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 |
|---|---|---|---|---|
1,212,336 | <p>Let R be the set of all real numbers. Is $\{\mathbb R^+,\mathbb R^−,\{0\}\}$ a partition of $\mathbb R$? Explain your answer.</p>
<p>My answer is no because of $\{0\}$. I am confused with $\{0\}$. please help.</p>
| Batman | 127,428 | <p>Draw a picture of $\log x$ and the left and right Riemann sums corresponding to the integral with interval width $1$. Which one matches the sum on the left hand side? </p>
|
74,347 | <blockquote>
<p>Construct a function which is continuous in $[1,5]$ but not differentiable at $2, 3, 4$.</p>
</blockquote>
<p>This question is just after the definition of differentiation and the theorem that if $f$ is finitely derivable at $c$, then $f$ is also continuous at $c$. Please help, my textbook does not h... | AD - Stop Putin - | 1,154 | <p>How about $f(x) = \max(\sin(n\pi x),0)$ or perhaps $g(x) = |\sin(n\pi x)|$?</p>
|
1,290,516 | <p>Find the values of $m$ if the line $y=mx+2$ is a tangent to the curve $x^2-2y^2=1$.</p>
<p>My working:</p>
<p>First we differentiate $x^2-2y^2=1$ with respect to $y$ to get the gradient. We get $y^2=\frac{1}{2}x^2-\frac{1}{2}\implies y=\pm\sqrt{\frac{1}{2}x^2-\frac{1}{2}}$.</p>
<p>We take the positive one for dem... | André Nicolas | 6,312 | <p>Let $(a,b)$ be a point of tangency. We have $2x-4y\frac{dy}{dx}=0$, so the slope of the tangent line at $(a,b)$ (if $b\ne 0$) is $\frac{a}{2b}$.</p>
<p>The tangent line has equation $y-b=(x-a)(a/2b)$. Simplifying , and comparing with $y=mx+2$, we find that $b-a^2/(2b)=2$. It follows that $2b^2-a^2=4b$. Since $a^2-2... |
1,701,176 | <p>The problem I'm having is with the logs. I go:</p>
<p>$$\lim_{n \to \infty} \Big( \frac{\log{(n+1)}}{\log{(n)}} \cdot \frac{n-2}{n-1} \Big)$$</p>
<p>$$=\lim_{n \to \infty} \Big( \frac{\log{(n+1)}}{\log{(n)}}\Big) \cdot \lim_{n \to \infty} \Big(\frac{n-2}{n-1} \Big)$$</p>
<p>and here I know that $$\lim_{n \to \inf... | Doug M | 317,176 | <p>I say, $\lim\limits_{n \to \infty} \Big( \frac{\log{(n+1)}}{\log{(n)}}\Big) = 1 $
In wich case, I must show that</p>
<p>$\forall \epsilon > 0, \exists N>0$ such that $n>N\implies |\Big( \frac{\log{(n+1)}}{\log{(n)}}\Big) -1|<\epsilon$</p>
<p>$|\Big( \frac{\log{(n+1)}-\log{(n)}}{\log{(n)}}\Big)|<\eps... |
1,131,622 | <p>The question itself is a very easy one:<br/></p>
<blockquote>
<p>Somebody has got two kids, one of whom is a girl. Then what's the probability that he's got <strong>at least</strong> one boy?</p>
</blockquote>
<p>My answer is that, since he's already got a girl, then "he's got at least one boy" amounts to "the o... | Community | -1 | <p>The a priori probabilities indeed follow a binomial distribution, and all pairs are equiprobable
$$P(BB)=P(BG)=P(GB)=P(GG)=\dfrac14.$$
The distribution of the number of boys follows $(\dfrac14,\dfrac12,\dfrac14)$.</p>
<p>Now you are told that $BB$ is excluded, then the a posteriori probabilities turn to
$$P(BB|\lno... |
394,517 | <p>How can I evaluate $\lim_{x \to \infty}\left(\sqrt{x+\sqrt{x}}-\sqrt{x-\sqrt{x}}\right)$?</p>
| lab bhattacharjee | 33,337 | <p>HINT:</p>
<p>As we are dealing with the limit in real numbers, $x>0\implies x\to+\infty$</p>
<p>Put $x=y^2$</p>
<p>$$\implies \sqrt{x+\sqrt{x}}-\sqrt{x-\sqrt{x}}$$</p>
<p>$$=\sqrt{y^2+y}-\sqrt{y^2-y}$$</p>
<p>$$=\frac{y^2+y-(y^2-y)}{\sqrt{y^2+y}+\sqrt{y^2-y}} \text{ (Rationalizing the numerator )}$$</p>
<p>... |
394,517 | <p>How can I evaluate $\lim_{x \to \infty}\left(\sqrt{x+\sqrt{x}}-\sqrt{x-\sqrt{x}}\right)$?</p>
| doubting thomas | 78,251 | <p>Rationalize. Then divide througout by $\sqrt x$. Then substitute limit.</p>
|
2,867,207 | <p>This question is a (perhaps naive) 'simplification' of a result in a paper, so the answer could be negative.</p>
<p>Define the cone $\Sigma(\theta)$ for $\theta\in(0,\pi/2]$,
$$\Sigma(\theta) = \left\{ z = x+iy : x>0, |y|<(\tan\theta)x\right\}, $$
and define the norm $\|f\|_\theta$ for functions $f$ analytic ... | DeepSea | 101,504 | <p>For starters such as yourself, you can begin by assuming otherwise. That is the fraction is reducible. So $\exists k \in \mathbb{N}, k > 1$ such that $k \mid 21n + 4, k \mid 14n+3\implies 21n+4 = ak, 14n+3 = bk$ for some natural numbers $a,b$. Thus: $42n+8 = 2ak, 42n+9 = 3bk\implies 3bk - 2ak = 1\implies k(3b-2a... |
3,208,613 | <p>We have a fair <span class="math-container">$3$</span> sided die <span class="math-container">$(a,b,c)$</span>, and we perform the following experiment:</p>
<p>Roll the die until we have seen <span class="math-container">$10$</span> of any of the sides, let <span class="math-container">$X$</span> be the number of t... | Jane Cooper | 647,608 | <p>Hint: Observe that by the fundamental theorem of algebra, every polynomial of degree 2 has 2 roots. Therefore, if we have 2 polynomials of degree 2, we need one of them to have a root with multiplicity and not the other one. The quadratic formula will be useful here.</p>
<p>Edit: Matthew pointed out that it could b... |
2,479,363 | <p>For every square matrix $A$, does there always exists a non diagonal matrix $B$ such that AB=BA</p>
| Will Jagy | 10,400 | <p>well, no, not if $A$ is diagonal with all distinct diagonal entries. </p>
<p>The simplest way to state the relevant theorem is this: for a square matrix $M,$ if the characteristic polynomial of $M$ and the minimal polynomial of $M$ are the same, then the only matrices that commute with $M$ are polynomials in $M,$ t... |
2,479,363 | <p>For every square matrix $A$, does there always exists a non diagonal matrix $B$ such that AB=BA</p>
| David314 | 493,155 | <p>In general, no.</p>
<p>However, when $A$ is not diagonal, the answer is yes, since $B=A$ satisfies $B$ is not diagonal and $AB=BA$.</p>
<p>Consider</p>
<p>$$A=
\begin{bmatrix}
0 & 0 \\
0 & 1
\end{bmatrix}
$$</p>
<p>and any</p>
<p>$$B=
\begin{bmatrix}
a & b \\
c & d
\end{bmatrix}
$$</p>
<p>Then<... |
1,119,563 | <p>Why is $\sec^{-1}(2/\sqrt{2}) = \sec^{-1}(\sqrt{2})$ true?</p>
| Workaholic | 201,168 | <p>Since </p>
<p>$$
\dfrac{2}{\sqrt{2}}=\dfrac{\sqrt{2}\sqrt{2}}{\sqrt{2}}=\sqrt{2}.
$$</p>
<p>In general, for $a\geqslant0$ we have</p>
<p>$$
\dfrac{a}{\sqrt{a}}=\sqrt{a}.
$$</p>
|
2,466,949 | <p>Room coordinates are following my walls, to use the guidance system I build the position from various other sensors & built a GPS position from it.</p>
<p>As I also need the a "fake" compass I'm trying to interface a moving robot with a sensor I made.</p>
<p>Robot expect compass to send him the values of a 3-a... | JMoravitz | 179,297 | <p>This can be explained as the following using multiplication principle:</p>
<ul>
<li>Pick which of the seven available spaces is occupied by the <code>n</code></li>
<li>Pick which of the six remaining available spaces is occupied by the <code>g</code></li>
<li>$\vdots$</li>
<li>Pick which of the four remaining space... |
2,708,891 | <p>Let $R$ be a ring and consider $f = r_nx^n + 1.x^{n-1} + \cdots + rx + r_0\; \in R[X]$ such that $r^n = 0$ for all $r \in R$. Then can I call $f$ a monic polynomial in $R[X]$ (assume $r_n$ is non-invertible)?</p>
| K B Dave | 534,616 | <p>I think the answer is <em>no</em> for the same reason that $x^p-x$ is not said to be the zero polynomial in $\mathbb{F}_p[x]$.</p>
<p>One "reason" that $x^p-x$ is not the zero polynomial in $\mathbb{F}_p[x]$ is that, even though it evaluates to zero in $\mathbb{F}_p$, it does not evaluate to zero in every $\mathbb{... |
10,974 | <p>Is the following true: If two chain complexes of free abelian groups have isomorphic homology modules then they are chain homotopy equivalent.</p>
| Mariano Suárez-Álvarez | 1,409 | <p>The natural functor $K^b(\mathbb Z\mathrm{-free})\to D^b(\mathbb Z)$ from the homotopy category of bounded complexes of finitely generated free abelian groups to the derived category of bounded complexes of finitely generated abelian groups is an equivalence. This means that a map of bounded complexes of finitely ge... |
1,239,211 | <p>I have been allowed to attend some preparatory lectures for a seminar on the Goodwillie Calculus of Functors. I found in my notes from one of the lectures two statements which I would like to ask about.</p>
<p>The first one is probably straightforward and I'm guessing is related to Whitehead-type theorems. Still, I... | Najib Idrissi | 10,014 | <p>I think it's easier to understand if you look at it the other way around. Singular homology preserves filtered colimits (exercise: prove it), but it does not preserve other types of colimits in general (exercise: find a counterexample, a very simply one in fact; if you're stuck, have a look <a href="https://math.sta... |
1,955,729 | <p>Are undecidable problems only those that have no algorithm to give a yes or no answer in a finite time or are there problems with no algorithm to give a yes no answer even in an infinite time? (If undecidability means there isn't a yes/no answer over a finite period, doesn't that mean given enough time these problem... | Robert Israel | 8,508 | <p>By definition, an algorithm must give an answer in a finite time (although that time may be arbitrarily long). </p>
<p>For example: the halting problem is undecidable. That is, there is no algorithm that, given a Turing machine and its input, decides whether or not the Turing machine given that input will halt.</... |
1,955,729 | <p>Are undecidable problems only those that have no algorithm to give a yes or no answer in a finite time or are there problems with no algorithm to give a yes no answer even in an infinite time? (If undecidability means there isn't a yes/no answer over a finite period, doesn't that mean given enough time these problem... | Mitchell Spector | 350,214 | <p>The hyperarithmetic sets of integers are precisely the ones for which membership in them is decidable algorithmically, with a finite set of instructions, if you are allowed to do an infinite sequence of steps in any positive time interval you want. (These sets are the effective analogue to the Borel sets.)</p>
<p>... |
589,309 | <p>Finding all sets of primes $p$ and $q$ such that $p$ divides $q^2 -4$ and $q$ divides $p^2-1$.</p>
| Community | -1 | <p>Hint :</p>
<p>$p$ divides $q+2$ or $q-2$ and $q$ divides $p+1$ or $p-1$</p>
<p>Consider one by one case : </p>
<p>$p=l(q+2),q=m(p+1)\Rightarrow p=l(m(p+1)+2)=lmp+lm+2l\Rightarrow p= ??$</p>
<p>try other cases...</p>
<ul>
<li>$p=l(q+2),q=m(p-1)$ </li>
<li>$p=l(q-2),q=m(p-1)$ </li>
<li>$p=l(q-2),q=m(p+1)$ </li>
<... |
2,200,034 | <p>Suppose the points A and B are connected by two roads of the length $S_1$ and $S_2$. Cars can drive from A to B on either of the two roads. They start at point A and must decide, <strong>one after the other</strong> which road to take. They know how many cars already chose to drive on each road.</p>
<p>The speed of... | Brian Tung | 224,454 | <p>I think you're better off looking for the long-run equilibrium result in the limit as time goes to infinity (assuming that interarrival times are small compared to the travel time), rather than as the arrival rate goes to infinity.</p>
<p>With that interpretation, at equilibrium, the travel times on the two roads a... |
366,844 | <p>Using the infinite product of $\sin(\pi z)$, one can find the Hadamard product for $e^z-1$:</p>
<p>$$e^z-1 =2ie^{z/2}\sin(-iz/2)= 2i e^{z/2} (-iz/2) \prod_n \left(1+\frac{z^2}{4\pi n^2}\right)\\= e^{z/2} z \prod_n \left(1+\frac{z^2}{4\pi n^2}\right).$$</p>
<p>I don't see a way to find the product for $\cos\pi z$.... | Ron Gordon | 53,268 | <p>Well, you can perform a logarithmic differentiation and get a series that may be summed using the residue theorem.</p>
<p>Let $p(z)$ be the product in question; we intend to prove that $p(z)=\cos{\pi z}$. </p>
<p>$$\log{p} = \sum_{n=0}^{\infty} \log{\left ( 1-\frac{4 z^2}{(2 n+1)^2}\right)}$$</p>
<p>$$\frac{d}{d... |
366,844 | <p>Using the infinite product of $\sin(\pi z)$, one can find the Hadamard product for $e^z-1$:</p>
<p>$$e^z-1 =2ie^{z/2}\sin(-iz/2)= 2i e^{z/2} (-iz/2) \prod_n \left(1+\frac{z^2}{4\pi n^2}\right)\\= e^{z/2} z \prod_n \left(1+\frac{z^2}{4\pi n^2}\right).$$</p>
<p>I don't see a way to find the product for $\cos\pi z$.... | Felix Marin | 85,343 | <p><span class="math-container">$\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\exp... |
2,429,769 | <p>I watched a <a href="https://www.youtube.com/watch?v=lXkRj6MKbZs" rel="nofollow noreferrer">great youtube video</a> about how to prove a limit of a multivariable function exists. It explained that one method is by substitution. For example, we can solve $$lim_{(x, y) \to 0,0} \frac{xy}{\sqrt{x^2 + y^2}}$$</p>
<p>By... | An aedonist | 143,679 | <p>Maybe the following visualisation could be helpful.</p>
<p>Draw a rectangle, with a base long $1$, and height long $v$, initial velocity.</p>
<p>Next to it, to the right, you can draw a rectangle whose base is still long $1$, while the height equals $v - d$, $d$ for drag.
You could continue, so that the $n$-th rec... |
1,208,323 | <p>I am trying to prove that a $n \times n$ matrix $A$ and $A^T$ have the same eigenvalues.</p>
<p>I can prove that $A$ and $A^T$ have the same entries on the diagonal, but I am not sure where to go from there.</p>
| PersonaA | 226,382 | <p>Hint: They will have the same eigenvalues if they have the same characteristic polynomial. (which can be shown that they do have the same easily)</p>
|
1,208,323 | <p>I am trying to prove that a $n \times n$ matrix $A$ and $A^T$ have the same eigenvalues.</p>
<p>I can prove that $A$ and $A^T$ have the same entries on the diagonal, but I am not sure where to go from there.</p>
| Community | -1 | <p>$\lambda$ is an eigenvalue of the $n \times n$ matrix $A$ iff $\det(A-\lambda I)=0$.</p>
<p>Remember that the determinant of a matrix is equal to the determinant of its transpose. Thus if $\det(A-\lambda I)=0$ then $\det([A-\lambda I]^T)=0$. But $[A-\lambda I]^T = A^T-\lambda I^T = A^T - \lambda I$. Therefore $\... |
1,301,522 | <p>Many texts will define a manifold as "a second-countable Hausdorff space that is locally homeomorphic to Euclidean space".
By definition of homeomorphism, shouldn't this really and officially read as "locally homeomorphic to a <em>subset</em> of Euclidean space"?</p>
| Rob Arthan | 23,171 | <p>Definitely not: "locally homeomorphic to an <em>open</em> subset of Euclidean space" would be equivalent to the stated (and standard) definition, but Euclidean $n$-space for any $n > 0$ has subsets that are not manifolds, e.g., $\{0\} \cup \{1/n : 0 < n \in \mathbb{N}\} \subseteq \mathbb{R}$ or the union of th... |
241,998 | <p>Consider a list of even length, for example <code>list={1,2,3,4,5,6,7,8}</code></p>
<p>what is the fastest way to accomplish both these operations ?</p>
<p><strong>Operation 1</strong>: two by two element inversion, the output is:</p>
<pre><code>{2,1,4,3,6,5,8,7}
</code></pre>
<p>A code that work is:</p>
<pre><code>... | kglr | 125 | <pre><code>PermutationList[Cycles[Partition[list,2]]]
</code></pre>
<blockquote>
<pre><code>{2, 1, 4, 3, 6, 5, 8, 7}
</code></pre>
</blockquote>
<p>Simply <code>Reverse</code> the output above to get your second list:</p>
<pre><code>Reverse @ %
</code></pre>
<blockquote>
<pre><code>{7, 8, 5, 6, 3, 4, 1, 2}
</code></p... |
3,772,399 | <p>I need help with the following question:</p>
<p>Let <span class="math-container">$X_i$</span> be independent, non-negative random variables, <span class="math-container">$i \in \{1,...,n\}$</span>. I want to show that for all <span class="math-container">$t > 0$</span>, <span class="math-container">$$P(S_n > ... | Eric Wofsey | 86,856 | <p>You can make a continuity argument to reduce to the case of diagonalizable matrices. The characteristic polynomial of <span class="math-container">$C_A$</span> varies continuously with <span class="math-container">$A$</span>, and diagonalizable matrices are dense in <span class="math-container">$GL_n(\mathbb{C})$</... |
1,989,182 | <p>Why does only one particular solution allow enough degrees of freedom for the general solution?</p>
| Artem | 29,547 | <p>This is an elementary but very important fact for any <em>linear</em> operators. That is, let $A$ be a linear operator, $A\colon U\longrightarrow V$. Consider the problem
$$
A(u)=v,\tag{1}
$$
that is, to find a $u\in U$ for a given $v\in V$. Assume that such solution exists. Then it is true that the general solution... |
2,221,897 | <p>Show that </p>
<p>$$\lim_{n \to \infty} \sum_{k=3}^n \frac{2k}{k^2+n^2+1} = \ln(2)$$</p>
<p>How many ways are there to prove it ?</p>
<p>Is there a standard way ?</p>
<p>I was thinking about making it a Riemann sum.
Or telescoping.</p>
<p>What is the easiest way ?
What is the shortest way ?</p>
| Claude Leibovici | 82,404 | <p><em>Just added for your curiosity.</em></p>
<p>Riemann sum is certainly the fastest way to do it but you can also do it differently using generalized harmonic numbers (after partial fraction decomposition) and obtain
$$S_n=\sum_{k=3}^n \frac{2k}{k^2+n^2+1}=-H_{2-\sqrt{-n^2-1}}+H_{n-\sqrt{-n^2-1}}-H_{\sqrt{-n^2-1}+2... |
2,221,897 | <p>Show that </p>
<p>$$\lim_{n \to \infty} \sum_{k=3}^n \frac{2k}{k^2+n^2+1} = \ln(2)$$</p>
<p>How many ways are there to prove it ?</p>
<p>Is there a standard way ?</p>
<p>I was thinking about making it a Riemann sum.
Or telescoping.</p>
<p>What is the easiest way ?
What is the shortest way ?</p>
| Felix Marin | 85,343 | <p>$\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
\... |
1,265,531 | <p>I understand the question but I am not sure how to solve it. For example, if we flip HHHTTTTT then the next three must be heads because of the question. This however seems counterintuitive. I believe that there are $2^{10}$ possible strings, but I am unsure of how to count all possible strings that begin with HHH.</... | ajotatxe | 132,456 | <p>If $A$ means three heads at the beginning, and $B$ means $5$ heads and $5$ tails, we want to compute
$$p(A/B)=\frac{p(A\cap B)}{p(B)}$$</p>
<p>And
$$p(A\cap B)=\frac{\binom72}{2^{10}}$$
$$p(B)=\frac{\binom{10}5}{2^{10}}$$</p>
|
33,153 | <p>Here is one definition of a differential equation:</p>
<blockquote>
<p>"An equation containing the derivatives of one or more dependent variables, with respect to one of more independent variables, is said to be a differential equation (DE)" <em>(Zill - A First Course in Differential Equations)</em></p>
</... | Sam Lisi | 8,343 | <blockquote>
<p><i>
"When I use a word," Humpty Dumpty said, in a rather a scornful tone, "it means just what I choose it to mean—neither more nor less."</i></p>
</blockquote>
<p>I think Arnol'd is correct, but I think he is being unnecessarily confrontational about it. All the books on your list that I am famili... |
378,966 | <p>$$A_t-A_{xx} = \sin(\pi x)$$
$$A(0,t)=A(1,t)=0$$
$$A(x,t=0)=0$$
Find $A$.</p>
<p>I know I need to find the homogeneous and particular solutions. Im just not sure on this PDE.</p>
| Ron Gordon | 53,268 | <p>The solution may be accomplished using a Laplace transform. Defining</p>
<p>$$\hat{A}(x,s) = \int_0^{\infty} dt \, A(x,t) \, e^{-s t}$$</p>
<p>and applying the initial condition, we get an ordinary differential equation in $x$:</p>
<p>$$\frac{d^2}{dx^2} \hat{A} - s \hat{A} = -\frac{1}{s} \sin{\pi x}$$</p>
<p>Th... |
163,640 | <p>Early in a course in Algebra the result that every group can be embedded as a subgroup<br>
of a symmetric group is introduced. One can further work on it to embed it as a subgroup of a suitable (higher degree) alternating group.</p>
<p>Inverting the view point we can say that the family of simple groups $A_n, n\... | Derek Holt | 35,840 | <p>In general, for the groups in a family of fixed Lie rank, there will be a bound on the degree of an alternating group that can occur as a subgroup, so the answer to your question is no. This is easily seen from the fact that they have representations of a fixed degree. For example $E_8(q)$ has a representation of de... |
163,640 | <p>Early in a course in Algebra the result that every group can be embedded as a subgroup<br>
of a symmetric group is introduced. One can further work on it to embed it as a subgroup of a suitable (higher degree) alternating group.</p>
<p>Inverting the view point we can say that the family of simple groups $A_n, n\... | DavidLHarden | 12,610 | <p>Another bit of information available from Cayley's Theorem: </p>
<p>It is possible to prove, without using the transfer homomorphism, that a finite group $G$ with a cyclic (and nontrivial) Sylow 2-subgroup has a normal 2-complement.<br>
First we show that having a cyclic, nontrivial Sylow 2-subgroup implies the gr... |
2,767,679 | <blockquote>
<p>Let $A$ be an $m \times n$ matrix and let $B, C$ be $n \times p$ matrices. Prove that $A(B + C) = AB + AC$</p>
</blockquote>
<p>I know it's obvious that it is and that every mathematician takes this for granted but I've been asked to prove it and I don't know how to do it without just multiplying out... | Joppy | 431,940 | <p>The $(i, j)$th entry of the left hand side is
$$ \sum_{k = 1}^n a_{ik}(b_{kj} + c_{kj})$$
while the $(i, j)$th entry of the right hand side is
$$ \sum_{k = 1}^n a_{ik}b_{kj} + \sum_{k = 1}^n a_{ik} c_{kj}$$
which are indeed equal. And since every entry is equal, the matrices must be equal.</p>
|
3,910,739 | <p>I am trying to find a pdf for a random variable <span class="math-container">$X$</span> where <span class="math-container">$X=-2Y+1$</span> and <span class="math-container">$Y$</span> is given by <span class="math-container">$N(4,9)$</span></p>
<p>Here is my attempt:</p>
<p>we know <span class="math-container">$\mu=... | Kolmogorov | 551,240 | <p>Alternatively, one may use the very well known (and easy) fact that</p>
<blockquote>
<p>If <span class="math-container">$S \sim N(0,1)$</span> , then <span class="math-container">$T = aS + b \sim N(b , a^2)$</span> .</p>
</blockquote>
<p>Here, as <span class="math-container">$Y \sim N(4,9)$</span> , so <span class="... |
870,240 | <p>Which number is larger? $\underbrace{888\cdots8}_\text{19 digits}\times\underbrace{333\cdots3}_\text{68 digits}$ or $\underbrace{444\cdots4}_\text{19 digits}\times\underbrace{666\cdots67}_\text{68 digits}$? Why? How much is it larger?</p>
| Martin Sleziak | 8,297 | <p>You have<br>
$\underbrace{888\cdots8}_\text{19 digits}\times\underbrace{333\cdots3}_\text{68 digits} =
8\cdot 3 \cdot (\underbrace{111\cdots1}_\text{19 digits}\times \underbrace{111\cdots1}_\text{68 digits}) =
24 \cdot (\underbrace{111\cdots1}_\text{19 digits}\times \underbrace{111\cdots1}_\text{68 digits})$.<br>
... |
4,115,069 | <p>I understand 'functionals' as functions of functions, for example:</p>
<p><span class="math-container">$$ S[y(x)]= \int_{t_1}^{t_2} \sqrt{1+(y')^2} dx$$</span></p>
<p>Which is the famous arc length integral</p>
<p>Now, in a similar way, a limit we can write as:</p>
<p><span class="math-container">$$L(a, [y(x)] ) = \... | user21820 | 21,820 | <p>It's not been pointed out yet, but your syntax is wrong. In "<span class="math-container">$L(0,e^x)$</span>" the variable "<span class="math-container">$x$</span>" is undefined, so the whole thing is meaningless if you want <span class="math-container">$L$</span> to be a function. The point is th... |
16,627 | <p>Yesterday, I wrote <a href="https://math.stackexchange.com/a/904777">this answer</a>, but then realized that the OP had considered breaking things into specific cases, so I deleted my answer. Right after I deleted my answer, I saw that the OP had accepted my answer. I undeleted my answer and commented to the OP, ask... | Community | -1 | <p>This appears to be a <a href="http://en.wikipedia.org/wiki/Race_condition" rel="nofollow noreferrer">race condition</a> between deletion and acceptance. In the example linked above, the answer's revision history shows it <a href="https://math.stackexchange.com/posts/904777/revisions">was deleted</a> at 6:23:36 on Au... |
2,193,550 | <p>Prove that $G$ acts faithfully on $X$ when there are no two elements of $G$ acting the same way on an element $X$.</p>
<p>So I don't have much of a proof, but here's what I'm thinking. I know that for $G$ to act faithfully on $X$, the identity is the only element that fixes every element in $X$, so $\forall x \in ... | Daniel | 150,142 | <p>You can easily verify that the order of the given element is 8: just compute all powers and check that the first one that yields 1 is the 8th power.</p>
<p>On the other hand, you're right when you say that the polynomial in the quotient is irreducible (there are no square roots of -2 in $\Bbb F_5$), but since the d... |
1,821,248 | <p>Which of the following are true?</p>
<ol>
<li><p>$\sigma \circ \sigma(j)=j~\forall j, 1 \leq j \leq 5$.</p></li>
<li><p>$\sigma^{-1}(j)=\sigma(j)~\forall j, 1 \leq j \leq 5$.</p></li>
<li><p>The set $ \{k: \sigma(k) \neq k \}$ has even number of elements.</p></li>
<li><p>The set $\{k:\sigma(k) =k \}$ has an odd num... | Virtuoz | 153,521 | <p>For $k=1,\ldots 5$ denote
$$
i_k = \sigma(k),\; k = \sigma^{-1}(i_k)
$$
Then
$$
k=\sigma^{-1}(i_k) \le \sigma(i_k)
$$
That's why $\sigma(i_5) = 5, \sigma(i_4) = 4, \ldots \sigma(i_1) = 1$.
Since $\sigma(i_k) = k$ and $\sigma^{-1}(i_k) = k$ we get $$\sigma = \sigma^{-1}$$
Next steps must be obvious :)</p>
|
1,821,248 | <p>Which of the following are true?</p>
<ol>
<li><p>$\sigma \circ \sigma(j)=j~\forall j, 1 \leq j \leq 5$.</p></li>
<li><p>$\sigma^{-1}(j)=\sigma(j)~\forall j, 1 \leq j \leq 5$.</p></li>
<li><p>The set $ \{k: \sigma(k) \neq k \}$ has even number of elements.</p></li>
<li><p>The set $\{k:\sigma(k) =k \}$ has an odd num... | drhab | 75,923 | <p>In general: If $\tau$ and $\sigma$ are permutations on $\left\{ 1,\dots,n\right\} $
with $\tau\left(j\right)\leq\sigma\left(j\right)$ for each $j\in\left\{ 1,\dots,n\right\} $
then with induction it can be shown that $\sigma^{-1}\left(i\right)=\tau^{-1}\left(i\right)$
for $i=1,\dots,n$ or equivalently $\tau=\sigma$.... |
1,300,273 | <p>I have a question about evaluating the limit:</p>
<p>$$\lim_{x \to\infty }\left(x^{f(x)}-x \right)$$</p>
<p>where:</p>
<p>$f(x)$ is a continuous map from the positive reals to the positive reals , and</p>
<p>$\lim_{x\rightarrow \infty }f(x)= 1$.</p>
<p>I attempted to apply L'Hôpital's rule by writing:</p>
<p>... | Community | -1 | <p>Let $L$ the desired limit. Take $f(x)=1$ we get trivially $L=0$. Now take $f(x)=1+\frac1{\ln x}$ we get</p>
<p>$$x^{f(x)}-x=x\left(x^{f(x)-1}-1\right)=x\left(e-1\right)\xrightarrow{x\to\infty}\infty$$
so we see that the result depends on the choice of $f$.</p>
|
14,340 | <blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="https://mathematica.stackexchange.com/questions/3247/consistent-plot-styles-across-multiple-mma-files-and-data-sets">Consistent Plot Styles across multiple MMA files and data sets</a> </p>
</blockquote>
<p>So, here's my problem; I have a lot of d... | J. M.'s persistent exhaustion | 50 | <p>Use <code>Sequence[]</code> for the purpose:</p>
<pre><code>optionPacket = Sequence[PlotStyle -> {RGBColor[1, 0, 0]}, Frame -> True,
BaseStyle -> {FontSize -> 20}]
{ListPlot[RandomVariate[NormalDistribution[], {7, 2}], optionPacket],
ListPlot[RandomVariate[WeibullDistribution[1... |
3,915,771 | <p>I'm given the series:</p>
<p><span class="math-container">$$\sum_{n=2}^{\infty} \frac{n^2}{n^4-n-3}$$</span></p>
<p>I know it converges, however I'm meant to show that by the comparison test. What would be a good choice here? <span class="math-container">$\frac{1}{k^2}$</span> and <span class="math-container">$\frac... | José Carlos Santos | 446,262 | <p>Since <span class="math-container">$\sum_{n=2}^\infty\frac1{n^2}$</span> converges and since<span class="math-container">$$\lim_{n\to\infty}\frac{\dfrac{n^2}{n^4-n-3}}{\dfrac1{n^2}}=1,$$</span>your series converges.</p>
|
672,707 | <p><img src="https://i.stack.imgur.com/Tr5Jy.gif" alt="enter image description here" /></p>
<p>2 How do I solve this equation involving a logarithm? 3</p>
| Newb | 98,587 | <p>$$\log_2\left(\frac{x}{2}\right) = \log_3\left(\frac{2+x}{3}\right)$$</p>
<p>$$\log_2(x)-\log_2(2) = \log_3(2+x)-\log_3(3)$$</p>
<p>$$\log_2(x)-1 = \log_3(2+x)-1$$</p>
<p>$$\log_2(x) = \log_3(2+x)$$</p>
<p>$$\log_3(2+x)=\frac{\log_{10}(2+x)}{\log_{10}(3) }$$</p>
<p>$$\log_2(x) = \frac{\log_{10}(x)}{\log_{10}(2)... |
15,237 | <p><a href="https://matheducators.stackexchange.com/questions/176/knowing-mathematics-does-not-translate-to-knowing-to-teach-mathematics-why">A question</a> has been asked about why great mathematicians are not necessarily great teachers. On the other hand, I am wondering if knowing more mathematics actually helps with... | guest | 11,935 | <p>My personal impression across many fields is that a certain amount of extra subject matter can help (especially with "top high school" classes like AP Calculus or AP Chemistry or AP Bio). but that by and large, the issues in teaching and learning intro topics like first year college calc, chem, physics, bio, and th... |
15,237 | <p><a href="https://matheducators.stackexchange.com/questions/176/knowing-mathematics-does-not-translate-to-knowing-to-teach-mathematics-why">A question</a> has been asked about why great mathematicians are not necessarily great teachers. On the other hand, I am wondering if knowing more mathematics actually helps with... | Jessica B | 4,746 | <p>There's a general point I can't see explicitly in the other answers: knowing more maths (and generally having spent time knowing/thinking about maths) helps you have a bigger picture. A lot of maths starts to fit together better as you know more for longer.</p>
<p>As a (not very good) analogy, suppose someone left ... |
285,227 | <p>I am trying to prove $\exp(x+y) = \exp(x) \exp(y)$.</p>
<p>I may use that $$\exp(x) = \sum_{n=0}^\infty \frac {x^n}{n!}$$
I further know how to multiply two power series in one point, i.e. if $f(x) = \sum_{n=0}^\infty c_n(x-a)^n$ and $g(x) = \sum_{k=0}^\infty d_n(x-a)^n$ then
$$
f(x)g(x) = \sum_{n=0}^\infty e_n(x-a... | Community | -1 | <p>My solution</p>
<p>Let $x,y \in \mathbb R$ and </p>
<p>$f(z) := \sum_{n=0}^\infty \left(\frac {x^n}{n!} \right )z^n$ and $g(z) := \sum_{n=0}^\infty \left(\frac {y^n}{n!} \right )z^n$. Then $\exp(x) \exp(y) = f(1)g(1)$. That is
$$
f(z)g(z) = \sum_{n=0}^\infty \left( \sum_{k=0}^m \frac {x^m y^{n-m}}{m! (n-m)!} \rig... |
285,227 | <p>I am trying to prove $\exp(x+y) = \exp(x) \exp(y)$.</p>
<p>I may use that $$\exp(x) = \sum_{n=0}^\infty \frac {x^n}{n!}$$
I further know how to multiply two power series in one point, i.e. if $f(x) = \sum_{n=0}^\infty c_n(x-a)^n$ and $g(x) = \sum_{k=0}^\infty d_n(x-a)^n$ then
$$
f(x)g(x) = \sum_{n=0}^\infty e_n(x-a... | nordmann | 59,392 | <p>$A(t)=\exp(x) = \sum_{n=0}^\infty \frac {x^n}{n!}t^n$<br>
$B(t)=\exp(y) = \sum_{n=0}^\infty \frac {y^n}{n!}t^n$<br>
$C(t) = A(t)*B(t)=\sum_{n=0}^\infty (\sum_{k+z=n}^\ \frac {x^k}{k!}*\frac {y^z}{z!})t^n=\sum_{n=0}^\infty \frac {(x+y)^n}{n!}t^n=exp(x+y)$</p>
<p>and use $t=1$</p>
<p>sry i was too late^^</p>
|
599,602 | <p>Please help with this calculus question. I'm asked to solve
$$(1+y^2) \,\mathrm{d}x = (\tan^{-1}y - x)\,\mathrm{d}y.$$</p>
| alexjo | 103,399 | <p>The ODE
$$
(1+y^2)+(x-\arctan y)y'=0\tag 1
$$
is not exact because, putting $M(x,y)=1+y^2$ and $N(x,y)=x-\arctan y$, $$M_y=\frac{\partial M}{\partial y}=2y\neq 1=\frac{\partial N}{\partial x}=N_x.$$
We have to find an integrating factor $\mu(y)$ such that
$$\frac{\partial (\mu M)}{\partial y}=\frac{\partial (\mu ... |
2,462,722 | <p><a href="https://i.stack.imgur.com/jtWGA.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/jtWGA.jpg" alt="enter image description here"></a></p>
<p>I got this from QFT Demystified in the author attempt to derive the Euler Lagrange equation. But isn't the Taylor expansion for $f(x+a)$ supposed to b... | Gerhard S. | 474,939 | <p>The confusion arises because of the notation you are using. Write (1) as
$$f(x+a)=f(a)+xf'(a).$$
Now proceed as you suggest and replace $x$ by $\epsilon$ and $a$ by $x$. This yields
$$f(\epsilon+x)=f(x)+\epsilon f'(x).$$
Is it clearer now?</p>
|
2,740,349 | <blockquote>
<p>A triangle has the side lengths of $3$, $5$, and $7$. Express $\cos(y)+\sin(y)$, where $y$ is the largest angle in the triangle.</p>
</blockquote>
<p>I have tried to apply pythagoras theorm, trying to express the other two angles in some way, split the triangle into smaller triangles, but all without... | Nico | 340,686 | <p>Yes, in fact this is sometimes taken as the definition of the determinant. You may have seen the determinant defined as the unique alternating, $n$-linear map such that $\det(e_1,\dots,e_n) = 1$. Rephrased, we can say this succinctly as "the unique $n$-form such that $\omega (e_1,\dots,e_n) = 1$. </p>
<p>One way to... |
3,692,435 | <p>prove the following identity:</p>
<p><span class="math-container">$\displaystyle\sum_{k=0}^{n}\frac{1}{k+1}\binom{2k}{k}\binom{2n-2k}{n-k} = \binom{2n+1}{n}$</span></p>
<p>what I tried:</p>
<p>I figured that: <span class="math-container">$\displaystyle\binom{2n+1}{n} = (2n+1) C_n$</span>
and <span class="math-con... | Angina Seng | 436,618 | <p>This is
<span class="math-container">$$\sum_{k=0}^n C_kA_{n-k}=B_n\tag{*}$$</span>
where
<span class="math-container">$$C_n=\frac1{n+1}\binom{2n}{n},$$</span>
<span class="math-container">$$A_n=\binom{2n}{n}$$</span>
and
<span class="math-container">$$B_n=\binom{2n+1}{n}.$$</span>
All we need to confirm (*) is to pr... |
2,098,882 | <p>How to show that:</p>
<p>$\forall k\in\mathbb{N}^*$ and $\forall x\in\mathbb{R}^*$, the inequality $\left(kx-1\right)e^{kx}>-1$ holds.</p>
<p>Thank you for your help.</p>
| Bernard | 202,857 | <p>It results from the variations of $f$ on $\mathbf R$: $f'(x)=k^2x\mathrm e^{kx}$, so $f$ decreases on $\mathbf R^-$, increases on $\mathbf R^+$ and has a minimum at $x=0$, hence for $x\ne 0$, $f(x)>f(0)=-1$.</p>
|
2,098,882 | <p>How to show that:</p>
<p>$\forall k\in\mathbb{N}^*$ and $\forall x\in\mathbb{R}^*$, the inequality $\left(kx-1\right)e^{kx}>-1$ holds.</p>
<p>Thank you for your help.</p>
| PMC1234 | 338,746 | <p>We consider the function $f$ defined on $\mathbb{R}$ such that $f(x)=(kx-1)e^{kx}$ with $k$ a natural integer (excluding zero).</p>
<p>Let $X=kx$. We then have $f(X)=(X-1)e^X$. Let's show that $f$ is strictly superior to $-1$. </p>
<p>$f$ is differentiable on $\mathbb{R}$ where:
\begin{align*}
\forall X\in\mathbb{... |
3,752,770 | <p>I tested this in python using:</p>
<pre><code>import numpy as np
import matplotlib.pyplot as plt
x = np.linspace(0, 10*2*np.pi, 10000)
y = np.sin(x)
plt.plot(y/y)
plt.plot(y)
</code></pre>
<p>Which produces:</p>
<p><a href="https://i.stack.imgur.com/pCwoV.png" rel="nofollow noreferrer"><img src="https://i.stack.img... | John Hughes | 114,036 | <p>Let's ask a simpler question: is <span class="math-container">$\frac{x}{x} = 1$</span> ?</p>
<p>The answer (which follows from the axioms for a field) is that <span class="math-container">$y = \frac{x}{x} = x \cdot x^{-1}$</span> is <em>undefined</em> if <span class="math-container">$x = 0$</span>, so while <span cl... |
202,719 | <p><code>Reduce</code> often provides a much fuller solution than <code>Solve</code>. But it's always in the form of a true statement rather than functions or replacement rules, e.g.</p>
<p>Input:</p>
<pre><code>Reduce[Sin[x^2] + Cos[a] == 0 && -π/2 <= x <= π/2, x]
</code></pre>
<p>Output:</p>
<pre><c... | AsukaMinato | 68,689 | <p>Convert it to string and solve it.</p>
<pre><code>convert[x_] :=
x /. (Or[(a_) && (b_)]) :> {b, a} // InputForm // ToString //
StringReplace[#, "||" -> ","] & // "Piecewise[{" ~~ # ~~ "}]" & //
ToExpression;
</code></pre>
<p>this works for easy situation, for example:</p>
<pre><... |
440,439 | <p>I was working more on the topic on my previous question when I have to know whether the following statement is true to circumvent the "exception" caused by division by singular matrices; again, long story short, the statement follows:</p>
<p>If two singular matrices <span class="math-container">$A, B$</spa... | Conrad | 133,811 | <p>This holds in general in any domain <span class="math-container">$U$</span>, while of course if <span class="math-container">$f=h^2$</span> is bounded, then <span class="math-container">$h$</span> bounded, so the extra condition is automatically satisfied (similarly note that <span class="math-container">$f$</span> ... |
4,527,300 | <p>An AP practice question asks:</p>
<p><span class="math-container">$$\lim_{h\to0} \frac{(1+h)^3 + \frac{8}{\sqrt{1+h}}-9}{h} $$</span></p>
<p>The answer should be -1. How did they get this without a calculator?</p>
| Lorago | 883,088 | <p>Let <span class="math-container">$f(x)=x^3+\frac{8}{\sqrt{x}}$</span>. Then <span class="math-container">$f(1)=9$</span>, and <span class="math-container">$f(1+h)=(1+h)^3+\frac{8}{\sqrt{1+h}}$</span>. This means that you limit can be written as</p>
<p><span class="math-container">$$\lim_{h\to0}\frac{f(1+h)-f(1)}{h}=... |
4,527,300 | <p>An AP practice question asks:</p>
<p><span class="math-container">$$\lim_{h\to0} \frac{(1+h)^3 + \frac{8}{\sqrt{1+h}}-9}{h} $$</span></p>
<p>The answer should be -1. How did they get this without a calculator?</p>
| binbni | 1,011,566 | <p>We can also calculate it this way.
<span class="math-container">\begin{align}
\lim_{h \to 0}\frac{{(1+h)^3}+\frac{8}{\sqrt{1+h}}-9}{h}
&=\lim_{h\to0}\frac{{h^3+3h^2+3h+8(\frac{1}{\sqrt{1+h}}-1)}}{h} \\
&=\lim_{h \to 0}h^2+3h+3+8\frac{1-\sqrt{1+h}}{h\sqrt{1+h}}\\
&=\lim_{h \to 0}h^2+3h+3+8\frac{1-(1+h)}{h... |
1,291,511 | <p>This may seem like a silly question, but I just wanted to check. I know there are proofs that if $f(x)=f'(x)$ then $f(x)=Ae^x$. But can we 'invent' another function that obeys $f(x)=f'(x)$ which is <strong>non-trivial</strong>?</p>
| Michael Hardy | 11,667 | <p>You have $\dfrac{dy}{dx}=y$. Often one writes $\dfrac{dy} y = dx$ and then evaluates both sides of $\displaystyle\int\frac{dy} y = \int dx$, etc.</p>
<p>However, for a question like this perhaps one should be more careful.</p>
<p>If $f(x)\ne 0$ for all $x$, then one has $\dfrac{f'(x)}{f(x)}=1$ for all $x$. This ... |
1,291,511 | <p>This may seem like a silly question, but I just wanted to check. I know there are proofs that if $f(x)=f'(x)$ then $f(x)=Ae^x$. But can we 'invent' another function that obeys $f(x)=f'(x)$ which is <strong>non-trivial</strong>?</p>
| Lukas Betz | 238,388 | <p>Consider $g(x) = f(x)\exp(-x)$. Then we have $g'(x) = f'(x)\exp(-x)-f(x)\exp(-x) = 0$. Thus, $g\equiv c$ for some constant $c$. Hence $f(x) = c\exp(x)$.</p>
|
1,822,008 | <p>Here are two functions:
$f\left(u,v\right)=u^{2}+3v^{2}$</p>
<p>$g\left(x,y\right)=\begin{pmatrix} e^{x}\cos y \\ e^{x}\sin y \end{pmatrix} $</p>
<p>I need to make Jacobian matrix of $f\circ g$. I found derivative of their composition:</p>
<p>$\frac{d\left(f\circ g\right) }{d\left(x,y\right) }=2e^{2x}\cos^{2}{y... | b00n heT | 119,285 | <p>Using the chain rule instead:
\begin{align*}D(f\circ g)(x,y)& =\color{red}{Df(g(x,y))}\cdot\color{blue}{ Dg(x,y)}\\
& = \color{red}{\begin{pmatrix} 2u&6v \end{pmatrix}\circ(g(x,y))}\cdot \color{blue}{ \begin{pmatrix}e^x\cos y & -e^x\sin y \\ e^x\sin y & e^x\cos y\end{pmatrix}}\\
& =\color{red... |
1,175,297 | <p>Note: The following definitions from my book, Discrete Mathematics and Its Applications [7th ed, 598].</p>
<p>This is my book's definition for a reflexive relation
<img src="https://i.stack.imgur.com/og5wE.png" alt="enter image description here"></p>
<p>This is my book's definition for a anti symmetric relation
<i... | Michael Burr | 86,421 | <p>No, antisymmetric is not the same as reflexive.</p>
<p>An example of a relation that is reflexive, but not antisymmetric is the relation
$$R={(1,1),(1,2),(2,2),(2,1)}$$ on $$A={1,2}.$$ It is reflexive because for all elements of $A$ (which are $1$ and $2$), $(1,1)\in R$ and $(2,2)\in R$. The relation is not anti-... |
154,757 | <p>I have this data:</p>
<ul>
<li><p>$a=6$</p></li>
<li><p>$b=3\sqrt2 -\sqrt6$ </p></li>
<li><p>$\alpha = 120°$</p></li>
</ul>
<p><strong>How to calculate the area of this triangle?</strong></p>
<p>there is picture:</p>
<p><img src="https://i.stack.imgur.com/hr2Cp.jpg" alt=""></p>
| zzzzzzzzzzz | 26,498 | <p>Assuming the diagram like so, where C = $\alpha=120$<img src="https://i.stack.imgur.com/I9Umc.gif" alt="enter image description here"></p>
<p>Then we have the equation </p>
<p>$$Area = \displaystyle\frac{a b\sin C}{2}$$</p>
<p>This is the same as the equation you probably know,</p>
<p>$$Area = \displaystyle\frac... |
3,222,871 | <p>Let <span class="math-container">$P(x, y, 1)$</span> and <span class="math-container">$Q(x, y, z)$</span> lie on the curves <span class="math-container">$$\frac{x^2}{9}+\frac{y^2}{4}=4$$</span> and <span class="math-container">$$\frac{x+2}{1}=\frac{y-\sqrt{3}}{\sqrt{3}}=\frac{z-1}{2}$$</span> respectively. Then find... | Vedant Chourey | 638,765 | <p>You can use the method of Lagrange's multipliers. The function formed by the distance between the two points <span class="math-container">$ (x,y,z)$</span> and <span class="math-container">$(x,y,1)$</span> is examined.
i.e. <span class="math-container">$\phi = \sqrt{(z-1)^2} $</span>
The constrains are respectively
... |
3,222,871 | <p>Let <span class="math-container">$P(x, y, 1)$</span> and <span class="math-container">$Q(x, y, z)$</span> lie on the curves <span class="math-container">$$\frac{x^2}{9}+\frac{y^2}{4}=4$$</span> and <span class="math-container">$$\frac{x+2}{1}=\frac{y-\sqrt{3}}{\sqrt{3}}=\frac{z-1}{2}$$</span> respectively. Then find... | Christian Blatter | 1,303 | <p>You can do it without using Lagrange's method. Consider the parametric representations
<span class="math-container">$$p(s):=\bigl(6\cos s,4\sin s,1\bigr)\qquad(s\in{\mathbb R}/(2\pi))$$</span>
and
<span class="math-container">$$q(t):=\bigl(t-2,\sqrt{3}(t+1),2t+1\bigr)\qquad(t\in{\mathbb R})\ .$$</span>
We have to de... |
3,222,871 | <p>Let <span class="math-container">$P(x, y, 1)$</span> and <span class="math-container">$Q(x, y, z)$</span> lie on the curves <span class="math-container">$$\frac{x^2}{9}+\frac{y^2}{4}=4$$</span> and <span class="math-container">$$\frac{x+2}{1}=\frac{y-\sqrt{3}}{\sqrt{3}}=\frac{z-1}{2}$$</span> respectively. Then find... | Claude Leibovici | 82,404 | <p>Starting from @Christian Blatter's answer, using <span class="math-container">$s=2 \tan ^{-1}(x)$</span> and expanding, we end with
<span class="math-container">$$2 \sqrt{3}\, x^4+70 \,x^3+72 \sqrt{3} \,x^2-274\, x-26 \sqrt{3}=0$$</span>
Let <span class="math-container">$x=t-\frac{35}{4 \sqrt{3}}$</span> to get the ... |
4,072,769 | <blockquote>
<p>How to evaluate this?
<span class="math-container">$$\prod_{k=1}^m \tan \frac{k\pi}{2m+1}$$</span></p>
</blockquote>
<p>My work</p>
<p>I couldn't figure out a method to solve this product.
I thought that this identity could help.
<span class="math-container">$$\frac{e^{i\theta}-1}{e^{i\theta}+1}=i\tan \... | Quanto | 686,284 | <p>Let</p>
<p><span class="math-container">$$f(x) = x^{n-1} + x^{n-2} + x^{n-3} \>\cdots \> +\>x +1
= \prod_{k=1}^{n-1}(x - e^{i\frac{2\pi k}n})
$$</span>
and set <span class="math-container">$n=2m+1$</span> to evaluate</p>
<p><span class="math-container">$$\frac{f(1)}{f(-1)}= 2m+1
= \prod_{k=1}^{2m}\frac{1 -... |
2,431,861 | <p>Let $P(z)=\displaystyle \sum_{0\le k\le n}a_kz^k$ a complex polynomial. What conditions must satisfy the coefficients $a_k$ to have $$P(z)=-\overline{P(\overline z)}\space \space ?$$</p>
| José Carlos Santos | 446,262 | <p>That's when and only whe every $a_k$ is purely imaginary.</p>
<p>If every $a_k$ is purely imaginary, then\begin{align}-\overline{P\bigl(\overline z\bigr)}&=-\overline{\sum_{k=0}^na_k\overline z^k}\\&=-\sum_{k=0}^n\overline{a_k}z^k\\&=\sum_{k=0}^na_kz^k\end{align}because $(\forall k\in\{0,1,\ldots,n\}):\... |
90,876 | <p>$$2x-\dfrac{x+1}{2} + \dfrac{1}{3}(x+3)= \dfrac{7}{3}$$</p>
<p>When I solve this I always end up with 11x = 5, which is wrong, no matter which way I solve it. Does anyone know how to solve it? Steps? (Because I know the answer should be x=1)</p>
| David Mitra | 18,986 | <p>$$\eqalign{&2x -{x+1\over 2}+{x+3\over 3 }={7\over 3}\cr
&\iff12x \color{red}{- 3}(x+1) +{2 (x+3)}={14}\cr
&\iff12x-3x\color{red}{-3}+2x+6 ={14}\cr
&\iff 11x =11\cr
&\iff x=1
}
$$</p>
<p>You most likely forgot to "distribute the negative" (since you said you obtained $11x=5$).</p>
<p><... |
928,644 | <blockquote>
<p>Let $f,g$ be $\mathcal E$-$\mathcal B(\mathbb R)$-measurable functions. I want to show piecewise function $h$ of $f$ and $g$ is also measurable.</p>
</blockquote>
<p>Suppose $(X, \mathcal E)$ is a measure space, let $f,g$ be $\mathcal E$-$\mathcal B(\mathbb R)$-measurable functions and let $A \in \ma... | PhoemueX | 151,552 | <p>Do you know that multiplication and addition of measurable functions are again measurable? If yes, simply note that</p>
<p>$$
h = f \cdot \chi_A + g \cdot \chi_{A^c},
$$</p>
<p>where $\chi_A$ is the characteristic function of $A$.</p>
|
198,739 | <p>I'm actually doing an exercise where I have to draw graphs of functions.
I understand r=|s| but not |r|=|s|.
Are they the same?</p>
| Brian M. Scott | 12,042 | <p>They are not the same. If $r=|s|$, then $r$ can never be negative, but $|r|=|s|$ is true if $r=-1$ and $s=1$ (or for that matter if $s=-1$). The statement that $|r|=|s|$ just says that $r=\pm s$, i.e., that $r=s$ or $r=-s$: in both cases $r$ and $s$ will have the same absolute value, regardless of their algebraic si... |
103,540 | <p>Suppose you have a triangular chessboard of size $n$, whose "squares" are ordered triples $(x,y,z)$ of nonnegative integers that add up to $n$. A rook can move to any other point that agrees with it in one coordinate -- for example, if you are on $(3,1,4)$ then you can move to $(2,2,4)$ or to $(6,1,1)$, but not to ... | Will Sawin | 18,060 | <p>For n=6 you can fit 5 rooks</p>
<p>(0,2,4)
(4,0,2)
(1,4,1)
(3,3,0)
(2,1,3)</p>
<p>For n=9 you can fit 7 rooks</p>
<p>(0,3,6)
(6,0,3)
(2,6,1)
(4,5,0)
(3,1,5)
(5,2,2)
(1,4,4)</p>
|
1,983,614 | <p>Consider a measurable space $(\Omega, \mathcal{F})$ and let $I$ be an arbitrary index set. </p>
<p>Is the following true?</p>
<blockquote>
<p>If $\left( A_i \right)_{i \in I}$ is a chain in $\mathcal{F}$ – that is, $\forall i \in I$, $A_i \in \mathcal{F}$ and for all $i, j \in I$, we have $A_i \subseteq A_... | Michael Hardy | 11,667 | <p>Within the known axioms of set theory you cannot disprove that $2^{\aleph_0} = \aleph_1.$ Recall that $\aleph_1$ is defined as the cardinality of the set of all countable ordinals and $2^{\aleph_0}$ is the cardinality of $[0,1]$.</p>
<p>Let $A$ be any non-measurable subset of $[0,1]$. Suppose $|[0,1]| = 2^{\aleph_... |
2,710,200 | <p>I need to find the norm of an operator from $l^2 \to l^1$, but I'm struggling because of the different norms on $l^2$ and $l^1$. </p>
<p>The operator is defined by $T:l^2 \to l^1, x_i \mapsto 2^{-i}x_i$. </p>
<p>Using the canonical basis, I have that $||T||\geq 1/2$, but I have a feeling this is not a very good lo... | Rigel | 11,776 | <p>Using the Cauchy-Schwarz inequality you get
$$
\|Tx\|_1 = \sum_{i=1}^\infty 2^{-i} |x_i| \leq
\left(\sum_{i=1}^\infty 4^{-i}\right)^{1/2}
\left(\sum_{i=1}^\infty|x_i|^2\right)^{1/2}
= \frac{1}{\sqrt{3}} \|x\|_2.
$$
On the other hand, if you choose $x=(x_i)$ with $x_i = 2^{-i}$, you check in a moment that you get eq... |
2,515,765 | <p>The following question is from an intermediate calculus book I am going through: </p>
<p>Find two sets in $\mathbb R^2$ that have the same interior, but whose complements have different interiors. </p>
<p>This seems like the kind of question that should be fairly straightforward, but I just can't think of an answe... | William Elliot | 426,203 | <p>{0} and {0,1}........................</p>
|
3,082,944 | <blockquote>
<p>Prove that space <span class="math-container">$X$</span> of all symmetric matrices in <span class="math-container">$GL_2(\mathbb R)$</span> with both the eigenvalues belonging to the interval <span class="math-container">$(0,2),$</span> with the topology inherited from <span class="math-container">$M_... | José Carlos Santos | 446,262 | <p>The space <span class="math-container">$SO_2(\mathbb{R})$</span> is connected. Now let<span class="math-container">$$\Lambda=\left\{\begin{bmatrix}\lambda_1&0\\0&\lambda_2\end{bmatrix}\,\middle|\,\lambda_1,\lambda_2\in(0,2)\right\}.$$</span>The set <span class="math-container">$\Lambda$</span> is connected t... |
1,520,643 | <p>If A² = I, prove that the matrix A is diagonalizable.</p>
<p>I have computed the eigenvalues to be 1 or -1 but I'm not sure how to proceed from here. </p>
<p>I'm thinking along the lines of "since rank(A + I) + rank(A - I) = n, therefore there exists n linearly independent vectors which corresponds to n eigenvecto... | skyking | 265,767 | <p>You're right the eigenvalues are $\pm1$. Next step is to observe that eigenvectors corresponding to a eigenvector form a linear subspace - therefore you can create a base for these linear subspace. Then you end up with a base $u_j$ and $v_k$ so that $Au_j = u_j$ and $Av_k=-v_k$. Express $A$ in this base.</p>
|
1,520,643 | <p>If A² = I, prove that the matrix A is diagonalizable.</p>
<p>I have computed the eigenvalues to be 1 or -1 but I'm not sure how to proceed from here. </p>
<p>I'm thinking along the lines of "since rank(A + I) + rank(A - I) = n, therefore there exists n linearly independent vectors which corresponds to n eigenvecto... | abel | 9,252 | <p>we know that the eigenvalues of $A$ are $\pm 1.$ suppose the dimension of the null space of $A - I$ is $k\ge 1$ and a basis for the null space is $\{x_1, x_2, \ldots, x_k\}.$ pick any $y$ not in the null space of $A-I.$ then $Ay - y \ne 0$ and $A(Ay-y) = A^2y-Ay = y-Ay = -1(Ay-y)$ that is $Ay-y$ is an eigenvector ... |
2,853,668 | <blockquote>
<p>Show that $$\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{x^n}$$ converges for every $x>1$.</p>
</blockquote>
<p>let $a(x)$ be the sum of the series. does $a$ continious at $x=2$? differentiable?</p>
<p>I guess the first part is with leibniz but I am not sure about it.</p>
| marty cohen | 13,079 | <p>Let's look at the
partial sums,
and let $y = -1/x$
so
$-1 < y < 0$..</p>
<p>$\begin{array}\\
s_m(y)
&=\sum_{n=1}^{m} (-1)^{n-1}(-y)^n\\
&=\sum_{n=1}^{m} (-1)^{n-1}(-1)^ny^n\\
&=-\sum_{n=1}^{m} y^n\\
&=-y\sum_{n=0}^{m-1} y^n\\
&=-y\dfrac{1-y^m}{1-y}\\
&=\dfrac{-y}{1-y}-\dfrac{-y^{m+1}}{... |
2,969,004 | <p>I have seen several references to "order" of an element in the Symmetric Group. Specifically, that the order of a cycle is the least common multiple of the lengths of the cycles in its decomposition.</p>
<p>But the Symmetric Group is not cyclic, and I'm only familiar with the concept of "order" for cyclic groups. S... | seamp | 606,999 | <p>The order of an element <span class="math-container">$g$</span> in a finite group <span class="math-container">$G$</span> is the smallest integer <span class="math-container">$n \in \mathbb{N}^*$</span> such that <span class="math-container">$g^n = e$</span> (the neutral element of the group). This is well-defined f... |
15,871 | <p>I would like to state something about the existence of solutions $x_1,x_2,\dots,x_n \in \mathbb{R}$ to the set of equations</p>
<p>$\sum_{j=1}^n x_j^k = np_k$, $k=1,2,\dots,m$</p>
<p>for suitable constants $p_k$. By "suitable", I mean that there are some basic requirements that the $p_k$ clearly need to satisfy... | fedja | 1,131 | <p>Normally your "defect" is called an "additional assumption"/"extra condition"/... and the typical phrase is "the inverse implication also holds under the additional assumption that...". Yes, the search for such things is something that mathematicians do on an everyday basis trying to bridge the gap between what is n... |
15,871 | <p>I would like to state something about the existence of solutions $x_1,x_2,\dots,x_n \in \mathbb{R}$ to the set of equations</p>
<p>$\sum_{j=1}^n x_j^k = np_k$, $k=1,2,\dots,m$</p>
<p>for suitable constants $p_k$. By "suitable", I mean that there are some basic requirements that the $p_k$ clearly need to satisfy... | Gerald Edgar | 454 | <p>In one field of mathematics, it's the "Tauberian condition".</p>
|
3,134,991 | <p>If nine coins are tossed, what is the probability that the number of heads is even?</p>
<p>So there can either be 0 heads, 2 heads, 4 heads, 6 heads, or 8 heads.</p>
<p>We have <span class="math-container">$n = 9$</span> trials, find the probability of each <span class="math-container">$k$</span> for <span class="... | Selene Routley | 10,549 | <p>Nine coins, so that two events </p>
<p><span class="math-container">$\mathscr{E}_1$</span> = #heads is even and </p>
<p><span class="math-container">$\mathscr{E}_2$</span> = #tails is even</p>
<p>are mutually exclusive (the number of tails is 9 - number of heads, so former is even iff latter odd) and comprise all... |
2,406,061 | <p>I am also confused about whether these are symbols or have some meaning of their own.
PS- I know that <span class="math-container">$\operatorname{d}y\over\operatorname{d}x$</span> geometrically represents the slope. But, I've come across <span class="math-container">$\operatorname{d}x\over\operatorname{d}y$</span> t... | Daniel Cunha | 355,450 | <p>You should be very careful, those are merely notations.</p>
<p>$\frac{d\,y(x)}{d\,x}$ is the derivative of a variable $y$ with respect to $x$. It represents how much $y$ varies for small variations of $x$. If you draw the curve of $y(x)$, the derivative will be the slope, as you said.</p>
<p>The opposite works as ... |
1,291,107 | <p>Let $X$ be random variable and $f$ it's density. How can one calculate $E(X\vert X<a)$?</p>
<p>From definition we have:</p>
<p>$$E(X\vert X<a)=\frac{E\left(X \mathbb{1}_{\{X<a\}}\right)}{P(X<a)}$$</p>
<p>Is this equal to:</p>
<p>$$\frac{\int_{\{X<a\}}xf(x)dx}{P(X<a)}$$</p>
<p>? If yes, then ho... | KittyL | 206,286 | <p>From the beginning:
$$(1-x)(x-5)^3=x-1\\
(1-x)(x-5)^3+1-x=0\\
(1-x)(x-5)^3+(1-x)=0\\
(1-x)[(x-5)^3+1]=0\\$$</p>
<p>This implies $1-x=0$ or $(x-5)^3=-1$. I believe you can solve these.</p>
|
97,130 | <p>I tried to prove that
$$(1-2x)^2=1/3+4/\pi^2\sum_1^\infty \cos(2n x \pi)/n^2$$ for $x \in [0,1)$ with Fourier analysis, but I just found a Fourier series which defines the function. I also found the fourier series of $\cos(2n x \pi)$.</p>
<p>I don't think these results are helpful.</p>
<p>Any suggestions on how to... | AD - Stop Putin - | 1,154 | <p><strong>Hint/Problems</strong></p>
<p>(<em>Changed since the previous was wrong -- the function is not even in it self..</em>)</p>
<ol>
<li><p>Note first that $x\mapsto e(x)=(1-2x)^2$ is even on $[0,1]$ in the sense $e(x)= e(1-x)$ (either visualise it or by computation $(1-2(1-x))^2 = (1-2+2x)^2 = (-1+2x)^2)$).</p... |
97,130 | <p>I tried to prove that
$$(1-2x)^2=1/3+4/\pi^2\sum_1^\infty \cos(2n x \pi)/n^2$$ for $x \in [0,1)$ with Fourier analysis, but I just found a Fourier series which defines the function. I also found the fourier series of $\cos(2n x \pi)$.</p>
<p>I don't think these results are helpful.</p>
<p>Any suggestions on how to... | robjohn | 13,854 | <p>Consider the equivalent problem using $y=x-\frac12$ on the interval $[-\frac12,\frac12]$: Prove that
$$
4y^2=\frac13+\frac{4}{\pi^2}\sum_{n=1}^\infty(-1)^n\frac{\cos(2\pi ny)}{n^2}\tag{1}
$$
Since $\pi\csc(\pi z)$ has residue $(-1)^n$ at each integer, let's consider $f_y(z)=\pi\csc(\pi z)\frac{\cos(2\pi zy)}{z^2}$... |
182,756 | <p>I'm trying to solve a non linear ODE numerically with <code>ParametricNDSolve</code>, but as far as I got is shown below. My problem is to set the find root correctly.
What I know is this: <code>x'[0] == 0, x[R] == 0, x'[R] == 0</code>.
Any help?
Here is my code: </p>
<pre><code>c = -0.7177;
r1 = 0.8;
r2 = 125;
R ... | Henrik Schumacher | 38,178 | <p>As Alex Trounev said, this is a second-order ODE with discontinuous right-hand side. You can use <code>Piecewise</code> to set up the forcing term:</p>
<pre><code>rhs = Piecewise[{
{c n0 Exp[-x[r]] + (3 h)/(a^3 - b^3), a <= r < b}
},
c n0 Exp[-x[r]]
]
</code></pre>
<blockquote>
<p><span class="math-c... |
182,756 | <p>I'm trying to solve a non linear ODE numerically with <code>ParametricNDSolve</code>, but as far as I got is shown below. My problem is to set the find root correctly.
What I know is this: <code>x'[0] == 0, x[R] == 0, x'[R] == 0</code>.
Any help?
Here is my code: </p>
<pre><code>c = -0.7177;
r1 = 0.8;
r2 = 125;
R ... | Alex Trounev | 58,388 | <p>This problem has a solution. It is given below</p>
<pre><code>c = 0.72;
h = 300;
a = 15;
b = 17;
R = 25;
f[r_] := Piecewise[{{0, 0 <= r <= a}, {(3 h)/(a^3 - b^3),
a < r <= b}, {0, b < r <= R}}]
ps = ParametricNDSolveValue[{x''[r] + 2 x'[r] ==
c n0 Exp[-x[r]] + ... |
64,130 | <p>This is an arithmetic follow-up to my previous question <a href="https://mathoverflow.net/questions/64112/does-there-exist-a-non-trivial-semi-stable-curve-of-genus-1-with-only-4-singular">Does there exist a non-trivial semi-stable curve of genus >1 with only 4 singular fibres</a> </p>
<p>Let $k$ be an algebraica... | JSE | 431 | <p>I have the opposite intuition -- I would think the answer would be yes for all g. In genus 1, you are asking (I think) whether there are elliptic curves with prime conductor. There are a lot of elliptic curves of prime conductor; I believe the question of whether there are infinitely many is open, and considered h... |
70,803 | <p>Let $S$ be the sphere in $\mathbb{R}^3$ and $C:[0,1]\to S$ a continuously differentiable curve on $S$. Let $T:[0,1]\to\mathbb{R}^3$ denote the tangent vector of $C$. Let $P(t)$ be the plane containing $C(t)$ and having normal vector $T(t)$.</p>
<p>Given a size $d$ of the "paint brush" we define the "brush" $b:[0,1]... | Jean-Marc Schlenker | 9,890 | <p>This question is somewhat related to <a href="https://mathoverflow.net/questions/69099/shortest-closed-curve-to-inspect-a-sphere">this recent one</a>. More precisely, the comment by Gjergji Zaimi in the earlier question gives a painting of length $2\sqrt{2}\pi$ for $d=\pi/4$, which, as explained in another comment t... |
4,510,384 | <p>In exercise 2.13 of page 43 of the book <a href="https://rads.stackoverflow.com/amzn/click/com/0134746759" rel="nofollow noreferrer" rel="nofollow noreferrer">Mathematical Proofs: A Transition to Advanced Mathematics</a> the reader is asked to state the logical negation of some statements. Of these, I find the autho... | fleablood | 280,126 | <p>Your difficulty arises from your interpreting the sentence "two sides are equal" as "two sides are equal (and the third side is a different length)". However that is not what "two sides are equal" means. "two sides are equal" means "there are two sides that are equal...... |
1,176,615 | <p>I am invited to calculate the minimum of the following set:</p>
<p>$\big\{ \lfloor xy + \frac{1}{xy} \rfloor \,\Big|\, (x+1)(y+1)=2 ,\, 0<x,y \in \mathbb{R} \big\}$.</p>
<p>Is there any idea?</p>
<p>(The question changed because there is no maximum for the set (as proved in the following answers) and I assume ... | Ishfaaq | 109,161 | <p>This needs further verification. </p>
<p><em>I believe that the maximum does not exist since the set is not bounded above.</em> </p>
<p>Suppose $x, y \gt 0$ satisfies $ (x + 1)(y + 1) = 2 $. Then we can conclude that </p>
<ol>
<li>$ xy + x+ y = 1 $</li>
<li>$ 1 + \dfrac{1}{x} + \dfrac{1}{y} = \dfrac{1}{xy} $ (di... |
451,722 | <p>I want to find the projection of the point $M(10,-12,12)$ on the plane $2x-3y+4z-17=0$. The normal of the plane is $N(2,-3,4)$.</p>
<p>Do I need to use Gram–Schmidt process? If yes, is this the right formula?</p>
<p>$$\frac{N\cdot M}{|N\cdot N|} \cdot N$$</p>
<p>What will the result be, vector or scalar?</p>
<p... | eccstartup | 26,947 | <p>Set the projection point on the plane as $P=(x,y,z)$.</p>
<p>You need three equations:</p>
<ol>
<li><p>Point $P$ on the plane.
$$2x-3y+4z=17$$</p></li>
<li><p>$\vec{MP}\perp plane$</p></li>
</ol>
<p>$$\vec{MP}\perp \vec{PQ_1}$$</p>
<p>$$\vec{MP}\perp \vec{PQ_2}$$</p>
<p>where $Q_1$ and $Q_2$ are two different p... |
75,880 | <p>Say $f:X\rightarrow Y$ and $g:Y\rightarrow X$ are functions where $g\circ f:X\rightarrow X$ is the identity. Which of $f$ and $g$ is onto, and which is one-to-one?</p>
| Brian M. Scott | 12,042 | <p>HINT: $$\begin{array}{}&&\bullet&&\\
&&&\searrow&\\
\bullet&\to&\bullet&\to&\bullet\\
X&f&Y&g&X
\end{array}$$</p>
|
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