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 |
|---|---|---|---|---|
2,133,769 | <p>Asymptotic notations are a little vague for me at the moment. Here I have a problem that asks me if two equations are equal to each other, and it involves the big O notation. I wrote my question in the image in magenta color.</p>
<p><a href="https://i.stack.imgur.com/hxDx3.jpg" rel="nofollow noreferrer"><img src="h... | Community | -1 | <p>Rewrite the system as</p>
<pre><code> x1 + x2 + x3 = 1 - x4 - x5
-x1 + x2 + x3 = 1 + 2x4 + 2x5
2x1 + 2x2 - x3 = 1 + x4 - x5
</code></pre>
<p>Then for any choice of <code>x4</code> and <code>x5</code> (possibly renamed <code>s</code> and <code>t</code>) you can evaluate the RHSs and find a solution in <code>x... |
240,139 | <p>Does a quadratic form always come from symmetric bilinear form ?
We know when $q(x)=b(x,x)$ where $q$ is a quadratic form and $b$ is a symmetric bilinear form.
But when we just take a bilinear form and $b(x,y)$ and write $x$ instead of $y$,does it give us a quadratic form ?</p>
| Will Jagy | 10,400 | <p>Sure. In finite dimension over the real numbers you do not even need the bilinear form symmetric, because any matrix is the sum of a symmetric and a skew-symmetric matrix, and these are easy to find. </p>
<p>Meanwhile, as long as characteristic of the underlying field is not $2,$ you can go back with $$ c(x,y) = \f... |
240,139 | <p>Does a quadratic form always come from symmetric bilinear form ?
We know when $q(x)=b(x,x)$ where $q$ is a quadratic form and $b$ is a symmetric bilinear form.
But when we just take a bilinear form and $b(x,y)$ and write $x$ instead of $y$,does it give us a quadratic form ?</p>
| Serkan Yaray | 50,727 | <p>If we have b symmetric bilinear form we can get q quadratic form<br>
$q\colon V \to \mathbb{R}$<br>
q(v)=b(v,v)</p>
<p>conversely if q is a quadratic form<br>
$q\colon V \to \mathbb{R}$<br>
we can define<br>
$\frac 12$(q(v+w)-q(v)-q(w)):=b(v,w)</p>
<p>the vital answer is you just get a bilinear form not always a s... |
2,167,541 | <p>Is this expression true?$$y(x) =\lim_{h \rightarrow 0}\frac{f(x+h)-f(x)}{h} = \lim_{h \rightarrow 0}\frac{f(x)-f(x-h)}{h}$$</p>
<p>I'm taking the limit of h goes to 0 for y(x) and I got the latter equation. Can I just write y(x) = df(x)/dx?</p>
| Stefano | 387,021 | <p>$$ 2\sin x \cos x =2 \left(\sum_{k=0}^\infty (-1)^k\frac{x^{2k+1}}{(2k+1)!}\right) \left(\sum_{k=0}^\infty (-1)^k\frac{x^{2k}}{(2k)!}\right)$$</p>
<p>$$= 2\sum_{k=0}^\infty \sum_{i=0}^k (-1)^i \frac{x^{2i+1}}{(2i+1)!} (-1)^{k-i} \frac{x^{2(k-i)}}{(2k-2i)!} = \sum_{k=0}^\infty (-1)^k x^{2k+1} \sum_{i=0}^k\frac{2}{(2... |
3,154,947 | <p>Hello MathStackExchange Community, </p>
<p>Confused about how to change the notation-formula for permutations when you have more complex processes. i.e. multiple independent action choices of different types of objects and orientation limitations. </p>
<ul>
<li><em>The generic formula is : <span class="math-contai... | Claude Leibovici | 82,404 | <p>Making a parametric plot, there is effectively a loop which is symmetric with respect to the <span class="math-container">$x$</span> axis; the points where the curve intersect the axis correspond to <span class="math-container">$x=33$</span> and <span class="math-container">$x=49$</span> as @Ross Millikan already an... |
1,326,652 | <p>What would be the nature of the roots of the equation $$2x^2 - 2\sqrt{6} x + 3 = 0$$</p>
<p>My book says that as the discriminant is 0 so the roots are rational and equal.
But discriminant can be used for determining the nature of roots only when the roots are rational numbers. Is the answer in the book wrong becau... | Robert Israel | 8,508 | <p>Rationality and multiplicity of roots are two separate questions. The discriminant tells you whether there are repeated roots. It offers no opinion on whether those repeated roots are rational. </p>
<p>Of course, for a polynomial whose roots are all rational the coefficients (divided by the leading coefficient) w... |
3,789,584 | <p>I tried to find the limit <span class="math-container">$$\mathop {\lim }\limits_{n \to \infty } \ln (n) \cdot \int_0^1 \ln ( n^{-t} + 1 ) \, \mathrm dt$$</span>, but I couldn't find a conclusive argument.</p>
<hr />
<p>By assuming that the sequence <span class="math-container">$\displaystyle x_n = \ln (n) \cdot \int... | Alessio K | 702,692 | <p>We have <span class="math-container">$\int ln(1+n^{-t})dt=\int \sum_{k=1}^{\infty}(-1)^{k+1}\frac{n^{-tk}}{k}dt=\sum_{k=1}^{\infty}(-1)^{k+1}\frac{1}{k} \int n^{-tk}dt$</span> by <a href="https://en.wikipedia.org/wiki/Fubini%27s_theorem" rel="nofollow noreferrer">Fubini/Tonelli theorems</a> (and using the Taylor ser... |
11,267 | <p>John Von Neumann once said to Felix Smith, "Young man, in mathematics you don't understand things. You just get used to them." This was a response to Smith's fear about the method of characteristics. </p>
<p>Did he mean that with experience and practice, one obtains understanding? </p>
| Cian Moriarty | 117,828 | <p>One thing he might have been referring to is that in mathematics you often have to learn to apply a method without actually understanding what it is all about.</p>
<p>Take for example matrix multiplication. You could (and many students do) beat themselves up about why it is so "weird" in comparison to say multiplic... |
11,267 | <p>John Von Neumann once said to Felix Smith, "Young man, in mathematics you don't understand things. You just get used to them." This was a response to Smith's fear about the method of characteristics. </p>
<p>Did he mean that with experience and practice, one obtains understanding? </p>
| Dahn | 48,999 | <p>This is quite an old post, but I choose to answer, because I feel that I offer a <strong>completely different understanding</strong> of this quote.</p>
<p>I am very surprised to find out my understanding of it is different from others, since when I first read it, I thought to myself: <em>exactly</em>!</p>
<p>To me, ... |
11,267 | <p>John Von Neumann once said to Felix Smith, "Young man, in mathematics you don't understand things. You just get used to them." This was a response to Smith's fear about the method of characteristics. </p>
<p>Did he mean that with experience and practice, one obtains understanding? </p>
| littleO | 40,119 | <p>One word frequently missing from this discussion is "joke". Von Neumann joked around a lot. I'd guess he was 50% joking when he said this. But it's one of those jokes that's funny because it has an element of truth to it. We all know the feeling of "getting used to" ideas that once seemed strange to us, so that ... |
11,267 | <p>John Von Neumann once said to Felix Smith, "Young man, in mathematics you don't understand things. You just get used to them." This was a response to Smith's fear about the method of characteristics. </p>
<p>Did he mean that with experience and practice, one obtains understanding? </p>
| BCLC | 140,308 | <p>My 2 interpretations:</p>
<ol>
<li><p>To the <strong>arrogant</strong>, don't think you know that much. You don't know what you don't know. As <a href="https://math.stackexchange.com/users/88078/nero">Nero</a> aka <a href="https://www.facebook.com/nntaleb/posts/10152256017673375" rel="nofollow noreferrer">Nassim Nic... |
11,267 | <p>John Von Neumann once said to Felix Smith, "Young man, in mathematics you don't understand things. You just get used to them." This was a response to Smith's fear about the method of characteristics. </p>
<p>Did he mean that with experience and practice, one obtains understanding? </p>
| Jos van Nieuwman | 545,640 | <p>I actually reject most claims on this page. Coming from an (astro)physical background, pure mathematics was a blessing for me. It took only a few months for me to cry out to all my physics friends that this pure maths thing, <strong>as opposed to physics</strong>, is something <em>"that can actually be understood!"... |
389,532 | <p>I'm working out the following problem form Ahlfors' Complex Analysis text:</p>
<p>"Let $X$ and $Y$ be compact sets in a complete metric space $(S,d)$. Prove that there exist $x \in X,y \in Y$ such that $d(x,y)$ is a minimum."</p>
<p>My attempt:</p>
<p>Define $E:=\{d(x,y): x \in X,y \in Y \}$. We will prove that $... | Martin Citoler | 73,096 | <p>I just wanted to mention that your characterization of compactness is not quite correct. Compact is equivalent to closed and bounded in $\mathbb{R}^n$, but it is not true in general. In the case of metric spaces,a set is compact if and only if it is complete and totally bounded. </p>
<p>However, in this case is eno... |
3,469,392 | <p>Let <span class="math-container">$a_n$</span> be a recursive sequence where <span class="math-container">$a_1=42$</span> and for all <span class="math-container">$n$</span>:</p>
<p><span class="math-container">$$a_{n+1}=a_n+\frac{(-1)^n(1+n!)}{2^nn!}$$</span></p>
<p>After plugging some values of <span class="math-... | Fabio Lucchini | 54,738 | <p>First note that
<span class="math-container">$$a_{N+1}=a_1+\sum_{n=1}^N\frac{(-1)^n(1+n!)}{2^nn!}$$</span>
and that
<span class="math-container">$$\left|\frac{(-1)^n(1+n!)}{2^nn!}\right|\sim 2^{-n}$$</span>
as <span class="math-container">$n\to+\infty$</span>, so that the serie converges absolutely.</p>
|
522,179 | <p>Everyone says that the dot product is interpreted as the projection of A onto B (if you are dot producting A and B), but isn't that length just equal to |A|$\cos \left( \theta \right)$? Why does the dot product have an extra |B|?</p>
| G. Khanna | 416,891 | <p>I have also heard people say "the dot product is as the projection of A onto B .." But, of course, that is not entirely correct. Yes -- the dot product is fundamentally a projection; but the actual computation of the dot product is given by the equation</p>
<p><span class="math-container">$$a \cdot b = \| a \| \|... |
2,607,449 | <p>I'm trying to get intuition about why gradient is pointing to the direction of the steepest ascent. I got confused because I found that directional derivative is explained with help of gradient and gradient is explained with help of directional derivative.</p>
<p>Please explain what are the exact steps that lead fr... | Alejandro Nasif Salum | 481,187 | <p>The limit <span class="math-container">$\lim_{t\to 0} \frac{f(x_0+tv)-f(x_0)}t$</span> gives the <em>definition</em> of the derivative in the direction of the unit vector <span class="math-container">$v$</span> at <span class="math-container">$x=x_0\in \mathbb R^n$</span>, that is <span class="math-container">$\frac... |
2,607,449 | <p>I'm trying to get intuition about why gradient is pointing to the direction of the steepest ascent. I got confused because I found that directional derivative is explained with help of gradient and gradient is explained with help of directional derivative.</p>
<p>Please explain what are the exact steps that lead fr... | BRSTCohomology | 474,595 | <p>This question is best answered by starting from the definition of the derivative.</p>
<p>Let $f: \mathbb{R}^3 \to \mathbb{R}$. (basically the functions you're thinking of in your multivariable class). The good definition of the derivative (read: the one I like and the one that generalizes well to higher dimensions... |
2,607,449 | <p>I'm trying to get intuition about why gradient is pointing to the direction of the steepest ascent. I got confused because I found that directional derivative is explained with help of gradient and gradient is explained with help of directional derivative.</p>
<p>Please explain what are the exact steps that lead fr... | Acccumulation | 476,070 | <p>In basic calculus, you deal with functions that have one-dimensional input and one-dimensional output. When you take the derivative, you get another 1D -> 1D function. Once you start dealing with functions over more dimensions, however, things get much more complicated. The directional derivative is a way to simplif... |
2,153,580 | <p>I'm trying to prove that $K((x))/K(x)$ is not an algebraic extension, where $K$ is a field of characteristic $p>0$, and $K((x))$ is the field of fractions of $K[[x]]:=\{\sum_{i=0}^{\infty}a_ix^i\mid a_i\in K\}$.</p>
<p>To make it more concrete, I'm considering the case $K=\mathbb{Z}/p\mathbb{Z}$ and $x=p$, so th... | Lubin | 17,760 | <p>My only suggestion would be to try to imitate the ordinary proof of Liouville’s Theorem, which can be used to show that $\sum_n10^{n!}$ is transcendental over $\Bbb Q$. This would mean showing that $\sum_n t^{n!}$ is transcendental over $\kappa(t)$, for (say) $\kappa$ a finite field.</p>
|
779,328 | <p>I am almost a beginner in the topics of Fourier transform. So, I am asking this question here.</p>
<p>Let <span class="math-container">$n=3$</span> and let <span class="math-container">$\mu_t$</span> denote surface measure on the sphere <span class="math-container">$|x|=t$</span>. Then how do we show that
<span cla... | Community | -1 | <p><span class="math-container">$\def\Rbf{\mathbf{R}}$</span>
<span class="math-container">$\def\inta{\int_0^{\pi}}$</span>
<span class="math-container">$\def\intb{\int_0^{2\pi}}$</span></p>
<p>This is a very nice exercise for multivariable calculus. </p>
<p>Denote the surface measure of the unit sphere <span class="... |
17,885 | <p>In most education systems, Mathematics is a compulsory subject from primary school all the way to the start of university. A common reason given is that essential concepts like addition and multiplication are taught to the children. </p>
<p>But for many high school students, especially those who are keen on pursuin... | johnnyb | 9,594 | <p>Math teaches reasoning using scenarios that are exact enough for there to be concrete right and wrong answers. Many areas of life require reason, but mathematics forces students to actually engage in the discipline of finding the <em>right</em> answer, because there is only one right answer.</p>
<p>For example, so... |
372,181 | <p>The theory of real closed fields is decidable. The <a href="https://en.wikipedia.org/wiki/Hyperreal_number" rel="noreferrer">hyperreals</a> satisfy that theory, so we can interpret statements in the theory of real closed fields as being about hyperreals.</p>
<p>If we add a unary predicate for "is a standard rea... | Matt F. | 44,143 | <p>Here are some comments on quantifier elimination for this theory, focusing on what expansion of the language might make it work. This may be parallel to identifying a model companion, or determining the structures to consider in a more model-theoretic proof, as Emil Jerabek was proposing.</p>
<p>First, we will need ... |
1,313,216 | <p>Its my first time on here and my maths is poor so please be kind. I am working on a Masters dissertation focused on document clustering methods in which I would like to apply a weight based on the time interval between two documents. </p>
<p>I am looking for some help coming up with a function to express a time int... | Klaus Draeger | 65,787 | <p>One standard approach in this kind of situation is to consider some discount factor $d<1$ such that the value of an example decreases by this factor for each time unit which has passed, i.e. for an example with initial value $v_0$ and age $t$, you would use the value $v = v_0 \cdot d^t$.</p>
|
2,243,646 | <p>Suppose a, b, c are positive real numbers such that </p>
<p>$$(1+a+b+c)\left(1+\frac 1a+\frac 1b+\frac 1c\right)=16$$
Then is it true that we must have $a+b+c=3$ ?</p>
<p>Please help me to solve this. Thanks in advance. </p>
| Michael Rozenberg | 190,319 | <p>By Cauchy-Schwarz $$(1+a+b+c)\left(1+\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)\geq\left(1\cdot1+\sqrt{a}\cdot\frac{1}{\sqrt{a}}+\sqrt{b}\cdot\frac{1}{\sqrt{b}}+\sqrt{c}\cdot\frac{1}{\sqrt{c}}\right)^2= 16$$
The equality occurs for $a=b=c=1$, which says that $a+b+c=3$.</p>
|
3,798,402 | <p>For any natural number <span class="math-container">$n$</span>, how would one calculate the integral</p>
<p><span class="math-container">$$ \int_{0}^{2 \pi} |1 - ae^{i\theta}|^n \ d \theta $$</span></p>
<p>where <span class="math-container">$a$</span> is a complex number such that <span class="math-container">$|a| ... | Felix Marin | 85,343 | <p><span class="math-container">$\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\exp... |
823,641 | <p>Find and classify the critical points of this function: $f(x,y)= (x^y)-(xy)$ in the domain $x>0, y>0$.</p>
<p>I am having trouble treating x and y as constants when taking partial derivatives.</p>
| AlainD | 67,021 | <p>Claude Leibovici approach is correct and elegant. However in the general case, you can avoid treating variables as constants, by using <em>derivatives</em> rather than <em>differentials</em>. </p>
<p>A derivative is an operator <em>d</em> such that $d(f+g) = d(f) + d(g) = df + dg$, and $d(fg) = f d(g) + g d(f) = f ... |
1,210,948 | <blockquote>
<p><strong>(Convergence of Types Theorem)</strong> Suppose that $F_n(u_nx+v_n) \Rightarrow F(x)$ and $F_n(a_nx+b_n) \Rightarrow G(x)$, where $u_n>0, a_n>0$ and $F$ an $G$ are non-degenerate. Then there exist $a>0$ an $b \in\mathbb R$ such that $a_n/u_n \to a$ and $(b_n-v_n)/u_n \to b$, and $F(ax... | ozoz | 226,985 | <p>It should be:
$$A^{-1} = \frac{1}{3} (A-2I)$$</p>
|
2,482,083 | <p>I want to know if the following is a true statement:</p>
<blockquote>
<p>If $A$ is real a symmetric and $V$ is orthogonal then $V^TAV=D$ where $D$ is a diagonal matrix. </p>
</blockquote>
<p>To prove it I can say </p>
<p>$$(V^TAV)^T=V^TA^TV = V^TAV.$$
So $V^TAV$ is itself symmetric. So if I know $V^TAV$ is tri... | egreg | 62,967 | <p>You're probably misunderstanding what your textbook or instructor is trying to tell you.</p>
<blockquote>
<p>If $A$ is <em>any</em> real square matrix with <em>all</em> real eigenvalues, there exists an orthogonal matrix $V$ such that
$$
V^TAV
$$
is triangular.</p>
</blockquote>
<p>This is done by induction ... |
2,204,834 | <p>I have to show by induction that this function is a multiple of 8. I have tried everything but I can only show that is multiple of 4, some hints? The function is
$$5^{n+1}+2\cdot 3^n+1 \hspace{1cm}\forall n\ge 0$$, because it is a multiple of 8, you can say that$$5^{n+1}+2\cdot 3^n+1=8\cdot m \hspace{1cm}\forall m\i... | Felix Marin | 85,343 | <p>$\newcommand{\bbx}[1]{\,\bbox[8px,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{\expo}[1]{\,\mathrm{e}^{#1}\,}
\n... |
2,561,245 | <p>Here is the question:
Let $f(x)$ be an integrable function on $[-1,1]$. $f(x)$ is continuous at $x=0$. Define $g(x)=\int_0^{\int_0^xf(n)dn}f(y)dy$ and prove $g'(0)=f(0)^2$.</p>
<p>My initial thought was to look at $g(0)$ and say when you take an integral from $a$ to $b$ and $a=b$ then the integral is $0$, but I do ... | Community | -1 | <p>We have, using differentiation under the integral sign, $$g’(x)=[\int_{0}^{\int_{0}^{x} f(n)\, dn} f(y) \, dy]’=f(\int_{0}^{x} f(n) \, dn)[\int_{0}^{x} f(n) \, dn]’ = f(\int_{0}^{x}f(n) \, dn) [f(x)]$$ </p>
<p>Thus, $$g’(0)=f(\int_{0}^{0} f(n)\, dn)[f(0)]=[f(0)][f(0)]=[f(0)]^2$$</p>
|
3,676,127 | <p>Is there a term in mathematics for the <strong>SET</strong> of all possible <span class="math-container">$n \times n$</span> zero-one matrices?</p>
<p>In other words, for a particular <span class="math-container">$n \in \mathbb{Z}^+$</span>, is there a term for the set of all possible <span class="math-container">$... | rschwieb | 29,335 | <p>I would call such a matrix an <a href="https://en.wikipedia.org/wiki/Adjacency_matrix" rel="nofollow noreferrer"><strong>adjacency matrix</strong></a> since basically that would be choosing to represent the relation as a graph with vertices in <span class="math-container">$S$</span>. (This was more relevant before <... |
1,887,315 | <p>$e$ is one of the most important numbers in our universe, it is everywhere. When I try to find out why the most common explanation is some reverse-engineered physics or finance problem. But these are just one off examples of why $e$ <em>is</em> important, they fall short of illuminating the origin of its significanc... | Michael Hardy | 11,667 | <p>\begin{align}
\frac d {dx} 10^x & = ( 10^x\cdot\text{constant}) \approx 10^x \cdot (2.3) \\[10pt]
\frac d {dx} 2^x & = ( 2^x \cdot \text{constant}) \approx 2^x\cdot (0.693)
\end{align}
etc. It is easy to show that the derivative of an exponential function is a constant multiple of the same exponential func... |
2,362,485 | <blockquote>
<p>Evaluate: $$\lim_\limits{x\to \pi/4} \frac {2-\csc^2 x}{1-\cot x}$$</p>
</blockquote>
<p>My Attempt:
\begin{align}\lim_\limits{x\to \pi/4} \frac {2-\csc^2 x}{1-\cot x}&=\lim_{x\to \pi/4} \frac {2-\csc^2 x}{1-\cot x}\\\\
&=\lim_{x\to \pi/4} \frac {2-\frac {1}{\sin^2 x}}{1-\frac {\cos x}{\sin x... | Math-fun | 195,344 | <p>Note first that $\sin^2 x+\cos^2=1$ hence we obtain $$2-\frac1{\sin^2x}=\frac{2\sin^2x-1}{\sin^2x}=\frac{\sin^2x-\cos^2x}{\sin^2x}=1-\cot^2x=(1-\cot x)(1+\cot x)$$</p>
<p>hence $$\lim_{x\to\frac{\pi}4}\frac{2-\frac1{\sin^2x}}{1-\cot x}=\lim_{x\to\frac{\pi}4}\left(1+\cot x\right)=2.$$</p>
<p>You could also use a <a... |
7,900 | <p>This semesters I am teaching a course to 12-13 year old kids in which they are supposed to learn the basics of spreadsheet usage.</p>
<p>I am having difficult in coming up with fun / interesting exercises to teach them the basic functions, such as average, max, minimum and the like. One of the biggest problems is h... | aparente001 | 4,956 | <ol>
<li><p>Write a formula that converts Centigrade to Fahrenheit or vice versa. Plug in some common temperatures for the climate where you live. They can print out the table of values, nice and big, and post it on the wall at home and at school. At your next meeting, have a little Conversion Bee, with little prize... |
4,048,223 | <blockquote>
<p><strong>Problem 20.3</strong>
If <span class="math-container">$R$</span> is a nonnegative random variable, then Markov’s Theorem gives an upper
bound on <span class="math-container">$Pr[R \geq x]$</span> for any real number <span class="math-container">$x \gt Ex[R]$</span>. If b is a lower bound on R, t... | peaceful breeze | 794,823 | <p>Since <span class="math-container">$R-b\ge 0$</span> by assumption, a simple application of the Markov's inequality gives <span class="math-container">$$\Pr(R\ge x)=\Pr(R-b\ge x-b)\le \frac{\text{Ex}[R-b]}{x-b}=\frac{\text{Ex}[R]-b}{x-b}.$$</span>
Now, observe that <span class="math-container">$$f(b):=\frac{\text{Ex... |
2,964,346 | <p>Suppose that <span class="math-container">$X_1,X_2,\cdots,X_n$</span> are iid sequence with pdf <span class="math-container">$\frac{2}{\pi (1+x^2)}\cdot 1_{(0,+\infty)}(x)$</span>. Denote <span class="math-container">$S_n$</span> as <span class="math-container">$S_n:=X_1+X_2+\cdots+X_n$</span>. Prove that there exit... | S.Wei | 389,872 | <p>Note that <span class="math-container">$\frac{S_n}{n\log n}$</span> can be divided into two parts:
<span class="math-container">$$\frac{S_n}{n\log n}=\frac{\sum_{k=1}^nX_k 1_{X_k\le n\log n}}{n\log n}+\frac{\sum_{k=1}^nX_k 1_{X_k> n\log n}}{n\log n}$$</span>
The second part converges to zero in probability, and o... |
2,964,346 | <p>Suppose that <span class="math-container">$X_1,X_2,\cdots,X_n$</span> are iid sequence with pdf <span class="math-container">$\frac{2}{\pi (1+x^2)}\cdot 1_{(0,+\infty)}(x)$</span>. Denote <span class="math-container">$S_n$</span> as <span class="math-container">$S_n:=X_1+X_2+\cdots+X_n$</span>. Prove that there exit... | Shiina | 607,080 | <p>Let <span class="math-container">$S_n' = \sum_{k=1}^n X_k 1_{\{|X_k|\le a_n\}}$</span>. Clearly,</p>
<p><span class="math-container">$$
\mathbb P\left(\left|\frac{S_n - b_n}{a_n}\right|>\varepsilon\right)\le\mathbb P(S_n\ne S_n') + \mathbb P\left(\left|\frac{S_n' - b_n}{a_n}\right|>\varepsilon\right).
$$</spa... |
663,218 | <p>Basic question in linear algebra here. $T$ is a linear transform from $\mathbb R^n$ to $\mathbb R^n$ defined by $T(v)=Av$, $A\in \mathrm{Mat}_n(\mathbb R)$. We are given some inner product $\langle ,\rangle$ of $\mathbb R^n$. Does not have to be the standard one, just some random inner product.</p>
<p>let $T^*$ be ... | Daniel Fischer | 83,702 | <p>In general, the adjoint of $T$ is not given by multiplication with the transpose of the matrix of $T$.</p>
<p>You can represent an inner product $\langle\,\cdot\,,\,\cdot\,\rangle$ by a matrix, let's call it $S$,</p>
<p>$$\langle x,y\rangle = {}^tx Sy.$$</p>
<p>Then let $A$ be the matrix representing $T$, and $B$... |
2,027,529 | <p>I want to show that $\pi_1(S^1\vee S^1) = \mathbb{Z}*\mathbb{Z}. $ I know that it follows from the Seifert-van Kampen theorem, but we haven't talked about that in class. </p>
<p>We are given the following hint:</p>
<blockquote>
<p>Define a group homomorphism $\mathbb{Z}*\mathbb{Z}\to\pi_1(S^1\vee S^1)$ via the t... | JHF | 50,427 | <p>The left hand side is the free group on two letters, say $a$, $b$. In terms of generators and relations, this free group is generated by two elements $a$ and $b$ subject to <em>no</em> nontrivial relations. A typical element of this group looks like $aba^{-2}b^3 a^{-1} b$, i.e., a word in $a$, $b$, $a^{-1}$, and $... |
1,593,899 | <p>Let $f = u + iv$ be a holomorphic function on unit (open) disk $\mathbb D^2$.</p>
<p>and Suppose $v$ is nonnegative and $v(0)=0$.</p>
<p>Can we determine all such $f$?</p>
<p>I wanted to use schwarz lemma, but I couldn't catch anything about bound condition.</p>
<p>I think it is about liouville's theorem.</p>
<... | Anubhav Mukherjee | 72,329 | <p>$ f(0)=r $ some real number. Then open mapping theorem says that if $f$ is non-constant , then there is a open nbd of $0$ which maps onto an open nbd of $r$ ... which implies there exists a point lets say $ r_1 -ir_2$ where $r_1, r_2$ +ve real number, is in the image. But $v$ is non-negative. contadiction. So $f$ is... |
2,523,581 | <p>I'm a bit rusty on my complex numbers, how would you solve the following problem on paper?</p>
<blockquote>
<p>Determine and sketch (graph) the set of all complex numbers of form:
$$z_n=\frac{2n+1}{n-i},n\in\mathbb R$$</p>
</blockquote>
<p>Rationalizing yields $$S=\left\{z_n\in\mathbb C : z_n=\frac{n + 2 n^2}{... | José Carlos Santos | 446,262 | <p>I'll use $x$ instead of $n$.</p>
<p>Note that$$\frac{2x^2+x}{x^2+1}=1+\frac{x^2+x-1}{x^2+1}=1+\frac{2x^2+2x-2}{2(x^2+1)}$$and that$$\frac{2x+1}{x^2+1}=\frac12+\frac{-x^2+4x+1}{2(x^2+1)}.$$So$$\frac{2x+1}{x-i}=1+\frac i2+\frac{2x^2+2x-2}{2(x^2+1)}+\frac{-x^2+4x+1}{2(x^2+1)}i.$$Now, observe that$$\left(\frac{2x^2+2x-... |
3,681,254 | <p><span class="math-container">$$\int_{0}^{\frac{\pi}{2}}{e^{-x}} \cos3xdx$$</span>
using the intergral by parts formula
<span class="math-container">$$\int{f(x).g(x)'}\mathrm{d}x = (f(x).g(x)) - \int{f'(x)g(x)dx}$$</span>
so it should be
<span class="math-container">$${{{e^{-x}}.\frac{1}{3}\cos(3x)}} - \int_{0}^{\fra... | Community | -1 | <p><strong>Hint:</strong> </p>
<p>By parts, twice,</p>
<p><span class="math-container">$$\int e^{-x}f(x)\,dx=-e^{-x}f(x)+\int e^{-x}f'(x)\,dx=-e^{-x}f(x)-e^{-x}f'(x)+\int e^{-x}f''(x)\,dx$$</span></p>
<p>so that</p>
<p><span class="math-container">$$\int e^{-x}(f(x)-f''(x))\,dx=-e^{-x}(f(x)+f'(x)).$$</span></p>
|
2,338,797 | <p>A bag contains $3$ red balls and $4$ blue balls. Find the probability of picking, at random, two balls of </p>
<p>a) the same color </p>
<p>b) different colors</p>
| David G. Stork | 210,401 | <p>b) Probability of different is probability of red on first (3/7) times probability of blue given a red has been chosen (4/6), plus the probability of blue on first (4/7) times probability of red given a blue has been chosen (3/6), or: 4/7.</p>
<p>You can do a) by yourself.</p>
|
267,161 | <p>I would like to write <code>x^5 + 1/x^5</code> in terms of <code>x^2 + 1/x^2</code>, <code>x^3 + 1/x^3</code> and <code>x + 1/x</code> or express <code>x^6 + 1/x^6</code> in terms of <code>x^2 + 1/x^2</code> and <code>x^4 + 1/x^4</code> etc.</p>
<p>How can I do that in <em>Mathematica</em>?</p>
| Artes | 184 | <p>There is a special approach with <code>Solve</code> eliminating appropriate variables, e.g. in the following <code>x</code> should be eliminated</p>
<pre><code>a/. Solve[{x^5 + 1/x^5 == a, x^3 + 1/x^3 == b, x^2 + 1/x^2 == c, x + 1/x == d},
{a, b, d}, {x}]
</code></pre>
<blockquote>
<pre><code>{-Sqrt[2 + 5 c - ... |
4,164,405 | <p>Grothendieck has proven that whenever <span class="math-container">$X\longrightarrow\operatorname{Spec}(A)$</span> is a proper morphism of Noetherian schemes, <span class="math-container">$F$</span> is coherent over <span class="math-container">$X$</span> and flat over <span class="math-container">$\operatorname{Spe... | The Thin Whistler | 241,032 | <p>The argument becomes easy if <span class="math-container">$A$</span> is assumed hereditary. In the construction given in [https://amathew.wordpress.com/2011/01/09/the-grothendieck-complex], all <span class="math-container">$K^{i}$</span> are free for <span class="math-container">$i\geq 1$</span> and therefore projec... |
650,291 | <p>I'm not so good at combinatorics, but I want to know if my answer for this question is right. Originally this question is written in spanish and it says:</p>
<blockquote>
<p>Se dispone de una colección de 30 pelotas divididas en 5 tamaños distintos y 6 colores diferentes de tal manera que en cada tamaño hay los s... | Davis Yoshida | 113,908 | <p>Not quite correct. Here's a breakdown of the choices you need to make.</p>
<ol>
<li>You will need to choose two sizes of ball, so do so: $\binom{5}{2}$</li>
<li>For the first size, choose two colors: $\binom{6}{2}$</li>
<li>For the second size, choose two colors: $\binom{6}{2}$</li>
</ol>
<p>Combine these to count... |
1,712,933 | <p>I am slightly ashamed to be asking this, but I have been recently reflecting on changing variables in very simple problems. If I missed a question that already discusses this please point it out to me and I will delete this one.
Anyhow writing this will probably be a learning experience.</p>
<p>Directly from the Wi... | Eric S. | 263,514 | <p>I must admit I never gave such much thought into this method as you did, and that I do not understand the formal introduction on the Wikipedia page (I assume you refered to <a href="https://en.wikipedia.org/wiki/Change_of_variables">this one</a>). But perhaps this may provide you with some more insight into why the ... |
1,712,933 | <p>I am slightly ashamed to be asking this, but I have been recently reflecting on changing variables in very simple problems. If I missed a question that already discusses this please point it out to me and I will delete this one.
Anyhow writing this will probably be a learning experience.</p>
<p>Directly from the Wi... | Community | -1 | <p>By the rules of algebra,</p>
<p>$$x^6 - 9 x^3 + 8 = 0$$</p>
<p>is strictly equivalent to</p>
<p>$$(x^3)^2 - 9 x^3 + 8 = 0.$$</p>
<p>Then it makes no harm to substitute $u=x^3$ and solve</p>
<p>$$u^2-9u+8,$$</p>
<p>leading to a set of solutions $u\in S=\{s_k\}$. And this is equivalent to $x^3\in S$, or $x\in\{\... |
4,619,641 | <p>I need to find the image of the function <span class="math-container">$f(z) = z^2$</span> whose domain is <span class="math-container">${z: Re(z) > 0}$</span>. I first let <span class="math-container">$z = x + iy$</span>. Then <span class="math-container">$$w = f(z) = z^2 = (x+iy)^2 = (x^2-y^2) + 2xyi$$</span> He... | joriki | 6,622 | <p>Nice idea, but unfortunately not.</p>
<p>If we choose <span class="math-container">$x_i=x_j=y$</span> and <span class="math-container">$x_k=x_l=z$</span>, the second and third terms vanish by symmetry, and we’re left with <span class="math-container">$\mathbb E\left[y^2z^2\right]=\mathbb E\left[y^2\right]\mathbb E\l... |
4,181,235 | <p>The topology on <span class="math-container">$\mathcal D$</span> is the locally convex topology induced by seminorms like <span class="math-container">$\|\partial_\alpha f\|_\infty$</span>, where <span class="math-container">$\alpha=(d_1,\ldots,d_n) $</span> and <span class="math-container">$\partial_\alpha=\partial... | Phicar | 78,870 | <p><strong>Hints:</strong>
Notice that if <span class="math-container">$a<b=a+c$</span> then it is the same thing as the case <span class="math-container">$a=b$</span> because
<span class="math-container">$$\sum _{j=0}^n\binom{a+j}{b}=\sum _{j=-c}^{n-c}\binom{a+c+i}{b}=\sum _{j=0}^{n-c}\binom{b+i}{b},$$</span>
can y... |
13,230 | <p>Introduction:</p>
<p>Let A be a subset of the naturals such that <span class="math-container">$\sum_{n\in A}\frac{1}{n}=\infty$</span>. The <a href="https://en.wikipedia.org/wiki/Erd%C5%91s_conjecture_on_arithmetic_progressions" rel="nofollow noreferrer">Erdos Conjecture</a> states that A must have arithmetic progr... | Pete L. Clark | 1,149 | <p>Most of the point of this answer is to promote a piece of terminology:</p>
<p>Three years ago I first taught a number theory course at UGA in which I made the following definition: a subset <span class="math-container">$A$</span> of the positive integers is <b>substantial</b> if</p>
<p><span class="math-container">$... |
13,230 | <p>Introduction:</p>
<p>Let A be a subset of the naturals such that <span class="math-container">$\sum_{n\in A}\frac{1}{n}=\infty$</span>. The <a href="https://en.wikipedia.org/wiki/Erd%C5%91s_conjecture_on_arithmetic_progressions" rel="nofollow noreferrer">Erdos Conjecture</a> states that A must have arithmetic progr... | gowers | 1,459 | <p>I've never much liked Erdős's conjecture. Of course, I don't mean by that that I wouldn't love to solve it, or that I wouldn't be fascinated to know the answer. However, I think the precise form in which it is stated, with sums of reciprocals, gives it a mystique it doesn't quite deserve. The real question, it ... |
13,230 | <p>Introduction:</p>
<p>Let A be a subset of the naturals such that <span class="math-container">$\sum_{n\in A}\frac{1}{n}=\infty$</span>. The <a href="https://en.wikipedia.org/wiki/Erd%C5%91s_conjecture_on_arithmetic_progressions" rel="nofollow noreferrer">Erdos Conjecture</a> states that A must have arithmetic progr... | John H. Johnson | 9,949 | <p>There is a way you can recast Erdős conjecture into a statement about certain inclusions among various compact left and two-sided ideals. Such topological-algebraic statements, and a few combinatorial statements, are proved by <a href="http://mysite.verizon.net/nhindman" rel="noreferrer">Neil Hindman</a> in</p>
<blo... |
2,880,377 | <p>As the title states, I need to find the limit for $x\left(x + 1 - \frac{1}{\sin(\frac{1}{1+x})}\right)$ as $x \rightarrow \infty$, as part of a larger proof I am working on.</p>
<p>I believe the answer is 0. I think that to start, I can show that $\frac{1}{\sin(\frac{1}{1+x})} \rightarrow x + 1$ for large x. By loo... | Nosrati | 108,128 | <p>Let $y=\dfrac{1}{x+1}$ then $y\to0$ in
\begin{align}
\lim_{xto\infty}x(x + 1 - \frac{1}{\sin(\frac{1}{1+x})})
&= \lim_{y\to0}\dfrac{(1-y)}{y}\left(\dfrac{1}{y}-\dfrac{1}{\sin y}\right)\\
&= \lim_{y\to0}\dfrac{(1-y)(\sin y-y)}{y^2\sin y}\\
&= \lim_{y\to0}\dfrac{(1-y)\left(y-\dfrac16y^3+O(y^5)-y\right)}{y^... |
2,779,945 | <p>Picture the setup:</p>
<p>A particle of mass $m$ attached with a string of length $R$ spins in a vertical circle around a fixed peg with constant velocity $v>\sqrt{Rg}$ in the $x$-$z$ plane, acting under its weight $mg$. In this model, the probability that the string snaps when the string is at an angle $\theta\... | Sam Spedding | 202,717 | <p>Okay I think I've solved my own question here. We have the tension $T(\theta)$ is given by
$$
T(\theta) = \frac{mv^2}{R} - mg\sin\theta.
$$
Then to find the pdf, we normalise $T$ by dividing by the total "mass" of $T$, the $\textbf{polar}$ integral from $0$ to $2\pi$:
$$
\int_0^{2\pi} \int_0^{T(\theta)} r\; \mathrm{... |
3,439,833 | <p><strong>Problem:</strong>
Let <span class="math-container">$f:\mathbb{R} \rightarrow \mathbb{R}$</span> be differentiable.
Suppose <span class="math-container">$f(0)=0$</span>, and <span class="math-container">$f'(x) < \frac{1}{2}$</span> for all <span class="math-container">$a$</span>. Show <span class="math-con... | Quanto | 686,284 | <p>Use the mean value theorem </p>
<p><span class="math-container">$$\frac{f(4)-f(0)}{4-0} = f’(c) < \frac12$$</span></p>
<p>where <span class="math-container">$0<c<4$</span>. Then <span class="math-container">$f(4)<2$</span> follows. </p>
|
322,598 | <p><a href="https://en.wikipedia.org/wiki/Partially_ordered_set" rel="noreferrer">Partially ordered sets</a> (<em>posets</em>) are important objects in combinatorics (with <a href="https://gilkalai.wordpress.com/2019/02/05/extremal-combinatorics-v-posets/" rel="noreferrer">basic connections to extremal combinatorics</a... | Sam Hopkins | 25,028 | <p>The 1/3-2/3 conjecture is probably considered one of the most significant open problems about finite posets; see the Wikipedia page: <a href="https://en.wikipedia.org/wiki/1/3%E2%80%932/3_conjecture" rel="noreferrer">https://en.wikipedia.org/wiki/1/3%E2%80%932/3_conjecture</a>.</p>
|
322,598 | <p><a href="https://en.wikipedia.org/wiki/Partially_ordered_set" rel="noreferrer">Partially ordered sets</a> (<em>posets</em>) are important objects in combinatorics (with <a href="https://gilkalai.wordpress.com/2019/02/05/extremal-combinatorics-v-posets/" rel="noreferrer">basic connections to extremal combinatorics</a... | Timothy Chow | 3,106 | <p>What about the Stanley–Stembridge conjecture on (3+1)-free posets? Given a poset <span class="math-container">$P$</span> with vertex set <span class="math-container">$V = \{1,2,\ldots,n\}$</span>, define the symmetric function <span class="math-container">$X_P$</span> in countably many indeterminates <span cla... |
2,565,932 | <p>Completion of rational number at primes is clear. But when it comes to quadratic field is little tricky.
Can any one provide a material or illustrate, how does a quadratic field, when completed at a prime ideal becomes a p-adic field.</p>
| Lubin | 17,760 | <p>We don’t need to restrict to quadratic fields. </p>
<p>Take any finite algebraic extension $k$ of $\Bbb Q$, and its ring of integers $R$. There’s unique factorization of ideals into products of primes here, so if $\mathfrak p$ is a prime ideal of $R$, you can define, for any $z\in R$, the $\mathfrak p$-absolute val... |
244,875 | <p>\begin{align}
\min_{x} c^Tx \\
s.t.~Ax=b
\end{align}
Note that here $x$ is unrestricted. I need to prove that the dual of this program is given by
\begin{align}
\max_{\lambda} \lambda^Tb \\
s.t.~\lambda^TA\leq c^T
\end{align}</p>
<p>But in the constraint, I always get an equality (using what I learnt)
\begin{align}... | Inquest | 35,001 | <p>\begin{align}
\min_{x} c^Tx \\
s.t.~Ax=b
\end{align}</p>
<p>Is the same as:
\begin{align}
\min_{x} c^T(x^+-x^-) \\
s.t.~A(x^+-x^-)=b\\
x^+,x^-\geq 0
\end{align}
Is the same as:
\begin{align}
\min_{x} [c^T|-c^T]z \\
s.t.~[A|-A]z=b\\
z\geq 0
\end{align}
$$z=[x^T|-x^T]^T$$
Dual of this is :
\begin{align}
\max \quad b^... |
313,508 | <p>If anyone is familiar with Horowitz and Hill... its exercise 1.19</p>
<p>Show that all the average power delivered to the preceding circuit winds up in the resistor. Do this by computing the value of $V^2/R$. What is the power, in watts for a series circuit of a 1$\mu F$ capacitor and a 1k resistor placed across th... | Romeo Lampa | 70,074 | <p>This is a simple series $RC$ across a $110$ volt $60$ HZ with $R=1k\Omega$ and $C=1 \mu F$ capacitor.</p>
<p>First solve for $Xc=1/2(3.1416)60(.000001)$, the capacitive reactance of $C$ which $2.65k\Omega$. Then solve for $Z=\sqrt{R^2 + Xc^2}$, vector sum of $R$ and $C$ which is $2.83k\Omega$.</p>
<p>Then solve f... |
221,137 | <p>What's the difference between Fourier transformations and Fourier Series? </p>
<p>Are they the same, where a transformation is just used when its applied (i.e. not used in pure mathematics)?</p>
| paul garrett | 12,291 | <p>Fourier transform and Fourier series are two manifestations of a similar idea, namely, to write general functions as "superpositions" (whether integrals or sums) of some special class of functions. Exponentials $x\rightarrow e^{itx}$ (or, equivalently, expressing the same thing in sines and cosines via Euler's ident... |
221,137 | <p>What's the difference between Fourier transformations and Fourier Series? </p>
<p>Are they the same, where a transformation is just used when its applied (i.e. not used in pure mathematics)?</p>
| chaohuang | 27,973 | <p>The Fourier series is used to represent a periodic function by a discrete sum of complex exponentials, while the Fourier transform is then used to represent a general, nonperiodic function by a continuous superposition or integral of complex exponentials. The Fourier transform can be viewed as the limit of the Fouri... |
2,258,281 | <p>I have the following relation symbolized as $\sim$, defined on $\mathbb{Z}\times\mathbb{Z}$: </p>
<p>$$x\sim y \iff |x-y|=1$$</p>
<p>I would like to know how the transitive-reflexive closure looks like in this case, because this relation isn't transitive or reflexive, is it just an empty set?</p>
<p>In that case,... | Ashwin Ganesan | 157,927 | <p>You can obtain the argument by just looking at the given statements. You need to prove $\neg q \implies s$. So suppose $\neg q$ is true. We need to prove $s$ is true. Identify where in the given three propositions can we find $\neg q$ is true. We can take the contrapositive of the first proposition. </p>
<p>Observe... |
1,148,624 | <p>This is for teaching math. I'm wondering if someone knows some striking near-equalities between simple arithmetic expressions. I vaguely remember that such things exist (e.g., numbers that look alike out 10 digits, but that are really different), but I don't remember where I saw them or what they are. I think there ... | Christian Blatter | 1,303 | <p>Imagine the following scenario: A physicist modeling a certain system finds out that the number
$$\alpha:={1\over10}\sum_{n=-\infty}^\infty e^{-(n/10)^2}$$
is relevant for his problem. He numerically computes partial sums $s_N:=\sum_{|n|\leq N}$ and obtains, e.g.,
$$s_{10}=1.529\ldots,\quad s_{20}=1.7659\ldots,\quad... |
1,589,964 | <p>Given the LFT for a complex $z$,
\begin{align*}
\phi:z\mapsto \frac{2z+1}{z+2}.
\end{align*}
I'm asked about the image under $\phi$ of $C:=\left\{\left\lvert z+\frac25\right\rvert = \frac25\right\}$. I've parametrized this as $\gamma: \frac25 \exp(it) - \frac25$ and computed the image
\begin{align*}
w(t) = 2-\frac{1... | Brevan Ellefsen | 269,764 | <p>This probably refers to the complex conjugate, which, for any complex number, has the same real part and magnitude of imaginary part but opposite sign on te imaginary part. For example, if $f(x) = 5+4i$, then $\overline{f(x)} = 5-4i$</p>
|
1,589,964 | <p>Given the LFT for a complex $z$,
\begin{align*}
\phi:z\mapsto \frac{2z+1}{z+2}.
\end{align*}
I'm asked about the image under $\phi$ of $C:=\left\{\left\lvert z+\frac25\right\rvert = \frac25\right\}$. I've parametrized this as $\gamma: \frac25 \exp(it) - \frac25$ and computed the image
\begin{align*}
w(t) = 2-\frac{1... | Henricus V. | 239,207 | <p>That refers to the complex conjugation $\overline{x+iy} = x-iy$. It ensures that
$$ \langle f,g\rangle_{L_2 (\mathbb{R})}= \int_\mathbb{R} f(x) \overline{g(x)} \, dx
$$
is an inner product since it must satisfy $\langle f, f\rangle \geq 0$ and $\langle f, g\rangle = \overline{\langle g, f\rangle}$</p>
|
2,948,673 | <p>I was thinking about the squeeze theorem here. We can denote the <span class="math-container">$\left[\frac{1}{|x|}\right] =n$</span>, and then try something like:</p>
<p><span class="math-container">$$x^2(1+2+3+...+(n-1)+(n-1)) \leq x^2 \frac{n(n+1)}{2} \leq x^2(1+2+3+...+(n-1)+(n+1))$$</span></p>
<p>But I don't k... | Bernard | 202,857 | <p><strong>Hint</strong></p>
<p>First we may suppose <span class="math-container">$x> 0$</span> as well. Next,
<span class="math-container">$$\biggl\lfloor\frac1x\biggr\rfloor=n\iff n\le\frac1x<n+1, \enspace\text{ so }\enspace x\dots$$</span></p>
|
650,381 | <p>the definition of variance is $V(X) = E((X-E(X))^2 )$</p>
<p>For a discrete random variable: if we have put $Y = g(X)$ , where $g$ is a real function</p>
<p>$E(Y) = E(g(X)) = \sum\limits_{k} g(k)p_X(k)$ , where $p_X(k) = P(X = k) $ </p>
<p>how can i use this formula to show that:</p>
<p>$V(X) = \sum\limits_{... | DonAntonio | 31,254 | <p>$$|xI-A|=\left(\frac i3\right)^3\begin{vmatrix}x-1&2&-1\\2&x-1&-1\\-1&-1&x+2\end{vmatrix}=$$</p>
<p>$$=-\frac i{27}(x-1)^2(x+2)+4-2(x-1)-4(x+2)=$$</p>
<p>$$=-\frac i{27}\left(x^3-3x+\color{red}2-2x+\color{red}2+\color{red}4-4x-\color{red}8\right)=-\frac i{27}(x^3-9x)=$$</p>
<p>$$=-\frac i{... |
121,760 | <p>I want to find the minimal polynomial (over $\mathbb{Q}$) of: $k:=\sqrt[3]{7-\sqrt{2}}$.</p>
<p>With simple 'tricks' I got that: $P=(x^3-7)^2+2$ is a polynomial such that $P(k)=0$.</p>
<p>But I don't know if, or how to prove that $P$ is the minimal polynomial. How can I prove this/find the minimal polynomial ? </p... | Jyrki Lahtonen | 11,619 | <p>The only point of this answer is to supplement Bill Dubuque's answer, and to give a short
argument proving that $f(x)$ is irreducible in the ring $F_5[x]$.</p>
<p>Here $f(x)\equiv x^6+x^3+2$. Because $g(x)=x^2+x+2$ is irreducible, it is not a factor of
$$x^8-1=(x^4-1)(x^4+1)=(x-1)(x-2)(x-3)(x-4)(x^2-2)(x^2+2)$$
in... |
3,619,835 | <blockquote>
<p>Find the maximum of the function <span class="math-container">$f(x,y) = (a + x)(b + y)$</span> under the constraint <span class="math-container">$d = x + y$</span>, where <span class="math-container">$a$</span>, <span class="math-container">$b$</span>, <span class="math-container">$d$</span> are known... | trancelocation | 467,003 | <p>Since you are not acquainted with the trick mentioned in the comment, I elaborate on this a bit.</p>
<p>You surely know the binomial formula</p>
<ul>
<li><span class="math-container">$(u+v)^2 = u^2+2uv+v^2$</span></li>
</ul>
<p>From this it follows
<span class="math-container">$$(u+v)^2-(u-v)^2 = 4uv \Rightarrow ... |
1,808,019 | <p>I'm trying to better understand how the following group action looks like on a torus. The group $\mathbb{R}$ acts on the torus $C\times C=\{(e^{ix},e^{iy}):x,y\in \mathbb{R}\}$ ($C$ being the unit circle in $\mathbb{C}$) via $$t\cdot (e^{ix},e^{iy})\mapsto (e^{i(x+t)},e^{iy})$$</p>
<p>I am able (I think) to visuali... | 5xum | 112,884 | <p>Your inductive step, and in fact your whole inductive hypothesis, is wrong. Don't worry too much about it, induction causes a lot of confusion when you first encounter it. You just have to take it slowly and not jump over too many steps. Always second-guess everything you do, and make sure you have an explanation fo... |
916,705 | <p>I came across an example in Chapter-2 of Dummit and Foote(page-47) which says :$D_6$ is not a subgroup of $D_8$ ,the former is not even a subset of latter.I can't understand why is it not the subset of $D_8$?
How do we define one group as a subset of another?</p>
| Kyle Miller | 172,988 | <p>The statement "$D_6$ is not a subgroup of $D_8$" means there is no subgroup $H\subset D_8$ such that $D_6$ is isomorphic to $H$. This is easy to verify as $D_6$ has an element of order $3$, but no subgroup of $D_8$ has an element of order $3$.</p>
<p>It is possible, though not particularly useful, to regard $D_6$ ... |
3,973,025 | <p>Does the structure of the category of models of a given first order theory where the arrows are embeddings tell us anything interesting about the theory itself?</p>
<p>Two models <span class="math-container">$M$</span> and <span class="math-container">$M'$</span> are isomorphic if and only if there exists an embeddi... | Mark Kamsma | 661,457 | <p>We can say <em><strong>a lot</strong></em> about the first-order theory by just looking at its category of models. Let's use <em>elementary embeddings</em> instead of just embeddings. They are the 'right' arrows to take because they respect first-order logic. You could also work with just embeddings to get some sort... |
1,320,659 | <p>Is the ratio test applicable for testing convergence of infinite products?</p>
<p>In other words, consider the sequence $(a_i)_{i=1}^\infty$ of non-zero real numbers.</p>
<p>Also, consider the product $\displaystyle\mathcal P=\prod_{k=1}^\infty a_k$</p>
<p>Are the following statements true?</p>
<blockquote>
<p... | Ant | 241,144 | <p>A simple way to test if an infinite product converges or diverges is to test the related sum:</p>
<p>$$P=\prod_{k=1}^\infty {(a_k)}$$</p>
<p>can be turned into:</p>
<p>$$\ln{(P)}=S = \sum_{k=1}^\infty {(\ln{a_k})}$$</p>
<p>Now, the product $P$ converges if the sum $S$ converges. If $S$ diverges, then $P$ diverge... |
2,790,910 | <blockquote>
<p>Let <span class="math-container">$X \sim N (0, 1)$</span> and <span class="math-container">$Y ∼ N (0, 1)$</span> be two independent random variables, and define <span class="math-container">$Z = \min(X, Y )$</span>. Prove that <span class="math-container">$Z^2\sim\chi^2(1),$</span> i.e. Chi-Squared with... | Mike Earnest | 177,399 | <p>Let $\Phi$ be the standard normal cdf. First, find the cdf of $Z^2$. For any $z>0$,</p>
<p>$$
\begin{align}
P(Z^2\le z)
&=P(Z>-\sqrt{z})-P(Z>\sqrt{z})\\
&=P(X>-\sqrt{z})P(Y>-\sqrt{z})-P(X>\sqrt{z})P(Y>\sqrt{z})\\
&=(1-\Phi(-\sqrt{z}))^2-(1-\Phi(\sqrt{z}))^2.
\\&=(\Phi(\sqrt{z}))... |
4,115,308 | <p>Hi I have a problem where I need to solve the following set of equations:</p>
<p><span class="math-container">$$
v = U u
$$</span></p>
<p><span class="math-container">$$
u = 1 -Uv
$$</span></p>
<p><span class="math-container">$$
U^2 = u^2 + v^2
$$</span></p>
<p>I have tried subbing <span class="math-container">$u$<... | Ted Shifrin | 71,348 | <p>You're wrong. Let <span class="math-container">$B$</span> be the <span class="math-container">$n\times k$</span> matrix consisting of those columns, and let <span class="math-container">$B'$</span> be the <span class="math-container">$k\times k$</span> submatrix of <span class="math-container">$B$</span> with nonzer... |
107,425 | <p>Problem 1:</p>
<p>If $n$ people are seated in a random manner in a row containing $2n$ seats, what is the probability that no two people will occupy adjacent seats?</p>
<p>I know that the probability that $n$ people are seated in $2n$ rows is ${2n\choose n}$</p>
<p>I also know that the answer to this problem is $... | Dev Kinger | 993,349 | <p>Let us consider <span class="math-container">$6$</span> people and <span class="math-container">$3$</span> seats. Here is how we can arrange such that no two people are adjacent:</p>
<p>Let 3 people sit on any of the <span class="math-container">$2n$</span> seats:</p>
<p><span class="math-container">$\#P_{1}\#P_{2}\... |
2,709,921 | <p>My attempt is:
$$\lim_{h \to 0} \frac{f((2k+1)π/2+h) - f((2k+1)π/2)}{h} = \lim_{h \to 0} \frac{\sin^{-1}(\sin((2k+1)π/2+h)-\sin^{-1}(\sin((2k+1)π/2)}{h} =
\lim_{h \to 0} \frac{kπ+π/2+h-(2k+1)π/2}{h} =
1$$
Am I right upto this ?</p>
| Eric Wofsey | 86,856 | <p>No, this is not true. For instance, let $k$ be your favorite field, let $R$ be the subring $k[x,y,x^{-1}y,x^{-2}y,x^{-3}y,\dots]\subset k(x,y)$ and let $I$ be the ideal generated by $x$. Then $I^\infty$ is the ideal generated by $x^{-n}y$ for all $n\in\mathbb{N}$. In particular, $y\in I^\infty$, but $y\not\in (I^... |
2,709,921 | <p>My attempt is:
$$\lim_{h \to 0} \frac{f((2k+1)π/2+h) - f((2k+1)π/2)}{h} = \lim_{h \to 0} \frac{\sin^{-1}(\sin((2k+1)π/2+h)-\sin^{-1}(\sin((2k+1)π/2)}{h} =
\lim_{h \to 0} \frac{kπ+π/2+h-(2k+1)π/2}{h} =
1$$
Am I right upto this ?</p>
| tomasz | 30,222 | <p>For a more "natural" example, if you don't mind noncommutative rings, pick a field $k$ and consider the non-commutative ring $k[X^{\omega_1}]$ of polynomials over $k$ with exponents in $\omega_1$ (so that for any ordinals $\alpha$ and $\beta$, you have $X^\alpha\cdot X^\beta=X^{\alpha+\beta}$).</p>
<p>This is a loc... |
191,262 | <p>I have to fit a Cos^2 function to data I measured. The function is <span class="math-container">$a \cos^2(\frac{bx\pi}{180}+c)+d $</span> and I tried the FindFit and Linear Model Function. I have 4 datasets which I have to fit, the first one worked. The other three only yielded not usable fits. I am pretty new to Ma... | Ulrich Neumann | 53,677 | <p>You can solve the problem with FindFit-Option <code>Method->"NMinimize</code></p>
<pre><code>FindFit[data45, a Cos[(b x Pi)/180 + c]^2 + d, {a, b, c, d}, x,
</code></pre>
<p>Method -> "NMinimize"]
(<em>{a -> 300.499, b -> 0.990388, c -> 65.0408, d -> 28.431}</em>)</p>
|
1,789,033 | <p>Let $X$ be an irreducible affine variety. Let $U \subset X$ be a nonempty open subset. Show that dim $U=$ dim $X$.</p>
<p>Since $U \subset X$, dim $U \leq$ dim $X$ is immediate. I also know that the result is not true if $X$ is any irreducible topological space, so somehow the properties of an affine variety have t... | paf | 333,517 | <p>You should note that any nonempty open subset of an irreducible variety (or topological space) is dense : <a href="https://en.wikipedia.org/wiki/Hyperconnected_space" rel="nofollow">https://en.wikipedia.org/wiki/Hyperconnected_space</a>.
Then you can use the definition of the dimension to conclude.</p>
|
3,720,856 | <p><strong>Question:</strong>
Prove that <span class="math-container">$\sin(nx) \cos((n+1)x)-\sin((n-1)x)\cos(nx) = \sin(x) \cos(2nx)$</span> for <span class="math-container">$n \in \mathbb{R}$</span>.</p>
<p><strong>My attempts:</strong></p>
<blockquote>
<p>I initially began messing around with the product to sum iden... | Integrand | 207,050 | <p>Use <span class="math-container">$\sin(a)\cos(b)=\frac{1}{2}(\sin(a-b)+\sin(a+b))$</span>:
<span class="math-container">$$
\sin(nx) \cos((n+1)x)-\sin((n-1)x)\cos(nx)
$$</span>
<span class="math-container">$$
=\frac{1}{2}\left(\sin(-x)+\sin((2n+1)x)-\sin(-x)-\sin((2n-1)x) \right)
$$</span>
<span class="math-containe... |
3,768,099 | <blockquote>
<p>How does the quotient ring <span class="math-container">$\Bbb Z[x]/(x^2-x,4x+2)$</span> look like?</p>
</blockquote>
<p>Normally to solve this you play around with generators until you get something you can work with. I was unsuccessful in reducing it to something neat.</p>
<p>I have proven that <span c... | GreginGre | 447,764 | <p>We have <span class="math-container">$8(x^2-x)-2x(4x+2)=-12x=-3(4x+2)+6$</span>, so <span class="math-container">$6\in (x^2-x,4x+2)$</span> ,and <span class="math-container">$(x^2-x,4x+2)=(6,x^2-x,4x+2)$</span>.</p>
<p>Hence, your ring <span class="math-container">$R$</span> is isomorphic to <span class="math-contai... |
3,768,099 | <blockquote>
<p>How does the quotient ring <span class="math-container">$\Bbb Z[x]/(x^2-x,4x+2)$</span> look like?</p>
</blockquote>
<p>Normally to solve this you play around with generators until you get something you can work with. I was unsuccessful in reducing it to something neat.</p>
<p>I have proven that <span c... | user26857 | 121,097 | <p>We have
<span class="math-container">$$(x^2-x,4x+2)=(x,4x+2)(x-1,4x+2)=(2,x)(6,x-1).$$</span></p>
<p>The ideals <span class="math-container">$(2,x)$</span> and <span class="math-container">$(6,x-1)$</span> are comaximal, and by CRT <span class="math-container">$$\mathbb Z[x]/(x^2-x,4x+2)\simeq\mathbb Z[x]/(2,x)\time... |
232,387 | <p>A discrete random variable $X$ of values in $\mathbb N$ verifies the property that
$$P(X=k)=\cfrac 23 (k+1)P(X=k+1)$$
What is the distribution of $X$?</p>
<p>I found that
$$P(X\ge 0)=\sum_{k=0}^\infty P(X=k)=\sum_{k=0}^\infty\cfrac 23 (k+1)P(X=k+1)=\cfrac 23\sum_{k=1}^\infty kP(X=k)=\cfrac 23\text E(X)=1$$
$\ \ \ ... | Ross Millikan | 1,827 | <p>No, take the vectors $(0,1)$ and $(1,0)$. The average is $(\frac 12, \frac 12)$. For an even simpler example, try $(1,0)$ and $(-1,0)$</p>
|
199,889 | <p>What are applications of the theory of Berkovich analytic spaces? The analytification $X \mapsto X^{\mathrm{an}}$</p>
| Adam Epstein | 15,819 | <p>Jan Kiwi (Duke Journal) used Berkovich spaces to give the first proof of my conjecture that any sequence of quadratic rational maps which divergence in moduli space (of Mobius conjugacy classes) has at most two "rescaling limits". </p>
|
1,873,825 | <p>I can see why linear regression is linear, i.e., because it is represented by a line, but what does regression have to do with the term as a whole? </p>
<p>What is the meaning this word contributes to the term?</p>
| Snufsan | 122,989 | <p>The idea of linear regression is to infer from dataset the dependence of some random variable in other variable, what the term regression mean in this context is that from the dataset we regress to some kind of dependency that brought us to that data.</p>
|
530 | <p>Tau ($\tau = 2 \pi$) has more merits in its application, but pi is the established standard in industry and education. Is the trade-off of teach-ability of circle concepts worth the subsequent confusion due to pi's omnipresence?</p>
<p>How can both be most effectively taught in order to maximize student's understan... | EuYu | 97 | <p>I feel that it is perhaps a little irresponsible to teach $\tau$ instead of $\pi$. As a first introduction, it is the norm which should be taught: teaching a rare alternative to $\pi$ only serves to confuse students, especially when almost all available resources use $\pi$ instead of $\tau$. Imagine a student's conf... |
240,038 | <p>How can I compute this integral:
$$I=\int\frac{1-(au+1)\exp(-au)}{u^{2}}du$$
$$$$Can anyone help me in this case?</p>
| DonAntonio | 31,254 | <p>Seems you have</p>
<p>$$I=\int\frac{1-axe^{-ax}-e^{-ax}}{x^2}dx=\int\frac{dx}{x^2}-a\int\frac{e^{-ax}}{x}du-\int\frac{e^{-ax}}{x^2}dx$$</p>
<p>making parts for the third integral:</p>
<p>$$u=e^{-ax}\;,\;u'=-ae^{-ax}\;\;;\;\;v'=\frac{1}{x^2}\;,\;v=-\frac{1}{x}$$</p>
<p>we get</p>
<p>$$\int\frac{e^{-ax}}{x^2}dx=-... |
2,922,290 | <p>Lemma 2.2</p>
<p>Let $(X,d)$ be a compact metric space and suppose that $U$ is a clopen subset of $X$. </p>
<p>There exists $\delta >0 $ s.t $d(x,y) \geq \delta $ $\forall x \in U $ and $y\in X-U $</p>
<p>Proof attempt</p>
<p>Suppose that it is not true then for all $\delta >0 $ $\exists x \in U $ and $\e... | Henno Brandsma | 4,280 | <p>Note that $U$ is itself compact (being closed in $X$). Also $\{U\}$ is an open cover of $U$, as $U$ is open. By compactness of $U$ this has a <a href="https://en.wikipedia.org/wiki/Lebesgue%27s_number_lemma" rel="nofollow noreferrer">Lebesgue number</a> $\delta>0$. This means that for all $x \in U$, $B(x,\delta)$... |
2,122,149 | <p>Using the standard topology on $\mathbb{R}^d$, interior of a subset $S$ of topological space $M$ is defined as the set of all elements of $S$ except the boundary of $S$. </p>
<p>However, I only understand the boundary in this sense to be defined with the standard topology on $\mathbb{R}^d$, so how would we define b... | 5xum | 112,884 | <p>By definition, the interior of a set $S$ is defined as the largest open set contained in $S$.</p>
<p>For this definition to "work", all you need is a topology, there is no need for it to be the standard topology on $\mathbb R^d$.</p>
<hr>
<p>Equivalently, you can define the interior of $S$ (denoted $\mathrm{int}(... |
560,792 | <p>Here it is, I know the answer is 49! (I found it by trial and error!) I just need to know how I can show this in Working!</p>
<p>Thanks,
Alex</p>
| Dan Shved | 47,560 | <p>Start with two equations:
$$
\begin{align}
\frac{a_1 + a_2 + \ldots + a_7}{7} &= 65 \\
\frac{a_1 + a_2 + \ldots + a_7 + a_8}{8} &= 63.
\end{align}
$$
Now multiply the first one by $7$, the second one by $8$, and subtract. You get $a_8 = 8 \cdot 63 - 7 \cdot 65 = 49$.</p>
|
4,068,493 | <p><a href="https://i.stack.imgur.com/Rzj9N.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/Rzj9N.jpg" alt="triangle diagram" /></a></p>
<p>Given that:</p>
<ol>
<li>[BM) is the bisector of the angle ABC.</li>
<li>(BM) and (AN) are parallel straight lines.</li>
</ol>
<p>I am trying to prove that the t... | Joffan | 206,402 | <p>Not using Thales's intercept theorem, but:</p>
<p><span class="math-container">$AN \parallel BM$</span>, so from <span class="math-container">$AB$</span> crossing <span class="math-container">$\color{#092}{\angle NAB} = \color{#02E}{\angle ABM}$</span> and from <span class="math-container">$CN$</span> crossing <span... |
14,167 | <p>I randomly pick a natural number <em>n</em>. Assuming that I would have picked each number with the same probability, what was the probability for me to pick <em>n</em> before I did it?</p>
| Shai Covo | 2,810 | <p>If you want to choose a natural number from the set $\lbrace 1,2,3,\ldots \rbrace$ of all natural numbers, then user3533 is right. But if you pick a natural number from a finite set, say $A = \lbrace 1,2,\ldots,N \rbrace$, each number with the same probability, then this probability must be the reciprocal of the num... |
14,167 | <p>I randomly pick a natural number <em>n</em>. Assuming that I would have picked each number with the same probability, what was the probability for me to pick <em>n</em> before I did it?</p>
| joriki | 6,622 | <p>There's an aspect that doesn't seem to have been mentioned in the other answers yet.</p>
<p>The question presupposes that it is possible to pick each natural number with the same probability. Since this is not possible, it raises the question what may have led to the assumption that it should be possible. It's inst... |
1,073,730 | <p>Is it possible to prove the proposition</p>
<p>if $P \land Q \Rightarrow R$</p>
<p>then</p>
<p>$P \land R \Rightarrow \lnot Q $</p>
<p>or in other words</p>
<p>($P \land Q \Rightarrow R) \Rightarrow (P \land R \Rightarrow \lnot Q )$</p>
| Julian Rachman | 200,808 | <p>First, I would like you to think about the relation between $\cup$ and $\land$, and $\cap$ and $\lor.$ Then, prove this statements using true and false propositional definitions. Once done, you will know how to prove this problem on your own.</p>
|
3,315,516 | <p>A convex function is locally Lipschitz on a closed and bounded set. This implies that a nonconvex function may not be locally Lipschitz. Is every nonconvex function not locally Lipschitz? </p>
| Community | -1 | <p>No, that would be ridiculous. For instance, every <span class="math-container">$C^1$</span> function is locally Lipschitz, but it needs not be convex.</p>
|
737,614 | <p>I'm confused as to how to finish answering this question. </p>
<p>Compute $$\sum_{i=28}^n (3i^2 - 4i + \frac{5}{7^i}) $$</p>
<p>I end up with
$$ 3[\frac{n(n+1)(2n+1)}{6}] - 4[\frac{n(n+1)}{2}] + [\frac{k^n(5(-log(7))^n}{n!}]$$</p>
<p>I'm not entirely sure if the last part of my answer is right but my main quest... | Michael Albanese | 39,599 | <p>As David mentions in the comments, you are correct. The rule you refer to is often called the degree sum formula or the <a href="http://en.wikipedia.org/wiki/Handshaking_lemma" rel="nofollow">handshaking lemma</a>. The same rule can be used to show the following more general statement: </p>
<blockquote>
<p>Any gr... |
2,165,066 | <p>I'm asked to show that if $X$ and $Y$ are independent exponential random variables with parameter <img src="https://latex.codecogs.com/gif.latex?%5Clambda&space;=1" title="\lambda =1" />, then <img src="https://latex.codecogs.com/gif.latex?X/(X+Y)" title="X/(X+Y)" /> has a Beta distribution.</p>
<p>Up to know ... | V. Vancak | 230,329 | <p>Define $ U = X/(X+Y) = g_u(X,Y) $ and $V = X+Y = g_v(X,Y)$, thus
$$
f_{U,V}(u,v) = f_X (g_u^{-1} (u,v))f_{Y} (g_v^{-1} (u,v))| \det\frac{\partial(X,Y)}{\partial(u,v)} |
$$
where $X =UV$ and $Y = V (1-U)$, so
$$
| \det\frac{\partial(X,Y)}{\partial(u,v)} |=v.
$$
As such,
$$
f_{U,V}(u,v)=e^{-uv}e^{-v(1-u)}v=ve^{-v... |
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