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 |
|---|---|---|---|---|
2,529,387 | <p><a href="https://i.stack.imgur.com/3M7iS.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/3M7iS.jpg" alt="enter image description here"></a></p>
<p>The fourth question is how can I do?</p>
<blockquote>
<p>Q: Given that $a, b \in \mathbb{Z}$ and $a$ is even, show that if $a^2b$ is not divisible ... | Dipak Kambale | 505,137 | <p>Since $a$ is even number, let us assume it as, $a=2n$. Hence $a^2\cdot b = (2n)^2\dot b = 4n^2\cdot b$.</p>
<p>Now $a^2\cdot b$ is divisible by $8$ if $a^2 \cdot b = 8k$ where $k \in \mathbb{Z}$. This is possible only if $b$ is even. Hence $a^2b$ is not divisible by $8$ if $b$ is odd.</p>
|
2,529,387 | <p><a href="https://i.stack.imgur.com/3M7iS.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/3M7iS.jpg" alt="enter image description here"></a></p>
<p>The fourth question is how can I do?</p>
<blockquote>
<p>Q: Given that $a, b \in \mathbb{Z}$ and $a$ is even, show that if $a^2b$ is not divisible ... | Mark Bennet | 2,906 | <p>HINT I would argue by contradiction as I think this is easier here: suppose both $a$ and $b$ are even ... can you show that $a^2b$ is then divisible by $8$? </p>
<p>So what gives if $a^2b$ is not divisible by $8$?</p>
|
3,091,090 | <p>I came across this question the other day and have been trying to solve it by using some simple algebraic manipulation without really delving into L'Hospital's Rule or the Power Series as I have just started learning limit calculations.
We needed to find :
<span class="math-container">$$\lim_{x \to 0} \frac {x\cos x... | user3482749 | 226,174 | <p>First, note that both of your answers are incorrect: the actual answer is <span class="math-container">$\frac{-1}{3}$</span>. </p>
<p>For your questions:</p>
<ol>
<li>The problem is actually earlier than that: it is not necessarily the case that <span class="math-container">$\lim \frac{f(x)}{g(x)} = \frac{\lim f(x... |
2,708,071 | <p>Question: Suppose $|x_n - x_k| \le n/k^2$ for all $n$ and $k$.Show that $\{x_n\}_{n=1}^{\infty}$ is cauchy. </p>
<p>Attempt : To prove this, I have to find $M \in N$ that for $\varepsilon >0$, $n/k^2 < \varepsilon$ for $n,k \ge M$. </p>
<p>Let $\varepsilon > 1/M$. </p>
<p>Then, $n/k^2 \le M/M^2$ (#) $= ... | Prasun Biswas | 215,900 | <p><strong>Hint:</strong></p>
<p>With $n$ fixed, the map $k\mapsto\dfrac n{k^2}$ is strictly decreasing with limit $0$ as $k\to\infty$. This means, given $\epsilon\gt 0$, for a fixed $n$, there exists $N\in\Bbb N$ such that $\dfrac n{k^2}\lt\epsilon~\forall~k\geq N$</p>
|
2,785,993 | <p>Let</p>
<ul>
<li><p><span class="math-container">$k(0)=11$</span></p>
</li>
<li><p><span class="math-container">$k(1)=1101$</span></p>
</li>
<li><p><span class="math-container">$k(2)=1101001$</span></p>
</li>
<li><p><span class="math-container">$k(3)=11010010001$</span></p>
</li>
<li><p><span class="math-container">... | Mr Pie | 477,343 | <p>This is not really an answer but might be of some help.</p>
<hr>
<p>According to the "<a href="http://www.softschools.com/math/topics/the_divisibility_rules_3_6_9/" rel="nofollow noreferrer">Divisibility by $3$ Rule</a>," if $n\equiv 1\pmod 3$ then $k(n)$ will not be prime as it will be divisible by $3$. And, it w... |
2,785,993 | <p>Let</p>
<ul>
<li><p><span class="math-container">$k(0)=11$</span></p>
</li>
<li><p><span class="math-container">$k(1)=1101$</span></p>
</li>
<li><p><span class="math-container">$k(2)=1101001$</span></p>
</li>
<li><p><span class="math-container">$k(3)=11010010001$</span></p>
</li>
<li><p><span class="math-container">... | XinBae | 562,933 | <p>Here is a Mathematica search to answer your question:</p>
<pre><code>b[1] = 1;
b[2] = 2;
b[n_] := b[n] = b[n - 1] + n - 1;
list[t_] := b /@ Range[t];
Reap@Do[a = ReplacePart[Array[0 &, b[t]], Transpose[{list[t]}] -> 1];
c = FromDigits[a, 2]; If[PrimeQ[c], Sow@c], {t, 100}]
</code></pre>
<p>which yields</... |
3,873,138 | <p>Since we have variable coefficients we will use the cauchy-euler method to solve this DE. First we substitute <span class="math-container">$y=x^m$</span> into our given DE. This then gives
"</p>
<p><span class="math-container">$9x(m(m-1)x^{m-2}) + 9mx^{m-1} = 0$</span></p>
<p>Note that:</p>
<p><span class="math... | E.H.E | 187,799 | <p>Hint:</p>
<p>you can multiply by <span class="math-container">$x$</span> to make it Euler Cauchy Differential Equation ,see <a href="https://en.wikipedia.org/wiki/Cauchy%E2%80%93Euler_equation" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Cauchy%E2%80%93Euler_equation</a></p>
|
2,452,084 | <ul>
<li><em>I'm having trouble understanding why the arbitrariness of $\epsilon$ allows us to conclude that $d(p,p')<0$. It seems we could likely
conclude a value such as $\frac {\epsilon}{100}$ couldn't we?? The
other idea that would normally work is the limit (as $n$ approaches
$\infty$, $p$ approaches $p'$) but ... | fleablood | 280,126 | <p>$d(p,p') \ge 0$.</p>
<p>Case 1: $d(p,p') > 0$.</p>
<p>Let $\epsilon = d(p,p') > 0$. Then $d(p,p') < \epsilon = d(p,p')$. That is a contradiction.</p>
<p>Case 2: $d(p,p') = 0$.</p>
<p>Then..... $d(p,p') = 0$.....</p>
|
2,312,096 | <p>How do I compute the integration for $a^2<1$,
$$\int_0^{2\pi} \dfrac{\cos 2x}{1-2a\cos x+a^2}dx=? $$
I think that:
$$\cos2x =\dfrac{e^{i2x}+e^{-2ix}}{2},
\qquad\cos x =\dfrac{e^{ix}+e^{-ix}}{2}$$
But I cannot. Please help me.</p>
| Chappers | 221,811 | <p>One can factorise the denominator:
$$ 1-2a\cos{x}+a^2 = (e^{ix}-a)(e^{-ix}-a). $$
Then using partial fractions gives
$$ \frac{1}{2}\frac{e^{2ix}+e^{-2ix}}{(e^{ix}-a)(e^{-ix}-a)} = -\frac{a^2+1}{2 a^2} + \frac{a^4+1}{2 a (a^2-1)}\left( -\frac{1}{e^{i x}-a}-\frac{1}{1-a e^{i x}} \right)-\frac{e^{i x}+e^{-i x}}{2 a} $$... |
949,052 | <p>I am having a hard time understanding the definitions of dimensions. Dimension of a finite-dimensional vector space is defined as the length of any basis of the vector space. The definition seems to be easy to understand. But I need examples to have a better understanding of dimensions so if anyone can explain me mo... | Andrew Maurer | 51,043 | <p>Dimension is actually quite a difficult notion to make rigorous -- we want to have a down-and-dirty definition, but it usually turns out that in a sufficiently general context,mathematical objects behave unexpectedly.</p>
<p>When dealing with real numbers, it seems natural to define the dimension of the unit interv... |
1,956,338 | <p>Let $f\in C_b(\mathbb{R})$ and $g$ any continuous function on $\mathbb{R}$ (or maybe there has to be a restiction on $g$?). Now let K be a compact subset of $\mathbb{R}$ and $U$ an open subset of $\mathbb{R}$. Then I want to show that the following set $$\Phi:=\{t\in\mathbb{R}_+:(e^{t\cdot g}f)(K)\subseteq U\}$$ is ... | s.harp | 152,424 | <p>Let $t\in\Phi$ and write $h=e^{tg}f$. Note $h$ is continuous so bounded on $K$. For $x\in K$:</p>
<p>$$|e^{(t+a) g(x)}f(x)-e^{tg(x)}f(x)|=|e^{ag(x)}h(x)-h(x)|≤\|h\|_K\cdot |1-e^{ag(x)}|≤\|h\|_K |1-e^{|a|\cdot \|g\|_K }|\tag{1}$$
So it is clear, for every $\epsilon>0$ one can choose a $\delta$ independent of $x$ ... |
217,429 | <blockquote>
<p>Let $A\subset\mathbb R$. Show for each of the following statements that it is either true or false.</p>
<ol>
<li>If $\min A$ and $\max A$ exist then $A$ is finite.</li>
<li>If $\max A$ exists then $A$ is infinite.</li>
<li>If $A$ is finite then $\min A$ and $\max A$ do exist.</li>
<li>If ... | fgp | 42,986 | <p>All points on the unit circle obey the equation $x^2 + y^2 = 1$. You know $y=-\frac{7}{25}$, and can thus solve for $x$.</p>
<p>In case you forgot, in general, the circle around $(x_c,y_c)$ with radius $r$ is described by $$
(x - x_c)^2 + (y - y_c)^2 = r^2
$$</p>
|
2,592,477 | <p>I'm trying to solve the following exercise but I appear to be stuck. </p>
<p>Prove that every finite solvable group $G$ contains a fully invariant abelian $p$-subgroup for some prime $p$.</p>
<p>I know that the next-to-last derived subgroup is abelian and fully invariant plus I know that every finite solvable grou... | Nicky Hekster | 9,605 | <p>This solution is for "characteristic" in stead of "fully-invariant", which is slightly different - see the remark by prof D. Holt. <br>The trick is to look at a <em>minimal</em> normal subgroups of $G$. <br>If $M$ is minimal normal and non-trivial, then because a subgroup of $G$ it is also solvable and we must have ... |
2,592,477 | <p>I'm trying to solve the following exercise but I appear to be stuck. </p>
<p>Prove that every finite solvable group $G$ contains a fully invariant abelian $p$-subgroup for some prime $p$.</p>
<p>I know that the next-to-last derived subgroup is abelian and fully invariant plus I know that every finite solvable grou... | Derek Holt | 2,820 | <p>All terms in the derived series of a solvable group are fully invariant. In particular, the last nontrivial term, $H$ say, in the derived series is abelian and fully invariant.</p>
<p>Now, if $H$ is finite, then it is easy to see that any Sylow $p$-subgroup of $H$ is fully invariant in $G$, and is an abelian $p$-gr... |
82,279 | <p>Consider a ten-digit sequence of positive integers 0 - 9.<br>
The 1st, 4th, and 5th digits are either 7 or 9<br>
3rd and tenth digits are either 2 or 4.<br>
Somewhere in the phone number are 2 zeros, and the sum of all the digits equate to 42.<br>
What are all possible such sequence of 10 digits??</p>
<p>Just from ... | Arturo Magidin | 742 | <p>For the first, it is true that $h$ is differentiable. But if the conclusion is true, then take an example in which <em>neither</em> $f$ nor $g$ are ever equal to $0$. Then the conclusion would be that $f/g$ and $g/f$ are <em>both</em> differentiable increasing. But if $f/g$ is increasing, then $g/f$ has to be decre... |
242,773 | <p>Did I tackle this implicit differentiation correctly?</p>
<p>$$5x^2+3xy-y^2=5$$</p>
<p>$$10x+3x\dfrac{dy}{dx}+3y-2y\dfrac{dy}{dx}=0$$
$$\dfrac{dy}{dx}(y-2y)=-10x-3z-3y$$</p>
<p>$$\dfrac{dy}{dx}=\dfrac{-10x-3x-3y}{y-2y}$$</p>
| CCC | 258,418 | <p>I am not sure where your $z$ is coming from. </p>
<p>Steps 1 and 2 are correct,</p>
<p>but step 3 onwards, you group the terms incorrectly.</p>
<p>It should be $\frac {dy}{dx}(3x-2y)$</p>
<p>Therefore, it should be $\frac{dy}{dx}=\frac{-10x-3y}{3x-2y}$.</p>
|
377,471 | <p>If $F[x]$ is a polynomial ring, and $f(x), g(x), h(x)$ and $r(x)$ are four polynomials in it, then is it always true that $f(x)=h(x)g(x)+r(x)$ where $deg(r(x))<deg(g(x))$, or is this true only when $F[x]$ is a Euclidean domain?</p>
<p>Please note that this question has been edited heavily to make it more coheren... | Will Orrick | 3,736 | <p>I think that your question gets the logic backwards. I'll assume that you mean for $F$ to be a field and for $F[x]$ to be the ring of polynomials over that field. That $F$ is a field guarantees that we can perform <a href="http://en.wikipedia.org/wiki/Euclidean_division_of_polynomials#Euclidean_division" rel="nofo... |
3,209,722 | <p>I saw in another post on the website a simple proof that <span class="math-container">$$\lim_{n\to\infty} \left( 1+\frac{x}{n} \right)^n = \lim_{m\to\infty} \left( 1+\frac{1}{m} \right)^{mx}$$</span></p>
<p>which consists of substituting <span class="math-container">$n$</span> by <span class="math-container">$mx$</... | fGDu94 | 658,818 | <p>For <span class="math-container">$x$</span> negative, we must substitute <span class="math-container">$n$</span> with <span class="math-container">$-mx$</span> with <span class="math-container">$m$</span> positive. We can use the fact that we know what the limit <span class="math-container">$(1-\frac{1}{m})^{-m}$</s... |
1,466,618 | <p>Find the equation of the locus of the intersection of the lines below<br>
$y=mx+\sqrt{m^2+2}$<br>
$y=-\frac{ 1 }{ m }x+\sqrt{\frac{ 1 }{ m^2 +2}}$</p>
<hr>
<p>By <a href="https://www.desmos.com/calculator/rpif4yiotx" rel="nofollow">graphing</a>, I have got an ellipse as locus : $x^2+\dfrac{y^2}{2}=1$.<br>
The give... | Claude Leibovici | 82,404 | <p>If you solve the intersection point of the two lines, you should get $$x=-\frac{m}{\sqrt{m^2+2}}$$ $$y=\frac{2}{\sqrt{m^2+2}}$$ and then the result you found.</p>
|
4,196,868 | <p><span class="math-container">$(f_n)$</span> is a sequence of continuous, real valued functions on a metric space <span class="math-container">$M$</span>.</p>
<p>It converges pointwise to a <strong>continuous</strong> function <span class="math-container">$f$</span>.</p>
<p>Suppose that <span class="math-container">$... | David | 471,540 | <p>As far as I know, you can only solve this numerically. It has infinite solutions, of course. It's equivalent to solving</p>
<p><span class="math-container">$$x = \tan x,$$</span></p>
<p>and in physics it arises in a number of contexts that we want to solve</p>
<p><span class="math-container">$$x = \tan k x.$$</span>... |
1,397,576 | <p>To me there is a hierarchy where vectors $\subset$ sequences $\subset$ functions $\subset$ operators</p>
<ul>
<li><p>All vectors are sequences, but not all sequences are vectors because
sequences are infinite dimensional</p></li>
<li><p>All sequences are functions, but not all functions are sequences
because functi... | user251257 | 251,257 | <p>To summarize the comments:</p>
<p>The concepts in their general meaning do not admit a linear hierarchy.</p>
<ul>
<li><p>Vectors are elements of a <em>vector space</em>. For example, $\mathbb R^n$, the set of $\mathbb R$ valued function over a non empty set $X$, $\mathbb R$ as $\mathbb Q$ vector space.</p></li>
<... |
2,668,275 | <p>I have two independent, exponential RVs $X$ and $Y$ that both have the same parameter. I am trying to find the distribution of $Y$ given that $X>Y$. So far, I have: </p>
<p>$$P(Y|X>Y) = P(Y=y|X>Y) = P(Y=y, X>Y)/P(X>Y)$$</p>
<p>and I don't know how to proceed at this point. I understand how to get th... | Pedro | 23,350 | <p>Let $S$ denote the topologist sine curve, which is a union of a sine graph and an interval. Adjoin to this interval a circle. The points $x$ on the interval and on the circle have $\pi(X,x)=\mathbb Z$, while those points $x'$ on the sine graph have $\pi(X,x') = 0$. </p>
|
2,852,020 | <p>Five identical(indistinguishable) balls are to be randomly distributed into $4$ distinct boxes. What is the probability that exactly one box remains empty?</p>
<p>we can choose the empty box in $4$ ways.
We should fit the $5$ balls now into $3$ distinct boxes in $^7C_5$ ways.
The correct answer must be: $\frac37$ ... | Phil H | 554,494 | <p>I couldn't see how the previous answer came up with a solution so I did it differently.</p>
<p>$0$ boxes empty has $1,1,1,2$ box contents times $\frac{5!}{2!}$ ball arrangements $= \frac{4!}{3!}\cdot \frac{5!}{2!} = 240$</p>
<p>$1$ box empty has $1,1,3$ and $1,2,2$ box contents and $\frac{5!}{3!}$ and $\frac{5!}{2... |
107,915 | <p>I randomly place $k$ rooks on an (arbitrarily sized) $N$ by $M$ chessboard. Until only one rook remains, for each of $P$ time intervals we move the pieces as follows:</p>
<p>(1) We choose one of the $k$ rooks on the board with uniform probability. </p>
<p>(2) We choose a direction for the rook, $(N, W, E, S)$, w... | Joseph O'Rourke | 6,094 | <p>To supplement Per Alexandersson's and Aaron Golden's data,
here are two distributions for $4$ rooks on a $4 \times 4$
board (mean: 21.5 moves), and $8$ rooks on an $8 \times 8$ board (mean: 73.6 moves):
<br /> <img src="https://i.stack.imgur.com/bokjE.jpg" alt="4x4"><br />
<br /> &... |
1,199,044 | <p>What is the general solution $h: \mathbb{R}^2 \rightarrow \mathbb{C}$ (or maybe $h: \mathbb{C}^2 \rightarrow \mathbb{C}$ if necessary), $(x, y) \mapsto h(x, y)$ to the non-linear partial differential equation
$$
h_x^2 + h_y^2 = -1
$$
where the subscript denotes a partial derivative. I tried to approach the problem b... | Chinny84 | 92,628 | <p>Changing coords
$$
x\to iv\\
y\to iu
$$
We find
$$
\partial_x = -i\partial_v\\
\partial_y = -i\partial_u
$$
Thus you get
$$
\left(-i\partial_vh\right)^2 + \left(-i\partial_uh\right)^2 = -\left(h_v^2 + h_u^2\right) = -1
$$
Thus you get
$$
h_v^2 + h_u^2 = 1
$$
Solutipns of the above have the form given by
$$
h^2 = (... |
1,199,044 | <p>What is the general solution $h: \mathbb{R}^2 \rightarrow \mathbb{C}$ (or maybe $h: \mathbb{C}^2 \rightarrow \mathbb{C}$ if necessary), $(x, y) \mapsto h(x, y)$ to the non-linear partial differential equation
$$
h_x^2 + h_y^2 = -1
$$
where the subscript denotes a partial derivative. I tried to approach the problem b... | doraemonpaul | 30,938 | <p>$h_x^2+h_y^2=-1$</p>
<p>$h_x^2=-h_y^2-1$</p>
<p>$h_x=\pm i\sqrt{h_y^2+1}$</p>
<p>$h_{xy}=\pm\dfrac{ih_yh_{yy}}{\sqrt{h_y^2+1}}$</p>
<p>Let $u=h_y$ ,</p>
<p>Then $u_x=\pm\dfrac{iuu_y}{\sqrt{u^2+1}}$</p>
<p>$u_x+\mp\dfrac{iuu_y}{\sqrt{u^2+1}}=0$</p>
<p>Follow the method in <a href="http://en.wikipedia.org/wiki/... |
4,261,763 | <p>I was working with this problem:</p>
<p><a href="https://i.stack.imgur.com/AgVpu.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/AgVpu.png" alt="enter image description here" /></a></p>
<p><span class="math-container">$f(x) = 1, x \in(-\infty,0)\cup(0,\infty)\\
f(x) = -1, x = 0 $</span></p>
<p>Th... | Neptune | 825,557 | <p>In general, anytime you have a function whose absolute value is already continuous (whether or not it is continuous other than abs), <span class="math-container">$g: x \in \Bbb X\mapsto g(x)$</span>; <span class="math-container">$g(x)$</span> is not identically <span class="math-container">$0$</span>, <span class="... |
4,528,489 | <p>I hope I could get clarification on a minor detail in the proof to theorem 3.11 (b) in Rudin's Principles of Mathematical Analysis. The theorem and proof are as follows.
<br>Theorem:</p>
<blockquote>
<p>If X is a compact metric space and if {<span class="math-container">$p_n$</span>} is a Cauchy sequence in X, then ... | Mark Bennet | 2,906 | <p>Well you might begin by observing that if <span class="math-container">$x^3\equiv y^3$</span> then <span class="math-container">$x^3-y^3=(x-y)(x^2+xy+y^2)\equiv 0$</span></p>
<p>Then <span class="math-container">$pq$</span> cannot be a factor of <span class="math-container">$x-y$</span> because <span class="math-con... |
976,617 | <p>Find supremum and infimum of the set:
$B={ \frac{x}{1+ \mid x \mid }} \ for \ x\in \mathbb{R}$
For me it is visible that it will be 1 and -1 respectively but how to prove it properly?</p>
| graydad | 166,967 | <p>You can do both in one move. Show for any $x \in \Bbb{R}$ that $$\left |\frac{x}{1+|x|} \right| \leq 1 $$ Then suppose there is some $c>0$ such that $$\left |\frac{x}{1+|x|} \right|\leq c < 1 $$ for all $x$. Show this case is a contradiction, as you can always find some $x$ where $$\left|\frac{x}{1+|x|} \righ... |
976,617 | <p>Find supremum and infimum of the set:
$B={ \frac{x}{1+ \mid x \mid }} \ for \ x\in \mathbb{R}$
For me it is visible that it will be 1 and -1 respectively but how to prove it properly?</p>
| CLAUDE | 118,773 | <p>It is completely apparent that the magnitude of this function is less than one. Now we want to prove that its supremum is $1$, for this issue, suppose that the supremum is $a$, which according to the previous statement it must be less than or equal to 1. Hence for all values of $x$, $\frac{x}{1+|x|}\leq a\rightarrow... |
3,496,594 | <p>I have to factorize the polynomial <span class="math-container">$P(X)=X^5-1$</span> into irreducible factors in <span class="math-container">$\mathbb{C}$</span> and in <span class="math-container">$\mathbb{R}$</span>, this factorisation happens with the <span class="math-container">$5$</span>th roots of the unity. <... | Dave | 334,366 | <p>In <span class="math-container">$\mathbb C[x]$</span> we can write it as <span class="math-container">$$p(x)=(x-1)(x-\color{blue}{e^{2\pi i/5}})(x-\color{blue}{e^{-2\pi i/5}})(x-\color{red}{e^{4\pi i/5}})(x-\color{red}{e^{-4\pi i/5}})$$</span> where the blue and red are conjugate root pairs. Multiplying these conjug... |
538,811 | <p>Suppose A(.) is a subroutine that takes as input a number in binary, and takes linear time (that is, O(n), where n is the length (in bits) of the number).
Consider the following piece of code, which starts with an n-bit number x.</p>
<p>while x>1:</p>
<p>call A(x)</p>
<p>x=x-1</p>
<p>Assume that the subtraction ... | ashley | 50,188 | <p>Supposing your code is as follows-- so that the inner-loop is both lines: </p>
<pre><code>while (x>1) {
call A(x)
x=x-1
}
</code></pre>
<p>the statements <strong>A(x)</strong> & <strong>x=x-1</strong> run <strong>at O(n)</strong> time each, at each step
of the loop-- asymptotically adding up to <stro... |
2,070,739 | <p>The following paragraph appears on page 42 in the book <em>Rational Number Theory in the 20th Century: From PNT to FLT</em> (Par Wladyslaw Narkiewicz):</p>
<blockquote>
<p>The fact that the strip <span class="math-container">$0<\Re{s}<1/2$</span> contains infinitely many zeros of the zeta-function follows from... | Semiclassical | 137,524 | <p>Looking at some other sources, it appears that to be a typo: It should be "$0<\Re s<1$" not "$0<\Re s<1/2$", i.e. the number of zeros in the critical strip. </p>
<p>In fact, no non-trivial zeros of the Riemann Zeta function occur outside the critical strip, so this restriction is superfluous i.e. the fo... |
2,070,739 | <p>The following paragraph appears on page 42 in the book <em>Rational Number Theory in the 20th Century: From PNT to FLT</em> (Par Wladyslaw Narkiewicz):</p>
<blockquote>
<p>The fact that the strip <span class="math-container">$0<\Re{s}<1/2$</span> contains infinitely many zeros of the zeta-function follows from... | Chappers | 221,811 | <p>Yes, if it were correct, the Riemann hypothesis would be false. It's definitely a typo, however. You can see, for example, <a href="http://www.math.umn.edu/~garrett/m/mfms/notes_2013-14/02c_counting_zeros_of_zeta.pdf" rel="nofollow noreferrer">here</a> for a fairly detailed proof; suffice to say that nothing can be ... |
633,799 | <p>I am a little confused about the basic definition of inclusion.</p>
<p>I understand that, for example, $\{4\}\subset\{4\}$.</p>
<p>I also understand that $4\in\{4\}$, and that it is false to say that $\{4\}\in\{4\}$.</p>
<p>However, is it possible to say that $4\subset\{4\}$?</p>
| Alex R. | 119,906 | <p>No, because $4$ is not a set, but an element.</p>
|
73,550 | <p>I am learning a bit about <a href="http://en.wikipedia.org/wiki/PGF/TikZ" rel="nofollow noreferrer">TikZ</a> and found a nice feature in its graphics, that I am having hard time duplicating with <code>Graphics3D</code>. It is making a <code>Cylinder</code>, where the bottom will have part of its edge, that is behind... | Michael E2 | 4,999 | <p>Well, it was fairly quick for such a simple figure. David Carraher's solution is simpler, except that the envelope is not drawn.</p>
<pre><code>DynamicModule[{vv = {0, 0, 1}, vp = {3.2, 2., 4.}},
Graphics3D[{{Opacity[.5], EdgeForm[], Yellow,
Cylinder[{{0, 0, 0}, {0, 0, 1}}, 1]},
{Directive[Thick, Dashed,... |
2,604,206 | <p>Can anyone provide links to a concrete proof? Intuitively, the two-dimensional real space is infinite. so there should be infinitely many subspaces. But how do I go about a proof?</p>
| Community | -1 | <p>You should use your intuition from Euclidean geometry.</p>
<p>If you fix a point $O$ in a geometrical ($2$-dimensional!) plane, and identify every geometrical vector $v=\vec{OA}$ with its ending point $A$, then:</p>
<ul>
<li>The whole $2$-dimensional vector space is identified with the whole plane;</li>
<li>Any $1... |
637,898 | <p>I have to find the number of irreducible factors of $x^{63}-1$ over $\mathbb F_2$ using the $2$-cyclotomic cosets modulo $63$.</p>
<p>Is there a way to see how many the cyclotomic cosets are and what is their cardinality which is faster than the direct computation?</p>
<p>Thank you.</p>
| user91500 | 91,500 | <p>Note that $x^{p^n}-x\in\mathbb{Z}_p[x]$ equals to product of all irreducible factors of degree $d$ such that $d|n$. Suppose $w_p(d)$ is the number of irreducible factors of degree $d$ on $\mathbb{Z}_p$, then we have
$$p^n=\sum_{d|n}dw_p(d)$$
now use <strong>Mobius Inversion Formula</strong> to obtain
$$w_p(n)=\frac... |
1,811,109 | <p>How can we cause this relation to be true?</p>
<blockquote>
<p>$$x \sin\theta + y \cos\theta = \sqrt{ x^2 + y^2 } \tag{$\star$}$$</p>
</blockquote>
<p>I know the identity</p>
<p>$$x \sin\theta + y \cos\theta = \sqrt{x^2+y^2}\; \sin\left(\theta + \operatorname{atan}\frac{y}{x}\right)$$
What can make the sine pa... | Ng Chung Tak | 299,599 | <p>The equality holds if $$(\sin \theta, \cos \theta)=\left( \frac{x}{\sqrt{x^2+y^2}},\frac{y}{\sqrt{x^2+y^2}} \right)$$</p>
|
304,209 | <p>I am trying to learn weak derivatives. In that, we call <span class="math-container">$\mathbb{C}^{\infty}_{c}$</span> functions as test functions and we use these functions in weak derivatives. I want to understand why these are called <em>test functions</em> and why the functions with these properties are needed. I... | liuyao | 133,582 | <p>Specifically <em>why</em> it is called test function. (Essentially not different from another answer.)</p>
<p>Strichartz liked to introduce the concept with physical intuition. </p>
<p><a href="https://books.google.com/books?id=T7vEOGGDCh4C&lpg=PP1&dq=strichartz%20distribution&pg=PA1#v=onepage&q=te... |
987,054 | <p>Prove that the sequence
$$b_n=\left(1+\frac{1}{n}\right)^{n+1}$$
Is decreasing.</p>
<p>I have calculated $b_n/b_{n-1}$ but it is obtain:
$$\left(1-\frac{1}{n^2}\right)^n \left(1+\frac{1}{n}\right)^n$$
But I can't go on.</p>
<p>Any suggestions please?</p>
| Eclipse Sun | 119,490 | <p>Since the inequality holds for any positive numbers:$$\sqrt[n]{{a_1}{a_2}\cdots {a_n}}\ge \frac{n}{a_1^{-1}a_2^{-1}\cdots a_n^{-1}},$$we have:$$\sqrt[n+2]{\underbrace{\frac{n+1}n\frac{n+1}n\cdots \frac{n+1}n}_{n+1\text{ times}}\cdot 1}\ge \frac{n+2}{\frac n{n+1}+\frac n{n+1}+\cdots +\frac n{n+1}+1}.$$Hence,$$\left(1... |
1,620,512 | <p>$\mathcal P_3(\mathbb R)$ is the set of polynomials with degree at most $3$ with coefficients in $\mathbb R$.</p>
<p>In the last paragraph is says $U$ cannot be extended to a basis of $\mathcal P_3(\mathbb R)$. I do not understand why not. Why would we get a list with length greater than $4$? Why can't we add $(x-5... | Andrew D'Angelo | 295,634 | <p>As we have already determined that $U$ is not equal to $\mathcal{P}_3(\mathbf{R})$, and that $U$ is a subspace of $\mathcal{P}_3(\mathbf{R})$, we know that $U$ cannot have dimension 4 (the same dimension as $\mathcal{P}_3(\mathbf{R})$).</p>
<p>The only way that $U$ would have the same dimension as $\mathcal{P}_3(\m... |
1,620,512 | <p>$\mathcal P_3(\mathbb R)$ is the set of polynomials with degree at most $3$ with coefficients in $\mathbb R$.</p>
<p>In the last paragraph is says $U$ cannot be extended to a basis of $\mathcal P_3(\mathbb R)$. I do not understand why not. Why would we get a list with length greater than $4$? Why can't we add $(x-5... | Pablo Herrera | 135,689 | <p>We know, If $\dim U = 4 $ and $U \subseteq \mathcal{P}_3$ then $U = \mathcal{P}_3$.</p>
<p>But, is easy to show a polynomial in $\mathcal{P}_3$. such that $p'(5)\neq 0 $ (for example $p(x)= x-3$).</p>
<p>that's mean $U$ isn't $\mathcal{P}_3$ finally $\dim U < 4$.</p>
|
911,075 | <p>This is one of my first proofs about fields.
Please feed back and criticise in every way (including style and details).</p>
<p>Let $(F, +, \cdot)$ be a field.
Non-trivially, $\textit{associativity}$ implies that any parentheses are meaningless.
Therefore, we will not use parentheses.
Therefore, we will not use $\te... | Adriano | 76,987 | <p>Your proof is fine. As the comments have pointed out, the proof can be shortened. If you still want to do it in a single chain of equalities, then you can do something like this:</p>
<p>\begin{equation*}
\begin{split}
0a &= 0a + 0 && \quad \text{by }\textit{identity element }(+ ) \\
&= ... |
1,990,670 | <blockquote>
<p>Assume that $0 < \theta < \pi$. Solve the following equation for $\theta$. $$\frac{1}{(\cos \theta)^2} = 2\sqrt{3}\tan\theta - 2$$ </p>
</blockquote>
<p><a href="https://i.stack.imgur.com/SoU8A.png" rel="nofollow noreferrer">Question and Answer</a></p>
<p>Regarding to the attached image, that... | Community | -1 | <p>Multiply both members by $\cos^2\theta$:</p>
<p>$$1=2\sqrt3\sin\theta\cos\theta-2\cos^2\theta=\sqrt3\sin2\theta-1-\cos2\theta.$$</p>
<p>Then</p>
<p>$$\frac{\sqrt3}2\sin2\theta-\frac12\cos2\theta=1,$$
$$\sin\left(2\theta-\frac\pi6\right)=1.$$</p>
|
3,850,320 | <p>If a graph is Eulerian (i.e. has an Eulerian tour), then do we immediately assume for it to be connected?</p>
<p>The reason I ask is because I came across this question:</p>
<p><a href="https://math.stackexchange.com/questions/1689726/graph-and-its-line-graph-that-both-contain-eulerian-circuits">Graph and its line G... | Arthur | 15,500 | <p>We are given that the original graph has an Eulerian circuit. So each edge must be connected to each other edge, regardless of whether the graph itself is connected. Thus the line graph must be connected. Technically this ought to have been pointed out in the answer post you linked, yes.</p>
|
508,059 | <p>How it is possible to considerably shorten the list of properties that define a vector space by using definitions from abstract algebra?</p>
| Loki Clock | 58,287 | <p>Let a $K$-algebra be a ring with a homomorphism from $K$. Then a $K$-vector space $V$ is the underlying addition of a $K$-algebra $R$, together with a product $K \times V \rightarrow V$ given by restriction of the multiplication of $R$.</p>
|
11,175 | <p>This question is a follow-up from <a href="https://mathematica.stackexchange.com/questions/11171/functional-programming-and-do-loops">here</a>. I have a function that generates a list of correlations between some random variables:</p>
<pre><code>varparams = Table[i, {i, 0.1, 0.9, 0.1}];
var[i_] :=
RandomChoice[{... | kglr | 125 | <p>A minor issue with your current test is that it ignores the case where <code>Mean[var[i]]=0</code>. You need to use <code>Variance[var[i]]</code> in your condition. </p>
<p>However, this does not fix the real problem. The real issue with the current formulation is that randomization is performed everytime <code>va... |
4,113,376 | <p>Given following predicates:</p>
<p><span class="math-container">$$
F_1 = (\forall x)(F(x) \leftrightarrow G(x)) \text{ and } F_2 = (\forall x)F(x) \leftrightarrow (\forall x)G(x)
$$</span></p>
<p>I think that they are not equivalent, but if it possible to prove that?</p>
| esoteric-elliptic | 425,395 | <p>Consider <span class="math-container">$a,b\in\mathbb R$</span>.
<span class="math-container">$$\forall x(ax + b = 0) \leftrightarrow \forall x (a = b = 0)$$</span>
certainly holds, but
<span class="math-container">$$\forall x (ax + b = 0 \leftrightarrow a = b = 0)$$</span>
does not!</p>
|
751,138 | <p>Show that if $3\mid(a^2+1)$ then $3$ does not divide $(a+1)$. </p>
<p>using proof of contradiction </p>
<p>can someone prove this using contradiction method please</p>
| mookid | 131,738 | <p><strong>Hint:</strong> just consider the 3 cases
$$a = 3k, a=3k+1, a=3k-1
$$</p>
|
293,921 | <p>The problem I am working on is:</p>
<p>An ATM personal identification number (PIN) consists of four digits, each a 0, 1, 2, . . . 8, or 9, in succession.</p>
<p>a.How many different possible PINs are there if there are no restrictions on the choice of digits?</p>
<p>b.According to a representative at the author’s... | Metin Y. | 49,793 | <p>For d): We have the case: $1$ * * $1$. </p>
<p>But the thief knows, by prohibition (i), it can not be $1111$. Thus he eleminates $1$ possibility.</p>
<p>Also he knows, by (iii), the second digit can not be $9$. There are exactly $10$ number of the form $19$ * $1$, namely, $1900$, $1910$, $1920$... So, at this stag... |
1,177,349 | <p>Let $\gamma = e^{2 \pi i/5} + (e^{2 \pi i/5})^4
%γ = e<sup>2πi / 5</sup> + (e<sup>2πi / 5</sup>) <sup>4</sup>
$.</p>
<p>I am looking for the basis for $[\mathbb{Q}(\gamma):\mathbb{Q}] = 2$, and then looking for a dependence between $\gamma^2,\gamma$, and $1$. </p>
<p>I've worke... | String | 94,971 | <p>Multiply $x<y$ by the positive numbers $x$ and $y$ in turn to get $xx<xy$ and $xy<yy$ so that
$$
xx<xy<yy
$$</p>
|
1,177,349 | <p>Let $\gamma = e^{2 \pi i/5} + (e^{2 \pi i/5})^4
%γ = e<sup>2πi / 5</sup> + (e<sup>2πi / 5</sup>) <sup>4</sup>
$.</p>
<p>I am looking for the basis for $[\mathbb{Q}(\gamma):\mathbb{Q}] = 2$, and then looking for a dependence between $\gamma^2,\gamma$, and $1$. </p>
<p>I've worke... | marty cohen | 13,079 | <p>Using the definition of "$>$":</p>
<p>If $y > x$,
there is a $d > 0$
such that
$y = x+d$.</p>
<p>Then
$y^2-x^2
=(x+d)^2-x^2
=x^2+2xd+d^2-x^2
= 2xd+d^2
> 0
$
since both
$2xd$ and $d^2$
are positive.</p>
|
4,114,034 | <p>In Linear Algebra Done Right by Axler, there are two sentences he uses to describe the uniqueness of Linear Maps (3.5) which I cannot reconcile. Namely, whether the uniqueness of Linear Maps is determined by the choice of 1) <em>basis</em> or 2) <em>subspace</em>. These two seem like very different statements to me ... | D_S | 28,556 | <p>Let <span class="math-container">$f$</span> be a function with domain <span class="math-container">$D$</span> and range <span class="math-container">$R$</span>. What does it mean for a point <span class="math-container">$x \in D$</span> to be a fixed point? It means that if you plug <span class="math-container">$x... |
1,271,935 | <p>Let $(e_n)$ (where $ e_n $ has a 1 in the $n$-th place and zeros otherwise) be unit standard vectors of $\ell_\infty$. </p>
<p>Why is $(e_n)$ not a basis for $\ell_\infty$?</p>
<p>Thanks.</p>
| ナウシカ | 213,845 | <p>A Schauder basis is a dense countable subset. But there does not exist any dense countable subset in $\ell^\infty$.</p>
<p>Let us prove that no such set can exist:</p>
<p>By contradiction, assume that $D$ was a dense countable subset of $\ell^\infty$. Now consider the set $S = \{s\mid s_n \in \{0,1\}\}$, that is, ... |
28,195 | <p>So yesterday I came across a question. Something seemed suspicious (a badly worded question and an incorrect answer accepted), so I did some snooping. It appears that every question from the OP has been answered by the same user within minutes of posting, and subsequently upvoted and accepted. </p>
<p>I suspect ... | Jay | 144,901 | <p>Assuming that you are correct about what happened, and it's not just a coincidence or some such ...</p>
<p>(a) This seems really pointless. So a person boosts his reputation score by cheating. Why? What did he gain? Unless there's something going on that I don't know about, there are no cash prizes for high reputat... |
4,409,290 | <p>Can I have a matrix <span class="math-container">$Q$</span> which is orthogonal because each of the column vectors dot products with each other is 0? Or must only satisfy <span class="math-container">$QQ^T=I$</span>. For example consider the following matrix <span class="math-container">$Q$</span>:</p>
<p><span cla... | A. P. | 1,027,216 | <p>By definition a matrix <span class="math-container">$Q\in \mathcal{M}_{n}(\mathbb{R})$</span> is orthogonal <strong>if</strong>:</p>
<ol>
<li><span class="math-container">$Q$</span> is invertible.</li>
<li><span class="math-container">$Q^{T}=Q^{-1}$</span>.</li>
</ol>
<p>Moreover, we know that <span class="math-cont... |
311,849 | <p>How to evaluate:
$$ \int_0^\infty e^{-x^2} \cos^n(x) dx$$</p>
<p>Someone has posted this question on fb. I hope it's not duplicate.</p>
| Ayman Hourieh | 4,583 | <p>Using the <a href="http://en.wikipedia.org/wiki/List_of_trigonometric_identities#Power-reduction_formula" rel="nofollow">power reduction formula</a> for $\cos$ for odd powers:</p>
<p>\begin{align}
I &= \int_0^\infty e^{-x^2} \cos^n(x) \, dx \\
&= \int_0^\infty e^{-x^2} \left( \frac{2}{2^n} \sum_{k=0}^{(n-1)... |
1,044,910 | <blockquote>
<p>Prove that $$\sum_{i = 2^{n-1} + 1}^{2^n}\frac{1}{a + ib} \ge \frac{1}{a + 2b}$$</p>
</blockquote>
<p>I tried to to prove the above statement using the AM-HM inequality:</p>
<p>$$\begin{align}\frac{1}{2^n - 2^{n-1}}\sum_{i = 2^{n-1} + 1}^{2^n}\frac{1}{a + ib} &\ge \frac{2^n - 2^{n-1}}{\sum_{i = ... | Kim Jong Un | 136,641 | <p>For $a,b,x>0$, the function $f(x)=\frac{1}{a+bx}$ is convex. So by Jensen's inequality
$$
\frac{1}{2^{n-1}}\sum_{i=2^{n-1}+1}^{2^n}f(i)\geq f\left(\frac{1}{2^{n-1}}\sum_{i=2^{n-1}+1}^{2^n}i\right)=f(0.5+3\times2^{n-2})=\frac{1}{a+(0.5+3\times2^{n-2})b}.
$$
It remains to show $2^{n-1}$ times the rightmost expressi... |
3,055,649 | <p>Can there be more than four different types of polygons meeting at a vertex? How?
(The polygons must be convex, regular and different)</p>
<p>There are two ways to fit 5 regular polygons around a vertex, what are they?
(The polygons must be regular, they may not be of different types)</p>
| user26872 | 26,872 | <p><span class="math-container">$\def\b{\beta}$</span><span class="math-container">\begin{align*}
\newcommand\cmt[1]{{\small\textrm{#1}}}
I_n(\b) &= \int_0^{\pi/2}
\left(\frac{\sin (2n+1)x}{\sin x}\right)^\b dx \\
&= \frac 1 4 \int_0^{2\pi}
\left(\frac{\sin (2n+1)x}{\sin x}\right)^\b dx
& \cmt{begin si... |
315,457 | <p>I am trying to evaluate $\cos(x)$ at the point $x=3$ with $7$ decimal places to be correct. There is no requirement to be the most efficient but only evaluate at this point.</p>
<p>Currently, I am thinking first write $x=\pi+x'$ where $x'=-0.14159265358979312$ and then use Taylor series $\cos(x)=\sum_{i=1}^n(-1)^n\... | dineshdileep | 41,541 | <p>The best place to look is this <a href="http://en.wikipedia.org/wiki/Perron%E2%80%93Frobenius_theorem" rel="nofollow noreferrer">wiki link</a>. To add to the other answer, another equivalent condition is that for every index $[i,j]$, there should be a $m$ such that $(A^m)_{ij}>0$ which is naturally satisfied if t... |
1,101,371 | <p>Any book that I find on abstract algebra is somehow advanced and not OK for self-learning. I am high-school student with high-school math knowledge. Please someone tell me a book can be fine on abstract algebra? Thanks a lot. </p>
| user153330 | 153,330 | <p>I'm astonished that this wasn't mentioned:</p>
<blockquote>
<p><a href="http://rads.stackoverflow.com/amzn/click/1939512018" rel="nofollow noreferrer"><em>Learning Modern Algebra</em>
<strong>From Early Attempts to Prove</strong>
<strong><em>Fermat’s Last Theorem</em></strong> By Al Cuoco and Joseph J. Rotman... |
291,684 | <p>Linear ODE systems $x'=Ax$ are well understood. Suppose I have a quadratic ODE system where each component satisfies $x_i'=x^T A_i x$ for given matrix $A_i$. What resources, textbooks or papers, are there that study these systems thoroughly? My guess is that they aren't completely understood, but it would be good to... | Artem | 29,547 | <p>You are right, these systems are not completely understood even in the simple case of dimension $n=2$ (i.e., on the plane). You can check out <a href="http://rads.stackoverflow.com/amzn/click/1441940243" rel="noreferrer">this book</a> on planar quadratic differential equations. In general you can consider systems of... |
701,122 | <p>Part 1
Let $f(x) = ax^n$, where $a$ is any real number. Prove that $f$ is even if $n$ is an even integer. (Integers can be negative too)</p>
<p>Part 2
Prove that if you add any two even functions, you get an even function</p>
<p>I'm confused as to how you would prove adding two even functions would get you an eve... | Community | -1 | <p>Part 1)
$$f(x)=ax^n$$
So, let $n=2k$. Then,
$$f(x)=ax^n=a(x^2)^k$$
Obviously, $x^2$ is an even function ($(-x)^2=x^2$). So,
$$f(x)=ax^n=a(x^2)^k$$
is an even function.</p>
<p>Part 2)
Let $f(x),g(x)$ be two even functions. Then,
$$f(x)=f(-x),\qquad g(x)=g(-x)$$
Adding the two gives
$$f(x)+g(x)=h(x)$$
Using the abov... |
3,055,904 | <p>Everyone help me please!</p>
<p>Find for all <span class="math-container">$x$</span> is positive
Such that
<span class="math-container">$x^2+5x+2$</span> is perfect square of integer
First I suppose <span class="math-container">$x^2+5x+2= y^2$</span> and substract <span class="math-container">$y^2$</span> from both... | clathratus | 583,016 | <p>Not yet a complete answer</p>
<p><span class="math-container">$$P=\prod_{n=1}^{\infty}(1-e^{-18\pi n})$$</span>
I don't think it is possible without special functions, but it is worthwhile to recall the definition of the Euler function
<span class="math-container">$$\phi(q)=\prod_{n=1}^{\infty}(1-q^n)$$</span>
Imme... |
3,055,904 | <p>Everyone help me please!</p>
<p>Find for all <span class="math-container">$x$</span> is positive
Such that
<span class="math-container">$x^2+5x+2$</span> is perfect square of integer
First I suppose <span class="math-container">$x^2+5x+2= y^2$</span> and substract <span class="math-container">$y^2$</span> from both... | Paramanand Singh | 72,031 | <p>You need to go step by step. Let <span class="math-container">$$F(q) =\prod_{n=1}^{\infty} (1-q^{2n})\tag{1}$$</span> where <span class="math-container">$0<q<1$</span>. Then you wish to seek the value of <span class="math-container">$F(q^9)$</span> where <span class="math-container">$q=e^{-\pi} $</span>. Fortu... |
1,007,309 | <p><img src="https://i.stack.imgur.com/xJKtU.png" alt="enter image description here"></p>
<p>I cannot understand part ii) in this solution. I cannot see the significance of arbitrarily close to 0 points for which $|sin(\frac{1}{x_n})|=1$</p>
| Mr. Barrrington | 73,431 | <p>Well, if $g^{-1}(B)$ was open, it should contain an $\varepsilon$-ball with center $0$ for some $\varepsilon>0$, since $0\in g^{-1}(B)$. But for all $\varepsilon>0$ you find a point in this ball, which maps to $1$, hence the ball cannot lie in $g^{-1}((-\frac{1}{2},\frac{1}{2})$!</p>
|
144,364 | <p>I have been in a debate over <a href="http://9gag.com/" rel="noreferrer">9gag</a> with this new comic: <a href="http://9gag.com/gag/4145133" rel="noreferrer">"The Origins"</a></p>
<p><img src="https://i.stack.imgur.com/BOEnP.jpg" alt=""-1 doesn't have a square root?" "Here come imaginary numbers&... | Kaz | 28,530 | <p>The square root of a number $k$ may be regarded as the root of a polynomial of the form $z^2 - k$. The fundamental theorem of algebra tells us that we can write $z^2 - k$ as $(z - m)(z - n)$, where $m$ and $n$ are the roots which this function necessarily has, being of degree two. $z$, $k$, $m$ and $n$ are all comp... |
995,029 | <p>I want to find the power series of $\frac{1}{3!}$ in the field $\mathbb{Q}_3$.</p>
<p>In order to do this, do I have to solve the congruence $3!x \equiv 1 \pmod{3^n} \Rightarrow 6x \equiv 1 \pmod 3$?</p>
<p>If so, for $n=1$, we get $0 \equiv 1 \pmod 3$, that is a contradiction.</p>
<p>What does this mean?</p>
| Daniel Fischer | 83,702 | <p>It's not a power series but a Laurent series. You can express elements of $\mathbb{Q}_p$ as series</p>
<p>$$\sum_{k=m}^\infty a_k p^k,$$</p>
<p>where $m\in\mathbb{Z}$ and each $a_k$ is an integer between $0$ and $p-1$ inclusive.</p>
<p>For $\frac{1}{3!}$ in $\mathbb{Q}_3$, the shortcut way to find the series is t... |
995,029 | <p>I want to find the power series of $\frac{1}{3!}$ in the field $\mathbb{Q}_3$.</p>
<p>In order to do this, do I have to solve the congruence $3!x \equiv 1 \pmod{3^n} \Rightarrow 6x \equiv 1 \pmod 3$?</p>
<p>If so, for $n=1$, we get $0 \equiv 1 \pmod 3$, that is a contradiction.</p>
<p>What does this mean?</p>
| Lubin | 17,760 | <p>Can I cut through the fog with a somewhat different approach? You can write elements of $\mathbb Z_p$ as infinite $p$-ary expansions going to the left, a typical one would look like
$$
\cdots a_4a_3a_2a_1a_0;
$$
which means $a_0+pa_1+p^2a_2+\cdots=\sum_{i\ge0}a^ip^i$, and as appropriate for a $p$-ary expansion, all ... |
962,287 | <p>I am trying to isolate x in the equation $$(x-20)^{2} = -(y-40)^{2} - 525.$$ How can I do it?</p>
| Deepak | 151,732 | <p>$$(x-20)^{2} = -(y-40)^{2} - 525$$</p>
<p>Square root both sides:</p>
<p>$$x-20= \pm \sqrt{-(y-40)^{2} - 525}$$</p>
<p>Rearrange:</p>
<p>$$x= 20 \pm \sqrt{-(y-40)^{2} - 525}$$</p>
<p>Or equivalently,</p>
<p>$$x= 20 \pm i\sqrt{(y-40)^{2} + 525}$$</p>
|
4,572,517 | <p>Given two 3x3 matrix:</p>
<p><span class="math-container">$$
V=
\begin{bmatrix}
1 & 0 & 9 \cr
6 & 4 & -18 \cr
-3 & 0 & 13 \cr
\end{bmatrix}\quad
W=
\begin{bmatrix}
13 & 9 & 3 \cr
-14 & -8 & 2 \cr
5 & 3 & -1 \cr
\end{bmatrix}
$$</span></p>
<p>Is there any way to predic... | egreg | 62,967 | <p>Your proof is fine. The hypothesis that <span class="math-container">$V$</span> is finite dimensional ensures the existence of the polynomial <span class="math-container">$p$</span>. But the argument doesn't need such existence. <em>If</em> such a polynomial exists, then its roots are eigenvalues.</p>
<p>Indeed, if ... |
19,586 | <p>I am looking for resources for teaching math modeling to high school teachers with rusty math background. It will be a 6-week course. Some tips/directions on simple projects would be helpful. I would want to introduce coding for numerically solving systems of ODEs.</p>
| ncr | 1,537 | <p>The information and handbooks at <a href="https://m3challenge.siam.org/resources" rel="nofollow noreferrer">SIAM Mathworks Modelling Challenge</a> pretty helpful. One handbook introduces the reader to the modeling process and the second one introduces the reader to the communication of computation related to modeli... |
2,666,772 | <blockquote>
<p>$W$ = $\begin{bmatrix} 1 & 1 & 1 & 1 \\ 1 & -1 & 1 & -1 \\ 1 & 1 & -1 & -1 \\ 1 & -1 & -1 & 1 \end{bmatrix}$. Use W to build an 8x8 matrix encoding an orthonormal basis in $R^8$ by scaling A = $\begin{bmatrix} W & W \\ W & -W \end{bmatrix}$ in t... | Martin Argerami | 22,857 | <p>You have
$$
A^TA=\begin{bmatrix} W^T & W^T \\ W^T & -W^T \end{bmatrix}\begin{bmatrix} W & W \\ W & -W \end{bmatrix}=\begin{bmatrix}2W^TW&0\\ 0&2W^TW \end{bmatrix}.
$$
Since $W^TW=4I$, we got $A^TA=8I$. Then
$$
\frac1{2\sqrt2}\,\begin{bmatrix} W & W \\ W & -W \end{bmatrix}
$$
is an or... |
4,005,522 | <p>Let <span class="math-container">$U$</span> be connected open set of <span class="math-container">$\mathbb{R}^{n}$</span>. Consider <span class="math-container">$C^{\infty}(U)$</span> with sup norm, we said <span class="math-container">$A\subset C^{\infty}(U)$</span> is a minimal dense subalgebra of <span class="mat... | Ulli | 821,203 | <p>Concerning the third bullet:
It should definitely read as "whenever X is extremally disconnected <strong>and compact</strong>". Otherwise, <span class="math-container">$\mathbb{N}$</span> is extremally disconnected, but <span class="math-container">$\mathbb{N} \setminus \{1\}$</span> is not almost compact.... |
335,929 | <p>Let $A=\{1, 2,\dots,n\}$
What is the maximum possible number of subsets of $A$ with the property that any two of them have exactly one element in common ?</p>
<p>I strongly suspect the answer is $n$, but can't prove it.</p>
| Sean Eberhard | 23,805 | <p>This is a well known type of problem in combinatorics. (Try googling "exact intersections".) The (slight) generalisation of your claim, in which we require $|A\cap B|=\ell>0$ whenever $A\neq B$, is apparently due to Fisher.</p>
<p>Let $\mathcal{A}$ be a family of subsets of $\{1,2,\dots,n\}$ such that for every ... |
564,195 | <p>Using continuity I was able to show the sequence $x_0 = 1$, $x_{n+1} = sin(x_n)$ converges to 0, but I was wondering if there was a way to prove it using only properties and theorems related to sequences and series, without using continuity.</p>
<p>So far, I know the sequence is monotonically decreasing and bounded... | Suraj M S | 85,213 | <p>$$\sin(\sin...sin(1)))....)) =L$$
$$ \sin(\sin...sin(1)))....)) =arcsin(L)$$
$$L=arcsin(L)$$
$$\sin(L) =L$$
$$L=0$$</p>
|
41,883 | <p>Let $G$ be an abelian group, $A$ a trivial $G$-module. We know that $\text{Ext}(G,A)$ classifies abelian extensions of $G$ by $A$, whereas $H^2(G,A)$ classifies central extensions of $G$ by $A$. So we have a canonical inclusion $\text{Ext}(G,A)\hookrightarrow H^2(G,A)$. Is there some naturally arising exact sequenc... | Francesco Polizzi | 7,460 | <p>Let $G$ and $A$ be groups and assume that $R \rightarrowtail F \twoheadrightarrow G$ is a presentation of $G$.</p>
<p>Then, by a general theorem of MacLane, there is an exact sequence</p>
<p>$\textrm{Hom}(F_{ab}, A) \to \textrm{Hom}(R/[R,F], A) \to H^2(G, A) \to 0$.</p>
<p>Moreover, if $G$ and $A$ are Abelian the... |
3,963,479 | <p>In a quadrilateral <span class="math-container">$ABCD$</span>, there is an inscribed circle centered at <span class="math-container">$O$</span>. Let <span class="math-container">$F,N,E,M$</span> be the points on the circle that touch the quadrilateral, such that <span class="math-container">$F$</span> is on <span cl... | Quanto | 686,284 | <p><a href="https://i.stack.imgur.com/ceiEW.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/ceiEW.png" alt="enter image description here" /></a></p>
<p>Construct line from C parallel to the chord MN, meeting the extension of AD at X. Then, XMNC is isosceles, implying XM = CN = CE. Note that the trian... |
1,697,206 | <p>In the figure, $BG=10$, $AG=13$, $DC=12$, and $m\angle DBC=39^\circ$.</p>
<p>Given that $AB=BC$, find $AD$ and $m\angle ABC$.</p>
<p>Here is the figure:</p>
<p><a href="https://i.stack.imgur.com/u05wa.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/u05wa.jpg" alt="enter image description here">... | assefamaru | 262,896 | <p>Since triangle $ABC$ is isosceles, with $AB=BC$,</p>
<ol>
<li>To find $\angle ABC$, notice that $\angle BCD=51^0=\angle DAB$, hence $\angle ABD=39^0$, and $\angle ABC=39^o+39^o=78^o $.</li>
<li>To find $AD$, since ABC is an isosceles triangle, and since $\angle ABD=\angle DBC$, the line $BD$ divides $AC$ into two e... |
3,112,682 | <p>I was looking at</p>
<blockquote>
<p><em>Izzo, Alexander J.</em>, <a href="http://dx.doi.org/10.2307/2159282" rel="nofollow noreferrer"><strong>A functional analysis proof of the existence of Haar measure on locally compact Abelian groups</strong></a>, Proc. Am. Math. Soc. 115, No. 2, 581-583 (1992). <a href="htt... | Bumblebee | 156,886 | <p>Assume the required quantity is <span class="math-container">$2\cos n\theta$</span> (you can see this squaring the given identity). Then apply the induction using the identity <span class="math-container">$$x^{n+1}+\dfrac{1}{x^{n+1}}=\left(x+\dfrac1x\right)\left(x^{n}+\dfrac{1}{x^{n}}\right)-\left(x^{n-1}+\dfrac{1}{... |
273,219 | <p>I have been studying for the AP BC Calculus exam (see this <a href="https://math.stackexchange.com/questions/272628/how-know-which-direction-a-particle-is-moving-on-a-polar-curve">previous question</a>) and most of the questions that deal with the first derivative in polar coordinates say that if ${dr\over d\theta}&... | Thomas Andrews | 7,933 | <p>Just looking at finite simple groups, the <a href="http://en.wikipedia.org/wiki/Classification_of_finite_simple_groups" rel="nofollow">classification is fairly complex</a>, and I doubt you'd call all of the examples, "familiar."</p>
<p>In particular, the odd examples are the so-called <a href="http://en.wikipedia.o... |
3,082,873 | <p>Consider the set of all boolean square matrices of order <span class="math-container">$3 \times 3$</span> as shown below where a,b,c,d,e,f can be either 0 or 1.</p>
<p><span class="math-container">$\begin{bmatrix}
a&b&c\\
0&d&e\\
0&0&f
\end{bmatrix}$</span></p>
<p>Out of all p... | Siong Thye Goh | 306,553 | <p>To be singular, we need <span class="math-container">$a=0$</span> or <span class="math-container">$d=0$</span> or <span class="math-container">$f=0$</span>.</p>
<p>To be non-singular, we need <span class="math-container">$a=d=f=1$</span>.</p>
<p>Hence, the probaility is <span class="math-container">$$1-\frac{1}{2^... |
1,229,467 | <p>I tried substituting the $4$ into the $3x-5$ equation, so my slope would be represented as $3(4)-5 = 7$. Then my equation for the line would be $y-3 = 7(x-4)$. That means the equation of the line would be $y = 7x - 25$. However, I'm trying to submit this answer online and it comes back as incorrect. Any help would b... | user46372819 | 86,425 | <p>You have the slope is $7$ at the point $(4,3)$ which is correct. But, what you found was the equation of the tangent line to the curve at the point $(4,3)$.</p>
<p><strong>SOLUTION:</strong> We need to integrate $\frac{dy}{dx}$ to get the function of the original curve. So,
$$y=\int (3x-5) \ dx=\frac{3}{2}x^2-5x+C.... |
682,741 | <p>Use the Mean Value Theorem to prove that if $p>1$ then $(1+x)^p>1+px$ for $x \in (-1,0)\cup (0,\infty)$</p>
<p>How do I go about doing this?</p>
| Paramanand Singh | 72,031 | <p><strong>Hint</strong>: Try $f(x) = (1 + x)^{p}$ and then use $f(x) - f(0) = xf'(c)$. Clearly $f'(x) = p(1 + x)^{p - 1}$.</p>
<blockquote class="spoiler">
<p> If $x > 0$ then clearly we have $0 < c < x$ and therefore $(1 + c) > 1$ and that implies $f'(c) = p(1 + c)^{p - 1} > p$ so that $f(x) - f(0) ... |
129,530 | <p>This book, which needs to be returned quite soon, has a problem I don't know where to start. How do I find a 4 parameter solution to the equation</p>
<p>$x^2+axy+by^2=u^2+auv+bv^2$</p>
<p>The title of the section this problem comes from is entitled (as this question is titled) "Numbers of the Form $x^2+axy+by^2$"... | Will Jagy | 10,400 | <p>Oh, well. This is from page 57 of <em>Binary Quadratic Forms</em> by Duncan A. Buell. As long as $$ \gcd(a_1, a_2, B) = 1 $$ we have
$$ (a_1 x_1^2 + B x_1 y_1 + a_2 C y_1^2) (a_2 x_2^2 + B x_2 y_2 + a_1 C y_2^2) = a_1 a_2 X^2 + B X Y + C Y^2, $$
where we take
$$ X = x_1 x_2 - C y_1 y_1, \; \; \; \; Y = a_1 x_1 y_... |
3,649,221 | <blockquote>
<p>Suppose <span class="math-container">$\{f_n\}$</span> is an equicontinuous sequence of functions defined on <span class="math-container">$[0,1]$</span> and <span class="math-container">$\{f_n(r)\}$</span> converges <span class="math-container">$∀r ∈ \mathbb{Q} \cap [0, 1]$</span>. Prove that <span cla... | Reveillark | 122,262 | <p>Step 1: For every <span class="math-container">$x\in [0,1]$</span>, <span class="math-container">$\{f_n(x)\}_n$</span> converges. </p>
<p>Fix <span class="math-container">$\varepsilon >0$</span>. Pick <span class="math-container">$\delta>0$</span> witnessing the definition of equicontinuity for <span class="m... |
3,649,221 | <blockquote>
<p>Suppose <span class="math-container">$\{f_n\}$</span> is an equicontinuous sequence of functions defined on <span class="math-container">$[0,1]$</span> and <span class="math-container">$\{f_n(r)\}$</span> converges <span class="math-container">$∀r ∈ \mathbb{Q} \cap [0, 1]$</span>. Prove that <span cla... | WannaBeRealAnalysist | 319,730 | <p>This is actually a simple case of applying the <em><strong>Arzela-Ascoli Propagation Theorem</strong></em> which states:</p>
<p><em>Point-wise convergence of an equicontinuous sequence of functions on a dense subset of the domain propagates to uniform
convergence on the whole domain.</em></p>
<p>The rational numbers... |
3,056,616 | <p><span class="math-container">$P(x) = 0$</span> is a polynomial equation having <strong>at least one</strong> integer root, where <span class="math-container">$P(x)$</span> is a polynomial of degree five and having integer coefficients. If <span class="math-container">$P(2) = 3$</span> and <span class="math-containe... | Eric Wofsey | 86,856 | <p>The assumption that <span class="math-container">$P$</span> has degree <span class="math-container">$5$</span> is irrelevant and unhelpful.</p>
<p>If <span class="math-container">$r$</span> is a root of <span class="math-container">$P$</span>, we can write <span class="math-container">$P(x)=(x-r)Q(x)$</span> for so... |
594,975 | <p>What will be the basis of vector space <span class="math-container">$\Bbb C$</span> over a field of rational numbers <span class="math-container">$\Bbb Q$</span>?</p>
<p>I think it will be an infinite basis! I think it will be <span class="math-container">$B=\{r_1+r_2i \mid r_1, r_2 \in \Bbb Q^{c}\}\cup\{1,i\}$</s... | tomasz | 30,222 | <p>Note first that for any ring $R$ and any $R$-module $M$, if the cardinality $\lvert M\rvert$ is infinite and greater than $\lvert R \rvert$, then any generating set of $M$ (in particular, any basis) must have cardinality equal to $\lvert M\rvert$ (this is simple combinatorics).</p>
<p>In particular, any basis of $\... |
2,752,073 | <p>Looking for a finite recursive formula for the constant e preferably using standard operators (ones a calculator could carry out)</p>
<p>i.e. a formula of the form $ x_{n+1}=f(x_n)$ where $\lim_{n\rightarrow\infty}(x_n)=e$ for some $x_1$</p>
<p>Bonus points if it works for any $x_1$</p>
| N. S. | 9,176 | <p>$$x_{n+1}= \left( 1+ \frac{\max\{ |x_n|,1 \} }{n}\right)^{\frac{\max\{ |x_n|,1 \}}{n}}$$</p>
|
2,752,073 | <p>Looking for a finite recursive formula for the constant e preferably using standard operators (ones a calculator could carry out)</p>
<p>i.e. a formula of the form $ x_{n+1}=f(x_n)$ where $\lim_{n\rightarrow\infty}(x_n)=e$ for some $x_1$</p>
<p>Bonus points if it works for any $x_1$</p>
| Brian Tung | 224,454 | <p>One approach is to use Newton's method on a function known to have a zero at $e$—for instance, $f(x) = \ln x - 1$. Then $f'(x) = 1/x$, and we can iterate as</p>
<p>$$
x_{n+1} = x_n (2-\ln x_n)
$$</p>
<p>Of course, this isn't very satisfying, since the natural log is sitting right there in the expression. (I also... |
4,015,203 | <p>How would you prove the following property about covariances?</p>
<p><a href="https://i.stack.imgur.com/y18wd.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/y18wd.png" alt="enter image description here" /></a></p>
<p>I found it here:</p>
<p><a href="https://www.probabilitycourse.com/chapter5/5_3_... | drhab | 75,923 | <p>It is enough to prove for suitable random variables <span class="math-container">$X,Y,Z$</span> and constant <span class="math-container">$a$</span> that:</p>
<ul>
<li><span class="math-container">$\mathsf{Cov}(X+Y,Z)=\mathsf{Cov}(X,Z)+\mathsf{Cov}(Y,Z)$</span></li>
<li><span class="math-container">$\mathsf{Cov}(aX,... |
2,368,179 | <p>Answer should be in radians
Like π/4 (45°) π(90°).
I used $\tan(A+B)$ formula and got $5/7$ as the answer, but that's obviously wrong.</p>
| MCCCS | 357,924 | <p>Another approach:</p>
<blockquote>
<p>$\tan a + \tan b = \frac{\sin(a+b)}{\cos a \cos b}$</p>
</blockquote>
<p>Then</p>
<p>$\tan a + \tan b = \frac{\sin(a+b)}{\cos a \cos b}$</p>
<blockquote>
<p>$\cos (\tan^{-1}x) = \frac{1}{\sqrt{x^2+1}}$</p>
</blockquote>
<p>$-\frac{1}{2} + -\frac{1}{3} = \frac{\sin(a+b)}... |
2,762,237 | <blockquote>
<p>Does the sequence $(x_n)_{n=1}^\infty$ with $x_{n+1}=2\sqrt{x_n}$ converge?</p>
</blockquote>
<p>I'm almost positive this converges but I am not entirely sure how to go about this. The square root is really throwing me off as I haven't dealt with it at all up until now.</p>
| marty cohen | 13,079 | <p>First, find the limit.</p>
<p>Suppose the sequence has a limit L.
Then $L=2\sqrt{L}$,
so $L=4$.
Note: this does not show
that the limit exists.
It shows that if the limit does exist then it must be 4.</p>
<p>Then, see how the limit is approached.</p>
<p>If
$y=2\sqrt{x}$,
then
$y-4=2\sqrt{x}-4
=2(\sqrt{x}-2)
$
so
... |
2,420,255 | <p>If the price of an article is increased by percent $p$, then the decrease in percent of sales must not exceed $d$ in order to yield the same income. The value of $d$ is:
$\textbf{(A)}\ \frac{1}{1+p} \qquad \textbf{(B)}\ \frac{1}{1-p} \qquad \textbf{(C)}\ \frac{p}{1+p} \qquad \textbf{(D)}\ \frac{p}{p-1}\qquad \textbf... | Intelligenti pauca | 255,730 | <p>Your construction can be rephrased in a more geometrical way as follows: given a triangle $ABC$, we construct a triangle $DBE$ having angle $\angle B$ in common with $ABC$ (in your case $\angle B$ is a right angle, but that is not necessary) and $AD=BD-BA=BC-BE=EC$ (see diagram below). Lines through $A$ and $D$, par... |
1,388,434 | <p>I was brushing up on some calculus and I was thinking about the following function:</p>
<blockquote>
<p>Let$$f(x) =
\begin{cases}
\frac{yx^{6} +y^{3}+x^{3}y}{x^{6}+y^{2}} & \text{for $(x,y)$ $\neq (0,0)$,} \\
0 & \text{for $(x,y)$ $=$ $(0,0)$. } \\
\end{cases}$$The function $f$ is continuous over the en... | ignoramus | 155,096 | <p><strong>Hint:</strong> consider approaching the origin along the path $(x,x^3)$</p>
|
73,912 | <p>For my work, I am examining the values of a complex function as I vary the input according to a real parameter, and I want to both give the general plot and the plot of specific points, with labels (so one sees the direction of increase). </p>
<p>I knew from the documentation that <code>Point</code> and <code>Epil... | kglr | 125 | <p>You are five keystrokes (<code>[[1]]</code>) close to something that works (See also: <a href="https://mathematica.stackexchange.com/a/71997/125">this</a> and <a href="https://mathematica.stackexchange.com/a/69940/125">this</a>)</p>
<pre><code>ParametricPlot[parts[ourF[x + I/4]], {x, -3, 3}, PlotRangeClipping ->... |
268,091 | <p><a href="https://i.stack.imgur.com/PAO6T.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/PAO6T.jpg" alt="enter image description here" /></a></p>
<p>I have to solve ODE x'(t)=1/2(x(t))-t, x(0)
The existence of solutions of this IVP is equivalent to finding a fixed point of integral operator T:C[0,... | I.M. | 26,815 | <p>Can be also solved with <code>AsymptoticDSolveValue</code></p>
<pre><code>AsymptoticDSolveValue[{x'[t] == 1/2 x[t] - t, x[0] == 0}, x[t], t, {t, 0, 4}]
(* -(t^2/2)-t^3/12-t^4/96 *)
</code></pre>
|
1,676,347 | <p>Suppose that $V$ and $W$ are vector spaces, $g:V\rightarrow W$ a linear map. Show that g is surjective $\iff$ For any vector space $U$ the map $g^{\ast}:Hom(W,U) \rightarrow Hom(V,U)$ defined by $g^{\ast}(f) = f\circ g$ is injective.</p>
<p>I've failed too much trying to solve this problem. Any hint could be useful... | Luiz Cordeiro | 58,818 | <p>I thinkg the notation can become quite heavy at some point, so let's try to make thinkgs easy. You can show that the map $g^*$ is always linear.</p>
<p>Suppose $g$ is surjective. Let $U$ be any vector space. Let's show that the map $g^*$ described above is injective. Suppose $f:W\to U$ satisfies $g^*(f)=0$, that is... |
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