus M|}\right]\prod_{x\in M}x \\
& = \sum_{M\subseteq S}\left[\sum_{\ell=|M|}^{|S|}{|S|-|... |
255,652 | <p>I came across a problem that says:</p>
<p>Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a function. If $|f'|$ is bounded, then which of the following option(s) is/are true?</p>
<blockquote>
<p>(a) The function $f$ is bounded.<br>
(b) The limit $\lim_{x\to\infty}f(x)$ exists.<br>
(c) The function $f$ is unifor... | Sugata Adhya | 36,242 | <p><strong>(a) & (b) are false:</strong> Consider $f(x)=x$ $\forall$ $x\in\mathbb R$;</p>
<p><strong>(c) is true:</strong> $|f'|$ is bounded on $\mathbb R\implies f'$ is bounded on $\mathbb R$ [<em>See: <a href="http://en.wikipedia.org/wiki/Uniform_continuity#Examples" rel="nofollow">Related result</a></em>];</p>
... |
502,160 | <p>Is there any representation of the exponential function as an infinite product (where there is no maximal factor in the series of terms which essentially contributes)? I.e.</p>
<p>$$\mathrm e^x=\prod_{n=0}^\infty a_n,$$</p>
<p>and by the sentence in brackets I mean that the $a_n$'s are not just mostly equal to $1$... | Thorgott | 422,019 | <p>This function is related to the one Pixel posted, but it is not the same:
<span class="math-container">$$e=\prod\limits_{n=1}^\infty\left(1-\frac{1}{\tau^n}\right)^{\frac{\mu(n)-\phi(n)}{n}}$$</span>
where <span class="math-container">$\tau$</span> denotes the golden ratio, <span class="math-container">$\mu(n)$</spa... |
355,421 | <p>I have two questions.</p>
<p><span class="math-container">$\bf 1.$</span> First, a reference request. Let <span class="math-container">$G\cong{\mathbb F}_p^r$</span> for some integer <span class="math-container">$r\geq 0$</span> and let <span class="math-container">$V=G^*={\rm Hom}(G,{\mathbb F}_p)$</span>. Then <s... | Constantin-Nicolae Beli | 140,180 | <p>I'm almost done with writing the paper. Here is the result.</p>
<p>If someone saw anything similar, then please let me know.</p>
<p>We have a basis <span class="math-container">$s_1,\ldots,s_r$</span> of <span class="math-container">$G$</span> over <span class="math-container">${\mathbb F}_p$</span> and a basis <s... |
355,421 | <p>I have two questions.</p>
<p><span class="math-container">$\bf 1.$</span> First, a reference request. Let <span class="math-container">$G\cong{\mathbb F}_p^r$</span> for some integer <span class="math-container">$r\geq 0$</span> and let <span class="math-container">$V=G^*={\rm Hom}(G,{\mathbb F}_p)$</span>. Then <s... | Constantin-Nicolae Beli | 140,180 | <p>I posted an article with this result on arXiv:
<a href="https://arxiv.org/abs/2005.11868" rel="nofollow noreferrer">https://arxiv.org/abs/2005.11868</a></p>
<p>Before sending it to be published, I want to make sure it is original. If anybody saw a similar result somewhere, then please let me know.</p>
<p>Also plea... |
2,035,454 | <p>I am an upcoming year $12$ student, school holidays are coming up in a few days and I've realised I'm probably going to be extremely bored. So I'm looking for some suggestions.</p>
<p>I want a challenge, some mathematics that I can attempt to learn/master. Obviously nothing impossible, but mathematics is my number ... | Itay.V | 386,757 | <p>Well my suggestion would be to learn about (in this particular order): </p>
<ol>
<li><p>Infinite series (converging and diverging tests). </p></li>
<li><p>Taylor form (Taylor-maclaurin approximation function). </p></li>
<li><p>Evaluation of different equations and the solution of these equations with Taylor forms... |
2,035,454 | <p>I am an upcoming year $12$ student, school holidays are coming up in a few days and I've realised I'm probably going to be extremely bored. So I'm looking for some suggestions.</p>
<p>I want a challenge, some mathematics that I can attempt to learn/master. Obviously nothing impossible, but mathematics is my number ... | Wildcard | 276,406 | <p>I was at a similar level of experience when I reached U.S. 12th grade. (I'm not sure what "Year 12" refers to.)</p>
<p>What I eventually found was that within the branches of math you have mentioned, <strong>you have had virtually zero experience with <em>discrete</em> mathematics.</strong></p>
<p>The mathematics... |
697,506 | <p>Assume the following summation,</p>
<p>$$
\sum_{i=0}^{1000}\left(-1\right)^{i}{1000 \choose i}\left(100 - i\right)^{500}.
$$</p>
<p>I know that this summation is zero, $0$ ( I've checked it with Maple, though ). But I cannot find any proof for that!. Can you provide any help ?.</p>
<p>P.S. This is not a homework ... | Felix Marin | 85,343 | <p>$\newcommand{\+}{^{\dagger}}
\newcommand{\angles}[1]{\left\langle #1 \right\rangle}
\newcommand{\braces}[1]{\left\lbrace #1 \right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack #1 \right\rbrack}
\newcommand{\ceil}[1]{\,\left\lceil #1 \right\rceil\,}
\newcommand{\dd}{{\rm d}}
\newcommand{\down}{\downarrow}
\new... |
2,047,579 | <p>Today I was thinking about how to generalize the derivative of $f(x) = x^n$. Of course, we know the following:</p>
<p>$$\frac{d}{dx}x^n = n\cdot x^{n-1}$$</p>
<p>But how can we show that the power rule behaves this way? I took to the limit process and came up with this:</p>
<p>$$\frac{d}{dx}x^n = \lim_{h\to 0}\fr... | David | 119,775 | <p>If it helps, you could retain the summation notation, assuming $n\ge2$:
$$\eqalign{
\Bigl(\sum_{i=0}^n\binom ni x^{n-i}h^i\Bigr)-x^n
&=x^n+nx^{n-1}h+\Bigl(\sum_{i=2}^n\binom ni x^{n-i}h^i\Bigr)-x^n\cr
&=nx^{n-1}h+h^2\sum_{i=2}^n\binom ni x^{n-i}h^{i-2}\cr}$$
and keep going from there.</p>
|
2,047,579 | <p>Today I was thinking about how to generalize the derivative of $f(x) = x^n$. Of course, we know the following:</p>
<p>$$\frac{d}{dx}x^n = n\cdot x^{n-1}$$</p>
<p>But how can we show that the power rule behaves this way? I took to the limit process and came up with this:</p>
<p>$$\frac{d}{dx}x^n = \lim_{h\to 0}\fr... | Michael L. | 153,693 | <p>It might be more elegant to prove the power rule for $n\in \mathbb{N}$ using induction. If we already know the product rule and that $\frac{\mathrm{d}x}{\mathrm{d}x} = 1$ (which serves as our base case), then we can deduce that if $\frac{\mathrm{d}}{\mathrm{d}x}x^{n-1} = (n-1)x^{n-2}$, then $$\frac{\mathrm{d}}{\math... |
3,707,880 | <p>Let <span class="math-container">$W_{1}$</span> and <span class="math-container">$W_{2}$</span> be subspaces of a vector space <span class="math-container">$V$</span>.</p>
<p>(a) Prove that <span class="math-container">$W_{1}+W_{2}$</span> is a subspace of <span class="math-container">$V$</span> that contains both ... | hdighfan | 796,243 | <p>Your solution is almost correct, except it seems you've only shown that <span class="math-container">$W_1+W_2$</span> is indeed a subspace in part a), when it also asks you to show it contains <span class="math-container">$W_1$</span> and <span class="math-container">$W_2$</span>. This is straightforward to show and... |
3,915,413 | <p>Let <span class="math-container">$A$</span> be a positive definite <span class="math-container">$n\times n$</span> matrix. We use the iteration of the mapping <span class="math-container">$$f(X)=0.5(X^2+B), \ X\in \mathbb{R}^{n\times n}$$</span> where <span class="math-container">$B=I-A$</span>.</p>
<p>Show that the... | VIVID | 752,069 | <p>We have <span class="math-container">$$f(X) = \frac 12X^2 + \frac 12 B$$</span>
Let's try to see what initial terms of <span class="math-container">$(X_n)$</span> look like (with <span class="math-container">$X_0=0$</span>):
<span class="math-container">$$\begin{align}X_1 &= f(0) = \frac 12 0^2 + \frac 12 B = \f... |
4,624,011 | <p>I am looking to minimize the value of:
<span class="math-container">$$g(t)=\mathrm{Tr}\left[\exp(X+tY)\right]$$</span>
where both <span class="math-container">$X$</span> and <span class="math-container">$Y$</span> are symmetrical matrices with real coefficients. In general, <span class="math-container">$X$</span> an... | Hyperplane | 99,220 | <p>The simple way to get it numerically is to use an <a href="https://en.wikipedia.org/wiki/Automatic_differentiation" rel="nofollow noreferrer">automatic differentiation</a> library such as <a href="https://jax.readthedocs.io/en/latest/index.html" rel="nofollow noreferrer">JAX</a>.</p>
<pre><code>from jax.config impor... |
87,636 | <p>I'm following the book <em>Measure and Integral</em> of Richard L. Wheeden and Antoni Zygmund. This is problem 4 of chapter 8.</p>
<p>Consider <span class="math-container">$E\subseteq \mathbb{R}^n$</span> a measurable set. In the following all the integrals are taken over <span class="math-container">$E$</span>, <sp... | Laurent Claessens | 294,061 | <p>I want to add some details around the answer by "community wiki". This is essentially an answer to <a href="https://math.stackexchange.com/q/4125105/294061">my own question</a>.</p>
<p>Disclaimer : I did not found these details anywhere. What you are going to read is the result of my own (lack of) understa... |
3,104,049 | <p>I have a set:</p>
<p><span class="math-container">$X = \{p | p \in P \wedge \forall a(a \in A \wedge p \in F(a) \wedge p \notin F'(a))\}$</span></p>
<p>This reads (to me): the set containing all p, where p is an element of P, and for all a, a is an element of A and p is in F(a) and not in F'(a).</p>
<p>1) Does th... | hmakholm left over Monica | 14,366 | <blockquote>
<p>2) Would it be better as <span class="math-container">$X = \{p | p \in P \wedge \forall a(a \in A \rightarrow p \in F(a) \wedge p \notin F'(a))\}$</span> </p>
</blockquote>
<p>Yes, you definitely want that formulation.</p>
<p>In standard set theory a formula of the shape
<span class="math-container"... |
1,432,729 | <p>I know that $\pi \approx \sqrt{10}$, but that only gives one decimal place correct. I also found an algebraic number approximation that gives ten places but it's so cumbersome it's just much easier to just memorize those ten places.</p>
<p>What's a good approximation to $\pi$ as an irrational algebraic number (or a... | Zach466920 | 219,489 | <p>How about,</p>
<p>$$ \sqrt[3] {31}=3.14138...$$</p>
<p>Where, $31$ is the length of a month.</p>
<p>If you want memorable, you could always use,</p>
<p>$$\pi \sim \sqrt{{{69} \over {7}}}=3.139...$$</p>
<p>Do I really need to explain this one?</p>
<p>You could also use,</p>
<p>$$\sqrt{{69 \cdot 1001} \over {7 ... |
1,432,729 | <p>I know that $\pi \approx \sqrt{10}$, but that only gives one decimal place correct. I also found an algebraic number approximation that gives ten places but it's so cumbersome it's just much easier to just memorize those ten places.</p>
<p>What's a good approximation to $\pi$ as an irrational algebraic number (or a... | Community | -1 | <p>I'm hardly the first to think of this, but I might be the first to say it in this thread: $\sqrt{10} \approx \pi$ suggests that we look at the powers of $\pi$ and see which come closest to integers. Then do floor or ceiling on $\pi^n$ and that gives you an approximation as an irrational algebraic integer of degree ... |
1,432,729 | <p>I know that $\pi \approx \sqrt{10}$, but that only gives one decimal place correct. I also found an algebraic number approximation that gives ten places but it's so cumbersome it's just much easier to just memorize those ten places.</p>
<p>What's a good approximation to $\pi$ as an irrational algebraic number (or a... | Glen O | 67,842 | <p>An interesting simple one is
<span class="math-container">$$
\pi\approx \frac{3(3+\sqrt{5})}{5} = \frac{6\varphi^2}{5} \approx 3.1416407
$$</span>
where <span class="math-container">$\varphi$</span> is the golden ratio.</p>
<p>If you don't mind the algebraic numbers expressed in terms of their polynomials, here are... |
1,407,183 | <p>I know this is trivial but I don't don't know if I'm right</p>
<p>Verify, by substitution, if the function is a solution of differential equation.</p>
<p>$y'=3x^2$ , $y=x^3+7$</p>
<hr>
<p>Differentiating the function $y=x^3+7$ you get $y'=3x^2$</p>
<p>Adding that resut to the $y'=3x^2$ I get</p>
<p>$3x^2+3x^2... | A. Thomas Yerger | 112,357 | <p>We don't know what the differential equation you are trying to solve is.</p>
<p>An (ordinary) differential equation is an expression $F(x, y, y', ..., y^{(n)}) = 0$. Now if you have one of these, and they give you some function $y=f(x)$, then you can take the appropriate derivatives, and substitute in to the equati... |
472,684 | <p>Consider the random walk $X_0, X_1, X_2, \ldots$ on state space $S=\{0,1,\ldots,n\}$ with absorbing states $A=\{0,n\}$, and with $P(i,i+1)=p$ and $P(i,i-1)=q$ for all $i \in S \setminus A$, where $p+q=1$ and $p,q>0$.</p>
<p>Let $T$ denote the number of steps until the walk is absorbed in either $0$ or $n$.</p>
... | Loic | 475,798 | <p>This is an appendix to the excellent answer of Did, giving an explicit formula for <span class="math-container">$u_k$</span> (whose derivation involves quite a lot of work).</p>
<p><span class="math-container">$u_k$</span>, that is <span class="math-container">$E_k(T:X_T=n)$</span>, can be expressed as:<span class=... |
322,363 | <p>The question is in the title. Fix a field <span class="math-container">$k$</span>. Let <span class="math-container">$P_n$</span> be the poset of proper nonempty affine subspaces of <span class="math-container">$k^n$</span> under inclusion. The geometric realization <span class="math-container">$|P_n|$</span> is <... | Aaron Meyerowitz | 8,008 | <p>There are very good reasons to think that for every even <span class="math-container">$g$</span> there are infinitely many primes <span class="math-container">$p$</span> with <span class="math-container">$p+g$</span> also prime. For this to be true for even one <span class="math-container">$g$</span> makes the answe... |
2,433,051 | <p>Following <a href="http://functions.wolfram.com/ElementaryFunctions/Sin/29/" rel="nofollow noreferrer">Wolfram Sine inequalities</a> I found that</p>
<p>$$|\sin(x)| \le |x| \quad \text{for} \quad x \in \mathbb{R}$$
How can I prove this relation?</p>
| Nosrati | 108,128 | <p><strong>Hint:)</strong> $\cos u\leq1$ then for $x\geq0$
$$\displaystyle\int_0^x\cos udu\leq\int_0^xdu$$
and for $x\leq0$
$$\displaystyle\int_x^0\cos udu\leq\int_x^0du$$</p>
|
2,946,451 | <p>Is there a reason why we call it differently?</p>
| Melody | 598,521 | <p>From (2) we get <span class="math-container">$-\epsilon<\beta-y$</span>, so <span class="math-container">$\alpha-\epsilon<\alpha+\beta-y$</span>. The proof needs a little work, but you can salvage it. You don't need to use (1).</p>
<p>Also, we have <span class="math-container">$\alpha-\epsilon<x$</span> an... |
2,284,376 | <p>Let H be a normal subgroup of G of index 4 show that there are either exactly 3 or exactly 5 subgroups of G containing H( including G and H themselves )</p>
<p>Where do we start from i am completly got no idea..</p>
| Nicky Hekster | 9,605 | <p>Hint: $G/H$ is a group of order $4$. And there are only two of them up to isomorphism: $C_4$ or $C_2 \times C_2$. The first has $3$ subgroups, the second $5$. So this boils down to classifying all groups of order $4$. Whether or not you can do that, depends on your group theory knowledge sofar.</p>
|
429,844 | <p>If I have a $4\times 4$ matrix $A$ with real entries that has all $1$'s on the main diagonal, $A$ is singular and we know one eigenvalue $k_{1}=3+2i$. What about the others three eigenvalues?</p>
<p>I think one should be $k_{2}=3-2i$ because they always come in pairs, right?</p>
<p>Then, since $A$ is singular I th... | Clement C. | 75,808 | <p>You have the sum of all eigenvalues being equal to the trace; hence $k_1+k_2+k_3+k_4=4$. This'll give you $k_4$.</p>
|
3,501,763 | <p>My question is:</p>
<blockquote>
<p>Is there a way to stack marbles by using only a single one-marble stacking operation such that an infinite 3-dimensional stack is constructed?</p>
</blockquote>
<p>For example:</p>
<p>In 1-dimension one can start with a stack such as</p>
<pre><code>-o-o-o-o
</code></pre>
<p... | Steven Stadnicki | 785 | <p>I'm going to offer a slightly different mathematical footing for this question that gives it both a positive (locally) and a negative (globally) answer.</p>
<p>One useful model for thinking about structures like this is the notion of a <em>Cayley graph</em> for (a presentation of) a group; the nodes of the graph ar... |
2,633,975 | <p>Let $u\in C^2(\Omega)\cap C^0(\overline{\Omega})$, $\Omega\subset\mathbb{R}^n$ open and bounded and $\lambda>0$ sufficiently small so that $2\lambda u<1$. Define $w$ by $w\leq\frac{1}{\lambda}$ and
$$
u(x)=w(x)-\frac{\lambda}{2}w(x)^2.
$$
I have to prove that $w\rightarrow u$ uniformly as $\lambda\rightarrow0$... | Fabian | 7,266 | <p>The beginning looks fine. However, note that you need one Lagrange multiplier per constraint. Thus you need a vector $\lambda$.</p>
<p>The function to minimize is then
$$\operatorname{tr}(A^T-B^T)(A-B) - \lambda^T (B x -v). $$</p>
<p>Taking the gradient with respect to $B$, we arrive at
$$2 (B-A)- \lambda x^T =0. ... |
2,633,975 | <p>Let $u\in C^2(\Omega)\cap C^0(\overline{\Omega})$, $\Omega\subset\mathbb{R}^n$ open and bounded and $\lambda>0$ sufficiently small so that $2\lambda u<1$. Define $w$ by $w\leq\frac{1}{\lambda}$ and
$$
u(x)=w(x)-\frac{\lambda}{2}w(x)^2.
$$
I have to prove that $w\rightarrow u$ uniformly as $\lambda\rightarrow0$... | Akababa | 87,988 | <p>One thing you could use is the fact that each row is independent, i.e. for row vectors $a_i,b_i$ you can minimize $(a_i-b_i)^2:b_i\cdot \textbf 1=nv_i$ separately.</p>
<p>We can use the lagrange method directly on each scalar element to get $$b_{ij}=a_{ij}+v_i-\overline{a_i}$$</p>
<p>where $\overline{a_i}$ is the ... |
4,154,851 | <p>Pls help me with this
<span class="math-container">$$2y''–y'–3y= 5e^{3x/2}$$</span>
I've solved the <span class="math-container">$yc$</span> part but the particular solution is given me zero
I also try to use <span class="math-container">$y=Axe^{3x/2}$</span> but I keep getting <span class="math-container">$x$</span... | user577215664 | 475,762 | <p><span class="math-container">$$2y''–y'–3y= 5e^{3x/2}$$</span>
Substitute <span class="math-container">$y=ve^{3x/2}$</span> the DE becomes:
<span class="math-container">$$2v''+5v'=5$$</span>
<span class="math-container">$$\implies v=c_1e^{-5x/2}+c_2+x$$</span>
This is easier to solve.
The particular solution you trie... |
4,154,851 | <p>Pls help me with this
<span class="math-container">$$2y''–y'–3y= 5e^{3x/2}$$</span>
I've solved the <span class="math-container">$yc$</span> part but the particular solution is given me zero
I also try to use <span class="math-container">$y=Axe^{3x/2}$</span> but I keep getting <span class="math-container">$x$</span... | HUK | 931,765 | <p><span class="math-container">$$y[x] =\dfrac 1{25} E^{3 x/2} (-2 + 5 x) + E^{3 x/2} C[1] + E^{-x} C[2]$$</span> is the correct answer according to Mathematica. Maybe the method of "variation of the constant" might help.</p>
|
2,168,524 | <p>I know a solution to this question having to do with the fact that the $\gcd(15, 21) = 3$, so the answer is no.</p>
<p>But I can't figure out what is the reasoning behind this. Any help would be really appreciated! </p>
| Student | 304,018 | <p>No: you can prove that $\text{gcd}(a, a +b) = \text{gcd}(a,b)$. Therefore, we have that $\text{gcd}(a,a+1) = \text{gcd}(a,1) =1$.</p>
<p>If we now consider multiples of $15$ and $21$, say $k \cdot 15, n \cdot 21$ with $k, n \in \mathbb{Z}$, such that $n \cdot 21 = k \cdot 15 + 1$, then we find that $3$ divides $\te... |
1,013,692 | <p>I want to determine which group $(\mathbb{Z}/24\mathbb{Z})^{*}$ is isomorphic to.</p>
<p>$\mathbb{Z}/24\mathbb{Z}$ contains the 24 residue classes $z + 24\mathbb{Z}$ of the division mod 24. For brevity, I will identify them with $z$, so $\mathbb{Z}/24\mathbb{Z} = \{ 0, 1, ..., 23\}$. For $(\mathbb{Z}/24\mathbb{Z})^... | Alex Wertheim | 73,817 | <p>Very nicely posed question, and some good work on your part. For the first part, identifying the elements of $(\mathbb{Z}/24\mathbb{Z})^{\star}$, note that an element of $\mathbb{Z}/24\mathbb{Z}$ is a unit if and only if it is relatively prime to $24$. You can see this by noting that if $\gcd(\alpha, 24) = 1$, then ... |
3,674,924 | <p>The angular momentum components in Cartesians are
<span class="math-container">$$\hat L_x=\hat y\hat p_z-\hat z\hat p_y$$</span>
<span class="math-container">$$\hat L_y=\hat z\hat p_x-\hat x\hat p_z$$</span>
<span class="math-container">$$\hat L_z=\hat x\hat p_y-\hat y\hat p_x$$</span></p>
<p>Starting from
<span cl... | CHAMSI | 758,100 | <p>Let <span class="math-container">$ n $</span> be a positive integer, we have : <span class="math-container">\begin{aligned}\left(\sum_{k=1}^{n}{\frac{1}{k^{2}}}\right)^{2}&=\sum_{1\leq i,j\leq n}{\frac{1}{i^{2}j^{2}}}\\ &=\sum_{1\leq i\leq j\leq n}{\frac{1}{i^{2}j^{2}}}+\sum_{1\leq j<i\leq n}{\frac{1}{i^{... |
23,003 | <p>I'm trying to illustrate the solutions to a textbook problem dealing with quadratic functions.</p>
<p>This will involve plotting a quadratic and overlaying the plot and the image.</p>
<p>Here is the textbook scan.....<img src="https://i.stack.imgur.com/La8Zs.jpg" alt="enter image description here"></p>
<p>The ide... | einbandi | 4,296 | <p><strong>EDIT:</strong> (see below for old version)</p>
<p>New version with alpha channels, the option to lock the graph at the ball, adjustable player position and a button to remove player and basket:</p>
<pre><code>setalpha[im_] :=
Module[{mask =
ChanVeseBinarize[im, TargetColor -> {1., 1., 1.},
... |
211,865 | <p>The given matrix is </p>
<p>$$
\begin{pmatrix}
2 & 2 & 2 \\
2 & 2 & 2 \\
2 & 2 & 2 \\
\end{pmatrix}
$$</p>
<p>so, how could i find the eigenvalues and eigenvector without computation?
Thank you</p>
| ASHWINI SANKHE | 335,009 | <p>If sum of all the elements in each rows or each columns are same then sum of elements in one row is nothing but one of the eigenvalue of that matrix.<br>
In above matrix sum of all the elements in each rows is 6.Hence 6 is the eigenvalue of that matrix. </p>
|
861,230 | <p>How to calculate:
$$\int_0^\infty \frac{x \sin(x)}{x^2+1} dx$$
I thought I should find the integral on the path
$[-R,R] \cup \{Re^{i \phi} : 0 \leq \phi \leq \pi\}$.</p>
<p>I can easily take the residue in $i$
$$
Res_{z=i} \frac{x \sin(x)}{x^2+1}
\quad = \quad
\frac{i (e^{ii}-e^{-ii})}{2i}
\quad = \quad
\frac{i... | Empy2 | 81,790 | <p>Remember that $\sin x=(\exp(ix)-\exp(-ix))/2i$, and $\exp(-ix)$ grows large in the upper half plane. You have to integrate the $\exp(ix)$ part in the upper half plane, and the $\exp(-ix)$ in the lower half plane. </p>
<p>Also, the upper half-plane is $0<\phi<\pi$, not $2\pi$.</p>
<p>The trick of completing... |
68,748 | <p>Among the theorems of early geometric model theory there is one by Lachlan stating that every totally categorical structure with a trivial pregeometry is intrepretable in a dense linear order.</p>
<p>That suggests in particular that there are totally categorical structures with a trivial pregeometry that are not in... | Szymon Toruńczyk | 87,983 | <p>An example of such a structure is given here: <a href="https://mathoverflow.net/questions/231791/%CF%89-categorical-%CF%89-stable-structure-with-trivial-geometry-not-definable-in-the-pur?noredirect=1#comment573324_231791">ω-categorical, ω-stable structure with trivial geometry not definable in the pure set</a>. Esse... |
1,328,799 | <p>I'm looking for modern graduate-level non-English-language differential geometry textbooks. I'm interested in original works by non-English-language speakers in their native languages, not translations into non-English languages. I'm also not interested in non-English language textbooks which are available in Englis... | Holonomia | 73,953 | <p>Time ago a read a really nice book "Geometria de espacios fibrados" (Spanish) writen by Roberto J. Miatello and Carlos E. Olmos in 1992 and published in Serie "B" Trabajos de Matematica, FaMAF, Universidad Nacional de Cordoba, Argentina. Prof. Olmos is well known expert in Riemannian Geometry that in 2005 gave a geo... |
1,328,799 | <p>I'm looking for modern graduate-level non-English-language differential geometry textbooks. I'm interested in original works by non-English-language speakers in their native languages, not translations into non-English languages. I'm also not interested in non-English language textbooks which are available in Englis... | Marta Oliveira | 457,851 | <p>Brazilian books.</p>
<p>www. im.ufal.br > posmat > index.php > downloads > category > 6-livros</p>
<p>www.sbm.org.br > wp-content > uploads > 2016/06 > Introdução-a-Geometria-Diferencial</p>
|
361,045 | <p>I'm trying to prove that <span class="math-container">$\operatorname{Aut}(\mathbb Z_8)$</span> is isomorphic to <span class="math-container">$\mathbb Z_2 \oplus\mathbb Z_2$</span>, but I have no idea how to prove it. First of all, I'm trying to prove that <span class="math-container">$\operatorname{Aut}(\mathbb Z_8)... | cigar | 1,070,376 | <p>Yeah well as a general rule you have the nice: <span class="math-container">$$\rm{Aut}(\Bbb Z_n)\cong \Bbb Z_n^×.$$</span></p>
<p>This is pretty straightforward to prove given the fact about the generators.</p>
<p>So in this case we just need <span class="math-container">$\Bbb Z_8^×$</span>, which it is well-known ... |
2,654,507 | <blockquote>
<p>Find the residue of $\dfrac{z^2}{(z-1)(z-2)(z-3)}$ at $\infty$.</p>
</blockquote>
<p>We know that $\text{Res} (f)_\infty +\text{Res} (f)_{\text{ at other poles}}=0$</p>
<p>Now $f$ has poles at $1,2,3$ of order $1$.</p>
<p>Sum of residues of $f$ at $1,2,3=\dfrac{1}{2}+(-4)+\dfrac{9}{2}=1\implies \t... | lab bhattacharjee | 33,337 | <p>$$(\sin^2t)^3+(\cos^2t)^3=(\sin^2t+\cos^2t)^3-3\sin^2t\cos^2t(\sin^2t+\cos^2t)=?$$</p>
|
369,585 | <p>Let <span class="math-container">$X$</span> be a smooth, projective ireducible scheme over an algebraically closed field <span class="math-container">$k$</span>. I'm trying to understand when there exists an abelian variety <span class="math-container">$A$</span> such that <span class="math-container">$X$</span> is ... | Francesco Polizzi | 7,460 | <p>An obvious class of counterexamples are uniruled varieties. In fact, abelian varieties contain no rational curves.</p>
<p>More generally, and for the same reason, if <span class="math-container">$X$</span> is any algebraic variety that contain a (possibly singular) rational curve, then <span class="math-container">$... |
369,585 | <p>Let <span class="math-container">$X$</span> be a smooth, projective ireducible scheme over an algebraically closed field <span class="math-container">$k$</span>. I'm trying to understand when there exists an abelian variety <span class="math-container">$A$</span> such that <span class="math-container">$X$</span> is ... | cgodfrey | 113,296 | <p>I just want to point out that "adjunction+translation" tells us quite a bit:</p>
<p>Let <span class="math-container">$A$</span> be an abelian variety, say of dimension <span class="math-container">$n>1$</span> and let <span class="math-container">$D \subset A$</span> be a (let's say smooth) divisor. Sin... |
1,488,831 | <p>Written in Abstract Algebra by T. W. Judson :</p>
<p><strong>Theorem</strong></p>
<blockquote>
<p>Let R be a ring with identity and suppose that I is an ideal in R such that 1 is in I. Since for any r∈R, r1=r∈I by the definition of an ideal, I=R. </p>
</blockquote>
<p>and considering the definition of the ideal... | Community | -1 | <p>Any $R$-multiple of an element of $I$ is again in $I$ (by definition). So let $r\in R$, then $r\cdot 1$ as a product of an element $r\in R$ and $1\in I$ is again in $I$, hence $I=R$.</p>
|
3,547,816 | <p>I have such an equation where I need to find x</p>
<p><span class="math-container">$$y =\frac{1}{x+1}$$</span></p>
<p>And I know that answer is
<span class="math-container">$$x = \frac{1}{y}-1$$</span></p>
<p>But I did not understand how to get it?</p>
| Ross Millikan | 1,827 | <p><span class="math-container">$$y=\frac 1{x+1}\\
(x+1)y=1\\
xy=1-y\\
x=\frac 1y-1$$</span>
Note that <span class="math-container">$x$</span> cannot equal <span class="math-container">$-1$</span> from the original equation and <span class="math-container">$y$</span> cannot equal <span class="math-container">$0$</span>... |
728,186 | <p>The following definition has been given in <a href="http://www.sciencedirect.com/science/article/pii/0304397585901355?via=ihub" rel="nofollow">this article</a>.</p>
<p>A term algebra is an algebra $ \langle \mathcal{S}, \mathcal{G} \rangle $ where every time that $g_\alpha, g_\beta \in \mathcal{G}$ and
$$ g_\alpha... | Giorgio Mossa | 11,888 | <p>The correct answer is the second one: functions of the algebras are influenced by the syntax, that's the special property of term algebras.</p>
<p>Usually term algebras are defined as algebras whose carrier are set formed by strings of a specified grammar. Usually they are build in such way:</p>
<ul>
<li><p>you st... |
611,230 | <p>I'm not a mathematician at all, but I'm looking for a formula that will help with the following problem:</p>
<p>It's coming up to Christmas and our employees want to take leave, but some employees will have accrued less leave than than the amount they wish to take in which case they are entitled to take all of thei... | r3mainer | 102,961 | <p>Yes. Multiply the number of hours by 1.04.</p>
<p>For every one hour of leave, an employee gains an additional $x$ hours ($x$ being 0.038462 in this case). And for these $x$ hours of additional leave, the employee gains an additional $x \times x = x^2$ hours of leave, and so on.</p>
<p>So an employee entitled to $... |
3,248,730 | <p>Consider the problem below:</p>
<blockquote>
<p>Let <span class="math-container">$S(A)$</span> represent the sum of elements in set <span class="math-container">$A$</span> of size <span class="math-container">$n$</span>. We shall
call it a special sum set if for any two non-empty disjoint subsets, <span class="... | Dan Uznanski | 167,895 | <p>Explicitly, the list of subset pairs are</p>
<pre><code>{1},{2}
{1},{3}
{1},{4}
{2},{3}
{2},{4}
{3},{4}
{1,2},{3}
{1,2},{4}
{1,3},{2}
{1,3},{4}
{1,4},{2}
{1,4},{3}
{2,3},{1}
{2,3},{4}
{2,4},{1}
{2,4},{3}
{3,4},{1}
{3,4},{2}
{1,2},{3,4}
{1,3},{2,4}
{1,4},{2,3}
{1,2,3},{4}
{1,2,4},{3}
{1,3,4},{2}
{2,3,4},{1}
</code><... |
4,373,267 | <p>In view of the paper "Forcing As A Computational Process" by J. Hamkins, R. Miller and K. Williams, I have revised my original question, part (a) about the computability of the Forcing Truth Definition, <a href="https://math.stackexchange.com/questions/3465785/looking-into-the-future-within-forcing">Lookin... | Acccumulation | 476,070 | <p>By the Pigeonhole Principle, one of the degree classes has at least 5 members. If that degree class is 6, we're done. So assume there are at least 5 vertices with degree 5. We can't have an odd number of odd degrees, so there must be at least 6 vertices with degree 5.</p>
<p>If you're confused by the last statement,... |
3,294,564 | <blockquote>
<p>There is only one real values of <span class="math-container">$k$</span> for which the quadratic equation <span class="math-container">$kx^2+(k+3)x+k-3=0$</span> has <span class="math-container">$2$</span> positive integer roots. Then the product of these two solutions is</p>
</blockquote>
<p>What i ... | Mick | 42,351 | <p>Let p and q be the two positive roots, then m, the average value of the two must also be positive. That is <span class="math-container">$m = \dfrac {p + q}{2} = \dfrac {- (k + 3)}{2k} > 0$</span>.</p>
<p>It should be clear that k must not be 0.</p>
<p>If k > 0 and after solving the above, we get k < -3. This... |
2,223,267 | <p>For<br>
$$e^{-j\pi n}$$</p>
<p>How does this become
$$(-1)^n$$</p>
<p>or is it actually
$$(-1)^{-n}$$
I have checked on calculator and values are all the same when the same n value is used</p>
| BLAZE | 144,533 | <p>It's best to simply sketch an Argand diagram for this. You will soon see that the value of $e^{n \pi j}$ just oscillates between $-1$ and $1$ depending on whether $n$ is odd or even hence it is equal to $(-1)^n$.</p>
<p>To see this explicitly just use the fact that $e^{j \pi n}=\cos nx +j \sin nx$ and noting that t... |
1,496,438 | <p>Let $A=[a_{ij}]$ be a square matrix of order $2$ where $a_{ij}\in\left\{0,1,2,3,4,6\right\}$.Find the number of matrices $A$ with distinct elements such that $AA^{-1}=I$,where $I$ is unit matrix of order $2$.<br></p>
<hr>
<p>My Attempt:<br>
Let $A=\begin{bmatrix}
a & b \\
c & d
\end{bmatrix}$<br>... | E.H.E | 187,799 | <p>Hint:
$$(x+1)^3=(x+1)(x+1)(x+1)$$
but
$$(x^3+1)=(x+1)(x^2-x+1)$$
so the roots are not repeated</p>
|
1,496,438 | <p>Let $A=[a_{ij}]$ be a square matrix of order $2$ where $a_{ij}\in\left\{0,1,2,3,4,6\right\}$.Find the number of matrices $A$ with distinct elements such that $AA^{-1}=I$,where $I$ is unit matrix of order $2$.<br></p>
<hr>
<p>My Attempt:<br>
Let $A=\begin{bmatrix}
a & b \\
c & d
\end{bmatrix}$<br>... | Lutz Lehmann | 115,115 | <p>No,
$$
r^3+1=(r+1)(r^2-r+1)=(r+1)((r-\tfrac12)^2+\tfrac34)
$$
has only a single root at $r=-1$ and a pair of complex conjugate roots that divide the unit circle equally in parts of 120°.</p>
|
206,421 | <p>If $4 \tan(\alpha - \beta) = 3 \tan \alpha $, then prove that
$$\tan \beta = \frac{\sin(2 \alpha)}{7 + \cos(2 \alpha)}$$</p>
<p>This is not homework and I've tried everything so I would just like a straight answer thank you in advance. </p>
| minthao_2011 | 40,146 | <p>We have $$4 \tan(\alpha - \beta) = 3 \tan \alpha$$
$$ \tan(\alpha - \beta) = 3 [\tan \alpha - \tan(\alpha - \beta)]$$
$$ \dfrac{\sin(\alpha - \beta)}{\cos(\alpha - \beta)} = \dfrac{3\sin \beta}{(\alpha - \beta)\cdot \cos \alpha}$$
$$\sin(\alpha - \beta) \cdot \cos \alpha = 3\sin \beta $$
$$\sin(2\alpha - \beta) -... |
208,694 | <p>Could someone explain how to correctly prove that $$\lim_{n\to\infty}\sin\frac{1}{n}$$ where $n=1,2,\cdots,n$ doesn't exist.
I have no problem with it if $\sin\frac{1}{x}$ where $x$ is real, because just taking values $x=\frac{2}{(2n-1)\pi}, x=\frac{1}{n\pi}, x=\frac{2}{(2n+1)\pi}$ it is clear, for example, by Cauch... | Community | -1 | <p>In my opinion this limit does exist. It is 0 because $\sin(1/n)$ is continuous and so we have
$$
\lim_{n \rightarrow \infty} \sin\left(\frac 1n\right ) = \sin \left(\frac 1 {\lim_{n \rightarrow \infty} n }\right) = \sin(0) = 0
$$</p>
|
3,507,695 | <p>Suppose that <span class="math-container">$A \in M_{5 \times 5}(\mathbb{C})$</span> such that <span class="math-container">$(A - 2I)^{5} = 0$</span>. Suppose that <span class="math-container">$B \in M_{5 \times 5}(\mathbb{C})$</span> such that the minimal polynomial of <span class="math-container">$B$</span> is <spa... | Robert Israel | 8,508 | <p><span class="math-container">$B$</span> has three eigenvalues <span class="math-container">$0$</span> and <span class="math-container">$\pm i$</span>. The generalized eigenspaces of <span class="math-container">$B$</span> must be invariant under <span class="math-container">$A$</span>: therefore <span class="math-c... |
3,507,695 | <p>Suppose that <span class="math-container">$A \in M_{5 \times 5}(\mathbb{C})$</span> such that <span class="math-container">$(A - 2I)^{5} = 0$</span>. Suppose that <span class="math-container">$B \in M_{5 \times 5}(\mathbb{C})$</span> such that the minimal polynomial of <span class="math-container">$B$</span> is <spa... | Bach | 497,335 | <p>Since the minimal polynomial of <span class="math-container">$B$</span> has no multiple roots, we know that we can find a set of eigenbasis with respect to <span class="math-container">$B$</span>. Note that there are only two possible combinations of dimensions of eigenspaces of <span class="math-container">$B$</spa... |
450,142 | <p>the equation $x^3-5x+1=0$ has a root in $(0,1)$. Using a proper sequence for which $$|a(n+1)-a(n)|\le c|(a(n)-a(n-1)|$$ with $0<c<1$ , find the root with an approximation of $10^{-4}$.</p>
| chris | 87,260 | <p>Wouldn't a bisect algorithm do something like this: </p>
<p>Let $f(x)=x^3-5x+1$. Then we find the root by:</p>
<p>f(0)>0, f(1)<0</p>
<p>f(0.5)<0 (hence, there is a root between 0 and 0.5)</p>
<p>f(0.25)<0</p>
<p>f(0.125)>0</p>
<p>f(0.1875)</p>
<p>and so on...
The sequence would be (0.5, 0.25,0.125, 0... |
3,269,080 | <p>There are a lot of functions that look wobbly.</p>
<p>For example <span class="math-container">$x^4 + x^3$</span> looks a little wobbly when it gets near the x axis. The function <span class="math-container">$\sin(x)$</span> is extremely wobbly. The function <span class="math-container">$\sin(x) + x$</span> is also... | Adam Latosiński | 653,715 | <p>You can use the <a href="https://en.wikipedia.org/wiki/Curvature" rel="nofollow noreferrer">curvature</a> of the graph to measure how non-straight the graph is at a given point. It is given by <span class="math-container">$$ \kappa(x) = \frac{|f''(x)|}{\big(1+(f'(x))^2\big)^\frac32}$$</span>
It's similar to your fi... |
869,892 | <p>The polynomials $p(x) = 5x^3 - 27x^2 + 45x - 21$ and $q(x) = x^4 - 5x^3 + 8x^2 - 5x + 3$ both interpolate the points $(1,2) , (2,1) , (3,6), (4,47)$. Even though these polynomials are of different degree, I do not understand how this is possible when the Lagrange interpolation theorem states there is only one polyno... | Ian | 83,396 | <p>Working in the Newton basis makes this clearer. The Lagrange interpolation theorem says that your cubic polynomial is the unique polynomial interpolant whose degree is at most $3$. It can be written in the Newton basis as:</p>
<p>$$c_1 + c_2 (x-1) + c_3 (x-1)(x-2) + c_4 (x-1)(x-2)(x-3)$$</p>
<p>for some $c_1 , \do... |
1,830,799 | <p>1There are $\frac{21!}{2!3!} = 120$ total positions (disregarding order within same colour). I imagine labelling the people Y (yellow) and NY (not yellow), so I imagine I have $4$ copies of the letter Y and $5$ of NY. So I draw out $9$ slots and want to arrange so that at least $2$ Y are together.
I get $30$ arrang... | N. F. Taussig | 173,070 | <p>As others have indicated, it is easier to count the number of cases in which no two people in yellow shirts are adjacent.</p>
<p>We focus only on the shirts, not who is wearing them, which reduces the problem to arranging red, blue, and yellow shirts.</p>
<p>For the denominator, we arrange the shirts in a row. We... |
1,830,799 | <p>1There are $\frac{21!}{2!3!} = 120$ total positions (disregarding order within same colour). I imagine labelling the people Y (yellow) and NY (not yellow), so I imagine I have $4$ copies of the letter Y and $5$ of NY. So I draw out $9$ slots and want to arrange so that at least $2$ Y are together.
I get $30$ arrang... | user84413 | 84,413 | <p>As pointed out in the other answers, it is easier to solve this problem by finding the probability that no two people wearing yellow are sitting together and then subtracting this from 1.</p>
<hr>
<p>Using your approach, though, with 4 Y's and 5 N's there are $\dbinom{9}{4}$ total arrangements, </p>
<p>and we can... |
355,262 | <p>Is there a closed-form expression for the sum $\sum_{k=0}^n\binom{n}kk^p$ given positive integers $n,\,p$? Earlier I thought of this series but failed to figure out a closed-form expression in $n,\,p$ (other than the trivial case $p=0$).</p>
<p>$$p=0\colon\,\sum_{k=0}^n\binom{n}kk^0=2^n$$</p>
<p>I know that $\sum_... | Aryabhata | 1,102 | <p>If you consider <span class="math-container">$p$</span> as fixed, then the below can be considered as closed form I suppose:</p>
<p><span class="math-container">$$\sum_{k=0}^{n} \binom{n}{k} k^p = \sum_{k=1}^{p} S(p,k) n(n-1)\dots(n-k+1) 2^{n-k} \quad \quad (1)$$</span></p>
<p>where <span class="math-container">$S(k... |
2,717,264 | <blockquote>
<p>$$\int{\theta \tan^3{(\theta^2)}\sec^4{(\theta^2)}d\theta}$$</p>
</blockquote>
<p>I thought of the method of splitting up the $\tan^3 \theta^2$ to $\tan^2 \theta^2$ and $\tan\theta^2$.</p>
<p>And then using trig identity $1+\tan^2\theta=\sec^2\theta$ to express the whole integral in terms of $\sec\t... | José Carlos Santos | 446,262 | <p>You can simply say that $(\forall n\in\mathbb{N}):\ln n<n$ and that therefore$$\sum_{n=2}^\infty\frac1{\ln n}\geqslant\sum_{n=2}^\infty\frac1n=+\infty.$$</p>
|
1,040,505 | <p>Apparently,
$$(1-\cot 37^\circ)(1-\cot 8^\circ)=2.00000000000000000\cdots$$<br>
Since it is a $2.0000000000\cdots$ instead of $2$, it isn't exactly $2$.<br>
Why is that?</p>
| Jason | 195,308 | <p>Using various trigonometric identities, you should be able to show the left hand side is equal to $2$. The ones you will need are:
$$\cos(A+B)=\cos A\cos B-\sin A\sin B,\quad \cos(-A)=\cos A,\quad \cos(45^\circ)=\sin(45^\circ)=\frac1{\sqrt{2}}$$
as well as an understanding of what $\cot A$ means. The reason you have... |
955,721 | <p>Okay, so last time I got help figuring out a simple binomial coefficient misunderstanding. Now I'm trying to figure out what happens if the following scenario occurs:</p>
<p>Player <span class="math-container">$1$</span> gets a <span class="math-container">$5$</span>-hand of poker cards and tells everyone that he di... | Lucian | 93,448 | <blockquote>
<p><em>I want to find an elementary solution for $\displaystyle\int\frac{1}{e^x + 1}$</em></p>
</blockquote>
<p>Normally, I wouldn't be upset at someone for writing $\displaystyle\int f(x)$ instead of $\displaystyle\int f(x)\color{red}{dx}$, but it would appear that in this particular case the lack of a... |
1,060,011 | <p>Given <strong>a, b, c</strong> 3 real numbers. Prove that if a < b < c, then |b|$\leqslant$max (|a|,|c|).</p>
<p>I orginally proved this by discussing four cases as </p>
<p>1) a,b,c<0 </p>
<p>2) a<0, b,c>0</p>
<p>3) a,b<0, c>0</p>
<p>4) a,b,c>0</p>
<p>Here I wonder if there is an easier way to d... | Wood | 196,858 | <p>It's always true that $b<c$ and $-b<-a$. Just use the first inequality if $b\ge 0$ and the second when $b<0$:
$$
b \ge 0 \Rightarrow |b|<|c|<\max\left(|a|,|c|\right)\\
b < 0 \Rightarrow |b|<|a|<\max\left(|a|,|c|\right)
$$</p>
|
3,852,436 | <p>I'm struggling on the following problem:</p>
<blockquote>
<p>You are given N boxes indexed from 1 to N. Each box contains either
no coins or one coin. The number of empty boxes and the number of
boxes with one coin are denoted by n0 and n1, respectively. You take
a random subset of the boxes where each subset has... | awkward | 76,172 | <p>Here is another proof that if <span class="math-container">$n_1 >0$</span>, then the number of subsets containing an even number of coins is the same as the number of subsets containing an odd number of coins.</p>
<p>Let's say <span class="math-container">$N_{even}$</span> is the number of subsets containing an e... |
945,489 | <p>Let $A \subset X$ be a subspace of $X$. Recall that a <strong>retraction</strong> of $X$ onto $A$ is a continuous map $r: X \to A$ such that $r(a) = a$ for every $a \in A$.</p>
<p>Let $X = \bf R$ endowed with the standard topology, and let $A$ be the closed interval $[0,1]$ endowed with the subspace topology. Clear... | Daniel Valenzuela | 156,302 | <p>You see in the answers and comments the existence of such a retraction. </p>
<p>I want to get to the point, which you have misunderstood:</p>
<p>Every topological space $Y$ is both, open and closed. And for EVERY function (in particular the continous ones) $f:X \to Y$, it holds that $f^{-1}$ is open and closed. Th... |
1,531,154 | <p>A question from my calculus book states,</p>
<blockquote>
<p>Which points on the graph $y=4-x^2$ are the closest to the point (0,2)?</p>
</blockquote>
<p>Using some of my notes, I have a formula as follows (not sure what it's actually called):</p>
<p>$$d=\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$$</p>
<p>So I plugged... | Paul Sinclair | 258,282 | <p>That formula is just called the "Distance formula", or sometimes the "Euclidean distance formula". It is just an application of the Pythagorean theorem.</p>
<p>Your problem is here:</p>
<blockquote>
<p>$$F'(x) = 2(x-0)(1) + 2(2-x^2)(-1)$$</p>
</blockquote>
<p>(But first of all, surely you know $x - 0 = x$. Why ... |
81,435 | <p>Normally we consider simple arithmetic to be related to the world of objects. So the sum $3+2=5$ means $3$ three apples and $2$ apples gives $5$ apples. But is there an alternative interpretation which does not have anything to do with discrete objects?</p>
| Newtom | 19,349 | <p>I dont think there is any physical unit that you can consistently do all algebra with, $5=2^1+3^1$, so if every number is an apple, what is say (2 apples)^(2 apples) ?</p>
<p>This is the real problem: Everything a human can do in math a computer can do given enough time, and computers are deterministic, so in this ... |
2,688,808 | <p><span class="math-container">$$\frac{1}{u^2}+\frac{1}{v^2}=\frac{1}{w^2}$$</span></p>
<p>I want to generate all primitive solutions up to <span class="math-container">$u \le N$</span>. Is there a parametric solution?</p>
<p>By brute force, I got these solutions:</p>
<p><span class="math-container">$(15, 20, 12),(20,... | K B Dave | 534,616 | <p>Wikipedia calls the projective curve
$$\{[a:b:c]| a^2b^2=c^2(a^2+b^2)\}$$
the <a href="https://en.wikipedia.org/wiki/Quartic_plane_curve#Cruciform_curve" rel="nofollow noreferrer">cruciform curve</a>.</p>
<p>Consider the birational map on the projective plane induced by the <a href="https://www.encyclopediaofmath.o... |
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