qid int64 1 4.65M | question large_stringlengths 27 36.3k | author large_stringlengths 3 36 | author_id int64 -1 1.16M | answer large_stringlengths 18 63k |
|---|---|---|---|---|
2,259,840 | <blockquote>
<p>Points $P$, $Q$, and $R$ lie on the same line. Three semi-circles with the diameters $PQ$, $QR$, and $PR$ are drawn on the same side of the line segment $PR$. (That is, suppose we have an <a href="https://en.wikipedia.org/wiki/Arbelos" rel="nofollow noreferrer">arbelos</a>.) The centers of the semi-ci... | CY Aries | 268,334 | <p>Hint:
Draw the figure and you will have $AC=a+c$, $BC=b+c$, $OA=b$, $OB=a$ and $OC=a+b-c$. Apply the cosine formula to find $\cos\angle BAC$ in</p>
<p>(1) $\triangle ABC$</p>
<p>(ii) $\triangle OAC$</p>
|
4,447,522 | <p>Show that <span class="math-container">$$\cot\left(\dfrac{\pi}{4}+\beta\right)+\dfrac{1+\cot\beta}{1-\cot\beta}=-2\tan2\beta$$</span>
I'm supposed to solve this problem only with sum and difference formulas (identities).</p>
<p>So the LHS is <span class="math-container">$$\dfrac{\cot\dfrac{\pi}{4}\cot\beta-1}{\cot\d... | Anamaria | 857,136 | <p><span class="math-container">$$\frac{4 \sin \beta \cos \beta}{\sin ^{2} \beta-\cos ^{2} \beta} = \frac{2 \sin2\beta}{-(\cos ^{2} \beta-\sin ^{2} \beta)}=-2\frac{\sin 2\beta}{\cos 2\beta}=-2 \tan(2\beta)$$</span></p>
|
1,341,440 | <p>I came across a claim in a paper on branching processes which says that the following is an <em>immediate consequence</em> of the B-C lemmas:</p>
<blockquote>
<p>Let $X, X_1, X_2, \ldots$ be nonnegative iid random variables. Then $\limsup_{n \to \infty} X_n/n = 0$ if $EX<\infty$, and $\limsup_{n \to \infty} X_... | Nicky Hekster | 9,605 | <p>Put $N=\langle x \rangle$. Then $|N|=4$ and index$[G:N]=2$ so, $G=N \cup gN$, with $g \notin N$. $N$ is normal, so $g^{-1}x^2g \in N$. But this element, being conjugate to $x^2$, has order equal to that of $x^2$, which is $2$. Since $x^2$ is the unique element of order $2$ in $N$, it follows that $g^{-1}x^2g=x^2$. S... |
627,871 | <p>Let $\mathbf{A}$ be an algebra (in the sense of universal algebra) of some signature $\Sigma$. By <em>quasi-identity</em> I mean the formula of the form</p>
<p>$$(\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... | J.-E. Pin | 89,374 | <p>Your idea was correct. Take the cancellation law in a commutative monoid. It is satisfied by $\mathbb N$ but not by its quotient $(\{0, 1\}, +)$ (obtained by identifying all positive numbers to $1$). In this latter monoid, you have $0 + 1 = 1 = 1 + 1$, but $0 \not= 1$.</p>
|
1,390,976 | <p>Similar to <a href="https://math.stackexchange.com/questions/54763/what-do-algebra-and-calculus-mean">What do Algebra and Calculus mean?</a>, what is the difference between a logic and a calculus?</p>
<p>I am learning about the different kinds of logics, and often when I look them up in a different resource, some p... | starseed_trooper | 515,191 | <p>I would elaborate on the answer above with respect to the etymology of logic. Logic is closely associated with philosophy since the days of Aristotle. Many philosophers are well-versed in logic, but not as something that you represent symbolically. This is what is known as informal logic and involves such things as ... |
276,329 | <p>I have a problem, from Gelfand's "Algebra" textbook, that I've been unable to solve, here it is:</p>
<p><strong>Problem 268.</strong> </p>
<p>What is the possible number of solutions of the equation $$ax^6+bx^3+c=0\;?$$</p>
<p>Thanks in advance.</p>
| Belgi | 21,335 | <p><strong>Hint:</strong> Set $t=x^3$ and solve for $t$</p>
|
276,329 | <p>I have a problem, from Gelfand's "Algebra" textbook, that I've been unable to solve, here it is:</p>
<p><strong>Problem 268.</strong> </p>
<p>What is the possible number of solutions of the equation $$ax^6+bx^3+c=0\;?$$</p>
<p>Thanks in advance.</p>
| Elias Costa | 19,266 | <p>The aplication of <a href="http://en.wikipedia.org/wiki/Quadratic_formula#Quadratic_formula" rel="nofollow">quadratic formula</a> in
$$
a\cdot (x^3)^2+b\cdot(x^3)+c=0
$$
give us that the possible roots enjoy<br>
$$
x^3 =\left[\frac{-b+\sqrt{b^2+4ac}}{2a}\right]
\quad
\mbox{ or }
\quad
x^{3} =\left[\frac{-b-\sqrt{b^2... |
2,098,395 | <p>Evaluate the following;</p>
<p>$$\sum_{r=0}^{50} (r+1) ^{1000-r}C_{50-r}$$</p>
<p>Using $^{n}C_{r}=^{n}C_{n-r}$ we get $\sum_{r=0}^{50} (r+1) ^{1000-r}C_{950}$</p>
<p>but I am not getting how to solve $\sum_{r=0}^{50} r \cdot \hspace{0.5 mm} ^{1000-r}C_{950}$</p>
| DXT | 372,201 | <p>$\displaystyle \sum^{50}_{r=0}r\cdot \binom{1000-r}{950}=0\cdot \binom{1000}{950}+1\cdot \binom{999}{950}+2\cdot \binom{998}{950}+\cdots \cdots \cdots +50\cdot \binom{950}{950}$</p>
<p>Using $\displaystyle \binom{n}{k} = $ Coefficients of $x^k$ in $(1+x)^n$</p>
<p>so coefficients of
$\displaystyle x^{950}$ in</... |
2,049,685 | <p>If a team 1 has a probability of p of winning against team 2. What is the probability "formula" that team one will win 7 games first. </p>
<p>There are no ties and the teams play until one t am wins 7 games </p>
| Canardini | 341,007 | <p>A state $(a,b)$ where $a$ is the number of victories from team1 and $b$ from team2.</p>
<p>Team 1 wins if they reach the states $(7,0), (7,1),...,(7,6)$.</p>
<p>Each state $(7,k)$ are tied to $\binom{6+k}{k}$ possible scenarios of same probability of occuring, that is $p^7(1-p)^{k}$. Indeed, if the result is $(7,k... |
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.... | JonathanZ supports MonicaC | 275,313 | <p>This isn't an infallible method, but every hyperbola has two <a href="https://en.wikipedia.org/wiki/Asymptote" rel="nofollow noreferrer">asymptotes</a>, whereas parabolas don't have even one. </p>
|
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... | Helge | 3,983 | <p>This is off-topic: But powerful techniques to show that gap exist, can be found in the following papers:</p>
<ul>
<li><p>Michael Goldstein, Wilhelm Schlag, <em><a href="https://arxiv.org/abs/math/0511392" rel="nofollow noreferrer">On resonances and the formation of gaps in the spectrum of quasi-periodic Schroedinger... |
4,118,149 | <p>Let's say I have the following matrix:
<span class="math-container">\begin{bmatrix}
\frac{1}{3} & \frac{2}{3} & 0 & \frac{2}{3} \\
\frac{2}{3} & -\frac{1}{3} & \frac{2}{3} & 0 \\
a & b & c & d \\
e & f & g & h
\end{bmatrix}</span></p>
<p>How do I find t... | Surb | 154,545 | <p>First, you can easily solve this equation and see what happen. But you can also see it without solving this equation. Let <span class="math-container">$g(x,y)=(y^2-1)\cos(x)$</span>. The function <span class="math-container">$g(x,\cdot )$</span> is locally-Lipschitz, and thus your IVP has a unique local solution. Le... |
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... | Nicky Hekster | 9,605 | <p>$A_5$ is simple so you will not find any interesting normal subgroups. Let me give you a hint: is $g,a,b \in G$, then $g^{-1}abg=g^{-1}ag \cdot g^{-1}bg$. This should help you proving that the subgroup generated by a conjugacy class is actually normal.</p>
|
637,199 | <p>If $K^T=K$, $K^3=K$, $K1=0$ and $K\left[\begin{matrix}1\\2 \\-3\end{matrix}\right]=\left[\begin{matrix}1\\2 \\-3\end{matrix}\right]$,</p>
<p>how can I find the trace of $K$ and the determinant of $K$?</p>
<p>I think for determinant of $K$, since $K^3-K=(K^2-I)K=0$, then $K^2=I$ since $K$ is nonzero. Then this impl... | Community | -1 | <p>The polynomial $x^3-x=x(x^2-1)$ annihilates the matrix $K$ which's diagonalizable since it's real <em>(not clear from the hypothesis)</em> symmetric or since the polynomial has simple roots, moreover $0$ and $1$ are eigenvalues of $K$ since $K1=0$ and since
$$K\left[\begin{matrix}1\\2 \\-3\end{matrix}\right]=\left[... |
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... | J.G. | 56,861 | <p>Yes, <span class="math-container">$k$</span> can be arbitrary. Define a sequence<span class="math-container">$$a_1:=3,\,a_{k+1}:=\frac12\left(a_k^2+1\right)$$</span> of odd positive integers (since <span class="math-container">$\frac12((2n+1)^2+1)=2(n^2+n)+1$</span>), so<span class="math-container">$$a_{k+1}^2-a_k^2... |
3,466,870 | <p>Suppose </p>
<p><span class="math-container">$$a^2 = \sum_{i=1}^k b_i^2$$</span> </p>
<p>where <span class="math-container">$a, b_i \in \mathbb{Z}$</span>, <span class="math-container">$a>0, b_i > 0$</span> (and <span class="math-container">$b_i$</span> are not necessarily distinct).</p>
<p>Can any positive... | David G. | 733,021 | <p>Solutions exist for every <span class="math-container">$k>0$</span>. The simplest forms use most of the <span class="math-container">$b_n$</span> values as <span class="math-container">$1$</span>. I will list them as "<span class="math-container">$b_*$</span>". I suspect there are infinitely many distinct answ... |
3,466,870 | <p>Suppose </p>
<p><span class="math-container">$$a^2 = \sum_{i=1}^k b_i^2$$</span> </p>
<p>where <span class="math-container">$a, b_i \in \mathbb{Z}$</span>, <span class="math-container">$a>0, b_i > 0$</span> (and <span class="math-container">$b_i$</span> are not necessarily distinct).</p>
<p>Can any positive... | ralphmerridew | 231,358 | <p>Yes.</p>
<p>For <span class="math-container">$k = 2$</span>:
<span class="math-container">$3^2 + 4^2 = 5^2$</span></p>
<p>For <span class="math-container">$k > 2$</span>:<br>
Start with a solution for <span class="math-container">$k-1$</span><br>
Multiply both sides by <span class="math-container">$5^2$</span><... |
3,466,870 | <p>Suppose </p>
<p><span class="math-container">$$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... | Piquito | 219,998 | <p>It is known enough the generalization of Phytagorean triples which let us to say YES. Just as for <span class="math-container">$k=2$</span> two parameters are needed, for <span class="math-container">$k\gt2$</span> we need <span class="math-container">$k$</span> arbitrary parameters <span class="math-container">$t... |
431,236 | <p>I have a cylinder of radius 4 and height 10 that is at a 30 degree angle. I need to find the volume.</p>
<p>I have no clue how to do this, I have spent quite a while on it and went through many ideas but I think my best idea was this.</p>
<p>I know that the radius is 4 so if I cut the cylinder in half from corner ... | chijioke | 116,927 | <p>assuming that the height giving in the question where to be the slanted height.Therefore the straight height H, must be found before proceeding for further solving using l multiplied the sine of the giving angle 30 degree to give 5, after that you substitute in normal volume formula to give ans. 251.36 </p>
|
384,006 | <p>Just came across the following question:</p>
<blockquote>
<p>Let $S=\{2,5,13\}$. Notice that $S$ satisfies the following property: for any $a,b \in S$ and $a \neq b$, $ab-1$ is a perfect square. Show that for any positive integer $d \not\in S$, $S \cup \{d\}$ does not satify the above property.</p>
</blockquote>
... | Matthew W. | 82,204 | <p>I have been trying to find a 4-element set that satisfies the property. I ran a computer program to check all 4-element sets of integers <= 20,000 and found nothing, so I am guessing that there are no such sets, although of course I have nothing conclusive.
<p>
The program also listed all of the 3-element sets t... |
2,238,614 | <p>For $n\ge3$ a given integer, find a Pythagorean Triple having n as one of its members.<br>
Hint: For n an odd integer, consider the triple $$\left(n, \frac 12\left(n^2-1\right), \frac 12(n^2+1)\right);$$ For n even, consider the triple $$\left(n, \left(\frac{n^2}{4}\right)-1, \left(\frac{n^2}{4}\right)+1 \right)$$</... | Micah | 30,836 | <p>You do want to add the squares of the first two and see if it equals the square of the last.</p>
<p>You also have to make sure all three numbers are integers, which is why you need to do something different depending on whether $n$ is even or odd.</p>
|
2,461,506 | <p>I am trying to derive / prove the fourth order accurate formula for the second derivative:</p>
<p>$f''(x) = \frac{-f(x + 2h) + 16f(x + h) - 30f(x) + 16f(x - h) - f(x -2h)}{12h^2}$.</p>
<p>I know that in order to do this I need to take some linear combination for the Taylor expansions of $f(x + 2h)$, $f(x + h)$, $f... | Vladimir F Героям слава | 134,138 | <p>You can easily derive the formula, if you do not know it, as a derivative of the Lagrange polynomial</p>
<pre><code>D[D[InterpolatingPolynomial[{(-2*h,y0),(-1*h,y1),(0*h,y2),(1*h,y3),(2*h,y4)},x],x],x] /. x=0
</code></pre>
<p><kbd> <a href="https://www.wolframalpha.com/input/?i=D%5BD%5BInterpolatingPolynomial%5B%7B%... |
3,424,687 | <blockquote>
<p>Let <span class="math-container">$n$</span> be a positive integer and a complex number with unit modulus is a solution of the equation <span class="math-container">$z^n+z+1=0$</span>. Prove that <span class="math-container">$n $</span> can't be <span class="math-container">$196$</span>. </p>
</blockqu... | Mohammad Riazi-Kermani | 514,496 | <p>Let <span class="math-container">$$z^{196}+z+1=0$$</span></p>
<p>Then we have <span class="math-container">$$z^{196}= -z-1$$</span> </p>
<p>Thus <span class="math-container">$$|z|^{196} =|-z-1|$$</span></p>
<p>Since <span class="math-container">$|z|=1$</span> we get <span class="math-container">$|z+1|=1$</span></... |
2,087,107 | <p>In the following integral</p>
<p>$$\int \frac {1}{\sec x+ \mathrm {cosec} x} dx $$</p>
<p><strong>My try</strong>: Multiplied and divided by $\cos x$ and Substituting $\sin x =t$. But by this got no result.</p>
| Dhanvi Sreenivasan | 332,720 | <p><strong>HINT</strong> </p>
<p>Use half-angle formulae to substitute for $\sec x$ and $\csc x$, and use the substitution $u = \tan \frac{x}{2}$, to get the following expression:</p>
<p>$$I = 4\int\frac{u(u^2-1)}{(u^2+1)^2(u^2-2u-1)}du$$</p>
<p>This should be easier to compute, using factorization and other basic t... |
350,747 | <p>Base case: $n=1$. Picking $2n+1$ random numbers 5,6,7 we get $5+6+7=18$. So, $2(1)+1=3$ which indeed does divide 18. The base case holds.
Let $n=k>=1$ and let $2k+1$ be true. We want to show $2(k+1)+1$ is true. So, $2(k+1)+1=(2k+2) +1$....</p>
<p>Now I'm stuck. Any ideas?</p>
| André Nicolas | 6,312 | <p>For clarity, we start by taking $n=4$. </p>
<p>Look at the following $2n+1=9$ consecutive integers:
$$-4, \quad-3,\quad -2, \quad -1,\quad 0,\quad 1,\quad 2,\quad 3,\quad 4.$$</p>
<p>The sum of these is $0$, obviously divisible by $2n+1$.</p>
<p>Now any $9$ consecutive integers can be obtained by adding a suitabl... |
733,675 | <p>A new question has emerged after this one was successfully answered by r9m: <a href="https://math.stackexchange.com/questions/731292/inequality-with-abcd-2/731930#731930">If $a+b+c+d = 2$, then $\frac{a^2}{(a^2+1)^2}+\frac{b^2}{(b^2+1)^2}+\frac{c^2}{(c^2+1)^2}+\frac{d^2}{(d^2+1)^2}\le \frac{16}{25}$</a>. I thought o... | Shane | 58,882 | <p>I'm afraid the inequality is wrong. Note that RHS $< 1$. However, if we take $x:=1$, then $x^2/(x^2+1)^2=1/4$. Thus, we choose a large $n$, and let as much $x_i$ as possible be $1$, then the inequality fails. But I believe that there exists a bound of $n$ to let the inequality hold.</p>
|
1,220,502 | <p>The problem is this. Given that $\int_0^a f(x) dx = \int_0^a f(a-x)dx$, evaluate $$\int_0^\pi \frac{x\sin x}{1+\cos^2x} dx$$</p>
<p>I write the integral as $$\int_0^\pi \frac{(\pi-x)\sin(\pi-x)}{1+\cos^2(\pi-x)} dx$$
but I don't see how that helps to do it.</p>
| Olivier Oloa | 118,798 | <p>If you set
$$
I=\int_0^\pi \frac{x\sin x}{1+\cos^2x} dx
$$then
$$\begin{align}
I&=\int_0^\pi \frac{(\pi-x)\sin(\pi-x)}{1+\cos^2(\pi-x)} dx\\\\
&=\int_0^\pi \frac{(\pi-x)\sin x}{1+\cos^2x} dx\\\\
&=\pi\int_0^\pi \frac{\sin x}{1+\cos^2x} dx-\int_0^\pi \frac{x\sin x}{1+\cos^2x} dx\\\\
&=\pi\int_0^\pi \f... |
281,450 | <p>How can I find the area bound by $\;x=0,\, x=1,\;$ the $\;x$-axis ($y = 0$) and $\;y=x^2+2x\;$ using Riemann sums? </p>
<p>I want to use the right-hand sum. Haven't really found any good resources online to explain the estimation of areas bounded by curves, hoping anyone here can help?</p>
<p>By the way, I would l... | Hagen von Eitzen | 39,174 | <p>The sequence for which $(1+x+x^{10})^{20}$ is the generating function, is
$1, 20, 190, 1140, 4845, 15504, 38760, 77520, 125970, 167960, 184776, 168340, 129390, 96900, 116280, 248064, 547485, 1008900, 1511830, 1847580, 1847751, 1515060, 1036830, 697680, 813960, 1705440, 3546540, 6049980, 8314400, 9237820, 8315160, 60... |
67,929 | <p>Joel David Hamkins in an answer to my question <a href="https://mathoverflow.net/questions/67259/countable-dense-sub-groups-of-the-reals">Countable Dense Sub-Groups of the Reals</a> points out that "one can find an uncountable chain of countable dense additive subgroups of $\mathbb{R}$ whose subset relation has the ... | Juris Steprans | 13,878 | <p>The result of Joel quoted above certainly shows a complicated structure, but does not actually provide continuum many non-isomorphic subgroups since there is no reason why $G$ being a subgroup of $H$ should imply that $G$ and $H$ are not isomorphic. Indeed, considering two groups generated by the rationals and two i... |
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... | David Quinn | 187,299 | <p>hint...if you only want to use induction, let <span class="math-container">$$f(k)=15k^7+21k^5+70k^3-k$$</span>
and consider <span class="math-container">$$f(k+1)-f(k)=$$</span></p>
<p>For the induction step you have to show this is divisible by <span class="math-container">$105$</span></p>
<p>So, for example, <spa... |
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... | Jean-Claude Arbaut | 43,608 | <p>You can use the <a href="https://en.wikipedia.org/wiki/Binomial_transform" rel="nofollow noreferrer">binomial transform</a> to prove that</p>
<p><span class="math-container">$$\frac{k^7}{7}+\frac{k^5}{5}+\frac{2k^3}{3}-\frac{k}{105}
\\={k\choose1}+28{k\choose2}+292{k\choose3}+1248{k\choose4}+2424{k\choose5}+2160{k\... |
3,075,979 | <p>Prove that <span class="math-container">$$\frac{k^7}{7}+\frac{k^5}{5}+\frac{2k^3}{3}-\frac{k}{105}$$</span> is an integer using mathematical induction.</p>
<p>I tried using mathematical induction but using binomial formula also it becomes little bit complicated.</p>
<p>Please show me your proof.</p>
<p>Sorry if t... | Cardioid_Ass_22 | 631,681 | <p>Base case for <span class="math-container">$k=1$</span>: <span class="math-container">$$\frac{1^7}{7}+\frac{1^5}{5}+\frac{2*1^3}{3}-\frac{1}{105}=1$$</span></p>
<p>Now, assume for some k that <span class="math-container">$\frac{k^7}{7}+\frac{k^5}{5}+\frac{2k^3}{3}-\frac{k}{105}$</span> is indeed in integer. </p>
... |
2,759,827 | <blockquote>
<p>Let $\{x_n\}$ be a bounded sequence and $s=\sup\{x_n|n\in\mathbb N\}.$
Show that if $s\notin \{x_n|n\in\mathbb N\}, $then there exists a
subsequence convereges to $s$.</p>
</blockquote>
<p>$s-1$ cannot be the upperbound and $s\notin \{x_n|n\in\mathbb N\}, \exists n_1\in \mathbb N:n_1\ge1:s-1<x... | user284331 | 284,331 | <p>One can do it like looking for $n_{k+1}\geq 1$ such that $\max\left\{s-\dfrac{1}{k+1},x_{1},...,x_{n_{k}}\right\}<x_{n_{k+1}}<s$.</p>
|
2,461,962 | <p>I'm looking to reproduce
\begin{align}
\partial_{j_m,j_n}\bigg|_{\bf{j}=0}\exp\left(\frac12\bf{j}^\top\bf{B}\bf{j}\right) = B_{mn}
\end{align}
where $B_{mn}=B_{nm}$ is a real, symmetric, positive-definite $N\times N$ matrix. I have tried the following, and I know this is incorrect due to the surplus of indices.
\beg... | Brethlosze | 386,077 | <p>Using doubled index sum notation the result is straightforward.
$$
\left.\frac{\partial}{\partial x_m}\frac{\partial}{\partial x_n} e^{\frac12 x_iB_{ij}x_j}\right|_{x=0}\\
=\left.\frac{\partial}{\partial x_m} \frac 12 (B_{nq}x_q+x_pB_{pn})e^{\frac12 x_iB_{ij}x_j}\right|_{x=0}\\
=\left.(\frac 12 (B_{nm}+B_{mn})+\frac... |
1,483,802 | <p>Take $B(0,1)$ the ball in $\mathbb{R}^2$ with the normalized Lebesgue measure $\lambda$ such that $\int_{B(0,1)} d \lambda=1.$</p>
<p>Now, I want to show, or give a counterexample that this is false, that for all $f \in H^1_0(B(0,1))$ we have
for fixed constants $a,b>0$ and any(!) $p \in (2,\infty)$
\begin{equat... | Community | -1 | <p>If such a $a, b$ is found for a $f\in W^{1,2}_0(B)$, then
$$\|f\|_p <C$$</p>
<p>for all $p >2$. In particular, this shows $\|f\|_\infty \le C$ as <a href="https://math.stackexchange.com/questions/242779/limit-of-lp-norm">$\|f\|_p \to \|f\|_\infty$ as $p\to \infty$</a>.</p>
<p>In particular, $f\in L^\infty$.... |
1,457,623 | <p>A floor is paved with rectangular marble blocks,each of length $a$ and breadth $b$.A circular block of diameter $c(c<a,b)$ is thrown on the floor at random.Show that the chance that it falls entirely on one rectangular block is $\frac{(a-c)(b-c)}{ab}$<br></p>
<p>I thought over this problem,i found total number o... | ewcz | 274,913 | <p>to have the circle contained entirely in a block, you need to put it within an inner block of size $(a-c)\times(b-c)$ since the center should be placed at a distance of at least $c/2$ from any of the edges. Now, dividing by the total area $a\times b$ yields the desired result.</p>
<p><a href="https://i.stack.imgur.... |
299,405 | <p>(a) In how many ways can the students answer a 10-question true false examination? </p>
<p>(b) In how many ways can the student answer the test in part (a) if it is possible to leave a question unanswered in order to avoid an extra penalty for a wrong answer</p>
<hr>
<p>For part (a) I've got the answer, it is $2^... | Andreas Caranti | 58,401 | <p>If the answer has to be $3^{10}$, then this means that in case (b) it is intended that <em>for each question</em> the student can choose 1`out of 3 possibilites: true, false, not telling. </p>
|
3,609,191 | <p>Actually I am not very comfortable with using blocks, I understand the definition that it is a maximal <span class="math-container">$2$</span>-connected graph, though.</p>
<p><strong>My attempt</strong> Suppose not. Then there exists a maximal graph <span class="math-container">$G$</span> which cannot be written as... | Aryak Sen | 768,532 | <p>A block of a graph G is a maximal connected subgraph of G that has no articulation point(cut-point or cut-vertex). If G itself is connected and has no cut-vertex, then G is a block.Two or more blocks of a graph can meet at a single vertex only, which must necessarily be an articulation point of the graph.Hence, any ... |
84,605 | <p>Let $w$ be a word in letters $x_1,...,x_n$. A value of $w$ is any word of the form $w(u_1,...,u_n)$ where $u_1,...,u_n$ are words. For example, $abaaba$ is a value of $x^2$. A word $u$ is called unavoidable if every infinite word in a finite alphabet contains a value of $u$ as a subword. There is a nice characteriza... | Gerhard Paseman | 3,402 | <p>Yes. However, the initial application is related to semigroup varieties, so it is likely very boring to you.</p>
<p>In studying the hyperidentity for associativity, one can look at its representation on algebras of type <2>, a.k.a. groupoids or magmas or sets with one binary operation. Such an algebra is hyper... |
32,021 | <p>On more than one occasion, always with an explicit disclaimer, I
have posted a comment of more than 600 characters as an "answer".
I have done this because I have quite often seen other people do it, and
I have never once, in 5 years in Maths.SE, seen anyone object to the
practice. But a comment I posted i... | Aloizio Macedo | 59,234 | <p>First I must point out that this question is somewhat unclear. What do you mean by "post a <em>comment</em> of more than 600 characters (...)"? (Emphasis mine.) If "comment" means that <em>I would post it in the comments below the question, but there is no space</em>, then obviously this has neve... |
3,337,440 | <p>Suppose that, in a memoryless way, an object A can suddenly transform into object B or object C. Once it transforms, it can no longer transform again (so if it becomes B, it cannot become C, and visa versa) </p>
<p>Suppose that the pdf of an object A becoming object B is </p>
<p><span class="math-container">$$\lam... | Grada Gukovic | 679,434 | <p>Altough the underlying processes are indepenedent the events <span class="math-container">$A \rightarrow B$</span> and <span class="math-container">$A \rightarrow C$</span> are not independent. <span class="math-container">$A \rightarrow B$</span> can occur only if <span class="math-container">$A \rightarrow C$</spa... |
2,189,123 | <p>There are 36 gangsters, and several gangs these gangsters belong to. No two gangs have identical roster, and no gangster is an enemy of anyone in their gang. However, each gangster has at least one enemy in every gang they are <strong>not</strong> in. What is the greatest possible number of gangs? </p>
| bof | 111,012 | <p>This question is the case $n=36$ of the general problem: find the maximum number of maximal cliques in a graph on $n$ vertices. The general solution is given by OEIS sequence <a href="http://oeis.org/A000792" rel="nofollow noreferrer">A000792</a>, namely: $a(3k)=3^k,$ $\ a(3k+1)=4\cdot3^{k-1},$ $\ a(3k+2)=2\cdot3^k,... |
3,169,668 | <p>If A and B don't commute are there counterexamples that AB is diagonalizable but BA not?</p>
<p>I read that if AB=BA then both AB and BA are diagonalizable. </p>
| Michael Biro | 29,356 | <p>Try <span class="math-container">$A = \begin{bmatrix}0&0\\1&0\end{bmatrix}$</span> and <span class="math-container">$B = \begin{bmatrix}0&0\\0&1\end{bmatrix}$</span></p>
|
1,926,423 | <p>My problem with this is the final step. Through iterative substitution, I come up with the following:
$$T(n) = T(n-4) + (n-3) + (n-2) + (n-1) + n$$</p>
<p>which leads to the general form:
$$T(n) = T(n-k) + kn - \frac{[k(k-1)]}{2}$$</p>
<p>The restrictions are $T(1)=1$ and $n=2^k-1$. What I am doing at this point i... | Olivier Oloa | 118,798 | <p>Alternatively, one may use a <em>telescoping sum</em>. From
$$
T(k)-T(k-1)=k, \qquad k=1,2,3,\cdots, \tag1
$$ one gets
$$
\sum_{k=1}^n \left(T(k)-T(k-1)\right)=T(n)-T(0)=\sum_{k=1}^n k, \qquad n\ge1,
$$ giving</p>
<blockquote>
<p>$$
T(n)=\frac{n(n+1)}2+T(0),\qquad n\ge1.
$$</p>
</blockquote>
|
1,926,423 | <p>My problem with this is the final step. Through iterative substitution, I come up with the following:
$$T(n) = T(n-4) + (n-3) + (n-2) + (n-1) + n$$</p>
<p>which leads to the general form:
$$T(n) = T(n-k) + kn - \frac{[k(k-1)]}{2}$$</p>
<p>The restrictions are $T(1)=1$ and $n=2^k-1$. What I am doing at this point i... | AlphaNumeric | 96,679 | <p>$\begin{array}{rcl}T(n) &=& T(n-1) + n \\
&=& T(n-2) + (n-1) + n \\
&=& \ldots \\
&=& T(n-[n-1]) + (n-[n-2]) + (n-[n-3]) + \ldots + n \\
&=& T(1) + 2 + 3 + \ldots + n \\
&=& 1 + 2 + 3 + \ldots \\
&=& \sum_{j=1}^{n}j \\
&=& \frac{1}{2}n(n+1)
\end{array}$... |
2,584,688 | <p>Consider normed spaces $X$ and $Y$. You can assume that they are Banach spaces if needed. Let $\mathcal{L}(X, Y)$ denote the spaces of bounded linear operators from $X$ to $Y.$ Now consider the set </p>
<p>$$\Omega=\{T \in L(X,Y): T \textrm{ is onto}\}.$$ Is $\Omega$ open with the norm topology? </p>
| mechanodroid | 144,766 | <p>$\Omega$ is not open in general, at least if $X$ and $Y$ are not assumed to be Banach.</p>
<p>Consider $c_{00}$, the space of all finitely-supported sequences, equipped with $\|\cdot\|_2$.</p>
<p>Define $A : c_{00} \to c_{00}$ as </p>
<p>$$A(x_1, x_2, x_3, \ldots) = \left(x_1, \frac12x_2, \frac13x_3\ldots\right)$... |
286,574 | <p>How to prove that each non-convex polygon with no self-intersecting parts, has at least one interior angle which size is less then $180$ degrees.</p>
| Gerry Myerson | 8,269 | <p>The formula for the sum of the angles of a polygon, as a function of the number of sides, holds for non-convex as well as for convex polygons, and the result follows immediately from that formula. </p>
|
286,574 | <p>How to prove that each non-convex polygon with no self-intersecting parts, has at least one interior angle which size is less then $180$ degrees.</p>
| Community | -1 | <p>Hint: Consider the vertex with the largest $x$-coordinate.</p>
|
4,142,540 | <p>Let <span class="math-container">$V$</span> a vector subspace of dimension <span class="math-container">$n$</span> on <span class="math-container">$\mathbb R$</span> and <span class="math-container">$f,g \in V^* \backslash \{0\}$</span> two linearly independent linear forms. I want to show that <span class="math-con... | Tsemo Aristide | 280,301 | <p>If <span class="math-container">$f,g$</span> are linearly independent, <span class="math-container">$Ker f +Ker g=\mathbb{R}^n$</span>.</p>
<p><span class="math-container">$dim(Ker f+Kerg) =dim Ker f +dim Ker g-dim(Ker f\cap Ker g)$</span> implies that</p>
<p><span class="math-container">$n = n-1+n-1-dim(Ker f\cap K... |
243,903 | <p>I have posted this question on mathstack echange but did not get any answer. It mam trying my luck here. </p>
<p>The only simple finite groups admitting an irreducible character of degree 3 are $\mathfrak{A}_5$
and $PSL(2,7)$. That seems to be a result coming from Blichfeldt's work on $GL(3,\mathbb{C})$, which I ca... | yakov | 92,209 | <p>A comprehensive report on groups possessing a faithful irreducible character of small degree is contained in Walter Feit's report `The current situation in the theory of finite simple groups, Actes Congr. Internat. Math. Nice 1970, vol. 1 Gauthier-Villars, Paris, 1971, 55-93.'</p>
|
3,453,175 | <p>If <span class="math-container">$y=\dfrac {1}{x^x}$</span> then show that <span class="math-container">$y'' (1)=0$</span></p>
<p>My Attempt:</p>
<p><span class="math-container">$$y=\dfrac {1}{x^x}$$</span>
Taking <span class="math-container">$\ln$</span> on both sides,
<span class="math-container">$$\ln (y)= \ln \... | lab bhattacharjee | 33,337 | <p><span class="math-container">$y(1)=?$</span></p>
<p><span class="math-container">$$-y_1=y(1+\ln x)$$</span></p>
<p><span class="math-container">$y_1(1)=?$</span></p>
<p><span class="math-container">$-y_2=y_1(1+\ln x)+y/x$</span></p>
<p><span class="math-container">$$y_2(1)=-y_1(1)-y(1)=?$$</span></p>
|
4,166,579 | <p>I'm trying to solve this proof but I'm not completely sure how to start. Discrete has been pretty rough for me so far so any help would be greatly appreciated!</p>
| Robert Lee | 695,196 | <p><em><strong>Hint:</strong></em> If <span class="math-container">$f(S) \subseteq f(T)$</span> then for every <span class="math-container">$s \in S$</span> there exists a <span class="math-container">$t \in T$</span> such that <span class="math-container">$f(s) = f(t)$</span>. Recalling that <span class="math-containe... |
1,703,120 | <p>So I have a vector <span class="math-container">$a =( 2 ,2 )$</span> and a vector <span class="math-container">$b =( 0, 1 )$</span>.<br />
As my teacher told me, <span class="math-container">$ab = (-2, -1 )$</span>.</p>
<p><span class="math-container">$ab = b-a = ( 0, 1 ) - ( 2, 2 ) = ( 0-2, 1-2 ) = ( -2, -1 )$</sp... | chandings | 203,901 | <p>I know I am late to answer. Also I do not have a lot of knowledge of Mathematics. But I will try and simplify. Please do not refrain from commenting if I get it horribly or event slightly wrong.</p>
<p>Now to simplify lets us first imagine things in one dimensions. That is a number line. Now we have 2 points on a nu... |
2,263,230 | <p>Let's say I wanted to express sqrt(4i) in a + bi form. A cursory glance at WolframAlpha tells me it has not just a solution of 2e^(i<em>Pi/4), which I found, but also 2e^(i</em>(-3Pi/4))</p>
<p>Why do roots of unity exist, and why do they exist in this case? How could I find the second solution? </p>
| Kaynex | 296,320 | <p>$$\sqrt{4i}$$
$$= \sqrt{4e^{\pi i /2 + 2\pi n}}$$
$$= 2e^{\pi i/4 + \pi n}$$
$$= 2e^{\pi i/4}, 2e^{5\pi i/4}$$</p>
<p>There are two solutions to the square root of any number. There are three cube roots, four fourth roots, ect.</p>
|
279,994 | <p>Note that S [n] is the sum of the first n terms of the sequence a [n]. It is known that a [1]==1, and the sequence {S [n]/a [n]} is an equal difference sequence with a tolerance of 1/3. Find the general term formula of sequence a [n]</p>
<p>Let b [n]==S [n]/a [n], first work out the general term formula of b [n], an... | Bob Hanlon | 9,362 | <pre><code>Clear[a, b, m, S];
b[n_] = RSolveValue[
{b[n + 1] == b[n] + 1/3, b[1] == 1}, b[n], n]
(* (2 + n)/3 *)
a[1] = 1;
S[n_] = Sum[a[k], {k, 1, n}];
m = 5;
sol = Solve[Table[b[n] == S[n]/a[n], {n, 2, m}],
Array[a, m - 1, 2]][[1]]
(* {a[2] -> 3, a[3] -> 6, a[4] -> 10, a[5] -> 15} *)
seq = Arra... |
3,353,826 | <p>All the vertices of quadrilateral <span class="math-container">$ABCD$</span> are at the circumference of a circle and its diagonals intersect at point <span class="math-container">$O$</span>. If <span class="math-container">$∠CAB = 40°$</span> and <span class="math-container">$∠DBC = 70°$</span>, <span class="math-c... | Bart Michels | 43,288 | <p>I have seen <span class="math-container">$\lfloor x \rceil$</span>. It must have been in the context of math olympiads, so I can't point to a book that uses it. Wikipedia suggest this notation, among others: <a href="https://en.wikipedia.org/wiki/Nearest_integer_function" rel="noreferrer">nearest integer function</a... |
1,480,179 | <p>Let $n ≥ 2$, and $A ∈ M_n$ be Hermitian, and let ${\rm{B }} \in {\rm{ }}{{\rm{M}}_{n - 1}}$ be a leading principal
submatrix of A.</p>
<p>If $B$ is positive semidefinite and $rank B = rank A$, why does $A$ is positive
semidefinite?</p>
| user1551 | 1,551 | <p>The size of $B$ doesn't matter (you don't need $B\in M_{n-1}$; $B$ can be smaller sized). As long as it is a positive semidefinite leading principal submatrix of a Hermitian matrix $A$ of the same rank, $A$ must be positive semidefinite.</p>
<p>Let $B$ be $r\times r$ and $A=\pmatrix{B&X\\ X^\ast&Y}$. Since ... |
3,977,281 | <p>Is it possible for a function to not have a maxima or a minima? (S.t. I can't find the decreasing and increasing interval.) If so, how do we show it mathematically?</p>
<p>I was practicing and found these two functions.</p>
<p><span class="math-container">$a. f(x) = x+\sqrt{x^2-1} $</span> and <span class="math-cont... | saulspatz | 235,128 | <p>For the first problem, <span class="math-container">$1 + \frac{x}{\sqrt{x^2-1}} = 0$</span> gives <span class="math-container">$$\frac{x}{\sqrt{x^2-1}}=-1\\x=-\sqrt{x^2-1}\\x^2=x^2-1$$</span> which has no solution, so the derivative never vanishes. That means that the function has no extremum at a point of differen... |
894,152 | <blockquote>
<p>Let $x_{i}\ge 0$ for $i\in\{1,2,\cdots,n\}$ and $x_{1}+x_{2}+\cdots+x_{n}=n$ for $n\ge 3$</p>
<p>Show that for all strictly positive integers $k\ge2$ the following inequality holds :
$$\sum_{i=1}^{n}x^k_{i}\ln{x_{i}}\ln{\dfrac{x_{i}}{n}}\le 0$$</p>
</blockquote>
<p>We consider
$$f(x)=x^k\ln{x}... | babbupandey | 92,239 | <p>I think you need to look at $\ln\frac{x_i}{n}$ in your equation. Since, $$x_1 + x_2 + \dots + x_n = n$$
therefore $$x_i < n$$
and notice that $$\ln\frac{x_i}{n} = \ln(x_i) - \ln(n) \implies \ln\frac{x_i}{n} < 0$$
hence the expression $${x_i}^k\ln{x_i}\ln\frac{x_i}{n} < 0$$
as ${x_i}^k$ and $\ln{x_i}$ are g... |
894,152 | <blockquote>
<p>Let $x_{i}\ge 0$ for $i\in\{1,2,\cdots,n\}$ and $x_{1}+x_{2}+\cdots+x_{n}=n$ for $n\ge 3$</p>
<p>Show that for all strictly positive integers $k\ge2$ the following inequality holds :
$$\sum_{i=1}^{n}x^k_{i}\ln{x_{i}}\ln{\dfrac{x_{i}}{n}}\le 0$$</p>
</blockquote>
<p>We consider
$$f(x)=x^k\ln{x}... | oknsnl | 169,453 | <p>While $x_i<=1$ or $x_i>=e$ then $x_i^k\ln^2{x_i}\ln{\dfrac{x_i}n}\ge x_i^k\ln{x_i}\ln{\dfrac{x_i}n}$</p>
<p>If we call $y$ $x$'s between $x_i \le 1$ or $x_i \ge e$ </p>
<p>and we call $z$ $x$'s between $x_i \ge 1$ or $x_i \le e$.</p>
<p>After that point we write greater function then $\sum_{i=1}^{n}x^k_{i}\... |
3,423,674 | <p>According to my calculus professor and MIT open coursework, taking the derivative of (x^2+4)^-1 is an application of the chain rule, not the power rule. The answer to the question is -(x^2+4)^-2, which makes sense to me, but I just don't understand why this is considered an application of the chain rule rather than ... | Graham Kemp | 135,106 | <p>The Power Rule was applied, but so too was the chain rule, and also a few unmentioned others.</p>
<p>The power rule may only be applied when the derivation 'numerator' is a power of the 'denominator' , so first you must use the Chain Rule to obtain such.</p>
<p>Thus taking everything one elementary step at a time:... |
2,428,009 | <p>I want to solve the equation $2^n=2k$ for $n$ even with $n,k \in \Bbb{N}$. I'm not sure how to go about this, using logarithm makes me enter the reals.</p>
| gammatester | 61,216 | <p><strong>The</strong> solution is $m=1+\log_2k$. If $m$ is an integer, you are done; otherwise use
$\lceil m \rceil$ or $\lfloor m \rfloor$ depending on the problem.</p>
|
1,224,180 | <p>Q: evaluate $\lim_{x \to \infty}$ $ (x-1)\over \sqrt {2x^2-1}$</p>
<p>What I did:</p>
<p>when $\lim_ {x \to \infty}$ you must put the argument in the form of $1/x$ so in that way you know that is equal to $0$</p>
<p>but in this ex. the farest that I went was</p>
<p>$\lim_{x \to \infty}$ $x \over x \sqrt{2}$ $1-(... | JMP | 210,189 | <p>$$\lim_\limits{x\to\infty}\dfrac{x-1}{\sqrt{2x^2}}\le
\lim_\limits{x\to\infty}\dfrac{x-1}{\sqrt{2x^2-1}}\le
\lim_\limits{x\to\infty}\dfrac{x-1}{\sqrt{2x^2-2}}$$</p>
<p>$$\lim_\limits{x\to\infty}\dfrac{x-1}{\sqrt{2}x}\le
\lim_\limits{x\to\infty}\dfrac{x-1}{\sqrt{2x^2-1}}\le
\lim_\limits{x\to\infty}\dfrac{x-1}{\sqrt{... |
3,551,030 | <p>Let the real sequence <span class="math-container">$x_n s.t. \Vert x_{n+2} - x_{n+1}\Vert = M \Vert x_{n+1} - x_{n} \Vert$</span> </p>
<p>If the <span class="math-container">$0< \vert M \vert <1$</span>, then <span class="math-container">$x_n$</span> surely convergent since it is a contractive. </p>
<p>But t... | Peter Szilas | 408,605 | <p>Assume <span class="math-container">$M >1$</span>;</p>
<p><span class="math-container">$b_n:=|x_{n+1}-x_n|$</span>;</p>
<p>Let <span class="math-container">$n_0$</span> be a positive integer, s.t. <span class="math-container">$b_{n_0} >0.$</span>
(Such a <span class="math-container">$n_0$</span> exists (Why?... |
2,933,383 | <p>Cauchy's theorem of limits states that if <span class="math-container">$\ \lim_{n \to \infty} a_n=L ,$</span> then <span class="math-container">$$ \lim_{n \to \infty} \frac{a_1+a_2+\cdots+a_n}{n}=L $$</span>
If I apply this in the series <span class="math-container">$$S = \lim_{n\to\infty} \dfrac{1}{n}[e^{\frac{1}... | Mostafa Ayaz | 518,023 | <p>You can't apply the theorem in the first case since <span class="math-container">$$a_n=e^{n\over n}=e$$</span>which doesn't make sense. Another reason is that <span class="math-container">$a_k=e^{k\over n}$</span> that says <span class="math-container">$a_k$</span> is dependent to <span class="math-container">$n$</s... |
3,693,196 | <p>This is known as the factor formula. It is used for the addition of sin functions. I don't understand how the two are equal though. How would you get to the right side of the equation using the left?</p>
<p><span class="math-container">$$\sin(s) + \sin(t) = 2 \sin\left(\frac{s+t}{2}\right) \cos \left(\frac{s-t}{2}\... | global05 | 696,211 | <p>P(A|B) = (∩)/() </p>
<p>P(∩) = 0.8</p>
<p>P (B) = 0.9</p>
<p>Hence P (A|B) = 0.8/0.9
= 8/9.</p>
|
1,004,303 | <p>Let $S=\{(x,0)\} \cup\{(x,1/x):x>0\}$. Prove that $S$ is not a connected space (the topology on $S$ is the subspace topology)</p>
<p>My thoughts: Now in the first set $x$ is any real number, and I can't see that this set in open in $S$. I can't find a suitable intersection anyhow.</p>
| Michael Hardy | 11,667 | <p>The set $\{(x,0) : x\in\mathbb R\}$ is open in $S$ because every point $(x,0)$ has an open neighborhood that does not intersect the graph of $y=1/x$. Just use $I\times\{0\}$ where $I$ is any open interval containing $x$.</p>
<p>Then do a similar thing with the set $\{(x,1/x): x>0\}$.</p>
|
2,141,082 | <p>Let me first put down a couple definitions, two of which have terminology I make up for this post. If you already know about sheaf theory, you can safely skip Definitions 1-3 and 7-8, and the Construction. Definitions 4-6 introduce notation and terminology that is probably nonstandard, so I recommend reading those i... | Roland | 113,969 | <p>That is a very long post. This is nice to give all the definition, but I bet that anyone who doesn't know them will not read through the whole post ;)</p>
<p>About all your wanna be facts : all of them are true, these are all very basic facts about sheaf theory. They are in every book on the topic. So let me point o... |
480,504 | <p>If <span class="math-container">$M$</span> is a symmetric positive-definite matrix, is it possible to get a <strong>positive</strong> lower bound on the smallest eigenvalue of <span class="math-container">$M$</span> in terms of a matrix norm of <span class="math-container">$M$</span> or elements of <span class="math... | Evan | 38,878 | <p>A quick comment: If you have diagonal dominance, then <a href="https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&cad=rja&ved=0CC4QFjAA&url=http://en.wikipedia.org/wiki/Gershgorin_circle_theorem&ei=pKUiUun7DJXQsATfnIDoBQ&usg=AFQjCNFgcxCvB-_iFubVENtxtz_JF025iw&... |
1,223,209 | <p>As part of another problem I am working on, I have the following product to work out. </p>
<p>$\begin{bmatrix} 1 & 2 & 3 \end{bmatrix} \cdot h $</p>
<p>where $h$ is a scalar. My question is, if I commute the row vector and the scalar then I can just multiply it through. If I think of the $h$ as a $1 \times... | Rellek | 228,621 | <p>Well, from your equation, I notice that it is only increasing, which can't be right, since the probability should keep going down. So, instead of having a constant denominator, there needs to be some variable there. Also, if I checked this right, there should be a 0 probability of getting a "new" ball on your 6th tr... |
1,291,050 | <p>I have been doing doing this problem $∇ × (\varphi∇\varphi)=0$</p>
<p>I am just having trouble applying the product result i get which is below.</p>
<p>$$i(( \frac {d}{dy} )(\varphi \frac {d}{dz} \varphi) - ((\frac {d}{dz})(\varphi \frac {d}{dy} \varphi)) )$$</p>
<p>if i take the first part </p>
<p>$$(\varphi \f... | Alex Fok | 223,498 | <p>Claim: $\det(A^{-1}A^\top+I)\geq 2^n$.</p>
<p>Proof: Note that $\det(A^{-1}A^\top)=1$. Let $\lambda_1, \cdots, \lambda_n$ be the eigenvalues of $A^{-1}A^\top$. Then $\lambda_1\cdots\lambda_n=1$ and the eigenvalues of $A^{-1}A^\top+I$ are $\lambda_1+1, \cdots, \lambda_n+1$. So
\begin{align*}
\det(A^{-1}A^\top+I)&... |
3,573,575 | <p>I'm trying to find the eigenvalues of a matrix <span class="math-container">$$A=\begin{bmatrix}2/3 & -1/4 & -1/4 \\ -1/4 & 2/3 & -1/4 \\ -1/4 & -1/4 & 2/3\end{bmatrix}$$</span></p>
<p>The eigenvalues of this matrix, are the roots <span class="math-container">$\lambda$</span> of the equation ... | Mohammad Riazi-Kermani | 514,496 | <p>Check for the rational roots of the characteristic polynomial.</p>
<p>The eigenvalues are <span class="math-container">$$\frac {11}{12}, \frac {11}{12}, \frac {1}{6} $$</span></p>
|
2,060,891 | <p>Number of solutions of $a^3=e$ in $C_9$</p>
<p>The solution goes: $a^3=e$ if and only if $a$ lies in the unique subgroup of $C_9$ of order $3$ thus there are $3$ solutions.</p>
<p>I'm questioning why? </p>
| MikeWasTaken | 392,196 | <p>The text book must be wrong then</p>
<p><a href="https://i.stack.imgur.com/c0ADa.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/c0ADa.png" alt="enter image description here"></a></p>
<p>6 Combinations (4,6), (5,5), (5,6), (6,4), (6,5), (6,6)</p>
|
176,059 | <p>I asked this question in MSE, but I did not received any answer, so I repeat it here:</p>
<p><a href="https://math.stackexchange.com/questions/858238/a-question-on-fixed-point-property">https://math.stackexchange.com/questions/858238/a-question-on-fixed-point-property</a></p>
<p>Assume that $0<k<n-1$, Note ... | user56137 | 56,137 | <p>I don't think $\mathbb{C}P^n/\mathbb{C}P^k$ (or $\mathbb{R}P^n/\mathbb{R}P^k$) ever has the fixed point property in the range you describe. I haven't thought about this very long so could be wrong, but on first look I think you can construct an endomorphism with no fixed points in the following way:</p>
<p>Conside... |
302,179 | <p>The question I am working on is:</p>
<blockquote>
<p>"Use a direct proof to show that every odd integer is the difference of two squares."</p>
</blockquote>
<p>Proof:</p>
<p>Let n be an odd integer: $n = 2k + 1$, where $k \in Z$</p>
<p>Let the difference of two different squares be, $a^2-b^2$, where $a,b \in Z... | P.K. | 34,397 | <p>$$2k + 1 = 1 \times (2k +1) = (\color{#C00}{k + 1} -{\bf k}) (\color{#C00}{k + 1} +\bf{ k})$$</p>
|
4,092,473 | <p>I am trying to prove if the following set is bounded <span class="math-container">$$S=\bigcup_{a\in(0,1)} M_{a},$$</span> where
<span class="math-container">$$M_{a}=\{(x,y) \in \mathbb{R}^2: ax+(1-a)y=b, x>0, y>0, b \text{ is a fixed positive real number}\}.$$</span>
I think this set is bounded since all lines... | Rahul Madhavan | 439,353 | <p>Claim: The set <span class="math-container">$S=\{(x,y) \in \mathbb{R}\times\mathbb{R}: ax+(1-a)y=b, x>0, y>0\}: 0<a<1\}$</span> is unbounded if <span class="math-container">$b>0$</span> and empty otherwise.</p>
<p>Proof:
<span class="math-container">$ax+(1-a)y$</span> is positive. So if <span class="m... |
317,601 | <blockquote>
<p>Let <span class="math-container">$F$</span> be a ring, let <span class="math-container">$f(x)=a_0+a_1x+\cdots+a_nx^n$</span> be in <span class="math-container">$F[x]$</span>, and <span class="math-container">$f'(x)$</span> be the regular derivative of <span class="math-container">$f(x)$</span>.</p>
<p>P... | Zev Chonoles | 264 | <p>There isn't really anything to prove; part of the axioms for a ring are that a ring is an abelian group under addition. If you feel the need to say anything about the issue at all, it should be fine to just say $F[x]$ is abelian group under addition by the ring axioms.</p>
<hr>
<p><em>Hint for finding the kernel o... |
1,464,143 | <p>$\lim_{n \to \infty} n\ln\left(1+\frac{1}{n}\right)$ using L'Hòpital rule show that this is $1$. Can you do this since there isn't a division and $n$ will obviously tend to infinity and $\ln\left(1+\frac{1}{n}\right)$ will tend to $0$? So there limits aren't matching?</p>
<p>So I set $u=n $</p>
<p>$du=1$</p>
<p>$... | Bernard | 202,857 | <p>Using L'Hospital's rule here is properly ridiculous: set $\;\dfrac1n=x$ and observe we have a rate of variation:
$$\lim_{n\to\infty}n\ln\Bigl(1+\frac1n\Bigr)=\lim_{x\to0}\frac{\ln(1+x)}x=\bigl(\ln(1+x)\bigr)'_{x=0}=1.$$</p>
|
302 | <p>I know that the Fibonacci numbers converge to a ratio of .618, and that this ratio is found all throughout nature, etc. I suppose the best way to ask my question is: where was this .618 value first found? And what is the...significance?</p>
| Larch | 31,272 | <p>Golden ration came first in nature long before humans evolved to think about Fibonacci numbers.</p>
|
34,671 | <p>Going through some old papers, I came up with a simple-looking problem I thought about 5 years ago or so. </p>
<p>MO wants motivation ... Associated to a probability measure on a metric space is something called "quantization dimension" ... this involves defining a function $D \colon (0,\infty) \to (0,\infty)$. E... | fedja | 1,131 | <p>All right. The hardest thing is to do all algebra right (I am always prone to misplacing + and -, so check everything I say in the first part. :)).</p>
<p>Let $\lambda=\log \frac 1p$ and $\Gamma=\log\frac 1s$. We want to create a situation when there are 3 points on the line $D=1+br$ ($b>0$). Let's write $(ps^r)... |
2,269,042 | <p>Consider $f(x) = \lfloor x \rfloor + \lfloor -x \rfloor $ . Now find value of $\lim_{x \to \infty} f(x) $ . I know that if $x_0 \in \mathbb{R}$ then $\lim_{x \to x_0} f(x) = -1$ but I don't know whether it is true or not in the infinity . </p>
| boaz | 83,796 | <p>It has no limit when $x\to\infty$. Consider the sequences
$$
x_n=n\qquad y_n=n+\frac{1}{2}
$$
Both sequence tend to $\infty$, but notice that $f(x_n)=0$ while $f(y_n)=-1$ for every $n$.</p>
|
2,612,416 | <p>Can you please help me with this limit? I can´t use L'Hopital rule.</p>
<p>$$\lim_{x\to \infty} \frac{\sqrt{4x^2+5}-3}{\sqrt[3]{x^4}-1} $$</p>
| Guy Fsone | 385,707 | <p>Since the OP seems to be doubtfully on the whether $x\to \infty$ or $x\to1$ I have included both answers:</p>
<hr>
<p>If $x\to \infty$ then we have $$\lim_{x\to \infty} \frac{\sqrt{4x^2+5}-3}{\sqrt[3]{x^4}-1} =\lim_{x\to \infty}{\sqrt{4+{5\over x^2}}-{3\over x}\over \sqrt[3]{x}-{1\over x}} =\lim_{x\to \infty}{\sq... |
2,612,416 | <p>Can you please help me with this limit? I can´t use L'Hopital rule.</p>
<p>$$\lim_{x\to \infty} \frac{\sqrt{4x^2+5}-3}{\sqrt[3]{x^4}-1} $$</p>
| Martín-Blas Pérez Pinilla | 98,199 | <p>Hint: use that
$$a^2 - b^2 = (a - b)(a + b)$$
and
$$a^3 - b^3 = (a - b)(a^2 + ab + b^2).$$</p>
|
227,311 | <p>It is known that the roots of Chebyshev polynomials of the second kind, denote it by $U_n(x)$, are in the interval $(-1,1)$. I have noticed that, by looking at the low values of $n$, the roots of $(1-x)U_n(x)+U_{n-1}(x)$ are in the interval $(-2,2)$. However, I don't have a clear idea how to start proving this, co... | John Jiang | 4,923 | <p>The key is to write $U_n(x)$ in terms of $x$ explicitly. Let $x = \cos t$, then $\sin t = \sqrt{1-x^2}$, where we fixed a branch of square root so that $\sqrt{-1} = i$; the value of $U_n$ is independent of the choice. Then $e^{it} = x + i \sqrt{1 - x^2}$. For $|x| \ge 1$, we can thus write $e^{it} = x - \sqrt{x^2 - ... |
1,902,138 | <p>It's common to see a plus-minus ($\pm$), for example in describing error
$$
t=72 \pm 3
$$
or in the quadratic formula
$$
x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}
$$
or identities like
$$
\sin(A \pm B) = \sin(A) \cos(B) \pm \cos(A) \sin(B)
$$</p>
<p>I've never seen an analogous version combining multiplication with div... | barak manos | 131,263 | <p>I think that this question is primarily opinion-based, so here is my opinion:</p>
<ul>
<li>The expression $[t=72\pm3]$ is equivalent to $[t=72+(+3)]\vee[t=72+(-3)]$</li>
<li>The expression $[t=72\frac{\times}{\div}3]$ would be equivalent to $[t=72\times3]\vee[t=72\times\frac13]$</li>
</ul>
<p>So the second operand... |
4,121,607 | <p>I want to find a function which satisfies certain following limits.</p>
<p>The question is:
Find a function which satisfies</p>
<p><span class="math-container">$$
\lim_{x\to5} f(x)=3, \text{ and } f(5) \text{ does not exist}
$$</span></p>
<p>I would think that because it says <span class="math-container">$f(5)$</spa... | Amaan M | 860,916 | <p>There are, generally, three types of discontinuities: removable, jump, and asymptotic. If you have a jump or asymptotic discontinuity, a finite two-sided limit won't exist.</p>
<p>If you have a jump discontinuity, your left-handed and right-handed limits aren't equal, so the limit doesn't exist.</p>
<p>If you have a... |
4,017,554 | <p>Simple question - how to prove that:</p>
<p><span class="math-container">$\sqrt {x\times y} = \sqrt x \times \sqrt y$</span> ?</p>
<p>If I use the exponentation the answer seems easy, because</p>
<p><span class="math-container">$(x\times y)^n = x^n \times y^n$</span> because I get</p>
<p><span class="math-containe... | uriyaba | 728,938 | <p>Since <span class="math-container">$n$</span> isn't a natural number, we can approach the proof from a different angle.</p>
<p>Change the square root and rewrite it as a power of a <span class="math-container">$0.5 $</span>, then use the <a href="https://mathinsight.org/exponentiation_basic_rules#power_product" rel=... |
1,043,956 | <p>Find a normal vector and a tangent vector to the curve given by the equation: $x^5 + y ^5 =2x^3$ at the point $P(1, 1)$. Find the equation of the tangent line. <br/>
Edit: The notes I have:<img src="https://i.stack.imgur.com/arOee.png" alt="enter image description here"></p>
<p>Taking $f(x, y) = x^5 - 2x^3 + y^5 = ... | HDE 226868 | 170,257 | <p>You can find the slope of the tangent line by differentiating. If you don't want to do implicit differentiation (which may be simpler in this case), you can just do some algebra beforehand:
$$x^5+y^5=2x^3 \to y^5=2x^3-x^5 \to y=\sqrt[5]{2x^3-x^5}$$
and differentiate according to the power rule. You have the slope of... |
97,672 | <p>Given that I have a set of equations about varible $x_0,x_1,\cdots,x_n$, which own the following style:</p>
<p>$
\left(
\begin{array}{cccccccc}
\frac{1}{6} & \frac{2}{3} & \frac{1}{6} & 0 & 0 & 0 & 0 & 0 \\
0 & \frac{1}{6} & \frac{2}{3} & \frac{1}{6} & 0 & 0 & 0... | Cesareo | 62,129 | <p>This script accepts the equations (consistent equations) with the unknowns in any order.</p>
<pre><code>vars = Variables[eqns]
A = Grad[eqns, vars]
B1 = Map[First, yValues]
B2 = Map[Last, yValues]
LinearSolve[A, B1]
LinearSolve[A, B2]
</code></pre>
|
268,185 | <p><strong>Question:</strong> </p>
<ol>
<li>Given a PDE, is there a general method to show that it is <em>not solvable</em> using the inverse scattering transform?</li>
<li>Specifically, for the perturbed 1D NLS or the 2D cubic NLS, where was it first shown that these equations can not be solved using <em>any</em> for... | Phil Harmsworth | 106,467 | <p>The prolongation structure method developed by Wahlquist and Estabrook is one method to show whether or not a PDE is solvable via the inverse scattering transform. (There are others - refer Y. Kosmann-Schwarzbach, B. Grammaticos, K. M. Tamizhmani (eds.), <em>Integrability of Nonlinear Systems</em>, Lect. Notes Phys... |
1,890,047 | <p>Consider two linear transformations $L_1, L_2: V \to W$.</p>
<p>Fix a basis of $V$, $W$, and consider $M_1$, $M_2$, the matrices of the aforementioned transformations w.r.t said basis.</p>
<p>Suppose you can obtain $M_2$ from swapping columns in $M_1$.</p>
<p>How are $L_1$ and $L_2$ related? (Besides having the s... | paf | 333,517 | <p>If $M_2$ is obtained from $M_1$ by swapping the first two columns (for example), and if we denote by $(v_1,\dots,v_n)$ the basis of $V$, then we have
$$L_2(v_1)=L_1(v_2)\text{ and }L_2(v_2)=L_1(v_1).$$</p>
<p>This is because in the matrix of a linear map $L:V\to W$, the $j$th column contains the coordinates (in th... |
102,963 | <p>Could someone please explain the difference between the group of all icosahedral symmetries and S5? I know that the former is a direct product, but don't they work the same? Say I have an icosahedron, why wouldn't S5 work as a description of its symmetries? Thank you very much.</p>
<p><strong>Added:</strong> When c... | Thomas Andrews | 7,933 | <p>Hint: If $g$ is the symmetry that send each point to it's opposite point in the icosahedron, then show $\{1,g\}$ is a normal subgroup of the group of symmetries. Show that $S_5$ does not have any normal subgroup of order $2$.</p>
|
360,464 | <p>Is there a publication containing this obvious fact: For any real <span class="math-container">$T>0$</span>, any natural <span class="math-container">$n$</span>, any complex <span class="math-container">$c_1,\dots,c_n$</span>, and any distinct complex <span class="math-container">$z_1,\dots,z_n$</span> such that ... | Alexandre Eremenko | 25,510 | <p>Let <span class="math-container">$y_k(t)=e^{tz_k}$</span>. Proving by contradiction, suppose that they are linearly dependent, that is
<span class="math-container">$$\sum_{k=1}^nc_ky_k\equiv 0.$$</span>
Differentiating <span class="math-container">$n-1$</span> times we obtain a homogeneous system of linear equation... |
360,464 | <p>Is there a publication containing this obvious fact: For any real <span class="math-container">$T>0$</span>, any natural <span class="math-container">$n$</span>, any complex <span class="math-container">$c_1,\dots,c_n$</span>, and any distinct complex <span class="math-container">$z_1,\dots,z_n$</span> such that ... | Todd Trimble | 2,926 | <p>I will recount the more general statement of linear independence of characters, given in Lang's Algebra book, and credited to Artin. Let <span class="math-container">$G$</span> be a group, and <span class="math-container">$K$</span> a field. Then distinct homomorphisms <span class="math-container">$\phi_1, \ldots, \... |
5,739 | <p>Hi,
I have recently got interested in multi-index (multi-dimensional) Dirichlet series, i.e. series of the form $F(s_1,...,s_k)=\sum_{(n_1,...,n_k)\in\mathbb{N}^k}\frac{a_{n_1,...,n_k}}{n_1^{s_1}...n_k^{s_k}}$. I found some papers suggesting that multi-index Dirichlet series are in fact a distinct subfield for itsel... | user3010 | 3,010 | <p>I don´t know about general multi-index Dirichlet series, but there is a good amount of theory on <em>multiple zeta-functions</em> (special cases of what you are asking for). There is plenty of stuff in MathSciNet on this.</p>
|
1,207,250 | <p>I have to calculate $\lim_{x \to \infty}{x-x^2\ln(1+\frac{1}{x})}$. I rewrote it as $\lim_{x \to \infty}{\frac{x-x^3\ln^2(1+\frac{1}{x})}{1 + x\ln(1+\frac{1} {x})}}$ and tried to apply L'Hôpital's rule but it didn't work. How to end this?</p>
| Prasun Biswas | 215,900 | <p>Make the substitution $t=\dfrac{1}{x}$. Then, $x\to\infty \implies t\to 0$</p>
<p>$$\lim_{x\to\infty}\left(x-x^2\ln\left(1+\frac{1}{x}\right)\right)\\ = \lim_{t\to 0}\left(\frac{1}{t}-\frac{1}{t^2}\ln(1+t)\right)\\= \lim_{t\to 0}\left(\frac{t-\ln(1+t)}{t^2}\right)$$</p>
<p>This comes out as $\frac{0}{0}$ on direct... |
1,207,250 | <p>I have to calculate $\lim_{x \to \infty}{x-x^2\ln(1+\frac{1}{x})}$. I rewrote it as $\lim_{x \to \infty}{\frac{x-x^3\ln^2(1+\frac{1}{x})}{1 + x\ln(1+\frac{1} {x})}}$ and tried to apply L'Hôpital's rule but it didn't work. How to end this?</p>
| tqviet | 378,056 | <p>The reason why L'Hôpital's rule didn't work is that you added on a determined-term into the limit, i.e.
$$
1 + x \ln \left( 1 + \frac{1}{x} \right)
=
1 + \ln \left( \left( 1 + \frac{1}{x} \right)^x \right)
\to
1 + \ln \left( e \right)= 2
$$
when $x \to \infty$. Another approach with the one from Prasun Biswas is u... |
182,346 | <p>Let's call a polygon $P$ <em>shrinkable</em> if any down-scaled (dilated) version of $P$ can be translated into $P$. For example, the following triangle is shrinkable (the original polygon is green, the dilated polygon is blue):</p>
<p><img src="https://i.stack.imgur.com/M0LOu.png" alt="enter image description here... | Stewart Hinsley | 76,354 | <p>If I understand the question correctly the requirement is for a figure, F, such there exists a translation T(c) for all contractions c, such that cF+T(c) lies within F. It seems to me that that criterion holds for monoconvex hexagons (chevrons) and biconvex hexagons (hourglasses), which are not stars.</p>
|
3,062,701 | <p>I want to solve this system by Least Squares method:<span class="math-container">$$\begin{pmatrix}1 & 2 & 3\\\ 2 & 3 & 4 \\\ 3 & 4 & 5 \end{pmatrix}\begin{pmatrix}x\\y\\z\end{pmatrix} =\begin{pmatrix}1\\5\\-2\end{pmatrix} $$</span> This symmetric matrix is singular with one eigenvalue <span ... | random | 513,275 | <p>Since the matrix has the eigenvector <span class="math-container">$\pmatrix{1&-2&1\cr}^t$</span> with eigenvalue <span class="math-container">$0$</span>, one has</p>
<p><span class="math-container">$$\begin{pmatrix}1 & 2 & 3\\\ 2 & 3 & 4 \\\ 3 & 4 & 5 \end{pmatrix}\begin{pmatrix}x\\y... |
1,515,776 | <p>How can I solve something like this?</p>
<p>$$3^x+4^x=7^x$$</p>
<p>I know that $x=1$, but I don't know how to find it. Thank you!</p>
| lab bhattacharjee | 33,337 | <p>HINT:</p>
<p>As for $0<a<1,$ $$a^m>a^n$$ if $m<n,$</p>
<p>$$\left(\dfrac37\right)^m+\left(\dfrac47\right)^m>\left(\dfrac37\right)^n+\left(\dfrac47\right)^n$$</p>
<p>if $m<n$</p>
|
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