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 |
|---|---|---|---|---|
354,365 | <p>Let $X_i$ be pairwise-uncorrelated random variables, $\forall\,i \in \mathbf{n} \equiv \{0,\dots,n-1\}$, with identical expectation value $\mathbb{E}(X_i)=\mu$, and identical variance $\mathrm{Var}(X_i)=\sigma^2$. Also, let $\overline{X}$ be their average, $\frac{1}{n}\sum_{i\in \mathbf{n}} X_i$.</p>
<p>Then, as s... | DBS | 192,092 | <p>For this example, let hypothetical matrix $A$ be 3x3. Let the columns of $A$ have a linear dependence relation satisfied by a non-zero vector $\mathbf{x}$: $ \ x_1\mathbf{a_1}+x_1\mathbf{a_2}+x_1\mathbf{a_3}=\mathbf{0}$. Here, $\mathbf{a_1}, \mathbf{a_2}, \mathbf{a_3}$ are column vectors of $A$ and $x_1, x_2, x_3$... |
247,335 | <p>Prove, formally that:
$\log_2 n! \ge n$ for all integers $n>3$. </p>
<p>Hint: first prove that $n! ≥2^n$, for all integers $n >3$.</p>
<p>So far what I have: </p>
<p>Base case, $n = 4$,</p>
<p>$4! = 24$</p>
<p>$2^4 = 16$.</p>
<p>Therefore, it is true when $n = 4$.</p>
<p>So how do I proceed from here?</... | Dirk Dinther | 47,913 | <p>Induction. You just stated the start case for $n=4$ already. The induction step then takes the statement for $n$ to $n+1$. So we have $n! > 2^n$ given. Looking at $n+1$ we get $(n+1)!=n!*(n+1)>n!*2>2^n*2 = 2^{(n+1)}$. The first inequality uses n>3. So in total $(n+1)!>2^{n+1}$ if already $n!>2^n$ and ... |
4,561,921 | <p>I'm studying newtonian dynamical systems and they can be described by the differential equation
<span class="math-container">$$1)\space m\ddot{x} = F(x)$$</span>
supposing <span class="math-container">$F$</span> sufficiently regular we could define the potential <span class="math-container">$V$</span> as its primit... | Enrico M. | 266,764 | <p>Consider a simple example: the gravitational potential <span class="math-container">$V = -mgx$</span> where <span class="math-container">$x$</span> is the height.</p>
<p>Then <span class="math-container">$$F = -\dfrac{\text{d}V}{\text{d}x} = mg$$</span> which is indeed the weight force, as we expect.</p>
<p>In this ... |
4,128,041 | <p>Assume <span class="math-container">$Z,B \in C^{2 \times 2}$</span> and that <span class="math-container">$c \in C$</span> is an eigenvalue of <span class="math-container">$Z$</span> and <span class="math-container">$u \in C$</span> is an eigenvalue of <span class="math-container">$B$</span>. Then <span class="math... | reema alhamdan | 860,262 | <p>Maybe if the two matrices were diagonal in respect with the same basis. Then the matrix which is the addition of these two matrices will have its Eigen values as the addition of the two earlier matrices.</p>
|
75,421 | <p>Can someone help me finish my solution?</p>
<p><strong>Question:</strong> Show that there are sets $A_{ij}$ for $i,j$ ∈ $\mathbb N$ such that for no <em>countable</em> $\space$H$\subseteq\mathbb N^{\mathbb N}$</p>
<p>$\bigcup_{i=0}^\infty\Bigg(\bigcap_{j=0}^\infty{A_{ij}}\Bigg)=\bigcap\Bigg\lbrace\Bigg(\bigcup... | user18096 | 18,096 | <p>If we define $A_{ij}$ as below then it will turn L.H.S into $\emptyset$ and $R.H.S$ empty for only function $g(i)$ but $\{x\}$ for any $h\in H$</p>
<p>$\forall i\in \mathbb N$</p>
<p>define $A_{ij}$ as below:-</p>
<p>$A_{i0}=\emptyset$ $\space$ if $g(i)=0$</p>
<p>$A_{i0}=\{x\}$ $\space$if $g(i)=1$</p... |
2,496,309 | <p>i am wondering if the unity elements of a ring form a ring ? In other words do they form an abelian group under addition ? I have tryied but i have not reached to a conclusive answer. Thanks for any comment.</p>
| Siminore | 29,672 | <p>Hint: as $h \neq 0$,
$$
\frac{h}{\sqrt{h+4}-2} = \frac{h(\sqrt{h+4}+2)}{h+4-4} = \sqrt{h+4}+2
$$</p>
|
84,036 | <p>I was doing an optimization but facing a problem getting what exactly Minimize function do. I run the following code:</p>
<pre><code> Log1[x_] := If[x == 0, 0, Log2[Abs[x]]];
VEntropy[x_] := -(x Log1[x] + (1 - x) Log1[1 - x]);
Prob[a_, b_, x_, y_] := 1 - 1/((a/x)^2 + (b/y)^2);
Cost[a_, b_, x_, y_] :=... | MarcoB | 27,951 | <p>I think I finally understand what you were asking (takes me a while to get in gear in the morning...), and I believe that you are simply running into numerical problems. </p>
<p>In fact, according to <a href="https://reference.wolfram.com/language/ref/Minimize.html" rel="nofollow noreferrer">its documentation</a>, ... |
468,291 | <p>I want to evaluate
$$\lim_{n \to \infty} n^{3/2}\int_0^1 \frac{x^2}{(1+x^2)^n}\ dx$$</p>
<p>All that I needed is an intergrable control function $g(\cdot)$ independent of $n \in \mathbb{N}$ such that $n^{3/2} \frac{x^2}{(1+x^2)^n}\leq g(x)$, but I do not find direct control function anyway....</p>
| Community | -1 | <p>By the change variable $x^2=\frac{u}{n} $ we find
$$n^{3/2}\int_0^1 \frac{x^2}{(1+x^2)^n}\ dx=\frac{1}{2}\int_0^n\frac{\sqrt{u}}{(1+\frac{u}{n})^n}du\to\frac{1}{2}\Gamma(\frac{3}{2})=\frac{\sqrt{\pi}}{4}$$</p>
|
468,291 | <p>I want to evaluate
$$\lim_{n \to \infty} n^{3/2}\int_0^1 \frac{x^2}{(1+x^2)^n}\ dx$$</p>
<p>All that I needed is an intergrable control function $g(\cdot)$ independent of $n \in \mathbb{N}$ such that $n^{3/2} \frac{x^2}{(1+x^2)^n}\leq g(x)$, but I do not find direct control function anyway....</p>
| 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... |
4,016,918 | <p><span class="math-container">$$_2F_1(\frac{1}{2},-\frac{1}{2};\frac{1}{2}; \sin^{2}(x))=\cos(x)$$</span></p>
<p>I plug these values into the definition of the hypergeometric function:</p>
<p><span class="math-container">$$_2F_1(a,b,c,x)=\displaystyle\sum_{n=0}^{+\infty}\dfrac{a^{\overline{n}}b^{\overline{n}}}{c^{\ov... | metamorphy | 543,769 | <p>Actually <span class="math-container">$_2F_1(a,b;a;z)=(1-z)^{-b}$</span> for <span class="math-container">$|z|<1$</span> and any <span class="math-container">$a,b$</span> (well, with <span class="math-container">$a\notin\mathbb{Z}_{\leqslant 0}$</span>). Indeed, both are equal to <span class="math-container">$\su... |
4,052,739 | <p>Given fundamental groupoid <span class="math-container">$\Pi_1(S^1)$</span> of the circle, how can one define a topology on it? The information on <a href="https://ncatlab.org/nlab/show/fundamental+groupoid#topologizing_the_fundamental_groupoid" rel="nofollow noreferrer">nlab</a> did little help other than the fact ... | Connor Malin | 574,354 | <p>For CW complexes (like <span class="math-container">$S^1$</span>), the topology on the fundamental groupoid will be discrete, i.e. it is just a usual groupoid. The idea for an arbitrary space is that the morphism sets in the fundamental groupoid is a quotient of the path space from point a to point b. Quotients have... |
388,014 | <p>You are offered a contract on a piece of land which is worth $1,000,000$ USD $70\%$ of the time, $500,000$ USD $20\%$ percent of the time, and $150,000$ USD $10\%$ of the time. We're trying to max profit.</p>
<p>The contract says you can pay $x$ dollars for someone to determine the land's value from which you can d... | zyx | 14,120 | <p>There is no unique arbitrage-free solution to the pricing problem with $3$ outcomes, so you will need to impose more assumptions to get a numerical value for the land.</p>
|
388,014 | <p>You are offered a contract on a piece of land which is worth $1,000,000$ USD $70\%$ of the time, $500,000$ USD $20\%$ percent of the time, and $150,000$ USD $10\%$ of the time. We're trying to max profit.</p>
<p>The contract says you can pay $x$ dollars for someone to determine the land's value from which you can d... | Marc Shivers | 24,972 | <p>To get a solution, you'll need to specify an objective that you are trying to maximize. </p>
<p>If you are just maximizing the expected profit, then your $\$15,000$ answer is correct. </p>
<p>Typically, however, in finance we assume people are risk-averse. A standard objective function in that case is $E[\text{... |
2,239,056 | <p>Trying to prove that for any two matrices $A,B$ representing the same symmetric bilinear form $f:V \times V \to \mathbb{R}$, there is an invertible matrix $P$, such that $B = P^{T}AP$ ?</p>
<p>I have a method for constructing such $P$ by repeated change of bases (when it is a quadratic form) but can't see a way to... | Exodd | 161,426 | <p>You can't prove it because it is false.</p>
<p>$$A = \begin{pmatrix}1 & 0\\ 0 &1\end{pmatrix}$$</p>
<p>$$B = \begin{pmatrix}-1 & 0\\ 0 &-1\end{pmatrix}$$</p>
<p>aren't congruent, since $PAP^T = PP^T$ is definite positive, so can't be $B$. </p>
<p>Another example? let $A$ be the identity matrix, a... |
3,064,458 | <blockquote>
<p>Let <span class="math-container">$f:[0,1]\rightarrow\mathbb{R}$</span> be a continuous function. </p>
<ol>
<li>Show that for each <span class="math-container">$\epsilon\in(0,1)$</span>, <span class="math-container">$\lim\limits_{n\rightarrow\infty}\int\limits_0^{1-\epsilon}f(x^n)dx=(1-\epsilon... | Martin Argerami | 22,857 | <p>As <span class="math-container">$f$</span> is continuous at <span class="math-container">$0$</span>, given <span class="math-container">$c>0$</span>, there exists <span class="math-container">$\delta>0$</span> such that <span class="math-container">$|f(x)-f(0)|<c$</span> whenever <span class="math-container... |
2,584,969 | <blockquote>
<p>Given the second-order ordinary differential equation:
$$
{y}''+y=f(x)
$$
prove that:
$$
y_p(x)=\int_{0}^{x}f(u)\sin(x-u)du
$$
is the particular solution of the equation.</p>
</blockquote>
<p>I know this is homework but I've been trying to solve it for the past few days and I can't. I even a... | Olivier Oloa | 118,798 | <p><strong>Hint</strong>. We assume our $f$ is nice enough to be allowed to use the <a href="https://en.wikipedia.org/wiki/Leibniz_integral_rule" rel="nofollow noreferrer">Leibniz rule</a>,
$$
\frac{d}{dx} \left (\int_{0}^{b(x)}f(x,u)\,du \right) = f\big(x,b(x)\big)\cdot \frac{d}{dx} b(x) + \int_{0}^{b(x)}\frac{\partia... |
3,426,704 | <p>How to interpret the results
<span class="math-container">$$
1^2+2^2+\ldots+n^2=\binom{n+1}{2}+2\binom{n+1}{3}
\\
1^3+2^3+\ldots+n^3=\binom{n+1}{2}+6\binom{n+1}{3}+6\binom{n+1}{4}
$$</span></p>
<p>I want to find a clear argument (combinatorial example,etc.) to prove this, other than induction or merely use the form... | bof | 111,012 | <p>Two different combinatorial interpretations of your first identity are given in the answers to <a href="https://math.stackexchange.com/questions/3113187/combinatorial-proof-of-sum-k-1n-k2-binomn13-binomn23">this question</a>. My answer (not the accepted one) was based on counting the number of ways you can place two... |
3,916,490 | <p>It's easy to get this:
<span class="math-container">$$\int \sqrt{1+\sin x}\, dx \\= \int \sqrt{ \sin^2{\frac{x}{2}} + \cos^2{\frac{x}{2}} + 2\sin{\frac{x}{2}}\cos{\frac{x}{2}}}\,\, dx \\ = \int \left | \sin{\frac{x}{2}} + \cos{\frac{x}{2}} \right |\, dx \\= \sqrt{2} \int \left | \sin{\left ( \frac{x}{2} + \frac{\pi... | Albus Dumbledore | 769,226 | <p>As zkutch has given how to calculate with absolute value here is another approach</p>
<p>To avoid confusion its better to do this we have <span class="math-container">$$\int \frac{\cos x}{\cos x}\sqrt{1+\sin x}dx=\int \frac{\cos x dx}{\sqrt{1-\sin x}}=-2\sqrt{1-\sin x}+C$$</span>
Here in above integral i had put <sp... |
1,493,874 | <p>How can solve that logarithms</p>
<p>$\log _{\frac{4}{x}}\left(x^2-6\right)=2$</p>
<p>It's look diffucult to solve </p>
<p>I was solve but stop with</p>
<p>$x^4−6x^2−16=0$</p>
<p>what is next?</p>
| Dr. Sonnhard Graubner | 175,066 | <p>it is equivalent to $$\frac{\ln(x^2-6)}{\ln(\frac{4}{x})}=2$$ and further we get
$$\ln(x^2-6)=\ln((\frac{4}{x})^2)$$ and this is equivalent to
$$x^4-6x^2-16=0$$ Now set $x^2=t$ and solve the quadratic equation.</p>
|
31,701 | <p>How do I write a chevron/circum (^) in MathJax? A backslash doesn't work as an escape character.</p>
<p>(Specific context: using <em>x</em>^<em>y</em> to mean XOR as in <span class="math-container">$x\oplus y$</span>. It's not my choice of notation so don't tell me that I can just use a different symbol.)</p>
| gen-ℤ ready to perish | 347,062 | <p>The first time you use MathJax in a post, use this command:</p>
<p><code>\newcommand{\^}{\text{^}}</code></p>
<p>If you apply it by itself, it will just make a tiny blank space: <span class="math-container">$\newcommand{\^}{\text{^}}$</span>.</p>
<p>Thenceforth, every time you use <code>\^</code>, you will get a ... |
328,197 | <p>Let $R$ be a commutative ring with unity, and let $S\subset R$ be any finite set. Then
$$ \sum_{L \subset S} \prod_{x \in L} (x-1) = \prod_{x \in S} x,$$
which is easy enough to show by induction.</p>
<p>Does this follow from any sort of general principle? Perhaps inclusion-exclusion, in some form?</p>
| Hagen von Eitzen | 39,174 | <p>If $S=\emptyset$, then $L=\emptyset$ is the only subset and by convention the empty product $\prod_{x\in L} (x-1)$ equals $1$. That shows
$$\tag1 \sum_{L\subseteq S}\prod_{x\in L}(x-1)=\prod_{x\in S}x$$
holds at least if $S=\emptyset$.
For an induction step, let $a\in S$ be any element and let $S'=S\setminus\{a\}$. ... |
328,197 | <p>Let $R$ be a commutative ring with unity, and let $S\subset R$ be any finite set. Then
$$ \sum_{L \subset S} \prod_{x \in L} (x-1) = \prod_{x \in S} x,$$
which is easy enough to show by induction.</p>
<p>Does this follow from any sort of general principle? Perhaps inclusion-exclusion, in some form?</p>
| Matemáticos Chibchas | 52,816 | <p>Clearly it suffices to show the result when $R=\mathbb Z[S]$ and $S$ is the (finite) set of indeterminates of the polynomial ring $R$ (just specialize). </p>
<p>Let $T$ be any nonempty subset of $S$, and let $x_T=\prod_{x\in T}x$. Given $L\subseteq S$, it follows that $x_T$ appears in the expansion of $\prod_{x\in ... |
123,759 | <p>Let $T^n$ be the $n$-dimensional torus and $g$ be a Riemannian metric on $T^n$. Let $\tilde g$ be the induced metric on the universal covering; using suitable coordinates, $\tilde g$ is therefore a $\mathbb{Z}^n$-periodic metric on $\mathbb{R}^n$ (I shall conflate the lattice $\mathbb{Z}^n$ with the fundamental grou... | Mikhail Katz | 28,128 | <p>This is true for n=2, but for n>2 one is unlikely to have such inequalities regardless of what the additive constant is. This is due to "tunneling" phenomena. Take a high multiple K of a primitive class, and represent it by an imbedded loop. Then take a thin tubular neighborhood of the loop. Here the metric can ... |
502,160 | <p>Is there any representation of the exponential function as an infinite product (where there is no maximal factor in the series of terms which essentially contributes)? I.e.</p>
<p>$$\mathrm e^x=\prod_{n=0}^\infty a_n,$$</p>
<p>and by the sentence in brackets I mean that the $a_n$'s are not just mostly equal to $1$... | Did | 6,179 | <p>If $x\geqslant0$ (or $x\ne-2^n$ for every $n\geqslant0$), one can use
$$a_0=1+x,\qquad a_{n+1}=\left(1+\frac{x^2}{2^{n+2}(x+2^n)}\right)^{2^n}
$$
If $x\leqslant0$ (or $x\ne2^n$ for every $n\geqslant0$), one can use
$$a_0=\frac1{1-x},\qquad a_{n+1}=\left(1-\frac{x^2}{(2^{n+1}-x)^2}\right)^{2^n}
$$
<em>Where does this... |
2,626,919 | <p>I'm attempting to calculate the first-order perturbation energy shift for the quantum harmonic oscillator with a perturbing potential of $V(x)=A\cos(kx)$. Omitting the relevant physical factors, I've gotten to the point where I need to calculate:
\begin{equation}
\int_{-\infty}^{\infty} e^{-x^2}(H_n (x))^2 e^{ikx}dx... | Tom Davis | 864,833 | <p>An alternative answer, based off setting n=m in the answer <a href="https://math.stackexchange.com/questions/3585572/fourier-transform-of-a-product-of-the-hermite-polynomials/4203025#4203025">Fourier transform of a product of the Hermite polynomials</a></p>
<p>There, it is shown that the integral
<span class="math-c... |
2,035,454 | <p>I am an upcoming year $12$ student, school holidays are coming up in a few days and I've realised I'm probably going to be extremely bored. So I'm looking for some suggestions.</p>
<p>I want a challenge, some mathematics that I can attempt to learn/master. Obviously nothing impossible, but mathematics is my number ... | Keith McClary | 252,672 | <p>You could look at some of the Questions tagged <a href="https://math.stackexchange.com/questions/tagged/recreational-mathematics">recreational-mathematics</a>, <a href="https://math.stackexchange.com/questions/tagged/soft-question?sort=votes&pageSize=15">soft-question</a> and <a href="https://math.stackexchange... |
2,035,454 | <p>I am an upcoming year $12$ student, school holidays are coming up in a few days and I've realised I'm probably going to be extremely bored. So I'm looking for some suggestions.</p>
<p>I want a challenge, some mathematics that I can attempt to learn/master. Obviously nothing impossible, but mathematics is my number ... | Mateen Ulhaq | 2,736 | <p>It depends on what catches your eye. Having fun is important! At your point in mathematics education, you have a variety of options available for self-study. Here's a possible list of topics, and your future path from then on:</p>
<ul>
<li>Multivariable calculus (followed by vector calculus, complex analysis, and F... |
1,130,876 | <p>If the derivatives $f'(x_0)$ and $g'(x_0)$ exist for the functions $f, g: (x_0 - d, x_0 + d)\to\mathbb{R}$, then for $g(x_0) \neq 0$, $\frac{f}{g}$ is also differentiable in $x_0$ and the following applies:</p>
<p>$(\frac{f}{g})'(x_0) = \frac{f'(x_0)g(x_0) - f(x_0)g'(x_0)}{(g(x_0))^2}$</p>
<p>How can I prove this ... | Mariano Suárez-Álvarez | 274 | <p>Use the definition!</p>
<p>We have $$\frac{f(x+h)/g(x+h)-f(x)/g(x)}{h} = \frac{f(x+h)g(x)-f(x)g(x+h)}{hg(x+h)g(x)}.\tag{1}$$ As $$f(x+h)g(x)-f(x)g(x+h) = (f(x+h)-f(x))g(x)-f(x)(g(x+h)-g(x)),$$ we can rewrite the right hand side of equation (1) in the form
\begin{multline}
\frac{(f(x+h)-f(x))g(x)-f(x)(g(x+h)-g(x))}... |
2,264,435 | <p>I need to solve this series:</p>
<p>$$\sum _{ k=2 }^{ \infty } (k-1)k \left( \frac{ 1 }{ 3 } \right) ^{ k+1 }$$</p>
<p>I converted it into $$\sum _{ k=0 }^{ \infty } \frac { { k }^{ 2 }-k }{ 3 } \left(\frac { 1 }{ 3 } \right)^{ k } -\frac { 4 }{ 3 } $$ with the idea, that $$\sum _{ k=0 }^{ \infty }{ { q }^{... | Jack D'Aurizio | 44,121 | <p>By <a href="https://en.wikipedia.org/wiki/Stars_and_bars_(combinatorics)" rel="nofollow noreferrer">stars and bars</a> the coefficient of $x^n$ in $\frac{1}{(1-x)^3}$, that is the number of ways of writing $n$ as the sum of three natural numbers (or the number of ways for writing $n+3$ as the sum of three positive n... |
153,772 | <p>I am reading a paper on Chiral Differential Operators</p>
<p><a href="http://arxiv.org/pdf/hep-th/0604179v3.pdf" rel="nofollow noreferrer">http://arxiv.org/pdf/hep-th/0604179v3.pdf</a></p>
<p>and it says on page 23 that a line bundle over a manifold C can be characterized completely by its restriction to a non-tri... | Danny Ruberman | 3,460 | <p>Alex is on the right track, but I don't think this is correct in general. It is true that a bundle over a manifold $C$ is characterized by its first Chern class $c_1 \in H^2(C;{\bf Z})$. It is not true that a cohomology class in $H^2$ is determined by its evaluation on all cycles; this would be tantamount to sayin... |
1,432,729 | <p>I know that $\pi \approx \sqrt{10}$, but that only gives one decimal place correct. I also found an algebraic number approximation that gives ten places but it's so cumbersome it's just much easier to just memorize those ten places.</p>
<p>What's a good approximation to $\pi$ as an irrational algebraic number (or a... | MathAdam | 266,049 | <p>How about $$\frac {3.1415926535}{1}$$ </p>
<p>It's fairly easy to memorize, and it's good to10 decimal places.</p>
|
1,738,050 | <p>I know that the series converges. My questions is to what. I tried seeing if it was a telescoping series:
$\sum_{n=2}^\infty \frac{2}{n^3-n} = 2\sum_{n=2}^\infty (\frac{1}{n^2-1}-\frac{1}{n})$ but it doesn't seem to cancel any terms. Thoughts?</p>
| almagest | 172,006 | <p>It is telescoping series: the term is $\frac{1}{n-1}-\frac{2}{n}+\frac{1}{n+1}$, so all the terms cancel except the initial $-\frac{2}{2}+\frac{1}{1}+\frac{1}{2}=\frac{1}{2}$.</p>
|
322,363 | <p>The question is in the title. Fix a field <span class="math-container">$k$</span>. Let <span class="math-container">$P_n$</span> be the poset of proper nonempty affine subspaces of <span class="math-container">$k^n$</span> under inclusion. The geometric realization <span class="math-container">$|P_n|$</span> is <... | user44191 | 44,191 | <p>The answer is <strong>yes</strong>, there are "recursion patterns" which result in sequences of primes for more than one choice of <span class="math-container">$c(0)$</span>. In fact, if you are given any two "starting points", it is possible to find such sequences. The proof comes from the pigeonhole principle.</p>... |
702,675 | <p>p = False, q = True and r = False.
Is $¬(p∨q)∧(¬p∨r)$ = false?</p>
<p><strong>My reasoning:</strong></p>
<p>$$(p∨q)=T \text{ as it is (F or T)}$$</p>
<p>but its the negation so $¬(p∨q)=F$? </p>
<p>Then, $(¬p∨r)$ as p is F but its the negation again so its T and r=F.
So its $(T\lor F)$ in this case only one has... | Kailas | 119,536 | <p>This question can be answered in two ways. </p>
<p>1) Using Binary Mathematics, Too simple, use </p>
<pre><code>1 for true,
0 for false,
+ for ∨,
. for ∧,
</code></pre>
<p>then </p>
<p>¬(p∨q)∧(¬p∨r) given that p = false, q = true, r = false
can be rewritten as:</p>
<pre><code>¬(0+1).(¬0+0) = ¬1.(1+0)
¬1.(1... |
215,846 | <p>Are proofs by induction limited to cases where there is an explicit dependance on a integer, like sums? I cannot grasp the idea of induction being a proof in less explicit cases. What if you have a function that suddenly changes behavior? If a function is positive up to some limit couldn't I prove by induction that ... | The Chaz 2.0 | 7,850 | <p><strong>The Principle of Mathematical Induction</strong> is equivalent to the <strong>Well-Ordering Principle</strong>, which states that every non-empty set of positive integers has a least element. You either <em>assume</em> (as an axiom of your number system) PMI and <em>prove</em> WOP from this assumption, or vi... |
215,846 | <p>Are proofs by induction limited to cases where there is an explicit dependance on a integer, like sums? I cannot grasp the idea of induction being a proof in less explicit cases. What if you have a function that suddenly changes behavior? If a function is positive up to some limit couldn't I prove by induction that ... | John Gowers | 26,267 | <p>Induction is normally defined for the case when you have an explicit dependence on an integer (or an ordinal for transfinite induction, but I'm not going to talk about that). I'm not sure what you're talking about with the function, but since induction requires you to prove that <span class="math-container">$P(k)\L... |
4,341,356 | <p>A French-suited cards pack consist of <span class="math-container">$52$</span> cards where <span class="math-container">$13$</span> are clovers. <span class="math-container">$4$</span> players play a game where every player has <span class="math-container">$13$</span> cards in his/her hands (, in other words the ful... | CHAMSI | 758,100 | <p>You made a small mistake :<span class="math-container">\begin{aligned}\sqrt[n]{\dfrac{20}{2^{2n+4}+2^{2n+2}}}&=\sqrt[n]{\dfrac{20}{2^{2n}\cdot2^4+2^{2n}\cdot2^{\color{red}{2}}}}\\&=\sqrt[n]{\dfrac{20}{2^{2n}\cdot\color{red}{20}}}=\sqrt[n]{\dfrac{1}{2^{2n}}}=\frac{1}{4}\end{aligned}</span></p>
|
2,168,524 | <p>I know a solution to this question having to do with the fact that the $\gcd(15, 21) = 3$, so the answer is no.</p>
<p>But I can't figure out what is the reasoning behind this. Any help would be really appreciated! </p>
| the_candyman | 51,370 | <p>Suppose that $15m > 21n$. Then you are asking if:
$$15m - 21n= 1.$$</p>
<p>This corresponds to:</p>
<p>$$15m = 21n + 1.$$</p>
<p>The term on the left is divisible by $3$. What about the term on the right?</p>
<p>Well, $21n$ is divisible by $3$ too. But, if you add $1$ to $21n$, it can't be divisible by $3$ an... |
81,221 | <p>Suppose that a hypothetical math grad student was pretty comfortable with first-year real variables and algebra, and had even studied some other things (algebraic geometry, Riemannian geometry, complex analysis, algebraic topology, algebraic number theory), but had miraculously never taken a differential equations c... | drbobmeister | 8,472 | <p>I really like <i>Ordinary Differential Equations</i> by Jack K. Hale. It's very rigorous
and thorough in the fundamentals, has a great section on periodic linear systems, and covers
some advanced stuff such as integral manifolds. Arnold, Abraham and Marsden, and Hirsch, Smale and Devaney are also nice, though the e... |
81,221 | <p>Suppose that a hypothetical math grad student was pretty comfortable with first-year real variables and algebra, and had even studied some other things (algebraic geometry, Riemannian geometry, complex analysis, algebraic topology, algebraic number theory), but had miraculously never taken a differential equations c... | Edward Dunne | 49,409 | <p>Teschl's book is a good, modern text on ODEs and is available from the AMS in their GSM series. It blends ODEs and dynamical systems. Bob Devaney updated the classic Hirsch and Smale book some years ago. It has an emphasis on dynamical systems. A real classic is the book by Ince available from Dover. It predates the... |
1,874,555 | <p>If $4x/3y = 7/2$, what is the value of $y/x$?</p>
<p>This is a multiple choice question, and the choices are as follows:</p>
<p>A. $3/14$ </p>
<p>B. $8/21$</p>
<p>C. $21/8$</p>
<p>D. $14/3$</p>
<p>I started off answering this by cross multiplying it down to $8x=21y$</p>
<p>From there, $x=21y/8$ and $y=8x/21$ ... | gt6989b | 16,192 | <p>multiply original equation by $3/4$ to get
$$
\frac{x}{y} = \frac{3}{4} \frac{7}{2} = \frac{21}{8}
$$
so inverting the fractions
$$
\frac{y}{x} = \frac{8}{21}.
$$</p>
|
206,026 | <p>Let $k$ be a field.</p>
<p>$G/k$ be a simply connected semisimple algebraic group. </p>
<p>Let $X/k$ be a smooth affine $k$-scheme. </p>
<p><strong>Question</strong>: Is every principal $G$ bundle on $X\times {\mathbb A}^1$ a pull back from $X$?</p>
| Roman Fedorov | 6,772 | <p>You may also want to look at <a href="http://arxiv.org/abs/1308.3078" rel="nofollow">http://arxiv.org/abs/1308.3078</a>. You will see that the answer is in general "no" even if you assume that $k$ is the field of complex numbers.</p>
|
3,422,095 | <p>I have been playing with Maclaurin series lately, I have been able to come across this:</p>
<p><span class="math-container">$\dfrac{1}{1+x}=1-x+x^2-x^3+x^4-x^5...$</span></p>
<p><span class="math-container">$\dfrac{1}{(1+x)^2}=1-2x+3x^2-4x^3+5x^4-6x^5+7x^6...$</span></p>
<p>I found out by accident that:</p>
<p><... | Kishalay Sarkar | 691,776 | <p>Given any analytic function <span class="math-container">$f ,f(x)=\sum_{n=0}^{\infty}f^n(0)x^n/n!$</span>.You are guaranteed to get a series representation iff <span class="math-container">$R_n(x)=f(x)-\sum_{k=0}^{n}f^k(0)x^k/k! \to 0$</span> pointwise as <span class="math-container">$n\to \infty$</span>.</p>
|
62,967 | <p><code>CoefficientRules</code> acts like the following.</p>
<pre><code>In[1]:= CoefficientRules[2 x^3 + 3 x^2 y + 4 x y^2 - 5 x + 1]
Out[1]= {{3, 0} -> 2, {2, 1} -> 3, {1, 2} -> 4, {1, 0} -> -5, {0, 0} -> 1}
</code></pre>
<p>My question is how one can "extend" this function so that it may allow the n... | ybeltukov | 4,678 | <p>There is nice undocumented function</p>
<pre><code>{c, v} = GroebnerBasis`DistributedTermsList[2 x^3 + 3 x^(-2) y + 4 x y^2 - 5 y^(-3) + 1]
(* {{{{2, 0, 0, 1}, 3}, {{0, 3, 0, 0}, 2}, {{0, 1, 0, 2},
4}, {{0, 0, 3, 0}, -5}, {{0, 0, 0, 0}, 1}}, {1/x, x, 1/y, y}} *)
</code></pre>
<p>Then one can simplify the resul... |
104,132 | <p>$$f''(x) \thickapprox\dfrac{1}{2h^2}[f(x+2h) - 2f(x) + f(x - 2h)]$$</p>
<p>I'm supposed to be deriving the above formula and establish an error formula in using them.</p>
<p>This is one of a series of problems like this, and I'm not quite too sure on how to get started on this. (This is in a chapter of Estimating ... | J. M. ain't a mathematician | 498 | <p>Taylor expansion does the trick:</p>
<p>$$\begin{align*}
f(x+2h)&=f(x)+2hf^\prime(x)+2h^2 f^{\prime\prime}(x)+\frac43 h^3 f^{(3)}(x)+\cdots\\
f(x-2h)&=f(x)-2hf^\prime(x)+2h^2 f^{\prime\prime}(x)-\frac43 h^3 f^{(3)}(x)+\cdots
\end{align*}$$</p>
<p>Combine these two appropriately to remove the term contai... |
4,021,746 | <p>Euclidean distance is not linear in high dimensions. However, in multiple regression the idea is to minimize square distances from data points to a hyperplane.</p>
<p>Other data analysis techniques have been considered problematic for their reliance on Euclidean distances (nearest neighbors), and dimensionality redu... | kalgoritmi | 886,239 | <p>The answer that the book provides you is the direct application of the
following formula:
<span class="math-container">$$
nP_k = {\displaystyle\prod_{i=0}^{k-1} (n - i)}=\dfrac{n!}{(n-k)!}
$$</span></p>
<p>If we plug <span class="math-container">$n=4$</span> (our pool of distinct flags) and <span class="math-contain... |
31,571 | <p>In his paper "Smooth models for elliptic threefolds" (In: The Birational Geometry of Degenerations, Progress in Mathematics, v. 29, Birkhauser, (1983), 85-133), Rick Miranda mentions in the example of section 8 (page 101-102) that it is an unfortunate fact of life that there are no small resolutions for the singular... | Hailong Dao | 2,083 | <p>It is worth noting the <a href="http://books.google.com/books?id=dJf0DQArhzkC&lpg=PA61&ots=z0OPkgQCng&dq=katz%20small%20resolution%20gorenstein&pg=PA61#v=onepage&q=katz%20small%20resolution%20gorenstein&f=false" rel="nofollow">following result</a> by S. Katz:</p>
<p>If the singularity $X$ d... |
3,135,085 | <p><span class="math-container">$\lim_{x\to 2} {x^2 - 4\over x^3 - 4x^2 +4x}$</span></p>
<p>I used L'Hospital's rule twice on this, and got a solution, but my textbook says it's an indeterminate form. Is using L'Hospital's rule twice wrong, and if yes, why so?</p>
| Rebellos | 335,894 | <p><strong>Edited answer after the correction of the OP :</strong></p>
<p>We have the limit :</p>
<p><span class="math-container">$$\lim_{x \to 2} \frac{x^2-4}{x^3-4x^2 + 4x} $$</span></p>
<p>Note that this is an indeterminate form, thus L'Hospital's can be applied :</p>
<p><span class="math-container">$$\lim_{x \t... |
74,036 | <p>I have a 3D model of a heart in Mathematica and I'm trying to create a plane so that the open surface (as seen in the image below) is cut off so that the heart can have a solid, level surface. How can I combine this plane with my 3D contour plot?</p>
<pre><code>heart = (2 x^3 + y^2 + z^2 - 1)^3 - (1/10) x^2 z^3 - ... | Michael E2 | 4,999 | <p>You can replace the boundary <code>Line</code> with a <code>Polygon</code>:</p>
<pre><code>heart = (2 x^3 + y^2 + z^2 - 1)^3 - (1/10) x^2 z^3 - y^2 z^3;
g = Show[
ContourPlot3D[heart == 0,
{x, 0., 1.5}, {y, -1.5, 1.5}, {z, -1.5, 1.5},
Mesh -> None, PlotPoints -> 40, ContourStyle -> Opacity[0.8, R... |
1,620,540 | <p>In topological space, does first countable+ separable imply second countable? If not, any counterexample?</p>
| Unit | 196,668 | <p>Check out the <a href="https://en.wikipedia.org/wiki/Lower_limit_topology" rel="nofollow">Sorgenfrey line</a>!</p>
|
198,204 | <p>Is complex valued function like $y(t) = t^2 + i\cdot t^2$ a periodic function?</p>
| Bombyx mori | 32,240 | <p>This is nothing but $(1+i)t^{2}$. Why you would think this is periodical?</p>
|
728,186 | <p>The following definition has been given in <a href="http://www.sciencedirect.com/science/article/pii/0304397585901355?via=ihub" rel="nofollow">this article</a>.</p>
<p>A term algebra is an algebra $ \langle \mathcal{S}, \mathcal{G} \rangle $ where every time that $g_\alpha, g_\beta \in \mathcal{G}$ and
$$ g_\alpha... | user27887 | 102,331 | <p>The definition of "term algebra" given in the article is incorrect: it gives a necessary condition, but not the full condition. The missing condition is that the set of what it calls "generators" must actually generate the algebra. For more details I wrote a presentation of term algebras in <a href="http://settheory... |
2,393,625 | <p>Any subgroup of order $2$ will be cyclic subgroup and so will be generated by single element of order $2$ in $S_4$, so to count number of subgroups of order $2$ we need to count number of elements of order $2$ in $S_4$, I tried counting them but I got answer $8$ but is $9$ actually.</p>
| ajotatxe | 132,456 | <p>Assume that $\sigma\in S_4$ has order two. Then $\sigma$ changes some element $x$. Let $y=\sigma(x)$. Then $\sigma(y)=x$.</p>
<p>This shows that $\sigma$ is the product of some disjoint $2$-cycles.</p>
<p>The possibilities are:
$$(1,2),(1,3),(1,4),(2,3),(2,4),(3,4),(1,2)(3,4),(1,3)(2,4),(1,4)(2,3)$$</p>
|
172,271 | <p>First a warm-up. Let $\ V\ $ be an arbitrary set of odd natural numbers. Let $\ S(V)\ $ be the generated multiplicative semi-group. What are the necessary and/or sufficient conditions on $\ V\ $ for the property: $\ \exists_{x\ y\in S(V)}\ y-x=2\ $?</p>
<p>Now real questions, all of them open to me. Let $\ \mathbb ... | Aaron Meyerowitz | 8,008 | <p>As far as $Q2$ , we don't know if there are infinitely many twin primes but expect that there are infinitely many <a href="https://oeis.org/A136720" rel="nofollow">prime quadruples</a> -- $p-2,p,q,q+2$ all prime with $q=p+4.$ Then you are looking at $V=\{{p,q\}}$ and it seems unlikely that even one such pair $p,q=p,... |
1,530,118 | <p>I was wondering if anyone could help with this $\epsilon–\delta$ definition of a limit. I have looked it up in my calculus book and online and I just don't understand how to do it.</p>
<p>Prove, using the $\epsilon–\delta$ definition of a limit that</p>
<p>$$\lim_{(x,y)\to(0,0)}\frac{(x^2-y)}{(4x^2+y^2)}$$</p>
| Rory Daulton | 161,807 | <p>Divide numerator and denominator by $x^4$, giving you</p>
<p>$$\lim_{x\to 0}\frac{\frac{\sin x^4}{x^4}-\cos x^4+x^{16}}{e^{2x^4}-1-2x^4}$$</p>
<p>Now expand the sine, cosine, and exponential into power series (Maclaurin series) with as many terms as needed.</p>
|
2,961,864 | <p><strong>Problem</strong></p>
<p>Prove using Stokes' theorem that
<span class="math-container">$$\int_C y dx +z dy + x dz = \pi a^2 3^.5,$$</span> where <span class="math-container">$C$</span> is the curve of intersection of the sphere <span class="math-container">$x^2+y^2+z^2=a^2$</span> and the plane <span class=... | user247327 | 247,327 | <p>S is the unit disk in the xy-plane and <span class="math-container">$\int_S\int dxdy$</span> is simply its area. </p>
|
3,098 | <p>This is a really newbie question, but it has me confused. Why does this code <strong>work without</strong> <code>// MatrixForm</code> and <strong>doesn't work with</strong> <code>// MatrixForm</code>?</p>
<pre><code>cov = {{0.02, -0.01}, {-0.01, 0.04}} // MatrixForm
W = {w1, w2}; FindMinimum[ W.cov.W, W]
</code></p... | rm -rf | 5 | <p><code>MatrixForm</code> is a wrapper that pretty-prints your matrices. When you do the following:</p>
<pre><code>cov = {{0.02, -0.01}, {-0.01, 0.04}} // MatrixForm
</code></pre>
<p>you're assigning the prettified matrix to <code>cov</code> (i.e., wrapped inside a <code>MatrixForm</code>). This is not accepted as a... |
3,294,564 | <blockquote>
<p>There is only one real values of <span class="math-container">$k$</span> for which the quadratic equation <span class="math-container">$kx^2+(k+3)x+k-3=0$</span> has <span class="math-container">$2$</span> positive integer roots. Then the product of these two solutions is</p>
</blockquote>
<p>What i ... | Maverick | 171,392 | <p>Look at it in an another way. The equation can be re-written as
<span class="math-container">$$x-1+\frac{k}{3}\left(x^2+x+1\right)=0$$</span>
which can be said to be a family of curves passing through points of intersection of the curves</p>
<p><span class="math-container">$y=x-1$</span> </p>
<p>and </p>
<p><sp... |
264,025 | <p>Suppose we start with a $n\times n$ matrix with entries sampled independently and uniformly at random from $[0,1]$. The weight of a set of entries will simply be the sum of those entries. A permutation refers to a set of $n$ entries, no two on the same row or column.</p>
<p>Pick a permutation whose corresponding en... | Chris Godsil | 1,266 | <p>What follows is a proof that semisymmetric graphs are walk-regular.</p>
<p>Say vertices $u$ and $v$ in a graph $X$ are <em>cospectral</em> if the graphs $X\setminus u$ and $X\setminus v$ are cospectral. If $X$ is semisymmetric, then the vertices in each colour class are cospectral. Two vertices $u$ and $v$ in a bip... |
3,163,067 | <p><strong>Definition 1</strong> (Formal Language). A <em>language</em> <span class="math-container">$L$</span> over an <em>alphabet</em> <span class="math-container">$\Sigma$</span> (any nonempty finite set) is a subset of the set of all finite sequences of elements of <span class="math-container">$\Sigma$</span>, i.e... | WhySee | 152,868 | <p>Let <span class="math-container">$\epsilon>0$</span> be given.</p>
<p>Now, <span class="math-container">$\left|g_n(y)-g(y)\right|=\left|f(x_n,y)-f(x,y)\right|$</span><br>
As <span class="math-container">$f$</span> is continuous, therefore for each <span class="math-container">$y\in\left[0,1\right]$</span>, <span... |
3,163,067 | <p><strong>Definition 1</strong> (Formal Language). A <em>language</em> <span class="math-container">$L$</span> over an <em>alphabet</em> <span class="math-container">$\Sigma$</span> (any nonempty finite set) is a subset of the set of all finite sequences of elements of <span class="math-container">$\Sigma$</span>, i.e... | Peter Szilas | 408,605 | <p>Credit to Kavi.</p>
<p><span class="math-container">$f$</span> is uniformly continuos on <span class="math-container">$K={x,x_1,x_2,...}×[0,y],$</span> compact.</p>
<p><span class="math-container">$\epsilon >0$</span> given, there exists a <span class="math-container">$\delta >0$</span> s.t.</p>
<p><span cl... |
208,694 | <p>Could someone explain how to correctly prove that $$\lim_{n\to\infty}\sin\frac{1}{n}$$ where $n=1,2,\cdots,n$ doesn't exist.
I have no problem with it if $\sin\frac{1}{x}$ where $x$ is real, because just taking values $x=\frac{2}{(2n-1)\pi}, x=\frac{1}{n\pi}, x=\frac{2}{(2n+1)\pi}$ it is clear, for example, by Cauch... | Thomas | 26,188 | <p>You are asking about the limit:
$$
\lim_{n\to \infty} \sin\left(\frac{1}{n}\right).
$$
For this limit (as others have already noted, $\frac{1}{n} \to 0$ as $n\to\infty$. I.e., as $n$ gets very large $\frac{1}{n}$ becomes very small. So this limit is the same as the limit
$$
\lim_{t \to 0} \sin(t)
$$
where $t = \frac... |
3,507,695 | <p>Suppose that <span class="math-container">$A \in M_{5 \times 5}(\mathbb{C})$</span> such that <span class="math-container">$(A - 2I)^{5} = 0$</span>. Suppose that <span class="math-container">$B \in M_{5 \times 5}(\mathbb{C})$</span> such that the minimal polynomial of <span class="math-container">$B$</span> is <spa... | Ben Grossmann | 81,360 | <p><strong>Hint:</strong> If the normal form of <span class="math-container">$A$</span> consists of <span class="math-container">$k$</span> Jordan blocks in total and <span class="math-container">$B$</span> commutes with <span class="math-container">$A$</span>, then <span class="math-container">$B$</span> can have at m... |
19,590 | <p><a href="http://www.xamuel.com/graphs-of-implicit-equations/" rel="noreferrer">Here</a> are several equations, it seems that Mathematica couldn't plot them well, although I set PlotPoints>100</p>
<pre><code> ContourPlot[Csc[1. - x^2] Cot[2. - y^2] - x*y == 0,
{x, -10, 10}, {y, -10, 10}, PlotPoints -> 120]
</... | carlosayam | 1,215 | <p>I got a plot with Mathemaica 9, but far from the one mentioned in the page you said. If one uses</p>
<pre><code>Plot3D[Csc[1. - x^2] Cot[2. - y^2] - x*y, {x, -10, 10}, {y, -10, 10}]
</code></pre>
<p>then you get this picture:</p>
<p><img src="https://i.stack.imgur.com/OjN36.png" alt="plot3d"></p>
<p>I don't see ... |
19,590 | <p><a href="http://www.xamuel.com/graphs-of-implicit-equations/" rel="noreferrer">Here</a> are several equations, it seems that Mathematica couldn't plot them well, although I set PlotPoints>100</p>
<pre><code> ContourPlot[Csc[1. - x^2] Cot[2. - y^2] - x*y == 0,
{x, -10, 10}, {y, -10, 10}, PlotPoints -> 120]
</... | cormullion | 61 | <p>I don't know what's meant by "good" or "well", but, on the assumption that it means "like a 1970s retro wallpaper design", here's a suggestion:</p>
<p><img src="https://i.stack.imgur.com/PJxJO.png" alt="retro"></p>
<pre><code>ContourPlot[Csc[1. - x^2] Cot[2. - y^2] - x y,
{x, -2 Pi, 2 Pi},
{y, -2 Pi, 2 Pi},
Col... |
3,269,080 | <p>There are a lot of functions that look wobbly.</p>
<p>For example <span class="math-container">$x^4 + x^3$</span> looks a little wobbly when it gets near the x axis. The function <span class="math-container">$\sin(x)$</span> is extremely wobbly. The function <span class="math-container">$\sin(x) + x$</span> is also... | David K | 139,123 | <p>You could consider when the <em>second</em> derivative of the function changes sign.
When the second derivative is negative you’re going over a hump and when it’s positive you’re inside a kind of bowl shape even if the function is increasing the whole time like <span class="math-container">$2x+\sin x.$</span></p>
|
3,928,842 | <p>Let <span class="math-container">$B$</span> and <span class="math-container">$W$</span> be two independent Brownian motions on <span class="math-container">$(\mathcal{F}_{t})$</span>. Is <span class="math-container">$\tau$</span> defined as:
<span class="math-container">$$\tau = \inf \{ t \ge 0: B_{t} \ge W_{t} + e^... | Wuestenfux | 417,848 | <p>If the matrix <span class="math-container">$A$</span> is invertible, there exists an inverse matrix <span class="math-container">$A^{-1}$</span> with <span class="math-container">$AA^{-1}=I=A^{-1}A$</span>, where <span class="math-container">$I$</span> is the identity matrix. The inverse matrix is uniquely determine... |
3,546,801 | <p>I have often heard (both online and in person) people say that "<span class="math-container">$\mathbb{R}^2$</span> can't be totally ordered." I would like to understand this statement. </p>
<p>Of course, on the face of it, this is false: Pick your favorite bijection <span class="math-container">$f:\mathbb{R}^2 \to ... | Daniel Schepler | 337,888 | <p>Suppose you have a point <span class="math-container">$x_0 \in X$</span> such that <span class="math-container">$X \setminus \{ x_0 \}$</span> is connected. Then under your definition of a "nice" total order, <span class="math-container">$\{ x \in X \mid x < x_0 \}$</span> and <span class="math-container">$\{ x ... |
49,015 | <p>This question might be astoundingly naive, because my understanding of modular forms is so meek. It occurred to me that the reason I was never able to penetrate into the field of modular forms, automorphic forms, the Langland's program and so forth was because my appeal is to things that have the feel of SGA1, and t... | Maxime | 64,184 | <p>In « Récoltes et Semailles » Grothendieck has a reflexion about the article of Langlands : « <strong>Automorphic representations, Shimura varieties, and motives</strong> » where he sees the influence of his ideas on the « <strong>motivic Galois group</strong> » that Langlands has m... |
615,614 | <h1>Question</h1>
<p>Given a square complex matrix $A$, what ways are there to define and compute $A^p$ for non-integral scalar exponents $p\in\mathbb R$, and for what matrices do they work?</p>
<h1>My thoughts</h1>
<h2>Integral exponents</h2>
<p>Defining $A^k$ for $k\in\mathbb N$ is easy in terms of repeated multi... | Philip Hoskins | 41,421 | <p>You are correct that your proposed definition for rational exponents can run into issues of uniqueness. Consider just the problem of trying to find the square root of a matrix. If $I$ is the 2x2 identity, then any matrix of the form </p>
<p>\begin{pmatrix}
\pm1 & a \\
0 & \mp1 \\
\end{pmatrix}</p>
<p>s... |
3,643,186 | <p>A <span class="math-container">$k^{th}$</span> order PDE is defined by <span class="math-container">$$F(D^ku(x),D^{k-1}u(x),\dotsc,Du(x),u(x),x)=0,$$</span> where <span class="math-container">$x$</span> is an element of <span class="math-container">$U$</span>. I know that <span class="math-container">$\mathbb{R}^n$... | Hagen von Eitzen | 39,174 | <p><span class="math-container">$D$</span> is the differential operator, <span class="math-container">$D^k$</span> is the <span class="math-container">$D$</span> applied <span class="math-container">$k$</span> times. In other words, <span class="math-container">$Du(x)=u'(x)$</span>, <span class="math-container">$D^2u(x... |
4,204,506 | <p>We can write <span class="math-container">$4x^2+8x+5$</span> in the form <span class="math-container">$a(x+b)^2+c$</span> as <span class="math-container">$4(x+1)^2+1$</span>. However, the question I am doing asks me to write it in the form <span class="math-container">$(ax+b)^2+c$</span>. How do I change it to that ... | Joe | 623,665 | <p>You can also solve the equation <span class="math-container">$(ax+b)^2+c\equiv 4x^2+8x+15$</span> (the <span class="math-container">$\equiv$</span> sign means "true for all values of <span class="math-container">$x$</span>"—the coefficients on both sides of the equation have to be the same). If we expand t... |
1,830,799 | <p>1There are $\frac{21!}{2!3!} = 120$ total positions (disregarding order within same colour). I imagine labelling the people Y (yellow) and NY (not yellow), so I imagine I have $4$ copies of the letter Y and $5$ of NY. So I draw out $9$ slots and want to arrange so that at least $2$ Y are together.
I get $30$ arrang... | Hagen von Eitzen | 39,174 | <p>Alright, we have 4 times Y and 5 times N. </p>
<p>There are $9\choose 4$ arrangements in total, including those with YY somewhere</p>
<p>If we replace each occurrance of N followed by Y with an X, then a <em>non-favourable</em> outcome is </p>
<ul>
<li>either Y followed by any combination of three X and two N; th... |
3,503,820 | <p>I have the following equation:</p>
<p><span class="math-container">$v(t) =\frac{-I_0}{C}\frac{e^{-at}-e^{-bt}}{b-a}$</span></p>
<p><span class="math-container">$a=\frac{L+\sqrt{L^{2}-16CLR^{2}}}{4CLR}$</span></p>
<p><span class="math-container">$b=\frac{L-\sqrt{L^{2}-16CLR^{2}}}{4CLR}$</span></p>
<p><span class=... | Community | -1 | <p><span class="math-container">$$\frac{e^{(-1/\tau+i\omega)t}-e^{(-1/\tau-i\omega)t}}{(-1/\tau+i\omega)-(-1/\tau-i\omega)}=e^{-t/\tau}\frac{e^{i\omega t}-e^{-i\omega t}}{2i\omega}=e^{-t/\tau}\sin\omega t.$$</span></p>
<p>This is a damped sinusoid.</p>
|
3,503,820 | <p>I have the following equation:</p>
<p><span class="math-container">$v(t) =\frac{-I_0}{C}\frac{e^{-at}-e^{-bt}}{b-a}$</span></p>
<p><span class="math-container">$a=\frac{L+\sqrt{L^{2}-16CLR^{2}}}{4CLR}$</span></p>
<p><span class="math-container">$b=\frac{L-\sqrt{L^{2}-16CLR^{2}}}{4CLR}$</span></p>
<p><span class=... | YNK | 587,353 | <p>No, you don't have to go anywhere. If you have continued simplifying the expression you got for <span class="math-container">$v\left(t\right)$</span> as shown below, you would have arrived at your goal without any difficulty, i.e.
<span class="math-container">$$v\left(t\right)=-\frac{I_0}{C}\frac{1}{de^{ct}}\frac{e^... |
2,717,264 | <blockquote>
<p>$$\int{\theta \tan^3{(\theta^2)}\sec^4{(\theta^2)}d\theta}$$</p>
</blockquote>
<p>I thought of the method of splitting up the $\tan^3 \theta^2$ to $\tan^2 \theta^2$ and $\tan\theta^2$.</p>
<p>And then using trig identity $1+\tan^2\theta=\sec^2\theta$ to express the whole integral in terms of $\sec\t... | Dr. Sonnhard Graubner | 175,066 | <p>Use that $$n>\log(n)$$ for $n>0$</p>
|
2,717,264 | <blockquote>
<p>$$\int{\theta \tan^3{(\theta^2)}\sec^4{(\theta^2)}d\theta}$$</p>
</blockquote>
<p>I thought of the method of splitting up the $\tan^3 \theta^2$ to $\tan^2 \theta^2$ and $\tan\theta^2$.</p>
<p>And then using trig identity $1+\tan^2\theta=\sec^2\theta$ to express the whole integral in terms of $\sec\t... | Jim Haddocc | 167,086 | <p>Use the fact that
$$n>\log(n)$$
So that we get
$$\sum_{n=2}^\infty\frac{1}{\ln n}> \sum_{n=2}^\infty\frac{1}{n}=\infty$$</p>
|
58,914 | <p>Question:</p>
<p>If a square matrix $A$ satisfies $A^2=I$ and $\det A>0$, show that $A+I$ is non-singular.</p>
<p>I have tried to suppose a non-zero vector $x$ s.t. $Ax=x$ but fail to make a contradiction.</p>
<p>And I tried to find the inverse matrix of $A+I$ directly, suppose $(A+I)^{-1}=\alpha I +\beta A$, ... | zyx | 14,120 | <p>$(I+A)(I-A)=0$, so $I+A$ is invertible (for $A$ satisfying $A^2=I$) if and only if $A=I$.</p>
<p>This works for $A$ in any ring with unit [in which $2$ is invertible], not only a ring of square matrices.</p>
|
1,040,505 | <p>Apparently,
$$(1-\cot 37^\circ)(1-\cot 8^\circ)=2.00000000000000000\cdots$$<br>
Since it is a $2.0000000000\cdots$ instead of $2$, it isn't exactly $2$.<br>
Why is that?</p>
| David H | 55,051 | <p>$$\begin{align}
1-\cot{\left(x\right)}
&=1-\frac{\cos{\left(x\right)}}{\sin{\left(x\right)}}\\
&=\frac{\sin{\left(x\right)}-\cos{\left(x\right)}}{\sin{\left(x\right)}}\\
&=-\sqrt{2}\frac{\sin{\left(\frac{\pi}{4}-x\right)}}{\sin{\left(x\right)}}.\\
\end{align}$$</p>
<p>Therefore,</p>
<p>$$\begin{align}
... |
4,616,445 | <p>Could anyone enlighten me on how to go about expanding the following function around <span class="math-container">$x_0 = 0$</span>:</p>
<p><span class="math-container">$$
f(x):= \log(1+x)e^{x}
$$</span></p>
<p>I have tried using Cauchy Product Series and bruteforce computation of the coefficients but I always find m... | Yiorgos S. Smyrlis | 57,021 | <p>We are looking of the McLaurin series
<span class="math-container">$$
f(x)=\log(1+x)e^x \quad\Longrightarrow\quad f(x)=\sum_{n=0}^\infty \frac{f^{(n)}(0)}{n!}x^n
$$</span>
of <span class="math-container">$f(x)=\log(1+x)e^x$</span>.</p>
<p>So
<span class="math-container">$$
f^{(n)}(x)=\sum_{k=0}^n
\binom{n}{k}\big(\... |
4,616,445 | <p>Could anyone enlighten me on how to go about expanding the following function around <span class="math-container">$x_0 = 0$</span>:</p>
<p><span class="math-container">$$
f(x):= \log(1+x)e^{x}
$$</span></p>
<p>I have tried using Cauchy Product Series and bruteforce computation of the coefficients but I always find m... | GEdgar | 442 | <p><span class="math-container">$$
\log(1+x) = \sum_{k=1}^\infty\frac{(-1)^{k+1}}{k} x^k
\\
e^x = \sum_{j=0}^\infty\frac{1}{j!}x^j
\\
\log(1+x)e^x = \sum_{n=1}^\infty\left(\sum_{k+j=n}\frac{(-1)^{k+1}}{k\,j!}\right)x^n
=\sum_{n=1}^\infty\left(\sum_{k=1}^n\frac{(-1)^{k+1}}{k\,(n-k)!}\right)x^n
$$</span>
So our Laurent s... |
161,780 | <p>I've written the code that follows:</p>
<pre><code>tails = Function[l, If[ l == {}, {}, Prepend[tails[Drop[l, 1]], l]]]
Te = Composition[AllTrue[PrimeQ], Map[FromDigits], tails, IntegerDigits]
Timing[Select[Range[2, 10^6], Te]]
</code></pre>
<p>This takes around 21seconds on my computer, but the equivalent Haske... | Daniel Lichtblau | 51 | <p>[Not really an answer but too long for a comment.]</p>
<p>Not sure if there is a direct Haskell-like way that can be made quite that fast (I guess it depends on exactly how equivalent is that equivalent Haskell code). Here is an approach that short-circuites when it hits a non-prime.</p>
<pre><code>Te4[n_?PrimeQ] ... |
1,292,836 | <p>I am trying to evaluate $$\oint _C \frac{-ydx+xdy}{x^2+y^2}$$</p>
<p>clockwise around the square with vertices (−1,−1), (−1,1), (1,1), and (1,−1).</p>
<p>So from the question,
$$\vec{F}=<\frac{-y}{x^2+y^2},\frac{x}{x^2+y^2}>$$</p>
<p>I first conducted the gradient test $\frac{\partial F_2}{\partial x}=\fra... | KittyL | 206,286 | <p>Since $\nabla \times \vec{F}=0$ is valid in $\mathbb{R}\backslash{(0,0)}$, you can reshape your curve to a circle centered at origin and use polar coordinates to evaluate it. </p>
|
1,032,331 | <p>I am trying to find the limit </p>
<p>$\large\lim_{n \to \infty} (n^{\frac{1}{2n}})$</p>
<p>WolframAlpha says that I can transform as follows</p>
<p>$\large\lim_{n \to \infty} (n^{\frac{1}{2n}})=e^{\lim_{n \to\infty} \frac{ln(n)}{2n}}$</p>
<p>However, I do not understand where $n$ in $ln (n)$ comes from. Could a... | hardmath | 3,111 | <p>Assuming that you want a critique of your work, here are a couple of notes:</p>
<p>You claim to be showing $M\cap N$ is a <em>normal</em> subgroup of $G$, but the steps (1),(2) only establish that the intersection is a subgroup. Showing normality needs a few more words.</p>
<p>The construction of the surjective h... |
3,934,819 | <p>I am wondering if we can find a linear transformation matrix <span class="math-container">$A$</span> of size <span class="math-container">$3\times 3$</span> over the field of two elements <span class="math-container">$\mathbb{Z}_2$</span> i.e. a matrix <span class="math-container">$A$</span> of zeros and ones s.t.</... | Martin Argerami | 22,857 | <p>There is no such <span class="math-container">$A$</span>, invertible or not. For instance your equations lead to
<span class="math-container">$$
\begin{bmatrix} 1 \\ 0\\ 0\end{bmatrix}=\begin{bmatrix} 1 \\ 1 \\ 1\end{bmatrix}+ \begin{bmatrix} 0 \\ 1 \\ 1\end{bmatrix} =A \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} +
A ... |
1,276,206 | <p>I am writing a computer program that involves generating 4 random numbers, a, b, c, and d, the sum of which should equal 100. </p>
<p>Here is the method I first came up with to achieve that goal, in pseudocode:</p>
<pre><code>Generate a random number out of 100. (Let's say it generates 16).
Assign this value as th... | Scott | 239,527 | <p>Are computational constraints really an issue? Do you intend to scale this up to higher numbers? Does this method need to be achieved using physical dice, a dice rolling program, Excel, or a programming language?</p>
<p>As Thomas Andrews points out, using that method will bias towards something like 50/25/12/12 com... |
1,276,206 | <p>I am writing a computer program that involves generating 4 random numbers, a, b, c, and d, the sum of which should equal 100. </p>
<p>Here is the method I first came up with to achieve that goal, in pseudocode:</p>
<pre><code>Generate a random number out of 100. (Let's say it generates 16).
Assign this value as th... | DJohnM | 58,220 | <p>Generate four random numbers between $0$ and $1$</p>
<p>Add these four numbers; then divide each of the four numbers by the sum, multiply by $100$, and round to the nearest integer.</p>
<p>Check that the four integers add to $100$ (they will, two thirds of the time). If they don't (rounding errors), try again...... |
1,276,206 | <p>I am writing a computer program that involves generating 4 random numbers, a, b, c, and d, the sum of which should equal 100. </p>
<p>Here is the method I first came up with to achieve that goal, in pseudocode:</p>
<pre><code>Generate a random number out of 100. (Let's say it generates 16).
Assign this value as th... | Gabeee | 736,508 | <p>The problem is that <span class="math-container">$x_2$</span> is dependent on <span class="math-container">$x_1$</span>, then <span class="math-container">$x_3$</span> is dependent on <span class="math-container">$x_2,x_1$</span> and so on. Right from the start, once you generate the first number, the rest will "oft... |
1,276,206 | <p>I am writing a computer program that involves generating 4 random numbers, a, b, c, and d, the sum of which should equal 100. </p>
<p>Here is the method I first came up with to achieve that goal, in pseudocode:</p>
<pre><code>Generate a random number out of 100. (Let's say it generates 16).
Assign this value as th... | jblood94 | 697,491 | <p>The method described will be random, as will any number of schemes. Whether the method is suitable depends on the application. (For example, we could choose <span class="math-container">$\{x_1,x_2,x_3,x_4\}$</span> by taking a simple random sample from <span class="math-container">$\{\{10,20,30,40\},\{25,25,25,25\},... |
1,767,751 | <p>I want to prove that the distance between incentre and orthocentre is $$\sqrt{2r^2-4R^2\cos A\cos B\cos C} $$here $r$ is inradius and $R$ is circumradius.
I considered $\triangle API$ ($P$ is orthocentre and $I$ is incentre). I could find $AP=2R\cos A$, $AI=4R\sin\frac{B}{2}\sin\frac{C}{2} $ and $\angle PAI=\angle ... | mathlove | 78,967 | <blockquote>
<p>So applying cosine rule I got $$\small PI^2=4R^2+16R^2\sin^2\frac{B}{2}\sin^2\frac{C}{2} -16R^2\cos A\sin\frac{B}{2}\sin\frac{C}{2}\Bigg(\cos\frac{B}{2}\cos\frac{C}{2}+\sin\frac{B}{2}\sin\frac{C}{2}\Bigg)$$</p>
</blockquote>
<p>I think that you have a typo (the red part) : </p>
<p>$$\begin{align}&am... |
2,012,371 | <p>I am new to calculus. Can you help me with this?</p>
<h2>$a+\sqrt{a}=4$</h2>
<p>$5a+a\sqrt{a}=? $</p>
| mfl | 148,513 | <p>$$a+\sqrt{a}=4\iff \sqrt a=4-a\underbrace{\implies}_{\mathrm{squaring}} a=a^2-8a+16.$$</p>
<p>$$a+\sqrt{a}\implies a=4-\sqrt a\underbrace{\implies}_{\times a} \color{red}{ a^2=4a-a\sqrt a}.$$ Substitute $a^2$ in the first equality and get </p>
<p>$$a=a^2-8a+16=\color{red}{4a-a\sqrt{a}}-8a+16.$$ This is equivalent... |
4,315,858 | <p>When tackling the <a href="https://math.stackexchange.com/a/4313831/732917">question</a>, I found that for any <span class="math-container">$a>1$</span>,</p>
<p><span class="math-container">$$
I_1(a)=\int_{0}^{\pi} \frac{d x}{a-\cos x}=\frac{\pi}{\sqrt{a^{2}-1}}.
$$</span>
Then I started to think whether there is... | Bertrand87 | 953,938 | <p>One way using complex analysis:</p>
<p>Since the integrand is even:
<span class="math-container">$$
I=\int_{0}^{\pi} \frac{d x}{(a-\cos x)^n} = \frac{1}{2} \int_{-\pi}^{\pi} \frac{d x}{(a-\cos x)^n}
$$</span></p>
<p>Since <span class="math-container">$a>1$</span>, we can expand the integrand with the generalized ... |
46,496 | <p>I'm tracing through solution for a question I was working on. I don't quite understand how they got to the line which I marked with an arrow (apologies for using an image, I don't have much time left and didn't wanna have to look up how to do sigma notation in LaTeX here)</p>
<p><img src="https://i.stack.imgur.com/... | Zev Chonoles | 264 | <p>$$7\sum_{k=1}^{n-1}\frac{1}{k}=7\left(1+\frac{1}{2}\right)+7\sum_{k=3}^{n-1}\frac{1}{k}$$
$$3\sum_{k=2}^{n}\frac{1}{k}=3\left(\frac{1}{2}+\frac{1}{n}\right)+3\sum_{k=3}^{n-1}\frac{1}{k}$$
$$4\sum_{k=3}^{n+1}\frac{1}{k}=4\left(\frac{1}{n}+\frac{1}{n+1}\right)+4\sum_{k=3}^{n-1}\frac{1}{k}$$
Do you see how to get the m... |
880,437 | <p>If the numerator of a fraction is increased by $2$ and the denominator by $1$, it becomes $\displaystyle \frac{5}{8}$ and if the numerator and the denominator of the same fraction are each increased by $1$, the fraction becomes equal to $\displaystyle \frac{1}{2}$. Find the fraction.</p>
<p>I tried,
Let the numerat... | SuperAbound | 140,590 | <p>We have the equations
$$\frac{x+2}{y+1}=\frac{5}{8}, \ \frac{x+1}{y+1}=\frac{1}{2}=\frac{4}{8}$$
Hence
$$x+1=4, \ y+1=8 \ \implies x=3, \ y=7$$
We are allowed to do so since $(x+2)-(x+1)=5-4$</p>
|
880,437 | <p>If the numerator of a fraction is increased by $2$ and the denominator by $1$, it becomes $\displaystyle \frac{5}{8}$ and if the numerator and the denominator of the same fraction are each increased by $1$, the fraction becomes equal to $\displaystyle \frac{1}{2}$. Find the fraction.</p>
<p>I tried,
Let the numerat... | Community | -1 | <p>$\dfrac{x+2}{y+1}=\dfrac{5}{8}$ and $\dfrac{x+1}{y+1}=\dfrac{1}{2}=\dfrac{4}{8}$</p>
<p>Dividing first equation by second equation you would get</p>
<p>$$\dfrac{x+2}{x+1}=\dfrac{5}{4}=\dfrac{3+2}{3+1}$$</p>
<p>This should be easy now..</p>
|
597,845 | <p>I am almost embarrassed writing this. But can someone tell me why this may not be true (so, please give me a counter example) for a power series where $x \in [-1,1]$</p>
<p>$|\sum_{n \geq 0} a_n x^n| \leq \sum_{n \geq 0} a_n $</p>
<p>Where $\sum_{n\geq 0} a_n $ is known to be convergent. </p>
<p>What if $a_n \geq... | QED | 91,884 | <p>Take all the $a_n$'s to be negative, say $a_n=-1/n^2$. Then $\sum_{n}a_n$ converges. However the right side of the inequality is negative while the left side of the inequality has to be non-negative. </p>
|
594,785 | <p>$f:\mathbb N\to\mathbb N$
such that
$f(x) = 2x$.</p>
<p>$f:\mathbb Z\to\mathbb Z$
such that
$f(x) = 2x$</p>
<p>How are these two different?</p>
<p>And also $h:\mathbb R\to\mathbb R$ where $h(x) = \sqrt x$</p>
<p>$f:\mathbb N\to\mathbb N$ where $f(x) = \sqrt n$</p>
| Haha | 94,689 | <p>Because $f:\Bbb N\to \Bbb N$ has only positive values in difference with $f:\Bbb Z\to \Bbb Z$ that has negative values too.</p>
<p>for $h:[0,\infty]\to \Bbb R$ (From $[0,\infty] $and not $\Bbb R$ because $\sqrt x$ is not defined in negative reals)we have that $h(x)>0$ for every $x$ in difference with $f:\Bbb N\... |
1,400,436 | <p>This is a question we asked on a second semester calculus test.</p>
<p>For what values of $p$ does this series converge?
$$\sum_{n=1}^{\infty}\frac{\sin(1/n)}{n^p}$$</p>
<p>I believe that it actually can be shown that $p> 0$ is a valid answer. </p>
<p>However. I am interested in finding a proof that is simple ... | twinkle twinkle little star | 155,261 | <p>Limit comparison test. </p>
<p>$$\lim \frac{\sin(1/n)/n^p}{1/n^{p+1}}=1,$$</p>
<p>$\sum\frac{1}{n^{p+1}}$ converges when $p>0$, diverges when $p\leq 0$.</p>
|
4,180,865 | <p>A family of subsets <span class="math-container">$\mathcal{G}$</span> is called intersecting if <span class="math-container">$G_{1} \cap G_{2} \neq \emptyset$</span> for all <span class="math-container">$G_{1}, G_{2} \in \mathcal{G}$</span>. Let <span class="math-container">$\mathcal{F}_{1}, \mathcal{F}_{2}, \ldots,... | Mengfan Ma | 443,861 | <p>Choose <span class="math-container">$S\in 2^{\{1,\cdots,n\}}$</span> uniformly at random, then</p>
<p><span class="math-container">$$
\begin{align}
\frac{|\bigcup_{i=1}^{k} \mathcal{F}_{i}|}{2^n}&=P(S\in \bigcup_{i=1}^{k} \mathcal{F}_{i}) \\
&=1-P(\bigvee_{i=1}^{k}S\notin \mathcal{F}_i) \\
&\le 1-\prod_{... |
24,990 | <p>Usually I write equations in questions/answers, but writing the equation in LaTeX takes up the most time. It takes me nearly three or four times as long to write them in LaTeX. Is there any way to make my equation writing faster? I already use a visual LaTeX editor to speed things up, but I'm wondering if there are ... | Alex M. | 164,025 | <p>I appreciate your work and am looking forward to using it; it seems that I am not alone in this, given that your post currently has 12 - 1 votes. There is one problem though: starting by the end of August, you have posted on MSE a number of answers that only advertise your search engine and point to duplicates found... |
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