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 |
|---|---|---|---|---|
2,985,256 | <blockquote>
<p>Let there are three points <span class="math-container">$(2,5,-3),(5,3,-3),(-2,-3,5)$</span> through which a plane passes. What is the equation of the plane in Cartesian form?</p>
</blockquote>
<p>I know how to find it in using vector form by computing the cross product to get the normal vector and p... | amd | 265,466 | <p>Remember that the implicit Cartesian equation for a plane is not unique. If you mutiply both sides of such an equation by a nonzero constant, you get another equation for the same plane. So, having an undetermined system for the coefficients of this equation is to be expected. Choose any convenient solution of the s... |
3,860,623 | <p>I'm trying to prove
<span class="math-container">$$\forall z\in\mathbb C-\{-1\},\ \left|\frac{z-1}{z+1}\right|=\sqrt2\iff\left|z+3\right|=\sqrt8$$</span>
thus showing that the solutions to <span class="math-container">$\left|(z-1)/(z+1)\right|=\sqrt2$</span> form the circle of center <span class="math-container">$-3... | user376343 | 376,343 | <p>For <span class="math-container">$r>0, a,b \in \mathbb{C}, a\neq b\;$</span> the equation <span class="math-container">$\left|\frac{z-a}{z-b}\right|=r$</span> defines a hyperbolic pencil of <a href="https://en.wikipedia.org/wiki/Apollonian_circles" rel="nofollow noreferrer">Apollonian circles</a>. Their centers l... |
747,816 | <p>1) Can a non-square matrix have eigenvalues? Why?</p>
<p>2) True or false: If the characteristic polynomial of a matrix A is p($\lambda$)=$\lambda$^2+1, then A is invertible.
Thank you!</p>
| user1357015 | 45,669 | <p>For 1) No, it has to be a square matrix by definition.</p>
<p>To see why, consider the following:</p>
<p>Recall that for an eigenvector $v$ and and an eigenvalue $\lambda$, you have that $Av$ = $\lambda v$. </p>
<p>Now suppose that dim(v) = n x 1. That means that dim(Av) = n x 1 and dim($\lambda v$) = n x 1. If A... |
1,148,043 | <p>Is $2\sqrt{12}$ or $4\sqrt{3}$ a better representation? Also, for $\sqrt{675}$, is $3\sqrt{75}$ or $15\sqrt{3}$ considered more simplified? Why is one more simplified than the others?</p>
| Community | -1 | <p>You would want to simplify the root as much as possible, removing all squares. You would want to use 4$\sqrt{3}$.</p>
<p>Further, the prime factorization of 48 is $2^4*3$, and that is what leads us to 4$\sqrt{3}$.</p>
|
398,371 | <p>How to calculate $$\lim_{t\rightarrow1^+}\frac{\sin(\pi t)}{\sqrt{1+\cos(\pi t)}}$$? I've tried to use L'Hospital, but then I'll get</p>
<p>$$\lim_{t\rightarrow1^+}\frac{\pi\cos(\pi t)}{\frac{-\pi\sin(\pi t)}{2\sqrt{1+\cos(\pi t)}}}=\lim_{t\rightarrow1^+}\frac{2\pi\cos(\pi t)\sqrt{1+\cos(\pi t)}}{-\pi\sin(\pi t)}$$... | N. S. | 9,176 | <p>$$\lim_{t\rightarrow1^+}\frac{\sin(\pi t)}{\sqrt{1+\cos(\pi t)}}=\lim_{t\rightarrow1^+}\frac{\sin(\pi t)}{\sqrt{1+\cos(\pi t)}}\frac{\sqrt{1-\cos(\pi t)}}{\sqrt{1-\cos(\pi t)}}=\lim_{t\rightarrow1^+}\frac{\sin(\pi t)\sqrt{1-\cos(\pi t)}}{\sqrt{\sin^2(\pi t)}}$$</p>
<p><strong>P.S.</strong> Pay attention to the sign... |
878,686 | <p>How do I derive the $m$ in the formula:
$$I=\left(1+\frac{r}{m}\right)^{mn} -1$$</p>
<p>all the values of the variables in the formula except $m$ is given and the question is find $m$.
I just don't know how to derive the formula using the knowledge of Algebra I have.</p>
| David Holden | 79,543 | <p>this looks like a compound interest question, so we see that the somewhat esoteric Lambert-W function may soon be part of the technical toolbox of chartered accountants! in order to gain a clearer idea of what is happening, using only elementary algebra, OP may find it useful to make an initial substitution, say $ x... |
3,652,879 | <p>If <span class="math-container">$\langle x_n\rangle $</span> is a sequence of positive real numbers such that <span class="math-container">$$x_{(n+2)}=\frac{(x_{n+1}+ x_{n})}{2}$$</span> for all <span class="math-container">$n \in \mathbb{N},\ $</span> let <span class="math-container">$x_1 <x_2$</span></p>
<p>th... | marty cohen | 13,079 | <p>(I'm pretty sure that
I and many others
have done this before
but I'll work it out again.)</p>
<p>If
<span class="math-container">$x(n+1) = ax(n)+(1-a)x(n-1)
$</span>
where
<span class="math-container">$0 < a < 2$</span>
then</p>
<p><span class="math-container">$\begin{array}\\
x(n+1)-x(n)
&= ax(n)+(1-a... |
855,329 | <p>$$
\mbox{Question: Evaluate}\quad
\tan^{2}\left(\pi \over 16\right) + \tan^{2}\left(2\pi \over 16\right) + \tan^{2}\left(3\pi \over 16\right) + \cdots + \tan^{2}\left(7\pi \over 16\right)
$$</p>
<p>What I did:
Well I know that $\tan^{2}\left(7\pi/16\right)$ is the same as
$\cot^{2}\left(\pi/16\right)$. Thus this wi... | Vishwa Iyer | 71,281 | <p><strong>HINT:</strong> Use the half angle formula with $\theta = \pi/4$ to find $\tan(\pi/8)$ and do the same with $\theta = \pi/8$ to find $\tan(\pi/16)$</p>
<p>EDIT: The half angle formula is:
$$\tan(a) = \frac{2\tan(\frac{a}{2})}{1- \tan^2(\frac{a}{2})}$$
So use this formula for $a = \pi/4$ and $a = \pi/8$</p>
|
2,605,208 | <p>Decide whether the given set of vectors is linearly independent in the indicated vector space:</p>
<p>$\{ x_1, x_1 +x_2, x_1 +x_2 +x_3, ..., x_1+\cdots+x_n\} $</p>
<p>if $\{x_1, x_2, x_3, ..., x_n\}$ is linearly
independent, in some vector space $V$.</p>
<hr>
<p>If $n=4:$</p>
<p>$x_1 - (x_1+x_2) + (x_1+x_2+x_3)... | Tsemo Aristide | 280,301 | <p>Suppose that such terminal object $e$ exists, you have a morphism of bundles $X\times G\rightarrow e$, implies that $e$ is isomorphic to $X\times G$, for every $G$-principal bundle $P$, you have a morphism $P\rightarrow X\times G$. This implies that $P$ is trivial. The terminal object exists if every $G$-principal b... |
2,775,087 | <blockquote>
<p>Without using a calculator, what is the sum of digits of the numbers from $1$ to $10^n$?</p>
</blockquote>
<p>Now, I'm familiar with the idea of pairing the numbers as follows:</p>
<p>$$\langle 0, 10^n -1 \rangle,\, \langle 1, 10^n - 2\rangle , \dotsc$$</p>
<p>The sum of digits of each pair is $9n$... | theREALyumdub | 175,429 | <p>Well, it's not too hard to figure out that the sum of digits of 0 or 1 to 9 is 45. That's very helpful, since that sequence will appear a bunch. Just add 1, and 46 is the sum of digits from 1 to 10 (or 0 to 10).</p>
<p>In a way, that and the base shifts are all you need I believe. Let's do 100. We're going to get 1... |
186,182 | <p>Suppose, for the sake of keeping things as simple as possible, that I have the following equation that I wish to simplify in Mathematica:</p>
<p><span class="math-container">$y = x x$</span></p>
<p>But suppose further that I also have a restriction, not directly on <span class="math-container">$x$</span>, but on w... | Carl Woll | 45,431 | <p>When a function has 2 arguments (not a single list argument), use:</p>
<pre><code>f[x_, y_] := x^4 + y^4
Derivative[1, 0][f][x, y]
Derivative[0, 1][f][x, y]
</code></pre>
<blockquote>
<p>4 x^3</p>
<p>4 y^3</p>
</blockquote>
|
323,971 | <p>I know basic things about cardinality (I'm only in High School) like that since $\mathbb{Q}$ is countable, its cardinality is $\aleph_0$. Also that the cardinality of $\mathbb{R}$ is $2^{\aleph_0}$.</p>
<blockquote>
<p>Are there any direct applications of these numbers outside of theoretical math?</p>
</blockquot... | eggcrook | 65,627 | <p>Quoted from Christian Marks blog(blog seems to be gone now):</p>
<blockquote>
<p>In an unexpected development for the depressed market for mathematical logicians, Wall Street has begun quietly and aggressively recruiting proof theorists and recursion theorists for their expertise in applying ordinal notations and or... |
4,616,155 | <p>This is a Question from an Analysis 1 exam. The question is as follows: Decide if the functions <span class="math-container">$f: \mathbb{R} \longrightarrow \mathbb{R}$</span> can be written as the difference of two monotonically increasing functions</p>
<p>a) <span class="math-container">$f(x) = \cos(x)$</span></p>
... | kandb | 1,072,546 | <p>The support of <span class="math-container">$f$</span> is <span class="math-container">$\mathbb{R}$</span> (note that the support of a function is the subset of its domain on which it does not vanish), which is closed by definition, but <span class="math-container">$\mathbb{R}$</span> is not a compact subset of <spa... |
112,147 | <p>I have a long vector and some of the values (19 out of 64) are complex. I got them using the Mathematica Rationalize function, so the complex ones are written in the a+bi form. Is there a function I can apply to the entire vector, that would change my complex numbers to the form A<em>Exp[I</em>phi]? </p>
| bill s | 1,783 | <p>Since the arguments are rational:</p>
<pre><code>v = {1/10, -1 - 2 I, 3 - 5/3 I, 7, 9/10 + I};
Abs[v] Exp[I Arg[v]]
{1/10, Sqrt[5] E^(I (-π + ArcTan[2])), 1/3 Sqrt[106] E^(-I ArcTan[5/9]), 7,
1/10 Sqrt[181] E^(I ArcTan[10/9])}
</code></pre>
|
3,215,556 | <p>I have a question in my paper, Express 4225 as the product of its prime factors in index notation. That was easy to answer, but the next question is express the square root of 42250000 using prime factorisation. Apparently there is a way to use my answer in the first question to do the second, but how do I?</p>
| CoffeeCrow | 227,228 | <p>Using that <span class="math-container">$\sqrt{x}=x^{\frac{1}{2}}$</span>, index laws and the prime factorisations of <span class="math-container">$1000$</span> and <span class="math-container">$4255$</span>, as well as that <span class="math-container">$42250000=4225\times10000$</span>, we have;</p>
<p><span class... |
3,215,556 | <p>I have a question in my paper, Express 4225 as the product of its prime factors in index notation. That was easy to answer, but the next question is express the square root of 42250000 using prime factorisation. Apparently there is a way to use my answer in the first question to do the second, but how do I?</p>
| drhab | 75,923 | <p>If <span class="math-container">$$n=p_1^{2k_1}\cdots p_m^{2k_m}$$</span> where the <span class="math-container">$k_i$</span> are non-negative integers then: <span class="math-container">$$\sqrt n=p_1^{k_1}\cdots p_m^{k_m}$$</span></p>
|
3,215,556 | <p>I have a question in my paper, Express 4225 as the product of its prime factors in index notation. That was easy to answer, but the next question is express the square root of 42250000 using prime factorisation. Apparently there is a way to use my answer in the first question to do the second, but how do I?</p>
| Bill Dubuque | 242 | <p><strong>Hint</strong> <span class="math-container">$\,f(ab) = f(a)f(b)\,$</span> where <span class="math-container">$\,f(n) := $</span> prime factorization of <span class="math-container">$n$</span> and, <a href="https://math.stackexchange.com/a/21637/242">furthermore,</a> if <span class="math-container">$\,a,b\,$<... |
10,807 | <p>Over the past few days, the Chrome browser page with Questions is crashing.</p>
<p>It has crashed while typing answers twice and several other times when just viewing and updating the questions listing.</p>
<p>Did something change in the past three days that could be affecting this as I have not seen this behavior... | Pedro | 23,350 | <p>I have been able to diminish the number of crashes by using Robjohn's "rendering off" tab. That seems to fix it a little, since it seems Chrome crashes when compiling and re-compiling the code over and over when we type. It seems every new character entered forces a recompilation of the code that as been already bee... |
41,183 | <p>Is it true that any manifold homotopy equivalent to a k-dimensional CW-complex admits a proper Morse function with critical points all of index <= k? I believe this is not true, so I would like to see a counterexample.</p>
| Tom Goodwillie | 6,666 | <p>Suppose that $M$ has a proper Morse function $f\ge 0$ with all critical points of index at most $k$. Then homotopically $M$ can be made of low-dimensional cells: it has homotopical dimension at most $k$. But also $M$, relative to its boundary $M^{\ge c}$, can be made by attaching <i>high</i>-dimensional cells. Thus ... |
4,634,797 | <p>I am currently trying to show that the sequence of functions defined by <span class="math-container">$f_n(x) = \frac{x}{1 + x^n}$</span> converges pointwise on <span class="math-container">$U = [0, \infty)$</span>. I have found the limits for the three specific cases and they are:
<span class="math-container">\begin... | Anne Bauval | 386,889 | <p>You have <span class="math-container">$x\in[0,1),$</span> <span class="math-container">$\epsilon>0,$</span> and you want <span class="math-container">$x^{n+1}<\left(1+x^n\right)\epsilon,$</span> i.e.
<span class="math-container">$$x^n\left(x-\epsilon\right)<\epsilon.$$</span></p>
<ul>
<li>If <span class="ma... |
69,225 | <p>Does anybody have suggestions on what to read to learn more about couplings pertaining to statistics?</p>
<p>I'm working on a research project on Poisson approximations and am looking to perform a coupling on the unknown distribution. However, I cannot find much material on how to perform a coupling and the general... | Omer | 9,422 | <p>Have you looked at Lindvall's "Lectures on the Coupling Method"?</p>
|
69,225 | <p>Does anybody have suggestions on what to read to learn more about couplings pertaining to statistics?</p>
<p>I'm working on a research project on Poisson approximations and am looking to perform a coupling on the unknown distribution. However, I cannot find much material on how to perform a coupling and the general... | Will Jagy | 3,324 | <p>My friend Marty suggests the Lindvall book as well as</p>
<p>H. Thorisson, Coupling, Stationarity, and Regeneration. Springer, New York, 2000. </p>
<p><a href="http://www.springer.com/mathematics/probability/book/978-0-387-98779-8" rel="nofollow">http://www.springer.com/mathematics/probability/book/978-0-387-98779... |
1,752,506 | <p>Question: $ \sqrt{x^2 + 1} + \frac{8}{\sqrt{x^2 + 1}} = \sqrt{x^2 + 9}$</p>
<p>My solution: $(x^2 + 1) + 8 = \sqrt{x^2 + 9} \sqrt{x^2 + 1}$</p>
<p>$=> (x^2 + 9) = \sqrt{x^2 + 9} \sqrt{x^2 + 1}$</p>
<p>$=> (x^2 + 9) - \sqrt{x^2 + 9} \sqrt{x^2 + 1} = 0$</p>
<p>$=> \sqrt{x^2 + 9} (\sqrt{x^2 + 9} - \sqrt{x^... | Deepak | 151,732 | <p>A suggested simplification. You should always look for simplifications to the algebra if they're easy to find.</p>
<p>Let $y = x^2 + 1$</p>
<p>Then you're solving $\sqrt y + \frac{8}{\sqrt y} = \sqrt{y + 8}$</p>
<p>$\frac{y + 8}{\sqrt y} = \sqrt{y + 8}$</p>
<p>$(y+8)^2 = y(y+8)$</p>
<p>$8(y+8) = 0$</p>
<p>From... |
4,389,997 | <p>In Enderton's <em>A Mathematical Introduction to Logic</em>, he defines <span class="math-container">$n$</span>-tuples recursively using ordered pairs, i.e. <span class="math-container">$\langle x_1,\dots,x_{n+1}\rangle=\langle\langle x_1,\dots,x_n\rangle, x_{n+1}\rangle$</span>. But he also notes,</p>
<blockquote>
... | Wuestenfux | 417,848 | <p>Actually, you don't need the definition of function by using the above recursive definition. For this, you start with induction base for <span class="math-container">$n=2$</span>:</p>
<p><span class="math-container">$(x,y) = \{\{x\},\{x,y\}\}$</span> (ordered pair) is a set and so all <span class="math-container">$n... |
246,571 | <p>How can I calculate the following limit epsilon-delta definition?</p>
<p>$$\lim_{x \to 0} \left(\frac{\sin(ax)}{x}\right)$$</p>
<p>Edited the equation, sorry...</p>
| anegligibleperson | 17,248 | <p>you can rewrite $\dfrac{\sin (ax)}{x} = a \dfrac{\sin (ax)}{ax}$, then take limits, as suggested by @rscwieb in the comments</p>
|
2,906,797 | <p>I want to express this polynomial as a product of linear factors:</p>
<p>$x^5 + x^3 + 8x^2 + 8$</p>
<p>I noticed that $\pm$i were roots just looking at it, so two factors must be $(x- i)$ and $(x + i)$, but I'm not sure how I would know what the remaining polynomial would be. For real roots, I would usually just d... | A. Pongrácz | 577,800 | <p>Yo can also do long division for complex polynomials. But in this case, I would suggest to pull out the two factors at once, i.e., divide by their product, which is $(x^2+1)$.</p>
<p>$(x^5+x^3+8x^2+8):(x^2+1)= x^3+8$.
You can easily factorize this polynomial. (Hint: $8=2^3$.)</p>
|
653,449 | <p>According to <a href="http://en.wikibooks.org/wiki/Haskell/Category_theory" rel="noreferrer">the Haskell wikibook on Category Theory</a>, the category below is not a valid category due to the addition of the morphism <em>h</em>. The hint says to " think about associativity of the composition operation." Bu... | Giorgio Mossa | 11,888 | <p>It not so clear what is they mean, but I guess what they mean is that if you consider the graph above (in which edges with different labels are different) then you cannot put on that graph a structure of a category.</p>
<p>To prove that you have to use reductio ab absurdum:
if there where any category structure on ... |
760,654 | <p>If $\lambda$ is the eigenvalue of matrix $A$,what is the eigenvalue of $A^TA$?I have no clue about it. Can anyone help with that?</p>
| Ted Shifrin | 71,348 | <p>The good exercise for you is to prove, in general, that
$$\lambda_{\text{max}}(A)\le \max_{\|x\|=1} \|Ax\|=\sqrt{\lambda_{\text{max}}(A^\top A)}.$$</p>
|
268,461 | <blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="https://math.stackexchange.com/questions/228080/operatornameimfz-leq-operatornamerefz-then-f-is-constant">$|\operatorname{Im}f(z)|\leq |\operatorname{Re}f(z)|$ then $f$ is constant</a> </p>
</blockquote>
<blockquote>
<p>Let $f\colon\mathbb C \t... | amWhy | 9,003 | <p>$\forall x\in K, (\text{and}\;\;\forall x \in X)$: $x\in F\cup F^c$. $\;x\in F$ or $x\in F^c \implies (x \in F$ or $x \notin F$). That is $x$ is in $F$, or it's not in $F$ (Law of excluded middle). If it's in $F$, it's covered by the open cover of $F$: $\{V_\alpha\}$. If it's in $F^c$ (i.e., if it's not in $F$), ... |
268,461 | <blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="https://math.stackexchange.com/questions/228080/operatornameimfz-leq-operatornamerefz-then-f-is-constant">$|\operatorname{Im}f(z)|\leq |\operatorname{Re}f(z)|$ then $f$ is constant</a> </p>
</blockquote>
<blockquote>
<p>Let $f\colon\mathbb C \t... | sureshs | 54,666 | <p>The open cover you're dealing with is $\{F^c\} \cup \{V_\alpha\}$. This covers the entire space $X$, not just $K$. To see this, let $x \in X$. Either $x \in F$ or $x \notin F$. If $ x \in F$, then</p>
<p>$$x \in F \subseteq \bigcup V_{\alpha}$$</p>
<p>Otherwise, $x \in F^c$, so $X \subseteq F^c \cup \bigcup V_{... |
1,655,884 | <blockquote>
<p>How many integer-sided right triangles exist whose sides are combinations of the form $\displaystyle \binom{x}{2},\displaystyle \binom{y}{2},\displaystyle \binom{z}{2}$?</p>
</blockquote>
<p><strong>Attempt:</strong></p>
<p>This seems like a hard question, since I can't even think of one example to ... | Tito Piezas III | 4,781 | <p>Solving $(1)$ for $z$, we have,</p>
<p>$$z = \frac{1\pm\sqrt{1\pm4w}}{2}\tag3$$</p>
<p>where,</p>
<p>$$w^2 = (x^2-x)^2+(y^2-y)^2\tag4$$</p>
<p>It can be shown that $(4)$ has infinitely many integer solutions. (<em>Update</em>: Also proven by Sierpinski in 1961. See link given by MXYMXY, <em><a href="http://www.f... |
901,357 | <p>Let there be $T:R^3 \rightarrow R^3$
<br>
$T(0,-1,1)=(3,3,3)$
<br>
$T(1,0,-1)=(0,1,1)$
<br>
$T(1,1,0)=(1,2,-1)$</p>
<p>Is (1,2,3) is the only image of the vector $(1, \frac{-7}{9}, \frac{-8}{9})$?</p>
<p>I have thought to create a matrix $[T]^T_E$*$[T]^E_C$=$[T]^T_C$
so I will have a matrix that does the transform... | robjohn | 13,854 | <p>As was noted by Adam Hughes, the series for $\frac1{1-z}$ about the point $5i$ is
$$
\begin{align}
\frac1{1-z}
&=\frac1{(1-5i)-(z-5i)}\\
&=\frac1{1-5i}\frac1{1-\color{#C00000}{\frac{z-5i}{1-5i}}}\\
&=\frac1{1-5i}\sum_{k=0}^\infty\left(\color{#C00000}{\frac{z-5i}{1-5i}}\right)^k\\
&=\sum_{k=0}^\infty\... |
627,575 | <p>There is a whiskey made up of 64% corn, 32% rye, and 4% barley that was made by blending other whiskies together. I am trying to figure out if there is a chance the ratio of this whiskey could be the result of blending two, maybe three whiskies of different ratios.</p>
<p>The possible whiskies:</p>
<p>Whiskey A is... | Henry | 6,460 | <p>You could any of</p>
<ul>
<li><p>Whiskey D made up of $\dfrac{17}{21}$ of Whiskey A and $\dfrac{4}{21}$ of Whiskey B </p></li>
<li><p>Whiskey E made up of $\dfrac{11}{15}$ of Whiskey A and $\dfrac{4}{15}$ of Whiskey C</p></li>
<li><p>Any blend of whiskeys D and E</p></li>
</ul>
<p>and you would have your 64%, 32% ... |
627,575 | <p>There is a whiskey made up of 64% corn, 32% rye, and 4% barley that was made by blending other whiskies together. I am trying to figure out if there is a chance the ratio of this whiskey could be the result of blending two, maybe three whiskies of different ratios.</p>
<p>The possible whiskies:</p>
<p>Whiskey A is... | Nick D. | 115,491 | <p>Yes, and in infinitely many different ways!</p>
<p>Row reduce $\begin{bmatrix}
60&81&75&64\\
36&15&21&32\\
4&4&4&4\\
\end{bmatrix}$ to get $\begin{bmatrix}
1&0&\frac{2}{7}&\frac{17}{21}\\
0&1&\frac{5}{7}&\frac{4}{21}\\
0&0&0&0\\
\end{bmatrix}$.... |
835,639 | <p>These days I have read many descriptions of a noncooperative game like the one below.</p>
<p>A noncooperative game is a game in which players are unable to make enforceable contracts outside of the rules/description of such a game.</p>
<p>As a graduate student majoring in math, I wonder if there is any mathematica... | Michael | 79,220 | <p>Two mathematicians who won Nobel prizes in economics:</p>
<ol>
<li><p>Nash</p></li>
<li><p>Shapley</p></li>
</ol>
<p>for work in non-cooperative and cooperative game theory respectively. Google and JSTOR are your friends.</p>
|
835,639 | <p>These days I have read many descriptions of a noncooperative game like the one below.</p>
<p>A noncooperative game is a game in which players are unable to make enforceable contracts outside of the rules/description of such a game.</p>
<p>As a graduate student majoring in math, I wonder if there is any mathematica... | Sergio Parreiras | 33,890 | <p>I don't think you will ever find a formal definition. The informal one is very good for all purposes. The closest you can get is the definition of a strategic game (= non-cooperative games) see Osborne and Rubinstein (section 2.1) and the definition of coalitional games with and without transferable payoff ( = coope... |
2,973,825 | <p>when you are checking to see if a sum of say <span class="math-container">$k^2$</span> from <span class="math-container">$k=1$</span> to to <span class="math-container">$k=n$</span> is equal to a sum of <span class="math-container">$(k+1)^2$</span> from <span class="math-container">$k=0$</span> to <span class="math-... | Johann Birnick | 974,190 | <p>Here is an explicit strategy that works for <span class="math-container">$n \leq 100$</span> cards.</p>
<p>As seen in previous answers, we use the order of the 4 transmitted cards to encode a number between 1 and 24. Now if the magician can reduce from the <em>set</em> of the 4 transmitted cards the possible 5th car... |
2,647,868 | <p>I'm very confused at the following question:</p>
<blockquote>
<p>Find the basis for the image and a basis of the kernel for the following matrix:
$\begin{bmatrix} 7 & 0 & 7 \\ 2 & 3 & 8 \\ 9 & 0 & 9 \\ 5 & 6 & 17 \end{bmatrix}$</p>
</blockquote>
<p>I just don't know how to do an... | Mathematician 42 | 155,917 | <p>So you have to find the basis for the image and kernel of the map $$T_A:\mathbb{R}^3\rightarrow \mathbb{R}^4:X\mapsto AX$$ where
$$A=\begin{bmatrix} 7 & 0 & 7 \\ 2 & 3 & 8 \\ 9 & 0 & 9 \\ 5 & 6 & 17 \end{bmatrix}.$$</p>
<p>Im not a fan of blindly following an algorithm someone showe... |
1,705,656 | <p>I've hit a wall on the above question and was unable to find any online examples that also contain trig in $f(g(x))$. I'm sure I am missing something blatantly obvious but I can't quite get it.</p>
<p>$$ g(x)=3x+4 , \quad f(g(x)) = \cos\left(x^2\right)$$</p>
<p>So far I've managed to get to the point where I have... | ThePortakal | 137,487 | <p><strong>Hint:</strong> $(f \circ g) \circ g^{-1} =f \circ (g \circ g^{-1}) = f$</p>
|
2,874,840 | <blockquote>
<p>If $P\left(A\right)=0.8\:$ and $P\left(B\right)=0.4$, find the maximum and minimum values of $\:P(A|B)$.</p>
</blockquote>
<p>My textbook says the answer is $0.5$ to $1$. But I think the answer should be $0$ to $1$.</p>
<p>The textbook claims $P(A∩B)$ is $0.2$ when $P(A'∩B')=0$</p>
<p>I think that ... | Rushabh Mehta | 537,349 | <p>For completeness, I'll show both bounds of $P(A|B)$.</p>
<h1>Lower Bound</h1>
<p>We note that $$P(A)+P(B)-P(A\cup B) = P(A\cap B)$$ which is a simple rearrangement of the standard probability summation formula. Note that the upper bound of $P(A\cup B)=1$, so when we substitute, we get $0.8+0.4-1=P(A\cap B) = 0.2$<... |
2,298,873 | <p><strong>Problem statement:</strong></p>
<p>There are three spheres. The one which will roll is $\textbf{X}=(x_1,x_2,x_3)$ with radius $R_X$. The other two spheres are $\textbf{A}=(a_1,a_2,a_3)$ with $R_A$ and $\textbf{B}=(b_1,b_2,b_3)$ with $R_B$. They are both below $X$, meaning $x_3>a_3$ and $x_3>b_3$. They... | Travis Willse | 155,629 | <p>By translating coordinates we may as well take ${\bf B} = {\bf 0}$, so that $\ell = \langle {\bf A} \rangle$. Then, there is a (unique) orthogonal decomposition $${\bf X} = {\bf X}^{\parallel} + {\bf X}^{\perp} ,$$
where ${\bf X}^{\parallel} \parallel \ell$ and ${\bf X}^{\perp} \perp \ell$. (Explicitly, ${\bf X}^{\p... |
118,486 | <p>I am seeking a deeper understanding of the representation of set-based objects in terms of Boolean algebras.</p>
<p>Let $\wp(A)$ be the set of subsets of a set $A$. A relation $R \subseteq A \times B$ generates two operators $pre: \wp(B) \to \wp(A)$ and $post: \wp(A) \to \wp(B)$ where $pre$ maps a set $X \subseteq ... | Nik Weaver | 23,141 | <p>Having a subset of a set is "the same" as having a function from that set into $\{0,1\}$ (namely, the function which is $1$ on the subset and $0$ on its complement). If I have a function $f: X \to Y$ I can compose it with functions from $Y$ to $\{0,1\}$ and thereby turn subsets of $Y$ into subsets of $X$. I guess th... |
118,486 | <p>I am seeking a deeper understanding of the representation of set-based objects in terms of Boolean algebras.</p>
<p>Let $\wp(A)$ be the set of subsets of a set $A$. A relation $R \subseteq A \times B$ generates two operators $pre: \wp(B) \to \wp(A)$ and $post: \wp(A) \to \wp(B)$ where $pre$ maps a set $X \subseteq ... | Andreas Blass | 6,794 | <p>For a function $f:X\to Y$, the operation "pre-image along $f$" from $\mathcal P(Y)$ to $\mathcal P(X)$ has adjoints on both sides. [I'm viewing the power sets as partially ordered by $\subseteq$ and then viewing these partially ordered sets as categories, so that "adjoint" makes sense.] The left adjoint is the fa... |
2,317,496 | <p><strong>Definition:</strong> An ordinal number $\alpha$ is called a <strong><em>limit ordinal number</em></strong> if there is no ordinal number immediately preceding $\alpha$. </p>
<p>Now my lecture notes say that $\omega, 2\omega, \omega^2, \omega^\omega$ are limit ordinal numbers whereas $\omega+3,2^\omega+5$ ar... | Stefan Mesken | 217,623 | <p>You already have a characterization of limit ordinals. In any specific case you just have to verify that this characterization is fulfilled. Consider for example $\omega^2$:</p>
<p>$$
\omega^2 = \omega \cdot \omega = \sup \{ \omega \cdot n \mid n < \omega \}.
$$</p>
<p>Hence, if $\alpha < \omega^2$, there is... |
338,099 | <p>Are there general ways for given rational coefficients <span class="math-container">$a,b,c$</span> (I am particularly interested in <span class="math-container">$a=3,b=1,c=8076$</span>, but in general case too) to answer whether this equation has a rational solution or not?</p>
| Max Alekseyev | 7,076 | <p>Multiplying by <span class="math-container">$x^2$</span> and denoting <span class="math-container">$X:=-abx^2$</span>, <span class="math-container">$Y:=ab^2xy$</span>, we get an elliptic curve:
<span class="math-container">$$Y^2 = X^3-ab^2cX.$$</span>
If it turns out that this curve has zero rank, then the number of... |
2,484,855 | <p>How can I prove that these three sets have no common values:</p>
<ul>
<li>A: {prime numbers} </li>
<li>B: {Fibonacci numbers}</li>
<li>C: {8|n+1} </li>
</ul>
<blockquote>
<ul>
<li>C: for example 15: 15+1 = 16 => 8|16 <= 16/8 = 2</li>
<li>C: for example 23: 23+1 = 24 => 8|24 <= 24/8 = 3</li>
</ul>
... | Dietrich Burde | 83,966 | <p>The claim is that Fibonacci primes $F_n$ (which implies that $n$ is prime, except for $n=4$) do not satisfy $F_p\equiv 7\bmod 8$. Let $a(n)$ denote the sequence of Fibonacci primes. Then they
satisfy $a(n)\equiv 1,5\bmod 8$, see <a href="http://oeis.org/A005478" rel="nofollow noreferrer">OEIS</a>, because of $a(n) \... |
21,182 | <p>In writing my senior thesis I met the following problem: Sometimes I have some intuition about some mathematical statement. Yet I find it extremely painful trying to put these intuition into precise form on paper. In particular it is very hard to specify the correct condition for statement.</p>
<p>Does anyone have ... | Angelo | 4,790 | <p>You think about it and try to clarify your ideas, till you can write them up precisely. If there is another way, I am not aware of it.</p>
|
21,182 | <p>In writing my senior thesis I met the following problem: Sometimes I have some intuition about some mathematical statement. Yet I find it extremely painful trying to put these intuition into precise form on paper. In particular it is very hard to specify the correct condition for statement.</p>
<p>Does anyone have ... | Igor Pak | 4,040 | <p>I see this in students all the time, and I always give the same advice: talk to somebody. Find a friend who would be willing to listen and challenge you on every point. Sit down with a piece of paper, and try to tell her/him the whole story, explain the theorem you are trying to prove, examples, counter-examples, e... |
21,182 | <p>In writing my senior thesis I met the following problem: Sometimes I have some intuition about some mathematical statement. Yet I find it extremely painful trying to put these intuition into precise form on paper. In particular it is very hard to specify the correct condition for statement.</p>
<p>Does anyone have ... | Cam McLeman | 35,575 | <p>In addition to the other answers...experiment! Write SAGE (or other) code to look at a hundred or a thousand examples of what you're trying to say something about. You'll probably see the pattern more clearly when it's sitting right in front of you in numerical form, and probably catch a class of exceptions you ha... |
2,495,176 | <p>For how many positive values of $n$ are both $\frac n3$ and $3n$ four-digit integers?</p>
<p>Any help is greatly appreciated. I think the smallest n value is 3000 and the largest n value is 3333. Does this make sense?</p>
| fleablood | 280,126 | <p>Some basic thoughts.</p>
<p>1) $\frac n3 < n < 3n$ </p>
<p>2) If $k$ has four digits then $1000 \le k \le 99999$.</p>
<p>So therefore</p>
<p>$1000 \le \frac n3 < n < 3n \le 9999$</p>
<p>So </p>
<p>$1000 \le \frac n3 \implies 3000 \le n$.</p>
<p>And $3n \le 9999\implies n \le 3333$.</p>
<p>So $300... |
2,299,678 | <p>Question:</p>
<p>Assume $x, y$ are elements of a field $F$. Prove that if $xy = 0$, then $x = 0$ or $y = 0$.</p>
<p>My thinking:</p>
<p>I am not sure how to prove this. <strong>I can only use basic field axioms.</strong> Should I assume that both x and y are not equal to 0 and then prove by contradiction or shoul... | fleablood | 280,126 | <p>First prove for any element $a $ that $0*a=0$:</p>
<p>$0*a = 0*a +0=$</p>
<p>$0a+0a -0a =(0+0)a-0a =$</p>
<p>$0a-0a=0$.</p>
<p>Let $x\ne 0$ and $y\ne 0$. Then $x^{-1},y^{-1} $ and $g= (y^{-1}x^{-1})$ exist.</p>
<p>$(xy)*g=(xy)* (y^{-1}x^{-1})=1\ne 0$.</p>
<p>Thus $xy\ne 0$</p>
<p>So $xy=0$ only if $x$ and $y... |
1,898,810 | <p>How do I integrate $\frac{1}{1-x^2}$ without using trigonometric identities or partial fractions? Thanks!</p>
| haqnatural | 247,767 | <p>Hint:
$$\frac { 1 }{ 1-{ x }^{ 2 } } =\frac { 1 }{ 2 } \left[ \frac { 1 }{ 1-x } +\frac { 1 }{ 1+x } \right] $$</p>
|
4,360,054 | <p><span class="math-container">$\textbf{Question}$</span>: Show that there exist an uncountable subset <span class="math-container">$X$</span> of <span class="math-container">$\mathbb{R}^{n}$</span> with property that every subset of <span class="math-container">$X$</span> with <span class="math-container">$n$</span> ... | Bart Michels | 43,288 | <p>The set of vectors <span class="math-container">$(1, x, x^2, \ldots, x^{n-1})$</span> with <span class="math-container">$x > 0$</span> works, because the condition amounts to the fact that a Vandermonde determinant is nonzero.</p>
|
4,026,795 | <blockquote>
<p>Let <span class="math-container">$V=\{(x,y,z)\in\mathbb{R^3}: x^2+y^2\le z, z\le x+2\}$</span></p>
<p>Then the volume of V is:</p>
<p>(A) Vol(V) = <span class="math-container">$\frac{75}{8}\pi$</span></p>
<p>(B) Vol(V) = <span class="math-container">$\frac{81}{32}\pi$</span></p>
<p>(C) Vol(V) = <span cl... | user2661923 | 464,411 | <p>Suppose that the problem was changed to minimizing <span class="math-container">$h(x) = \int_x^2 g(t)dt$</span> and further suppose that you were (temporarily) only considering values of <span class="math-container">$x$</span> such that <span class="math-container">$x < 1$</span>. It is clear that</p>
<ol>
<li><... |
622,883 | <p>I'm finding maximum and minimum of a function $f(x,y,z)=x^2+y^2+z^2$ subject to $g(x,y,z)=x^3+y^3-z^3=3$.</p>
<p>By the method of Lagrange multiplier, $\bigtriangledown f=\lambda \bigtriangledown g$ and $g=3$ give critical points. So I tried to solve these equalities, i.e.</p>
<p>$\quad 2x=3\lambda x^2,\quad 2y=3\... | MJD | 25,554 | <p>Since you suggest that you know how to calculate how many even numbers are in a set that <em>does</em> start with 1, why not calculate how many even numbers are in $\{1, 2, \ldots, 456\}$, then how many even numbers are in $\{1, 2, \ldots, 44\}$, and then subtract?</p>
|
2,291,175 | <p>I know that the domain is: $(-\infty,0) \cup (0,\infty)$, or all Reals except Zero.</p>
<p>But if a take the "nearest negative number to zero" and then, the "nearest positive number to zero", my function will hugely increase (Y will approach +infinity).</p>
<p>And my domain have all numbers that approach Zero in b... | Jacob Manaker | 330,413 | <p>There are two phenomena occurring here. This first is that, in an informal sense, the region where $x\mapsto\frac{1}{x}$ is increasing is also where it is infinite, so we traditionally exclude that from the domain. The second is that this region has all been squashed together, so that $x\mapsto\frac{1}{x}$ has no ... |
2,716,036 | <p>In reviewing some old homework assignments, I found two problems that I really do not understand, despite the fact that I have the answers.</p>
<p>The first is: R(x, y) if y = 2^d * x for some nonnegative integer d. What I do not understand about this relation is how it can possibly be transitive (according to my n... | 57Jimmy | 356,190 | <p>For the first problem: the integer $d$ is not fixed. And indeed, if $y = 2^d x$ and $x = 2^e z$ you get $y = 2^f z$ for $f=d+e$.</p>
<p>For the second problem: reflexivity means that for all $x$, $R(x,x)$. But if $x$ is not divisible by $17$, this does not hold!</p>
|
2,716,036 | <p>In reviewing some old homework assignments, I found two problems that I really do not understand, despite the fact that I have the answers.</p>
<p>The first is: R(x, y) if y = 2^d * x for some nonnegative integer d. What I do not understand about this relation is how it can possibly be transitive (according to my n... | drhab | 75,923 | <p>What your missing where it concerns transitivity of $R$ is the fact that $d$ is not necessarily the same for each pair $x,y$ with $\langle x,y\rangle\in R$.</p>
<p>If $y=2^dx$ and $x=2^{d'}z$ for nonnegative integers $d,d'$ then $y=2^{d+d'}z$ for nonnegative integer $d+d'$.</p>
<hr>
<p>The second relation you men... |
1,248,329 | <p>Let $R$ be an integral domain, and let $r \in R$ be a non-zero non-unit. Prove that $r$ is irreducible if and only if every divisor of $r$ is either a unit or an associate of $r$.</p>
<p>Proof. ($\leftarrow$) Suppose $r$ is reducible then $r$ can be expressed as $r = ab$ where $a$, $b$ are not units. This contradic... | user39082 | 97,620 | <p>The Statement is not true. Für example $X=\overline{X}={\mathbb R}^2$, which is simply connected, p the Identity, and $A=S^1$. The Fundamental Group of A is nontrivial, so $i_\sharp$ is not injective, but $p^{-1}A=A$ is connected.</p>
<p>By the way, the Statement as above has no meaning anyway. The Path component i... |
2,727,237 | <p>$$
\begin{matrix}
1 & 0 & -2 \\
0 & 1 & 1 \\
0 & 0 & 0 \\
\end{matrix}
$$</p>
<p>I am told that the span of vectors equal $R^m$ where $m$ is the rows which has a pivot in it. So when describing the span of the above vectors, is it correct it saying that they don't span $... | Disintegrating By Parts | 112,478 | <p>If $X$ is a real (complex) n-dimensional inner product space, then you choose an orthonormal basis $\{ e_1,e_2,\cdots,e_n \}$ of $X$ and establish a map
$$
U_e : X \rightarrow \mathbb{C}^n
$$
given by $U_ef = (\langle f,e_1\rangle,\langle f,e_2\rangle,\cdots)$. This will map $e_1$ to $(1,0,0,\cdots)$, $... |
1,642,427 | <p>If the stem of a mushroom is modeled as a right circular cylinder with diameter $1$, height $2$, its cap modeled as a hemisphere of radius $a$ the mushroom has axial symmetry, is of uniform density,and its center of mass lies at center of plane where the cap and stem join, then find $a$.</p>
<p>I really need help.<... | xoxox | 308,875 | <p>$\lim \limits_{n\to\infty}{2^n*Pi/2^n}$</p>
<p>This obviously is $Pi$. The circles may seem very small, but can't get $0$ (otherwise, the two points at $-1$ and $1$ wouldn't be connected). Consider their radius' as infinitesimal, so the curve length always will stay $Pi$ and will never get $2$.</p>
<p>Also, the th... |
3,861,820 | <p>Given the following <span class="math-container">$\frac{1}{2^2}+
\frac{1}{2^3}+\frac{1}{2^4}+...+\frac{1}{3^2}+\frac{1}{3^3}+...+$</span></p>
<p>Can this be symbolized as:
<span class="math-container">$$\sum_{n=2}^{\infty}2^{-n}+(n+1)^{-n}$$</span></p>
<p>and if so, are the following values for <span class="math-con... | user | 505,767 | <p>Your expression seems uncorrect, we can use a double series</p>
<p><span class="math-container">$$\sum_{n=2}^{\infty}\left(\sum_{k=2}^{\infty} \frac1{n^k}\right)=\sum_{n=2}^{\infty}\left(\frac{1}{1-\frac1n}-1-\frac1{n}\right)=\sum_{n=2}^{\infty}\frac{1}{n(n-1)} =\sum_{n=2}^{\infty}\left(\frac1{n-1}-\frac1{n}\right)=... |
46,726 | <p>In many proofs I see that some variable is "fixed" and/or "arbitrary". Sometimes I see only one of them and I miss a clear guideline for it. Could somebody point me to a reliable source (best a well-known standard book) which explains, when and how to use both in proofs?</p>
<p>EDIT: A little add-on to the question... | ryang | 21,813 | <ol>
<li><p>The proof of a universally-quantified (“for each <span class="math-container">$x$</span>”) statement might begin with one of these:</p>
<ul>
<li>Let <span class="math-container">$x$</span> be <strong>an arbitrary</strong> (i.e., <strong>any</strong>) real number greater than <span class="math-container">$7.... |
1,330,376 | <p>In textbooks and online tutorials I see that the remainder is calculated by using a new unknown variable on the same interval. For example we take the Taylor polynomial $T_n(a)$ but find the remainder $R(x)$ with a new variable $z$ inside it. See <a href="http://www.millersville.edu/~bikenaga/calculus/remainder-term... | Christian Blatter | 1,303 | <p>Taylor's theorem with remainder term is not meant to allow the <em>exact</em> computation of $f(x)$ for some $x$ near $a$, using a hell of a detour. The remainder term is just intended to be a help in <em>estimating</em> the error when you replace the exact value $f(x)$ by its $n$th Taylor approximation $j_{\>a}^... |
1,619,679 | <p>If the series $\sum\limits_{n=1}^{\infty} u_n $ is convergent then the sequence $u_n \rightarrow 0$ as $n \rightarrow \infty$. Therefore if the ratio test $R=\frac{u_{n+1}}{u_n}$ gives $R<1$ then we can conclude that $(u_{n})_{n\in \mathbb{N}}$ is convergent, right?</p>
<p>Generally, if we can find that a series... | Logan Clark | 305,671 | <p>If I'm understanding you correctly, then yes, you're right. The ratio test states that if the limit is less than one, the sum converges. I think you might be getting caught up in the terms.</p>
|
1,651,452 | <p>I am trying to solve a modular arithmetic system and I got to the point where I have to solve $22t \equiv 9 \pmod{7}$ for $t$. I researched on the internet and found that there are many ways to solve this, such as using linear diophantine equations, the euclidean algorithm or using inverses.</p>
<p>Can someone show... | Phillip Hamilton | 312,810 | <p>Without learning more theory, the basic identity here is that $$ a \equiv b\; (mod\; c)$$ if and only if $$ \; c\; | \;a - b $$ </p>
<p>So for $22t\; \equiv 9\;(mod \;7)$, we see $7\;|\;22t - 9$, which means we can rewrite this as a linear combination (where the Euclidean Algorithm would come into play) as $$22t - ... |
1,518,258 | <p>I haven't been able to come up with a counterexample so far. </p>
| Teoc | 190,244 | <p>Take $n=4$, $a=3$, $b=6$. Then you have a counterexample, which shows that the claim is invalid.</p>
|
2,006,993 | <p>Let $(a_{n,k})_{n, k \in \mathbb N} \subset \mathbb C$ be a series satisfying</p>
<p>$$
\sum_{n=0}^\infty \left| \sum_{k=0}^\infty a_{n,k}\right| \lt \infty
$$
and
$$
\sum_{k=0}^\infty \left|a_{n,k}\right| \lt \infty \qquad \forall n\in \mathbb N.
$$
Does this imply that $\sum_{k=0}^\infty \sum_{n=0}^\infty a_{n,k... | zhw. | 228,045 | <p>Consider the double series</p>
<p>$$1-1+0+0+0+0 +\cdots$$ $$0+2-2+0+0+0+\cdots $$ $$0+0+3-3+0+0 +\cdots $$ $$0+0+0 +4 -4+ 0 + \cdots$$ $$0+ 0+0+0+5-5+\cdots$$</p>
<p>where of course we continue the rows in this pattern on out to $\infty.$ We have absolutely convergent series in each row and column. But the interat... |
33,743 | <p>I have a lot of sum questions right now ... could someone give me the convergence of, and/or formula for, $\sum_{n=2}^{\infty} \frac{1}{n^k}$ when $k$ is a fixed integer greater than or equal to 2? Thanks!!</p>
<p>P.S. If there's a good way to google or look up answers to these kinds of simple questions ... I'd lo... | Henry | 6,460 | <p>You are looking for the <em>Riemann zeta function</em> $\zeta(k)$ (or close to it: the sum usually starts at $n=1$). </p>
<p>Since you are supposed to be doing research for a class project, perhaps you should search for it.</p>
|
854,671 | <p>So I'm a bit confused with calculating a double integral when a circle isn't centered on $(0,0)$. </p>
<p>For example: Calculating $\iint(x+4y)\,dx\,dy$ of the area $D: x^2-6x+y^2-4y\le12$.
So I kind of understand how to center the circle and solve this with polar coordinates. Since the circle equation is $(x-3)^2... | rogerl | 27,542 | <p>Figure out where the 13th day of each month falls relative to January 1st (remember there are two possibilities, corresponding to leap year or non leap year). There are only seven possible values: 0, 1, 2, 3, 4, 5, 6 days after the day on which Jan. 1 falls. If each of those values occurs at least once, then at leas... |
854,671 | <p>So I'm a bit confused with calculating a double integral when a circle isn't centered on $(0,0)$. </p>
<p>For example: Calculating $\iint(x+4y)\,dx\,dy$ of the area $D: x^2-6x+y^2-4y\le12$.
So I kind of understand how to center the circle and solve this with polar coordinates. Since the circle equation is $(x-3)^2... | Rebecca J. Stones | 91,818 | <p>In fact, every year will contain a Friday the 13-th between March and October (so leap years don't enter into it).</p>
<p>If March 13 is assigned $0 \pmod 7$, then the other moduli occur as indicated below:
$$(\underbrace{\underbrace{\underbrace{\underbrace{\underbrace{\underbrace{\overbrace{31}^{\text{March}}}_{3 ... |
611,198 | <p>A corollary to the Intermediate Value Theorem is that if $f(x)$ is a continuous real-valued function on an interval $I$, then the set $f(I)$ is also an interval or a single point.</p>
<p>Is the converse true? Suppose $f(x)$ is defined on an interval $I$ and that $f(I)$ is an interval. Is $f(x)$ continuous on $I$? <... | lhf | 589 | <p>Here is one converse:</p>
<blockquote>
<p>If $f$ is monotone and $f(I)$ is an interval, then $f$ is continuous.</p>
</blockquote>
|
3,429,350 | <p>If we a sample of <span class="math-container">$n$</span> values from a given population and if <span class="math-container">$X$</span> is the variable of the sample, then the mean of <span class="math-container">$X$</span> is just <span class="math-container">$\dfrac{ \sum x }{n}$</span></p>
<p>Now, suppose <span ... | Angela Pretorius | 15,624 | <p>This is a birds-eye view of a torus with <span class="math-container">$K_6$</span> embedded on it's surface. You can only see the three vertices on the top of the torus, but I'm sure you can imagine how they connect to the three vertices on the bottom of the torus.
<a href="https://i.stack.imgur.com/zF2V9.png" rel="... |
2,990,642 | <p><span class="math-container">$\lim_{n\to \infty}(0.9999+\frac{1}{n})^n$</span></p>
<p>Using Binomial theorem:</p>
<p><span class="math-container">$(0.9999+\frac{1}{n})^n={n \choose 0}*0.9999^n+{n \choose 1}*0.9999^{n-1}*\frac{1}{n}+{n \choose 2}*0.9999^{n-2}*(\frac{1}{n})^2+...+{n \choose n-1}*0.9999*(\frac{1}{n})... | farruhota | 425,072 | <p>Alternatively, consider the series: <span class="math-container">$\sum_{n=1}^{\infty} (0.9999+\frac{1}{n})^n$</span>, which converges by the root test:
<span class="math-container">$$\lim_{n\to \infty} a_n^{1/n}=\lim_{n\to \infty} (0.9999+\frac1n)=0.9999<1.$$</span>
Hence:
<span class="math-container">$$\lim_{n\t... |
556,807 | <p>What is the sum of this series ?</p>
<p>$(n-1)+(n-2)+(n-3)+...+(n-k)$ </p>
<p>$(n-1)+(n-2)+...+3+2+1 = \frac{n(n-1)}{2}$</p>
<p>So how can we find the sum from $n-1$ to $n-k$ ?</p>
| Ron Gordon | 53,268 | <p>$ n k $ minus the sum from $1$ to $k$ equals </p>
<p>$$ n k - \frac12 k (k+1)$$</p>
|
556,807 | <p>What is the sum of this series ?</p>
<p>$(n-1)+(n-2)+(n-3)+...+(n-k)$ </p>
<p>$(n-1)+(n-2)+...+3+2+1 = \frac{n(n-1)}{2}$</p>
<p>So how can we find the sum from $n-1$ to $n-k$ ?</p>
| Nick Peterson | 81,839 | <p><strong>Hint:</strong></p>
<p>Try writing:
$$
\sum_{k=1}^{n-1}k=\sum_{k=1}^{n-k-1}k+\sum_{k=n-k}^{n-1}k.
$$
Your formula allows you to find the first two sums; subtraction should do the rest!</p>
|
2,467,095 | <p>I'm considering the original coupon collector problem with a small modification. For the sake of completeness I shall state the original problem again first, where <strong>my question is at the end</strong>. </p>
<p>say there is a coupon inside every packet of wafers, for the moment let's assume there are only tw... | Especially Lime | 341,019 | <p>I'm not sure I understand the problem either, but do you mean "how many packets (on average) do I have to buy to get a $C_2$ followed by a $C_1$"? </p>
<p>If so, the answer is $4$. You want the next $C_1$ after the first $C_2$. The expected number of packets up to and including the first $C_2$ is $2$ (expectation o... |
4,285,448 | <p>For training I have decided to solve this limit of a succession</p>
<p><span class="math-container">$$\lim\limits_{n\to \infty }\left(3^{n+1}-3^{\sqrt{n^2-1}}\right).$$</span></p>
<p><strong>My first attempt</strong>:</p>
<p><span class="math-container">\begin{split}
\lim_{n\to \infty }\left(3^{n+1}-3^{\sqrt{n^2-1}}... | Servaes | 30,382 | <p>For <span class="math-container">$n>1$</span> you have
<span class="math-container">$$3^{n+1}-3^{\sqrt{n^2-1}}>3^{n+1}-3^{\sqrt{n^2}}=3^{n+1}-3^n=2\cdot3^n.$$</span></p>
|
1,841,173 | <blockquote>
<p>Consider the symmetric group of$S_{20}$ and it's subgroup $A_{20}$ consisting of all even permutations. Let $H$ be a $7$-Sylow subgroup of$A_{20}$. Is $H$ cyclic? And is correct the statement which says that any $7$-Sylow subgroup of $S_{20}$ is subset of $A_{20}$?</p>
</blockquote>
<p>I know that o... | p Groups | 301,282 | <p>In $|S_{20}|$, the highest power of $7$ which divides $20!$ is $7^2$. So it is clear that the Sylow-$7$ subgroup of $S_{20}$ is of order $7^2$. </p>
<p>Group of order $7^2$ is either cyclic or isomorphic to $Z_7\times Z_7$. </p>
<p>If it is cyclic, then $S_{20}$ will have an element of order $49$, and it should b... |
108,297 | <p>I've been told that strong induction and weak induction are equivalent. However, in all of the proofs I've seen, I've only seen the proof done with the easier method in that case. I've never seen a proof (in the context of teaching mathematical induction), that does the same proof in both ways, and I can't seem to f... | Arturo Magidin | 742 | <p>Statements that say that two propositions are equivalent have to be done carefully, because the background theory is important.</p>
<p>Specifically, you are talking about two statements about the natural numbers:</p>
<ol>
<li><p><strong>Induction</strong> (or "weak" induction): Let $S\subseteq \mathbb{N}$ be such ... |
2,032,501 | <p>Let A and B be subsets of $R^n$ Define</p>
<p>$A+B=\{a+b\ |\ a\in\, A , b\in \,B\}$</p>
<p>Consider the sets
$W=\{(x,y) \in\,R^2\ |\ x>0 , y>0\}
\\ X=\{(x,y) \in\,R^2\ |\ x\in\,R , y=0\}
\\ Y=\{(x,y) \in\,R^2\ |\ xy=1\}
\\Z=\{(x,y) \in\,R^2\ |\ |x|\le 1,|y|\le 1\}$<br>
Which of the following statements are... | Aweygan | 234,668 | <p>Here are a few hints for each one.</p>
<p>$(1)$ Observe that $A$ is open (Why?), that the translation of any open set is open (Why?), and
$$ W+X=\bigcup_{x\in X}(W+x). $$</p>
<p>$(2)$ Find a sequence in $X+Y$ convergent in $\mathbb{R}^2$, that does not converge in $X+Y$.</p>
<p>$(3)$ Observe that $Z$ is compact... |
1,523,230 | <p>Prove that for any primitive Pythagorean triple (a, b, c), exactly one of a and b must be a multiple
of 3, and c cannot be a multiple of 3.</p>
<p><strong>My attempt:</strong></p>
<p>Let a and b be relatively prime positive integers.</p>
<p>If $a\equiv \pm1 \pmod{3}$ and $b\equiv \pm1 \pmod{3}$, </p>
<p>$c^2=a^2... | Sam Weatherhog | 258,916 | <p>Your proof is complete. You are asked to show that one of $a,b,c$ is divisible by $3$. In the first part you show that $a$ and $b$ can't both be non-divisible by $3$. In the second part, you assume that one of $a,b$ is divisible by $3$ and show that $c^2\equiv 1$ (mod $3$) which implies that $c$ is not divisible by ... |
2,574,962 | <p>An even graph is a graph all of whose vertices have even degree. </p>
<p>A spanning subgraph $H$ of $G$, denoted by $H \subseteq_{sp} G$, is a graph obtained by $G$ by deleting <em>only</em> edges of $G$.</p>
<p>I want to show that if $G$ is a connected graph, then
$\big|\{H \subseteq_{sp} G | H$ $is$ $even\}\big|... | Hw Chu | 507,264 | <p>Let $v_1, \cdots, v_n$ be the vertices of the graph. Fixing the number of vertices, let us do induction on $e$, the number of edges.</p>
<p>Since $G$ is connected, The base case is $e = n-1$, which happens when $G$ is a tree. If $H \subseteq_{sp}G$ is even and contains at least one edge, then $H$ contains a cycle, ... |
986,494 | <p>Could somebody please show me how to integrate the following:</p>
<p>$dA/dt = -kA$</p>
<p>I'm told that the answer is:</p>
<p>$A(t) = A(0)e^-kt$</p>
<p>but I do not know why. Could you be explicit in your answer and explain precisely why it works? </p>
| Dr. Sonnhard Graubner | 175,066 | <p>if we have $f(x)=-5x^2+12x-7$ so we get $f'(x)=-10x+12$ and now must be $-10x+12\geq 0$ or $-10x+12\le 0$</p>
|
986,494 | <p>Could somebody please show me how to integrate the following:</p>
<p>$dA/dt = -kA$</p>
<p>I'm told that the answer is:</p>
<p>$A(t) = A(0)e^-kt$</p>
<p>but I do not know why. Could you be explicit in your answer and explain precisely why it works? </p>
| Prakhar Nagpal | 453,679 | <p>So your first clue should be that it is a function of degree <span class="math-container">$2$</span> so it will have only one maximum or minima, in this case, maxima since it is a quadratic opening downward<span class="math-container">$\rightarrow a \lt 0$</span>. I am including the reason for this as well. Let us s... |
713,626 | <p>If $X$ is $Beta\left(\dfrac{ \alpha_1}{ 2 }, \dfrac{\alpha_2}{2}\right)$ then $\dfrac{\alpha_2 X}{\alpha_1(1-X)}$ is $F(\alpha_1, \alpha_2)$? </p>
<p>Any help is appreciated I don't know where to start. I'm assuming I need the pdf's of each distribution?</p>
| Maverick Meerkat | 342,736 | <p>I'm not 100% sure this way is valid, but I'm gonna give it a try using the CDF's:</p>
<p><span class="math-container">$Y =\dfrac{\alpha_2 X}{\alpha_1(1-X)}$</span></p>
<p>CDF of <span class="math-container">$Y = F_{Y}(y) = \mathbb {P}(Y < y) = \mathbb {P}(\dfrac{\alpha_2 X}{\alpha_1(1-X)} < y) = \mathbb {P}(... |
2,218,341 | <p>Like in topic, you have 6 dice. You sum their values. What is the probability you get 9? How do I calculate it?</p>
| Brian Tung | 224,454 | <p><strong>General approach.</strong> There aren't that many ways to get $9$ from six dice. (I assume these are ordinary six-sided dice.) Up to reordering, there are only the following three combinations:</p>
<p>$$
1, 1, 1, 1, 1, 4
$$</p>
<p>$$
1, 1, 1, 1, 2, 3
$$</p>
<p>$$
1, 1, 1, 2, 2, 2
$$</p>
<p>Calculate th... |
2,218,341 | <p>Like in topic, you have 6 dice. You sum their values. What is the probability you get 9? How do I calculate it?</p>
| N. F. Taussig | 173,070 | <p>Let's assume the dice are six-sided and distinct (each of a different color, say). Then there are $6^6$ possible outcomes in the sample space. Let $x_k$ be the outcome on the $k$th die. Then
$$x_1 + x_2 + x_3 + x_4 + x_5 + x_6 = 9 \tag{1}$$
Equation 1 is an equation in the positive integers subject to the restri... |
1,558,665 | <p>I am looking for examples of finitely generated solvable groups that are not polycyclic. In <a href="http://groupprops.subwiki.org/wiki/Finitely_generated_and_solvable_not_implies_polycyclic#Some_examples_based_on_the_general_construction_and_otherwise" rel="nofollow">Wikipedia</a> Baumslag-Solitar group $BS(1,2)$ i... | Dietrich Burde | 83,966 | <p>For a polycyclic group, every subgroup is finitely generated, in particular the commutator subgroup is finitely generated. Now assume that $BS(1,2)$ is polycyclic. Hence its derived subgroup is finitely generated. However, the derived subgroup is isomorphic to the group of $2$-adic rationals, i.e., the group of all... |
471,561 | <p><img src="https://ukpstq.bn1.livefilestore.com/y2p5St1yRZbdxBvzMbBTYjqjqtwDvBaoWtc7YRGZXCwBTax0XseUIh_l_O92NO6XAbLeGaqU67bkBI4lroIlcD2ade_rxfDast52B_7ECcMd68/question3.png?psid=1" alt="question"></p>
<p>I have attempted these few simple questions, can someone let me know if this is correct please? If not please pro... | J.-E. Pin | 89,374 | <p>Your answers:<br>
(i) is OK<br>
(ii) There are four questions, you need to give four answers.<br>
(iii) <strong>Hint</strong>. Give a regular expression for the languages accepted by $FA_1$ and $FA_2$ and then convert them to plain English.</p>
|
236,546 | <p>$f(x)=x^4-16x^2+4$, the root of $f(x)$ is $a= \sqrt{3} + \sqrt{5}$</p>
<p>Factorise $f(x)$ as a product of irreducible polynomials over $\mathbb{Q}$, over $\mathbb{R}$ and over $\mathbb{C}$.</p>
<p>I am really confused as to how to start.</p>
| Brenin | 29,125 | <p>Yes. You are asking whether the origin is a nonsingular point of $C=\textrm{Spec}\,A\subset \mathbb A^2_k$. Write the homogeneous decomposition $F=\sum_{d\geq 1}f_d$, where $f_1=aX+bY\neq 0$. Let us show that $P$ is a regular point of $C$. If $P=(0,0)$ were singular, then (by <em>definition</em>) the two partial de... |
236,546 | <p>$f(x)=x^4-16x^2+4$, the root of $f(x)$ is $a= \sqrt{3} + \sqrt{5}$</p>
<p>Factorise $f(x)$ as a product of irreducible polynomials over $\mathbb{Q}$, over $\mathbb{R}$ and over $\mathbb{C}$.</p>
<p>I am really confused as to how to start.</p>
| Mariano Suárez-Álvarez | 274 | <p>Your ring is an integrally closed noetherian local ring with Krull dimension one, and such a thing is a DVR.</p>
|
236,546 | <p>$f(x)=x^4-16x^2+4$, the root of $f(x)$ is $a= \sqrt{3} + \sqrt{5}$</p>
<p>Factorise $f(x)$ as a product of irreducible polynomials over $\mathbb{Q}$, over $\mathbb{R}$ and over $\mathbb{C}$.</p>
<p>I am really confused as to how to start.</p>
| Makoto Kato | 28,422 | <p><strong>Lemma</strong>
Let $A$ be a Noetherian local domain.
Let $\mathfrak{m}$ be its unique maximal ideal.
Suppose $\mathbb{m}$ is a non-zero principal ideal.
Then $A$ is a discrete valuation ring.</p>
<p>Proof:
Let $t$ be a generator of $\mathfrak{m}$.
We claim that $\bigcap_n \mathfrak{m}^n = 0$.
Let $x \in \bi... |
226,249 | <p>I am struggling to convert a base64 string to a list of UnsignedInteger16 values.
I have limited experience with mathematica, so please excuse me if this should be obvious.</p>
<p>I read the base64 string from an XML file and ultimately into a variable base64String.
This is a long string with 7057 UnsignedInteger16 ... | kglr | 125 | <pre><code>k = {6, 4, 2};
lbls = Row[{Subscript["N", "B"] , #}, " = "] & /@ k;
cols = {Red, Orange, Black};
</code></pre>
<p>You can use the option <code>LegendLayout</code> as follows:</p>
<pre><code>linesperlabel = 2;
labels = Flatten[Thread[{lbls, ##& @@ ConstantArray[SpanFromAb... |
1,948,634 | <blockquote>
<p>Is it possible to find $6$ integers $a_1,a_2,\ldots,a_6 \geq 2$ such that $$a_1+a_1a_2+a_1a_2a_3+a_1a_2a_3a_4+a_1a_2a_3a_4a_5+a_1a_2a_3a_4a_5a_6 = 248?$$</p>
</blockquote>
<p>I was wondering how we could establish the existence of such numbers. Is there a way to do it without finding the actual $6$ n... | HarrySmit | 332,761 | <p>I will provide the first step, from which you can hopefully complete the proof yourself. As @snulty mentioned, we can factor this as follows:
$$
a_1(1 + a_2(1 + a_3(1 + a_4(1 + a_5(1 + a_6))))) = 2^3 \cdot 31.
$$
This implies that $a_1 = 2$ or $a_1 \geq 4$. However, if $a_1 \geq 4$, we have
$$
a_1(1 + a_2(1 + a_3(1... |
1,859,652 | <p>I've just been studying cyclic quads in geometry at school and I'm thinking see seems pretty interesting, but where would I actually find these in the real world? They seem pretty useless to me...</p>
| preferred_anon | 27,150 | <p>I can't think of any applications, and I doubt any satisfactory ones exist - for example, as noted in the comments there may well have been connections to astronomy, but I think it's fair to suggest that almost no-one who is being taught circle theorems is going to use them in their life at any point. </p>
<p>Thus... |
1,859,652 | <p>I've just been studying cyclic quads in geometry at school and I'm thinking see seems pretty interesting, but where would I actually find these in the real world? They seem pretty useless to me...</p>
| Joseph O'Rourke | 237 | <p>Theorem 3 of the Bern-Eppstein paper cited below proves that any polygon of $n$ vertices
may be partitioned into $O(n)$ cyclic quadrilaterals.
A hint of how this might be achieved can be glimpsed in the figure below, where
all the white quadrilaterals are cyclic.
<hr />
<img src="... |
19,478 | <p>Let $K$ and $L$ be two subfields of some field. If a variety is defined over both $K$ and $L$, does it follow that the variety can be defined over their intersection?</p>
| Bjorn Poonen | 2,757 | <p>Yes, if varieties are interpreted as <strike>subvarieties</strike> closed subschemes of base extensions of a fixed ambient <strike>variety</strike> scheme (e.g., affine space or projective space).</p>
<p>More precisely, suppose that $k \subseteq F$ are fields and <strike>the variety</strike> $X$ is <strike>an $F$-s... |
1,164,040 | <p>Let $A$ be a $m \times n$ matrix. Determine whether or no the set $W= \{y : Ay=0\}$ is a vector space.
This proof involves nullspace work and another way of asking it is also proving that $W$ is the nullspace of $A$.</p>
<p>I think you can solve this with the use of the subspace theorem being that you can just pro... | Brian Fitzpatrick | 56,960 | <p>We wish to show that
$$
W=\{\vec x\in\Bbb R^n:A\vec x=\vec 0\}
$$
is a subspace of $\Bbb R^n$. To do so, we use the <a href="https://proofwiki.org/wiki/One-Step_Vector_Subspace_Test" rel="nofollow">one-step vector subspace test</a>. To do so, let $\vec x,\vec y\in W$ and let $\lambda\in\Bbb R$. Then
$$
A(\vec x+\la... |
2,240,756 | <p>I tried rewriting $(1+x+x^2)^\frac{1}{x}$ as $e^{\frac{1}{x}\ln(1+x+x^2)}$ and then computing the taylor series of $\frac{1}{x}$ and $\ln(1+x+x^2)$ but I'm still not getting the correct answer..</p>
| robjohn | 13,854 | <p>This appears to be the approach you used, but without seeing your work, it is hard to tell where you were having trouble.</p>
<p>Use the series $\log(1+x)=x-\frac{x^2}2+\frac{x^3}3+\cdots$
$$
\begin{align}
\frac1x\log\left(1+x+x^2\right)
&=\frac1x\left(x+x^2-\frac{x^2+2x^3+x^4}2+\frac{x^3+3x^4+3x^5+x^6}3+\cdots... |
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