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 |
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1,041,134 | <p>I need to show if $a$ is in $\mathbb{R}$ but not equal to $0$, and $a+\dfrac{1}{a}$ is integer, $a^t+\dfrac{1}{a^t}$ is also an integer for all $t\in\mathbb N$.
Can you provide me some hints please?</p>
| Adhvaitha | 191,728 | <p>Use induction on $$a^{t+1} + \frac1{a^{t+1}} = \left(a^t+\frac1{a^t}\right)\left(a+\frac1a\right) - \left(a^{t-1}+\frac1{a^{t-1}}\right)$$</p>
|
771,997 | <p>I'm working on a matrix extension of Fermat's Little Theorem, but I'm stuck on trying to show that if $A^p \equiv A$ mod $p$, then $A$ does not have to be diagonalizable.</p>
<p>Any help would be appreciated!</p>
<p><strong>Edit::</strong> I would like to either find a reason why $A$ does not have to be diagonaliz... | Pedro | 23,350 | <p>You haven't been clear about your base field. Let $k=\Bbb R$. Consider $p=2$. Then $\begin{pmatrix}1&2\\0&1\end{pmatrix}$ is congruent to its square modulo two. However, it is not diagonalizable over $\Bbb R$, say, since it is a Jordan block.</p>
|
963,125 | <p>I'm not exactly sure where to start on this one. Any help would be greatly appreciated.</p>
<p>Show that $4$ does not divide $12x+3$ for any $x$ in the integers.</p>
<p>Here's what I have so far:</p>
<p>There exist c in the integers such that 4 | 12x+3. Then 12x+3 = 4y for some y in the integers. This is a contra... | C.S. | 95,894 | <ul>
<li>A number which is divisible by $4$ is <strong>always even</strong> where as $12x+3$ is always <strong>Odd</strong>.</li>
</ul>
|
1,435,189 | <p>A map $T:X \rightarrow Y$ is $A, B$- measurable if the pre-image of every set in $B$ is a set in $A$, where $A,B$ are $\sigma$-algebras in $X,Y$. </p>
<p>But I was thinking that isn't any map $T: X \rightarrow Y$ measurable then? Because since pre-images behave so nicely, the pre-image of all sets from $B$ make up... | Giuseppe Negro | 8,157 | <p>It is true that preimages of $B$ form a sigma-algebra in $X$. However, there is no a priori reason why this sigma-algebra should be contained in $A$.</p>
|
1,435,189 | <p>A map $T:X \rightarrow Y$ is $A, B$- measurable if the pre-image of every set in $B$ is a set in $A$, where $A,B$ are $\sigma$-algebras in $X,Y$. </p>
<p>But I was thinking that isn't any map $T: X \rightarrow Y$ measurable then? Because since pre-images behave so nicely, the pre-image of all sets from $B$ make up... | Community | -1 | <p>That's not true, of course. </p>
<p>Here's an example which doesn't require us to be in any special setting: Let $X$ be any set and let $\mathcal A$ be a $\sigma$-algebra on $X$ that is a strict subset of $\mathcal P(X)$. Let $A \in \mathcal P(X) -\mathcal A$. Then: $1_A - 1_{A^C}$ is not measurable. </p>
<p>A par... |
1,160,076 | <p>Consider $f:\left[a,b\right]\rightarrow \mathbb{R}$ continuous at $\left[a,b\right]$ and differentiable at $\left(a,b\right)$.<br>
$\forall x\:\ne \frac{a+b}{2}$, $f'\left(x\right)\:>\:0$, but at $x\:=\frac{a+b}{2}$, $f'\left(x\right)\:=\:0$.<br>
Show that $f$ is <strong>strictly increasing</strong> at $\left[a,b... | MooS | 211,913 | <p>For sure $f$ is increasing in the sense that $f(x) \geq f(y)$ whenever $x > y$. Now assume $f(x)=f(y)$ for $x > y$. Clearly, then $f$ must be constant on $[y,x]$, contradicting the fact that the derivative vanishes at only one single point.</p>
|
24,912 | <p>I have the category-theoretic background of the occasional stroll through MacLane's text, so excuse my ignorance in this regard. I was trying to learn all that I could on the subject of tensor algebras, and higher exterior forms, and I ran into the notion of cohomological determinants. Along this line of inquiry, I ... | David Ben-Zvi | 582 | <p>Determinants are discussed (in a language relevant to this current question) in <a href="https://mathoverflow.net/questions/7124/determinant-of-a-perfect-complex/">this MO question</a>.</p>
<p>One place Picard categories naturally appear is as fundamental (aka Poincare) groupoids -- specifically those of infinite l... |
24,912 | <p>I have the category-theoretic background of the occasional stroll through MacLane's text, so excuse my ignorance in this regard. I was trying to learn all that I could on the subject of tensor algebras, and higher exterior forms, and I ran into the notion of cohomological determinants. Along this line of inquiry, I ... | Community | -1 | <p>Regarding the transcription of H X Sinh’s thesis <a href="https://agrothendieck.github.io/divers/GCS.pdf" rel="nofollow noreferrer">[pdf]</a> (still in progress), professor J Baez pointed out
<a href="https://categorytheory.zulipchat.com/#narrow/stream/260000-practice.3A-translation/topic/Sinh.20thesis.2E.20grothend... |
4,629,824 | <p>Let <span class="math-container">$k$</span> be a given positive integer. I want to solve the following system of Diophantine equations: <span class="math-container">$$\begin{cases} a^2 + b^2 + c^2 = k^2 \\ b^2 = ac \end{cases}$$</span>
where <span class="math-container">$a, b, c \in \mathbb{N}$</span> are non-zero.<... | Tomita | 717,427 | <p>According to Will Jagy's <a href="https://math.stackexchange.com/questions/2445717/how-to-solve-the-diophantine-equation-x2xyy2-r2-x-y-r-in-bbb-z?noredirect=1">answer</a>, a parametric solution of <span class="math-container">$a^2+ ac +c^2=k^2$</span> is given as follows.<br />
<span class="math-container">$(a,c)=(... |
3,291,303 | <p>this very same problem appeared in a different thread but the questions was slightly different. In my case, I'm looking precisely for the answer.</p>
<p>This is how I solved it, I only need confirmation of whether this actually is correct:</p>
<p>So I assumed that 1111... (100 ones) is going to be exactly divided ... | Robert Israel | 8,508 | <p>The first seven <span class="math-container">$1$</span>'s gives you a multiple of <span class="math-container">$1111111$</span>, same for the next seven, etc.
Since <span class="math-container">$100 = 14 \cdot 7 + 2$</span> after removing <span class="math-container">$14$</span> groups of seven <span class="math-con... |
156,769 | <p>Let $\{a_{n}\}$ be a sequence of real numbers, where $0<a_{n}<1$, such that $\lim_{n\to \infty} a_{n}=0$, (then every subsequence will converges to zero). Is there any way to find a subsequence of $a_{n}$ which is decreasing to 0?</p>
| ncmathsadist | 4,154 | <p>Yes. Put $n_1 = 1$. Suppose $n_1<n_2 < \cdots n_k$ are chosen. Choose $n_{k+1}> n_k$ so that $a_{n_{k+1}} < \min\{a_{n_j}, 1\le j \le k\}$. Such a sequence will fulfill your specification.</p>
|
3,276,124 | <p>Consider the equation</p>
<p><span class="math-container">$$-242.0404+0.26639x-0.043941y+(5.9313\times10^{-5})\times xy-(3.9303\times{10^{-6}})\times y^2-7000=0$$</span></p>
<p>with <span class="math-container">$x,y>0$</span>. If you plot it, it'll look like below:</p>
<p><a href="https://i.stack.imgur.com/Oqf... | Adrian Keister | 30,813 | <p>We first simplify the expression, and then solve for <span class="math-container">$x:$</span>
<span class="math-container">\begin{align*}
-242.0404+0.26639x-0.043941y+\left(5.9313\times10^{-5}\right)xy-\left(3.9303\times{10^{-6}}\right) y^2-7000&=0 \\
0.26639x-0.043941y+\left(5.9313\times10^{-5}\right)xy-\left(3... |
29,323 | <p>You're hanging out with a bunch of other mathematicians - you go out to dinner, you're on the train, you're at a department tea, et cetera. Someone says something like "A group of 100 people at a party are each receive hats with different prime numbers and ..." For the next few minutes everyone has fun solving the p... | Pietro Majer | 6,101 | <p>Countably many little dwarfs are going to their everyday work to the mine. They are marching and singing in a well-ordered line (by natural numbers), so that number 1 watches the backs of all the other ones, and, in general, number <em>n</em> watches the backs of all the others from <em>n+1</em> on. Suddenly, an ev... |
29,323 | <p>You're hanging out with a bunch of other mathematicians - you go out to dinner, you're on the train, you're at a department tea, et cetera. Someone says something like "A group of 100 people at a party are each receive hats with different prime numbers and ..." For the next few minutes everyone has fun solving the p... | BlueRaja | 2,883 | <p><em>Cryptography riddle - that's a branch of mathematics, right? :)</em></p>
<p>Jan and Maria have fallen in love (via the internet) and Jan wishes to mail her a ring. Unfortunately, they live in the country of Kleptopia, where anything sent through the mail will be stolen unless it is enclosed in a padlocked box. ... |
29,323 | <p>You're hanging out with a bunch of other mathematicians - you go out to dinner, you're on the train, you're at a department tea, et cetera. Someone says something like "A group of 100 people at a party are each receive hats with different prime numbers and ..." For the next few minutes everyone has fun solving the p... | BlueRaja | 2,883 | <p>Most of us know that, being deterministic, computers cannot generate true <a href="http://en.wikipedia.org/wiki/Pseudorandom_number_generator">random numbers</a>.</p>
<p>However, let's say you have a box which generates truly random binary numbers, but is biased: it's more likely to generate either a <code>1</code... |
29,323 | <p>You're hanging out with a bunch of other mathematicians - you go out to dinner, you're on the train, you're at a department tea, et cetera. Someone says something like "A group of 100 people at a party are each receive hats with different prime numbers and ..." For the next few minutes everyone has fun solving the p... | Richard Stanley | 2,807 | <ol>
<li><p>Alice shuffles an ordinary deck of cards and turns the cards face
up one at a time while Bob watches. At any point in this process
before the last card is turned up, Bob can guess that the next
card is red. Does Bob have a strategy that gives him a probability
of success greater that .5?</p></li>
<li><p>Let... |
29,323 | <p>You're hanging out with a bunch of other mathematicians - you go out to dinner, you're on the train, you're at a department tea, et cetera. Someone says something like "A group of 100 people at a party are each receive hats with different prime numbers and ..." For the next few minutes everyone has fun solving the p... | Ian Agol | 1,345 | <p>Suppose 100 ants are placed randomly (with random orientations) along a yard stick. Each ant walks at a pace of an inch a minute. Each time two ants meet, they instantaneously reverse direction, and if an ant meets the end of the yardstick, it instantaneously reverses direction.<br>
Do the ants ever return to their ... |
29,323 | <p>You're hanging out with a bunch of other mathematicians - you go out to dinner, you're on the train, you're at a department tea, et cetera. Someone says something like "A group of 100 people at a party are each receive hats with different prime numbers and ..." For the next few minutes everyone has fun solving the p... | Andrej Bauer | 1,176 | <p>It is very important that you tell these two puzzles in the correct order, i.e., first the first puzzle and then the second one. The first puzzle is very easy but messes with people's minds in just the right way. In my experience some mathematicians are driven crazy by the second puzzle.</p>
<p><strong>Puzzle 1:</s... |
29,323 | <p>You're hanging out with a bunch of other mathematicians - you go out to dinner, you're on the train, you're at a department tea, et cetera. Someone says something like "A group of 100 people at a party are each receive hats with different prime numbers and ..." For the next few minutes everyone has fun solving the p... | villemoes | 4,747 | <p>A magician places $N = 64$ coins in a row on a table, then leaves the room. A person from the audience is then asked to flip each coin however he likes [so there are $2^{64}$ possible states]. He is also asked to mention a number between 1 and $N$. After this, the magician's assistant flips exactly one coin. The mag... |
29,323 | <p>You're hanging out with a bunch of other mathematicians - you go out to dinner, you're on the train, you're at a department tea, et cetera. Someone says something like "A group of 100 people at a party are each receive hats with different prime numbers and ..." For the next few minutes everyone has fun solving the p... | doetoe | 8,216 | <p>You and your adversary have a sufficiently large bag of identical coins, and are seated on opposite sides of a rectangular table. You take turns placing coins on the table. The first one that cannot put a coin on the table without overlapping any other coin loses. What is your strategy to always win if you're allowe... |
29,323 | <p>You're hanging out with a bunch of other mathematicians - you go out to dinner, you're on the train, you're at a department tea, et cetera. Someone says something like "A group of 100 people at a party are each receive hats with different prime numbers and ..." For the next few minutes everyone has fun solving the p... | Steven Landsburg | 10,503 | <p>Let $I$ be the set of irrational numbers, with the usual topology. Is $I$ homeomorphic to $I\times I$?</p>
<p>Edited to add: In fact, it's an even better puzzle if you replace $I$ with $Q$, the rational numbers.</p>
|
29,323 | <p>You're hanging out with a bunch of other mathematicians - you go out to dinner, you're on the train, you're at a department tea, et cetera. Someone says something like "A group of 100 people at a party are each receive hats with different prime numbers and ..." For the next few minutes everyone has fun solving the p... | Jim Ward | 12,534 | <p>There was a puzzle in the Journal of Recreational Mathematics. I apologize if I am telling it incorrectly. A wire is stretched between two telephone poles. A flock of crows lands simultaneously on the wire. When they land, each crow looks at his nearest neighbor. What percentage of the flock are looking at a crow th... |
4,159,227 | <p>Given the problem of finding the Laplace transform of the function <span class="math-container">$$f(t)=t^ne^{at}$$</span> with <span class="math-container">$n\in\mathbb{N}$</span> and <span class="math-container">$a\in\mathbb{R}$</span>, I realize it can be shown the transform is <span class="math-container">$$\frac... | Gaurang | 879,859 | <p>My attempt :</p>
<p><span class="math-container">$$ \mathcal{L}\{t^ne^{at}\}= \frac{n!}{(s-a)^{n+1}}$$</span></p>
<p>For <span class="math-container">$P(k):$</span></p>
<p><span class="math-container">$$ \tag{1}\mathcal{L}\{t^ke^{at}\}= \frac{k!}{(s-a)^{k+1}}$$</span></p>
<p>Need to prove for <span class="math-conta... |
1,194,440 | <p>How do I prove if the following formulas are consistent?</p>
<p>∀x$\neg$S(x,x)</p>
<p>∃x P(x)</p>
<p>∀x∃y S(x,y)</p>
<p>∀x(P(x)$\to$∃y S(y,x))</p>
<p>I think I proved part of it...</p>
<p>There is at least one value for x such that P(x) is <em>True</em></p>
<p>This means that for ∀x(P(x)$\to$∃y S(y,x)) ... t... | Daniel H | 224,434 | <p>The easiest way to show that it is consistent is to explicitly construct a model for this theory. (Then we're done: recall that $T$ is consistent $\leftrightarrow$ $T$ has a model)</p>
<p>Due to axioms 1 and 3, we need to have at least 2 elements: the simplest model is then $A=\{0,1\}$, and make this into an L-Stru... |
1,582,544 | <p>$X_1,\dots,X_n$ and $Y_1,\dots,Y_n$ are random variables that take values in $\{0,1\}$. Their distribution is unknown, each variable may have a different distribution and they might be dependent. The only known facts are:</p>
<ul>
<li>$\sum_{i=1}^n X_i = \sum_{i=1}^n Y_i$</li>
<li>For all $i$, $\Pr[X_i=0] \geq \alp... | JimmyK4542 | 155,509 | <p>Thanks to linearity of expectation, this is indeed true. Taking the expectation of both sides yields:</p>
<p>$$\mathbb{E}\left(\sum_{i = 1}^{n}X_i\right) = \mathbb{E}\left(\sum_{i = 1}^{n}Y_i\right)$$</p>
<p>$$\sum_{i = 1}^{n}\mathbb{E}(X_i) = \sum_{i = 1}^{n}\mathbb{E}(Y_i)$$</p>
<p>$$\sum_{i = 1}^{n}\left(1-\Pr... |
4,245,205 | <p>Suppose <span class="math-container">$X$</span> is a CW-complex of dimension <span class="math-container">$n$</span>. If <span class="math-container">$e_i$</span> is an <span class="math-container">$n$</span>-cell then is <span class="math-container">$H_n(X) \to H_n(X, X \setminus e_i)$</span> the zero map? If not, ... | kamills | 497,007 | <p>I have an answer that doesn't require you to know much about the homology of CW complexes but requires a bit more work. We're going to look at the long exact sequences for the pairs you mention.</p>
<p>First, we have a simple example to use for your first question that might illuminate the general case. Let <span cl... |
3,530,285 | <blockquote>
<p>Let <span class="math-container">$\text{char}(\mathbb{K}) = 0$</span>. It then follows that <span class="math-container">$AB-BA \ne 1 \, (A, B \in \Bbb K^{n \times n})$</span>.</p>
</blockquote>
<p>I first showed that <span class="math-container">$\text{trace}(AB) = \text{trace}(BA)$</span> for every... | Henno Brandsma | 4,280 | <p><span class="math-container">$0 \in \Bbb Q$</span> but <span class="math-container">$0\notin S$</span> while it <strong>is</strong> a limit point of <span class="math-container">$S$</span>. So <span class="math-container">$S$</span> does not contain all its limit points so is not closed. </p>
|
78,946 | <p>I want to find min of the function
$$\frac{1}{\sqrt{2
x^2+\left(3+\sqrt{3}\right)
x+3}}+\frac{1}{\sqrt{2
x^2+\left(3-\sqrt{3}\right)
x+3}}+\sqrt{\frac{1}{3}
\left(2 x^2+2 x+1\right)}.$$
I know, the exact value minimum is $\sqrt{3}$ at $x = 0$. With <em>Mathematica</em>, I tried </p>
<pre><code>A = 1... | Coolwater | 9,754 | <p>You could use <code>FindInstance</code> to find some of the exact stationary points, when mathematica can't solve the equation:</p>
<pre><code>FindInstance[Evaluate[D[A, x]] == 0, x]
(*{{x -> 0}}*)
</code></pre>
|
1,930,933 | <blockquote>
<p>Does there exist an $n \in \mathbb{N}$ greater than $1$ such that $\sqrt[n]{n!}$ is an integer?</p>
</blockquote>
<p>The expression seems to be increasing, so I was wondering if it is ever an integer. How could we prove that or what is the smallest value where it is an integer?</p>
| P Vanchinathan | 28,915 | <p>EDIT: The comment below by bof points out the flaw in my argument.
I am not deleting this answer. This proves a much weaker statement version that instead of "never happening" this shows it cannot happen for two consecutive integers. </p>
<hr>
<p>Bertrand's postulate is the shortest way to prove this as has bee... |
786,301 | <p>Is there a systematic way to express the sum of two complex numbers of different magnitude (given in the exponential form), i.e find its magnitude and its argument expressed in terms of those of the initial numbers?</p>
| John Hughes | 114,036 | <p>Convert to rectangular form, add. convert back. </p>
<p>$$\begin{eqnarray}
[r, \theta] + [s, \phi]& \to&
(r \cos \theta, r \sin \theta) + (s \cos \phi, s \sin \phi) \to
(r \cos \theta + s \cos \phi, r \sin \theta + s \sin \phi)\\
&\to&
\bigg[\sqrt{(r \cos \theta + s \cos \phi)^2 + ( r \sin \theta... |
72,651 | <p>$G$ is a group and $H$ is a subgroup of $G$ such that $\forall a, b$ in $G, ab\in H\implies ba\in H$. Show that $H$ is normal in $G$</p>
| Sugata Adhya | 36,242 | <p>Choose $x\in G$ and $h\in H.$ Let $a=x^{-1}.$</p>
<p>Suppose $ha=b.$ </p>
<p>Then $ba^{-1}=h$</p>
<p>$\implies a^{-1}b=h_1$ for some $h_1\in H$ (by the given condition)</p>
<p>$\implies b=ah_1$</p>
<p>$\implies ha=ah_1$</p>
<p>$\implies a^{-1}ha=h_1\in H$</p>
<p>$\implies xhx^{-1}\in H.$</p>
<p>Consequently ... |
3,831,702 | <p>One can prove that for <span class="math-container">$x\in \mathbb{R}$</span>, the sequence
<span class="math-container">$$
u_0=x\text{ and } \forall n\in \mathbb{N},\qquad u_{n+1}=\frac{e^{u_n}}{n+1}
$$</span>
converges to <span class="math-container">$0$</span> if <span class="math-container">$x \in ]-\infty,\delta... | Community | -1 | <p>My numerics agree with Simply's result. Here's a little Maple code:</p>
<p>Define <span class="math-container">$u_n(x)$</span>:</p>
<p><a href="https://i.stack.imgur.com/BeYzO.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/BeYzO.png" alt="enter image description here" /></a></p>
<p>At the <span c... |
220,618 | <p>The cyclic group of $\mathbb{C}- \{ 0\}$ of complex numbers under multiplication generated by $(1+i)/\sqrt{2}$</p>
<p>I just wrote that this is $\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}i$ making a polar angle of $\pi/4$. I am not sure what to do next. My book say there are 8 elements. </p>
<p>Working backwards, maybe... | Stefan Geschke | 16,330 | <p>Draw a sketch. Do you know how to multiply complex numbers in polar coordinates?
Add the angles, multiply the lengths. Now compute the powers of your number until you reach 1.</p>
|
2,046,957 | <p>The random variable $X$ is $N(5,2)$ and $y=2X+4$.
Find:</p>
<p>a) $\eta_y$</p>
<p>b) $\sigma_y$</p>
<p>c) $f_Y(y)$</p>
<p>My attempt:</p>
<p>I have solved a and b as follow:</p>
<p>a) $\eta_y = 2\eta_X+4 = 14$</p>
<p>b) $\sigma_y^2 = 4\sigma_{x}^2 = 16, \sigma_y = 4$</p>
<p>c) how can I solve $f_Y(y)$?</p>
| Gono | 384,471 | <p>By definition:
$$\begin{align*}f_Y(y) &= P'(Y \le y) \\ &= P'(2X + 4 \le y) \\ &= P'\left( X \le \frac{y - 4}{2}\right) \\ &= \frac{1}{2}f_X\left(\frac{y - 4}{2}\right)\end{align*}$$</p>
<p>And $f_X$ you should know…</p>
|
866,885 | <p>I try to write math notes as clearly as possible. In practice, this means using letters and notation similar to what the reader is already familiar with.</p>
<p>A real function is often $f(x)$, an angle is often $\theta$, a matrix has size $m\times n$, and $i$ is often an index. The full theoretical list is long an... | Joel | 85,072 | <p>In my own experience the letters $a,b,c,d,e$ are reserved for coefficients. $f, g, h$ are functions. $i,j,k$ are indices. (Sometimes $i=\sqrt{-1}$) $l,m,n$ are also indicies (and in particular natural numbers or integers)</p>
<p>$o$ isn't often used, since it can be confused with zero. Though it can be "little"-o f... |
1,003,020 | <p>Without recourse to Dirichlet's theorem, of course.
We're going to go over the problems in class but I'd prefer to know the answer today.</p>
<p>Let $S = \{3n+2 \in \mathbb P: n \in \mathbb N_{\ge 1}\}$</p>
<p>edit:</p>
<p>The original question is "the set of all primes of the form $3n + 2$, but I was only consid... | mookid | 131,738 | <p><strong>Hint:</strong>
Assume there are only $m$ such primes. Consider
$$P=(3n_1+2)(3n_2+2)...(3n_m+2) = (-1)^m \mod 3$$</p>
<p>Now if $2|m, P+2 = -1\mod 3$.</p>
<p>If $2\nmid m$ then $P=1\mod 3$.</p>
|
465,945 | <p>I just need some verification on finding the basis for column spaces and row spaces.</p>
<p>If I'm given a matrix A and asked to find a basis for the row space, is the following method correct?</p>
<p>-Reduce to row echelon form. The rows with leading 1's will be the basis vectors for the row space.</p>
<p>When l... | elbeardmorez | 15,263 | <p>yes you're correct.</p>
<p>note that row echelon form doesn't necessarily result in 'leading 1s'. it's '<strong>reduced/canonical</strong> row echelon form' that requires that form.</p>
<p>having reduced your matrix to the set of the linearly independent rows/columns via the row transformations, you can choose eit... |
1,103,478 | <p>$ r = 2\cos(\theta)$ has the graph<img src="https://i.stack.imgur.com/yvLb1.png" alt="enter image description here"></p>
<p>I want to know why the following integral to find area does not work $$\int_0^{2 \pi } \frac{1}{2} (2 \cos (\theta ))^2 \, d\theta$$</p>
<p>whereas this one does:</p>
<p>$$\int_{-\frac{\pi}{... | colormegone | 71,645 | <p>The families of polar curves $ \ r \ = \ a \ \cos \ n \theta \ $ and $ \ r \ = \ a \ \sin \ n\theta \ $ are called "rosettes", because they form symmetrical "flower-petal" arrangements. Those with $ \ n \ $ even trace their respective curves once in a period of $ \ 2 \pi \ $ and have $ \ 2n \ $ "petals", while tho... |
2,000,556 | <p>Suppose $N>m$, denote the number of ways to be $W(N,m)$</p>
<h3>First method</h3>
<p>Take $m$ balls out of $N$, put one ball at each bucket. Then every ball of the left the $N-m$ balls can be freely put into $m$ bucket. Thus we have: $W(N,m)=m^{N-m}$.</p>
<h3>Second method</h3>
<p>When we are going to put $N$... | Hypergeometricx | 168,053 | <p>First put $1$ ball into each of $m$ buckets. </p>
<p>Then you are left with $N-m$ balls. </p>
<p>Use <a href="https://en.wikipedia.org/wiki/Stars_and_bars_(combinatorics)" rel="nofollow noreferrer">stars and bars</a> method. Arrange the remaining balls in a row. Add $m-1$ dividers. Compute number of ways to arran... |
66,719 | <p>I know some elementary proofs of this fact. I was wondering if there's some short slick proof of this fact using the structure of the $2$-adic integers? I'm looking for a proof of this fact that's easy to remember.</p>
| Plop | 2,660 | <p>There is indeed a $2$-adic proof.
$(1+u)^{1/2} = 1+\sum_{k \geq 1} \binom{1/2}{k} u^k$ where $\binom{1/2}{k}= \frac{1/2(1/2-1) \ldots (1/2-k+1)}{k!} = 2^{-k}(-1)^{k-1}\frac{(2k-3) (2k-5) \ldots 3}{k!}$.
We would like the sum to converge, and for this we only need the general term to go to $0$.</p>
<p>$v_2\left(\bin... |
913,998 | <p>What can we do with this function, so the function will be continuous in $(0,0)$?</p>
<p>$f:\mathbb{R}^2\rightarrow\mathbb{R}:(x,y) \mapsto \frac{x^2+y^2-x^3y^3}{x^2+y^2}$</p>
<p>What I think we should do, is:</p>
<p>approximate $(0,0)$ via the line $y=x$, so substitute $y=x$ and take the limit of that function,... | idm | 167,226 | <p>We can observe that
$$x^4+2x^2\geq 0$$
for all $x\in\mathbb R$ and so,
$$x^4+2x^2+2\geq 2$$
for all $x\in \mathbb R$. Moreover, for $x=0$, we have that
$$0^4+2\cdot 0^2+2=2$$
and so $2$ is the minimum.</p>
|
913,998 | <p>What can we do with this function, so the function will be continuous in $(0,0)$?</p>
<p>$f:\mathbb{R}^2\rightarrow\mathbb{R}:(x,y) \mapsto \frac{x^2+y^2-x^3y^3}{x^2+y^2}$</p>
<p>What I think we should do, is:</p>
<p>approximate $(0,0)$ via the line $y=x$, so substitute $y=x$ and take the limit of that function,... | Khosrotash | 104,171 | <p>$$ x^4+x^2+2=(x^2+1)^2+1\\x^2\geq 0\\x^2+1 \geq 0+1 \\(x^2+1)^2 \geq (1)^2\\so\\(x^2+1)^2+1\geq 1+1 $$</p>
|
301,724 | <p>There's an example in my textbook about cancellation error that I'm not totally getting. It says that with a $5$ digit decimal arithmetic, $100001$ cannot be represented.</p>
<p>I think that's because when you try to represent it you get $1*10^5$, which is $100000$. However it goes on to say that when $100001$ is r... | Tpofofn | 4,726 | <p>I think the point is about the precision of the number that can be stored, not so much the exponent. In other words you cannot store $1.00001*10^5$ because it cannot store five decimal places of precision. Equally it would not be able to store $1.00001*10^{15}$ or $1.00001*10^{-5}$.</p>
|
458,472 | <p>Just doing some revision for ODEs and came across this problem. Find the general solution to $$u''+4u=0.$$</p>
<p>So far I've applied the characteristic polynomial:
$$\begin{array}{r c l}
\lambda^2 +4 & = & 0 \\
\lambda^2 & = & -4 \\
\lambda & = & i\sqrt{4} \\
\lambda & = & 2i, -2i. ... | Dan | 79,007 | <p>$i$ is a constant, and so is included in $C_2$. So you're both right! :)</p>
|
6,340 | <p>Just to clarify, I want to show that:</p>
<p>If $f$ is entire and $\int_{\mathbb{C}} |f|^p dxdy <\infty$, then $f=0$.</p>
<p>I think I can show that this is the case for $p=2$, but I'm not sure about other values of $p$...</p>
| Mariano Suárez-Álvarez | 274 | <p>Use Hölder's inequality and Cauchy's integral formula to show the function and its derivatives all vanishes at zero.</p>
|
2,438,795 | <blockquote>
<p>Why are any two initial objects of a category equivalent?</p>
</blockquote>
<p>By definition:</p>
<p>If $A,B$ are initial objects of a category $C$, then for each $X \in \text{obj } C$, there exists a unique morphisms $f : A \rightarrow X, g : B \rightarrow X$.</p>
<p>How can these be equivalent? ... | Mr. Chip | 52,718 | <p>Any two initial objects are <em>isomorphic</em>, even though they may not literally be equal.</p>
<p>Proof: Call them $A$ and $A'$. By assumption, there are unique morphisms $a: A \to A'$, $a': A' \to A$. Then $aa': A' \to A'$ has to be $1_{A'}$, because $A'$ is initial. Similarly $a'a = 1_A$. Thus $A \cong A'$.</p... |
1,423,728 | <p>The definition of a limit in <a href="http://rads.stackoverflow.com/amzn/click/0321888545" rel="nofollow noreferrer">this book</a> stated like this </p>
<p><a href="https://i.stack.imgur.com/tKjaa.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/tKjaa.png" alt="enter image description here"></a><... | R.N | 253,742 | <p>Domain $f$ here is $\mathbb{Q}$, you can talk about limit of functon, and $\lim f=1$</p>
|
131,435 | <p>Wikipedia is a widely used resource for mathematics. For example, there are hundreds of mathematics articles that average over 1000 page views per day. <a href="http://en.wikipedia.org/wiki/Wikipedia:WikiProject_Mathematics/Popular_pages" rel="noreferrer">Here is a list of the 500 most popular math articles</a>. The... | Neil Strickland | 10,366 | <p>In general, I find the wikipedia maths pages to be very comprehensive and useful, and I have not found any serious errors. However, they are often not very well organized or clearly explained. I agree that it would be good if more professional mathematicians tried to help with this.</p>
<p>I rewrote the article <... |
131,435 | <p>Wikipedia is a widely used resource for mathematics. For example, there are hundreds of mathematics articles that average over 1000 page views per day. <a href="http://en.wikipedia.org/wiki/Wikipedia:WikiProject_Mathematics/Popular_pages" rel="noreferrer">Here is a list of the 500 most popular math articles</a>. The... | Aaron Meyerowitz | 8,008 | <p>Some articles have contributors who are very possessive. It is amazing and impressive that the Wikipedia model works so well. I find it a great first pass on a variety of topics. Often I feel no need to look further. Other times I do. The price of that is that contributing is not worth it unless one is willing to wa... |
131,435 | <p>Wikipedia is a widely used resource for mathematics. For example, there are hundreds of mathematics articles that average over 1000 page views per day. <a href="http://en.wikipedia.org/wiki/Wikipedia:WikiProject_Mathematics/Popular_pages" rel="noreferrer">Here is a list of the 500 most popular math articles</a>. The... | Gerhard Paseman | 3,528 | <p>I will expand on one of Aaron Meyerowitz's remarks.</p>
<p>I am somewhat possessive about what I have written.
Indeed one of the hurdles I had to overcome to
participate on MathOverflow was to accept that what
I submitted could be changed by many other people.
I was upset when it happened, and the major thing
that ... |
131,435 | <p>Wikipedia is a widely used resource for mathematics. For example, there are hundreds of mathematics articles that average over 1000 page views per day. <a href="http://en.wikipedia.org/wiki/Wikipedia:WikiProject_Mathematics/Popular_pages" rel="noreferrer">Here is a list of the 500 most popular math articles</a>. The... | Jérémy Blanc | 23,758 | <p>One difference between MO and wikipedia is that here you get very quickly answers or remarks on what you have written, and it is then very much more attractive. Going to wikipedia for finding something for oneself, you directly have the answer you were looking for (or dont find it), but really writing new stuff take... |
361,060 | <blockquote>
<p>Consider the ring of Gaussian integers $D=\lbrace a+bi\mid a,b \in \mathbb{Z \rbrace}$, where $i \in \mathbb{C}$ such that $i^2=-1$. Consider the map $f$ from $D$ to $\mathbb{Z}[x]/(x^2+1)$ sending $i$ to the class of $x$ modulo $x^2+1$. Show that $f$ is a ring isomorphism.</p>
</blockquote>
<p>I got... | Ittay Weiss | 30,953 | <p>The ring $\mathbb Z[x]/(x^2+1)$ is a quotient ring. Its elements are equivalence classes modulo the ideal $(x^2+1)$. Thus, a typical element in $\mathbb Z[x]/(x^2+1)$ is of the form $f(x)+(x^2+1)$ where $f(x)\in \mathbb Z[x]$ and $(x^2+1)$ is the ideal in $\mathbb Z[x]$ generated by $x^2+1$. So, the map $D\to \mathb... |
361,060 | <blockquote>
<p>Consider the ring of Gaussian integers $D=\lbrace a+bi\mid a,b \in \mathbb{Z \rbrace}$, where $i \in \mathbb{C}$ such that $i^2=-1$. Consider the map $f$ from $D$ to $\mathbb{Z}[x]/(x^2+1)$ sending $i$ to the class of $x$ modulo $x^2+1$. Show that $f$ is a ring isomorphism.</p>
</blockquote>
<p>I got... | AlexM | 72,471 | <p>Im not sure if I'm quite correctly interpreting your question, but I'll answer to the best of my ability. Start out with $\mathbb{Z}[x]$; the image of $x$ under the map $\pi: \mathbb{Z}[x] \to \mathbb{Z}[x]/(x^2 + 1)$, which let's denote $\overline{x}$, is such that $\overline{x}^2 + 1 = 0$, or that $\overline{x}^2 ... |
2,507,565 | <p>I'm failing to find any reasonable solution to this given problem.</p>
<p>Find explicit formula of $n$-th element from given sequence:
$$\begin{cases}
a_1 = 1\\
a_2 = 2\\
a_{n+2} = 4 a_{n+1} + 4 a_n + 2^n
\end{cases}$$</p>
<p>I have tried finding some clues whilst typing out few first expressions to no avail. I al... | Cye Waldman | 424,641 | <p>I propose what is possibly a more direct solution. With problems of this type I try to reduce the original recurrence to a more familiar form, usually what I call the generalized Fibonacci form, say, $f_n=af_{n-1}+bf_{n-2}$. In this instance, let us assume that</p>
<p>$$a_n=f_n+A2^n$$</p>
<p>Then,</p>
<p>$$f_{n+2... |
566 | <h3>We all love a good puzzle</h3>
<p>To a certain extent, any piece of mathematics is a puzzle in some sense: whether we are classifying the homological intersection forms of four manifolds or calculating the optimum dimensions of a cylinder, it is an element of investigation and inherently puzzlish intrigue that driv... | mau | 89 | <p>From <em>Mathematical Puzzles</em> by Peter Winkler:</p>
<p>Divide an hexagon in equilateral triangles, like in the figure. Now fill all the hexagon with the three kinds of diamonds made from two triangles, also shown in the figure. Prove that the number of each kind of diamond is the same.</p>
<p><img src="https:... |
566 | <h3>We all love a good puzzle</h3>
<p>To a certain extent, any piece of mathematics is a puzzle in some sense: whether we are classifying the homological intersection forms of four manifolds or calculating the optimum dimensions of a cylinder, it is an element of investigation and inherently puzzlish intrigue that driv... | Community | -1 | <p>If you have 32 2x1 dominoes then you can cover an 8x8 board easily enough; if you throw away a domino and cut a 1x1 square from each end of a diagonal of the board, can you cover the remaining shape with the remaining dominoes?</p>
<p>I heard this decades ago at university.</p>
|
655,591 | <p>$S=(v_1, \cdots v_n)$ of vectors in $\mathbb{F^n}$ is a basis iff the matrix obtained by forming a matrix (call it A) of the co-ordinate vectors of $v_i$ is invertible</p>
<p>My Idea: I was able to prove the reverse direction wherein we can show that $AX =0$ has trivial solutions so linearly independent and also sp... | Leo Azevedo | 120,820 | <p>HINT: The determinant of a matrix is zero iff the vectors which form it are linearly independent.</p>
<p>See also:</p>
<p><a href="https://math.stackexchange.com/questions/79356/using-the-determinant-to-verify-linear-independence-span-and-basis">Using the Determinant to verify Linear Independence, Span and Basis</... |
844,887 | <p>I am trying to prove by induction that every non-zero natural number has at least one predecessor. However, I don't know what to use as a base case, since 0 is not non-zero and I haven't yet established that 1 is the number following zero.</p>
<p>My axioms are: </p>
<ol>
<li>$0$ is a natural number. </li>
<li>if $... | Foster Antony | 337,532 | <p>Proof of Existence: Let us assume that it was already shown that $(\forall n)((n\ \in\ \omega)\ \ \rightarrow\ \ (n^+\
\neq\ 0))$ which is another way of saying that the natural number $0$ has no immediate predecessor in
$\omega$. So,</p>
<pre><code> what about the nonzero natural numbers---Do they have an im... |
724,302 | <p>I have read the page about category theory in wikipedia carefully, but i don't really get what this theory is.</p>
<p>Is category theory a content in ZFC-set theory? (Just like measure theory, group theory etc.) If not, is it just another formal logic system independent from the standard ZFC-set theory?</p>
<p>Fol... | Peter Smith | 35,151 | <p>A useful resource is <a href="http://plato.stanford.edu/entries/category-theory/">http://plato.stanford.edu/entries/category-theory/</a> which aims to explain something of the role and nature of category theory at an introductory but not merely arm-waving level. </p>
|
724,302 | <p>I have read the page about category theory in wikipedia carefully, but i don't really get what this theory is.</p>
<p>Is category theory a content in ZFC-set theory? (Just like measure theory, group theory etc.) If not, is it just another formal logic system independent from the standard ZFC-set theory?</p>
<p>Fol... | Giorgio Mossa | 11,888 | <p>That's a question. Well for start as shown in Mac Lane <em>Categories for the working mathematician</em> there are two different way to approach categories, functor and natural transformations:</p>
<ul>
<li>you can either regard categories as some family of sets and operation between them (eventually adding some ax... |
2,703,323 | <p>How can one show that the limit of the following is $1$?</p>
<p>$$\lim_{x\to 0}\frac{\frac{1}{1-x}-1}{x}=1$$</p>
| Galc127 | 111,334 | <p>$$\frac{\dfrac{1}{1-x}-1}{x}=\frac{\dfrac{1-(1-x)}{1-x}}{x}=\frac{1}{1-x}\xrightarrow{x\to 0}1$$</p>
|
549,585 | <p>Let $a_1=1$ and $a_{n+1}=ma_n+10$ for $n\geq 2$. Determine for which value of m, the sequence $(a_n)$ is convergent and find the limit.</p>
<p>I know that if if we simplify we will get $0=L(M-1)+10$, which is equal to $10/(1-M)=L$ but I have no clue what to do from there</p>
| 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{\dd}{{\rm d}}%
\newcommand{\isdiv}{\,\left.\right\vert\,}%
\newcommand{\ds}[1]{\displaystyle{#1}}%
\... |
3,170,629 | <p>I am trying to define a sequence.
The first few terms of the sequence are:</p>
<p><span class="math-container">$2,5,13,43,61$</span></p>
<p>Not yet found other terms because I am working with paper and pen, no software.</p>
<p>Why the first term is <span class="math-container">$5$</span>?</p>
<p>Let be <span cla... | Ross Millikan | 1,827 | <p>I believe your sequence continues forever but grows quickly. If <span class="math-container">$n$</span> is large, the density of primes around <span class="math-container">$n$</span> is <span class="math-container">$\log n$</span>. Since <span class="math-container">$\log n$</span> is so much smaller than <span cl... |
3,372,305 | <blockquote>
<p>When it is allowed to pull a r.v. out of the expectation, i.e.</p>
</blockquote>
<p><span class="math-container">$E(XY|Y)\overset?=YE(X|Y)\tag1$</span>.</p>
<p>or even </p>
<p><span class="math-container">$E(Y^2|Y)\overset?=Y^2\tag2$</span></p>
<p>in the computation rules it is written that if, </... | grand_chat | 215,011 | <p>I don't think the textbook explanation is correct. Equation (2) doesn't apply here, since we're mixing matrices and vectors, and equation (1) doesn't get us far enough.</p>
<p>It may be easier to derive the gradient directly from scratch. For brevity write <span class="math-container">$u:=x-\mu$</span>, and <span c... |
2,885,420 | <p>Show that V is infinite dimensional vector space if and only if there is a sequence $v_1,v_2,...$ of vectors in V such that $\{v_1,v_2,...,v_n\}$ is linearly independent for every positive integer n.</p>
| Aniruddha Deshmukh | 438,367 | <p>Dimension of a vector space is defined as the number of elements in the basis, provided the basis has a finite number of elements. In such cases, the vector space is said to be finite dimensional.</p>
<p>Therefore, a vector space is said to be infinite dimensional if the basis of the vector space has infinitely man... |
2,885,420 | <p>Show that V is infinite dimensional vector space if and only if there is a sequence $v_1,v_2,...$ of vectors in V such that $\{v_1,v_2,...,v_n\}$ is linearly independent for every positive integer n.</p>
| DanielWainfleet | 254,665 | <p>Take an arbitrary $v_1\ne 0.$ Now for $n\in \Bbb N$ suppose that $S(n)=\{v_j:j\leq n\}$ is a set of $n$ linearly independent vectors. Since $V$ is not finite dimensional we can take some $v_{n+1}$ that does not belong to the linear span of $S(n),$ so $S(n+1)=\{v_{n+1}\}\cup S(n)$ is a linearly independentset </p>
... |
173,745 | <p>I am trying to prove the following: 'If $(X_1,d_1)$ and $(X_2,d_2)$ are separable metric spaces (that is, they have a countable dense subset), then the product metric space $X_1 \times X_2$ is separable.' It seems pretty straightforward, but I would really appreciate it if someone could verify that my proof works.<... | DanielWainfleet | 254,665 | <p>You can bypass a lot of details in proving that $Y=X_1\times X_2$ is separable by observing that for $ z=(p_1,p_2) \in Y$ and for $r>0$, we have $I_1\times I_2 \subset B_d(z.r)$, where $I_1=B_{d_1}(p_1,r/2)$ and $I_2=B_{d_2}(p_2,r/2)$. So if $D_1$ is dense in $X_1$ and $D_2$ is dense in $X_2$ then $$\phi\ne (D_1... |
3,699,303 | <p>Let <span class="math-container">$R$</span> be a commutative ring with <span class="math-container">$1$</span>. This is a simple question, but I can't see whether it is true or not.</p>
<p>Suppose <span class="math-container">$R$</span> does not have a "unit" of (additive) order <span class="math-container">$2$</sp... | David Popović | 549,692 | <p>This is not true. For example, consider the ring <span class="math-container">$\mathbb{Z}[X]/(2X)$</span>. The units in this ring are only <span class="math-container">$\pm 1$</span>, but <span class="math-container">$X+X=0$</span>.</p>
|
175,576 | <p><a href="https://i.stack.imgur.com/wtjxT.png" rel="noreferrer"><img src="https://i.stack.imgur.com/wtjxT.png" alt="enter image description here"></a></p>
<p>This should work? But it is giving me the wrong t values</p>
| Bill | 18,890 | <p>If you didn't have a vast knowledge of all the internal functions provided by Mathematica then you might try a more brute force approach.</p>
<p>Trying a variety of starting values for t1 and t2 seems to usually come up with the same value for t1 and t2. Thus it looks like <code>FindRoot</code> might not have suffi... |
175,576 | <p><a href="https://i.stack.imgur.com/wtjxT.png" rel="noreferrer"><img src="https://i.stack.imgur.com/wtjxT.png" alt="enter image description here"></a></p>
<p>This should work? But it is giving me the wrong t values</p>
| Ulrich Neumann | 53,677 | <p>Without special knowledge NMinimize solves the problem straight forward: </p>
<pre><code>NMinimize[{1/(t2 - t1)^2 (*force t1!=t2*), r[t1] == r[t2], 0 < t1 < t2 < 1}, {t1, t2}]
(*{7.86584, {t1 -> 0.511377, t2 -> 0.867933}}*)
</code></pre>
|
164,103 | <p>I have some questions. </p>
<hr>
<p>The first one is about the product of Prikry's forcing.
Let $\kappa$ be a measurable cardinal, $U_1, U_2$ be normal measures on $\kappa$ and let $\mathbb{P}_{U_1}, \mathbb{P}_{U_2}$ be the corresponding Prikry forcings. Let $G\times H$ be $\mathbb{P}_{U_1}\times \mathbb{P}_{U_2... | Joel David Hamkins | 1,946 | <p>Concerning question 2, Yair Hayut's comment is exactly right, and this kind of situation is just the kind of situation that I had aimed to analyze with those results.</p>
<p>The basic fact is that if one performs forcing $\mathbb{P}_0$ of size $\delta$ followed by nontrivial strategically $\leq\delta$-closed forcin... |
1,144,540 | <p>I'm looking to determine the convergence of $\{e^{in}\}_{n=1}^\infty$ in $\mathbb{C}$, where $n \in \mathbb{N}$. Using the definition of convergence that iff $\exists \ \epsilon > 0: \forall \ n_0 \in \mathbb{N}, \exists \ n \ge n_0, |z_n - z| \le \epsilon$. </p>
<p>Using $z_n = e^{in} = \cos{n} + i\sin{n}$, a... | Chival | 118,564 | <p>A calculation of <em>Laplace transformation</em> of Poisson $\big(\lambda_i\big)$ shows us , $\forall u \in\mathbb{R}_+$,<br>
$$\mathbb{E}\Big[ \, e^{-u\cdot X_i} \, \Big] = \exp\big[\, -\lambda_i\cdot\big(1- e^{-u}\big) \,\big] \, \,\Longrightarrow\quad \mathbb{E}\Big[\, s^{X_i} \, \Big] = \exp\big[\, -\lambd... |
3,176,482 | <p>In my Econometrics class yesterday, our teacher discussed a sample dataset that measured the amount of money spent per patient on doctor's visits in a year. This excluded hospital visits and the cost of drugs. The context was a discussion of generalized linear models, and for the purposes of Stata, he found that a g... | Henry | 6,460 | <p>There is a minor problem with using the normal distribution here as it has a positive probability of giving negative values</p>
<p>That being said, your idea of what the density in your model might look like is reasonable. Try the following R function:</p>
<pre><code>plotdensity <- function(meanvisitcost, sdvi... |
2,441,488 | <p>We are not unfamiliar to imaginary angles, so what can be $\tan(a+ib)$</p>
<blockquote>
<p>If$$\tan(a+ib)=x+iy$$then find $$x,y$$</p>
</blockquote>
<p>My attempt,</p>
<p>Let $x$ and $z$ be two complex bombers such that,
$$\tan(z)=x$$
$$z=\tan^{-1} (x)$$
$$z=\tan^{-1}(re^{i\theta}).$$
But this doesn't seem to ge... | Bernard | 202,857 | <p>Use the relations between complex circular and hyperbolic functions, mainly
$$\tan iz=i\tanh z.$$
From the addition formula, we obtain
$$\tan(a+ib)=\frac{\tan a+\tan ib}{1-\tan a\tan ib}=\frac{\tan a+i\tanh b}{1-i\tan a\tanh b}$$
If you want a cartesian form of the result, multiplying by the conjugate expression f t... |
1,728,910 | <p>Whenever I get this question, I have a hard time with it. </p>
<p>An example of a problem:</p>
<p>In the fall, the weather in the evening is <em>dry</em> on 40% of the days, <em>rainy</em>
on 58% of days and <em>snowy</em> 2% of the days. </p>
<p>At noon you notice clouds in the sky. </p>
<p>Clouds appear
at noo... | Rubicon | 318,714 | <p>Using Bayes' Theorem and the law of total probability: $P(S | C) = 0.0365$</p>
|
13,432 | <p>I want to turn a sum like this</p>
<pre><code>sum =a-b+c+d
</code></pre>
<p>Into a List like this: </p>
<pre><code>sumToList[sum]={a,-b,c,d}
</code></pre>
<p>How can I achieve this?</p>
| b.gates.you.know.what | 134 | <p>Try :</p>
<pre><code>a - b + c + d /. Plus -> List
</code></pre>
<p>You can have a look at <code>a - b + c + d //FullForm</code> to see why this works.</p>
|
4,169,445 | <p>I'm getting stuck on perhaps a simple step in the Hille-Yosida theorem from 13.37 in Rudin's functional analysis.
I wonder if someone has had this same difficulty before or knows how to get around it -</p>
<p><strong>Setup:</strong> <span class="math-container">$A$</span> is a densely defined operator with domain <s... | user247327 | 247,327 | <p>A product of differentials is <em>anti-symmetric</em>. That is, <span class="math-container">$dr\,d\theta= -d\theta \,dr$</span>. It is also follows that <span class="math-container">$dr\,dr= -dr\,dr$</span> so that <span class="math-container">$dr\,dr= 0$</span> and <span class="math-container">$d\theta \,d\theta=... |
1,206,528 | <p>Find the matrix $A^{50}$ given</p>
<p>$$A = \begin{bmatrix} 2 & -1 \\ 0 & 1 \end{bmatrix}$$ as well as for $$A=\begin{bmatrix} 2 & 0 \\ 2 & 1\end{bmatrix}$$</p>
<p>I was practicing some questions for my exam and I found questions of this form in a previous year's paper.</p>
<p>I don't know how to ... | DanielV | 97,045 | <p>To diagonalize a matrix, you first need to find the eigenvalues, solve:</p>
<p>$$\det(A - \lambda I ) = 0$$
$$\det \begin{bmatrix} 2 - \lambda & 0 \\ 2 & 1 - \lambda \end{bmatrix} = 0$$</p>
<p>$$(1 - L)(2 - L) = 0$$
$$L \in \{1, 2\}$$</p>
<p>So the diagonal values are $1$ and $2$.</p>
<p>Then you need to... |
637,379 | <p>So I have a $10$ gallon aquarium slightly salty aquarium...</p>
<p>When I add water the water is at $.7$% salt. ($7$ part per thousand)</p>
<p>I let $1$ gallon evaporate, at which point I have a $\frac{7}{9}$% salinity
I then drain $1$ additional gallon. now I have $8$ gallons of $\frac{7}{9}$% salinity water.</p>... | John | 7,163 | <p>Let's say that your 10 gallons of (pure) water weighs 37.85 kg. Then to reach 0.7% salinity you'd add $0.26495$ kg of salt.</p>
<p>Let's call this amount of salt $x$. If you add two gallons water with this salinity, you'll be adding $x/5$. If the initial salinity is different, then it will be some non-negative f... |
1,358,927 | <p>I have to solve this problem using integration by parts. I am new to integration by parts and was hoping someone can help me.</p>
<p>$$\int\frac{x^3}{(x^2+2)^2} dx$$</p>
<p>Here is what I have so far:</p>
<p>$$\int udv = uv-\int vdu $$</p>
<p>$$u=x^2+2$$ Therefore, $$xdx=\frac{du}{2}$$
$$dv=x^3$$
Therefor, $$v=3... | MCT | 92,774 | <p>Though it isn't perfect, you can create a "saw-tooth" function which is pretty much a periodic linear function. For example, if you want to take $\mod 3$, you have the function $f$ which has period $3$ and looks like $f(x) = x$ for $x \in [0, 3)$. </p>
<p>You can approximate this saw-tooth function using a Fourier ... |
2,431,052 | <p>Let $M$ be a an oriented riemannian manifold.
I have seen the following definition for the Hodge-star operator acting on a differential form. Starting with $\beta\in \Omega^p(M)$ we have
$$\alpha \wedge \star \beta = \left<\alpha,\beta\right>\text{vol} ~~\forall \alpha \in \Omega^p(M)$$</p>
<p>where $ \left&l... | Isao Sauzedde | 483,390 | <p>If I'm not misteaking, the first expression you propose vanishes as the trace is conjugaison invariant. Moreover YM action is not suppose to measure only an error of commutation (with G=U(1), YM should be different from 0).</p>
|
1,571,083 | <p>Given the 2 terms
$$ \frac{k + a}{k + b}$$
and $$\frac{a}{b}$$
with $a, b, k \in \mathbb{R^+}$ and $a > b$</p>
<p>I want to show, that the first term is always bigger than the second one.</p>
<p><strong>My try</strong>
$$
\frac{k + a}{k + b} > \frac{a}{b} \\
\frac{k + b + (a - b)}{k + b} > \frac{b + ... | paw88789 | 147,810 | <p>We might also consider what is the chance that someone (not a specific individual chosen in advance) wins twice in a row. This is a bit tricky since there are different numbers of entrants. But assuming that all $25$ people in the second contest were also in the first contest, the probability of someone winning bo... |
1,429,000 | <p>Prove that for any integer $n$, the integer $n^2 + 7n + 1$ is odd.</p>
<p>I have $n=2k+1$ for some $k\in Z$</p>
<p>I really do not how to do this problem. any help in understanding would be greatly appreciated.</p>
| lhf | 589 | <p>$n^2 + 7n + 1= n(n+1)+6n+1$</p>
<p>$n(n+1)=2\binom{n}{2}$ and so is even.</p>
<p>$6n+1=2(3n)+1$ and so is odd.</p>
<p>So the sum is odd.</p>
|
2,579,137 | <p>According to the definition of harmonic number $H_n = \sum\limits_{k=1}^n\frac{1}{k}$.</p>
<p>How we can define $H_{n+1}$ and $H_{n+\frac{1}{2}}$? </p>
| Robert Israel | 8,508 | <p>$$H_{n+1} = \sum_{k=1}^{n+1} \dfrac{1}{k} = H_n + \frac{1}{n+1}$$</p>
<p>$H_{n+1/2}$ is not defined by the formula for $H_n$, unless you want to just say it's the same as $H_n$. However, one function whose restriction to the positive integers is $H_n$ is $\psi(n+1)+\gamma$, where $\psi$ is the <a href="https://en... |
1,283,541 | <p>Find the least value of $f(x)=3^{-x+1} + e^{-x-1}$.</p>
<p>I tried to use the maxima/minima concept but it was of no use. Please help.</p>
| tomi | 215,986 | <p>Are you happy to start from:</p>
<p>$\sin x=x-\frac {x^3}{3!}+\frac{x^5}{5!}-...$</p>
<p>$\frac{\sin x}x=1-\frac {x^2}{3!}+\frac{x^4}{5!}-...$</p>
<p>$\cos x=1-\frac {x}{2!}+\frac{x^4}{4!}-...$</p>
<p>$\cos x-\frac{\sin x}x=-x^2\left(\frac 1{2!}-\frac 1{3!}\right)+x^4\left(\frac 1 {4!}-\frac 1{5!}\right)+...$</p... |
1,283,541 | <p>Find the least value of $f(x)=3^{-x+1} + e^{-x-1}$.</p>
<p>I tried to use the maxima/minima concept but it was of no use. Please help.</p>
| Jack D'Aurizio | 44,121 | <p>Both:
$$\frac{\sin x}{x}\leq 1,\qquad\frac{\tan x}{x}\geq 1$$
can be seen as convexity inequalities. The first one follows from the fact that $\frac{d}{dx}\sin x = \cos x$ is decreasing on $[0,\pi]$ and the second one follows from the fact that $\frac{d}{dx}\tan x=\frac{1}{\cos^2 x}$ is increasing over the same inte... |
214,520 | <p>I'm trying to make tables like in the image to calculate max and min of the expressions.
Whenever I change the max and min of <code>a</code> and <code>b</code> in the upper table then the <code>MixEx</code>and <code>MaxEx</code> are updated automatically.</p>
<p>How can I do that in Mathematica?
<a href="https://i.... | Nasser | 70 | <p>One way is to use <code>Minimize</code> and <code>Maximize</code> with constraint.</p>
<p>Need to make sure min is smaller than max, else you'll get unexpected results. </p>
<p><a href="https://i.stack.imgur.com/cI7R6.gif" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/cI7R6.gif" alt="enter image de... |
2,245,113 | <p>I want to calculate:</p>
<p>$2^{33} \mod 25$</p>
<p>Therefore I calculate first the Euler's phi function of 25:</p>
<p>$\phi(25) = 5^{2-1} (5-1)$ = 20</p>
<p>Now I can continue with Fermat's little problem:</p>
<p>$2^{33} \mod 25$ = $(2^{20} * 2^{13}) \mod 25 \equiv 2^{13} \mod 25$</p>
<p>How can I proceed to ... | Luiz Cordeiro | 58,818 | <p>Your approach is right, but the next-to-last steps on both your calculations are wrong: in the first approach, $2^3\bmod{25}=8\bmod25$, and not $7$, so $$7^2*2^3\bmod25=24*8\bmod25=192\bmod25=17\bmod25$$.</p>
<p>In the second approach you susbsituted $7^2*2^3$ by 1568, which I don't understand.</p>
|
2,245,113 | <p>I want to calculate:</p>
<p>$2^{33} \mod 25$</p>
<p>Therefore I calculate first the Euler's phi function of 25:</p>
<p>$\phi(25) = 5^{2-1} (5-1)$ = 20</p>
<p>Now I can continue with Fermat's little problem:</p>
<p>$2^{33} \mod 25$ = $(2^{20} * 2^{13}) \mod 25 \equiv 2^{13} \mod 25$</p>
<p>How can I proceed to ... | Steven Alexis Gregory | 75,410 | <p>What I would do....</p>
<p>Express $13$ as a sum of powers of $2$: $13 = 8 + 4 + 1$</p>
<p>$2^4 \equiv 16 \pmod{25}$</p>
<p>$2^8 \equiv 256 \equiv 6 \pmod{25}$</p>
<p>$2^{13} \equiv 6 \times 16 \times 2 \equiv -4 \times 2 \equiv 17 \pmod{25}$</p>
|
804,963 | <p>Let $f,g:\mathbb{R}\longrightarrow \mathbb{R}$ be Lebesgue measurable. If $f$ is Borel measurable, then $f\circ g$ is Lebesgue mesuarable. In general, $f\circ g$ is not necessarily Lebesgue measurable. Is there any counterexample?</p>
| Poor Math Guy | 136,786 | <p>Recall that a function <span class="math-container">$f$</span> is Lebesgue (reps. Borel measurable) if <span class="math-container">$f^{-1}(B)$</span> is a Lebesgue (resp. Borel measurable) set whenever <span class="math-container">$B$</span> is a Borel set.</p>
<p>Observe from <span class="math-container">$(f\circ ... |
2,249,036 | <p>I was watching a video on PDEs and when arriving at the part of Fourier Series, the professor said:</p>
<blockquote>
<p>And one of the most fascinating reads I ever had was a paper by Riemann on the history of this [Fourier Series].</p>
</blockquote>
<p>I tried looking for it but didn't succeed, and I was wonder... | Zain Patel | 161,779 | <p>Draw two position vectors, $\mathbf{v}_1$ and $\mathbf{v}_2$ with unit magnitude and at angles $\alpha, \beta$ to the positive $x$-axis. Then the angle between the two is $\alpha - \beta$ (assuming $\alpha > \beta$ w.l.o.g). But $\mathbf{v}_1 \cdot \mathbf{v}_2$ is the cosine of the angle between them. So $\cos (... |
2,786,138 | <blockquote>
<p>Find all triples of prime numbers <span class="math-container">$(p,q,r)$</span> such that <span class="math-container">$$p^q+q^r=r^p.$$</span></p>
</blockquote>
<p>I proved that when <span class="math-container">$r=2$</span>, the equation becomes <span class="math-container">$$p^q+q^2=2^p.$$</span> Then... | Piquito | 219,998 | <p>So $$p^q+q^2=2^p \qquad(*)$$ It is easy to show that $\boxed{p\gt q}$ or just look at the graph of the function $x ^ y + y ^ 2 = 2 ^ x$ for $x,y\gt1$.</p>
<p>If $p=4n+1$ then from $(*)$,
$$q=4m\pm1\Rightarrow4N+1+16m^2\pm8m+1=2N+8m^2\pm4m+1=2^{p-1}\space\text{ absurde}.$$
Then $p$ must be of the form $\boxed{p=4n... |
519,560 | <p>I need prove that:</p>
<p>$$\int_{0}^{2\pi} \frac{R^{2}-r^{2}}{R^{2}-2Rr\cos \theta +r^{2}} d\theta= 2\pi$$</p>
<p>By deformation theorem, with $0<r<R$.</p>
<p>Professor gave us the hint to use the function $f(z)= \frac{R+z}{z(R-z)}$, and define an adequate $\gamma : [a,b]\rightarrow \mathbb{C}$ circular cu... | Dan | 79,007 | <p>The integrand is the famous <a href="http://mathworld.wolfram.com/PoissonKernel.html">Poisson kernel</a> function! Here's a sketch of how to integrate it using the Residue Theorem:</p>
<p>Show that</p>
<p>$$
\frac{R^2-r^2}{R^2-2Rr\cos\theta +r^2} = \text{Re}\left(\frac{R+re^{i\theta}}{R-re^{i\theta}}\right)
$$</p>... |
1,267,644 | <p>Having a circle of radius <span class="math-container">$R$</span> with the center in <span class="math-container">$O(0, 0)$</span>, a starting point on the circle (e.g. <span class="math-container">$(0, R)$</span>) and an angle <span class="math-container">$\alpha$</span>, how can I move the point on the circle with... | Mann | 126,204 | <p><img src="https://i.stack.imgur.com/Aa8QG.jpg" alt="enter image description here"></p>
<p>As you can see from the image, The point you require is at $-45 $ Degree from $0$ . So your required point is $R \cos (-\pi /4), R \sin (-\pi/4)$</p>
<p>Which can be equivalently written as?</p>
|
136,570 | <p>Good day everyone. I was reading the more advanced lectures on complex analysis and encountered a lot of questions, concerning the determination of complex logarithm. As far I don't even understand the concept of it, but I'll do provide you with several practical questions concerning the topic. </p>
<p>First of all... | Gerry Myerson | 8,269 | <p>I think your first question is, why is there no continuous logarithm function defined in the complex plane, omitting $0$. The answer is, trace out a circle of radius $1$ centered at the origin, starting at $1$, and evaluate the logarithm function as you go around. Go ahead, do it! You will find that you either make ... |
3,346,543 | <p>I am aware this is a pretty big topic, but the attempts at layman's explanations I have seen either barely provide commentary on the formal proofs, or fail to provide an explanation (e.g "it gets too complex" does not really say anything)</p>
<p>Is there a good intuitive explanation as to why we fail to obtain a ge... | Dmitry Ezhov | 602,207 | <p>Formula for roots of equation <span class="math-container">$\displaystyle z^m-az^n-1=0$</span> with definite integration:</p>
<p><span class="math-container">$\displaystyle z_j=e^{2j\pi i/m}+\frac{1}{2\pi i}\left(e^{(2j+1)\pi i/m}\int_0^\infty log\left(1+a\frac{t^n}{1+t^m}e^{(2j+1)\pi in/m}\right)dt \\- e^{(2j-1)\p... |
3,150,645 | <p>Suppose <span class="math-container">$\dim V = \dim W$</span>. Given <span class="math-container">$\phi:V\rightarrow V$</span> and <span class="math-container">$\psi:W\rightarrow W$</span> when do we have bijective linear maps <span class="math-container">$a,b:V\rightarrow W$</span> such that
<span class="math-cont... | user647486 | 647,486 | <p>Then that condition is just equivalence of <span class="math-container">$\phi$</span> and <span class="math-container">$\psi$</span>, which is equivalent to having the same rank. </p>
<p>To show this, take a basis <span class="math-container">$e_i^V$</span>, <span class="math-container">$i\in I_1$</span> of the ker... |
455,302 | <p>Given: $x + \sqrt x = \sqrt 3$<br>
Evaluate: $x^3 - 1 / x^3$</p>
| Matt Groff | 2,626 | <p>HINT: we can solve $x+\sqrt{x}=\sqrt{3}$, using the quadratic formula. Just use $\sqrt{x}$ as the variable.</p>
<p>From there, it's just plugging in those values for $x$ into the second equation.</p>
|
4,005,463 | <p>In the problem I am solving, table values are given for function and it says to find <span class="math-container">$f''(0.5)$</span> using second order central difference formula. I know the formula which is
<span class="math-container">$\frac{f(x+\triangle x)-2f(x)+f(x-\triangle x)}{{\triangle x}^2}$</span>. But the... | mathcounterexamples.net | 187,663 | <p><strong>Let's prove that <span class="math-container">$F$</span> is totally bounded</strong></p>
<p>Take <span class="math-container">$\epsilon \gt 0$</span>. As <span class="math-container">$\mathcal F$</span> is supposed to be equicontinuous, for all <span class="math-container">$x \in X$</span>, it exists an open... |
299,056 | <p>Let $\overline{\mathbb{Q}}$ be the algebraic closure of $\mathbb{Q}$. Let $K = \mathbb{Q}(\sqrt{d})$ and $\overline{K}$ defined to be the algebraic closure of $K$. Is it true that $\overline{K} \cong \overline{\mathbb{Q}}$?</p>
| Chris Eagle | 5,203 | <p>Algebraic closures are only defined up to isomorphism, so it doesn't make sense to ask if $\overline K$ and $\overline{\Bbb{Q}}$ are equal. It does make sense to ask if they are isomorphic, which they are: $\overline K$ is an algebraic extension of $\Bbb Q$ (since it's an algebraic extension of an algebraic extensio... |
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