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 |
|---|---|---|---|---|
3,032,950 | <p>I have the following Cauchy problem. I do not know where to start, so I would appreciate any help and tips.
<span class="math-container">$$\frac{\partial^2 Y(t, x)}{\partial t^2} = 9\frac{\partial^2 Y(t,x)}{\partial x^2} - 2Z(t,x)$$</span>
<span class="math-container">$$\frac{\partial^2 Z(t, x)}{\partial t^2} = 6\fr... | Frits Veerman | 273,748 | <p>Your equation is linear, so you could try expanding the solutions in a Fourier series. Moreover, as the initial condition is given in terms of <span class="math-container">$\cos(x)$</span>, it's a good idea to write
<span class="math-container">\begin{align}
Y(t,x) &= \sum_{k=0}^\infty a_Y(k,t) \cos(k x) + b_Y(... |
879,640 | <p>Does a matrix have only one inverse matrix (like the inverse of an element in a field)? If so, does this mean that</p>
<p>$A,B \text{ have the same inverse matrix} \iff A=B$?</p>
| vadim123 | 73,324 | <p>Specific counterexample to the non-square case: Let $A=\left(\begin{smallmatrix}1&0&0\\0&1&0\end{smallmatrix}\right)$. </p>
<p>Then there is a matrix $B$ such that $AB=I$, namely $B=\left(\begin{smallmatrix}1&0\\0&1\\x&y\end{smallmatrix}\right)$. Note that $x,y$ can each be anything, s... |
902,653 | <p>I have recently been learning some hyperbolic geometry and the professor briefly mentioned spherical geometry. From a modern, naive point of view, it seems quite easy to show that spherical geometry is an example of non euclidean geometry.</p>
<p>Clearly, this is not true since it took over a 1000 years to do this.... | Mikhail Katz | 72,694 | <p>The key modern insight here is the distinction between axioms and models, in other words between syntax and semantics. Euclidean axioms were once thought to characterize a unique <em>space</em> whatever that may be. The idea that truth of a proposition may be relative to a model is a revolutionary idea that we take ... |
1,598,545 | <p>Maybe I am not well versed with the actual definition of mean, but I have a doubt. On most resources, people say that arithmetic mean is the sum of $n$ observations divided by n. So my first question: </p>
<blockquote>
<p>How does this formula work? Is there any derivation to it? If not,
then while creating thi... | Aloizio Macedo | 59,234 | <p>Let me give the bureaucratic answer first:</p>
<p>No, there is no "derivation" (supposing that you mean something like a "theorem") that concludes that $E(X)$ is what it is. $E(X)$ is <em>defined</em> that way.</p>
<p>Now to properly answer your question:</p>
<p>At a very naive level, you can think of the mean as... |
1,726,187 | <p>Recently in class our teacher told us about the evaluating of the sum of reciprocals of squares, that is $\sum_{n=1}^{\infty}\frac{1}{n^2}$. We began with proving that $\sum_{n=1}^{\infty}\frac{1}{n^2}<2$ by induction. However, we actually proved a stronger result, namely that $\sum_{n=1}^{\infty}\frac{1}{n^2}<... | Mark Viola | 218,419 | <p>To arrive at the result without induction, we note that (<a href="https://en.wikipedia.org/wiki/Integral_test_for_convergence#Proof" rel="nofollow">See this for a proof</a>) an upper bound for the sum is given by</p>
<p>$$\begin{align}
\sum_{n=1}^N\frac{1}{n^2}&\le 1+\int_1^N\frac{1}{x^2}\,dx\\\\
&=2-\frac1... |
2,721,836 | <p>I recently found a different method to compute prime number in $\mathcal O(\log(\log n))$ complexity. At present, that logic working fine for $300$ digits prime number, which I found on websites.I need to validate whether that logic will be working fine for a higher number of digits. At present, I have computed a pr... | Arnaud Mortier | 480,423 | <p>From what you write I understand that you want to prove that your method is fine. It probably is if you checked up to 300, digits. But the only way to validate a method is by analysing it step by step and actually <em>prove</em> that it works. No matter how many digits you try, that will not be a proof that your met... |
18,960 | <p>I am making this post in regards to the ongoing delete/undelete skirmish (let's at least change the monotonicity of the use of "war"). The old version of the question is <a href="https://math.stackexchange.com/revisions/172652/3">here</a>, the current version (after edits today) <a href="https://math.stackexchange.c... | Jyrki Lahtonen | 11,619 | <p>For my part I will add the following hopefully clarifying comments:</p>
<ul>
<li>The question showed no effort, and was a wrapper for five separate although closely related questions. I believe that the community at large disproves of questions of this type.</li>
<li>IMO the question was sufficiently non-trivial t... |
920,782 | <p>How do I find the number of integral solutions to the equation - </p>
<p>$$2x_1 + 2x_2 + \cdots + 2x_6 + x_7 = N$$</p>
<p>$$x_1,x_2,\ldots,x_7 \ge 1$$</p>
<p>I just thought that I should reduce this a bit more, so I replace $x_i$ with $(y_i+1)$, so we have:</p>
<p>$$y_1 + y_2 + \cdots + y_6 = \tfrac{1}{2}(N + 13... | Khosrotash | 104,171 | <p>$$x_{7}=2k+1 , x_{7}=2k \\ partition - by - x_{7}\\2x_{1}+2x_{2}+2x_{3}+...+x_{7}=N\\(1) x_{7}=2k+1 , 2x_{1}+2x_{2}+2x_{3}+...+2k+1=N\\2x_{1}+2x_{2}+2x_{3}+...+2K=N-1\\if -(N-1) - was- even -divide - by -2\\x_{1}+x_{2}+x_{3}+...+K=\frac{N-1}{2}\\\binom{\frac{N-1}{2}-1}{7-1}\\ \\(2) x_{7}=2k , 2x_{1}+2x_{2}+2x... |
813,716 | <p>I am supposed to calculate the following as simple as possible.</p>
<p>Calcute:
$$1 + 101 + 101^2 + 101^3 + 101^4 + 101^5 + 101^6 + 101^7$$
Tip: $$ 101^8 = 10828567056280801$$</p>
<p>I have absolutely no idea how this tip is supposed to help me.<br>
Do I still have to calculate each potency?<br>
Can I somehow solv... | lhf | 589 | <p><em>Hint:</em> Let $S=1 + 101 + 101^2 + 101^3 + 101^4 + 101^5 + 101^6 + 101^7$ and consider $101S$.</p>
|
337,930 | <p>Given two polynomials</p>
<p>$$
p(x) = a_0 + a_1 x + a_2 x^2 + \ldots + a_{n-1}x^{n-1} \\
q(x) = b_0 + b_1 x + b_2 x^2 + \ldots + b_{n}x^{n}
$$</p>
<p>And the series expansion from their rational polynomial</p>
<p>$$
\frac{p(x)}{q(x)} = c_0 + c_1 x + c_2 x^2 + \ldots
$$</p>
<p>is it possible to recover the the o... | riemann_lebesgue | 539,416 | <p>Here's a different way of doing it! Unfortunately I don't really know how to use latex, so here is the outline</p>
<p>Using the residue theorem, we know that <span class="math-container">${n \choose k}$</span> equals the contour integral of </p>
<p><span class="math-container">$(1+z)^N / z^{k+1}) {/}(2*pi*i)$</sp... |
622,076 | <p>Continuity $\Rightarrow$ Intermediate Value Property. Why is the opposite not true? </p>
<p>It seems to me like they are equal definitions in a way. </p>
<p>Can you give me a counter-example? </p>
<p>Thanks</p>
| Andrés E. Caicedo | 462 | <p><strong>I.</strong></p>
<p>Some of the answers reveal a confusion, so let me start with the definition. If $I$ is an interval, and $f:I\to\mathbb R$, we say that $f$ has the <em>intermediate value property</em> iff whenever $a<b$ are points of $I$, if $c$ is between $f(a)$ and $f(b)$, then there is a $d$ <em>bet... |
3,009,345 | <p>I ran into this question which hints me to use Cauchy's Integral Theorem for Derivatives, however I don't seem to be able to fit this integral into the form of the Integral Formula</p>
<p><span class="math-container">$$\displaystyle \int_{|z|=2} \frac{\cos(z)}{z(z^2+8)}dz$$</span> I tried using the fact that <span ... | Felix Marin | 85,343 | <p><span class="math-container">$\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\exp... |
871,542 | <p>I have the following theorem:</p>
<blockquote>
<p>Let <span class="math-container">$\rho$</span> be the traffic intensity.</p>
<p>a) If <span class="math-container">$\rho<1$</span>, then <span class="math-container">$X$</span> is positive recurrent.</p>
<p>b) If <span class="math-container">$\rho>1$</span>, t... | Community | -1 | <p>I disagree with <a href="https://math.stackexchange.com/users/88052/mark-fantini">@MarkFantini</a>'s <a href="https://math.stackexchange.com/a/871548">answer</a>.</p>
<p>In this section (<span class="math-container">$\S$</span>1.II.2 <em>Moving Particle Argument</em>), Needham is giving a heuristic argument in suppo... |
1,499,423 | <p>Are there a group of numbers whose squares are made up of squares? For example, $7$ would be one because $7^2$ is $49$ which has $2^2$ and $3^2$. $20$ would be another example.</p>
<p>What are these numbers called?</p>
<p>Please help me find good <strong>tags</strong> for this question.</p>
| Piquito | 219,998 | <p>I am not agree with taking $20$ as a number of that kind ( or trivial in whose case $10, 30,40,50,60,70,80,90$ are too). </p>
<p>We show here that the only two-digit number with four-digit square of this kind is $41$. Looking at the algorithm of extraction of the square root in the figure below we can write</p>
<p... |
1,262,305 | <p>$$f(x)=\int_0^x\left(\frac{t^3-2t^2-4}{t^2+1}\right)\ dt$$
I need to find the $x$ and $y$ intercepts, and the inflection points of the function $f(x)$ (with both $x$ and $y$ coordinates). I need to find it through the calculator and explain my answer. How do I find the antiderivative?</p>
| mathreadler | 213,607 | <p>So you already have how to calculate the integral explicitly from Olivier. However you can find the maximums and minimums with the fundamental theorem of calculus together with finding zeros to the polynomial $t^3 - 2t^2 - 4$. At least one $t$ root is easy to guess and then continue with polynomial division and comp... |
1,598,006 | <p>(Here, $B$ is relatively compact means the closure of $B$ is compact.)</p>
<blockquote>
<ol>
<li><p>$\hat A$ is compact.</p></li>
<li><p>$\hat A=\hat {\hat A}$.</p></li>
<li><p>$\hat A$ is connected.</p></li>
<li><p>$\hat A=X$.</p></li>
</ol>
</blockquote>
<p>I try to eliminate the options by using an ... | Forever Mozart | 21,137 | <p>I think $\hat A=A$ for your example. The closure in your $X$ of $(1,2)$ equals $(1,2)$, which is not compact, and similarly for $(2,3)$. But you can still mark off 1 and 4 because of this.</p>
<p>How about $X=\{1\}\cup \{2\}$ and $A=\{1\}$. Then $\hat A=X$, which is not connected. So you can rule out option 3.</p>... |
1,341,486 | <p>Problem:
Find the sum to $n$ terms of
\begin{eqnarray*}
\frac{1}{1\cdot 2\cdot 3} + \frac{3}{2\cdot 3\cdot 4} + \frac{5}{3\cdot 4\cdot 5} +
\frac{7}{4\cdot 5\cdot 6}+\cdots \\
\end{eqnarray*}
Answer:
The way I see it, the problem is asking me to find this series:
\begin{eqnarray*}
S_n &=& \sum_{i=1}^... | Masacroso | 173,262 | <p>You can write it as $\sum_{k\ge0}\frac{2k+1}{(k+3)_3}=\sum_{k\ge0}(2k+1)(k)_{-3}$ and now it seems easy to solve by summation by parts:</p>
<p>$$\sum (2k+1)(k)_{-3}\delta k=(2k+1)\frac{(k)_{-2}}{-2}+\sum(k+1)_{-2}=\frac{2k+1}{-2(k+2)_2}-\frac{1}{k+2}=\frac{4k+3}{-2(k+2)_2}$$</p>
<p>And taking limits we have that t... |
872,657 | <p>For $1 \leq r < p < \infty$ prove the continuous injection of $L^p([0, 1])$ into $L^r([0, 1])$. </p>
<p>I am having a hard time starting. Any suggestions. I tried a straight forward approach. That is, given $\epsilon > 0$, I tried to find a $\delta >0$ such that $||f - g||_p < \delta$ implies tha... | Vítězslav Štembera | 663,062 | <p>Interesting! The fulltext of the paper is here:</p>
<p><a href="https://cowles.yale.edu/sites/default/files/files/pub/cdp/m-0403.pdf" rel="nofollow noreferrer">https://cowles.yale.edu/sites/default/files/files/pub/cdp/m-0403.pdf</a></p>
|
1,876,287 | <p><strong>Question:</strong></p>
<p>Let P be a point where the normal (in the point where the x-coordinate is h) to the curve</p>
<p>$$y = e^{2x} - 2x$$</p>
<p>cuts the y-axis. Determine the y-coordinates of P when h goes to 0.</p>
<p><strong>Attempted solution:</strong></p>
<p>I first decided to draw the followi... | Jean Marie | 305,862 | <p>@Claude Leibovici @Ahmed Hussein </p>
<p>There is an alternative answer. </p>
<p>Let us first recall that the envelope of the normals to a curve (called its <a href="https://en.wikipedia.org/wiki/Evolute" rel="nofollow noreferrer">evolute</a>) is the locus of the centres of curvature for this curve.</p>
<p>Here, ... |
2,481,046 | <p>I have a question that asks to show that $S^2 = \{(x,y,z) \in \mathbb{R}^3|x^2+y^2+z^2=1\}$ is a differentiable manifold. My professor says that one way to do this is to define the following 6 parametrizations of the sphere, which cover the entire sphere.</p>
<p>$\vec{\phi_{i}}:V \to \mathbb{R}^3$ where $V = \{(u,v... | Fred | 380,717 | <p>Convergence: Since $ \int_0^1 1 dx$ converges and $|\sin(x+1/x)| \le 1$, the integral $\int_{0}^{1} \sin(x+\frac{1}{x})dx$ converges absolutely by the comparison test.</p>
|
891,575 | <p>The circumference of a circle has length 90 centimeters, Three points on the circle divide the circle into three equal lengths. Three ants A, B, and C start to crawl clockwise on the circle, with starting from one of the three points. Initially A is ahead of B and B is ahead of C. Ant A crawls 3 centimeters per seco... | paw88789 | 147,810 | <p>First focus on A and B. Since B goes 2 cm/sec faster than A, B first catches A after 15 seconds; and then every 45 seconds thereafter. </p>
<p>Now look at B and C. Since C goes 5 cm/sec faster than B, C first catches B after 6 seconds, and every 18 seconds there after.</p>
<p>So A and B coincide at times $15 + ... |
2,694,740 | <p>$$\frac{2.10^{-7} - 0,4.10^{-6}}{10^{-8}} = ? $$</p>
<p>These questions are making me confused because we're dealing with the terms like $10^x$. What are your professional tips? </p>
<p><strong>My attempt:</strong></p>
<p>$$\frac{2.10^{-7} - 4.10^{-7}}{10^{-8}} \tag{1} $$
$$\frac{ -8.10^{-7}}{10^{-8}} \tag{2} $$<... | CY Aries | 268,334 | <p>$(2)$ is incorrect.</p>
<p>From $(1)$, $\displaystyle \frac{2\cdot10^{-7} - 4\cdot10^{-7}}{10^{-8}}=\frac{(2-4)\cdot 10^{-7}}{10^{-8}}=\frac{-2\cdot 10^{-7}\cdot 10^8}{10^{-8}\cdot 10^8}=\frac{-2\cdot 10}{1}=-20$</p>
|
2,012,532 | <p>The following is all confirmed to be true:</p>
<p>Matrix A =
$
\begin{bmatrix}
0 & 1 & -2 \\
-1 & 2 & -1 \\
2 & -4 & 3 \\
1 & -3 & 2 \\
\end{bmatrix}
$</p>
<p>U =
$
\begin{bmatrix}
-1 & 2 & -1 \\
0 &am... | Siong Thye Goh | 306,553 | <p>Multiplying the first row of $L$ with the first column of $U$ gives us $-1$, which is not the $(1,1)$ entry of $A$. Hence, you have made a mistake in computation of LU decomposition. </p>
<p>There seems to be a missing permutation matrix being involved in your computation.</p>
<p>Your procedure to solve the linear... |
87,466 | <p>I have some code which involves tiny numbers being put to the power of very large numbers. The function I'm looking at is</p>
<p>$\varphi = \omega(T) \left(1 - (1 - \epsilon)^{n_{e}(T)} \right)$</p>
<p>when $\epsilon $ is very small (~$10^{-16}$) and $n_{e}$ is large ($> 10^{10}$). Both $n_{e}$ and $\omega$ are... | Bob Hanlon | 9,362 | <p>You can use <code>Rationalize</code> to convert numbers to exact numbers</p>
<pre><code>gammaex = 0.2506 // Rationalize[#, 0] &;
omega[t_] =
2.43163218375*10^7*Exp[1700*(1/298.15 - 1/(273.15 + t))] //
Rationalize[#, 0] &;
w[t_] = (3.414105049212413*10^12)/(omega[t]) // Rationalize[#, 0] &;
v[t_]... |
362,062 | <p>If you didn't know anything about stabilization phenomena in algebraic topology and were trying to discover/prove theorems about the homotopy theory of spaces, what clues would point you toward results such as Freudenthal suspension or the existence of stable homotopy groups of spheres? </p>
<p>References suggest t... | Nicholas Kuhn | 102,519 | <p>For those whose German is shaky or non-existent, it is fun to copy and paste a couple of paragraphs of Freudenthal's paper into Google translate. The answer to your question emerges. His paper is concerned with the interplay of the then new Hopf invariant and "suspension" - "Einhängung" in Germ... |
362,062 | <p>If you didn't know anything about stabilization phenomena in algebraic topology and were trying to discover/prove theorems about the homotopy theory of spaces, what clues would point you toward results such as Freudenthal suspension or the existence of stable homotopy groups of spheres? </p>
<p>References suggest t... | Arun Debray | 97,265 | <p><span class="math-container">$\newcommand{\R}{\mathbb R}\newcommand{\inj}{\hookrightarrow}$</span>This will be an anachronistic answer, because it was discovered a bit later, but: Whitney proved that every
<span class="math-container">$n$</span>-manifold embeds in <span class="math-container">$\R^N$</span> for <span... |
345,735 | <p>If <strong>two planes</strong> are <strong>intersected</strong> <em>by making a straight line, like <span class="math-container">$AB$</span></em> then</p>
<blockquote>
<p>Does the angle between two planes (see figure) <strong>always</strong> given by the
angle between normal vectors (<span class="math-container">$n_... | Phil Wang | 67,822 | <p>In your figure, thing is right. If you reverse vector n1, the angle between two planes plus that between two normal vectors will be 2pi. There will be some difference.</p>
|
3,518,221 | <p>So I had this complex integral </p>
<blockquote>
<p>If <span class="math-container">$0 \leq y \leq 1$</span>, find the maximum value of the integral
<span class="math-container">$$
\int_0^y \left(x^4 + (y-y^2) \right)^{1/2}\, dx
$$</span></p>
</blockquote>
<p>I differentiated the integral using the leibniz rul... | jacky | 14,096 | <p>I am assuming that you mean <span class="math-container">$\max$</span> of</p>
<p><span class="math-container">$$\displaystyle \int^{y}_{0}\sqrt{x^4+(y-y^2)^2}dx$$</span> subjected to <span class="math-container">$0 \leq y\leq 1$</span></p>
<p><span class="math-container">$\bullet\; $</span> From <span class="math-... |
3,537,654 | <p><span class="math-container">$$\lim_{x\to 0^{+}} (\tan x)^x$$</span></p>
<p><span class="math-container">$$\lim_{x\to 0^{+}} e^{\ln((\tan x)^x)}=\lim_{x\to 0^{+}} e^{x\ln(\tan x)}=\lim_{x\to 0^{+}} e^{x[\ln(\sin x)-\ln(\cos x)]}$$</span></p>
<p>We can continue to create an expression that may help us use L'Hospita... | Kavi Rama Murthy | 142,385 | <p><span class="math-container">$\lim_{x \to 0+} x \ln (\tan x)=\lim_{x \to 0+} \frac {\ln (\tan x)} {1/x}=-\lim_{x \to 0+} \frac {\sec^{2}x} {\tan x /x^{2}}$</span>. You can write this as <span class="math-container">$-\lim_{x \to 0+} \frac {x^{2}} { \sin x \cos x}$</span> and this limit is <span class="math-container... |
2,453,126 | <p>I'm at a loss here. I tried everything I could think of, can't seem to get the correct answer after plugging in the new boundaries (the teacher wants us to use the new boundaries instead of the ones given). How do I find where the radical values of $\sin(\arctan(x/3))$? I am supposed to evaluate the integral using t... | Michael Rozenberg | 190,319 | <p>Let $AB=1$.</p>
<p>Thus, by law if sines for $\Delta FEB$ we obtain:
$$\frac{\sin\measuredangle BFE}{BE}=\frac{\sin\measuredangle FBE}{FE}$$ or
$$\frac{\sin\measuredangle BFE}{2\sin54^{\circ}}=\frac{\sin18^{\circ}}{1}.$$
Id est,
$$\sin\measuredangle BFE=2\sin54^{\circ}\sin18^{\circ}=2\cos36^{\circ}\cos72^{\circ}=... |
2,453,126 | <p>I'm at a loss here. I tried everything I could think of, can't seem to get the correct answer after plugging in the new boundaries (the teacher wants us to use the new boundaries instead of the ones given). How do I find where the radical values of $\sin(\arctan(x/3))$? I am supposed to evaluate the integral using t... | Edward Porcella | 403,946 | <p><a href="https://i.stack.imgur.com/GVdg7.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/GVdg7.jpg" alt="angle outside pentagon=30^o"></a>Using only elementary geometry, in the given figure, join $EB$. From the nature of the regular pentagon, $\angle ABE=36^o$. Through $E$ draw $EH$ perpendicular ... |
556,054 | <p>Please help me to prove the inequality
$$
\sqrt{a^2 + b^2} \geq \frac{|a-b|}{\sqrt{2}}.
$$</p>
| Community | -1 | <p><strong>Hint</strong>: Square both sides and multiply by $2$, and you'll find that this is equivalent to proving that</p>
<p>$$2(a^2 + b^2) \ge |a - b|^2 = (a - b)^2$$</p>
|
2,208,755 | <p>I got stuck on this question: find all solutions $x$ for $a\in R$:</p>
<p>$$\frac{(x^2-x+1)^3}{x^2(x-1)^2}=\frac{(a^2-a+1)^3}{a^2(a-1)^2}$$</p>
<p>I see that if we simplify we get:</p>
<p>$$\frac{(x^2-x+1)^3}{x^2(x-1)^2}=\frac{[(x-{\frac 12})^2+{\frac 34}]^3}{[(x-{\frac 12})^2-{\frac 14}]^2}$$</p>
<p>From the ex... | Dr. Sonnhard Graubner | 175,066 | <p>factorizing the given equation and cancel the denominators we get
$$(ax-a-x)(ax-x+1)(x-1+a)(-x+a)(ax-1)(ax-a+1)=0$$</p>
|
4,080,385 | <p>Would you please compute the behavior of the following composed generalized function?</p>
<p><span class="math-container">$g(t) = $</span> <span class="math-container">$\delta(e^t)$</span></p>
<p><strong>Is it even a valid generalized function?</strong></p>
<p>Thank you very much for your time.</p>
| Mohammad Ali Dastgheib | 559,629 | <p>Thank you, everyone. I think I am ready now to write the answer to my question. Let me write a synopsis first:</p>
<p>In Distribution Theory (or theory of Generalized Functions), a generalized function is considered not a function in itself but only in relation to how it affects other functions when "integrated... |
78,641 | <p>I am interested in the relation between the property of countable chain condition (ccc) and the property of separable. Could someone recommend some papers or books about this to me? thanks in advance.</p>
| Nathan | 18,662 | <p>Lest there be any question on this point, ZFC alone implies the existence of some non-separable ccc spaces. The tables in the back of Counterexamples in Topology list four of them. Probably the most interesting is #63, the Countable Complement Extension Topology. This is the minimal extension of the Euclidean topolo... |
3,964,429 | <p>Zeckendorf : <em>Every positive integer N can be expressed uniquely as a sum of distinct non-consecutive Fibonacci numbers</em></p>
<p>I was wondering if this theorem can be applied with the extended Fibonacci numbers, and especially I am looking for a way to <strong>find the Zeckendorf-like representation of <span ... | wendy.krieger | 78,024 | <p>The negative fibbonacci numbers run as F(-n)=F(n) for 2|n and F(-n)=-F(n) for n odd.</p>
<p>So we get for fibonacci numbers</p>
<pre><code> -21 13 -8 5 -3 2 -1 1
-1
-3 1
-3
-3 -1
-8 2 1
... |
4,206,039 | <p>Find the radius of convergence of the following power series <span class="math-container">$$\sum_{n=1}^\infty \frac{(-1)^n z^{n(n+1)}}{n}$$</span></p>
<p>Here's my working
<span class="math-container">$$\lim_{n\to \infty}| \frac{(-1)^{n+1} z^{(n+1)(n+2)}}{n+1} \frac{n}{(-1)^nz^{n(n+1)}}|$$</span>
<span class="math... | Oliver Díaz | 121,671 | <p>The series in the posting can be expressed as
<span class="math-container">$$
\sum^\infty_{n=1}\frac{(-1)^n}{n} z^{n(n+1)}=\sum^\infty_{m=0}a_mz^m$$</span></p>
<p>where <span class="math-container">$a_m=0$</span> unless <span class="math-container">$m\in\{n(n+1):n\in\mathbb{N}\}$</span>, in which case <span class="m... |
2,624,498 | <p>Evaluate $$\lim_{n \rightarrow\infty} \sqrt[n]{3^{n} +5^{n}}$$</p>
<p>Attempt:</p>
<p>The only sort of manipulation that has come to mind is: $$e^{\frac{1}{n}ln(e^{n\ln(3)} + e^{n\ln(5)})}$$</p>
<p>So what is the trick to successfully evaluate this?</p>
| marty cohen | 13,079 | <p>By Bernoulli's inequality,
if $x > 0$
and $n \ge 1$,
$(1+x)^n
\ge 1+nx$.</p>
<p>Therefore
$(1+x/n)^n
\ge 1+x$
so that
$(1+x)^{1/n}
\le 1+x/n
$.</p>
<p>Therefore,
if $0 < a < b$
then
$\sqrt[n]{a^n+b^n}
=b\sqrt[n]{1+(a/b)^n}
\le b(1+\frac{(a/b)^n}{n})
= b+\frac{b(a/b)^n}{n}
\lt b+\frac{b}{n}
$
since
$a/b ... |
204,612 | <blockquote>
<p>Is it possible to verify the following <code>lhs,rhs</code> involving the sums
are equal, with Mathematica?</p>
</blockquote>
<p>I can verify it for individual values of <span class="math-container">$d$</span> variable:</p>
<pre><code>ClearAll[d, q, h, eq1, eq2, x, lhs, rhs];
eq1[d_: d, q_: q, h_:... | Chris K | 6,358 | <p>The function is <a href="https://reference.wolfram.com/language/ref/Expectation.html" rel="nofollow noreferrer"><code>Expectation</code></a> not <code>ExpectedValue</code>. Unfortunately,</p>
<pre><code>Expectation[b*x*(1 + ω*x^ρ)^κ, x \[Distributed] LogNormalDistribution[μ, σ]]
</code></pre>
<p>does not yield an... |
204,612 | <blockquote>
<p>Is it possible to verify the following <code>lhs,rhs</code> involving the sums
are equal, with Mathematica?</p>
</blockquote>
<p>I can verify it for individual values of <span class="math-container">$d$</span> variable:</p>
<pre><code>ClearAll[d, q, h, eq1, eq2, x, lhs, rhs];
eq1[d_: d, q_: q, h_:... | mikado | 36,788 | <p>The expression whose expectation you seek can be expanded as a power series in <code>x</code>. (You might need to worry about whether this converges).</p>
<pre><code>expr = b*x*(1 + ω*x^ρ)^κ;
coeff = Assuming[κ > 0 && n >= 0,
SeriesCoefficient[expr /. x^ρ -> z, {z, 0, n}]];
</code></pre>
<p>W... |
3,442,862 | <blockquote>
<p>Let <span class="math-container">$F $</span> be a subset family of the set {<span class="math-container">$ 1, 2, ..., 2017 $</span>} such that
for any <span class="math-container">$ A, B \in F $</span>, worth that <span class="math-container">$A \cap B$</span> has exactly one
element. Determine a... | oshill | 651,041 | <p>How is your linear algebra? It is saying we are constructing vectors in a 2017-dimensional space. I will give the example in <span class="math-container">$3$</span>-D. Let <span class="math-container">$\{1,2,3\}$</span> be the base set, so the subset <span class="math-container">$\{1,3\}$</span> maps to the vector ... |
1,028,371 | <p>I have been trying to prove this, but I am having trouble understanding how to prove the following mapping I found is injective and surjective. Just as a side note, I am trying to show the complex ring is isomorphic to special $2\times2$ matrices in regard to matrix multiplication and addition. Showing these hold is... | AlexR | 86,940 | <p>You could prove injectivity simply by showing
$$z\ne w \Rightarrow \phi(z) \ne \phi(w)$$
wich is next to obvious.<br>
Regarding surjectivity: A function is always surjective on its range. All you need to show here is that any Matrix $\pmatrix{a&-b\\b&a}$ is in the range of $\phi$, explicitly $a+bi$ is the re... |
448 | <p>Let's say, I have 4 yellow and 5 blue balls. How do I calculate in how many different orders I can place them? And what if I also have 3 red balls?</p>
| bryn | 106 | <p>For some reason I find it easier to think in terms of letters of a word being rearranged, and your problem is equivalent to asking how many permutations there are of the word YYYYBBBBB. </p>
<p>The formula for counting permutations of words with repeated letters (whose reasoning has been described by Noldorin) give... |
2,677,584 | <p>I have the following question:</p>
<blockquote>
<p>Find the real values of $a$ for which the equation
$$(1+\tan^2\theta)^2 + 4a\tan\theta(\tan^2\theta + 1) + 16\tan^2\theta = 0$$
has four distinct real roots in $\left(0, \dfrac{\pi}{2}\right)$.</p>
</blockquote>
<p>I tried to solve the above equation by div... | giuseppe mancò | 191,154 | <p>The four roots are:</p>
<p><span class="math-container">$tan(\theta_{1})=-a+\sqrt{a^{2}-4}-\sqrt{-5+2a^{2}-2a\sqrt{a^2-4}}$</span></p>
<p><span class="math-container">$tan(\theta_{2})=-a+\sqrt{a^{2}-4}+\sqrt{-5+2a^{2}-2a\sqrt{a^2-4}}$</span></p>
<p><span class="math-container">$tan(\theta_{3})=-a-\sqrt{a^{2}-4}-\sqr... |
1,984,178 | <p>I have a problem with the following exercise:</p>
<p>We have the operator $T: l^1 \to l^1$ given by</p>
<p>$$T(x_1,x_2,x_3,\dots)=\left(\left(1-\frac11\right)x_1, \left(1-\frac12\right)x_2, \dots\right)$$ for $(x_1,x_2,x_3,\dots)$ in $l^1$. Showing that this operator is bounded is easy, but I am really desperate... | J.R. | 44,389 | <p>First observe that $\|T\|\le 1$.</p>
<p>Fix a large $n$ and let $x_n=(\dots,0,1,0,\dots)$ with $1$ exactly at the $n$th position and $0$ everywhere else. We have $Tx_n = (1-\frac1n) x_n$, $\|x_n\|=1$ and $\|Tx_n\| = 1-\frac1n.$
This shows $\|T\|\ge 1- \frac1n$. Now let $n\to\infty$ to conclude $\|T\|\ge 1$.</p>
|
1,662,226 | <p>Find a sufficient statistic for $σ^2$ with $μ$ known, where $X_i$ is a random sample from $N(μ,σ^2)$</p>
<p>I was able to find a sufficient statistic for $μ$ with $σ^2$ known, but I'm stuck on finding one for $σ^2$ when $μ$ is known. Can anyone give me some help? </p>
<p>I was using the factorization method before... | dbanet | 220,258 | <p>This absolutely is the worst way to solve this problem, but I though it might be useful to you, or at least interesting (I bet ability to evaluate things numerically on paper <em>is</em> something interesting).</p>
<p>From \begin{align}\sqrt{0.016}&=\sqrt{16\cdot10^{-3}}=4\sqrt{10^{-1}10^{-2}}=4\cdot10^{-1}\sqr... |
3,403,855 | <p>Construct an example of a set of real numbers E that has no points of accumulation and yet has the property
that for every ε > 0 there exist points x, y ∈ E so that 0 < |x − y| < ε.</p>
<p>so i know we need a convergent sequence to show that the difference between two elements can be as small as we like, also... | pancini | 252,495 | <p>Consider</p>
<p><span class="math-container">$$E=\left\{\sum_{k=1}^n k^{-1}:n\in \Bbb N\right\}.$$</span></p>
|
1,200,919 | <p>Let $x$ be the solution of the equation $x^x=2$. Is $x$ irrational? How to prove this?</p>
| Thomas Andrews | 7,933 | <p>If $x$ is rational, $p/q$, then $2^{q/p}$ is rational. That's only possible if $p=1$ and $p/q$ is an integer.</p>
<p>A quick way to write it, assuming $p,q$ are relatively prime:</p>
<p>$$\left(\frac pq\right)^{p/q}=2\implies p^p=2^qq^p$$</p>
<p>But $p$ and $q$ are relatively prime, using unique factorization, we... |
1,384,053 | <p>Which number is bigger? $1.01^{101}$ or $2$? and how about $e^{\pi}$ or $\pi^e$?</p>
<p>Tried some algebraic manipulations to no end, so would love some suggestions or some different ways to approach those kind of problem</p>
| Khosrotash | 104,171 | <p>$$x \to 0 , (1+x)^n \approx 1+nx \\(1.01)^101=(1+\frac{1}{100})^101 \approx 1+ 101(\frac{1}{100}) >2$$ for the second one
Consider this function $$x^{\frac{1}{x}}$$.
$f'=x^{\frac{1}{x}}(\frac{1}{x^2})(1-\ln x)$,
function has global maximum at $x=e$.</p>
<p>so $e^{\frac{1}{e}} \geq \pi^{\frac{1}{\pi}} \t... |
2,436,268 | <p>My problem is evaluating the following limit:
$$\lim_{(x,y)\to(0,0)}\frac{x^5+y^2}{x^4+|y|}$$
The answer should be 0. I tried to convert the limit into polar form, but it didn't help because I couldn't isolate the $r$ and $\theta$-variables of the expression. My "toolbox" for solving problems like these is very limi... | Peter Szilas | 408,605 | <p>$0\le |\dfrac{x^5+y^2}{x^4 +|y|}| \le$</p>
<p>$\dfrac{|x^5| +|y^2|}{|x^4+|y||} \le$</p>
<p>$\dfrac{|x^5|}{x^4} + \dfrac{y^2}{|y|} =$</p>
<p>$|x| + |y|.$</p>
<p>And now?</p>
|
3,549,072 | <p>The following are given</p>
<p><span class="math-container">$$ \lim_{x \to \infty}{\log(x)} = \infty$$</span></p>
<p><span class="math-container">$$ \lim_{x \to \infty}{\cosh(x)} = \infty$$</span></p>
<p><span class="math-container">$$ \lim_{x \to \infty}{\sinh(x)} = \infty$$</span></p>
<p><span class="math-cont... | Axion004 | 258,202 | <p><span class="math-container">\begin{align}\frac{\tanh (x)-1}{e^{-2 x}}&=e^{2x}\big(\tanh (x)-1\big)\\&=
e^{2x}\left( \frac{e^{x}-e^{-x}}{e^{x}+e^{-x}}-1\right)\\&=
e^{2x}\left(\frac{-2e^{-x}}{e^{x}+e^{-x}}\right)\\&=
\frac{-2e^{x}}{e^{x}+e^{-x}}\\&=\frac{2}{e^{2x}+1}-2
\end{align}</span></p>
|
4,347,308 | <p><em><strong>Definition:</strong></em></p>
<p>Let <span class="math-container">$(X,\mathscr{A},\mu)$</span> be a measurable space, an atom of the measure <span class="math-container">$\mu$</span> is a set <span class="math-container">$A \in\mathscr{A}$</span> with the property that
<span class="math-container">$\mu(A... | Bonnaduck | 92,329 | <p>The answer you are conjecturing can be written as</p>
<p><span class="math-container">$$\frac{1+x^3}{1+x^2}=1-x^2+\sum_{n=0}^\infty (-1)^n(x^{2n+3}+x^{2n+4})$$</span></p>
<p>Multiplying both sides by <span class="math-container">$1+x^2$</span> gives us</p>
<p><span class="math-container">\begin{align*}
1+x^3&=(1... |
1,533,745 | <p>$\lim_{x \to 0}\frac{\sin^2x-x^2}{x^2\sin^2x}$</p>
<p>I just can't do anything with this besides l'Hospital's rule (which doesn't seem to be a good idea). Can you help me, please?</p>
| lisyarus | 135,314 | <p>Since $x \approx \sin x$ when $x \rightarrow 0$, the denominator is of order $x^4$.</p>
<p>The numerator needs more careful analysis of $\sin x$. Using $\sin x = x - \frac{x^3}{6} + o(x^3)$, we get $\sin^2 x = x^2 - \frac{x^4}{3} + o(x^4)$, so the numerator is $\sin^2 x - x^2 \approx -\frac{x^4}{3}$.</p>
<p>Dividi... |
1,533,745 | <p>$\lim_{x \to 0}\frac{\sin^2x-x^2}{x^2\sin^2x}$</p>
<p>I just can't do anything with this besides l'Hospital's rule (which doesn't seem to be a good idea). Can you help me, please?</p>
| Idris Addou | 192,045 | <p>Hint:
\begin{equation*}
\frac{\sin ^{2}x-x^{2}}{x^{2}\sin ^{2}x}=\left( \frac{\sin x-x}{x^{3}}%
\right) \left( \frac{\sin x+x}{x}\right) \left( \frac{x}{\sin x}\right) ^{2}
\end{equation*}
\begin{eqnarray*}
\lim_{x\rightarrow 0}\left( \frac{\sin x-x}{x^{3}}\right) &=&-\frac{1}{6} \\
\lim_{x\rightarrow 0}\le... |
1,087,874 | <p>I want to understand how I can count the terms of the expression $x^{m-1} + x^{m-2} +\ldots+ x^0$ when $x=1$.</p>
<p>The result is $m$, I dont know how to count them formally, any advice would be helpful. I'm desperated, not because it is required to do the above, but how can be done, I need to understand the subje... | hmakholm left over Monica | 14,366 | <p>When you set $x=1$, then the value of each of the terms is $1$! (Note that there are no explicit coefficients). Adding the ones then simply tells you how many ones there are.</p>
|
1,087,874 | <p>I want to understand how I can count the terms of the expression $x^{m-1} + x^{m-2} +\ldots+ x^0$ when $x=1$.</p>
<p>The result is $m$, I dont know how to count them formally, any advice would be helpful. I'm desperated, not because it is required to do the above, but how can be done, I need to understand the subje... | Mark Bennet | 2,906 | <p>The terms of the polynomial can be indexed by the exponents of $x$ which are $m-1, m-2, \dots 0$ and these are simply $m$ consecutive integers in reverse order. Each term has a different exponent and no exponent is omitted.</p>
|
795 | <p>Please observe the following thread <a href="https://math.stackexchange.com/questions/4489/proving-that-the-given-diophantine-equation-has-a-solution">Proving that the given Diophantine equation has a solution</a>.</p>
<p>There is a long boring argument/discussion about whether it should be posted, who should post ... | Larry Wang | 73 | <p>I removed those comments that were not related to <a href="https://math.stackexchange.com/questions/4489/proving-that-the-given-diophantine-equation-has-a-solution">the question Chandru1 posted</a>, and reproduce them here so that they will not be lost. Both on-topic and off-topic comments have been provided for con... |
169,097 | <p>In the beginning of chapter two in The HoTT Book there is a discussion about synthetic vs. analytic geometry:</p>
<blockquote>
<p>An important difference between homotopy type theory and classical homotopy theory is that homotopy type theory provides a <em>synthetic</em> description of spaces, in the following se... | Mike Shulman | 49 | <p>Francois' answer is good; let me add a bit more. Homotopy type theory is synthetic <em>homotopy theory</em>, which means that the "spaces" in question are not the same sort of "spaces" that you find in point-set topology: it's better to think of them as "homotopy types" or "$\infty$-groupoids". They are not "geome... |
271,824 | <p>I have a list= {4, 8, 10, 11, 12, 14, 16, 7, 9}</p>
<p>How can i partition the list by group of Arithmetic Progression with common difference 1 :</p>
<p>{{4}, {8}, { 10, 11, 12}, {14}, {16}, {7}, {9}}</p>
| lericr | 84,894 | <p>Something like this, maybe?</p>
<pre><code>Map[If[MemberQ[li2, #], #, x] &, li1]
</code></pre>
|
4,547,918 | <p>Given the torus and given the point p <span class="math-container">$\in$</span> M corresponding to the parameters <span class="math-container">$s=\frac{\pi }{4}$</span> and <span class="math-container">$t=\frac{\pi }{3}$</span>.
Determine the cartesian equation of the tangent plane to M in p.</p>
<p><span class="mat... | P. Lawrence | 545,558 | <p>Use the "tan half-ange" substitution, <span class="math-container">$viz$</span> <span class="math-container">$$u=\tan (x/2)$$</span> to convert the integral to the integral of a rational function.</p>
|
204,560 | <p>I am a computer programmer interested in prime numbers. I have implementations of several algorithms related to prime numbers at my <a href="http://programmingpraxis.com" rel="nofollow">blog</a>. I want to add an implementation of the AKS primality prover to my collection, but I am having trouble, and my knowledge o... | user448810 | 20,808 | <p>Ok, I've got it. The calculation of r was incorrect; it should be 191. Working code, including the full AKS prover, is at <a href="http://programmingpraxis.codepad.org/6ZHrsEmx" rel="nofollow">http://programmingpraxis.codepad.org/6ZHrsEmx</a>. I'll have a full write-up at my blog later this week: <a href="http://pro... |
204,560 | <p>I am a computer programmer interested in prime numbers. I have implementations of several algorithms related to prime numbers at my <a href="http://programmingpraxis.com" rel="nofollow">blog</a>. I want to add an implementation of the AKS primality prover to my collection, but I am having trouble, and my knowledge o... | Copperfield | 326,292 | <blockquote>
<p>The calculation of r was incorrect; it should be 191.</p>
</blockquote>
<p>wait what?, that is wrong the r for 89 is not 191, is 43</p>
<p>first star with the definition of r in the context of the AKS test:</p>
<blockquote>
<p>find the smallest r such that such that: ord<sub>r</sub>(n) > (log<sub... |
1,817,035 | <p>Gradient descent reduces the value of the objective function in each iteration. This is repeated until convergence happens.</p>
<p>The question is if the norm of gradient has to decrease as well in every iteration of gradient descent?</p>
<hr>
<p><strong>Edit:</strong> How about when the objective is a convex fun... | Rodrigo de Azevedo | 339,790 | <p>Suppose we have the following objective function</p>
<p>$$f (x) := \frac{1}{2} x^T Q x - r^T x$$</p>
<p>where $Q \in \mathbb R^{n \times n}$ is symmetric and positive definite and $r \in \mathbb R^n$. From the symmetry of $Q$, we conclude that its eigenvalues are real. From the positive definiteness of $Q$, we con... |
289,864 | <p>Let $C_{1}$ be a circle of unit radius. Let A and B be two points inside $C_{1}$. Now I want to construct another circle $C_{2}$ such that A and B lie on $C_{2}$ and $C_{2}$ is orthogonal to $C_{1}$ at their point of intersection(I want $C_{2}$ in such a way that it intersects $C_{1}$). I tried and failed to find a ... | Neal | 20,569 | <p>If I understand your question correctly, it is answered in Construction 2.1 of this paper: <a href="http://comp.uark.edu/~strauss/papers/hypcomp.pdf" rel="nofollow">http://comp.uark.edu/~strauss/papers/hypcomp.pdf</a></p>
<p>Edit upon request: Given two points $A$ and $B$ in the interior of a circle $C_1$, invert $... |
2,166,075 | <blockquote>
<p>Prove $a_1+\cdots+a_n=\dfrac{(a_1+a_n)n}{2}$ inductively.</p>
</blockquote>
<p>Where $a_i=a_{i+1}-r$.</p>
<p>I tried to start proving it inductively, but any try lead to a bad conclusion, so I ended up proving it by making $a_n$ depend on $a_i$.</p>
<p>But I didn't know how to prove it inductively,... | Bram28 | 256,001 | <p>No induction needed ... just use a simple trick famously used by Gauss when he was 10 years old:</p>
<p>Take two of these series, one going from $a_1$ to $a_n$, and the other one going back from $a_n$ to $a_1$, put them under each other, and add them up by entry (that is, add the first entries of the two series, th... |
2,166,075 | <blockquote>
<p>Prove $a_1+\cdots+a_n=\dfrac{(a_1+a_n)n}{2}$ inductively.</p>
</blockquote>
<p>Where $a_i=a_{i+1}-r$.</p>
<p>I tried to start proving it inductively, but any try lead to a bad conclusion, so I ended up proving it by making $a_n$ depend on $a_i$.</p>
<p>But I didn't know how to prove it inductively,... | Stefano | 387,021 | <p>Statement true for $n=2$. Assume</p>
<p>$$ a_1 + \dots + a_n = \frac{(a_1+a_n)n}{2}.$$</p>
<p>Then, using the inductive hypothesis and adding and subtracting terms, we get</p>
<p>$$a_1 + \dots+ a_n + a_{n+1} = \frac{(a_1+a_n)n}{2}+ a_{n+1}$$</p>
<p>$$= \frac{(a_1+ a_{n+1})(n+1) }{2} + \frac{a_nn-a_1}{2}+ a_{n+1}... |
4,047,601 | <p>I did a question <span class="math-container">$\int_{0}^{1}\frac{1}{x^{\frac{1}{2}}}\,dx$</span>, and evaluating this is divergent integral yes? Then as a general form <span class="math-container">$\int_{0}^{1} \frac{1}{x^p}\,dx$</span>, <span class="math-container">$p \in \mathbb{R}$</span>, what values of <span cl... | José Carlos Santos | 446,262 | <p>Actually,<span class="math-container">$$\int_0^1x^{-1/2}\,\mathrm dx=\left[2x^{1/2}\right]_{x=0}^{x=1}=2.$$</span></p>
<p>On the other hand,<span class="math-container">$$\int_0^1x^{-p}\,\mathrm dx=\left[\frac{x^{1-p}}{1-p}\right]_{x=0}^{x=1}=\frac1{1-p}.$$</span>So, take <span class="math-container">$p=\frac14$</sp... |
4,047,601 | <p>I did a question <span class="math-container">$\int_{0}^{1}\frac{1}{x^{\frac{1}{2}}}\,dx$</span>, and evaluating this is divergent integral yes? Then as a general form <span class="math-container">$\int_{0}^{1} \frac{1}{x^p}\,dx$</span>, <span class="math-container">$p \in \mathbb{R}$</span>, what values of <span cl... | Obsessive Integer | 864,339 | <p><span class="math-container">$\int_0^1{\frac{1}{x^\frac{1}{2}}}dx$</span> is not divergent.</p>
<p><span class="math-container">$\int_0^1{\frac{1}{x^\frac{1}{2}}}dx=\int_0^1 x^{-\frac{1}{2}}dx=2x^\frac{1}{2}|_0^1=2$</span></p>
<p>Similarly solving: <span class="math-container">$\int_0^1 \frac{1}{x^p}dx=\frac{4}{3}$<... |
3,005,100 | <p>Given the following formula
<span class="math-container">$$
\sum^n_{k=0}\frac{(-1)^k}{k+x}\binom{n}{k}\,.
$$</span>
How can I show that this is equal to
<span class="math-container">$$
\frac{n!}{x(x+1)\cdots(x+n)}\,?
$$</span></p>
| ajotatxe | 132,456 | <p>Induction step:</p>
<p><span class="math-container">$$\begin{align}
\sum_{k=0}^{n+1}&\frac{(-1)^k}{x+k}\binom{n+1}k=\frac1x+\frac{(-1)^{n+1}}{x+n+1}+\sum_{k=1}^{n}\frac{(-1)^k}{x+k}\left[\binom nk+\binom n{k-1}\right]
\\&=\frac{n!}{x(x+1)\cdots(x+n)}+\frac{(-1)^{n+1}}{x+n+1}+\sum_{k=1}^{n}\frac{(-1)^k}{x+k}... |
2,965,989 | <p>Why <span class="math-container">$$p(y)=\int_0^\infty x\delta (y-x)dx=y\ \ ?$$</span></p>
<p>For me, <span class="math-container">$$p(y)=\int_0^\infty x\delta (y-x)dx=\int_{\{y\}}xdx=0.$$</span></p>
<p>If it would be written <span class="math-container">$\int_0^\infty xd\delta _y$</span>, then I would be agree wit... | Masacroso | 173,262 | <p>The <span class="math-container">$\delta_y$</span> of Dirac is a measure that is defined as</p>
<p><span class="math-container">$$
\delta_y(A)=\begin{cases}1,& y\in A\\ 0,&y\notin A\end{cases}
$$</span></p>
<p>for any subset <span class="math-container">$A$</span> of the measure space (in your case the mea... |
947,770 | <p>Here I have a diophantine equation featuring a homogeneous polynomial:</p>
<p>$$x^2+5y^2+34z^2+2xy-10xz-22yz=0; x, y, z\in\mathbb{Z}$$</p>
<p>I have no idea how to approach this, I've tried various substitutions like $x=py, x=qz$ but then I get a non-homogeneous polynomial of 2 variables which is no better than th... | Mahmoud. A .Solomon | 361,251 | <p>assuming </p>
<p>$x^2+5y^2+2xy=c$</p>
<p>and solving the equation,</p>
<p>$34z^2-(10x+22y)z+c=0$</p>
<p>one can find that </p>
<p>$z=(10x+22y∓(√(-4(3x-7y)^2 ))/(2*34)$</p>
<p>which mean that the discriminant D is negative and z is complex
but z∈Z then $d=0$</p>
<p>or ,$3x=7y$</p>
<p>and $34z=5x+11y$</p>
... |
4,069,499 | <p>If we let <span class="math-container">$x = 0$</span>.</p>
<p><span class="math-container">\begin{align*}
3(0+7)-y(2(0)+9) \\
21-9y \\
\end{align*}</span></p>
<p>Then <span class="math-container">$9y$</span> should always equal <span class="math-container">$21$</span>?
Solving for <span class="math-container">$y$</... | Royi | 33 | <p>You have the function:</p>
<p><span class="math-container">$$ f \left( x, y \right) = 3 \left( x + 7 \right) - y \left( 2 x + 9 \right) $$</span></p>
<p>Since the function is constant for <span class="math-container">$ x $</span> then it means:</p>
<p><span class="math-container">$$ \nabla_{x} f \left( x, y \right) ... |
7,130 | <p>I'm looking for an explanation on how reducing the Hamiltonian cycle problem to the Hamiltonian path's one (to proof that also the latter is NP-complete). I couldn't find any on the web, can someone help me here? (linking a source is also good).</p>
<p>Thank you.</p>
| Prashant | 96,758 | <p>The answer mentioned is correct for the purpose of showing reduction, but I have this idea which is faster than the idea presented. Please correct me if I am wrong.
If ham path says no for even a single vertex then ham cycle says no.
This means that all vertices should now say yes for Ham path(hence one method id t... |
3,248,123 | <p>I have this statement:</p>
<blockquote>
<p>If <span class="math-container">$a$</span> belongs to the interval <span class="math-container">$[- 4, - 1]$</span> and <span class="math-container">$b$</span> belongs to the
interval <span class="math-container">$[- 2, 3]$</span>, what interval does it contain? all po... | herb steinberg | 501,262 | <p>Max of <span class="math-container">$2a=-2$</span>, min of <span class="math-container">$b=-2$</span>, max of <span class="math-container">$2a-b=0$</span>.</p>
<p>Min of <span class="math-container">$2a=-8$</span>, max of <span class="math-container">$b=3$</span>, min of <span class="math-container">$2a-b=-11$</spa... |
3,248,123 | <p>I have this statement:</p>
<blockquote>
<p>If <span class="math-container">$a$</span> belongs to the interval <span class="math-container">$[- 4, - 1]$</span> and <span class="math-container">$b$</span> belongs to the
interval <span class="math-container">$[- 2, 3]$</span>, what interval does it contain? all po... | DanielWainfleet | 254,665 | <p>The error is assuming that <span class="math-container">$A\le B$</span> and <span class="math-container">$C\le D$</span> imply <span class="math-container">$A-B\le C-D.$</span> E.g. <span class="math-container">$4\le 6$</span> and <span class="math-container">$2\le 5$</span> but <span class="math-container">$\neg (4... |
2,189,029 | <p>Can all numbers $a, b, c$ that $a^2+b^2=c^2$ be sides of a right triangle? ($c$ is the hypotenuse)
I saw this problem:</p>
<p>Find the number of right triangles that the hypotenuse and one side are both prime numbers. </p>
<p>I said that we don't have any property about the other side, so we can assume it is a re... | Claude Leibovici | 82,404 | <p><em>As Friedrich Philipp commented.</em></p>
<p>If you apply the fundamental theorem of calculus $$\begin{equation}
g(z-z_0)=\int_{z_0}^{z}~f(t-z_0)\,g(z-t)\,dt
\end{equation}$$
$$\begin{equation}
\frac{dg(z-z_0)}{dz}= f(z-z_0)\,g(0)+\int_a^z f(t-z_0)\, g'(z-t) \, dt
\end{equation}$$</p>
|
2,189,029 | <p>Can all numbers $a, b, c$ that $a^2+b^2=c^2$ be sides of a right triangle? ($c$ is the hypotenuse)
I saw this problem:</p>
<p>Find the number of right triangles that the hypotenuse and one side are both prime numbers. </p>
<p>I said that we don't have any property about the other side, so we can assume it is a re... | Zaid Alyafeai | 87,813 | <p>$$g(z-z_0)=\int_{z_0}^{z}f(t-z_0)g(z-t)dt
$$</p>
<p>Let $z-t = y$
$$g(z-z_0)=\int_{0}^{z-z_0} f(z-z_0 -y)g(y)dy $$</p>
<p>This implies that for $z-z_0 = t \geq 0$ </p>
<p>$$g(t) = (g*f)(t)$$</p>
<p>Then apply Laplace to both sides </p>
<p>$$G(s) = G(s) F(s)$$</p>
<p>I think you can go from here. </p>
|
154,173 | <p>The cross product $a \times b$ can be represented by the determinant</p>
<p>$$\mathbf{a}\times\mathbf{b}= \begin{vmatrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
a_1 & a_2 & a_3 \\
b_1 & b_2 & b_3 \\
\end{vmatrix}.$$</p>
<p>Does the <em>matrix</em> whose determinant is this have any signifi... | Phira | 9,325 | <p>You can let the matrix act by ordinary matrix multiplication on ordinary vectors in three-dimensional space.</p>
<p>This will transform a vector in a triple containing the original vector and the lengths of the two projections on $a$ and $b$.</p>
<p>While I feel that this counts as "any significance", it isn't ver... |
154,173 | <p>The cross product $a \times b$ can be represented by the determinant</p>
<p>$$\mathbf{a}\times\mathbf{b}= \begin{vmatrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
a_1 & a_2 & a_3 \\
b_1 & b_2 & b_3 \\
\end{vmatrix}.$$</p>
<p>Does the <em>matrix</em> whose determinant is this have any signifi... | TheEmptyFunction | 609,020 | <p>You could think of the entries as all being Quaternions, i.e. the matrix is an element of <span class="math-container">$M_3(\mathbb{H})$</span>. This gives you a genuine way to act on vectors and do matrix multiplication. But you need to be careful about converting between the two different interpretations <span cla... |
4,618,433 | <p>Just a heads up: "<span class="math-container">$a$</span>" and "<span class="math-container">$α$</span>" are different</p>
<p>Let <span class="math-container">$a,b \in \Bbb R$</span> and suppose <span class="math-container">$a^2 − 4b \neq 0$</span>. Let <span class="math-container">$\alpha$</span... | user2661923 | 464,411 | <p>Instead of factoring, you can apply the formula for the quadratic equation directly.</p>
<p>If <span class="math-container">$Ax^2 + Bx + C = 0$</span>, then the two roots are given by</p>
<p><span class="math-container">$$\frac{1}{2A} \left[-B \pm \sqrt{B^2 - 4AC}\right].$$</span></p>
<p>This means that (when the ro... |
4,618,433 | <p>Just a heads up: "<span class="math-container">$a$</span>" and "<span class="math-container">$α$</span>" are different</p>
<p>Let <span class="math-container">$a,b \in \Bbb R$</span> and suppose <span class="math-container">$a^2 − 4b \neq 0$</span>. Let <span class="math-container">$\alpha$</span... | Andrew Chin | 693,161 | <p>Solving the equation <span class="math-container">$x^2+ax+b=0$</span>, we have <span class="math-container">$$x=\frac{-a\pm\sqrt{a^2-4b}}{2},$$</span>
or more specifically, wlog, we can write <span class="math-container">$$
\alpha=\frac{-a+\sqrt{a^2-4b}}{2},\quad \beta=\frac{-a-\sqrt{a^2-4b}}{2}.$$</span>
The differ... |
2,022,423 | <p>You are asked to <strong>permute the neighboring sub-sequence</strong> of the sequence $n,n-1,n-2,\cdots,1$ until the sequence is brought to the increasing order. </p>
<p>By <em>permute the neighboring sub-sequence</em> I mean for example:
$5,4,3,2,1 \to 5,3,4,2,1 $ or $5,4,3,2,1\to 5,2,4,3,1$ or $5,4,3,2,1\to5,2,... | Dap | 467,147 | <p>You need at least <span class="math-container">$\lceil (n+1)/2\rceil.$</span> Consider the number <span class="math-container">$G$</span> of terms whose right neighbor is greater. <span class="math-container">$G$</span> starts at zero and ends as <span class="math-container">$n-1.$</span></p>
<p>For example
<span c... |
1,942,854 | <blockquote>
<p>Does there exist an $n \not \equiv 3 \pmod{4}$ where $n \in \mathbb{N}$ and is greater than $1$ such that there exists a prime $p >5$ such that \begin{cases}3^{n^2-1} &\equiv 1 \pmod{p}\\2^{n^2-1} &\equiv 1 \pmod{p}?\end{cases}</p>
</blockquote>
<p>I tried finding such a prime but I could... | DonAntonio | 31,254 | <p>$$(-5)^{-\frac43}=\left[(-5)^{-4}\right]^{1/3}=\frac1{625^{1/3}}=\frac1{\underbrace{5\sqrt[3]5}_{=5^{4/3}}}>0$$</p>
<p>Or also</p>
<p>$$(-5)^{-\frac43}=\left[(-5)^{1/3}\right]^{-4}=\frac1{(\sqrt[3]{-5})^4}=\frac1{(\sqrt[3]{-5})^3(\sqrt[3]{-5})}=\frac1{(-5)\cdot(-5^{1/3})}=\frac1{5^{4/3}}>0$$</p>
|
1,611,052 | <p>Let H an infinite-dimensional Hilbert space in $\mathbb{R}$</p>
<p>If $x_1, x_2, \ldots x_n \in H$, how to prove: </p>
<p>$\sum_{1\leq i,j\leq n} {\lvert\lvert x_i - x_j \rvert\rvert}^2 \leq \sum_{1\leq i,j\leq n} ({\lvert\lvert x_i \rvert\rvert}^2 + {\lvert\lvert x_j \rvert\rvert}^2)$</p>
| mkk030572 | 210,906 | <p>The triangle inequality yields only $\|x_i - x_j\|^2 \le (\|x_i\| + \|x_j\|)^2 \le \|x_i\|^2 + 2 \|x_i\| \|x_j\| + \|x_j\|^2 \le 2 (\|x_i\|^2 + \|x_j\|^2)$...</p>
|
142,105 | <p>In trying to deduce the lower bound of the ramsey number R(4,4) I am following my book's hint and considering the graph with vertex set $\mathbb{Z}_{17}$ in which $\{i,j\}$ is colored red if and only if $i-j\equiv\pm2^i,i=0,1,2,3$; the set of non-zero quadratic (mod 17) and blue otherwise. This graph shows that $R(4... | Han Altae-Tran | 28,011 | <p>We know that $N(3,4;2)=9$. By symmetry, we have that $N(4,3;2)=9$.
Thus we have the following: $
\lbrack
N(4,4;2)\leq N(3,4;2)+N(4,3;2)=18
\rbrack
$</p>
<p>Thus we create a counterexample, $K_{17}$ that does not contain any
red $K_{4}$ or green $K_{4}$. To do this, we label the vertices
$v_{1}=1,\dots,v_{17}=17$. T... |
761,616 | <p>How do you integrate $\sqrt{(x^4 + x^2)}$? </p>
| user103816 | 103,816 | <p>Hint: Put $x^2+1=t$ and and notice $2xdx=dt.$</p>
|
761,616 | <p>How do you integrate $\sqrt{(x^4 + x^2)}$? </p>
| Anastasiya-Romanova 秀 | 133,248 | <p>\begin{align}
\int\sqrt{x^4+x^2}\,dx&=\int\sqrt{x^2(x^2+1)}\,dx\\
&=\int x\sqrt{x^2+1}\,dx\\
&=\int \sqrt{x^2+1}\,xdx\\
\end{align}
Let $u=x^2+1$, then $du=2x\,dx$ or $x\,dx=\frac{1}{2}du$. Hence
\begin{align}
\int\sqrt{x^4+x^2}\,dx
&=\int \sqrt{x^2+1}\,xdx\\
&=\int \sqrt{u}\frac{1}{2}du\\
&=... |
761,616 | <p>How do you integrate $\sqrt{(x^4 + x^2)}$? </p>
| Andrew Thompson | 92,231 | <p>Note that you may write, as David pointed out, the simplification $$\int \sqrt{x^4 + x^2}\space dx = \int x\sqrt{x^2+1}\space dx$$</p>
<p>Now, notice the substitution $u = x^2+1$, $du = 2x\space dx$. So we have $$\frac{1}{2}\int\sqrt{u}\space du = \frac{1}{2}\int u^{1/2}\space du = \frac{1}{2}\cdot\frac{2}{3}u^{3/2... |
3,939,620 | <p>Given a polynomial of the form <span class="math-container">$R(z):=\frac{P(z)}{Q(z)}$</span> such that <span class="math-container">$R(z)$</span> has no real roots and <span class="math-container">$deg(Q) \geq deg(P) + 2$</span>, then the integral can be expressed as</p>
<p><span class="math-container">$$\int_{-\inf... | Tryst with Freedom | 688,539 | <p>There is perhaps a better way to approach, consider:</p>
<p><span class="math-container">$$ f= \sin^2 x + \sin x$$</span></p>
<p>Now, we can see there are zero set for this function: <span class="math-container">$ \sin x = \{0,-1 \}$</span>, in the interval <span class="math-container">$[0, 2 \pi]$</span> the zero ... |
3,939,620 | <p>Given a polynomial of the form <span class="math-container">$R(z):=\frac{P(z)}{Q(z)}$</span> such that <span class="math-container">$R(z)$</span> has no real roots and <span class="math-container">$deg(Q) \geq deg(P) + 2$</span>, then the integral can be expressed as</p>
<p><span class="math-container">$$\int_{-\inf... | The 2nd | 751,538 | <p>Let <span class="math-container">$t = \sin x \,\, (-1 \leq t \leq 1)$</span></p>
<p>The domain of the function will become:</p>
<p><span class="math-container">$$t^2 + t \geq 0$$</span></p>
<p><span class="math-container">$$\implies t \in (-\infty, -1] \cup [0, +\infty)$$</span></p>
<p>But <span class="math-containe... |
574,614 | <p>if $\gamma:[0,2\pi]\mapsto\Bbb C,\quad \gamma(t)=1+e^{it}$ then show that $|\int_\gamma\frac{dz}{z-\frac{3}{2}}|\le4\pi$ (without computing)</p>
<p>I tried :
$ |\int_\gamma\frac{dz}{z-\frac{3}{2}}| \le \int_\gamma|\frac{1}{z-\frac{3}{2}}|dz$ and $ |z-\frac{3}{2}|\ge||z|-\frac{3}{2}|$ we should find its max va... | MichalisN | 8,432 | <p>Without replacing $z$ with $|z|$ you need to find the minimal value of $|e^{it}-1/2|$ which is $1/2$ (on the unit circle $1$ is closest to 1/2). The length of your path is $2\pi$ so the integral is smaller than $\pi$.</p>
|
3,059,571 | <p><span class="math-container">$$\lim_{x\to \frac\pi2} \frac{(1-\tan(\frac x2))(1-\sin(x))}{(1+\tan(\frac x2))(\pi-2x)^3}$$</span></p>
<p>I only know of L'hopital method but that is very long. Is there a shorter method to solve this?</p>
| A.Γ. | 253,273 | <p>Another trick is to multiply both the numerator and denominator by
<span class="math-container">$(1+\tan(x/2))(1+\sin x)$</span> and use that
<span class="math-container">\begin{align}
1-\sin^2x&=\cos^2x,\\
1-\tan^2(x/2)&=\frac{\cos^2(x/2)-\sin^2(x/2)}{\cos^2(x/2)}=\frac{\cos x}{\cos^2(x/2)}.
\end{align}</s... |
376,517 | <p>Let <span class="math-container">$U$</span> be a smooth variety, and <span class="math-container">$U\hookrightarrow X$</span> an smooth compactification with snc boundary <span class="math-container">$D=X\setminus U$</span>. Suppose that <span class="math-container">$\omega\in H^0(U,\Omega^n_U)$</span> is global al... | AG learner | 74,322 | <p>This rarely happens. For example, when <span class="math-container">$U$</span> is an affine smooth variety, then by Grothendieck's algebraic de Rham theorem (or degeneration of Hodge spectral sequence at <span class="math-container">$E_2$</span>),</p>
<p><span class="math-container">$$H^n(U,\mathbb C)\cong\{\alpha\i... |
1,203,269 | <p>I am trying to compute the hitting time of a linear Brownian motion on a two-sided boundary. More specifically, let $W_t$ be a (one-dimensional) Wiener process. Let $T = \inf \{t: |W_t| = a \}$ for some $ a > 0$. I want to find $\mathbb{P}\{ T > t\}$. </p>
<p>I know that probability distribution hitting time ... | C.Koca | 426,569 | <p>The distribution of hitting times for a Brownian Motion, <span class="math-container">$S$</span> starting at <span class="math-container">$0$</span> with barriers at <span class="math-container">$c$</span> and <span class="math-container">$-c$</span> and step size <span class="math-container">$l=1$</span> is given b... |
945,395 | <p>Let $a_1$ be real, and define $$a_{n+1}=\frac{2a_n^3}{1+a_n^4}$$ How can I prove that this $\{a_n\}$ to have limit. </p>
<p>I find it is hard to track. What I can do is just when $a_1=1$ then $a_n=1$; when $a_1=-1$, then $a_n=-1$; when $|a_1|<1$, $a_n\to 0$. When $|a_1|>1$, I have not find any idea.</p>
| Macavity | 58,320 | <p>It is sufficient to consider $a_n > 0$, as otherwise you could do the same analysis with $b_n = -a_n$ and $a_1=0 \iff a_n=0$, so that's trivial.</p>
<p>Note that $1+x^4 \ge 2x^2$ by AM-GM with equality iff $x=1$, so we have $a_{n+1} < a_n$, unless $a_k=1 \implies a_{n> k}=1$. Thus we have a decreasing, po... |
2,884,785 | <p>Find all integral pairs (x,y) such that - $$( xy - 1)^2 = (x +1)^2 + ( y+1)^2$$</p>
<hr>
<p><strong>My Approach :</strong></p>
<p>I just expanded this equation and wrote it in another form - $$\frac{(xy+1)(xy-1)}{(x+y)}-2=x+y$$ and from this we can say that $(x+y)|(xy+1) \ \mathrm{or}\ (x+y)|(xy-1) $ . But i don'... | nonuser | 463,553 | <p>Or $$x^2y^2 = (x+y)^2+2(x+y)+1 \implies x^2y^2 = (x+y+1)^2$$</p>
<p><strong>1. case:</strong> $$xy =x+y+1\implies (x-1)(y-1)=2\implies ....$$</p>
<p><strong>2. case:</strong> $$xy =-x-y-1\implies (x+1)(y+1)=0\implies ....$$</p>
|
257,623 | <p>Consider the following ellipse, generated by the bounding region of the following points</p>
<pre><code>ps = {{-11, 5}, {-12, 4}, {-10, 4}, {-9, 5}, {-10, 6}};
rec = N@BoundingRegion[ps, "FastEllipse"];
Graphics[{rec, Red, Point@ps}]
</code></pre>
<p><a href="https://i.stack.imgur.com/gvtUB.png" rel="nofol... | halmir | 590 | <p>Here's the alternate way using SingularValueDecomposition:</p>
<pre><code>pt = rec[[1]]; mat = rec[[2]];
{u, s, v} = SingularValueDecomposition[(mat + Transpose[mat])/2];
func = Composition[AffineTransform[{u, pt}], ScalingTransform[Sqrt[Diagonal[s]]]];
</code></pre>
<p>and the length:</p>
<pre><code>EuclideanDista... |
7,981 | <p>I've read so much about it but none of it makes a lot of sense. Also, what's so unsolvable about it?</p>
| Sniper Clown | 21,855 | <p>You have to...have to....read <a href="http://www.math.jhu.edu/~wright/RH2.pdf" rel="noreferrer">this friendly introduction</a>. <em>(Making it CW since it's just a link.)</em></p>
|
188,102 | <p>I have the following list: </p>
<pre><code>m={{14, "extinguisher"}, {54, "virgule"}, {55, "turnoff"}, {51,
"sofa"}, {77, "beachcomber"}, {61, "stoic"}, {6,
"isomorphism"}, {34, "leftist"}, {84, "spline"}, {42,
"heartiness"}, {35, "postnatal"}, {41, "stratified"}, {66,
"silkworm"}, {95, "conformance"}, {... | OkkesDulgerci | 23,291 | <p><strong>Edit:</strong></p>
<pre><code>m = Sort[{{14, "extinguisher"}, {54, "virgule"}, {55, "turnoff"}, {51,
"sofa"}, {77, "beachcomber"}, {61, "stoic"}, {6,
"isomorphism"}, {34, "leftist"}, {84, "spline"}, {42,
"heartiness"}, {35, "postnatal"}, {41, "stratified"}, {66,
"silkworm"}, {95, "co... |
2,267,005 | <p>I have been asked to evaluate the $\int{Fdr}$ over a curve $C$ where $F = yz\mathbf{i} + 2xz\mathbf{j} + e^{xy}\mathbf{k}$ and $C$ is the curve $x^2 + y^2 = 16, z =5$ with downward orientation</p>
<p>I want to use Stokes theorem, so I am thinking of parametrizing this surface as $(x, y, z)=(4 \cos t, 4\sin t,z)$ bu... | marshal craft | 167,793 | <p>Ok if I interpret the question so that it is interesting (as currently phrased it is trivial that answer is, it doesnt matter the max distance is 450) then we ask if the front wears through 100% of the tire in 450 km and back in 600 km then what is the maximum distance one can travel on good tires provided they swap... |
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