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 |
|---|---|---|---|---|
796,564 | <p><img src="https://i.stack.imgur.com/rLmA6.jpg" alt="enter image description here"></p>
<p>I don't get the answer to this problem, can somebody please tell me what the answer is. </p>
| afedder | 29,604 | <p><strong>Hint</strong>: Suppose some extension of a line segment has equation $y=mx+b$. Then any line that is perpendicular to this line (in other words, at a $90$ degree angle with it), has slope $-\frac{1}{m}$.</p>
|
276,310 | <p>I'm trying to eliminate variables in some fairly simple sets of equations. A typical example is: </p>
<p>$$ 9 x^2 + 18 xy + 9 y^2 - 32 = 256z$$
$$ 9 x^2 + 6 xy - 3 y^2 - 8 = 376z$$
$$ 9 x^2 - 6 xy + y^2 = 512z$$</p>
<p>I'd like to eliminate $x$ and $y$. Mathematica tells me that the answer is $ 161 z^2 -162z + 1 ... | bubba | 31,744 | <p><strong>Clever Observant Solution (as in Mark's comment)</strong></p>
<p>Denote the right-hand sides by $a$, $b$, $c$. In other words, $a = 32+256z$, $b = 8 + 376z$, $c = 512z$. Also, let $p = 3x + 3y$ and $q = 3x - y$. Then the equations become simply:
$$ p^2 = a \quad ; \quad pq = b \quad ; \quad q^2 = c$$
From t... |
140,294 | <p>Generative adversarial networks (GAN) is regarded as one of "the most interesting idea in the last ten years in machine learning" by Yann LeCun. It can be used to generate photo-realistic images that are almost indistinguishable from the real ones.</p>
<p>GAN trains two competing neural networks: a generator networ... | Taliesin Beynon | 7,140 | <p>Yes it is possible. You can do alternating training manually by literally following the algorithm, so that you have a Do loop whose body contains two calls to NetTrain, but that suffers from overhead at each alternation (this could be overcome with clever caching, but we haven't done that yet). An approximation of t... |
140,294 | <p>Generative adversarial networks (GAN) is regarded as one of "the most interesting idea in the last ten years in machine learning" by Yann LeCun. It can be used to generate photo-realistic images that are almost indistinguishable from the real ones.</p>
<p>GAN trains two competing neural networks: a generator networ... | Anton Antonov | 34,008 | <p>I see the generation of images from classification models as a trivial idea, largely, because this kind of processes and algorithms are standard in natural language processing. </p>
<p>More to the point of the question, the following posts show generation of images with classification derived bases:</p>
<ul>
<li>... |
140,294 | <p>Generative adversarial networks (GAN) is regarded as one of "the most interesting idea in the last ten years in machine learning" by Yann LeCun. It can be used to generate photo-realistic images that are almost indistinguishable from the real ones.</p>
<p>GAN trains two competing neural networks: a generator networ... | Michael Curry | 26,564 | <p>I wrestled with this for a while and got some kind of results, but nowhere near the great performance for which GANs are famous. Ultimately, they're absurdly sensitive to hyperparameters and initialization, and if you don't <em>exactly</em> imitate the published settings, you are unlikely to get good results.</p>
<... |
23,674 | <p>Let $v$ be the 3-adic valuation on $\mathbb{Q}$ and consider the subring $\mathbb{Z}_{(3)}$ of $\mathbb{Q}$ defined by
$$
\mathbb{Z}_{(3)} = \{ x \in \mathbb{Q} : v(x) \geq 0 \}.
$$
That is, $\mathbb{Z}_{(3)}$ is the ring of rational numbers that are integral with respect to $v$. $\mathbb{Z}_{(3)}$ is also the l... | Pete L. Clark | 299 | <p>Here is a different and (perhaps) somewhat more elementary approach than Andrea's.</p>
<p>Let <span class="math-container">$R$</span> be the integral closure of <span class="math-container">$\mathbb{Z}_{(3)}$</span> in <span class="math-container">$\mathbb{Q}(\sqrt{-5})$</span>. Clearly <span class="math-container"... |
136,363 | <p>Consider the following case:</p>
<pre><code>(a^3*b) //. {a^2 -> c, a*b -> d}
</code></pre>
<p>instead of <code>c d</code> the output is:</p>
<pre><code>(*a^3*b*)
</code></pre>
<p>How can I get what I want?</p>
| corey979 | 22,013 | <p>How about</p>
<pre><code>Last@PolynomialReduce[a^3*b, {a^2 - c, a*b - d}, {a, b}]
</code></pre>
<blockquote>
<p>c d</p>
</blockquote>
<p>Or a more tricky</p>
<pre><code>a^3*b //. {a^n_ :> c*Quotient[n, 2]*a^Mod[n, 2],
Times[r1___, a, b, r2___] :> Times[d, r1, r2]}
</code></pre>
<blockquote>
<p>c d<... |
2,583,454 | <p>Consider for instance the linear system:</p>
<p>$$\left(
\begin{array}{cc}
1 & 2 \\
3 & 4 \\
5 & 6 \\
\end{array}
\right).\left(
\begin{array}{c}
x \\
y \\
\end{array}
\right)=\left(
\begin{array}{c}
1 \\
2 \\
4 \\
\end{array}
\right)$$</p>
<p>This is over determined and thus has no solution. Y... | dxiv | 291,201 | <blockquote>
<p>I assumed that the solutions to $\textbf{A}x = \textbf{B}$ were the same as the solutions to $\textbf{P}\textbf{A}x = \textbf{P}\textbf{B}$.</p>
</blockquote>
<p>The direct implication $\textbf{A}x = \textbf{B} \implies \textbf{P}\textbf{A}x = \textbf{P}\textbf{B}$ holds for any $\textbf{P}$. However... |
24,230 | <p>$f$ is continuous between $[0,1]$, and $f(0)=f(1)$.</p>
<p>I want to prove that there is an $a \in [0,0.5]$ such that $f(a+0.5)=f(a)$.</p>
<p>ok, so Rolle's theorem can be useful here, but I can't see the connection to the derivative,</p>
<p>(Weierstrass, Uniform continuity?) I'll be glad to instructions.</p>
<p... | Yuval Filmus | 1,277 | <p>Consider the function $g(a) = f(a+0.5) - f(a)$ on the interval $[0,0.5]$, and use the intermediate value theorem.</p>
|
3,339,780 | <p>The popular definition of a vector is </p>
<blockquote>
<p>A vector is an object that has both a magnitude and <strong>a</strong> direction.</p>
</blockquote>
<p>We know that zero vector has no specific <strong>single</strong> direction.</p>
<p>Then how can it be a vector?</p>
| David G. Stork | 210,401 | <p><span class="math-container">$0 \cdot (a,b,c)$</span> (for arbitrary <span class="math-container">$a$</span>, <span class="math-container">$b$</span>, and <span class="math-container">$c$</span>) has a magnitude (<span class="math-container">$0$</span>) and a direction <span class="math-container">$(a,b,c)$</span>. ... |
2,474,355 | <p>I have this system of linear equations.</p>
<p><em>2x<sub>1</sub> - 4x<sub>2</sub> - x<sub>3</sub> = 1</em></p>
<p><em>x<sub>1</sub> - 3x<sub>2</sub> + x<sub>3</sub> = 1</em></p>
<p><em>3x<sub>1</sub> - 5x<sub>2</sub> - 3x<sub>3</sub> = 1</em></p>
<p>What is the best way or is there any special way to solve this... | DanielWainfleet | 254,665 | <p>A topology $T$ on a set $S$ has a closure operator $Cl_T(A)$ for $A\subset S$, which can be regarded as a function from the set of all subsets of $S$ to the set of all $T$-closed sets. If $T_1,T_2$ are topologies on $S$ and $Cl_{T_1}=Cl_{T_2}$ then the set of $T_1$-closed sets equals the set of $T_2$-closed sets, ... |
501,512 | <p>The lines CD and EF are perpendicular with points $C(1,2)$, $D(3,-4)$, $E(-2,5)$, and $F(k,4)$. Find the value of the constant $k$.</p>
| njguliyev | 90,209 | <p>Hint: $(2, -6)\cdot(k+2,-1)=0$.</p>
|
3,984,480 | <p>Show that <span class="math-container">$-\vec{0} = \vec{0}$</span> in any vector space.</p>
<p>I know this is a seemingly obvious statement but is the following justification correct:</p>
<p>Assume <span class="math-container">$-\vec{0} \neq \vec{0}$</span>.</p>
<p><span class="math-container">$$(4): \vec{u} + \vec{... | Moishe Kohan | 84,907 | <p>For every finite generating set <span class="math-container">$S$</span> of <span class="math-container">$G=Z^n$</span>, the generating function of its growth
<span class="math-container">$$
\sum_{n=0}^\infty \beta_S(n)t^n
$$</span>
is rational. Here <span class="math-container">$\beta_S(n)$</span> is the number of g... |
1,371,580 | <p>i'm reading "A concise introduction ti pure mathematics" by Liebeck and in the exercises of the second chapter i found this question:</p>
<p>"Show that the decimal expression for $\sqrt 2 $ is not periodic"</p>
<p>If i write $\sqrt 2$ in its decimal form, i should obtain something like:</p>
<p>$\sqrt 2 = {a_o}.{... | DanielWainfleet | 254,665 | <p>If x has an expansion ,in any base B, which is eventually periodic, with period length P, then multiplying x by the Pth power of B and subtracting x, yields a rational value for (B**P - 1)x. Hence x is rational. Conversely,if x is rational, then computing its representation in base B by long division eventually giv... |
1,339,875 | <p>Suppose we are given a function
$$g\left ( x \right )= \sum_{n=1}^{\infty}\frac{\sin \left ( nx \right )}{10^{n}\sin \left ( x \right )},x\neq k\pi , k\in\mathbb{Z}$$
and
$$g\left ( k\pi \right )=\lim _{x\rightarrow k\pi}g\left ( x \right )$$
I found that $\lim _{x\rightarrow k\pi}g\left ( x \right )= \sum_{n=1}^{... | Community | -1 | <p>You should first examine the series
<p>
$\sum\limits_{n = 1}^\infty {\frac{{\sin (nx)}}{{{{10}^n}}}} $
and check that this is a continuous function.<p>
Looking at the form of the series, think of geometric series
<p>
$f(z) = \sum\limits_{n = 1}^\infty {\frac{{{z^n}}}{{{{10}^n}}}} $ .
Then by the theory of power ... |
1,339,875 | <p>Suppose we are given a function
$$g\left ( x \right )= \sum_{n=1}^{\infty}\frac{\sin \left ( nx \right )}{10^{n}\sin \left ( x \right )},x\neq k\pi , k\in\mathbb{Z}$$
and
$$g\left ( k\pi \right )=\lim _{x\rightarrow k\pi}g\left ( x \right )$$
I found that $\lim _{x\rightarrow k\pi}g\left ( x \right )= \sum_{n=1}^{... | fho | 225,297 | <p>First check that function converge uniformly. It sure does, on this union: $x \in I = \bigcup_{k=-\infty}^{\infty} (\pi k,\, \pi (k+1))$ because:
$$
\sum_{n=1}^{\infty} \frac{1}{10^n} \frac{\sin(nx)}{\sin(x)} \le \sum_{n=1}^{\infty} \frac{1}{10^n} \frac{|\sin(nx)|}{|\sin(x)|} \le \sum_{n=1}^{\infty} \frac{n}{10^n}
$... |
72,669 | <p>I encountered this site today <a href="https://code.google.com/p/google-styleguide/">https://code.google.com/p/google-styleguide/</a> regarding the programming style in some languages. What would be best programming practices in Mathematica, for small and large projects ?</p>
| Albert Retey | 169 | <p>I think this is a very relevant question as I think it is agreed standard that having "a" coding styleguide for every project where several people write code is a very good (inevitable?) thing. It also seems to be agreed that it is more important to have a styleguide/standard than how excatly that looks li... |
72,669 | <p>I encountered this site today <a href="https://code.google.com/p/google-styleguide/">https://code.google.com/p/google-styleguide/</a> regarding the programming style in some languages. What would be best programming practices in Mathematica, for small and large projects ?</p>
| Shredderroy | 9,257 | <p>In the excellent responses above, I find that one of my favourite guidelines is missing. It concerns the case when one has to apply one function after another.</p>
<p>Instead of writing</p>
<pre><code>f[g[h[k[q[x]]]]]
</code></pre>
<p>write either</p>
<pre><code>x // (q /* k /* h /* g /* f)
</code></pre>
<p>or... |
980,941 | <p>How can I calculate $1+(1+2)+(1+2+3)+\cdots+(1+2+3+\cdots+n)$? I know that $1+2+\cdots+n=\dfrac{n+1}{2}\dot\ n$. But what should I do next?</p>
| Hypergeometricx | 168,053 | <p>$$\begin{align}
&1+(1+2)+(1+2+3)+\cdots+(1+2+3+\cdots+n)\\
&=n\cdot 1+(n-2)\cdot 2+(n-3)\cdot 3+\cdots +1\cdot n\\
&=\sum_{r=1}^n(n+1-r)r\\
&=\sum_{r=1}^n {n+1-r\choose 1}{r\choose 1}\\
&={n+2\choose 1+2}\\
&={n+2\choose 3}\\
&=\frac16 n(n+1)(n+2)
\end{align}$$</p>
|
479,551 | <p>A store carries three types of donuts: Strawberry, Chocolate and Glazed</p>
<p>Suppose you bought $4$ of each kind and in addition, you have the option to apply sprinkles on your donuts. How many ways are there to eat the donuts if you never eat two donuts in a row that both have sprinkles? </p>
<p>The idea I have... | ShreevatsaR | 205 | <p>I'll put hints below; ask me if you need help with any step.</p>
<p>First, count the number of ways of ordering the $12$ donuts: which $4$ are strawberry, which $4$ are chocolate, and which $4$ are glazed. This is a straightforward counting problem.</p>
<p>Second, count the number of ways of sprinkling (or not spr... |
1,027,807 | <p>So I have this question that looks like</p>
<p>$$ \frac{x^3 + 3x^2 - x - 8}{x^2 + x - 6} $$</p>
<p>and first I got the partial fraction so getting </p>
<p>$$ x + 2 + \frac{3x + 4}{x^2 + x -6} $$</p>
<p>but now I'm trying to integrate it and I cannot remember for the life of me how I should integrate the fraction... | John | 7,163 | <p>You can factor the denominator of the last term and decompose again:</p>
<p>$$\frac{3x+4}{x^2+x-6} = \frac{3x+4}{(x+3)(x-2)} = \frac{1}{x+3} + \frac{2}{x-2}.$$</p>
<p>Can you take it from there?</p>
|
3,669,080 | <p>I would love to get some insight on how to solve <span class="math-container">$\int_0^{\frac\pi4}\log(1+\tan x)\,\mathrm dx$</span> using Leibniz rule of integration. I know it can be solved using the property<span class="math-container">$$\int_a^bf(x)\,\mathrm dx=\int_a^bf((a+b)-x)\,\mathrm dx,$$</span>but I find t... | Eeyore Ho | 747,376 | <p>You should let <span class="math-container">$ I(t)=\int_{0}^{\frac{\pi}{4}}\ln(1+t \tan x) dx,I=I(1),I(0)=0 $</span>
<span class="math-container">$$ I'(t)=\int_{0}^{\frac{\pi}{4}}\frac{\tan x}{1+t \tan x} dx=\int_{0}^{\frac{\pi}{4}}\frac{\sin x}{\cos x+t \sin x} dx=\int_{0}^{\frac{\pi}{4}}\left( \frac{t}{1+t^2}+\fra... |
487,084 | <p>I need to know if every group whose order is a power of a prime $p$ contains an element of order $p$? Should I proceed by picking an element $g$ of the group and proving that there is an element in $\langle g \rangle$ that has order $p$?</p>
| N. S. | 9,176 | <p>This follows immediately from Lagrange theorem, you don't need any stronger result.</p>
<p>If the order of the group is $p^k$ with $k \neq 0$, then by Lagrange Theorem, the order of any element divides $p^k$.</p>
<p>Pick some $x \in G, x \neq e$. Then the order of $x$ is $p^m$ with $1 \leq m \leq k$. Let</p>
<p>$... |
3,256,767 | <p>So I'm trying to understand a solution made by my teacher for a question that asks me to determine whether the following is true. I'm having trouble understanding where some values in the steps are coming from.</p>
<p>Like for the first part, I don't really get where n≥5 came from. My guess is getting 16n^2 + 25 to... | DonAntonio | 31,254 | <p>Simply define <span class="math-container">$\;T:\Bbb R^4\to\Bbb R^3\;$</span> by </p>
<p><span class="math-container">$$T(1,1,1,1)=(1,2,1)\,,\,\,T(1,0,1,0):=(2,1,0)$$</span></p>
<p>and then complete <span class="math-container">$\;\{v_1=(1,1,1,1)\,,\,\,v_2=(1,0,1,0)\}\;$</span> to a basis of <span class="math-co... |
1,840,924 | <p>My teacher showed this proof using the dominated convergence theorem or Fourier analysis, but I wonder if there is an elementary proof of this problem. My teacher said it is difficult to solve this in an elementary way, but do you know how?</p>
| stewbasic | 197,161 | <p>Here is a proof using measure theory (so unfortunately not elementary).</p>
<p>Fix the subsequence $n_1<n_2<\ldots$ and let $X$ be the set of $x\in\mathbb R$ for which $\sin(n_kx)$ converges. For any $\epsilon\in(0,1)$ let
$$\begin{eqnarray*}
Y_k^\epsilon&=&\{x\in[0,4\pi]\mid|\sin(n_kx)-\sin(n_{k+1}... |
1,840,924 | <p>My teacher showed this proof using the dominated convergence theorem or Fourier analysis, but I wonder if there is an elementary proof of this problem. My teacher said it is difficult to solve this in an elementary way, but do you know how?</p>
| stewbasic | 197,161 | <p>Here is an elementary proof inspired by <a href="https://gowers.wordpress.com/2008/07/25/what-is-deep-mathematics/" rel="nofollow noreferrer">this proof</a>.</p>
<p>Fix the subsequence $1\leq n_1<n_2<\ldots$. I will inductively construct sequences $a_1\leq a_2\leq\ldots$ and $b_1\geq b_2\geq\ldots$ such that:... |
976,881 | <p>Simple and quick question. These two have to do with 90 degree angles.</p>
<p>This is the picture of the two words I have.</p>
<blockquote>
<p>Perpendicular is strictly restricted to lines.</p>
</blockquote>
<ul>
<li><p>"Line A and B are perpendicular to each other."</p></li>
<li><p>"v=(1, 1) and w=(-1, 1) -> c... | Karthik Upadhya | 98,696 | <p>Yep. Perpendicular to each other in a higher dimensional space ( 4 dimensions in your example ).</p>
<p>In fact, you can write out the angle $\theta$ between two vectors $x,y$ as</p>
<p>$$cos(\theta) = \frac{x^T y}{\sqrt{x^T x} \sqrt{y^T y}} $$</p>
|
638,566 | <p>I came across a question yesterday about combinations, and I wanted to know what the correct answer was. The question states as follows:</p>
<p>There are 8 spaces that are alternately black and white. There is one king, one queen, two identical rooks, two identical bishops, and two identical knights. The king needs... | Ross Millikan | 1,827 | <p>Hint: how many ways are there to select the leftmost square of the RKR series (I am taking surrounded by as one rook is immediately next to the king on each side)? Now place the bishops-how many ways to do that? Now place the queen-how many ways? The knights are now fixed.</p>
|
2,203,907 | <p>I am trying to show that:
$$\mathcal{L}\{erfc( \frac{k}{2\sqrt t})\} = \frac{1}{s}e^{-k\sqrt s}$$
The hint given for this question is the Laplace Transform of an integral (from convolution):
$$\mathcal{L}\{\int_{0}^{t}f(u) \, du \} = \frac{1}{s} \mathcal{L}\{f(u)\} \tag{1}$$</p>
<p>I have read in a different text t... | Fabian | 7,266 | <p>Let us denote with $$F(t) = \mathop{\rm erfc}(k/2\sqrt{t})$$
and also $$f(t)= \frac{d}{dt}F(t).$$</p>
<p>Now let us look at the expression
$$\mathcal{L}\{F(t) \}=\mathcal{L}\{\int_{0}^{t}f(u) \, du \} = \frac{1}{s} \mathcal{L}\{f(t)\} \tag{1}.$$
In order to calculate $\mathcal{L}\{F(t) \}$, you need to evaluate
$$\... |
3,416,019 | <p>If a function <span class="math-container">$f:A\longrightarrow B\times C, f(x)=(g(x),h(x))$</span> is injective, does it imply that <span class="math-container">$h(x)$</span> and <span class="math-container">$g(x)$</span> are also injective? I think this is straight forward but just want to confirm. </p>
<p>Suppose... | John Hughes | 114,036 | <ol>
<li><p>Your question should be about <span class="math-container">$g$</span> and <span class="math-container">$h$</span>, not <span class="math-container">$f$</span> and <span class="math-container">$g$</span>. </p></li>
<li><p>Your argument is mistaken. For instance, <span class="math-container">$h(x) = 1, g(x) =... |
262,003 | <p>Is there a positive integer $N$, besides 1 and 2, such that there is a permutation $a_1=1,a_2,a_3,\dots,a_N$ of $1,2,3,\dots,N$ in which for each $k>1$, $a_k=a_{k-1}\div k,a_k=a_{k-1}-k,a_k=a_{k-1}+k,\textrm{or }a_k=a_{k-1}\times k$?</p>
| Gerhard Paseman | 3,402 | <p>It may be of interest to see how far it can be continued, but it cannot be completed. Suppose we ignore the restriction that the first term must be 1. The last term must be the previous term times or divided by N. So the last two terms are 1,N or N,1. Now unless N is 2, there is no way to use N-1 to generate the... |
2,113,596 | <p>Questions with likely obvious answers, but I don't have the required intuition to go with the flow.</p>
<p>Consider $a+be^x + ce^{-x} = 0$. To solve it for the constants, we can try out different values of $x$ to get a system of $3$ equations and simply use a calculator (given technique). Why are we allowed to stic... | Community | -1 | <p>Let $F'=F+1$ denote the new temperature in Fahrenheit where $F$ is the old temperature. Let $C'$ and $C $ similarly represent the new and old temperature in Celsius. </p>
<p>Then we get $$C' =\frac {5}{9}((F+1)-32) =\frac {5}{9}(F-32) +\frac{5}{9} = C +\frac {5}{9} $$ Hope it helps. </p>
|
2,801,406 | <blockquote>
<p>Find the coordinates of the points where the line tangent to the curve $$x^2-2xy+2y^2=4$$ is parallel to the $x$-axis, given that $$\frac{dy}{dx}=\frac{y-x}{2y-x}$$</p>
</blockquote>
<p>By letting $dy/dx = 0$ I get $y=x$ which is no help... what do I do?</p>
<p>Thanks</p>
| tien lee | 557,074 | <p>$$x^2-2xy+2y^2=4$$
$$\frac{d}{dx}(x^2-2xy+2y^2)=\frac{d}{dy}(4)$$
$$2x-2(y+x\frac{dy}{dx})+4y\frac{dy}{dx}=0$$
$$2x-2y-2x\frac{dy}{dx}+4y\frac{dy}{dx}=0$$
$$\frac{dy}{dx}=\frac{y-x}{2y-x}$$
Now equate $\frac{dy}{dx}=0$
$$\frac{y-x}{2y-x}=0$$
$$x=y,x=2y$$
Solve by substitution, I substituted $y=x$ in $x^2-2xy+2y^2=4$... |
6,562 | <p>I want to make some button shaped graphics that would essentially be a rectangular shape with curved edges. In the example below I have used <code>Polygon</code> rather than <code>Rectangle</code> so as to take advantage of <code>VertexColors</code> and have a gradient fill. The code below illustrates the sort of th... | M.R. | 403 | <p>Vitaliy had a great answer. I guess another way to do this is to simply make the curves using many lines:</p>
<p><img src="https://i.stack.imgur.com/Vv8z8.png" alt="enter image description here"></p>
<p>In the following code, resolution is the number of lines used to make the curve and m is how big the corners are... |
6,562 | <p>I want to make some button shaped graphics that would essentially be a rectangular shape with curved edges. In the example below I have used <code>Polygon</code> rather than <code>Rectangle</code> so as to take advantage of <code>VertexColors</code> and have a gradient fill. The code below illustrates the sort of th... | Mr.Wizard | 121 | <p>The raster method I alluded to in a comment was requested.</p>
<pre><code>g1 = Graphics[{
Polygon[{{0, 0}, {3, 0}, {3, 1}, {0, 1}}, VertexColors -> {Red, Red, Blue, Blue}]
}]
g2 = Graphics[{Rectangle[{0, 0}, {3, 1}, RoundingRadius -> 0.5]}]
ImageAdd[g1, g2]
</code></pre>
<p><img src="https://i... |
6,562 | <p>I want to make some button shaped graphics that would essentially be a rectangular shape with curved edges. In the example below I have used <code>Polygon</code> rather than <code>Rectangle</code> so as to take advantage of <code>VertexColors</code> and have a gradient fill. The code below illustrates the sort of th... | Carl Woll | 45,431 | <p>I think you can achieve what you want using <a href="http://reference.wolfram.com/language/ref/Texture.html" rel="nofollow noreferrer"><code>Texture</code></a>. Basically, create a <a href="http://reference.wolfram.com/language/ref/FilledCurve.html" rel="nofollow noreferrer"><code>FilledCurve</code></a> version of a... |
1,829,975 | <p>Find three different systems of linear equation whose solutions are $x_1 = 3, x_2 = 0, x_3 = -1$</p>
<p>I'm confused, how exactly can I do this?</p>
| Doug M | 317,162 | <p>come up with an a linear equation that includes the indicated point.</p>
<p>e.g. $x + y + 3z = 0$
Now come up with two more. If you want your solution to be unique you will need to make sure that the planes you have chosen are "linearly independent."</p>
|
134,001 | <p>what is the basic difference between the Discrete Fourier Transform and the Wavelet Transform ? and why does JPEG2000 preferred DWT over DCT or DFT ? </p>
| Christian Blatter | 1,303 | <p>The basic difference is the following: Assume you have a data vector ${\bf x}=(x_1,\ldots, x_N)$ of length $N:=2^n$ that models a function of a real variable $t$ on some finite interval.</p>
<p>In DFT you compute $N$ complex Fourier coefficients of this data vector according to the formula
$$\hat x_j:={1\over N}\s... |
2,391,624 | <p>This question pertains to Mosteller's classic book <em>Fifty Challenging Problems in Probability</em>. Specifically, this in regards to an algebraic operation Mosteller performs in the solution to the first question, entitled "The Sock Drawer."</p>
<p>Mosteller writes:</p>
<blockquote>
<p>Then we require the pro... | Donald Splutterwit | 404,247 | <p>\begin{eqnarray*}
\frac{r}{r+b} \color{blue}{\frac{r}{r+b}} > \frac{r}{r+b} \color{blue}{\frac{r-1}{r+b-1}}=\frac{1}{2} =\color{orange}{\frac{r}{r+b}} \frac{r-1}{r+b-1}>\color{orange}{\frac{r-1}{r+b-1}} \frac{r-1}{r+b-1}.
\end{eqnarray*}</p>
|
679,135 | <p><img src="https://i.stack.imgur.com/uJEMx.png" alt="enter image description here">The question is find the δ by The maximum likelihood estimation?
My answer is δ=0 but I am not sure whether it is correct and how tho show its biasness?</p>
| user44197 | 117,158 | <p>Assume both number are nonzero.</p>
<p>The condition is equivalent to
$$
(z_1-z_2)(\bar z_1 - \bar z_2)=(z_1+z_2)(\bar z_1 + \bar z_2)
$$
Multiply out to get
$$
\frac{z_1}{\bar z_1} = -\frac{z_2}{\bar z_2}
$$
or
$$
2 \angle z_1 = \pm\pi + 2 \angle z_2
$$
which gives
$$
\angle z_1 -\angle z_2 = \pm\frac{\pi}{2}
$$<... |
1,266,210 | <p>Hello everybody my query is regarding the number of positive integral solution.</p>
<blockquote>
<p>In the sport of cricket, find the number of ways in which a batsman can score $14$ runs in $6$ balls not scoring more than $4$ runs in any ball.</p>
</blockquote>
| Mann | 126,204 | <p>I just realized that this game is cricket, and actually this question has insufficient information ,because in cricket you usually can't make scores of $2$ or $3$ but only $0,1,4,6$ most of time. Well it depends on condition, but i am here going to assume that all outcome are possible from $0\to4$.</p>
<p>So our si... |
1,771,727 | <p>Prove that </p>
<p>$$
\frac{1}{\sqrt2} \sum_{i=1}^m \|s_i - P\|_1
\leq \sum_{i=1}^m \|s_i - P\|_2
\leq \sum_{i=1}^m \|s_i - P\|_1
$$</p>
<p>Where $m$ is a number of points in 2d plane, e.g $s_i = (s_{x_i}, s_{y_i})$ and $p = (p_x, p_y)$</p>
<p>Note that
$$
\begin{split}
\|s_i - p\|_1 &= |s_{x_i} - p_x... | Bérénice | 317,086 | <p>You can use the exponential form.</p>
<p>$z=2(1+ \sqrt3 i)^{1/2}=2\sqrt2(\frac{1}{2}+ \frac{\sqrt3}{2} i)^{1/2}=2\sqrt2(e^{\frac{i\pi}{3}})^{\frac{1}{2}}=2\sqrt2(e^{\frac{i\pi}{6}})=2\sqrt2(\frac{\sqrt3}{2}+i\frac{1}{2})=\sqrt2\sqrt3+i\sqrt2=\sqrt6+i\sqrt2$</p>
|
1,771,727 | <p>Prove that </p>
<p>$$
\frac{1}{\sqrt2} \sum_{i=1}^m \|s_i - P\|_1
\leq \sum_{i=1}^m \|s_i - P\|_2
\leq \sum_{i=1}^m \|s_i - P\|_1
$$</p>
<p>Where $m$ is a number of points in 2d plane, e.g $s_i = (s_{x_i}, s_{y_i})$ and $p = (p_x, p_y)$</p>
<p>Note that
$$
\begin{split}
\|s_i - p\|_1 &= |s_{x_i} - p_x... | Emilio Novati | 187,568 | <p>Hint:</p>
<p>As noted in the comment you have $(4+4 \sqrt {3} i)^{1/2}=2(1+\sqrt {3} i)^{1/2}$. Now use the <a href="https://en.wikipedia.org/wiki/Complex_number#Polar_form" rel="nofollow">polar form</a> for the complex number:
$
u=(1+\sqrt {3} i)= re^{i\theta}
$
where:</p>
<p>$r=|u|=\sqrt{1+3}=2$ and $\theta=\ar... |
2,877,578 | <p>Yesterday, I asked the question: <a href="https://math.stackexchange.com/questions/2876740/prove-that-if-a-b-are-closed-then-exists-u-v-open-sets-such-that-u-cap?noredirect=1#comment5938458_2876740">Prove that if $A,B$ are closed then, $ \exists\;U,V$ open sets such that $U\cap V= \emptyset$</a>. </p>
<p>Here is th... | qualcuno | 362,866 | <p>Let </p>
<p>$$
f(X,d) \longrightarrow [0,1] \\
x \mapsto \frac{d(x,A)}{d(x,A) + d(x,B)}
$$</p>
<p>Note that $f$ is continuous, because $d(\cdot,A)$ is, and that it is in fact well defined, because $d(x,A) + d(x,B) = 0$ if and only if $d(x,A) = 0$ and $d(x,B) = 0$ which cannot happen because it would imply $x \in \... |
1,679,615 | <p>From what I have read about a transitive relation is that if xRy and yRz are both true then xRz has to be true. </p>
<p>I'm doing some practice problems and I'm a little confused with identifying a transitive relation. </p>
<p>My first example is a "equivalence relation"
$S=\{1,2,3\}$ and $R = \{(1,1),(1,3),(2,2),... | Eric Wofsey | 86,856 | <p>To be transitive, $xRz$ needs to hold <em>whenever</em> you have $x$, $y$, and $z$ such that $xRy$ and $yRz$. You're only testing the case $x=y=1$ and $z=3$, but there might be other cases. For instance, in your first example, you can let $x=1$, $y=3$, and $z=2$, and you find that $1R3$ and $3R2$, but $1R2$ is not... |
50,680 | <p>Similar question to <a href="https://math.stackexchange.com/questions/50675/logical-squabbles">this</a> question (This question was actually the motivation for the other question, which - if I haven't got it wrong - generalizes this one): I have to prove a proposition of the following form: $$\forall T \subseteq V: ... | Brian M. Scott | 12,042 | <p>The proposition $(\forall T \in \{F \in \mathcal P(V):P(F)\})Q(T)$ is <em>not</em> vacuously true: it says that if $P(T)$ is true, then so is $Q(T)$. It's equivalent to $(\forall T \subseteq V)[P(T) \to Q(T)]$, not to $(\forall T \subseteq V)[P(T) \land Q(T)]$</p>
|
50,680 | <p>Similar question to <a href="https://math.stackexchange.com/questions/50675/logical-squabbles">this</a> question (This question was actually the motivation for the other question, which - if I haven't got it wrong - generalizes this one): I have to prove a proposition of the following form: $$\forall T \subseteq V: ... | Arturo Magidin | 742 | <p>The reason the kind of "rewriting" you are doing is not in general valid is that the universal quantifier does not specify a domain; when we write something like "$\forall T\in X (U(T))$", we are actually abusing notation, because the $\forall$ only quantifies over <em>all</em> objects, not over <em>some</em> object... |
1,803,205 | <p>I am trying to find $\int_0^{\infty} \frac{dx}{1 + x^n}$ using contour integration. I did the computation by taking the contour $[0,R] \cup \gamma_R \cup [R e^{2i\pi/n}, 0]$, with $\gamma_R$ the arc joining $R$ to $Re^{2i\pi/n}$, and found that (for $n \ge 2$):</p>
<p>$$\int_0^{\infty} \frac{dx}{1 + x^n} = \frac{2\... | Marco Cantarini | 171,547 | <p>If I have understood correctly, you want to see why the integral is also equal to $$\frac{\pi/n}{\sin\left(\pi/n\right)}
$$ (if I am wrong tell me and I will delete the answer). A way to see it to observe that, taking $1+x^{n}=u^{-1}
$ $$\int_{0}^{\infty}\frac{1}{1+x^{n}}dx=\frac{1}{n}\int_{0}^{1}u^{-1/n}\left(1-x... |
3,366,158 | <blockquote>
<p>find a linear transformation such that <span class="math-container">$M^2 (1- M) =0$</span> but <span class="math-container">$M$</span> is not idempotent ?</p>
</blockquote>
<p>My attempt : i take take the vector space generated by the base <span class="math-container">$\{e_1, e_2\}$</span> and def... | TomGrubb | 223,701 | <p>Hint: you can assume with no loss of generality that <span class="math-container">$M$</span> is in Jordan Normal form.</p>
|
2,161,911 | <p>Find the limit :</p>
<p>$$\lim_{ n\to \infty }\sqrt[n]{\prod_{i=1}^n \frac{1}{\cos\frac{1}{i}}}=\,\,?$$</p>
<p>My try :</p>
<p>$$\lim_{ n\to \infty }\sqrt[n]{a} =1\,\, \text {for} \,\,a>0$$</p>
<p>and;</p>
<p>$$\prod_{i=1}^n \frac{1}{\cos\frac{1}{i}}>0$$</p>
<p>so :</p>
<p>$$\lim_{ n\to \infty }\sqrt[n... | Claude Leibovici | 82,404 | <p><em>Much less elegant than RRL'answer.</em></p>
<p>Consider $$P_n=\left(\prod _{i=1}^n \sec \left(\frac{1}{i}\right)\right){}^{\frac{1}{n}}$$ $$\log(P_n)=\frac 1n \sum _{i=1}^n \log\left(\sec \left(\frac{1}{i}\right)\right)$$ Now, by Taylor for large values of $i$ $$\sec \left(\frac{1}{i}\right)=1+\frac{1}{2 i^2}+\... |
124,498 | <p>When I produce a simple plot like so:</p>
<pre><code>Plot[Sin[x], {x, 0, 2 Pi}, GridLines -> Automatic, GridLinesStyle -> Directive[Red]]
</code></pre>
<p>It looks like this:</p>
<p><a href="https://i.stack.imgur.com/KcyZo.png" rel="noreferrer"><img src="https://i.stack.imgur.com/KcyZo.png" alt="enter imag... | halirutan | 187 | <p>The grid-lines are directly connected to the <code>PlotRange</code> that you use. Although you only plot from 0 to <code>2*Pi</code>, <em>Mathematica</em> adds a little space around your plot. This little space, called <code>PlotRangePadding</code> is the source of this issue:</p>
<pre><code>Plot[Sin[x], {x, 0, 2 P... |
734,248 | <p>Example of two open balls such that the one with the smaller radius contains the one with the larger radius.</p>
<p>I cannot find a metric space in which this is true. Looking for hints in the right direction. </p>
| user2345215 | 131,872 | <p>Have you considered the trivial $1$ point metric space? Every ball is the same there.</p>
<p>In general, pick any space with an isolated point. Then you can have equal balls around it with different radii.</p>
<p>Also note that every metric space can be bounded by cutting off the distance, i.e. $d'(x,y)=\min(d(x,y... |
734,248 | <p>Example of two open balls such that the one with the smaller radius contains the one with the larger radius.</p>
<p>I cannot find a metric space in which this is true. Looking for hints in the right direction. </p>
| Henno Brandsma | 4,280 | <p>Look at the plane ($\mathbb{R}^2$) in the so-called post-office metric. Denoting the usual norm on the plane by $\|x\|$, we define this metric for two distinct points $x$ and $y$ in the plane as $d(x,y) = \|x\| + \|y\|$, and $d(x,x) = 0$, of course. One easily checks that this is a a metric.</p>
<p>The intuition is... |
2,105,963 | <p>Suppose ${{A_1}}$=[1,3] and ${{A_2}}$=[2,4], then ${{A_1} \cup {A_2}}$=[1,4] now $\sup \left( {{A_1} \cup {A_2}} \right)$ is clearly 4. so, $\sup \left( {{A_1} \cup {A_2}} \right) = \max \left( {\sup {A_1},\sup {A_2}} \right)$ is true.</p>
<p>Confusion with definition: <br/>
<em>s</em> is least upper bound for a se... | Ansel B | 397,059 | <p>It will be the case that $s1\leq s_2$ or $s_2\leq s_1$, but we can take wlog $s_1\leq s_2$ as the argument will be mirrored in the dual case. So then $s_2$ is a uper bound (ub) of both $A_1$ and $A_2$ and the union as well. So then if u is a ub of $A_1\cup A_2$ it is a ub of $A_2$ as well so then it must be that $s_... |
373,357 | <p>I've tought using split complex and complex numbers toghether for building a 3 dimensional space (related to my <a href="https://math.stackexchange.com/questions/372747/what-are-the-uses-of-split-complex-numbers?noredirect=1">previous question</a>). I then found out using both together, we can have trouble on the pr... | Anixx | 2,513 | <p>Here is the Mathematica code for your proposed system, split-quaternions, using <span class="math-container">$2\times2$</span> real matrices <span class="math-container">$i=\left(
\begin{array}{cc}
0 & 1 \\
-1 & 0 \\
\end{array}
\right),j=\left(
\begin{array}{cc}
0 & 1 \\
1 & 0 \\
\end{array}
\ri... |
272,173 | <p>I'm looking for several references on the spectral analysis of the Laplacian operator. It is such a well-known topic, but I'm a bit struggling to locate modern systematic expositions in the literature. </p>
<p>I'd appreciate multiple suggestions that explore the topic using different approaches too.</p>
<p>I'm par... | Bombyx mori | 18,850 | <p>I took a reading course based on Sogge's book "<a href="http://press.princeton.edu/titles/10290.html" rel="noreferrer">Hangzhou Lectures on Eigenfunctions of the Laplacian</a>" a long time ago. This may serve as a standard reference because most of the results mentioned in this book are well-known among researchers.... |
820,878 | <p>I am not quite familiar with the concept of correlation.
The Pearson's correlation coefficient is defined as:</p>
<p><span class="math-container">$$\rho_{X,Y}=\mathrm{corr}(X,Y)={\mathrm{cov}(X,Y) \over \sigma_X \sigma_Y} ={E[(X-\mu_X)(Y-\mu_Y)] \over \sigma_X\sigma_Y}$$</span></p>
<p>which makes use of Mean and Sta... | Anatoly | 90,997 | <p>The Pearson's correlation can be used only for normal distributions. If you have some non-normal distribution, you can use the Spearman's correlation, which does not require the normality assumption.</p>
|
9,990 | <p>Consider the following problem: </p>
<ul>
<li>Maria always buys ice-cream when she goes to the beach. She bought ice-cream today. So, she must have gone to the beach. </li>
</ul>
<p>Obviously this statement is wrong. Maria could have gone to other place and bought an ice-cream. You don't need any math tool to ar... | Syntax Junkie | 5,465 | <p>There is a paradox here. Implication IS tricky. Part of the reason is it's REALLY HARD to avoid implicit assumptions in human scenarios. The only absolute way to avoid assumptions is to translate the English into meaningless symbols (p, q) and rigorously apply logic rules. The rules themselves are tricky. You pr... |
9,990 | <p>Consider the following problem: </p>
<ul>
<li>Maria always buys ice-cream when she goes to the beach. She bought ice-cream today. So, she must have gone to the beach. </li>
</ul>
<p>Obviously this statement is wrong. Maria could have gone to other place and bought an ice-cream. You don't need any math tool to ar... | Dan Fox | 672 | <p>This is a subtle issue. It goes to the heart of the difference between math and physics.</p>
<p>That A implies B does not entail that B implies A. One encounters frequently the errant reasoning that it does even among engineering students in the university (yesterday a student told me that because a matrix was diago... |
1,653,416 | <p>We know that:
<a href="https://www.youtube.com/watch?v=w-I6XTVZXww" rel="nofollow">https://www.youtube.com/watch?v=w-I6XTVZXww</a>
$$S=1+2+3+4+\cdots = -\frac{1}{12}$$</p>
<p>So multiplying each terms in the left hand side by $2$ gives:
$$2S =2+4+6+8+\cdots = -\frac{1}{6}$$
This is the sum of the even numbers</p>
... | Travis Willse | 155,629 | <p>Interpreted literally (i.e., using the usual sense of limits of infinite series), the first line,
$$1 + 2 + 3 + \cdots = -\frac{1}{12} ,$$
is simply false, as the series on the l.h.s. diverges.</p>
<p>What's true, for example, is that there's a natural way to extend the function $$Z(s) := \sum_{k = 1}^{\infty} k^{-... |
2,660,595 | <p>Ten people are sitting in a row, and each is thinking of a negative integer no smaller than $-15$. Each person subtracts, from his own number, the number of the person sitting to his right (the rightmost person does nothing). Because he has nothing else to do, the rightmost person observes that all the differences w... | quasi | 400,434 | <p>Proceed by induction on $n$.
<p>
For $n=1$,
$$x^{2^1}-1 = x^2-1 = (x+1)(x-1)$$
which is a multiple of $2^{1+1}=4$, since $x+1$ and $x-1$ are both even.
<p>
Thus, the base case is verified.
<p>
Next, suppose the claim holds for a fixed integer $n \ge 1$.
<p>
By the inductive hypothesis, we have $2^{n+1}{\,\mid\,}\lef... |
1,206,195 | <p>I am trying to find the maximum of $x^{1/x}$. I don't know how to find the derivative of this. I have plugged in some numbers and found that $e^{1/e}$ seems to be the maximum at around 1.44466786. I don't know if this is the maximum, and I would like an explanation of why it is/what the maximum is. essentially, how ... | evinda | 75,843 | <p>$$x^{\frac{1}{x}}=e^{\ln{x^{\frac{1}{x}}}}=e^{\frac{1}{x} \ln x}$$</p>
<p>Can you find now the derivative?</p>
|
191,796 | <blockquote>
<p>I met with the following difficulty reading the paper <a href="http://www.cnki.com.cn/Article/CJFDTotal-ZZDZ198801008.htm" rel="nofollow">Li, Rong Xiu "The properties of a matrix order column" (1988)</a>:</p>
<p>Define the matrix $A=(a_{jk})_{n\times n}$, where
$$a_{jk}=\begin{cases}
j+k\cdot ... | math110 | 38,620 | <p>I use Christian Remling idea,In fact,I can find the matrix $$B_{ij}=\min{\{i,j\}}$$eigenvalue is
$$\dfrac{1}{4\sin^2{\dfrac{j\pi}{2(n+1)}}},j=1,2,\cdots,n$$
proof:
then we have
$$B=\begin{bmatrix}
1&1&1&\ddots&1&1\\
1&2&2&\ddots&\ddots&2\\
1&2&3&3&\ddots&... |
1,083,841 | <p>I have extracted the below passage from the wikipedia webpage - Point (geometry): </p>
<blockquote>
<p>In particular, the geometric points do not have any length, area, volume, or any other dimensional attribute. </p>
</blockquote>
<p>I think the above passage imply\ies that the point is zero dimensional. If... | Community | -1 | <p>The trick is that there's more to a line than just being made up of points -- the line is also known to live in some sort of <em>topological space</em> or some richer structure. e.g. the axioms of Euclidean geometry talk not just of points lying on lines, but that one point on a line may be between others, that line... |
1,083,841 | <p>I have extracted the below passage from the wikipedia webpage - Point (geometry): </p>
<blockquote>
<p>In particular, the geometric points do not have any length, area, volume, or any other dimensional attribute. </p>
</blockquote>
<p>I think the above passage imply\ies that the point is zero dimensional. If... | Simon S | 21,495 | <p>It's a good question. Here's one approach that is broadly consistent with modern measure theory:</p>
<p>Start with a line segment of length $1$. If we halve its length $n$ times, then the resulting line segment has length of $1/2^n$ and that is always greater than the length of a point in the line. Write $L(point)$... |
1,083,841 | <p>I have extracted the below passage from the wikipedia webpage - Point (geometry): </p>
<blockquote>
<p>In particular, the geometric points do not have any length, area, volume, or any other dimensional attribute. </p>
</blockquote>
<p>I think the above passage imply\ies that the point is zero dimensional. If... | Tyler Durden | 63,397 | <p>I assume you mean a line segment, not a line.</p>
<p>A line segment is not a "set of points". Euclid defines a line segment as a length without width. In other words, a line segment is defined as its length, not as a set of points.</p>
|
1,299,127 | <p>Can someone please help me answer this question as I cannot seem to get to the answer.
Please note that the Cauchy integral formula must be used in order to solve it.</p>
<p>Many thanks in advance!
\begin{equation*}
\int_{|z|=3}\frac{e^{zt}}{z^2+4}=\pi i\sin(2t).
\end{equation*}</p>
<p>Also $|z| = 3$ is given the ... | Alex Fok | 223,498 | <p>Since all the poles of the integrand are enclosed by the contour, we have
\begin{eqnarray}
\int_{|z|=3}\frac{e^{zt}}{z^2+4}dz=2\pi i\text{Res}_{z=0}\frac{1}{z^2}\frac{e^{\frac{t}{z}}}{\frac{1}{z^2}+4}=2\pi i\text{Res}_{z=0}\frac{e^{\frac{t}{z}}}{1+4z^2}
\end{eqnarray}
Note that
$\displaystyle \frac{e^{\frac{t}{z}}... |
1,299,127 | <p>Can someone please help me answer this question as I cannot seem to get to the answer.
Please note that the Cauchy integral formula must be used in order to solve it.</p>
<p>Many thanks in advance!
\begin{equation*}
\int_{|z|=3}\frac{e^{zt}}{z^2+4}=\pi i\sin(2t).
\end{equation*}</p>
<p>Also $|z| = 3$ is given the ... | Kaj Hansen | 138,538 | <p>You should know that using the residue theorem would be easier, but if we are to restrict ourselves to <a href="http://en.wikipedia.org/wiki/Cauchy%27s_integral_formula" rel="nofollow noreferrer"><strong>Cauchy's integral formula</strong></a>, then here's one way of attacking it:</p>
<p>First, note that the integra... |
2,873,474 | <p>My staff room is having a debate about the construction of sample spaces.</p>
<blockquote>
<p>When you toss a coin twice, do you consider the sample space to be
$$\{H,H\}, \{H,T\}, \{T,T\}$$ or $$\{H,H\}, \{H,T\}, \{T,H\},\{T,T\}$$</p>
</blockquote>
<p>In my humble opinion, I feel there is no single correct an... | Patrick | 317,074 | <p>It depends what the sample space is <strong>for</strong>. </p>
<p>If you are tossing 2 coins and just counting the number of heads/tails that happened then your first sample space would be correct. I.e you <strong>don't</strong> care about the order that the events occurred.</p>
<p>If you are tossing 2 coins and r... |
3,493,151 | <p>This is a calculus problem from a high school math contest in Greece,from 2012.</p>
<p>I wish to know some solutions for this. I attempted to solve it.</p>
<blockquote>
<p>Let <span class="math-container">$f:\Bbb{R} \to \Bbb{R}$</span> differentiable such that <span class="math-container">$\lim_{x \to +\infty}f(... | Peter Szilas | 408,605 | <p>Credit to user622002.</p>
<p>Show <span class="math-container">$\lim_{x \rightarrow \infty} \dfrac{f(x)}{x^n}=\infty$</span>, <span class="math-container">$n \in \mathbb{N}$</span>.</p>
<p>Induction:</p>
<p>Base case: <span class="math-container">$n=0$</span> √.</p>
<p>Hypothesis:</p>
<p><span class="math-conta... |
4,054,428 | <p>The question is</p>
<p>Let <span class="math-container">$X$</span> be a discrete random variable with probability mass function</p>
<p><a href="https://i.stack.imgur.com/DhJVI.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/DhJVI.png" alt="enter image description here" /></a></p>
<p>(a) Find <span... | RobertTheTutor | 883,326 | <p>You forgot to square the x's before multiplying and then adding.
<span class="math-container">$$E[X^2] = (-2)^2\cdot 0.3 + (-1)^2 \cdot 0.6 + (12)^2\cdot 0.1 = 1.2 + 0.6 + 14.4 = 16.2$$</span>
When you multiply a variable by <span class="math-container">$c$</span>, you multiply the mean of it by <span class="math-co... |
2,894,376 | <blockquote>
<p>$2$ different History books, $3$ different Geography books and $2$ different Science books are placed on a book shelf. How many different ways can they be arranged? How many ways can they be arranged if books of the same subject must be placed together?</p>
</blockquote>
<p>For the first part of the ... | Ary Jazz | 587,160 | <p>How many ways can the books be arranged? As you said = $(2+3+2)! = 7!$</p>
<p>If the books of the same subject need to be arranged together you need to calculate de permutations for the groups and multiply them by the permutations within every category.</p>
<p>$3! (2! \times 3! \times 2!) = 144$ ways</p>
<p>Group... |
118,029 | <p>It is well known that a generic hypersurface of degree $2n-3$ in $\mathbb CP^n$ has finite number of lines. I would like to ask a couple of questions about lines on Fermat hypersurfaces and their symmetries: </p>
<p>$$\sum_{i=1}^{n+1}x_i^{2n-3}=0.$$</p>
<p>Fermat hypersurfaces have a group of automorphisms of ord... | Johannes Nordström | 13,061 | <p>Regarding question 1, any line in the Fermat cubic $C = \{X_0^3 + X_1^3 + X_2^3 + X_3^3 = 0\}$ must meet the coordinate hyperplane $H_0 = \{X_0 = 0\}$. So which points $x \in (C \cap H_0)$ can lie on lines? If $Y, Z$ are homogenous coordinates on $T_x(C \cap H_0) \cong \mathbb{P}^1$, then the restriction of $X_0^3 +... |
275,820 | <p>I have a question concerning the definition of the square root of bounded linear operators. To introduce some notation: tr denotes the trace of linear operators and $\mathcal{L}(H)$ denotes the set of bounded linear operators, from H to H, where H symbolizes a Hilbert space. L' stands for the adjoint operator of L.
... | Martin Argerami | 22,857 | <p>It is a well-known fact that every bounded positive operator on a Hilbert space (which are exactly those of the form $LL'$ for some bounded $L$) has a unique positive square root. This is a consequence of the <em>continuous functional calculus</em>; you can find about in any Functional Analysis book that talks about... |
215,531 | <p>I must solve following inequation:</p>
<p>$\frac{x-3}{1-2x}<0$</p>
<p>Now the text says that I have to solve the inequation "direct" without solving the according equations.</p>
<p>What does that mean?</p>
<p>I would say that I have to multiply by $(1-2x)$ then I get</p>
<p>$x-3<0$ and </p>
<p>$L_1 = [x... | Berci | 41,488 | <p>If you mulitply an inequality by a <em>negative</em> number, it will exchange the signs $>$ and $<$ (because for example $1<2$ but $-1>-2$...)
So it is not the best to multiply by the denominator unless you're sure it is positive!</p>
<p>I think, by 'direct', they meant as starting point to think about,... |
215,531 | <p>I must solve following inequation:</p>
<p>$\frac{x-3}{1-2x}<0$</p>
<p>Now the text says that I have to solve the inequation "direct" without solving the according equations.</p>
<p>What does that mean?</p>
<p>I would say that I have to multiply by $(1-2x)$ then I get</p>
<p>$x-3<0$ and </p>
<p>$L_1 = [x... | Mikasa | 8,581 | <p>Besides to above solutions see the table below:</p>
<p><img src="https://i.stack.imgur.com/73aXH.jpg" alt="enter image description here"></p>
<p>It shows that your desire interval would be $(-\infty, 1/2)\cup (3,+\infty)$</p>
|
35,230 | <p>This happens frequently, both on the main site and on meta: An old question pops back up on the front page, I open it, the text under the title says "Modified today", but when I check the timeline the last event is years ago, even if I show vote summaries.</p>
<p>Here's the <a href="https://math.stackexcha... | Eric Wofsey | 86,856 | <p>A question will be labelled as "modified [time]" on the front page if either the question or one of its <em>answers</em> was modified. For instance, in the example you linked <a href="https://math.stackexchange.com/a/924288/86856">this answer</a> was edited today.</p>
|
136,453 | <p>For every $k\in\mathbb{N}$, let
$$
x_k=\sum_{n=1}^{\infty}\frac{1}{n^2}\left(1-\frac{1}{2n}+\frac{1}{4n^2}\right)^{2k}.
$$
Calculate the limit $\displaystyle\lim_{k\rightarrow\infty}x_k$.</p>
| Brett Frankel | 22,405 | <p>I would start by saying that much of number theory, to the extent that you can really describe number theory in a single sentence, is devoted to solving equations with integer solutions. And it turns out that understanding how these equations behave with respect to primes is often the key to understanding how they b... |
285,841 | <p>I would like to solve the following problem:</p>
<p>$$\begin{array}{ll} \text{minimize} & \mathbf{x}^T \mathbf{A} \mathbf{x}\\ \text{subject to} & \mathbf{x}^T\mathbf{B}\mathbf{x} = 0\\ & \mathbf{x}^T \mathbf{x} = 1\end{array}$$</p>
<p>where $\bf x$ is a vector, $\bf A, \bf B$ are square matrices, and ... | Igor Rivin | 11,142 | <p>Generically, your system will have no solution, since $\mathbf{x}^t B \mathbf{x}$ is rarely zero for full-rank matrices. In the special case where $B$ is a degenerate symmetric matrix, then $x$ is in the null-space of $B,$ but then your Lagrange multiplier equation seems to indicate that $x$ has to <em>also</em> be ... |
198,132 | <p>Putting the equation $x^2 - x \sin(x) - \cos (x)$ into Wolfram Alpha, I am surprised that it has a nice <a href="http://www.wolframalpha.com/input/?i=roots%20of%20x%5E2%20-%20x%20sinx%20-%20cos%20x" rel="nofollow">parabolic shape</a>. Also, it has two complex roots.</p>
<p><strong>Question</strong></p>
<p>Is it po... | Bombyx mori | 32,240 | <p>I think you are confused by treating this as a standard quadratic which can only have two real roots or a pair of conjugate complex roots. But this is not. As others commented it is more appropriate to use calculus to detect the distribution of roots on the real line. In the complex case your equation become $$z^{2}... |
198,132 | <p>Putting the equation $x^2 - x \sin(x) - \cos (x)$ into Wolfram Alpha, I am surprised that it has a nice <a href="http://www.wolframalpha.com/input/?i=roots%20of%20x%5E2%20-%20x%20sinx%20-%20cos%20x" rel="nofollow">parabolic shape</a>. Also, it has two complex roots.</p>
<p><strong>Question</strong></p>
<p>Is it po... | copper.hat | 27,978 | <p>$f(x) = x^2-x\sin x-\cos x$ is even, $f(0) < 0$ and $\lim_{x \to \infty} f(x) = \infty$, hence the equation has at least two real roots. Also, $f'(x) = x(2-\cos x)$ satisfies $f'(x) \geq x \geq 0$ for $x \geq 0$, hence these are the only real roots.</p>
|
2,617,621 | <p>$$z^4 =\lvert z \lvert , z \in \mathbb{C}$$</p>
<p>Applying the formula to calculate $ \sqrt[4]{z} $, I find that solutions have to have this form:</p>
<p>$$z=\sqrt[4]{\lvert z \lvert}$$
$$z=\sqrt[4]{\lvert z \lvert} \ e^{i \frac{\pi}{2}}=i \ \sqrt[4]{\lvert z \lvert}$$
$$z=\sqrt[4]{\lvert z \lvert} \ e^{i \fra... | Arthur | 15,500 | <p>You are right that those are all the solutions, but I think there are easier ways to get there. For instance, I generally try to hold off on substituting $z = a+bi$ for as long as possible, since it rarely makes things nicer.</p>
<p>First off, begin by taking absolute values on both sides, giving $|z| = |z|^4$. Thi... |
178,666 | <p>For every natural number n, let:</p>
<ul>
<li><p>Gn be the number of distinct group structures with at most n elements;</p></li>
<li><p>An be the number of distinct abelian group structures wit at most n elements;</p></li>
<li><p>Sn be the number of distinct solvable group structures with at most n elements.</p></l... | Geoff Robinson | 14,450 | <p>(Edited following Emil Jerabek's coment below) From results of L. Pyber (and implicitly, C. Sims) it appears likely that $\frac{f(n)}{g(n)} \to 1$ as $n \to \infty,$ where $f(n)$ is the number of isomorphism types of nilpotent groups of order $n$ and $g(n)$ is the number of isomorphism types of all groups of order $... |
52,480 | <p>The question comes from a statement in Concrete Mathematics by Graham, Knuth, and Patashnik on page 465.</p>
<p>$$\sum_{k \geq n} \frac{(\log k)^2}{k^2} = O \left(\frac{(\log n)^2}{n} \right).$$</p>
<p>How is this calculated?</p>
| Shai Covo | 2,810 | <p>Note that the function $f$ defined by
$$
f(x) = \frac{{(\log x)^2 }}{{x^2 }}
$$
is decreasing for $x > x_0$ (for some $x_0 > 0$), and that
$$
\frac{{\frac{d}{{dx}}\int_x^\infty {\frac{{(\log t)^2 }}{{t^2 }}dt} }}{{\frac{d}{{dx}}\frac{{(\log x)^2 }}{x}}} = \frac{{ - \frac{{(\log x)^2 }}{{x^2 }}}}{{\frac{{2\lo... |
135,012 | <p>How to prove (or to disprove) that all the roots of the polynomial of degree $n$ $$\sum_{k=0}^{k=n}(2k+1)x^k$$ belong to the disk $\{z:|z|<1\}?$ Numerical calculations confirm that, but I don't see any approach to a proof of so simply formulated statement. It would be useful in connection with an irreducibility p... | Emil Jeřábek | 12,705 | <p>Let $f$ denote your polynomial. Roots of the polynomial
$$g(x)=\sum_{k=0}^nx^{2k+1}=x\frac{x^{2(n+1)}-1}{x^2-1}$$
are $0$ and roots of unity. By the <a href="http://en.wikipedia.org/wiki/Gauss%E2%80%93Lucas_theorem">Gauss–Lucas theorem</a>, the roots of $g'(x)=f(x^2)$ lie in their convex hull, and a fortiori in the ... |
123,202 | <blockquote>
<p>Let $X,Y$ be vectors in $\mathbb{C}^n$, and assume that $X\ne0$. Prove
that there is a symmetric matrix $B$ such that $BX=Y$.</p>
</blockquote>
<p>This is an exercise from a chapter about bilinear forms. So the intended solution should be somehow related to it.</p>
<p>Pre-multiplying both sides by... | Vedran Šego | 78,926 | <p>Here is a short, constructive proof:</p>
<ol>
<li><p>If $y = 0$, then $B = 0$.</p></li>
<li><p>If $y^Tx \ne 0$, then $B = \frac{1}{y^Tx} yy^T$.</p></li>
<li><p>If $y^Tx = 0$, then $B = \frac{\|y\|}{\|x\|} H$, where $H$ is a <a href="http://en.wikipedia.org/wiki/Householder_transformation" rel="nofollow noreferrer">... |
1,085,702 | <p>It's said that a computer program "prints" a set <span class="math-container">$A$</span> (<span class="math-container">$A \subseteq \mathbb N$</span>, positive integers.) if it prints every element of <span class="math-container">$A$</span> in ascending order (even if <span class="math-container">$A$</span... | Jihad | 191,049 | <p>There are countably many programs but the number of subsets of $\mathbb{N}$ is uncountable.</p>
|
1,085,702 | <p>It's said that a computer program "prints" a set <span class="math-container">$A$</span> (<span class="math-container">$A \subseteq \mathbb N$</span>, positive integers.) if it prints every element of <span class="math-container">$A$</span> in ascending order (even if <span class="math-container">$A$</span... | QuestionC | 187,599 | <p>There is a really elegant proof for this that requires virtually no math background at all.</p>
<p>All computer programs are a finite sequence of bytes, which is just a number in base 256. So each computer program can be represented as a unique natural number. <strong>This statement is elaborated in detail below ... |
1,274,717 | <p>Say that $V$ is a finite dimensional vector space over a field and and $f : V \to V$ a linear map. There is an integer $i$ such that $\text{ker}(f^n) = \text{ker}(f^{n+1})$ for all $n \geq i$. You see that by noting that $\text{ker}(f^n) \subseteq \text{ker}(f^{n+1})$ for all $n$ and since $V$ is finite dimensional,... | Pierrev | 235,744 | <p>You are right about your proof. How to do it ? </p>
<p>It's only using rank–nullity theorem many times.</p>
<p>I take your notation :</p>
<p>If $ker(f^{n-1})=ker(f^n)$, using rank-nullity theorem : $$dim(im(f^{n}))=dim(V)-dim(ker(f^n))=dim(V)-dim(ker(f^{n-1}))=dim(im(f^{n-1}))$$and the fact that $im(f^{n})$ is in... |
4,438,512 | <p>Recently I learned about dividing and forking of formula / partial types:</p>
<p>We say that <span class="math-container">$φ(x,b)$</span> divides over <span class="math-container">$C$</span> (where <span class="math-container">$b$</span> in the monster and <span class="math-container">$C$</span> is small) if there e... | Alex Kruckman | 7,062 | <p>Some years ago I wrote an answer about the motivation for the definitions of dividing and forking. You may find it helpful: <a href="https://math.stackexchange.com/a/1613033/7062">https://math.stackexchange.com/a/1613033/7062</a></p>
<p>But I'll add to what I wrote there, addressing some specific points in your ques... |
2,135,228 | <p>Find</p>
<p>(a) $P\{A \cup B\}$</p>
<p>(b) $P\{A^c\}$</p>
<p>(c) $P\{A^c \cap B\}$</p>
<p>This is what I have right now:</p>
<p>(a) $P\{A \cup B\}=0.4+0.5=0.90$</p>
<p>(b) $P\{A^c\}= 1-0.4=0.60$</p>
<p>(c) $P\{A^c \cap B\}= (0.6)\cdot(0.5)=0.30$</p>
<p>Am I doing it correctly?</p>
| victoria | 412,473 | <p>You can't see how this property can be satisfied because with this definition, no, it can't be satisfied. You have just proved that this thing fails to be a vector space. That is how you do it.</p>
|
2,441,359 | <p>It's an example given in my book after monotone convergence theorem and dominated convergence theorem (without explanation) : </p>
<p>Find an equivalent of $$\int_0^{\pi/2}\dfrac {dx} {\sqrt{\sin^2(x)+\epsilon \cos^2(x)}}$$</p>
<p>when $\epsilon\to 0^{+}$.</p>
<p>Inspired of the theorems, I naturally think of the... | user21820 | 21,820 | <p>The answer is "no" even in intuitionistic propositional logic, which is what I assume you mean by "constructive logic".
$
\def\imp{\Rightarrow}
\def\eq{\Leftrightarrow}
\def\from{\leftarrow}
$</p>
<p>Let $P \overset{def}\equiv ( A \lor \neg A ) \eq ( B \lor \neg B )$. Then $P$ is a classical tautology. But let $K$ ... |
226,346 | <p>I have the three dimensional Laplacian <span class="math-container">$\nabla^2 T(x,y,z)=0$</span> representing temperature distribution in a cuboid shaped wall which is exposed to two fluids flowing perpendicular to each other on either of the <span class="math-container">$z$</span> faces i.e. at <span class="math-co... | bbgodfrey | 1,063 | <p><strong>Problem Statement</strong></p>
<p>For notational simplicity, use the non-dimensional formulation described near the end of the question. (Doing so also facilitates comparison with results of a <a href="https://mathematica.stackexchange.com/a/235979/1063">2D approximation</a> solved earlier.) The PDE is giv... |
242,203 | <p>What's the derivative of the integral $$\int_1^x\sin(t) dt$$</p>
<p>Any ideas? I'm getting a little confused.</p>
| Amr | 29,267 | <p>Using the fundamental theorem of calculus we know that the answer is $\sin(x)$</p>
|
2,306,570 | <p>I'm new to this topic and trying to solve system of equations over the field $Z_{3}$:
$$\begin{array}{rcr} x+2z & = & 1 \\ y+2z & = & 2 \\ 2x+z & = & 1 \end{array}$$</p>
<p>I solved the system but I have roots:
$$x=1/3,
y=4/3, z=1/3$$ and it's probably not right. Can you help with this one?<... | Ahmed S. Attaalla | 229,023 | <p>If,</p>
<p>$$AX+B=CX$$</p>
<p>Then by existence of an additive inverse ,</p>
<p>$$AX+B-AX=CX-AX$$</p>
<p>And by commutativity and associative properties of matrix addition,</p>
<p>$$B=CX-AX$$</p>
<p>And by distributive property of matrix multiplication,</p>
<p>$$B=(C-A)X$$</p>
<p>If an inverse to $C-A$ exist... |
2,306,570 | <p>I'm new to this topic and trying to solve system of equations over the field $Z_{3}$:
$$\begin{array}{rcr} x+2z & = & 1 \\ y+2z & = & 2 \\ 2x+z & = & 1 \end{array}$$</p>
<p>I solved the system but I have roots:
$$x=1/3,
y=4/3, z=1/3$$ and it's probably not right. Can you help with this one?<... | Bill T | 451,681 | <p>This is the easy way to understanding this question, however it needs lots of work.</p>
<p><a href="https://i.stack.imgur.com/Iv781.jpg" rel="nofollow noreferrer">enter image description here</a></p>
<p><a href="https://i.stack.imgur.com/rZaFj.jpg" rel="nofollow noreferrer">enter image description here</a></p>
|
2,563,402 | <p>For which values of $a$ does the system
$$x_1 + x_2 + x_3 = 1$$
$$x_1 + 2x_2 + ax_3 = 2$$
$$2x_1 + ax_2 + 4x_3 = a^2$$
have (i) a unique solution, (ii) no solution, (iii) infinitely many solutions? Where the system has infinitely many solutions, write the solutions in parametric form.</p>
<p>So I tried to row reduc... | Community | -1 | <p><strong>Hint:</strong></p>
<p>The discussion depends on the zeroes of the main determinant. Developing with the first column,</p>
<p>$$\Delta=2-(a-2)(a-1).$$</p>
<p>The rest is yours.</p>
|
1,732,526 | <p>I start by expanding the denominator and separating the real and imaginary but get stuck when deciding what my $u$ and $v$ should be.</p>
<p>Thanks.</p>
| Olivier Oloa | 118,798 | <p><strong>Hint</strong>. One may observe that, for $a \neq0$ and for $|z|<|a|$, we have</p>
<blockquote>
<p>$$
z \mapsto \frac{1}{a+z}=\sum_{n=0}^\infty\frac{(-1)^nz^n}{a^{n+1}} \tag1
$$ </p>
</blockquote>
<p>and the considered function is analytic over $|z|<|a|$.</p>
<p>Now, by partial a fraction decompos... |
34,294 | <p>Let $M$ be a 2-dimensional Riemannian manifold of non-positive curvature everywhere, of genus > 1. Let $\textbf{D} \subset \textbf{C}$ be the open unit disc in the complex plane, the universal cover of $M$. Let $\gamma \subset \textbf{D}$ be a curve representing a geodesic in $M$ which is entirely in a region of z... | Sergei Ivanov | 4,354 | <p>I am answering the question as clarified in your Aug 3 comment. No it is not always possible to identify the universal cover with $D$ so that all zero-curvature geodesics become straight lines.</p>
<p>Begin with a metric in a small region where some 9 zero-curvature geodesics intersect each other but violate the De... |
3,059,676 | <blockquote>
<p>Why is the sum of all external angles in a convex polygon <span class="math-container">$360^\circ$</span>? </p>
</blockquote>
<p>From my understanding, for each vertex in a convex polygon, there exist exactly <span class="math-container">$2$</span> exterior angles corresponding to it, which are both ... | punk4me | 863,609 | <p>I recommend you look at Geometry by Jurgensen under the angles of a polygon section.</p>
<p>Draw a picture, it will help. Start out with pentagon to get some intuition. Now separate the inside of the polygon into non-overlapping triangles, and observe how you found the total sum of the interior angles of the polygon... |
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