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 |
|---|---|---|---|---|
167,221 | <p>I'm wondering if this is possible for the general case. In other words, I'd like to take $$\int_a^b{g(x)dx} + \int_c^d{h(x)dx} = \int_e^f{j(x)dx}$$ and determine $e$, $f$, and $j(x)$ from the other (known) formulas and integrals. I'm wondering what restrictions, limitations, and problems arise.</p>
<p>If this is ... | Matt Groff | 2,626 | <p>Here's a method that should allow one a large degree of freedom as well as allowing Rieman Integration (instead of Lesbegue Integration or some other method):</p>
<p>Let $\tilde{g}$ be such that: $$\int_a^b{g(x)dx} = \int_e^f{\tilde{g}(x)dx}$$
...and $\tilde{h}$ follows similarly. Then they can be both added insid... |
167,221 | <p>I'm wondering if this is possible for the general case. In other words, I'd like to take $$\int_a^b{g(x)dx} + \int_c^d{h(x)dx} = \int_e^f{j(x)dx}$$ and determine $e$, $f$, and $j(x)$ from the other (known) formulas and integrals. I'm wondering what restrictions, limitations, and problems arise.</p>
<p>If this is ... | JL344 | 34,774 | <p>Certainly. In fact $e$ and $f$ can be anything you want, as long as they are not equal. An affine transformation is one way to do it. Namely if
$$j(x)=\frac{b-a}{f-e}g\left(\frac{b-a}{f-e}(x-e)+a\right)
+\frac{d-c}{f-e}h\left(\frac{d-c}{f-e}(x-e)+c\right),$$ then
$$\int_a^bg(u)du+\int_c^dh(v)dv=\int_e^fj(x)d... |
748,442 | <p>I have $\lceil x \rceil = -\lfloor -x \rfloor$, but I can't figure out how to rely on this in order to get $\lfloor x \rfloor$ from $\lceil x−1 \rceil$.</p>
<p>For the record, I am only interested in non-negative real values.</p>
<p>I wish to avoid the use of $modulo$, $abs$, $round$ and $if$.</p>
<p>The motivati... | user2345215 | 131,872 | <p>You know that $\lfloor x\rfloor=-\lceil-x\rceil$, so if you replace $x$ by $-x$ and multiply the equation by $-1$, you get
$$x-\frac{1}{2}-\frac{\arctan(\tan(\pi(x-\frac{1}{2})))}{\pi}=-\lceil -x \rceil=\lfloor x\rfloor$$</p>
|
2,403,851 | <p>Here is Proposition 3 (page 12) from Section 2.1 in <em>A Modern Approach to Probability</em> by Bert Fristedt and Lawrence Gray. </p>
<blockquote>
<p>Let $X$ be a function from a measurable space $(\Omega,\mathcal{F})$ to another measurable space $(\Psi,\mathcal{G})$. Suppose $\mathcal{E}$ is a family of subsets... | CiaPan | 152,299 | <p>For any applicable base $b$ you have $$\log_b x = \frac 1{\ln b}\cdot\ln x$$
so $O(\log_b f(n))$ is exactly the same as $O(\ln f(n))$.</p>
<p>Sometimes $\log$ is used as a 'general' logarithm with the base unspecified, although constant and greater than $1$ (which is useful when multiplicative constant does not mat... |
3,533,023 | <p>We have <span class="math-container">$x_1+x_2+...+x_k=n$</span> for some integers <span class="math-container">$k,n$</span>. We have that <span class="math-container">$0 \leq x_1,...,x_k$</span> and individually <span class="math-container">$x_1 \leq a_1$</span>, <span class="math-container">$x_2 \leq a_2$</span>, .... | Narasimham | 95,860 | <p>Area above horizontal line of radius length r is only the outer area left out near periphery marked yellow on one of six minor segments.</p>
<p><span class="math-container">$$ A_{segment}=\pi r^2/6- r^2 \sqrt{3}/4= r^2(\pi/6-\sqrt {3}/4) $$</span>
Area of an equilateral triangle is known <span class="math-containe... |
3,427,794 | <p>Let's see I have the following equation</p>
<p><span class="math-container">$$
x=1
$$</span></p>
<p>I take the derivate of both sides with respect to <span class="math-container">$x$</span>:</p>
<p><span class="math-container">$$
\frac{\partial }{\partial x} x = \frac{\partial }{\partial x}1
$$</span></p>
<p>The... | cemsicles | 531,285 | <p>Let me throw my hat as well; according to fundamental theorem of calculus, which I assume we all know, derivative is the result of integration, </p>
<p><span class="math-container">$$\int_0^1\frac{d}{dx}xdx=\int_0^1dx=[x-x]_0^1=1-0=1$$</span></p>
<p>I prefer to think integration and derivatives as what they are, a... |
1,900,640 | <p>I have a function $$f(x)=(x+1)^2+(x+2)^2 + \dots + (x+n)^2 = \sum_{k=1}^{n}(x+k)^2$$
for some positive integer $n$. I started wondering if there is an equivalent expression for $f(x)$ that can be calculated more directly (efficiently). </p>
<p>I began by expanding some terms to look for a pattern.
$$
(x^2+2x+1) + ... | TonyK | 1,508 | <p>It means this:</p>
<p>Let $n$ be a positive integer, and define $s_n$ to be the number of square-free integers between $1$ and $n$ inclusive. Then
$$\lim_{n\to\infty}\frac{s_n}{n} = \frac{6}{\pi^2}$$</p>
<p>(Equivalently, you can allow negative integers, and define $t_n$ to be the number of square-free integers be... |
4,104,364 | <p>I got the following exercise:<br />
Let <span class="math-container">$W$</span> be a finite-dimensional <span class="math-container">$\Bbb{R}$</span>-vector space. Let <span class="math-container">$\Bbb{R}_W=\Bbb{R}\times W$</span>. Define addition and multiplication by <span class="math-container">$(r,w)+(s,v)=(r+s... | rschwieb | 29,335 | <p>You can also observe simply that it is an <span class="math-container">$n+1$</span> dimensional <span class="math-container">$\mathbb R$</span> algebra, where <span class="math-container">$n$</span> is the dimension of <span class="math-container">$W$</span>. That is a hard bound on the length of any chain of ideals... |
4,624,421 | <p>I need to solve <span class="math-container">$x'=x^2$</span>, <span class="math-container">$x(0)=1$</span> using the power series <span class="math-container">$\sum_{n=0}^{\infty}a_n t^{n}$</span> and show that it has a solution on <span class="math-container">$(-1,1)$</span> that can be extended to <span class="mat... | user170231 | 170,231 | <p>You made a mistake in replacing <span class="math-container">$x^2$</span> with its series form. Looks like all you did was swap out <span class="math-container">$t^n$</span> for <span class="math-container">$t^{2n}$</span>. There's <a href="https://en.wikipedia.org/wiki/Multinomial_theorem" rel="nofollow noreferrer"... |
4,624,421 | <p>I need to solve <span class="math-container">$x'=x^2$</span>, <span class="math-container">$x(0)=1$</span> using the power series <span class="math-container">$\sum_{n=0}^{\infty}a_n t^{n}$</span> and show that it has a solution on <span class="math-container">$(-1,1)$</span> that can be extended to <span class="mat... | A. P. | 1,027,216 | <ul>
<li><p>If you suppose that there exists a solution in power series of the form <span class="math-container">$y=\sum_{n=0}^{+\infty}a_n t^n$</span>, then <span class="math-container">$y'=\sum_{n=1}^{+\infty}na_{n}t^{n-1}=\sum_{n=0}^{+\infty}(n+1)a_{n+1}t^n$</span></p>
</li>
<li><p>By Cauchy product <span class="mat... |
2,111,734 | <p>a) Show that $f^2 - (f')^2 = 0$</p>
<p>I tried to solve by doing $f=\pm f'$ not sure what to do from here.</p>
| Nitin Uniyal | 246,221 | <p>$(f-f')(f+f')=0$</p>
<p>$\implies f'=f $ or $f'=-f$.</p>
<p>Try solving the two IVP's:</p>
<p>$1$. $f'(x)-f(x)=0$ with $f(0)=f'(0)=0$.</p>
<p>$2$. $f'(x)+f(x)=0$ with $f(0)=f'(0)=0$.</p>
<p>Both have a unique solution which $f(x)=0$.</p>
|
250,651 | <p>(<a href="https://math.stackexchange.com/questions/1940157/coend-of-mathscrdf-bullet-g-bullet?noredirect=1#comment3982852_1940157">Crosspost</a> from stack)</p>
<p>Given categories $\mathscr{C}$ and $\mathscr{D}$ and functors $F,G: \mathscr{C} \to \mathscr{D}$, we can form a bifunctor
$$\mathscr{D}(F(\bullet), G(\... | Todd Trimble | 2,926 | <p>I'm not sure I have much to add beyond my comment, but I might add a point of view (which could in turn provide some search terms). </p>
<p>I'd situate this construction within the bicategory of (small) categories, profunctors/bimodules, and transformations between them. Recall that a profunctor $R: C \nrightarrow ... |
29,016 | <p>Suppose I want to compute $f(1)\vee f(2) \vee \ldots \vee f(10^{10})$, but I know <em>a priori</em> that $f(n)$ is <code>True</code> for some $n \ll 10^{10}$ with high probability. For example, <code>f = PrimeQ</code>.</p>
<p>One way to do this is to write: <code>Or[f/@Range[1,10^10]]</code>, but that would involve... | Leonid Shifrin | 81 | <h3>Streams and iterators</h3>
<p>Using streams would certainly be one of the most elegant ways to do this. In any case, you will at least need an iterator for your sequence of numbers. The reason why an abstraction of an iterator is useful is because it separates the iteration over your sequence from the stuff you wa... |
106,560 | <p>Mochizuki has recently announced a proof of the ABC conjecture. It is far too early to judge its correctness, but it builds on many years of work by him. Can someone briefly explain the philosophy behind his work and comment on why it might be expected to shed light on questions like the ABC conjecture?</p>
| Olaf Teschke | 100,979 | <p>For the sake of completeness, let me add the references of the published version in Publ. RIMS that appeared earlier in March this year (should be rather a comment, but the references are too long for that):</p>
<p><em>Mochizuki, Shinichi</em>, <a href="http://dx.doi.org/10.4171/PRIMS/57-1-1" rel="nofollow noreferre... |
155,237 | <p>The axiom of constructibility $V=L$ leads to some very interesting consequences, one of which is that it becomes possible to give explicit constructions of some of the "weird" results of AC. For instance, in $L$, there is a definable well-ordering of the real numbers (since there is a definable well-ordering of the ... | Noah Schweber | 8,133 | <p>Yes, but it's not particularly nice: since $L$ has a definable well-ordering of the sets of reals (in fact, of all of $L$) coming from the $L$-hierarchy itself, there is a formula defining the "least" (in that well-ordering) ultrafilter, $U$.</p>
<p>(This $U$ isn't the only naturally definable ultrafilter in $L$; w... |
320,019 | <p>Assume $Y$ is non negative random variable. Prove that $X+Y$ is stochastically greater than $X$ for any random variable $X$.</p>
<p>We have to prove there that $\Pr(X+Y > x) \geq \Pr(X>x) $ for all $x$</p>
| jdods | 212,426 | <p>Stochastic ordering is equivalent to the existence of a coupling which strictly preserves the order. I.e. there is a random variable $u$ study that $X(u) +Y(u) \geq X(u)$. The statements about probabilities fall out of this readily.</p>
|
2,486,334 | <p>What is the plane graph of $|z-1|+|z-5| < 4$ ?</p>
<p>What i know is that there is nothing for $y\geq4$ or $y \leq -4$ or $x \geq 5$ or $x \leq 1$.</p>
<p>Trying to let $z=x+y i$ such that $x,y \in \mathbb{R}$ did not help either.</p>
| Raffaele | 83,382 | <p>$\left| x+i y-5\right| +\left| x+i y-1\right| <4$</p>
<p>$\sqrt{(x-5)^2+y^2}+\sqrt{(x-1)^2+y^2}<4$</p>
<p>$2 \sqrt{(x-5)^2+y^2} \sqrt{(x-1)^2+y^2}+2x^2-12 x+2 y^2+26<16$</p>
<p>$\sqrt{(x-5)^2+y^2} \sqrt{(x-1)^2+y^2}<-x^2+6 x-y^2-5$</p>
<p>$\left((x-5)^2+y^2\right) \left((x-1)^2+y^2\right)<\left(-x... |
3,841,266 | <p>Let <span class="math-container">$E$</span> and hilbert space and <span class="math-container">$f(x)=\| x\|$</span> for all <span class="math-container">$x\in E$</span>. Study the differentiability of <span class="math-container">$f$</span> on <span class="math-container">$0$</span> and find <span class="math-conta... | Claude Leibovici | 82,404 | <p>In terms of elementary function, it is sure that you must stay with the implicit form. Howver, sooner or later, you will learn that the solution is given by
<span class="math-container">$$y=\sqrt{\frac 12 W\left(2 e^{2(1+ \sin (x))}\right)}$$</span> where <span class="math-container">$W(.)$</span> is Lambert functio... |
2,441,660 | <p>Suppose I have $n$ observations, which are all normally distributed with the same mean (which is unknown) but each has a different variance (the different variances could be called $v_1,...,v_n$ for example, which are all assumed to be known).</p>
<p>What is the best estimate of the mean? And further, how does one ... | paulinho | 474,578 | <p>In these types of problems, it is not too hard to just consider case by case. Let's list out all the possibilities and how many ways to reorganize them:
$$1,2,6 \rightarrow 3!=6 \text{ ways}$$
$$1,3,5 \rightarrow 3!=6 \text{ ways}$$
$$1,4,4 \rightarrow 3!/2!=3 \text{ ways}$$
$$2,2,5 \rightarrow 3!/2!=3 \text{ ways}$... |
2,938,372 | <p>Seems to me like it is. There are only finitely many distinct powers of <span class="math-container">$x$</span> modulo <span class="math-container">$p$</span>, by Fermat's Little Theorem (they are <span class="math-container">$\{1, x, x^2, ..., x^{p-2}\}$</span>), and the coefficient that I choose for each of thes... | Arkady | 23,522 | <p>If <span class="math-container">$p=3$</span>, it's not true that <span class="math-container">$x=x^4$</span> in this ring <span class="math-container">$\mathbb Z_3[x]$</span>. Only the coefficients are mod <span class="math-container">$p$</span>. While <span class="math-container">$x^{p-1}$</span> might be the same ... |
2,938,372 | <p>Seems to me like it is. There are only finitely many distinct powers of <span class="math-container">$x$</span> modulo <span class="math-container">$p$</span>, by Fermat's Little Theorem (they are <span class="math-container">$\{1, x, x^2, ..., x^{p-2}\}$</span>), and the coefficient that I choose for each of thes... | nguyen quang do | 300,700 | <p>I think that you'll get a clearer view if you come back to the definition of a ring structure (commutative ring for simplification). The definitions being granted, let <span class="math-container">$A$</span> be a subring of a ring <span class="math-container">$B$</span>. For <span class="math-container">$x\in B, A[x... |
2,335,275 | <p><img src="https://i.stack.imgur.com/hZN4z.jpg" alt="image 1">
<img src="https://i.stack.imgur.com/TbcGP.jpg" alt="image 2"></p>
<p>I am asking that the convergence in probability explained in the first image shows that the probability that Y-bar in the range (uY - c) to (uY + c) becomes arbitrarily close to 1 for a... | spaceisdarkgreen | 397,125 | <p>The standard definition (in my experience) of $X_n\to_P X$ is that for any $\epsilon > 0,$ $P(|X_n-X|>\epsilon) \to 0,$ i.e. your second definition. (Whether it's $>\epsilon$ or $\ge \epsilon$ is immaterial.) </p>
<p>However, that definition is equivalent to this one: For any $\epsilon>0,$ $P(|X_n-X| \l... |
2,335,275 | <p><img src="https://i.stack.imgur.com/hZN4z.jpg" alt="image 1">
<img src="https://i.stack.imgur.com/TbcGP.jpg" alt="image 2"></p>
<p>I am asking that the convergence in probability explained in the first image shows that the probability that Y-bar in the range (uY - c) to (uY + c) becomes arbitrarily close to 1 for a... | Dhruv Kohli | 97,188 | <p>The second image is the standard definition of convergence in probability which is: A sequence of random variables $X_1, X_2, \ldots, X_n$ is said to converge to the random variable $X$ in probability if for every $\epsilon > 0$, </p>
<p>$$\lim\limits_{n \rightarrow \infty} P(|X_n-X|\geq \epsilon)=0 \equiv \lim\... |
256,138 | <p>I need to generate four positive random values in the range [.1, .6] with (at most) two significant digits to the right of the decimal, and which sum to exactly 1. Here are three attempts that do not work.</p>
<pre><code>x = {.15, .35, .1, .4}; While[Total[x] != 1,
x = Table[Round[RandomReal[{.1, .6}], .010], 4]];... | Domen | 75,628 | <p>Because you have very tight constraints, the number of allowed points is not very large, so you can generate all of them and then sample.</p>
<pre><code>list = Flatten[
Table[If[10 <= (100 - i - j - k) <= 60, {i, j, k, 100 - i - j - k}/
100., Nothing], {i, 10, 60}, {j, 10, 60}, {k, 10, 60}], 2]
Length@... |
4,014,756 | <p>I was reading the book "Quantum Computing Since Democritus".</p>
<blockquote>
<p>"The set of ordinal numbers has the important property of being well
ordered,which means that every subset has a minimum element. This is
unlike the integers or the positive real numbers, where any element
has another tha... | Kavi Rama Murthy | 142,385 | <p><span class="math-container">$C_1\times C_2\times C_3...=\bigcap_{n=1}^{\infty} (C_1\times C_2\times C_3...\times C_n\times \mathbb R\times \mathbb R\times ...)$</span> and intersection of closed sets is always closed.</p>
|
4,057,959 | <p>What is the number of N step random walks starting from a point (x0,y0) to a point (x1,y1) assuming each direction (right,left,up,down) has equal probability. I know the expression for the one dimension case is:</p>
<p><span class="math-container">$$ \binom{N}{\frac{N+(y1-y0)}{2}} $$</span></p>
<p>is there a similar... | Mike Earnest | 177,399 | <p>The linked answer has a formula, but I wanted to provide a different perspective. I am generalizing the method used <a href="https://math.stackexchange.com/a/516043/177399">by Brian M. Scott in this answer</a>.</p>
<p>Suppose the vertices your walk visits are <span class="math-container">$(x_0,y_0),(x_1,y_1),\dots,(... |
2,602,410 | <p>$$\int_{0}^{\pi /2} \frac{\sin^{m}(x)}{\sin^{m}(x)+\cos^{m}(x)}\, dx$$</p>
<p>I've tried dividing by $\cos^{m}(x) $, and subbing out the $\ 1+\cot^{m}(x) $ with $\csc^{n}(x) $ for some $n$, but to no avail. I've also tried adding and subtracting $\cos^{m}(x)$ to the numerator, and substituting $x$ by $\pi-y$, but ... | Anurag A | 68,092 | <p>Let $$I=\int_{0}^{\pi /2} \frac{\sin^{m}(x)}{\sin^{m}(x)+\cos^{m}(x)} \, dx.$$ Then using the substitution $u=\frac{\pi}{2}-x$, we get
$$I=\int_{0}^{\pi /2} \frac{\cos^{m}(u)}{\sin^{m}(u)+\cos^{m}(u)} \, du=\int_{0}^{\pi /2} \frac{\cos^{m}(x)}{\sin^{m}(x)+\cos^{m}(x)} \, dx.$$
Thus
$$2I=\int_{0}^{\pi /2} \frac{\sin... |
175,535 | <p>I have <code>data1</code> and a target point <code>targetPts</code>, and want to find the closest point from the data. As you can see from below, the 29th point from the data is the closest point and thus the value of <code>data1[[29]]</code> is same as the desired value, which is <code>{0.67033, 0.84245}</code>.</p... | Coolwater | 9,754 | <p>You could pass a <code>Rule</code> to <code>Nearest</code>:</p>
<pre><code>First[Nearest[data1 -> data1data2, targetPts, 1]]
</code></pre>
<blockquote>
<p>{0.67032993, 0.84245042, 0.17769644, 0.49872995}</p>
</blockquote>
|
1,873,648 | <p>Let $A=\{1,2,3,...,2^n\}$. Consider the greatest odd factor (not necessarily prime) of each element of A and add them. What does this sum equal? </p>
| KaliMa | 50,707 | <p>For positive integer $k$, let $L(k)$ be the largest odd factor of $k$. Then we have:</p>
<p>$L(k) = L(k/2)$ if $k$ is even, and $L(k) = k$ if $k$ is odd.</p>
<p>We compute:</p>
<p>$$S(n) = \sum_{k=1}^{2^n} L(k)$$</p>
<p>$$S(n) = \sum_{k=1}^{2^{n-1}} L(2k) + \sum_{k=1}^{2^{n-1}} L(2k-1)$$</p>
<p>$$S(n) = \sum_{... |
393,280 | <p>I am reading about the construction of the Affine Grassmannian in Dennis Gaitsgory's seminar <a href="http://www.math.harvard.edu/%7Egaitsgde/grad_2009/SeminarNotes/Oct13(AffGr).pdf" rel="noreferrer">notes</a>
and there are some commutative algebra facts that I am not able to figure out by myself apparently, like th... | KotelKanim | 28,129 | <p><strong>New attempt:</strong></p>
<p>Lemma 24.6.6 in Vakil's AG notes states:</p>
<blockquote>
<p>Suppose $N$ is an $R$-module and $t\in R$ is not a zero divisor on $N$. Then for any $R/(t)$-module $M$, we have
$$
\operatorname{Tor}_i^R(M,N)=\operatorname{Tor}_i^{R/(t)}(M,N/(t)).
$$
(Actually it is stated w... |
3,460,749 | <p>I'm an undergraduate student currently studying mathematical analysis. </p>
<p>Our professor uses Zorich's Mathematical Analysis, but I found the text too difficult to understand. </p>
<p>After exploring some textbooks, I found that Abbott was easier to follow, so I studied Abbott until I realized that there's a s... | Grothendix | 884,032 | <p>True that Zorich is difficult to understand.
BUT I still think if you go through with it, you will learn a lot in the end.
However, I think there’s also an other book you could consider about, which is <strong>Mathematical Analysis</strong> written by Apostol.Or, consider Courant’s book.
They are both good books.
Go... |
1,901,302 | <p>Let $R$ be a principal ideal ring, meaning that every ideal is generated by one element. Given a subset $A\subseteq R$, it is generally not possible to choose one element $x\in A$ so that $I(\{x\})=I(A)$, i.e. the ideal generated by $A$ need not be generated by a single element of $A$.</p>
<p>As an example one can ... | Keith Kearnes | 310,334 | <p>"My question is whether one can choose a finite subset $X$ of $A$ so that $I(A)=I(X)$."</p>
<p>Yes. If $I(A) = (p)$, then $p$ is a linear combination of elements of $A$, say
$p=r_1a_1+r_2a_2+\cdots +r_ka_k$ with $r_i\in R$ and $a_i\in A$. Now take $X = \{a_1,\ldots,a_k\}$.</p>
|
4,283,063 | <p>For positive real numbers <span class="math-container">$a,b ,c$</span> prove that: <span class="math-container">$a^{b+c}b^{c+a}c^{a+b} ≤ (a^ab^bc^c)^2$</span></p>
<p>My working</p>
<p><span class="math-container">$c^c≥b^c≥b^a$</span></p>
<p><span class="math-container">$(c^c)^2 ≥ b^{a+c}$</span></p>
<p>Similarly
<sp... | user10354138 | 592,552 | <p><strong>Hint</strong>: Consider <span class="math-container">$g(x):=x^\alpha\sin(x^\beta)$</span> for <span class="math-container">$x\neq 0$</span>, suitable <span class="math-container">$\alpha,\beta$</span>.</p>
|
4,310,529 | <p>I need a formula that will give me all points of random ellipse and circle intersection (ok, not fully random, the center of circle is laying on ellipse curve)</p>
<p>I need step by step solution (algorithm how to find it) if this is possible.</p>
| Blabbo the Verbose | 994,827 | <p>Choose a coordinate system where the <span class="math-container">$x$</span> axis is parallel to the ellipse major semiaxis, <span class="math-container">$y$</span> axis parallel to the ellipse minor semiaxis, and origin where they intersect ("center of ellipse"). If <span class="math-container">$a$</span... |
598,838 | <p><span class="math-container">$11$</span> out of <span class="math-container">$36$</span>? I got this by writing down the number of possible outcomes (<span class="math-container">$36$</span>) and then counting how many of the pairs had a <span class="math-container">$6$</span> in them: <span class="math-container">$... | waj cheema | 92,867 | <p>The probability of rolling a certain number, can be easily calculated by listing the options available, or listing the probability of not rolling that specific number and subtracting the answer from 1. The probability of a certain event occurring is always a number between 0 and 1.</p>
|
815,963 | <p>I am trying to understand how to do proof by induction for inequalities. The step that I don't fully understand is making an assumption that n=k+1. For equations it is simple. For example:</p>
<blockquote>
<p><strong>Prove that 1+2+3+...+n = $ \frac {n(n+1)}{2} $ is valid for $ i \ge 1 $</strong> </p>
</blockquo... | doppz | 48,746 | <p>For the first part, you haven't done anything, you just rewrote everything. Most importantly, you didn't prove the statement. Induction works by first showing that $P(1)$ holds then assuming $P(n)$ and proving $P(n+1)$. From there, it follows that the $n$ such that $P(n)$ is true must be all of $\mathbf{N}$. </p>
<... |
345,260 | <p>I'm trying to prove that there exists a multiplicative linear functional in $\ell_\infty^*$ that extends the limit funcional that is defined in $c$ (i.e., im looking for a linear functional $f \colon \ell_\infty \to \mathbb K$ such that $f( (x_n * y_n) ) = f (x_n) f(y_n)$, for every $(x_n), (y_n) \in \ell_\infty$, a... | Berci | 41,488 | <p>$\lim_{\mathcal F}$ is of course continuous, and has norm $1$, as
$$|\lim_{\mathcal F}(x_n)|\le \|(x_n)\|_\infty\,,$$
and on constant sequences it holds with equality.</p>
|
2,389,324 | <p>For a continuous Function $f$, prove that:</p>
<p>$$\lim_{x\to0^+}\int_{x}^{2x} \frac{1}{t} f(t) dt = ln(2)f(0)$$ </p>
<p>I have already concluded that since f is continuous, it's therefore integrable.Moreover I assumed there is a function $F$
$$F(x)=\int_{x}^{2x} f(s)ds$$
in order to simplify the limit express... | mouse_wheel | 471,024 | <p>you can use epsilons:
since f is continuous, for a given $\epsilon $ , you can find $ \eta $ such that for all $x \leq \eta$ , we have $ f(0)-\epsilon <=f(x) <= f(0) + \epsilon $
Then you integrate this (divided by t) between x and 2x for x sufficiently small and you get a bounding between $ln(2)(f(0) -\epsilo... |
2,389,324 | <p>For a continuous Function $f$, prove that:</p>
<p>$$\lim_{x\to0^+}\int_{x}^{2x} \frac{1}{t} f(t) dt = ln(2)f(0)$$ </p>
<p>I have already concluded that since f is continuous, it's therefore integrable.Moreover I assumed there is a function $F$
$$F(x)=\int_{x}^{2x} f(s)ds$$
in order to simplify the limit express... | tattwamasi amrutam | 90,328 | <p>$$\int_x ^{2x} \frac{1}{t} f(t) dt-\int_x^{2x}\frac{1}{t}f(0)dt=\int_x^{2x}\frac{1}{t}\left(f(t)-f(0)\right)dt$$
Since $f$ is continuous at $0$, given $\epsilon \gt 0$, there is $\delta \gt 0$ such that $|x| \lt \delta \implies |f(x)-f(0)| \lt \frac{\epsilon}{\ln 2}$. For $|x| \lt \delta$, since $|t| \lt |x| \lt \de... |
3,717,172 | <blockquote>
<p>The random variable <span class="math-container">$X$</span> has an exponential distribution function given by density:
<span class="math-container">$$
f_X(x) =
\begin{cases}
e^{-x}, & x\ge 0,\\
0, & x<0.
\end{cases}
$$</span>
Find the distribution and density function of the random variable... | tommik | 791,458 | <p>A graphic approach</p>
<p>Note: as already note in a comment, <span class="math-container">$f(x)=e^{-x}\mathbb{1}_{[0;+\infty)}(x)$</span></p>
<p>First of all note that <span class="math-container">$X\sim Exp(1)$</span> and so</p>
<ul>
<li><p><span class="math-container">$\mathbb{P}[X \leq x]=F_X(x)=1-e^{-x}$</span>... |
4,563,725 | <p>Is it possible for two non real complex numbers a and b that are squares of each other? (<span class="math-container">$a^2=b$</span> and <span class="math-container">$b^2=a$</span>)?</p>
<p>My answer is not possible because for <span class="math-container">$a^2$</span> to be equal to <span class="math-container">$b$... | whoisit | 1,094,230 | <p>Let <span class="math-container">$a = re^{i \theta}$</span> <br />
Then, <span class="math-container">$a = b^2 = (a^2)^2 = a^4 = r^4e^{i (4 \theta)}$</span></p>
<p>What we need is, first <span class="math-container">$r=r^4$</span> <br />
That is, <span class="math-container">$r =0$</span> or <span class="math-contai... |
3,326,310 | <p>From Serge Lang's Linear Algebra:</p>
<blockquote>
<p>Let <span class="math-container">$x_1$</span>, <span class="math-container">$x_2$</span>, <span class="math-container">$x_3$</span> be numbers. Show that:</p>
<p><span class="math-container">$$\begin{vmatrix} 1 & x_1 & x_1^2\\ 1 &x_2 & x_2^2\\ ... | Monadologie | 669,687 | <p>"Since determinant is a multilinear alternating function, it can be seen that <strong>adding</strong> a scalar multiple of one column (resp. row) to other column (resp. row) does not change the value (I omitted the proof to avoid too much text)
" is right. But
<span class="math-container">$$
\begin{vmatrix} 1 & ... |
3,326,310 | <p>From Serge Lang's Linear Algebra:</p>
<blockquote>
<p>Let <span class="math-container">$x_1$</span>, <span class="math-container">$x_2$</span>, <span class="math-container">$x_3$</span> be numbers. Show that:</p>
<p><span class="math-container">$$\begin{vmatrix} 1 & x_1 & x_1^2\\ 1 &x_2 & x_2^2\\ ... | DIEGO R. | 297,483 | <p>Using induction on <span class="math-container">$n$</span> is maybe the most convenient way. Let's proof that if we assume that the result is true for <span class="math-container">$n - 1$</span> then the result is true for <span class="math-container">$n$</span>. So, consider the Vandermonde matrix
<span class="math... |
3,267,216 | <p>An urn contains equal number of green and red balls. Suppose you are playing the following game. You draw one ball at random from the urn and note its colour. The ball is then placed back in the urn, and the selection process is repeated. Each time a green ball is picked you get 1 Rupee. The first time you pick a re... | Michael Rozenberg | 190,319 | <p>Let <span class="math-container">$a$</span> and <span class="math-container">$b$</span> be positives and <span class="math-container">$a=bx$</span>.</p>
<p>Thus, <span class="math-container">$$\frac{\frac{a+b}{2}}{\sqrt{ab}}=\frac{m}{n}$$</span> or
<span class="math-container">$$x+1=\frac{2m\sqrt{x}}{n}.$$</span>
N... |
1,266,674 | <p>There are $2^{10} =1024$ possible $10$ -letters strings in which each letter is either an $A$ or a $B$. Find the number of such strings that do not have more than $3$ adjacent letters that are identical.</p>
| Jack D'Aurizio | 44,121 | <p>Any allowed string can be seen as a sequence of blocks made of $A$ or $B$ only, whose length is between $1$ and $3$. For instance:</p>
<p>$$ ABBABABBBA \longrightarrow (A)(BB)(A)(B)(A)(BBB)(A)$$
can be associated with the identity: $10=1+2+1+1+1+3+1$. Hence we just have to count in how many ways we can write $10$ a... |
1,965,040 | <p>Find the cyclic subgroups $<\rho_1>, <\rho_2>, and <\mu_1>$ of $S_3$.<a href="https://i.stack.imgur.com/AOnTC.png" rel="nofollow noreferrer"> Elements of $S_3$.</a></p>
<p>I know the answer is suppose to be $<\rho_1> = <\rho_2> = \{\rho_0, \rho_1, \rho_2 \}$ and $<\mu_1> = \{ \rh... | CyclotomicField | 464,974 | <p>The importance of ideals comes from the fact that they're the kernels of a ring homomorphism as consequence of the <a href="https://en.wikipedia.org/wiki/Isomorphism_theorems#Theorem_A_(rings)" rel="nofollow noreferrer">first isomorphism theorem</a>. From perspective if I have a ring homomorphism <span class="math-c... |
1,461,311 | <p>In axiomatic approach to real numbers, that is by defining them to be the complete ordered field, one is expected to prove every theorem and solve every problem by using ultimately only the axioms. I was trying to solve a Spivak's calculus problem that asked to show that the sum of any number of real numbers is stil... | Domates | 8,065 | <blockquote>
<p>The fundamental principle behind induction is that if $S$ is some
subset of $\mathbb{N}$ with the property that <br>
1. $S$ contains $1$<br>
2. whenever $S$ contains a natural number $n$, it also contains $n+1$, then it must be that $S = \mathbb{N}$.(Abbot, Understanding Analysis)</p>
</blockq... |
2,227,135 | <p>I have this information from my notes:<span class="math-container">$\def\rk{\operatorname{rank}}$</span></p>
<p>Let <span class="math-container">$A ∈ \mathbb{R}^{m\times n}$</span>. Then </p>
<ul>
<li><span class="math-container">$\rk(A) = n$</span></li>
<li><span class="math-container">$\rk(A^TA) = n$</span></li>... | copper.hat | 27,978 | <p>Note that $v v^T v = \|v\|^2 v \neq 0$, hence $\operatorname{rk} (v v^T) \ge 1$.</p>
<p>Note that $v v^T x = (v^T x) v \in \operatorname{sp} \{ v \}$ for all $x$. Hence ${R (v v^T)} = \operatorname{sp} \{ v \}$ and hence
$\operatorname{rk} (v v^T) = 1$.</p>
|
2,965,993 | <p>Suppose you have the surface <span class="math-container">$\xi$</span> defined in <span class="math-container">$\mathbb{R}^3$</span> by the equation:
<span class="math-container">$$ \xi :\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1 $$</span>
For <span class="math-container">$ x \geq 0$</span> , <span clas... | Cesareo | 397,348 | <p>Calling </p>
<p><span class="math-container">$$
\begin{cases}
p_0 = (x_0,y_0,z_0)\\
p_1 = (x_1,y_1,z_1)\\
\Lambda = \mbox{diag}[\frac{1}{a^2},\frac{1}{b^2},\frac{1}{c^2}]\\
\vec n_0 = 2\Lambda\cdot p_0\\
f(p_1^*) = \frac 16 x_1^* y_1^* z_1^*\\
x_1^* = \frac{a^2}{x_0}\\
y_1^* = \frac{b^2}{y_0}\\
z_1^* = \frac{c^2}{z... |
1,215,926 | <p>Can someone highlight what is the connection between the transition probability of a continuous time stochastic process $X_t$, i.e. $p(x,t\vert x_0,0)$ and the stochastic differential equation of the evolution of a trajectory of the process $X_t$, i.e. $dX_t=bdt+\sigma dB_t$. How to get from $p$ to the SDE and what ... | tcby_wang | 226,098 | <p>It will be a long story. Simply speaking, the connection is given by Kolmogorov equations. For more comprehensive understanding, I refer you to the book by Daniel Stroock, "Partial differential equations for probabilists"</p>
|
1,215,926 | <p>Can someone highlight what is the connection between the transition probability of a continuous time stochastic process $X_t$, i.e. $p(x,t\vert x_0,0)$ and the stochastic differential equation of the evolution of a trajectory of the process $X_t$, i.e. $dX_t=bdt+\sigma dB_t$. How to get from $p$ to the SDE and what ... | stochastic | 491,395 | <p>From Wikipedia: </p>
<p><a href="https://en.wikipedia.org/wiki/Fokker%E2%80%93Planck_equation#One_dimension" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Fokker%E2%80%93Planck_equation#One_dimension</a></p>
<p>The stochastic differential equation
$$
dX_t = \mu(X_t, t)dt+ \sigma(X_t,t) dW_t
$$</p>
<p>ha... |
3,454,395 | <p>So I’m working on this equation <span class="math-container">$z^{10} + 2z^5 + 2 = 0$</span> to find all complex solutions, and I think I managed to solve it, but I can’t find solution manual for it, since it is really old exam task. The thing that makes me uncomfortable with my solution is that, shouldn’t I get just... | Ahmad Bazzi | 310,385 | <p>solve for <span class="math-container">$$w^2 + 2w + 2=0$$</span>
which gives <span class="math-container">$$w_{1,2} = -1 \pm i = \sqrt{2} e^{i(\pi \pm \frac{\pi}{4})}$$</span>
Now you got two equations to solve
<span class="math-container">\begin{align}
z_1^5 &= \sqrt{2} e^{i(\pi + \frac{\pi}{4})} \\
z_2^5 &... |
1,691,825 | <p>My textbook goes from</p>
<p>$$\frac{\left( \frac{6\ln^22x}{2x} \right)}{\left(\frac{3}{2\sqrt{x}}\right)}$$</p>
<p>to:</p>
<p>$$\frac{6\ln^22x}{3\sqrt{x}}$$</p>
<p>I don't see how this is right. Could anyone explain?</p>
| Newb | 98,587 | <p>$$\frac{\frac{6ln^22x}{2x}}{\frac{3}{2\sqrt{x}}} = \frac{6ln^22x}{2x} \cdot \frac{2\sqrt{x}}{3} = \frac{3ln^22x}{x} \cdot \frac{2\sqrt{x}}{3} = \frac{6\sqrt{x}ln^22x}{3x} = \frac{6\ln^22x}{3} \cdot \frac{\sqrt{x}}{x}$$</p>
<p>$$\frac{6\ln^22x}{3} \cdot \frac{\sqrt{x}}{x} = \frac{6\ln^22x}{3} \cdot x^{1/2}/x^1 = \f... |
3,714,995 | <blockquote>
<p>Using sell method to find the volume of solid generated by revolving the region bounded by <span class="math-container">$$y=\sqrt{x},y=\frac{x-3}{2},y=0$$</span> about <span class="math-container">$x$</span> axis, is (using shell method)</p>
</blockquote>
<p>What I try:</p>
<p><a href="https://i.sta... | Chrystomath | 84,081 | <p>For any inner product space, complete or not, let <span class="math-container">$v$</span> be any vector and consider the finite-dimensional subspace generated from <span class="math-container">$v,Av,\ldots,A^{k-1}v$</span>. Then this space is <span class="math-container">$A$</span>-invariant, so <span class="math-co... |
47,561 | <p>The Hilbert matrix is the square matrix given by</p>
<p>$$H_{ij}=\frac{1}{i+j-1}$$</p>
<p>Wikipedia states that its inverse is given by</p>
<p>$$(H^{-1})_{ij} = (-1)^{i+j}(i+j-1) {{n+i-1}\choose{n-j}}{{n+j-1}\choose{n-i}}{{i+j-2}\choose{i-1}}^2$$</p>
<p>It follows that the entries in the inverse matrix are all i... | J. M. isn't a mathematician | 7,934 | <p>It's a bit circuitous, but I'd like to point out this <a href="http://dx.doi.org/10.1007/BF03167904" rel="noreferrer">paper</a> by Hitotumatu where he derives explicit expressions for the Cholesky triangle of a Hilbert matrix. From the expressions for the Cholesky triangle, you should be able to derive explicit expr... |
2,983,199 | <p>I am looking for an example demonstrating that <span class="math-container">$\lim\inf x_n+\lim \inf y_n<\lim \inf(x_n+y_n)$</span> but for the life of me i can't find one. any suggestions?</p>
| user2820579 | 141,841 | <p>Choose <span class="math-container">$(x_n) = (1,-1,1,-1,\dots)$</span> and <span class="math-container">$(y_n) = (-1,1,-1/2,1,-1/3,\dots)$</span>. Then <span class="math-container">$(x_n+y_n) = (0,0,1/2,0,2/3,\dots)$</span>. This means <span class="math-container">$\liminf x_n = -1$</span>, <span class="math-contain... |
3,252,076 | <p>I'm doing some Galois cohomology stuff (specifically, trying to calculate <span class="math-container">$H^1(\mathbb{Q}_3,E[\varphi])$</span>, where <span class="math-container">$\varphi:E\to E'$</span> is an isogeny of elliptic curves), and it involves calculating <span class="math-container">$\mathbb{Q}_3(\sqrt{-6}... | Lubin | 17,760 | <p>First, you needn’t have worried about what parameter you used: <span class="math-container">$\sqrt{-6}$</span> is just as good as <span class="math-container">$\sqrt3$</span>. Indeed, if <span class="math-container">$\mathfrak o$</span> is a complete discrete valuation ring with fraction field <span class="math-cont... |
164,328 | <p>What are good introductory textbooks available on Cohomology of Groups?</p>
| Community | -1 | <p>Also, if you are interested in <em>number theory</em>, <a href="http://www.mathi.uni-heidelberg.de/~schmidt/NSW2e/index-de.html" rel="nofollow">http://www.mathi.uni-heidelberg.de/~schmidt/NSW2e/index-de.html</a> (Jürgen Neukirch, Alexander Schmidt, Kay Wingberg:
Cohomology of Number Fields, second edition) and <a hr... |
1,787,460 | <blockquote>
<p>Suppose the n<em>th</em> pass through a manufacturing process is modelled by the linear equations <span class="math-container">$x_n=A^nx_0$</span>, where <span class="math-container">$x_0$</span> is the initial state of the system and</p>
<p><span class="math-container">$$A=\frac{1}{5} \begin{bmatrix} 3... | Emilio Novati | 187,568 | <p>Diagonalize the matrix $A$:
$$
A=\begin{bmatrix}
\frac{3}{5}&\frac{2}{5}\\
\frac{2}{5}&\frac{3}{5}
\end{bmatrix}=
\begin{bmatrix}
-1&1\\
1&1
\end{bmatrix}
\begin{bmatrix}
\frac{1}{5}&0\\
0&1
\end{bmatrix}
\begin{bmatrix}
-\frac{1}{2}&\frac{1}{2}\\
\frac{1}{2}&\frac{1}{2}
\end{bmatrix}... |
2,965,756 | <p>I want to intuitively say tht the answer is yes, but if it so happens that <span class="math-container">$|\mathbf{a}|=|\mathbf{b}|cos(\theta)$</span>, where <span class="math-container">$\theta$</span> is the angle between the two vectors, then the equation will be satisfied without the two vectors being the same.</... | Hans Lundmark | 1,242 | <p>The equation <span class="math-container">$\mathbf{a} \cdot \mathbf{a} = \mathbf{a} \cdot \mathbf{b}$</span>
is equivalent to
<span class="math-container">$\mathbf{a} \cdot (\mathbf{a}-\mathbf{b})=0$</span>,
which is the same as saying that <span class="math-container">$\mathbf{a}$</span> is orthogonal to <span clas... |
135,308 | <p>Let $G$ be a compact Lie group and $M$ be a smooth manifold on which $G$ acts smoothly and effectively. Then the orbit space $M/G$ admits an Whitney stratification as follows
$$M/G=\bigsqcup_{H<G}(M_{(H)}/G).$$
Where $H$ is a closed subgroup of $G$, and $M_{(H)}$ is the set of the points in $M$ such that the isot... | Peter Michor | 26,935 | <p>A codimension 1 stratum needs a reflection in the isotropy group; better, in the isotropy representation $G_x:T_xM\to T_xM$ for x over a codimension 1 stratum. </p>
|
2,119,971 | <p>If $a\mid c$ and $b\mid c$, must $ab$ divide $c$? Justify your answer.</p>
<p>$a\mid c$, $c=ak$ for some integer $k$</p>
<p>$b\mid c$, $c=bu$ for some integer $u$</p>
<p>From here I wanted to try to check if there were counter examples I could use,</p>
<p>$c\ne(ab)w$ for some integer $w$</p>
<p>From here I got ... | E. Joseph | 288,138 | <p>This is <strong>not</strong> true.</p>
<p>Take for instance $a=b=2$ and $c=2$.</p>
<p>Then $a\mid c $ and $b\mid c$, but</p>
<p>$$ab=4\nmid 2=c.$$</p>
|
2,119,971 | <p>If $a\mid c$ and $b\mid c$, must $ab$ divide $c$? Justify your answer.</p>
<p>$a\mid c$, $c=ak$ for some integer $k$</p>
<p>$b\mid c$, $c=bu$ for some integer $u$</p>
<p>From here I wanted to try to check if there were counter examples I could use,</p>
<p>$c\ne(ab)w$ for some integer $w$</p>
<p>From here I got ... | Carlos Seda | 411,249 | <p>No it must not, you can give a simple counter-example like <span class="math-container">$3\mid 9$</span> and <span class="math-container">$9\mid 9$</span> but obviously <span class="math-container">$$ab=9\cdot3=27 \nmid9$$</span> Hope it helps you out! </p>
|
2,319,766 | <p>Lets say ,I have 100 numbers(1 to 100).I have to create various combinations of 10 numbers out of these 100 numbers such that no two combinations have more than 5 numbers in common given a particular number can be used max three times.
E.g.</p>
<ol>
<li>Combination 1: 1,2,3,4,5,6,7,8,9,10</li>
<li>Combination 2: ... | layabout | 352,341 | <p>This is a <a href="https://en.wikipedia.org/wiki/Stars_and_bars_(combinatorics)" rel="nofollow noreferrer">stars and bars problem</a>. </p>
<p><a href="https://en.wikipedia.org/wiki/Stars_and_bars_(combinatorics)#Theorem_one" rel="nofollow noreferrer">Theorem 1</a>: in this case, we require that we must include at ... |
2,893,388 | <p>My textbook is confusing me a little. Here is a worked example from my textbook:</p>
<blockquote>
<p>Line <span class="math-container">$l$</span> has the equation <span class="math-container">$\begin{pmatrix}3\\ -1\\ 0\end{pmatrix}+\lambda \begin{pmatrix}1\\ -1\\ 1\end{pmatrix}$</span> and point <span class="math-c... | Umberto P. | 67,536 | <p>It doesn't mean that a basis $\cal B$ of $\cal T$ is countable. It means that <em>there exists</em> a basis $\cal B$ of $\cal T$ that is countable.</p>
|
52,677 | <p>Why people have to find quadratic formula,isn't that the formula cannot solve a polynomial with 2 and 1/2 degree?
and just curious, how many roots does a polynomial with 2 and 1/2 degree have and how to solve them all(by formula)?</p>
| Jay | 9,814 | <p>As Ross points out in his comment polynomials are defined to have exponents that are positive integers. If you find the roots of a polynomial $p(x)$ using this definition than, in certain cases, one can find the roots by algebraic means. A polynomial of degree $n$ has $n + 1$ different coefficients, the constant ter... |
4,316,771 | <p>I'm asked to compute <span class="math-container">$$\sum_{k=-3}^{10} 2k^4$$</span>
I looked up for<a href="https://en.m.wikipedia.org/wiki/Bernoulli_number" rel="nofollow noreferrer">Bernoulli Number</a> on Wikipedia and found a general formula for that. But my teacher has asked me to evaluate this by breaking the s... | p_square | 920,884 | <p><span class="math-container">$$\sum_{k=-3}^{10} 2k^4 = \sum_{n=1}^{14} 2(n-4)^4$$</span>
now expand <span class="math-container">$2(n-4)^4$</span> which comes out to be : <span class="math-container">$$2n^4 - 32n^3 + 192n^2 - 512n + 512$$</span> Now our sum becomes :
<span class="math-container">$$\sum_{n=1}^{14} 2n... |
4,402,613 | <p>I know that the limits of integration in spherical coordinates are these, but I can't find the reason why that 2 appears,but I can't find a way to go further, and evaluate r in a <span class="math-container">$a \sin \theta$</span>,</p>
<p><span class="math-container">$$V = \int_0^{2\pi}\int_0^{a\sin{\theta}}\int_{-\... | Átila Correia | 953,679 | <p>The number two appears because you have to consider the integral corresponding to the upper and lower parts of the sphere, which are equal due to the symmetry of the problem.</p>
<p>As far as I have understood, you are not applying spherical change of coordinates. Instead, you are applying the cylindrical change of ... |
398,409 | <p>I was asked to help someone with this problem, and I don't really know the answer why. But I thought I'd still try.</p>
<p>$$\lim_{t \to 10} \frac{t^2 - 100}{t+1} \cos\left( \frac{1}{10-t} \right)+ 100$$</p>
<p>The problem lies with the cos term. What can I do with the cos term to remove divide by 0 ? </p>
<p>I f... | Cantlog | 66,760 | <p>A necessary condition is <span class="math-container">$m\mid n$</span> and <span class="math-container">$u$</span> is an <span class="math-container">$m$</span>-th power in <span class="math-container">$\mathbb F_p$</span>. So the real question is:</p>
<blockquote>
<p>Let <span class="math-container">$a\in \mathbb Z... |
1,540,412 | <p>Let x be a (right) eigenvector of A corresponding to an eigenvalue λ and let y be a left eigenvector of A corresponding to a different eigenvalue µ, where λ $\neq$ µ. Show that x∗y = 0. Hint : Ax = λx and y'A = µy'</p>
| YVLM | 292,316 | <ul>
<li>Step 1) <span class="math-container">$Ax=λx$</span></li>
<li>Step 2) <span class="math-container">$y'Ax=λy'x$</span></li>
<li>Step 3) <span class="math-container">$y'Ax-λy'x=0$</span></li>
<li>Step 4) <span class="math-container">$(y'A-λy')x=0$</span></li>
<li>Step 5) <span class="math-container">$(\mu y'-λy'... |
2,605,208 | <p>Decide whether the given set of vectors is linearly independent in the indicated vector space:</p>
<p>$\{ x_1, x_1 +x_2, x_1 +x_2 +x_3, ..., x_1+\cdots+x_n\} $</p>
<p>if $\{x_1, x_2, x_3, ..., x_n\}$ is linearly
independent, in some vector space $V$.</p>
<hr>
<p>If $n=4:$</p>
<p>$x_1 - (x_1+x_2) + (x_1+x_2+x_3)... | Qiaochu Yuan | 232 | <p>The category of principal bundles is a groupoid, so the only way it could have a terminal object is if it were contractible, meaning both that</p>
<ol>
<li>There is only one isomorphism class of principal bundle, and</li>
<li>Its automorphism group is trivial.</li>
</ol>
<p>However, you can verify that the automor... |
186,182 | <p>Suppose, for the sake of keeping things as simple as possible, that I have the following equation that I wish to simplify in Mathematica:</p>
<p><span class="math-container">$y = x x$</span></p>
<p>But suppose further that I also have a restriction, not directly on <span class="math-container">$x$</span>, but on w... | Alan | 19,530 | <p>Here are two approaches. The first requires perhaps more tolerance for "noisy" notation. Note that I did not use a vector argument. If you must, the notation will be correspondingly "noisier".</p>
<pre><code>Clear[f, x, y]
Dt[f[x, y]]
Grad[f[x, y], {x, y}].{dx, dy}
</code></pre>
|
266,834 | <p>I know the values for a function v[x,y] on an irregular grid of (x,y) points. Call the table storing all these points xyvtriples. Because of the irregular grid, the Mathematica function Interpolation only works as</p>
<p><code>interpolatedvfunc = Interpolation[xyvtriples, InterpolationOrder -> 1];</code></p>
<p... | bmf | 85,558 | <pre><code>Part[CoefficientArrays[
1/4 (-1 - Subscript[φ, 49, 50] - Subscript[φ, 50, 49]) +
Subscript[φ, 50, 50]], 1]
</code></pre>
<blockquote>
<p>-(1/4)</p>
</blockquote>
|
3,774,400 | <p>Just like the title says, I don't know how to write an example matrix here to look like matrix. If this makes sense, <span class="math-container">$A$</span> can be <span class="math-container">$[1 0 0;0 1 0;0 1 0]$</span> (like in MATLAB syntax) then if we find determinants of <span class="math-container">$A-\lambda... | Ennar | 122,131 | <p>Let <span class="math-container">$A = (a_{ij}),\ i,j=1,\ldots,n$</span>. From the definition of determinant, it's clear that <span class="math-container">$\det(A-\lambda I)$</span> is a polynomial in <span class="math-container">$\lambda$</span>. Moreover, it is of degree <span class="math-container">$n$</span>:</p>... |
324,219 | <p>Given an urn with $M$ unique balls, how many times do I need to draw with replacement before the probability that I have seen each ball at least once is greater than $\epsilon$?</p>
| Christian Blatter | 1,303 | <p>Any sequence of $n$ drawings results in a word $w$ of length $n$ over the alphabet $[M]$. There are $M^n$ such words.</p>
<p>How many of these words $w$ are <em>admissible</em>, meaning that $w$ contains each letter $\ell\in [M]$ at least once? Any admissible word can be fabricated in the following way: Choose a pa... |
4,634,797 | <p>I am currently trying to show that the sequence of functions defined by <span class="math-container">$f_n(x) = \frac{x}{1 + x^n}$</span> converges pointwise on <span class="math-container">$U = [0, \infty)$</span>. I have found the limits for the three specific cases and they are:
<span class="math-container">\begin... | geetha290krm | 1,064,504 | <p>I will asume that <span class="math-container">$\epsilon <1$</span> and let you see what heppens if <span class="math-container">$\epsilon \geq 1$</span>.</p>
<p><span class="math-container">$\frac{x^{n + 1}}{1 + x^n} < \epsilon$</span> if <span class="math-container">$\frac{x^{n }}{1 + x^n} < \epsilon$</sp... |
1,752,506 | <p>Question: $ \sqrt{x^2 + 1} + \frac{8}{\sqrt{x^2 + 1}} = \sqrt{x^2 + 9}$</p>
<p>My solution: $(x^2 + 1) + 8 = \sqrt{x^2 + 9} \sqrt{x^2 + 1}$</p>
<p>$=> (x^2 + 9) = \sqrt{x^2 + 9} \sqrt{x^2 + 1}$</p>
<p>$=> (x^2 + 9) - \sqrt{x^2 + 9} \sqrt{x^2 + 1} = 0$</p>
<p>$=> \sqrt{x^2 + 9} (\sqrt{x^2 + 9} - \sqrt{x^... | Murtuza Vadharia | 132,945 | <p>Try a different method.</p>
<p>Take $\sqrt{x^2 +1}$ on RHS Then rationalise $\sqrt{x^2+9}$ - $\sqrt{x^2 +1}$ by it's conjugate.</p>
<p>So your next step would be :</p>
<p>$\dfrac{8}{\sqrt{x^2+1}}$[$\sqrt{x^2+9}$ + $\sqrt{x^2 +1}$] = 8</p>
<p>So 8 gets cancelled. and next step is as follows:
$\dfrac{\sqrt{x^2+9}... |
419,802 | <p>Let <span class="math-container">$A$</span> be a C<span class="math-container">$^*$</span>-algebra with closed two-sided ideal <span class="math-container">$I$</span>. Set <span class="math-container">$B=A/I$</span> and let <span class="math-container">$\pi:A\to B$</span> be the quotient map. Suppose that <span clas... | Konstantinos Kanakoglou | 85,967 | <p>Far from being a specialist on the topic, section 4 of the article <a href="https://kconrad.math.uconn.edu/blurbs/grouptheory/SL(2,Z).pdf" rel="nofollow noreferrer"><span class="math-container">$\text{SL}_2(\mathbb{Z})$</span></a>, by K. Conrad discusses non-congruence subgroups.</p>
|
419,802 | <p>Let <span class="math-container">$A$</span> be a C<span class="math-container">$^*$</span>-algebra with closed two-sided ideal <span class="math-container">$I$</span>. Set <span class="math-container">$B=A/I$</span> and let <span class="math-container">$\pi:A\to B$</span> be the quotient map. Suppose that <span clas... | Will Sawin | 18,060 | <p><span class="math-container">$\DeclareMathOperator\SL{SL}\DeclareMathOperator\PSL{PSL}\DeclareMathOperator\GL{GL}$</span>The reason that there are uncountably many irreducible representations is not so bad, and gets at an important point: You shouldn't think of irreducible representations of a group like <span class... |
419,802 | <p>Let <span class="math-container">$A$</span> be a C<span class="math-container">$^*$</span>-algebra with closed two-sided ideal <span class="math-container">$I$</span>. Set <span class="math-container">$B=A/I$</span> and let <span class="math-container">$\pi:A\to B$</span> be the quotient map. Suppose that <span clas... | Andy Putman | 317 | <p><span class="math-container">$\DeclareMathOperator\SL{SL}$</span>The issue is that <span class="math-container">$\SL(2,\mathbb{Z})$</span> is very close to a free group, so it is not hard to map it to other groups, and in particular to produce lots of varied representations of it.</p>
<p>You might be interested to k... |
2,482,564 | <p>Let $(\Omega,\Sigma)$ be a given measurable space and let $f$ be a $\Sigma$-measurable function. If $h:[-\infty,\infty]\rightarrow[-\infty,\infty]$ is a continuous function, then the composite function $hf$ is measurable.</p>
<p>This can be proven easily if every continuous function is measurable as I just need to ... | H.Vogel | 475,363 | <p>As the composition $hf$ needs to be well-defined, I assume the following domain and target sets of $f$ and $h$: $f:\Omega \rightarrow \mathbb{R}$ and $h: \mathbb{R}\rightarrow \mathbb{R}$ (as $(-\infty,\infty)=\mathbb{R}$).<br>
Thus we have a measurable function $h$ from $(\Omega, \Sigma) \rightarrow (\mathbb{R},\ma... |
2,420,435 | <p>I'm learning logic for computer science and came across the question:</p>
<blockquote>
<p>If $\ n$ is a real number, $\frac{1}{n}$ is the reciprocal of $\ n$. Prove that all
numbers have a unique reciprocal.</p>
</blockquote>
<p>I came up with the following method, but it seems so simple that I doubt it'll wor... | Raffaele | 83,382 | <p>Your conclusion is false. Not all $x\in\mathbb{R}$ has reciprocal. $0$ as no reciprocal. </p>
<p>To show unicity. </p>
<p>If $a\ne 0$ then $a^{-1}a=aa^{-1}=1$. If $a$ had another reciprocal $b$ such that $ab=1$ then left multiply by $a^{-1}$ you get $a^{-1}ab=a^{-1}$ hence $b=a^{-1}$. Same procedure for $ba=1$ rig... |
20,463 | <p>I'm new here and I don't know how things work here and don't have any knowledge about commenting or answering questions. How do I get to know if my question has been commented or answered?</p>
| Ivo Terek | 118,056 | <p>Check your <em>inbox</em>:</p>
<p><img src="https://i.stack.imgur.com/L8ibr.png" alt="enter image description here"></p>
<p>Like this:</p>
<p><img src="https://i.stack.imgur.com/EGY2E.png" alt="enter image description here"></p>
<p>The number of unread messages will show up in red:</p>
<p><img src="https://i.st... |
2,361,275 | <p>$$\neg(A\land B)\land(B\lor\neg C)\Leftrightarrow(\neg A\land B)\lor(\neg B\land\neg C)$$
Checked their truth tables are the same, but can you show the steps how to transform one to the other?</p>
| Bram28 | 256,001 | <p>As an addendum to @Koepi's Answer:</p>
<p>To go from $(\neg A \land B) \lor (\neg A \land \neg C) \lor (\neg B \land \neg C)$ to $(\neg A \land B) \lor (\neg B \land \neg C)$ is an immediate application of the Consensus theorem:</p>
<p><strong>Consensus</strong></p>
<p>$$(P \land Q) \lor (\neg Q \land R) \lor (P ... |
627,575 | <p>There is a whiskey made up of 64% corn, 32% rye, and 4% barley that was made by blending other whiskies together. I am trying to figure out if there is a chance the ratio of this whiskey could be the result of blending two, maybe three whiskies of different ratios.</p>
<p>The possible whiskies:</p>
<p>Whiskey A is... | Claude Leibovici | 82,404 | <p>For simplification, without any lost of information, I shall consider that the mixture you will make will be in the ratio 1:x:y. </p>
<p>Now, let us forget about percentage and consider what were percentages as quantities and let us mix 1 liter of A plus "x" liters of B plus "y" liters of C. So, the total volume o... |
3,757,972 | <p>If <span class="math-container">$\frac{ab} {a+b} = y$</span>, where <span class="math-container">$a$</span> and <span class="math-container">$b$</span> are greater than zero, why is <span class="math-container">$y$</span> always smaller than the smallest number substituted?</p>
<p>Say <span class="math-container">$a... | Michael Rozenberg | 190,319 | <p>For positives <span class="math-container">$a$</span> and <span class="math-container">$b$</span> let <span class="math-container">$a=kb$</span>, where <span class="math-container">$k\geq1$</span>.</p>
<p>Thus, <span class="math-container">$a\geq b$</span> and <span class="math-container">$$\frac{ab}{a+b}=\frac{kb}{... |
2,647,868 | <p>I'm very confused at the following question:</p>
<blockquote>
<p>Find the basis for the image and a basis of the kernel for the following matrix:
$\begin{bmatrix} 7 & 0 & 7 \\ 2 & 3 & 8 \\ 9 & 0 & 9 \\ 5 & 6 & 17 \end{bmatrix}$</p>
</blockquote>
<p>I just don't know how to do an... | Connington | 530,877 | <p>You're absolutely correct with your basis for the image of A, aka its row space. The basis for the kernel of A is found similarly: you must solve the homogeneous system $ A \mathbf x = \mathbf 0 $, where $\mathbf x$ in your case is the 3x1 column vector $\mathbf x = (x_1, x_2, x_3)^T $. The solution to this is found... |
757,702 | <p>I am in my pre-academic year. We recently studied the Remainder sentence (at least that's what I think it translates) which states that any polynomial can be written as <span class="math-container">$P = Q\cdot L + R$</span></p>
<p>I am unable to solve the following:</p>
<blockquote>
<p>Show that <span class="mat... | lab bhattacharjee | 33,337 | <p>$$(x+1)^{2n+1}+x^{n+2}=(x+1)\{(x+1)^2\}^n+x^{n+2}=(x+1)(x^2+2x+1)^n+x^{n+2}$$</p>
<p>Now as $x^2+2x+1\equiv x\pmod{x^2+x+1}$</p>
<p>$$(x+1)(x^2+2x+1)^n+x^{n+2}\equiv (x+1)(x)^n+x^{n+2}\pmod{x^2+x+1}$$</p>
<p>$$\equiv x^n(x+1+x^2)$$</p>
|
4,351,725 | <p>This is what the solution says:</p>
<p>Since each string of 4 digits are independent, having 2018 in a string has probability of <span class="math-container">$(1/10)^4$</span></p>
<p>By geometric distribution, expected value of digits to obtain 2018 in a string would be <span class="math-container">$10^4$</span></p>... | Salcio | 821,280 | <p>Since <span class="math-container">$log_{10}2$</span> is irrational <span class="math-container">$\{n*log_{10}2\}$</span>, <span class="math-container">$n \in N$</span> is uniform distributed on <span class="math-container">$[0,1]$</span>. In particular it is dense on <span class="math-container">$[0,1]$</span>. Thu... |
2,298,873 | <p><strong>Problem statement:</strong></p>
<p>There are three spheres. The one which will roll is $\textbf{X}=(x_1,x_2,x_3)$ with radius $R_X$. The other two spheres are $\textbf{A}=(a_1,a_2,a_3)$ with $R_A$ and $\textbf{B}=(b_1,b_2,b_3)$ with $R_B$. They are both below $X$, meaning $x_3>a_3$ and $x_3>b_3$. They... | amd | 265,466 | <p>The axis of rotation is, as you’ve already determined, the line through the center of the two fixed spheres, which can be represented parametrically as $(1-\lambda)\mathbf A+\lambda\mathbf B$. This line has direction vector $\mathbf B-\mathbf A$, which is normalized to $\mathbf w=\|\mathbf B-\mathbf A\|$. When a poi... |
338,099 | <p>Are there general ways for given rational coefficients <span class="math-container">$a,b,c$</span> (I am particularly interested in <span class="math-container">$a=3,b=1,c=8076$</span>, but in general case too) to answer whether this equation has a rational solution or not?</p>
| Sam | 144,949 | <p>Above equation shown below:</p>
<p><span class="math-container">$ax^4+by^2=c$</span> --------<span class="math-container">$(1)$</span></p>
<p>For equation <span class="math-container">$(1)$</span>, Seiji Tomita has given parametric solution. </p>
<p>For given <span class="math-container">$(a,b,c) =[(2),(1),(3*33... |
1,898,810 | <p>How do I integrate $\frac{1}{1-x^2}$ without using trigonometric identities or partial fractions? Thanks!</p>
| Jan Eerland | 226,665 | <p>Use, the series representation:</p>
<p>$$\frac{1}{1-x^2}=\sum_{n=0}^{\infty}\frac{x^n(1+(-1)^n)}{2}$$</p>
<p>So, we get:</p>
<p>$$\text{I}=\int\frac{1}{1-x^2}\space\text{d}x=\int\sum_{n=0}^{\infty}\frac{x^n(1+(-1)^n)}{2}\space\text{d}x=\sum_{n=0}^{\infty}\frac{1+(-1)^n}{2}\int x^n\space\text{d}x$$</p>
<p>Now, us... |
1,898,810 | <p>How do I integrate $\frac{1}{1-x^2}$ without using trigonometric identities or partial fractions? Thanks!</p>
| Community | -1 | <p>$$\frac1{1-x^2}=\frac{1-x+x}{1-x^2}=\frac1{1+x}+\frac x{1-x^2}$$ hence</p>
<p>$$\log(1+x)-\frac12\log(1-x^2).$$</p>
<p>(Not much different from a decomposition in simple fractions.)</p>
|
1,621,302 | <blockquote>
<p>Assume $a,n\in\mathbb{N}$ such that $\gcd{(a,n)}=1$. We say $n$ is prime if $a^{n-1}\equiv 1\mod{n}$ and $a^x\not\equiv1\mod{n}$ for any divisor $x$ of $n-1$.</p>
</blockquote>
<p>I am presented with the following proof but there are jumps within it that I don't understand.</p>
<hr>
<p><em>Proof</e... | Bernard | 202,857 | <p>Assertion 1 is false in general, except if $n$ is a prime number (in the ordinary sense). The definition here given is that of a <em>(Fermat) pseudoprime relative to base $a$</em> (first condition) $+$ an extra condition on the order..</p>
<p>A counter-example: $\gcd(3,8)=1$, but the order of $3$ mod. $8$ est $2$ –... |
1,414,690 | <p>This is a followup to my question <a href="https://math.stackexchange.com/questions/1414636/eigenvalues-of-matrix-with-all-1s">here</a>.</p>
<p>Let $A$ be the $n \times n$ matrix over a field of characteristic 0, all of whose entries are 1. Is $A$ diagonalizable?</p>
| Will Jagy | 10,400 | <p>Here is a matrix <span class="math-container">$P$</span> that I made up some time ago. Note that <span class="math-container">$P$</span> is not orthogonal, although the columns are pairwise orthogonal.</p>
<p>10:
<span class="math-container">$$
\left(
\begin{array}{rrrrrrrrrr}
1 & -1 & -1 &... |
2,733,142 | <p>This question has been asked a <a href="https://math.stackexchange.com/questions/207029/a-b2-for-which-matrix-a">few</a> <a href="https://math.stackexchange.com/questions/583442/square-root-of-nilpotent-matrix">times</a>. In the former case I noticed that there was some argument trending towards using the Jordan fo... | Will Jagy | 10,400 | <p>just calculations over some field. If
$$
\left(
\begin{array}{cc}
a & b \\
c & d
\end{array}
\right)^2 =
\left(
\begin{array}{cc}
0 & 1 \\
0 & 0
\end{array}
\right) \; ,
$$
then
$$
\left(
\begin{array}{cc}
a^2 + bc & b(a+d) \\
c(a+d) & d^2 + bc
\end{array}
\right) =
\left(
\begin{array}{cc}
... |
202,111 | <p>Consider the following PDE
$$ \frac{\partial \Phi}{\partial t} - \frac{1}{2}y \frac{\partial \Phi}{\partial x} + \alpha \beta y^{3/2} \frac{\partial^2 \Phi}{\partial x \partial y} + \frac{1}{2} y \frac{\partial^2 \Phi}{\partial x^2} + \frac{1}{2} \alpha^2 y^2 \frac{\partial^2 \Phi}{\partial y^2} = 0 $$
What is a goo... | Mhenni Benghorbal | 35,472 | <p>Here is a solution to the pde</p>
<p>$$ \Phi \left( x,y,t \right) = A\left( 2\,{\frac {\ln \left( y \right) }{
{\alpha}^{2}}} + t \right) + B\left( {\frac {y\ln \left(
y \right) }{{\alpha}^{2}}} - {\frac {y}{{\alpha}^{2}}}+x \right)+{\it C_1}+C_2{{\rm e}^{x}}+{\it C_3}\,y\,,$$</p>
<p>where $A,B,C_i\,,i=1,2,3 \... |
202,111 | <p>Consider the following PDE
$$ \frac{\partial \Phi}{\partial t} - \frac{1}{2}y \frac{\partial \Phi}{\partial x} + \alpha \beta y^{3/2} \frac{\partial^2 \Phi}{\partial x \partial y} + \frac{1}{2} y \frac{\partial^2 \Phi}{\partial x^2} + \frac{1}{2} \alpha^2 y^2 \frac{\partial^2 \Phi}{\partial y^2} = 0 $$
What is a goo... | doraemonpaul | 30,938 | <p>Case $1$ : $\alpha=0$</p>
<p>Then $\dfrac{\partial\Phi}{\partial t}-\dfrac{y}{2}\dfrac{\partial\Phi}{\partial x}+\dfrac{y}{2}\dfrac{\partial^2\Phi}{\partial x^2}=0$</p>
<p>Case $1$a : $\text{Re}(yt)\geq0$</p>
<p>Let $\Phi(x,y,t)=X(x)T(y,t)$ ,</p>
<p>Then $X(x)\dfrac{\partial T(y,t)}{\partial t}-\dfrac{yT(y,t)}{2... |
2,716,036 | <p>In reviewing some old homework assignments, I found two problems that I really do not understand, despite the fact that I have the answers.</p>
<p>The first is: R(x, y) if y = 2^d * x for some nonnegative integer d. What I do not understand about this relation is how it can possibly be transitive (according to my n... | Thomas Bakx | 545,960 | <p>Recall that the first relation has to hold for $\textit{some}$ integer $d$. Can you find it, given the ones that relate x to y and y to z? </p>
<p>For the second: x is related to x if and only if x and x are divisible by 17, which is the same as x being divisible by 17. This need not be true, right? </p>
<p>Hope t... |
1,187,706 | <p>I'd imagine this is a duplicate question, but I can't find it:</p>
<p>How many quadratic residues are there $\pmod{2^n}$. </p>
<p>I tried small $n$: $n=1: 2, n=2:2, n=3: 3, n=4: 4, n=5: $not 5: 0, 1, 4, 9, 16, 25</p>
<p>No pattern :/</p>
| Jef | 188,361 | <p>There is a basic solution that only uses modular arithmetic.
Let $p(n)$ be the number of quadratic residues modulo $2^n$. We know that $p(1) = p(2) = 2$. Now suppose $n \geq 3$. We will proof 3 small lemmas.</p>
<p><strong>Lemma 1.</strong> It suffices to consider the residues of the numbers $0,1,...,2^{n-2}$</p>
... |
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