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 |
|---|---|---|---|---|
10,352 | <p>I teach high school geometry and see many of my students fall in to the trap of "it looks like it, so it's true" -- a <a href="https://en.wikipedia.org/wiki/Van_Hiele_model" rel="noreferrer">Van Hiele Level 0 to 1</a> thought process. For instance, when talking about parallel line angle relationships, once they real... | mweiss | 29 | <p>I think one problem with this whole approach is that one of the fundamental tenets of the van Hiele model is what the van Hieles called "the principle of <em>separation</em>":</p>
<blockquote>
<p>a teacher who is reasoning at one level speaks a different "language"
from a student at a lower level, preventing un... |
10,537 | <p>In Finnish middle school (yläkoulu) the students learn to measure distances and angles, draw geometric figures and do certain calculations (area, volume, surface measure, trigonometry). There are also geometric constructions.</p>
<p><strong>As a teacher</strong>, to what extent is it important and useful for me to ... | Amir Asghari | 1,217 | <p><strong>First</strong>. Since you have naturally come to the question you asked, this paper of Fischbein, <a href="http://www.google.co.uk/url?sa=t&rct=j&q=&esrc=s&source=web&cd=2&ved=0ahUKEwil6vfptoLMAhWGFiwKHcQUAHkQFggkMAE&url=http%3A%2F%2Fweb.math.unifi.it%2Fusers%2Fdolcetti%2FFischbei... |
674,564 | <p>I came up with a problem that's been bugging me:</p>
<p><em>How many ways can you tile an NxM rectangle with L-polyominos?</em></p>
<p>The L shapes can be any size, so long as they aren't lines.</p>
<p>For clarification:<br>
$L(1,m) = 0$,<br>
$L(2,2) = 0$,<br>
$L(2,3) = 2$,<br>
$L(2,4) = 2$,<br>
$L(3,3) = 0$...</... | amaurs | 87,713 | <p>The curve is the image of a continuous function over the set $[0,1]$ as you can see in the <a href="https://math.stackexchange.com/questions/154382/is-koch-snowflake-a-continuous-curve">answer</a> that you post. Therefore, as compactness is preserved under continuous functions, the Koch curve is compact.</p>
|
2,230,507 | <p>I want to compute the smallest nonzero singular value of the matrix A, which is defined as follows. </p>
<p>Let $B = rand(500, 250)$, $A = B*B^t$, where $t$ denotes the transpose of the matrix. </p>
<p>I found the following matlab code to compute singular values of the matrix A which is based on the Singular val... | Alireza Kazemi | 550,587 | <p>Thanks to all answers/comments, I suggest a method based on @Rahul's method (which has converted the problem from SVD to EVD).</p>
<p><strong>Summary</strong></p>
<p>The solution in short: <strong><code>s = 1/eigs(inv(B'*B),1,'lm');</code></strong></p>
<p><strong>Description</strong></p>
<p>There exist a <a href... |
190,914 | <p>In <a href="https://math.stackexchange.com/questions/38350/n-lines-cannot-divide-a-plane-region-into-x-regions-finding-x-for-n">this</a> post it is mentioned that $n$ straight lines can divide the plane into a maximum number of $(n^{2}+n+2)/2$ different regions. </p>
<p>What happens if we use circles instead of lin... | Marc van Leeuwen | 18,880 | <p>By a simple induction one shows that if every pair among $n$ circles intersects each other transversally, without any three circles ever being concurrent, then they divide the plain into
$$1+1+2+4+6+\cdots+2(n-1)=1+[n>0]+\sum_{i=1}^{n-1}2i=1+[n>0]+n(n-1)
$$
different regions; this is indeed equal to the formul... |
849,838 | <blockquote>
<p>Consider <span class="math-container">$\displaystyle f(z)=\sum_{k=0}^\infty a_k z^k$</span> and suppose that <span class="math-container">$\displaystyle\sum_{n=0}^\infty f^{(n)}(0)$</span> converges.</p>
<p>Prove that <span class="math-container">$\forall z \in \mathbb C, \displaystyle\sum_{n=0}^\infty ... | PhoemueX | 151,552 | <p>My answer is more of a hint than an answer because it applies only if the convergence of $\sum_{n=0}^\infty f^{(n)}(0)$ is assumed to be absolute.</p>
<p>In that case, calculate
\begin{eqnarray*}
\sum_{n=0}^{\infty}f^{\left(n\right)}\left(z\right) & = & \sum_{n=0}^{\infty}\sum_{\ell=n}^{\infty}a_{\ell}\frac... |
849,838 | <blockquote>
<p>Consider <span class="math-container">$\displaystyle f(z)=\sum_{k=0}^\infty a_k z^k$</span> and suppose that <span class="math-container">$\displaystyle\sum_{n=0}^\infty f^{(n)}(0)$</span> converges.</p>
<p>Prove that <span class="math-container">$\forall z \in \mathbb C, \displaystyle\sum_{n=0}^\infty ... | Kelenner | 159,886 | <p>Put $R_n=\sum_{k\geq n} k! a_k$. We have $R_n\to 0$ as $n$ goes to infinity.</p>
<p>Let $N>0$. We have for $n\geq 0$</p>
<p>$$f^{(n)}(z)=\sum_{k\geq n} k! a_k \frac{z^{k-n}}{(k-n)!}=\sum_{l\geq 0} (n+l)! a_{n+l}\frac{z^l}{l!}$$
and hence
$$\sum_{n=0}^N f^{(n)}(z)=\sum_{n=0}^N (\sum_{l\geq 0} (n+l)! a_{n+l})\fr... |
3,425,499 | <ul>
<li><p><span class="math-container">$\sqrt{1+x} - \sqrt{1-x}$</span> is a subtractive cancellation when <span class="math-container">$x \approx 0$</span>, and</p>
</li>
<li><p><span class="math-container">$1 - \cos(x)$</span> is a catastrophic cancellation when <span class="math-container">$x \approx \pi$</span>.<... | Lutz Lehmann | 115,115 | <p>The terms in the first expression have floating point errors of about <span class="math-container">$(1+|x|)\mu$</span> where <span class="math-container">$\mu$</span> is the machine constant, <span class="math-container">$1{\rm ulp}$</span> or relative precision of the floating point data type. So in the difference ... |
94,209 | <p>Consider the following region:</p>
<pre><code>eq1 = 4 x;
eq2 = 1/x;
eq3 = 1;
cover = y^2/x^2;
bottom = eq3 < y < eq1 && eq3 < y < eq2;
reg3D = ImplicitRegion[bottom && 0 <= z <= cover, {{x, 1/4, 1}, {y, 1, 2}, {z, 0, 15}}];
RegionPlot3D[bottom && z < cover, {x, 1/4, 1},... | Michael E2 | 4,999 | <p>Is this sufficient?</p>
<pre><code>mesh = DiscretizeGraphics[
RegionPlot3D[
bottom && z < cover, {x, 1/4, 1}, {y, 1, 2}, {z, 0, 15},
PlotPoints -> 100, Mesh -> None,
AxesLabel -> Automatic] /. _Directive :> {}];
Show[mesh, BoxRatios -> {1, 1, 1}]
</code></pre>
<p><img src=... |
2,057,788 | <p><strong>Is the following true?</strong></p>
<p>If $k \geq 0$...</p>
<p>$x^2 > k \implies$</p>
<p>$|x| > \sqrt k \implies$</p>
<p>$x > \sqrt k$ or $x < -\sqrt k$</p>
<p>If $k < 0$...</p>
<p>$x^2 > k \implies$</p>
<p>$x^2 \geq 0$</p>
| Mark | 310,244 | <p>An easy way to understand this is, when $x\neq y$, we can divide by $|x-y|$ to get that:
$$\frac{|f(x)-f(y)|}{|x-y|}\leq |x-y|$$
Now, one characterization of the derivative of $f$ is:
$$f'(x) = \lim_{y\to x}\frac{f(x)-f(y)}{x-y}$$
As out inequality is valid for all $x,y$, we can pick a sequence $y_k$ with $y_k\to x$... |
3,408,320 | <p>Let <span class="math-container">$T : R^{3} \to R^{3}$</span> be the linear transformation defined by <span class="math-container">$T(x,y,z) = (x+3y+2z,3x+4y+z,2x+y-z)$</span>. Find the rank of <span class="math-container">$T^{2}$</span> and <span class="math-container">$T^{3}$</span>.</p>
<p>I formed the matrix of... | cangrejo | 86,383 | <p>Consider you want to group the numbers <span class="math-container">$1,2,3,4$</span> into two groups with 2 numbers each. Then the two following configurations are equal:</p>
<ul>
<li>One group contains <span class="math-container">$1,2$</span> and another group contains <span class="math-container">$3,4$</span></l... |
1,783,841 | <p>If a function $f(x)$ has a derivative $f'(x)$ then where $f'(x_0) = 0$ there is an extreme point at $x=x_0$.
And where $f''(x_0)=0$ there is an inflection point at $x=x_0$.</p>
<p>I am asking are there any significance or applications where $f(x_0)=f'(x_0)$ at some point $x=x_0$?</p>
<p>I am referring to a case li... | Luísa Borsato | 296,426 | <p>Well, $f(x) = f'(x)$, for every $x \implies f(x) = Ce^x$, C is a constant that can be determined if you have a inicial condition. </p>
<p>There are many applications, such as radioactive decay or growth of some population!</p>
|
3,555,250 | <p>I was learning differential calculus when I came with a doubt that why do we need to find derivative of polynomial. How is it useful?</p>
| Henry Lee | 541,220 | <p>The derivative in any case can be useful in many ways. The most obvious comparison is if you are given a continuous function to represent displacement (<span class="math-container">$s$</span>) then we can calculate velocity and acceleration by differentiating with respect to time:
<span class="math-container">$$v=\f... |
3,555,250 | <p>I was learning differential calculus when I came with a doubt that why do we need to find derivative of polynomial. How is it useful?</p>
| Jean Marie | 305,862 | <p>An example one can find <a href="https://math.stackexchange.com/q/743223">here</a>, but in a simplified presentation.</p>
<p>Suppose we have a have two walls at right angle and we want to delimitate a rectangular area with a <span class="math-container">$3$</span> meters long barbed wire fence, knowing that 2 sides... |
197,470 | <p>I'm new in Mathematica software, and I need help with a simple exercise. </p>
<p>Given the list</p>
<pre><code>list = {x, 3.2, y, 1, 2.679, 3, z, 4, -7.9};
</code></pre>
<p>I need to make some rules to generate a new list with only three elements.</p>
<ul>
<li>first element is the product of its symbols </li>
<l... | Shadowray | 47,416 | <p>Using rules for this task may be not very efficient, but here is a rule-based solution:</p>
<pre><code>list /. {
{List -> Times, _Real -> 1, _Integer -> 1},
{List -> Times, _Symbol -> 1, _Real -> 1},
{List -> Times, _Symbol -> 1, _Integer -> 1}
}
</code></pre>
<bloc... |
896,079 | <blockquote>
<p>for real numbers like <span class="math-container">$a,b,c,x,y,z$</span> that <span class="math-container">$ax-2by+cz=0$</span> and <span class="math-container">$ac-b^2>0$</span> Prove:<span class="math-container">$$zx-y^2\leq0$$</span></p>
<p>Additional info: The Proof should be by contradiction.we c... | lab bhattacharjee | 33,337 | <p>If $x,y$ are of opposite sign, $y^2>zx,$ and we are done</p>
<p>Again $ac-b^2>0\iff ac>b^2\ge0\implies a,c$ are of same sign</p>
<p>So, we need to consider $a,c$ & $x,z$ in pair are same sign</p>
<p>Case $(1):$ $a,c>0; z,x>0$</p>
<p>Case $(4):$ $a,c<0; z,x<0$</p>
<p>$$2by=ax+cz\ge2\sqr... |
896,079 | <blockquote>
<p>for real numbers like <span class="math-container">$a,b,c,x,y,z$</span> that <span class="math-container">$ax-2by+cz=0$</span> and <span class="math-container">$ac-b^2>0$</span> Prove:<span class="math-container">$$zx-y^2\leq0$$</span></p>
<p>Additional info: The Proof should be by contradiction.we c... | Ewan Delanoy | 15,381 | <p>Notice that $by=\frac{ax+cz}{2}$, so that</p>
<p>$$
acy^2 \geq b^2y^2=\bigg(\frac{ax+cz}{2}\bigg)^2 \geq ax\times cz
$$</p>
<p>and hence $y^2 \geq xz$ (notice that $ac>0$ because $ac>b^2$).</p>
|
2,554,060 | <p>The book said: "if $f$ is a bounded linear functional then ker $f$ is a closed subspace of the Hilbert space $H$."</p>
<p>But I do not know why, could anyone clarify this for me please? </p>
| Community | -1 | <p>Your start is correct. We thus have, $$\text{ Discriminant } =0 \implies (b(c-a))^2 = 4ac(b-c)(a-b)$$ $$\implies b^2c^2+2ab^2c+a^2b^2-4a^2bc+4a^2c^2-4abc^2=0$$ Dividing by $a^2b^2c^2$, we get, $$\frac{1}{a^2}+\frac{2}{ac}+\frac{1}{c^2}+\frac{4}{b^2}=\frac{4}{bc}+\frac{4}{ab}$$ $$\implies (\frac{1}{a}+\frac{1}{c})^2+... |
3,174,732 | <p>Let <span class="math-container">$(X,\mathcal{A}, \mu)$</span> be of finite measure and <span class="math-container">$f$</span> integrable on the measure space. Show that: </p>
<p><span class="math-container">$\lim_{n \to \infty} \int_{X}|f|^{\frac{1}{n}}d\mu=\mu(\{f\neq0\})$</span></p>
<p>My ideas:</p>
<p>I init... | Umberto P. | 67,536 | <p>Try using <span class="math-container">$1+|f|$</span> as a majorant instead of <span class="math-container">$|f|$</span>.</p>
|
468,437 | <p>If $A\times B$ is closed in $X\times Y$, then are $A$ and $B$ closed in $X$ and $Y$ respectively?</p>
| Hagen von Eitzen | 39,174 | <p>You write that your random numbers are from the range from $0$ to $1.999999$ with six decimal places.
Hnece there are $2000000=2\times 10^6$ possible numbers. Then the final result is much bigger of course:
$$\approx 1.0748601772107342002865544942320363407 \times 10^{-2867}$$</p>
|
1,710,095 | <p>I know that, for $x_i ∈ \mathbb R$, </p>
<p>$ 0 = \sum_{i=0}^n x_i^2$ </p>
<p>iff $x_0=x_1=x_2=...=x_n=0$</p>
<p>However, if we consider $z_i ∈ \mathbb C$, does the same principle hold? Or rather, under what conditions would that principle hold?</p>
<p><strong>EDIT</strong>: Just realized that... </p>
<p>For th... | user247327 | 247,327 | <p>Since the equation itself contains y', y must be at least once differentiable.</p>
<p>Now, differentiate both sides of the equation: $(y'(t))'= (y^2+ t)'$ so $y''(t)= 2yy'+ 1$. Since y is differentiable, yy' exists so y is at least twice differentiable. Differentiate that: $y'''(t)= 2y'^2+ 2yy''$. Since y is twi... |
62,002 | <p>I have a two-part question:</p>
<p>(1) First and foremost: I have been going through the paper by Dijkgraaf and Witten "Group Cohomology and Topological Field Theories." (See <a href="https://projecteuclid.org/journals/communications-in-mathematical-physics/volume-129/issue-2/Topological-gauge-theories-and... | Aaron Bergman | 947 | <p>They are trying to define the Chern-Simons action over a manifold $M$ by writing it as the integral of $\int F \wedge F$ over a bounding manifold $B$. When the bundle is nontrivial, they consider a more general cochain and show that there exists a $B$ over which the bundle extends such that $\partial B$ = $n$ copies... |
62,002 | <p>I have a two-part question:</p>
<p>(1) First and foremost: I have been going through the paper by Dijkgraaf and Witten "Group Cohomology and Topological Field Theories." (See <a href="https://projecteuclid.org/journals/communications-in-mathematical-physics/volume-129/issue-2/Topological-gauge-theories-and... | charris | 13,377 | <p>I don't think they actually find the constant you refer to in the update. As they remark, the choice of $\omega$ which maps to $\frac{k}{8\pi^2}Tr(F\wedge F)$
under $H^4(BG,\mathbb Z)\rightarrow H^4(BG, \mathbb R)$ is only defined up to torsion. The Chern-Simons invariant depends on the choice of $\omega$. At lea... |
2,500,723 | <blockquote>
<p>A price $p$ (in dollars) and demand $x$ for a product are related by $$\left(2x^2\right)-2xp+50p^2 = 20600$$ If the price is increasing at a rate of $2$ dollars per month when the price is $20$ dollars, find the rate of change of the demand.</p>
</blockquote>
<p>I was a little confused on how to proc... | Siong Thye Goh | 306,553 | <p>Guide:</p>
<p>$$2x^2-2xp+50p^2=20600$$</p>
<p>Differentiate with respect to $t$.</p>
<p>We have information about $x, p, \frac{dp}{dt}$, and you are interested in finding $\frac{dx}{dt}$.</p>
<p>Note that when $p=20$, you can compute the corresponding $x$.</p>
|
28,670 | <p><strong>The problem:</strong> Sometimes, an user begins acting adversarially and is unresponsive to improvement of their post. This is bad behaviour in general and part of the community feels urged to address it.</p>
<p>We know that, in practice, the close reason "off-topic because ... missing context or other ... | Community | -1 | <p>I do not think this is a good proposal, because the reason is either inaccurate or inappropriate.</p>
<p>I do not currently believe closing a user's question as retaliation for bad behavior is appropriate. Any punishment a user gets for bad behavior should come through flagging the user's posts and letting diamond ... |
561,210 | <p>I have a simple part of a question to solve. The problem is my answer is different to the solution in my textbook.</p>
<p>The equation is: $$\frac{5v}{6} = \frac{(\frac{1}{2}a+b+\frac{1}{2} c)v}{a+b+c}$$</p>
<p>I am supposed to get $$\frac{2}{3}(a+b+c) = b$$</p>
<p>But I simply get: $$b=2a +2c$$</p>
<p>I get my... | Zafer Sernikli | 98,237 | <p>What you get is equivalent to what you're supposed to get.</p>
<p>$ b= 2a+2c \implies 3b=2a+2c+2b \implies b= 2(a+b+c)/3$</p>
|
404,372 | <p>There are $7$ empty seats on a bus and four people get on. How many different ways can they be seated?</p>
<p>Would it be ${}^7C_{4}$ or ${}^7P_{4}$?</p>
| Austin Mohr | 11,245 | <p>Here is how you can distinguish between permutations and combinations. Suppose four people sit down on the bus in one way, then those <em>same four people</em> get up and change seats. Does this count as a different seating for you? If yes, you should use permutations (7P4). If not, you should use combinations (7C4)... |
127,645 | <p>In Lay's Linear Algebra and Its Applications textbook, he defines the matrix
$$A=\begin{bmatrix}
4 & 11 & 14\\8 & 7 & -2
\end{bmatrix}$$
and claims that the transformation $T(x)=A x$ maps the sphere to an ellipse.</p>
<p><a href="https://i.stack.imgur.com/grGkv.png" rel="nofollow noreferrer"><img sr... | Community | -1 | <p><strong>Visualization</strong></p>
<p>From a <code>GeometricTransformation</code> on the unit sphere:</p>
<pre><code>unitsphere = Sphere[];
transformTo2D = {{4, 11, 14}, {8, 7, -2}};
transformIn3D = Append[transformTo2D, {0, 0, 0}];
g = Graphics3D[
{Black, GeometricTransformation[unitsphere, transformIn3D]},
... |
3,360,515 | <p>EDIT: Added some clarifications.
EDIT 2: More clarifications and 1 more example.</p>
<p>Here's the basis of the problem:</p>
<p>I have 2 numbers, (they are integers that can be separated by 0.5 or 1, i.e. 1; 1.5; 2; 2.5; ...) let's say as an example these numbers are 22.5 and 10.</p>
<p>Now you must turn the bigg... | user7440 | 692,525 | <p>Note that <span class="math-container">$X^T X = R^T Q^T Q R = R^T R$</span> (as <span class="math-container">$Q^T Q = I$</span>).</p>
<ul>
<li><p>Assume <span class="math-container">$R$</span> is invertible. Then you have
<span class="math-container">$$
H = Q R \left( R^T R \right)^{-1} R^T Q^T
= Q R R^{-1} R^{-T... |
1,348,714 | <p>Is there always a square between two consecutive cubes? </p>
<p>I just thought of this question. It seems really simple and the answer is probably yes.</p>
<p>Edit: I should have given this more than 2 seconds of thought.</p>
| Noah Schweber | 28,111 | <p>We have $$(n+1)^2-n^2=2n+1$$ and $$(n+1)^3-n^3=3n^2+3n+1.$$ Let $f(x)=2x^{1\over 2}+1$ and $g(x)=3x^{2\over 3}+3x^{1\over 3}+1$. Then:</p>
<ul>
<li><p>When do we have $f(x)>g(x)$? (HINT: look at $h(x)=g(x)-f(x)=x^{1\over 3}(3x^{2\over 6}-2x^{1\over 6}+3)$.)</p></li>
<li><p>So?</p></li>
</ul>
|
31,113 | <p>Zagier has a very short proof (<a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1041893" rel="noreferrer">MR1041893</a>, <a href="https://www.jstor.org/stable/2323918" rel="noreferrer">JSTOR</a>) for the fact that every prime number $p$ of the form $4k+1$ is the sum of two squares.
The proof defines an inv... | Franz Lemmermeyer | 3,503 | <p>It's been a while since I read Elsholtz's article, but after doing so I felt none the wiser. Below I have translated Heath-Brown's proof into the language of binary quadratic forms; Zagier's proof looks more interesting from this point of view (the connections to Gauss reduction are much closer), but when working ou... |
4,295,884 | <p>The challenge is to solve this equation <span class="math-container">$2^{x}+7^{y}=9^{z}$</span> in positive integers. The obvious solution is <span class="math-container">$x=y=z=1$</span>. Using brute force, I found <span class="math-container">$3$</span> possible solutions:
<span class="math-container">\begin{eqnar... | Will Jagy | 10,400 | <p>here is a method I learned from <a href="https://math.stackexchange.com/questions/1551324/exponential-diophantine-equation-7y-2-3x/1551383#1551383">Exponential Diophantine equation $7^y + 2 = 3^x$</a> and polished. I have answered many questions of the form <span class="math-container">$A^u - B^v = C$</span> for g... |
275,558 | <p>I recently had a homework problem asking to prove that the direct product of rings (or rings with identity) are still rings (with identity), and it seemed really silly to go through all the steps in order to conclude what was extremely trivial. </p>
<p>My question is, are there instances when this is not the case? ... | Ittay Weiss | 30,953 | <p>The question requires care when interpreting what "taking direct products mean". One way to interpret this is: given two algebraic structures $A,B$ of the same kind, where $A,B$ are sets with extra structure, when will the same structure not be defined naturally on the product $A\times B$ as sets". This interpretati... |
2,029,326 | <p>In the latest episode of Survivor there were 6 people and they had to draw rocks from a bag. In the bag were 6 rocks, 5 white rocks and 1 black rock.
First person drew a rock and kept it in his hand.
Then the second person drew a rock.
Last person took the only rock that left. </p>
<p>Then they all showed what the... | PattuX | 390,101 | <p>Yes.</p>
<p>For the first person it's obviously $1/6$.</p>
<p>For the second it is either $1/5$ if the first person drew a white ($5/6$), or $0$ if the first person drew black already ($1/6$). So $p($Person 2 draws black$)=5/6 \times 1/5+1/6 \times 0=1/6$.</p>
<p>Logic continues and gives $1/6$ for each person.</... |
834,451 | <p>Simple question from a calc 3 beginner. Visually I cannot imagine the span of two vectors, what does this necessarily mean? For example my text mentions if two vectors are parallel their span is a line, otherwise a plane. Can anyone elaborate?</p>
| Jacob Bond | 120,503 | <p>Well, the span of a single vector is all scalar multiples of it. For example, if you have $\mathbf{v} = (1,1)$, $\text{span}(v)$ is all multiples of $(1,1)$. So $2v = (2,2)$ is in the span, $-3.75v = (-3.75,-3.75)$ is in the span, and so on. What you end up with is the whole line $y=x$, which is what you get if y... |
2,410,118 | <p>Edit: Excuse me, the numerator of the given solution should've been a rising factorial. That said, I still don't understand where it comes from?</p>
<p>The question is as follows :</p>
<blockquote>
<p>n friends visit k exhibitions, each person visits only 1 exhibition. Find the number of possibilities if</p>
<p>b) O... | copper.hat | 27,978 | <p>If you are willing to prepend a zero then there is a straightforward way:</p>
<p>If $n=0,1,2,3,4,5,...$ then
$\lfloor {n \over 2}\rfloor =0,0,1,1,2,2,...$ and
$n-2\lfloor {n \over 2}\rfloor = 0,1,0,1,0,1,...$
and so the sequence can be written as
$k \lfloor {n \over 2}\rfloor + n-2\lfloor {n \over 2}\rfloor$.</p>
|
2,154,295 | <p>Consider $(1)$</p>
<blockquote>
<p>$$\int_{0}^{\infty}{\sin^2bx\over x^2}\cdot{\mathrm dx\over e^{2ax}}=I\tag1$$
$(a,b)>0$</p>
</blockquote>
<p>How does one show that $$I=\color{blue}{b\tan^{-1}\left({b\over a}\right)-{a\over 2}\ln{\left(a^2+b^2\over a^2\right)}}?$$</p>
<p>An attempt:</p>
<p>Using $\sin^2... | Aditya Narayan Sharma | 335,483 | <p>Consider $\displaystyle F(a)=\int\limits_0^\infty e^{-2ax}\dfrac{\sin^2 bx}{x^2}\; dx\\$</p>
<p>Double differentiating w.r.t 'a' we have,</p>
<p>$\displaystyle F''(a)=4\int\limits_0^\infty e^{-2ax}\sin^2 bx \; dx$</p>
<p>And a little integration by parts gives the last integral which simplifies as,</p>
<p>$\disp... |
1,345,503 | <p>I have no idea about the answer, but I'm viewing the question this way;</p>
<p>If the smallest score obtained from the any three throws of the die is $1$, then largest among the other two throws must be $3$, and likewise
if the smallest score obtained from any of the three throw of the die is $2$ then a $6$ must a... | true blue anil | 22,388 | <p>It is small enough to enumerate, but since you want formulas, divide into cases with subdivisions</p>
<p><em>Case 1:</em> Extreme #s are 1 & 3</p>
<p><strong>(a)</strong> 3rd # is a duplicate: 2 ways $\times \frac{3!}{2!}$ permutations = 6</p>
<p><strong>(b)</strong> 3rd # is distinct: 1 way $\times 3!$ permu... |
3,168,214 | <blockquote>
<p>Suppose that <span class="math-container">$x$</span> and <span class="math-container">$y$</span> are related by the equation <span class="math-container">$\displaystyle x=\int_0^y\frac{1}{\sqrt{1+4t^2}}\,dt$</span>. Show that <span class="math-container">$d^2y/dx^2$</span> is proportional to <span cla... | Kavi Rama Murthy | 142,385 | <p>The constant is <span class="math-container">$4$</span>: <span class="math-container">$\frac {dx }{dy} =(1+4y^{2})^{-1/2}$</span> so <span class="math-container">$\frac {dy }{dx} =(1+4y^{2})^{1/2}$</span>. This gives <span class="math-container">$\frac {d^{2}y }{dx^{2}} =4y (1+4y^{2})^{-1/2}\frac {dy }{dx}=4y$</span... |
219,793 | <p>I have the following function </p>
<p><code>f[z_] := Piecewise[{{Cos[z] - 17 Sin[z]/z, z > 0}, {Cosh[z] - 17 Sinh[z]/z, z <= 0}}]</code></p>
<p>I am interested in the regions where <span class="math-container">$\vert f(z)\vert \leq 1$</span>, I tried to shade that region using <code>Filling</code>, but my re... | user1066 | 106 | <p>One possible interpretation?</p>
<pre><code>Permutations[Range[0,9,1],{3}]//condition1//condition2//condition3
</code></pre>
<blockquote>
<p>{{0, 4, 2}}</p>
</blockquote>
<pre><code> condition1:=Cases[#, {6, Except[8|2], Except[2|8]}|
{Except[6|2], 8, Except[2|6]} |
{Except[6|8], Except[8|6], 2}]&am... |
57,261 | <p>In Wellin's <a href="http://rads.stackoverflow.com/amzn/click/1107009464" rel="nofollow noreferrer">Programming with Mathematica</a> book, here's one of his implementations of the Newton method, where the iteration runs until the error tolerance is reached.</p>
<p><img src="https://i.stack.imgur.com/EnGoj.png" alt=... | murray | 148 | <p>Here is one approach to using a more reasonable stopping criterion. For clarity we separately define a function that effects one step of the Newton-Raphson procedure. The first argument will be a function rather than a function expression.</p>
<pre><code> newtonStep[{f_, x_}] := {f, x - f[x]/f'[x]}
newton[f_, s... |
2,164,338 | <p>I am stuck up with this simple problem related to averages. Please help me out in explaining the complete solution to this problem.</p>
<blockquote>
<p>Consider a class of <span class="math-container">$40$</span> students whose average weight is <span class="math-container">$40\ \mathrm{kg}$</span>. <span class="... | Arnaldo | 391,612 | <p>The new average is given by:</p>
<p>$$NA=\frac{40\cdot 40+m\cdot n}{40+m}$$</p>
<p>Once $m+n=50→n=50-m$ then</p>
<p>$$NA=f(m)=\frac{1600+50m-m^2}{40+m}=90-m-\frac{2000}{40+m}$$</p>
<p>In order to find the maximum you can do $f'(m)=0$.</p>
<p>Can you finish?</p>
|
1,552,292 | <p>I would like to share a point on which I was really stuck with you, in order for you to help me to see my mistake:</p>
<p>Here is the theorem:</p>
<p><strong>A set X is infinite if and only if it is equipotent (equivalent) to a proper subset of itself</strong></p>
<p>And here is my question.
The set of positive i... | Asaf Karagila | 622 | <p>The fact you found <em>one</em> subset which is not equipotent with $\Bbb N$ does not mean there are no others. For example $\{2n\mid n\in\Bbb N\}$ is equipotent with $\Bbb N$.</p>
|
245,464 | <p>Here's the statement that I am to prove/disprove:</p>
<blockquote>
<p>If the minimum polynomial of $A$ is $A^5 = 5 A^2$, then $A$ is not
diagonalizable.</p>
</blockquote>
<p>First, I'm confused about how this can actually be a minimum polynomial. From my understanding, the minimum polynomial is a specific poly... | Community | -1 | <p>There is a well known criterion for diagonalizability. A matrix A is diagonalizable</p>
<p>if and only if its minimum polynomial is of the form</p>
<p><span class="math-container">$P_{min}(x)=(x-\lambda _{1})(x-\lambda _{2})...(x-\lambda _{k})$</span></p>
<p>where <span class="math-container">$\lambda_{1},..\lambda ... |
2,041 | <p>I'd like to process data like the following, from my most recent seven-question exam.</p>
<pre><code> Item 1 2 3 4 5 6 7
PtsPossible 9 13 11 12 12 11 12
Median 8 9 8 7 10 10 12
Mean 7.10 9.10 7.25 6.65 9.27 9.39 10.57... | Alecos Papadopoulos | 689 | <p>To answer your direct question, not much. But, assuming you have available the analytical grades and not just the statistics: </p>
<p>In the specific example, questions are distinguished by their <em>maximum value</em> "PtsPossible". So they are different objects, because they have a possibly different range -the... |
871,221 | <blockquote>
<p>Let $F$ be a field. Show that the field of quotients of $F$ is ring isomorphic to $F$.</p>
</blockquote>
<p><strong>Attempt:</strong> Let $F'$ be the field of Quotients of the field $F$. </p>
<p>Let $\Phi:F \rightarrow F'$ such that $\Phi(x)=x/1$ </p>
<p>$x/1$ refers to the equivalence class contai... | dantopa | 206,581 | <p>The answer by @TrialAndError is a nice discussion and is amplified with a toy example.</p>
<p>$$
\begin{align}
A x &= b \\
\left[
\begin{array}{cc}
1 & 0 \\
0 & 0
\end{array}
\right]
\left[
\begin{array}{cc}
x_{1} \\
x_{2}
\end{array}
\right]
&=
... |
2,617,388 | <p>I've an interesting question on my hand which I've approach several others but all of them gives me different insights to this probability question. </p>
<p>Here it is,</p>
<blockquote>
<p>The incidence of a suspicious transaction in a bank is <code>1 in 149</code>. They are able to correctly identify a legitima... | Graham Kemp | 135,106 | <blockquote>
<p>The incidence of a suspicious transaction in a bank is 1 in 149. They are able to correctly identify a legitimate transaction 92% of the time. However, this bank is also able to correctly pinpoint a suspicious transaction 92% of the time. One day, the bank identify a transaction as suspicious. What is... |
4,321,423 | <p>Let <span class="math-container">$(X, \mathcal{A}, \mu)$</span> be a finite measure space. Fix <span class="math-container">$1 < p, q < \infty$</span> with <span class="math-container">$\frac{1}{p} + \frac{1}{q} = 1$</span>. Let <span class="math-container">$\{f_n\}$</span> be a sequence in <span class="math-c... | AlanD | 356,933 | <p>Clearly, <span class="math-container">$\phi(g)=\lim \int_X f_ng\,d\mu$</span> is well-defined and linear. Moreover, <span class="math-container">$$|\phi(g)|\leq M\|g\|_q$$</span> so <span class="math-container">$\|\phi\|\leq M$</span>.</p>
<p>By Riesz Representation, there exists a unique <span class="math-container... |
1,414,717 | <p>Given a probability space $(\Omega,\cal F,\Bbb P)$, when $\sigma$-algebra $\cal F_0$$\subseteq \cal F$ is finite (which is generated by a finite partition $\Gamma \subseteq \cal F_0$), the conditional expectation $\Bbb{E}(X\mid\cal F)$ is a function that maps each set $A$ in $\Gamma$ to the mean value of $X$ on $A$,... | Tony Piccolo | 71,180 | <p>If you want to try yourself, translate into algebra the fact the triangle of vertices $(x_1,y_1)$, $(x_2,y_2)$ and $(x_3,y_3)$ is equilateral. Two equations are enough since congruence is transitive: don't use distance but orthogonality.
Then you need some algebraic manipulations to get the result.</p>
<p>Otherwise... |
1,414,717 | <p>Given a probability space $(\Omega,\cal F,\Bbb P)$, when $\sigma$-algebra $\cal F_0$$\subseteq \cal F$ is finite (which is generated by a finite partition $\Gamma \subseteq \cal F_0$), the conditional expectation $\Bbb{E}(X\mid\cal F)$ is a function that maps each set $A$ in $\Gamma$ to the mean value of $X$ on $A$,... | V.G | 746,921 | <p>Writing the solution from Gelca,Andreescu <em>Putnam and Beyond</em> (2007), pp. 213-214 as Tony suggested:</p>
<hr />
<blockquote>
<p>Let us determine first some algebraic conditions that the coordinates <span class="math-container">$(x_i, y_i)$</span>,
<span class="math-container">$i = 1, 2, 3$</span>, of the vert... |
415,743 | <p><strong>Given Problem</strong> is to solve this separable differential equation:</p>
<p>$$y^{\prime}=\frac{y}{4x-x^2}.$$</p>
<p><strong>My approach</strong>: was to build the integral of y':</p>
<p>$$\int y^{\prime} = \int \frac{y}{4x-x^2}dy = \frac{y^2}{2(4x-x^2)}.$$</p>
<p>But now i am stuck in differential eq... | Michael Hardy | 11,667 | <p>Separate variables:
$$
\frac{dy}{dx} = \frac{y}{4x-x^2}
$$
$$
\frac{dy}{y} = \frac{dx}{4x-x^2} = \frac{dx}{x(4-x)}
$$
Then integrate both sides, using partial fractions on the right side.</p>
|
415,743 | <p><strong>Given Problem</strong> is to solve this separable differential equation:</p>
<p>$$y^{\prime}=\frac{y}{4x-x^2}.$$</p>
<p><strong>My approach</strong>: was to build the integral of y':</p>
<p>$$\int y^{\prime} = \int \frac{y}{4x-x^2}dy = \frac{y^2}{2(4x-x^2)}.$$</p>
<p>But now i am stuck in differential eq... | MyUserIsThis | 51,373 | <p>That's not the way to solve separable equations, this is the general procedure:</p>
<p>$$\frac{dy}{dx}=\frac{y}{4x-x^2}$$</p>
<p>$$\frac{dy}{y}=\frac{dx}{4x-x^2}$$</p>
<p>Now that's what you integrate:</p>
<p>$$\int\frac{dy}{y}=\int\frac{dx}{4x-x^2}$$</p>
<p>The left one is immediate, the second one can be done... |
415,743 | <p><strong>Given Problem</strong> is to solve this separable differential equation:</p>
<p>$$y^{\prime}=\frac{y}{4x-x^2}.$$</p>
<p><strong>My approach</strong>: was to build the integral of y':</p>
<p>$$\int y^{\prime} = \int \frac{y}{4x-x^2}dy = \frac{y^2}{2(4x-x^2)}.$$</p>
<p>But now i am stuck in differential eq... | Shuhao Cao | 7,200 | <p>It is separable in that you can separate everything that has $y$ in it from everything that has $x$ in it, i.e.,
$$
\frac{y'}{y} = \frac{1}{4x-x^2},
$$
and this is:
$$
(\ln y)' = \frac{1}{4x-x^2}.
$$
Integrating both sides with respect to $x$:
$$
\int (\ln y)'\,dx = \int \frac{1}{4x-x^2}\,dx,
$$
gives:
$$
\ln y = \... |
164,007 | <p>I didn't find it in any book, although it seems that this should be standard: Endow the space $C^\infty_c(\mathbb{R})$ of compactly supported functions with the inductive topology coming from the embeddings
$$ \mathcal{D}_K \longrightarrow C^\infty_c(\mathbb{R}).$$
(Here $\mathcal{D}_K$ is the set of all smooth func... | Peter Michor | 26,935 | <p>The inductive topology you describe in the category of topological spaces is not locally convex -- it equals the final topology with respect to all smooth curves in <span class="math-container">$C^\infty_c(\mathbb R)$</span>; there are also many other descriptions. See
section 4 in</p>
<ul>
<li><a href="http://www.m... |
3,051,452 | <p>How can I prove that <span class="math-container">$\mathbb Q \times \mathbb Q$</span> is connected? I was trying to prove that it isn't because <span class="math-container">$\mathbb Q$</span> is not connected but I can't find two open <span class="math-container">$A,B$</span> to prove it.</p>
| Noah Schweber | 28,111 | <p>Your initial intuition was correct - <span class="math-container">$\mathbb{Q}^2$</span> is indeed <strong>not connected</strong>.</p>
<p>Think about it intuitively. <span class="math-container">$\mathbb{Q}^2$</span> is basically a bunch of points sprinkled in the plane <span class="math-container">$\mathbb{R}^2$</s... |
30,339 | <p>I need to construct a vector similar to:</p>
<pre><code>v[x_]:={0, 0, x, 0, 0, 2*x, 0, 0, 3*x, ....., 0, 0, n*x};
</code></pre>
<p>where <code>n=10^9</code>. How can I make such a vector wisely?</p>
| Jacob Akkerboom | 4,330 | <p>When n is large and x is known, using a PackedArray may be a good option. </p>
<pre><code>ar3=ConstantArray[0,3n];
ar3[[3;;;;3]]=Range[x, n x, x];
ar3
</code></pre>
<p>To see that the result is a <code>PackedArray</code>, we see that</p>
<pre><code><< Developer`;
PackedArrayQ[ar]
</code></pre>
<blockquote>... |
30,339 | <p>I need to construct a vector similar to:</p>
<pre><code>v[x_]:={0, 0, x, 0, 0, 2*x, 0, 0, 3*x, ....., 0, 0, n*x};
</code></pre>
<p>where <code>n=10^9</code>. How can I make such a vector wisely?</p>
| chyanog | 2,090 | <pre><code>(1 - Unitize[#~Mod~3]) # &@Range[15] x/3
(*{0, 0, x, 0, 0, 2 x, 0, 0, 3 x, 0, 0, 4 x, 0, 0, 5 x}*)
Clip[#~Mod~3, {1, 0}] #/3 &@Range[15] x
(*{0, 0, x, 0, 0, 2 x, 0, 0, 3 x, 0, 0, 4 x, 0, 0, 5 x}*)
</code></pre>
|
864,823 | <p>given the following horses, each with its chance of winning:</p>
<p>Horse 1 -> 0.29<br>
Horse 2 -> 0.34<br>
Horse 3 -> 0.11<br>
Horse 4 -> 0.07<br>
Horse 5 -> 0.14<br>
Horse 6 -> 0.05<br>
Sum -> 1</p>
<p>At the moment, in order to calculate the straight forecast of 1-4, that is the probability that horse 1 wins th... | Calvin Lin | 54,563 | <p>Hurkyl's solution is nice, but the change of variables obscures the inherent closed orbit property, which is the crucial part of the problem.</p>
<p>Observe that the functional equation only involves $x$ and $ -x$. In particular, if $h(x) = -x$, then the orbit of $x \neq 0$ is $\{x, -x\}$ and the orbit of $0$ is $\... |
3,060,789 | <blockquote>
<p>Prove that a group with <span class="math-container">$3$</span> elements is cyclic.</p>
</blockquote>
<p>I tried the case where <span class="math-container">$G=\{e,a,b\}
$</span>
and I kept trying multiplication
and finally I found that
<span class="math-container">$a^2$</span> must equal to <span ... | egreg | 62,967 | <p>Instead of “keeping multiplying” it's easier to fill the Cayley diagram: in every row and column every element must appear.</p>
<p><span class="math-container">\begin{array}{c|ccc}
& e & a & b \\ \hline
e & e & a & b \\
a & a & \\
b & b & \\
\end{array}</span>
In the slot c... |
1,046,828 | <p>Suppose we have 2 lines $$l_1 : x = 5 , \frac{y}{3-\alpha}=\frac{z}{-2}$$ and $$ l_2: x= \alpha , \frac{y}{-1}= \frac{z}{2-\alpha}$$ so what will be value of $\alpha$ for lines to be coplaner ? </p>
<p>I believe that this is unsymmetrical form of lines equation and we have lines from intersecting planes in each of ... | mfl | 148,513 | <p>The parametric equations of $l_1$ are $$\left\{\begin{array}{l} x=5 \\ y=\lambda \\ z=-\frac{2}{3-\alpha}\lambda\end{array}\right.$$ and the parametric equations of $l_2$ are $$\left\{\begin{array}{l} x=\alpha \\ y=-\frac{1}{2-\alpha}\lambda \\ z=\lambda\end{array}\right.$$</p>
<p>Thus, we have
$P_1(5,0,0)\in l_1,P... |
240,894 | <p>I am importing an excel sheet into Mathematica. The result is a rectangular array. The array contains sequences of commas. Whenever a cell is empty in the Excel sheet consecutive commas appear in the imported array. How to get rid of them? So I am looking for a function <span class="math-container">$f$</span> such t... | James Stein | 48,962 | <pre><code>f[L_List] := Map[If[# === Null, Nothing, #] &, L]
</code></pre>
|
1,597,483 | <p>I'm trying to prove that for every $x,y>0$, $x>y$, and $n \in
\mathbb N$, $x^{1/n}>y^{1/n}$. Just some direction will be fine. We haven't learned derivatives yet. Thanks in advance.</p>
| ASKASK | 136,368 | <p>I don't know how much tools you have access to, but if you can use derivatives, then $\frac{d}{dx} x^n = nx^{n-1} $. Thus if n>0 and x>0, then the derivative of $x^n$ is positive, and thus the function is monotonic. </p>
|
350,239 | <p>How to prove that if polynomial $W(x)$ has $n$ real roots then
$\forall a \in \mathbb{R}$</p>
<p>$a W(x)+W'(x)$ has more than $n-1$ roots</p>
<p>I have no idea how to solve. Please some hint.</p>
| muzzlator | 60,855 | <p>Posted an incorrect solution first, but I want to write @Marvis's answer in another way (so full credit goes to him):</p>
<p>Let $f(x) = e^{ax} W(x)$, then $f'(x) = e^{ax} (a W(x) + W'(x))$</p>
<p>$f(x)$ has exactly $n$ real roots and by mean value theorem between each of the roots, $f'(x)$ must have $n-1$ real ro... |
350,239 | <p>How to prove that if polynomial $W(x)$ has $n$ real roots then
$\forall a \in \mathbb{R}$</p>
<p>$a W(x)+W'(x)$ has more than $n-1$ roots</p>
<p>I have no idea how to solve. Please some hint.</p>
| Community | -1 | <p>First note that
$$f(x) = aW(x) + W'(x) = e^{-ax} \dfrac{d \left(e^{ax} W(x)\right)}{dx} \implies g(x) = e^{ax} f(x) = \dfrac{d \left(e^{ax} W(x)\right)}{dx}$$
Now any real root of $g(x)$ is also a real root of $f(x)$ and vice-versa.</p>
<p>The main idea to be used is between any two zeros of a smooth function, ther... |
1,516,482 | <p>I need to proof that $\sqrt{3} + \sqrt{2}$ is irrational, without using the fact that an irrational number plus a rational number equals irrational. also, i can't use the rational root theorem. that's why i posted a new question..</p>
<p>thanks for help! </p>
| Alexander | 185,590 | <p>Assume that $\sqrt2 + \sqrt3 =$$ p \over q $$p,q \in \mathbb{Z}$.
Then $\sqrt3 = $$p \over q$ $-\sqrt2$.</p>
<p>Squaring gives $\sqrt2 = $$p^2-q^2 \over 2pq$ which is a contradiction since $\sqrt2 \notin \mathbb{Q}$</p>
|
4,039,241 | <p>Let <span class="math-container">$\mathcal{C}([-1, 1]) = \{f: [-1, 1] \to \mathbb{R}\mid f \text{ is continuous}\}$</span>, <span class="math-container">$\mathcal{C}^{1}([-1, 1]) = \{f: [-1, 1] \to \mathbb{R}\mid f' \text{ is continuous}\}$</span>, where both sets are equipped with the supremum norm <span class="mat... | MikeG | 579,758 | <p>If you want this to hold for all <span class="math-container">$k$</span>, then the answer is <strong>NO</strong>, even if <span class="math-container">$X$</span> takes value in integers, unless <span class="math-container">$X$</span> is constant. For example we take <span class="math-container">$k=2$</span>, what yo... |
2,279,581 | <blockquote>
<p>Let $a,b$ be primes. Find all primes such that $a^2+b^2=9ab-13$. </p>
</blockquote>
<p>Whatever I've done is from parity checking. But I can't proceed with the case when both $a,b$ are odd primes.
I tried with some modulo-chasing but couldn't complete.</p>
| Will Jagy | 10,400 | <p>unlikely that this can be finished... ALL integer solutions are in two families, related by the mapping
$$ (x,y) \mapsto (9x-y, x) $$
beginning with two "seeds,"
$$ (1,2), \; \; (7,1) , \;\; ( 62,7), \;\; ( 551, 62), \;\; $$
$$ (2,1), \; \; (17,2) , \;\; ( 151,17), \;\; ( 1342, 151), \;\; (11927, 1342), \;\; $$
... |
2,125,327 | <p>How can we solve these questions using limit comparison test, specifically how can we choose $b_n$ for any series,
(The $n^{th}$ terms are given)</p>
<p><strong>(1)</strong> $\dfrac{1.2.3.....n}{1.3.5....(2n-1)}$</p>
<p><strong>(2)</strong> $\dfrac{(2n-1)}{n!}$</p>
<p><strong>(3)</strong> $\dfrac{(n+1)(1)}{(n)(4^... | John W. Smith | 397,485 | <p>In the first case, let's make it simple and consider 2 balls and 2 urns. Have the balls be labeled X and Y, respectfully. Since in the first case the balls are not identical, this means that the probability of choosing X is not equal to choosing Y. The these events are not equally probable, no matter in what urns th... |
2,125,327 | <p>How can we solve these questions using limit comparison test, specifically how can we choose $b_n$ for any series,
(The $n^{th}$ terms are given)</p>
<p><strong>(1)</strong> $\dfrac{1.2.3.....n}{1.3.5....(2n-1)}$</p>
<p><strong>(2)</strong> $\dfrac{(2n-1)}{n!}$</p>
<p><strong>(3)</strong> $\dfrac{(n+1)(1)}{(n)(4^... | epi163sqrt | 132,007 | <p><em>Hint:</em> We should also consider if <em>urns</em> are distinguishable or not. If this is not explicitly stated, we typically assume they are <em>indistinguishable</em> (but should state this assumption in an answer). Here is an example with $n=3$ balls and $r=4$ urns.</p>
<blockquote>
<p>We think of $N=\{1... |
3,134,179 | <p>Let <span class="math-container">$f \colon R \to R $</span> a continuous and periodic function, with period <span class="math-container">$T$</span>.
Show that <span class="math-container">$\lim_{n \to \infty} (\int_{a}^{b} f(nx)dx)=\frac{b-a}{T}\int_0^T f(x)dx$</span>, con <span class="math-container">$a < b$</s... | jmerry | 619,637 | <p>Use a substitution: <span class="math-container">$\int_a^b f(nx)\,dx = \frac1n\int_{na}^{nb}f(t)\,dt$</span>.</p>
<p>Now, if <span class="math-container">$m=\left\lfloor\frac{n(b-a)}{T}\right\rfloor$</span> is the number of full periods that can fit into an interval of length <span class="math-container">$nb-na$</s... |
167,463 | <p>How many permutations of the letters of the word "DEFEATED" are there in which the E's are separated from each other?</p>
<p>I relied on the fact that: (The number of permutations of the letters in this word with NO restriction) - (The number of permutations of the letters in this word with the 3 E's are s... | M.Samuel | 34,810 | <p>Back in the days, I solved it by another method which is computing the total number of permutations without any restriction and then subtracting away the unwanted cases, which in our problem are two cases. The first case is when the the 3 E's are all together, and the second case is when 2 E's are together and the r... |
1,531,488 | <p>I hope to know the singularity and pole of $\frac{z^2 e^z}{1+e^{2z}}$.</p>
<p>I try $\frac{z^2 e^z}{1+e^{2z}} = \frac{z^2}{e^{-z}+e^{z}}$ and observe that the denominator seems like cosine function. So I think the singularites are i(2n+1)$\pi$/2. But when I try to evaluate the poles, I fail and different values of ... | okrzysik | 246,433 | <p>Ok, so to evaluate the residue of a pole of order $m$, at $z_0$ you can use the formula</p>
<p>\begin{equation}
Res(z_0) = \frac{1}{(m-1)!} \lim_{z \rightarrow z_0} \frac{d^{m-1}}{d z^{m-1}} [(z - z_0)^m f(z)].
\end{equation}</p>
<p>Now in this case, the poles are of order 1, i.e. $m = 1$, so the formula just beco... |
3,079,696 | <p>Let <span class="math-container">$ f : \mathbb{R} \rightarrow \mathbb{R} $</span> be a function which takes a convergent sequence and gives us a convergent sequence.</p>
<p>Show that <span class="math-container">$f$</span> is continuous.</p>
<p>So I saw a proof for this but I don't get it.</p>
<p><strong>proof</s... | José Carlos Santos | 446,262 | <p>The sequence <span class="math-container">$\bigl(f(b_{2n+1})\bigr)_{n\in\mathbb N}$</span> is a subsequence of the sequence <span class="math-container">$\bigl(f(b_n)\bigr)_{n\in\mathbb N}$</span> and therefore, since the limit <span class="math-container">$\lim_{n\in\mathbb N}f(b_n)$</span> exists, you have<span cl... |
285,965 | <p>I was told to assume $f(x)$ is a polynomial with degree $d\geq 1$ with integer coefficients and positive leading coefficient.</p>
<p>(i) I need to show that there are infinitely many $x$ such that $f(x)$ isn't prime.</p>
<p>(ii) I also need to show that if $f(x_0) = m$, where $m>0$, then $f(x) \equiv 0 \pmod m$... | Ishan Banerjee | 52,488 | <p>$f(kc)$ where $k\in \mathbb{N}$ and $c$ is the constant term of the polynomial is nonprime. Not sure what to do if $c=1$ or $c=-1$ though.</p>
|
850,095 | <p>How to compute absolute value of the complex number $(a+bi)/(c+di)$ is it right $(a^2+b^2)/(c^2+d^2)$? is there another way (without use of polar form)?</p>
| user99680 | 99,680 | <p>You can work it out by rationalizing *</p>
<p>You can use a half-angle formula for Tan, i.e., a formula for Tan(B/2)</p>
<p>$tan(B/2) = (1 − cos B) / sin B = sin B / (1 + cos B)$</p>
<p>For CosB=SinB =$\sqrt \frac{2}{2}$, then , $Tan \pi/8= \frac{\frac{\sqrt{2}}{2}}{1+\frac{\sqrt{2}}{2}}=\frac{\sqrt2}{2+\sqrt... |
4,185,702 | <p>I could find three good approaches to find maximum of the function <span class="math-container">$y=x\sqrt{100-x^2}$</span>. I will explain them briefly :</p>
<p>First: Finding <span class="math-container">$x$</span> satisfies <span class="math-container">$y'=0$</span> then plugging it in the function.</p>
<p>Second:... | Toby Mak | 285,313 | <p>Square both sides to get <span class="math-container">$y^2 = x^2 (100-x^2) \implies -x^4+100x^2-y^2=0$</span>. This is a quadratic in <span class="math-container">$x^2$</span>: when <span class="math-container">$\Delta = 0$</span>, <span class="math-container">$100^2 - 4(-1)(-y^2) = 0 \implies y = ±50$</span>.</p>
<... |
152,707 | <p>I have a simple equation (if you are interested it is originated from the field of hydrology):</p>
<pre><code>S^0.5 (1-(1-S^6)^0.166667)^2 == 1/1000
</code></pre>
<p>with <code>S</code> certainly being between 0 and 1. The LHS of the equation is strictly growing function of <code>S</code>. So, the equation can be ... | jiaoeyushushu | 35,486 | <p>The best answer is coming - (。・`ω´・)</p>
<pre><code>With[{a = {2*(1 + Cos[θ]), 2*(1 - Cos[θ])}},
Show[
PolarPlot[a, {θ, 0, 2 π}],
ParametricPlot[r Min[a] {Cos[θ], Sin[θ]}, {θ, 0, 2 Pi}, {r, 0, 1}],
PlotRange -> All
]
]
</code></pre>
|
3,454,810 | <blockquote>
<p>Study the following sequence of numbers:
<span class="math-container">$$\forall n\in\mathbb{N}, u_n=\sum_{k=n}^{2n}\frac{k}{\sqrt{n^2+k^2}}$$</span></p>
</blockquote>
<p>I tried to calculate <span class="math-container">$u_{n+1}-u_n$</span>, but I couldn't simplify the expression.</p>
<p>Plotting ... | Robert Israel | 8,508 | <p>Using the <a href="https://en.wikipedia.org/wiki/Euler%E2%80%93Maclaurin_formula" rel="nofollow noreferrer">Euler-Maclaurin formula</a> we can get more detailed asymptotics, e.g.:</p>
<p><span class="math-container">$$ u_n = \left( \sqrt {5}-\sqrt {2} \right) n
+ \frac{\sqrt{5}}{5} + \frac{\sqrt{2}}{4}
+\left(\fra... |
1,807,070 | <p>Suppose company {X} represents clients in buying inventory during automated (online) auctions. The client chooses the bid value beforehand, and {X} repeatedly makes that bid in the automated auctions which are constantly occurring throughout the day. (Bid value is constant). Consider that for any given auction, {X} ... | Hypergeometricx | 168,053 | <p>Here's another non-induction and non-combinatorial proof.</p>
<p>$$\begin{align}
\sum_{j=0}^{s}(n-s+j)!\binom{s}{j}(s-j)! &=s! \sum_{j=0}^{s} \frac{(n-s+j)!}{j!} \cdot \color{lightgrey}{\frac{(n-s)!}{(n-s)!}}\\
&=s!(n-s)!\sum_{j=0}^s\binom {n-s+j}{n-s}\\
&=s!(n-s)!\binom {n+1}{n-s+1}
&&\tiny\te... |
1,532,331 | <p>Define $f:[0,1]\to \Bbb R$ by</p>
<p>$$f(x)=\sum_{n=1}^{\infty}x^2(1-x^2)^{n-1}$$</p>
<p>Check that $f$ is continuous or not.</p>
<p>Attempt: $$f(x)=\sum_{n=1}^{\infty}x^2(1-x^2)^{n-1}\\
=\lim\int_{0}^{1}x^2(1-x^2)^{n-1}$$
Now, Putting $x=\sin\theta$, then $\mathrm dx=\cos\theta\mathrm d\theta$, therefore integra... | Alan | 175,602 | <p>To see that the series is geometric, hold x constant, compute $\frac {a_{n+1}} {a_n}=\frac {x^2(1-x^2)^n}{x^2(1-x^2)^{n-1}}=1-x^2$, which you can see is independant of $n$. At $x=0$, all your terms are trivially 0. At $x=1$, the first term is 1 (if you accept $0^0=1$ as the set theorists would have it!) and all t... |
1,880,396 | <p>According to article, author wrote "since $|b_n|\to\infty$, we may and do assume that $|b_n|\ge 1$ for every $n\ge 1$". Can you explain for me this? And what will $\sum\limits_{1\le |b_n| \le |X_1|} \dfrac{1}{|b_n|} $ become? (with $X_1$ be a random variable). Can I assum that $|b_n|\ge a$ for all $n>1$? And th... | Vincent | 358,123 | <p>If the author was only interested in the limit of the sequence, only what happens when $n$ is sufficiently great is of interest. Here, $|b_n|\rightarrow \infty$ implies that there exists some $N$ such that for $n>N$, $|b_n|>1$. You can now consider the sequence $c_n = b_{n+N}$, which will verify the same prope... |
79,991 | <p>I'm attempting to MapThread a function of two lists that requires the index of the list values. For example,</p>
<pre><code>MapThread[#1*i+#2*j&,{{a,b,c},{e,f,g}}]
</code></pre>
<p>Where #1 represents the value from list 1, #2 the value from list 2, i the index of list 1, and j the index of list 2. The expecte... | penguin77 | 27,290 | <p>You may consider this </p>
<pre><code>MapIndexed[Times, #] & /@ {{a, b, c}, {e, f, g}} // Plus @@ # &
(* {{a + e}, {2 b + 2 f}, {3 c + 3 g}} *)
</code></pre>
<p><code>MapIndexed</code> applies a function (in your example <code>Times</code> to all elements of the list giving part specification (in your e... |
79,991 | <p>I'm attempting to MapThread a function of two lists that requires the index of the list values. For example,</p>
<pre><code>MapThread[#1*i+#2*j&,{{a,b,c},{e,f,g}}]
</code></pre>
<p>Where #1 represents the value from list 1, #2 the value from list 2, i the index of list 1, and j the index of list 2. The expecte... | ciao | 11,467 | <p>Not sure why you'd need differing variable names, since the index would always be same, but, that aside:</p>
<pre><code>imt[f_, l_, v_] := Module[Evaluate@v, MapThread[(v = ConstantArray[#3, Length@v]; f) &,
Append[l, Range@Length@l[[1]]]]];
</code></pre>
<p>Your example (note last entry i... |
1,609,845 | <p><strong>Problem</strong>: Let $\alpha, \beta$ be non-negative numbers.
Suppose the number of strictly increasing sequences $a_0, a_1, a_2 \cdots a_{2014}$ satisfying $0 \leq 3m$ is $2^{\alpha}(2\beta+1)$.
Find $\alpha$.</p>
<p>My attempt:</p>
<p>I tried to find an explicit form for the $nth$ of the sequence using... | Dave | 337,025 | <p>There is also a quick derivation available by utilizing Binet's first integral for the digamma function
<span class="math-container">$$ \psi(z) =\log z -\int_0^{\infty} \left(\frac{1}{e^t-1}+1-\frac{1}{t} \right) e^{-zt} \, dt, \hspace{0.5cm} \text{Re}(z)>0 $$</span>
and the integral
<span class="math-container">... |
3,500,910 | <p>*If there exists the same question with answer anywhere, plz delete this one, i didn't find one.</p>
<p>If <span class="math-container">$f$</span> is continuous on <span class="math-container">$\Bbb R$</span> and uniformly continuous on every <span class="math-container">$(a,b)$</span> so it is also uniformly conti... | Jacky Chong | 369,395 | <p>The statement is false. </p>
<p>Consider <span class="math-container">$f(x) = x^2$</span>. Observe <span class="math-container">$f$</span> is continuous on <span class="math-container">$\mathbb{R}$</span> and
<span class="math-container">\begin{align}
|x^2-y^2|\leq |x+y||x-y|\leq 2\max(|a|, |b|)|x-y|
\end{align}</... |
2,662,747 | <p>I want to prove that a connected component of a locally Euclidean space X is open in this space.</p>
<p>I start the proof taking a point y in the connected component Y of X. In particular, y is a element of X and have an open neighborhood U, and there is an open subset in an euclidean space and a homeomorphism. I'm... | Bram28 | 256,001 | <p>First, you need to get your set-up straight!</p>
<p>The base is fine, but for the step you write:</p>
<blockquote>
<p>Assume that for every integer k ≥ 0, $\sum_{i=0}^n (f_i)^2 = f_n f_{n+1}$</p>
</blockquote>
<p>That's not good: you make no refernce to $k$, and for the step you do not want to assume anything a... |
123,793 | <p>I tried the following two ways:</p>
<pre><code>Animate[Plot3D[Sin[x*y^2], {x, -2, 2}, {y, -2, 2},
Axes -> True,
Boxed -> True, PlotRange -> {{-2, 2}, {-2, 2}, {-1, 1}},
AxesLabel -> {"x1", "x2", "x3"}, Boxed -> True, PlotPoints -> 100,
BoxStyle -> Directive[Gray, Thickness[0.005]], ... | JimB | 19,758 | <p>The jumpiness of the first <code>Animate</code> occurs when the axes labels switch from on side to another. Adding in <code>ImagePadding -> 50</code> makes this <code>Animate</code> as "smooth" as your second Animate. If you need additional smoothness, then something else would need to be done but at least addi... |
123,793 | <p>I tried the following two ways:</p>
<pre><code>Animate[Plot3D[Sin[x*y^2], {x, -2, 2}, {y, -2, 2},
Axes -> True,
Boxed -> True, PlotRange -> {{-2, 2}, {-2, 2}, {-1, 1}},
AxesLabel -> {"x1", "x2", "x3"}, Boxed -> True, PlotPoints -> 100,
BoxStyle -> Directive[Gray, Thickness[0.005]], ... | MaTECmatica | 4,320 | <p>You need to use the Plot3D option AxesEdge->{{-1,-1},{-1,-1},{-1,-1}} with the Plot3D option SphericalRegion->True, next one has no jumpling nor rescaling: </p>
<pre><code>Animate[Plot3D[Sin[x*y^2], {x, -2, 2}, {y, -2, 2},
Axes -> True, Boxed -> True, PlotRange -> {{-2, 2}, {-2, 2}, {-1, 1}},
AxesLabel ... |
3,194,279 | <p>I'd like to find a smooth function to approximate <span class="math-container">$$f(x)=\begin{cases}0&\text{if}\;x<0\\x&\text{if}\;x \geq 0 \end{cases}$$</span>
This function should be differentiable everywhere. Thanks.</p>
| Community | -1 | <p>This is the ReLu function. You can approximate it with the softplus function which is given by </p>
<p><span class="math-container">$$ f_{t} = \frac{1}{t} \ln(1+e^{tx}) $$</span></p>
<p>As <span class="math-container">$\lim_{t \to \infty} f_{t} = f(x) $</span></p>
<p>Here is some visualization with Python</p>
<... |
1,366,737 | <p>Demonstrate by induction: $3^{2n} + 7 = 4k$ is true, for any $n\in \mathbb N$. I need to demonstrate this using the induction principle.</p>
<p>So far I have:</p>
<p><strong>$n = 1$</strong></p>
<p>$$3^{2\cdot 1} + 7 = 4\cdot k $$
$$9 + 7 = 4k$$
$$16 = 4k$$
$$k = 4$$</p>
<p>So it checks for $n=1$.</p>
<p><stron... | DeepSea | 101,504 | <p>Observe that $3^{2n} = 9^n = (9^n-1^n) + 1 = 8\cdot(...)+1=0+1 =1\pmod 4$, and $7 = 3 \pmod 4$, hence $3^{2n} + 7 = 1+3 = 4 = 0 \pmod 4$.</p>
|
1,366,737 | <p>Demonstrate by induction: $3^{2n} + 7 = 4k$ is true, for any $n\in \mathbb N$. I need to demonstrate this using the induction principle.</p>
<p>So far I have:</p>
<p><strong>$n = 1$</strong></p>
<p>$$3^{2\cdot 1} + 7 = 4\cdot k $$
$$9 + 7 = 4k$$
$$16 = 4k$$
$$k = 4$$</p>
<p>So it checks for $n=1$.</p>
<p><stron... | Steven Alexis Gregory | 75,410 | <p>$3^{2n} + 7 \equiv (-1)^{2n} + 7 \equiv 1^n+7 \equiv 0 \pmod 4$</p>
|
414,120 | <p>Somebody can to help me in such exercices:</p>
<blockquote>
<p>(1) A ring R such that $a^2 = a$ for all $a\in R$ is called a Boolean ring. Prove that every
Boolean ring R is commutative and $a + a = 0$ for all $a \in R$. </p>
<p>(2) Let R be a ring with more than one element such that for each nonzero $a\... | Jared | 65,034 | <p>Suppose $a$ is a zero divisor. So there is $c\ne 0$ with $ac=0$. Notice that $a(c+b)a=a$, contradicting uniqueness of $b$.</p>
|
414,120 | <p>Somebody can to help me in such exercices:</p>
<blockquote>
<p>(1) A ring R such that $a^2 = a$ for all $a\in R$ is called a Boolean ring. Prove that every
Boolean ring R is commutative and $a + a = 0$ for all $a \in R$. </p>
<p>(2) Let R be a ring with more than one element such that for each nonzero $a\... | KirkLand | 285,478 | <p>For part (a):
Assume that $d$ is a zero divisor, i.e., $d\neq 0$. Thus, there exists a unique $c \in R$ such that $dcd = d$, and $dc = 0$. </p>
<p>Since $d$ is a zero divisor, $d = dcd = (dc)d = 0\cdot d = 0$, i.e., $d=0$, which contradicts the assumption that $d$ is a zero divisor. </p>
<p>Therefore, R has no zer... |
3,946,152 | <p>In general, almost sure convergence and convergence in <span class="math-container">$L^1$</span> are independent modes of convergence in the sense that neither implies the other. For martingales, at least one implication, namely that almost sure convergence implies convergence in <span class="math-container">$L^1$</... | user6247850 | 472,694 | <p>Yes, for a martingale convergence in <span class="math-container">$L^1$</span> implies almost sure convergence. Suppose <span class="math-container">$(M_n) \rightarrow M_\infty$</span> in <span class="math-container">$L^1$</span> where <span class="math-container">$(M_n)$</span> is a martingale. Then <span class="... |
4,455,969 | <p>The following question is part of a chemistry problem that came in the Dhaka University admission exam 2013-14.</p>
<hr />
<blockquote>
<p>What is <span class="math-container">$-\log\left({\sqrt{1.8\times 10^{-5}\times 0.1}}\right)$</span>? <br> <br>
(a) 2.672 <br> <br>
(b) 2.772 <br> <br>
(c) 2.872 <br> <br>
(d) 2.... | CrackedBauxite | 867,032 | <p>Write (all logs are w.r.t base 10) <span class="math-container">$-\log(\sqrt{1.8}\cdot 10^{-3})=3-\log(1.8)/2$</span> using properties of the logarithm. Now, one sees <span class="math-container">$1.8<2<10^{1/3}$</span>. Also, by just multiplying one finds <span class="math-container">$1.8^4=(1.8^2)^2=10.24$<... |
1,558,850 | <p>I'm solving the 1D wave equation
\begin{equation}
\frac{\partial^2 \eta}{\partial t ^2} - \frac{\partial^2 \eta}{\partial x ^2} = 0
\end{equation}
with boundary conditions
\begin{equation}
\frac{\partial \eta}{\partial x} = 0 \qquad \qquad \text{on} \qquad \qquad x = 0, 1
\end{equation}
using Finite Element Method.... | mathreadler | 213,607 | <p>The approach I am most used to solve these types of problems would be to rewrite it as a linear optimization problem, replacing the <em>equality to zero</em> with minimizing the 2-norm: $$\underset{\bf v}{\min}\{ \|({\bf D_t}^2-{\bf D_x}^2) {\bf v}\|_2^2 + {\bf C_b}\|{\bf D_xv}\|_2^2\}$$
where $\bf v$ is a vectoriza... |
2,132,863 | <p>I am studying trigonometry in my school and learned that in a triangle the side opposite to angle theta should be taken as perpendicular side - hypotenuse remains the same and the third remaining side is the base.</p>
<p>The same is required for calculating the sine / cosine etc of the angle theta for below formula... | embedded_dev | 371,741 | <p>One could think as follows:
Since the side opposite to the angle B is the hypotenuse, then: $$ sinB = \frac{hypotenuse}{hypotenuse} = 1$$</p>
<p>However, one must always look carefully and rigorously to definitions. The sine of an angle is defined as the division of the opposite side to the angle by the hypotenuse.... |
2,132,863 | <p>I am studying trigonometry in my school and learned that in a triangle the side opposite to angle theta should be taken as perpendicular side - hypotenuse remains the same and the third remaining side is the base.</p>
<p>The same is required for calculating the sine / cosine etc of the angle theta for below formula... | Deepak | 151,732 | <p>The short answer is that you can't construct a right angle triangle to visualise $\sin 90^{\circ}$.</p>
<p>The "opposite" (or "perpendicular" as per your nomenclature) side has to be distinct from the hypotenuse, it cannot be the same side. But in a plane figure with two right angles opposite one another, you will ... |
1,852,826 | <p>Let $P_n$ be the $(n+1) \times (n+1)$ matrix that contains the numbers of Pascal's triangle in the upper triangle. For example in the case of $n=3$
$$
P_3 =
\begin{pmatrix}
1 & 1 & 1 & 1 \\
0 & 1 & 2 & 3 \\
0 & 0 & 1 & 3 \\
0 & 0 & 0 & 1 \\
\end{pmatrix}
$$
or in gene... | epi163sqrt | 132,007 | <p>In the following it's somewhat more convenient to equivalently consider the <em>transpose</em> $P_n^T$ of the matrix $P_n$ and its inverse.</p>
<p>We obtain for $n=3$</p>
<p>\begin{align*}
P_3^T&=
\begin{pmatrix}
1 & & & \\
1 & 1 & & \\
1 & 2 & 1 & \\
1 & 3 & 3 &... |
2,837,230 | <p>When using vector notation in coordinate systems (Cartesian coordinates) we see that the magnitude of a vector in two dimensions is equal to the square root of its Y component squared added to its X component squared (Pythagorean theorem).<br>
But the same calculation is done for a three dimensional vector that has ... | eternalGoldenBraid | 225,419 | <p>This is similar to Mr Z.'s answer, but I want to contribute my own version of the figure.<a href="https://i.stack.imgur.com/JtpBU.png" rel="noreferrer"><img src="https://i.stack.imgur.com/JtpBU.png" alt="enter image description here"></a></p>
<p>We are visualizing a vector $x = (x_1,x_2,x_3)$. From the Pythagorean ... |
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