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 |
|---|---|---|---|---|
745,613 | <p>I've been pondering this since yesterday. I</p>
<blockquote>
<p>Is it true that given two irreducible polynomials <span class="math-container">$f(x)$</span> and <span class="math-container">$ g(x)$</span> will <span class="math-container">$f(g(x))$</span> or <span class="math-container">$g(f(x))$</span> be irreducib... | M.darwish | 111,150 | <p>I am extending @Pipicito's answer for the case when both polynomials have degree greater than 1. The answer is also no. For example, consider <span class="math-container">$f(x)=x^2-\frac{4}{3}$</span>; it is irreducible over <span class="math-container">$\mathbb{Q}$</span>. However, <span class="math-container">$f(f... |
1,666,396 | <p>I can show the convergence of the following infinite product and some bounds for it:</p>
<p>$$\prod_{k \geq 2}\sqrt[k]{1+\frac{1}{k}}=\sqrt{1+\frac{1}{2}} \sqrt[3]{1+\frac{1}{3}} \sqrt[4]{1+\frac{1}{4}} \cdots<$$</p>
<p>$$<\left(1+\frac{1}{4} \right)\left(1+\frac{1}{9} \right)\left(1+\frac{1}{16} \right)\cdo... | robjohn | 13,854 | <p>I don't know if a closed form exists, but to get geometric convergence, we can use the following.</p>
<p>Since the product starts at $k=2$, we compute the sum
$$
\begin{align}
\sum_{k=2}^\infty\frac1k\log\left(1+\frac1k\right)
&=\sum_{k=2}^\infty\frac1k\sum_{n=1}^\infty\frac{(-1)^{n-1}}{nk^n}\\
&=\sum_{n=1}... |
4,128,050 | <p><a href="https://i.stack.imgur.com/ybNTh.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/ybNTh.jpg" alt="enter image description here" /></a></p>
<p>I think we should use the corollary to solve this problem.</p>
<p>For the first question, I think we can use the property that <span class="math-con... | J. Darné | 611,408 | <p>Since <span class="math-container">$J$</span> is a free <span class="math-container">$\mathbb Z[G]$</span>-module of rank two (the two generators being <span class="math-container">$(s-1)$</span> and <span class="math-container">$(t-1)$</span>), the tensor product <span class="math-container">$J \otimes_{\mathbb Z[G... |
4,103,366 | <p>I have <span class="math-container">$A \in \mathbb{R}^{q\times n }, B \in \mathbb{R}^{n \times p} $</span> with <span class="math-container">$\text{rank}(A)=q~$</span> and <span class="math-container">$~\text{rank}(B)=p$</span>.</p>
<p>Additionally there is the condition: <span class="math-container">$n\geq p \geq q... | Ben Grossmann | 81,360 | <p>No, the equality does not necessarily hold. Consider the case of <span class="math-container">$n = 3,p=2,q=1$</span>. Take
<span class="math-container">$$
A = \pmatrix{1&0&0},\quad B = \pmatrix{0&0\\1&0\\0&1}.
$$</span>
We have <span class="math-container">$\operatorname{rank}(AB) = 0 < q = 1... |
3,429,489 | <blockquote>
<p>Let <span class="math-container">$\mathcal F = \{f \mid f : \mathbb R \rightarrow \mathbb R\}$</span> and
define relationship <span class="math-container">$R$</span> on <span class="math-container">$\mathcal F$</span> as follows:</p>
<p><span class="math-container">$$R = \{(f,g) \in \mathcal F... | Community | -1 | <p>The only problem that I see is that you have assumed that <span class="math-container">$f^{-1}$</span> has a domain of <span class="math-container">$\mathbb R$</span> in the last part. That is only true if <span class="math-container">$f$</span> is also surjective. It is true that every injective function has a le... |
4,008,488 | <p>While looking for the answer for my question I came across <a href="https://math.stackexchange.com/questions/1840801/why-is-ata-invertible-if-a-has-independent-columns?rq=1">this</a> post. It may be a silly idea, but if <span class="math-container">$A^{t}$</span> has independent rows can I just transpose it and get ... | Ali | 251,307 | <p>For every matrix <span class="math-container">$A$</span> with independent rows, we have <span class="math-container">$det A \neq 0$</span> also <span class="math-container">$det A= det A^t$</span>so both <span class="math-container">$A$</span> and <span class="math-container">$A^t$</span> are invertible. Then <span... |
4,351,990 | <p>I have just finished my undergrad and while I haven't studied much in representation theory I find it a very fascinating subject. My current interest is in differential equations, and I am wondering is there any ongoing research that combines these two areas?</p>
| paul garrett | 12,291 | <p>The wide variation in interpretation of your question should be a convincing indicator that your question is not going to get a useful answer. At extremes, the answer is "obviously yes" or "obviously no", depending on interpretations of many of the words... :)</p>
<p>That is, on one hand, if peop... |
4,722 | <p>Is there a necessary and sufficient condition for the boundary of a planar region to be a finite union of Jordan curves?</p>
| Harald Hanche-Olsen | 802 | <p>I am going to throw caution to the wind and suggest an Answer, based on the previous comments: No, there is no useful characterization of the regions you seek, other than the requirement stated. On the one hand, there is the region $\{(x,y):0\lt x\lt 1, -2\lt y\lt \sin x^{-1}\}$, and on the other, you can make part ... |
4,722 | <p>Is there a necessary and sufficient condition for the boundary of a planar region to be a finite union of Jordan curves?</p>
| Clinton Curry | 1,743 | <p>Local connectivity of the boundary provides much of the topological structure that such a domain would have. If you specify the following two conditions, you have that the boundary of a domain $U$ is a finite union of simple closed curves, I think.</p>
<ol>
<li>$\partial U$ is locally connected.</li>
<li>$\overlin... |
2,368,827 | <p>I would like to know how a piecewise function and its derivative would look like under these circumstances. Suppose that the function is continuous (and also nice like poly, trig etc) but defined differently for points $\le a$ and point $\gt a $</p>
<p>1) The function is differentiable at $a$. Then the derivative w... | Emilio Novati | 187,568 | <p>Consider the function:
$$
y= \begin{cases}
x^2 \quad \mbox{for} \quad x\ge0\\
x^3 \quad \mbox{for} \quad x<0\\
\end{cases}
$$
this function is differentiable at $x=0$ ( and for any $x\in \mathbb{R}$) but its derivative is continuous but not differentiable at $x=0$</p>
|
14,712 | <p>I have matrix <code>in</code> as shown, consisting of real numbers and 0. How can I sort it to become <code>out</code> as shown?</p>
<pre><code>in ={
{0, 0, 3.411, 0, 1.343},
{0, 0, 4.655, 2.555, 3.676},
{0, 3.888, 0, 3.867, 1.666}
};
out ={
{1.343, 3.411, 0, 0, 0},
{2.555, 3.676, 4.655, 0, 0},
... | Rojo | 109 | <pre><code>in = {{0, 0, 3.411, 0, 1.343}, {0, 0, 4.655, 2.555, 3.676}, {0, 3.888,
0, 3.867, 1.666}};
</code></pre>
<p>A possible solution</p>
<pre><code>Sort /@ (in I - Unitize[in]) // Im
</code></pre>
|
14,712 | <p>I have matrix <code>in</code> as shown, consisting of real numbers and 0. How can I sort it to become <code>out</code> as shown?</p>
<pre><code>in ={
{0, 0, 3.411, 0, 1.343},
{0, 0, 4.655, 2.555, 3.676},
{0, 3.888, 0, 3.867, 1.666}
};
out ={
{1.343, 3.411, 0, 0, 0},
{2.555, 3.676, 4.655, 0, 0},
... | Mr.Wizard | 121 | <p>You might use:</p>
<pre><code>SortBy[#, # /. 0 -> {} &] & /@ in
</code></pre>
<p>This works because <code>{}</code> will be placed after atomic elements such as real numbers. If your zeros may not always be precise (head Integer) you may use:</p>
<pre><code>SortBy[#, # /. x_ /; x == 0 -> {} &] ... |
2,262,167 | <p>The question in title has been considered for finite groups $G$ and $H$, but I do not know its situation, how far it is known whether $G$ and $H$ could be isomorphic. I have two simple questions regarding it.</p>
<p><strong>Q.0</strong> If $\mathbb{Z}[G]\cong \mathbb{Z}[H]$ then $|G|$ should be $|H|$; because, $G$ ... | Jeremy Rickard | 88,262 | <p>You're right about the zeroth question.</p>
<p>For the first, a counterexample was constructed by Hertweck in 2001. There are two non-isomorphic groups $G$ and $H$ of order $2^{21}97^{28}$ with $\mathbb{Z}[G]\cong\mathbb{Z}[H]$.</p>
|
2,905,022 | <p>I recently stumbled upon the problem $3\sqrt{x-1}+\sqrt{3x+1}=2$, where I am supposed to solve the equation for x. My problem with this equation though, is that I do not know where to start in order to be able to solve it. Could you please give me a hint (or two) on what I should try first in order to solve this equ... | Rakibul Islam Prince | 551,644 | <p><strong>Hint :</strong></p>
<p>Let,$$x=t^2+1$$
So, $$3t+\sqrt{3t^2+4}=2$$
$$\implies 3t^2+4=(2-3t)^2$$</p>
<p>Now,it's your turn to go on...</p>
<p>The answer should be $$x=1,if~t=0$$</p>
<p>$$and$$ $$x=5,if~t=2$$</p>
|
2,905,022 | <p>I recently stumbled upon the problem $3\sqrt{x-1}+\sqrt{3x+1}=2$, where I am supposed to solve the equation for x. My problem with this equation though, is that I do not know where to start in order to be able to solve it. Could you please give me a hint (or two) on what I should try first in order to solve this equ... | user579102 | 579,102 | <p><span class="math-container">$$3\sqrt{x-1}+\sqrt{3x+1}=2$$</span>
this equation is defined for <span class="math-container">$$x\geq 1$$</span>
<span class="math-container">$$9(x-1)+6\sqrt{(x-1)(3x+1)}+3x+1=4$$</span>
<span class="math-container">$$12x+6\sqrt{(x-1)(3x+1)}=12$$</span>
<span class="math-container">$$2x... |
99,506 | <p>I am trying to show that how the binary expansion of a given positive integer is unique.</p>
<p>According to this link, <a href="http://www.math.fsu.edu/~pkirby/mad2104/SlideShow/s5_3.pdf" rel="nofollow">http://www.math.fsu.edu/~pkirby/mad2104/SlideShow/s5_3.pdf</a>, All I see is that I can recopy theorem 3-1's pro... | N. S. | 9,176 | <p>Assume by contradiction that a number $n$ has two different binary expansions.</p>
<p>Then $n=a_m...a_0=b_k...b_0$.</p>
<p>Let $l$ be the smallest index so that $a_l \neq b_l$. It follows that the binary numbers $a_{l-1}...a_1a_0=b_{l-1}...b_1b_0$. By subtracting these from $n$ we get</p>
<p>$$a_m...a_l00000..0=b... |
2,971,313 | <blockquote>
<p>Prove that every even degree polynomial function <span class="math-container">$f$</span> has maximum or minimum in <span class="math-container">$\mathbb{R}$</span>. (without direct using of derivative and making <span class="math-container">$f'$</span>)</p>
</blockquote>
<p>The problem seems very easy a... | José Carlos Santos | 446,262 | <p>Let <span class="math-container">$f(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$</span>. Let us assume that <span class="math-container">$a_n>0$</span> (the case in which <span class="math-container">$a_n<0$</span> is similar). Then<span class="math-container">\begin{align}\lim_{x\to\pm\infty}f(x)&=\lim_{x\to... |
2,971,313 | <blockquote>
<p>Prove that every even degree polynomial function <span class="math-container">$f$</span> has maximum or minimum in <span class="math-container">$\mathbb{R}$</span>. (without direct using of derivative and making <span class="math-container">$f'$</span>)</p>
</blockquote>
<p>The problem seems very easy a... | G Cab | 317,234 | <p>In an even degree polynomial <span class="math-container">$p_{\,2n}(x) \quad |\; 0<n$</span>, we have that
<span class="math-container">$$
\mathop {\lim }\limits_{x\; \to \, + \infty } p_{\,2n} (x) = \mathop {\lim }\limits_{x\; \to \, - \infty } p_{\,2n} (x) = \pm \infty
$$</span>
and it does not have other <em... |
547,050 | <p>Which trigonometric formulas are used for these problems?
<img src="https://i.stack.imgur.com/TVBCx.png" alt="enter image description here"></p>
| lab bhattacharjee | 33,337 | <p>HINT:</p>
<p>Use $$\sin C+\sin D=2\sin\frac{C+D}2\cos\frac{C-D}2$$</p>
<p>and $$\cos C+\cos D=2\cos\frac{C+D}2\cos\frac{C-D}2$$</p>
|
2,195,197 | <blockquote>
<p>A circle goes through $(5,1)$ and is tangent to $x-2y+6=0$ and $x-2y-4=0$. What is the circle's equation?</p>
</blockquote>
<p>All I know is that the tangents are parallel, which means I can calculate the radius as half the distance between them: $\sqrt5$. So my equation is
$$(x-p)^2+(y-q)^2=5$$
How ... | lab bhattacharjee | 33,337 | <p>As the radius $=$ the perpendicular distance of a tangent from the center.</p>
<p>If $(h,k)$ is the center, radius $r=\dfrac{|h-2k+6|}{\sqrt{1^2+2^2}}=\dfrac{|h-2k-4|}{\sqrt{1^2+2^2}}$</p>
<p>Squaring we get, $$(h-2k+6)^2=(h-2k-4)^2\iff h=2k-1$$</p>
<p>$r=\dfrac{|-1+6|}{\sqrt{1^2+2^2}}=\sqrt5$</p>
<p>Finally $$(... |
360,663 | <p>Let <span class="math-container">$\mathcal{D}$</span> be a triangulated category and a <span class="math-container">$t$</span>-structure <span class="math-container">$(\mathcal{D}^{\leq 0},\mathcal{D}^{\geq 0})$</span> on <span class="math-container">$\mathcal{D}$</span>. The heart of the <span class="math-containe... | Maxime Ramzi | 102,343 | <p>Assume <span class="math-container">$\mathcal D$</span> is a presentable stable <span class="math-container">$\infty$</span>-category with a <span class="math-container">$\mathrm t$</span>-structure (which is accessible and compatible with filtered colimits), and let <span class="math-container">$\mathcal A$</span> ... |
360,663 | <p>Let <span class="math-container">$\mathcal{D}$</span> be a triangulated category and a <span class="math-container">$t$</span>-structure <span class="math-container">$(\mathcal{D}^{\leq 0},\mathcal{D}^{\geq 0})$</span> on <span class="math-container">$\mathcal{D}$</span>. The heart of the <span class="math-containe... | Dan Petersen | 1,310 | <p>I had reason to think about this a few years ago. When <span class="math-container">$\mathcal D$</span> arises as the derived category of an abelian category (with a possibly exotic <span class="math-container">$t$</span>-structure), a construction of a realization functor <span class="math-container">$D^b(A) \to \m... |
633,858 | <p>If G is cyclic group of 24 order, then how many element of 4 order in G?
I can't understand how to find it, step by step. </p>
| DonAntonio | 31,254 | <p>Hints:</p>
<p>1) A cyclic group of order $\;n\;$ has exactly $\;\varphi(n)\;$ generators.</p>
<p>(2) A cyclic group of order $\;n\;$ has exactly one unique subgroup of order $\;d\;$ for <strong>any</strong> divisor $\;d\;$ of $\;n\;$</p>
|
3,298,445 | <p>A random variable is defined by it distribution function. The density function is the derivative of the distribution function. Thus the density function exisst iff the distribution function is absolutely continuous. However, can we construct a distribution function without a density function, except for the finite d... | Masacroso | 173,262 | <p>Of course we can, there are the mixed distributions. </p>
<p>In short: any increasing and right continuous function <span class="math-container">$F$</span> with <span class="math-container">$\lim_{x\to -\infty} F(x)=0$</span> and <span class="math-container">$\lim_{x\to+\infty}F(x)=1$</span> defines a probability d... |
321,230 | <p>Suppose $Z$ is a topological space; and $X$ is dense in $Z$. Then do we have $W(X)= W(Z)$?
Where $W(X)$, $W(Z)$ denote the weight of the $X$ and $Z$ respectively. </p>
<p><strong>What I've tried:</strong> On one hand, $W(X)\le W(Z)$, clearly; On the other hand, for any open set $U$ of $Z$, we have $U\cap X$, an op... | user642796 | 8,348 | <p>The weight of a space does not necessarily equal the weight of a dense subspace. </p>
<p>As an example, note that $\mathbb N$ is clearly second-countable ($w(\mathbb{N}) = \aleph_0$), but its Stone–Čech compactification $\beta \mathbb{N}$ has weight $2^{\aleph_0}$. This can be generalised for the Stone–Čech compa... |
831,472 | <p>I am learning about Karnaugh maps to simplify boolean algebra expressions. I have this:</p>
<p>$$\begin{bmatrix}
& bc & b'c & bc' & b'c' \\
a & 0 & 1 & 1 & 0\\
a' & 1 & 1 & 0 & 1
\end{bmatrix}$$</p>
<p>There are no groups of four, so I am now looking for groups of tw... | thewillix | 156,545 | <p>Karnaugh maps require a particular ordering of the variables different from a normal truth table. Your K-map is ordered like a truth table: bc b'c bc' b'c' (or 11 01 10 00) whereas it has to be ordered such that only one variable changes going from one column (or row) to the next, and it is usually written with 00 o... |
1,617,462 | <p>Is this a line or a plane, I thought it would be a plane where z=0 always so it will be the xy plane.</p>
<p>Also: what will be the normal vector for this if it is a plane?</p>
| David Quinn | 187,299 | <p>In $\mathbb{R}^3$, this represents a plane with normal vector $$\left(\begin{matrix}2\\-1\\0\end{matrix}\right)$$ and is therefore perpendicular to the $z$ plane</p>
|
2,136,937 | <p>Find $$\lim_{z \to \exp(i \pi/3)} \dfrac{z^3+8}{z^4+4z+16}$$</p>
<p>Note that
$$z=\exp(\pi i/3)=\cos(\pi/3)+i\sin(\pi/3)=\dfrac{1}{2}+i\dfrac{\sqrt{3}}{2}$$
$$z^2=\exp(2\pi i/3)=\cos(2\pi/3)+i\sin(2\pi/3)=-\dfrac{1}{2}+i\dfrac{\sqrt{3}}{2}$$
$$z^3=\exp(3\pi i/3)=\cos(\pi)+i\sin(\pi)=1$$
$$z^4=\exp(4\pi i/3)=\cos(4\... | A. Fenzry | 295,901 | <p>I studied Physics and would definetely recommend you Apostol's. It is a good book, rigorous enough and complete, while at the same time keeps a casual layout with a lot of explanative text, which makes it a good transition between High School and college.</p>
<p>As a personal reference I can tell you it was the rec... |
94,440 | <p>In Sean Carroll's <em>Spacetime and Geometry</em>, a formula is given as
$${\nabla _\mu }{\nabla _\sigma }{K^\rho } = {R^\rho }_{\sigma \mu \nu }{K^\nu },$$</p>
<p>where $K^\mu$ is a Killing vector satisfying Killing's equation ${\nabla _\mu }{K_\nu } +{\nabla _\nu }{K_\mu }=0$ and the convention of Riemann curvatu... | AstoundingJB | 93,907 | <p>@C.R. This is my 'simpler proof'; I'm pretty sure it's correct, and simpler than Zhen's one as well.</p>
<p>From the <em>first Bianchi identity</em> [Carroll, (3.132)]
$$R_{\mu\nu\rho\sigma}+R_{\mu\rho\sigma\nu}+R_{\mu\sigma\nu\rho}=0$$
we have that, for every vector $V^\rho$,
$$\nabla_{[\mu}\nabla_\nu V_{\rho]}=\t... |
1,677,868 | <p>The sequence is:</p>
<p>$$a_n = \frac {2^{2n} \cdot1\cdot3\cdot5\cdot...\cdot(2n+1)} {(2n!)\cdot2\cdot4\cdot6\cdot...\cdot(2n)} $$</p>
| parsiad | 64,601 | <p>A direct approach is to try to show that is the same as $a_n=(2n+1)!/(2(n!)^{3})$, which goes to zero as $n\rightarrow\infty$. Alternatively, use the ratio test as suggested in the comments above.</p>
|
2,468,326 | <p>I want to read <a href="https://www.amazon.co.uk/Introduction-Cyclotomic-Fields-Graduate-Mathematics/dp/0387947620" rel="noreferrer">Lawrence Washington's An <em>Introduction to Cyclotomic Fields</em></a>. However, my knowledge of algebraic number theory doesn't extend farther than what's found in <a href="https://w... | paul garrett | 12,291 | <p>I think you'd be happier to have a full introduction to algebraic number theory, like S. Lang's, or equivalent. Then you'll be able to see many of the features of cyclotomic fields as special cases of what would happen more generally, rather than having those special cases appear as novelties.</p>
<p>That is, I thi... |
330,991 | <p>Many things in math can be formulated quite differently; see the list of statements equivalent to RH <a href="https://mathoverflow.net/questions/39944/collection-of-equivalent-forms-of-riemann-hypothesis">here</a>, for example, with RH formulated as a bound on lcm of consecutive integers, as an integral equality, et... | Sam Hopkins | 25,028 | <p>I think "Geometric Complexity Theory" is roughly speaking an attempt to do what you're talking about: formulate P vs. NP in very different language. See <a href="https://en.wikipedia.org/wiki/Geometric_complexity_theory" rel="noreferrer">https://en.wikipedia.org/wiki/Geometric_complexity_theory</a>. I think that tec... |
330,991 | <p>Many things in math can be formulated quite differently; see the list of statements equivalent to RH <a href="https://mathoverflow.net/questions/39944/collection-of-equivalent-forms-of-riemann-hypothesis">here</a>, for example, with RH formulated as a bound on lcm of consecutive integers, as an integral equality, et... | Ryan O'Donnell | 658 | <p>There is the descriptive complexity formulation: </p>
<p>P = NP is equivalent to the statement that every property expressible by a second order existential statement is also expressible in first order logic with a least fixed point operator.</p>
<p>See, e.g., Immerman's survey here: <a href="https://people.cs.um... |
330,991 | <p>Many things in math can be formulated quite differently; see the list of statements equivalent to RH <a href="https://mathoverflow.net/questions/39944/collection-of-equivalent-forms-of-riemann-hypothesis">here</a>, for example, with RH formulated as a bound on lcm of consecutive integers, as an integral equality, et... | none | 140,370 | <p>"<a href="https://en.wikipedia.org/wiki/P_versus_NP_problem#Polynomial-time_algorithms" rel="nofollow noreferrer">This version of</a> Levin's universal search algorithm solves SUBSET-SUM in polynomial time" is equivalent to P=NP.</p>
|
1,499,583 | <p>If $a=b\log b$, how does $b$ grow asymptotically in terms of $a$?</p>
<p>I think the answer should be $b=\Theta\left(\frac{a}{\log a}\right)$. I tried taking logs to get $\log a=\log b+\log\log b$, but it's not clear how to separate $b$.</p>
| Deepak | 151,732 | <p>There is no contradiction, only a misconception. When working with complex numbers, $1^{\frac{25}{4}}$ has multiple values, not just $1$. (Or, to put it more generally, real positive numbers to fractional exponents can return multiple complex values).</p>
<p>A correct manipulation would be $i^{25} = ({i^4})^6 \cdot... |
1,499,583 | <p>If $a=b\log b$, how does $b$ grow asymptotically in terms of $a$?</p>
<p>I think the answer should be $b=\Theta\left(\frac{a}{\log a}\right)$. I tried taking logs to get $\log a=\log b+\log\log b$, but it's not clear how to separate $b$.</p>
| Paul Sinclair | 258,282 | <p>When dealing with complex functions, you often run into multi-valued functions, because when you circle around a singularity, it will pick up some constant value. There are three ways in which this is often handled:</p>
<ol>
<li>You can restrict the function to domains, called branches, that do not completely circl... |
275,371 | <p>I was wondering if it is possible to decompose any symmetric matrix into a positive definite and a negative definite component. I can't seem to think of a counterexample if the statement is false.</p>
| Arin Chaudhuri | 404 | <p>If $X$ is symmetric then $X = (X + \lambda I) - \lambda I$. Since the eigenvalues of $X + \lambda I $ are $ \lambda_i + \lambda$ where $\lambda_i$'s are the eigenvalues of X we can find a positive $\lambda$ such that $(X + \lambda I)$ is positive definite.</p>
|
364,278 | <p>Let <span class="math-container">$X$</span> be a variety over a number field <span class="math-container">$K$</span>. Then it is known that for any topological covering <span class="math-container">$X' \to X(\mathbb{C})$</span>, the topological space <span class="math-container">$X'$</span> can be given the structur... | SashaP | 39,304 | <p>Let's assume that <span class="math-container">$X$</span> admits a <span class="math-container">$K$</span>-point <span class="math-container">$x$</span> and use the corresponding geometric point as the base point. The existence of a rational point is in fact necessary for a positive answer, as explained by S. carmel... |
3,453,408 | <p>I'm reading through some lecture notes and see this in the context of solving ODEs:
<span class="math-container">$$\int\frac{dy}{y}=\int\frac{dx}{x} \rightarrow \ln{|y|}=\ln{|x|}+\ln{|C|}$$</span> why is the constant of integration natural logged here?</p>
| Quantum_Magnet | 383,722 | <p>See this:</p>
<p><span class="math-container">$\int\frac{dy}{y}=\int\frac{dx}{x} \Rightarrow \int\frac{dy}{y}=\int\frac{dx}{x} + c$</span></p>
<p><span class="math-container">$\ln |y| = \ln |x| + c \Rightarrow \ln |y/x| = c \Rightarrow y/x = \pm e^c \Rightarrow y= \pm e^c x \Rightarrow y = C x $</span> [Where ... |
238,547 | <p>I have a PDE like</p>
<pre><code>D[h[x1, x2], x1]*a[x1,x2]+D[h[x1,x2], x2]*b[x1,x2] + c[x1,x2] == h[x1,x2]
s.t. gradient(h(0,0))==0
</code></pre>
<p>where a,b,c are known functions of x1 and x2, and h are the function to be solved. x1 and x2 are both in [-2, 2].
For some selected a,b,c, DSolveValue can give me perfe... | bbgodfrey | 1,063 | <p>It also is possible to obtain an approximate symbolic solution, valid for small <code>x1</code>. Expand the PDE in <code>x1</code> to obtain</p>
<pre><code>DSolve[(x1 + x2)*D[h[x1, x2] + x2*D[h[x1, x2] == h[x1, x2], h[x1, x2], {x1, x2}]
</code></pre>
<p>Although <code>DSolve</code> cannot solve this PDE directly, i... |
2,631,220 | <p>If $z$ is a variable complex number , and $a$ is a fixed complex number , is it true that if $z$ , $a$ satisfy the following condition </p>
<p>$|z+a| = |z-a|$ </p>
<p>Then the locus of $z$ is the perpendicular bisector of $a$ and $-a$ ?</p>
| egreg | 62,967 | <p>Of course, you have to assume $a\ne0$. Write $a=ru$, with $|u|=1$ and $r>0$; then, writing $z=wu$, the equation becomes
$$
|w-r|=|w+r|
$$
Notice that this corresponds to a rotation around the origin by the negative of the angle determined by $u$.</p>
<p>By squaring,
$$
(w-r)(\bar{w}-r)=(w+r)(\bar{w}+r)
$$
that s... |
4,021,994 | <p>I was taught in high school algebra to translate word problems into algebraic expressions. So when I encountered <a href="https://artofproblemsolving.com/wiki/index.php/2016_AMC_10A_Problems/Problem_3" rel="nofollow noreferrer">this</a> problem I tried to reason out an algebra formula for it</p>
<blockquote>
<p>For ... | fleablood | 280,126 | <p>I have to admit I'm having a hard time seeing how you "imagine a series of comparisons " and how you "loop" them.</p>
<p>As a mathematician, I simply view this as Ben did a <em>single</em> transaction where in one go he spent <span class="math-container">$x$</span> dollars on bagels.</p>
<p>There... |
319,262 | <p>If the first 10 positive integer is placed around a circle, in any order, there exists 3 integer in consecutive locations around the circle that have a sum greater than or equal to 17? </p>
<p>This was from a textbook called "Discrete math and its application", however it does not provide solution for this question... | Gerry Myerson | 8,269 | <p>EDIT: it has been pointed out that this answer only gives $\ge17$, while the question asks for $\gt17$. More work is needed. </p>
<p>Let $A_1=a_1+a_2+a_3$, $A_2=a_2+a_3+a_4$, and so on, $A_{10}=a_{10}+a_1+a_2$. Then $A_1+A_2+\cdots+A_{10}=3(a_1+a_2+\cdots+a_{10})=(3)(55)=165$, so some $A_i\ge165/10=16.5$, so some $... |
319,262 | <p>If the first 10 positive integer is placed around a circle, in any order, there exists 3 integer in consecutive locations around the circle that have a sum greater than or equal to 17? </p>
<p>This was from a textbook called "Discrete math and its application", however it does not provide solution for this question... | joriki | 6,622 | <p>Original answer:</p>
<blockquote>
<p>To have all sums $\le17$, all four numbers from $7$ to $10$ would have to be separated by at least two numbers; but it would take at least $12$ slots to space them like that.</p>
</blockquote>
<p>As has been pointed out in the comments, this is wrong, but Gerry showed how to ... |
1,092,665 | <p>My question is really simple, how can I write symbolically this phrase: </p>
<blockquote>
<p>$x=\sum a_mx^m$ where $m$ range over
$\{1,\ldots,g\}\setminus\{t_1,\ldots,t_u\}$</p>
</blockquote>
<p>Being more specific, I would like to know how to write with mathematical symbols this part: "range over $\{1,\ldots,... | Kez | 201,782 | <p>I'd suggest
$$\Large x=\sum_{\substack{m=1\\[0.1cm] m\,\notin\, \{t_1,\,\ldots\,,\,t_u\}}}^g a_mx^m$$</p>
|
599,126 | <p>Question is to check which of the following holds (only one option is correct) for a continuous bounded function $f:\mathbb{R}\rightarrow \mathbb{R}$.</p>
<ul>
<li>$f$ has to be uniformly continuous.</li>
<li>there exists a $x\in \mathbb{R}$ such that $f(x)=x$.</li>
<li>$f$ can not be increasing.</li>
<li>$\lim_{x... | Dan | 79,007 | <p>For the third point, consider $f(x) = \arctan(x)$. For the fourth point, you've already found a counterexample in one of your other points!</p>
|
599,126 | <p>Question is to check which of the following holds (only one option is correct) for a continuous bounded function $f:\mathbb{R}\rightarrow \mathbb{R}$.</p>
<ul>
<li>$f$ has to be uniformly continuous.</li>
<li>there exists a $x\in \mathbb{R}$ such that $f(x)=x$.</li>
<li>$f$ can not be increasing.</li>
<li>$\lim_{x... | Hayden | 27,496 | <p>Here is an incredibly non-interesting trivial example: $f(x)=a$ for $a$ being some real number.</p>
|
1,924,033 | <blockquote>
<p><strong>Question.</strong> Let $\mathfrak{g}$ be a real semisimple Lie algebra admitting an invariant inner-product. Is every connected Lie group with Lie algebra $\mathfrak{g}$ compact?</p>
</blockquote>
<p>I know that the converse is true: If $G$ is a compact connected Lie group, then the Haar meas... | Community | -1 | <p>I will assume that gluing data is also meant to include the condition $U_{ii} = U_i$. </p>
<p>I'm too tired to organize this all in a narrative, so this will be fairly disjointed.</p>
<hr>
<p>It is fairly common in category theory to consider two families of objects $X_k$ and $X_{ij}$, families of maps $f_{ij} : ... |
370,058 | <p>How can I take this integral?</p>
<p>$$\int_{0}^{x} (z- u)_+^2 du $$</p>
<p>which <code>+</code> means If $z$ is bigger than u its equal $z - u$ and else it's equal zero.</p>
| Ron Gordon | 53,268 | <p>Wouldn't this integral just be</p>
<p>$$\int_{\tau}^t du \: (\tau-u)^2 = \frac{1}{3} (t-\tau)^3$$</p>
|
627,871 | <p>Let $\mathbf{A}$ be an algebra (in the sense of universal algebra) of some signature $\Sigma$. By <em>quasi-identity</em> I mean the formula of the form</p>
<p>$$(\forall x_1) (\forall x_2) \dots (\forall x_n) \left(\left[\bigwedge_{i=1}^{k}t_i(x_1, \dots, x_n)=s_i(x_1, \dots, x_n)\right]\rightarrow t(x_1, \dots, x... | bof | 111,012 | <p>The quasi-identity $\forall x(x+x=0\to x=0)$ holds in $\mathbb Z$ but not in $\mathbb Z/2\mathbb Z$.</p>
|
2,574,117 | <p>For a matrix $A$, define the operator $\ell_p$-norm of $A$ to be
$$
\|A\|_p = \sup_{x \neq 0} \frac{\|Ax\|_p}{\|x\|_p}.
$$
Here $\|x\|_p$ denotes the $\ell_p$ norm of the vector $x$.</p>
<p>For $1 \le p \le q \le 2$ and $x \in \mathbb{R}^n$, we know that $\|x\|_q \le \|x\|_p \le n^{1 / p - 1 / q} \|x\|_q$. </p>
<p... | daw | 136,544 | <p>Using H\"older's inequality, we can prove that these norms are equivalent. Let $1\le p\le q\le +\infty$. Then
$$
\|Ax\|_p \le n^{1/p-1/q} \|Ax\|_q \le n^{1/p-1/q} \|A\|_q\|x\|_q\le n^{1/p-1/q} \|A\|_q\|x\|_p,
$$
hence
$$
\|A\|_p \le n^{1/p-1/q}\|A\|_q.
$$
Similarly,
$$
\|A\|_q \le m^{1/p-1/q}\|A\|_p.
$$
Equality ho... |
313,794 | <p>I am not able to calculate extremums for the given function:
$u = 3x^2 - 3xy + 3x +y^2 + 5y$</p>
<p>I am able to calculate
$
u_x = 6x - 3y+ 3
$</p>
<p>$
u_{xx} = 6
$</p>
<p>$
u_{xy} = -3 = u_{yx}
$</p>
<p>$
u_y = -3x + 2y + 5
$</p>
<p>$
u_{yy} = 2
$</p>
<p>But what is next? Where are extremum points?</p>
| wj32 | 35,914 | <p>Since $u$ is differentiable, any local min/max point $(x,y)$ must satisfy $Du(x,y)=0$. You've already done the calculations, so just solve \begin{align}
6x-3y+3&=0 \\
-3x+2y+5&=0
\end{align} to get a single point $(x_0,y_0)$. The matrix representing $D^2u(x_0,y_0)$ (i.e. the "Hessian matrix") is $$H=\begin{b... |
313,794 | <p>I am not able to calculate extremums for the given function:
$u = 3x^2 - 3xy + 3x +y^2 + 5y$</p>
<p>I am able to calculate
$
u_x = 6x - 3y+ 3
$</p>
<p>$
u_{xx} = 6
$</p>
<p>$
u_{xy} = -3 = u_{yx}
$</p>
<p>$
u_y = -3x + 2y + 5
$</p>
<p>$
u_{yy} = 2
$</p>
<p>But what is next? Where are extremum points?</p>
| Bob | 48,443 | <p>You're almost there.</p>
<p>$$u_x=6x-3y+3=0 \to y= 2x+1$$
$$u_y=-3x+2y+5=0$$</p>
<p>Substitution leads to
$$-3x+2(2x+1)+5=0 \to x=-7, y=-13$$</p>
<p>Which is the global minimum since both $u_{xx},u_{yy}>0$</p>
|
2,096,408 | <p>My Physics book has many graphs. Some are straight lines, some parabolas while others are hyperbolas. I have not studied these curves (conic sections) yet and to me parabola and hyperbola look just the same. Is there any way of knowing whether a line is a parabola or a hyperbola just by seeing the graph of the line.... | Edward Porcella | 403,946 | <p>As you go out from the vertex (turning-point) of a parabola, tangents to opposite sides of the curve approach parallelism. With a hyperbola there's a limit to how small an angle the tangents can make with each other--the angle of the "asymptotes". Parabolas are more u-shaped, hyperbolas more v-shaped.</p>
|
2,780,403 | <p>I've tried to solve this but I don't seem to get anywhere.</p>
<p>The question states:</p>
<blockquote>
<p>Tom's home is $1800$ m from his school. One morning he walked part of the way to school and then ran the rest. If it took him $20$ mins or less to get to school, and he walks at $70$ m/min and runs at $210$... | Dylan | 135,643 | <p>The answer you were given has a several mistakes. First, the gradient should be</p>
<p>$$ \nabla T(1,1,1) = \langle 2,4,4 \rangle $$</p>
<p>The direction vector is in the wrong direction
$$ \hat{u} = -\frac{\nabla T}{|\nabla T|} = -\left\langle \frac13, \frac23, \frac23 \right\rangle $$</p>
<p>And the directional... |
20,726 | <p>The following situation is ubiquitous in mathematical physics. Let
$\Lambda_N$
be a finite-size lattice with linear size
$N$. An typical example would be the subset of
$\mathbb{Z}\times\mathbb{Z}$
given by those pairs of integers
$(j,k)$ such that
$j,k \in$ {
$0,\ldots,N-1$}. On each vertex
$j$ of the latt... | Steve Flammia | 1,171 | <p><em>Bounding angles in projector-valued Hamiltonians</em></p>
<p>Suppose that each of the
<span class="math-container">$h_k$</span> are projectors, and suppose that we shift the total energy so that
<span class="math-container">$\lambda_1 = 0$</span>. Further suppose that this eigenspace is known to be non-degenera... |
2,129,830 | <p>I am wondering if this is generally true for any topology. I think there might be counter examples, but I am having trouble generating them. </p>
| user404961 | 404,961 | <p>The set $(0,1) \cup (1,2)$ is a counterexample.</p>
|
2,129,830 | <p>I am wondering if this is generally true for any topology. I think there might be counter examples, but I am having trouble generating them. </p>
| Eric Wofsey | 86,856 | <p>No. A quick way to verify counterexamples is the following observation: if $U$ is the interior of a closed set $C$, then $U$ is also the interior of $\overline{U}$. Indeed, since $C$ is closed and $U\subseteq C$, $\overline{U}\subseteq C$, so the interior of $\overline{U}$ is contained in the interior of $C$. But... |
2,129,830 | <p>I am wondering if this is generally true for any topology. I think there might be counter examples, but I am having trouble generating them. </p>
| gerw | 58,577 | <p>Every open, dense set is a counterexample. In $\mathbb{R}$ you can use
$$\bigcup_{n \in \mathbb{N}} (q_n - 1/2^n, q_n + 1/2^n),$$
where $\mathbb{Q} = \{q_0, q_1, \ldots\}$ are the rationals.</p>
|
1,034,335 | <p>I'm preparing for my calculus exam and I'm unsure how to approach the question: "Explain the difference between convergence of a sequence and convergence of a series?" </p>
<p>I understand the following:</p>
<p>Let the sequence $a_n$ exist such that $a_n =\frac{1}{n^2}$ </p>
<p>Then $\lim_{n\to\infty} a_n=\lim_{n... | Gyu Eun Lee | 52,450 | <p>If we are talking about sequences and series of real or complex numbers, or of vectors in a real (or complex) normed vector space, then convergence of sequences and series are equivalent concepts.</p>
<p>Convergence of a series $\sum_{n=1}^\infty a_n$ is simply the convergence of the sequence of partial sums $S_N =... |
50,002 | <p>a general version: connected sums of closed manifold is orientable iff both are orientable.
I think this can be prove by using homology theory, but I don't know how.Thanks.</p>
| gary | 6,595 | <p>Consider the perspective of simplicial homology, for manifolds M,M'. Assume WOLG that M,M' are both connected: if an m-manifold M is orientable (I think that there is a result that all manifolds can be made into simplicial complexes), this means that the top cycle --call it m'-- can be assigned a coherent orientatio... |
942,470 | <p>I am trying to count how many functions there are from a set $A$ to a set $B$. The answer to this (and many textbook explanations) are readily available and accessible; I am <strong>not</strong> looking for the answer to that question and <strong>please do not post it</strong>. Instead I want to know what fundamen... | Stijn Hanson | 145,554 | <p>A function requires assignment so you cannot just not return anything. What you're calling a function is, in fact, a partial function.</p>
|
2,049,207 | <p>The question is this: <strong><em>How many ways are there to put 5 different balls into 3 different boxes so that none of the boxes is empty?</em></strong> </p>
<p>The correct answer as per my lecturer's notes is <strong>150</strong>, and I would like to know where I am going wrong in my approach.</p>
<p><strong>H... | mesel | 106,102 | <p>A subgroup generated by conjugacy classes is always normal.</p>
<p><strong>Hint:</strong> Let $N$ be a subgroup generated by $<g^x|x\in G>$. Take a $n\in N$.</p>
<p>Then $n=h_1h_2...h_k$ for $h_i=g^x$ for some $x\in G$. Now what can you say about $n^y$ for some $y\in G$ ?</p>
|
2,542,056 | <p>Baire's Category Theorem states that a meager subset of a complete metric space has empty interior. </p>
<p>Are there examples of meager subsets of non-complete metric spaces which do not have empty interior?<br>
In particular, are the rationals numbers as a subset of themselves an example?</p>
| Asinomás | 33,907 | <p>Yes they do because $\mathbb Q \setminus \{a\}$ is open and dense for each $a\in \mathbb Q$. When you take the intersection of these sets for every $a$ you get $\varnothing$, notice that this is a countable intersection since there ate countably many rationals.</p>
|
2,651,537 | <p>Prove that $(5m+3)(3m+1)=n^2$ is not satisfied by any <strong>positive</strong> integers $m,n$.</p>
<p>I have been staring at this for some time (it's the difficult part of a competition problem, I won't spoil it by naming the problem). I tried looking at it modulo 3,4,5,7,8,16 for a contradiction, as well as looki... | Macavity | 58,320 | <p>You can rewrite that equation as $(15m+7)^2-15n^2=4$. Looking at this $\pmod 4$, we can find both $15m+7$ and $n$ need to be even. Hence let $15m+7=2a, \,n=2b$, so that we have the more familiar Pell equation $a^2-15b^2=1$ to solve. </p>
<p>Note the smallest positive solution to the Pell equation is $(a_1, b_1) ... |
3,255,654 | <p>In a multiple choice question, there are five different answers, of which only one is correct. The probability that a student will know the correct answer is 0.6. If a student does not know the answer, he guesses an answer at random.</p>
<p>a) What is the probability that the student gives the correct answer?</p>
... | yurnero | 178,464 | <p>Slightly different notation. Let <span class="math-container">$A$</span> be the event that the student knows the answer and <span class="math-container">$B$</span> is as you defined.
<span class="math-container">$$
P(B)=P(B|A)P(A)+P(B|\neg A)P(\neg A)=1\times 0.6+0.2\times0.4=0.68.
$$</span>
Next,
<span class="math-... |
73,410 | <p>Gromov proved that if $$
f,g:\left[ {a,b} \right] \to R
$$
are integrable functions, such that the function $$
t \to \frac{{f\left( t \right)}}
{{g\left( t \right)}}
$$
is also integrable, and decreasing. Then the function $$
r \to \frac{{\int\limits_a^r {f\left( t \right)dt} }}
{{\int\limits_a^r {g\left( t ... | AD - Stop Putin - | 1,154 | <p><em>Extra assumption 1: $g$ is non-negative or non-negative. (Thanks robjohn)</em></p>
<p><em>Extra assumption 2: $f$ and $g$ are absolute continuous (e.g. they are strictly increasing/decreasing). (Thanks Mariano Suárez-Alvarez and t.b.)</em></p>
<p>Fix $r$. Since $f/g$ is decreasing we have
$$\frac{f(x)}{g(x)}\... |
4,368,464 | <p>How to solve <span class="math-container">$\sum_{i=1}^{n} \frac{P_i}{1+(d_i-d_1)x/365} = 0$</span> in spreadsheet?</p>
<p>We have already known that in Excel,</p>
<p>XIRR() find the root of the equation:
<span class="math-container">$\sum_{i=1}^{n} \frac{P_i}{(1+x)^{(d_i-d_1)/365}} = 0$</span>, which is the IRR (Int... | Greg Martin | 16,078 | <p>Yes, the formula is the same, since
<span class="math-container">$$
\sum_{i=0}^{n} i = 0+1+\cdots+n = 0+(1+\cdots+n) = 0 + \sum_{i=1}^n i.
$$</span></p>
|
3,112,043 | <p>The first part of the problem is:</p>
<p>Prove that for all integers <span class="math-container">$n \ge 1$</span> and real numbers <span class="math-container">$t>1$</span>,
<span class="math-container">$$ (n+1)t^n(t-1)>t^{n+1}-1>(n+1)(t-1)$$</span></p>
<p>I have done the first part by induction on <span... | G Cab | 317,234 | <p>Hint:</p>
<p>Note that <span class="math-container">$x^n$</span> is convex when <span class="math-container">$1 \le n$</span>.<br>
Thus you can profitably demonstrate the inequality through a Riemann sum of the
integral of <span class="math-container">$x^n$</span> (histogram below and above the continuous curve).</... |
3,466,870 | <p>Suppose </p>
<p><span class="math-container">$$a^2 = \sum_{i=1}^k b_i^2$$</span> </p>
<p>where <span class="math-container">$a, b_i \in \mathbb{Z}$</span>, <span class="math-container">$a>0, b_i > 0$</span> (and <span class="math-container">$b_i$</span> are not necessarily distinct).</p>
<p>Can any positive... | Keith Backman | 29,783 | <p>"Is any <span class="math-container">$k$</span> possible?" An easy route to "Yes": You know from the Pythagorean theorem that two squares can add to a perfect square. <span class="math-container">$$c^2=a^2+b^2$$</span></p>
<p><span class="math-container">$c^2$</span> must be either odd or even. If odd, it is the di... |
3,466,870 | <p>Suppose </p>
<p><span class="math-container">$$a^2 = \sum_{i=1}^k b_i^2$$</span> </p>
<p>where <span class="math-container">$a, b_i \in \mathbb{Z}$</span>, <span class="math-container">$a>0, b_i > 0$</span> (and <span class="math-container">$b_i$</span> are not necessarily distinct).</p>
<p>Can any positive... | poetasis | 546,655 | <p>Yes, any number of squares can be a perfect square. Take the example of Pythagorean triples generated by the formula with a table of triples shown below it where <span class="math-container">$n$</span> is a set and where <span class="math-container">$k$</span> is a set member.</p>
<p><span class="math-container">\be... |
3,466,870 | <p>Suppose </p>
<p><span class="math-container">$$a^2 = \sum_{i=1}^k b_i^2$$</span> </p>
<p>where <span class="math-container">$a, b_i \in \mathbb{Z}$</span>, <span class="math-container">$a>0, b_i > 0$</span> (and <span class="math-container">$b_i$</span> are not necessarily distinct).</p>
<p>Can any positive... | Erick Wong | 30,402 | <p>Many of the other answers (based on extending smaller values of <span class="math-container">$k$</span>) yield a fairly large value for the LHS. It’s worth pointing out that we actually know a great deal about the set of integers representable as the sum of <span class="math-container">$k$</span> nonzero squares.</... |
649,502 | <p>What do we mean when we talk about a topological <em>space</em> or a metric <em>space</em>? I see some people calling metric topologies metric spaces and I wonder if there is some synonymity between a topology and a space? What is it that the word means, and if there are multiple meanings how can one distinguish t... | Thomas Andrews | 7,933 | <p>A metric space is a set with a metric.</p>
<p>A topological space is a set with a topology.</p>
<p>Both metric spaces and topologies are useful for defining "continuity." </p>
<p>Every metric space can be made a topological in a useful way, so that the notion of continuity in metric spaces agrees with the notion ... |
2,461,506 | <p>I am trying to derive / prove the fourth order accurate formula for the second derivative:</p>
<p>$f''(x) = \frac{-f(x + 2h) + 16f(x + h) - 30f(x) + 16f(x - h) - f(x -2h)}{12h^2}$.</p>
<p>I know that in order to do this I need to take some linear combination for the Taylor expansions of $f(x + 2h)$, $f(x + h)$, $f... | Paul Aljabar | 435,819 | <p>$$
f(x+h) =
f(x) + h f'(x) + \frac{h^2}{2} f''(x)
+ \frac{h^3}{6} f'''(x)
+ O(h^4)
$$</p>
<p>$$
f(x-h) =
f(x) - h f'(x) + \frac{h^2}{2} f''(x)
- \frac{h^3}{6} f'''(x)
+ O(h^4)
$$</p>
<p>$$
f(x+2h) =
f(x) + 2h f'(x) + 2 h^2 f''(x)
+ \frac{4 h^3}{3} f'''(x) + O(h^4)
$$</p>
<p>$$
f(x-2h) =
f(x) - 2h f'(x) +... |
2,087,107 | <p>In the following integral</p>
<p>$$\int \frac {1}{\sec x+ \mathrm {cosec} x} dx $$</p>
<p><strong>My try</strong>: Multiplied and divided by $\cos x$ and Substituting $\sin x =t$. But by this got no result.</p>
| DXT | 372,201 | <p>$\displaystyle I = \int\frac{1}{\sec x+\csc x}dx = \frac{1}{2}\int\frac{\sin 2x}{\sin x+\cos x}dx = \frac{1}{2}\int\frac{(\sin x+\cos x)^2-1}{\sin x+\cos x}dx$</p>
<p>$\displaystyle I = \frac{1}{2}\int (\sin x+\cos x)dx - \frac{1}{2}\int \frac{1}{\sin x+\cos x}dx$</p>
<p>$\displaystyle I =\frac{1}{2}(\sin x-\cos x... |
2,087,107 | <p>In the following integral</p>
<p>$$\int \frac {1}{\sec x+ \mathrm {cosec} x} dx $$</p>
<p><strong>My try</strong>: Multiplied and divided by $\cos x$ and Substituting $\sin x =t$. But by this got no result.</p>
| Animesh Ashish | 401,839 | <p>Since I don't know LaTex, I am going to add a <a href="http://m.meritnation.com/ask-answer/question/what-is-the-integration-of-1-secx-cosecx/integrals/489125" rel="nofollow noreferrer">link</a> so that you can understand with your best of abilities. Let me know if you find a problem/bug.</p>
<p>Since, I could only ... |
350,747 | <p>Base case: $n=1$. Picking $2n+1$ random numbers 5,6,7 we get $5+6+7=18$. So, $2(1)+1=3$ which indeed does divide 18. The base case holds.
Let $n=k>=1$ and let $2k+1$ be true. We want to show $2(k+1)+1$ is true. So, $2(k+1)+1=(2k+2) +1$....</p>
<p>Now I'm stuck. Any ideas?</p>
| robjohn | 13,854 | <p>The sum of a sequence of consecutive integers is the average of the sequence, which is also the average of the largest and smallest terms, times the number of terms in the sequence. If the smallest term of a sequence of length $2n+1$ is $k$, the largest is $k+2n$. The average is $k+n$. Thus, the sum is $(2n+1)(k+n)$... |
733,675 | <p>A new question has emerged after this one was successfully answered by r9m: <a href="https://math.stackexchange.com/questions/731292/inequality-with-abcd-2/731930#731930">If $a+b+c+d = 2$, then $\frac{a^2}{(a^2+1)^2}+\frac{b^2}{(b^2+1)^2}+\frac{c^2}{(c^2+1)^2}+\frac{d^2}{(d^2+1)^2}\le \frac{16}{25}$</a>. I thought o... | Jimmy R. | 128,037 | <p><em>Alternative way to reach the result, that is already proposed in the other answers:</em></p>
<p>Consider the non-linear maximization problem $$\max_{x_i}\sum_{i=1}^{n}\frac{x_i^2}{(x_i^2+1)^2}$$ subject to $x_1+x_2+\ldots+x_n=\sqrt{n}$ and $x_i\ge0$. In that case the <a href="http://en.wikipedia.org/wiki/Lagran... |
2,928,196 | <p>I thought that I could take all points with rational coordinates, but this space is not discrete</p>
| Henno Brandsma | 4,280 | <p>In <a href="http://at.yorku.ca/p/a/c/a/26.pdf" rel="nofollow noreferrer">this note</a> a classic fact is shown that in a metric space the boundary of an open set is the closure of a discrete set. So e.g. the circle in the plane, being the boundary of the open unit disk, is such a set.</p>
|
2,476,865 | <p>As the title suggests, I'm trying to establish a good bound on</p>
<p>\begin{equation}
S(n) = \sum_{k = 2}^n (en)^k k^{-Cn/\log{n} - k - 1/2},
\end{equation}</p>
<p>where $C$ is some reasonably large positive constant. In particular I would like to have $S(n) = o(1)$, i.e., </p>
<p>\begin{equation}
\lim_{n \t... | Nate River | 279,404 | <p>In spite of my previous comment, I do think your intuition is correct.</p>
<p>Attempted proof (sorry but it is very ugly): Again, let $r(d)=\frac{s_2}{s_{2+d}}$. As you noted,
$$
r(d)=\left(\frac{2+d}{en}\right)^d\left(1+\frac{d}{2}\right)^{Cn/\log n + 2 + 1/2}
$$</p>
<p>Taking logarithm, we get
$$
\log r(d)=d\l... |
1,955,591 | <p>I have to prove that ' (p ⊃ q) ∨ ( q ⊃ p) ' is a tautology.I have to start by giving assumptions like a1 ⇒ p ⊃ q and then proceed by eliminating my assumptions and at the end i should have something like ⇒(p ⊃ q) ∨ ( q ⊃ p) but could not figure out how to start.</p>
| Hagen von Eitzen | 39,174 | <p>As a quick overview: You should be able to show $p\vdash q\supset p$ as well as $\neg p\vdash p\supset q$, and then by case distinction $p\lor\neg p\vdash (p\supset q)\lor (q\supset p)$.</p>
|
96,468 | <blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="https://math.stackexchange.com/questions/22537/how-many-fixed-points-in-a-permutation">How many fixed points in a permutation</a> </p>
</blockquote>
<p>Suppose we have a collection of n objects, numbered from 1 to n. These objects are placed in ... | Elvis | 21,435 | <p>Let’s denote $P$ the random variable defined by $P = {}$ number of objects in correct position. We compute $\mathbb P(P = p)$ by counting the number of permutations "with $p$ fixed points" (just an other way to say "$p$ objects in correct position).</p>
<ul>
<li><p>You have a total of $n!$ permutations ;</p></li>
<... |
121,546 | <p>Consider the 2-Wasserstein distance between probability measures $\mu$ and $\nu$ (on $\mathbb{R}^d$), defined as
$$
d_{W_2}(\mu,\nu) = \inf_{\gamma} \Big[\int \|x-y\|^2 d\gamma(x,y)\Big]^{1/2}
$$
where the $\inf$ is over all couplings $\gamma$ of $\mu$ and $\nu$. Can we define a norm (or something norm-like) on th... | Vladimir Zolotov | 32,454 | <p>(I guess you missed a square in your definition.)</p>
<p>2-Wasserstein distance doesn't respect the convex structure on measures. Consider two points $x_1 \ne x_2$ and Dirac measures $\delta(x_1), \delta(x_2)$. The measure $\frac{\delta(x_1)+\delta(x_2)}{2}$ is not a midpoint between $\delta(x_1)$ and $\delta(x_2)$... |
816,088 | <blockquote>
<p>The sum of two variable positive numbers is $200$.
Let $x$ be one of the numbers and let the product of these two numbers be $y$. Find the maximum value of $y$.</p>
</blockquote>
<p><em>NB</em>: I'm currently on the stationary points of the calculus section of a text book. I can work this out in my... | Andrés E. Caicedo | 462 | <p>You call one of your numbers $x$. Write $x$ in the form $100-k$, so the other number is $100+k$ (it does not matter here whether $k$ is zero, or positive, or negative). We have $$z=(100-k)(100+k)=100^2-k^2\le 100^2,$$ with equality happening if and only if $k=0$.</p>
|
606,431 | <p>Can someone explain to me how to solve this using inverse trig and trig sub?</p>
<p>$$\int\frac{x^3}{\sqrt{1+x^2}}\, dx$$</p>
<p>Thank you. </p>
| lab bhattacharjee | 33,337 | <p>$$ \frac{x^3}{\sqrt{1+x^2}}=\frac{x^3+x-x}{\sqrt{1+x^2}}=x\sqrt{1+x^2}-\frac x{\sqrt{1+x^2}}$$</p>
<p>Set $1+x^2=u$ in each case</p>
|
606,431 | <p>Can someone explain to me how to solve this using inverse trig and trig sub?</p>
<p>$$\int\frac{x^3}{\sqrt{1+x^2}}\, dx$$</p>
<p>Thank you. </p>
| Alraxite | 61,039 | <p>Set $u=1+x^2$. Then $\dfrac{du}{dx}=2x$ and so,</p>
<p>$\displaystyle\int \dfrac{x^3}{\sqrt{1+x^2}}dx=\int \dfrac12\dfrac{u-1}{\sqrt{u}}du=\dfrac12\int (u^{1/2}-u^{-1/2})\, du$.</p>
|
17,713 | <p>I am a bit perplexed in trying to find values <span class="math-container">$a,b,c$</span> so that the approximation is as precise as possible:</p>
<p><span class="math-container">$$\sum_{k=n}^{\infty}\frac{(\ln(k))^{2}}{k^{3}} \approx \frac{1}{n^{2}}[a(\ln (n))^{2}+b \ln(n) + c]$$</span></p>
<p>I can see from Wolf... | Aryabhata | 1,102 | <p>You can try using elementary estimates (upper and lower bounds) with integrals (see the end of the answer), or can use the more general and quite useful <a href="http://en.wikipedia.org/wiki/Euler%E2%80%93Maclaurin_formula" rel="nofollow">Euler Mclaurin Summation Formula</a>, which gives us</p>
<p>$$\sum_{k=1}^{n} ... |
1,871,103 | <blockquote>
<p>Given that $a_{1}=0$, $a_{2}=1$ and
$$a_{n+2}=\frac{(n+2)a_{n+1}-a_{n}}{n+1}$$
prove that $\lim\limits_{n\to\infty} a_n=e$</p>
</blockquote>
<p>What I did:</p>
<p>It was hinted to prove that $a_{n+1}-a_{n}=\frac{1}{n!}$ which I did inductively.
But then using this information now I get:</p>
<... | user84413 | 84,413 | <p>Now you can use induction to show that $\displaystyle a_n=1+\frac{1}{2!}+\frac{1}{3!}+\cdots+\frac{1}{(n-1)!}$, </p>
<p>so then $\displaystyle\lim_{n\to\infty}a_n=\sum_{k=1}^{\infty}\frac{1}{k!}=\color{red}{e-1}$</p>
|
3,075,979 | <p>Prove that <span class="math-container">$$\frac{k^7}{7}+\frac{k^5}{5}+\frac{2k^3}{3}-\frac{k}{105}$$</span> is an integer using mathematical induction.</p>
<p>I tried using mathematical induction but using binomial formula also it becomes little bit complicated.</p>
<p>Please show me your proof.</p>
<p>Sorry if t... | Michael Rozenberg | 190,319 | <p>Because
<span class="math-container">$$\frac{k^7}{7}+\frac{k^5}{5}+\frac{2k^3}{3}-\frac{k}{105}=\frac{k^7-k}{7}+\frac{k^5-k}{5}+\frac{2(k^3-k)}{3}+k.$$</span></p>
|
170,830 | <p>I have two circles with the same radius and I want to calculate the points of tangency. </p>
<p>For example, in the picture below, I want to calculate $(x_3, y_3)$ and $(x_4,y_4)$. I have the radius and the distance between the two circles as shown below:</p>
<p><img src="https://i.stack.imgur.com/tQ2qu.png" alt="... | Barun Dasgupta | 34,858 | <p>The gradient of the tangent to any circle is given by $$-\frac{(x'-a)}{(y'-b)}$$ Where (x',y') is the point of tangency and (a,b) is the center of the circle. Now the gradient of the line joining the centers of the two circles is same as the gradient of the tangent. Hence in this case this essentially translates to ... |
1,955,212 | <p>I couldn't find this on the whole internet. My life depends on solving this. Please help.
I must write a formula for this sequence
<span class="math-container">$-8, -14, 8, 14, -8, -14, 8, 14$</span>.</p>
| IntegrateThis | 270,702 | <p>Consider for an indice $n$ that if n is divisible by 2 the absolute value of your sequence is 14, else the absolute value is 8.</p>
<p>Next if an indice $n$ can be represented by $4k-1$ or $4k$ for $k \in \Bbb N$ then the sequence value is positive, else the sequence value is negative.</p>
|
1,368,899 | <p>I am fairly new to statistics and just recently encountered queueing theory.</p>
<p>I have programmed a simulation for a $M/M/1$ queue in which I specify the inter-arrival times and service times. I input say, an exponential distribution with a mean of $1$ for both the inter-arrival and service time.</p>
<p>I also... | Vinod Kumar Punia | 255,098 | <p>$3\alpha=\frac{-a}{a-b},
2\alpha^2=\frac{1}{a-b}$
divide
$\frac{3\alpha}{2\alpha^2}=-a$,
$a\alpha=\frac{-3}{2}...............(1)$,
As $\alpha$ is a root of $(a-b)x^2+ax+1=0$,
$(a-b)\alpha^2+a\alpha+1=0$
$(a-b)\frac{9}{4a^2}-\frac{3}{2}+1=0$
$2a^2-9a+9b=0$
this is quadratic in $a$ and as $a$ being real
$81-4(2)(9b)\g... |
2,792,770 | <p>I found the following question in a test paper:</p>
<blockquote>
<p>Suppose $G$ is a monoid or a semigroup. $a\in G$ and $a^2=a$. What can we say
about $a$?</p>
</blockquote>
<p>Monoids are associative and have an identity element. Semigroups are just associative. </p>
<p>I'm not sure what we can say about $a... | BCLC | 140,308 | <p>Let $(\Omega',\mathcal{F}',\mathbb{P}')$ be a probability space where $\Omega'$ is countable. It can be shown that any $X$ in the probability space must be discrete. Thus, for any non-discrete random variable $Y$ on the probability space $(\Omega,\mathcal{F},\mathbb{P})$, $Y$ is not $\mathcal{F}$-measurable but not ... |
2,919,266 | <p>Let $(x_n)$ be a sequence in $(-\infty, \infty]$. </p>
<p>Could we define the sequence $(x_n)$ so that limsup$(x_n) = -\infty$? </p>
<p>My intuitive thought is no, but I’m not 100% sure. </p>
| parsiad | 64,601 | <p>You are interested in the sum
$$
S_{N}\equiv\sum_{n=1}^{N}\frac{n}{\left(n+1\right)\left(n+2\right)}2^{n}.
$$
As Donald suggests, you can use the partial fraction expansion
$$
\frac{n}{\left(n+1\right)\left(n+2\right)}=\frac{2}{n+2}-\frac{1}{n+1}
$$
to get
\begin{multline*}
S_{N}=\sum_{n=1}^{N}\frac{2}{n+2}2^{n}-\su... |
1,831,134 | <p><a href="https://i.stack.imgur.com/cQEeH.jpg" rel="nofollow noreferrer">Worked examples</a></p>
<p>Can somebody please explain to me how the generator matrix is obtained when we are given the codewords of the binary code in the examples attached.</p>
<p>I tried arranging the codes in a matrix with each row being a... | Dietrich Burde | 83,966 | <p>$C_1$ is a $2$-dimensional vector space over the finite field $\mathbb{F}_2$ with basis $e_1=(0,1),e_2=(1,0)$. So we have $C_1=\{\lambda e_1+\mu e_2\mid \lambda,\mu\in \mathbb{F}_2\}=\{(0,0),(1,0),(0,1),(1,1)\}$. Of course the generator matrix $G$ is formed by $e_1$ and $e_2$, which is the canonical basis for the li... |
2,285,202 | <p>So I want to find a field extension that has the galois group $Z_{3} \times Z_{3} \times Z_{3} $. Now if the 3's where changed to 2's then I guess for example $(x^2-2)(x^2-3)(x^2-5)$ would suffice but I don't see how any clever way to do it with $Z_{3}$. I tried a bit with cyclotomic extensions but came up empty han... | Chickenmancer | 385,781 | <blockquote>
<p>First: <strong>Your example won't work.</strong> For example take $\theta=\sqrt[3]{2}.$ </p>
</blockquote>
<p>Then $$x^3-2=(x-\theta)(x^2+\theta x+\theta^2).$$</p>
<p>Since You can see that $(x^2+\theta x+\theta^2)$ has roots $$x=\theta\left(\dfrac{1}{2}\pm\dfrac{\sqrt{3}}{2}i\right).$$</p>
<p>Tak... |
2,584,688 | <p>Consider normed spaces $X$ and $Y$. You can assume that they are Banach spaces if needed. Let $\mathcal{L}(X, Y)$ denote the spaces of bounded linear operators from $X$ to $Y.$ Now consider the set </p>
<p>$$\Omega=\{T \in L(X,Y): T \textrm{ is onto}\}.$$ Is $\Omega$ open with the norm topology? </p>
| David C. Ullrich | 248,223 | <p>We've seen in another answer that this is false if $X$ is not a Banach space.</p>
<blockquote>
<blockquote>
<p><strong>Lemma</strong> Suppose $X$ is a Banach space, $T:X\to Y$ is linear and bounded, $0<\epsilon<1$, $\delta>0$, and for every $y\in Y$ there exists $x\in X$ with $||x||\le\delta ||y||$ a... |
1,849,608 | <p>Given the irreducible fraction $\frac a b$, with $a, b \in \mathbb N$, what is the expression that enumerates all the irreducible fractions of integers that add up to $\frac a b$? Namely, an expression (in terms of $a$ and $b$) for all the $\frac c d$ and $\frac e f$, with $c,d,e,f \in \mathbb N$, such that $\frac c... | G Cab | 317,234 | <p>Every couple of fractions $c/a$ and $d/b$ in the Stern-Brocot tree can be represented (in the inverse notation according to <em>Concrete Mathematics</em>) by the matrix
${\bf M} = \left\| {\,\begin{array}{*{20}c}
a & b \\
c & d \\
\end{array}\,} \right\|$, where, iff the fractions are the generator... |
3,796,216 | <p>Use a group-theoretic proof to show that <span class="math-container">$\mathbb{Q}^*$</span> under multiplication is
not isomorphic to <span class="math-container">$\mathbb{R}^*$</span> under multiplication.</p>
<p><strong>I have tried this:</strong></p>
<p>Suppose <span class="math-container">$$ \phi: \mathbb{Q}^*\t... | lhf | 589 | <p>Your idea is in the right direction but <span class="math-container">$x\mapsto x^2$</span> is not onto. However, <span class="math-container">$x\mapsto x^3$</span> is onto and works. More precisely:</p>
<p>The map <span class="math-container">$\mathbb R^* \to \mathbb R^*$</span> given by <span class="math-container"... |
265,067 | <p>$$\lim_{n\to\infty}\frac{(2n-1)!}{3^n(n!)^2}$$</p>
<p>How can I associate limit problem with series? And how can i find limits from series?
Can anyone help?</p>
| André Nicolas | 6,312 | <p><strong>Hint:</strong> Let $a_n=\dfrac{(2n-1)!}{3^n(n!)^2}$. </p>
<p>It is useful to look at the ratio $\dfrac{a_{n+1}}{a_n}$ for large $n$. </p>
|
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