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 |
|---|---|---|---|---|
658,758 | <p>How do you plot $$f(x,y) = \frac{x}{1-y} \text{with}~ x^2+y^2<1$$ in Mathematica or Maple?</p>
| heropup | 118,193 | <p>In <em>Mathematica</em> (version 6 or newer)</p>
<pre><code>Plot3D[x/(1 - y), {x, -1, 1}, {y, -1, 1},
RegionFunction -> Function[{x, y}, x^2 + y^2 < 1]]
</code></pre>
|
1,301,937 | <p>A curve $C$ is said to be <em>trigonal</em> if it admits a rational function $f: C \to \mathbb{CP}^1$ of degree $3$. Is any nonsingular plane curve of degree four trigonal? Can the map be chosen so that it has exactly $10$ ramification points?</p>
| Alex Fok | 223,498 | <p>For the first question, the answer is yes. We can choose a point $P$ on the nonsingular plane curve $C$ of degree 4. Project $C$ from $P$ onto a hyperplane (i.e. a copy of $\mathbb{P}^1$) in $\mathbb{P}^2$. Noting that any line passing through $P$ cuts $C$ at other three points, this projection is a rational map of ... |
3,340,107 | <p>In the solution to justify why <span class="math-container">$E_{\mathbb Q}[\vert X_{n}-X\vert ]\xrightarrow{n \to \infty} 0$</span> does not necessarily hold, given <span class="math-container">$\mathbb Q << \mathbb P$</span> and <span class="math-container">$E_{\mathbb P}[\vert X_{n}-X\vert ]\xrightarrow{n \t... | Simon | 649,092 | <p>Since <span class="math-container">$(Y_n)$</span> may not be uniformly upper bounded (i.e. <span class="math-container">$(-Y_n)$</span> may not be uniformly lower bounded), your statement that <span class="math-container">$$
E\left[\limsup_{n\to\infty} Y_n\right]\ge \limsup_{n\to\infty} E[Y_n]
$$</span> is not deduc... |
3,340,107 | <p>In the solution to justify why <span class="math-container">$E_{\mathbb Q}[\vert X_{n}-X\vert ]\xrightarrow{n \to \infty} 0$</span> does not necessarily hold, given <span class="math-container">$\mathbb Q << \mathbb P$</span> and <span class="math-container">$E_{\mathbb P}[\vert X_{n}-X\vert ]\xrightarrow{n \t... | mbartczak | 699,276 | <p>One can construct a counterexample such as Simon suggested, e.g. <span class="math-container">$$\Omega = (0,1),\ \mathbb{P}(dx) = dx,\ \mathbb{Q}(dx) = \frac{dx}{2\sqrt{x}},\ X_n(x) = n^{3/4}\ 1_{(0,1/n)}(x).$$</span>
Then
<span class="math-container">$$\mathbb{E_P}|X_n| = \frac{n^{3/4}}n\rightarrow 0$$</span>
and
<... |
18,413 | <p>The thought came from the following problem:</p>
<p>Let $V$ be a Euclidean space. Let $T$ be an inner product on $V$. Let $f$ be a linear transformation $f:V \to V$ such that $T(x,f(y))=T(f(x),y)$ for $x,y\in V$. Let $v_1,\dots,v_n$ be an orthonormal basis, and let $A=(a_{ij})$ be the matrix of $f$ with respect ... | Listing | 3,123 | <p>I did some numerical search for higher dimensions:</p>
<p>$n=3:$</p>
<p>$\left(
\begin{array}{ccc}
1 & 1 & 0 \\
1 & 1 & 1 \\
0 & 1 & 1
\end{array}
\right).\left(
\begin{array}{ccc}
-383 & 13 & -13 \\
-36 & -445 & -36 \\
-13 & 13 & -383
\end{array}
\... |
3,082,944 | <blockquote>
<p>Prove that space <span class="math-container">$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_... | Community | -1 | <p>We can prove much more (in <span class="math-container">$S_n(\mathbb{R})$</span> , the set of <span class="math-container">$n\times n$</span> real symmetric matrices).</p>
<p><span class="math-container">$\textbf{Proposition 1}$</span>. The set <span class="math-container">$Z_I=\{A\in S_n(\mathbb{R});spectrum(A)\su... |
3,082,944 | <blockquote>
<p>Prove that space <span class="math-container">$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_... | user1551 | 1,551 | <p>One-line proof: every <span class="math-container">$A\in X$</span> is path-connected to <span class="math-container">$I$</span> by the line segment <span class="math-container">$\{tA+(1-t)I: 0\leq t\leq 1\}\subseteq X$</span>.</p>
<p>Your original approach also works. Let <span class="math-container">$A,B\in X$</sp... |
3,763,744 | <p>The helix is a curve <span class="math-container">$x(t) \in \mathbb{R}^3$</span> defined by:</p>
<p><span class="math-container">$$
x(t) = \begin{bmatrix}
\sin(t) \\
\cos(t) \\
t
\end{bmatrix}
$$</span></p>
<p>and it takes the classic shape:</p>
<p><a href="https://en.wikipedia.org/wiki/File:Rising_circular.gif" rel... | Mike F | 6,608 | <p>Any answer to this question is necessarily going to be a bit arbitrary, but here are a few thoughts:</p>
<ul>
<li>We have an interesting map <span class="math-container">$\theta \mapsto (\cos \theta, \sin \theta) : \mathbb{R} \to S^1$</span>. The helix is the graph of this map.</li>
<li>In this spirit, we might cons... |
2,839,802 | <p>Consider the following system of equations</p>
<p>$$\begin{cases}
\dot x=y-x^2-x \\
\dot y=3x-x^2-y \\
\end{cases}
$$
Then, the equilibriums are $(0,0)$ and $(1,2)$.
Using linearization around $(1,2)$ one can obtain
$$
\begin{pmatrix}
\dot x \\
\dot y \\
\end{pmatrix}
=
\begin{pmatrix}
-3... | Alex Jones | 350,433 | <p>Linear stability analysis comes to a useful conclusion as long as there are no eigenvalues with zero real part. If all eigenvalues have nonzero real part, the stability of the nonlinear system will match that of the linear system. Simply put, since you have two positive real eigenvalues, the fixed point is unstable ... |
2,839,802 | <p>Consider the following system of equations</p>
<p>$$\begin{cases}
\dot x=y-x^2-x \\
\dot y=3x-x^2-y \\
\end{cases}
$$
Then, the equilibriums are $(0,0)$ and $(1,2)$.
Using linearization around $(1,2)$ one can obtain
$$
\begin{pmatrix}
\dot x \\
\dot y \\
\end{pmatrix}
=
\begin{pmatrix}
-3... | Cesareo | 397,348 | <p>Near the point $(0,0)$ the DE system can be approximated as</p>
<p>$$
\dot x = y-x\\
\dot y = 3x-y
$$</p>
<p>or</p>
<p>$$
3 x\dot x = 3 x y - 3 x^2\\
y \dot y = 3 x y - y^2
$$</p>
<p>now subtracting we have</p>
<p>$$
\frac 12\frac{d}{dt}(y^2-3x^2)+(y^2-3x^2) = 0
$$</p>
<p>so locally we have</p>
<p>$$
\frac 12... |
2,538,305 | <p>Here is the question I am struggling with:</p>
<p>A box has 16 Balls, of which 8 are Green, 6 are Red, and 2 are Blue. If you draw 2 Balls with replacement, what is the probability of getting 1 Green Ball and 1 Blue Ball in no particular order?</p>
<p>I see three different ways to get an answer to this problem. Pl... | Xander Henderson | 468,350 | <p>Another way of thinking of the problem is to actually write down the sample space and compute the probabilities. In this case, there are 9 possible outcomes:
$$ \{ GG, GR, GB, RG, RR, RB, BG, BR, BB \}, $$
where, for example $RG$ denotes the event of first drawing a red ball then a green ball. There are two "favor... |
163,534 | <p>Consider a set of $N$ points in $n$-dimensional space, i.e.
\begin{align*}
\{x_1, \dots, x_N\} \subset \mathbb R^n.
\end{align*}
Let us be given a finite family of non-injective matrices
\begin{align*}
\{M_j \in \mathbb R^{m \times n} : j = 1, \dots, J\},
\end{align*}
e.g. $m<n$.</p>
<p>In a nutshell, the ... | Tommi | 1,445 | <p>I don't provide an actual answer (I don't have one), but do provide some musings that might be helpful to others who would like to consider this problem.</p>
<p>First, if there is only one point, then the condition
$$\bigcap_{j = 1,\dots, J} \ker M_j = \{0\}$$
is both necessary and sufficient. It is clearly necessa... |
163,534 | <p>Consider a set of $N$ points in $n$-dimensional space, i.e.
\begin{align*}
\{x_1, \dots, x_N\} \subset \mathbb R^n.
\end{align*}
Let us be given a finite family of non-injective matrices
\begin{align*}
\{M_j \in \mathbb R^{m \times n} : j = 1, \dots, J\},
\end{align*}
e.g. $m<n$.</p>
<p>In a nutshell, the ... | Liviu Nicolaescu | 20,302 | <p>Denote by $S$ your finite collection of $N$ points in $\newcommand{\bR}{\mathbb{R}}$ $\bR^n$. Here is how you can recover $S$ from the knowledge of its images via a finite collections of linear maps of rank $<n$. More precisely one can use a universal family consisting of roughly $\frac{N^4}{2}$ matrices of ... |
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="... | Remellion | 639,782 | <p>There's a way to do it with barely any maths:</p>
<p>It's clear that if there's an odd number of heads, there's an even number of tails and vice versa, so P(even number of heads) + P(even number of tails) = 1.</p>
<p>Formally rename "heads" to "tails". The problem remains unchanged.</p>
<p>So P(even number of hea... |
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... | Michael Hardy | 11,667 | <p><a href="https://math.stackexchange.com/questions/200393/what-is-dx-in-integration/200403#200403">Here</a> $\leftarrow$ is something I wrote about that.</p>
<p>$dx$ is thought of as an infinitely small but nonzero increment of $x$, just as $\Delta x$ is a finite increment of $x$.</p>
<p>$dy$ is the corresponding i... |
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$$ for $x \in [0,1)$ with Fourier analysis, but I just found a Fourier series which defines the function. I also found the fourier series of $\cos(2n x \pi)$.</p>
<p>I don't think these results are helpful.</p>
<p>Any suggestions on how to... | Christian Blatter | 1,303 | <p>The series on the right hand has terms of the form $a_n\cos(2\pi n x)$ with $a_n\ne0$ for all $n$. This indicates a fundamental period of length $L=1$. When the sum of such a series is claimed to be equal to the function $f\!: x\mapsto (1-2x)^2$ on the interval $[0,1]$ then it has to be the Fourier series of the fun... |
455,979 | <p>Suppose we have three 6-sided die that all share the same common bias:</p>
<p>For a single dice: let the probability of rolling a 2 or $P(2) = 2{\times}P(1$), let the probability of rolling a 3 or $P(3) = 3{\times}P(1)$, and so on...</p>
<p>Such that:
$P(2) = 2P(1), P(3) = 3P(1), P(4) = 4P(1), P(5)=5P(1), P(6)=6P... | Henry | 6,460 | <p>For (A) and two biased dice, $P(S=3)=\frac{1\times 2 + 2\times 1}{21^2} = \frac{4}{441}$ and similarly $P(S=6)= \frac{1\times 5 + 2\times 4 + 3\times 3 + 4\times 2 + 5\times 1}{441}$ (which you can simplify).</p>
<p>For (B) and three biased dice, you cannot get a sum above $18$.</p>
<p>The probability mass functi... |
4,437,921 | <p>Given <span class="math-container">$X$</span>~<span class="math-container">$N(0,\sigma^2_X)$</span> and <span class="math-container">$Y$</span>~<span class="math-container">$N(0,\sigma^2_Y)$</span> (<span class="math-container">$X$</span> and <span class="math-container">$Y$</span> are normally distributed random va... | heropup | 118,193 | <p>Independence of <span class="math-container">$X$</span> and <span class="math-container">$Y$</span> implies <span class="math-container">$$\operatorname{E}[Z] = \operatorname{E}[\sin X] \operatorname{E}[\cos Y] = 0,$$</span> as you already observed.</p>
<p>The variance is more complicated. You'd need to compute <sp... |
2,736 | <p><a href="https://mathoverflow.net/questions/18989/generating-classical-groups-over-finite-local-rings">Generating Classical Groups over Finite Local Rings</a> asks a question that, according to the poster's own 'answer' <a href="https://mathoverflow.net/a/19098/2383">https://mathoverflow.net/a/19098/2383</a>, is not... | Todd Trimble | 2,926 | <p>Since there was a moderator flag, I went ahead and performed the edit to the question based on asm's "answer" and evident intentions. </p>
|
203,673 | <p>Let $X$ be a locally compact Hausdorff space. Does there exist a locally finite open covering consisting of relatively compact sets?</p>
| user642796 | 13,653 | <p>Not necessarily. The ordinal space $\omega_1 = [ 0 , \omega_1 )$ provides a counterexample. </p>
<p>To see that there is no locally finite cover by relatively compact sets, note that every compact subset — and therefore every relatively compact subset — is bounded. So if $\mathcal{A}$ is a cover by rela... |
2,275,016 | <p>I am currently going through Dummit/Foote's <em>Abstract Algebra</em>, and was asked to prove the above for a specific case but was wondering if it holds in the general case.</p>
<p>I have a feeling it might be false but I am bad at coming up with counterexamples so I tried to think of some simple contradictions if... | Lukas Heger | 348,926 | <p>For any $y \in G$, we have $xy = yx$. Multiplying from the right and left each with $x^{-1}$, we get $yx^{-1}=x^{-1}y$. One can say even more, the set of commutative elements forms a subgroup, called the center of $G$.</p>
|
108,409 | <p>Given a commutative ring <span class="math-container">$A$</span> with unity, Grothendieck used universal polynomials to define a <em>special</em> <span class="math-container">$\lambda$</span>-ring structure on <span class="math-container">$\Lambda(A):=1+t\:A[[t]]$</span>. Suppose <span class="math-container">$A$</sp... | darij grinberg | 2,530 | <p>This is not an answer, as I don't exactly know what Fulton and Lang are trying to achieve with the $\lambda$-ring structure on $\Lambda^{\circ}\left(A\right)$ (I must admit that, while I had the quixotic intent to read and rewrite Fulton-Lang's Chapter I in the notes that you cited, I never found the resolve to walk... |
108,409 | <p>Given a commutative ring <span class="math-container">$A$</span> with unity, Grothendieck used universal polynomials to define a <em>special</em> <span class="math-container">$\lambda$</span>-ring structure on <span class="math-container">$\Lambda(A):=1+t\:A[[t]]$</span>. Suppose <span class="math-container">$A$</sp... | JBorger | 1,114 | <p>As others have said, the definition of the Chern ring there is wrong. But if memory serves, the only mistake is that they forgot to introduce the right multiplication law on the sets of power series they consider. The usual one in the theory is given by the universal formulas for exterior powers of tensor products $... |
108,409 | <p>Given a commutative ring <span class="math-container">$A$</span> with unity, Grothendieck used universal polynomials to define a <em>special</em> <span class="math-container">$\lambda$</span>-ring structure on <span class="math-container">$\Lambda(A):=1+t\:A[[t]]$</span>. Suppose <span class="math-container">$A$</sp... | John Baez | 2,893 | <p>Just to save people some work, here are some problems with of <em>Riemann-Roch Algebra</em> pointed out by K. R. Coombes in his review on MathSciNet:</p>
<blockquote>
<p>The beginner, however, may find the going rough at first. Chapters I and III in particular could have been written more carefully. Nowhere is there... |
879,640 | <p>Does a matrix have only one inverse matrix (like the inverse of an element in a field)? If so, does this mean that</p>
<p>$A,B \text{ have the same inverse matrix} \iff A=B$?</p>
| Andreas Blass | 48,510 | <p>More generally, in any situation where the associative law holds, if some $x$ has both a left-inverse $l$ and a right inverse $r$, then $l=r$. The reason is that $l=l(xr)=(lx)r=r$. In particular, if $x$ has a $2$-sided inverse, then that's unique. On the other hand, it is entirely possible for some $x$ to have man... |
3,243,503 | <p>If <span class="math-container">$x + y = 2c$</span>, find minimum value of
<span class="math-container">$ \sec x +\sec y $</span> if <span class="math-container">$x,y\in(0,\pi/2)$</span>, in terms of <span class="math-container">$c$</span>.</p>
<p>I was able to solve by differentiating the equation and got the ans... | DeepSea | 101,504 | <p>The function <span class="math-container">$f(x) = \sec x$</span> has <span class="math-container">$f''(x) = \sec x\tan^2 x + \sec^3 x > 0$</span> on <span class="math-container">$(0,\frac{\pi}{2})$</span>. Thus <span class="math-container">$f(x)$</span> is convex on the indicated domain and it follows that <span ... |
902,653 | <p>I have recently been learning some hyperbolic geometry and the professor briefly mentioned spherical geometry. From a modern, naive point of view, it seems quite easy to show that spherical geometry is an example of non euclidean geometry.</p>
<p>Clearly, this is not true since it took over a 1000 years to do this.... | Per Erik Manne | 33,572 | <p>To an ancient geometer (or one from the 18th century), spherical geometry would seem to violate Euclid's second postulate, which says that any finite straight line can be extended indefinitely.
It is also a problem that two great arcs intersect in <em>two</em> points, rather than one, which would be natural for str... |
1,507,526 | <p>Let $S$ be the portion of the sphere $x^2+y^2+z^2=9$, where $1\leq x^2+y^2\leq4$ and $z\geq0$. Calculate the surface area of $S$</p>
<p>Ok i'm really confused with this one. I know i have to apply the surface area formula but and possibly spherical coordinates but i can't seem how to get the integral out.</p>
<p>T... | Christian Blatter | 1,303 | <p>The surface $S$ in question is a spherical zone. Its area can be found by elementary means: If $R$ is the radius of the sphere and $h$ is the $z$-height of the zone then the area $\omega(S)$ is given by $$\omega(S)=2\pi R h\ .$$
As $R=3$ and $h$ is easily computed as $h=\sqrt{9-1}-\sqrt{9-4}$ we obtain $$\omega(S)=6... |
761,286 | <p>let $G$ be an infinite group of the form $G_1 \oplus G_2 \oplus \dots \oplus G_n$ where each $G_i$ is a <strong>non trivial</strong> group and $n>1$. Prove that $G$ is not cyclic.</p>
<p><strong>Attempt</strong> : Let $G = G_1 \oplus G_2 \oplus \dots \oplus G_n$ be cyclic.</p>
<p>then $\exists ~g =(g_1,g_2,....... | Vector_13 | 96,276 | <p>If the inner loop runs $$\sqrt n$$ times and outer loop runs n times as you indicated in you comments then you get: $$n\cdot \sqrt n = n^{3/2}$$ Since $$f=O(g)$$ means that your f grows no faster than g, it follows that $$n^{3/2}=O(n^{3/2})$$ i.e. there is a constant C that makes g grow faster than f.</p>
|
2,299,466 | <p>For the first-order language with vocabulary $(E)$ (the binary relation $E$ which holds if two vertices have an edge) together with a set $G$ of vertices, I've been told that the property "a symmetric graph is connected" cannot be axiomatized by any set of first-order sentences. </p>
<p>I think the proof involved t... | Daron | 53,993 | <p>Suppose out language is $(V,E,a,b)$ where $V$ is the set of vertices; $E(\cdot,\cdot)$ is the edge relation, and $a$ and $b$ are distinct constants.</p>
<p>Let each $T_n$ encode "There is no path of length $n$ between $a$ and $b$"</p>
<p>And suppose conversely there is a sentence $T$ that encodes "The graph is con... |
2,721,836 | <p>I recently found a different method to compute prime number in $\mathcal O(\log(\log n))$ complexity. At present, that logic working fine for $300$ digits prime number, which I found on websites.I need to validate whether that logic will be working fine for a higher number of digits. At present, I have computed a pr... | gammatester | 61,216 | <p>Check it with a table of Mersenne primes (e.g. <a href="http://oeis.org/A000043" rel="nofollow noreferrer">http://oeis.org/A000043</a>) or use
provable primes (e.g. Maurer's method, see Alg. 4.62 in the <a href="http://cacr.uwaterloo.ca/hac/" rel="nofollow noreferrer">Handbook of Applied Cryptography</a>).</p>
|
320,557 | <p>let $S$ be the set of pairs $(x,y)$ where x,y are orthogonal unit vectors in $\mathbb R^3$. i am trying to show this is a topological manifold. for starters one needs to define a suitable topology on it. i was thinking let a set $U$ be open in $S$ iff $U \cap S^2$ (intersection with sphere) is open in $S^2$ in the s... | Ryan Budney | 642 | <p>Your set $S$ is a subset of $\mathbb R^6$, so give it the subspace topology. That ensures it's 2nd countable and Hausdorff.</p>
<p>To show it's a manifold, notice $S$ is the pairs $(x,y) \in \mathbb R^3 \times \mathbb R^3$ such that:</p>
<p>$$ |x|^2 =1,\ \ |y|^2=1, \ \ x\cdot y = 0 $$</p>
<p>This is the same as... |
30,653 | <p>I have a data file which contains thousands of lines and each line has eight elements. Here is a small sample of the data file</p>
<pre><code>-4.00 -0.80 0.1886024468848907E+01 0.1467147621657460E+01 -.1217067274319363E+01 0.7206100000000000E+03 0.7693457688734395E-12 5
-4.00 -0.70 0.143035... | Kuba | 5,478 | <p>There are many ways to acheive this:</p>
<pre><code>SetDirectory@NotebookDirectory[];
data = Import["list.txt", "Table"] (*I saved your sample in txt file*)
ntot = Length@data;
temp = SortBy[Tally@data[[;; , -1]], 1];
temp[[;; , 2]] = temp[[;; , 2]] 100/ntot // N;
(*I've multiplied by 100 to get % value not the ... |
3,930,659 | <blockquote>
<p>Evaluate: <span class="math-container">$$ \int \frac{x^2}{\sqrt{1-x^2}}\,dx$$</span></p>
</blockquote>
<p>The solution I came across does a <span class="math-container">$u$</span>-substitution by letting <span class="math-container">$x = \sin(t)$</span>. But why <span class="math-container">$\sin(t)$</s... | DMcMor | 155,622 | <p>Draw a triangle! When looking at the term <span class="math-container">$\sqrt{1 - x^{2}}$</span> in the integrand, you should immediately be reminded of the Pythagorean Theorem. If you draw a right triangle with hypotenuse length <span class="math-container">$1$</span>, and side lengths <span class="math-container... |
18,960 | <p>I am making this post in regards to the ongoing delete/undelete skirmish (let's at least change the monotonicity of the use of "war"). The old version of the question is <a href="https://math.stackexchange.com/revisions/172652/3">here</a>, the current version (after edits today) <a href="https://math.stackexchange.c... | Jonas Meyer | 1,424 | <p>Now that it has been edited to be reasonable, I would prefer for it to be unlocked and reopened.</p>
|
920,782 | <p>How do I find the number of integral solutions to the equation - </p>
<p>$$2x_1 + 2x_2 + \cdots + 2x_6 + x_7 = N$$</p>
<p>$$x_1,x_2,\ldots,x_7 \ge 1$$</p>
<p>I just thought that I should reduce this a bit more, so I replace $x_i$ with $(y_i+1)$, so we have:</p>
<p>$$y_1 + y_2 + \cdots + y_6 = \tfrac{1}{2}(N + 13... | user2566092 | 87,313 | <p>Note $x_7$ must be either odd or even depending on whether $N$ is odd or even. So if you just subtract $1$ from $N$ if necessary and assume $x_7 = 2y$ where $y \geq 0$, then you get that the number of solutions is the same as the number of solutions to $x_1 + \ldots + x_7 = N/2$ where $x_i \geq 1$ for $1 \leq i \leq... |
504,524 | <p>I'm trying to learn probability and statistics but I can't really get my head around this one. I realize after drawing the first card there will only be 51 cards in the deck but I'm having trouble calculating the chance that the second one is an Ace if I don't know what the first card is?</p>
<p>Assuming that the i... | drmortimer | 34,817 | <p>Break the answer into two scenarios:</p>
<ol>
<li>If the first card is an ace and the second card is also an ace</li>
</ol>
<p>The probability, in this case, is <span class="math-container">$$\dfrac{4}{52}\dfrac{3}{51}.$$</span></p>
<ol start="2">
<li><p>If the first card is not an ace but the second card is an ace<... |
2,898,390 | <p>Is there any algorithm or a technique to calculate how many prime numbers lie in a given closed interval [a1, an], knowing the values of a1 and an, with a1,an ∈ ℕ?</p>
<p>Example: </p>
<p>[2, 10] --> 4 prime numbers {2, 3, 5, 7}</p>
<p>[4, 12] --> 3 prime numbers {5, 7, 11}</p>
| Doesbaddel | 587,094 | <blockquote>
<p>$(A \implies C) \wedge \neg (B \wedge C \wedge D).$ </p>
<p>Truth table:</p>
<p>\begin{array}{| c | c | c | c | c | c | c |} \hline A & B & C & D &
> \underbrace{A \implies B}_{E} & \underbrace{\neg (B \wedge C \wedge
D)}_{F} & \underbrace{E \wedge F}_{G} \\ \hlin... |
1,527,197 | <p>So in the case where data points have the same variance $\sigma^2$, the estimator (in normal equation form) can be written as </p>
<p>$$\theta=(X^TX)^{-1}X^TY$$</p>
<p>I'm not sure how to derive a similar formula when the data points have different variances, and thus the covariance matrix would be</p>
<p>$$\Sigm... | As soon as possible | 288,506 | <p>The series $\sum a_n$ is divergent if $(a_n)$ hasn't the limit $0$. So the series in 1. is divergent.</p>
<p>For 3.the sequence $\frac{1}{\sqrt n+1}$ is decreasing and has the limit $0$ so you can apply AST.</p>
|
3,488,405 | <blockquote>
<p>Let <span class="math-container">$L(n)$</span> denote the number of positive divisors of a number <span class="math-container">$n$</span>. Prove that <span class="math-container">$\sum_{n=1}^N L(n)=\lfloor{\sqrt N}\rfloor\pmod 2$</span>.</p>
</blockquote>
<p>I wanted to prove that by induction.
For <... | Will Jagy | 10,400 | <p>Your evaluation of <span class="math-container">$L$</span> using the Euler phi function is wrong, it works only for primes and 1 and 4.</p>
<pre><code>Thu Dec 26 13:34:20 PST 2019
1 = 1 divisor count: 1 n - phi + 1: 1 EQUAL
2 = 2 divisor count: 2 n - phi + 1: 2 EQUAL
3 = 3 divisor ... |
2,523,342 | <p>Assume $f\in L^p(\Bbb R^d) $ and $g\in L^q(\Bbb R^d) $
Where, $1<p<\infty$ and $1<q<\infty$ are dual ecxponents namely, $$\frac1p+\frac1q =1$$
Then for every $s\in\Bbb R$ such that, $sp\le d$ show that,
$$\lim_{j\to\infty} \int_{\Bbb R^d}f_j(x)g(x)dx = 0$$</p>
<p>Where, $$f_j(x) = j^sf(jx)~~~$$</p>
... | David C. Ullrich | 248,223 | <p>Note it's not true for $p=1$. Assume $p>1$. Come to think of it, it's also false for $p=\infty$. Assume $p<\infty$.</p>
<p>Let $\epsilon>0$. Choose $A>0$ so that $$\left(\int_{|x|\le A}|g(x)|^q\right)^{1/q}<\epsilon.$$(This is where $q<\infty$ is needed.)
An explicit calculation shows that $$\left... |
422,225 | <p>The proof uses this lemma which I understand: </p>
<p>$\mathbf {Lemma}$: Suppose $x$ and $y$ are positive real numbers such that $x>y$. If we decrease $x$ and increase $y$ by some positive quantity $E$ such that $x-E \ge y+E$, then $(x-E)(y+E) \gt xy$ . $\;$Hence, by subtracting $E$ from $x$ and adding it to $y$... | Hagen von Eitzen | 39,174 | <p>1) I'd suggest to choose <em>any</em> indices $j,k$ with $a_j<A<a_k$ (which exist unless all $a_i$ are equal) and let $E=\min\{a_k-A,A-a_j\}$ in the lemma.</p>
<p>2) $E$ is specifically chosen so that at least one of $a_j+E$, $a_k-E$ equals $A$. The text explicitly makes $a_j=A$, i.e. chooses $E=A-a_j$, and d... |
3,009,345 | <p>I ran into this question which hints me to use Cauchy's Integral Theorem for Derivatives, however I don't seem to be able to fit this integral into the form of the Integral Formula</p>
<p><span class="math-container">$$\displaystyle \int_{|z|=2} \frac{\cos(z)}{z(z^2+8)}dz$$</span> I tried using the fact that <span ... | Math Lover | 348,257 | <p>Note that <span class="math-container">$Y<X^3$</span> and <span class="math-container">$XY<z \implies Y<z/X$</span>; i.e., <span class="math-container">$Y < \min\{X^3,z/X\}$</span>. </p>
<p>For <span class="math-container">$X^3<z/X$</span>, or <span class="math-container">$X < z^{1/4}$</span>, we ... |
2,064,643 | <p>Let $a_n,b_n > 0, \sum_{n=1}^{\infty}a_n < \infty, \sum_{n=1}^{\infty}b_n = \infty$. Is it possible for the Cauchy product of the two series to converge?</p>
| Tsemo Aristide | 280,301 | <p>Hint: $a_n={1\over{n^3}}$, $b_n=n$</p>
|
2,780,731 | <p>In school, I have recently been learning about simple differential equations. We know that the solution of $y'=y$ is $y=Ae^x$, where $A$ is a constant. But how can we know that it is the <strong>only</strong> solution? The only thing I can figure out is that $y$ is continuously differentiable. Help me, please.</p>
| Artem | 29,547 | <p>Let $z$ be a solution to $y'=y$. Consider $z(t)e^{-t}$. We have
$$
\frac{d}{dt}(z(t)e^{-t})=z'(t)e^{-t}-z(t)e^{-t}=0,
$$
therefore
$$
z(t)e^{-t}=Const\implies z(t)=Ae^{t}.
$$</p>
|
4,315,449 | <p>Is it correct to say that <span class="math-container">$\mathbb{R}^n \subset \mathbb{R}^n$</span> ?</p>
<p><strong>EDIT</strong>: The context of may question is that I am having a function that is defined as <span class="math-container">$f \colon D \subset \mathbb{R}^n \to \mathbb{R}^n $</span> and I am wondering if... | Lorenzo Pompili | 884,561 | <p>The symbol “<span class="math-container">$\subset$</span>” could mean different things depending on the author.</p>
<p>In your case, I suggest trying to figure it out from the context. If there are no other restrictions on <span class="math-container">$D$</span> (e.g. bounded,compact,…), then it is very likely that ... |
1,341,486 | <p>Problem:
Find the sum to $n$ terms of
\begin{eqnarray*}
\frac{1}{1\cdot 2\cdot 3} + \frac{3}{2\cdot 3\cdot 4} + \frac{5}{3\cdot 4\cdot 5} +
\frac{7}{4\cdot 5\cdot 6}+\cdots \\
\end{eqnarray*}
Answer:
The way I see it, the problem is asking me to find this series:
\begin{eqnarray*}
S_n &=& \sum_{i=1}^... | Michael Galuza | 240,002 | <p>There is more simple way (for me). You have
$$a_n=\frac{2n-1}{n(n+1)(n+2)}=-\frac52\frac{1}{n+2} + \frac{3}{n+1} - \frac{1}{2n};$$
hence
$$S_N = \sum_{n=1}^N a_n = -\frac52\left(H_{N+2}-1-\frac12\right) + 3(H_{N+1}-1) - \frac12 H_N,$$
where $H_N$ is $N$-th harmonic number. Simplify it:
$$S_N = -\frac52\left(H_N + \f... |
872,657 | <p>For $1 \leq r < p < \infty$ prove the continuous injection of $L^p([0, 1])$ into $L^r([0, 1])$. </p>
<p>I am having a hard time starting. Any suggestions. I tried a straight forward approach. That is, given $\epsilon > 0$, I tried to find a $\delta >0$ such that $||f - g||_p < \delta$ implies tha... | Mohammad Khosravi | 87,886 | <p>M. Slater, "Lagrange Multipliers Revisited," Cowles Commission Discussion
Paper No. 403, November, 1950</p>
|
858,576 | <p>Prove that the union of three subspaces of V is a subspace iff one of the subspaces contains the other two.</p>
<p>I can do this problem when I am working in only two subspaces of $V$ but I don't know how to do it with three. </p>
<p>What I tried is:
If one of the subspaces contains the other two, Then their union... | W.Leywon | 300,668 | <p>Gina gave an excellent answer,in fact,we can have : If $V$ is a vector space over the field $F$ and there is a collection of finite number of subspaces of $V$, $\{U_1,U_2,U_3,\cdots ,U_n\}$,and $n$,the number of the elements of the collection above,is not more than the cardinality of $F$,when $F$ is finite,or $F$ is... |
858,576 | <p>Prove that the union of three subspaces of V is a subspace iff one of the subspaces contains the other two.</p>
<p>I can do this problem when I am working in only two subspaces of $V$ but I don't know how to do it with three. </p>
<p>What I tried is:
If one of the subspaces contains the other two, Then their union... | Jia Cheng Sun | 669,184 | <p>With regard to Gina and Jeff's answers, I believe the simplification at the start can be made easier (at least in my opinion).</p>
<p>Suppose that subspaces <span class="math-container">$U_1,U_2,U_3$</span> union to form a subspace.
We have 2 cases.</p>
<p>Case 1: Suppose <span class="math-container">$U_i\subseteq \... |
76,853 | <p>I have a list of stock symbols and related information containing some entries <code>Missing["NotAvailable"]</code>. I would like to delete all nested lists which contain a NotAvaiable entry, as <em>Mathematica</em> obviously does not support these instruments anymore (see also <a href="http://reference.wolfram.com/... | kglr | 125 | <pre><code>indexMaster= {"^RDM-SO", Missing["NotAvailable"], "AMEX"}
Select[indexMaster, Internal`LiterallyAbsentQ[#, "NotAvailable"] &]
(* {"^RDM-SO", "AMEX"} *)
indexMaster2 = {"^RDM-SO", Missing["NotAvailable"], Missing[], "AMEX", "NotAvailable"};
Select[indexMaster2, Internal`LiterallyAbsentQ[#, "NotAvailable"... |
891,575 | <p>The circumference of a circle has length 90 centimeters, Three points on the circle divide the circle into three equal lengths. Three ants A, B, and C start to crawl clockwise on the circle, with starting from one of the three points. Initially A is ahead of B and B is ahead of C. Ant A crawls 3 centimeters per seco... | David | 119,775 | <p><strong>Hint</strong>. Suppose that the ants meet after $t$ seconds, and measure distance around the circle from where C starts. Then A has travelled $3t$ metres, but had a $60$ metre start for a total of $60+3t$ from the initial point. Likewise B will be a distance 30+5t from the initial point. However B may ha... |
1,521,779 | <p>I have a homework question that I want to make sure I'm getting it right.</p>
<p>This is a joint probability table for the proportions of survey respondents who smoke and who have had heart attacks.</p>
<p><kbd> &n... | N. S. | 9,176 | <p>Hopefully I didn't make any mistake.</p>
<p>Note that the squares modulo <span class="math-container">$9$</span> are <span class="math-container">$0,1,4,7$</span>.</p>
<p>Now
<span class="math-container">$$2017 = a^2 + b^2 \Rightarrow \\
1 \equiv a^2+b^2 \pmod{9}$$</span></p>
<p>This gives that, WLOG we have
<span c... |
2,012,532 | <p>The following is all confirmed to be true:</p>
<p>Matrix A =
$
\begin{bmatrix}
0 & 1 & -2 \\
-1 & 2 & -1 \\
2 & -4 & 3 \\
1 & -3 & 2 \\
\end{bmatrix}
$</p>
<p>U =
$
\begin{bmatrix}
-1 & 2 & -1 \\
0 &am... | dantopa | 206,581 | <p>$$
\begin{align}
\mathbf{P} \mathbf{A} &= \mathbf{L} \mathbf{U} \\
% P
\left[
\begin{array}{cccc}
0 & \boxed{1} & 0 & 0 \\
\boxed{1} & 0 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 \\
\end{array}
\right]
% A
\left[
\begin{array}{rrr}
0 & 1 & -2 \\
-1 ... |
2,867,404 | <p>I have a problem how to get the area from the picture.
Some ideas I got are not good enough to get the correct value of the whole element.</p>
<p><a href="https://i.stack.imgur.com/PeQ5O.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/PeQ5O.jpg" alt="enter image description here"></a></p>
| 高田航 | 407,845 | <p><strong>Hint</strong>: You know the width is $13.62$, and the maximum height it reaches is $1.65$, so it must fit within a rectangle of those dimensions.</p>
|
2,867,404 | <p>I have a problem how to get the area from the picture.
Some ideas I got are not good enough to get the correct value of the whole element.</p>
<p><a href="https://i.stack.imgur.com/PeQ5O.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/PeQ5O.jpg" alt="enter image description here"></a></p>
| Servaes | 30,382 | <p><strong>Hint:</strong> Draw straight lines between the upper left corner and the midline, and upper right corner and the midline. Now you have two rectangles and two triangles. Can you compute their areas?</p>
|
1,311,367 | <p>Recently, I am considering a question, as is well known, Cauchy's inequality is a famous and useful inequality. $$\left|\int_{a}^{b}f(x)dx\right|^2\leq|b-a| \int_{a}^{b}f^2(x)dx.$$ My question is: can we obtain a inequality such that
$$\left|\int_{a}^{b}f(x)dx\right|^2\ge |A|\times \left|\int_{a}^{b}f(x)^2dx\right... | robjohn | 13,854 | <p>Your inequalities are reversed. If you reverse both, you get the following:
$$
\left|\int_a^bf(x)\,\mathrm{d}x\right|^2\le(b-a)\int_a^b|f(x)|^2\,\mathrm{d}x
$$
This inequality follows from <a href="http://en.wikipedia.org/wiki/Cauchy%E2%80%93Schwarz_inequality" rel="nofollow">Cauchy-Schwarz</a> or <a href="http://en... |
1,419,315 | <p>I have a particular scenario.</p>
<p>In this scenario, we have the standard cubic equation,</p>
<pre><code>ax^3 + bx^2 + cx + d = y
</code></pre>
<p>as well as 3 points that are graphed, <a href="https://i.imgur.com/VCZKuGW.png" rel="nofollow noreferrer">as can be seen in this graph</a>. (The line is irrelevant ... | Amitai Yuval | 166,201 | <p>Find $A$'s eigenvalues first. Once you know them, you know everything you need about $A$.</p>
<p>More explicitly, you can start by calculating $A$'s characteristic polynomial. A straightforward calculation shows that its roots are $0$ and $3$. These are $A$'s eigenvalues, and hence, with respect to an appropriate o... |
272,057 | <p>Let $\mu$ and $\nu$ be two probability measures on $\mathbb R^n$ with finite first moment. Denote by $d:=W_1(\mu,\nu)$, where $W_1(\cdot,\cdot)$ stands for the Wasserstein distance of order $1$. </p>
<p>My question is the following: Let $X$ be a random variable defined on some probability space (rich enough) with l... | MB2009 | 111,097 | <p>I've a solution but it's not perfectly satisfying. Assume that </p>
<p>$$V~~~:=~~~\int |x|^pd\mu(x)~+~\int |x|^pd\nu(x)~~~<~~~+\infty$$</p>
<p>for some fixed $p>1$. It follows from <strong>Thought 1</strong> that, there exists $f_0$ and $G$ s.t. </p>
<p>$$Y_0~:=~f_0(X,G)~\sim~\nu~~~~\mbox{ and }~~~~\mathbb ... |
4,500,928 | <p>How many ten-digit positive integers are there such that all of the
following conditions are satisfied:</p>
<p>(a) each of the digits 0, 1, ... , 9 appears exactly once;</p>
<p>(b) the first digit is odd;</p>
<p>(c) five even digits appear in five consecutive positions?</p>
<p>From Combinatorics by Pavle Mladenovic<... | Johnson | 1,071,001 | <p>Focussing on the 5 even numbers first is a good way to go. There are 5 different positions the set of them may go in (2,3,4,5 or 6 for the first even digit).</p>
<p>For each of the 5! * 5 arrangements of the even digits, there will be 5! different orders for the odd digits.</p>
<p>This is exactly the same as your me... |
736,684 | <p>I'm trying to figure the probability that <span class="math-container">$X < Y$</span> with:</p>
<p><span class="math-container">$$X, Y \in \mathbb R^+;\ X\in [0,5] ; \ Y \in [0,2]$$</span>
What is the law to use?</p>
| Sergio Parreiras | 33,890 | <p>Assuming independence:
$$\Pr(X<Y)= \int_{0}^5 Pr(x<Y) \cdot f_X(x)dx= \int_{0}^5 (1-F_Y(x)) \cdot f_X(x)dx$$</p>
|
1,263,865 | <p>So I have that $700=7\cdot2^2\cdot5^2$ and I got that $3^2\equiv1\pmod2$ so then $3^{1442}\equiv1\pmod2$ also $3^2\equiv1\pmod{2^2}$ so $3^{1442}\equiv1\pmod{2^2}$ which covers one of the divisors of $700$. Im not sure if I'm supposed to use $2$ or $2^2$ and I was able to find that $3^2\equiv-1\pmod5$ so $3^{1442}\e... | lab bhattacharjee | 33,337 | <p><a href="http://en.wikipedia.org/wiki/Carmichael_function" rel="nofollow">Carmichael function</a> $\lambda(700)=60$</p>
<p>As $(3,700)=(3,3\cdot233+1)=(3,1)=1,3^{60}\equiv1\pmod{700}$</p>
<p>and $1442=24\cdot60+2\equiv2\pmod{60}\implies3^{1442}\equiv3^2\pmod{700}$</p>
|
2,792,651 | <p>I want to prove that $h_K=2$ if $K=\mathbb{Q}[\sqrt{-6}]$. Using Minkowski Theorem I have that $Cl_K=\{(1),(3,\sqrt{-6}),(2,\sqrt{-6})\}$, and I thought it was a good idea to use Lagrange Theorem (order of an element divides order of the gorup).</p>
<p>The main problem is that I can't reduce $(2,\sqrt{-6})^2$: </p>... | dan_fulea | 550,003 | <p>I will display the answer as an answer, rather than as a simple comment.
The above computation line is "almost ok", correctly:</p>
<p>$$(2,\sqrt{-6})(2,\sqrt{-6})=(4,2\sqrt{-6},2\sqrt{-6},-6)=(2)\ .$$</p>
<p>For the last equality, we have to show the double inclusion.
It is clear that from $4,-6$ we can exhibit t... |
556,054 | <p>Please help me to prove the inequality
$$
\sqrt{a^2 + b^2} \geq \frac{|a-b|}{\sqrt{2}}.
$$</p>
| André Nicolas | 6,312 | <p><strong>Hint:</strong> Note that $(a-b)^2+(a+b)^2=2(a^2+b^2)$.</p>
|
16,982 | <p>one can obtain solutions to the <a href="http://en.wikipedia.org/wiki/Laplace%27s_equation" rel="noreferrer">Laplace equation</a>
<span class="math-container">$$\Delta\psi(x) = 0$$</span></p>
<p>or even for the <a href="http://en.wikipedia.org/wiki/Poisson%27s_equation" rel="noreferrer">Poisson equation</a> <span cl... | Raskolnikov | 3,567 | <p>It does not seem that anybody has a more complete reply, so I might as well summarize what I have said in the comments. The technique you are refering to in the OP is an application of the <a href="http://en.wikipedia.org/wiki/Feynman%E2%80%93Kac_formula" rel="noreferrer">Feynman-Kac formula</a>.</p>
<p>This techni... |
2,204,944 | <p>A line is a collection of infinitely many points. By definition, a point has no dimensions. But, how can infinitely many dimensionless points give rise to a line with a dimension. This is the same case with planes, solids and higher dimensions too...</p>
<p>Thanks in advance for any help..!!</p>
| uniquesolution | 265,735 | <p>The answer is that $X_i$ are copies of one and the same random variable, namely, the random variable that returns the number on a card randomly drawn from your card collection.</p>
|
2,677,823 | <p>How can I precisely prove the existence of a continuous function $\rho(x)$ such that $0 \leq \rho(x) \leq 1 \forall x \in R^d $ such that $g(x) \rho(x)$ is bounded and continuous for $g(x)$ continuous?Both $g(x)$ and $\rho(x)$ are defined on $R^d$.</p>
<p>My idea was that we can choose $\rho(x)$ such that $\rho(x)g... | user | 505,767 | <p>Let $f=g+ih$ then</p>
<p>$$\overline{\int_{E}f}=\overline{\int_{E}g+ih}=\overline{\int_{E}g}+\overline{\int_{E}ih}=\int_{E}g-i\int_{E}h=\int_{E}\bar f,$$</p>
|
2,677,823 | <p>How can I precisely prove the existence of a continuous function $\rho(x)$ such that $0 \leq \rho(x) \leq 1 \forall x \in R^d $ such that $g(x) \rho(x)$ is bounded and continuous for $g(x)$ continuous?Both $g(x)$ and $\rho(x)$ are defined on $R^d$.</p>
<p>My idea was that we can choose $\rho(x)$ such that $\rho(x)g... | Community | -1 | <p>The Lebesgue integral of <em>complex</em>-valued functions is not (that I know of) defined with the same measure-theoretic machinery that you use for $\Bbb R$ (the notions of which, starting from <em>positivity</em> or <em>increasing convergence</em>, would actually be moot if $f$ could take non-real values): it's ... |
99,799 | <p>I have a <code>Solve</code> similar to the following:</p>
<pre><code>Solve[e^2 - c^2 == -15, {e, c}, Integers]
(* {{e -> -7, c -> -8}, {e -> -7, c -> 8}, {e -> -1, c -> -4},
{e -> -1, c -> 4}, {e -> 1, c -> -4}, {e -> 1, c -> 4},
{e -> 7, c -> -8}, {e -> 7, c -... | Jason B. | 9,490 | <p>The <code>Interval</code> seems to be the problem, it returns an "Interval Object" and, rather than figure out what that is, just use the <code><=</code> operator to state the conditions explicitly</p>
<pre><code>Solve[
e^2 - c^2 == -15 && 0 <= e <= 4 && 0 <= c <= 4, {e, c}, Integers... |
250,454 | <p>Is there a <code>ReplaceOnce</code> which does only one replacement if possible by trying the rules sequentially in order. Consider the following as an example:</p>
<pre><code>ReplaceOnce[{"May","5","May","5"},{"May"->1,"5"->2}]
</code></pre>
<p>shoul... | kglr | 125 | <pre><code>ClearAll[replace1ce]
replace1ce = Block[{<span class="math-container">$done = False},
Fold[ReplaceAll, #, # :> RuleCondition[$</span>done = True; #2, ! $done] & @@@ #2]] &;
</code></pre>
<p><em><strong>Examples:</strong></em></p>
<pre><code>replace1ce[{"May", "5", "Ma... |
834,228 | <p>$$u_{1}=2, \quad u_{n+1}=\frac{1}{3-u_n}$$
Prove it is decreasing and convergent and calculate its limit.
Is it possible to define $u_{n}$ in terms of $n$?</p>
<p>In order to prove it is decreasing, I calculated some terms but I would like to know how to do it in a more "elaborated" way.</p>
| Jacob Bond | 120,503 | <p>After seeing that $u_{1} > u_{2}$, the result follows since $u_{n} > u_{n + 1}$ implies
$$\frac{1}{3 - u_{n}} > \frac{1}{3 - u_{n + 1}}.$$</p>
<p>Once you know that there is a limit, say $L$, you have
$$\lim_{n\rightarrow\infty} \frac{1}{3-u_{n}} = L.$$
But as $n \rightarrow \infty$, $u_{n} \rightarrow L$,... |
4,206,039 | <p>Find the radius of convergence of the following power series <span class="math-container">$$\sum_{n=1}^\infty \frac{(-1)^n z^{n(n+1)}}{n}$$</span></p>
<p>Here's my working
<span class="math-container">$$\lim_{n\to \infty}| \frac{(-1)^{n+1} z^{(n+1)(n+2)}}{n+1} \frac{n}{(-1)^nz^{n(n+1)}}|$$</span>
<span class="math... | José Carlos Santos | 446,262 | <p>Now, use the fact that<span class="math-container">$$\lim_{n\to\infty}\left|\frac{-nz^{2n+2}}{n+1}\right|=\lim_{n\to\infty}\frac n{n+1}|z|^{2n+2}=\begin{cases}0&\text{ if }|z|<1\\1&\text{ if }|z|=1\\\infty&\text{ if }|z|>1.\end{cases}$$</span>So, your series converges if <span class="math-container... |
4,206,039 | <p>Find the radius of convergence of the following power series <span class="math-container">$$\sum_{n=1}^\infty \frac{(-1)^n z^{n(n+1)}}{n}$$</span></p>
<p>Here's my working
<span class="math-container">$$\lim_{n\to \infty}| \frac{(-1)^{n+1} z^{(n+1)(n+2)}}{n+1} \frac{n}{(-1)^nz^{n(n+1)}}|$$</span>
<span class="math... | ancient mathematician | 414,424 | <p>As so often straightforward comparisons will give the result.</p>
<p>The series <span class="math-container">$\sum z^n$</span> has radius of convergence <span class="math-container">$1$</span>. The series <span class="math-container">$\sum (n+1)$</span> diverges.</p>
<p>(a) For <span class="math-container">$|z|<1... |
2,624,498 | <p>Evaluate $$\lim_{n \rightarrow\infty} \sqrt[n]{3^{n} +5^{n}}$$</p>
<p>Attempt:</p>
<p>The only sort of manipulation that has come to mind is: $$e^{\frac{1}{n}ln(e^{n\ln(3)} + e^{n\ln(5)})}$$</p>
<p>So what is the trick to successfully evaluate this?</p>
| user577215664 | 475,762 | <p>With the well known limit of the exponential $\lim\limits_{n \rightarrow\infty} \frac 1 {e^{n}}=0 $</p>
<p>$$\lim_{n \rightarrow\infty} \sqrt[n]{3^{n} +5^{n}}=5\lim_{n \rightarrow\infty} \sqrt[n]{ \left(\frac 35\right)^{n} +1}=5\lim_{n \rightarrow\infty} \sqrt[n]{ \frac 1 {e^{n(\ln 5 -\ln 3)}} +1}=5(0+1)^0=5$$</... |
2,624,498 | <p>Evaluate $$\lim_{n \rightarrow\infty} \sqrt[n]{3^{n} +5^{n}}$$</p>
<p>Attempt:</p>
<p>The only sort of manipulation that has come to mind is: $$e^{\frac{1}{n}ln(e^{n\ln(3)} + e^{n\ln(5)})}$$</p>
<p>So what is the trick to successfully evaluate this?</p>
| user | 505,767 | <p>The correct maniplulation by exponential is</p>
<p>$$\sqrt[n]{3^{n} +5^{n}}=(3^{n} +5^{n})^\frac1n=e^{\frac{\log (3^n+5^n)}{n}}\to5$$</p>
<p>indeed</p>
<p>$$\frac{\log (3^n+5^n)}{n}=\frac{\log 5^n+\log \left(1+\left(\frac{3}{5}\right)^n\right)}{n}=\frac{n\log 5+\log \left(1+\left(\frac{3}{5}\right)^n\right)}{n}=$... |
91,302 | <p>So, we represent numbers usually in a form of a sequence of digits where each one of them multiplies the power of a base:</p>
<p>$13.2 = 1 * 10^1 + 3 * 10^0 + 2 * 10^{-1}$</p>
<p>So that much is clear, perfectly. But what interests me is the "symmetry" between the left and right of the radix point which separates ... | lhf | 589 | <p>Zero digits do not count, ever. A zero digit in position $k$ corresponds to a term $0\cdot 10^k$. For instance, $502.03= 5 \cdot 10^2 + 0 \cdot 10^1 + 2 \cdot 10^0 + 0 \cdot 10^{-1} + 3 \cdot 10^{-2}$. There is no deeper reason.</p>
|
31,099 | <p>I was wondering what "anti-optimization" is about? Is it related to optimization? What topics does it cover? </p>
<p>All I can find out from Google is <a href="http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TJ4-42WP6GH-M&_user=10&_coverDate=07/31/2001&_rdoc=1&_fmt=high&_orig=gate... | Henry | 6,460 | <p><em>Anti-optimization</em> is a term popularized by Isaac Elishakoff for an approach to safety factors in engineering structures which he describes as pessimistic and searching for least favourable responses, in combination with optimistization techniques but in contrast to probabilistic approaches.</p>
<p>You migh... |
448 | <p>Let's say, I have 4 yellow and 5 blue balls. How do I calculate in how many different orders I can place them? And what if I also have 3 red balls?</p>
| Noldorin | 56 | <p>This is a standard problem involving the <a href="http://en.wikipedia.org/wiki/Combination" rel="noreferrer">combinations of sets</a>, though perhaps not very obvious intuitively.</p>
<p>Firstly consider the number of ways you can rearrange the entire set of balls, counting each ball as indepndent (effectively igno... |
2,677,584 | <p>I have the following question:</p>
<blockquote>
<p>Find the real values of $a$ for which the equation
$$(1+\tan^2\theta)^2 + 4a\tan\theta(\tan^2\theta + 1) + 16\tan^2\theta = 0$$
has four distinct real roots in $\left(0, \dfrac{\pi}{2}\right)$.</p>
</blockquote>
<p>I tried to solve the above equation by div... | CY Aries | 268,334 | <p>You have $y^2+4ay+16=0$, which gives $y=-2a\pm2\sqrt{a^2-4}$. Note that $\tan\theta>0$ for $\displaystyle \theta\in\left(0,\frac{\pi}{2}\right)$. The equation has four real roots in $\displaystyle \left(0,\frac{\pi}{2}\right)$ if $-2a+2\sqrt{a^2-4}>2$ and $-2a-2\sqrt{a^2-4}>2$. </p>
<p>As $\sqrt{a^2-4}$ i... |
3,419,276 | <p>I'm reading about the directional derivative:</p>
<blockquote>
<p>Let <span class="math-container">$(E,\|\cdot\|)$</span> and <span class="math-container">$(F,\|\cdot\|)$</span> be Banach spaces over the field <span class="math-container">$\mathbb{K}$</span>, and <span class="math-container">$X$</span> an open su... | Allawonder | 145,126 | <p>Sometimes we find it useful to express a given positive number as a power of some fixed positive number different from <span class="math-container">$1,$</span> usually called a base. Then if I expressed <span class="math-container">$u>0$</span> as a power of <span class="math-container">$1\ne b>0,$</span> then... |
1,234,471 | <p>Given two sequences $(a_k),(b_k)$ with $a_k\geq0,b_k>0$ such that the power series $\sum_{k=0}^\infty a_k b_kr^{k}$ and $\sum_{k=0}^\infty a_kr^k$ converge for each $r>0$. My question now is: Does there exist a constant $c$ (depending only on $(b_k))$ such that
\begin{align*}
\sum_{k=0}^\infty a_kb_kr^{k}\geq ... | wlad | 228,274 | <p>Let $S$ be this set, and $\epsilon$ be the empty string.</p>
<p>Define $S_0 ::= 0S_*$ and $S_* ::= \epsilon \mid 1S_0 \mid 1S_*$. </p>
<p>$S ::= S_0 \mid S_*$.</p>
<p>This is BNF notation.</p>
<hr>
<p>Let $u_n$ be the number of strings of length $n$ in $S_0$, and $v_n$ be the number of strings of length $n$ in ... |
4,092,643 | <p>The solutions manual says</p>
<p><span class="math-container">$$\lim_{x \to \infty}e^{-x^2}\int_{x}^{x+\frac1x}e^{t^2}dt=\lim_{x \to \infty}\frac{e^{(x+\frac1x)^2}-e^{x^2}}{2xe^{x^2}}$$</span></p>
<p>I'm trying to understand how they arrived there. Using L'Hôpital's rule rule, I have</p>
<p><span class="math-contain... | Aatmaj | 769,348 | <p>Hint-- use <a href="https://mathworld.wolfram.com/LeibnizIntegralRule.html" rel="nofollow noreferrer">Leibniz rule of integration</a>.</p>
<hr />
<p>What they have done is basically use l's hospital rule. To take the derivative they have used the leibniz rule, and approximated <span class="math-container">$1/x^2$</s... |
4,092,643 | <p>The solutions manual says</p>
<p><span class="math-container">$$\lim_{x \to \infty}e^{-x^2}\int_{x}^{x+\frac1x}e^{t^2}dt=\lim_{x \to \infty}\frac{e^{(x+\frac1x)^2}-e^{x^2}}{2xe^{x^2}}$$</span></p>
<p>I'm trying to understand how they arrived there. Using L'Hôpital's rule rule, I have</p>
<p><span class="math-contain... | Peter Szilas | 408,605 | <p>MVT for integrals as an option:</p>
<p><span class="math-container">$f(x) = e^{-x^2}e^{z^2}\int_{x} ^{x+1/x}dx=$</span></p>
<p><span class="math-container">$e^{-x^2}e^{z^2}(1/x)$</span>, where <span class="math-container">$z \in [x, x+1/x];$</span></p>
<p><span class="math-container">$1/x \le f(x) \le e^{-x^2}e^{(x+... |
474,632 | <p>Let $M$ be a set with three elements: $a$, $b$, and $c$. Define $D\colon M\times M\to[0,\infty)$ so that $D(x, x) = 0$ for all $x$, $D(x, y) = D(y, x)$ for $x \ne y$. Say $D(a, b) = r$, $D(a, c) = s$, $D(b, c) = t$, and $r \le s \le t$. </p>
<p>Prove that $D$ makes $M$ a metric space iff $t \le r + s$.</p>
<p>I ha... | Don Larynx | 91,377 | <p>P1) $t <= r + s$ implies $D(b, c) <= D(a, b) + D(a, c)$.</p>
<p>P2) Either $D(x, y) = 0$ or $r$.</p>
<p>P3) Suppose $D(a, b) = D(a, c) = 0$. Then a = b = c = 0. So the triangle inequality is satisfied.</p>
<p>P4) In case $D(a, b) = D(a, c) = r, b = c$. This the triangle inequality is trivially satisfied.</p... |
474,632 | <p>Let $M$ be a set with three elements: $a$, $b$, and $c$. Define $D\colon M\times M\to[0,\infty)$ so that $D(x, x) = 0$ for all $x$, $D(x, y) = D(y, x)$ for $x \ne y$. Say $D(a, b) = r$, $D(a, c) = s$, $D(b, c) = t$, and $r \le s \le t$. </p>
<p>Prove that $D$ makes $M$ a metric space iff $t \le r + s$.</p>
<p>I ha... | Stefan Hamcke | 41,672 | <p>Your argument is a bit complicated. It is much easier:</p>
<p>We want to verify the triangle inequality $d(x,y)\le d(x,z)+d(y,z)$.</p>
<p>If $x,y,z$ are not all distinct, then it is satisfied as shown in your previous question. So let's assume they are all distinct.</p>
<p>There are three possibilities for $x,y$:... |
3,846,891 | <p>Let me know if this proof is correct. This is in french.</p>
<p>Translation French --> English</p>
<p><strong>prémisse = premise</strong></p>
<p><strong>supposition = assumption</strong></p>
<p><a href="https://i.stack.imgur.com/Ey8bH.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/Ey8bH.png" a... | Bram28 | 256,001 | <p>No, not correct.</p>
<p>Line 5 cannot refer to line 3, since that is part of a subproof that was closed right after line 3</p>
<p>Line 6, 7, and 8 are all not correct: you introduce a conditional by <em>closing</em> a subproof, inferring a conditional with as its antecednt the assumption of that subproof, and as its... |
3,846,891 | <p>Let me know if this proof is correct. This is in french.</p>
<p>Translation French --> English</p>
<p><strong>prémisse = premise</strong></p>
<p><strong>supposition = assumption</strong></p>
<p><a href="https://i.stack.imgur.com/Ey8bH.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/Ey8bH.png" a... | Mauro curto | 781,761 | <p>Your derivation is incorrect. Your supposition in step 2 is not the adequate supposition. After you colse a supposition you can't use the supposition anymore so your step 7 is wrong, nad therefore the step 8 is also wrong.</p>
<p>Since the main connective of the consequent is a conditional you have to use the "... |
3,846,891 | <p>Let me know if this proof is correct. This is in french.</p>
<p>Translation French --> English</p>
<p><strong>prémisse = premise</strong></p>
<p><strong>supposition = assumption</strong></p>
<p><a href="https://i.stack.imgur.com/Ey8bH.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/Ey8bH.png" a... | Graham Kemp | 135,106 | <p><span class="math-container">$\def\fitch#1#2{~~\begin{array}{|l}#1\\\hline#2\end{array}}$</span></p>
<blockquote>
<p><img src="https://i.stack.imgur.com/r090i.png" alt="no" /></p>
</blockquote>
<p>Not quite.</p>
<p>Conditional Introduction deduces a conditional statement whose antecedent is the assumption of a subpr... |
3,688,680 | <p>I know cantor set and rational numbers in <span class="math-container">$\mathbb{R}$</span> are meagre. But they are all zero measure.</p>
<p>So is there any meagre set that is non-zero measure?</p>
| 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... |
632,891 | <p>I'm trying to solve this limit, for which I already know the solution thanks to Wolfram|Alpha to be $\sqrt[3]{abc}$:</p>
<p>$$\lim_{n\rightarrow\infty}\left(\frac{a^\frac{1}{n}+b^\frac{1}{n}+c^\frac{1}{n}}{3}\right)^n:\forall a,b,c\in\mathbb{R}^+$$</p>
<p>As this limit is an indeterminate form of the type $1^\inft... | Hagen von Eitzen | 39,174 | <p>While $\sqrt[n]a\to 1$, it is not correct to say $(n\sqrt[n]a-n)\to 0$. Actually, $\sqrt[n]{1+\epsilon}\approx 1+\frac1n\epsilon$ so $n\sqrt[n]a-n\approx a$.</p>
|
51,390 | <p>Say $g$ is a matrix which is given as, $g = g_0 + xg_2 + x^2 g_4 .. +x^{d/2 -1}g_{d-2}+ x^{d/2}(g_d + h_d(\log (x)))$ where $d$ is an even number and each $g_i$ is a matrix (same dimension as $g$) and $h_d$ is another matrix. </p>
<ul>
<li>For such a set of arbitrary matrices, how can one power-series expand $\sqrt... | Dr. Wolfgang Hintze | 16,361 | <p>For any square matrix M which is the sum of two similar matrices M = A + B the determinant can be written as a sum of determinants as follows (example for two dimensions):</p>
<pre><code>det(M) = det( ( A11 + B11, A12 + B12), (A21 + B21, A22 + B22) )
= det( ( A11 + 0, A12 + 0), (A21 + B21, A22 + B22) )
+ det( ( 0... |
2,436,268 | <p>My problem is evaluating the following limit:
$$\lim_{(x,y)\to(0,0)}\frac{x^5+y^2}{x^4+|y|}$$
The answer should be 0. I tried to convert the limit into polar form, but it didn't help because I couldn't isolate the $r$ and $\theta$-variables of the expression. My "toolbox" for solving problems like these is very limi... | Nosrati | 108,128 | <p><strong>Hint:)</strong></p>
<p>For paths $x=0$ as $y\to0$ and $y=0$ as $x\to0$ the result of limit clearly is zero, furthermore
$$\Big|\frac{x^5+y^2}{x^4+|y|}\Big|\leq\Big|\frac{x^5}{x^4+|y|}\Big|+\Big|\frac{y^2}{x^4+|y|}\Big|\leq|x|+|y|$$</p>
|
3,640,298 | <p><span class="math-container">$$ f(x,y,z)= \int_{-\infty}^{+\infty} e^ {-t(t-x)(t-y)(t-z)}\;dt$$</span> </p>
<p><span class="math-container">$$t=p+x$$</span>
<span class="math-container">$$ f(x,y,z)= \int_{-\infty}^{+\infty} e^ {-p(p+x)(p-(y-x))(p-(z-x))}\;dp$$</span> </p>
<p><span class="math-container">$$ ... | Asinomás | 33,907 | <p>Because for each prime <span class="math-container">$p$</span> that does not divide <span class="math-container">$b$</span> we have <span class="math-container">$v_p(a)= v_p(ab) = v_p(m^3)=3v_p(m)$</span>.</p>
<p>We can clearly make the part that is not <span class="math-container">$dp_1^{r_1}\dots p_r^{r_t}$</span... |
550,188 | <p>Okay so I have an equation in my book which is as follows..
$$
\frac {a}{s(s+a)}
$$
it says "using partial fractions this can be expanded to
$$
\frac {1}{s} + \frac {-1}{s+a}
$$</p>
<p>My usual method would be to cross multiply and do something like this
$$
\frac {a}{s(s+a)} = \frac {A(s+a)}{s(s+a)} + \frac {B(s)}... | Amzoti | 38,839 | <p>We have:</p>
<p>$$\dfrac{a}{s(s+a)} = \dfrac{A}{s}+\dfrac{B}{s+a}$$</p>
<p>So, </p>
<p>$$a = A(s+a) + Bs = (A+B)s + A a$$</p>
<p>we have $A = 1, B = -1$</p>
<p>Final result:</p>
<p>$$\dfrac{a}{s(s+1)} = \dfrac{1}{s}-\dfrac{1}{s+a}$$</p>
|
323,109 | <p>Could someone help with the following integration:
$$\int^0_\pi \frac{x \sin x}{1+\cos^2 x}$$</p>
<p>So far I have done the following, but I am stuck:</p>
<p>I denoted $ y=-\cos x $ then:
$$\begin{align*}&\int^{1}_{-1} \frac{\arccos(-y) \sin x}{1+y^2}\frac{\mathrm dy}{\sin x}\\&= \arccos(-1) \arctan 1+\a... | Lai | 732,917 | <p>Let <span class="math-container">$y=x-\frac{\pi}{2}$</span>.</p>
<p><span class="math-container">\begin{array}{l}\displaystyle \int_{0}^{\pi} \frac{x \sin x}{1+\cos ^{2} x} d x&=\displaystyle \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{\left(y+\frac{\pi}{2}\right) \cos y}{1+\sin ^{2} y} d y\\&=\displaystyle... |
3,024,169 | <p>First, my apologies. This question may have been asked many times before but I do not know the correct terms to search on..... and my school trigonometry is many years ago. Pointing me to an appropriate already-answered question would be an ideal solution for me.</p>
<p>I am writing a program to do 3D view from a... | goulding | 621,218 | <p>I would look to solve this using utility methods contained in 3D Vectors. I'm not sure what you're using for creating the application, but I've found Unity 3D's documentation on 3D vector Math to be very easy to follow (regardless of whether you're working in Unity).</p>
<p><a href="https://unity3d.com/learn/tutori... |
946,973 | <p>After completing the square, what are the solutions to the quadratic equation below?
<span class="math-container">$$x^2 + 2x = 25$$</span></p>
<p><img src="https://i.stack.imgur.com/AoFhV.png" alt="enter image description here" /></p>
<p>Honstely I think it's B. But I'm not sure.</p>
| Timbuc | 118,527 | <p>$$25=x^2+2x=(x+1)^2-1\implies (x+1)^2=26\implies x+1=\pm\sqrt{26}$$</p>
|
578,487 | <p>Maybe this is a well know result, however, I could not find it. Before stating it, let me write here a well know result (at least for me)</p>
<blockquote>
<p>Assume that $\Omega\subset\mathbb{R}^N$ is a open domain and $f:\Omega\to\mathbb{R}$. If there is constants $L>0$ and $\alpha>1$ such that $$|f(x)-f(y... | Post No Bulls | 111,742 | <h3>Counterexample</h3>
<p>As usual, studiosus is right: the answer is negative. A natural parametrization of an arc of the <a href="http://en.wikipedia.org/wiki/Koch_snowflake" rel="noreferrer">von Koch snowflake</a> gives a topological embedding $g:[0,1]\to \mathbb R^2$ such that $$|g(x)-g(y)|\ge C|x-y|^{p},\quad p=... |
643,918 | <blockquote>
<p>Let $G$ be a group and $a, b \in G$. Show that $(a*b)' = a' * b'$ if and only if $a*b = b*a$.</p>
</blockquote>
<p>While this is simple to see by intuition, I am having a hard time expressing this formally. It seems as if I want to show that $(a*b)' = a' * b'$ strictly implies $a*b = b*a$, but I'm no... | voldemort | 118,052 | <p>Assuming by $a'$ you mean $a^{-1}$ :</p>
<p>Suppose $(ab)=(ba)$. Then we know $(ab)^{-1}=b^{-1}a^{-1}$. But $ab=ba$ implies that $(ab)^{-1}=(ba)^{-1}$.</p>
<p>Conversely assume that $(ab)^{-1}=a^{-1}b^{-1}=(ba)^{-1}$. Then $ab=ba$</p>
|
643,918 | <blockquote>
<p>Let $G$ be a group and $a, b \in G$. Show that $(a*b)' = a' * b'$ if and only if $a*b = b*a$.</p>
</blockquote>
<p>While this is simple to see by intuition, I am having a hard time expressing this formally. It seems as if I want to show that $(a*b)' = a' * b'$ strictly implies $a*b = b*a$, but I'm no... | user44197 | 117,158 | <p>From your equation
$$
(a * b) = (a' *b')' = b'' * a'' = b * a$$</p>
|
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