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 |
|---|---|---|---|---|
1,736,341 | <p>As stated in the title: Find the eigenvalues and eigenfunctions for $y''+\lambda y=0$, where $y'(1)=0$ and $y'(2)=0$.</p>
<p>So I have already eliminated the cases for $\lambda=0$ and $\lambda<0$ and I'm focused on the case for $\lambda>0$ now.
The characteristic equation for this case is $r^2+\lambda=0$, so... | user5713492 | 316,404 | <p>If $y_n(x)=\sin(n\pi x)$, then $y_n^{\prime}(1)=(-1)^n\cdot n\pi$, so that's a problem. Let's go back and eliminate cases. If $\lambda<0$, then the solutions are
$$y(x)=k_1\cosh(\sqrt{-\lambda}(x-1))+k_2\sinh(\sqrt{-\lambda}(x-1))$$
$$y^{\prime}(x)=\sqrt{-\lambda}k_1\sinh(\sqrt{-\lambda}(x-1))+\sqrt{-\lambda}k_2\... |
233,075 | <p>I am trying to solve the following equation in the Natural Numbers, with the condition <span class="math-container">$a\ge1$</span>, <span class="math-container">$b\ge1$</span>, and <span class="math-container">$r\ge3$</span>:</p>
<p><span class="math-container">$$\frac{a(a + 3)(a(r - 5) + (12 - r))}{9}=\frac{b (9 + ... | Roman | 26,598 | <p>Much faster to solve for <span class="math-container">$r$</span> and check that it's an integer: (it's always rational; no square-roots involved)</p>
<pre><code>Solve[(a(a+3)(a(r-5)+(12-r)))/9 == (b(9+b(-14+r)-r))/3, r] // FullSimplify
(* {{r -> (a(3+a)(-12+5a)+3(9-14b)b)/((-1+a)a(3+a)-3(-1+b)b)}} *)
R = ... |
377,925 | <p>This post comes from the suggestion of <a href="https://mathoverflow.net/users/18698/joel-moreira">Joel Moreira</a> in a <a href="https://mathoverflow.net/questions/377706/an-alternative-to-continued-fraction-and-applications#comment958452_377706">comment</a> on <a href="https://mathoverflow.net/q/377706/34538">An a... | Ben Burns | 51,771 | <p>I want to leave a few elementary comments, maybe they will be helpful.</p>
<p>The question asks about recurrence relation</p>
<p><span class="math-container">$$
u_{n+1}= \lfloor u_n \rfloor (u_n − \lfloor u_n \rfloor + 1)
$$</span></p>
<p>Suppose you write rational <span class="math-container">$u$</span> in terms of... |
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>
| George Tomlinson | 92,587 | <p>The base case is correct. </p>
<p>Inductive step:</p>
<p>Assume that the result is true for $n = k$, which is to say that everybody in a group of $k$ people has the same hair colour. </p>
<p>For the proof to work, you now have to prove that $P(k)$ implies $P(k+1)$, but it doesn't. Just because it's true that fo... |
2,692,341 | <blockquote>
<p>Is it possible for a commutative ring with 1 to be isomorphic to the
power series ring over itself? $$R \cong R[[x]].$$ Bonus marks if you
can answer for $n$ variables more generally, $R \cong R[[x_1, \dots, x_n]]$.</p>
</blockquote>
<p>Obviously, $R$ would have to be huge (infinite Krull dimensi... | Pierre-Yves Gaillard | 660 | <p>As I said in the comments, the zero ring is a first example. Another example is $A[[X_1,X_2,...]]$ where $A$ is a commutative ring with one and the $X_i$ are indeterminates.</p>
<p><strong>Edit.</strong> Here is a brief reminder.</p>
<p>If $A$ is a commutative ring with one, if $(X_i)_{i\in I}$ is a family of inde... |
2,692,341 | <blockquote>
<p>Is it possible for a commutative ring with 1 to be isomorphic to the
power series ring over itself? $$R \cong R[[x]].$$ Bonus marks if you
can answer for $n$ variables more generally, $R \cong R[[x_1, \dots, x_n]]$.</p>
</blockquote>
<p>Obviously, $R$ would have to be huge (infinite Krull dimensi... | jgon | 90,543 | <p>Let $M$ be a monoid which is "locally finite" by which we mean, for all $m\in M$, $\{(a,b)\in M\times M : ab =m \}$ is finite.</p>
<p>For such a monoid, for a ring $R$, we can make $R^M$ into a ring with the convolution product instead of $R^{\oplus M}$. Lets denote $R^M$ with the convolution product by $R[[M]]$. N... |
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... | user8268 | 8,268 | <p>You know that convex subsets of $\mathbb{R}^n$ have trivial fundamental group. Check the proof to see that it proves also this: if $X\subset \mathbb{R}^n$ and $a\in X$ are such that for every $b\in X$ the segment connecting $a$ and $b$ is in $X$ (i.e. if $X$ is "star-shaped") then $\pi_1(X,a)$ is trivial. (Since $X$... |
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... | Thomas Andrews | 7,933 | <p>If $T$ is your space, then $T$ has the property that if $(x,y,z)\in T$ and $t >0$ then $(x,y,z+t)\in T$.</p>
<p>This lets you find a homotopy between any loop in $T$ to a loop in $T_0=\{(x,y,z): z>=1\}$ But $T_0$ is convex, so it is simply connected.</p>
|
2,331,191 | <p>Use either direct proof, proof by contrapositive, or proof by contradiction.</p>
<p>Using proof by contradiction method</p>
<blockquote>
<p>Assume n is a perfect square and n+3 is a perfect square (proof by
contradiction)</p>
<p>There exists integers x and y such that <span class="math-container">$n = x^2$</span> an... | Ross Millikan | 1,827 | <p>You have proved that if $n$ and $n+3$ are perfect squares, $n=1=1^2, n+3=4=2^2$ The statement you were trying to prove is false. Maybe you were also given $n \gt 1$?</p>
|
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... | Dave Rusin | 463 | <p>You can understand the squares purely algebraically: let $R = F_2[x_1, x_2, \ldots, x_n]$ be a polynomial ring over the field $F_2$ of $2$ elements.
A ring homomorphism $f$ on $R$ is completely determined by the values of $f(x_i)$, and those values are unrestricted, right? So define $f \colon R \to R$ by $... |
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... | Andrew Ranicki | 732 | <p>Section 4L of Hatcher's <a href="http://www.math.cornell.edu/~hatcher/AT/ATpage.html">Algebraic Topology</a> is a downloadable reference.</p>
|
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 $a\i... | Robert Bruner | 6,872 | <p>I give an intuitive account of the origin of Steenrod operations in cohomology, Dyer-Lashof operations in homology of infinite loop spaces, and Steenrod operations in the cohomology of cocommutative Hopf algebras in Chapter 3 of my (unfinished)
<a href="http://www.rrb.wayne.edu/papers/adams.pdf" rel="nofollow norefe... |
2,153,421 | <p>I want to find curvatures and torsions for the following curves but get stuck with their natural parametrizations ($s$ is natural if $|\dot{\gamma}(s)| = 1$). Can anyone help me?</p>
<p>(a) $e^t(\cos t,\sin t,1)$</p>
<p>(b) $(t^3+t,t^3-t,\sqrt{3}t^2)$</p>
<p>(c) $3x^2+15y^2=1, z=xy$</p>
<p><strong>Update</strong... | Ted Shifrin | 71,348 | <p>You do not <em>need</em> an arclength or "natural" parametrization to do these computations. It's just a matter of using the chain rule. I've posted about this numerous times on here. Rather than write out another lecture, I'll just refer you to my (free) <a href="https://math.franklin.uga.edu/sites/default/files/in... |
4,069,053 | <blockquote>
<p>If a finite field has characteristic 2 why is every element a square?</p>
</blockquote>
<p>I've attempted this problem by calculating the square root as follows: <span class="math-container">$\mathbb{F}$</span> has <span class="math-container">$q = 2^m$</span> elements, so if <span class="math-container... | Aryaman Maithani | 427,810 | <p>Consider the function <span class="math-container">$f : \Bbb F \to \Bbb F$</span> given by <span class="math-container">$f(x) = x^2$</span>.<br />
Then, <span class="math-container">$$f(x) = f(y) \iff x^2 = y^2 \iff x^2 - y^2 = 0 \color{red}{\iff} (x - y)^2 = 0 \iff x - y = 0.$$</span>
The <span class="math-containe... |
2,232,390 | <p>$X$ is a subset of $\mathbb{R}$. It contains $\sqrt{3}$ and $\sqrt{2}$ and is closed under addition, subtraction and multiplication.</p>
<ol>
<li>Prove that $X$ contains $\sqrt{8}$.</li>
<li>Prove that $X$ contains $1$.</li>
<li>Prove that $X$ contains $\frac{1}{\sqrt{2}+1}$.</li>
<li>Is it true that $X$ is necessa... | Ian | 83,396 | <p>Heuristically, if $X_1,X_2$ are independent and have one sign (as in this case) you should expect $f_{X_1/X_2}(r)=\int_{-\infty}^\infty f_{X_1}(rx) f_{X_2}(x) dx$, because if $X_1/X_2=r$ and $X_2=x$ then $X_1=rx$. </p>
<p>Plugging in the relevant densities in this case gives $\int_0^\infty 9 e^{-3(rx)} e^{-3x} dx$.... |
3,355,544 | <p><a href="https://i.stack.imgur.com/oYf7f.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/oYf7f.jpg" alt="enter image description here"></a></p>
<p>Given the following equations:</p>
<p><span class="math-container">$$x=t^2$$</span></p>
<p><span class="math-container">$$y=t^2$$</span></p>
<p><sp... | Intelligenti pauca | 255,730 | <p>If a curve <span class="math-container">$u(t)$</span> lies on a plane, then the plane defined by any three points <span class="math-container">$A$</span>, <span class="math-container">$B$</span>, <span class="math-container">$C$</span> on the curve must be always the same, i.e. the vector product
<span class="math-... |
2,269,917 | <p>I've come across this question in a university exam paper. It's causing me a huge headache due to the fact that it goes from one vector space to another ($\def\R{\Bbb R}\R^3 \to \R^2$), otherwise it would be fairly standard. If anyone could shed some light on what I'm missing it would be much appreciated.</p>
<p>A4... | egreg | 62,967 | <p>A standard method is to begin with the matrix representation of $T$: denoting by $e_1$, $e_2$ and $e_3$ the vectors in the standard basis of $\mathbb{R}^3$, we have
$$
T(e_1)=\begin{bmatrix}2\\1\end{bmatrix}
\qquad
T(e_2)=\begin{bmatrix}3\\-2\end{bmatrix}
\qquad
T(e_3)=\begin{bmatrix}-5\\1\end{bmatrix}
$$
so we have... |
569,927 | <p>$p^{\; \left\lfloor \sqrt{p} \right\rfloor}\; -\; q^{\; \left\lfloor \sqrt{q} \right\rfloor}\; =\; 999$</p>
<p>How do you find positive integer solutions to this equation?</p>
| hmakholm left over Monica | 14,366 | <p>Calculate $x^{\lfloor x^{1/2}\rfloor}$ for the first dozen or so integers. Soon the function starts to increase so quickly that there's no hope of getting 999 as the difference between its values. Check whether any of the values you get until that happens happen to be an earlier value plus 999.</p>
<p>(There are tw... |
699,872 | <p>Compute the double integral of f(x, y) = 8 x^3 y over the given shaded domain in the following Figure :</p>
<p><img src="https://i.stack.imgur.com/RHimo.png" alt="enter image description here"></p>
<p>I am confused about the limits,
my work so far for the limits is x from 0 to 4 and y from 0 to 2 </p>
| MPW | 113,214 | <p>Note that $f$ doesn't enter into it.</p>
<p>It's easiest in this case to let $y$ range between $0$ and $2$. Then for a fixed $y$, $x$ ranges between $y$ and $4$.</p>
<p>You could do it the other way around, but you would have to break it into two cases. First let $x$ range between $0$ and $2$; for a fixed $x$ in t... |
699,872 | <p>Compute the double integral of f(x, y) = 8 x^3 y over the given shaded domain in the following Figure :</p>
<p><img src="https://i.stack.imgur.com/RHimo.png" alt="enter image description here"></p>
<p>I am confused about the limits,
my work so far for the limits is x from 0 to 4 and y from 0 to 2 </p>
| David | 119,775 | <p>The important thing to realise about an iterated integral is that there are implied brackets:
$$\int\!\!\int f(x,y)\,dx\,dy=\int\left(\int f(x,y)\,dx\right)dy\ .$$
To evaluate the integral we have to do the part in brackets first. This means integrating with respect to $x$, taking $y$ to be a fixed value. Therefor... |
13,030 | <p>At work, we were discussing when is it the best time to change to winter tires for bikes and/or cars.</p>
<p>Using <code>WeatherData[]</code> and <code>DateListPlot[]</code>, it was fairly straightforward for me to create the diagram below:</p>
<p><img src="https://i.stack.imgur.com/Y5wNT.png" alt="Mean temperatur... | chris | 1,089 | <p>Following @J.M. if you are after the likelihood of icy days as a function of month</p>
<pre><code>ff[x_] := Select[cityTemp, #[[1, 2]] == x &] // (Count[#, {d_List, t_?Negative}]/Length[#]) &
{Range[12] - 1/2, ff /@ Range[12]} // Transpose //
ListLinePlot[#, InterpolationOrder -> 0,AxesLabel -> {"Mo... |
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 $... | egreg | 62,967 | <p>That's the intuition; but the powers are properly defined by recursion from
$$
a^0=1,\quad
a^{n+1}=a\cdot a^n
$$
so $a^0=1$ <em>by definition</em>. It is a sound definition, because it agrees with the property $a^{m+n}=a^m\cdot a^n$ for any natural $m$ and $n$.</p>
<hr>
<p>Think to what you do when you have a heap... |
126,074 | <p>I would like to know some applications of Anick's resolution in non-commutative algebras.</p>
| Noah Schweber | 8,133 | <p>The first paragraph of David Anick's paper, "On the Homology of Associative Algebras" (<a href="http://www.jstor.org/stable/2000383" rel="nofollow">http://www.jstor.org/stable/2000383</a>): </p>
<blockquote>
<p>Let $k$ be a field and let $G$ be an associative augmented $k$-algebra. For many purposes one wishes to... |
1,407,700 | <p>I am stuck on solving the following systems of equations with 3 variables. The textbook asks to use the addition method so can we please stick to that.</p>
<p>${5x -y = 3}$</p>
<p>${3x + z = 11}$</p>
<p>${y - 2z = -3}$</p>
<p>I am used to systems of equations where each equation has at least one instance of the... | pisco | 257,943 | <p>It is not uniformly continuous on $(0,+\infty)$. We apply the following simple theorem, which immediately says the function not uniform continuous on $[1,+\infty)$</p>
<hr>
<p><strong>Theorem</strong>: Suppose $f(x)$ is uniformly continuous on $[1,+\infty)$, then $f(x)\over x$ is bounded on $[1,+\infty)$.</p>
<p>... |
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... | Jonathan Hebert | 120,932 | <p>There are ultimately a lot of different ways we use the term "linear". In the context of linear transformations, $L(x) = ax+b$ is linear if and only if $b=0$. Perhaps you could explore a proof for that instead.</p>
|
1,402,953 | <ol>
<li>How can one intuitively understand the definition of a bilinear map? Is there some way of looking at it geometrically? I found the following definition:</li>
</ol>
<p>Let $\mathit{A}$,$\mathit{B}$,$\mathit{C}$ be vector spaces. A map $f:\mathit{A}\times \mathit{B}\to C$ is said to be bilinear if for each fixe... | Luckyluck63 | 206,839 | <p>For a definition of a bilinear algorithm, please look at:</p>
<p>Michael Kaminski, David G Kirkpatrick, Nader H Bshouty, Addition requirements for matrix and transposed matrix products, Journal of Algorithms, Volume 9, Issue 3, 1988, Pages 354-364.</p>
<p>(<a href="http://www.sciencedirect.com/science/article/pii/... |
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... | Noah Schweber | 28,111 | <p>"$\vee$" by itself is <strong>inclusive or</strong>. To express <strong>exclusive or</strong>, we need to write a more complicated expression.</p>
<p>Remember that "$A$ xor $B$" just means "$A$ or $B$, but not $A$ and $B$." With that in mind, "$A$ xor $B$" can be expressed as $$(A\vee B)\wedge\neg(A\wedge B).$$ Ano... |
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... | N. S. | 9,176 | <p>It is easy to see that $2^{2n}-1 \equiv 0 \mod 3$. This shows that for $n \geq 2$, $2^{2n}-1$ is never prime.</p>
<p>Also, since $2^3 \equiv 1 \mod 7$ it follows that $2^{3n} -1$ is always divisible by $7$.</p>
<p>Hence $2^{6n+2}-1, 2^{6n+3}-1, 2^{6n+4}-1$ are always composite. </p>
|
1,641,076 | <p>Let's say I have the following decomposition: </p>
<p>$$\{100,10011,00110\}^*$$</p>
<p>How would I determine if the decomposition is ambiguous or unambiguous?</p>
| Maverick | 171,392 | <p>Putting $x=0$ and solving the two equations we obtain a point which satisfies both equations i.e.$(0,-1,3)$.</p>
<p>Therefore this point would lie on the line of intersection of the two planes. Let equation of line be
$$ \frac{x-0}{l}=\frac{y+1}{m}=\frac{z-3}{n}$$.</p>
<p>Angle between this line and both the plan... |
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... | Kuba | 5,478 | <p>When you define your function like this, then e.g. <code>ffd[.5, .5, .5]</code> is really <code>D[ff[.5, .5, .5], {{.5, .5, .5}}]</code>.</p>
<p>to avoid scoping issues with </p>
<pre><code>x=5;
ffd[x_, y_, z_] = D[ff[x, y, z], {{x, y, z}}]
</code></pre>
<p>you can use</p>
<pre><code>ffd = Evaluate[D[ff[#, #2, ... |
191,307 | <pre><code>roots = NSolve[Sin[2z] == Cos[Sin[z]] + 1 && -2π <= Re[z] <= 2π && -2π <= Im[z] <= 2π, z];
w = z\.roots;
ListPlot[{Re[w], Im[w]}, Ticks -> {{-2π, -π, 0, π, 2π}, {-2π, -π, 0, π, 2π}}]
</code></pre>
<p>In this code, <code>Ticks</code> command is not working properly, that i... | Greg Hurst | 4,346 | <p>Hard to know exactly what you want without the explicit data, but maybe you'd like to bin the data?</p>
<pre><code>SeedRandom[1234];
pts = RandomReal[{0, 1}, {1000, 3}];
centers = Tuples[Range[.1, .9, .2], 3];
nf = Nearest[centers];
ListDensityPlot3D[KeyValueMap[Append, CountsBy[pts, First@*nf]]]
</code></pre>
... |
275,310 | <p>I am a bit confused. What is the difference between a linear and affine function? Any suggestions will be appreciated</p>
| mdp | 25,159 | <p>A linear function fixes the origin, whereas an affine function need not do so. An affine function is the composition of a linear function with a translation, so while the linear part fixes the origin, the translation can map it somewhere else.</p>
<p>Linear functions between vector spaces preserve the vector space ... |
275,310 | <p>I am a bit confused. What is the difference between a linear and affine function? Any suggestions will be appreciated</p>
| Francis Zerbib | 853,307 | <p>Answer for any confused French reader.</p>
<p>In France/French, the distinction between linear and affine appears to be different from other countries.</p>
<p>An affine function in the French sense is a any function that may be written</p>
<p><span class="math-container">$f(x) = a + bx$</span></p>
<p>where <span cla... |
3,580,293 | <blockquote>
<p>The value of <span class="math-container">$$\lim\limits_{x \rightarrow \infty} \left(5^x + 5^{3x}\right)^{\frac{1}{x}}$$</span> is...</p>
</blockquote>
<p>My approach :</p>
<blockquote>
<p><span class="math-container">$$\lim\limits_{x \rightarrow \infty} \left(5^x + 5^{3x}\right)^{\frac{1}{x}}$$</... | bjorn93 | 570,684 | <p>Hint:
<span class="math-container">$$\left(5^x+5^{3x}\right)^{1/x}=5^3\left(1+5^{-2x}\right)^{1/x}$$</span>
Can you finish now? The limit of the last expression in parentheses is not an indeterminate form now.</p>
|
1,234,661 | <p>I lost my baby Rudin book on real analysis book but I recall a pair of results in homework exercises that he seemed to indicate that there is no "boundary" between convergent and divergent series of positive decreasing terms. One result was that if $a_n$ is positive decreasing, and $\sum_n a_n$ is divergent, then $\... | Will Jagy | 10,400 | <p>Never seen it, but I can repeat the standard collection of examples that exhibit this behavior. The series with terms</p>
<p>$$ \frac{1}{n}, $$
$$ \frac{1}{n \log n}, $$
$$ \frac{1}{n \log n \log \log n}, $$
$$ \frac{1}{n \log n \log \log n \log \log \log n}, $$
all diverge (we need to start at $n = n_0$ la... |
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... | symplectomorphic | 23,611 | <p>Here's a more general way of asking the question. Let <span class="math-container">$c\neq0$</span>.</p>
<p>In order for <span class="math-container">$f(x)$</span> and <span class="math-container">$g(x):=f(cx)$</span> to have the same slope for all <span class="math-container">$x$</span>, we need <span class="math-c... |
102,357 | <p>Let $G=PSL(2,q)$ where $q$ is prime power. What is Aut$(G\times G)$ and Aut$(G\times G\times G)$? Also if $G=A_{n}$ where $A_{n}$ is the alternating group of degree $n$, then what is Aut$(G\times G)$? </p>
<p>Thanks in advance</p>
| Geoff Robinson | 14,450 | <p>In general, if $S$ is a finite non-Abelian simple group, and $E$ is a direct product of $n$ copies of $S$, then ${\rm Out}(E) = {\rm Aut}(E)/E$ is isomorphic to ${\rm Out}(S) \wr S_{n}.$ This is because every minimal normal subgroup of $E$ is isomorphic to $S$ (in fact, is one of the obvious simple direct factors of... |
1,742,409 | <blockquote>
<p>Let <span class="math-container">$(Y, \langle .,. \rangle )$</span> be an inner product space with <span class="math-container">$Y=C[0,1]$</span> and <span class="math-container">$$\langle f,g \rangle = \int_0^1 f(x)g(x)dx$$</span></p>
<p>In <span class="math-container">$C[0,1]$</span>, let <span class=... | Hagen von Eitzen | 39,174 | <p>Your $g$ is not continuous. Also, you will have $\langle f,g\rangle = 5\int_0^1f(x)\,\mathrm dx\ne 0$ in general.</p>
<hr>
<p>If $f=0$, any $g$ will do, for example $g(t)=5$.</p>
<p>Otherwise, pick $a\in(0,1)$ with $f(a)\ne 0$. Wlog. $f(a)>0$. Then for some small $r>0$ we have $f(x)>0$ for all $x\in(a-r,... |
1,742,409 | <blockquote>
<p>Let <span class="math-container">$(Y, \langle .,. \rangle )$</span> be an inner product space with <span class="math-container">$Y=C[0,1]$</span> and <span class="math-container">$$\langle f,g \rangle = \int_0^1 f(x)g(x)dx$$</span></p>
<p>In <span class="math-container">$C[0,1]$</span>, let <span class=... | Travis Willse | 155,629 | <p>The candidate function $g$ you mention is not continuous, i.e., not in $C[0, 1]$.</p>
<p>One simple approach is the following: Choose a simple $1$-parameter family of functions $g$ satisfying the given condition $g(0) = 5$, compute $\langle f, g \rangle$, and then find a parameter value (if one exists) such that th... |
360,063 | <p>Let <span class="math-container">$\mathbb{N}$</span> denote the set of positive integers. For any prime <span class="math-container">$p\in\mathbb{N}$</span> let <span class="math-container">$p\mathbb{N} = \{np: n\in \mathbb{N}\}$</span>. Is there a partition <span class="math-container">${\cal P}$</span> of <span cl... | Jan Kyncl | 24,076 | <p>Place each <span class="math-container">$n$</span> that is not a prime power into its own <span class="math-container">$B=B(n)$</span>. Then fill the rest of <span class="math-container">$B(n)$</span> with powers of primes that do not divide <span class="math-container">$n$</span>.</p>
|
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... | copper.hat | 27,978 | <p>Suppose that there is an isomorphism $\phi$.</p>
<p>Then you must have $\phi(0 + 0) = \phi(0) * \phi(0)$, so
from the table we must have $\phi(0) = b$.</p>
<p>Now there are only two possibilities for $\phi(1),\phi(2)$.</p>
<p>If we try $\phi(1) = a,\phi(2) = c$, it is easy to check that
$\phi$ is an isomorphism (... |
659,254 | <p>Say $X_1, X_2, \ldots, X_n$ are independent and identically distributed uniform random variables on the interval $(0,1)$.</p>
<p>What is the product distribution of two of such random variables, e.g.,
$Z_2 = X_1 \cdot X_2$?</p>
<p>What if there are 3; $Z_3 = X_1 \cdot X_2 \cdot X_3$?</p>
<p>What if there are $n$ ... | Julian Wergieluk | 731 | <p>If $X_1$ is uniform, then $-\log X_1 \sim \textrm{Exp}(1)$. Therefore,
$$- \log X_1 \dots X_n = -\log X_1 + \dots -\log X_n$$
is a sum of independent exponential random variables and has Gamma distribution with parameters $(n,1)$ and density $g(y) = \frac{1}{(n-1)!} y^{n-1}e^{-y}$ for $y\geq 0$. Let $f$ be the den... |
3,742,517 | <p>Let <span class="math-container">$f(x)=x^3-1$</span>. To approximate the root <span class="math-container">$x^{\star}=1$</span>, we consider the sequence <span class="math-container">$(x_n)$</span> that we get if we apply Newton's method with <span class="math-container">$x_0>0$</span>. Show that the sequence con... | Robert Israel | 8,508 | <p>You can try doing it in two stages. First solve the problem with the original objective to minimize <span class="math-container">$\sum x_i$</span> and original constraints. Suppose the optimal solution has <span class="math-container">$\sum x_i = s$</span>. Then solve the problem with additional constraint <span c... |
942,651 | <p>Let $n,r,k$ be non negative integers such that $r,k\leq n$.</p>
<p>If $r^{n-r}=k^{n-k}$, when is this true other than $r=k$ ?</p>
<p>For example it is holds for $n=6,r=2,k=4$.</p>
| SPK.z | 171,119 | <p>This equation has many solutions. I have no general form to find all these solutions, but if you're just interested in finding some other values for which it holds, you can do the following:
$$r^{n-r} = k^{n-k} \quad\Longrightarrow\quad 1=\frac{k^{n-k}}{r^{n-r}}=\left(\frac{k}{r}\right)^{n-r}k^{r-k} \quad\Longrighta... |
1,134,510 | <p>Regarding My Background I have covered stuff like </p>
<p>1.Single Variable Calculus</p>
<p>2.Multivariable Calculus (Multiple Integration,Vector Calculus etc) (Thomas Finney)</p>
<p>3.Basic Linear Algebra Course (Containing Vector spaces,Linear Transformation)</p>
<p>4.Ordinary Differential Equation</p>
<p>5.R... | JMP | 210,189 | <p>There are $\binom{7}{3}$ ways to choose 3 from 7, which is 35. The first letter can be one of 26, the second one of 25 (unless both letters are allowed to be the same!). So the answer is $35*26*25=22750$.</p>
|
152,582 | <p>I have tried to use <code>OpenRead</code> for my application as it appears to be less of a burden on the memory.</p>
<p>I <code>OpenRead</code> a .csv file to extract coordinates in the form {x,y} in the following way</p>
<pre><code>ClearAll[f];
f = OpenRead["mathfile.csv"];
g = ReadList[f, String, RecordL... | ZachB | 1,043 | <p>Mathematica doesn't have configurable separators for <code>Read</code> and <code>ReadList</code>, except when reading <code>Word</code>s and <code>Record</code>s, so importing a CSV of numbers is trickier. Some workarounds:</p>
<ol>
<li><p>Use TSV format, which will "just work."</p></li>
<li><p>Read a <code>Charact... |
3,753,839 | <p>Upon inspection, I found out that: <span class="math-container">$3!\cdot5! =6! $</span> and <span class="math-container">$6!\cdot7! =10!.$</span> (couldn't dare to search for more such triplets)</p>
<p>How to check whether there exist more such triplets or not? Please show me a proper way to handle such problems.</p... | Henry | 6,460 | <p>Your <span class="math-container">$3!\cdot 5! =6!$</span> is an example of the case for general <span class="math-container">$n$</span>:</p>
<p><span class="math-container">$$n! \cdot (n!-1)! = (n!)!$$</span></p>
<p>and when <span class="math-container">$n=4$</span> this gives <span class="math-container">$4! \cdot... |
3,753,839 | <p>Upon inspection, I found out that: <span class="math-container">$3!\cdot5! =6! $</span> and <span class="math-container">$6!\cdot7! =10!.$</span> (couldn't dare to search for more such triplets)</p>
<p>How to check whether there exist more such triplets or not? Please show me a proper way to handle such problems.</p... | hamam_Abdallah | 369,188 | <p>For example, you can observe that</p>
<p><span class="math-container">$$(n!-1)!.(n!)=(n!)!$$</span></p>
<p>If <span class="math-container">$ n=3$</span>, it gives</p>
<p><span class="math-container">$$5!.3!=6!$$</span>
for <span class="math-container">$ n=4 $</span>, we get</p>
<p><span class="math-container">$$23!.... |
4,621,390 | <p>I'm studying Linear Algebra and have come to think of a column vector as an ordered bunch of objects, where each object is the product of a scalar and a basis vector, vis:</p>
<p><span class="math-container">$\begin{bmatrix}
a \\
b \\
\vdots \\
\end{bmatrix} = a \hat{i} + b \hat{j} + \dots$</span></p>
<p>Where <span... | Seeker | 1,050,393 | <p>That’s not exactly correct. The entries in a column vector come from a linear combination of the basis vectors.</p>
<p>This would be the case if you use the basis vectors <span class="math-container">$$\begin{bmatrix}1 \\ 0 \end{bmatrix}, \begin{bmatrix}0 \\ 1 \end{bmatrix} $$</span> The entries of a vector written ... |
4,651,364 | <p>I am trying to evaluate the following contour integral by evaluating the residue at <span class="math-container">$z=0$</span> of the integrand.</p>
<p><span class="math-container">$$I=\oint_{|z|=1} \frac{z^n+z^{-n}}{2iz(1-rz)(1-rz^{-1})}dz.$$</span></p>
<p>We can manipulate the integrand (which we denote by <span cl... | ecrin | 629,202 | <p>Let <span class="math-container">$x,y\in X$</span>. Since <span class="math-container">$f(X_0)$</span> is dense, there exists two sequence <span class="math-container">$z_n,w_n$</span> in <span class="math-container">$f(X_0)$</span> converging to <span class="math-container">$f(x),f(y)$</span>, respectively.</p>
<p>... |
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>
| E.H.E | 187,799 | <p>The inequality equivalent to
$$\tan^{-1}(x)< x$$</p>
<p>take the derivative for both sides to get
$$\frac{1}{1+x^2}< 1$$ </p>
|
2,623,924 | <p>My textbook explains the proof, which I don't understand:</p>
<p>"Consider two bases $v_1...v_p$ and $w_1...w_q$ of V. Since the vectors $v$ are linearly independent and the vectors $w$ span V..."</p>
<p>How exactly does $w$ span V? </p>
<p>The book then says the same for vectors $v$, that $v$ spans V and hence $... | dxiv | 291,201 | <p>Take for example $\,f(x) = g(x) = \dfrac{1}{2}\,$, $\,c=2\,$, then:</p>
<ul>
<li><p>$f(x) \le c g(x)\,$ holds because $\,\dfrac{1}{2} \le 2 \cdot \dfrac{1}{2}$</p></li>
<li><p>$\log\big(f(x)\big) \gt c \cdot \log\big(g(x)\big)\,$ holds because $\log\big(\dfrac{1}{2}\big) \gt 2 \cdot \log\big(\dfrac{1}{2}\big) = \lo... |
1,023,463 | <p>I know $x \sin(1/x)$ is not Lipschitz on $[0,1]$, but some experimentation makes me conjecture that it is $1/2$-Holder. What is a good way to prove this?</p>
| Gaultier | 193,240 | <p>I will sketch the proof that $f(x)=x \sin(x^{-1})$ is 1/2-Holder on $[0,1/2\pi]$.</p>
<p>There are two cases.</p>
<ol>
<li>If $x,y\in [\frac{1}{2\pi (n+1)} , \frac{1}{2\pi n}]$.</li>
</ol>
<p>Write $x= \frac{1}{2\pi n+ \xi}$ and $y=\frac{1}{2\pi n+\zeta}$ where $0 \le \xi,\zeta \le 2\pi$.
There exists a constant ... |
300,944 | <p>Show that there are no intergers $x$ and $y$ such that</p>
<p>$P(x,y)=x^2-5y^2=2$</p>
<p>Hint from professor:</p>
<p>Consider the equation in a convenient $\mod (n)$ so that you end up with a polynomial in a single variable. Then proceed as solving number of congruence.</p>
<hr>
<p>Im not sure how to approach t... | Math Gems | 75,092 | <p>Following the hint: $\rm\ mod\ 5\!:\ x^2\! - 5y^2 \equiv x^2\in \{0, \pm1, \pm 2\}^2\! \equiv \{0, 1, 4\},\:$ so is not $\equiv 2.$</p>
|
3,159,884 | <p>Prove that if <span class="math-container">$|z+w|=|z-w|$</span> then <span class="math-container">$z\overline{w}$</span> is purely imaginary.</p>
<p>To start off, I said let <span class="math-container">$z=a+bi$</span> and let <span class="math-container">$w=p+qi$</span>. Not sure where to go from here after subbin... | 1123581321 | 444,046 | <p><span class="math-container">$z + w = (a + p) + (b + q)i$</span> </p>
<p><span class="math-container">$|z + w|^2 = (a + p)^2 + (b + q)^2$</span> </p>
<p><span class="math-container">$z - w = (a - p) + (b - q)i$</span> </p>
<p><span class="math-container">$|z - w|^2 = (a - p)^2 + (b - q)^2$</span> </p>
<p><span c... |
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... | Mozibur Ullah | 26,254 | <p>Seeing how software has infiltrated our lives in so many ways it seems that category theoretic applications to programming is pretty much 'real world' stuff. </p>
<p>Category theory is more geared up to clarifying conceptual structures, so I imagine that there isn't likely to be real world applications in a very di... |
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... | Dan Piponi | 80 | <p>I recently wrote some software to simulate a real world physical system to show that a hardware technology has a chance of doing what we want it to do. That project has grown to a pretty substantial piece of engineering with sizable budget. I wrote the simulation in a programming language whose syntax could be descr... |
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... | Harley Eades | 208,820 | <p>This answer is also very late. There is a recent book by David Spivak titled "Category Theory for the Sciences." Its sole purpose is to connect the "real world" with category theory. You can find an older draft <a href="http://math.mit.edu/~dspivak/CT4S.pdf">here</a> from the authors homepage.</p>
|
148,809 | <p>Let $M$ be a complete Riemannian manifold, does there exists a positive non-constant harmonic function $f \in L^1(M)$? Who can answer me or give me a counter example? Thank you very much!</p>
| Willie Wong | 1,543 | <p>Consider the surface of revolution (so topologically we are dealing with $\mathbb{R}\times\mathbb{S}$) with standard coordinates $(z,\theta)$. Let the metric be
$$ \mathrm{d}s^2 = \mathrm{d}z^2 + h^2(z) \mathrm{d}\theta^2$$
This manifold is clearly complete (it is a warped product of two geodesically complete manifo... |
2,164,823 | <p>Could someone please explain mathematical explanation behind this?</p>
<p>You can win three kinds of basketball points, 1 point, 2 points, and 3 points. Given a total score $n$, print out all the combination to compose $n$.</p>
<p>Examples:<br>
For n = 1, the program should print following:<br>
1 </p>
<p>For n =... | benguin | 121,903 | <p>Consider the set of sequences of points that add up to $n$ and let's call it $S(n)$. There are three different cases to consider regarding the type of sequence we might have.</p>
<ol>
<li><p>The first point value is 1. This can only happen if the sum of the remaining points is $n-1$ so the rest of the subsequence m... |
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}^... | Gary | 83,800 | <p>Proof sketch:
<span class="math-container">\begin{align*}
\frac{{{\rm d}^3 F_1 }}{{{\rm d}\alpha ^3 }} & = -\int_0^1 {x^2 \int_{ - 1}^1 {{\rm e}^{ - \alpha \sqrt {1 + x^2 + 2xy} } (xy + 1){\rm d}y} \,{\rm d}x} \\ & = -\frac{{{\rm e}^{ - 2\alpha } (4\alpha ^4 + 14\alpha ^3 + 27\alpha ^2 + 30\alpha + 15)}... |
179,659 | <p>I want to assign each cell in my current notebook a tag so that I can rerun cells with specific tags later in the notebook <a href="https://mathematica.stackexchange.com/a/71333/44420">(Using a method similar to this answer)</a>.</p>
<p>My questions are</p>
<ul>
<li>How do I assign cells in my current notebook tag... | AzJ | 44,420 | <p>For thoose looking at this question in the future if you want to label the first 3 input cells cell tags use the following</p>
<pre><code>{cell1, cell2, cell3} = Cells[CellStyle -> {"Input"}][[1 ;; 3]]
SetOptions[cell1, CellTags -> {"FOO"}]
SetOptions[cell2, CellTags -> {"FOO2"}]
SetOptions[cell3, CellTags... |
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... | user23201 | 23,201 | <p>How about Ravi Agarwal and Donal O'Regan with 445 joint journal publications on MathSciNet?</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... | Gon | 969,491 | <p>This is the kind of exercise that is perfect to use the spherical coordinates.
Lets start with</p>
<p><span class="math-container">$f(x,y,z)=z^2 \rightarrow f(r,\theta,\phi)=(rcos(\phi))^2=r^2cos^2(\phi)$</span></p>
<p><span class="math-container">$C=\{(x,y,z)\in\mathbb{R^3} | 1\leq x^2+y^2+z^2\leq4\} \rightarrow C=... |
606,843 | <p><strong>Definitions:</strong></p>
<ol>
<li><p>$[x]$ is the integer value of $x$. For example: $[4.3]=4$.</p></li>
<li><p>$\{x\}=x-[x]$.</p></li>
</ol>
<p>Can someone help me calculate the derivative of the following functions, and determine where the derivative are not defined?</p>
<ol>
<li><p>$[x^2]\sin^2(\pi x)... | Parth Thakkar | 70,311 | <p>For the first one:</p>
<p>Consider 2 intervals $ A = [ \sqrt n, \sqrt{n+1} )$ and $ B = [ \sqrt {n+1}, \sqrt{n+2} )$. Of course, $n$ is an integer.</p>
<p>Now, in the first interval, your function becomes $ n\sin^2(\pi x) $. It's derivative is $ n\pi\sin(2\pi x) $. Call it $ f'_1(x) $.</p>
<p>Consider the second ... |
1,808,881 | <p>I am struggling with finding all roots of unity in $\mathbb{Q}(i)$. I know that if $a+bi$ is a root of unity in $\mathbb{Q}(i)$, then $a^2+b^2=1$, and I know how to find all $a, b \in \mathbb{Q}$ that satisfy that equation. However, I do not know how to filter out the roots of unity. I think there should be a somewh... | Eric Towers | 123,905 | <p>This might help:</p>
<p>Dresden, G., "<a href="http://home.wlu.edu/~dresdeng/papers/rattan.pdf" rel="nofollow">A New Approach to Rational Values of Trigonometric Functions</a>", (preprint).</p>
<p>This paper attacks: "for $c,d$ relatively prime integers (with $d > 0$), the primitive $d^\text{th}$ root of unity ... |
4,516,356 | <p>In Ch. 20 of Spivak's <em>Calculus</em>, he shows that the remainder terms for <span class="math-container">$\arctan$</span> and <span class="math-container">$\log{(1+x)}$</span> become large with the order of the Taylor polynomial used to approximate these functions. Thus these approximations</p>
<blockquote>
<p>ar... | Suzu Hirose | 190,784 | <p>With base ten logarithms, we have <span class="math-container">$\log_{10}(1234)=4+\log_{10}(0.1234)$</span>, and in general,
<span class="math-container">$$\log_{10} x=n+\log_{10} (x/10^n),$$</span>
where <span class="math-container">$n$</span> is the number of digits of <span class="math-container">$x$</span> befor... |
685,918 | <p>I'm doing exercises in Real Analysis of Folland and got stuck on this problem. I don't know how to calculate limit with the variable on the upper bound of the integral. Hope some one can help me solve this. I really appreciate.</p>
<blockquote>
<blockquote>
<p>Show that $\lim\limits_{k\rightarrow\infty}\int_0... | Felix Marin | 85,343 | <p>$\newcommand{\+}{^{\dagger}}%
\newcommand{\angles}[1]{\left\langle #1 \right\rangle}%
\newcommand{\braces}[1]{\left\lbrace #1 \right\rbrace}%
\newcommand{\bracks}[1]{\left\lbrack #1 \right\rbrack}%
\newcommand{\ceil}[1]{\,\left\lceil #1 \right\rceil\,}%
\newcommand{\dd}{{\rm d}}%
\newcommand{\down}{\downarrow}... |
226,449 | <p>Many counting formulas involving factorials can make sense for the case $n= 0$ if we define $0!=1 $; e.g., Catalan number and the number of trees with a given number of vetrices. Now here is my question:</p>
<blockquote>
<p>If $A$ is an associative and commutative ring, then we can define an
unary operation on ... | GEdgar | 442 | <p>In general, we want the "associative" law to hold in the form:
$$
\times(A \cup B) = (\times A) (\times B)
$$
whenever $A$ and $B$ are disjoint. What would this mean when $B = \emptyset$ ??</p>
|
226,449 | <p>Many counting formulas involving factorials can make sense for the case $n= 0$ if we define $0!=1 $; e.g., Catalan number and the number of trees with a given number of vetrices. Now here is my question:</p>
<blockquote>
<p>If $A$ is an associative and commutative ring, then we can define an
unary operation on ... | Michael Hardy | 11,667 | <p>Multiplying by $1$ is the same as not multiplying by anything at all.</p>
|
2,464,641 | <p>Let we have an equation $x^2+4x+2$ and we want to divide it with $x-4$. By remainder theorem it gives the value of $34$. But when we put the value of $x=6$ in equations it gives zero remainder. And quotient is $31$. How's that? Where I am mistaking?</p>
| Sagar Chand | 477,029 | <p>You are getting remainder 34. This is because you haven't placed value of x yet. If you place value of x, you will see that divisor is (6-4) = 2. Now obviously you can't get remainder of 34 when divisor is 2.(remainder can't be greater than divisor) You get remainder 0.</p>
<p>The equation you solved earlier was ge... |
2,464,641 | <p>Let we have an equation $x^2+4x+2$ and we want to divide it with $x-4$. By remainder theorem it gives the value of $34$. But when we put the value of $x=6$ in equations it gives zero remainder. And quotient is $31$. How's that? Where I am mistaking?</p>
| Francisco José Letterio | 482,896 | <p>$x^2 + 4x +2=(x+8)(x-4)+34$. I think you're mistaking what dividing a polynomial means</p>
|
1,851,084 | <p>I have to solve the following problem:
find the matrix $A \in M_{n \times n}(\mathbb{R})$ such that:
$$A^2+A=I$$ and $\det(A)=1$.
How many of these matrices can be found when $n$ is given?
Thanks in advance.</p>
| Mark Fischler | 150,362 | <p>Let the eigenvalues of $A$ be $\{e_i\}$ and let $P$ be a similarity transformation that diagonalizes $A$. Then $P$ also diagonalizes $A^2$ and the equation becomes
$$
\forall i: e_i^2 + e_i = 1
$$</p>
<p>Since there are two real roots to that equation, there are $2^n$ candidates for $A$, before imposing that the ... |
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... | Mozibur Ullah | 35,706 | <p>The machinery of Jets were introduced by Ehresmann as an geometrically invariant way of describing PDEs and their solutions in analogy to the way that vector fields and integral curves invariantly describe ODEs and their solutions.</p>
<p>For example, there is a way of geometrically describing symmetries here and so... |
1,356,932 | <blockquote>
<p><strong>Problem.</strong> Let $h\in C(\mathbb{R})$ be a continuous function, and let
$\Phi:\Omega:=[0,1]^{2}\rightarrow\mathbb{R}^{2}$ be the map defined
by \begin{align*}
\Phi(x_{1},x_{2}):=\left(x_{1}+h(x_{1}+x_{2}),x_{2}-h(x_{1}+x_{2})\right) \tag{1}
\end{align*}
What is the (Lebesgue) meas... | Disintegrating By Parts | 112,478 | <p>Perform the Gram-Schmidt process on $\{1,x,x^{2},\cdots\}$ to obtain an orthonormal sequence of polynomials $\{p_0,p_1,p_2,\cdots\}$, where $p_n$ is a polynomial of order $n$. Note that $p_n$ is orthogonal to $1,x,\cdots,x^{n-1}$. This sequence must be a complete orthonormal sequence because the polynomials are dens... |
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</... | Keen-ameteur | 421,273 | <p>Following the answer I wrote in <a href="https://math.stackexchange.com/questions/4633639/when-is-a-borel-measure-locally-finite/4635045?noredirect=1#comment9780717_4635045">your follow up thread</a>, I would like to try and propose a counter example for this more restrictive question. I think this is in the same sp... |
3,104,051 | <p>I have the work of the proof done, but at the end after showing
<span class="math-container">$3^{2(n+1)} - 1=9(3^{2n} - 1)+8$</span>
I make the statement that since <span class="math-container">$9(3^{2n} - 1)$</span> is a multiple of 8 and 8 is a multiple of 8 then <span class="math-container">$3^{2n} - 1$</span> is... | Dr. Sonnhard Graubner | 175,066 | <p>Note that <span class="math-container">$$3^{2n}-1=9^n-1$$</span> and <span class="math-container">$$9\equiv 1 \mod 8$$</span></p>
|
231,360 | <p>Let $A,B\subseteq\mathbb R^d$ with $A$ closed and $B$ open and such that $A\cap B\neq\emptyset$. Assume that there exists a sequence $(x_k)_{k\in\mathbb N}\subseteq\mathbb R^d$ conveging in Euclidean norm to some $x\in A\cap B$. Do we have $x_k\in A\cap B$ for some $k\in\mathbb N$? If $x$ is an interior point of $A$... | P.. | 39,722 | <p>For a counterexample take $A=[-1,0], \ B=(-1,1) \subseteq \mathbb{R}$ and $x_n=\frac{1}{n}$. </p>
<p>This sequence $(x_n)\to 0 \in A \cap B$ but $x_n \not \in (-1,0]=A \cap B, \ \forall n \in \mathbb{N} $ </p>
|
231,360 | <p>Let $A,B\subseteq\mathbb R^d$ with $A$ closed and $B$ open and such that $A\cap B\neq\emptyset$. Assume that there exists a sequence $(x_k)_{k\in\mathbb N}\subseteq\mathbb R^d$ conveging in Euclidean norm to some $x\in A\cap B$. Do we have $x_k\in A\cap B$ for some $k\in\mathbb N$? If $x$ is an interior point of $A$... | fgp | 42,986 | <p>Let $x_i = (1 + \frac{1}{i},0)$, let $A$ be the closed unit ball and $B$ some small open ball around $(1,0)$. $(x_i)_{i\in\mathbb{N}}$ the converges to $(1,0) \in A \cap B$, yet none of the $x_i$ lie in $A$ and hence not in $A \cap B$. The answer is therefore no.</p>
|
3,390,979 | <blockquote>
<p>Let <span class="math-container">$\sum_i^na_i=n$</span>, <span class="math-container">$a_i>0$</span>. Then prove that <span class="math-container">$$ \sum_{i=1}^n\left(\frac{a_i^3+1}{a_i^2+1}\right)^4\geq n $$</span></p>
</blockquote>
<p>I have tried AM-GM, Cauchy-Schwarz, Rearrangement etc. but n... | Batominovski | 72,152 | <p><strong>A Hint for an Alternative Solution.</strong></p>
<p>We want to show that <span class="math-container">$$\left(\frac{x^3+1}{x^2+1}\right)^4\geq 2x-1$$</span> for every <span class="math-container">$x\in\mathbb{R}$</span>. By the AM-GM Inequality,
<span class="math-container">$$\left(\frac{x^3+1}{x^2+1}\righ... |
2,329,600 | <p>I haven't studied any maths since I was at university 20 years ago. Yesterday, however, I came across a pair of equations in an online article about gaming and I couldn't understand how they'd been derived. </p>
<p>Here's the scenario. If we make a single trial of generating a number between 1 and 20, there's an ev... | praveen kr | 453,426 | <p>Given $P$ is the probability of getting a number $N$ or greater with one trial.</p>
<p>1)For two trials, if you are taking the lower number, then both results should he greater than $N$. Since the two trials can be assumed to be independent, we have to multiply the two probabilities. Thus we get $P\times P=P^2$</p>... |
2,942,879 | <p>Suppose that <span class="math-container">$lim_{n\rightarrow \infty} a_n = L$</span> and <span class="math-container">$L \neq 0$</span>. Prove there is some <span class="math-container">$N$</span> such that <span class="math-container">$a_n \neq 0$</span> for all <span class="math-container">$n \geq N$</span>.</p>
... | Peter Szilas | 408,605 | <p>For fun.</p>
<p>Given :</p>
<p><span class="math-container">$\lim_{n \rightarrow \infty}a_n=L \not = 0$</span>.</p>
<p>Statement :</p>
<p>There is a <span class="math-container">$N \in \mathbb{Z^+}$</span> s.t. for</p>
<p><span class="math-container">$ n \ge N$</span> <span class="math-container">$a_n \not =0$<... |
307,545 | <p>If $\gcd(a,b)=1$, how can I find the values that $\gcd(a+b,a^2+b^2)$ can possibly take? I can't find a way to use any of the elemental divisibility and gcd theorems to find them. </p>
| Community | -1 | <p>$$\begin{align}
\gcd(a+b, a^2 + b^2)
&= \gcd(a+b, a^2 + b^2 - a(a+b))
\\&= \gcd(a+b, b^2 - ab)
\\&= \gcd(a+b, b^2 - ab + b(a+b))
\\&= \gcd(a+b, 2b^2)
\end{align}
$$</p>
<p>Now, $\gcd(a+b,b) = \gcd(a,b) = 1$, so we can get rid of the factors of $b$ and have</p>
<p>$$ \gcd(a+b, a^2 + b^2) = \gcd(a+b,... |
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... | Eric Wofsey | 86,856 | <p>To a first approximation, there is a spoonful of wine in the cup and a spoonful of tea in the barrel. How much are each of these approximations off by? Well, there is a bit less than a spoonful of wine in the cup, since a bit of the wine was removed in the second step. And, there is a bit less than a spoonful of ... |
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... | Kinro | 850,504 | <p>After step 1, there is always one spoonful of wine in your cup.</p>
<p>If, on step two:</p>
<p>You place one spoonful of wine back into your barrel, then there is 0 tea in your wine and 0 wine in your barrel.</p>
<p>You place one spoonful of tea back into in your barrel, then there is 1 tea in your wine and 1 tea in... |
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... | Dr. Michael W. Ecker | 438,184 | <p>Imagine the tea in the wine in the form of a tiny ball within the wine. Then that ball must be exactly the amount of wine that is missing from the wine. Ergo, it is the amount of wine that is in the tea. So, the two amounts are the same, with exactly as much tea in the wine as wine in the tea.</p>
|
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... | Bram28 | 256,001 | <p>Huh. I must be a 5 or 6 year old since I thought this was completely trivial. I have noticed though that I tend to reason visually far more often than algebraically.</p>
<p>Before:</p>
<p><a href="https://i.stack.imgur.com/IQgGB.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/IQgGB.png" alt="enter... |
3,446,663 | <blockquote>
<p>Let <span class="math-container">$\sigma\in S_{14}$</span> which is an even permutation of the order of <span class="math-container">$28$</span>.<br> Prove that exist <span class="math-container">$x\in \left\{ 1,...,14 \right\}$</span> such that <span class="math-container">$\sigma(x)=x$</span>.</p>
<... | lhf | 589 | <p>Let <span class="math-container">$n_k$</span> be the number of <span class="math-container">$k$</span>-cycles in the disjoint cycle decomposition of <span class="math-container">$\sigma$</span>. Then</p>
<ul>
<li><p><span class="math-container">$k \in \{1,2,4,7,14,28\}$</span></p></li>
<li><p><span class="math-cont... |
35,393 | <p>I have some data that represents small-scale surface topography. The data is planar but not flat because it's impractical to level this microscope to a tolerance of microns. It looks like this:</p>
<p><img src="https://i.stack.imgur.com/nQcLB.jpg" alt="An example of 3D profilometric data with a nonzero planar slope... | bobthechemist | 7,167 | <p>This solution is giving some artifacts, I believe, due to the downsampling, but the problem can be solved by using <a href="http://reference.wolfram.com/mathematica/ref/RotationTransform.html" rel="nofollow noreferrer"><code>RotationTransform</code></a> similar to <a href="https://mathematica.stackexchange.com/quest... |
4,181,442 | <p>I know this is a dumb question but...</p>
<p>Is <span class="math-container">$x ≠ 1,3,5$</span> the same as <span class="math-container">$x$</span> does not belong to {<span class="math-container">$1,3,5$</span>}, for example ?</p>
<hr />
<p>Sorry for the formatting.</p>
<p>Btw, anyone has a link with mathjax's comm... | Soham | 242,402 | <p>Using the set notation has benefits of its own. For eg. once you define how to want an arbitrary set <span class="math-container">$S$</span> to look like, the name <span class="math-container">$S$</span> becomes an alias for that definition forever. So, you are saved from a ton of writing by referring to <span class... |
1,484,736 | <p>[<img src="https://i.stack.imgur.com/i5iPj.jpg" alt="enter image description here">]</p>
<p>$$\lim_{x\to 0}f\big(f(x)\big)$$</p>
<p>(Original image <a href="https://imgur.com/a/Uogqk" rel="nofollow noreferrer">here</a>.)</p>
<p>I don't need an answer, I just want to know how I can calculate the limit based on the... | Stefan | 280,501 | <p>The limit at a point exists in general if the right hand and left hand limit at that point equal the actual value of the function at that point. For example at $x=2$, we see that the function maps to $-1$. So if the left hand and right hand limit as $x$ approaches $x=2$ both equal $-1$, then the function is continuo... |
3,518,719 | <p>Evaluate <span class="math-container">$$\lim_{n \to \infty} \sqrt[n^2]{2^n+4^{n^2}}$$</span></p>
<p>We know that as <span class="math-container">$n\to \infty$</span> we have <span class="math-container">$2^n<<2^{2n^2}$</span> and therefore the limit is <span class="math-container">$4$</span></p>
<p>In a more... | TeM | 247,735 | <p>Given a <em>regular curve</em> <span class="math-container">$\boldsymbol{\alpha}(t)$</span> (therefore of class <span class="math-container">$C^1$</span> and with <span class="math-container">$\boldsymbol{\alpha}'(t) \ne \boldsymbol{0}$</span>), <em>tangent</em>, <em>normal</em> and <em>binormal versors</em> are def... |
76,420 | <p>The following code segment shows what I'd like to do. I'm a procedural programmer trying to learn the Mathematica functional style. Any help on this would be appreciated.</p>
<pre><code>B = {{1, 3}, {1, 5}, {4, 2}, {5, 2}, {5, 5}}
u = SparseArray[{{1, 2} -> 5, {1, 3} -> 9, {1, 4} -> 6,
{1, 5} ->... | Simon Woods | 862 | <p>You could also use</p>
<pre><code>Function[{i, j},
r[[i]] -= u[[i, j]];
r[[j]] += u[[i, j]]
] @@@ B;
r
</code></pre>
<p>or</p>
<pre><code>With[{e = Extract[u, B]},
r[[B[[All, 1]]]] -= e;
r[[B[[All, 2]]]] += e];
r
</code></pre>
|
119,696 | <p>Let us suppose i have two graphs for sequences A and B as follows</p>
<pre><code>a1 := {{0.9, 0.086133}, {0.086133, 0.0082432}, {0.0082432,
0.0007889}, {0.0007889, 0.0000755}, {0.0000755,
7.2256*10^-6}, {7.2256*10^-6, 6.9151*10^-7}, {6.9151*10^-7,
6.618*10^-8}, {6.618*10^-8, 6.3336*10^-9}, {6.3336*10... | gwr | 764 | <p>Actually the answer is already given in the documentation for <a href="http://reference.wolfram.com/language/ref/SwatchLegend.html" rel="nofollow noreferrer"><code>SwatchLegend</code></a>. All you need to do is to wrap <a href="http://reference.wolfram.com/language/ref/Legended.html" rel="nofollow noreferrer"><code>... |
619,370 | <p>Let $C,Q$ is complex numbers Field and Rational number
Field,respectively,if $f(x),g(x)\in Q[x]$,</p>
<p>if $g(x)|f(x)$ on $C[x]$,show that
$$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>... | Bill Dubuque | 242 | <p><strong>Hint</strong> $\ $ It is an immediate consequence of the <a href="https://math.stackexchange.com/a/168853/242"><em>uniqueness</em> of the quotient (and remainder)</a> in the Division Algorithm (which is the same in $\rm\:\Bbb Q[x]\:$ and $\rm\:\Bbb C[x]).$ Since dividing $\rm\:f\:$ by $\rm\:g\:$ in $\rm\:\Bb... |
474,807 | <p>I am studying a text on permutation groups, which has the following example in a section on regular normal subgroups: </p>
<blockquote>
<p>If $Z(N)=1$, then $N \cong \mathrm{Inn}(N)$, the group of inner automorphisms of $G$, and the semidirect product $N \rtimes \mathrm{Inn}(N)$ is the diagonal group $N^*=N \tim... | spin | 12,623 | <p>If $N \trianglelefteq H$, then $H$ acts on $N$ by conjugation. If this action induces only inner automorphisms (ie. for all $h \in H$, the map $n \mapsto n^h$ is an inner automorphism of $N$), it follows that $H = N C_H(N)$. If also $Z(N) = 1$, then $H = N \times C_H(N)$.</p>
<p>Thus in your case of $H = N \rtimes ... |
23,192 | <p>The question if there is an upper bound known for Brun's constant was discussed briefly here: <a href="http://gowers.wordpress.com/2009/05/22/what-is-wolfram-alpha-good-for/" rel="nofollow">http://gowers.wordpress.com/2009/05/22/what-is-wolfram-alpha-good-for/</a> but no sure answer was given. </p>
<p>So I thought ... | David Hansen | 1,464 | <p>This seems to me like a simple matter of enumerating all the small twin primes, and then estimating the resulting error using a sieve bound. In particular, if $\pi_2(x)$ is the number of twin primes $\leq x$ then we know $\pi_2(x) \ll x (\log x)^{-2}$. By a simple summation-by-parts exercise, this gives</p>
<p>$B... |
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>
| André Nicolas | 6,312 | <p>The first problem is quite a bit harder than the second. </p>
<p>For $1$), use <a href="http://en.wikipedia.org/wiki/K%C3%B6nig%27s_theorem_%28set_theory%29" rel="nofollow">König's theorem.</a> We still then need to show that, for example, there are infinitely many cardinals of cofinality say $\omega$, but that pa... |
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)^{... | DMcMor | 155,622 | <p>Here's one way to look at it:</p>
<p>When you write $x = \lbrace -2, -1, \dotsc ,3\rbrace$ you are saying "$x$ is equal to the set consisting of the integers $-2$ through $3$". If $x$ were really a set then you'd be fine, but if you want to say that the set consists of the possible values for $x$, that is it's the... |
3,499,352 | <p>Define a function <span class="math-container">$f:X\to Y$</span> to be a strict contraction, if there exists a <span class="math-container">$c\in [0,1[$</span> such that
<span class="math-container">$$\forall x,y \in X: d_Y\left( f(x),f(y)\right)\le cd_X(x,y)$$</span>
Now, consider a metric space <span class="math-c... | bign | 735,454 | <p>It seems to me you don't need the completeness of <span class="math-container">$X$</span>. As you pointed out, all you need is uniform convergence of the sequence of contractions mappings <span class="math-container">$(f_{n})$</span>.</p>
<h1>Here's my argument, using much of what you've already written</h1>
<p>Sinc... |
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