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 |
|---|---|---|---|---|
3,608,114 | <p>Consider the example where I have a matrix <span class="math-container">$\mathbf{D}$</span> in <span class="math-container">$-1/1$</span> coding with <span class="math-container">$5$</span> columns,</p>
<p><span class="math-container">$$D = \begin{bmatrix}-1&-1&-1&1&1\\1&-1&-1&-1&1\\... | P. Lawrence | 545,558 | <p>For an <span class="math-container">$m \times n$</span> matrix <span class="math-container">$A$</span> let <span class="math-container">$A'$</span> be the unique row-reduced echelon matrixx that is row-equivalent to <span class="math-container">$A.$</span> The relations of linear dependence among the columns of <spa... |
575,513 | <p>Can someone help me find the density function $f_X$ for $X$ and hence find $E(X)$ and $Var(X)$ of the following distribution function $F_X$ given by:</p>
<p>$F_X(x)=\begin{cases}
1-(1+x)e^{-x} & x>0 \\
0 & otherwise.
\end{cases}$</p>
<p>$X$ is a continuous random variable.</p>
<p>From memory, do I h... | tomasz | 30,222 | <p>An example for a commutative ring which is not a domain: $R=\{0,a,1-a,1\}$ with $a^2=a,a+a=1+1=0$ and $A=R$. The elements $a,1-a$ are torsion, but $a+(1-a)=1$ isn't.</p>
|
575,513 | <p>Can someone help me find the density function $f_X$ for $X$ and hence find $E(X)$ and $Var(X)$ of the following distribution function $F_X$ given by:</p>
<p>$F_X(x)=\begin{cases}
1-(1+x)e^{-x} & x>0 \\
0 & otherwise.
\end{cases}$</p>
<p>$X$ is a continuous random variable.</p>
<p>From memory, do I h... | Aufenthaltsraum | 238,486 | <p>actually, the Torsion subset is a submodule for all $R$-modules $M$.
Recall that $m\in M$ is called torsion, if there is $r\in R$ which is regular (i.e. not a zero divisor) such that $r.m=0$.</p>
<p>Assume $m$ and $m'$ are torsion with corresponding regular elements $r$ and $r'$. Then $rr'$ is non-zero and still no... |
239,863 | <p>I've to study this series:</p>
<p>$$\sum_{n=1}^\infty e^{\sqrt n\,x}$$ </p>
<p>My teacher wrote that with the asymptotic comparison with this series:</p>
<p>$$\sum_{n=1}^\infty\frac{1}{n^2}$$<br>
My series converges for every </p>
<p>$$x<0$$</p>
<p>I don't understand the motivation, hoping for someone to... | Norbert | 19,538 | <p>There is a more general result.</p>
<p><strong>Theorem.</strong> Let $E$ be a normed space. Let $\{x_n:n\in\mathbb{N}\}\subset E$ and $x\in E$, then the following conditions are equivalent:</p>
<ul>
<li>$\{x_n:n\in\mathbb{N}\}$ weakly converges to $x\in E$</li>
<li>$\{x_n:n\in\mathbb{N}\}$ is bounded and for all $... |
64,881 | <p>I am having trouble with this problem from my latest homework.</p>
<p>Prove the arithmetic-geometric mean inequality. That is, for two positive real
numbers $x,y$, we have
$$ \sqrt{xy}≤ \frac{x+y}{2} .$$
Furthermore, equality occurs if and only if $x = y$.</p>
<p>Any and all help would be appreciated.</p>
| André Nicolas | 6,312 | <p>Note that
$$\frac{x+y}{2}-\sqrt{xy}=\frac{(\sqrt{x}-\sqrt{y})^2}{2}.$$</p>
|
64,881 | <p>I am having trouble with this problem from my latest homework.</p>
<p>Prove the arithmetic-geometric mean inequality. That is, for two positive real
numbers $x,y$, we have
$$ \sqrt{xy}≤ \frac{x+y}{2} .$$
Furthermore, equality occurs if and only if $x = y$.</p>
<p>Any and all help would be appreciated.</p>
| Mongol-genius | 111,192 | <p>$$0\le ({\sqrt x}-{\sqrt y})^{2}$$
$$0\le x-2{\sqrt {xy}}+y$$
$$2{\sqrt {xy}}\le x+y$$
$${\sqrt {xy}}\le {x+y\over2}$$</p>
|
64,881 | <p>I am having trouble with this problem from my latest homework.</p>
<p>Prove the arithmetic-geometric mean inequality. That is, for two positive real
numbers $x,y$, we have
$$ \sqrt{xy}≤ \frac{x+y}{2} .$$
Furthermore, equality occurs if and only if $x = y$.</p>
<p>Any and all help would be appreciated.</p>
| Daniel W. Farlow | 191,378 | <p>I am surprised no one has given the following very straightforward algebraic argument:
\begin{align}
0\leq(x-y)^2&\Longleftrightarrow 0\leq x^2-2xy+y^2\tag{expand}\\[0.5em]
&\Longleftrightarrow 4xy\leq x^2+2xy+y^2\tag{add $4xy$ to both sides}\\[0.5em]
&\Longleftrightarrow xy\leq\left(\frac{x+y}{2}\right)... |
69,961 | <p>I want to determine the set of natural numbers that can be expressed as the sum of some non-negative number of 3s and 5s.</p>
<p>$$S=\{3k+5j∣k,j∈\mathbb{N}∪\{0\}\}$$</p>
<p>I want to check whether that would be:
0,3, 5, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, and so on.</p>
<p>Meaning that it would include 0, 3, 5,... | robjohn | 13,854 | <p>This doesn't tell you exactly which numbers can be written as $3k+5j$ with $j,k\ge0$, but it might be the best that can be said in general. These are two Theorems that usually accompany <a href="http://en.wikipedia.org/wiki/B%C3%A9zout%27s_identity" rel="nofollow">Bezout's Identity</a>.
<hr/>
<strong>Theorem $\bolds... |
252,767 | <p>I'm looking for a tangible example of a free abelian group whose quotient with a subgroup is not free abelian. There's a theorem that says that every abelian group is a quotient of some free group, but I'm looking for a more exact example.</p>
| Bombyx mori | 32,240 | <p>The confusion maybe because every subgroup of a free abelian group is free abelian, while for the quotient group this is not necessarily true. The canoical example maybe $\mathbb{Z}/p\mathbb{Z}$, where $p$ is prime. Here $\mathbb{Z},p\mathbb{Z}$ are both free but the above group is not free. The wiki article probab... |
252,767 | <p>I'm looking for a tangible example of a free abelian group whose quotient with a subgroup is not free abelian. There's a theorem that says that every abelian group is a quotient of some free group, but I'm looking for a more exact example.</p>
| pepa.dvorak | 85,466 | <p>The fact that you mention is a more general fact, i.e. every module is factor of a free module - you can imagine the construction in the following way:</p>
<p>take "enough" generators and create a free module over them, then, since different modules differ in "which elements are the same", i.e. in relations between... |
90,940 | <p>It seems known that the category of hypergraphs is a topos.
I am looking for any reference here, or just a statement of this in the literature, but can't find anything. There is one paper </p>
<blockquote>
<p>A category-theoretical approach to hypergraphs,
W. Dörfler and D. A. Waller, ARCHIV DER MATHEMATIK, Vol... | James Cranch | 14,901 | <p>One can reinterpret a hypergraph as a span-shaped diagram of sets where the left leg of the span is a finite map (meaning, all preimages are finite). Indeed, given a hypergraph, consider the span
$$V\leftarrow\lbrace(v,e)\in V\times E\mid v\in h(e)\rbrace\rightarrow E;$$
it is clear that this gives a correspondence.... |
186,240 | <p>I need some notion about topology(I'm very interested in boundary points, open sets) and few examples of solved exercises about limits of functions($f:\mathbb{R}^{n}\rightarrow \mathbb{R}^m$) using $\epsilon, \delta$ and also some theory for continous functions.
Please give me some links or name of the books which ... | Carl Wienecke | 38,256 | <p>$Topology$ by James Munkres is an excellent book for that sort of thing. </p>
|
1,278,860 | <p>Use the process of implicit differentiation to find $dy/dx$ given that:</p>
<p>$$x^2e^y − y^2e^x=0 $$</p>
<p>I am trying first to find $y$, </p>
<p>$$y^2e^x = x^2e^y$$</p>
<p>$$y^2 = (x^2e^y)/e^x$$</p>
<p>$$y = \sqrt{(x^2e^y)/e^x}$$</p>
<p>Is this correct? I have the feeling it is not.</p>
| architectpianist | 141,199 | <p>The math is right, but if you are using implicit differentiation the point is <em>not</em> to solve for $y$. Instead you would differentiate each term with respect to $x$, assuming that $y$ is some function of $x$ whose derivative is $dy/dx$. For instance, the term $y^2e^x$ yields</p>
<p>$$2y\frac{dy}{dx}e^x+y^2e^x... |
253,359 | <p>I'm trying to prove by induction the following statement without success:<br>
$$\forall n \ge 2, \;\forall d \ge 2 : d \mid n(n+1)(n+2)...(n+d-1) $$</p>
<p>For the base case: $n = 2$, $d = 2$<br>
$2\mid 2(2+1)$ which is true.<br></p>
<p>Now, the confusion begins! I assume I would need to use the second induction p... | N. S. | 9,176 | <p><strong>Hint</strong></p>
<p>$$ (n+1)(n+2)...(n+d-1)(n+d)= \left[(n+1)(n+2)(n+3)...(n+d-1)\right]n + \left[(n+1)(n+2)(n+3)...(n+d-1) \right]d$$</p>
<p>$P(n)$ tells you that the first term on RHS is divisible by $d$, while the second one is clearly divisible by $d$...</p>
|
253,359 | <p>I'm trying to prove by induction the following statement without success:<br>
$$\forall n \ge 2, \;\forall d \ge 2 : d \mid n(n+1)(n+2)...(n+d-1) $$</p>
<p>For the base case: $n = 2$, $d = 2$<br>
$2\mid 2(2+1)$ which is true.<br></p>
<p>Now, the confusion begins! I assume I would need to use the second induction p... | Brian M. Scott | 12,042 | <p>Note that $n(n+1)\dots(n+d-1)$ is the product of $d$ consecutive integers. Thus, it suffices to prove that if $n,n+1,\dots,n+d-1$ are any $d$ consecutive integers, then $d$ divides one of these integers. I would prove this by induction on $n$, simultaneously for all $d$.</p>
<p>First, it’s clearly true for $n=1$, s... |
306,011 | <p>Does anyone have a proof for $$\int_0^{\infty}\frac{\sin(x^2)}{x^2}\,dx=\sqrt{\frac{\pi}{2}}.$$
I tried to get it from contour integrating $$\frac{e^{iz^2}-1}{z^2},$$ but failed.
Thanks.</p>
| hunminpark | 54,833 | <p><strong>Another solution.</strong> <em>(which does not use complex analysis)</em><br>
Substitute $u=x^2$, then the integral becomes
$$A:=\int_{0}^{\infty}\frac{\sin (x^2)}{x^2}dx=\frac{1}{2}\int_{0}^{\infty}u^{-3/2}\sin u du$$
Now we'll consider more general one;
$$f(p):=\int_{0}^{\infty}\frac{\sin u}{u^p}du\phantom... |
1,458,975 | <p>I'm having a issue with solving this problem. I know that the answer is $ a=3, b=1 $. But i'm not sure how to get to that conclusion.</p>
<p>Given that $(a+i)(2-bi)=7-i$, find the value of $a$ and of $b$, where $a,b \in \mathbb{Z}$.</p>
| Asinomás | 33,907 | <p>If that were possible then we would have $(a+b\sqrt{2})^2)=a^2+2b^2+2ab\sqrt{2}=3$, this would imply $2ab\sqrt{2}\in\mathbb Q$. So we would have $ab=0$.</p>
<p>So you are left with two cases:</p>
<p>$a=0,b=0$.</p>
<p>The first case gives us $a^2=3$ which is clearly imposible in the rationals. The second one gives... |
2,161,294 | <p>I was wondering... $1$, $\phi$ and $\frac{1}{\phi}$, they have something in common: they share the same decimal part with their inverse. And here it comes the question:</p>
<p>Are these numbers unique? How many other members are in the set if they exist? If there are more than three elements: is it finite or infin... | Amin235 | 324,087 | <p>If you want to find these numbers, you should search in the limit values of sequence numbers like Fibonacci numbers. For example, Consider the following sequence</p>
<p>\begin{equation}
a_n=
\left\{
\begin{array}{cc}
a_{n-3} & n=1(mod \hspace{1mm}2)~,\\ \\
a_{n-3}+ a_{n-2} & n=0(mod \hspace{1mm}2)~.
\end{ar... |
2,406,043 | <p>Let the triangle $\triangle ABC$ have sides $a,b,c$ and be inscribed in a circle with radius $R$. If $p=\frac{a+b+c}{2}$ The radius of the circle can be expressed as</p>
<p>a) $$R=\frac{\sqrt{p(p-a)(p-b)(p-c)}}{4abc}$$</p>
<p>b) $$R=\frac{4\sqrt{p(p-a)(p-b)(p-c)}}{abc}$$</p>
<p>c) $$R=\frac{abc}{4\sqrt{p(p-a)(p-b... | Michael Rozenberg | 190,319 | <p>Another way.</p>
<p>We need to prove that
$$a^3+b^3+c^3-3abc\geq0.$$
Let $a+b+c=3u$, $ab+ac+bc=3v^2$ and $abc=w^3$.</p>
<p>Hence, our inequality is a linear inequality of $v^2$ because it's third degree.</p>
<p>Thus, it remains to prove our inequality for an extremal value of $v^2$,</p>
<p>which happens for equ... |
2,489,498 | <p>A={a,b,c,d}</p>
<p>R={(a,b),(a,c),(c,b)}</p>
<p>According to the definition for transitive relation, if there is (a,b) and (b,c) there should be (a,c)</p>
<p>In the above relation there is (a,c),(c,b) as well as (a,b). Shouldn't it be transitive?</p>
| Jaideep Khare | 421,580 | <p>For quickly finding the limit; apply L'Hospital's rule to $$\lim_{x \to 0} \frac{\ln(x)}{\frac{1}{\sqrt x}}$$</p>
<p>To get $$\lim_{x \to 0} \frac{\ln(x)}{\frac{1}{\sqrt x}} =\lim_{x \to 0} \frac{\frac 1x}{\frac{-1}{2 x \sqrt x}}=\lim_{x \to 0} \left(-2\sqrt x \right)=0$$</p>
|
2,505,863 | <p>I have to find one affine transformation that maps the point P=(1,1,1) to P'=(-1,-1,-1), the point P=(-1,-1,-1)' to P=(1,1,1) and the point Q=(0,0,0) to Q'=(2,2,2).
I started with a sketch and think that it is not possible to map both points with one affine transformation, but I must somehow prove that.
So I take th... | Guy Fsone | 385,707 | <p>Hint use the definition of derivative at x=4 as follows
$$\lim_{x\to 4} \frac{\sqrt{2x+1}-3}{\sqrt{x-2}-\sqrt{2}} =
\lim_{x\to 4} \frac{\sqrt{2x+1}-3}{x-4}
\cdot\lim_{x\to 4} \frac{x-4}{\sqrt{x-2}-\sqrt{2}}=(\sqrt{2x+1})'
\frac{1}{(\sqrt{x-2})'}\Big|_{x=4}
= \frac{1}{3}\frac{2\sqrt2}{1}$$</p>
|
3,778,024 | <p>Let <span class="math-container">$(\Omega, \mathcal{F}, P)$</span> be a probability space, <span class="math-container">$X$</span> a random variable and <span class="math-container">$F(x) = P(X^{-1}(]-\infty, x])$</span>. The statement I am trying to prove is</p>
<blockquote>
<p>The distribution function <span class... | Michael Hardy | 11,667 | <p>The probability assigned to an interval is certainly not bounded by its length. For example, discrete distributions assign positive probability to intervals of length <span class="math-container">$0.$</span></p>
<p>To prove right-continuity you need countable additivity.</p>
<p><span class="math-container">\begin{al... |
2,243,083 | <p>I'm writing an advanced interface, but I don't yet have a concept of derivatives or integrals, and I don't have an easy way to construct infinite many functions (which could effectively delay or tween their frame's contributing distance [difference between B and A] over the next few frames).</p>
<p>I can store valu... | Narasimham | 95,860 | <p>Elastic motion obeys a time differential equation representing a dynamic system of order two or higher in which elasticity constants like $m,k$ are fixed. The simplest harmonic motion $ m \ddot x + k x=0 $ enforces distances and you have no further control except on the imposed boundary conditions.</p>
|
3,636,667 | <blockquote>
<p>Evaluate
<span class="math-container">$$\lim_{n\to\infty}\frac{1}{n^{p+1}}\cdot \sum_ \limits{i=1}^{n} \frac{(p+i)!}{i!} , p \in N$$</span> </p>
</blockquote>
<p>Now, I found this problem while doing some practice and I am curious on how to solve it . I have no good ideas yet, so I will appreciate... | Gary | 83,800 | <p>A lower bound is given by
<span class="math-container">$$
\mathop {\lim }\limits_{n \to + \infty } \frac{1}{{n^{p + 1} }}\sum\limits_{i = 1}^n {\frac{{(p + i)!}}{{i!}}} \ge \mathop {\lim }\limits_{n \to + \infty } \frac{1}{{n^{p + 1} }}\sum\limits_{i = 1}^n {i^p } \\ = \mathop {\lim }\limits_{n \to + \infty } \f... |
1,309,670 | <p>Suppose $D \subset \mathbb{R}$ is open, $f : D \to \mathbb{R}$ is a smooth (not necessarily real analytic) function, $x_0 \in D$, and $T_n$ is the degree $n$ Taylor polynomial of $f$ centered at $x_0$. Let $S=\{ x \in D : f(x)=T_n(x) \}$. It is not hard to see that $S$ is closed and contains $x_0$. What else can be ... | zhw. | 228,045 | <p>I add this because I'm not sure if Robert Israel is giving the same answer. Let $E \subset \mathbb {R}$ be closed, with $0\in E.$ Then there exists $g\in C^\infty(\mathbb {R})$ such that $g=0$ on $E$ and $g>0$ on $\mathbb {R}\setminus E.$ Let $P$ be a polynomial of degree $n.$ Define</p>
<p>$$f(x) = e^{-1/x^2}g(... |
1,419,897 | <blockquote>
<p><strong>Theorem:</strong> Let $A$ be a bounded infinite subset of $\mathbb{R}^l$. Then
it has a limit point.</p>
</blockquote>
<p>So this is the Euclidean version of the Bolzano-Weierstrass theorem, the thing is that I was trying to prove it by induction, but it doesn't help because in the case $l=... | principal-ideal-domain | 131,887 | <p>The closure of $A$ is compact. In compact metric spaces each sequence has a convergent subsequence. If you may use that result you are done.</p>
<p><strong>Elaboration:</strong> Since $A$ is infinite there is a injective sequence $(a_n)_{n\in\mathbb N}\subseteq A$. Since $\overline{A}$ is compact $(a_n)_{n\in\mathb... |
909,228 | <p>I'm trying to find a closed form for the following sum
$$\sum_{n=1}^\infty\frac{H_n}{n^3\,2^n},$$
where $H_n=\displaystyle\sum_{k=1}^n\frac{1}{k}$ is a harmonic number.</p>
<p>Could you help me with it?</p>
| Cleo | 97,378 | <p>$$\sum_{n=1}^\infty\frac{H_n}{n^3\,2^n}=\frac{\pi^4}{720}+\frac{\ln^42}{24}-\frac{\ln2}8\zeta(3)+\operatorname{Li}_4\left(\frac12\right).$$</p>
|
997,602 | <blockquote>
<p>Prove that the function <span class="math-container">$x \mapsto \dfrac 1{1+ x^2}$</span> is uniformly continuous on <span class="math-container">$\mathbb{R}$</span>.</p>
</blockquote>
<p>Attempt: By definition a function <span class="math-container">$f: E →\Bbb R$</span> is uniformly continuous iff for ... | Tom | 103,715 | <p>Hint: $$\frac{|x|}{(1+x^2)(1+a^2)} \leq \frac{|x|}{1+x^2} < 1$$
and
$$\frac{|a|}{(1+x^2)(1+a^2)} \leq \frac{|a|}{1+a^2} < 1$$</p>
|
86,800 | <p>I am curious about how the Heegaard genus changes after a finite covering. </p>
<p>Is there anyone constructing an closed hyperbolic 3-manifold $N$ such that </p>
<p>the Heegaard genus of a finite covering of $N$ is less than the Heegaard genus of $N$? </p>
<p>Thank you!</p>
<p>Note: Heegaard genus of a 3-manifo... | Yo'av Rieck | 22,631 | <p>Hyam Rubinstein and me have results about the behavior of the Heegaard genus under double covers for non-Haken manifolds, see <a href="http://arxiv.org/abs/math/0607145">http://arxiv.org/abs/math/0607145</a>. Essentially, we show that the Heegaard genera of the two manifolds bound each other linearly. (The stateme... |
2,745,570 | <p>Use the mathematical Induction show that $H_{2^n}\le n+1$</p>
<p>here $H$ is harmonic numbers ie. $H_n=1+\frac{1}{2}+\frac{1}{3}+.....\frac{1}{2^n}$</p>
<p><strong>my idea</strong></p>
<p>so for $n=0$ L.H.S=R.H.S</p>
<p>Suppose this is true for $n$</p>
<p>we prove for $n+1$</p>
<p>So $H_{2^{n+1}}=1+\frac{1... | user061703 | 515,578 | <p>We have already assumed that $$H_n=1+\frac{1}{2}+\frac{1}{3}+.....\frac{1}{2^n}\le n+1$$</p>
<p>We need to prove that:</p>
<p>$$H_{n+1}=1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{2^n}+\frac{1}{2^n+1}+...+\frac{1}{2^n+2^n}\le n+2$$</p>
<p>This is true because $$\frac{1}{2^n+1}+\frac{1}{2^n+2}+...+\frac{1}{2^n+2^n}<... |
801,081 | <p>I was doing some school work and got bored so I started messing with k-gonal numbers. I started with the triangular numbers, square numbers and looked for patterns. I noticed something.</p>
<p>Let $n^{(k)}$ denote the $n$-th $k$-gonal number. For example, $3^{(3)}$ is the third triangular number, 6.</p>
<p>I f... | Janaka Rodrigo | 1,043,137 | <p>Let <span class="math-container">$n$</span> th <span class="math-container">$r$</span>-gonal number be <span class="math-container">$u(r,n)$</span></p>
<p>By the patterns of terms up
to heptagonal numbers it can be observed that,
<span class="math-container">$$u(r,n) = u(r-1,n) + u(3,n-1)$$</span></p>
<p>That is, <... |
644,163 | <p>The question asks:
Find the line through $(3,1,-2)$ that intersects and is perpendicular to</p>
<p>$$x = -1 + t, y = -2 + t, z = -1 + t.$$</p>
<p>My thoughts:
Say the point of intersection is $(x_0,y_0,z_0)$, then my line can be of the form</p>
<p>$$L(s) = (3,1,-2) + (x_0- 3,y_0- 1,z_0+ 2)s$$</p>
<p>Then I tried... | David Park | 99,469 | <p>These kind of problems are especially appropriate to Grassmann algebra so, if I may, I would like to show this approach using the Mathematica code of John Browne. First, some nomenclature:</p>
<p><img src="https://i.stack.imgur.com/TqphR.png" alt="enter image description here"></p>
<p>Load the package, define a 3-... |
4,442,223 | <p>How does one show this?
<span class="math-container">$$
\exp(-x) \sum_{k=0}^\infty x^k \frac{(k+m)!}{(k!)^2} = L_m(-x) m!,
$$</span> where <span class="math-container">$m$</span> is a positive integer, and <span class="math-container">$L_{m}(x)$</span> is the <span class="math-container">$m$</span>th order Laguerre ... | Lee Mosher | 26,501 | <p>What you heard is correct: for any simplicial complex <span class="math-container">$X$</span> endowed with the simplicial topology, if we let <span class="math-container">$X^{(0)}$</span> denote its set of vertices endowed with the subspace topology, then <span class="math-container">$X^{(0)}$</span> is indeed a dis... |
1,690,715 | <p>I have this space $E=\mathcal{C}([0,1],\mathbb{R})$ and the inner product $d(f,g)=\int_0^1 |f(x)-g(x)|\,{\rm d}x$.</p>
<p>Who have an idea about a simple sequence $\{f_n\}_{n=1}^\infty$ which is Cauchy but not convergent in $(E,d)$?</p>
| nullUser | 17,459 | <p>Take $f= 1_{[1/2,1]}$. Approximate it by $f_n$ which agrees with $f$ outside of $[1/2-1/n,1/2+1/n]$ and interpolates linearly inbetween. By dominated convergence $f_n \to f \in L^1([0,1])$ and hence $f_n$ is a Cauchy sequence in the $d$-metric. Now let $g \in \mathcal{C}([0,1],\mathbb{R})$ and assume for contradicti... |
3,631,648 | <p>Suppose <span class="math-container">$X_1, ..., X_n \stackrel{iid}{\sim}$</span> Exponential(rate = <span class="math-container">$\lambda$</span>) independent of <span class="math-container">$Y_1, ..., Y_n \stackrel{iid}{\sim}$</span> Exponential<span class="math-container">$(1)$</span>. </p>
<p>Define <span class=... | heropup | 118,193 | <p>If you observe both <span class="math-container">$Z_i$</span> and <span class="math-container">$Y_i$</span>, then when they are equal, you know <span class="math-container">$X_i > Y_i$</span>. When they are not, you know <span class="math-container">$X_i = Z_i$</span>. Therefore, your likelihood function is <sp... |
172,292 | <p>I am trying to find the residue of the function $$f(z)=(z+1)^2e^{3/z^2}$$
at $z=0$.
I tried the following in Mathematica</p>
<pre><code>Residue[(z+1)^2*Exp[3/z^2],{z,0}]
</code></pre>
<p>which remains unevaluated. Computing this by hand gives the value of $6$. What is going on?</p>
<p>I’ve noticed that Mathemati... | Carl Woll | 45,431 | <p>You could use <a href="http://reference.wolfram.com/language/ref/SeriesCoefficient" rel="nofollow noreferrer"><code>SeriesCoefficient</code></a> instead:</p>
<pre><code>SeriesCoefficient[(z+1)^2 Exp[3/z^2], {z, 0, -1}]
</code></pre>
<blockquote>
<p>6</p>
</blockquote>
<p><strong>Addendum</strong></p>
<p>Anothe... |
208,802 | <p>Is there a continuous increasing function $ f : [0, \pi] \to [0, e] $ such that $ f(0) = 0, f(\pi) = e $ and $ f (q ) \in \mathbb{Q} $ for $ q \in \mathbb{Q} $ and $ f (q ) \in \mathbb{Q}^c $ for $ q \in \mathbb{Q}^c $? I think there should be, but I am unable to construct one. </p>
| Hagen von Eitzen | 39,174 | <p>Find a suitable strictly ascending sequence $(c_n)_n$ of rationals and define $f(c_1)=0$ and recursively for $x\in[c_{n-1},c_n]$ by $f(c_n)=f(c_{n-1})+\frac12 (x-c_{n-1})$ if $n$ is even and $f(c_n)=f(c_{n-1})+c_n-c_{n-1}$ if $n$ is odd.
Let $d_n=c_{n+1}-c_n$.
In order to make this an example you are looking for, w... |
1,144,695 | <p>I'm currently trying to solve this problem. </p>
<blockquote>
<p>Let $f: R \rightarrow S$ be a surjective ring homomorphism. Let $K = \ker(f)$. Assume $P$ is a prime ideal s.t. $K \subset P$. Show $f(P)$ is a prime ideal in $S$.</p>
</blockquote>
<p>I solved the ideal part. </p>
<p>Let $y \in f(P)$, by definiti... | MooS | 211,913 | <p>You should show $R/P \cong S/f(P)$. To that extend, consider the composition $$R \to S \to S/f(P).$$</p>
|
1,144,695 | <p>I'm currently trying to solve this problem. </p>
<blockquote>
<p>Let $f: R \rightarrow S$ be a surjective ring homomorphism. Let $K = \ker(f)$. Assume $P$ is a prime ideal s.t. $K \subset P$. Show $f(P)$ is a prime ideal in $S$.</p>
</blockquote>
<p>I solved the ideal part. </p>
<p>Let $y \in f(P)$, by definiti... | Slade | 33,433 | <p>$f:R\to S$ induces an isomorphism $\overline{f}:R/K \to S$. So it is enough to show that $P/K$ is a (completely) prime ideal of $R/K$.</p>
<p>Every ideal of $R/K$ can be written uniquely in the form $I/K$ for some ideal $I\supset K$ of $R$ (take the preimage under the projection $R\to R/K$). But if $A/K\cdot B/K ... |
1,722,964 | <p>Expression :$$(p\rightarrow q)\leftrightarrow(\neg q\rightarrow \neg p)$$
What does the symbol $\leftrightarrow$ mean ? Please explain by drawing the truth table for this expression and also with other examples if possible. <strong>I'm in a desperate situation so I'd really appreciate a quick response !</strong></p>... | Matthias | 164,923 | <p>$$A\leftrightarrow B$$ is the same as </p>
<p>$$(A\rightarrow B) \land (B\rightarrow A)$$</p>
|
2,516,123 | <p>Problem 11985, by Donald Knuth, <em>American Mathematical Monthly</em>, June-July, 2017:</p>
<blockquote>
<p>For fixed $s,t \in \mathbb{N}$. with $s\leq t$. let $a_{n}=\sum\limits_{k=s}^{t}$ $ {n}\choose{k}$. Prove that this sequence is log-concave, namely that $a_{n}^{2}\geq a_{n-1}a_{n+1} \ \forall n\geq 1$. </... | Dap | 467,147 | <p>This follows from the log-concavity of binomial coefficients. Using the identity $\binom nk=\binom{n-1}{k-1}+\binom{n-1}{k}$ we can express the desired inequality $a_n^2\geq a_{n-1}a_{n+1}$ in terms of binomial coefficients of $n-1:$ we need to show</p>
<p>$$\sum_{i=s}^t\sum_{j=s-2}^{t-2}\binom{n-1}{i}\binom{n-1}{j... |
2,516,123 | <p>Problem 11985, by Donald Knuth, <em>American Mathematical Monthly</em>, June-July, 2017:</p>
<blockquote>
<p>For fixed $s,t \in \mathbb{N}$. with $s\leq t$. let $a_{n}=\sum\limits_{k=s}^{t}$ $ {n}\choose{k}$. Prove that this sequence is log-concave, namely that $a_{n}^{2}\geq a_{n-1}a_{n+1} \ \forall n\geq 1$. </... | Sil | 290,240 | <p>Solution by Roberto Tauraso <a href="http://www.mat.uniroma2.it/~tauraso/AMM/AMM11985.pdf" rel="nofollow noreferrer">http://www.mat.uniroma2.it/~tauraso/AMM/AMM11985.pdf</a> (who by the way has solutions to many of AMM's problems):</p>
<blockquote>
<p>Let $$F_n(x):=\sum_{k=s}^{t}\binom{n}{k}x^k.$$</p>
<p>The... |
3,611,072 | <p>Show that if a prime <span class="math-container">$p ≠ 3$</span> is such that <span class="math-container">$p≡1$</span> (mod 3) then p can be written as <span class="math-container">$a^2-ab+b^2$</span> where a and b are integers. </p>
<p>I have no idea how to approach this question, so any help much appreciated! T... | Piquito | 219,998 | <p>COMMENT.- I fear it is a problem not too elementary. It can be solved using the theory of representation of integers by quadratic forms. In short, consider the discriminant of the form <span class="math-container">$x^2-xy+y^2$</span> which is equal to <span class="math-container">$\Delta=-3$</span>. </p>
<p>One can... |
2,710,681 | <p>If I have a function of three variables and I want to create a new function in which it equals the other function squared, could I literally just square the other function or does this violate any rules? Would this also mean its gradient vector is just squared at a certain point?</p>
| gt6989b | 16,192 | <p><strong>HINT</strong></p>
<p>Say you have $f(x,y,z)$ and you would like to define $$g(x,y,z) = f(x,y,z)^2.$$ Then,
$$
\vec{\nabla} g =
\begin{pmatrix}
\partial g/\partial x \\
\partial g/\partial y \\
\partial g/\partial z
\end{pmatrix}
$$
For example, chain rule implies
$$
\frac{\partial g(x,y,z)}{\partial x}
=... |
76,683 | <p>How do I force mathematica to display the below expression as a sum <code>a+b</code> with a scaling factor of <code>1/r</code>.</p>
<p>(a+b)/r</p>
<p>I would like Mathematica to display (1/r) (a+b), ie. I want it to show 1/r as a scaling factor. </p>
<p>currently, it shows (a+b)/r , with r as a common denomi... | Nasser | 70 | <pre><code>expr = (a + b)/r
1/Denominator[expr]
</code></pre>
<p><img src="https://i.stack.imgur.com/LNEqO.png" alt="Mathematica graphics"></p>
|
1,621,269 | <p>I have tried everything in my knowledge and no, I cannot state it.
I have tried a factorizor online which tells me that it is not factorizable hence irreducible. But I cannot reason why.</p>
<p>I looked at Eisenstein's criteria but obviously, there is no prime $q$ that fits the criteria so this is useless.</p>
<p>... | lulu | 252,071 | <p>Let $\phi(x)$ denote your polynomial. Then we note that $$\phi(x+1)=x^5+3x^2+9x+3$$ and we can invoke Eisenstein's criterion.</p>
|
1,621,269 | <p>I have tried everything in my knowledge and no, I cannot state it.
I have tried a factorizor online which tells me that it is not factorizable hence irreducible. But I cannot reason why.</p>
<p>I looked at Eisenstein's criteria but obviously, there is no prime $q$ that fits the criteria so this is useless.</p>
<p>... | Travis Willse | 155,629 | <p>One option is to reduce the given polynomial modulo $11$, in which case it factors (over $\Bbb F_{11}$) as
$$(x - 5)(x^4 - x^2 - x - 3).$$
So, if the polynomial is reducible over $\Bbb Q$, it has one linear factor and one irreducible quartic factor there.</p>
<p>On the other hand, checking the short list, $\pm 1, \... |
476,899 | <p>Does someone know a proof that $\{1,e,e^2,e^3\}$ is linearly independent over $\mathbb{Q}$?</p>
<p>The proof should not use that $e$ is transcendental.</p>
<p>$e:$ Euler's number.</p>
<p><a href="http://paramanands.blogspot.com/2013/03/proof-that-e-is-not-a-quadratic-irrationality.html#.Uhv87tJFUnl">$\{1,e,e^2\}... | Paramanand Singh | 72,031 | <p>I thought to add an answer instead of giving long comments.</p>
<p>From <a href="http://en.wikipedia.org/wiki/Proof_that_e_is_irrational" rel="nofollow noreferrer">Wikipedia</a> we have the following quote</p>
<p>"In 1891, Hurwitz explained how it is possible to prove along the same line of ideas that $e$ is not a... |
4,064,760 | <p><a href="https://i.stack.imgur.com/tDP8G.png" rel="nofollow noreferrer">image shows the solution for the differential equation y double prime minus 4 y prime plus 5 y equal to e powered to the minus x</a></p>
<p>I solved this <span class="math-container">$y'' - 4y' + 5y = e^{-x}$</span> equation with the guess of:</... | 19aksh | 668,124 | <p>The auxiliary equation of the given ODE is,</p>
<p><span class="math-container">$m^2-4m+5 = 0 \Rightarrow (m-2)^2 +1 = 0 \Rightarrow \boxed{m = 2 \pm i}$</span></p>
<p>So the solution (complementary function) will be,</p>
<p><span class="math-container">$$y_{CF}(x) = k_1 e^{(2 + i)x} + k_2 e^{(2 - i)x} =e^{2x}(k_1 e... |
4,064,760 | <p><a href="https://i.stack.imgur.com/tDP8G.png" rel="nofollow noreferrer">image shows the solution for the differential equation y double prime minus 4 y prime plus 5 y equal to e powered to the minus x</a></p>
<p>I solved this <span class="math-container">$y'' - 4y' + 5y = e^{-x}$</span> equation with the guess of:</... | Henry Lee | 541,220 | <p>okay lets solve it, first:
<span class="math-container">$$y''-4y'+5y=0$$</span>
lets make an educated guess that the solutions will be of the form:
<span class="math-container">$$y=Ae^{\lambda x}$$</span>
now sub in:
<span class="math-container">$$Ae^{\lambda x}(\lambda^2-4\lambda+5)=0$$</span>
solving for <span cla... |
4,064,760 | <p><a href="https://i.stack.imgur.com/tDP8G.png" rel="nofollow noreferrer">image shows the solution for the differential equation y double prime minus 4 y prime plus 5 y equal to e powered to the minus x</a></p>
<p>I solved this <span class="math-container">$y'' - 4y' + 5y = e^{-x}$</span> equation with the guess of:</... | Community | -1 | <p>Let us pretend that we know nothing about the linear ODE with constant coefficients nor complex exponentials.</p>
<p>We will try by factoring <span class="math-container">$y$</span> and get some simplification.</p>
<p><span class="math-container">$$y=zh,\\y'=z'h+zh',\\y''=z''h+2z'h'+zh''.$$</span></p>
<p>We plug thi... |
2,960,501 | <p><span class="math-container">$(0^n 1)^* \ \ , n\geq 0 $</span></p>
<p>According to wiki</p>
<blockquote>
<p>If V is a set of strings, then V* is defined as the smallest superset of V that contains the empty string ε and is closed under the string concatenation operation</p>
<p>If V is a set of symbols or characters,... | J.-E. Pin | 89,374 | <p>If I understand correctly (and no, your definition is neither clear nor correct since
<span class="math-container">$\{(0^n1)^* \mid n \geqslant 0\}$</span> does not make any sense), your language is
<span class="math-container">$\{0^n1 \mid n \geqslant 0\}^*$</span>, which can be rewritten as <span class="math-cont... |
2,960,501 | <p><span class="math-container">$(0^n 1)^* \ \ , n\geq 0 $</span></p>
<p>According to wiki</p>
<blockquote>
<p>If V is a set of strings, then V* is defined as the smallest superset of V that contains the empty string ε and is closed under the string concatenation operation</p>
<p>If V is a set of symbols or characters,... | rici | 59,314 | <p>My reading of this question (which I think is the natural reading, notwithstanding other possibilities) is that the language being defined is:</p>
<p><span class="math-container">$$L = \bigcup\limits_{n\geq 0}^{} (0^n1)^*$$</span></p>
<p>which is, roughly speaking, the language of all strings in <span class="math-... |
3,527,919 | <p>I've tried to prove this property of Bessel function but I don't seem to be going anywhere</p>
<p><span class="math-container">$$\sqrt{\frac 12 \pi x} J_\frac 32 (x) = \cfrac{\sin x}{x} - \cos x$$</span></p>
<p>I have tried substituting <span class="math-container">$\frac 32$</span> for <span class="math-container... | Gary | 83,800 | <p>Using the series expansion of <span class="math-container">$J_{3/2}(x)$</span> and the Legendre duplication formula for the gamma function, we find
<span class="math-container">$$
\sqrt {\frac{{\pi x}}{2}} J_{3/2} (x) = \sqrt {\frac{{\pi x}}{2}} \left( {\tfrac{1}{2}x} \right)^{3/2} \sum\limits_{n = 0}^\infty {( - 1... |
1,023,575 | <p>How would one factor a number, say $9+4\sqrt{2}$ in $\mathbb{Z}[\sqrt{2}]$?</p>
<p>This is what I've attemped to do:
$$(a_1+b_1\sqrt{2})(a_2+b_2\sqrt{2}) $$
$$a_1a_2+a_1b_2\sqrt{2}+a_2b_1\sqrt{2}+2b_1b_2$$
Thus,
\begin{eqnarray}
a_1a_2+2b_1b_2&=&9 \\
a_1b_2+a_2b_1 &=& 4.
\end{eqnarray}</p>
<p>But t... | Kevin Arlin | 31,228 | <p>The point is, of course, that you want to factor into <em>primes</em>. The norm in $\mathbb{Z}[\sqrt{2}]$ is $N(a+b\sqrt 2)=a^2-2b^2$, so $N(9+4\sqrt 2)=49$ and we only have to worry about primes of norm $\pm 7$. So, when does $a^2-2b^2=\pm 7$ with $a,b$ integers? Well, $(3,1)$ looks tempting, but doesn't work. So w... |
1,931,754 | <p>I am trying to show that the interval $[0,1)$ is a closed subset of $(-1,1)$ by using the definition that a closed subset contains all of its limit points.
So for a convergent sequence $\{x_n\}$ in $[0,1)$ we have that $0 \leq x_{n} < 1$ for all $n \in \mathbf{N}$. How can I show that $\lim_{n \rightarrow \infty... | Mark Viola | 218,419 | <p>Herein, we present a way forward that does not rely on differential calculus, but rather uses an elementary pair of inequalities and the squeeze theorem. To that end we proceed.</p>
<blockquote>
<p><strong>PRIMER:</strong></p>
<p>In <a href="https://math.stackexchange.com/questions/1589429/how-to-prove-that-logxx-w... |
1,745,136 | <p>Show that among every consecutive 5 integers one is coprime to the others<br>
I considered these 5 numbers as: $5k,5k+1,5k+2,5k+3,5k+4$<br>
It's seen that for example $5k+1$ is coprime to $5k$ and $5k+2$,now it remains to show $5k+1$ is coprime to $5k+3,5k+4$<br>
Let $\gcd(5k+1,5k+3)=d\Rightarrow\ d|2\Rightarrow\ d... | user133281 | 133,281 | <p>Among any $6$ consecutive integers, there are two that are coprime to $6$. So among any $5$ consecutive integers, there is at least one that is coprime to $6$. This number if also coprime to the others, because the only possible common prime divisors are $2$ and $3$.</p>
|
1,745,136 | <p>Show that among every consecutive 5 integers one is coprime to the others<br>
I considered these 5 numbers as: $5k,5k+1,5k+2,5k+3,5k+4$<br>
It's seen that for example $5k+1$ is coprime to $5k$ and $5k+2$,now it remains to show $5k+1$ is coprime to $5k+3,5k+4$<br>
Let $\gcd(5k+1,5k+3)=d\Rightarrow\ d|2\Rightarrow\ d... | Patrick Da Silva | 10,704 | <p>Suppose the prime $p$ divides two of the integers $5k,\cdots,5k+4$. Then there are two integers $i,j \in \{0,\cdots,4\}$ such that $i \equiv j \pmod p$, so that $p = 2$ or $p=3$. The integer $6$ can obviously never divide two integers whose distance is less than $5$, so the gcd of two such integers is either $1,2$, ... |
1,745,136 | <p>Show that among every consecutive 5 integers one is coprime to the others<br>
I considered these 5 numbers as: $5k,5k+1,5k+2,5k+3,5k+4$<br>
It's seen that for example $5k+1$ is coprime to $5k$ and $5k+2$,now it remains to show $5k+1$ is coprime to $5k+3,5k+4$<br>
Let $\gcd(5k+1,5k+3)=d\Rightarrow\ d|2\Rightarrow\ d... | MathWiz | 323,681 | <p>Let $n$ be a natural number. Consider five consecutive numbers $(n-2),(n-1),(n),(n+1),(n+2)$.</p>
<ul>
<li><p>If $n$ is even then $n-1$ and $n+1$ only can be coprime to all others.
since these two are consecutive odd numbers, thus they are coprime .
now the largest odd number less than $ 5$ is $3$.
if $n-1$ is a mu... |
1,355,133 | <p>A while ago I asked a question about probability here <a href="https://math.stackexchange.com/questions/1353044/why-is-binomial-probability-used-here/">Why is binomial probability used here?</a></p>
<p>I get that you can find how many ways of choosing the $6$ correct out of $10$ questions.</p>
<p>But why do we <st... | Conrado Costa | 226,425 | <p>How many ways can you get 6 questions right?</p>
<p>1->6 right and 7->10 wrong is an event. But you need to count the others. For instance 1->3 wrong and 4->10 right. </p>
<p>How many ways can you get 6 questions out of 10 right? Choose $6$ out of $10$ to get right: ${10\choose 6}$. The rest will follow if you un... |
1,951 | <p>In <a href="https://matheducators.stackexchange.com/a/1949/704">this answer</a>, user <a href="https://matheducators.stackexchange.com/users/942/robert-talbert">Robert Talbert</a> stated that</p>
<blockquote>
<p>There are some amazing things you can do pedagogically with clickers.</p>
</blockquote>
<p>I'd like t... | Adrienne | 1,207 | <p>Warning. Biologist is answering.</p>
<p>Our instructors are often in very large, very sleepy lecture halls. Clickers provide a stimulus for student discussion and trigger learning through testing effects. </p>
<p>Common uses:</p>
<ol>
<li><p>The instructor is about to begin a new subject. She opens a clicker
que... |
2,440,802 | <p>The number of positive integers that $n$ can take in between the range $100$ to $200$.</p>
<p>I tried a lot using the prime factorization method but no use. </p>
| lab bhattacharjee | 33,337 | <p>HINT:</p>
<p>$$n^2-n-2=(n-2)(n+1)$$</p>
<p>$$n^2+2n-3=(n+3)(n-1)$$</p>
<p>As $n+1-(n-2)=3,n-2,n+1$ are of opposite parity, exactly one of them must be divisible by $8$</p>
<p>As $n+3-(n-1)=4,$ exactly one of them must be divisible by $27$</p>
<p>Now use <a href="http://mathworld.wolfram.com/ChineseRemainderThe... |
1,860,459 | <blockquote>
<p>Prove that $4k < 2^k$ by induction.</p>
</blockquote>
<p>It holds for $k = 5$. Assume $ k = n + 1 $. Then</p>
<p>$4(n+1) < 2^{(n+1)}$</p>
<p>$4n + 4 < 2^n * 2$</p>
<p>$2n + 2 \leq 2^n$</p>
<p>Now I just need to show that</p>
<p>$2n + 2 \leq 4n$</p>
<p>$n + 1 \leq 2n$</p>
<p>$1 \leq n$... | Daniel W. Farlow | 191,378 | <p>Consider the following (see if you can determine how one step relates to another):
\begin{align}
4(k+1)&=4k+4\\[1em]
&< 2^k+4\tag{why?}\\[1em]
&< 2^k+2^k\tag{why?}\\[1em]
&= 2\cdot2^k\\[1em]
&=2^{k+1}.
\end{align}</p>
|
2,801,936 | <p>To me, it seems obvious that the binary quadratic form $x^2+8y^2$ does not properly represent 3. However, I have managed to prove that it does so I think I must be doing something stupid.
I have used the following:</p>
<p><strong>Let f be a a binary quadratic form and n an integer. We say that f <em>properly repres... | user328442 | 328,442 | <p>A little more general:</p>
<p>Theorem: Suppose $f$ is a multiplicative function. Then $$\sum_{d|n} \mu(d) f(d) = \prod_{p|n} (1-f(p)).$$</p>
<p>Proof: Let $$g(n) = \sum_{d|n} \mu(d) f(d).$$ Then $g$ is multiplicative (the product $\mu f$ is obviously multiplicative so the Direchlet convolution $\mu f * u$ where $u... |
1,221,158 | <p>I'm interested in knowing whether $a^0 = 1$ ('$a$' not zero) is a definition.
If not, can anyone please help me with proving this?</p>
| Daniel | 150,142 | <p>It's a definition. A convenient definition. </p>
<p>We know from our early encounter with mathematics that $a^m\times a^n=a^{m+n}$ if $m,n\in \mathbb{N}$ (not including zero) because it's very natural: "Multiplying $m$ times and multiplying $n$ times, and then multiplying those values should be the same as multiply... |
1,221,158 | <p>I'm interested in knowing whether $a^0 = 1$ ('$a$' not zero) is a definition.
If not, can anyone please help me with proving this?</p>
| Chenkodan | 146,844 | <p>$${a^b\over a^c} = a^{b-c} $$and vice versa. {Index rule}</p>
<p>Therefore, $$a^0 = a^{x-x}$$ for any x</p>
<p>$$= {a^x\over a^x} ,$$ using the aforementioned index rule.</p>
<p>$$= 1 $${since any thing divided by itself is 1 except 0)</p>
|
551,662 | <p>I am reading "What Is Mathematics? An Elementary Approach to Ideas and Methods"
And I am stuck here, I don't get it. I have posted a screen shot underlining what my doubt is..</p>
<p>I dont get it when the author says while the pythagoras theorem is : $a^2 + b^2 = c^2$
and then he says $x=a/c$ and $y=b/c$
and then... | Olivier | 45,622 | <p>For the first part of your question: the author divides both sides of the Pythagorean equation by $c^2$. This yields: $\frac{a^2}{c^2} + \frac{b^2}{c^2} = 1$. Now, he defines $x = a/c$ and $y = b/c$. The equation can than be rewritten to: $x^2 + y^2 = 1$. Substracting $x^2$ from both sides yields $y^2 = 1 - x^2$, or... |
391,333 | <p>It is well known that $\sum_{k = 1}^{n}k^3 =\Big [\sum_{k=1}^{n}k^1\Big]^2$. My question is very simple.</p>
<blockquote>
<p>There are $3$-tuples $(p, q, \alpha) \in
\mathbb{N}\times\mathbb{N}\times\mathbb{N}$, in addition to $(3,1,2)$,
such that $\alpha\geq 2$ and $$\sum_{k = 1}^{n}k^{\,p} =\Big [\sum_{k... | Ivan Loh | 61,044 | <p>Let $n=2$, so $(1+2^p)=(1+2^q)^{\alpha}$, so $(1+2^q)^{\alpha}-2^p=1$. If $p=1$, then $q=\alpha=1$, contradicting $\alpha \geq 2$. Otherwise $1+2^q, \alpha, 2, p>1$, so by <a href="http://en.wikipedia.org/wiki/Catalan%27s_conjecture" rel="nofollow">Mihailescu's theorem</a> $1+2^q=3, \alpha=2, p=3$. This gives $(3... |
847 | <p>Apologies in advance if this is obvious.</p>
| Ben Webster | 66 | <p>By the way, <a href="http://www.ams.org/mathscinet-getitem?mr=1155753" rel="nofollow">this paper</a> may be of interest. It shows that for solvable groups, one doesn't have to do the Hilbert class extension moonface suggests, but for some non-solvable ones you do. Also <a href="http://www.ams.org/mathscinet-getitem... |
982,780 | <p>I have the following system of <span class="math-container">$M$</span> linear equations in <span class="math-container">$N$</span> unknowns.</p>
<p><span class="math-container">$$
\begin{bmatrix}
3 & 0 & 1 & 0 & -1 & -3 & 2\\
1 & 2 & 0 & 4 & 0 & 0 & -1\\
1 & 1 &a... | Henno Brandsma | 4,280 | <p>Prove by induction on $\beta$ that $L(\beta) = \{x: x \le \beta \}$ is compact, for all ordinals $\beta$.</p>
<p>This is clear for $\beta = 0$, where $L(0) = \{0\}$ and if $\beta+1$ is a successor, then $L(\beta+1) = L(\beta) \cup \{\beta+1\}$, so if $L(\beta)$ is compact, so is $L(\beta+1)$.</p>
<p>So assume $L(\... |
1,282,419 | <p>Let $\Delta\subset\Bbb C$ be the open unitary disk. Let $\varphi:\Delta\to\Bbb R$ defined as follows: $\varphi(z)=1$ if $\Re z\ge0$, $\varphi(z)=0$ otherwise. So $\varphi$ is upper semicontinous.</p>
<p>In order to prove $\varphi$ is NOT subharmonic, I've to find a compact subset $K\Subset\Delta$ and a real valued ... | Daniel Fischer | 83,702 | <p>Take a disk $\lvert z\rvert \leqslant r$ for some $0 < r < 1$, and define the boundary values by</p>
<p>$$h_0(re^{i\vartheta}) = l(\vartheta),$$</p>
<p>where</p>
<p>$$l(t) = \begin{cases} \frac{2}{\pi}(t+\pi) &, -\pi \leqslant t \leqslant -\frac{\pi}{2}\\ \quad 1 &, -\frac{\pi}{2} \leqslant t \leqsl... |
3,995,492 | <p>I have no clue how to do this, I manage to get I get that <span class="math-container">$11^{36} \equiv 1 \hspace{0.1cm} \text{mod} (13)$</span> but I can't get anywhere from there.</p>
| Olivier Roche | 649,615 | <p>All you need to know for this is that, since <span class="math-container">$13$</span> is a prime number, <span class="math-container">$\mathbb{Z} / 13 \mathbb{Z}$</span> is a <strong>field</strong>.</p>
<p>In particular, every non zero element has a unique inverse for multiplication. Constating that
<span class="mat... |
2,476,973 | <p>A fair six-sided die carries $1$ on one face, $2$ on two of its faces, and<br>
$3$ on the remaining three faces. </p>
<p>Suppose the die is rolled twice, and let $X$ be the random variable ’total score'. Find the probability distribution of $X$.</p>
| A. M. | 123,356 | <p>For example, $P(T=2)$ is the probability that we get $1$ at both tries. Since the trials are independent, this means:</p>
<p>$P(T=2) = P(1~on~first~try)\times P(1~on~second~try)=1/6\times 1/6=1/36$.
You can work through all other situations as it is suggested.</p>
|
3,695,127 | <p>Before the moderators close my question, I cant think of any starting approach to the question. </p>
<p>Another question of the similar type I am having trouble with is: 12 balls are distributed at random among 3 boxes. What is the probability that the first box will contain 3 balls?
For the second question I can f... | Kevin.S | 724,407 | <p>Geometrically, <span class="math-container">$S^1\times\{1\}\cup\{1\}\times S^1\cong S^1\vee S^1$</span>, and <span class="math-container">$S^1\times S^1\setminus\{(-1,-1)\}$</span> is the punctured torus (<span class="math-container">$=T^2\setminus \{p\}\cong([-1,1]^2/\sim)\setminus\{p\}$</span>).</p>
<p>I think th... |
134,987 | <blockquote>
<p>$$3x^2 + 2y^4 = z^4$$</p>
</blockquote>
<p><em>How do I solve this??</em> I would like to use so-called "elementary number theory", not abstract algebra (e.g. $\mathbb{Z} ( \sqrt d)$) or elliptic curves.</p>
<p>Note: I'm not asking <em>what</em> the solutions are, but rather <em>how</em> to find the... | Will Jagy | 10,400 | <p>Maybe what you need is Legendre's Theorem. Certainly it covers this situation. It tells you exactly what needs to be checked. It is presented in Ireland and Rosen, A Classical Introduction to Modern Number Theory, chapter 17, section 3. A very similar treatment is in <a href="http://alpha.math.uga.edu/%7Epete/4400ra... |
2,332,750 | <p>At the end of chapter 5 of stein's book <a href="http://wstein.org/books/ant/ant.pdf" rel="nofollow noreferrer">A Computational Introduction to Algebraic Number Theory</a> he proves proposition 5.2.4 which states that:</p>
<p>Given a prime ideal $\mathfrak{p}$ in a Dedekind domain $R$ we have the isomorphism
$$
\fr... | nguyen quang do | 300,700 | <p>I don't know much about effective calculation in ANT, but it seems to me that your question is rather a general one about the "purpose" of the natural isomorphisms ¤ $\mathfrak p^n /\mathfrak p^{n+1} \cong R/\mathfrak p := k_\mathfrak p$ (the residue field at $\mathfrak p$, viewed as an additive group). Moreover it... |
2,612,134 | <p>One of the exercise in Artin's algebra gives an eigenvector of an element of $SO(3)$, in one possible case. Namely, it is asked to show that </p>
<blockquote>
<p>If $A=[a_{ij}]$ is a rotation in $SO(3)$, then the vector
$$v=\begin{bmatrix} (a_{23}+a_{32})^{-1}\\ (a_{13}+a_{31})^{-1} \\ (a_{12}+a_{21})^{-1}\end... | Widawensen | 334,463 | <p>Let me use letter $R$ instead of $A$ for the considered matrix. </p>
<p>Generally for a rotation matrix $R(v,\theta)$ you can calculate the axis unit vector from the formula:
$v= {\dfrac {1}{2sin(\theta)}}\begin{bmatrix}
r_{32}-r_{23} \\
r_{13}-r_{31} \\
r_{21} -r_{12}
\end{bmatrix}$ where $r_{ij}$ are appropria... |
2,612,134 | <p>One of the exercise in Artin's algebra gives an eigenvector of an element of $SO(3)$, in one possible case. Namely, it is asked to show that </p>
<blockquote>
<p>If $A=[a_{ij}]$ is a rotation in $SO(3)$, then the vector
$$v=\begin{bmatrix} (a_{23}+a_{32})^{-1}\\ (a_{13}+a_{31})^{-1} \\ (a_{12}+a_{21})^{-1}\end... | Widawensen | 334,463 | <p>I would like to propose another approach to the problem. As it is substantially other that given above it will be presented as a separate answer.</p>
<p>Let $R(u,\theta)$ be rotation matrix with the unit axis vector $u=[x,y,z]^T$ and rotation angle $\theta$.<br>
In this situation we have <a href="https://en.wik... |
114,122 | <p>I am trying to figure out the maximum possible combinations of a (HEX) string, with the following rules:</p>
<ul>
<li>All characters in uppercase hex (ABCDEF0123456789)</li>
<li>The output string must be exactly 10 characters long</li>
<li>The string must contain at least 1 letter</li>
<li>The string must contain a... | J. Kyle | 221,466 | <p>I had a similar question. Basically I think I figured out how to do this.</p>
<p>First of all, if you are dealing with only numbers and you have a base 10 number system, it's pretty easy to figure out the number of combinations. If you have a 3 digit code and only use numbers, you have 999 possible combinations, ri... |
784,753 | <p>In spherical coordinates, we have</p>
<p>$ x = r \sin \theta \cos \phi $;</p>
<p>$ y = r \sin \theta \sin \phi $; and </p>
<p>$z = r \cos \theta $; so that</p>
<p>$dx = \sin \theta \cos \phi\, dr + r \cos \phi \cos \theta \,d\theta – r \sin \theta \sin \phi \,d\phi$;</p>
<p>$dy = \sin \theta \sin \phi \,dr + r ... | Omish | 486,107 | <p>After 3.5 year there needs to be an answer to this for searchers :D
First of all there's no need for complicated calculations. You can obtain that expressions just by looking at the picture of a spherical coordinate system.
The only thing you have to notice is that there are two definitions for unit vectors of ... |
2,206,247 | <p><strong>Question:</strong> Consider the following non linear recurrence relation defined for $n \in \mathbb{N}$:</p>
<p>$$a_1=1, \ \ \ a_{n}=na_0+(n-1)a_1+(n-2)a_2+\cdots+2a_{n-2}+a_{n-1}$$</p>
<p>a) Calculate $a_1,a_2,a_3,a_4.$</p>
<p>b) Use induction to prove for all positive integers that:</p>
<p>$$a_n=\dfra... | marty cohen | 13,079 | <p>Just square both sides
and see what matches on each side.</p>
|
1,238,210 | <p>How we can solve that $\lim _{_{x\rightarrow \infty }}\int _0^x\:e^{t^2}dt$ ?</p>
<p>P.S: This is my method as I thought:
$\int _0^x\:\:e^{t^2}dt>\int _1^x\:e^tdt=e^x-e$ which is divergent, so all your answers, helped me to think otherwise, maybe my method help something else :D</p>
| egreg | 62,967 | <p>Since
$$
e^{x^2}=1+x^2+\frac{x^4}{2!}+\dotsb
$$
we have that, for $x\ge0$, $e^{x^2}\ge1+x^2$. So
$$
\int_{0}^{x}e^{t^2}\,dt\ge\int_{0}^x(1+t^2)\,dt=x+\frac{x^3}{3}
$$
Can you finish?</p>
|
1,238,210 | <p>How we can solve that $\lim _{_{x\rightarrow \infty }}\int _0^x\:e^{t^2}dt$ ?</p>
<p>P.S: This is my method as I thought:
$\int _0^x\:\:e^{t^2}dt>\int _1^x\:e^tdt=e^x-e$ which is divergent, so all your answers, helped me to think otherwise, maybe my method help something else :D</p>
| zhw. | 228,045 | <p>Or just use $e^{x^2} \ge 1$ on $[0,\infty)$ to see $\int_0^x e^{t^2}dt \ge x \to \infty.$</p>
|
64,643 | <blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="https://math.stackexchange.com/questions/4467/a1-2-is-either-an-integer-or-an-irrational-number">$a^{1/2}$ is either an integer or an irrational number</a> </p>
</blockquote>
<p>I know how to prove $\sqrt 2$ is an irrational number. Who can tell ... | Ragib Zaman | 14,657 | <p>Suppose $\sqrt{3} = a/b$ where $a$ and $b$ have no common factor (and note $b\neq 1$). Then $ 3 = a^2/b^2$, but $a^2$ and $b^2$ no common factors to cancel to produce an integer, so we have a contradiction.</p>
|
64,643 | <blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="https://math.stackexchange.com/questions/4467/a1-2-is-either-an-integer-or-an-irrational-number">$a^{1/2}$ is either an integer or an irrational number</a> </p>
</blockquote>
<p>I know how to prove $\sqrt 2$ is an irrational number. Who can tell ... | marty cohen | 13,079 | <p>Another variation on a theme:</p>
<p>If $\sqrt 3 = m/n$, where $n$ is as small as possible, then
$$ \frac{m}{n} = \sqrt 3 \frac{\sqrt 3 - 1}{\sqrt 3 - 1} = \frac{3-\sqrt 3}{\sqrt 3 - 1}
= \frac{3-m/n}{m/n-1} = \frac{3 n - m}{m-n}$$
and the right side has a smaller denominator, since $m < 2n$ (i.e., $\sqrt 3 <... |
12,057 | <p>Let $p,q$ belong to $\mathbb{N}$ and are relatively prime to each other. If $\alpha,\beta$ belong to $\mathbb{N}$, are also relatively prime to each other,then are $(p\beta+q\alpha)$ and $q\beta$ always relatively prime ?</p>
| Alex B. | 3,212 | <p>The answer is clearly no: take $p=\alpha$, $q=\beta$. Then $p\beta+q\alpha=2pq$ and $q\beta=pq$.</p>
<p>If however $\beta$ is co-prime to $q$, then the answer is equally clearly yes, since no divisor of $q$ divides $p\beta$ and no divisor of $\beta$ divides $q\alpha$.</p>
|
12,057 | <p>Let $p,q$ belong to $\mathbb{N}$ and are relatively prime to each other. If $\alpha,\beta$ belong to $\mathbb{N}$, are also relatively prime to each other,then are $(p\beta+q\alpha)$ and $q\beta$ always relatively prime ?</p>
| TCL | 3,249 | <p>No, e.g. $p=3,q=5,\alpha=3,\beta=5$ is a counterexample.</p>
|
4,519,106 | <p>After I learned about the existence of such a concept as a contrapositive, I always try to translate any statements into a contrapositive. And every time I fail. I haven't found a general technique for this yet. I think that if I know the statement and its contrapositive form, it will give me a better understandin... | ryang | 21,813 | <p>Elaborating on Nitin’s answer: <span class="math-container">$A$</span> isn’t a proper subset of <span class="math-container">$B$</span> means precisely that<br><em>EITHER</em> some element of <span class="math-container">$A$</span> isn’t in <span class="math-container">$B\;$</span> <em>OR</em> <span class="math-cont... |
308,329 | <p>I need help with writing $\sin^4 \theta$ in terms of $\cos \theta, \cos 2\theta,\cos3\theta, \cos4\theta$.</p>
<p>My attempts so far has been unsuccessful and I constantly get developments that are way to cumbersome and not elegant at all. What is the best way to approach this problem?</p>
<p>I know that the answe... | muzzlator | 60,855 | <p>$$\begin{align}\sin^4 \theta &= (\sin^2\theta)^2\\ &= \left(\frac12-\frac12\cos(2\theta)\right)^2\\ &= \frac14 \left(1 - \cos(2\theta)\right)^2\\ &= \frac14\left(1 - 2 \cos(2\theta) + \cos^2(2 \theta)\right)\\ &= \frac14\left(1 - 2 \cos(2 \theta) + \frac12(\cos (4\theta) + 1)\right)\\ &= \fra... |
308,329 | <p>I need help with writing $\sin^4 \theta$ in terms of $\cos \theta, \cos 2\theta,\cos3\theta, \cos4\theta$.</p>
<p>My attempts so far has been unsuccessful and I constantly get developments that are way to cumbersome and not elegant at all. What is the best way to approach this problem?</p>
<p>I know that the answe... | Barbara Osofsky | 59,437 | <p>Hint: Start by noting $\sin^4 (\theta)=\left(\sin^2(\theta)\right)\cdot\left(\sin^2(\theta)\right)$. Then use the double angle formula derived by $\cos(2\theta)=\cos^2(\theta)-\sin^2(\theta)=1-2\sin^2 (\theta)$ so $$\sin^2(\theta)={1\over 2}\cdot\left(1-\cos(2\theta)\right)$$ and you now plug into the factors of $... |
1,807,479 | <blockquote>
<p>I recently took a test and was confused about a question. I feel that
the answer is B. Could anyone please elucidate it. Thanks!</p>
</blockquote>
<p>The point $(−4, 3)$ is on the terminal side of angle $\theta$ as sketched below. Find $\cos\theta$.</p>
<p><a href="https://i.stack.imgur.com/BiOiI.... | N. F. Taussig | 173,070 | <p>The number you computed is $\tan\theta = -\frac{3}{4}$. If an angle is in standard position (vertex at the origin, initial side on the positive $x$-axis) and $(x, y)$ is the point where the terminal side of the angle intersects the circle with radius $r$ with center at the origin, then
\begin{align*}
\sin\theta &... |
28,877 | <p>Since I self-study mathematical analysis without <em>formal</em> teacher, I can only appeal to help from out site most of the time. It's obvious that to grasp the underlying concepts in mathematics, we must roll the sleeves and solve problems.</p>
<p>It's clear that there are actually mistakes and misunderstanding ... | Arnaud Mortier | 480,423 | <p>Sometimes how attractive a question is is also a matter of luck, of who is connected at the time you ask. But in general here are a few tips:</p>
<ul>
<li>Avoid asking ten questions in one (I've seen that)</li>
<li>Try to <em>emphasize the point</em> - even if the context requires some terminology and perhaps an un... |
2,464,890 | <p>Here is link to some limit questions:</p>
<p><a href="https://i.stack.imgur.com/2rM9f.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/2rM9f.png" alt="Example" /></a>
Can anyone explain how has answers were derived? In (a), how can we cancel out <span class="math-container">$(x-2)$</span>? And how ... | Tanmay | 687,415 | <p>If there be a given limit like the one given underneath:</p>
<p><span class="math-container">$$\lim_{x \to 2}~ \frac{x^2-4}{x-2}$$</span></p>
<p>As it can be seen that as the denominator tends to <span class="math-container">$~0~$</span> the limit approaches to infinity,and therefore we cancel out both <span class... |
3,865,954 | <p>Suppose that I have the following sum:
<span class="math-container">$\sum_{m=0}^{\infty}(e^{it}(1-p))^{m}$</span>, where <span class="math-container">$i^2 = -1$</span>.</p>
<p>This is a geometric series, but involving the complex number <span class="math-container">$i$</span>. Can I just apply the geometric series f... | J.G. | 56,861 | <p>To prove that if <span class="math-container">$|z|<1$</span> then <span class="math-container">$\sum_{m\ge0}z^m=\tfrac{1}{1-z}$</span>, note that <span class="math-container">$\tfrac{1}{1-z}-\sum_{m=0}^{n-1}z^m=\frac{z^n}{1-z}$</span> has <span class="math-container">$n\to\infty$</span> limit <span class="math-co... |
1,807,456 | <p>I don't know asymptotic behaviour of the integral $$\int_{0}^{\infty}\frac{du}{\sqrt{4\pi u^{3}}}\left(1-\frac{e^{-\Omega u}}{\sqrt{\frac{1-\exp\left(-2u\right)}{2u}}}\right),$$
when I read a physics paper. It says that the integral have asymptotic behaviour $\log\left(\pi\Omega B\right)/\sqrt{2\pi}$, when $\Omega\t... | tired | 101,233 | <p>Please view this as a supplement to @Jack's Answer which avoids the use of special functions and adds another constant contribution which might explain the differences between numerics and former asymptotic calculations.</p>
<p>To make the analysis simpler, let us rescale $\Omega u=x$ to obtain (we drop the $4\pi$ ... |
2,612,308 | <p>Obviously we can rearrange for <span class="math-container">$x$</span> in a polynomial of degree 2. </p>
<p>Let <span class="math-container">$y=ax^2+bx+c$</span></p>
<p>then </p>
<p><span class="math-container">$x=\frac{-b\pm\sqrt{b^2-4ac+4ay}}{2a}$</span></p>
<p>Similarly, for <span class="math-container">$y=ax... | Martin Argerami | 22,857 | <p>You cannot expect in general to be able to solve for $x$. For instance, consider
$$
x^5-4x+2=0.
$$
One can easily show, using calculus (or by just plotting) that it has three real roots. One can, however, <strong>prove</strong> using <a href="https://en.wikipedia.org/wiki/Galois_theory" rel="nofollow noreferrer">G... |
2,647,194 | <p>show that $p(x)=x^3-x^2-4x+5$ is irreducible in $\mathbb{Q}[x]$ </p>
<p>How do we decide if a polynomial $p (x)$ in $\mathbb{Q}[x]$ is irreducible?</p>
| Dietrich Burde | 83,966 | <p>For a non-constant polynomial $f(x)$ of degree $n\le 3$ over a field we know that $f(x)$ is irreducible if and only if it has no root. This can be decided by the <a href="https://en.wikipedia.org/wiki/Rational_root_theorem" rel="nofollow noreferrer">rational root test</a>. The divisors of $5$ are $\pm 1$ and $\pm 5$... |
1,234,093 | <p>Given that $ e= \frac{a^2-b^2}{b^2} $ , and $L$ is the length of the perimeter, which equals $4aE(e, \pi/2)$, find the length of the perimeter up to $e^2$ in terms of $a$ and $b$.</p>
<p>How does one begin this?</p>
| Jack D'Aurizio | 44,121 | <p>If our ellipse is given by the equation $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, its length is given by:
$$ L(a,b)=4a\int_{0}^{\pi/2}\sqrt{1+e^2\sin^2\theta}\,d\theta. \tag{1}$$
Since:
$$ \int_{0}^{\pi}\sqrt{1+x^2+2x\cos(2\theta)}\,d\theta = \pi\sum_{n\geq 0}\left(\frac{1}{(2n-1)4^n}\binom{2n}{n}\right)^2 x^{2n}\tag{2} ... |
2,138,916 | <p>My question read: </p>
<p>Show that $S_{10}$ contains elements of orders $10,20,30$. Does it contain an element of order $40$? </p>
<p>I am not too sure what the question is asking. Would I have to explicitly write out all the permutations in $S_{10}$ first and then find the orders for all of them? </p>
<p>Update... | Ron Gordon | 53,268 | <p>Suppose $a \gt b$ for now. Consider the contour integral in the complex plane</p>
<p>$$\oint_C dz \frac{\log{\left ( z^2+a^2 \right )}}{z^2+b^2} $$</p>
<p>where $C$ is a semicircle of radius $R$ in the upper half-plane with a detour down and up the imaginary axis about the branch point $z=i a$. In the limit as $... |
1,454,344 | <p>Does span=(2,-1,1,2), (-2,1,-1,-2) represent a line, plane or hyperplane in R4?</p>
<p>We haven't learned matrices yet either </p>
| jimbo | 115,363 | <p>How $(2,-1,1,2)=-(-2,1,-1,-2)$ are dependent, then represent a line in $\mathbb{R}^4$</p>
|
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