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 |
|---|---|---|---|---|
464,426 | <p>Find the limit of $$\lim_{x\to 1}\frac{x^{1/5}-1}{x^{1/3}-1}$$</p>
<p>How should I approach it? I tried to use L'Hopital's Rule but it's just keep giving me 0/0.</p>
| Eric Jablow | 70,913 | <p>Or, and this will lead you to the chain rule, </p>
<p>$$
\lim_{x\to 1}\frac{x^{1/5}-1}{x^{1/3}-1} = \lim_{x\to 1}\frac{x^{1/5}-1}{x-1}\lim_{x\to 1}\frac{x-1}{x^{1/3}-1},
$$</p>
<p>provided both the limits on the right exist. Continue with</p>
<p>$$
\lim_{x\to 1}\frac{x^{1/5}-1}{x-1}\lim_{x\to 1}\frac{x-1}{x^{1/3... |
147,661 | <p>Let $R$ be a (noncommutative) ring and $a \in R$ such that $a(1-a)$ is nilpotent. Why is $1+a(t-1)$ a unit in $R[t,t^{-1}]$? Probably one just has to write down an inverse element, but I could not find it. Perhaps there is a trick related to the geometric series which motivates the choice of the inverse element, whi... | Bill Dubuque | 242 | <p>A simple local argument works more generally. Note that $\rm\: c = 1 + a(t-1)\:$ is a unit both $\rm\:mod\ a\:$ and $\rm\: mod\ b = a\!-\!1,\:$ since $\rm\: c\equiv 1\pmod a,\:$ and $\rm\:c\equiv t\pmod{a\!-\!1}.\:$ Now apply</p>
<p><strong>Theorem</strong> $\rm\ \ ab\:$ nilpotent $\rm\: \Rightarrow\ (U\!+\!aR)\cap... |
55,076 | <p>Let $T: \mathbb{R}^3 \rightarrow \mathbb{R}^3 $ be the linear operator </p>
<p>$$
A = \begin{pmatrix}
1 & 0 & 0 \\
1 & 2 & 0 \\
0 & 0 & 3 \end{pmatrix}
$$</p>
<p>The problem I am trying to understand is the following. </p>
<p>True or False? If $W$ is a $T$-invariant subspace of $\ma... | Dennis Gulko | 6,948 | <p>The answer is true for the given matrix $A$, since it is semi-simple (diagonalizable), as Geoff explained in his answer, but this is not true in general: for example, if you take $$B = \begin{pmatrix}
1 & 0 & 0 \\
1 & 1 & 0 \\
0 & 0 & 1 \end{pmatrix}$$
Then $W=Span\{(0,1,0),(0,0,1)\}$ is $... |
55,076 | <p>Let $T: \mathbb{R}^3 \rightarrow \mathbb{R}^3 $ be the linear operator </p>
<p>$$
A = \begin{pmatrix}
1 & 0 & 0 \\
1 & 2 & 0 \\
0 & 0 & 3 \end{pmatrix}
$$</p>
<p>The problem I am trying to understand is the following. </p>
<p>True or False? If $W$ is a $T$-invariant subspace of $\ma... | Pierre-Yves Gaillard | 660 | <p>Let $a$ be an endomorphism of a finite dimensional vector space $V$ over a field $K$. Assume that the eigenvalues of $a$ are in $K$. Let $\lambda$ be such an eigenvalue, let $f\in K[X]$ be the minimal polynomial, and let $d$ its degree. Then there is a unique polynomial $g_\lambda$ of degree $ < d$ such that $g_\... |
1,681,205 | <p>I would like a <strong>hint</strong> for the following, more specifically, what strategy or approach should I take to prove the following?</p>
<p><em>Problem</em>: Let $P \geq 2$ be an integer. Define the recurrence
$$p_n = p_{n-1} + \left\lfloor \frac{p_{n-4}}{2} \right\rfloor$$
with initial conditions:
$$p_0 = P ... | KonKan | 195,021 | <p>Let us start with the solution of the homogeneous recurrence
$$\phi_n = \phi_{n-1} + \frac{\phi_{n-4}}{2}$$
its characteristic equation is
$$x^4 - x^3 - \frac{1}{2} = 0$$
this equations has $4$ solutions, two of them are complex and the other two are a negative and a positive real number. Their approximate values,... |
1,746,180 | <p>I already solved a few integrals with substitution but in this case I have no idea how to start. How to solve the integral $$\int_0^{\pi/2} \frac{\sqrt{\sin(x)}}{\sqrt{\sin(x)}+\sqrt{\cos(x)}} dx,$$where $x=\frac{\pi}{2}-t$ with substitution, can you tell me how to start?
It would be great!</p>
| Dr. Sonnhard Graubner | 175,066 | <p>after the substitution we have the integral
$$-\int_{\pi/2}^0{\frac {\sqrt {\cos \left( t \right) }}{\sqrt {\cos \left( t \right) }
+\sqrt {\sin \left( t \right) }}}dt$$</p>
|
335,116 | <p>As a PhD student, if I want to do something algebraic / linear-algebraic such as representation theory as well as do PDEs, in both the theoretical and numerical aspects of PDEs, would this combination be compatible and / or useful? Is it feasible?</p>
<p>I'd be grateful for an online resource to look into.</p>
<p>... | Dima Pasechnik | 11,100 | <p>There is e.g. a book
<a href="https://singer.math.ncsu.edu/papers/dbook2.ps" rel="noreferrer">Differential Galois Theory</a> by M. van der Put and M. F. Singer, where in Appendix D one can find things on the PDE case (the book is mostly about ODEs).</p>
<p>In mathematical physics there are topics such as <a href="... |
2,764,381 | <p>one thing I don't understand is what is sin(0) and sin(1) exactly? I am alright with the concept of radian (pi) but don't understand 0 and 1. What does it mean?</p>
| Karn Watcharasupat | 501,685 | <p>Generally, unless stated, the interval $[0,1]$ means the range of $x$ is from $0$ radian to $1$ radian.</p>
<p>$\sin 0$ is basically $\sin x|_{x=0}=0$. You can think of $\sin 1 = \sin \frac\pi\pi\approx0.0175$.</p>
<p>This is how it looks like:
<a href="https://i.stack.imgur.com/l8sYr.jpg" rel="nofollow noreferrer... |
290,903 | <p>I am unable to understand the fundamental difference between a Gradient vector and a Tangent vector.
I need to understand the geometrical difference between the both. </p>
<p>By Gradient I mean a vector $\nabla F(X)$ , where $ X \in [X_1 X_2\cdots X_n]^T $</p>
<p>Note: I saw similar questions on "Difference betw... | dekuShrub | 556,042 | <p>The gradient is like the derivative of a function for multiple variables. It shows the rate of change depending on all the given variables of the function, and it is also a vector that points in the same direction as the normal to the function at a given point.</p>
<p>The tangent vector points along the surface of ... |
2,104,984 | <p>For every $x\in \mathbb{R}$, $f(x+6)+f(x-6)=f(x)$ is satisfied. What may be the period of $f(x)$?
I tried writing several $f$ values but I couldn't get something like $f(x+T)=f(x)$.</p>
| Mark Bennet | 2,906 | <p>Here is a big hint: apply the identity you have been given to $f(x+6)$ to obtain $f(x+6)=f(x+12)+f(x)$ and then ...</p>
|
2,104,984 | <p>For every $x\in \mathbb{R}$, $f(x+6)+f(x-6)=f(x)$ is satisfied. What may be the period of $f(x)$?
I tried writing several $f$ values but I couldn't get something like $f(x+T)=f(x)$.</p>
| Bumblebee | 156,886 | <p>If $f(x+6)+f(x-6)=f(x),$ then $f(x+12)+f(x)=f(x+6)$ and therefore $f(x+12)=-f(x-6)=-(-f(x-24)).$ Hence you have $$f(x)=f(x-36).$$</p>
|
772,315 | <p>Statement: $\limsup\limits_{n\to\infty} c_n a_n = c \limsup\limits_{n\to\infty} a_n$</p>
<p>Please help find a counterexample to this statement if $c<0$.</p>
<p>Edit: also suppose $c_n \to c$ and $\limsup a_n$ is finite</p>
| Joe | 107,639 | <p>$c_n\equiv-1$, $a_n=\sin\left(\frac{n\pi}{6}\right)$.</p>
<p>Then $\limsup(c_na_n)=1$ but $\limsup a_n=1$ too hence $-\limsup a_n=-1\neq\limsup(c_na_n)=1$.</p>
|
2,173,918 | <p>Let $f(z)=\sum\limits_{k=1}^\infty\frac{z^k}{1-z^k}$. I want to show that this series represents a holomorphic function in the unit disk. I'm, however, quite confused. For example, is $f(z)$ even a power series? It doesn't look as such. Here's what I have so far come up with.</p>
<blockquote>
<p>Proof:</p>
</bloc... | Jacky Chong | 369,395 | <p>Hint: Observe
\begin{align}
f_k(z) = \frac{z^k}{1-z^k}
\end{align}
is definitely holomorphic on the open unit disc for all $k$. In particular, we have that
\begin{align}
g_n(z) = \sum^n_{k=1}f_k(z)
\end{align}
is also holomorphic on the unit disc. Now, show that $g_n$ is a normally convergent sequence, i.e. show tha... |
4,041,758 | <p>If you have a line passing through the middle of a circle, does it create a right angle at the intersection of the line and curve?</p>
<p>More generally, is it valid to define an angle created between a line and a curve? Is the tangent to the curve at the point of intersection a valid interpretation (I.e a semi circ... | Narasimham | 95,860 | <p>Yes. The semi-circle can even be referred to sometimes as a ( Curvilinear) <em>Diangle</em>, sum of the two shown right angles is <span class="math-container">$\pi$</span>.</p>
<p>If Q and P are opposite points of a diameter in a circle then the tangent at Q makes a right angle to the line PQ. Similarly tangent at P... |
964,989 | <p>Question: Find the order of $(1/2)^{1/2}$, $(1/e)^{1/e}$, $(1/3)^{1/4}$ without using calculator.</p>
<p>Extra constraint: You only have about 150 seconds to do it, failing to do so will eh... make you run out of time on the exam which affects the chance of admitting into graduate school!</p>
<p>Back to the questi... | Pieter21 | 170,149 | <p>I'd first raise to the power $4$, to find $1/4 < 1/3$. Or if this is a typo, raise to the power of 6 to find $1/8 < 1/9$.</p>
<p>For the comparisons with $1/e$, I'd take the logarithm, and do comparisons, knowing $ln(2)$ is about $0.69$, and $ln(3)$ is more than $1$.</p>
|
539,363 | <p>Two horses start simultaneously towards each other and meet after $3h 20 min$. How much time will it take the slower horse to cover the whole distance if the first arrived at the place of departure of the second $5 hours$ later than the second arrived at the departure of the first.</p>
<p><strong>MY TRY</strong>::<... | BaronVT | 39,526 | <p>First, let's identify what you actually want to solve for, which is $\frac{d}{b}$. Solve for $a$ in your first equation: $a = 3/10 d - b$ and substitute into the second equation</p>
<p>$$
\frac{d}{\frac{3}{10} d - b} - \frac{d}{b} = 5\\
db- d\left(\frac{3}{10} d - b\right) = 5b\left(\frac{3}{10} d - b\right)\\
d\le... |
539,363 | <p>Two horses start simultaneously towards each other and meet after $3h 20 min$. How much time will it take the slower horse to cover the whole distance if the first arrived at the place of departure of the second $5 hours$ later than the second arrived at the departure of the first.</p>
<p><strong>MY TRY</strong>::<... | copper.hat | 27,978 | <p>Let $s_i$ be the speeds of the horses, with $s_1>s_2$ for definiteness. Let $d$ be the total distance. Let $t_i$ be the time taken to cover $d$, that is, $t_i = \frac{d}{s_i}$.</p>
<p>Clearly we have $s_i >0, d>0$.</p>
<p>Since the horses meet after 200 mins., we have the total distance travelled to be $d... |
4,417,896 | <p>I have only found information regarding doing this by integration by parts. By differentiating under the integral sign, I let
<span class="math-container">$$I_n = \int_0^\infty x^n e^{-\lambda x} dx $$</span>
and get <span class="math-container">$\frac{dI_n}{d\lambda} = -I_{n+1} $</span> and therefore <span class="m... | Eugene | 726,796 | <p>Let us differentiate a function
<span class="math-container">$$
\begin{aligned}
I_n(\lambda) = \int_0^{+\infty}x^ne^{-\lambda x}dx
\end{aligned}
$$</span></p>
<p>with respect to <span class="math-container">$\lambda$</span>:</p>
<p><span class="math-container">$$
\begin{aligned}
\frac{d}{d\lambda}I_n(\lambda) &=... |
1,383,380 | <p>On page 12 of Stein, Shakarchi textbook 'Complex analysis', the authors state that the <em>Cauchy-Riemann equations link complex and real analysis</em>. I have completed courses on real and complex analysis, but I feel that this is somewhat of an over-statement. But perhaps it is just me which doesnt have a good eno... | Augustin | 241,520 | <p>There could be both!</p>
<p>The existence of such a point is a consequence of the intermediate value theorem.</p>
|
464,489 | <p>We are given $H = \{(1),(13),(24),(12)(34),(13)(24),(14)(23),(1234),(1432)\}$ is a subgroup of $S_4$. Also assume $K = \{(1),(13)(24)\}$ is a normal subgroup of $H$. Show $H/K$ isomorphic to $Z_2\oplus Z_2$. </p>
<p>This is just a practice question (not assignment). So I have tried finding $H/K$ explicitly.</p>
<p... | Prahlad Vaidyanathan | 89,789 | <p>No - be careful. $(1234)^2 = (13)(24) \in K$. Hence, $(1234)K^2 = K$, so $(1234)K$ has order 2 in $H/K$. In fact, you can just check by hand that all the elements of $H$ either square to $(1)$ or to $(13)(24)$ [You need to use the fact that disjoint cycles commute]. Hence, every element of $H/K$ has order $\leq 2$, ... |
1,602,312 | <p>Can we prove that $\sum_{i=1}^k x_i$ and $\prod_{i=1}^k x_i$ is unique for $x_i \in R > 0$?<br>
I stated that conjecture to solve CS task, but I do not know how to prove it (or stop using it if it is false assumption). </p>
<p>Is the pair of Sum and Product of array of k real numbers > 0 unique per array? I do ... | Community | -1 | <p>Euclidean vectors are the easiest to visualize. Let's go through this in steps. We'll skip a set of only one vector until the end and start with a set of two vectors.</p>
<p>If a set of two Euclidean vectors is linearly <em>dependent</em> then there exists a line containing all both vectors and the origin. If a s... |
1,602,312 | <p>Can we prove that $\sum_{i=1}^k x_i$ and $\prod_{i=1}^k x_i$ is unique for $x_i \in R > 0$?<br>
I stated that conjecture to solve CS task, but I do not know how to prove it (or stop using it if it is false assumption). </p>
<p>Is the pair of Sum and Product of array of k real numbers > 0 unique per array? I do ... | fosho | 166,258 | <p>A set of <strong>linearly independent</strong> vectors: pick any vector in this set and you cannot write it as a linear combination of the other vectors in this set. For example suppose $\{x_1,x_2,x_3\}$ is a set of linearly independent vectors. Let's say you want to write $x_2$ as a linear combination of $x_1$ and ... |
1,602,312 | <p>Can we prove that $\sum_{i=1}^k x_i$ and $\prod_{i=1}^k x_i$ is unique for $x_i \in R > 0$?<br>
I stated that conjecture to solve CS task, but I do not know how to prove it (or stop using it if it is false assumption). </p>
<p>Is the pair of Sum and Product of array of k real numbers > 0 unique per array? I do ... | bhbr | 160,776 | <p>Two parallel vectors are linearly dependent. Three vectors lying in a plane are linearly dependent. Four vectors lying in the same three-dimensional hyperplane are linearly dependent.</p>
<p>In n-dimensional space, you can find at most n linearly independent vectors.</p>
<p>Think of the vectors as rods with which ... |
891,370 | <p>I got the function $8.513 \times 1.00531^{\Large t} = 10$. The task is to solve $t$. The correct answer is $t = 31$. How do I get there ?.</p>
| Felix Marin | 85,343 | <p>$\newcommand{\angles}[1]{\left\langle\, #1 \,\right\rangle}
\newcommand{\braces}[1]{\left\lbrace\, #1 \,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\, #1 \,\right\rbrack}
\newcommand{\ceil}[1]{\,\left\lceil\, #1 \,\right\rceil\,}
\newcommand{\dd}{{\rm d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcomma... |
3,499,862 | <p>What is the equivalent of <span class="math-container">$\neg (\forall x) (P(x) \vee Q(x))$</span>? Will <span class="math-container">$P(x) \vee Q(x)$</span> be negated too? Or is just <span class="math-container">$\forall x$</span> negated?</p>
| N. Bar | 641,014 | <p><span class="math-container">$$\neg (\forall x)(Px \lor Qx)$$</span></p>
<p>By quantifier negation (QN) rules,
<span class="math-container">$$(\exists x)\neg(Px \lor Qx)$$</span></p>
<p>By DeMorgan's Law</p>
<p><span class="math-container">$$(\exists x)(\neg Px \land \neg Qx)$$</span></p>
|
4,510,697 | <p><a href="https://i.stack.imgur.com/HKvBy.png" rel="nofollow noreferrer">How do I find explicit solutions of x and y in this system?</a></p>
| John Omielan | 602,049 | <p>You've got the right idea, but as <a href="https://math.stackexchange.com/questions/4510682/find-postive-integer-x-with-x1-and-dfracx6-1x-1-is-perfect-square#comment9470655_4510682">Kenta S's comment</a> indicates, it's <span class="math-container">$d \mid 3$</span> (since <span class="math-container">$x \equiv -1 \... |
2,032,241 | <p>In Euler's (number theory) theorem one line reads: since $d|ai$ and $d|n$ and $gcd(a,n)=1$ then $d|i$. I've been staring at this for over an hour and I am not convinced why this is true could anyone explain why? I have tried all sorts of lemma's I've seen before but I honestly just can't see it and I feel I'm going ... | Sean Haight | 273,590 | <p>Since $d|n$ and gcd$(a,n) = 1$ we must have gcd$(d,a) = 1$, since if $a$ and $d$ shared some common divisor, then $a$ and $n$ would share some common divisor. Now since $d|ai$ and gcd$(a,d) = 1$ , $d|i$. This last statement is a straight forward Theorem that says that if $d|bc$ and gcd$(d,b) = 1$ then $d|c$. Below i... |
972,281 | <p>I have to find the inverse laplace transform of:</p>
<p>$\mathcal{L}^{-1}(\frac{s}{-8+2s+s^2})$</p>
<p>I found it was </p>
<p>$\frac{2}{3}e^{-4t}+\frac{1}{3}e^{2t}$</p>
<p>But the question I'm asked is, determine $A,B,C,D$ such that $e^{At}(Bcosh(Ct)+Dsinh(Ct))$ is a solution of the inverse laplace transform.</p... | DLV | 160,304 | <p>I've solved this.</p>
<p>One creates a system of equations out of this expression. And equate it to my simplified answer.</p>
|
239,653 | <p>It is usually told that Birch and Swinnerton-Dyer developped their famous conjecture after studying the growth of the function
$$
f_E(x) = \prod_{p \le x}\frac{|E(\mathbb{F}_p)|}{p}
$$
as $x$ tends to $+\infty$ for elliptic curves $E$ defined over $\mathbb{Q}$, where the product is defined for the primes $p$ where $... | Community | -1 | <p>Question 1: The $L$-functon $L(E/\mathbf{Q},s)$ is a product over $L_p(E/\mathbf{Q},s) := 1/(1-a_pp^{-s}+p^{1-2s})$. Now plug in $s=1$ and use $p+1 - |E(\mathbf{F}_p)| = a_p$. (This is only a heuristic.)</p>
|
4,431,185 | <p>Let <span class="math-container">$ABC$</span> be a triangle and denote its area by <span class="math-container">$k = \mathrm{area}(ABC)$</span>. I want to divide <span class="math-container">$ABC$</span> into two sub-triangles <span class="math-container">$ABE$</span> and <span class="math-container">$AEC$</span> su... | nkhedekar | 803,212 | <p>I'm not sure what you mean by "compute this exactly". To find the point <span class="math-container">$E$</span> on the side <span class="math-container">$BC$</span>, you can compare the formulae for the areas of the new and original triangles <span class="math-container">$ABE$</span> and <span class="math-... |
25,485 | <p>Not sure if this is more appropriate for here or for Math.SE, but here goes:</p>
<p>How does one who is self-studying mathematics determine if a textbook is too hard for you?</p>
<p>Math is hard in general, but when does a textbook cross that line from being challenging to being nearly intractable?</p>
<p>Sometimes ... | Sue VanHattum | 60 | <p>I don't know how to decide, in general, whether a book is the right one or not. In particular cases, it might make sense to ask here, with the details.</p>
<p>But I have a great answer for the related question, <em><strong>"How do I approach reading a math book?"</strong></em></p>
<p>It's very different th... |
1,022,342 | <p>Assume we have a function $y=f(x)$ that is $\textit{C}^\infty$ and the function has a number of local maximums. Assume there are $k$ such maximums $\{m_1, m_2, m_3, \ldots , m_k\}$ where $f'(m_i)=0$. </p>
<p>If I performed a gradient ascent algorithm for any point $x$ where $f(x)$ is defined, it would get to one of... | Community | -1 | <p>It sounds like a zone of attraction or neighborhood of an optimum.</p>
<p>If you think of the gradient search as a dynamic process then these zones are limit sets. </p>
|
1,022,342 | <p>Assume we have a function $y=f(x)$ that is $\textit{C}^\infty$ and the function has a number of local maximums. Assume there are $k$ such maximums $\{m_1, m_2, m_3, \ldots , m_k\}$ where $f'(m_i)=0$. </p>
<p>If I performed a gradient ascent algorithm for any point $x$ where $f(x)$ is defined, it would get to one of... | Anthony Carapetis | 28,513 | <p>At least in the context of Morse theory these are known as the <em>(un)stable manifolds</em> of the critical points $m_i$.</p>
<p>If $\phi(x,t)$ is the gradient flow of $f$ (or more generally any flow):</p>
<p>$$ \phi(x,0) = x \\ \frac{\partial}{\partial t} \phi(x,t) = \nabla f(\phi(x,t))$$
then the stable and uns... |
4,359,019 | <p>Let <span class="math-container">$P \in \mathbb{R}^{N\times N}$</span> be an orthogonal matrix and <span class="math-container">$f: \mathbb{R}^{N \times N} \to \mathbb{R}^{N \times N}$</span> be given by <span class="math-container">$f(M) := P^T M P$</span>. I am reading about random matrix theory and an exercise i... | F_M_ | 1,012,495 | <p>Consider the map <span class="math-container">$\psi \colon E' \to F'$</span> given by <span class="math-container">$\psi(f) = \left. f \right|_F.$</span> For surjectivity, let <span class="math-container">$f \in F'$</span> be given. Then we can extend <span class="math-container">$f$</span> to <span class="math-cont... |
839,043 | <p>I have been studying power functions, and started to think about imaginary powers. Take the function $x^i$. Because I don't know how to multiply a number $i$ times, I tried to simplify the equation</p>
<p>$x^i = x^{\sqrt{-1}} = x^{(-1)^{1/2}}$</p>
<p>Then, using the property of exponents that states an exponent to... | user99680 | 99,680 | <p>Your definition works for Real numbers. But $(-1)^{1/2}$ is not a Real number. You need to use the complex logz, meaning you need to choose a branch of logz, so that you can define: $$x^i=e^{ilogx} $$;let $x=a+ib$, then this is equal to: $$e^{i(a+ib)}=e^{-b+ia} $$. If you do not select a branch of logz, you have a m... |
2,357,899 | <p>I want to prove that the cartesian product of a finite amount of countable sets is countable. I can use that the union of countable sets is countable. </p>
<p><strong>My attempt:</strong></p>
<p>Let $A_1,A_2, \dots, A_n$ be countable sets. </p>
<p>We prove the statement by induction on $n$</p>
<p>For $n = 1$, th... | H. H. Rugh | 355,946 | <p>The phrase 'the union of countable sets is countable' is wrong unless you add the word 'countable' before 'union'.</p>
<p>The zig-zag argument is nothing but a graphical way to describe a bijection between $ {\Bbb N}\times{\Bbb N}$ and $ {\Bbb N}$.
You may also give an explicit formula for such a bijection. Usin... |
2,348,909 | <p>Example 7.3 of Baby Rudin states that the sum
\begin{align}
\sum_{n=0}^{\infty}\frac{x^2}{(1 + x^2)^n}
\end{align}
is, for $x \neq 0$, a convergent geometric series with sum $1 + x^2$. This confuses me. As far as I know, a geometric series is a series of the form
\begin{align}
\sum_{n=0}^{\infty}x^n,
\end{align}
a... | Mark Viola | 218,419 | <p>$$\begin{align}
\sum_{n=0}^\infty \frac{x^2}{(1+x^2)^n}&=x^2\sum_{n=0}^\infty \left((1+x^2)^{-1}\right)^n\\\\
&=\frac{x^2}{1-(1+x^2)^{-1}}\\\\
&=\frac{x^2(1+x^2)}{1+x^2-1}\\\\
&=1+x^2
\end{align}$$</p>
<p>for $x\ne0$.</p>
|
81,267 | <p>I have the following problem: I have a (a lot)*3 table, meaning that I have 3 columns, say X, Y and Z, with real values. In this table some of the rows have the same (X,Y) values, but with different value of Z. For instance</p>
<pre><code>{{12.123, 4.123, 513.423}, {12.123, 4.123, 33.43}}
</code></pre>
<p>have th... | alephalpha | 6,652 | <pre><code>Min /@ Transpose@# & /@ GatherBy[data, Most]
</code></pre>
<p>Or if you're using Mathematica 10:</p>
<pre><code>First@*MinimalBy[Last] /@ GatherBy[data, Most]
</code></pre>
|
81,267 | <p>I have the following problem: I have a (a lot)*3 table, meaning that I have 3 columns, say X, Y and Z, with real values. In this table some of the rows have the same (X,Y) values, but with different value of Z. For instance</p>
<pre><code>{{12.123, 4.123, 513.423}, {12.123, 4.123, 33.43}}
</code></pre>
<p>have th... | 2012rcampion | 21,750 | <p>Using the new <code>Association</code> method <code>GroupBy</code>:</p>
<pre><code>Append @@@ Normal[GroupBy[data, Most -> Last, Min]]
</code></pre>
<p>This method is slightly slower than Bob Hanlon's and alephalpha's (and produces identical results), but I thought I'd show off one of the v10 features.</p>
|
3,609,906 | <p>I need to compute <span class="math-container">$$\lim_{n \to \infty}\sqrt{n}\int_{0}^{1}(1-x^2)^n dx.$$</span>
I proved that
for <span class="math-container">$n\ge1$</span>,
<span class="math-container">$$\int_{0}^{1}(1-x^2)^ndx={(2n)!!\over (2n+1)!!},$$</span>
but I don't know how to continue from here.</p>
<p>I a... | marty cohen | 13,079 | <p>Redoing what has been done
many many many times before.</p>
<p><span class="math-container">$\begin{array}\\
I_n
&=\int_0^1 (1-x^2)^n dx\\
I_0
&=\int_0^1 dx\\
&= 1\\
I_1
&=\int_0^1 (1-x^2) dx\\
&=1-\dfrac13\\
&=\dfrac23\\
I_n
&=\int_0^1 (1-x^2)^n dx\\
&=x(1-x^2)^n|_0^1+\int_0^1 2x^2... |
353 | <p>One of the challenges of undergraduate teaching is logical implication. The case by case definition, in particular, is quite disturbing for most students, that have trouble accepting "false implies false" and "false implies true" as true sentences. </p>
<blockquote>
<p><strong>Question:</strong> What are good poi... | Jim Hefferon | 213 | <p>My approach is like that of others but I like to use math instead of everyday language. I get them to agree that we want this statement to be true: "if <em>x</em> is a perfect square then <em>x</em> is not prime" simply because <em>x=y*y</em> is a factorization. Then we use various <em>x</em>'s to get the differen... |
353 | <p>One of the challenges of undergraduate teaching is logical implication. The case by case definition, in particular, is quite disturbing for most students, that have trouble accepting "false implies false" and "false implies true" as true sentences. </p>
<blockquote>
<p><strong>Question:</strong> What are good poi... | Clive Long | 7,954 | <p>This may add to the detailed discussions and explanations above.
I had problems with mathematical implication until I read Keith Devlin's "Introduction to Mathematical Thinking". Hopefully what follows is a faithful summary of some of the ideas in the book.</p>
<ol>
<li>Implication is not the conditional</li>
<li>T... |
353 | <p>One of the challenges of undergraduate teaching is logical implication. The case by case definition, in particular, is quite disturbing for most students, that have trouble accepting "false implies false" and "false implies true" as true sentences. </p>
<blockquote>
<p><strong>Question:</strong> What are good poi... | Dan Christensen | 6,103 | <p>If you will eventually be teaching the basic methods of proof (conditional proof, proof by contradiction, etc.) in your course for math majors, you might consider starting with the truth tables for NOT, AND and OR and postpone the truth table for IMPLIES until they understand some of those basic methods of proof. Th... |
353 | <p>One of the challenges of undergraduate teaching is logical implication. The case by case definition, in particular, is quite disturbing for most students, that have trouble accepting "false implies false" and "false implies true" as true sentences. </p>
<blockquote>
<p><strong>Question:</strong> What are good poi... | Dan Christensen | 6,103 | <p>My favour example is the implication, "If it is raining, then it is cloudy."</p>
<p>This does not mean that rain <em>causes</em> cloudiness. (There is no causality in mathematics.) In the usual sense of if-then statements in mathematics, it means only that it is not the case (present tense) that it is both... |
1,897,538 | <p>Next week I will start teaching Calculus for the first time. I am preparing my notes, and, as pure mathematician, I cannot come up with a good real world example of the following.</p>
<p>Are there good examples of
\begin{equation}
\lim_{x \to c} f(x) \neq f(c),
\end{equation}
or of cases when $c$ is not in the doma... | Slade | 33,433 | <p>If $f(t)$ is the number of humans alive at time $t$, then there is a discontinuity of $f$ at every time a human is born or dies.</p>
<p>The same logic holds for bank accounts, disk space, the number of molecules of caffeine in your body, and all other discrete phenomena.</p>
|
3,133,328 | <p>I am currently working on a question about proving the sum of eigenvalues and I have been searching for the solution <a href="https://www.youtube.com/watch?v=OLl_reBXY-g" rel="nofollow noreferrer">from YouTube</a>.</p>
<p>However, I do not understand why the teacher uses the diagonal method to show that the sum of ... | DonAntonio | 31,254 | <p>If we already know that the determinant of a square matrix is , up to sign, the sum of some products of the matrix's entry, all of which are formed by taking <strong>exactly one</strong> entry from each row and <strong>eactly one</strong> entry from each column, then it is not that hard: observe the coefficient of <... |
937,912 | <p>I'm looking for a closed form of this integral.</p>
<p>$$I = \int_0^1 \frac{\operatorname{Li}_2\left( x \right)}{\sqrt{1-x^2}} \,dx ,$$</p>
<p>where $\operatorname{Li}_2$ is the <a href="http://mathworld.wolfram.com/Dilogarithm.html" rel="noreferrer">dilogarithm function</a>.</p>
<p>A numerical approximation of i... | GEdgar | 442 | <p>Another one... replacing the Meijer G in Juan's answer
$$
{\mbox{$_4$F$_3$}(1/2,1/2,1,1;\,3/2,3/2,3/2;\,1)}+
\frac{\pi \,
{\mbox{$_4$F$_3$}(1,1,1,3/2;\,2,2,2;\,1)}}{16}
\\
\approx 1.3913072075067666818109648381255138301541952863
$$
and with user's comment
$$
{\mbox{$_4$F$_3$}(1/2,1/2,1,1;\,3/2,3/2,3/2;\,1)}+\frac{\p... |
907,851 | <p>I am really new into math, why is $-2^2 = -4 $ and $(-2)^2 = 4 $? </p>
| Community | -1 | <p>The <a href="http://en.wikipedia.org/wiki/Order_of_operations" rel="nofollow">order of operations</a> is a convention that give us what operation should we do before. In the given examples there are three operations <em>(given in the priority)</em>:operation in parenthesis, power,multiplication and subtraction so w... |
251,028 | <p>I want to make <span class="math-container">$S[\{,\cdots, \}]$</span> as follows</p>
<p>First input of <span class="math-container">$S$</span> is given list <span class="math-container">$\{1,2,3,\cdots, n\}$</span> and it produces <span class="math-container">$s_{123\cdots n}$</span></p>
<p>Further, if the ordering ... | Acus | 18,792 | <p>First, look at <a href="https://mathematica.stackexchange.com/questions/187654/how-to-build-a-templatebox-with-dynamic-length-gridbox%5Btemplatebox-with-dynamic-length-gridbox%5D%5B1%5D">https://mathematica.stackexchange.com/questions/187654/how-to-build-a-templatebox-with-dynamic-length-gridbox[templatebox-with-dyn... |
210,849 | <p>Let $p$ be an odd prime and $a$ be an integer with $\gcd(a, p) = 1$. Show that $x^2 - a \equiv 0 \mod p$ has either $0$ or $2$ solutions modulo $p$</p>
<p>I am clueless with this one. Hints please.</p>
| Bill Dubuque | 242 | <p><strong>Hint</strong> $\ $ In a field, $\rm\ x_2^2 = x_1^2 \iff 0 = x_2^2-x_1^2 = (x_2\!-x_1\!)\,(x_2\!+x_1\!)\iff x_2 = \pm\, x_1$</p>
<p><strong>Remark</strong> $\ $ Generally a nonzero polynomial over a domain <a href="https://math.stackexchange.com/a/173483/242">has no more roots than its degree.</a> </p>
|
3,404,673 | <p>Let <span class="math-container">$G$</span> be a unipotent connected linear algebraic group over a field <span class="math-container">$F$</span>. Then <span class="math-container">$G$</span> is called <em>split</em> if there is a series of closed subgroup schemes <span class="math-container">$1 = G_0 \subset G_1 \su... | D_S | 28,556 | <p>Tricky business if you were using the definition of <span class="math-container">$G = \operatorname{Res}_{E/F}(\mathbb G_a)$</span> as a form of <span class="math-container">$\mathbb G_a \times \mathbb G_a$</span> and you didn't recollect Hilbert's Theorem 90... </p>
<p>Worse, I actually did my PhD thesis on the r... |
2,780,043 | <p>In the Navy we had bunk beds with a locker and lock both built in. You could gain access with a combination lock on the side that you could reprogram the code for. The lock was nothing more than several push buttons. Ive wondered for a long time now how many possible combinations there were. I believe the lock h... | String | 94,971 | <p><strong>UPDATE</strong>: I have since learned (thanks to N. Shales in the comment section and the other answer in part) that $f(n,k)$ are called <a href="https://en.m.wikipedia.org/wiki/Stirling_numbers_of_the_second_kind" rel="nofollow noreferrer">Stirling numbers of the second kind</a>. Furthermore
$$
a_n=\sum_{k=... |
10,666 | <p>My question is about <a href="http://en.wikipedia.org/wiki/Non-standard_analysis">nonstandard analysis</a>, and the diverse possibilities for the choice of the nonstandard model R*. Although one hears talk of <em>the</em> nonstandard reals R*, there are of course many non-isomorphic possibilities for R*. My question... | Simon Thomas | 4,706 | <p>The following was useful in a recent paper on asymptotic cones with Kramer, Shelah and Tent. How many ultraproducts $\prod_{\mathcal{U}} \mathbb{N}$ exist up to isomorphism, where $\mathcal{U}$ is a non-principal ultrafilter over $\mathbb{N}$? If $CH$ holds, then obviously just one ... if $CH$ fails, then $2^{2^{\al... |
496,488 | <p>I have few questions to the proof on Universality of linearly ordered sets. Could anyone advise please? Thank you.</p>
<p>Lemma: Suppose $(A,<_{1})$ is a linearly ordered set and $(B,<_{2})$ is a dense linearly ordered set without end points. Assume $F\subseteq A$ and $E\subseteq B$ are finite and $h:F \to E... | Brian M. Scott | 12,042 | <p>First I’ll explain the proof and some of the motivation for it in greater detail, and then I’ll answer your specific questions.</p>
<p>For greater readability let $a_k=f(k)$ and $q_k=g(k)$ for each $k\in\Bbb N$, so that $F=\{a_k:k<n\}$, the subset of $A$ on which $h$ has already been defined. $E=\{h(a_k):k<n\... |
18,895 | <p>I know that the answer is $C(8,2)$, but I don't get, why.
Can anyone, please, explain it?</p>
| Yochai Timmer | 5,881 | <p>$C(8,2)$ is kinda self explanatory:<br>
you have $8$ bits, and each time you choose $2$ of them to be ones.</p>
|
1,981,014 | <blockquote>
<p>Given the function $f(x)=x^{2}-4x-5$, $A=[0,3)$ and $B=[0,1]$, find $f(A)$ and $f^{-1}(B)$.</p>
</blockquote>
<p>I found $f(A)$ by looking at the graph $f([0,3))=[-9,-5]$ but how would I calculate this without the graph? If I plug in $0$ into $f$ I get $-5$, and if I plug in $3$ I get $-8$, when I s... | Community | -1 | <p>More general if $f: \mathbb{R} \rightarrow \mathbb{R}, f(x) = ax^2 + bx + c, a \gt 0$ then $f$ is decreasing on $(-\infty, - \frac b {2a}]$ and increasing on $[- \frac b {2a}, + \infty)$. You can use this together with the fact that $f$ is continuous to get the images you want.</p>
|
1,981,014 | <blockquote>
<p>Given the function $f(x)=x^{2}-4x-5$, $A=[0,3)$ and $B=[0,1]$, find $f(A)$ and $f^{-1}(B)$.</p>
</blockquote>
<p>I found $f(A)$ by looking at the graph $f([0,3))=[-9,-5]$ but how would I calculate this without the graph? If I plug in $0$ into $f$ I get $-5$, and if I plug in $3$ I get $-8$, when I s... | Parcly Taxel | 357,390 | <p>$$f(x)=x^2-4x-5=(x-2)^2-9$$
The vertex of $f$ lies at $(2,-9)$ and 2 lies in $[0,3)$, so $f[A]$ has a closed lower endpoint at $-9$. 0 is farther from 2 than 3 is, so $f(0)=-5$ constitutes the upper endpoint, which is also closed. Hence $f[A]=[-9,-5]$.</p>
<p>As for $f^{-1}[B]$, solve for the places where $f(x)$ at... |
309,395 | <p>I am currently learning about <em>Direct Proofs</em>. I am struggling trying to find a starting point to prove the Statement: <em>For all real numbers $c$, if $c$ is a root of a polynomial with rational coefficients, then c is a root of a polynomial with integer coefficients</em>. Based on a definition given by the ... | Emily | 31,475 | <p>Example (as a hint):</p>
<p>$$P(x) = \frac{3}{2}x^2 + 2x - \frac{1}{3}$$</p>
<p>Multiply by 6:</p>
<p>$$ 6P(x) = \frac{18}{2}x^2 + 12x - \frac{6}{3}$$
$$ 6P(x) = 9x^2 + 12x - 2$$</p>
<p>If $c$ is a zero of $P(x)$, then $6P(c) = 0$, so $0 = 9c^2+12c-2$, and so $c$ is a zero of of a polynomial with integer coeffic... |
1,776,726 | <p>I'm trying to determine whether or not </p>
<blockquote>
<p>$$\sum_{k=1}^\infty \frac{2+\cos k}{\sqrt{k+1}}$$ </p>
</blockquote>
<p>converges or not. </p>
<p>I have tried using the ratio test but this isn't getting me very far. Is this a sensible way to go about it or should I be doing something else?</p>
| egreg | 62,967 | <p>Write
$$
\frac{a^2+1}{a^2+2}=1-\frac{1}{a^2+2}
$$
Minimizing this is the same as maximizing $1/(a^2+2)$ which, in turn, is the same as minimizing $a^2+2$ or, as well, minimizing $a^2$.</p>
<p>Since
$$
a=-\frac{x^2-3x-2}{x-1}
$$
the minimum value for $a^2$ is obtained when $x^2-3x-2=0$.</p>
|
1,098,438 | <p>This problem is really bothering me for some time, I appreciate if you have some idea and insight.</p>
<blockquote>
<p>Prove that</p>
<p>$$2^{2^n}+5^{2^n}+7^{2^n}$$</p>
<p>is divisible by $39$ for all natural numbers $n$.</p>
</blockquote>
<p>There was a suggestion that this should be done by mathemati... | Empy2 | 81,790 | <p>$2^{2^{n+1}}=2^{2^n2}=(2^{2^n})^2$ so each term in this sequence is the square of the previous one.<br>
Look at the remainders when you divide them by $13$:
$$2^{2^1}=4\\
2^{2^2}=4^2=16=3\pmod{13}\\
2^{2^3}=3^2=9\pmod{13}\\
2^{2^4}=9^2=81=3\pmod{13}$$
So the remainders repeat after that, and are $4,3,9,3,9,3,9$.<br>... |
1,098,438 | <p>This problem is really bothering me for some time, I appreciate if you have some idea and insight.</p>
<blockquote>
<p>Prove that</p>
<p>$$2^{2^n}+5^{2^n}+7^{2^n}$$</p>
<p>is divisible by $39$ for all natural numbers $n$.</p>
</blockquote>
<p>There was a suggestion that this should be done by mathemati... | VividD | 121,158 | <p>Following similar patterns of proof from other answers, one can also prove:</p>
<ul>
<li>$2^{2^n}+3^{2^n}+5^{2^n}$ is always divisible by $19$</li>
<li>$2^{2^n}+4^{2^n}+6^{2^n}$ is always divisible by $7$</li>
<li>$3^{2^n}+4^{2^n}+7^{2^n}$ is always divisible by $37$</li>
</ul>
|
1,037,068 | <p>If two sequences converge equally, we have
$$\lim_{n\rightarrow \infty }\left ( a_{n} \right )=\lim_{n\rightarrow \infty }\left ( b_{n} \right )$$</p>
<p>As a follow up, is the following equality also true?
$$\lim_{n\rightarrow \infty }\left ( \ln a_{n} \right )=\lim_{n\rightarrow \infty }\left ( \ln b_{n} \right ... | Disintegrating By Parts | 112,478 | <p>You have probably seen how Parseval's equality for a given $x$ is equivalent to convergence of $\sum_{j}\langle x,v_j\rangle v_j$, but have forgotten:
$$
x = \left(x-\sum_{j=1}^{N}\langle x,v_j\rangle v_j\right)+\sum_{j=1}^{N}\langle x,v_j\rangle v_j
$$
This is an orthogonal decomposition, meaning t... |
3,562,202 | <p>Show that <span class="math-container">$$\int_0^{2\pi}\arctan\Bigl( \frac {\sin x} {\cos x -2 }\Bigl) \, dx=0.$$</span> I can write this as <span class="math-container">$$\mathrm {Im}\int_0^{2\pi}\log( e^{ix}-2)\,dx.$$</span> Making the substitution <span class="math-container">$z =e^{ix} $</span> leads to <span c... | Random Variable | 16,033 | <p>I initially thought the issue was that <span class="math-container">$\operatorname{Arg}(a+ib)=\arctan \left(\frac{b}{a} \right)$</span> only if <span class="math-container">$a>0$</span>.</p>
<p>But <span class="math-container">$$ \begin{align} \int_{0}^{2 \pi} \arctan \left(\frac{\sin x}{\cos x -2} \right) \, \m... |
820,015 | <p>Suppose that you have $k$ dice, each with $N$ sides, where $k\geq N$. The definition of a straight is when all $k$ dice are rolled, there is at least one die revealing each number from $1$ to $N$. </p>
<p>Given the pair $(k,N)$, what is the probability that any particular roll will give a straight?</p>
| Graham Kemp | 135,106 | <p>An equivalent problem is fitting $N-1$ bars between $k$ stars so that there is one star between each bar. Here the stars represent the $k$ dice, and the bars are the borders of boxes which represent the $N$ values the dice can take.</p>
<p>There are ${k-1\choose N-1}$ ways to do this.</p>
<p>There are ${N+k-1\cho... |
54,541 | <p>Apparently, Mathematica has no real sprintf-equivalent (unlike any other high-level language known to man). <a href="https://mathematica.stackexchange.com/questions/970/sprintf-or-close-equivalent-or-re-implementation">This has been asked before</a>, but I'm wondering if the new <code>StringTemplate</code> function ... | Niki Estner | 242 | <p>This is my first attempt:</p>
<pre><code>Clear[formatPattern, formatPatternQ, applyFormat, applyFormatToValue, \
builtinFormatFunction]
formatPattern =
StartOfString ~~ "%" ~~
format : (DigitCharacter ... ~~ ("." ~~ DigitCharacter ..) ...) ~~
formatName : WordCharacter ... ~~ rest : ___;
formatPatternQ =... |
3,413,253 | <p>Can anyone give me some examples and non examples of Lindelöf or second countable space and spaces that is Lindelöf but not second countable? And I understand the definition but find it is hard to visualize and imagine.
I have tried google it but it turns out I only found some silly examples like finite set or emp... | Wlod AA | 490,755 | <p>An example of a Lindelöf non-second countable space, which has some additional nice properties, was constructed/discovered during the Prague 1961 Topological Conference (by wh). The point-set is the unit disc</p>
<p><span class="math-container">$$\ B(\mathbf 0\,\ 1)\ :=
\ \{p\in\mathbb R^2: |p|\le 1\} $$</span><... |
635,301 | <p>I need some help with the following problem: </p>
<blockquote>
<p>Let $f:\Bbb C \to \Bbb C$ be continuous satisfying that $f(\Bbb C)$ is an open set and that $|f(z)| \to \infty$ as $z\to \infty$. Prove that $f(\Bbb C)=\Bbb C$. </p>
</blockquote>
<p>My idea on this one is to prove by contradiction and assume that... | Community | -1 | <p>Your conditions imply that we can <em>continuously extend</em> $f$ to give a continuous function $\hat{f}$ from the Riemann sphere to itself given below</p>
<p>$$ \hat{f}(x) = \begin{cases}
f(x) & x \neq \infty \\ \infty & x = \infty \end{cases}$$</p>
|
465,001 | <p>$\mathbf{h}_i\in\mathbb{C}^{M}$ are column vectors $\forall i=\{1, 2, \cdots, K\}$.</p>
<p>$q_i\in\mathbb{R}_+$ are scalars $\forall i=\{1, 2, \cdots, K\}$</p>
<p>$\lvert\bullet\rvert$ denotes determinant of a square matrix or Euclidean norm of a vector according to the context. </p>
<p>From Sylvester's theorem, ... | amWhy | 9,003 | <p>Actually, for both functions $\cos\left(\frac 1x\right)$ and $\sin \left(\frac 1x\right)$, the limits as $x\to 0$ do not exist, and for the same reasons, irregardless of whether we consider the limit as $x \to 0^+$ or $x\to 0^{-1}$.</p>
<p>Look, for example, of the behavior of $\cos \left(\frac 1x\right)$ on the in... |
8,013 | <p>Given a set $S$, one can easily find a set with greater cardinality -- just take the power set of $S$. In this way, one can construct a sequence of sets, each with greater cardinality than the last. Hence there are at least countably infinite many orders of infinity.</p>
<p><strong>But do there exist uncountably in... | W Warren | 32,183 | <p>...lots of the math on Destiny's wall (as Mika McKinnon describes them) are just
guesses about energy inflow and outflow...particle densities...space-time configurations inside either a wormhole spacetime..or related space-times (accretion disk anomalous events,.. solar flare events..etc). In the original
pilot epi... |
34,204 | <p>I have several contour lines and one point. How can I find a point in one of those contour lines which is nearest to the given point?</p>
<pre><code>(*Create the implicit curves*)
Data={{10,20,1},{10,40,2},{10,60,3},{10,80,4},{20,25,2},{20,45,3},{20,65,4},{30,30,3},{30,50,4},{40,35,4},{40,55,5},{50,20,4},{50,40,5},... | PlatoManiac | 240 | <p>For a general purpose & more accurate solution I can suggest the following.</p>
<p><strong>Update</strong>(the missing <code>data</code> part)</p>
<pre><code>(* Create the implicit curves analytically *)
conts = ((a x^b (y^(1 - b)) + c /. U["BestFitParameters"]) == #) & /@Range[5];
(* Given x val function... |
34,204 | <p>I have several contour lines and one point. How can I find a point in one of those contour lines which is nearest to the given point?</p>
<pre><code>(*Create the implicit curves*)
Data={{10,20,1},{10,40,2},{10,60,3},{10,80,4},{20,25,2},{20,45,3},{20,65,4},{30,30,3},{30,50,4},{40,35,4},{40,55,5},{50,20,4},{50,40,5},... | C_S | 10,062 | <p>I combined my data and answer from PlatoManiac, as follows;</p>
<pre><code>(*Input Data*)
Data={{10,20,1},{10,40,2},{10,60,3},{10,80,4},{20,25,2},{20,45,3},{20,65,4},{30,30,3},{30,50,4},{40,35,4},{40,55,5},{50,20,4},{50,40,5},{60,25,5}};
U = NonlinearModelFit[Data, a x^b (y^(1 - b)) + c, {a, b, c}, {x, y}];
(*Creat... |
683,149 | <p>So I am just learning about bijections and I am having difficulty figuring out if these three problems are bijections and how to prove them. </p>
<ol>
<li>$f(x)=x/2$</li>
<li>$f(x)=2x^2$</li>
<li>$f(x)=\lfloor x\rfloor$</li>
</ol>
<p>Sorry I forgot to add the entire question. It is "Determine whether each of these... | Community | -1 | <p>Since you're new to bijections, I'll work out the details of one of them and leave the others for you to try. To show that $f(x) = x/2$ is a bijection, we need to check two things: Whether $f$ is $1-1$, and onto.</p>
<ul>
<li><p>To see if $f$ is $1-1$, we need to see if
$$f(a) = f(b) \implies a = b$$
(That is, whet... |
1,878,806 | <p>I am a graduate school freshman.</p>
<p>I did not take a probability lecture.</p>
<p>So I don't have anything about Probability.</p>
<p>Could you suggest Probability book No matter What book level?</p>
| Wavelet | 336,350 | <p>This is a surprisingly difficult question to answer. My natural instinct as a mathematician at heart (though no PhD, yet) is to echo Justin and tell you to go straight into measure theory and then worry about probability. Classical probability theory is indeed best understood in a measure theoretic context, though i... |
848,415 | <p>If the limit of one sequence $\{a_n\}$ is zero and the limit of another sequence $\{b_n\}$ is also zero does that mean that $\displaystyle\lim_{n\to\infty}(a_n/b_n) = 1$?</p>
| TZakrevskiy | 77,314 | <p>Let $a_n=1/n$, $b_n=1/n^2$. What can you say on the ratio?</p>
|
2,825,237 | <p>\begin{cases}
4x \equiv 14 \pmod m \\
3x \equiv 2 \pmod 5
\end{cases}</p>
<p>I want to prove that for $m \in 4\mathbb{Z}$ there are no solutions(1). Moreover, I want to determine all m for which I have solutions(2). </p>
<p>First of all, the second equation is equivalent to $ x \equiv 4$ (mod 5).</p>
<p>If $m... | lhf | 589 | <p>If $m$ is a multiple of $4$, then $4x \equiv 14 \bmod m$ implies $0 \equiv 4x \equiv 14 \equiv 2 \bmod 4$, a contradiction. </p>
|
136,996 | <p>What is known about $f(k)=\sum_{n=0}^{k-1} \frac{k^n}{n!}$ for large $k$?</p>
<p>Obviously it is is a partial sum of the series for $e^k$ -- but this partial sum doesn't reach close to $e^k$ itself because we're cutting off the series right at the largest terms. In the full series, the $(k+i-1)$th term is always at... | Aryabhata | 1,102 | <p>This appears as problem #96 in Donald J Newman's excellent book: A Problem Seminar. </p>
<p>The problem statement there is:</p>
<blockquote>
<p>Show that</p>
<p>$$ 1 + \frac{n}{1!} + \frac{n^2}{2!} + \dots + \frac{n^n}{n!} \sim
\frac{e^n}{2}$$</p>
</blockquote>
<p>Where $a_n \sim b_n$ mean $\lim \frac{a_n}... |
4,105,854 | <blockquote>
<p>Solve for integers <span class="math-container">$x, y$</span> and <span class="math-container">$z$</span>:</p>
<p><span class="math-container">$x^2 + y^2 = z^3.$</span></p>
</blockquote>
<p>I tried manipulating by adding and subtracting <span class="math-container">$2xy$</span> , but it didn't give me a... | RobertTheTutor | 883,326 | <p>First I search for trivial solutions: <span class="math-container">$(0,0,0)$</span> works. Likewise <span class="math-container">$(0,1,1)$</span> and <span class="math-container">$(1,0,1)$</span>.</p>
<p><span class="math-container">$z$</span> has to be non-negative, as its cube is a sum of squares. Likewise if a... |
1,582,275 | <p>Suppose that $B = S^{-1}AS$ for some $n \times n$ matrices $A$, $B$, and $S$.</p>
<ol>
<li>Show that if $x \in \ker(B)$ then $Sx \in \ker(A)$.</li>
</ol>
<p>Proof: $B = S^{-1}AS$ implies that $SB = AS$ which implies that $SBx = ASx = 0$, that is $Sx \in \ker(A)$.</p>
<ol start="2">
<li>Show that the linear transf... | Justpassingby | 293,332 | <p>A similar relationship exists with the roles of $A$ and $B$ reversed, and with $S$ replaced with its own inverse:</p>
<p>$$A=SBS^{-1}=(S^{-1})^{-1}BS^{-1}$$</p>
<p>This means that the conclusion from point 1 can be applied to this situation, as well: $S^{-1}$ maps the kernel of $A$ into the kernel of $B.$</p>
|
4,063,337 | <p>In an exercise I'm asked the following:</p>
<blockquote>
<p>a) Find a formula for <span class="math-container">$\int (1-x^2)^n dx$</span>, for any <span class="math-container">$n \in \mathbb N$</span>.</p>
<p>b) Prove that, for all <span class="math-container">$n \in \mathbb N$</span>: <span class="math-container">$... | Ayoub | 536,671 | <p>I assume integration is done between <span class="math-container">$0$</span> and <span class="math-container">$1$</span>.</p>
<ol>
<li>For <span class="math-container">$n\in\mathbb{N}$</span>, let's have <span class="math-container">$I_n=\int_{0}^1(1-x^2)^{n}\,dx$</span>.</li>
</ol>
<p>Then for any <span class="math... |
4,063,337 | <p>In an exercise I'm asked the following:</p>
<blockquote>
<p>a) Find a formula for <span class="math-container">$\int (1-x^2)^n dx$</span>, for any <span class="math-container">$n \in \mathbb N$</span>.</p>
<p>b) Prove that, for all <span class="math-container">$n \in \mathbb N$</span>: <span class="math-container">$... | Henry Lee | 541,220 | <p><span class="math-container">$$I(n)=\int_0^1(1-x^2)^ndx$$</span>
<span class="math-container">$u=x^2\Rightarrow dx=\frac{du}{2x}=\frac{du}{2u^{1/2}}$</span> and so:
<span class="math-container">$$I(n)=\frac12\int_0^1(1-u)^nu^{-1/2}du=\frac12 B\left(\frac12,n+1\right)=\frac{\Gamma\left(\frac12\right)\Gamma(n+1)}{2\Ga... |
4,063,337 | <p>In an exercise I'm asked the following:</p>
<blockquote>
<p>a) Find a formula for <span class="math-container">$\int (1-x^2)^n dx$</span>, for any <span class="math-container">$n \in \mathbb N$</span>.</p>
<p>b) Prove that, for all <span class="math-container">$n \in \mathbb N$</span>: <span class="math-container">$... | Trevor Gunn | 437,127 | <p>My idea here is to use the <a href="https://en.wikipedia.org/wiki/Beta_function" rel="nofollow noreferrer">Beta function</a>:</p>
<p><span class="math-container">$$ \int_0^1 t^{m}(1 - t)^n \;dt = \frac{\Gamma(m+1)\Gamma(n+1)}{\Gamma(m+n+2)}. $$</span></p>
<p>Where the <a href="https://en.wikipedia.org/wiki/Gamma_fun... |
638,164 | <p>Let $\textbf{F}\left ( x, y \right )=\left ( -\frac y{x^2+y^2},\frac x{x^2+y^2} \right )$ be a vector field in $\mathbb{R}^2-\left \{ \textbf{0} \right \}$.</p>
<p>I know that the potential function of $\textbf{F}$ on $x>0$ is $\arctan \left ( \frac yx \right )$.</p>
<p>But I want to know the potential function... | Joe Z. | 24,644 | <p>As the person from Math Overflow hinted, your question is best viewed in the paradigm of the base $n$ expansion of $1/(n-1)^2$.</p>
<p>Consider the following sum: $\displaystyle \sum_{k=1}^\infty \frac{k}{n^{k+1}}$. This is equal to $\displaystyle \frac{1}{n^2} + \frac{2}{n^3} + \frac{3}{n^4} + \frac{4}{n^5} + \cdo... |
2,725,831 | <blockquote>
<p>If we define $$f(x)=\left\lfloor \frac {x^{2x^4}}{x^{x^2}+3}\right\rfloor$$ and we have to find unit digit of $f(10)$</p>
</blockquote>
<p>I had tried approximation, factorization and substitutions like $x^2=u$ but it proved of no use. Moreover the sequential powers are feeling the hell out of me. C... | Ross Millikan | 1,827 | <p>First substitute in $10$ for $x$
$$f(10)=\left\lfloor \frac {10^{2\cdot 10^4}}{10^{10^2}+3}\right\rfloor\\
=\left\lfloor \frac {10^{20000}}{10^{100}+3}\right\rfloor$$
Now ask <a href="http://www.wolframalpha.com/input/?i=10%5E20000%2F(10%5E100%2B3)%20mod%2010" rel="nofollow noreferrer">Alpha</a>
<a href="https://i.s... |
151,076 | <p>If A and B are partially-ordered-sets, such that there are injective order-preserving maps from A to B and from B to A, is there necessarily an order-preserving bijection between A and B ?</p>
| Martin Brandenburg | 1,650 | <p>One needs additional assumptions:</p>
<p>If $f : A \to B$ is an order-embedding of partial orders with the property that no element of $f(A)$ is comparable with an element of $B \setminus f(A)$, and $g : B \to A$ is a map with the same properties, then $A$ and $B$ are isomorphic. The usual proof of Schröder-Bernste... |
2,933,753 | <p>Given two finite groups <span class="math-container">$G, H$</span>, we are going to say that <span class="math-container">$G<_oH$</span> if either</p>
<p>a. <span class="math-container">$|G|<|H|$</span></p>
<p>or </p>
<p>b. <span class="math-container">$|G|=|H|$</span> and <span class="math-container">$\dis... | Olexandr Konovalov | 70,316 | <p>To give an answer which uses open source software - one could do this in <a href="https://gap-system.org" rel="nofollow noreferrer">GAP</a>.</p>
<p>First, the GAP code based on the one from the answer by @Travis. It looks quite similar.</p>
<pre><code>gap> sumOfOrders := G -> Sum(List(G,Order));
function( G ... |
3,163,317 | <p>I am given the question "Find the equation of the line in standard form with slope <span class="math-container">$m = -3$</span> and passing through the point <span class="math-container">$(1, \frac{1}{3})$</span>"</p>
<p>The solution is provided as <span class="math-container">$x + 3y = 2$</span></p>
<p>I arrived ... | Alessio Del Vigna | 639,470 | <p>When you multiply both sides by <span class="math-container">$3$</span>, you made a mistake in the RHS.</p>
|
1,513,056 | <p>Why is a limit of an <strong>integer</strong> function $f(x)$ also integer? For example, a function that's defined on interval $[a, \infty)$ and the limit is $L$</p>
| Hippalectryon | 150,347 | <p>If the limit of $F$ is $L$, then a direct consequence is that for $x$ big enough, $F$ will be as close to its limit as you want, which is to say that $\epsilon=|f(x)-L|$ becomes as small as you want.</p>
<p>Hint : what happens if $\epsilon<\min(L-\lfloor L\rfloor,\lfloor L\rfloor+1-L)$ ?</p>
|
477,483 | <p>I'm trying to solve this question but yet I don't manage to find an answer...</p>
<p>Does it exist a topology with cardinality $\alpha, \> \forall \> \alpha \ge 1$ ?</p>
| bubba | 31,744 | <blockquote>
<p>is there any trivial quad tessellation that minimizes distortion in this case</p>
</blockquote>
<p>As indicated in a comment, a cube is a tesselation of a sphere using squares as the tiles. Not a very interesting/useful tesselation, from a graphics point of view, though, unless the spherical object i... |
791,020 | <p>From my textbook. </p>
<p>$$\sum\limits_{k=0}^\infty (-\frac{1}{5})^k$$</p>
<p>My work:</p>
<p>So a constant greater than or equal to $1$ raised to ∞ is ∞.</p>
<p>A number $n$ for $0<n<1$ is $0$. So when taking the limit of this series you get 0 but when formatting the problem a different way $(-1)^k/(5^k)... | Marc van Leeuwen | 18,880 | <p>The basic binomial formula $(1+x)^\alpha=\sum_k\binom\alpha kx^k$ is valid as a formal power series in $x$, or for concrete numbers$~x$ with $|x|<1$ (supposing that like is the case here, $\alpha$ is not a natural number). This cannot be used directly in the example, but you could bring the expression into a simi... |
2,180,102 | <p>If you have 3 labeled points on a surface of a paper. Like </p>
<pre><code> 1
2 3
</code></pre>
<p>This makes a perfect equilateral triangle.</p>
<p>From this perspective I can say that the camera is on top of the paper looking down. We can say the camera is at coordinate $(0,0,100)$. Which is 0 degree... | Adren | 405,819 | <p><strong>Remark</strong></p>
<p>I think that the multiplicativeness of $\phi$ should rather be considered as a <em>consequence</em> of the fact that the groups of units of $\mathbb{Z}/n\mathbb{Z}$ and $\displaystyle{\prod_{i=1}^j\left(\mathbb{Z}/p_i^{k_i}\mathbb{Z}\right)}$ are isomorphic and hence equipotent.</p>
... |
3,708,243 | <p>Equation: </p>
<blockquote>
<p><span class="math-container">$x^2-x-6=0$</span></p>
</blockquote>
<p>The two roots of this equation are <span class="math-container">$3$</span> and <span class="math-container">$-2$</span>. When writing the answer can I also write it as <span class="math-container">$-2, 3$</span> o... | swordlordswamplord | 593,999 | <p>For min z I got 5 where
x1=1, x2=2, and x5=2 </p>
<p>I forgot what the relationship between min and max are when finding solution to LP... I know the question wants to max z but the first constraint is tricky because it wants a negative number</p>
|
26,259 | <p>I've been reading <em>generationgfunctionology</em> by Herbert S. Wilf (you can find a copy of the second edition on the author's page <a href="http://www.math.upenn.edu/~wilf/DownldGF.html" rel="nofollow">here</a>).</p>
<p>On page 33 he does something I find weird. He wants to shuffle the index forward and does so... | Qiaochu Yuan | 232 | <p>The ring of formal power series is an integral domain, so you can cancel factors from two sides of an equation. (This is equivalent to saying that it embeds into a field, the ring of formal Laurent series.) This is the same thing you do when you say that $2 = \frac{6}{3}$ even though $3$ isn't invertible in $\mathbb... |
26,259 | <p>I've been reading <em>generationgfunctionology</em> by Herbert S. Wilf (you can find a copy of the second edition on the author's page <a href="http://www.math.upenn.edu/~wilf/DownldGF.html" rel="nofollow">here</a>).</p>
<p>On page 33 he does something I find weird. He wants to shuffle the index forward and does so... | Mitch | 1,919 | <p>The concept of ring formalizes all the allowed operations, but I think this can be presented without all reference to the machinery.</p>
<p>The original function is $f_n$. The generating function is $f(x)$ where $x$ is the variable ($X$ in the OP's question): </p>
<p>$$f(x) = \sum_{n\ge0} f_n x^n$$</p>
<p>Then w... |
15,124 | <p>Let $X \geq 1$ be an integer r.v. with $E[X]=\mu$. Let $X_i$ be a sequence of iid rvs with the distribution of $X$. On the integer line, we start at $0$, and want to know the expected position after we first cross $K$, which is some fixed integer. Each next position is determined by adding $X_i$ to the previous posi... | Shai Covo | 2,810 | <p>Let $\tau = \min \{ n \geq 1:X_1 + \cdots + X_n > K \}$. Then $\tau$ is an integer-valued random variable, bounded from above by $K+1$ (since $X_i \geq 1$). Note that $\tau = n$ if and only if $\sum\nolimits_{i = 1}^{n - 1} {X_i } \le K$ and $\sum\nolimits_{i = 1}^{n} {X_i } > K$. Thus, the event $\lbrac... |
1,332,420 | <p>Solution of this task is ${7\pi\over2}$ but I don't know how to get to this solution.I used the formula for double angle for $\sin{2x}$ and $\cos{2x}$ and moved everything on one side of the equation and made $2\cos{x}$ as common factor to get this :$$2\cos{x}(\sin{x} - {\sqrt{2}\over2} - \cos{x}) = 0$$
Then I can g... | DeepSea | 101,504 | <p><strong>hint</strong>:$$\sin x - \cos x = \dfrac{\sqrt{2}}{2} \Rightarrow \sin\left(x-\frac{\pi}{4}\right)=\dfrac{1}{2}$$. Can you continue ?</p>
|
1,332,420 | <p>Solution of this task is ${7\pi\over2}$ but I don't know how to get to this solution.I used the formula for double angle for $\sin{2x}$ and $\cos{2x}$ and moved everything on one side of the equation and made $2\cos{x}$ as common factor to get this :$$2\cos{x}(\sin{x} - {\sqrt{2}\over2} - \cos{x}) = 0$$
Then I can g... | Community | -1 | <p>$$(2\sin x-\sqrt{2}-2\cos x)=0$$
$$(2\sin x-2\cos x)=\sqrt{2}$$
$$4\sin^2 x-8\sin x \cos x+4\cos^2 x=2$$
$$4-8\sin x \cos x=2$$
$$\sin x\cos x=\frac{1}{4}$$
$$0.5\sin 2x=\frac{1}{4}$$
$$\sin 2x=\frac{1}{2}$$
$$x=\frac{5\pi}{12}$$</p>
<p>see the graph of function. </p>
<p>But I think there is a problem in your answ... |
3,156,962 | <blockquote>
<p>Find general term of <span class="math-container">$1+\frac{2!}{3}+\frac{3!}{11}+\frac{4!}{43}+\frac{5!}{171}+....$</span></p>
</blockquote>
<p>However it has been ask to check convergence but how can i do that before knowing the general term. I can't see any pattern,comment quickly!</p>
| lab bhattacharjee | 33,337 | <p>I'll write <span class="math-container">$\alpha=m,\beta=n$</span> for the ease of typing</p>
<p>If <span class="math-container">$d(\ge1)$</span> divides <span class="math-container">$m^3-3mn^2,n^3-3m^2n$</span></p>
<p><span class="math-container">$d$</span> will divide <span class="math-container">$n(m^3-3mn^2)+3m... |
4,016,921 | <p>Let <span class="math-container">$X, Y, Z$</span> be three random variables. Is the equality
<span class="math-container">$$
P(X=x, Y=y |Z=z) = P(X=x|Y=y,Z=z) P(Y=y|Z=z)
$$</span>
true?
Note that <span class="math-container">$P(X,Y)$</span> denotes the joint probability of random variables <span class="math-containe... | metamorphy | 543,769 | <p>Actually <span class="math-container">$_2F_1(a,b;a;z)=(1-z)^{-b}$</span> for <span class="math-container">$|z|<1$</span> and any <span class="math-container">$a,b$</span> (well, with <span class="math-container">$a\notin\mathbb{Z}_{\leqslant 0}$</span>). Indeed, both are equal to <span class="math-container">$\su... |
1,610,616 | <blockquote>
<p>$6$ letters are to be posted in three letter boxes.The number of ways
of posting the letters when no letter box remains empty is?</p>
</blockquote>
<p>I solved the sum like dividing into possibilities $(4,1,1),(3,2,1)$ and $(2,2,2)$ and calculated the three cases separately getting $90,360$ and $90... | Abhinav | 671,238 | <p>Ways of dividing in 3 and 3 are 6!/3!3!=20.
3 are directly given in the letter box . 3 are left number of ways from that is 3*3*3=27 as there are three options for each letter.
Total ways =27*20=540</p>
|
2,151,491 | <p>For which prime numbers $p$ does the decimal for $\frac{1}{p}$ have cycle length $p-1$? I started with some simple examples to find an idea to solve:</p>
<p>$\frac{1}{2}=0.5,\frac{1}{3}=\overline{3},\frac{1}{5}=0.2,\frac{1}{7}=0.\overline{142857},\frac{1}{11}=0.\overline{09},\frac{1}{13}=0.\overline{0769230}$</p>
... | Peter | 82,961 | <p>Let $p$ be a prime number different from $2$ and $5$. This implies that $p$ is coprime to $10$.</p>
<p>Fermat's little theorem states that $$10^{p-1}\equiv 1\mod p$$ is satisfied. If for every prime factor $q$ of $p-1$, we have $$10^{\frac{p-1}{q}}\ne 1\mod p$$ we can conlude that $p-1$ is the smallest exponent suc... |
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