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 |
|---|---|---|---|---|
129,261 | <p>I need to prove several inequalities trivially. (i.e. without using AM-GM, re-arrangement etc). I just keep hitting a blank. Could anyone help?</p>
<p>$$x^{4}+y^{4}+z^{4}\geq x^{2}yz+xy^{2}z+xyz^{2}$$</p>
| Rijul Saini | 27,729 | <p>I'm guessing Sum-of-Squares representation would amount to a 'trivial' proof, right? How about expanding the following (this further proves that your inequality holds for all reals, not just positive ones..): $$\sum_{cyclic}(x^2 - y^2)^2 + \sum_{cyclic}x^2(y-z)^2 \ge 0$$</p>
|
1,978,935 | <p>Let $f,g : [a,b] \rightarrow \mathbb{R}$ be continuous. We know that $f$ and $g$ have maximal values, as they are continuous on a closed interval. Let $M_f$ be the maximal value of $f$, and $M_g$ the maximal value of $g$. Show that if $M_f$ = $M_g$, then there exists $\psi \in [a,b]$ with $f(\psi) = g(\psi)$</p>
<p... | ec92 | 34,552 | <p>Suppose $f(x_1) = M_f$ and $g(x_2) = M_f = M_g$. </p>
<p>If $x_1 = x_2$, you're done. </p>
<p>Otherwise, consider the interval $[x_1, x_2]$ (or $[x_2, x_1]$ if $x_2 < x_1$).
The function $f-g$ is continuous on this interval, it's nonnegative at $x_1$, and nonpositive at $x_2$. Thus there must be a point in the... |
136,264 | <p>I have a question concerning the stability analysis for a kind of differential equation taking the form
$$\dot x=Ax+Bw,$$
where $A\in \mathbb{R}^{n \times n}$, $B\in \mathbb{R}^{n \times m}$ are constant matrices and
$w \in \mathbb{R}^m$ is a normal random variable, i.e., $w\sim \mathcal{N}(0,W)$ with $W$ ... | Nick Gill | 801 | <p>The Australian Mathematical Society have produced a ranking:</p>
<p><a href="http://www.austms.org.au/Rankings/0101_AustMS_final_ranked.html">http://www.austms.org.au/Rankings/0101_AustMS_final_ranked.html</a></p>
<p>It is widely used (for instance, by my own institution in the UK).</p>
<p>When choosing where to ... |
164,629 | <p>Probably this is well known to those who know it.</p>
<p>Got an argument and numerical support that over
number fields elliptic curves in minimal models
might have unbounded number of integral points,
the number depending on the degree of the field.</p>
<p>Set $f(x)=x^3+ax+b$ and consider the curve
$E: y^2=f(x)$.<... | JSE | 431 | <p>If the 5 points had a linear dependence, their coordinates could not generate a (Z/2Z)^5 - extension of Q. But they visibly do.</p>
|
93,063 | <p>There is a story I read about tiling the plane with convex pentagons.</p>
<p>You can read about it in this <a href="http://www.ivanrival.com/docs/picturepuzzling_2.pdf">article</a> on pages 1 and 2.</p>
<p>Summary of the story:
A guy showed in his doctorate work all classes of tiling the plane with convex pentagon... | Zarrax | 3,035 | <p>I think Carl Mummert's answer is spot-on. One other thing worth mentioning is that more often than being wrong, or even having a recognizably flawed proof, are papers which are just really hard to understand. (This is sometimes compounded by language barriers... more than once I've had to deal with papers that were ... |
735,101 | <p>Let $X_1$ and $X_2$ be independent random variables having the uniform density with $\alpha = 0$ and $\beta = 1$. Find expressions for the function $Y =X_1 + X_2$.</p>
<p>(a)$y \le 0$</p>
<p>(b)$0<y<1$</p>
<p>(c)$1<y<2$</p>
<p>(d)$y\ge2$</p>
<p>I'm thinking $f(x_1)=f(x_2) = 1$ for $0 \le x_1\le 1$ a... | user111013 | 111,013 | <p>If you find $x'$ and $y'$ such that $47x' + 64y' = gcd(47,64)$ then solve $gcd(47,64)\cdot t + 70z = 1$. Then $x = x't$ and $y = y't$. This works since $gcd(47,64,70) = gcd(gcd(47,64),70)$.</p>
|
735,101 | <p>Let $X_1$ and $X_2$ be independent random variables having the uniform density with $\alpha = 0$ and $\beta = 1$. Find expressions for the function $Y =X_1 + X_2$.</p>
<p>(a)$y \le 0$</p>
<p>(b)$0<y<1$</p>
<p>(c)$1<y<2$</p>
<p>(d)$y\ge2$</p>
<p>I'm thinking $f(x_1)=f(x_2) = 1$ for $0 \le x_1\le 1$ a... | Rodney Coleman | 73,128 | <p>Notice that $gcd(x,y,z)=gcd(x,gcd(y,z))$. First we find $a$, $b$ such that $gcd(x,gcd(y,z))=ax+bgcd(y,z)$, then $c$, $d$ such that $gcd(y,z)=cy+dz$. Finally we obtain $gcd(x,y,z)=ax+bcy+bdz$. </p>
|
2,180,398 | <p>I am not sure if I got the correct answers to these basic probability questions.</p>
<blockquote>
<p>You and I play a die rolling game, with a fair die. The die is equally likely to land on any of its $6$ faces. We take turns rolling the die, as follows. </p>
<p>At each round, the player rolling the die w... | N. F. Taussig | 173,070 | <blockquote>
<p>What is the probability that the player who rolls first wins in the first round?</p>
</blockquote>
<p>Your answer of $4/6 = 2/3$ is correct.</p>
<blockquote>
<p>What is the probability that the player who rolls first wins the game if the game lasts at most three rounds?</p>
</blockquote>
<p>The p... |
1,921,879 | <blockquote>
<p>Find all positive integers $n$ for which $\dfrac{x^n + y^n + z^n}2$ is a perfect square, whenever $x$, $y$, and $z$ are integers such that $x + y + z = 0$.</p>
</blockquote>
<p>I don't even know where to start.</p>
| ShakesBeer | 168,631 | <p>Leading on from Dietrich's answer, let's take $(x,y,z)=(1,1,-2)$ and consider even $n$ of the form $n=2k$, $k \geq 2$.</p>
<p>$$\frac{x^n+y^n+z^n}{2}=\frac{2+(-2)^n}{2}=1+2^{2k-1}$$</p>
<p>Suppose $1+2^{2k-1}$ is a perfect square. It is odd, so </p>
<p>$$1+2^{2k-1}=(2l+1)^2$$</p>
<p>$$\iff 2^{2k-1}=4l^2+4l$$</p>... |
541,890 | <p>Prove or disprove:
For all integers $m$ and $n$, if $m+n$ is even then so is $m-n$.
Would you just set them even to each other because you are given $m+n$ is even?</p>
| user103074 | 103,074 | <p>The problem is when you go from 1 to 2 persons.</p>
<p>Assuming S(K) valid for k=1: that in any group of 1 human being, everybody has the same hair colour</p>
<p>Now you should be able to prove the validity of S(K+1): the equivalent to any group of 2 human beings.</p>
<p>Ok. Let's try. If you have any group of 2... |
656,458 | <p>If $A$ is an $n\times n$-matrix, $A^H$ is a Hermitian Matrix and $A^S$ is a Skew Hermitian, show $A=A^H+A^S$.</p>
<p>I am having trouble working with these so far and really cannot find many characteristics except the definitions. A Hermitian is made up of reals on the diagonal and is $A^*=A$. It is skew hermitian... | Kathleen | 89,461 | <p>If $A$ is an $n\times n$-matrix, $A^H$ is a Hermitian Matrix and $A^S$ is a Skew Hermitian, show $A=A^H+A^S$.</p>
<p>$A^H=1/2(A+A*)$</p>
<p>$A^S=1/2(A-A*)$</p>
<p>$A^H+A^S=1/2(A+A*)+1/2(A-A*)$</p>
<p>$=1/2A+1/2A*+1/2A-1/2A*$</p>
<p>$=A$ </p>
<p>QED</p>
|
924,555 | <p>My homework question:</p>
<blockquote>
<p>From the order axioms for $\mathbb{R}$, show that $0 < 1$. [<em>Hint:</em> From the field axioms, $0 \not=1$. By the trichotomy property, either $0<1$ or $4<0$. Assuming $1 < 0$, get $0 < -1$. Now use Exercise 4.]</p>
</blockquote>
<p>Exercise 4 from my te... | Avrham Aton | 171,799 | <p>By contradiction assume $1<0$ then $0<-1 $ ,so $ -1 $ is a positive number so(by the fact that positive reals is closed under multiplication) $(-1)(-1) $must also be positive , but by assumption 1 is not positive . contradiction</p>
|
31,100 | <p>I have just begun to learn about the fundamental group.
An exercise asks me to prove that $$X=\{(x,y,z): z \ge 0\}-\{(x,y,z): y=0,0\leq z \leq 1\}$$ has trivial fundamental group.</p>
<p>What I know is:</p>
<p>1) the definition of the fundamental group.</p>
<p>2) X has trivial fundamental group iff any loop in X... | Sam Nead | 1,307 | <p>Suggestion - break $H$ (your set) into two pieces using the half plane $P = \{ (x,y,z) : y = 0, z > 1 \}$. Show that the two pieces $K^\pm$ of $H - P$ each have trivial fundamental group, as does $P$. Now assemble. </p>
<p>Two more suggestions: 1. If you want to talk about stuff, it helps to give the stuff nam... |
1,657,664 | <p>Struggling with a homework problem here and can't understand logically which one would be correct (each has different truth tables). I need to express the following statement using quantifiers, variables, and the predicates M(s), C(s), and E(s) </p>
<blockquote>
<p>"No computer science students are engineering st... | Graham Kemp | 135,106 | <blockquote>
<p>"No computer science students are engineering students" </p>
</blockquote>
<ul>
<li>$\neg \exists s {\in} D ~(C(s)\wedge E(s))$ "there is not a student who is both".</li>
<li>$\forall s {\in} D ~(\neg C(s)\vee \neg E(s))$ "all students are either not comsci. or not eng. student"</li>
<li>$\forall s {... |
1,515,775 | <p>So, everyone knows the famous <a href="https://en.wikipedia.org/wiki/Lagrange%27s_four-square_theorem" rel="nofollow">Lagrange's four-square theorem</a>, which states, that every positive integer can be written down as the sum of $4$ square numbers. Since $4=2^2$, and $2$ represents the square numbers, could this be... | Kieren MacMillan | 93,271 | <p>Because a square is both a <a href="https://en.wikipedia.org/wiki/Polygonal_number" rel="nofollow noreferrer">polygonal number</a> and a <a href="https://en.wikipedia.org/wiki/Perfect_power" rel="nofollow noreferrer">perfect power</a>, <a href="https://en.wikipedia.org/wiki/Lagrange%27s_four-square_theorem" rel="nof... |
461 | <p>There is a function on $\mathbb{Z}/2\mathbb{Z}$-cohomology called <em>Steenrod squaring</em>: $Sq^i:H^k(X,\mathbb{Z}/2\mathbb{Z}) \to H^{k+i}(X,\mathbb{Z}/2\mathbb{Z})$. (Coefficient group suppressed from here on out.) Its notable axiom (besides things like naturality), and the reason for its name, is that if $a\i... | Elizabeth S. Q. Goodman | 1,198 | <p>I second the references to Hatcher and to Mosher & Tangora, though you can also find Steenrod's original paper. At least the first two of those start out by listing the various axioms of Steenrod squares and then construct them.</p>
<p>The reason to axiomatize the properties of Steenrod squares is that it is h... |
461 | <p>There is a function on $\mathbb{Z}/2\mathbb{Z}$-cohomology called <em>Steenrod squaring</em>: $Sq^i:H^k(X,\mathbb{Z}/2\mathbb{Z}) \to H^{k+i}(X,\mathbb{Z}/2\mathbb{Z})$. (Coefficient group suppressed from here on out.) Its notable axiom (besides things like naturality), and the reason for its name, is that if $a\i... | Robert Bruner | 6,872 | <p>If I interpret the request a bit differently, I would say that the Steenrod operations in the cohomology of a spectrum tell you about the attachments of the cells. If $Sq^1 x = y$, then a cell dual to $y$ is attached by a map of degree 2 mod 4 to a cell dual to $x$. Similarly, $Sq^2 x = y$ tells us the attaching m... |
388,766 | <p>I need to show that every element in $\Bbb Z/p\Bbb Z$ can be written as a sum of two squares. The case $p=2$ is trivial and $0$ is always $0^2 + 0^2$. So all I have to do is show that every element of $(\Bbb Z/p\Bbb Z)^\times$ (the group of units) can be expressed as a sum of two squares. The question hints that I s... | Community | -1 | <p>For finite groups, to show that a subset is a subgroup if you check closure and non-emptyness you don't need to check inverses because each element has finite order: $g^{-1} = g^{o(g) - 1}$ is in the subset by closure assumption.</p>
|
3,974,394 | <p>I am relatively new to isomorphisms and I don't understand how <span class="math-container">$\varphi$</span> is surjective in this proof. I have searched online, but I still don't understand. If anyone could straight up tell me because I feel like I'm being a bit dumb.</p>
<p><a href="https://i.stack.imgur.com/oIB4A... | azif00 | 680,927 | <p>A function <span class="math-container">$f : A \to B$</span> is <em>surjective</em> if every element of <span class="math-container">$B$</span> can be written as <span class="math-container">$f(\cdot)$</span>, that is, for any <span class="math-container">$b \in B$</span> there is <span class="math-container">$a \in... |
544,779 | <p>So here are the contextual statements:
1) Maya either listens to music or does her homework. If she listens to music she feels happy.If she does her homework she feels unhappy. Therefore she will not do her homework while listening to music.</p>
<p>Let P be the statement "Maya listens to Music".
Q "Maya does homewo... | Henno Brandsma | 4,280 | <p>If you start with a Hausdorff space $X$ and want to consider a compactification, which by definition is a pair $(Y, f)$ where $Y$ is a compact space (let's be the most general and not require Hausdorff) and $f: X \to Y$ is an embedding and $f[X]$ is dense in $Y$.</p>
<p>Trivial example: if $X$ is already compact (a... |
184,219 | <p>I know that it is the standard functionality of <code>Merge</code> to combine the values of the same keys among associations.</p>
<p>Now I would like to deal with a situation in which, in my associations, the keys are strings (English words). And I want to define the sameness as two words having the same result fro... | Henrik Schumacher | 38,178 | <p>What about this?</p>
<pre><code>data = <|"effect" -> {5, {2, 3}}, "effects" -> {4, {1, 3, 5}}|>;
Merge[
KeyValueMap[{key, value} \[Function] WordStem[key] -> value, data],
mergeFunc
]
</code></pre>
<blockquote>
<p><|"effect" -> {9, {1, 2, 3, 5}}|></p>
</blockquote>
|
1,449,776 | <p>I have always known that $a^n=a*a*a*.....$(n times)</p>
<p>Then what exactly is the meaning if $a^0$ and why will it be equal to $1$?</p>
<p>I have checked it in the internet but everywhere the solution is based on the principle that $a^m*a^n=a^{m+n}$ and when $n=0$ it will be $a^m$ and clearly $a^0$ is equal to $... | Paul G | 273,789 | <p>The definition you give only applies when <em>n</em> is an integer greater than zero. It does not apply to <em>n</em> = 0 or negative <em>n</em> because it doesn't make sense to talk about multiplying together zero <em>a</em>'s or a negative number of <em>a</em>'s.</p>
<p>However, using the principle that you menti... |
1,217,557 | <p>I was tasked with drawing the contour lines of $ z = \sqrt{xy} $, which I find a bit problematic since I can see no way in which one can plot (by hand, and not with wolfram and others....) the $ z = \sqrt{xy} $ graph in $R^2( x-, y- $ projection} to begin with for this surface...</p>
<p>How can one draw this conto... | Community | -1 | <p>That function is what's called AFFINE (<a href="http://en.wikipedia.org/wiki/Affine_transformation" rel="nofollow">http://en.wikipedia.org/wiki/Affine_transformation</a>), it's not linear.</p>
|
707,193 | <p>The inequality to solve:
$$\left[\frac{-K^2+13K+44}{14-K}\right] > 0$$</p>
<p>How do I solve this?
I tried this:
$$
-K^2+13K+44 > 0 \quad \text{(multiply both sides by $14-K$)}\\
K^2-13K < 44\\
K(K-13) < 44
$$
Is this correct? Any way to get a more precise $K$ value? Thanks. </p>
| Alijah Ahmed | 124,032 | <p><strong>Hint:</strong></p>
<p>In order to maintain the same direction of the inequality you can multiply both sides by the positive term $(14-K)^2$.</p>
<p>Thus you will obtain the following inequality</p>
<p>$(14-K)(-K^2+13K+44)>0$</p>
<p>If you solve the quadratic $-K^2+13K+44=0$, you will obtain two roots ... |
501,678 | <p>Let $f(x)=x-\cos(x)$. Find all points on the graph of $y=f(x)$ where the tangent line has slope 1. (In each answer $n$ varies among all integers).</p>
<p>So far I've used the Sum derivative rule for which I have $1+\sin(x)$. So do I put in 1 in for $x$ for sin$(x)$.</p>
<p>Please Help!!</p>
| njguliyev | 90,209 | <p>Hint: $$\sqrt{\frac {1}{x}+2}-\sqrt{\frac {1}{x}} = \dfrac{2}{\sqrt{\dfrac {1}{x}+2}+\sqrt{\dfrac {1}{x}} }.$$</p>
|
501,678 | <p>Let $f(x)=x-\cos(x)$. Find all points on the graph of $y=f(x)$ where the tangent line has slope 1. (In each answer $n$ varies among all integers).</p>
<p>So far I've used the Sum derivative rule for which I have $1+\sin(x)$. So do I put in 1 in for $x$ for sin$(x)$.</p>
<p>Please Help!!</p>
| DonAntonio | 31,254 | <p>$$\sqrt{\frac1x+2}-\sqrt\frac1x=\frac2{\sqrt{\frac1x+2}+\sqrt\frac1x}=\frac{2\sqrt x}{\sqrt{2x+1}+1}\xrightarrow[x\to 0]{}\frac 0{1+1}=0$$</p>
|
374,881 | <p>I'd like to know how I can recursively (iteratively) compute variance, so that I may calculate the standard deviation of a very large dataset in javascript. The input is a sorted array of positive integers.</p>
| Schatzi | 182,154 | <p>There are two problems in the preceding answer, the first being the formula for the variance is incorrect(see the formula below for the correct version) and the second is that the formula for the recursion ends up subtracting large, nearly equal, numbers.</p>
<p>The definition for unbiased estimates of mean($\bar x... |
2,679,821 | <p>$K=\{(x,y,z)\in \mathbb{R}^{3}|1\leq x^{2}+y^{2}+z^{2}\leq 2,x+y\geq0,\sqrt3x-y\leq0,z\geq0\} \rightarrow \\
\rightarrow\{spherical\quad coordinates\}\rightarrow K:1\leq r\leq \sqrt2,\quad0\leq \theta\leq \pi/2,\quad\pi/3\leq\phi\leq3\pi/4.$</p>
<p>My question is how we arrive at these inequalities/intervals for th... | Bram28 | 256,001 | <p>If you are allowed to use biconditionals, you can do:</p>
<p>$$\forall x (Person(x) \rightarrow (LikesPizza(x) \leftrightarrow \neg Irrational(x)))$$</p>
|
3,333,924 | <p>I am wondering about solutions to the following differential equation:
<span class="math-container">$f(x)=C_1 \cdot f'(x+C_2) \; \forall x \in \mathbb{R} \; \exists \; C_1, C_2 \in \mathbb{R}$</span>. With <span class="math-container">$C_1, C_2$</span> being constant. Are the solutions uniquely in the family of sin/... | trula | 697,983 | <p>a transcendental function f(x) gives transcendental results for most rational x
example: e^x, sin(x) etc.
the simple seaming equation e^x=x or cos(x)=x have no formula for x as result, but must be calculated numerically.
also you cn not rewrite e^x as a polynomial or a fraction of polynoms
trula</p>
|
19,996 | <p>In 1556, Tartaglia claimed that the sums<br>
1 + 2 + 4<br>
1 + 2 + 4 + 8<br>
1 + 2 + 4 + 8 + 16<br>
are alternative prime and composite. Show that his conjecture is false. </p>
<p>With a simple counter example, $1 + 2 + 4 + 8 + 16 + 32 + 64 + 128 = 255$, apparently it's false. However, I want to prove it in gen... | Ross Millikan | 1,827 | <p>It is known that $2^n-1$ can only be prime if $n$ is prime. This is because if $jk=n$, $2^n-1=\sum_{i=0}^{n-1}2^i=\sum_{i=0}^{j-1} 2^i \sum_{i=0}^{k-1} 2^{ij}$ So they will only continue to alternate at twin primes. In particular, $2^{6k+2}-1, 2^{6k+3}-1$ and $2^{6k+4}-1$ will all be composite</p>
|
1,921,562 | <p>Couldn't solve this indefinite integral, can someone help me? $$\int \frac {x^3+4x^2+6x+1}{x^3+x^2+x-3} dx$$</p>
| GoodDeeds | 307,825 | <p>$$x^3+4x^2+6x+1=(x^3+x^2+x-3)+3x^2+5x+4$$
Thus,
$$\int\frac{x^3+4x^2+6x+1}{x^3+x^2+x-3}dx=\int\left(1+\frac{3x^2+5x+4}{x^3+x^2+x-3}\right)dx=\int\left(1+\frac{3x^2+5x+4}{(x-1)(x^2+2x+3)}\right)dx$$</p>
<p>Let
$$\frac{3x^2+5x+4}{(x-1)(x^2+2x+3)}=\frac{A}{x-1}+\frac{Bx+C}{x^2+2x+3}$$</p>
<p>Find A,B, and C and solve... |
103,960 | <pre><code>tt = Flatten[Table[{x, y, z, btot[x, y, z]}, {x, -1, 1, 0.1}, {y, -1, 1,0.1}, {z, -1, 1, 0.1}], 2];
ff = Interpolation[tt]
</code></pre>
<p>Till here it is working fine as it is returning the values of the interpolated function at various <code>{x,y,z}</code> points.</p>
<p>Then I want to find the gradient... | Coolwater | 9,754 | <p>With</p>
<pre><code>ffd[x_,y_,z_]:= D[ff[x,y,z],{{x,y,z}}]
</code></pre>
<p>the values of x, y, and z are substituted as arguments causing differentation wrt. numbers, i.e. nonsense. Moreover, you are using SetDelayed, which differentiate once for every call, which rather should be once for all time.</p>
<p>The s... |
367,116 | <p>I have a question about vacuous true and it always make me confused. If I want to prove that the empty set is the subset of all the set A, the proof is as following:
if x is in empty set, then x is in A. since x is in empty set is always false,, so the conditional statement is always true~
my question is why x is in... | Peter Smith | 35,151 | <p>@AustinMorh makes the key point. But a footnote about unicorns. We must distinguish two different claims.</p>
<p>Suppose $U$ is the predicate satisfied by all and only unicorns, then</p>
<blockquote>
<p>$\forall x(Ux \to x \in \emptyset)$</p>
</blockquote>
<p>indeed comes out true for sensible domains (since no... |
2,774,792 | <p>How would you go about proving the recursion
$$T(n) = T\left(\frac n4\right) + T\left(\frac{3n}4\right) + n$$is $\mathcal O(n\log n)$ using induction?</p>
<p>Thanks!</p>
| Henry | 6,460 | <p>Let's count all desired possibilities out the $6^3$ total potential results:</p>
<ul>
<li>${3 \choose 3} \times 3!=6$ ways of getting all three original values in some order</li>
<li>${3 \choose 2} \times {6-3 \choose 1} \times 3!= 54$ ways of getting two different original values and one non-original value in som... |
510,732 | <p>I am trying to think of a case where this is not true:</p>
<p>$f(n) = O(g(n))$ and $f(n) \neq \Omega(g(n))$, does $f(n) = o(g(n))$?</p>
<p>I suspect that it has to do with the varying $c$ and $n_{0}$ constants but am not completely sure. </p>
<p>Thanks!</p>
| mjqxxxx | 5,546 | <p>You can read the relations this way.</p>
<ul>
<li>$f \in O(g)$ means that $f$ eventually stays below some multiple of $g$.</li>
<li>$f \in \Omega(g)$ means that $f$ eventually stays above some multiple of $g$.</li>
<li>$f \in o(g)$ means that $f$ eventually stays below <em>any</em> multiple of $g$.</li>
</ul>
<p>S... |
3,684,048 | <p>I understand how to prove that algebraically using implicit differentiation:
<a href="https://i.stack.imgur.com/wvANe.png" rel="nofollow noreferrer">1</a></p>
<p>However, when I hope to gain an understanding through the graphs, I had a hard time wrapping my head around why the transformed lnx function still has the... | Community | -1 | <p>It's because of the identity <span class="math-container">$\ln(ab)=\ln a+\ln b$</span>, which holds for all <span class="math-container">$a,b>0$</span>.</p>
|
1,406,535 | <p>Let $ f$ be a function such that $|f(u)-f(v)|\leq|u-v|$ for all real $u$ and $v$ in an interval $[a,b]$.Then:<br>
$(i)$Prove that $f$ is continuous at each point of $[a,b]$.<br></p>
<p>$(ii)$Assume that $f$ is integrable on $[a,b]$.Prove that,$|\int_{a}^{b}f(x)dx-(b-a)f(c)|\leq\frac{(b-a)^2}{2}$,where $a\leq c \leq... | Sempliner | 122,727 | <p>This occurs because for Carmichael numbers like $N$ we have that, by the Chinese remainder theorem, $\mathbb{Z}/N\mathbb{Z} \cong \mathbb{Z}/p_1\mathbb{Z} \times \mathbb{Z}/p_2\mathbb{Z} \times \dots \times \mathbb{Z}/p_n\mathbb{Z}$ where the $p_i$ are its prime factors, and because for all these primes we have that... |
181,855 | <p>In the latest <a href="http://what-if.xkcd.com/113/" rel="noreferrer">what-if</a> Randall Munroe ask for the smallest number of geodesics that intersect all regions of a map. The following shows that five paths of satellites suffice to cover the 50 states of the USA:
<img src="https://i.stack.imgur.com/gyfYt.png" al... | The Masked Avenger | 35,626 | <p>I think the easiest route to a lower bound is to pick four states such as Hawaii, Alaska, Rhode Island, and Florida, and show that any geodesics cutting them leave too many states uncovered, or are five or more in number.
It should be possible to enumerate the maximal cutting geodesics for each pair of states, and t... |
1,580,270 | <p>Consider the groups $G = \{0,1,2\} = \mathbb Z_3$ and $H = \{a,b,c\}$
given by the following multiplication tables:</p>
<p><a href="https://i.stack.imgur.com/hXgBb.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/hXgBb.jpg" alt="enter image description here"></a></p>
<p>The first one isn't really... | Bernard | 202,857 | <p>First check the second table indeed defines a group law. Looking at the second row and second column in the right-hand side table, you can see <span class="math-container">$b$</span> is the neutral element. Then you can check <span class="math-container">$a^{-1}=b$</span> and <span class="math-container">$b^{-1}=a$... |
2,955,780 | <p>The midpoint of a chord of length <span class="math-container">$2a$</span> is at a distance <span class="math-container">$d$</span> from the midpoint of the minor arc it cuts out from the circle. Show that the diameter of the circle is <span class="math-container">$\frac{a^2+d^2}{d}$</span> .</p>
<p>I know I have t... | Phil H | 554,494 | <p><a href="https://i.stack.imgur.com/Gcnxa.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/Gcnxa.jpg" alt="enter image description here"></a></p>
<p><span class="math-container">$r = \sqrt{r^2 - a^2} + d$</span></p>
<p><span class="math-container">$r - d = \sqrt{r^2 - a^2}$</span></p>
<p><span cl... |
3,184,802 | <p>Are there any <span class="math-container">$C^\infty$</span> real functions except the exponential family and gamma function family which has all the derivatives of same sign on an interval [a,<span class="math-container">$\infty$</span>) with a<span class="math-container">$\gt$</span>0 ?
I speculate the function is... | Martin R | 42,969 | <p>Choose an arbitrary sequence <span class="math-container">$(a_n)$</span> with <span class="math-container">$a_n \ge 0$</span> and <span class="math-container">$\lim_{n\to \infty} \sqrt[n]{a_n} = 0$</span>. Then the power series
<span class="math-container">$$
f(x) = \sum_{n=0}^\infty a_n x^n
$$</span>
converges on ... |
2,698,357 | <p><strong>Draw a graph on $10$ vertices with no more than $20$ edges that contains no independent set of size $3$.</strong></p>
<p>So I was trying to draw the above graph but I was kind of stuck. What I basically did was draw a bipartite graph with $5$ vertices on each side and then $20$ edges randomly connecting bet... | Mk Utkarsh | 532,893 | <p>Take <a href="https://commons.wikimedia.org/wiki/File:Complete_bipartite_graph_K4,4.svg" rel="nofollow noreferrer">K4,4</a> and 2 vertices connected with an edge. That will result in 2 independent sets of size 4 and 2 independent sets of size 1</p>
|
43,743 | <p>An alternative title is: When can I homotope a continuous map to a smooth immersion?</p>
<p>I have a simple topology problem but it's outside my area of expertise and I worry may be rather subtle. Any help would be appreciated.</p>
<p>The set-up is the following:
Let $M$ be some (closed say) $n$ dimensional mani... | András Szűcs | 36,950 | <p>The answer is known in the special case you mentioned (curves in a 3-manifold)
Actually it is true more generally: Namely
if k = n-2 and your k-dimensional oriented submanifolds $\Sigma_1$ and $\Sigma_2$ represent the same INTEGER homology class in the ambient ORIENTED n-manifold M, and they are disjoint and embedde... |
1,134,510 | <p>Regarding My Background I have covered stuff like </p>
<p>1.Single Variable Calculus</p>
<p>2.Multivariable Calculus (Multiple Integration,Vector Calculus etc) (Thomas Finney)</p>
<p>3.Basic Linear Algebra Course (Containing Vector spaces,Linear Transformation)</p>
<p>4.Ordinary Differential Equation</p>
<p>5.R... | ganbustein | 198,393 | <p>$26^7\cdot 7!$ is clearly way too big. The $26^7$ factor all by itself counts <em>all</em> 7-letter passwords made up from a 26-letter alphabet. Multiplying by $7!$ can only make the number bigger (a lot bigger), when the correct number should be a lot smaller because of all the incorrect combinations.</p>
<p>There... |
2,952,028 | <p>The question asks: Find the values of k for which the line</p>
<p><span class="math-container">$y=2x-k$</span> is tangent to the circle with equation <span class="math-container">$x^2+y^2=5$</span></p>
<p>So I started by substituting,</p>
<p><span class="math-container">$x^2+(2x-k)^2=5$</span></p>
<p><span class... | TonyK | 1,508 | <p>Here is how I did it:</p>
<p>Differentiating <span class="math-container">$x^2+y^2=5$</span> gives <span class="math-container">$2x+2y\;dy/dx=0$</span>, or <span class="math-container">$dy/dx=-x/y$</span>. </p>
<p>The slope of the line <span class="math-container">$y=2x-k$</span> is <span class="math-container">$... |
67,630 | <p>I know of a theorem from Axler's <em>Linear Algebra Done Right</em> which says that if $T$ is a linear operator on a complex finite dimensional vector space $V$, then there exists a basis $B$ for $V$ such that the matrix of $T$ with respect to the basis $B$ is upper triangular.</p>
<p>The proof of this theorem is b... | Robert Israel | 8,508 | <p>The proof breaks down because $T|_U$ can't be made upper triangular, where in this case
$U = \{(0, x, y): x, y \in {\mathbb R}\}$.</p>
|
367,669 | <p><img src="https://i.stack.imgur.com/zQFyC.jpg" alt="enter image description here"></p>
<p>This is probably a very simple questions but I am not clear on Möbius transformations and how to solve this problem. I'd appreciate if somebody can point me towards a method to do these sort of questions or a webpage that expl... | xyzzyz | 23,439 | <p>Let $f(n) = k$ if $n = 2^k \cdot (2l -1)$ for natural numbers $k, l$ -- in other words, if $2^k$ is the highest power of $2$ that divides $n$. Then $f^{-1}(k) = \{2^k \cdot 1, 2^k \cdot 3, 2^k \cdot 5, 2^k \cdot 7\ldots \}$.</p>
|
84,249 | <p>I have a data set with evenly spaced data points. The plot is frequency vs. intensity. The overall shape of the plot is an upwards curve into a plateau, this cannot be seen in the data as this is an unimportant feature. There is also an oscillation in this curve. This can be seen in the plot.</p>
<p><img src="https... | Histograms | 29,371 | <p>For problem #2, I recommend a combination of <code>TimeSeries</code> and <code>TimeSeriesResample</code> to obtain a regular sampling of your data, followed by an eyeball inspection of the <code>Periodogram</code>. You'll see that there is a spike around 0.015-ish in this power spectrum. </p>
<p>You can then use a ... |
1,765,946 | <p>$\newcommand{\Sig}{\Sigma}$
Let $\Sig$ be a diagonal matrix with strictly positive entries on the diagonal. Define $V=\{B \in M_n\mid B\Sig +\Sig B^T=\Sig B +B^T \Sig \}$ (where $M_n$ is the vector space of $n \times n$ real matrices).</p>
<p><strong>What is the dimension of $V$? Is there a "nice" basis for it?</st... | b2coutts | 335,797 | <p>Let $\lambda_1, \dots, \lambda_n$ be the diagonal entries of $\Sigma$; note that $\Sigma$ is invertible, and the diagonal entries of $\Sigma$ are $\lambda_1^{-1}, \dots, \lambda_n^{-1}$. Now, we can rewrite your condition on $B$ as $B-B^T = \Sigma^{-1}(B-B^T)\Sigma$.</p>
<p>Let $B \in M_n$. For $1 \leq i,j \leq n$,... |
945,334 | <p>Here is a lemma whose proof is as under:</p>
<blockquote>
<p>If $S \in L(X,Y)$ and lim$_{r \to 0}\frac{\|Sr\|}{\|r\|}=0$,then $S=0$.</p>
</blockquote>
<p>Proof:</p>
<p>The condition lim$_{r \to 0}\Big(\frac{\|Sr\|}{\|r\|}\Big)=0$means that for each $\epsilon \gt 0$ there is a $\delta \gt 0$ such that
... | Petite Etincelle | 100,564 | <p>Suppose $\{x_n\}$ is increasing and has a subsequence $\{x_{n_k}\}$ which converges to $L$. We will prove that $\{x_n\}$ itself converges to $L$.</p>
<p>For any $\epsilon > 0$, we want to find an integer $N_\epsilon$ such that $|x_n - L| \leq \epsilon$ for any $n \geq N_\epsilon$. </p>
<p>Since $\{x_{n_k}\}$ is... |
945,334 | <p>Here is a lemma whose proof is as under:</p>
<blockquote>
<p>If $S \in L(X,Y)$ and lim$_{r \to 0}\frac{\|Sr\|}{\|r\|}=0$,then $S=0$.</p>
</blockquote>
<p>Proof:</p>
<p>The condition lim$_{r \to 0}\Big(\frac{\|Sr\|}{\|r\|}\Big)=0$means that for each $\epsilon \gt 0$ there is a $\delta \gt 0$ such that
... | Rafael Vergnaud | 301,493 | <p>Suppose <span class="math-container">$(x_n)$</span> is monotone and contains a convergent subsequence <span class="math-container">$(x_{n_i}).$</span> </p>
<p>Given that <span class="math-container">$(x_{n_i})$</span> is convergent, it is bounded above by some upper bound <span class="math-container">$b \in \mathbb... |
1,098,253 | <p>I have got some trouble with proving that for $x\neq 0$:
$$
\frac{\arctan x}{x }< 1
$$
I tried doing something like $x = \tan t$ and playing with this with no success.</p>
| idm | 167,226 | <p>$$\frac{\arctan x}{x}<1\iff \begin{cases}\arctan x<x&if\ x>0\\ \arctan x> x&if\ x<0\end{cases}$$</p>
<p>$$(\arctan x-x)'=\underbrace{\frac{1}{x^2+1}}_{< 1\ if\ x\neq 0}-1< 0$$ therefore $$f:x\longmapsto\arctan x -x$$ is strictly decrasing. Moreover $$\arctan 0-0=0,$$ therefore $(\arctan... |
298,912 | <p>I was reading some basic information from Wiki about category theory and honestly speaking I have a very weak knowledge about it. As it sounds interesting, I will go into the theory to learn more if it is actually useful in practice.</p>
<p>My question is to know if category theory has some applications in practice... | Manos | 11,921 | <p>Category theory is far from the engineering textbook level, for now. On the research level, there are a lot of instances where category theory is applied in engineering context, from electrical to biomedical engineering. Beware though: these usually come from people who try to apply category theory, rather than from... |
298,912 | <p>I was reading some basic information from Wiki about category theory and honestly speaking I have a very weak knowledge about it. As it sounds interesting, I will go into the theory to learn more if it is actually useful in practice.</p>
<p>My question is to know if category theory has some applications in practice... | Idempotent | 159,957 | <p>This too is a late answer, but in case anyone is still interested, <a href="https://golem.ph.utexas.edu/category/2007/11/category_theory_and_biology.html" rel="noreferrer">here</a> is a discussion with links about the use of category theory in biology/bioinformatics and genetics. Also, while not specifically a book... |
1,765 | <p>Can anyone explain to me this behaviour? I've been having more than a couple of similar doubts these last weeks. </p>
<p>For example</p>
<pre><code>f[_?NumericQ] := 8;
</code></pre>
<p>Now, if I do</p>
<pre><code>With[{a = f[a]}, HoldForm@Block[{NumericQ = True &}, a]]
</code></pre>
<p>I get</p>
<pre><code... | FJRA | 495 | <p>You can use <code>TracePrint</code> to try to understand how evolves the evaluation step by step:</p>
<pre><code>In[23]:= TracePrint[With[{b=f[a]},Block[{NumericQ=True&},b]]]
During evaluation of In[23]:= With[{b=f[a]},Block[{NumericQ=True&},b]]
During evaluation of In[23]:= With
During evaluation of In[... |
1,765 | <p>Can anyone explain to me this behaviour? I've been having more than a couple of similar doubts these last weeks. </p>
<p>For example</p>
<pre><code>f[_?NumericQ] := 8;
</code></pre>
<p>Now, if I do</p>
<pre><code>With[{a = f[a]}, HoldForm@Block[{NumericQ = True &}, a]]
</code></pre>
<p>I get</p>
<pre><code... | Szabolcs | 12 | <p>You were asking why</p>
<pre><code>f[_?NumericQ] := 8
With[{a = f[a]}, Block[{NumericQ = True &}, a]]
</code></pre>
<p>outputs <code>f[a]</code>.</p>
<p>This is because of <a href="http://reference.wolfram.com/mathematica/ref/Update.html">caching</a> of the result of <code>Condition</code>s and <code>Pattern... |
4,575,165 | <p>Let <span class="math-container">$\alpha$</span> be an arbitrary positive real number in:
<span class="math-container">$$
F_1 = \int_0^1 x^2 \left[ \int_{-1}^{+1} \frac{e^{-\alpha\sqrt{1+x^2+2xy}}(xy+1)}{(1+x^2+2xy)^{3/2}}dy\right]dx $$</span> <span class="math-container">$$
F_2 = \int_1^\infty x^2 \left[ \int_{-1}^... | Simeon Albert L.M. | 1,121,514 | <p>Define
<span class="math-container">$$ F(\alpha) = \int_0^{\infty} \int_0^{\pi} {\rm e}^{-\alpha r} \sin \theta \, \cos \theta \, {\rm d}\theta \, {\rm d}r $$</span></p>
<p>Integrating yields
<span class="math-container">$$ F(\alpha) = -\frac{1}{\alpha} \left. {\rm e}^{-\alpha r} \right|_{r=0}^\infty \; \cdot \; \fr... |
4,942 | <p>I'd like to control more aspects of a <code>DateListPlot</code>, for example: shading for weekend days, and/or indicators for daytime/nighttime areas. </p>
<p>By way of illustration, here's a simple example of a set of time data points (recent questions on mathematica.stackexchange):</p>
<pre><code>questions =
... | image_doctor | 776 | <p>Here is an alternative approach which uses colour coding of date labels to indicate Weekday/Weekend ( Green/Red ) and colour shade to indicate daytime/nightime ( Lighter/Darker ).</p>
<p>I've dropped some of the points and applied a minimum separation to help make the plot more legible in the limited display space ... |
28,892 | <p>I was searching on MathSciNet recently for a certain paper by two mathematicians. As I often do, I just typed in the names of the two authors, figuring that would give me a short enough list. My strategy was rather dramatically unsuccessful in this case: the two mathematicians I listed have written 80 papers toget... | Victor Protsak | 5,740 | <blockquote>
I.M.Gelfand and M.I.Graev: 119
</blockquote>
<p><b>Disclaimer:</b> For the purposes of this answer, the paper count is the number given by MathSciNet, which includes book translations.</p>
|
4,272,755 | <p>Calculate the triple integral using spherical coordinates: <span class="math-container">$\int_C z^2dxdydz$</span> where C is the region in <span class="math-container">$R^3$</span> described by <span class="math-container">$1 \le x^2+y^2+z^2 \le 4$</span></p>
<p>Here's what I have tried:</p>
<p>My computation for <s... | Joshua Pasa | 689,689 | <p>From the conditions you have stated above <span class="math-container">$C: 1 \leq x^2 + y^2 + z^2 \leq4$</span> by using a substitution we can see that <span class="math-container">$1 \leq r \leq 2$</span>. However there are no conditions on the other spherical components so we can see that
<span class="math-contain... |
3,760,450 | <p>I have a class in numerical mathematics, and I received several tasks I should answer. I am not a mathematician, and this is a bit out of my mind range, and I would be grateful for answers.
Question is as follows:</p>
<p>Let <span class="math-container">$x, y \in \Bbb{C}^n$</span> be arbitrary vectors. Show what can... | Community | -1 | <p><span class="math-container">$f(-x)=x^{2}-|x|=f(x)$</span></p>
<p>Hence <span class="math-container">$x^{2}-|x| \quad$</span> is an even function.</p>
<p>For <span class="math-container">$x<0$</span>
<span class="math-container">$f(x)=x^{2}+x$</span> (By symmetry we can plot the the total graph)</p>
<p>f(x) is a ... |
85,343 | <p>I am looking for a reference to study classical (i.e., not quantized) Yang-Mills theory. </p>
<p>Most of the sources I find focus on mathematical aspects of the theory, like Bleecker's book <em>Gauge theory and variational principles</em>, or Baez & Muniain's <em>Gauge fields, knots and gravity</em>.</p>
<p>Bu... | Jon | 19,520 | <p>I would add: Atiyah, Michael F. (1979), <em>Geometry of Yang–Mills fields</em> and then the book of the same author about gauge theories: Atiyah, Michael F. (1988e), <em>Collected works. Vol. 5 Gauge theories</em></p>
|
1,064,091 | <p>I am asked to generate 200 and 1000 points from a bi-variate normal mixture densities.
I am trying to understand the algorithm, not just the matlab code (I have to write it, not use an existing function). I found a code on mathworks: <a href="http://www.mathworks.com/help/matlab/ref/randn.html#bufqioz-2" rel="nofoll... | Suzu Hirose | 190,784 | <p>Let $f$ be the uniform distribution on $[0,1]$ then $P(S>x)=(1-x)$ and so according to your formula we should have $P(S>t)=\int_0^t P(S>x)dx=t-1/2t^2$. Plug in $t=1/3$ and you get $P(S>1/3)=1/3-1/18$. But clearly $P(S>1/3)=2/3$ so the equation is complete nonsense.</p>
|
226,449 | <p>Many counting formulas involving factorials can make sense for the case $n= 0$ if we define $0!=1 $; e.g., Catalan number and the number of trees with a given number of vetrices. Now here is my question:</p>
<blockquote>
<p>If $A$ is an associative and commutative ring, then we can define an
unary operation on ... | N. S. | 9,176 | <p>I think that the definition $0!=1$ is the one which makes most of the formulas work nicely.</p>
<p>For example $(n+1)!=n! (n+1)$ also works for $n=0$, as long as $0!=1$.</p>
<p>But most importantly, $\binom{n}{k}$ also works in the case $k=0$. Note that $\binom{n}{0}$ has to be $1$, if you want to have a nice form... |
300,253 | <p>I'm interested in invertible matrices that are built out of invertible sub-blocks. For example, four sub-blocks from $GL_n(F)$ (i.e. the group of $n \times n$ invertible matrices over a field $F$) can be assembled into a $2n \times 2n$ matrix, which may or may not be invertible.</p>
<p>Suppose that a $kn \times kn... | Michael Hardy | 11,667 | <p>"Did" has posted a guess. Here's another. Some processes $\{X_t\}_{t\ge0}$ satisfy $\operatorname{var}(X_t)=ct$ (Poisson processes, Wiener processes, many others). Possibly it could mean $c=1$.</p>
<p>(But if something like the <a href="http://en.wikipedia.org/wiki/Ornstein%E2%80%93Uhlenbeck_process" rel="nofoll... |
377,266 | <p>My question is very direct:</p>
<blockquote>
<p>What are the motivations for the name "jet"(subjet, superjet) in the context of viscosity solutions for second order fully nonlinear elliptic PDE?</p>
</blockquote>
<p>The definition of which can be seen in Crandall, Ishii, Lions:</p>
<p><em>Crandall, Michael... | YCor | 14,094 | <p>Strictly speaking, he answer is that there is no motivation for the name "jet" in the context of viscosity solutions for second order fully nonlinear elliptic PDE, because it was initially introduced in the more basic framework of differential calculus/geometry.</p>
<p>Still one can wonder about when and w... |
4,035,080 | <p>Let <span class="math-container">$E$</span> be a Polish space, and let <span class="math-container">$\mu$</span> be a measure on <span class="math-container">$E$</span>. Define the following properties:</p>
<ul>
<li><span class="math-container">$E$</span> is <span class="math-container">$\sigma$</span>-<em>compact</... | Henno Brandsma | 4,280 | <p>I think that indeed the idea from the common thread works:</p>
<p>Let <span class="math-container">$e_n, n \in \Bbb N$</span> be the standard orthonormal base of Hilbert space <span class="math-container">$\ell^2$</span>. Let <span class="math-container">$X = \{\lambda e_n \mid n \in \Bbb N, \lambda \in \Bbb R\}$</s... |
548,902 | <p>I need to find the limit:
$\mathop {\lim }\limits_{n \to \infty } {1 \over n}\left[ {{{(a + {1 \over n})}^2} + {{(a + {2 \over n})}^2} + ... + {{(a + {{n - 1} \over n})}^2}} \right]$ </p>
<p>any ideas here? I've tried to use "squeeze theorem" but with no luck.. </p>
| Daniel Robert-Nicoud | 60,713 | <p>Once you have shown that the function is continuous, you can certainly prove that the function is differentiable by showing that the partial derivatives agree. Indeed you have that the derivative of $f$ in direction $\mathbf{v}$ (a unit vector) at $(x,y)$ is given by
$$(\nabla f)_{(x,y)}\cdot\mathbf{v}$$
(you can tr... |
231,360 | <p>Let $A,B\subseteq\mathbb R^d$ with $A$ closed and $B$ open and such that $A\cap B\neq\emptyset$. Assume that there exists a sequence $(x_k)_{k\in\mathbb N}\subseteq\mathbb R^d$ conveging in Euclidean norm to some $x\in A\cap B$. Do we have $x_k\in A\cap B$ for some $k\in\mathbb N$? If $x$ is an interior point of $A$... | Harald Hanche-Olsen | 23,290 | <p>No. Take $A=\{0\}$ and $B=\mathbb{R}^n$, $x_k=(1/k,0,\ldots,0)$.</p>
<p>More generally, let $x$ be any non-interior point in $A$. For each $n$, pick $x_n$ with $|x_n-x|<1/n$, $x_n\notin A$, and $x_n\in B$ (the latter is automatic when $n$ is large enough).</p>
|
2,964,897 | <blockquote>
<p>Using Leibniz on <span class="math-container">$\sum_{n=1}^\infty \sin(\pi \sqrt{n^2+1})$</span></p>
</blockquote>
<p>So the question actually is how to rewrite <span class="math-container">$\sin(\pi\sqrt{n^2+1})$</span> in the form of <span class="math-container">$(-1)^n\times a_n$</span> so that I c... | marty cohen | 13,079 | <p>Let
<span class="math-container">$\sqrt{n^2+1}
=n+d(n)$</span>.
Then</p>
<p><span class="math-container">$\begin{array}\\
\sin(\pi\sqrt{n^2+1})
&=\sin(\pi(n+d(n))\\
&=\sin(\pi n)\cos(\pi d(n))
+\cos(\pi n)\sin(\pi d(n))\\
&=(-1)^n\sin(\pi d(n))\\
\end{array}
$</span></p>
<p>Also</p>
<p><span class="ma... |
3,910,623 | <p>There is a problem that appears in an interview<span class="math-container">$^\color{red}{\star}$</span> with <a href="https://en.wikipedia.org/wiki/Vladimir_Arnold" rel="nofollow noreferrer">Vladimir Arnol'd</a>.</p>
<blockquote>
<p>You take a spoon of wine from a barrel of wine, and you put it into your cup of tea... | C.F.G | 272,127 | <ol>
<li>First We have a <span class="math-container">$B_{wine}$</span> and a <span class="math-container">$C_{tea}$</span> and a <span class="math-container">$S$</span>poon</li>
<li>Now we have <span class="math-container">$B_{wine}-S_{wine}$</span> and <span class="math-container">$C_{tea}+S_{wine}$</span></li>
<li>T... |
2,247,900 | <p>Prove the following inequality</p>
<p>$$\ln \frac{\pi + 2}{2} \cdot \frac{2}{\pi} < \int \limits_0^{\pi/2} \frac{\sin\ x}{x^2 + x} < \ln \frac{\pi + 2}{2}$$</p>
<p>I can prove that $\frac{\sin\ x}{x^2 + x} < \frac{1}{x + 1} \ \forall x \in (0, +\infty) \Rightarrow \int \limits_0^{\pi/2} \frac{\sin\ x}{x^2... | DXT | 372,201 | <p>Using $$\frac{\sin x}{x}>\frac{2}{\pi}\forall x\in \left(0,\frac{\pi}{2}\right)$$</p>
<p><a href="https://math.stackexchange.com/questions/842978/proving-frac2-pi-x-le-sin-x-le-x-for-x-in-0-frac-pi-2">Proving $\frac2\pi x \le \sin x \le x$ for $x\in [0,\frac {\pi} 2]$</a></p>
|
160,169 | <p>Consider the following implementation of the complex square root:</p>
<pre><code>f[z_]:=Sqrt[(z - I)/(z + I)]*(z + I);
</code></pre>
<p>This implementation has branch points at $\lambda=\pm i$ and a (vertical) branch cut connecting them.</p>
<p>Then</p>
<pre><code>g[z_]:=Sinc[f[z]];
</code></pre>
<p>(recalling ... | Carl Woll | 45,431 | <p>I think it's worth reporting this issue to support. A workaround is to use something like:</p>
<pre><code>fSeries[e_, {z_,p_,n_}] := Series[
e /. z->z+p,
{z, 0, n}
] /. Verbatim[SeriesData][x_, 0, r__] :> SeriesData[x, p, r]
</code></pre>
<p>For your example:</p>
<pre><code>fSeries[Sinc[f[z]], {z, I... |
2,163,494 | <p>Let $f: A\to B; \ g,h:B\to A$ and $f\circ g = I_B$ and $h \circ f = I_A$</p>
<p>I want to simply state that for any function $f$ if $f \circ h = I_A$ then it must be that $h = f^{-1}$ but that seems incomplete to me. What can I do for fixing this?</p>
| E. Joseph | 288,138 | <p><strong>Yes</strong>, this is true.</p>
<p>Since $f$ is defined on $\mathbb R^n$, and $A\subset \mathbb R^n$, let's take $F$ a close subset containing $A$ (you can take $F=\overline A$ the closure of $A$).</p>
<p>Then since $f$ is continuous on $F$, $f(F)$ is bounded because $F$ is closed.</p>
<p>So $f$ is bounde... |
1,315,641 | <p>I rewrite $f(z)$ using partial fractions to get $f(z)=\frac{1}{z}+\frac{2}{1-2z}$.</p>
<p>We need powers of $\left(z-\frac{1}{2}\right)$
So how do I rewrite $\frac{1}{z}$?</p>
<p>So I rewrite it as $\frac{1}{\frac{1}{2}+\left(z-\frac{1}{2}\right)}$
And I write it as $2\left(z-\frac{1}{2}\right)$</p>
<p>How do I g... | Aleksandar | 240,930 | <p>The series is the following:</p>
<p>$\sum _{k=1}^\infty(-2)^{k}(z-\frac{1}{2})^{n}$.</p>
<p>That is the Laurent series for $f(z)$.</p>
<p>Hope this helped.</p>
<p>By the way:</p>
<p>Here is how to find the disk of convergence:</p>
<p>We'll find the pole of $f(z)$. To do that we must set:</p>
<p>$z(1-2z)=0$</p... |
3,348,780 | <p>I worked through <span class="math-container">$\int \frac{e^x}{(1-e^x)^2}dx$</span> using u-substitution, but my answer, <span class="math-container">$(1-e^x)^{-1}+C$</span> is incorrect. It should be <span class="math-container">$- \ln|1-e^x|+C$</span></p>
<blockquote>
<p><span class="math-container">$$\int \fra... | DonAntonio | 31,254 | <p>The result is false: in fact, </p>
<p><span class="math-container">$$\int\frac{e^x}{(1-e^x)^2}dx=\frac1{1-e^x} +C$$</span></p>
<p>This is: what you did is completely correct. There is not <span class="math-container">$\;\ln\;$</span> here.</p>
|
3,348,780 | <p>I worked through <span class="math-container">$\int \frac{e^x}{(1-e^x)^2}dx$</span> using u-substitution, but my answer, <span class="math-container">$(1-e^x)^{-1}+C$</span> is incorrect. It should be <span class="math-container">$- \ln|1-e^x|+C$</span></p>
<blockquote>
<p><span class="math-container">$$\int \fra... | mrtaurho | 537,079 | <p>Your answer is perfectly fine and agrees with what <a href="https://www.wolframalpha.com/input/?i=int+e%5Ex%2F%281-e%5Ex%29%5E2" rel="nofollow noreferrer">WolframAlpha</a> returns. You should give a source as the source seems to be wrong in its given answer.</p>
|
2,452,777 | <p>Let X be a topological space, $\mathcal{U} = \{U_\alpha\}_\alpha$ an open cover of $X$ and $\mathcal{F}$ a presheaf of abelian groups on $X$. Then one can define the Čech cohomology groups of $\mathcal{U}$ with values in $\mathcal{F}$:
\begin{equation}
\check{H}^k(\mathcal{U}, \mathcal{F})
\end{equation}
The Čech co... | paeolo | 406,116 | <p>I found my answer in <a href="http://www.seas.upenn.edu/~jean/sheaves-cohomology.pdf" rel="nofollow noreferrer">A Gentle Introduction to Homology, Cohomology, and. Sheaf Cohomology</a> by Jean Gallier and Jocelyn Quaintance, on page 238.
As they said: "Most textboook presentations of Čech cohomology ignore this subt... |
619,370 | <p>Let $C,Q$ is complex numbers Field and Rational number
Field,respectively,if $f(x),g(x)\in Q[x]$,</p>
<p>if $g(x)|f(x)$ on $C[x]$,show that
$$g(x)|f(x)$$ on $Q[x]$</p>
<p>My try: since
$g(x)|f(x)$,then we have
$$f(x)=g(x)h(x)$$
where $h(x)\in C[x]$.
Then I can't prove also have
$$g(x)|f(x)$$ on $Q[x]$.</p>
<p>... | Robert Lewis | 67,071 | <p>A generalization of the stated problem from $Q$ a subfield of $C$ to arbitrary fields $F \subset E$ is resolved by means of the following </p>
<p><em><strong>Proposition:</em></strong> Let $E$ and $F$ be fields with $F$ a subfield of $E$. Let $f(x), m(x) \in F[x]$ with $f(x) = q(x)m(x)$ in $E[x]$. Then in fact $... |
3,607,430 | <blockquote>
<p>Given that <span class="math-container">$a$</span>, <span class="math-container">$b$</span>, <span class="math-container">$c$</span> are the angles of a right-angled triangle, prove that:
<span class="math-container">$$\begin{align}
\sin a\sin b\sin(a-b) &+\sin b\sin c\sin(b-c)+\sin c\sin a\sin(... | Matt | 1,042,546 | <p>So I came across a similar issue and I couldn't find any answers which were satisfactory. So I dug up as much information as I could and I found a solution. Here goes...</p>
<p>We have a known point, <span class="math-container">$(6,-1)$</span>. For any line that passes through this point, it must have an x-intercep... |
2,485,997 | <p><a href="https://i.stack.imgur.com/OkhoU.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/OkhoU.png" alt="enter image description here"></a></p>
<p>Question: What if we consider (1,4,5) and (1,5,4) as non-distinct possibilities, then what should we do?</p>
<p>$${{9}\choose{2}}-2\cdot\frac{{9}\cho... | N. F. Taussig | 173,070 | <p>Since $10$ is not a multiple of $3$, it is not possible for all three numbers to be the same.</p>
<p>However, it is possible for two numbers of the same. Since each number is positive, the repeated number must be $1$, $2$, $3$, or $4$. There are $\binom{3}{2}$ ways to choose the locations of the repeated number. ... |
1,722,443 | <p>I am unsure how to solve the following problem. I was able to find similar questions, but had trouble understanding them since they did not show full solutions.</p>
<p>The question:</p>
<p>Find ALL solutions (between $1$ & $40$) to the equation $25x \equiv 10 \pmod{40}$.</p>
| Peter | 82,961 | <p>It suffices to solve the equation $25x\equiv 10\ (\ mod\ 8\ )$ because modulo $5$, the equation holds no matter what $x$ is.</p>
<p>This gives $x\equiv 2\ (\ mod\ 8\ )$.</p>
<p>I think you can easily find out the solutions now.</p>
|
159,965 | <blockquote>
<p>Find limits of a function $f:\mathbb{R}^3\rightarrow \mathbb{R}$ given by formula $f(x,y,x)=x+y+z$ on set $M=\left\{ (x,y,z)\in\mathbb{R}^3:x^2+y^2\le z\le 1 \right\}$. Does $f$ reaches all its limits?</p>
</blockquote>
<p>To answer the last question I need to know if $M$ is a closed and bounded set.... | Brian M. Scott | 12,042 | <p>Here’s an approach that minimizes the calculus and relies mostly on the geometry.</p>
<p>Let $R$ be the region in the $xz$-plane bounded by the curves $z=x^2$ and $z=1$; $M$ is the solid of revolution obtained by revolving $R$ about the $z$-axis. $R$ is a closed set in the plane, so $M$ is closed, and the minimum a... |
103,970 | <p>Here's a very bizarre inconsistency I've just struggled with and I'm wondering why it exists or if I'm missing something.</p>
<p>I have some noisy data and I wish to make a framed plot of the data but allow the data to extend outside the vertical limits of the frame (for stylistic reasons). Like so:</p>
<pre><code... | Alexey Popkov | 280 | <p>It is another gedanken functionality in <em>basic</em> plotting functions. Using new in version 10 <code>Region*</code> functionality here is a workaround:</p>
<pre><code>xs = Range[0, 1, 0.005];
data = Transpose[{xs, Sin[Pi xs]^2 + 0.05 RandomReal[{-1, 1}, Length[xs]]}];
inf = 10;
RegionPlot[RegionIntersection[Li... |
103,970 | <p>Here's a very bizarre inconsistency I've just struggled with and I'm wondering why it exists or if I'm missing something.</p>
<p>I have some noisy data and I wish to make a framed plot of the data but allow the data to extend outside the vertical limits of the frame (for stylistic reasons). Like so:</p>
<pre><code... | Edmund | 19,542 | <p>When using <code>PlotRange -> s</code> you are only specifying the range on the y-axis. Look at the <strong>Details</strong> section of the <a href="http://reference.wolfram.com/language/ref/PlotRange.html" rel="nofollow noreferrer"><code>PlotRange</code> documentation</a>. This is why only the y-axis range is ... |
133,794 | <p>Prove the existence of infinite number of infinite cardinals </p>
<p>1) $\alpha$, such that $\alpha<\alpha^\aleph$ </p>
<p>2) $\beta$, such that $\beta=\beta^\aleph$</p>
| Asaf Karagila | 622 | <p>Using $\aleph$ as a general cardinal might be ambiguous (there are places where it is used particularly for $2^{\aleph_0}$), the answer remains the same regardless to the intended use of $\aleph$.</p>
<ol>
<li><p>Recall that for every infinite $\kappa$ we have $\kappa<\kappa^{\operatorname{cf}(\kappa)}$. Simply s... |
2,333,857 | <p>I have this problem that I have worked out. Will someone check it for me? I feel like it is not correct. Thank you!</p>
<p>Rotate the graph of the ellipse about the $x$-axis to form an ellipsoid. Calculate the precise surface area of the ellipsoid. </p>
<p>$$\left(\frac{x}{3}\right)^{2}+\left(\frac{y}{2}\right)^{... | orlp | 5,558 | <p>If you really want to be formal you can say that the domain of $f(x)$ is $\{x \in \mathbb{Z} \mid -2 \leq x \leq 3\}$.</p>
|
220,139 | <p>Is there a way to either make <code>FindMinimum</code> do an exact computation or <code>Minimize</code> find also the local minima? Or other ideas to find local minima exactly?</p>
<p><strong>Example:</strong> find all local minima (exact values, not approximations) of <span class="math-container">$f(x,y)=x^2 − x +... | Rom38 | 10,455 | <p>The exact minimization of a function can be done using <code>Minimize</code>:</p>
<pre><code> f[x_, y_] := x^2 - x + 2 y^2
sol = Minimize[f[x, y], {x, y}]
(*{-(1/4), {x -> 1/2, y -> 0}} <= output of Minimize*)
Show[
Plot3D[f[x, y], {x, y} \[Element] Disk[{0, 0}, 1]],
Graphics3D[{Red, PointSi... |
239,863 | <p>I've to study this series:</p>
<p>$$\sum_{n=1}^\infty e^{\sqrt n\,x}$$ </p>
<p>My teacher wrote that with the asymptotic comparison with this series:</p>
<p>$$\sum_{n=1}^\infty\frac{1}{n^2}$$<br>
My series converges for every </p>
<p>$$x<0$$</p>
<p>I don't understand the motivation, hoping for someone to... | Davide Giraudo | 9,849 | <p>Let $L\in(\ell^p)'$ a linear continuous functional. We can assume that $x_n^{(k)}\to 0$ for all $k$. $L$ can be represent by an element $v$ of $\ell^q$, where $p^{-1}+q^{-1}=1$. Fix $\delta>0$ and $v'$ of finite support such that $\lVert v-v'\rVert_{\ell^q}\leqslant \delta$. Then
$$\lVert L(x_n)\rVert\leqslant\d... |
3,547,989 | <p><strong>An ordinary deck of cards is dealt randomly to four players so that each player receives 13 cards.
Find the probability that each player is dealt exactly one ace.</strong></p>
<p>I was wondering if I could compare this scenario with the following:
Given 4 indistinguishable balls and 4 distinguishable boxes,... | Saaqib Mahmood | 59,734 | <p>Let us put
<span class="math-container">$$ A \colon= X \left(X^\prime X\right)^{-1} X^\prime. $$</span>
Then we note that
<span class="math-container">$$
\begin{align}
A^2 &= AA \\
&= \left[ X \left(X^\prime X\right)^{-1} X^\prime \right] \left[ X \left(X^\prime X\right)^{-1} X^\prime \right] \\
&= X \l... |
3,335,892 | <p>If I have two injective functions <span class="math-container">$f : A \to B$</span> and <span class="math-container">$g : B \to A$</span>, as Schröder-Bernstein (SB) says, then there is a function <span class="math-container">$h : A \to B$</span> which is bijective.</p>
<p>As for a proof, my reasoning goes somethin... | BallBoy | 512,865 | <p>The conceptual order here must be a little different than you present it. If <span class="math-container">$A$</span> and <span class="math-container">$B$</span> are <em>finite</em> sets, then we can conclude from <span class="math-container">$|A|\leq|B|$</span> and <span class="math-container">$|B|\leq|A|$</span> th... |
2,864,585 | <p>I tried to calculate the Hessian matrix of linear least squares problem (L-2 norm), in particular:</p>
<p>$$f(x) = \|AX - B \|_2$$
where $f:{\rm I\!R}^{11\times 2}\rightarrow {\rm I\!R}$</p>
<p>Can someone help me?<br>
Thanks a lot.</p>
| Mauricio Cele Lopez Belon | 485,657 | <p>Calculate first the gradient vector: use the chain rule and calculate the partial derivatives of $f(x)$ w.r.t $x \in R^n$. You will get a function that eats a vector and produce other "vector" $g(x) \in R^n$ (well this is an abuse of notation and terminology, $g(x)$ produces a vector of functions not a vector in $R^... |
4,627,334 | <p>To my understanding that a primitive triple <span class="math-container">$x$</span> and <span class="math-container">$y$</span> can be written as <span class="math-container">$x = q^2 - p^2$</span> while <span class="math-container">$y=2pq$</span> for relatively prime opposite parity <span class="math-container">$q ... | Dstarred | 955,900 | <p>We start off with the fact that any pythagorean triples can only consist of all even numbers <strong>or</strong> <span class="math-container">$2$</span> odd numbers and <span class="math-container">$1$</span> even number.</p>
<p>Let the numbers are denoted <span class="math-container">$x, y, z \in \mathbb{N}$</span>... |
4,302,855 | <p>I have tried setting up multiple systems of equations using many known volumes but I always seem to come up short. My last attempt was a hollow cylinder but that leaves you with three unknowns in only two sim. equations (for V and S.A). Can anyone help?</p>
| bubba | 31,744 | <p>You can use the basic idea described by @PC1, based on Pappus’ theorems. But use a cone instead of a torus. I think that will work.</p>
|
2,308,770 | <p>I have an equation of the form $A*i*j + B*i +C*j = N$
where I have the values of $A,B,C$ and $N$ and I want to solve for integer values of $i$ and $j$.</p>
<p>How would I approach this? I could try trial and error but the numbers I'm working with are relatively large (eg $>10^{40}$). But I'm also happy to work ... | Michael Burr | 86,421 | <p>Here's an idea, not a fully formed approach.</p>
<p>You wish to solve
$$
Aij+Bi+Cj=N.
$$</p>
<p>If we complete the square on the LHS, we get
$$
A\left(i+\frac{C}{A}\right)\left(j+\frac{B}{A}\right)-\frac{BC}{A}=N.
$$</p>
<p>Multiplying through by $A$, we get
$$
(Ai+C)(Aj+B)=AN+BC.
$$</p>
<p>Therefore, both $(Ai... |
1,817,542 | <p><strong>Problem:</strong> Let $(X, Y)$ be uniformly distributed on the unit disk $\{ (x,y) : x^2 + y^2 \le 1\}$. Let $R = \sqrt{X^2 + Y^2}$. Find the CDF and PDF of $R$.</p>
<p><strong>Attempted Solution:</strong> First note that $r \in R = \sqrt{X^2 + Y^2}$ represents a point on $\mathbb{R}^2$ with radius $r$ abou... | Community | -1 | <p>Your reasoning is incorrect.
First of all, by the definition of CDF,
$$
F_R(r)=P(R\leq r)=P(X^2+Y^2\leq r^2)\neq 0
$$
when $0<r<1$. </p>
<p>Second, for a probability density $f_R(r)$, one must have
$$\int_{\mathbb{R}} f_R(r)\ dr=1$$
which contradicts your calculation. </p>
|
253,359 | <p>I'm trying to prove by induction the following statement without success:<br>
$$\forall n \ge 2, \;\forall d \ge 2 : d \mid n(n+1)(n+2)...(n+d-1) $$</p>
<p>For the base case: $n = 2$, $d = 2$<br>
$2\mid 2(2+1)$ which is true.<br></p>
<p>Now, the confusion begins! I assume I would need to use the second induction p... | Bill Dubuque | 242 | <p><strong>Hint</strong> $\ $ Any sequence of $\rm\,d\,$ consecutive naturals has an element divisible by $\rm\,d.\,$ This has a very simple proof by induction: shifting such a sequence by one does not change its set of remainders mod $\rm\,d,\,$ since it simply replaces the old least element $\rm\:\color{#C00}n\:$ b... |
306,011 | <p>Does anyone have a proof for $$\int_0^{\infty}\frac{\sin(x^2)}{x^2}\,dx=\sqrt{\frac{\pi}{2}}.$$
I tried to get it from contour integrating $$\frac{e^{iz^2}-1}{z^2},$$ but failed.
Thanks.</p>
| Felix Marin | 85,343 | <p>$\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
\... |
1,735,910 | <p>In <a href="https://www.youtube.com/watch?v=aHU-L3BLd_w">a recent video</a> the legendary Matt Parker claimed he kept flipping a two-sided (fair) coin untill he scored a sequence of ten consecutive 'switch flips', i.e. letting $T$ denote a tail and $H$ a head, then a sequence of ten switch flips is defined to be eit... | paw88789 | 147,810 | <p>Intuitively it seems like you should bet on this event occurring in the first 10 flips. </p>
<p>Reasoning: If you pick any particular 10 flips you have a $\frac{1}{2^9}$ probability of those particular ten flips being a sequence of interest. </p>
<p>However, we specifically want the first such sequence, which me... |
2,814,703 | <p>I am reading <a href="https://en.wikipedia.org/wiki/Lower_limit_topology" rel="nofollow noreferrer">lower limit topology</a> on Wikipedia, which states that the lower limit topology </p>
<blockquote>
<p>[...] is the topology generated by the basis of all half-open intervals $[a,b)$, where a and b are real numbers... | nonuser | 463,553 | <p>If $M = (a,b)\cap \mathbb{Q}$ then $$(a,b) = \bigcup _{c\in M} [c,b)$$</p>
|
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