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 |
|---|---|---|---|---|
2,799,439 | <blockquote>
<p>Prove that if $p$ is a prime in $\Bbb Z$ that can be written in the form $a^2+b^2$ then $a+bi$ is irreducible in $\Bbb Z[i]$ .</p>
</blockquote>
<p>Let $a+bi=(c+di)(e+fi)\implies a-bi=(c-di)(e-fi)\implies a^2+b^2=(c^2+d^2)(e^2+f^2)\implies p|(c^2+d^2)(e^2+f^2)\implies p|c^2+d^2 $ or $p|e^2+f^2$
since... | Jonathan Dunay | 538,622 | <p>Use different divides relations that you get from $a^2+b^2=(c^2+d^2)(e^2+f^2)$. You get that $c^2+d^2|p$ and $e^2+f^2|p$ which implies that one of $e+fi$ or $c+di$ them is a unit.</p>
|
3,485,441 | <p>I don't quite understand why Burnside's lemma
<span class="math-container">$$
|X/G|=\frac1{|G|}\sum_{g\in G} |X_g|
$$</span>
should be called a "lemma". By "lemma", we should mean there is something coming after it, presumably a theorem. However, I could not find a theorem which requires Burnside as a lemma. In ever... | User7238 | 736,765 | <p>Suppose G acts on H where <span class="math-container">$|G|,|H|<\infty$</span>. For each <span class="math-container">$g\in G$</span>, consider the permutation character associated with the action. That is, <span class="math-container">$\chi(g)=|\{\alpha\in H:\alpha\cdot g=\alpha\}|$</span>.<br>
Then, <span cla... |
1,748,001 | <p>I need to find a relation between $\sqrt{x+ia}$ and $\sqrt{\sqrt{x^2+a^2}+x}$</p>
<p>where $a>0$ $x\in \mathbb{R}$</p>
<p>Thank you</p>
| Ross Millikan | 1,827 | <p>When you multiply terms, you add exponents, so $x \cdot x^{\frac 23}=x^{\frac 53}$ They multiplied top and bottom of the first by $x$ to put the two over a common denominator.</p>
|
1,748,001 | <p>I need to find a relation between $\sqrt{x+ia}$ and $\sqrt{\sqrt{x^2+a^2}+x}$</p>
<p>where $a>0$ $x\in \mathbb{R}$</p>
<p>Thank you</p>
| ervx | 325,617 | <p>Recall that $x^{a}\cdot x^{b}=x^{a+b}$. Thus, to simplify your sum of fractions, we bring both of the original fractions to a common denominator using the aforementioned rule.</p>
<p>Note that $3x^{\frac{2}{3}}\cdot x=3x^{\frac{2}{3}+1}=3x^{\frac{5}{3}}$. Thus, </p>
<p>$$
\frac{1}{3x^{\frac{2}{3}}}+\frac{2}{3x^{\f... |
1,251,914 | <p>I do not understand how to set up the following problem:</p>
<p>"Forces of 20 lb and 32 lb make an angle of 52 degrees with each other. find the magnitude of the resultant force."</p>
<p>An actually picture would really help.</p>
| Lee Mosher | 26,501 | <p>Multidimensional spaces occur naturally all around us. Take the possible positions of your arm, for example. </p>
<ol>
<li>You can rotate your shoulder with two degrees of freedom;</li>
<li>You can bend your elbow with one degree of freedom;</li>
<li>You can rotate your wrist with two degrees of freedom</li>
</ol>
... |
1,251,914 | <p>I do not understand how to set up the following problem:</p>
<p>"Forces of 20 lb and 32 lb make an angle of 52 degrees with each other. find the magnitude of the resultant force."</p>
<p>An actually picture would really help.</p>
| dtldarek | 26,306 | <p>There are text explanations, so I will post some pictures.</p>
<p>We will try to build some intuition by comparing to zero-, one-, two-, three- and four-dimensional tables/arrays.</p>
<p><strong>Introduction:</strong></p>
<p>What's zero-dimensional array you ask? It is a single cell, as single point is a zero-dimens... |
1,251,914 | <p>I do not understand how to set up the following problem:</p>
<p>"Forces of 20 lb and 32 lb make an angle of 52 degrees with each other. find the magnitude of the resultant force."</p>
<p>An actually picture would really help.</p>
| N. Owad | 85,898 | <p>I am a firm believer in the idea that if something in math doesn't make sense to you, you just haven't found the proper way of thinking about it yet. And in some (or maybe most?) cases, there isn't anybody on earth who has found the right way. And this way is probably different for different people.</p>
<p>Now, t... |
114,895 | <blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="https://math.stackexchange.com/questions/21282/show-that-every-n-can-be-written-uniquely-in-the-form-n-ab-with-a-squa">Show that every $n$ can be written uniquely in the form $n = ab$, with $a$ square-free and $b$ a perfect square</a> </p>
</blockq... | Greg Martin | 16,078 | <p>For a proof using minimal machinery, I prefer:</p>
<p>Consider all representations of $n$ in the form $n=a^2b$, where $a$ and $b$ can be any positive integers. (These can be indexed by the set $A$ of positive integers $a$ such that $a^2\mid n$.) You can show that $A$ consists exactly of all divisors of a special in... |
2,443,496 | <blockquote>
<p>Can someone point me in the right direction as to how to take the derivative of this function:
$$ f(x) = 2 \pi \sqrt{\frac{x^2}{c}} $$</p>
</blockquote>
<p>Thank you</p>
| valer | 403,899 | <p>$f(x) = 2\pi\sqrt\frac{x^2}{c}$</p>
<p>$f'(x) = 2\pi*\frac{1}{2\sqrt{\frac{x^2}{c}}}*({\frac{x^2}{c}})'=2\pi\frac{\frac{x}{c}}{\sqrt\frac{x^2}{c}}=2\pi*\frac{\sqrt\frac{x^2}{c}}{x}$</p>
|
3,251,754 | <p>Let <span class="math-container">$M$</span> be the set of all <span class="math-container">$m\times n$</span> matrices over real numbers.Which of the following statements is/are true??</p>
<ol>
<li>There exists <span class="math-container">$A\in M_{2\times 5}(\mathbb R)$</span> such that the dimension of the nulls... | 5xum | 112,884 | <p>Your are correct, and the proof is rather simple (not requiring the wall of text you wrote :)</p>
<p><span class="math-container">$$\begin{align}P(\neg B|C)&=\frac{P(\neg B \land C)}{P(C)} &\text{by definition}
\\&= \frac{P(C) - P(B\land C)}{P(C)} & \text{Because $B\land C$ and $\neg B\land C$ form ... |
21,141 | <p>Is there a way to extract the arguments of a function? Consider the following example:</p>
<p>I have this sum</p>
<pre><code>g[1] + g[2] + g[3] + g[1]*g[3] + 3*g[1]*g[2] + 6*g[1]*g[2]*g[3]
</code></pre>
<p>Now, what I want to do is exctract the function arguments and apply them to another function <code>func</cod... | kglr | 125 | <pre><code>expr = g[1] + g[2] + g[3] + g[1]*g[3] + 3*g[1]*g[2] + 6*g[1]*g[2]*g[3];
expr /. Times[x___:1 , p__g] :> x func[{p}[[All, 1]]]
(* func[{1}] + func[{2}] + func[{3}] + 3 func[{1, 2}] + func[{1, 3}] + 6 func[{1, 2, 3}]*)
</code></pre>
|
3,181,502 | <p>We have <span class="math-container">$\tan(x)=\dfrac{\sin(x)}{\cos(x)}$</span>. I was wondering why <span class="math-container">$\tan(x+{\pi/2})=\tan(x)$</span>?</p>
<p>I wanted to Show </p>
<p><span class="math-container">$$\frac{\sin(x+\pi/2)}{\cos(x+\pi/2)}=\frac{\sin(x)}{\cos(x)}\iff\frac{\sin(x+\pi/2)\cos(x)... | MPW | 113,214 | <p>No.</p>
<p><span class="math-container">$$\tan(x+\pi/2)=\frac{\sin(x+\pi/2)}{\cos(x+\pi/2)}$$</span>
<span class="math-container">$$=\frac{\sin x\cos \pi/2 + \cos x \sin \pi/2}{\cos x \cos \pi/2 - \sin x \sin \pi/2}$$</span>
<span class="math-container">$$=\frac{\cos x}{-\sin x}$$</span>
<span class="math-container... |
2,588,968 | <p>I have the double integral</p>
<p>$$\int^{10}_0 \int^0_{-\sqrt{10y-y^2}} \sqrt{x^2+y^2} \,dx\,dy$$</p>
<p>And I am asked to evaluate this by changing to polar coordinates.</p>
| Community | -1 | <p>Set $x$ equal to the lower bound of your inner integral. This line defines a boundary of the region of integration. Plot it, along with the other boundaries $x=y=0$ and $y=10$ and see if you can express the region more naturally in polar coordinates. Make the necessary transformation and it should become clear how t... |
3,831,073 | <p>Let <span class="math-container">$\alpha = \sqrt[3]{4+\sqrt{5}}$</span>. I would like to prove that <span class="math-container">$\left[ \mathbb{Q} \left( \alpha \right ) : \mathbb{Q} \right] = 6$</span>. We have <span class="math-container">$\alpha^3 = 4 + \sqrt{5}$</span>, and so <span class="math-container">$(\al... | Hagen von Eitzen | 39,174 | <p>As <span class="math-container">$\Bbb Q(\sqrt 5)$</span> is a subfield, its automorphism <span class="math-container">$\sqrt 5\to-\sqrt 5$</span> can be extended, which means that <span class="math-container">$\sqrt[3]{4-\sqrt 5}\in\Bbb Q(\sqrt[3]{4+\sqrt 5})$</span> and also
<span class="math-container">$$ \sqrt[3]... |
2,544,864 | <p>I have been trying to prove the continuity of the function:
$f:\mathbb{R}\to \mathbb{R}, f(x) =x \sin(x) $ using the $\epsilon -\delta$ method. </p>
<p>The particular objective of posting this question is to understand <strong>the dependence of $\delta$ on $\epsilon$ and $x$</strong>. I know that $f(x) =x \sin(x) $... | Christian Blatter | 1,303 | <p>Forget about $2n\pi$ and stuff. The essential point is that $|\sin x-\sin y|\leq |x-y|$ for arbitrary $x$, $y\in{\mathbb R}$.</p>
<p>Let an $x\in{\mathbb R}$ be given and consider an increment $h$ with $|h|\leq1$ applied at $x$. Then
$$\eqalign{\bigl|(x+h)\sin(x+h)-x\sin x\bigr|&=\bigl|(x+h)(\sin(x+h)-\sin x)+... |
521,500 | <p>Today we proofed the (simple) Markov property for the Brownian motion. But I really don't get a crucial step in the proof.
The theorem states in particular that for $s\geq0$ fixed, the process $(C_t:=B_{t+s}-B_{s}, t\geq0)$ is independent from $\mathcal{F}_s=\sigma(B_u, 0\leq u\leq s)$.</p>
<p>The proof starts with... | Davide Giraudo | 9,849 | <p>If $D$ is a countable subset of $[0,s]$ and $S\in\mathcal B(\mathbb R^\infty)$, then $\{\omega,(B_u(\omega))_{u\in D}\in S)$ belongs to $\sigma(B_u,0\leqslant u\leqslant s)$. The collection of sets of this form is a $\sigma$-algebra. </p>
<p>So, after passing from countable intersections to finite ones, it's enough... |
9,085 | <p>So as the title says I am trying to make a list where each element is determined by a users choice of an element in a PopupMenu.</p>
<p>My first attempt:</p>
<pre><code>test = Table["A", {5}];
Table[PopupMenu[Dynamic[test[[n]]], {"A", "B", "C"}], {n, 5}]
</code></pre>
<p>Returned the following error</p>
<pre><co... | b.gates.you.know.what | 134 | <p>This seems to work :</p>
<pre><code>test = Table["A", {5}];
PopupMenu[Dynamic[test[[#]]], {"A", "B", "C"}, "A"] & /@ Range[5] // Row
Dynamic @ test
(* {"C", "A", "B", "A", "C"} *)
</code></pre>
|
1,797,712 | <p>Let $G = \Bbb{Z}_{360} \oplus \Bbb{Z}_{150} \oplus \Bbb{Z}_{75} \oplus \Bbb{Z}_{3}$</p>
<p>a. How many elments of order 5 in $G$</p>
<p>b. How many elments of order 25 in $G$</p>
<p>c. How many elments of order 35 in $G$</p>
<p>d. How many subgroups of order 25 in $G$</p>
<p>I think I have done a,b,c correctly ... | Quang Hoang | 91,708 | <p>Using the presentation theorem of finite abelian groups, one can restrict the problem to the $5$-primary part of $G$, which is
$$H=\mathbb Z_5\times \mathbb Z_{25}\times \mathbb{Z}_{25}.$$</p>
<p>$H$ has $5\times 25\times 25=3125$ elements. There should be $\color{red}{124}$ elements of order $5$, not $24$. </p>
<... |
1,357,247 | <p>I'm sorry if my question is repeated.</p>
<p>Let $E$ be the set of all $x\in [0,1]$ whose decimal expansion contains only the digits $4$ and $7$. Is $E$ contable? Is $E$ dense in $[0,1]$? Is $E$ compact? Is $E$ perfect?</p>
<p><strong>Proof:</strong> It is easy to see that $E$ is uncountable. </p>
<p>Also $E$ is ... | ajotatxe | 132,456 | <p>$E$ is closed in $[0,1]$ and in $\Bbb R$. The number $z=0,\bar 9$ (which is $1$), is in $E^c$ (<em>to check whether $E$ is closed in $\Bbb R$, you must take the complementary respect to $\Bbb R$</em>) and the interval $(0.9,1.1)$ is contained in $E^c$.</p>
|
1,357,247 | <p>I'm sorry if my question is repeated.</p>
<p>Let $E$ be the set of all $x\in [0,1]$ whose decimal expansion contains only the digits $4$ and $7$. Is $E$ contable? Is $E$ dense in $[0,1]$? Is $E$ compact? Is $E$ perfect?</p>
<p><strong>Proof:</strong> It is easy to see that $E$ is uncountable. </p>
<p>Also $E$ is ... | nombre | 246,859 | <p>Compactness is an absolute notion -meaning that it is defined for a topological space regardless of the ambient space whence it may be a subset - contrary to the notion of being closed, for instance. </p>
<p>Now, if a space $X$ is compact, and if $F$ is a closed subset of $F$, then $F$ seen as a topological space (... |
131,206 | <p>According to the wiki of <a href="http://en.wikipedia.org/wiki/Kakutani_fixed-point_theorem">Kakutani's fixed-point theorem</a>, A set-valued mapping $\varphi$ from a topological space $X$ into a powerset $\wp(Y)$ called upper semi-continuous if for every open set $W \subseteq Y$, $\lbrace x| \varphi(x) \subseteq W ... | Steven Landsburg | 10,503 | <p>$\phi$ is upper semicontinuous if, for every open $W\subset Y$, the set $\lbrace x | \phi(x)\subset W\rbrace $ is open in $X$.</p>
<p>$\phi$ is lower semicontinuous if, for every open $W\subset Y$, the set $\lbrace x | \phi(x)\cap W\neq \emptyset\rbrace$ is open in $X$.</p>
<p>$\phi$ is continuous if it is both up... |
2,761,509 | <p>I hope it's not a duplicate but I've been searching about this problem for some time on this site and I couldn't find anything. My problem is why a number $\in(-1,0)$ raised to $\infty$ is $0$. For example let's take
$$\lim_{n\to \infty} \left(\frac{-1}{2}\right)^n$$
Which is equivalent to
$$\left(\frac{-1}{2}\righ... | IDC | 422,221 | <p>Your assertion that $(\frac{-1}{2})^\infty=(-1)^\infty(\frac{1}{2})^\infty$ is wrong because the first limit on the right side does not exist. The limit of a product is only equal to the product of limits when all the limits exist.</p>
|
8,052 | <p>I wonder how you teachers walk the line between justifying mathematics because of
its many—and sometimes surprising—applications, and justifying it as the study
of one of the great intellectual and creative achievements of humankind?</p>
<p>I have quoted to my students G.H. Hardy's famous line,</p>
<bl... | MintChocolateIceCream | 5,141 | <p>Most of what you learn in school isn't directly useful. When I was in primary school I was taught the difference between warm and cold-blooded animals. I've never used that information; should I not have been taught it? Another example is that in any English-speaking country you still have to take English courses... |
8,052 | <p>I wonder how you teachers walk the line between justifying mathematics because of
its many—and sometimes surprising—applications, and justifying it as the study
of one of the great intellectual and creative achievements of humankind?</p>
<p>I have quoted to my students G.H. Hardy's famous line,</p>
<bl... | Kyle | 5,152 | <p>I'm no teacher, but my position is if you're only studying what has some useful application now, how are you ever going to discover the new applications that we don't know about? I think your Hardy example sums that up perfectly.</p>
<p>I know, that still probably somewhat misses the point of your question because... |
8,052 | <p>I wonder how you teachers walk the line between justifying mathematics because of
its many—and sometimes surprising—applications, and justifying it as the study
of one of the great intellectual and creative achievements of humankind?</p>
<p>I have quoted to my students G.H. Hardy's famous line,</p>
<bl... | Community | -1 | <p>You are not learning math.</p>
<p>You are learning the very useful, highly transferable skills of abstraction and problem solving. It's not that math can solve many problems, it's the skill you learn in order to do math are useful in so many other problems.</p>
|
8,052 | <p>I wonder how you teachers walk the line between justifying mathematics because of
its many—and sometimes surprising—applications, and justifying it as the study
of one of the great intellectual and creative achievements of humankind?</p>
<p>I have quoted to my students G.H. Hardy's famous line,</p>
<bl... | CaptainObvious | 5,172 | <p>I'm not a teacher/professor but I'd like to give my input as a person who hated math for a big part of his life, especially at high school. </p>
<p>I'd like first to explain why I hated math that much and for so long. At high school we were basically receiving some input and had to spit out some output. I'm referr... |
8,052 | <p>I wonder how you teachers walk the line between justifying mathematics because of
its many—and sometimes surprising—applications, and justifying it as the study
of one of the great intellectual and creative achievements of humankind?</p>
<p>I have quoted to my students G.H. Hardy's famous line,</p>
<bl... | John | 6,433 | <p>My simplest answer to any student asking "Why study math?" is this</p>
<blockquote>
<p>Math gives you power to get what you want.</p>
</blockquote>
<p>This is not the most "politically correct" way of "selling math", but it's both true and often persuasive to students and adults who want more power (which ends u... |
25,158 | <p>I'm trying to derive the LTE for CN applied to the linear heat equation;
$u_t = u_{xx}$.</p>
<p>The problem is that I end up with terms of the form $\frac{{\Delta t}^k}{{\Delta x}^2}$
when using a two dimensional Taylor expansion around $(x,t)$ for the term:</p>
<p>${\Delta x}^2 {\delta^2_x} = (u_{i+1}^{n+1} - 2 ... | Community | -1 | <p>your taylor expansions should be around $\left(x_i,t_{n+1}\right)$</p>
|
412,482 | <p>I have a few (semi-)related questions regarding certain Hilbert space representations of locally compact groups that come up in the theory of automorphic forms. </p>
<p>Let $G$ be a unimodular locally compact Hausdorff group, $\Gamma$ a discrete (hence closed) subgroup of $G$, and $Z$ a closed subgroup of the cente... | Bravo | 24,451 | <p>Consider <a href="http://en.wikipedia.org/wiki/Multivariate_normal_distribution#Conditional_distributions" rel="noreferrer">this theorem</a> on conditional expectation of multivariate Gaussian distribution. Let $x=X_u$ and $Y=\left[X_s \quad X_t\right]^\top$. Let $X_0=0$, so that $\mu_X=0$ and $\mu_Y=\left[0 \quad 0... |
1,201,942 | <p>I apologize if this question gets down-voted ahead of time.</p>
<p>I've been working on the Collatz Conjecture all day with Python, because that is the language I'm most familiar with (I'm not a CS student, just majoring in math). Below is the function I'm using for your reference during my comments:</p>
<p>\begin... | hmakholm left over Monica | 14,366 | <p>Yes, you're right: If there are any counterexample, then the <em>smallest</em> counterexample must be odd. And all successors and predecessors of a counterexample are themselves counterexamples.</p>
<p>There are two conceivable kinds of counterexample.</p>
<p>The first is a finite cyclic sequence that differs from... |
1,746,363 | <p>I got maybe easy problem. I am not sure if it is true that [$\mathbb Z_2[x]/f\mathbb Z_2[x]: \mathbb Z_2$]=deg $f$ where $f \in \mathbb Z_2[x]$ irreducible. Can anybody help me ? Thanks</p>
| Eman Yalpsid | 94,959 | <p>The rate of change at $x=\frac{5\pi}{6}$ is simply $h'(\frac{5\pi}{6}) = 12 \sin^2(\frac{5\pi}{6})\cos(\frac{5\pi}{6})$.</p>
<p>Your confusion probably arises from the fact that you are used to dealing with degrees instead of radians.</p>
<p>The period of $\sin, \cos$ is $2\pi$ or $360$ degrees. One radian is $\fr... |
1,578,783 | <p>A friend of mine found a book in which the author said that the dual space of $L^\infty$ is $L^1$, of course not with the norm topology but with the weak-* topology. Does anyone know where I can find this result? Thanks.</p>
| Community | -1 | <p>There is a general fact from duality of linear spaces (see Proposition 4.28 in Fabian-Habala-Hajek-Montesinos-Pelant-Zizler, Functional Analysis and Infinite-Dimensional Geometry): If we consider a linear subspace $F$ in the space of linear functionals on $E$, then the space of linear functionals on $E$ continuous i... |
3,547,529 | <p>I did the following: I set <span class="math-container">$3^m+3^n+1=x^2$</span> where <span class="math-container">$x\in\Bbb{N}$</span> and assumed it was true for positive integer exponents and for all whole numbers x so that I can later on prove it's invalidity with contradiction. Since <span class="math-container"... | Bart Michels | 43,288 | <p>It is clear that <span class="math-container">$A(y), B(y) \sim \pm (y/a)^{1/2n}$</span>, and the problem is to determine the lower order term in their expansion. Wlog suppose <span class="math-container">$B(y) < 0 < A(y)$</span>. By the mean value theorem,
<span class="math-container">$$(A(y) - (y/a)^{1/2n}) \... |
2,571,909 | <p>$$\left|\frac{-10}{x-3}\right|>\:5$$</p>
<ul>
<li>Find the values that $x$ can take. </li>
</ul>
<p>I know that</p>
<p>$$\left|\frac{-10}{x-3}\right|>\:5$$
and
$$\left|\frac{-10}{x-3}\right|<\:-5$$</p>
| Community | -1 | <p>First note that $x=3$ cannot be a part of the solution. That in mind, we have: $$|\frac{-10}{x-3}|>5 \implies \frac{10}{5}> |x-3| \implies -2< x -3 < 2 \implies x \in (1,5) - \{3\}$$</p>
|
1,808,258 | <p>I was reading about orthogonal matricies and noticed that the $2 \times 2$ matrix
$$\begin{pmatrix} \cos(\theta) & \sin(\theta) \\ -\sin(\theta) & \cos(\theta) \end{pmatrix} $$
is orthogonal for every value of $\theta$ and that every $2\times 2$ orthogonal matrix can be expressed in this form. I then wonder... | Travis Willse | 155,629 | <p>In general, (3) is not true. Since $\left\vert v \right\vert^2 = x(t)^2 + y(t)^2 = 1$ , we can write $$x(t) = \cos \theta(t), \quad y(t) = \sin \theta(t)$$
for some function $\theta(t)$. But computing gives $\left\vert\dot v\right\vert^2 = \dot\theta(t)^2$, so the condition $|\dot v|^2 = 1$ (which just says that $v$... |
1,200,358 | <blockquote>
<p>Assume the $n$-th partial sum of a series $\sum_{n=1}^\infty a_n$ is the following:
$$S_n=\frac{8n-6}{4n+6}.$$
Find $a_n$ for $n > 1$.</p>
</blockquote>
<p>I'm really stuck on what to do here.</p>
| kobe | 190,421 | <p>Hint. Note $a_n = S_n - S_{n-1}$ for $n > 1$.</p>
|
446,326 | <blockquote>
<p>Let $Q$ be a $3\times3$ special orthogonal matrix. Show that $Q(u\times v)=Q(u)\times Q(v)$ for any vectors $u, v\in\mathbb R^3$.</p>
</blockquote>
<p>I have no idea how to start. I'm not sure if $Q(u)\cdot Q(V)=Q(u\cdot v)$ would helps. Please give me some help. Thanks.</p>
| Julien | 38,053 | <p>The cross-product $u\times v$ is the unique vector such that
$$
\det(u,v,w)=(u\times v)\cdot w\qquad \forall w
$$
where $\det(u,v,w)$ is the determinant of the $3\times 3$ matrix whose columns are $u,v,w$ in this order, that is the determinant of the linear map that sends the canonical basis to $(u,v,w)$. That's a ... |
4,506,026 | <p>Consider the set of equations:
<span class="math-container">$$
\begin{cases}
x^2 &= -4y-10\\y^2 &= 6z-6\\z^2 &= 2x+2\\
\end{cases}$$</span></p>
<p>With <span class="math-container">$x,y,z$</span> being real numbers.</p>
<p>By adding the three equations, after simple manipulations, we easily obtain
<span ... | pierrecurie | 616,347 | <p>What you found is a necessary but not sufficient condition. This is what happens in general when you take linear combinations of your initial equations.</p>
<p>If you back substitute, you'll end up with a degree 8 polynomial with 8 complex roots. The manipulations you did aren't wrong - there are indeed 0 real roo... |
3,182,802 | <p>Show that if <span class="math-container">$ \sigma $</span> is a solution to the equation <span class="math-container">$ x^2 + x + 1 = 0 $</span> then the following equality occurs:</p>
<p><span class="math-container">$$ (a +b\sigma + c\sigma^2)(a + b\sigma^2 + c\sigma) \geq 0 $$</span></p>
<p>I looked at the solu... | Bernard | 202,857 | <p>I suppose <span class="math-container">$a,b,c$</span> are real numbers. Now observe <span class="math-container">$\sigma$</span> is a non-real cubic root of unity, so the other non-real root is <span class="math-container">$\bar \sigma=\sigma^2$</span>. Thus
<span class="math-container">\begin{align}
(a +b\sigma + c... |
1,088,734 | <p>It's possible the integral bellow. What way I must to use for solve it.</p>
<p>$$\int \sin(x)x^2dx$$</p>
| heropup | 118,193 | <p>If $p$, $q$, $r$ are distinct positive integer roots of $P(x) = x^3 + ax^2 + bx - 26$, then we must have $$(x-p)(x-q)(x-r) = x^3 - (p+q+r)x^2 + (pq+qr+rp)x - pqr = P(x),$$ hence equating the constant coefficient, $$pqr = 26 = 2 \cdot 13.$$ Since the prime factorization of $26$ contains only $2$ and $13$, it immedi... |
582,283 | <p>$H$ is subgroup of $G$ with $H$ not equal $G$.</p>
<p>Be $S=G-H$. I am being asked to prove that $\langle S \rangle=G$.</p>
<p>Some tip to solve this? I think in $S_3$ is possible but I can´t prove.</p>
| egreg | 62,967 | <p>Hint. You have to use the hypothesis that $H\ne G$ and show that any element of $G$ can be written as a product of elements in $S$ or inverse thereof.</p>
<p>If $g\in S$, then of course $g\in\langle S\rangle$; if $g\in H$, pick an element $s\in S$; then $t=sg\notin H$ (why?); but $s^{-1}t=\dots$</p>
|
35,688 | <p>I'm looking for a fun (not too many tedious calculations) calculus one problem that uses the concept that, after subsitution, you have two integrals of diffent functions with different limits, but equal area. For example:</p>
<p><a href="http://www.wolframalpha.com/input/?i=int%20%28sin%28%28pi%5E2%29/x%29%29/%28x%... | André Nicolas | 6,312 | <p>The following is part of the standard machinery for proving the basic properties of logarithms, if one chooses to introduce them by using an integral.</p>
<p>For any number $c \ge 1$, define $L(c)$ by saying that $L(c)$ is the area under the curve $y=1/x$, above the $x$-axis, from $x=1$ to $x=c$.</p>
<p>Then $L(ab... |
3,613,235 | <p>I know such integral: <span class="math-container">$\int_0^{\infty}\frac{\ln x}{e^x}\,dx=-\gamma$</span>. What about the integral <span class="math-container">$\int_0^{\infty}\frac{\ln x}{e^x+1}\,dx$</span>? </p>
<p>The answer seems very nice: <span class="math-container">$-\frac{1}{2}{\ln}^22$</span> but how it co... | Andronicus | 528,171 | <p>We can rewrite the sum as:</p>
<p><span class="math-container">$$\sum_{n=1}^{99} \frac{2}{\sqrt{n} + \sqrt{n+2}}\frac{\sqrt{n+2} - \sqrt{n}}{\sqrt{n+2} - \sqrt{n}}=
\sum_{n=1}^{99} \sqrt{n+2} - \sqrt{n}$$</span></p>
<p>It's a telescopic sum, where most of the terms are cancelled.</p>
<p>We can expand it:</p>
<p>... |
382,293 | <p>Not sure where to go with this one. Clearly will have to use the axiom of choice at some point. I haven't been able to think of a good example for the set $A.$ Once we've got that, it'd be a matter of showing that a representation $($as a sum, $q+a)$ exists for each real number $($which should be the case by constru... | Asaf Karagila | 622 | <p>If you already know the axiom of choice has to be involved (and it does), there is little hope to being able to write $A$ explicitly.</p>
<p>Here is a hint: consider the quotient $\Bbb{R/Q}$.</p>
|
1,649,907 | <p>Please kindly forgive me if my question is too naive, i'm just a <em>prospective</em> undergraduate who is simply and deeply fascinated by the world of numbers.</p>
<p>My question is: Suppose we want to prove that $f(x) > \frac{1}{a}$, and we <em>know</em> that $g(x) > a$, where $f,g$ and $a$ are all positive... | User1 | 305,659 | <p>Unless i'm missing something horribly wrong, it seems the <em>current</em> form of the question is straightforward (though this appears to contradict with fleablood's answer):</p>
<p>We know that $g(x)<a$, let $f(x)\leq t$. Therefore $f(x)g(x)<at$. But we also know that $f(x)g(x)<1$, hence we should have ... |
2,315,739 | <p>I have an irregular quadrilateral.
I know the length of three sides (a, b and c) and the length of the two diagonals (e and f).
All angles are unknown
How do I calculate the length of the 4th side (d)?</p>
<p>Thank you for your help.
Regards,</p>
<p>Mo</p>
<p><a href="https://i.stack.imgur.com/Jtdzv.jpg" rel="nof... | Plus Twenty | 453,093 | <p>Hint : You could try re-arranging the Cosine Rule: $a^2 = c^2+b^2-2bc\cos A$ to try and find some of the angles of the triangles.</p>
|
3,394,277 | <p>I am new to this site and not familiar with how to type out math notation so I will do my best. I have a problem I am working on regarding the volume of a circle wrapped around a cylinder of variable radius. For the first part of the problem I had no issue creating a function to represent the cross sectional area. U... | Kavi Rama Murthy | 142,385 | <p>Note that <span class="math-container">$X$</span> is a non-decreasing function. For any non-decreasing function <span class="math-container">$f$</span> observe that <span class="math-container">$\{x: f (x) <a\}$</span> is an interval for any real number <span class="math-container">$a$</span>. Hence <span class="... |
2,473,780 | <p>So I have the limit $$\lim_{x\rightarrow \infty}\left(\frac{1}{2-\frac{3\ln{x}}{\sqrt{x}}}\right)=\frac{1}2,$$ I now want to motivate why $(3\ln{x}/\sqrt{x})\rightarrow0$ as $x\rightarrow\infty.$ I cam up with two possibilites:</p>
<ol>
<li><p>Algebraically it follows that $$\frac{3\ln{x}}{\sqrt{x}}=\frac{3\ln{x}}{... | Siong Thye Goh | 306,553 | <p>When the numerator and denominator both go to $\infty$, it is in <a href="https://en.wikipedia.org/wiki/Indeterminate_form" rel="nofollow noreferrer">indeterminate form</a>, we can use <a href="https://en.wikipedia.org/wiki/L%27H%C3%B4pital%27s_rule" rel="nofollow noreferrer">L'Hospital's rule</a>.</p>
<p>$$\lim_{... |
362,881 | <p>I am going to try to explain this as easily as possible. I am working on a computer program that takes input from a joystick and controls a servo direction and speed. I have the direction working just fine now I am working on speed. To control the speed of rotation on the servo I need to send it so many pulses per s... | ncmathsadist | 4,154 | <p>The $\delta_0$ functional is representable as a measure. You have
$$\delta_0(E) = \cases{1 & if $0 \in E$\cr 0 & otherwise }$$
Then the linear functional here is represented as
$$f \mapsto f(0) = \int_{R^d} f(\xi)d\delta_0(\xi). $$
It is not hard to show this relation will hold for any continuous function. ... |
2,487,234 | <p>I want to prove that the following only have one solution, for $\zeta\in[0,1]$, at $\zeta =1$.</p>
<p>$$f(\zeta)=\frac{1}{1+\zeta}$$</p>
<p>$$g(\zeta) =
\frac{
(1-\zeta)(2-\zeta)\zeta
-
(2-\zeta)^2\log\left(2 - \zeta \right)
}
{\zeta(\zeta-4)(\zeta-1)^2}$$</p>
<p>These are plotted below. Note that $f(0)=1$, $\lim... | Daniel Pol | 481,873 | <p>Cutting places of this 2 functions are in zeros of equation $f(\zeta)-g(\zeta)=0$. I put this in form $f(\zeta)=g(\zeta)$. After this i put all powers together and try to separate $\log{(2-s)}$ in one side. I come to next expression :
$$
\log{(2-s)} = {{2(2s^3-5s^2+3s)} \over {(s-2)^2(s+1)}}
$$
Now i idea would to m... |
1,519,952 | <p>Show that
$$S(n,k) = \sum_{m = k-1}^{n-1} {n-1 \choose m} S(m,k-1) $$</p>
<p>-I was having trouble with this proof in class and my professor suggested to look at it as another proof of the following theorem which states:</p>
<p>-For all $n\ge1$
$$B(n) = \sum_{k=0}^{n-1} {n-1 \choose k} B(k) $$
-Unfortunately I s... | Brian M. Scott | 12,042 | <p>$S(n,k)$ is the number of partitions of the set $[n]=\{1,\ldots,n\}$ into exactly $k$ non-empty parts. Suppose that $\mathscr{P}$ is such a partition; then there must be a $P\in\mathscr{P}$ that contains the number $n$. </p>
<ul>
<li>The other $k-1$ parts must contain at least $k-1$ and at most $n-1$ elements of $[... |
2,375,736 | <p>If the Ratio of the roots of $ax^2+bx+c=0$ be equal to the ratio of the roots of $a_1x^2+b_1x+c_1=0$, then how one prove that $\frac{b^2}{b^2_1}=\frac{ac}{a_1 c_1}$?</p>
| David Quinn | 187,299 | <p>Let the roots of the first quadratic be $\alpha$ and $t\alpha$ and the roots of the second be $\beta$ and $t\beta$</p>
<p>Then $$\alpha(t+1)=\frac ca$$ and $$\alpha^2t=\frac ca$$</p>
<p>Then $$\frac {b^2}{a^2}=\alpha(t+1)^2=\frac{c}{at}(t+1)^2$$</p>
<p>In exactly the same way, $$\frac {b_1^2}{a_1^2}=\beta(t+1)^2... |
248,267 | <p>It is known that the transformation rule when you change coordinate frames of the Christoffel symbol is:</p>
<p>$$ \tilde \Gamma^{\mu}_{\nu\kappa} = {\partial \tilde x^\mu \over \partial x^\alpha} \left [ \Gamma^\alpha_{\beta \gamma}{\partial x^\beta \over \partial \tilde x^\nu}{\partial x^\gamma \over \partial \ti... | Andrews | 557,551 | <p>Let <span class="math-container">$(U, x^1, \ldots, x^n)$</span> be a chart of Riemannian manifold <span class="math-container">$M$</span>, smooth coordinate chang
<span class="math-container">$(U, g_{\mu\nu}, x^1, \ldots, x^n) \to (U, g'_{\mu'\nu'}, y^1, \ldots, y^n)$</span>.
Then <span class="math-container">$\pa... |
3,807,900 | <p>I have just met this exercise in functional analysis, asking us to determine if these two subspaces of the Hilbert space <span class="math-container">$\ell^2$</span> of square-summable complex sequences are closed:</p>
<blockquote>
<ol>
<li>The set of all sequences <span class="math-container">$\{x_n\}_{n=1}^{\infty... | Brian Moehring | 694,754 | <p>For (1), as has been noted by past answers, we can use Cauchy-Schwarz to see that <span class="math-container">$f(x)= \sum_{n=1}^\infty \frac{1}{n}x_n$</span> is continuous <span class="math-container">$\ell^2 \to \mathbb{C}$</span>, so the set <span class="math-container">$\{x \in \ell^2 \mid f(x) = 0\} = f^{-1}(0)... |
176,488 | <p><strong>Summary:</strong> My question, in a nutshell, is how we should intuitively imagine a generic real number (as opposed to a random one), and whether we can construct numbers which empirically behave like generic numbers in the same way that $e$ or $\pi$ behave empirically like random ones. I hope this is not ... | Robert Israel | 13,650 | <p>You can't really have a "given" empirically-generic number, because any explicit description of $x$ can be turned into an explicit description of the meagre closed set $\{x\}$ to which it belongs.</p>
|
2,003,916 | <p>Probably a very simple question:</p>
<p>Suppose a hospital orders defibrillators from a manufacturer. It is well known that defibrillations are often not effective, even when the defibrillators themselves are working properly. Suppose research shows that only 15% percent of defibrillations are effective. Over the n... | Bill Dubuque | 242 | <p>It's a special case of the following homogeneous generalization of the Euler-Fermat theorem.</p>
<p><strong>Theorem</strong> <span class="math-container">$ $</span> Suppose that <span class="math-container">$\rm\ n\in \mathbb N\ $</span> has the prime factorization <span class="math-container">$\rm\:n = p_1^{e_{\:1... |
3,811,498 | <p>Solve, by bringing the equation to Bernoulli form:</p>
<p><span class="math-container">$$
y’ = \frac{2-xy^3}{3x^2y^2}
$$</span></p>
<hr />
<p>Therefore we want to bring it to a form like:</p>
<p><span class="math-container">$$
y’ + p(x)y = q(x)y^n
$$</span></p>
<p>So working with the equation i get:</p>
<p><span cla... | Lutz Lehmann | 115,115 | <p>You selected the wrong term. Exchange the position of terms formerly on the right side to get
<span class="math-container">$$
y'+\frac1{3x}y = \frac2{3x^2}y^{-2}.
$$</span></p>
<p>With some contemplation of the formula you could also directly detect that
<span class="math-container">$$
(xy^3)'=3xy^2y'+y^3=\frac2x.
$... |
4,539,043 | <p>I have tried to prove this statement by utilizing the proof by cases method. My cases are (1)<span class="math-container">$x=6$</span>, (2)<span class="math-container">$x>6$</span> and (3)<span class="math-container">$x<6$</span>.</p>
<p>For (3) for some reason it's not true</p>
<p>Case (1):
For <span class="m... | Bob Dobbs | 221,315 | <p>For your case <span class="math-container">$(3)$</span>, notice that if <span class="math-container">$x<6$</span> then <span class="math-container">$|x-6|=6-x$</span>. Then, your inequality becomes
<span class="math-container">$$x^2+6-x>5$$</span>
<span class="math-container">$$x^2-x+1>0$$</span>
<span clas... |
876,310 | <p>So I <em>think</em> I understand what differentials are, but let me know if I'm wrong.</p>
<p>So let's take $y=f(x)$ such that $f: [a,b] \subset \Bbb R \to \Bbb R$. Instead of defining the derivative of $f$ in terms of the differentials $\text{dy}$ and $\text{dx}$, we take the derivative $f'(x)$ as our "primitive"... | user121955 | 121,955 | <p>Differentials are infinitely small changes in x or y. For instance, the concept of the integral is the sum of the areas of an infinite number of rectangles under a curve. The height of each is f(x) and the width is dx.</p>
|
1,794,072 | <p>My attempt :</p>
<p>If $n$ is odd, then the square must be 2 (mod 3), which is not possible.</p>
<p>Hence $n =2m$</p>
<p>$2^{2m}+3^{2m}=(2^m+a)^2$</p>
<p>$a^2+2^{m+1}a=3^{2m}$</p>
<p>$a (a+2^{m+1})=3^{2m} $</p>
<p>By fundamental theorem of arithmetic, </p>
<p>$a=3^x $</p>
<p>$3^x +2^{m+1}=3^y $</p>
<p>$2^{... | S.C.B. | 310,930 | <p>Your method is slightly wrong, as you have to deal with the $x=0$ case. However, the remaining diophantine equation is simple. </p>
<p>Here, I provide you with an alternative approach, though admittedly yours seems better. </p>
<p>For the case $m \equiv 0 \pmod 2$, note that $$2^{2m}+3^{2m}=n^2$$ implies that $2^m... |
1,794,072 | <p>My attempt :</p>
<p>If $n$ is odd, then the square must be 2 (mod 3), which is not possible.</p>
<p>Hence $n =2m$</p>
<p>$2^{2m}+3^{2m}=(2^m+a)^2$</p>
<p>$a^2+2^{m+1}a=3^{2m}$</p>
<p>$a (a+2^{m+1})=3^{2m} $</p>
<p>By fundamental theorem of arithmetic, </p>
<p>$a=3^x $</p>
<p>$3^x +2^{m+1}=3^y $</p>
<p>$2^{... | Ant | 66,711 | <p>This is more convoluted than your way, but I already wrote it so I'll just leave it here :-) </p>
<p>$4^m = (a-3^m)(a+3^m) \implies a-3^m = 2^p, a+3^m = 2^q$ with $p+q = 2m$. Note that $q \ge p$</p>
<p>Then $a = 2^p + 3^m = 2^q - 3^m \iff 2^p (2^{q-p}-1) = 2\cdot 3^m$</p>
<p>But then it must be $p=1$, $q = 2m-1$... |
105,750 | <p>Given a <code>ContourPlot</code> with a set of contours, say, this:</p>
<p><a href="https://i.stack.imgur.com/cKoyo.jpg"><img src="https://i.stack.imgur.com/cKoyo.jpg" alt="enter image description here"></a></p>
<p>is it possible to get the contours separating domains with the different colors in the form of lists... | Jason B. | 9,490 | <p>I don't know how to do this in an automated way, but here is something at least:</p>
<p>Make your plot, extract the lines, convert them to regions, and then take the <code>RegionDifference</code> between them</p>
<pre><code>plot = ContourPlot[x*Exp[-x^2 - y^2], {x, 0, 3}, {y, -3, 3},
PlotRange -> {0, 0.5}, C... |
718,748 | <p>How can I calculate the volume of the solid under the surface $z = 6x + 4y + 7$ and above the plane $z = 0$ over a given rectangle $R = \{ (x, y): -4 \leq x \leq 1, 1 \leq y \leq 4 \}$?</p>
<p>I know I have to integrate some function, but since the surface takes both positive and negative values I don't know what t... | gt6989b | 16,192 | <p><strong>Hint</strong></p>
<ul>
<li>Find the equation of the curve of the surface where you have $z=0$ over your region (that will be a curve in $x,y$).</li>
<li>Now draw the region with your curve there and indicate where you have $z>0$ and where $z<0$.</li>
<li>Set up the double integral just over the piece ... |
718,748 | <p>How can I calculate the volume of the solid under the surface $z = 6x + 4y + 7$ and above the plane $z = 0$ over a given rectangle $R = \{ (x, y): -4 \leq x \leq 1, 1 \leq y \leq 4 \}$?</p>
<p>I know I have to integrate some function, but since the surface takes both positive and negative values I don't know what t... | kokocijo | 136,606 | <p><strong>Hint</strong></p>
<ul>
<li>Graph the $x$, $y$, and $z$ intercepts of the given surface (plane) to see where in the $xy$ plane you have $z>0$.</li>
<li>Adjust the rectangle you are integrating over to only cover area where $z>0$. It may help to draw the rectangle in the $xy$ plane. (It looks like in th... |
909,734 | <p>I have answered this question to the best of my knowledge but somehow I feel as if I am missing something? Can I further prove this statement or add anything to it? </p>
<p>Question: </p>
<p>Let $m \in \mathbb N$. Prove that the congruence modulo $m$ relation on $\mathbb Z$ is transitive. </p>
<p>My attempt:</p>
... | Bill Cook | 16,423 | <p>In some sense, in your proof, you are assuming what you are trying to prove. </p>
<p>When showing a basic property like the transitivity of a congruence relation, it's important to keep in mind what you already "know". </p>
<p>For example: It seems obvious that $a \equiv b$ (mod $m$) implies that $b \equiv a$ (mod... |
875,458 | <p>Is it possible to draw a triangle, if the length of its medians $(m_1, m_2, m_3)$ are given only?</p>
<p>Someone asked me this question, but I can not see it. Is it really possible?</p>
<p><strong>UPDATE</strong></p>
<p>Apart from the algebraic solution given by <em>JimmyK4542</em>, can anyone give me a direct co... | JimmyK4542 | 155,509 | <p>The formulas for the lengths of the <a href="http://en.wikipedia.org/wiki/Median_(geometry)" rel="nofollow">medians</a> of a triangle given the sidelengths are: </p>
<p>$m_a^2 = \dfrac{2b^2+2c^2-a^2}{4}$</p>
<p>$m_b^2 = \dfrac{2c^2+2a^2-b^2}{4}$</p>
<p>$m_c^2 = \dfrac{2a^2+2b^2-c^2}{4}$</p>
<p>Solving for $a,b,c... |
619,477 | <blockquote>
<p>Alice opened her grade report and exclaimed, "I can't believe Professor Jones flunked me in Probability." "You were in that course?" said Bob.
"That's funny, i was in it too, and i don't remember ever seeing you there."
"Well," admitted Alice sheepishly, "I guess i did skip class a lot." "Yeah, ... | nadia-liza | 113,971 | <p>Let $n$ be number all lectures.</p>
<p>Let $k$ be number of the lectures that Bob missed. if $k<n/2$ then $-k>-n/2$</p>
<p>Let $m$ be number of the lectures that Alice missed.Then $m>=n-k>n/2$</p>
|
619,477 | <blockquote>
<p>Alice opened her grade report and exclaimed, "I can't believe Professor Jones flunked me in Probability." "You were in that course?" said Bob.
"That's funny, i was in it too, and i don't remember ever seeing you there."
"Well," admitted Alice sheepishly, "I guess i did skip class a lot." "Yeah, ... | Martin | 34,537 | <p>Suppose, to the contrary, that neither Alice nor Bob missed at least half of the classes. Hence, both must have attended more than half of the classes. </p>
<p>Then clearly, out of the $n \geq 1$ classes total, it cannot be the case that Alice and Bob were never in the same class. </p>
<p>Why? <em>More</em> than h... |
1,957,304 | <p>I'm proving the compact-to-Hausdorff lemma (probably not a universal name for it) which is stated as:</p>
<blockquote>
<p>If $X$ is compact, $Y$ Hausdorff, $f:X \rightarrow Y$ a continuous bijection, then $f$ is a homeomorphism.</p>
</blockquote>
<p>However, the following line has popped up in a proof of it:</p>... | Moritz | 215,955 | <p>My pleasure: </p>
<p>How about $f(U) \cup f(X\setminus U) = f(U\cup (X\setminus U)) = f(X) = Y$? Since $f(U)$ and $f(X\setminus U)$ are disjoint, you get $f(U) = (f(U) \cup f(X\setminus U))\setminus f(X\setminus U) = Y\setminus f(X\setminus U)$.</p>
|
197,730 | <blockquote>
<p>Prove that the states of the 8-puzzle are divided into two disjoint sets such that any
state in one of the sets is reachable from any other state in that set, but not from any state in the other set. To do so, you can use the following fact: think of the board as a one-dimensional array, arranged i... | Brian M. Scott | 12,042 | <p>You’re working too hard: every state has an associated value of $N$, and that value is either even or odd.</p>
|
66,463 | <p>Hi,</p>
<p>Let $\Gamma$ be a free subgroup of rank 2 in $\mathbb{G}_m^2(\mathbb{Q})$. For all but finitely many primes p we can reduce $\Gamma$ modulo p. Let $S$ be the of primes for which $\Gamma$ does not reduce modulo p, and for any $p$ not in $S$, let $\gamma_p$ be the size of $\Gamma \mod p$. My question is wh... | Tzanko Matev | 421 | <p>I would just like to give a small update for the question. In my thesis <a href="https://epub.uni-bayreuth.de/1721/1/thesis.pdf" rel="nofollow">https://epub.uni-bayreuth.de/1721/1/thesis.pdf</a> I showed that the group $\Gamma\ \mod{p}$ has two generators for almost all primes p. So I would conjecture that on averag... |
3,578,357 | <p>The problem is like this : How do you solve <span class="math-container">$$ \lim _{x\to 0}\:\:\frac{x^m-sin^n(x)}{x^{n+2}} $$</span> for different values of <span class="math-container">$ n \in \Bbb N $</span>
Now, what i've started doing is to add <span class="math-container">$$ \lim _{x\to 0}\:\:\frac{x^m-x^n+x^n... | Fred | 380,717 | <p>Suppose that there is <span class="math-container">$x_0 \in \mathbb R$</span> such that <span class="math-container">$f(x_0) \ne 0.$</span> Then there is an intervall <span class="math-container">$I$</span> such that <span class="math-container">$x_0 \in I$</span> and <span class="math-container">$f(x) \ne 0$</span>... |
3,578,357 | <p>The problem is like this : How do you solve <span class="math-container">$$ \lim _{x\to 0}\:\:\frac{x^m-sin^n(x)}{x^{n+2}} $$</span> for different values of <span class="math-container">$ n \in \Bbb N $</span>
Now, what i've started doing is to add <span class="math-container">$$ \lim _{x\to 0}\:\:\frac{x^m-x^n+x^n... | Peter Szilas | 408,605 | <p>Partial answer.</p>
<p>FTC:</p>
<p>Assume there exist <span class="math-container">$x_2 >x_1$</span> s.t. <span class="math-container">$f(x_2)=f(x_1)$</span>.</p>
<p>Then <span class="math-container">$f(x_2)-f(x_1)=\displaystyle{\int_{x_1}^{x_2}}f'(x)dx=$</span></p>
<p><span class="math-container">$\displayst... |
268,360 | <p>Why is $\log_xy=\frac{\log_zy}{\log_zx}$? Can we prove this using the laws of exponents?</p>
| latiff | 54,820 | <p>By definition,
log<sub>z</sub> x<sup>log<sub>x</sub>y</sup> = log<sub>z </sub>y, also by definition we have, log<sub>z</sub> x<sup>log<sub>x</sub>y</sup> = log<sub>x</sub>y*log<sub>z</sub>x. So log<sub>z</sub>y = log<sub>x</sub>y*log<sub>z</sub>x. With division this gives us the result: log<sub>z</sub>y/log<sub>z</s... |
882,590 | <p><img src="https://i.stack.imgur.com/BMapu.png" alt="enter image description here"></p>
<p>long method: Determine an equation for each and solve using average value formula</p>
<p>alternative methods? </p>
<p>How could you prove the average value to be C over an interval [a,b] if you are given a graph.... looking ... | Petite Etincelle | 100,564 | <p>Graph A:
The part for $x\in[0,2)$ compensates with the part for $x\in (2,4]$, giving an average of 2 over the interval $[0,4]$. But the part for $x\in(4,6]$ is above the line $y=2$. So A is not good.</p>
<p>GraphB:
Define a new function $g$ whose graph is composed of one straight line between $(0,0)$ and $(3,4)$ an... |
99,237 | <p>If we have a directed graph $G = (V,E)$ and we want to find if there is such node $s \in V$ that we can reach all other nodes of $G$</p>
<p>What is a good algorithm to solve this problem and what is its execution time?</p>
| Gerry Myerson | 8,269 | <p>Perhaps the section about algorithms at the wikipedia article on <a href="http://en.wikipedia.org/wiki/Reachability" rel="nofollow">reachability</a> has what you need. Why not have a look and let us know? </p>
<p>You can also search for "Warshall's algorithm," although this may do more than you need. </p>
|
1,615,883 | <p>A cubic polynomial with real coefficients, $a x^3 + b x^2 + c x + d$, has either three real roots, or one real root and a pair of complex conjugate ones. If the latter happens, what is the explicit formula for this real solution, and what conditions can be placed on $a,b,c$ and $d$ to guarantee that the real root is... | Asinomás | 33,907 | <p>No matter where we place the first book, the probability each of the other two is on the same shelf as the first book is $\frac{1}{5}$. I agree, the probability is $\frac{1}{25}$.</p>
|
11,266 | <p>I have a list of time durations, which are strings of the form: <code>"hh:mm:ss"</code>. Here's a sample for you to play with:</p>
<pre><code>durations = {"00:09:54", "00:31:24", "00:40:07", "00:11:58", "00:13:51", "01:02:32"}
</code></pre>
<p>I want to convert all of these into numbers in seconds, so that I can a... | VLC | 685 | <p>A method based on <code>AbsoluteTime</code>:</p>
<pre><code>AbsoluteTime /@ durations - AbsoluteTime["00:00:00"]
(* {594, 1884, 2407, 718, 831, 3752} *)
</code></pre>
|
12,114 | <p>I retired after 25 years of teaching and moved to Israel a year ago. My Hebrew is okay, but before moving here, I had no experience talking about math in Hebrew. I have been learning Hebrew math vocabulary by reading math textbooks and taking an online math course in Hebrew. </p>
<p>I recently started volunteering... | kjetil b halvorsen | 122 | <p>Well, I was in just such a situation when I had to learn Spanish and learn to teach in Spanish simultaneously. What I did was sitting every evening with the books and dictionaries and translating, and writing down, word for word, what I had to say next day. Timeconsuming, yes, but it worked. It did help to have so... |
2,555,399 | <p>The question is to find out the coefficient of $x^3$ in the expansion of $(1-2x+3x^2-4x^3)^{1/2}$</p>
<p>I tried using multinomial theorem but here the exponent is a fraction and I couldn't get how to proceed.Any ideas?</p>
| nonuser | 463,553 | <p>Say $$(1-2x+3x^2-4x^3)^{1/2} =a+bx+cx^2+dx^3...$$
then </p>
<p>$$1-2x+3x^2-4x^3 =(a+bx+cx^2+dx^3...)^2$$
but $$(a+bx+cx^2+dx^3...)^2 = a^2+2abx+(2ac+b^2)x^2+2(ad+bc)x^3+...$$</p>
<p>So $a=\pm 1$. </p>
<p>If $a=1$ then $b=-1$ and $c=1$ and $d=-1$</p>
|
4,034,709 | <p>What will be the operator norm of the matrix <span class="math-container">$A = \begin{pmatrix} a & b \\ c & d \end{pmatrix},$</span> where <span class="math-container">$a,b,c,d \in \Bbb C\ $</span>?</p>
<p>According to the definition of the operator norm it turns out that <span class="math-container">$$\|A\|... | donaastor | 251,847 | <p>Yes, there is. Please, keep in mind that I do not know the definition of operator norm, but if you did that part of job correctly, then there is a simplification.</p>
<p>Firstly, focus on the last half of your expression, the one with real parts. That sum is equal to <span class="math-container">$\Re(C_3\cdot z\over... |
867,209 | <p>I tried to do the implication part. Please, see what I need to do to fix it.</p>
<p>claim: $n|a – b → n|a^2 – b^2$.</p>
<p>claim: $nk = a – b$ for some $k \in \mathbb Z \to nk' = a^2 – b^2$ for some $k' \in Z$.</p>
<p>$(a + 1)^2 – (b +1)^2$</p>
<p>$= a^2 + 2a + 1 -(b^2 +2b + 1)$</p>
<p>$= a^2 + 2a -b^2 -2b$</... | Adam | 82,101 | <p>Any set $$R \subseteq A \times A = \{(x,y) \mid x \in A ,\,y \in A\} $$ is a relation on $A $. Since $$\{(b,c), (b,d)\} \subseteq A \times A$$ holds, it is indeed a relation on $ A$. </p>
|
3,598,476 | <p>I have proven that if <span class="math-container">$|x|<\varepsilon,\forall\varepsilon>0$</span>, then <span class="math-container">$x=0$</span>. Further I have proven that ,<span class="math-container">$L=\displaystyle\lim_{n\to\infty}\frac{1}{n} = 0$</span> so that by definition <span class="math-container... | Kavi Rama Murthy | 142,385 | <p>There is no contradiction. You are getting <span class="math-container">$|\frac 1n |<\epsilon$</span> only under the extra condition <span class="math-container">$n >N$</span>. If you had <span class="math-container">$|\frac 1n |<\epsilon$</span> without any precondition you could say <span class="math-c... |
588,802 | <p>The problem is: $\sum_{n=1}^{\infty} \frac{1}{n(n+3)}$</p>
<p>The first thing I did was use the divergence test which didn't help since the result of the limit was 0.</p>
<p>If I multiply it through, the result is $\sum_{n=1}^{\infty} \frac{1}{n^2+3n}$</p>
<p>I'm wondering if I can consider this as a p-series and... | Alec Teal | 66,223 | <p>Bound it above! Note $n(n+3)=n^2+3n>n^2$</p>
<p>so $\frac{1}{n(n+3)}<\frac{1}{n^2}$</p>
<p>Each term is clearly > 0 btw.</p>
<p>So! $\sum\frac{1}{n(n+3)}<\sum\frac{1}{n^2}$ which you ought to know (but can trivially show) converges.</p>
<p>Finally a question I can answer here!</p>
|
471,153 | <p>Let $k$ be a number field. Define a prime of $k$ to be an equivalence class of absolute values on $k$. If $\sigma:k\hookrightarrow \mathbb{C}$ is an embedding of $k$ into the complex numbers then we know that $|a|=|\sigma(a)|$ defines an absolute value on $k$. </p>
<p>My question is: </p>
<p>Why is there exactly ... | anon | 90,978 | <p>Fix a generator $a \in k$, so that $k = \mathbb Q(a)$. Let $f \in \mathbb Q[X]$ be its minimal polynomial, and let
$$ \alpha_1, \dots, \alpha_n \in \mathbb C $$
be the complex roots of $f$; then any embedding $\sigma \colon k \to \mathbb C$ is determined by where it sends $a$, which is one of the $\alpha_i$'s. </p>
... |
471,153 | <p>Let $k$ be a number field. Define a prime of $k$ to be an equivalence class of absolute values on $k$. If $\sigma:k\hookrightarrow \mathbb{C}$ is an embedding of $k$ into the complex numbers then we know that $|a|=|\sigma(a)|$ defines an absolute value on $k$. </p>
<p>My question is: </p>
<p>Why is there exactly ... | walcher | 89,844 | <p>There is the so called "Extension Theorem".<br><br>
<strong>Theorem:</strong> Let $\mid \;\mid_v$ be an absolute value on a field $K$ and $L$ an algebraic extension. Denote by $K_v$ the completion of $K$ w.r.t $\mid \;\mid_v$ and by $\bar K_v$ its algebraic closure. Note that $\mid\;\mid_v$ extends uniquely to $\bar... |
2,369,431 | <p>I took a course some years ago and in it was a treatment of how to associate a field to an abstract geometry. </p>
<p>I would very much appreciate some reading on this, as I have been unsuccessful on where to find any resources on such ideas, and I have since lost my notes!</p>
<blockquote>
<p>Is there a book wh... | xxxxxxxxx | 252,194 | <p>For the most part, you can start with a geometric object such as a projective plane, and coordinatize. This leads to a structure called a planar ternary ring. The ring you get is not unique up to isomorphism, but rather up to a relation known as <em>isotopism</em>.</p>
<p>I'm not sure about other objects, you can... |
151,425 | <p>I've considered the following spectral problems for a long time, I did not kow how to tackle them. Maybe they needs some skills with inequalities.</p>
<p>For the first, suppose $T:L^{2}[0,1]\rightarrow L^{2}[0,1]$ is defined by
$$Tf(x)=\int_{0}^{x} \! f(t) \, dt$$</p>
<p>How can I calculate:</p>
<ul>
<li>the rad... | yaoxiao | 21,273 | <p>thanks PZZ's answer for the first problem. Now I know why I did realize why I am failed in the second problem.</p>
<p>$$Tf(x)=\int_{0}^{1-x}f(t)dt$$</p>
<p>It is trival that this operator is is linaer compact operator according to Arzela-Ascoli theorem, so we get $\sigma(T)/{0} \subset \sigma_{p}(T)$, use simple c... |
4,187,238 | <p>How many points are common to the graphs of the two equations <span class="math-container">$(x-y+2)(3x+y-4)=0$</span> and <span class="math-container">$(x+y-2)(2x-5y+7)=0$</span>?</p>
<p><span class="math-container">\begin{align*}
(x-y+2)(3x+y-4) &= 0\tag{1}\\
(x+y-2)(2x-5y+7) &= 0\tag{2}
\end{align*}</s... | lab bhattacharjee | 33,337 | <p>If <span class="math-container">$x-y+2=0,y=?$</span></p>
<p>Replace the value of<span class="math-container">$y$</span> in terms of <span class="math-container">$x,$</span> in the second equation to find</p>
<p><span class="math-container">$$0=(x+x+2-2)(3x+x+2-4)=2x(4x-2)$$</span></p>
<p><span class="math-container"... |
1,990,804 | <p>I know that we can define the exponential by a function $f: \mathbb{N}^2 \rightarrow \mathbb{N}$ by letting:</p>
<p>$f(m,o) = 1$ and $f(m,n+1) = f_x(m^n,m)$ where $f_x$ is the multiplication function, which we know is recursive.</p>
<p>I would then let $g = s(z)$ and hence $g(n) = 1$ for each $n \in \mathbb{N}$ , ... | Noah Schweber | 28,111 | <p>Your usual sources of examples - $\mathbb{C}$ and its many variations - won't help here: this operation is <em>non-associative</em>. For example, $$a*(a*b)=a*a=b\quad\mbox{but}\quad (a*a)*b=b*b=a.$$ (In particular, I don't understand your claim that you get semigroups out of this.) Indeed it's not even <a href="http... |
617,389 | <p>How to sketch $y = \frac1{\sqrt{x-1}}$</p>
<p>My way:(which does not work here)</p>
<p>I normally solve these problems by squaring and converting them to equations of 2 degree curves(such as parabola, hyperbola, etc.) which I can easily plot. But this seems to go 3 degree as $xy^2$ term is coming.</p>
<p>Please h... | mathlove | 78,967 | <p>HINT : What is the domain of $x$? What happens if you make $x$ larger, larger, to infinity? What happens if you make $x$ closer to $1$ from the right side?</p>
|
28,811 | <p>There are lots of statements that have been conditionally proved on the assumption that the Riemann Hypothesis is true.</p>
<p>What other conjectures have a large number of proven consequences?</p>
| Jim Humphreys | 4,231 | <p>What is usually referred to as <em>Lusztig's Conjecture</em> in the modular representation theory of semisimple algebraic groups has been enormously influential, as seen in Jantzen's treatise <em>Representations of Algebraic Groups</em>. It is actually a series of closely related conjectures, from 1979 on, inspire... |
118,406 | <p>I have a single flat directory with over a million files. I just wanted to take a sample of the first few files but <code>FileNames</code> doesn't include a "only the first n" option, and so it took over a minute:</p>
<p><a href="https://i.stack.imgur.com/s5cBS.png" rel="nofollow noreferrer"><img src="https://i.sta... | MarcoB | 27,951 | <p>Using OS shells commands seems to be much faster, although their output will need some massaging to obtain only the file names. </p>
<p>For instance, the following works quite well on my system (Win7-64):</p>
<pre><code>Import["!dir /b /a-d C:\\Windows\\*", "Text"] // StringSplit[#, "\n"] &
</code></pre>
<p>T... |
889,719 | <p>Example $5.9$ on page $103$ of John Lee's Smooth Manifolds says the following:</p>
<p>The intersection of $S^n$ with the open subset $\{x:x^i>0\}$ is the graph of the smooth function
$$
x^i=f(x^1,\dots,x^{i-1},x^{i+1},\dots,x^{n+1})
$$
where $f\colon B^n\to\mathbb{R}$ is $f(u)=\sqrt{1-|u|^2}$. The intersection o... | PhoemueX | 151,552 | <p>The chart is</p>
<p>$$
\{(x_1, \dots , x_{n+1}) \mid (x_1, \dots x_{i-1}, x_{i+1}, \dots x_{n+1}) \in B^n \text{ and } x_i > 0\} \to \Bbb{R}^{n+1}, (x_1, \dots,x_{n+1}) \mapsto (x_1, \dots, x_{i-1},x_i - f(x_1, \dots x_{i-1}, x_{i+1}, \dots x_{n+1}), x_{i+1}, \dots x_{n+1}) .
$$</p>
|
1,032,714 | <p>'Let $X$ be a topological space and let $(U_i)_{i \in I}$ be a cover of $X$ by connected subspaces $U_i$. Supposed for all $i,j \in I$ there exists some $n \geq 0$ and $k_0,...,k_n \in I$ such that $k_0 = i, k_n = j$ and $$U_{k_0} \cap U_{k_1} \neq \emptyset, U_{k_1} \cap U_{k_2} \neq \emptyset, ..., U_{k_{n-1}} \ca... | user178543 | 178,543 | <p>If $\omega \in B$, then $\{\omega\} \cap B = \{\omega\}$.</p>
|
1,456,407 | <p><a href="https://i.stack.imgur.com/oy6T7.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/oy6T7.jpg" alt="enter image description here"></a></p>
<p>We need to find the area of the shaded region , where curves are in polar forms as $r = 2 \sin\theta$ and $r=1$.</p>
<p>I formulated the double integ... | Yiyuan Lee | 104,919 | <p>Note that $$\begin{align}42^n - 1 &\equiv 1 - 1 \\&\equiv 0 \pmod{41}\end{align}$$</p>
<p>so the only way for $42^n - 1$ to be a prime is for $n$ to be $1$.</p>
<p>In general, for $a^n - 1$ to be a prime, where $a, n \in\mathbb{Z}^+$, either <a href="https://en.wikipedia.org/wiki/Mersenne_prime">$a = 2$</a... |
467,609 | <blockquote>
<p>Find the value of
$$\int _0 ^ \pi \dfrac{x}{1+\sin^2(x)} dx $$</p>
</blockquote>
<p>I have tried using $\int_a ^bf(x) dx=\int_a^b f(a+b-x)dx$</p>
<p>$\displaystyle \int _0 ^ \pi \dfrac{x}{1+\sin^2(x)} dx=\int _0 ^ \pi \dfrac{\pi-x}{1+\sin^2(x)} dx=I$</p>
<p>I couldn't go any further with that!</p... | Suraj M S | 85,213 | <p>$I$ =$\int _0 ^ \pi \dfrac{x}{1+\sin^2(x)} dx$</p>
<p>also $\int _0 ^ \pi \dfrac{\pi-x}{1+\sin^2(x)} dx=I$</p>
<p>add both to get,
$2I$ = $\int _0 ^ \pi \dfrac{\pi}{1+\sin^2(x)} dx$</p>
<p>AND FURTHER SOLVE IT.</p>
|
1,269,738 | <p>I'm looking for problems that due to modern developments in mathematics would nowadays be reduced to a rote computation or at least an exercise in a textbook, but that past mathematicians (even famous and great ones such as Gauss or Riemann) would've had a difficult time with. </p>
<p>Some examples that come to min... | Oscar Cunningham | 1,149 | <p>That there exist transcendental numbers. This was first shown by Liouville, who proved that Liouville's number:
<span class="math-container">$$\sum_{i=0}^\infty10^{-i!}$$</span>
is transcendental.</p>
<p>The "modern" proof would be due to Cantor:</p>
<blockquote>
<p>There are countably many algebraic numbers and... |
25,917 | <p>$\sqrt{2+\sqrt{2+\sqrt{2+\dots}}}$ </p>
<p>$\dots\sqrt{2+\sqrt{2+\sqrt{2}}}$</p>
<p>Why they are different?</p>
| Arturo Magidin | 742 | <p>To prove something by strong induction, you have to prove that</p>
<blockquote>
<p>If all natural numbers strictly less than $N$ have the property, then $N$ has the property.</p>
</blockquote>
<p>is true for all $N$. </p>
<p>So: our induction hypothesis is going to be:</p>
<blockquote>
<p>Every natural numbe... |
576,553 | <p>Please, forgive me if this is an elementary question, as well as my the sloppy phrasing and notation.</p>
<p>Suppose we have two discrete probability distributions $p = {\lbrace p_i \rbrace}$ and $q={\lbrace q_i \rbrace}$, $i=1,\dots,n$, where $p_i=P(p=p_i)$ and $q_i=P(q=q_i)$. Let's represent them as vectors $\bol... | Nick | 110,656 | <p>Nate, first of all thank you for your time. However, I am afraid I am not persuaded by your example.</p>
<p>We have $||\boldsymbol{q}||_a = (q_{2}^a)^{1/a} = q_2 = 1, \forall a$.</p>
<p>On the other hand, we will have $||\boldsymbol{p}||_1 = (1-x) + x = 1 = ||\boldsymbol{q}||_1, \forall x$. The 1-norm case is triv... |
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