qid int64 1 4.65M | question large_stringlengths 27 36.3k | author large_stringlengths 3 36 | author_id int64 -1 1.16M | answer large_stringlengths 18 63k |
|---|---|---|---|---|
1,549,181 | <p><strong>Calculate Using Polar Coordinates</strong></p>
<p>This is a drawing I made to illustrate the problem. <a href="http://tube.geogebra.org/m/ZzvL0a38" rel="nofollow">http://tube.geogebra.org/m/ZzvL0a38</a></p>
<p>$$\int_{\frac 12}^{1} \int_{0}^{\sqrt{1-x^2}} 1 \quad dydx $$</p>
<p>What I am confused about in... | Michael | 293,919 | <p>$\frac{1}{2}≤x≤1$</p>
<p>$0≤y≤\sqrt{1-x^2}$</p>
<p>Sketch the region which satisfies the above. Then change the region from $dy dx$ to $dx dy$. The limits become $\frac{1}{2}≤x≤\sqrt{1-y^2}$ and $0≤y≤\frac{\sqrt3}{2}$</p>
<p>As suggested above integrating it without changing limits is probably easier.</p>
|
1,549,181 | <p><strong>Calculate Using Polar Coordinates</strong></p>
<p>This is a drawing I made to illustrate the problem. <a href="http://tube.geogebra.org/m/ZzvL0a38" rel="nofollow">http://tube.geogebra.org/m/ZzvL0a38</a></p>
<p>$$\int_{\frac 12}^{1} \int_{0}^{\sqrt{1-x^2}} 1 \quad dydx $$</p>
<p>What I am confused about in... | Jan Eerland | 226,665 | <p>HINT:</p>
<p>$$\int_{\frac{1}{2}}^{1}\left[\int_{0}^{\sqrt{1-x^2}}1\space\text{d}y\right]\space\text{d}x=\int_{\frac{1}{2}}^{1}\left[\left[y\right]_{0}^{\sqrt{1-x^2}}\right]\space\text{d}x=\int_{\frac{1}{2}}^{1}\sqrt{1-x^2}\space\text{d}x=$$</p>
<hr>
<p>For the integrand $\sqrt{1-x^2}$, substitute $x=\sin(u)$ and... |
1,549,181 | <p><strong>Calculate Using Polar Coordinates</strong></p>
<p>This is a drawing I made to illustrate the problem. <a href="http://tube.geogebra.org/m/ZzvL0a38" rel="nofollow">http://tube.geogebra.org/m/ZzvL0a38</a></p>
<p>$$\int_{\frac 12}^{1} \int_{0}^{\sqrt{1-x^2}} 1 \quad dydx $$</p>
<p>What I am confused about in... | Emilio Novati | 187,568 | <p>From elementar trigonometry we see that the area is $A=\dfrac {\pi}{6}-\dfrac{\sqrt{3}}{8}$ where $ \dfrac {\pi}{6}$ is the area of the circular segment of central angle $\theta =\dfrac{\pi}{3}$ and $\dfrac{\sqrt{3}}{8}$ is the area of the triangle of basis $\dfrac{1}{2}$ and height $\dfrac{\sqrt{3}}{2}$.</p>
<p>W... |
4,474,803 | <p>Let <span class="math-container">$f$</span> be a differentiate function and consider two operators
<span class="math-container">$$
A(f(x))=\int_0^1 \frac{ d}{dx}f(x t^\mu) dt,\\
B(f(x))=\mu xf(x)+\int_0^x f(t) dt,
$$</span>
where <span class="math-container">$\mu $</span> is a parameter.</p>
<p>I need to prove that<... | ryang | 21,813 | <blockquote>
<p>Example:
<span class="math-container">$$\phi_1 := \forall x\; P(x )\rightarrow \exists y\; T(y) \\
\phi_2 := \color{green}{\Big(}\forall x\;\exists y\; P(x) \color{green}{\Big)}\rightarrow T(y) \\$$</span></p>
</blockquote>
<p><span class="math-container">$$\phi_3 := \forall x\;\exists y \Big( P(x )\ri... |
176,240 | <p>I <a href="https://math.stackexchange.com/questions/2832912/sum-infty-infty-frac-exp-n21-4n2-in-closed-form/2832965#2832965">came across</a> a possible bug in Mathematica's evaluation of integrals that I thought I'd share. Consider the code:</p>
<pre><code>zmax = 4;
a = Integrate[Exp[-x^2] Abs[Sin[x]], {x, -z , ... | halirutan | 187 | <p>The bad news is that your code spiked at about 60 GB RAM on my machine before the result is returned. The problem is clearly the <code>Import</code> routine. I would like to offer an alternative if you have more than 22 GB of RAM available.</p>
<pre><code>data = Import[file, {"GIF", "RawData"}];
</code></pre>
<p>T... |
3,916,261 | <p>In the book Optimal Transport for Applied Mathematicians, Theorem 1.37, the authors states an inequality, but I'm having trouble seeing how one shows that the inequality is indeed true.</p>
<p><strong>Definition</strong>: Given a function <span class="math-container">$c:X\times Y \to \mathbb R$</span>, we say that <... | Jair Taylor | 28,545 | <p>There is a difference between the <em>range</em>, or the <em>image</em>, of a function, and the 'codomain'.</p>
<p>When we write <span class="math-container">$f: A \rightarrow B$</span>, this means that for every <span class="math-container">$a \in A$</span> there is a unique value <span class="math-container">$b \i... |
281,325 | <p>I am not very familiar with Stone-Čech compactification, but I would like to understand why the remainder $\beta\mathbb{R}\backslash\mathbb{R}$ has exactly two connected components.</p>
| Brian M. Scott | 12,042 | <p><span class="math-container">$\newcommand{\cl}{\operatorname{cl}}$</span>For any Tikhonov space <span class="math-container">$X$</span> let <span class="math-container">$X^*=\beta X\setminus X$</span>. Let <span class="math-container">$\Bbb H=[0,\to)$</span> and <span class="math-container">$\Bbb L=(\leftarrow,0]$</... |
1,726,540 | <p>How should $x^{\frac{1}{x}}$ be differentiated? I know the answer is
$$\frac{1-\ln(x)}{x^{2-\frac{1}{x}}}$$</p>
<p>but I do not understand how to get there.</p>
<h2>Attempt at solution.</h2>
<p>I believe the following is true:</p>
<p>$$
\begin{aligned}\frac{d}{dx}x^u&=ux^{u-1}\cdot u^\prime\\
\frac{d}{dx}a^... | Community | -1 | <p>As there is a rule for the sum and product, there is a rule for powers:</p>
<p>$$\left(u^v\right)'=\left(e^{v\ln u}\right)'=(v\ln u)'e^{v\ln u}=(u'v+v'u\ln u)u^{v-1}.$$</p>
|
2,636,679 | <p>For which values of x do the vectors $(x,0,x + 1),(1,x,1),(0,x,−x)$ form a basis for R3?</p>
<p>Is 1 a correct answer to this. The vectors would each be linearly independent, however this seems too simple and I'm guessing theres an error somewhere.</p>
<p>If 1 works then it would seem any real number would work si... | Siong Thye Goh | 306,553 | <p>Guide:</p>
<ul>
<li><p>Put those vectors as rows of a matrix, compute the determinant.</p></li>
<li><p>If they form a basis, the determinant would not take value zero, hence you just have to find the root of the charactheristic polynomial and avoid those root to make it a basis.</p></li>
<li><p>Remark: Find all val... |
3,025,375 | <p>What is<span class="math-container">$$\lim_{n→∞}n^3(\sqrt{n^2+\sqrt{n^4+1}}-n\sqrt2)?$$</span>So it is<span class="math-container">$$\lim_{n→∞}\frac{n^3(\sqrt{n^2+\sqrt{n^4+1}})^2-(n\sqrt{2})^2}{\sqrt{n^2+\sqrt{n^4+1}}+n\sqrt{2}}=\lim_{n→∞}\frac{n^3(n^2+\sqrt{n^4+1}-2n^2)}{\sqrt{n^2+\sqrt{n^4+1}}+n\sqrt{2}}.$$</span... | user | 505,767 | <p><strong>HINT</strong></p>
<p>You only need to apply the trick twice, indeed we have that</p>
<p><span class="math-container">$$\sqrt{n^2+\sqrt{n^4+1}}-n\sqrt{2}=(\sqrt{n^2+\sqrt{n^4+1}}-n\sqrt{2})\cdot\frac{\sqrt{n^2+\sqrt{n^4+1}}+n\sqrt{2}}{\sqrt{n^2+\sqrt{n^4+1}}+n\sqrt{2}}=$$</span><span class="math-container">... |
1,944,630 | <p>whether every positive definite matrix has to be symmetric? If not, what will be the example?</p>
| angryavian | 43,949 | <p>By <a href="https://en.wikipedia.org/wiki/Positive-definite_matrix" rel="nofollow"><strong>definition</strong></a> it is assumed to be symmetric.</p>
|
1,234,320 | <p>It is given that $4 x^4 + 9 y^4 = 64$.</p>
<p>Then what will be the maximum value of $x^2 + y^2$?</p>
<p>I have done it using the sides of a right-angled triangle be $2x , 3y $ and hypotenuse as 8 .</p>
| Simon S | 21,495 | <p>One more stab at the problem. Instead of working with $x$, let's work with $u = x^2$ as it makes the algebra slightly easier. </p>
<p>As $4x^4 + 9y^4 = 64$, we have $y^2 = \frac{1}{3}\sqrt{64-4u^2}$, taking only the positive square roots as $y^2 \geq 0$. Hence want to maximize</p>
<p>$$f(u) = x^2 + y^2 = u + y^2 =... |
2,558,619 | <p>Compute
$$
\int_{-\infty}^{\infty} t^2 \delta(\sin(t)) e^{-|t|} \mathrm dt
$$
In closed form, where $\delta(t)$ is the <a href="http://mathworld.wolfram.com/DeltaFunction.html" rel="nofollow noreferrer">Dirac Delta function </a>.</p>
<p>My attempt: </p>
<p>$$
\int_{-\infty}^{\infty} t^2 \delta(\sin(t)) e^{-|t|}... | Enrico M. | 266,764 | <p>Actually Wolfram Alpha is wrong.</p>
<p>The series</p>
<p>$$\sum_{n = 0}^{+\infty} n^2 e^{-n\pi}$$</p>
<p>Does converge to </p>
<p>$$\frac{e^{\pi } \left(1+e^{\pi }\right)}{\left(e^{\pi }-1\right)^3}$$</p>
<p>Which is easy provable by using differentiation under the summation sign together with the geometric se... |
1,797,243 | <p>Find the relative extrema of the function by applying the first-derivative test:</p>
<p>$$f(x)=x^5-5x^3-20x-2$$</p>
<p>So I found the $f'(x)$</p>
<p>$$f'(x) = 5x^4-15x^2-20$$</p>
<p>Now, I'm trying to find the critical values, which $x=0$ or undefined, so I can apply the first-derivative test. However, I can't s... | Laila | 316,954 | <p>To find the critical values, set $f'(x)=0$<br/>This function certainly can be simplified:$f'(x)=5x^4−15x^2−20=5(x^4-3x^2-4)=5(x^2-4)(x^2+1)=5(x+2)(x-2)(x^2+1)$
<br/></p>
|
2,738,023 | <p>We know we can check if two vectors are 'orthogonal' by doing an inner product.</p>
<p>$a*b=0$</p>
<p>tells us that these two vectors are orthogonal</p>
<p>here comes the question:</p>
<p>if there a way to compute if they are 'parallel'? i.e., they are pointing at the same direction. </p>
| Michael Hoppe | 93,935 | <p>Two vectors are parallel iff their Gram-determinant, that is the area of the parallelogram spanned by the vectors, vanishes.</p>
|
4,240,892 | <p><strong>My attempt</strong>:
<span class="math-container">$f(x+y)=f(x)f(y)$</span></p>
<p>Differenting wrt <span class="math-container">$y$</span>,</p>
<p><span class="math-container">$$f'(x+y)=f(x)f'(y)$$</span></p>
<p>Putting <span class="math-container">$y=0$</span>,</p>
<p><span class="math-container">$$f'(x)=f(... | Elias Costa | 19,266 | <p>An alternative way. First find <span class="math-container">$f(0)$</span>. Note that, for <span class="math-container">$y=0$</span> we have
<span class="math-container">$$
f(x)=f(0)\cdot f(x), \mbox{ for all } x\in \mathbb{R}
$$</span></p>
<ul>
<li><p>If there is <span class="math-container">$x_{0}\in \mathbb{R}$</s... |
633,482 | <p>Consider a cube that exactly fills a certain cubical box. As in Examples 8.7 and 8.10, the ways in which the cube
can be placed into the box corresponds to a certain group of permutations of the vertices of the cube. This is the group
of <strong>group of rigid motions (or rotations) of the cube</strong>. (It should ... | Christian Blatter | 1,303 | <p>Look at the following figure (from an analysis textbook). For more background, in particular in connection with Lie algebras/groups see the Wikipedia article about the <a href="http://en.wikipedia.org/wiki/Exponential_map" rel="noreferrer">exponential map</a>. The exponential series does indeed appear there.</p>
<p... |
463,814 | <p>I am struggling to find the values of these integrals after trying many substitution it did't worked for me </p>
<p>1) $$ \int_{0}^{a}\frac{dx}{\sqrt{ax-x^2}} $$
2) $$ \int_{}^{}\frac{3x^5dx}{1+x^{12}} $$</p>
| Ron Gordon | 53,268 | <p>For the first integral, write</p>
<p>$$ax = \left (\frac{a}{2}\right)^2 - \left ( \frac{a}{2}-x\right)^2$$</p>
<p>and substitute $y = a/2-x$, then $y=(a/2) \sin{\theta}$. The result I get is $\pi$, independent of $a$.</p>
<p>For the second integral, substitute $y=x^6$, and use </p>
<p>$$\int \frac{dy}{1+y^2} = ... |
21,257 | <p>I am interested in examples where the <a href="http://en.wikipedia.org/wiki/Shooting_method" rel="nofollow">Shooting Method</a> has been used to find solutions to systems of ordinary differential equations that are either </p>
<ul>
<li>reasonably large systems, or </li>
<li>the search algorithm in the shooting para... | Yossi Farjoun | 3,578 | <p>I don't know about state-of-the-art and I'm not sure if this is the kind of thing you were looking for...however in my two first papers I've used the shooting method in a parameter space that was originally too big (4 and 6 dimensions if I recall correctly) and the problem was that with randomly chosen parameters th... |
3,579,739 | <p>Its also mentioned that this question is an example of <strong>Sampling without replacement</strong></p>
<p>My question is , by the general method of how I do these kind of problems , I would assume that there are two possibilities</p>
<ol>
<li><p>You chose a defective bottle and then a non defective which will h... | Haran | 438,557 | <p>Your logic is perfectly sound. The answer is <span class="math-container">$\frac{2}{5}$</span>. I think you find it odd that this is more than the probability of choosing one bottle and it being fractured, but this is actually what you must expect. </p>
<p>Think of the bottles trying to 'clear an exam'. If the exam... |
3,287,067 | <p>I'm reading a solution to the following exercise:</p>
<p>"Assume that <span class="math-container">$\lim_{x\to c}f\left(x\right)=L$</span>, where <span class="math-container">$L\ne0$</span>, and assume <span class="math-container">$\lim_{x\to c}g\left(x\right)=0.$</span> Show that <span class="math-container">$\li... | azif00 | 680,927 | <p>Should be
<span class="math-container">$$\int_{-a}^a \cos \left( \frac{\pi x}{a}\right) dx =0$$</span> for all <span class="math-container">$a\neq 0$</span>.</p>
|
4,207,823 | <p>I came across the following question in a set of lecture notes:</p>
<p>If <span class="math-container">$(G, *)$</span> is a group, when, in general, is it true that there exists <span class="math-container">$X \subset \mathbb{R}^n$</span> such that <span class="math-container">$G \cong \text{Sym}(X)$</span>? Or, les... | Moishe Kohan | 84,907 | <p>Even the question about existence of <span class="math-container">$n$</span> and <span class="math-container">$X\subset {\mathbb R}^n$</span> such that <span class="math-container">$G< Sym(X)$</span> has negative answer. The reason is that <span class="math-container">$Sym(X)$</span> is a subgroup of the group of... |
3,740,501 | <p>This question builds upon a previous question I asked <a href="https://math.stackexchange.com/questions/3548807/is-at-least-1-of-4-non-concyclic-points-contained-in-the-circle-through-the-othe">here</a></p>
<p>The answers helped me understand that for <span class="math-container">$4$</span> points that are not concy... | Antonio Hernandez | 796,809 | <p>Another way to do, only with the formal definition of the Laplace transform. Then <br />
<span class="math-container">$\mathcal{L}\lbrace f\rbrace=\displaystyle\int_{0}^{4} 0dt+\int_{4}^{\infty}e^{-st}(t^{2}-8t+23)dt$</span>.</p>
|
159,948 | <p>If I have this function (just and example) </p>
<pre><code> Plot3D[-Abs[Cos[ y] Sqrt[x] Sqrt[ y] Cos[ x]], {x, 0, 20}, {y, 0, 20}]
</code></pre>
<p>I am interested in finding the locations of the smallest (minimum) 6 values of this function? How can I do that. Apparently, Findminimum and its likes give me one va... | aardvark2012 | 45,411 | <p>Depending on how similar your actual function is to your example, you could try giving <a href="http://reference.wolfram.com/language/ref/FindMinimum.html" rel="nofollow noreferrer"><code>FindMinimum</code></a> (see below) or <a href="http://reference.wolfram.com/language/ref/FindArgMin.html" rel="nofollow noreferre... |
159,948 | <p>If I have this function (just and example) </p>
<pre><code> Plot3D[-Abs[Cos[ y] Sqrt[x] Sqrt[ y] Cos[ x]], {x, 0, 20}, {y, 0, 20}]
</code></pre>
<p>I am interested in finding the locations of the smallest (minimum) 6 values of this function? How can I do that. Apparently, Findminimum and its likes give me one va... | José Antonio Díaz Navas | 1,309 | <p>Mathematically, it's simple, I think. The function is separable in both axis, so finding the minima along one axis, let say along the $x$-axis, you will have the corresponding ones along the other axis. All the minima in the $xy$-plane will be the combination of both.</p>
<p>Now, along one axis, one function is an ... |
1,426,264 | <p>Let $H$, $K$ be groups, and suppose that $H \cong K \times H$. Does it necessarily follow that $K$ is trivial?</p>
| joriki | 6,622 | <p>I'll answer the first question. (Asking more than one question per post is discouraged.)</p>
<p>Since each edge is incident at two vertices and adjacent to two faces, your conditions imply</p>
<p>$$e\ge\frac{4v}2=2v\;,$$
$$e\ge\frac{4f}2=2f\;,$$</p>
<p>where $v$, $e$, $f$ are the numbers of vertices, edges and fa... |
3,397,606 | <p>Find <span class="math-container">$[z^N]$</span> for <span class="math-container">$\frac{1}{1-z} \left(\ln \frac{1}{1-z}\right)^2$</span>. Here <em>generalized harmonic numbers</em> should be used. </p>
<p><span class="math-container">$$H^{(2)}_n = 1^2 + \frac{1}{2}^2 + \dots + \frac{1}{N}^2.$$</span></p>
<p>For n... | Ali Shadhar | 432,085 | <p>Using the generalization:</p>
<blockquote>
<p><span class="math-container">$$\sum_{n=1}^\infty a_nx^n=\frac1{1-x}\sum_{n=1}^\infty (a_n-a_{n-1})x^n,\quad a_{0}=0$$</span></p>
</blockquote>
<p>Let <span class="math-container">$a_{n}=H_n^2$</span> to have</p>
<p><span class="math-container">\begin{align}
\sum_{n=... |
783 | <p>In a category I have two objects $a$ and $b$ and a morphism $m$ from $a$ to $b$ and one $n$ from $b$ to $a$. Is this always an isomorphism? Why is it emphasized that this has to be true, too: $m \circ n = \mathrm{id}_b$ and $n \circ m = \mathrm{id}_a$?</p>
<p>I am looking for an example in which the id-part is not ... | Doug Spoonwood | 11,300 | <p>To make this perhaps even clearer, say we have the objects $a=\{x, y\}$ and $b=\{z\}$. Define $m\colon a\to b$ by $x\mapsto z$, $y\mapsto z$ (the only possible morphism here). Define $n\colon b\to a$ by $z\mapsto x$. But, of course, $n\circ m$ doesn't equal $\mathrm{id}_a$, since $(n\circ m)(y)=n(m(y))=n(z)=x$, $... |
1,660,231 | <blockquote>
<p>Show that the function $g(x)=\sqrt{x^2+x+2}$
is defined and is continuous on $\mathbb{R}$.</p>
</blockquote>
<p>I have tried completion of square for $$x^2+x+2=\left(x+\frac{1}{2}\right)^2+\frac{7}{4}$$
This means that range, $r\ge 7/4$ in domain $\mathbb{R}$.</p>
<p>Cannot find any more logic in... | Kamil Jarosz | 183,840 | <p>You have already shown that the inner function $x^2+x+2\geqslant 7/4$. The function $\sqrt t$ is defined for $t\geqslant 0$ and continuous for $t>0$. If you have
$$t=x^2+x+2$$ then the function $\sqrt t$ is defined and continuous as $t$ is continuous and $t\geqslant 7/4$ (here $t$ is a function of $x$).</p>
|
1,660,231 | <blockquote>
<p>Show that the function $g(x)=\sqrt{x^2+x+2}$
is defined and is continuous on $\mathbb{R}$.</p>
</blockquote>
<p>I have tried completion of square for $$x^2+x+2=\left(x+\frac{1}{2}\right)^2+\frac{7}{4}$$
This means that range, $r\ge 7/4$ in domain $\mathbb{R}$.</p>
<p>Cannot find any more logic in... | DonAntonio | 31,254 | <p>You need the expression within the square root is non-negative, and you already showed this since $\;x^2+x+2=\left(x+\frac12\right)^2+\frac74\;\leftarrow$ this is the sum of two non-negative expressions and thus is always non-negative, so your function's defined in all of $\;\Bbb R\;$.</p>
<p>And it is continuous i... |
664,347 | <p>For each of the following values of ($a,b$), find the largest number that is not of the form $ax+by$ with $x\geq 0$ and $y \geq 0$.</p>
<p>$(i) (a,b) = (3,7)$</p>
<p>$(ii) (a,b) = (5,7)$</p>
<p>$(iii) (a,b) = (4,11)$</p>
<p>I know how to compute numbers of these forms (clear), but have no idea how to generate on... | D.L. | 95,150 | <p>For the first one, for example, you can see that if $n$ is an integer, $n$, $n-7$ or $n-14$ is a multiple of 3 (by looking what happens modulo 3). So, if $n\geq 14$, you can write $n-7.x=3.y$ with $x=0, 1,$ or $2$ and $y\geq 0$. Then, the number you look for is between $1$ and $13$ and you can look one by one and ta... |
553,040 | <p>The question is: A card is drawn from an ordinary pack(52 cards) and a gambler bets that either a spade or an ace is going to appear. The probability of his winning are?
I think the answer is $\frac{16}{52} = \frac{4}{13}$. Did I go "probably" go wrong somewhere?</p>
| Cheyne H | 54,816 | <p>There are 16 `good' cards (the 13 spades and the three other aces), out of 52 total, so you're correct. </p>
|
4,562,006 | <p>Prove that <span class="math-container">$\text{Ker}(A)=\text{Ker}(A^2)$</span> if <span class="math-container">$A$</span> is a normal operator.</p>
<p>I know that that <span class="math-container">$\text{Ker}(A)\subseteq \text{Ker}(A^2)$</span>, which I proved as follows:
<span class="math-container">$x\in \text{Ker... | Just a user | 977,740 | <p>While this should be clear from the spectrum theorem, it can be established directly.</p>
<p>If <span class="math-container">$A^2x=0$</span>, then <span class="math-container">$$0=(A^2x, A^2x)=(Ax, A^*A^2x)=(Ax, AA^*Ax)=(A^*Ax, A^*Ax)$$</span></p>
<p>Thus <span class="math-container">$A^*Ax=0$</span>, hence <span cl... |
479,851 | <p>I toss a coin a many times, each time noting down the result of the toss. If at any time I have tossed more heads than tails I stop.
I.e. if I get heads on the first toss I stop.
Or if toss T-T-H-H-T-H-H I will stop.
If I decide to only toss the coin at most 2n+1 times, what is the probability that I will get more ... | Arthur | 15,500 | <p>Let $C_n$ be the $n$-th Catalan number, defined by $\frac{1}{n+1}\binom{2n}{n}$. Let's also say that you keep tossing even after you had more heads than tails, for good measure. Then there are $2^{2n + 1}$ different sequences of tosses you could've made. How many of them will at some point have more heads than tails... |
18,136 | <p>I introduced the hypercube (to undergraduate students in the U.S.) in the
context of generalizations of the Platonic solids, explained its
structure, showed it rotating.
I mentioned <a href="https://matheducators.stackexchange.com/a/1824/511">Alicia Stott</a>, who discovered the <span class="math-container">$6$</spa... | Pyrhos | 13,764 | <p>The best explanation of the general concept I've encountered so far is the introduction to a 4D game, Miegakure.</p>
<p>The idea of extra dimensions is described in the jump from 2D to 3D first, which makes it much easier to visualize and extrapolate.</p>
<p><a href="https://www.youtube.com/watch?v=9yW--eQaA2I" re... |
18,136 | <p>I introduced the hypercube (to undergraduate students in the U.S.) in the
context of generalizations of the Platonic solids, explained its
structure, showed it rotating.
I mentioned <a href="https://matheducators.stackexchange.com/a/1824/511">Alicia Stott</a>, who discovered the <span class="math-container">$6$</spa... | Helder | 5,848 | <p>For more than four dimensions, I would consider treating each pixel in a gray scale image as a "dimension" and its brightness as the value of the corresponding "coordinate". Then, ℝ<sup><i>m</i> x <i>n</i></sup> is just the set of all images (including photos) of <i>m</i> pixels by <i>n</i> pixels.</p>
<p>There is ... |
18,136 | <p>I introduced the hypercube (to undergraduate students in the U.S.) in the
context of generalizations of the Platonic solids, explained its
structure, showed it rotating.
I mentioned <a href="https://matheducators.stackexchange.com/a/1824/511">Alicia Stott</a>, who discovered the <span class="math-container">$6$</spa... | Ari | 13,775 | <p>Here's how i understood it, am 18 so correct me if I'm wrong</p>
<p>This site was cool --> <a href="https://4dtoys.com/" rel="nofollow noreferrer">https://4dtoys.com/</a></p>
<p>I think its important to have a physical understanding/visualization of the dimensions.</p>
<p>1D point would be like a bead on a string... |
4,038,685 | <p>Suppose that <span class="math-container">$u(x)$</span> is continuous and satisfies the integral equation</p>
<p><span class="math-container">\begin{equation}\label{1.4.1}
u(x) = \int_0^x \sin(u(t))u(t)^p dt
\end{equation}</span></p>
<p>on the interval <span class="math-container">$0 \leq x \leq 1$</span... | xpaul | 66,420 | <p>Use @Matthew Pilling's setting. Assume there exists <span class="math-container">$t\in(0,1)$</span> such that <span class="math-container">$u(t)>0$</span>. Because <span class="math-container">$u$</span> is continuous and <span class="math-container">$u(0)=0$</span>, we can identify <span class="math-container">$... |
648,066 | <p>Let $\gamma(z_0,R)$ denote the circular contour $z_0+Re^{it}$ for $0\leq t \leq 2\pi$. Evaluate
$$\int_{\gamma(0,1)}\frac{\sin(z)}{z^4}dz.$$</p>
<p>I know that
\begin{equation}
\int_{\gamma(0,1)}\frac{\sin(z)}{z^4}dz = \frac{1}{z^4}\left(z-\frac{z^3}{3!}+\frac{z^5}{5!}-\cdots\right)
= \frac{1}{z^3}-\frac{1}{6z}+\c... | mkl314 | 123,304 | <p>The function $f(z)=\frac{\sin{z}}{z^4}$ has a third order pole $z=0$ inside $\gamma$. By Cauchy's residue theorem, the integral
$$ \int_{\gamma}f(z)\,dz=2{\pi}i\cdot\underset{z=0}{\rm res}\, f(z),$$
where the residue at the third order pole $z=0$ can be calculated using formula
$$\underset{z=0}{\rm res}\, f(z)=\frac... |
2,254,025 | <p><strong>Please verify my answer to the following differential equation:</strong>
$$y''-xy'+y=0$$
Let $y = {\sum_{n=0}^\infty}C_nx^n$, then $y' = {\sum_{n=1}^\infty}nC_nx^{n-1}$ and $y''={\sum_{n=2}^\infty}n(n-1)C_nx^{n-2}$</p>
<p>Substituting this to the equation we get</p>
<p>$${\sum_{n=2}^\infty}n(n-1)C_nx^{n-2}... | TurlocTheRed | 397,318 | <p>You should be able to do a change of variable to transform this equation into Hermite's equation then compare the resulting coefficients to Hermite's. </p>
<p>Alternatively, you can determine the Rodrigue's Operator formulation of solutions of the equation and verify the operators have the desired effect on your se... |
391,509 | <p>We have $$\dfrac{1+2+3+...+ \space n}{n^2}$$</p>
<p>What is the limit of this function as $n \rightarrow \infty$?</p>
<p>My idea:</p>
<p>$$\dfrac{1+2+3+...+ \space n}{n^2} = \dfrac{1}{n^2} + \dfrac{2}{n^2} + ... + \dfrac{n}{n^2} = 0$$</p>
<p>Is this correct?</p>
| amWhy | 9,003 | <p>Recall that $$1 + 2 + \cdots + n =\sum_{i = 1}^n i = \frac{n(n+1)}{2}$$</p>
<p>So you want $$\lim_{n\to \infty}\dfrac{n(n+1)}{2n^2} = \lim_{n\to\infty}\frac{n^2+ n}{2n^2} = \lim_{n\to \infty} \left(\frac 12 + \frac 1{2n}\right)$$</p>
|
391,509 | <p>We have $$\dfrac{1+2+3+...+ \space n}{n^2}$$</p>
<p>What is the limit of this function as $n \rightarrow \infty$?</p>
<p>My idea:</p>
<p>$$\dfrac{1+2+3+...+ \space n}{n^2} = \dfrac{1}{n^2} + \dfrac{2}{n^2} + ... + \dfrac{n}{n^2} = 0$$</p>
<p>Is this correct?</p>
| egreg | 62,967 | <p>Your method is wrong. Consider the first values of the expression:</p>
<ul>
<li>for $n=1$ you get $1$;</li>
<li>for $n=2$ you get $1/4+2/4=3/4$;</li>
<li>for $n=3$ yoy get $1/9+2/9+3/9=6/9=2/3$.</li>
</ul>
<p>One could make the conjecture that these terms are always bigger than $1/2$, which can be proved by induct... |
3,249,391 | <p>By Mathematica, we find <span class="math-container">$$\sum_{n=1}^\infty \frac{4^n}{n^3\binom{2n}{n}}=\pi^2\log(2)-\frac{7}{2}\zeta(3).$$</span></p>
<blockquote>
<p>How to find the closed form for general series:
<span class="math-container">$$\sum_{n=1}^\infty \frac{4^n}{n^p\binom{2n}{n}}? \ \ (p\ge 3)$$</spa... | Zacky | 515,527 | <p>We can make use of the following representation
<span class="math-container">$$\sf 2\arcsin^2z=\sum\limits_{n\geq1}\frac {(2z)^{2n}}{n^2\binom {2n}n}, \ z\in[-1,1]$$</span>
Which gives integrating once with respect to <span class="math-container">$\sf z$</span> from <span class="math-container">$\sf 0$</span> to <... |
3,249,391 | <p>By Mathematica, we find <span class="math-container">$$\sum_{n=1}^\infty \frac{4^n}{n^3\binom{2n}{n}}=\pi^2\log(2)-\frac{7}{2}\zeta(3).$$</span></p>
<blockquote>
<p>How to find the closed form for general series:
<span class="math-container">$$\sum_{n=1}^\infty \frac{4^n}{n^p\binom{2n}{n}}? \ \ (p\ge 3)$$</spa... | Dr. Wolfgang Hintze | 198,592 | <p>Setting <span class="math-container">$$n^{-p} = \frac{1}{\Gamma(p)} \int_{0}^{\infty} t^{p-1} e^{- n t}$$</span> </p>
<p>I find for the sum</p>
<p><span class="math-container">$$s(p) = \zeta(p) + \frac{1}{\Gamma(p)} \int_0^\infty t^{p-1}\frac{e^{- t/2} }{( 1-e^{-t} )^{\frac{3}{2}}} \arcsin(e^{-t/2})$$</span> </p... |
2,821,308 | <p>Consider below complex function
$$H(\omega) = \dfrac{1}{i\omega} \left(e^{i\omega} - e^{-i\omega}\right)$$</p>
<p>If I replace $i$ by $-i$ in $H(\omega)$, I get back the same $H(\omega)$.<br>
Easy to see that $H(\omega) = \dfrac{2}{\omega}\sin(\omega t)$. </p>
<p>Long back I heard somebody claim this, but I coul... | Rodrigo de Azevedo | 339,790 | <p>Using <strong>polar</strong> coordinates,</p>
<p>$$\begin{aligned} x_1 &= \cos (\theta)\\ x_2 &= \sin (\theta)\end{aligned}$$</p>
<p>we obtain the <strong>unconstrained</strong> $1$-dimensional maximization problem in $\theta$</p>
<p>$$\text{maximize} \quad 1 + 2 \sin (2\theta)$$</p>
<p>Differentiating t... |
2,821,308 | <p>Consider below complex function
$$H(\omega) = \dfrac{1}{i\omega} \left(e^{i\omega} - e^{-i\omega}\right)$$</p>
<p>If I replace $i$ by $-i$ in $H(\omega)$, I get back the same $H(\omega)$.<br>
Easy to see that $H(\omega) = \dfrac{2}{\omega}\sin(\omega t)$. </p>
<p>Long back I heard somebody claim this, but I coul... | Michael Rozenberg | 190,319 | <p>$(x_1-x_2)^2\geq0$ gives $$x_1x_2\leq\frac{x_1^2+x_2^2}{2}=\frac{1}{2}.$$
The equality occurs for $x_1=x_2$.</p>
<p>Thus,
$$-x_1^2-4x_1x_2-x_2^2=-1-4x_1x_2\geq-1-4\cdot\frac{1}{2}=-3.$$
The equality occurs for
$$(x_1,x_2)\in\left\{\left(\frac{1}{\sqrt2},\frac{1}{\sqrt2}\right),\left(-\frac{1}{\sqrt2},-\frac{1}{\sqr... |
4,621,227 | <p>This is a very soft and potentially naive question, but I've always wondered about this seemingly common phenomenon where a theorem has some method of proof which makes the statement easy to prove, but where other methods of proof are incredibly difficult.</p>
<p>For example, proving that every vector space has a ba... | Dietrich Burde | 83,966 | <ol>
<li><p>Yes, there are many other examples of this. More or less every famous result (well, you wanted a "simple proof" for it) will be "incredibly challenging", or even impossible with a different proof. I think of Fermat's last theorem, the Poincare conjecture, or the weak Goldbach conjecture,... |
4,311,731 | <p>I was thinking about solutions of equation <span class="math-container">$x^2=i$</span> . First thought coming to my mind was <span class="math-container">$x=\pm \sqrt{i}$</span> . ( I know it's wrong ) .
Then I thought if we solve this equation like real problem then
<span class="math-container">$|x|=\sqrt i$</span>... | Rebellos | 335,894 | <p>As squid correctly pointed out in the comments, the distance is by definition <strong>real</strong>. We induce and construct norms over spaces that help us measure distance for this exact reason, regardless their elements.</p>
<p>As of such, the expression <span class="math-container">$|z| = i$</span> doesn't stand.... |
237,960 | <p>How could I solve this problem?</p>
<blockquote>
<p>Find the first digit of $2^{4242}$ without using a calculator.</p>
</blockquote>
<p>I know how to find the last digit with modular arithmetic, but I can't use that here.</p>
| Old John | 32,441 | <p>This is probably not the answer you are looking for, and wil probably only be appreciated by people of my age ...</p>
<p>I can still remember from school days that $\log_{10} 2 = 0,30102999$ (I always thought it was noteworthy that it is so close to $0.30103$) - people who went to school in the 1950's can probably ... |
1,626,840 | <blockquote>
<p>If I've played the lottery a certain $N$ number of times (and I didn't look into the results of each game), what this $N$ number must be to get me a $50\%$ chance of have already won at least one of my games? (imagine I 'll look up at all the results at once, after acheiving this chance) and the chanc... | Jimmy R. | 128,037 | <p>The number $X$ of successes in $N$ trials is binomially distributed with parameters $N$ and $p=\frac{1}{1000000}$. So, by complementarity the probability of at least one win, i.e. $P(X\ge 1)$, is equal to $1$ minus the probability of $0$ wins, i.e. $P(X=0)$. By the formula of the binomial distribution we get $$P(X\g... |
3,783,186 | <p>I am trying to prove that <span class="math-container">$$2≤\int_{-1}^1 \sqrt{1+x^6} \,dx ≤ 2\sqrt{2} $$</span> I learned that the equation <span class="math-container">$${d\over dx}\int_{g(x)}^{h(x)} f(t)\,dt = f(h(x))h'(x) - f(g(x))g'(x) $$</span> is true due to Fundamental Theorem of Calculus and Chain Rule, a... | Mojbn | 764,562 | <p><span class="math-container">$ \int_{-1}^1 \sqrt{1+x^6}dx=2\int_{0}^1 \sqrt{1+x^6} dx$</span> because <span class="math-container">$\sqrt{1+x^6}$</span> is an even function. so we must show:
<span class="math-container">$$2≤2\int_{0}^1 \sqrt{1+x^6} dx ≤ 2\sqrt{2}$$</span>
or we must show:
<span class="math-containe... |
3,048,381 | <p>We found the orthonormal basis for the eigen spaces.</p>
<p>We got <span class="math-container">$C$</span> to be the matrix</p>
<pre><code>[ 1/squareroot(2) 1/squareroot(6) 1/squareroot(3)
-1/squareroot(2) 1/squareroot(6) 1/squareroot(3)
0 -2/squareroot(6) ... | Digitalis | 515,828 | <p>Notice that <span class="math-container">$ A = \begin{bmatrix}4 & 2 & 2 \\ 2 & 4 & 2 \\ 2 & 2 & 4 \end{bmatrix}$</span>
has eigenvalues <span class="math-container">$2$</span> and <span class="math-container">$8$</span>. Since there exists a basis of eigenvectors <span class="math-contain... |
3,176,593 | <p>I'm having a hard time understanding how to find all solutions of the form <span class="math-container">$a_n = a^{(h)}_n+a_n^{(p)}$</span></p>
<p>I show that <span class="math-container">$a_n=n2^n \to a_n=2(n-1)2^{n-1} +2^n=2^n(n-1+1)=n2^n$</span>.</p>
<p>I can show that <span class="math-container">$a_n^{(h)}$</s... | lhf | 589 | <p>Here is another take.</p>
<p>Let <span class="math-container">$b_n=2^n$</span>. Then
<span class="math-container">$$
a_n=2a_{n-1}+2b_{n-1}, \quad b_n=2b_{n-1}, \quad b_0=1
$$</span>
and so
<span class="math-container">$$
\pmatrix{
a_n \\ b_n
}
=
\pmatrix{
2 & 2 \\ 0 & 2
}
\pmatrix{
a_{n-1} \\ b_{n-1}
}
=
2
... |
2,537,311 | <p>Question is, from finding a basis of e-vectors, determine an invertible matrix <strong>P</strong> such that: <img src="https://latex.codecogs.com/gif.latex?F=P%5E%7B-1%7DLP" title="F=L^{-1}AL" /> is diagonal. (write down the matrix F)</p>
<p><img src="https://latex.codecogs.com/gif.latex?L=%5Cbegin%7Bpmatrix%7D&am... | user284331 | 284,331 | <p>Mean value Theorem: $|f(x)-f(0)|=|f'(\xi_{x})||x|\leq|x|$, then...</p>
|
1,063,154 | <p>I Ran into this question and I can't find the right way to approach it.</p>
<p>We have $n$ different wine bootles numbered $i=1...n$. the first is 1 year old, the second is 2 years old ... the $n$'th bottle is $n$ years old.</p>
<p>Each bottle is still good at probability of $1/i$.</p>
<p>We pick out a random bot... | Henry | 6,460 | <p>You are interested in the age of the bottle (the random variable), conditioned on it being good. </p>
<p>The probability the bottle selected is of age $i$ given it is good is $\dfrac{\frac{1}{i}}{\sum_1^n \frac{1}{j}} = \dfrac{1}{iH(n)}$ where $H(n)$ is a harmonic number.</p>
<p>So the expected age given it is goo... |
416,407 | <blockquote>
<p>What examples are there of habitual but unnecessary uses of the axiom of
choice, in any area of mathematics except topology?</p>
</blockquote>
<p>I'm interested in standard proofs that use the axiom of choice, but where
choice can be eliminated via some judicious and maybe not quite obvious
rephrasing. ... | Pace Nielsen | 3,199 | <p>Many uses of Zorn's lemma really only need transfinite recursion, without any use of AC. Sometimes you don't even need transfinite recursion, but just normal recursion, or even less.</p>
<p>This is especially applicable in specific examples. For instance, you don't need AC to get an algebraic closure of <span clas... |
416,407 | <blockquote>
<p>What examples are there of habitual but unnecessary uses of the axiom of
choice, in any area of mathematics except topology?</p>
</blockquote>
<p>I'm interested in standard proofs that use the axiom of choice, but where
choice can be eliminated via some judicious and maybe not quite obvious
rephrasing. ... | Ege Erdil | 131,052 | <p>It's common to use the axiom of choice to prove that nonzero commutative rings have the invariant basis number property: in other words, that for a nonzero commutative ring <span class="math-container">$ R $</span>, the <span class="math-container">$ R $</span>-modules <span class="math-container">$ R^m $</span> and... |
416,407 | <blockquote>
<p>What examples are there of habitual but unnecessary uses of the axiom of
choice, in any area of mathematics except topology?</p>
</blockquote>
<p>I'm interested in standard proofs that use the axiom of choice, but where
choice can be eliminated via some judicious and maybe not quite obvious
rephrasing. ... | daw | 48,485 | <p>The supremum of an arbitrary set of measurable functions from a <span class="math-container">$\sigma$</span>-finite measure space into <span class="math-container">$\mathbb R\cup \{\pm\infty\}$</span> exists in the following sense:</p>
<p>Let <span class="math-container">$F$</span> be a set of such measurable functi... |
416,407 | <blockquote>
<p>What examples are there of habitual but unnecessary uses of the axiom of
choice, in any area of mathematics except topology?</p>
</blockquote>
<p>I'm interested in standard proofs that use the axiom of choice, but where
choice can be eliminated via some judicious and maybe not quite obvious
rephrasing. ... | Gro-Tsen | 17,064 | <p>It is a theorem of ZF that every sequentially continuous function <span class="math-container">$\mathbb{R}\to\mathbb{R}$</span> is continuous. The proof is usually given in ZFC (and indeed, Choice <em>is</em> necessary to assert that sequential continuity <em>at a point</em> implies continuity at that point), but a... |
836,753 | <p>$\{X_1, X2, \ldots, X_{121}\}$ are independent and identically distributed random variables such that $E(X_i)= 3$ and $\mathrm{Var}(X_i)= 25$. What is the standard deviation of their average? In other words, what is the standard deviation of $\bar X= {X_1+ X_2+ \cdots + X_{121} \over 121}$?</p>
| beep-boop | 127,192 | <p>Let $\bar X=\frac{1}{n}\sum\limits_{i=1}^{n}X_i$.</p>
<p>We want $\rm{SD}(\bar X)$ (standard deviation of $\bar X$).</p>
<p>Hints:</p>
<ul>
<li><p>$\rm{Var}[\bar X]=\frac{\sigma^2}{n}$</p></li>
<li><p>$\rm{SD}(\bar X)=\sqrt{\rm{Var}[\bar X]}$</p></li>
</ul>
<p>where $n=$ number of random variables, $\sigma^2=$ p... |
3,921,259 | <p>I need to calculate the operator Norm of the linear operator defined as:
<span class="math-container">$$ T:(C([0,1]),||\cdot||_\infty)\rightarrow\mathbb{R} \text{ where } Tf=\sum_{k=1}^n a_kf(t_k)$$</span>
for <span class="math-container">$0\leq t_1<t_2<...<t_n\leq 1$</span> and <span class="math-container"... | Tito Eliatron | 84,972 | <p>WLOG, one can assume that <span class="math-container">$a_k\ne0$</span> for all <span class="math-container">$k$</span> (if not, just drop all the zero-values, and in the case that all are zero, the problem is trivial).</p>
<p>Take <span class="math-container">$f(x)$</span> a continuous function with these propertie... |
2,306,122 | <p>Show $X=\{n \in \mathbb{N}: \text{n is odd and} \ n = k(k+1) \text{for some} \ k \in \mathbb{N}\}=\emptyset$</p>
<p>My proof is as follow, please point if I have made any mistake. </p>
<p><strong>proof:</strong></p>
<p>we have $\emptyset \subseteq X$
suppose $X≠\emptyset$ pick $n \in X$</p>
<p>Then there are 2 ... | JMP | 210,189 | <p>You are taking a very long-winded approach.</p>
<p>Your $X$ can be expressed as:</p>
<p>$$X=\{n\in\mathbb N|\text{n is odd} \land \text{n is even}\}$$</p>
<p>and we need to prove $X=\emptyset$.</p>
|
3,841,535 | <p>I'm having a lot of trouble solving this question via the differentiating with respect to a parameter method. I can get the correct result for the integral containing sine, but I'm totally lost when it comes to evaluating the integral containing cosine. Here's the problem statement:</p>
<p>Given:</p>
<p><span class=... | Felix Marin | 85,343 | <p><span class="math-container">$\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\exp... |
674,621 | <p>I am trying to figure out what the three possibilities of $z$ are such that </p>
<p>$$
z^3=i
$$</p>
<p>but I am stuck on how to proceed. I tried algebraically but ran into rather tedious polynomials. Could you solve this geometrically? Any help would be greatly appreciated.</p>
| EpicMochi | 95,992 | <p>Using Euler's formula, which states
$$
e^{i \theta} = \cos \theta + i \sin \theta
$$
we will see that
$$
i = 0 + i \cdot 1 = \cos \left( \frac{\pi}{2} + 2n \pi \right) + i \sin \left( \frac{\pi}{2} + 2n \pi \right) = e^{i \left(\frac{ \pi}{2} + 2n \pi \right)}
$$
for all integers $n$. Thus, if $z^3 = i$, then
$$z ... |
530,605 | <blockquote>
<p>Let <span class="math-container">$A$</span> be open in <span class="math-container">$\mathbb{R}^m$</span>; let <span class="math-container">$g:A\rightarrow\mathbb{R}^n$</span>. If <span class="math-container">$S\subseteq A$</span>, we say that <span class="math-container">$S$</span> satisfies the <stron... | copper.hat | 27,978 | <p>Or $x \mapsto \sqrt{|x|}$: ${}{}{}{}$</p>
<p><img src="https://i.stack.imgur.com/5u0xa.png" alt="enter image description here"></p>
|
1,443,471 | <p>I have this conjecture: Let <em>a</em> and <em>b</em> be integers and <em>n</em> and <em>m</em> natural numbers. </p>
<p>$$ a \equiv b \bmod n \Rightarrow a^m \equiv b^m \bmod n$$</p>
<p>I think I got the induction proof, but I'm having difficulties on how to proof this with well-ordering principle.</p>
| Kulisty | 170,765 | <p>Notice that</p>
<p>$a^m-b^m=(a-b)\sum\limits_{k=0}^{m-1}a^{m-k-1}b^{k}=(a-b)(a^{m-1}+a^{m-2}b+\ldots+ab^{m-2}+b^{m-1})$</p>
<p>So </p>
<p>$a-b | a^m-b^m $</p>
|
729,441 | <p>This question comes up after going over Arzela-Ascoli theorems.</p>
<p>For a set of continuous functions $\mathbb F$ from $\mathbb R$ to $\mathbb R$ that is equicontinuous. How do I show that if sup{$|f(0)|:f \in F$} $< \infty$ , then $\mathbb F$ is pointwise bounded?</p>
<p>I know I need to show that since su... | Brian Fitzpatrick | 56,960 | <p>If you don't know what an eigenvalue is, and if you're not worried about elegance, then here is a more direct approach (assuming you're working over a field not of characteristic two).</p>
<p>There exist scalars $\lambda_{ij}$ for $1\leq i,j\leq n$ such that for every basis $\{v_1,\dotsc,v_n\}$ of $V$ we have
$$
\b... |
615,396 | <p>What is considered a good format for writing problem sets in mathematics? Are there any good examples of problem sets that are well-written and formatted that you can show me?</p>
| nomen | 62,468 | <p>There are a few common styles. Personally, I like ink on printer paper, formatted either in two columns or one, depending on how wide the wide formulas get. I cross out "mistakes" instead of erasing.</p>
<p>Some people like using graph (grid) paper. Most people probably use pencils, but I find the ease of using ... |
767,474 | <blockquote>
<p>If $a,b,c\in\mathbb R^+$ prove that:
$$\frac{a^2+1}{b+c}+\frac{b^2+1}{a+c}+\frac{c^2+1}{a+b}\ge 3$$</p>
</blockquote>
| Calvin Lin | 54,563 | <p>Apply Am-GM to the numerator of each fraction. </p>
<p>You get the statement of Nesbit inequality. </p>
|
767,474 | <blockquote>
<p>If $a,b,c\in\mathbb R^+$ prove that:
$$\frac{a^2+1}{b+c}+\frac{b^2+1}{a+c}+\frac{c^2+1}{a+b}\ge 3$$</p>
</blockquote>
| Michael Rozenberg | 190,319 | <p>By C-S and AM-GM we obtain
$$\sum_{cyc}\frac{a^2+1}{b+c}=\sum_{cyc}\frac{a^2}{b+c}+\sum_{cyc}\frac{1}{b+c}\geq$$
$$\geq\frac{(a+b+c)^2}{2(a+b+c)}+\frac{9}{2(a+b+c)}=\frac{1}{2}\left(a+b+c+\frac{9}{a+b+c}\right)\geq$$
$$\geq\frac{1}{2}\cdot2\sqrt{\frac{9(a+b+c)}{a+b+c}}=3.$$
Done!</p>
|
238,809 | <p><a href="https://i.stack.imgur.com/W5ILn.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/W5ILn.jpg" alt="enter image description here" /></a>
How to construct a tree like <a href="https://i.stack.imgur.com/XKYl9.png" rel="nofollow noreferrer">this</a>? I was looking <code>CompleteKaryTree</code> i... | kglr | 125 | <p>You can use <a href="https://reference.wolfram.com/language/ref/ExpressionGraph.html" rel="nofollow noreferrer"><code>ExpressionGraph</code></a> to draw the tree</p>
<pre><code>expr = ConstantArray[x, Reverse @ Range[4]];
ExpressionGraph[expr, GraphLayout -> "LayeredEmbedding", ImageSize -> Large]
... |
238,809 | <p><a href="https://i.stack.imgur.com/W5ILn.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/W5ILn.jpg" alt="enter image description here" /></a>
How to construct a tree like <a href="https://i.stack.imgur.com/XKYl9.png" rel="nofollow noreferrer">this</a>? I was looking <code>CompleteKaryTree</code> i... | kglr | 125 | <p>Using a slight modification of the code in <a href="https://demonstrations.wolfram.com/PermutationTree/" rel="nofollow noreferrer">Wolfram Demonstrations >> Permutation Tree</a> (linked by George Varnavides in comments) and adding edge labels:</p>
<pre><code>ClearAll[permTree]
permTree[n_, opts : OptionsPatter... |
878,816 | <p>Solve inequality: $-5 < \frac{1}{x} < 0$ </p>
<p>I thought about how I can solve this. If I multiply all sides by $x$ I'm afraid I'm removing the answer, cause $\frac{x}{x}=1$. And when $x$ 'leaves' the inequality I'm left with no letter. </p>
<p>How do I get just $x$ in the middle without adding $x$ to ot... | amWhy | 9,003 | <p>Note that given $-5\lt \frac 1x \lt 0$, we know that $$\frac 1x < 0 \implies x< 0$$. So when you multiply by $x$ to remove it from the denominator, you need to reverse the directions of the inequalities.</p>
<p>$$-5 \lt \frac 1x \iff -5x \gt 1\iff x \lt -\frac 15 $$ and $$\frac 1x <0 \iff x\lt 0$$ The seco... |
878,816 | <p>Solve inequality: $-5 < \frac{1}{x} < 0$ </p>
<p>I thought about how I can solve this. If I multiply all sides by $x$ I'm afraid I'm removing the answer, cause $\frac{x}{x}=1$. And when $x$ 'leaves' the inequality I'm left with no letter. </p>
<p>How do I get just $x$ in the middle without adding $x$ to ot... | Ruslan | 64,206 | <p>You don't have to necessarily put $x$ to the center directly. One way to solve such double inequality is to split it to a system of two inequalities:</p>
<p>$$\begin{cases}-5<\frac1x\\
\frac1x<0\end{cases}$$</p>
<p>From the second one you get $x<0$, and from the first one you can obtain $x<-\frac15$ (n... |
2,619,344 | <p>I just struck with a doubt today</p>
<blockquote>
<p>Why do most of the standard inequalities require the variables to be positive.</p>
</blockquote>
<p>For example if we want to find minimum value of a certain expression say <span class="math-container">$a+b+c$</span> the very first thought that comes in our mind ... | Guy Fsone | 385,707 | <p>See that $f$ is bounded as continuous function on $[0,1]$ and
$$\frac{1}{n}\int_{0}^{1}e^{-nx}dx = \frac{1}{n^2}(1-e^{-n})\le \frac{1}{n^2}$$
we have
$$\sum_{n=1}^{\infty}\left|\frac{1}{n}\int_{0}^{1}f(x)e^{-nx}dx\right| \le M\sum_{n=1}^{\infty}\frac{1}{n}\int_{0}^{1}e^{-nx}dx \le M\sum_{n=1}^{\infty}\frac{1}{n^2}... |
269,062 | <p>Consider the following integral
$$
pv\int_0^{\infty}e^{N(-2Ax+A\log x)}\frac{e^{-B\log x}}{1-2x}dx
$$
where $A,B>0$ and we take the Cauchy principal value at $x=1/2$. I am interested in obtaining the asymptotics when $N$ is very big. The first thing I thought of was some variant of Laplace's method but I am unsur... | Igor Rivin | 11,142 | <p>Mathematica tells us that the principal value can be evaluated in closed form thus:</p>
<p>$$e^{-A} 2^{-A N+B-1} \left((-1)^{A N-B+1} \Gamma (-B+A N+1) \Gamma (B-A N,-A)-i \pi
\right).$$</p>
<p>(OK, the $i \pi$ is a little suspicious, but the appearance of incomplete gammas is clearly correct, once you replace ... |
1,524,109 | <p>Could anyone help me with this proof without using determinant? I tried two ways. </p>
<blockquote>
<p>Let $A$ be a matrix. If $A$ has the property that each row sums to zero, then there does not exist any matrix $X$ such that $AX=I$, where $I$ denotes the identity matrix. </p>
</blockquote>
<p>I then get stuck.... | Ilmari Karonen | 9,602 | <p>Assuming $A$ is an $n \times n$ matrix, let $v_0 = (0,0,\dots,0)$ and $v_1 = (1,1,\dots,1)$ be $n$-element column vectors. Since each row of $A$ sums to zero, it follows that $$A v_1 = (0,0,\dots,0) = A v_0,$$ showing that $A$ cannot have a (left) inverse.</p>
|
1,260,722 | <blockquote>
<p>Prove that <span class="math-container">$f=x^4-4x^2+16\in\mathbb{Q}[x]$</span> is irreducible.</p>
</blockquote>
<p>I am trying to prove it with Eisenstein's criterion but without success: for <strong>p=2</strong>, it divides <strong>-4</strong> and the constant coefficient 16, don't divide the leading ... | Barry Cipra | 86,747 | <p>As Jyrki Lahtonen observed in his answer, $p(x)=x^4-4x^2+16=(x^2-2)^2+12$ has no real roots, so the only possible factorization is of the form</p>
<p>$$(x^2+ax+b)(x^2+cx+d)$$</p>
<p>which expands to</p>
<p>$$x^4+(a+c)x^3+(ac+b+d)x^2+(ad+bc)x+bd$$</p>
<p>We conclude that $c=-a$ (from the $x^3$ term), hence* $d=b$... |
1,036,907 | <p>For <span class="math-container">$n$</span> in the natural numbers let</p>
<p><span class="math-container">$$a_n = \int_{1}^{n} \frac{\cos(x)}{x^2} dx$$</span></p>
<p>Prove, for <span class="math-container">$m ≥ n ≥ 1$</span> that <span class="math-container">$|a_m - a_n| ≤ \frac{1}{n}$</span> and deduce <span class... | Yao Zhao | 889,365 | <p>We only need to show <span class="math-container">$\{b_j\}$</span> is Cauchy, where <span class="math-container">$b_j=\int_{1}^{j} \frac{\cos x}{x^2} dx$</span></p>
<p>Proof: <span class="math-container">$b_j-b_k=\int_{1}^{j} \frac{\cos x}{x^2} dx-\int_{1}^{k} \frac{\cos x}{x^2} dx=\int_{k}^{j} \frac{\cos x}{x^2} dx... |
43,382 | <p>Let <span class="math-container">$C$</span> be a coalgebra and <span class="math-container">$\Delta: C \to C\otimes C$</span> a co-multiplication map. Then, due the co-associative property we can consider <span class="math-container">$\Delta^m$</span>. But how is defined <span class="math-container">$\Delta^{m}: C \... | Ralph | 10,194 | <p>When working with coalgebras I often find it helpful to dualize and to consider the corresponding situation for multiplication $\mu: A \otimes A \to A$ with an algebra $A$. </p>
<p>In this point of view $\Delta^m$ corresponds to $\mu^m: A^{m+1} \to A$, given by
$$\mu^m(a_0 \otimes ... \otimes a_m) = (a_0 ... a_{m-... |
3,719,575 | <p>For <span class="math-container">$\lambda =0$</span> and <span class="math-container">$\lambda <0$</span> the solution is the trivial solution <span class="math-container">$x\left(t\right)=0$</span></p>
<p>So we have to calculate for <span class="math-container">$\lambda >0$</span></p>
<p>The general solution... | Alessio K | 702,692 | <p><span class="math-container">$\lambda >0$</span> the general solution is <span class="math-container">$x(t)=C_{1}cos(\sqrt{\lambda}t)+C_{2}sin(\sqrt{\lambda}t)$</span> then the condition <span class="math-container">$x(0)=0$</span> gives <span class="math-container">$C_{1}=0$</span> as you've computed.</p>
<p>The... |
1,453,682 | <p>I'm having trouble classifying the solution set of the systems $Ax = b$ and $Ax = 0$.</p>
<p>Let A be an ($m \times n$) matrix.</p>
<p><strong>Case I:</strong></p>
<p><em>For $Ax = 0$, the system has a unique solution (the trivial one) when A is invertible, and infinitely many solutions when A is not.</em> </p>
... | Ant | 66,711 | <p>1) yes, there is always the zero - vector solution $x = 0$</p>
<p>2) Yes, you need to compute the rank of the augmented matrix.</p>
<p>3) No. If $Ay = b$ and $Az=b$ then $$A(y+z) = Ay + Az = b + b = 2b \neq b$$</p>
<p>except the case when $b= 0$. If $b=0$ then it is true and that is why $Ker(A)$ is a vector space... |
742,160 | <p>Answer true or false to each of the following questions. If a statement is true, prove it. If a statement is false, give a counterexample.</p>
<ol>
<li>For all sets $A$,$B$ and $C$: IF $A ⊆ B$ and $A ⊆ C$, Then $A ⊆ (B ∩ C)$</li>
<li>For all sets $A$ and $B$, if $|A| \le |B|$, then $A ⊆ B$</li>
</ol>
| fgp | 42,986 | <p>An <em>injection</em> $A \to B$ maps $A$ <em>into</em> $B$, i.e. it allows you to find a copy of $A$ inside $B$.</p>
<p>A <em>surjection</em> $A \to B$ maps $A$ <em>over</em> $B$, in the sense that the image covers the whole of $B$. The syllable "sur" has latin origin, and means "over" or "above", as for example in... |
35,877 | <p>I got stucked while I was working out whether
$$\frac{4n-3}{n+43}$$
converges. I would be pleased if I could a hint of the above question.</p>
| lhf | 589 | <p><em>Hint</em>: $$\frac{4n+3}{n+43} = 4-\frac{175}{n+43} $$</p>
|
2,095,349 | <p>I test my skills in statistics and probabilities and I decided to work with distributions. So, I tried to solve the below problem </p>
<p><strong>Problem</strong></p>
<p>Suppose that a hospital serves in average $80$ citizens daily from a city with $11000$ citizens.
In a random day, what is the probability that t... | BruceET | 221,800 | <p><strong>Binomial:</strong> If $X \sim Binom(n, p),$ then $E(X) = np = 80,$ and $n = 11000.$
So $p = 80/11000 = 0.00727.$ You seek $P(X \le 8).$ In R statistical
software this is about $6.6 \times 10^{-25}.$</p>
<pre><code>n = 11000; p = 80/n
pbinom(8, n, p)
## 6.572574e-25
</code></pre>
<p><strong>Poisson:</stron... |
3,703,011 | <p>So, I am an absolute beginner in mathematics; only being knowledgeable in some basic ideas of the subject. My interest in math started only recently, while reading about set theory and cardinality (particularly the concept of higher infinities) in some other forums. Can you guys recommend me any farily accessible bo... | Doug | 436,926 | <p>I was in a similar boat! I really liked Daniel Cunningham's <em>Set Theory: A First Course</em> (<a href="https://rads.stackoverflow.com/amzn/click/com/1107120322" rel="nofollow noreferrer" rel="nofollow noreferrer">https://www.amazon.com/Set-Theory-Cambridge-Mathematical-Textbooks/dp/1107120322</a>). The text sta... |
4,260,645 | <p>I can't find any insights online on how useful the graph of function <span class="math-container">$f(x)$</span> (on the <span class="math-container">$y$</span> axis) versus it's derivative <span class="math-container">$f'(x)$</span>? (on the <span class="math-container">$x$</span> axis) does it provide some useful i... | emacs drives me nuts | 746,312 | <p>There is this really good popular video from <em>MinutePhysics</em> titled <a href="https://youtu.be/54XLXg4fYsc" rel="nofollow noreferrer">How To Tell If We're Beating COVID-19</a>.</p>
<p>It plots "total number of cases" against "rate of change" for different countries and over time.</p>
<p>Eve... |
2,518,213 | <p>We work in the vector space $\mathbb{R}^k$ over $\mathbb{R}$. We define the (usual) infinite norm on $\mathbb{R}$:</p>
<p>$$\|x\|_{\infty} = \max\{|x_1|,|x_2|,\dots,|x_n|\}$$ where $x_1, x_2, \dots, x_n $ are the coordinates of of $x$ in $\mathbb{R}^k$. </p>
<p>We would like to prove the following fact:
$$\exists... | DanielWainfleet | 254,665 | <p>Suppoe no such $c$ exists. Then for every $r>0$ there exists $x_r$ such that $\|x_r\|_{\infty}>r\|x\|.$ </p>
<p>So for $n\in \Bbb N$ let $\|x_{n^2}\|_{\infty}>n^2\|x_{n^2}\|.$ This requires $\|x_{n^2}\| \ne 0.$ </p>
<p>Let $y_n=\frac {x_{n^2}}{n\|x_{n^2}\|}.$ $$\text { Then }\quad \|y_n\|=\frac {1}{n}$$ $... |
401,967 | <p>This question is about logical complexity of sentences in third order arithmetic. See <a href="https://en.wikipedia.org/wiki/Arithmetical_hierarchy" rel="nofollow noreferrer">Wikipedia</a> for the basic concepts.</p>
<p>Recall that the Continuum Hypothesis is a <span class="math-container">$\Sigma^2_1$</span> sente... | Elliot Glazer | 109,573 | <p>The Suslin hypothesis is <span class="math-container">$\Pi^2_2,$</span> and <span class="math-container">$T = ZFC + GCH + LC$</span> (LC an arbitrary large cardinal axiom) does not prove it to be equivalent to any <span class="math-container">$\Sigma^2_2$</span> sentence. Suppose toward contradiction <span class="ma... |
1,251,334 | <blockquote>
<p>$||x-2|-3| >1$, then $x$ belongs to:<BR> (a) $(-\infty, -2) \cup (0,4) \cup(6,\infty)$<br> (b) $(-1,1)$<br> (c) $(-\infty, 1)\cup (1,
> \infty)$ <br> (d) $(-2,2)$</p>
</blockquote>
<p><strong><em>Answer:(a)</em></strong></p>
<p>My solution:</p>
<p>let $|x-2| = p,$<br>
$|p-3|>1$ and finally... | Prahlad Vaidyanathan | 89,789 | <p>Just do it with two simultaneous inequalities:
$$
||x-2| - 3| > 1 \Leftrightarrow |x-2|-3 > 1 \text{ or } |x-2|-3 < -1
$$
$$
\Leftrightarrow |x-2| > 4 \text{ or } |x-2| < 2
$$
$$
\Leftrightarrow x-2 > 4 \text{ or } x-2 < -4 \text{ or } -2 < x-2 < 2
$$
$$
\Leftrightarrow x>6 \text{ or } ... |
1,315,922 | <blockquote>
<p>show that
$$\sum_{k=2}^{n}\left(\dfrac{2}{k}+\dfrac{H_{k}-\frac{2}{k}}{2^{k-1}}\right)\le 1+2\ln{n}$$where $ n\ge 2,H_{k}=1+\dfrac{1}{2}+\cdots+\dfrac{1}{k}$</p>
</blockquote>
<p>Maybe this $\ln{k}<H_{k}<1+\ln{k}$?</p>
| Henry | 106,067 | <p>It seems that it doesn't have anything to do with primes. </p>
<p>Let $0=r_0 < r_1 < \cdots <r_n$ be integers. I claim that $P_n(x)=\sum_{i=0}^n a_i x^{r_i}$ has at most $n$ positive zeros. </p>
<p>Let's proceed by induction. When $n=0$, it's obvious. </p>
<p>Suppose the statement holds for $n=k$. </p>
... |
108,890 | <p>Consider $V_{(n-1, 1)}$, the $n-1$ dimensional irreducible representation of $S_n$, i.e. the "standard" or "defining" representation. Is there a nice formula for how the $k$-th tensor power of $V_{(n-1, 1)}$ decomposes into irreps?</p>
| Bruce Westbury | 3,992 | <p>You want the "partition algebras". Some references to get you started are:</p>
<p>MR1317365 (97b:82023) Jones, V. F. R. The Potts model and the symmetric group.
Subfactors (Kyuzeso, 1993),
259--267, World Sci. Publ., River Edge, NJ, 1994. </p>
<p>MR1399030 (98g:05152) Martin, Paul . The structure of the par... |
440,534 | <p>Given two <em>origin symmetric convex</em> polytopes <span class="math-container">$P_1$</span> and <span class="math-container">$P_2$</span> (that is <span class="math-container">$P_i=-P_i$</span>) with the same edge-graph, but potentially of different dimensions and combinatorial types.
Let <span class="math-contai... | David E Speyer | 297 | <p><span class="math-container">$\def\RR{\mathbb{R}}$</span>
The answer is no! I will give an example of a centrally symmetric polytope in <span class="math-container">$\RR^4$</span> with <span class="math-container">$12$</span> vertices where there is a symmetry of the edge graph interchanging two non-antipodal vertic... |
2,458,613 | <p>Say that we have $k$ red balls and $n-k$ blacks balls for $n$ balls total. Then, say we partition the balls into equal sized groups of size $m$. What is the expected number of groups with a red ball?</p>
<p>It seems clear that I should use linearity of expectation of some sort. I tried calculating the probability t... | Marko Riedel | 44,883 | <p>If I understand this correctly we take one of the ${n\choose k}$
arrangements of these balls in a line and partition into groups of
size $m$ of consecutive balls starting at the left and then ask about
the expected number of groups containing at least one red ball.
Alternatively we may start t... |
3,183,931 | <p>I approached this way:</p>
<p>Having equal sides <span class="math-container">$1,2,3,4$</span> then the possible <span class="math-container">$3$</span> digit numbers are <span class="math-container">$111,112,221,222,223,...,448$</span></p>
<p>Then total number of <span class="math-container">$3$</span> digit numb... | Reinhard Meier | 407,833 | <p>I can only find <span class="math-container">$1+3+5+7 = 16$</span> triangles with equal sides <span class="math-container">$a$</span> and <span class="math-container">$b$</span> of lengths <span class="math-container">$1,\,2,\,3,\,4$</span> which respect the triangle inequality. From those <span class="math-containe... |
3,183,931 | <p>I approached this way:</p>
<p>Having equal sides <span class="math-container">$1,2,3,4$</span> then the possible <span class="math-container">$3$</span> digit numbers are <span class="math-container">$111,112,221,222,223,...,448$</span></p>
<p>Then total number of <span class="math-container">$3$</span> digit numb... | dan_fulea | 550,003 | <p>Using <a href="https://www.sagemath.org" rel="nofollow noreferrer">sage</a>, we have </p>
<pre><code>sage: R = [1..9]
sage: RRR = cartesian_product([R,R,R])
sage: S = [ (a,b,c) for (a,b,c) in RRR if len({a,b,c}) <= 2 and a < b+c and b < a+c and c < a+b ]
sage: T = [ 100*a + 10*b + c for (a,b,c) in S ]
s... |
3,267,110 | <p>This integral
<span class="math-container">$$\int_{-\infty}^{\infty}\frac{ a^2x^2 dx}{(x^2-b^2)^2+a^2x^2}=a\pi, ~ a, b \in \Re$$</span>
looks suspiciously interesting as it is independent of the parameter <span class="math-container">$b$</span>. The question is: What is the best way of proving or disproving this?</... | clathratus | 583,016 | <p>Glasser's master theorem states that for arbitrary constants <span class="math-container">$\alpha$</span>, <span class="math-container">$(\alpha_n)_{n=1}^{N}$</span>, <span class="math-container">$(\beta_n)_{n=1}^N$</span>, the function
<span class="math-container">$$\phi(x)=|\alpha|x-\sum_{n=1}^{N}\frac{|\alpha_n|... |
4,380,559 | <p><span class="math-container">$P$</span> is a <span class="math-container">$\mathbb{R}^{2\times2}$</span> positive semidefinite matrix satisfying <span class="math-container">$P=P^2$</span>, <span class="math-container">$P \neq 0$</span>, and <span class="math-container">$P \neq I$</span>. Show that
<span class="math... | Glorious Nathalie | 948,761 | <p>Since we're talking about positive semi-definite matrices, then it is implicitly assumed that <span class="math-container">$P$</span> is symmetric. So take</p>
<p><span class="math-container">$P = \begin{bmatrix} a && b \\ b && c \end{bmatrix} $</span></p>
<p>From <span class="math-container">$P = P... |
292,176 | <p>I'm having a difficult time finding any theory on an inverse problem I've come up against. Let's say I have an unknown function $f:[0,1] \rightarrow \mathbb{R}$, and I know $\int_{a}^{b} f$ for some collection $A$ of pairs $(a,b)\in[0,1]^2$. I'm looking for pointers to any material that discusses condtions on $f$... | Iosif Pinelis | 36,721 | <p>In general, it appears that hardly anything interesting can be said. E.g., let $A=\{(1/5,3/5),(2/5,4/5)\}$; here, it will be convenient to think of $A$ as a set of (say) open intervals, rather than a set of pairs of endpoints of intervals. </p>
<p>However, note first that, without loss of generality, for each open ... |
292,176 | <p>I'm having a difficult time finding any theory on an inverse problem I've come up against. Let's say I have an unknown function $f:[0,1] \rightarrow \mathbb{R}$, and I know $\int_{a}^{b} f$ for some collection $A$ of pairs $(a,b)\in[0,1]^2$. I'm looking for pointers to any material that discusses condtions on $f$... | Aaron Meyerowitz | 8,008 | <p>If $A$ is infinite and dense then you can find the function exactly by taking limits. I'll look at the finite case. In the event that the function is a polynomial of degree $n$ then knowing the integral over $n+1$ internally disjoint intervals is enough for recovery. Without knowing that the function is a polynomi... |
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