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 |
|---|---|---|---|---|
419,205 | <p>My question is: is it possible to find pentagonal numbers which are also tetrahedral?
A pentagonal number is obtained by the formula:
$$P_k=\frac{1}{2}k(3k-1)$$
The equivalent formula for the tetrahedral number $T_n$
is:
$$T_n=\frac{1}{6}n(n+1)(n+2)$$
So the problem is to find a $T_n=P_k$ that means to solve:
$$n(n+... | Kns | 27,579 | <p>Characteristic equation of any $2\times 2$ matrix is given by $\alpha^{2}-S_{1}\alpha+S_{2}=0$
, where $S_{1}=$ sum of principle diagonal element and $S_{2}=$ determinant of given matrix,
using this we can easily get eigen values of matrix.
So, here we have,
$$\alpha^{2}-7\alpha+12=0$$
$$\Rightarrow (\alpha-4)(\alp... |
354,365 | <p>Let $X_i$ be pairwise-uncorrelated random variables, $\forall\,i \in \mathbf{n} \equiv \{0,\dots,n-1\}$, with identical expectation value $\mathbb{E}(X_i)=\mu$, and identical variance $\mathrm{Var}(X_i)=\sigma^2$. Also, let $\overline{X}$ be their average, $\frac{1}{n}\sum_{i\in \mathbf{n}} X_i$.</p>
<p>Then, as s... | Berci | 41,488 | <p>Let's start out from the standard basis $e_1,..,e_n$. Let $a_1,..,a_k$ be the column vectors of $A$.</p>
<p>Check that the step on rows $r_i':=r_i+\lambda\,r_j$ corresponds to the basis transformation $e_j':=e_j-\lambda\,e_i$, that is, for a vector $v$ we have
$$v=\sum_i\alpha_ie_i=\sum_i\alpha_i'e_i'$$
where the r... |
4,298,602 | <p><a href="https://i.stack.imgur.com/jX5H1.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/jX5H1.png" alt="enter image description here" /></a></p>
<p>The prime <strong>Cauchy</strong> product is <span class="math-container">$C_n = (a_1b_1) + (a_1b_2+a_2b_1) + (a_1b_3+a_2b_2+a_3b_1)+...$</span>，
In ... | Booker ML | 833,312 | <p>I think I was understand，<span class="math-container">$\sum_n^N an$</span> is a finite series, it can rearrange</p>
|
117,558 | <p>Is there anything known about the existence of <a href="http://mathworld.wolfram.com/HeronianTriangle.html" rel="nofollow noreferrer">Heronian triangles</a> ABC (i.e. with rational side lengths and rational area) that can be decomposed into three Heronian triangles ABD, BCD, CAD? Equivalently, is there a degenerate ... | paul Monsky | 6,214 | <p>There should be many examples where the interior vertex lies on the perpendicular bisector of one of the sides. It's best to redefine a Heronian triangle to be one with rational sides and rational area. I'll call such a T "standard" if its vertices are at $(-2,0), (2,0)$ and $(r,s)$
where $r$ and $s$ are rational. E... |
2,496,309 | <p>i am wondering if the unity elements of a ring form a ring ? In other words do they form an abelian group under addition ? I have tryied but i have not reached to a conclusive answer. Thanks for any comment.</p>
| Vidyanshu Mishra | 363,566 | <p>Hint: Rationalise the denominator by multiplying the numerator and denominator by $\sqrt{h+4} +2$ and cancel the $h's$.</p>
<p>Then you are only left with $\lim_{h \to 0}{\sqrt{h+4}+2}$</p>
|
3,541,869 | <p>I was reading from <a href="https://books.google.com.gh/books/about/Ordinary_Differential_Equations.html?id=iU4zDAAAQBAJ&source=kp_book_description&redir_esc=y" rel="nofollow noreferrer"><em>Ordinary Differential Equations</em></a> <strong>(Lesson 13 Example 13.3 page 110)</strong> and came across this quest... | Christian Blatter | 1,303 | <p>You want to prove that <span class="math-container">$f(x)+g(x)$</span> is near <span class="math-container">$b+c$</span> when <span class="math-container">$x$</span> is near <span class="math-container">$a$</span>. This means that you have to look at the quantity
<span class="math-container">$$(f(x)+g(x))-(b+c)=(f(x... |
2,744,496 | <p>Consider the vector space $\Bbb{R}^3$ with coordinates $(x_1, x_2, x_3)$ equipped with the inner product $$\langle(a_1, a_2, a_3),(b_1, b_2, b_3)\rangle= 2(a_1b_1 + a_2b_2 + a_3b_3) − (a_1b_2 + a_2b_1 + a_2b_3 + a_3b_2).$$</p>
<p>Write down all vectors in $\Bbb{R}^3$ which are orthogonal to the plane $x_1 − 2x_2 + ... | Wouter | 89,671 | <p>Note that your plane contains $(0,0,0)$, which is handy.</p>
<p>Find 2 linearly independent vectors in the plane, for example
$$v_1=(0,1,1)$$
and
$$v_2=(2,0,-1)$$
All we have to do is find a vector $v=(x_1,x_2,x_3)$ perpendicular to $v_1$ and $v_2$
$$\langle(x_1,x_2,x_3),v_1\rangle=0$$
$$\langle(x_1,x_2,x_3),v_2\r... |
4,052,739 | <p>Given fundamental groupoid <span class="math-container">$\Pi_1(S^1)$</span> of the circle, how can one define a topology on it? The information on <a href="https://ncatlab.org/nlab/show/fundamental+groupoid#topologizing_the_fundamental_groupoid" rel="nofollow noreferrer">nlab</a> did little help other than the fact ... | Matthew Kvalheim | 322,545 | <p>Given a connected, locally path-connected, and semi-locally simply connected space <span class="math-container">$M$</span> (so that <span class="math-container">$M$</span> has a universal covering space), the fundamental groupoid <span class="math-container">$\Pi(M)$</span> (viewed here as simply being the set of pa... |
1,461,919 | <p>I am asked to calculate $\displaystyle \lim_{n\rightarrow\infty} \int_n^{n+1}\frac{\sin x} x \, dx$.</p>
<p>Before letting the $\lim$ to confuse me I used integration by parts, but it didn't get me far.</p>
<p>Any hint?</p>
| Brevan Ellefsen | 269,764 | <p>If you permit special functions, $\int \frac{\sin x}{x}dx = \text{Si}(x)$. Further, there is the well known identity $\displaystyle\lim_{x \to \infty} \text{Si}(x) = \frac{\pi}{2}$. From this, we get
$$\lim_{n\rightarrow\infty} \int_n^{n+1}\frac{\sin x} x \, dx = \lim_{n \to \infty} [\text{Si}(n+1)-\text{Si}(n)]$$
W... |
1,461,919 | <p>I am asked to calculate $\displaystyle \lim_{n\rightarrow\infty} \int_n^{n+1}\frac{\sin x} x \, dx$.</p>
<p>Before letting the $\lim$ to confuse me I used integration by parts, but it didn't get me far.</p>
<p>Any hint?</p>
| Jason | 195,308 | <p>There are a couple of answers already but I'll provide the simple one that was alluded to in comments. For all $x\in[n,n+1]$, $|\frac{\sin x}x|\le\frac1n$ and so $|\int_n^{n+1}\frac{\sin x}x\,\mathrm dx|\le\frac1n\to0$.</p>
|
328,811 | <p>I already asked a question (<a href="https://math.stackexchange.com/questions/328680/order-of-operations-in-rotation-matrix-notation">Order of operations in rotation matrix notation.</a>) about the order in which a particular equation is "processed" and now I need to generalise that and learn the rules of math notat... | Alexander Gruber | 12,952 | <p>You are exactly right that the expression in J.M.'s <a href="https://math.stackexchange.com/revisions/142831/3">(original)</a> answer contains ambiguity. Mathematicians can distinguish the correct meaning easily because we are used to seeing things written this way, but it can be difficult for the uninitiated. Let... |
2,239,056 | <p>Trying to prove that for any two matrices $A,B$ representing the same symmetric bilinear form $f:V \times V \to \mathbb{R}$, there is an invertible matrix $P$, such that $B = P^{T}AP$ ?</p>
<p>I have a method for constructing such $P$ by repeated change of bases (when it is a quadratic form) but can't see a way to... | user1551 | 1,551 | <p>This is just a change of basis. Let $A$ and $B$ be the matrices of the same bilinear form, but with respect to two ordered bases $\{x_1,\ldots,x_n\}$ and $\{y_1,\ldots,y_n\}$ respectively. Let $P$ be the matrix such that $y_j=\sum_ip_{ij}x_i$, i.e. the matrix such that $\sum_jc_jy_j=\sum_i(Pc)_ix_i$ for any $c\in\ma... |
3,064,458 | <blockquote>
<p>Let <span class="math-container">$f:[0,1]\rightarrow\mathbb{R}$</span> be a continuous function. </p>
<ol>
<li>Show that for each <span class="math-container">$\epsilon\in(0,1)$</span>, <span class="math-container">$\lim\limits_{n\rightarrow\infty}\int\limits_0^{1-\epsilon}f(x^n)dx=(1-\epsilon... | Matematleta | 138,929 | <p>Another approach uses the mean value theorem for integrals. Indeed, set <span class="math-container">$g_n(x)=f(x^n).$</span> Then, there is a sequence <span class="math-container">$(c_n)\subseteq (0,1-\epsilon)$</span> such that <span class="math-container">$\int\limits_0^{1-\epsilon}g_n(x)dx=g_n(c_n)(1-\epsilon)=f(... |
596,598 | <p>A graph with three vertices has a beta index no greater than 1. A beta index of a graph is the ratio of number of edges to the number of vertices. </p>
<p>The answer key says true but I think it's false. If loops are allowed and vertex $A$ connects to vertex $A$ then there could be 6 edges in a graph and 3 vertices... | M.B. | 2,900 | <p>There is an equivalent definition of continuity: a function $f: X\to Y$ between topological spaces is continuous at $x$ if there for any neighbourhood $V$ of $f(x)$ exists a neighbourhood $U$ of $x$ such that $f(U) \subseteq V$. The function is continuous if it is continuous at every $x\in X$.</p>
<p>In your case:<... |
724,462 | <p>I can not solve this question
Find a compound proposition logically equivalent to $p \to q$ using only the logical operator $\downarrow$.</p>
| Mauro ALLEGRANZA | 108,274 | <p>It's quite simple; let start with :</p>
<blockquote>
<p>$\lnot p \equiv p \downarrow p$.</p>
</blockquote>
<p>Then :</p>
<blockquote>
<blockquote>
<p>$p \rightarrow q \equiv ((p \downarrow p) \downarrow q) \downarrow ((p \downarrow p) \downarrow q)$.</p>
</blockquote>
</blockquote>
<p>In order to chec... |
704,457 | <p>from the set $\{a, b, c, d\}$?</p>
<p>Of the one's I have tried, it at best is two of the three, but never all.</p>
| NasuSama | 67,036 | <p>Consider the ordered pair, such that $\{(x,y)\,|\,x,y\in\{a,b,c,d\}\}$. The following relation satisfies those conditions:</p>
<p>$$\{(a,b) ,(b,c), (c,d), (d,a)\}$$</p>
<p>Clearly, this relation is not reflexive since there is no ordered pair with same members i.e. $(x,x)$. This relation is anti-symmetric since ... |
704,457 | <p>from the set $\{a, b, c, d\}$?</p>
<p>Of the one's I have tried, it at best is two of the three, but never all.</p>
| zyx | 14,120 | <p>If non-transitive means "never transitive for any triple", that is impossible for a complete tournament (irreflexive, never-symmetric relation) with $4$ or more players. The question sounds like an exercise of rediscovering that fact. </p>
<p>If partial tournaments are allowed then it can be done with any number... |
4,311,986 | <p>Here is the question:</p>
<p>If:</p>
<p><span class="math-container">$$125*(3^x) = 27*(5^x)$$</span></p>
<p>Then find the value of <span class="math-container">$x$</span>.</p>
<p>Here is what I have done so far:</p>
<p>I found that I can manipulate the numerical values to get the same bases:</p>
<p><span class="math... | Toni Mhax | 468,334 | <p>I don't know the proof of 'Here, it is necessary...' but any way a proof by induction is one that would be almost similar in any case, a direct proof may be:
Divide the two members by <span class="math-container">$\sqrt{n}$</span> then we prove <span class="math-container">$$\dfrac{S_n}{\sqrt{n}}\ge 1$$</span> or <s... |
4,311,986 | <p>Here is the question:</p>
<p>If:</p>
<p><span class="math-container">$$125*(3^x) = 27*(5^x)$$</span></p>
<p>Then find the value of <span class="math-container">$x$</span>.</p>
<p>Here is what I have done so far:</p>
<p>I found that I can manipulate the numerical values to get the same bases:</p>
<p><span class="math... | Alakey | 995,617 | <p>The part <span class="math-container">$\frac{\sqrt{n}}{n+1-\sqrt{n^2+n}}>S_n$</span> requires induction.</p>
<p>Consider the inequality above when n=1:
<span class="math-container">$$
\frac{1}{2-\sqrt{2}}>1
$$</span>
<span class="math-container">$$
-1>-\sqrt{2} - true
$$</span>
Let the inequality be correct... |
1,089,593 | <p>How to solve $\dfrac{dy}{dx}=\cos(x-y)$ ? How do I separate x and y here ?</p>
<p>Please advise.</p>
| Arian | 172,588 | <p>Set $u=x-y$ then
$$\frac{du}{dx}=1-\frac{dy}{dx}$$
and the original differential equation could be rewritten as
$$1-\frac{du}{dx}=\cos(u)\Rightarrow \frac{du}{dx}=1-\cos(u)$$
Using direct integration
$$\int\frac{1}{1-\cos(u)}\,du=\int\,dx\Leftrightarrow -\cot(\frac{u}{2})=x+c$$
In other words
$$u=2\cot^{-1}(-x-c)$$... |
2,148,976 | <p>I currently work on a problem where I consider a model which requires calculation of the term</p>
<p>$$\frac{1}{U'(0)}$$</p>
<p>with $U'(x)=x^{-0.5}$.</p>
<p>Therefore $U'(0)$ is undefined.</p>
<p>However, $\frac{1}{U'(x)}=x^{0.5}$ is defined for $x=0$ so I get</p>
<p>$$\frac{1}{U'(0)}=0^{0.5}=0$$</p>
<p>Is th... | GEdgar | 442 | <p>This will depend on your model. In some cases, you know the final result is defined and continuous (for reasons based on the model). If that is true, then having a few points where the calculation is undefined will not matter.</p>
|
4,222,999 | <p>I want to know a general way to express the PDF of <span class="math-container">$Y = X^2$</span> for <span class="math-container">$X\sim U(a,b)$</span> arbitrary <span class="math-container">$a$</span> and <span class="math-container">$b$</span> such that <span class="math-container">$a < b$</span> and <span clas... | tommik | 791,458 | <p>if <span class="math-container">$a,b \in R$</span> your formula doesn't work. In fact, in this case, <span class="math-container">$(a,b)=(-1;2)$</span> using the definition of CDF you get</p>
<p><span class="math-container">$$F_Y(y)=P(Y\leq y)=P(-\sqrt{y}\leq X\leq \sqrt{y})$$</span></p>
<p>that is</p>
<p><span clas... |
123,759 | <p>Let $T^n$ be the $n$-dimensional torus and $g$ be a Riemannian metric on $T^n$. Let $\tilde g$ be the induced metric on the universal covering; using suitable coordinates, $\tilde g$ is therefore a $\mathbb{Z}^n$-periodic metric on $\mathbb{R}^n$ (I shall conflate the lattice $\mathbb{Z}^n$ with the fundamental grou... | Sergei Ivanov | 4,354 | <p>If you allow an additive term $C(n)diam(g)$ rather than $2diam(g)$, then yes, the statement is true. In the paper D.Burago, "Periodic metrics", Adv. Soviet Math. 9, (1992), 205-210, he proves that for every periodic metric on $\mathbb R^n$ there is a constant $C$ such that
$$
| d(x,y)-\|x-y\|_{st} | \le C
$$
for al... |
502,160 | <p>Is there any representation of the exponential function as an infinite product (where there is no maximal factor in the series of terms which essentially contributes)? I.e.</p>
<p>$$\mathrm e^x=\prod_{n=0}^\infty a_n,$$</p>
<p>and by the sentence in brackets I mean that the $a_n$'s are not just mostly equal to $1$... | user91500 | 91,500 | <p>There exists an infinite product for $e$ as follows:</p>
<p>If we define a sequence $\lbrace e_n\rbrace$ by $e_1=1$ and $e_{n+1}=(n+1)(e_n+1)$ for $n=1,2,3,...;$ e.g.
$$e_1=1,e_2=4,e_3=15,e_4=64,e_5=325,e_6=1956,...$$
then
$$e=\prod_{n=1}^\infty\frac{e_n+1}{e_n}=\frac{2}{1}.\frac{5}{4}.\frac{16}{15}.\frac{65}{64}.\... |
502,160 | <p>Is there any representation of the exponential function as an infinite product (where there is no maximal factor in the series of terms which essentially contributes)? I.e.</p>
<p>$$\mathrm e^x=\prod_{n=0}^\infty a_n,$$</p>
<p>and by the sentence in brackets I mean that the $a_n$'s are not just mostly equal to $1$... | Mohammad Al Jamal | 23,373 | <p>For any <span class="math-container">$x\in \mathbb{C}/{\mathbb{N}^{-}}$</span>, we have :
<span class="math-container">$$e^{x}=(1+x)\prod_{n=1}^{\infty}\left(1+\frac{x}{n}\right)^{-n}\left(1+\frac{x}{n+1}\right)^{n+1}$$</span></p>
|
355,421 | <p>I have two questions.</p>
<p><span class="math-container">$\bf 1.$</span> First, a reference request. Let <span class="math-container">$G\cong{\mathbb F}_p^r$</span> for some integer <span class="math-container">$r\geq 0$</span> and let <span class="math-container">$V=G^*={\rm Hom}(G,{\mathbb F}_p)$</span>. Then <s... | Chris Gerig | 12,310 | <p>They're essentially exercises, compute it for <span class="math-container">$r=1$</span> and then invoke Künneth, and I'd expect every book to include it: Adem--Milgram's classic book for example (Corollary II.4.3 and Theorem II.4.4).</p>
<p>You can look at the identical homology case in Brown's classic book (Theore... |
2,035,454 | <p>I am an upcoming year $12$ student, school holidays are coming up in a few days and I've realised I'm probably going to be extremely bored. So I'm looking for some suggestions.</p>
<p>I want a challenge, some mathematics that I can attempt to learn/master. Obviously nothing impossible, but mathematics is my number ... | Pax | 47,926 | <p>This question probably has as many answers as there are mathematicians. I'll plug the following list of books that make a great introduction to a field:</p>
<p><a href="http://rads.stackoverflow.com/amzn/click/0387900934" rel="nofollow noreferrer">Finite-Dimensional Vector Spaces, Halmos</a></p>
<p><a href="http:/... |
2,035,454 | <p>I am an upcoming year $12$ student, school holidays are coming up in a few days and I've realised I'm probably going to be extremely bored. So I'm looking for some suggestions.</p>
<p>I want a challenge, some mathematics that I can attempt to learn/master. Obviously nothing impossible, but mathematics is my number ... | Simply Beautiful Art | 272,831 | <p>Notice to the reader: at the end on list item #8</p>
<hr>
<p>Start by proving this 'simple' sum:</p>
<p>$$\frac{1-r^{n+1}}{1-r}=1+r+r^2+r^3+\dots+r^n$$</p>
<p>After you've proven that, differentiate it and solve the following sum</p>
<p>$$1+2+3+\dots+n=\lim_{r\to1}\dots$$</p>
<p>See if you can derive what <a h... |
3,915,413 | <p>Let <span class="math-container">$A$</span> be a positive definite <span class="math-container">$n\times n$</span> matrix. We use the iteration of the mapping <span class="math-container">$$f(X)=0.5(X^2+B), \ X\in \mathbb{R}^{n\times n}$$</span> where <span class="math-container">$B=I-A$</span>.</p>
<p>Show that the... | user1551 | 1,551 | <p>By mathematical induction, we see that each <span class="math-container">$X_k$</span> is a polynomial in <span class="math-container">$A$</span>. Since <span class="math-container">$A$</span> is diagonalisable, it suffices to prove the problem statement in the scalar case.</p>
<p>Presumably, the matrix norm in quest... |
1,425,042 | <p>Let B be the set of all irrational numbers together with the numbers 0, 1, and -1. Let addition and multiplication be defined on B in the same way they are defined for real numbers. Determine the field properties that are satisfied by B. Is B a field?</p>
| snulty | 128,967 | <p>As an alternative you have that $2-\sqrt{2}, \sqrt{2}$ are both irrational and their sum is $2$ which is not in $B$ so it is not closed under addition either.</p>
|
946,902 | <p>Is it possible in some cases that using the ILATE rule does not yield an explicit antiderivative but making another choice does yields one? If so, please give examples.</p>
| StefanS | 164,635 | <p>Yes, it's easy for the rule to fail if the proposed derivative is not integrable.
For example in the integral</p>
<p>$$\int x^3 e^{x^2} \mathrm{d}x$$</p>
<p>the rule would propose $u=x^3$ and $dv=e^{x^2}$. The latter cannot be integrated and you are therefore stuck.</p>
<p>To solve the above integral use $u=x^2$ ... |
2,264,435 | <p>I need to solve this series:</p>
<p>$$\sum _{ k=2 }^{ \infty } (k-1)k \left( \frac{ 1 }{ 3 } \right) ^{ k+1 }$$</p>
<p>I converted it into $$\sum _{ k=0 }^{ \infty } \frac { { k }^{ 2 }-k }{ 3 } \left(\frac { 1 }{ 3 } \right)^{ k } -\frac { 4 }{ 3 } $$ with the idea, that $$\sum _{ k=0 }^{ \infty }{ { q }^{... | Community | -1 | <p>Hint:</p>
<p>$$\frac d{dx} \sum_{k=0}^{+\infty} x^k = \sum_{k=1}^{+\infty} kx^{k-1}$$
and
$$\frac d{dx} \sum_{k=1}^{+\infty} kx^{k-1} = \sum_{k=2}^{+\infty} (k-1)kx^{k-2}$$</p>
<p>First rewrite your given series as
$$\sum_{k=2}^{+\infty} (k-1)k\left(\frac13\right)^{k+1} = \frac1{27} \sum_{k=2}^{+\infty}(k-1)k\left... |
3,104,049 | <p>I have a set:</p>
<p><span class="math-container">$X = \{p | p \in P \wedge \forall a(a \in A \wedge p \in F(a) \wedge p \notin F'(a))\}$</span></p>
<p>This reads (to me): the set containing all p, where p is an element of P, and for all a, a is an element of A and p is in F(a) and not in F'(a).</p>
<p>1) Does th... | Alberto Takase | 146,817 | <p>Going off of what you want to create, have you considered the following?</p>
<p><span class="math-container">$$P\cap\bigcap_{a\in A}F(a)\setminus F'(a)$$</span></p>
<p>This equates to your number 2.</p>
|
1,432,729 | <p>I know that $\pi \approx \sqrt{10}$, but that only gives one decimal place correct. I also found an algebraic number approximation that gives ten places but it's so cumbersome it's just much easier to just memorize those ten places.</p>
<p>What's a good approximation to $\pi$ as an irrational algebraic number (or a... | Aditya Agarwal | 217,555 | <p>It is not a low degree polynomial, but easy to remember for sure. <br>
We know that $$\frac{1}{1+x}=1-x+x^2-x^3+\cdots$$
Substituting $x^2$, $$\frac1{1+x^2}=1-x^2+x^4-x^6+\cdots$$
But we also know that $\int\frac1{1+x^2}dx=\arctan x$.
<br>
So let us integrate both sides (from $x=0$ to $x=y$), $$\arctan y=y-\frac{y^... |
1,432,729 | <p>I know that $\pi \approx \sqrt{10}$, but that only gives one decimal place correct. I also found an algebraic number approximation that gives ten places but it's so cumbersome it's just much easier to just memorize those ten places.</p>
<p>What's a good approximation to $\pi$ as an irrational algebraic number (or a... | Jaume Oliver Lafont | 134,791 | <p>Dalzell's integral is related to the rational approximation $\pi\approx \frac{22}{7}$.</p>
<p>$$\pi=\frac{22}{7}-\int_0^1 \frac{x^4(1-x)^4}{1+x^2}dx\approx\frac{22}{7}$$</p>
<p>Similar small integrals are related to simple irrational approximations using $\sqrt{2}$ and $\sqrt{3}$.</p>
<p>$$\pi=\frac{20\sqrt{2}}{9... |
472,684 | <p>Consider the random walk $X_0, X_1, X_2, \ldots$ on state space $S=\{0,1,\ldots,n\}$ with absorbing states $A=\{0,n\}$, and with $P(i,i+1)=p$ and $P(i,i-1)=q$ for all $i \in S \setminus A$, where $p+q=1$ and $p,q>0$.</p>
<p>Let $T$ denote the number of steps until the walk is absorbed in either $0$ or $n$.</p>
... | Did | 6,179 | <blockquote>
<p>The usual approach:</p>
</blockquote>
<p>For every $0\leqslant k\leqslant n$, let $u_k=E_k(T:X_T=n)$ and $v_k=P_k(X_T=n)$, then $(u_k)$ and $(v_k)$ are determined by the fact that $u_0=u_n=v_0=0$, $v_n=1$, and, for every $1\leqslant k\leqslant n-1$,
$$
u_k=v_k+pu_{k+1}+qu_{k-1},\qquad
v_k=pv_{k+1}+q... |
2,522,255 | <p>I'm having difficulties to understand how to approach to this question:</p>
<p>Rolling a dice for 2 times.</p>
<ul>
<li>X - Result of the first time.</li>
<li>Y - The highest results from both of the rolls. </li>
</ul>
<p>Example: if in the first we'll have 5, and in the second 2, Y will be 5.</p>
<ol>
<li>Show ... | gt6989b | 16,192 | <p><strong>HINT</strong></p>
<p>the first question's intuition is, you have to show that if you know something about $X$, you will affect what $Y$ is, or vice versa. The simplest way to see this is, if $X$ is anything, $Y$ can be any number from 1 to 6. But if $X=6$, you know $Y=6$, so information about $X$ changes wh... |
2,946,451 | <p>Is there a reason why we call it differently?</p>
| Theo Bendit | 248,286 | <p>You fix <span class="math-container">$\varepsilon > 0$</span> and choose <span class="math-container">$x$</span> and <span class="math-container">$y$</span> such that <span class="math-container">$\alpha - \varepsilon < x \le \alpha$</span> and <span class="math-container">$-\beta - \varepsilon < -y \le -\b... |
429,844 | <p>If I have a $4\times 4$ matrix $A$ with real entries that has all $1$'s on the main diagonal, $A$ is singular and we know one eigenvalue $k_{1}=3+2i$. What about the others three eigenvalues?</p>
<p>I think one should be $k_{2}=3-2i$ because they always come in pairs, right?</p>
<p>Then, since $A$ is singular I th... | martini | 15,379 | <p><strong>Hint</strong>: Note that the trace is invariant under similarity transformations, that is for any $S \in GL(4)$ we have
$$ \mathrm{tr}(S^{-1}AS) = \mathrm{tr}(A) = 4 $$
Now choose an $S$ such that $S^{-1}AS$ has the eigenvalues on the diagonal.</p>
|
2,770,523 | <p>I'm struggling with the following problem,</p>
<p>Let $g(z)=\sum^k_1 m_{\alpha}(z-z_{\alpha})^{-1}$. Show that if $g(z)=0$, then $z_1,\cdots,z_k$ cannot all lie on the same sie of a straight line through $z$.</p>
<p><strong>What I did:</strong></p>
<p>The book says that I should use the fact that if $z_1,\cdots,z... | John Doe | 399,334 | <p>No - any two coprime odd numbers (e.g any two primes $\ne 2$) provide a counterexample.</p>
|
4,341,356 | <p>A French-suited cards pack consist of <span class="math-container">$52$</span> cards where <span class="math-container">$13$</span> are clovers. <span class="math-container">$4$</span> players play a game where every player has <span class="math-container">$13$</span> cards in his/her hands (, in other words the ful... | José Carlos Santos | 446,262 | <p>Since <span class="math-container">$2^{2n+4}+2^{2n+2}=2^{2n}(2^4+2^2)=20\times4^n$</span>,<span class="math-container">$$\sqrt[n]{\frac{20}{2^{2n+4}+2^{2n+2}}}=\sqrt[n]{\frac{20}{20\times4^n}}=\frac14.$$</span></p>
|
2,659,001 | <p>I would like to compute the subdifferential of the function</p>
<p>$$ f(x)=a^\text{T}x+\alpha\sqrt{x^\text{T}Bx} $$
where $\alpha>0$ and $B$ is symmetric positive definite. </p>
<p><strong>Attempted Solution</strong> (I am brand new to subdifferentiability)</p>
<p>Since subderivatives, like normal derivatives,... | nonuser | 463,553 | <p>Hint: write $t=u^{1/3}$ so $t^2-5t+7=0$. Plug now $u=t^3$ in </p>
<p>$$f(u) = u^3-20u+343 = t^9-20t^3+343 = ... $$ You must get $f(u)=0$. Don't forget to use $t^2 = 5t-7$</p>
|
2,659,001 | <p>I would like to compute the subdifferential of the function</p>
<p>$$ f(x)=a^\text{T}x+\alpha\sqrt{x^\text{T}Bx} $$
where $\alpha>0$ and $B$ is symmetric positive definite. </p>
<p><strong>Attempted Solution</strong> (I am brand new to subdifferentiability)</p>
<p>Since subderivatives, like normal derivatives,... | robjohn | 13,854 | <p>Cube both sides of $u^{2/3}=5u^{1/3}-7$ to get
$$
\begin{align}
u^2
&=125u\overbrace{-525u^{2/3}+735u^{1/3}}^{-105u}-343\\
&=20u-343
\end{align}
$$
where $-105u=-105\overbrace{\left(5u^{1/3}-7\right)}^{u^{2/3}}u^{1/3}$</p>
<p>Therefore,
$$
u^2-20u+343=0
$$</p>
<hr>
<p><strong>In General</strong></p>
<p>S... |
81,221 | <p>Suppose that a hypothetical math grad student was pretty comfortable with first-year real variables and algebra, and had even studied some other things (algebraic geometry, Riemannian geometry, complex analysis, algebraic topology, algebraic number theory), but had miraculously never taken a differential equations c... | Sean | 19,350 | <p>Probably not quite right but a GTM book you might be interested in is Olver's book "Applications of Lie Groups to Differential Equations" given your background.</p>
|
81,221 | <p>Suppose that a hypothetical math grad student was pretty comfortable with first-year real variables and algebra, and had even studied some other things (algebraic geometry, Riemannian geometry, complex analysis, algebraic topology, algebraic number theory), but had miraculously never taken a differential equations c... | just-learning | 11,216 | <p>If you happen to like Arnold's ODE book mentioned by Julien Puydt it could also be a good idea to look at his more advanced book <a href="https://doi.org/10.1007/978-1-4612-1037-5" rel="nofollow noreferrer">Geometrical methods in the theory of ordinary differential equations</a>. </p>
|
1,874,555 | <p>If $4x/3y = 7/2$, what is the value of $y/x$?</p>
<p>This is a multiple choice question, and the choices are as follows:</p>
<p>A. $3/14$ </p>
<p>B. $8/21$</p>
<p>C. $21/8$</p>
<p>D. $14/3$</p>
<p>I started off answering this by cross multiplying it down to $8x=21y$</p>
<p>From there, $x=21y/8$ and $y=8x/21$ ... | barak manos | 131,263 | <p>Take the equation:</p>
<p>$8x=21y$</p>
<p>Divide each side by $x$:</p>
<p>$8=\frac{21y}{x}$</p>
<p>Divide each side by $21$:</p>
<p>$\frac{8}{21}=\frac{y}{x}$</p>
|
1,013,692 | <p>I want to determine which group $(\mathbb{Z}/24\mathbb{Z})^{*}$ is isomorphic to.</p>
<p>$\mathbb{Z}/24\mathbb{Z}$ contains the 24 residue classes $z + 24\mathbb{Z}$ of the division mod 24. For brevity, I will identify them with $z$, so $\mathbb{Z}/24\mathbb{Z} = \{ 0, 1, ..., 23\}$. For $(\mathbb{Z}/24\mathbb{Z})^... | rogerl | 27,542 | <p>First, the fact that $\left(\mathbb{Z}/24\mathbb{Z}\right)^\times$ contains exactly the primes is an accident of the number $24$. For example, $(\mathbb{Z}/32\mathbb{Z})^\times$ contains lots of non-primes ($9$, $15$ for example). The elements of $(\mathbb{Z}/n\mathbb{Z})^\times$ are exactly the integers less than $... |
62,771 | <p>I was just going through a past exam paper for my intro graphs module and the following question came up, and I can't find any notes on it:</p>
<p>Let G = (V,E) be a simple graph. Show that:</p>
<p>$2|E| \leq |V|^2 - |V|$</p>
<p>any ideas?</p>
| Adam Saltz | 14,626 | <p>Equality holds for complete graphs. Since $G$ is simple, it's the subgraph of some complete graph $K_n$. So $|E(G)| \leq |E(K_n)|$ and the inequality follows.</p>
|
4,250,632 | <p>Calculating <span class="math-container">$81^{3/2}$</span>, I got <span class="math-container">$729$</span> (not saying it is correct, but I am trying :) ). Would <span class="math-container">$-81^{3/2}$</span> just be the opposite (<span class="math-container">$-729$</span>) and does it make a difference if <span ... | ndhanson3 | 808,617 | <p>We can think of the <span class="math-container">$-$</span> sign as the following words: "the opposite of".</p>
<p>In this case, <span class="math-container">$-1$</span> is the opposite of <span class="math-container">$1$</span>, which just means it's the number that, when added to <span class="math-contai... |
3,422,095 | <p>I have been playing with Maclaurin series lately, I have been able to come across this:</p>
<p><span class="math-container">$\dfrac{1}{1+x}=1-x+x^2-x^3+x^4-x^5...$</span></p>
<p><span class="math-container">$\dfrac{1}{(1+x)^2}=1-2x+3x^2-4x^3+5x^4-6x^5+7x^6...$</span></p>
<p>I found out by accident that:</p>
<p><... | epi163sqrt | 132,007 | <p>We are looking for polynomials <span class="math-container">$p_k(x)$</span> with
<span class="math-container">\begin{align*}
\frac{p_k(x)}{(1+x)^{k+1}}=\sum_{j=0}^\infty (-1)^j(j+1)^kx^j\qquad\qquad k\geq 0
\end{align*}</span></p>
<blockquote>
<p>We can find <span class="math-container">$p_k(x)$</span> as follows:
<... |
69,471 | <p>Suppose I am looking at $GL(4,K)$ acting on a cubic form in say four variables $x,y,z,w$ over $K$ via the usual induced action on a polynomial. Does anyone know what is/where I can find how to compute the ring of invariants? The case of personal interest is when $K$ is a finite field but the the answer over $\mathbb... | David Wehlau | 16,684 | <p>Don't be misled into thinking that the answer over $\Bbb C$ tells you very much about the answer over your finite field $K$. The space of cubic forms in 4 variables is 20 dimensional. The group $GL(4,K)$ is a finite group and so the ring of invariants you seek has Krull dimension 20. In particular, it has at leas... |
2,943,999 | <p>If <span class="math-container">$a,b,c \in \mathbb{R+, }$</span> Then Prove that <span class="math-container">$$\frac{a^2}{3^3}+\frac{b^2}{4^3}+\frac{c^2}{5^3} \ge \frac{(a+b+c)^2}{6^3}$$</span></p>
<p>My try:</p>
<p>Consider <span class="math-container">$$P=\frac{a}{3\sqrt{3}}+\frac{b}{4\sqrt{4}}+\frac{c}{5\sqrt{... | herb steinberg | 501,262 | <p>If <span class="math-container">$X_n$</span> and <span class="math-container">$Y_n$</span> have the same density function i.e. <span class="math-container">$(f_n=g_n)$</span> then <span class="math-container">$\int|f_n-g_n|=0$</span> for all
<span class="math-container">$n$</span>. Your condition that the random va... |
4,204,053 | <p>For any <span class="math-container">$t>0$</span> suppose that <span class="math-container">$f_t$</span> is a continuous function on <span class="math-container">$\mathbb{R}$</span> and uniformly bounded in <span class="math-container">$t$</span> : <span class="math-container">$\|f_t\|_\infty \leq C$</span>. Supp... | Medo | 496,598 | <p>This is a well-known theorem:
Let <span class="math-container">$1\leq p\leq \infty$</span>. If <span class="math-container">${h_n}$</span> converges in <span class="math-container">$L^p$</span> norm to <span class="math-container">$h$</span>, then
there exists a subsequence <span class="math-container">$h_{n_k}$</s... |
104,132 | <p>$$f''(x) \thickapprox\dfrac{1}{2h^2}[f(x+2h) - 2f(x) + f(x - 2h)]$$</p>
<p>I'm supposed to be deriving the above formula and establish an error formula in using them.</p>
<p>This is one of a series of problems like this, and I'm not quite too sure on how to get started on this. (This is in a chapter of Estimating ... | Ehsan M. Kermani | 8,346 | <p>Hint: $f''(x) \simeq \dfrac{f'(x+h)-f'(x-h)}{2h}.$ Try to replace $f'$ by $f$ in the definition.</p>
|
2,821,413 | <p>$$f'(x) = 2\frac{x^{1/3}-1}{x^{1/3}}$$</p>
<p>critical numbers: x = 1,0</p>
<p>What does it mean by this function is continuous at zero, but not differentiable at zero.</p>
| hesim | 469,664 | <p>The function is $f(x)=2x-\frac{6}{2}x^{\frac{2}{3}}+c$ for some constant $c$. So it is continuous at $0$ and have a derivation equals $f'(x)=2(\frac{x^{\frac{1}{3}}-1}{x^{\frac{1}{3}}})$. But $f'(x)$ is not defined at zero so $f(x)$ is not differentiable at zero.</p>
|
43,231 | <p>I'm trying to see why my textbook's solution is correct and mine isn't.</p>
<p>"Find an expression in terms of $x$ and $y$ for $\displaystyle \frac{dy}{dx}$, given that $x^2+6x-8y+5y^2=13$</p>
<p>First, the textbook's solution, which I understand and agree with fully:
<img src="https://i.stack.imgur.com/uyDlo.png... | Raeder | 8,246 | <p>Your answers agree. Note that:
$$\frac{-x-3}{5y-4} = \frac{-(x+3)}{-(-5y+4)} = \frac{x+3}{4-5y}.$$</p>
|
827,467 | <p>How do you prove that $n^5 \equiv n\pmod {10}$ Hint given was- Fermat little theorem. Kindly help me out. This is applicable to all positive integers $n$</p>
| Deepak | 151,732 | <p>By Fermat's Little Theorem, $n^5 \equiv n \pmod 5$</p>
<p>Also $n \equiv 0 \ or \ 1 \pmod 2 \implies n^5 \equiv n \pmod 2$</p>
<p>Chinese Remainder Theorem guarantees the presence of a solution $\pmod {10}$ as $(2,5) = 1$</p>
<p>From the second congruence, $n^5 = 2k + n$</p>
<p>Substitute into the first congruen... |
2,043,418 | <p>Suppose $1 \leq p < \infty$. Let $f: \mathbb{R} \to \mathbb{R}$ be continuous and has compact support and let $t \in \mathbb{R}$. Define $f_t(x) = f(x-t).$</p>
<p>Prove: $\lim_{t \to 0} ||f-f_t||_p = 0$.</p>
<p>Edit: I already got the answer. I'll post it as a formal answer soon!</p>
| kayak | 348,589 | <p>Let's use dominated convergence theorem.</p>
<p>Clearly, $|f-f_{t}|$ is bounded by $2M$ if we let $|f|\leq M$ and of course $2M$ is in $L^p$.
And $f-f_{t}$ converges to 0 with respect to $t$.
So the convergence theorem implies that $\|f-f_{t}\|_{p}$ goes to 0.</p>
|
261,144 | <p>Consider the following decomposition list of <span class="math-container">$\{a1,a2,a3,a4\}$</span> into three parts as <span class="math-container">$t$</span>,</p>
<blockquote>
<p>t={{{a1, a2}, {a3}, {a4}}, {{a1}, {a2, a3}, {a4}}, {{a1}, {a2}, {a3,
a4}}, {{a1, a2}, {a4}, {a3}}, {{a1}, {a2,
a4}, {a3}}, {{a1}, {a2}, {... | Nasser | 70 | <blockquote>
<p>In this case the list would be just list2={a,c,Xi,0}</p>
</blockquote>
<p>For this example you show, this should it do it. But I do not know if this will fail or not for other examples. If you can provide more examples, it will help test it. These parsing things can be tricky depending on input</p>
<pre... |
261,144 | <p>Consider the following decomposition list of <span class="math-container">$\{a1,a2,a3,a4\}$</span> into three parts as <span class="math-container">$t$</span>,</p>
<blockquote>
<p>t={{{a1, a2}, {a3}, {a4}}, {{a1}, {a2, a3}, {a4}}, {{a1}, {a2}, {a3,
a4}}, {{a1, a2}, {a4}, {a3}}, {{a1}, {a2,
a4}, {a3}}, {{a1}, {a2}, {... | kglr | 125 | <p>This looks like a case for the (often-overlooked) two-argument form of <a href="https://reference.wolfram.com/language/ref/First.html" rel="nofollow noreferrer"><code>First</code></a>:</p>
<blockquote>
<p><a href="https://i.stack.imgur.com/hUdBs.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/hUdB... |
3,135,085 | <p><span class="math-container">$\lim_{x\to 2} {x^2 - 4\over x^3 - 4x^2 +4x}$</span></p>
<p>I used L'Hospital's rule twice on this, and got a solution, but my textbook says it's an indeterminate form. Is using L'Hospital's rule twice wrong, and if yes, why so?</p>
| Jossie Calderon | 346,651 | <p>L'hopital rule only applies if the limit is indeterminate.</p>
<p>Here it is simply 0/(-4) = 0.</p>
|
74,036 | <p>I have a 3D model of a heart in Mathematica and I'm trying to create a plane so that the open surface (as seen in the image below) is cut off so that the heart can have a solid, level surface. How can I combine this plane with my 3D contour plot?</p>
<pre><code>heart = (2 x^3 + y^2 + z^2 - 1)^3 - (1/10) x^2 z^3 - ... | ubpdqn | 1,997 | <p>You can also use <code>RegionPlot3D</code>:</p>
<pre><code>RegionPlot3D[
reg = (2 x^3 + y^2 + z^2 - 1)^3 - (1/10) x^2 z^3 - y^2 z^3 <= 0 &&
x >= 0, {x, -2, 2}, {y, -2, 2}, {z, -2, 2}, Mesh -> None,
Boxed -> False, Axes -> None, PlotPoints -> 40, PlotStyle -> Red,
Background -&g... |
1,620,540 | <p>In topological space, does first countable+ separable imply second countable? If not, any counterexample?</p>
| Brian M. Scott | 12,042 | <p>It’s false in general. A simple counterexample is the <a href="https://en.wikipedia.org/wiki/Lower_limit_topology" rel="noreferrer">Sorgenfrey line</a>, also known as $\Bbb R$ with the lower limit topology: $\Bbb Q$ is still a dense set, and each point $x$ has a countable local base of sets of the form $\left[x,x+\f... |
198,204 | <p>Is complex valued function like $y(t) = t^2 + i\cdot t^2$ a periodic function?</p>
| Erick Wong | 30,402 | <p>No non-constant polynomial $p(x)$ can ever be periodic. If it were, there would be infinitely many solutions to $p(x)-p(0) = 0$.</p>
|
2,393,625 | <p>Any subgroup of order $2$ will be cyclic subgroup and so will be generated by single element of order $2$ in $S_4$, so to count number of subgroups of order $2$ we need to count number of elements of order $2$ in $S_4$, I tried counting them but I got answer $8$ but is $9$ actually.</p>
| amWhy | 9,003 | <p>The permutations (elements) of order two in $S_4$ are the following:</p>
<p>$$\underbrace{(12), (13), (14), (23), (2 4), (34)}_{6 \;\;\text{two-cycles}}, \underbrace{(12)(34), (13)(24), (14)(23)}_{3\;\; \text{products of disjoint two-cycles.}}$$</p>
<p>As Fransesco mentions in a comment above: We see "six transpo... |
489,848 | <p>So, I have the following points: $\left( \begin{matrix} 5 \\ 0 \\ 0 \end{matrix} \right), \left( \begin{matrix} 0 \\ 4 \\ -1 \end{matrix} \right), \left( \begin{matrix} -4 \\ 4 \\ 3 \end{matrix} \right)$ and I need to find the equation of the circle passing through them.</p>
<p>Here is how I solved it, but it wa... | GivAlz | 46,458 | <p>I noticed the points are linearly indipendent so I can use them as a base. With such a base the equation I need is $X_1^2+X_2^2+X_3^2=1$. In matrix form the equations is $X^t I_3 X = 1$ where $X=(X_1,X_2,X_3)$.</p>
<p>We now indicate with $Y=(Y_1,Y_2,Y_3)$ the coordinates in the canonical base. Let's call $P=\left(... |
34,600 | <p>Searching by <a href="/questions/tagged/closest" class="post-tag" title="show questions tagged 'closest'" rel="tag">closest</a> and <a href="/questions/tagged/position" class="post-tag" title="show questions tagged 'position'" rel="tag">position</a> I wasn't able to find an answer—but found <a ... | Dr. belisarius | 193 | <pre><code>SeedRandom[42];
haystack = RandomReal[1, 6300];
AbsoluteTiming[
f = Nearest[haystack -> Range@Length@haystack];
{f[.3, 1], haystack[[f[.3, 1]]]}]
(*
-> {0.015625, {{3123}, {0.300033}}}
*)
</code></pre>
<p>Of course in this case most of the time is expended calculating the nearest function. If you... |
1,445,561 | <p>Let us say we have the indicator function $\chi_{\{|x|\leq 1\}}$ in $\mathbb{R}^2$.</p>
<p>How can I write out the weak derivative of this indicator function?</p>
<p>Is it $\delta_{|x|=1}$? Or it should be vector valued measure like $\bigg(\frac{\partial}{\partial x_1} \chi_{{|x|\leq 1}},\frac{\partial}{\partial x... | Community | -1 | <p>The weak gradient of the characteristic function of a domain $\Omega$ with a smooth boundary is the vector-valued measure $\nu(x)d\sigma(x)$ where $\nu(x)$ is the outward unit normal at $x\in\partial\Omega$ and $d\sigma$ is the surface measure. </p>
<p>Why so? By the divergence theorem. Recall that by definition, t... |
67,043 | <p>Hi,</p>
<p>I'm trying to evaluate the following integral:</p>
<p>$\int_{-\infty}^\infty \phi(x)\Phi(a+bx)^2dx$</p>
<p>where Phi is the cdf of a std Normal random variable, and phi is the pdf $(1\sqrt(2pi))exp(-x^2\2)$.</p>
<p>I have that $\int_{-\infty}^\infty \phi(x)\Phi(a+bx)dx = Phi(\frac{a}{\sqrt(1+b^2)})$, ... | John D. Cook | 136 | <p>If a = 0 and b = 1, Mathematica evaluates the integral to 1/3 but otherwise it doesn't make any progress. Perhaps the integral could be evaluated using contour integration techniques.</p>
|
67,043 | <p>Hi,</p>
<p>I'm trying to evaluate the following integral:</p>
<p>$\int_{-\infty}^\infty \phi(x)\Phi(a+bx)^2dx$</p>
<p>where Phi is the cdf of a std Normal random variable, and phi is the pdf $(1\sqrt(2pi))exp(-x^2\2)$.</p>
<p>I have that $\int_{-\infty}^\infty \phi(x)\Phi(a+bx)dx = Phi(\frac{a}{\sqrt(1+b^2)})$, ... | Robert Israel | 13,650 | <p>The equation for $\int_{-\infty}^\infty \phi(x) \Phi(a+b x) \ dx$ comes from interpreting this as $P(X > Y)$ where $X \sim N(0,1)$ and $Y \sim N(-a/b, 1/b^2)$
are independent, so that $X - Y \sim N(a/b, 1 + 1/b^2)$.
Similarly $\int_{-\infty}^\infty \phi(x) \Phi(a+bx)^2 \ dx$ would be $P(X > \max(Y,Z))$ where $... |
1,836,538 | <p>In the figure, the ratio of AD to DC is 3 to 2. If area of $\Delta ABC$ is 40 $cm ^ {2}$ , what is the area of $\Delta BDC $ </p>
<p><a href="https://i.stack.imgur.com/wa1bW.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/wa1bW.png" alt="Note - Image NOT drawn to scale"></a></p>
| awllower | 6,792 | <p><strong>Hint:</strong><br>
The area of a triangle is equal to the height times the base. Now the two triangles $\Delta ABD$ and $\Delta BDC$ have the same height, so their areas are proportional to their bases. </p>
<p>Hope this helps.</p>
|
460,587 | <p>Two cards are drawn without replacement from an ordinary deck, find the probability that the second is a red card, given the first is a red card.</p>
<p>P (2nd Red Card / 1st Red Card) = 13/52 * 12 * 51 = 1/17 - is this correct?</p>
| bobsfm | 603,285 | <p>Can the alternate form of conditional probability be used here i.e.</p>
<p><span class="math-container">$$P(R_2 | R_1) = \frac{n(R_1 \bigcap R_2)}{n(R_1)}$$</span> </p>
<p>We've promoted this method since the product rule hasn't been taught yet.</p>
<p>thanks</p>
|
4,306,339 | <p>(I have already prove that <span class="math-container">$\mathbb{Q}(\sqrt{3},i\sqrt{5})=\mathbb{Q}(\sqrt{3}+i\sqrt{5})$</span> in case is useful); now I am asked to prove that if <span class="math-container">$v\in \mathbb{Q}(\sqrt{3},i\sqrt{5})$</span> has the property that every image of <span class="math-container... | David | 651,991 | <p>You can easily prove that <span class="math-container">$A:=\{(x,y) \in \mathbb{R}^2 | y\neq 1/x\}$</span> is open:</p>
<p>For a given <span class="math-container">$(x,y) \in A$</span>, let <span class="math-container">$d:=|y-1/x|$</span>, then the ball of center <span class="math-container">$(x,y)$</span> and radius... |
343,894 | <p>I've been helping my siblings with their GCSE and A Level maths and I've come across a question where they have just taken the positive square root. It's a pure maths question and there's no (obvious) reason to ignore the negative square root.</p>
<p>I always thought that the square root always gave two values, a p... | Jim | 56,747 | <p>There are always two numbers that square to a given number. By convention the symbol $\sqrt{x}$ represents the positive of the two numbers that square to $x$. The other number is given by putting in the negative sign yourself: $-\sqrt{x}$.</p>
<p>This is why, when manipulating equations you always put in a $\pm$ ... |
3,294,564 | <blockquote>
<p>There is only one real values of <span class="math-container">$k$</span> for which the quadratic equation <span class="math-container">$kx^2+(k+3)x+k-3=0$</span> has <span class="math-container">$2$</span> positive integer roots. Then the product of these two solutions is</p>
</blockquote>
<p>What i ... | Allawonder | 145,126 | <p><em>There is no such <span class="math-container">$k.$</span></em></p>
<hr>
<p>Since the product we're looking for is <span class="math-container">$$\frac {k-3}{k},$$</span> first we must have <span class="math-container">$k\ne 0.$</span> Also, since the roots are positive integers, it follows that their product i... |
2,223,267 | <p>For<br>
$$e^{-j\pi n}$$</p>
<p>How does this become
$$(-1)^n$$</p>
<p>or is it actually
$$(-1)^{-n}$$
I have checked on calculator and values are all the same when the same n value is used</p>
| Mark Pineau | 432,691 | <p>Consider the power series of $e^x$, that is:</p>
<p>$$e^x=\sum_{n=0}^{\infty}\frac{x^n}{n!}=1+\frac x{1!}+\frac{x^2}{2!} +\frac{x^3}{3!}+\frac{x^4}{4!}+\ ...$$</p>
<p>Now consider Euler's constant $e$ raised to the power $ix$, such that $i:=\sqrt{-1}$</p>
<p>$$e^{ix}=\sum_{n=0}^{\infty}\frac{(xi)^n}{n!}=1+ix+\fra... |
206,421 | <p>If $4 \tan(\alpha - \beta) = 3 \tan \alpha $, then prove that
$$\tan \beta = \frac{\sin(2 \alpha)}{7 + \cos(2 \alpha)}$$</p>
<p>This is not homework and I've tried everything so I would just like a straight answer thank you in advance. </p>
| Jonathan | 37,832 | <p>$$4\sin(\alpha-\beta)=3\tan\alpha\cos(\alpha-\beta)$$
so
$$4\sin\alpha\cos\beta-4\cos\alpha\sin\beta=3\tan\alpha\cos\alpha\cos\beta+3\tan\alpha\sin\alpha\sin\beta.$$
Divide through by $\cos\beta$ to find
$$4\sin\alpha-4\cos\alpha\tan\beta=3\tan\alpha\cos\alpha+3\tan\alpha\sin\alpha\tan\beta,$$
and solve for $\tan\be... |
3,163,067 | <p><strong>Definition 1</strong> (Formal Language). A <em>language</em> <span class="math-container">$L$</span> over an <em>alphabet</em> <span class="math-container">$\Sigma$</span> (any nonempty finite set) is a subset of the set of all finite sequences of elements of <span class="math-container">$\Sigma$</span>, i.e... | Kavi Rama Murthy | 142,385 | <p>Let <span class="math-container">$K=\{x,x_1,x_2,\cdots\}$</span>. Then <span class="math-container">$K$</span> is a compact set and <span class="math-container">$f$</span> is continuous, hence uniformly continuous on <span class="math-container">$K \times [0,1]$</span>. Now just write down the definition of uniform ... |
3,625,000 | <p>I have read that an n-dimensional cube has <span class="math-container">$2^n$</span> vertices, but I can't find a proof for that. What is the explanation to why that's true?</p>
| Costa Eladogra | 772,634 | <p>This is best proved by induction. Let <span class="math-container">$\gamma_n$</span> denote the general hypercube. The hypercube <span class="math-container">$\gamma_{n+1}$</span> is <span class="math-container">$\gamma_n \times \gamma_1$</span>, where <span class="math-container">$\gamma_1$</span> is the line segme... |
1,863 | <p>Problem / background: consider the following code snippet.</p>
<pre><code>pnt[fig_, n_] := {fig[[1, n]], fig[[2, n]]}
hor := {.025, .1, .25, .4, .475, .525, .6, .75, .9, .975}
ver := {.05, .20, .80, .95}
fig4 := Transpose[{
{hor[[1]], ver[[2]], 1}, {hor[[2]], ver[[1]], 1}, {hor[[4]],
ver[[1]], 1}, {hor[[5]]... | Mr.Wizard | 121 | <p>I presume you are looking for <a href="http://reference.wolfram.com/mathematica/ref/FilledCurve.html" rel="nofollow noreferrer">FilledCurve</a>.</p>
<p><img src="https://i.stack.imgur.com/hmjFV.gif" alt="enter image description here"></p>
<p><img src="https://i.stack.imgur.com/kBg09.gif" alt="enter image descripti... |
215,774 | <p>Let $G$ be the set of bijections $\mathbb{R} \to \mathbb{R}$ which preserve the distance between pairs of points, and send integers to integers. Then $G$ is a group under composition of functions. The following two elements are obviously in $G$: the function $t$ (translation) where $t(x)=x+1$ for each $x \in \mathbb... | PAD | 27,304 | <p>This is not an answer but a different construction of this group (which shows that it is a Coxeter group).</p>
<p>Let $M=\{1, -1 \}$. Then $M$ acts on $\mathbb{Z}$ by $(-1) x=-x$. </p>
<p>$D_{\infty}$ is the semidirect product of $M$ with $\mathbb{Z}$. The group law is given by</p>
<p>$$(\epsilon, x ) (\epsilon... |
4,204,506 | <p>We can write <span class="math-container">$4x^2+8x+5$</span> in the form <span class="math-container">$a(x+b)^2+c$</span> as <span class="math-container">$4(x+1)^2+1$</span>. However, the question I am doing asks me to write it in the form <span class="math-container">$(ax+b)^2+c$</span>. How do I change it to that ... | coffeemath | 483,139 | <p>If one has already found <span class="math-container">$a(x+b)^2+c$</span> form, then <em>provided</em> <span class="math-container">$a>0$</span> one can start by re-writing <span class="math-container">$a$</span> as <span class="math-container">$[\sqrt{a}]^2$</span> and use that <span class="math-container">$u^2v... |
1,367,966 | <p>How do you find the exact values of the following without using a calculator?</p>
<p>$$\tan(105^\circ) \qquad \tan(11\pi/12)$$</p>
| wythagoras | 236,048 | <p><strong>Hint:</strong> Use the sum-angle formula for $\tan$:</p>
<p>$$\tan(\alpha+\beta)=\frac{\tan \alpha + \tan \beta}{1-\tan \alpha\tan \beta}$$</p>
<p>for some nice values of $\alpha,\beta$ of which you know the tangent. </p>
|
1,367,966 | <p>How do you find the exact values of the following without using a calculator?</p>
<p>$$\tan(105^\circ) \qquad \tan(11\pi/12)$$</p>
| John_dydx | 82,134 | <p>Hint: $$ \tan(105^\circ) = \tan(60^\circ+45^\circ) $$</p>
<p>$$ \tan \theta = \frac{\sin\theta}{\cos\theta}$$
$$ \sin(A+B) = \sin A\cos B + \cos A \sin B $$
$$\cos(A+B) = \cos A\cos B - \sin A \sin B$$</p>
<p>Alternatively, you can use the double angle formula for $\tan (A + B)$ as wythagoras suggested.</p>
|
1,830,799 | <p>1There are $\frac{21!}{2!3!} = 120$ total positions (disregarding order within same colour). I imagine labelling the people Y (yellow) and NY (not yellow), so I imagine I have $4$ copies of the letter Y and $5$ of NY. So I draw out $9$ slots and want to arrange so that at least $2$ Y are together.
I get $30$ arrang... | Noble Mushtak | 307,483 | <p>You are thinking about this problem in a very complicated way which will probably lead to missed cases and the wrong answer. I think there is a better way to approach this problem.</p>
<p>There are $1260$ arrangements in all. Therefore, we can find the number of arrangements such that all of the $Y$s are separated ... |
355,262 | <p>Is there a closed-form expression for the sum $\sum_{k=0}^n\binom{n}kk^p$ given positive integers $n,\,p$? Earlier I thought of this series but failed to figure out a closed-form expression in $n,\,p$ (other than the trivial case $p=0$).</p>
<p>$$p=0\colon\,\sum_{k=0}^n\binom{n}kk^0=2^n$$</p>
<p>I know that $\sum_... | Sungjin Kim | 67,070 | <p>Let
$$
f(x)=(e^x+1)^n=\sum_{k=0}^n \binom{n}{k}e^{kx}.
$$</p>
<p>Then
$$
\left(\frac{d}{dx}\right)^p f(x)=\sum_{k=0}^n\binom{n}{k}k^pe^{kx}.$$</p>
<p>Plug in $x=0$. </p>
|
214,007 | <p>I have an experimental data set: </p>
<pre><code>data1 = {{71.6`, 0.41`}, {27.2`, 4.96`}, {59.3`, 0.18`}, {46.`,2.72`},
{42.2`, 1.06`}, {89.1`, 3.75`}, {88.6`, 1.9`}, {62.3`,1.8`},
{35.5`,1.84`}}
</code></pre>
<p>In order to eliminate unrealistic data points step by step automatically, I fit and ... | kglr | 125 | <h3>LinearModelFit - Influence Measures</h3>
<p>You can use <a href="https://reference.wolfram.com/language/ref/LinearModelFit.html" rel="nofollow noreferrer"><code>LinearModelFit</code></a> and make use of the properties <code>"CookDistances"</code>, <code>"FitDifferences"</code> or <code>"SingleDeletionVariances"</c... |
1,040,505 | <p>Apparently,
$$(1-\cot 37^\circ)(1-\cot 8^\circ)=2.00000000000000000\cdots$$<br>
Since it is a $2.0000000000\cdots$ instead of $2$, it isn't exactly $2$.<br>
Why is that?</p>
| Paul | 17,980 | <p>$2.\dot 0$ is exacyly equal to 2</p>
|
108,404 | <p>Let $H$ be a semisimple Hopf algebra. One of the Kaplansky's conjectures states that the dimension of any irreducible $H$-module divides the dimension of $H$. </p>
<p>In which cases the conjecture is known to be true?</p>
| Leandro Vendramin | 17,845 | <p>Yorck Sommerhäuser has a very nice <a href="http://www.mathematik.uni-muenchen.de/~sommerh/Publikationen/KaplConjwww/KaplConjRev.ps">survey</a> about Kaplansky's conjectures. Section 6 is devoted to Kaplansky's 6th conjecture. </p>
<p>In Sommerhäuser's survey it is mentioned that Richmond and Nichols proved that
th... |
156,511 | <p>Suppose I have a simple function like $\cos(kx)$, and I am plotting a number of such functions with different values for $k$ as in:</p>
<pre><code>Plot[Table[Cos[k x], {k, 1, 4, 1}], {x, 0, π}]
</code></pre>
<p>How can I color them depending on the $k$ parameter without setting it manually myself, as in:</p>
<pre... | Michael E2 | 4,999 | <p>This seems to be the desired form:</p>
<pre><code>Plot[
Evaluate[Table[Style[Cos[k x], Blend[{Blue, Red}, k/4]], {k, 1, 4, 1}]],
{x, 0, π}]
</code></pre>
<p><img src="https://i.stack.imgur.com/Ud6EJ.png" alt="Mathematica graphics"></p>
|
1,032,331 | <p>I am trying to find the limit </p>
<p>$\large\lim_{n \to \infty} (n^{\frac{1}{2n}})$</p>
<p>WolframAlpha says that I can transform as follows</p>
<p>$\large\lim_{n \to \infty} (n^{\frac{1}{2n}})=e^{\lim_{n \to\infty} \frac{ln(n)}{2n}}$</p>
<p>However, I do not understand where $n$ in $ln (n)$ comes from. Could a... | Martin Brandenburg | 1,650 | <p>Your proof that $M \cap N$ is normal is not complete since you only show that $M \cap N$ is a subgroup. There are some typos in the proof that $f$ is a homomorphism.</p>
<p>It is a very good idea to use the Isomorphism Theorem, but for this you have to show that the kernel of $f$ is $M \cap N$. And then you can als... |
4,346,363 | <p>I'm interested in the equation <span class="math-container">$$AX=B$$</span> where <span class="math-container">$A, X,$</span> and <span class="math-container">$B$</span> are rectangular matrices with suitable dimensions, for example <span class="math-container">$m\times n$</span>, <span class="math-container">$n\tim... | Hyperplane | 99,220 | <p>You can think of <span class="math-container">$AX=B$</span> as a collection of <span class="math-container">$p$</span> equations <span class="math-container">$Ax_1 = b_1$</span>, <span class="math-container">$Ax_2 = b_2$</span>, ..., <span class="math-container">$Ax_p = b_p$</span> where <span class="math-container"... |
4,186,977 | <p>The challenge is to prove <span class="math-container">$$\left(1-\dfrac{a}{b}\right)\left(1+\dfrac{c}{d}\right)=4.$$</span></p>
<p><a href="https://i.stack.imgur.com/mTFF9.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/mTFF9.jpg" alt="enter image description here" /></a></p>
<p>Apart from the Pyt... | Math Lover | 801,574 | <p>While we wait for an easier method, my answer is based on system of equations using Pythagoras and solving them by hand.</p>
<p><a href="https://i.stack.imgur.com/Sjf4k.png" rel="noreferrer"><img src="https://i.stack.imgur.com/Sjf4k.png" alt="enter image description here" /></a></p>
<p>We will use <span class="math-... |
3,934,819 | <p>I am wondering if we can find a linear transformation matrix <span class="math-container">$A$</span> of size <span class="math-container">$3\times 3$</span> over the field of two elements <span class="math-container">$\mathbb{Z}_2$</span> i.e. a matrix <span class="math-container">$A$</span> of zeros and ones s.t.</... | Mo145 | 857,217 | <p>A linear transformation can be defined by its values on the basis. So consider the standard basis <span class="math-container">$e_1,e_2,e_3$</span> of the <span class="math-container">$3$</span> dimensional vector space over the integers modulo <span class="math-container">$2$</span>. Then your <span class="math-con... |
2,593,392 | <p>The corollary is given below:</p>
<blockquote>
<p><a href="https://i.stack.imgur.com/7R4vx.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/7R4vx.png" alt="enter image description here"></a>
<a href="https://i.stack.imgur.com/cNDji.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur... | Henno Brandsma | 4,280 | <p>The corollary says that if $A \in \mathcal{O}$, (i.e. $A$ is invertible, so that $\|A^{-1}\| > 0$), then the ball with radius $r = \frac{1}{\|A^{-1}\|}$ with centre $A$ lies inside $\mathcal{O}$ (all $B$ with $d(A,B) =\|A-B\| < r$ are all invertible too). The exact formula is irrelevant for this question, th... |
1,736,405 | <p>I was told to consider the degrees but I'm not sure how the degrees of the polynomial so can help me here. </p>
| DonAntonio | 31,254 | <p>We have </p>
<p>$$\sqrt p=\left(\sqrt[6]p\right)^3\;,\;\;\sqrt[3]p=\left(\sqrt[6]p\right)^2\implies \Bbb Q(\sqrt p,\,\sqrt[3]p)\subset\Bbb Q(\sqrt[6]p)$$</p>
<p>But on the other hand</p>
<p>$$\frac16=\frac12-\frac13\implies\sqrt[6]p=\frac{\sqrt[2]p}{\sqrt[3]p}\implies\Bbb Q(\sqrt p,\,\sqrt[3]p)\supset\Bbb Q(\sqrt... |
3,098,350 | <p>Recently I was able to find a result to a common definite integral:
<span class="math-container">\begin{equation}
J(n_1, k_1, m_1) = \int_0^{\infty} \frac{x^{k_1}}{\left(x^{n_1} + a_1 \right)^{m_1}}\:dx = \frac{a^{\frac{k_1 + 1}{n_1} - m_1}}{n_1}\Gamma\left(m_1 - \frac{k_1 + 1}{n_1}\right)\Gamma\left(\frac{k_1 + 1}... | Peter | 220,102 | <p>To long for the comment.</p>
<p>Maybe it's too trite, but </p>
<p>In the case <span class="math-container">$n_1=n_2$</span> this integral can be rewrite through the hypergeometric function <span class="math-container">$_2F_1$</span>.
I do not want to pay attention to the coefficient. I this case the integral can b... |
1,025,088 | <p>I have been working on a problem as follows:
Do there exist 100 consecutive natural numbers none of which is prime?
I know that the answer is 'yes', by considering 101!, and noting the sequence 101! + 2, 101! + 3, ... , 101! + 101.</p>
<p>This approach generalises nicely by considering (n+1)!</p>
<p>However, whils... | Robert Israel | 8,508 | <p>All you need is that the density of prime numbers goes to $0$. If there are fewer than $M = \lfloor N/100 \rfloor$ primes $\le N$, then at least one of the $M$ intervals $[1,100], [101, 200], \ldots, [100(M-1)+1, 100 M]$ has no primes.</p>
|
4,343,792 | <p>Suppose that <span class="math-container">$$</span> and <span class="math-container">$ + 2$</span> are both prime numbers.</p>
<p><strong>a) Is the integer between <span class="math-container">$$</span> and <span class="math-container">$ + 2$</span> odd or even? Explain your answer.</strong></p>
<p>All prime numbers... | Tiago Cavalcante | 830,526 | <p><strong>(b)</strong> is straightforward: <span class="math-container">$p + 1$</span> can be written as <span class="math-container">$x ^ 2$</span>, so <span class="math-container">$p \ne \pm 2$</span> otherwise that perfect square couldn't be written as <span class="math-container">$(2n)^2$</span>. As you said <span... |
17,074 | <p>My exams are nearing and I have been posting question on vector calculus which are not answered yet because I'm writing in simple English. Many people have edited my questions. Kindly guide me on how to learn that...</p>
<p>Thanks</p>
| Community | -1 | <p>No, such acronyms should never be used in comments directed at the OP. </p>
<p>If you don't have the time to write a <strong>helpful</strong> comment, leave commenting to someone who does.</p>
|
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