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This is a follow up on another MO question.
Question. For $n\geq2$, the following is always an integer. Is it not?
$$\frac{(2^n-2)(2^{n-1}-2)\cdots(2^3-2)(2^2-2)}{n!}.$$ | Here is a direct number theoretic proof. First of all note that
$$(2^n-2)(2^{n-1}-2)\cdots(2^2-2)=2^{n-1}(2^{n-1}-1)(2^{n-2}-1)\cdots(2^1-1).$$
Now for any prime $p$ we have, by Legendre's formula and by Fermat's little theorem,
$$v_p(n!)\leq\left\lfloor\frac{n-1}{p-1}\right\rfloor\leq v_p\bigl(2^{n-1}(2^{n-1}-1)(2^{n-... |
A great many functions can be expressed as a series of the form
$$ U_0(x) + U_1(x) x + U_2(x) \frac{1}{2!}x(x-1) +... $$
Where $U_r(x)$ are integrable periodic functions with period $1$. Call such functions "1 periodic normal" functions. Note that the $U_r(x)$ being periodic can be decomposed into their fourier seri... | Consider a function $f$ defined at a set $D$ with the property that if a point $p$ is present in $D$ then $p+1, p+2,p+3.... $ are also present in $D$.
Thus we can form a discrete-difference expansions of $f$ around each point $p\in D$ by evaluating
$$ f(p) + D[f](p)*(x-p) + \frac{1}{2!}D^2[f](p)*(x-p)(x-p-1) + \fra... |
Let $r$ be the number of distinct prime factors of $m$. Show that there are exactly $2^r$ integers x such that $0\leq x | If $m$ is prime is true ($x=0,1$ are the only solutions).
If $m$ is a power of prime again true ($x=0,1$ are the only solutions).
Now use the Chinese remainder theorem. |
Is there any way to highlight the whole selected cell?
some times if I want to delete a certain cell and if the code is long and cells are without output, it is hard to know which cell that you are selecting.
For example, in this simple file, I want to select a certain cell as seen:
If I want to select the second cell... | You want to delete a certain cell, but you are unsure whether you selected the right bracket:
Click the bracket, select Style and then Title. If you hit the wrong bracket, select Input.
Another way (if you deleted the wrong cell) is to immediately press Ctrl + z |
let $ \kappa<\lambda $ and assume $ \aleph_{0}\leq\lambda $
prove that: $ \kappa\times\lambda=\lambda $
So, my attempt, based on the fact that i already proved for infinite cardinals $ \lambda $ that it follows: $ \lambda\times\lambda=\lambda $
if $ \kappa\neq0 $
choose $ \kappa,\lambda $ to be ordinals (we can ... | You are right, the case $\kappa = 0$ requires special attention. In that case we are looking at $0 \times \lambda$, which is just the cardinality of $\emptyset \times \lambda = \emptyset$. So $0 \times \lambda = 0$.
In particular, the inequality $\lambda \leq \kappa \times \lambda$ will not hold if $\kappa = 0$. But f... |
I'm having trouble solving the following equality:
$$x^5 + 25x^4 + 250x^3 + 1250x^2 + 3124x + 3120 $$
$$=$$
$$ax^5 + (b+c)x^4 +(10c+3d)x^3 + (d(b+c))x^2 + (24a+62d)x + 104e$$
It's supposed to end up in an augmented matrix form, so I can solve it with gaussian elimination and the likes, but it has been forever since I'... | If you know French, I have posted an elementary algorithm on:
https://fr.quora.com/Comment-puis-je-r%C3%A9soudre-x-40320-math%C3%A9matiquement
If you don't, use the following version of Stirling's formula (six terms):
$\ln x! = x F (x)$, so $F (x) = \ln (x) -1 + (0,5 \ln x + 0,921894) / x + 1/(12 n^2) – 1/(360 n^4)$
... |
There is a deck of $30$ cards, each card labeled with a unique number from $1$ to $30$. If you draw a hand of $10$ cards, what is the probability that both the $1$ and $2$ cards are in your hand after drawing this?
I just want to check my work is correct:
There is only one way to choose each card since they are uniqu... | Proof by induction on number of nodes $|V|$. The case where $|V| \leq 2$ is trivial. For $|V| = k \geq 3$, consider any node that is either on a cycle or has degree $\text d (v) = 1$ (prove the existence of such a node first), then apply the induction step. |
In the following link here I found the integral & the evaluation of
$$\displaystyle \int^1_0 \frac{\text{Li}_2(x)^3}{x}\,dx$$
I'll also include a simpler version together with the question: is it possible to find some easy
ways of computing both integrals without using complicated sums that require multiple zeta
fo... | By series expansion
$$\displaystyle \int^1_0 \frac{\text{Li}_2(x)^2}{x}\,dx=\sum_{k,n\geq 1}\frac{1}{(nk)^2}\int^1_0x^{n+k-1}\,dx =\sum_{k,n\geq 1}\frac{1}{(nk)^2(n+k)}$$
By some manipulations
$$\sum_{k\geq 1}\frac{1}{k^3}\sum_{n\geq 1}\frac{k}{n^2(n+k)}= \sum_{k\geq 1}\frac{1}{k^3}\sum_{n\geq 1}\frac{1}{n^2}-\sum_{... |
I've been reading a reaserch in Stanford's website about Intuitionism and I can't really understand what is a weak counter example and how does the intuitionistic continuum lookes like and it's realtaion to the reals.
Basiclly I think I just can't understand the example in the article.
The article's link: https://plat... | Assume that we know that either $r>0$, $r=0$ or $r<0$, where $r$ is a real number defined by $\langle r_n\rangle$. We can see that $r<0$ is impossible. If $r>0$, then $r_n$ stops to decrease at some point, that is, there is some $N$ such that $r_n = 2^{-N}$ for any $n\ge N$. Thus we have $\lnot P(N)$, which implies $\e... |
This is the problem:
Suppose you are given a finite set of coins in the plane, all with different diameters. Show that one of the coins is tangent to at most $5$ of the others.
I can easily imagine a coin with more than $5$ coins tangent to it, so I'm obviously missing something. | All the circles have distinct diameters (and henceforth, radii) hence there must be a smallest circle. This is known as the Extreme Principle. Now, consider the smallest circle. At most, how many circles can you draw tangent to it, like you actually can, when all of those tangent circles have bigger radius than the sm... |
Determine for which values of $x$ the sequence $\left(\frac{x^n}{1+x^n}\right)_{n=1}^{\infty}$ converges.
Workings:
I have to consider multiple $x$.
$x = 0$:
$\left(\frac{0^n}{1+0^n}\right)_{n=1}^{\infty} = \left(0\right)_{n=1}^{\infty} = (0, 0, 0, 0,...)$
This is convergent.
$x = 1$:
$\left(\frac{1^n}{1+1^n}\ri... | Let $P$ be a parallelepiped of dimensions $a,b,c$, its volume is given by $V=abc$ and its surface is $S=2ab+2bc+2ac$. Let us assume that $V>0$, then $c=\frac{V}{ab}$ and thus $S=2ab+2\frac{V}{a}+2\frac{V}{b}$. Now, we look for a maximum of $(a,b)\mapsto S=S(a,b)$ on $\{(x,y)\mid x,y>0\}$. Note that every maximum of $S$... |
If I have for example a heat map:
DensityPlot[x^2 + y^2, {x, -1.0, 2.0}, {y, -1.0, 1.0},ColorFunction ->ColorData["SunsetColors"], PlotLegends -> True]
and I want to highlight a particular point $(x,y)$ on the 2D place where the heat map is shown (for example with a red dot), how can I do that? Also is it possible to... | I draw a red point at the origin, a line between random points as well as a shortest tour between the same points shown in yellow.
orig = {0, 0};
pts = RandomReal[{-1, 1.5}, {10, 2}];
st = FindShortestTour[pts];
DensityPlot[x^2 + y^2, {x, -1.0, 2.0}, {y, -1.0, 2.0},
ColorFunction -> ColorData["SunsetColors"], PlotLe... |
If I have an equation, like this:
$k^{m} = m^{k}$,
I have been taught that I must apply natural logarithm, but others have taught me that I must apply the common logarithm.
Then, what should I use?
$m*ln(k)= k*ln(m)$, Natural logarithm (base $e$)?
$m*log(k)= k*log(m)$, Common logarithm (base $10$)?
I guess, the o... | $k^m = m^k \implies m\log_b k = k\log_a m$ will be true no matter what base $b$ you choose. And as $\log_b k$ an $\log_b m$ will always be in the same proportion ($\frac {\log_b k}{\log_b m} = \frac {\log_a k}{\log_a m}$ for all legitimate $a,b$) it doesn't matter which you pick.
Unless you are doing scientific notat... |
For this question, I am in the smooth finite-dimensional category. All objects are $\mathcal C^\infty$ manifolds and all maps are smooth. Recall the notion of groupoid, and let's restrict our attention to those for which the source map (and hence also the target map) is a (necessarily surjective) submersion. If $G\r... | I'm not sure whether it helps here, but your question reminds me of the Freiheitssatz. As you probably know, this is the theorem of Magnus that says that if $G=\langle x_1,\ldots,x_n|r\rangle$ and $r$ involves the generator $x_n$, then the elements $x_1,\ldots,x_{n-1}$ generate a free group of rank $n-1$ inside $G$. Ce... |
I just started Laurant series, I have this function
$$f(z) = {z^4 + 1 \over {(z-1)(z+2)}}$$ in $1<|z|<2$ ring, I know the theorems,but still cant figure it out, Any help with idea or transformation, welcome! | The sphere
$$x^2+y^2+z^2+2x-4y-2z+1=0 \implies$$
$$(x+1)^2+(y-2)^2+(z-1)^2=5$$
and the plane
$$2x-4y-2z+1=0 \implies $$
$$(-1)x+(2)y+(1)z=\frac{1}{2}$$
since $\frac{1}{2} \neq (-1)^2+(2)^2+(1)^2$ the plane doesn't pass through the center of the circle. However the plane is perpendicular to the vector from the origi... |
Is it possible to combine two convolution kernels (convolution in terms of image processing, so it's actually a correlation) into one, so that covnolving the image with the new kernel gives the same output as convolving it with the first, and then the second kernel?
For example, if I want to convolve an image with 1st... | Let's assume the original image is $x[n_1,n_2]$ and two kernels are $h_1[n_1,n_2]$ and $h_2[n_1,n_2]$. Convolution associative law says that:
$$(x[n_1,n_2]*h_1[n_1,n_2])*h_2[n_1,n_2] = x[n_1,n_2]*(h_1[n_1,n_2]*h_2[n_1,n_2])$$
So covnolving the image with the new kernel will always produce the same output as convolvin... |
Let $T$ be a linear operator on a vector space $V$, and let $x$ be a non-zero vector in $V$.
The subspace,
$$W = \operatorname{span}(\{x,T(x),T^2(x),\ldots\})$$
I have to prove that $W$ is a $T$-invariant subspace of $V$ and also that it is the smallest such subspace of $V$ containing $x$.
I don't really know how t... | As has been said elsewhere, there are loads of applications to classical mechanics and electromagnetism (both statics and dynamics). tylerc0816 mentioned statistics, which is great, but an awful lot of that looks like black magic without more advanced knowledge (that I don't have yet). |
Here is the Lemma $2.1$ of chapter $5$ from Sheldon Ross probability book. link
For a nonnegative random variable $Y$:
$$E[Y] = \int_0^\infty P\left \{ Y > y \right \}\,dy $$
They have used the following interchange of integrals. Please help to clarify this step!
$$
\int_0^\infty \int_y^\infty f_Y(x) \, dx\,dy = ... | $$\frac{1}{\tan3x-\tan{x}}-\frac{1}{\cot3x-\cot{x}}=$$
$$=\frac{1}{\frac{\sin3x}{\cos3x}-\frac{\sin x}{\cos x}}-\frac{1}{\frac{\cos3x}{\sin3x}-\frac{\cos x}{\sin x}}=$$
$$=\frac{\cos3x\cos x}{\sin3x\cos{x}-\cos3x\sin{x}}+\frac{\sin3x\sin{x}}{\sin3x\cos{x}-\cos3x\sin{x}}=$$
$$=\frac{\cos3x\cos x}{\sin2x}+\frac{\sin3x\si... |
Given $z^2-1\mid x^2z^2-1$, prove that $\frac{x^2z^2-1}{z^2-1}$ can never be prime, assuming $x$, $z$ are integers such that $x>z>1$.
So far I have tried taking mod a lot of different numbers, but I cannot find solution. I also tried writing as a quadratic and using quadratic formula, but that doesn't work either. Ple... | Answered for my own benefit, as I already had seen the above two answers first******
Note that $x^2z^2-1 = (xz-1)(xz+1)$. Then, as $\frac{x^2z^2-1}{z^2-1} = \frac{(xz-1)(xz+1)}{z^2-1}$ is an integer, it follows that $z^2-1$ can be written as follows: $z^2-1 = ab$ for some positive integers $a$ and $b$, that satisfy th... |
I am a physicist and I am trying to get a grasp on the following terms from functional analysis:
As I understand, an operator is Hermitian if it is symmetric and bounded (domains of A and A* don't need to be equal in this case.)
An operator is selfadjoint if it is symmetric and the domains of A and A* are equal, D(A... | In the finite case every operator is bounded and since the domain is the whole Hilbert space, every hermitian operator is self-adjoint. For the infinite case, the good definition of a hermitian operator is a densely defined operator $T$ such that $T\subset T^*$, i.e., it is symmetric and the adjoint $T^*$ is an extensi... |
I was solving an improper integral, but I am stuck in here
$$\lim_{b\rightarrow +\infty} b^{1-\theta} $$ could you write, please, how to solve this limit?
Thank you | Given
$$\int \limits_{k}^{\infty}\frac{1}{y}^{\theta+1}y{\theta}k^{\theta}dy$$
note that
$$\frac{1}{y}^{\theta+1}y{\theta}k^{\theta}\sim \frac{{\theta}k^{\theta}}{y^{\theta}}$$
thus by limit comparison test with $\int \frac{1}{y^{\theta}}$ the integral converges for $\theta >1$ |
I got this question on my homework in Calculus 3 class. This exercise is about Lagrange multiplications and I can't get around it.
The closed unit ball is compacted and $f$ is $C^1$ hence by Weierstrass theorem $f$ gets a minimum and a maximum over the ball.
Consider the constraint $g(x,y,z)=x^2+y^2+z^2-K$ where $K\in... | Kind of simpler argument would be like this - it is easy to see that at the point where maximum is attained (say, $x_0, y_0, z_0$ ) coordinates are not negative. Also, the expression is symmetric so we can assume that $x_0 \ge y_0 \ge z_0$.
Say, we have $x_0 > y_0$. We want to find an $\epsilon >0$ such that the point... |
I am stuck on a simple exercise. Let $(a_n)_{_n\in\mathbb N}$ be a sequence of real numbers. Prove or disprove the following statement:
$$\lim_{n\to \infty}\left(a_{n+1} - \frac 12 a_n\right)=0 \Rightarrow \lim_{n\to \infty}a_n=0.$$
I tried to prove it directly but didn't make any progress. So I considered proving the ... | We prove the following general claim, since doing so does not harm the essence of the idea:
Tauberian Theorem for Nørlund means. Let $(b_n)$ and $(c_n)$ be sequences such that
$b_n > 0$ for all $n \geq 1$ and $\frac{b_n}{b_1 + \cdots + b_n} \to 0$ as $n\to\infty$.
$c_n \to \ell$ as $n\to\infty$ for some $\ell$.
Then... |
There are several examples and questions regarding Map, but I couldn't find what I need.
This is a minimal working example.
I have two functions
$\qquad f=x^4+y^4-1$,
and
$\qquad g=x^2-4$.
One way to find the solution set would be to find the values of $x$ from $g=0$ and then substitute those values into $f$. Th... | Howzabout
Remove[solveAndBacksubstitute];
solveAndBacksubstitute[{pair: {_, _}}, opts___]:=
NSolve[pair[[1]] == 0, pair[[2]], opts]
solveAndBacksubstitute[pairs: {{_, _}..}, opts___]:= Module[{
partlsoln, recurse },
partlsoln = NSolve[#[[1]] == 0, #[[2]], opts] &[First[pairs]];
recurse[varValue_]:= Join[var... |
$|2x + 2| + |x - 1| > 3$
why can't $x>\frac 23$ be a part of the solution?
Thanks for your help! | Suppose $x>2/3$, then if $x \ge 1 $, then $2x+2 + x -1 = 3x + 1 \ge 3 + 1 > 3$.
If $2/3 < x < 1$, then $2x+2 - x +1 = x + 3 > 3$.
This is, however, not a complete solution (probably why it is not a solution!?).
Consider $-1 \le x \le 2/3$, then $2x+2 - x +1 = x + 3 >3 $. This is only true if $x>0$.
If $x<-1$, then $-... |
Are there any known examples of sets that are definitely infinite, but where we don't know whether or not they're countable? I haven't heard of anything like this before, but it seems that there should be some example of a set like this.
As mentioned in the comments, it's possible to construct examples of the form
$$... | Here's a wonderful open problem in set theory, which can be translated to a statement which you might be looking for.
Suppose that $\aleph_\omega$ is a strong limit cardinal. Is $2^{\aleph_\omega}<\aleph_{\omega_1}$?
We can prove that under the assumption that $\aleph_\omega$ is a strong limit cardinal, it is necessa... |
If $4^{y+3x} = 64$ and $\log_x(x+12)- 3 \log_x4= -1$ so $x + 2y=?$
I've tried this far, and I'm stuck
$$\begin{align}4^{y+3x}&= 64 \\
4^{y+3x} &= 4^3 \\
y+3x &= 3 \end{align}$$
$$\begin{align}\log_x (x+12)- 3 \log_x 4 &= -1 \\
\log_x (x+12)- \log_x 4^3 &= -1 \\
\log_x(x+12)- \log_x 64 &= -1 \end{align}$$
then I subs... | You're right up to $y+3x=3$.
Now consider the other statement $\log_x(x+12)-3\log_x 4=-1$
$\log_x{x+12 \over 64 }=-1$
${x+12 \over 64 }={1 \over x}$ |
"Introduction to Algorithm" C.2-6
Describe a procedure that takes as input two integers a and b such that $0 < a < b$
and, using fair coin flips, produces as output heads with probability $a / b$ and tails
with probability $(b - a) /b$. Give a bound on the expected number of coin flips,
which should be O(1) (Hi... | Basically, what Michael says is it like this:
We express a/b using say 64 binary digits.
Now, flip 64 coins. Represent the outcome of this as a binary using head=1 and tail=0.
The best approximation of the problem is:
If this number is larger than the 64 bit binary representation of a/b
then say "HEADS".
If this nu... |
I'm not a mathematician so I hope I ask this question properly; I apologize for anyone who is annoyed with how I ask it (I will try my best to be precise).
Say I have a point cloud in $\Re^3$. I wish to fit a line through to this point cloud. However - and this is the kicker - I wish to force the line through a spec... | This notion makes sense. You still have two degrees of freedom for the line, which can be two angles we will call elevation $(E)$ and azimuth $(A)$. It is easiest if you move the origin to the anchor point by subtracting the coordinates of the anchor points from each of the points in the cloud. Then your line has pa... |
Is there any way to add my own annotations to values or functions in Mathematica? Imagine, for example, that I wanted to annotate a List specifically as a List of Strings.
I don't think Mathematica will really do anything to enforce such a thing, but what I'd like to use this for is for connecting Mathematica to.NET g... | You could use UpValues:
mylist = {"Alice", "Bob", "Carol"};
numlist = {1, 2, 5, 3};
SetAttributes[NETType, HoldAll]
NETType[mylist] ^:= String
NETType[numlist] ^:= Integer
{NETType[mylist], NETType[numlist]}
(* Returns:
{String, Integer}
*)
Of course, this does not perform any checks that the elements in the list... |
Why can't π be expressed as a fraction?
Going by the above question, if it is not possible to accurately determine the circumference or diameter of a circle, how did they figure out that the ratio will result in an infinite non repeating value? | What was so special about C/D as compared to any other fraction?
That fraction is special because it has many practical uses. There are physical situations where measuring diameter is easier than measuring circumference. Then, if you know the value of $\pi$, you can calculate a circumference (or an area) without actua... |
I don't know if this is appropriate for math.stackexchange, or whether philosophy.stackexchange would have been a better bet, but I'll post it here because the content is somewhat technical.
In ZFC, we can prove that the (second-order) Peano axioms have a model, call it $\mathbb{N} = (N,0,S)$. Furthermore, $\mathbb{N}... | If you are a Platonist, or in any other sense believe that there is a "true" set of natural numbers, then you have pinned them down. If you believe that there is one concrete universe of sets, and suppose it even satisfies the axioms of $\sf ZF$, then that's universe $\omega$ can be thought of as the one true set of na... |
My students, when presented with an integral (source) like
$$\int (2x+2)e^{x^2+2x+3} \ dx$$
are apt to recognize derivative patterns like $u' e^{u}$ and reverse-engineer anti-derivatives rather than to utilize $u$-substitution.
With something like
$$\int x \cdot \cos(5x^2) \ dx$$
they would rewrite to get
$$ \frac{1}{... | Differentiation is a science; integration is an art.
When you integrate, you have to guess what function differentiated to give the one you are looking at. Substitution is one of the methods available for re-writing the function in the hope of finding something more recognisable.
Students who skip the substitution ar... |
Is it true that if $a\equiv b \bmod n$, the $c^a \equiv c^b \bmod n$? I'm not quite sure how to prove this myself. | A counterexample:
Consider when $n = 15$, and $2 \equiv 17 \bmod 15$. Then $2^2 \equiv 4 \bmod 15$ compared to $2^{17} \equiv 2 \bmod 15$ (and this last one is easy because $2^8 \equiv 1 \bmod 15$ by Euler's theorem). |
Recently I have come across a statement saying that if $x$ and $y$ are divisible by $a$, then $x - y$ is also divisible by $a$.
How can I prove this? Does it also apply to sum of $x$ and $y$? | If $x=aq$ and $y=ap$ then $x-y=aq-ap=a(q-p)$ by the distributive law. |
I have the equation
$$R\frac{dq(t)}{dt}+\frac{q(t)}{C}-V_0=0$$
And am asked to find the general solution using the integrating factor.
I am a bit confused as I have been shown two ways to do it. The first is using the form
$$y(x)=e^{-\int{p(x)}}*\int{e^{\int{p(x)}}}q(x)dx$$
The first step was to get it into the r... | Now, make a substitution: $e^\frac{t}{RC}q(t)=z$. Hope you can do the rest easily. |
For space pairs $(X_1,A_1)$ and $(X_2,A_2)$,is there a groups M,such that $\pi_{k}(X_1\vee X_2,A_1\vee A_2)\cong \pi_k(X_1,A_1)\oplus M$? | There are some cases when one can give complete information. We know that the relative homotopy group
$\pi_k(X_i,A_i)$ is a $\pi_1(A_i)$-module for $k >2$, crossed $\pi_1(A_i)$-module if $k=2$.
Suppose that the spaces are connected, well pointed, and the pairs
$(X_1,A_1), (X_2,A_2)$ are $(k-1)$-connected. This ... |
I literally just downloaded MMA11 and upgraded from MMA 10.4, and PlotLegends seems to not work the way it did previously... Kind of annoying. Anyone knows why?
Clear[var, expr, blah, g, derivatives];
var = Input["Type a variable, such as x, in this window."];
expr = Input["Type an expression to be differentiated i... | If you add Evaluate to the expression to be plotted there it works. Must be some bug in Plot. |
In class my professor said that
$$
\int_{-\infty}^{\infty}\frac{e^{iax}}{x^2 - b^2}dx = -\frac{2\pi}{b}\sin(ab)
$$
where $a,b > 0$.
However, since the poles are on the real axis, isn't the integral equal to
$$
\pi i\sum_{\text{real axis}}\text{Res}(f(z); z_j)\mbox{?}
$$
If that is the case, the integral is
$$
-\frac{\p... | $\newcommand{\+}{^{\dagger}}%
\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{\dd}{{\rm d}}%
\newcommand{\isdiv}{\,\left.\right\vert\,}%
\newcommand{\ds}[1]{\displaystyle{#1}}%
\new... |
I have a large amount of data available (in excel), which is not very organised. I can do this by copy paste, but I am trying to do this using mathematica. Hopefully someone can help me.
I imported the data, and it is now organised in a matrix that looks like this:
test = {{{dog, 5}, {"", ""}, {"", ""}}, {{cat, 7}, {... | Here's a one-liner....
Table[First[Cases[test[[i]], {animal, ct_}:> ct] /. {} -> {0}],
{i, Length@test}, {animal, {dog, cat, horse}}]~Prepend~{dog, cat, horse} // Grid |
I have the quadratic function
$$y=\frac{1}{294}(x-84)^2-24$$
that represents the shape of the cables of a certain bridge. I am suppose to be determining the vertical height of the cables above the minimum at a point that is $35$m horizontally from one of the towers.
I keep getting an answer of $-15.83$m, when I shou... | This is called simultaneous diagonalisation by congruence. Presumably the notation $X'$ means the conjugate transpose of $X$ in the complex case, or simply transpose in the real case. If this is the case, consider a matrix of the form $C=UB^{-1/2}$, where $U$ is unitary, $B^{1/2}$ is the unique positive definite square... |
I have to find an expression in terms of n using standard results for $$\sum_{r=n+1}^{2n} r(r+1)$$
And have found the general equation
$$\sum_{r=n+1}^{2n} r(r+1) = \frac{2n^3+6n^2+4n}{6}$$
However evaluating it as $$\frac{2(2n)^3+6(2n)^2+4(2n)}{6} - \frac{2(n+1)^3+6(n+1)^2+4(n+1)}{6}$$
does not yield the correct ... | Another approach: it's clear that the result is a polynomial of $n$ of degree $3$, let $$\sum\limits_{r=n+1}^{2n}(r^2+r)=An^3+Bn^2+Cn+D=P(n)$$
thus
\begin{align*}P(n)-P(n-1)&=\sum\limits_{r=n+1}^{2n}(r^2+r)-\sum\limits_{r=n}^{2n-2}(r^2+r)\\
&=-(n^2+n)+((2n-1)^2+(2n-1))+((2n)^2+2n)\\
&=7n^2-n\\
&\equiv A(3n^2-3n+1)+B(2n... |
Consider a morphism of commutative monoids $u\colon M\rightarrow N$. We say that $u$ is flat, if the tensor product functor $\bullet\otimes_MN$ from the category of $M$-modules to the category of $N$-modules commutes with finite projective limits.
Let $R$ be a non-zero commutative ring, and suppose that the induced mo... | No. Take $R$ to be the field of rationals $\mathbb Q$. Let $M$ be the multiplicative monoid of the two element field and $N$ the additive group of this field. Let $u$ be the trivial homomorphism. Both monoid algebras are isomorphic to $\mathbb Q\times \mathbb Q$ so semisimple. Thus all modules over both rings are proj... |
Are there two models $M$ and $N$ for $\text{ZFC}$ such that:
(1) $M\subseteq N$
(2) $\aleph_{1}^{N}=\aleph_{1}^{M}$
(3) $\aleph_{2}^{N}=\aleph_{\omega +1}^{M}$
Update: According to Peter's useful answer it seems this problem is still open. So my question is about "partial" or "similar" results with different (but n... | Regarding your question about the situation at other $\aleph$s: Lemma 3.1 from Cummings' paper "Collapsing successors of singulars," together with the fact (due to Shelah, I believe) that, if $\overrightarrow{f}$ is a scale of length $\aleph_{\omega+1}$, then there is a club $C$ in $\aleph_{\omega+1}$ such that every $... |
Let $X$ be a topological space and $N$ a subset of $X$. I want to show that $\partial \bar N\subset \partial N$.
I know that since $\bar N$ is closed then $\partial \bar N\subset \bar N$. By definition $\bar N=N \cup \partial N$, now if $x\in \partial\bar N$ then $x\in \bar N$ but why $x$ must lie precisely in $\parti... | By definition, the boundary of a set is the intersection of its closure and its complement's closure:
$$
\partial N = \overline{N} \cap \overline{C(N)}
$$
Apply the same definition to $\overline{N}$ to get:
$$
\partial \overline{N} = \overline{N} \cap \overline{C(\overline{N})}
$$
Since $N \subset \overline{N}$, $C(\... |
I have to prove that for $a \in N, c > 0$ (constants), this statement holds:
$\log_a(n + c) \in O(\log_a(n))$
So if I use the definition, the following should hold:
$\log_a(n + c) \leq d\log_a(n) = \log_a (n^d), \forall n \geq n_0, d > 0$
I have to find $n_0, d$ now
$n + c \leq n^d$
But how can I express $d$ so i... | You can't hope for it to equal $\mathbb{C}$: the most you could possibly hope for is the algebraic closure of $\mathbb{Q}$.
However, you don't get that: the algebraic closure of a field is a finite-degree extension if and only if the field itself is either real closed or is already algebraically closed.
Every real cl... |
I need to prove that $\lim\limits_{n\to\infty} x^{1/(2n-1)}= \begin{cases}
1, & \text{if}\ x>0 \\
0, & \text{if}\ x=0\\
-1, & \text{if}\ x<0
\end{cases}$.
But I have no idea where to start. I don't know how a definition of the limit could be applied here, if at all. Could the Monotone Converge... | I think it is wrong problem. Because the function $x^\alpha$ ($\alpha \in \mathbb R\backslash \mathbb Z$) is not well-defined if $x<0$. |
I have a huge list = {l1, l2, l3,...}, which I want to output by labeled rows. My attempt was to
Do[Print["list[i]:"];Print[list[i]], {i, 1, Length[list]}]
Which is not working very well. I would like to know how i can "label" every line of my output like so:
list[1] = l1
list[2] = l2
list[3] = l3... | You should use StringTemplate.
fmt = StringTemplate["list[``] = ``"];
Do[Print[fmt[i, lst[[i]]]], {i, 1, Length[lst]}]
This is still ugly: it creates a cell for every call to Print. You say the list is "huge" (justifying Do), but then you want to print it. Hmm; perhaps it is not so huge? If you don't mind using a ... |
I would like to know how to prove that
$$\frac{d}{dx}e^x=e^x$$
An argument I've seen used is that this is the defnition of the number $e$; we can say that for some number, let's call it $a$, the derivative of $a^x$ is $a^x$ and we've just called this number $e$. But a problem I have with this approach is that we are as... | Comment:
$y=e^x$
Due to definition of derivative we can write:
$$y'=\lim_{\Delta x\rightarrow 0}\frac{e^{x+\Delta x}-e^x}{\Delta x}=\lim_{\Delta x\rightarrow 0}e^x(\frac {e^{\Delta x}-1}{\Delta x})=e^x$$
Provided we prove that:
$\lim_{\Delta x\rightarrow 0} \frac {e^{\Delta x}-1}{\Delta x}=1$ |
How can I compute this limit:
$\lim_{n \to \infty} \frac{n!}{n^n}$.
I have tried taking the n-th root but then I have problem managing the factorial.
Thanks to everyone who will help! | HINT
By ratio test
$$\frac{(n+1)!}{(n+1)^{n+1}}\frac{n^n}{n!} =\frac{1}{\left(1+\frac1n\right)^n}$$
and recall that $\left(1+\frac1n\right)^n\to e$. |
I am a bit confused on what this question is asking me to prove:
Prove $$
\exists z\forall x\in\mathbb{R}^{+}[\exists y(y - x = y/x)\leftrightarrow x \neq z]
$$
Am I asked to prove that there exists a z where the bi-conditional statement is true? Or is z a given and I should just prove the statement in the brackets ... | It is conventional that quantifiers bind from left to right. Therefore, to prove this statement, you must first choose a $z$. Then, your adversary picks an $x$, and regardless of this choice, you must now prove that the biconditional holds.
To do so you must prove that under certain circumstances $y$ exists, and und... |
Using LOTUS:
$$\text{E}(Y) = \int_{-\infty}^{\infty}dx \Big( x^{6} \cdot \frac{1}{\sqrt{2 \pi}} \text{exp}(-\frac{1}{2}x^{2}) \Big)$$
I have two functions multiplied inside the integral, so use Integration by Parts:
$$\int_{a}^{b} f g' = [fg]_{a}^{b} - \int_{a}^{b} f'g$$
$$f = x^{6} \rightarrow f' = 6x^{5}$$
$$g' = \f... | Given that the integrand is even in a symmetric interval, then:
\begin{equation}
E(Y)=\sqrt{\frac{2}{\pi}}\int\limits_{0}^{+\infty} x^{6}e^{-\frac{x^{2}}{2}} \,dx
\end{equation}
Let $s=\frac{x^{2}}{2}$, which implies that: $x=\sqrt{2s} \Leftrightarrow dx=\frac{1}{\sqrt{2}}s^{-\frac{1}{2}}\,ds$. The limits do not cha... |
When asked to prove if this is true or not $\exists x \in\Bbb R$, $x^2 + 29x + 209 \leq 0$, it can easily be done by saying when $x = -15$ it works. But how to find that $x = -15$?
For any "there exists" problem that shows up on a test, how to find one that just works? | $y=x^2+29x +209$, a parabola, opening upward, has a minimum at the vertex.
Complete the square:
$y = (x+29/2)^2-(29/2)^2 +209.$
Choose $x=-29/2$ (why?).
P.S. Check if $-(29/2)^2+209 \le 0.$ |
So I'm reading through Serre's "Linear Representations of Finite Groups," and I'm a bit confused by what's probably a fairly minor point. However, subsequent proofs are hinging on it, so I figure I'll turn to you guys for clarification.
So he defines the direct sum (I guess in this case the internal one), and then in... | You just need to understand the following: Namely let $p: V \to V$ such that $p^2 = p.$ Then
$$V = \textrm{Im} \hspace{1mm} p \oplus \textrm{Ker} \hspace{1mm} p. $$
Assuming this result, what Serre is saying is this. Suppose you have a subspace $W$ of $V$. Then you can always extend the basis of $W$ to a basis of $V... |
A person, with a certain given amount, can buy either 8 apples or 14 oranges or 3 water melons. The weight of either 10 apples is equal to the weight of 12 oranges which is in turn equal to the weight of 4 watermelons. Find the number of oranges she has to buy with $6$ times the original amount if she wants to maximize... | Let: $M$:= 'certain given amount'
$o$:= # of oranges purchased
$a$:= # of apples purchased
$w$:= # watermelons purchased
To deal with the weights, it's probably easiest to assign a single unit for all of them. So let's say that 1 apple = 1 unit weight. This implies 1 orange has weight $\frac{10}{12}$; 1 watermelon ... |
Knowing that
$P(Y=1\mid X=5)=1/3$,
$P(Y=5\mid X=5)=2/3$.
Calculate $E(Y\mid X=5)$ and $E(XY^2\mid X=5)$. How to solve this question? I have no idea whatsoever. | Since the conditional probabilities sum up to $1$ then that is all for $Y\mid X=5$. So $$E[Y\mid X=5]=1\cdot P(Y=1\mid X=5)+5\cdot P(Y=5\mid X=5)=1\cdot\frac13+5\cdot\frac23=\frac{11}3$$ and
\begin{align}E[XY^2\mid X=5]&=E[5Y^2\mid X=5]=5E[Y^2\mid X=5]\\[0.2cm]&=5\left((1)^2\cdot P(Y=1\mid X=5)+(5)^2\cdot P(Y=5\mid X=... |
This is for a project which I've been trying to find some information for Covariance matrix and correlation matrix.
I understand that for a $n \times n$ matrix $A, AA^T$ will give me the covariance matrix.
Is there any relationship between the covariance and correlation matrix?
Sorry maybe I wasn't clear.
I wanted ... | From a matrix algebra point of view the answer is fairly simple. Assume your covariance matrix is $\Sigma$ and let
$$
D =\sqrt{ \text{diag}\left( {\Sigma} \right)}
$$
then the correlation matrix is given by
$$
\varrho = D^{-1}\Sigma D^{-1}
$$
Edit: fixed to include square root |
Thanks in advance for any helpful comments! As this is all theory and abstract, please don't leave comments about the practicality of this (unless you feel that it is really relevant). Comments like "infinity is an impossibility anyway" don't really help here. Some people accept and some reject infinity. So for the sak... | There are a number of frustrating things with the question of what happens after having done something for an infinite amount of time.
Consider the hypothetical scenario that the hotel is continuing to be built at a rate of one room per day and you are changing rooms once per day and you stay in the first room the day... |
I'm currently having a problem with my reputation. The chat says it is 2484, but SE shows 2267 which seems to be wrong given the last uptades on the Q&As I've made. Any suggestions?
Thanks. | This is just to support Willie Wong's answer. Here are your reputations for your various accounts:
which gives a total of 2753.
Here is what chat says you have:
So the chat total is a bit behind your actual total. |
Let $\left(P,\le\right)$ denote a poset.
Statement: if every sequence
$p_{1}\leq p_{2}\leq\cdots$ in $P$ stabilizes (in the sense that for some $n$ we have $k>n\Rightarrow p_k=p_n$) then every non-empty
subset of $P$ contains a maximal element.
Own effort:
Let $S$ be a non-empty subset of $P$ that contains no
maxi... | Here is an example when this doesn't work.
Suppose that $A_i$ for $i\in\Bbb N$ is a non-empty set, and there is choice function on them, meaning $\prod_{i\in\Bbb N}A_i$ is non-empty.
Now consider the partial order $P=\bigcup_{n\in\Bbb N}\prod_{i
If we had an infinite [strictly increasing] sequence, then its union wo... |
Evaluate:
$$\int x(x^2-16)dx$$
I have noticed that this integral can be solved using two different methods, but I am not sure which one is the correct one.
Way 1: Using $u$-subtitution
Let $u=x^2-16, du = 2xdx$
Then, we have $$\int x(x^2-16)dx$$ $$= \frac{1}{2}\int udu$$ $$= \frac{1}{2}(\frac{1}{2}u^2)+C$$
$$= \frac{... | Alternatively, use hyperbolic trigonometric substitution.
Let $x = 4\cosh t\Rightarrow x^2 - 16 = 16\sinh^2 t$ and $\frac{dx}{dt} = 4\sinh t$.
The integral becomes
$$\begin{array} {r c l }
\displaystyle \int 4\cosh t \cdot (16\sinh^2 t)(4\sinh t) \, dt
&=& 4^4 \displaystyle \int (\cosh t)(\sinh t)^3 \, dt = 4^3 \si... |
Question:
let matrix $$X=\begin{bmatrix}
a&b\\
c&d
\end{bmatrix},e^{X}=\begin{bmatrix}
-1&2\\
0&-1
\end{bmatrix}$$
and such $a+d=0$,
Find the matrix $X$
my idea
$$e^{Tr{(X)}}=det{(e^{X})}$$
so
$$1=1$$ is true
I think we can find the matrix function: $e^X$,but I have find this is ver ugly,so I think this pr... | Here's a sketch. Since $d = -a$, you can write $X = \left(\begin{array}{rr} a & b \\ c & -a\end{array}\right)$, which gives
$$X^2 = \left(\begin{array}{cc} a^2 + bc & 0 \\ 0 & a^2 + bc\end{array}\right) = (a^2+bc)I.$$
This allows you to write a general expression for $X^n$: $X^{2n} = (a^2+bc)^n I$ and likewise $X^{2n... |
Let $x,y \in \mathbb{R}^n$ and $\epsilon > 0$.
How can one prove that
$$U_\epsilon (x) \cap U_\epsilon(y) \neq 0 \Leftrightarrow \Vert x-y\Vert < 2\epsilon$$
I tried to search for a proof online via google and wasn't able to find a proof via Approach0 either.
Can someone show us how to prove this? | Since $x^{\frac{1}{\log_2 x}} = x^{\log_x 2} = 2$ then for $x = \frac{2}{3}$ we get
$$(\frac{2}{3})^{\frac{1}{\log_2 \frac{2}{3}}} = 2$$
So
$$1 - (\frac{2}{3})^{\frac{1}{\log_2 \frac{2}{3}}} = -1$$ |
For $0
But the problem is that the choice of $a_0$ does not matter.
The question is why? | We are considering a question equivalent to $f(x) = \sqrt{1 - x}$ for $x \in (0,1)$. It is not so hard to show that this function has a fixed point, and exactly one fixed point in the interval $(0,1)$. (In fact, this fixed point is our limit).
In your question, you seem to know that the limit exists and are able to ca... |
Show that for any natural number n>24 we have:
$n=5p+7q$ such that $p$ and $q$ are natural.
I tried using induction
1) for $n=24$ we have $n=(7 \cdot 2)+(5 \cdot 2)$
2) we suppose that $n=5p+7q$
We prove that:
Show that for any $n>24$, $n+1=5p'+7q'$
If $n$ is even then $p$ and $q$ should be also even! Same if... | Actually, it is possible using induction. Notice that $1 = 3\times 5 - 2\times 7 = -4\times 5 + 3\times 7$.
So, suppose that $n\ge 24$, with $n = 5p + 7q$ for $p,q\in\mathbb{N}$. Then you just have to show that we can pick $(p,q)$ such that either $$q\ge 2$$
so that we can add $3\times 5 - 2\times 7$ and get $n+1 = 5(... |
Wilkie's well known question asks whether $I\Delta_{0}$ proves the unboundedness of primes. We know that by adding a sentence to $I\Delta_{0}$ which says "the exponential function is total", it is possible to prove the unboundedness of primes. This sentence is $\Pi_{2}$. Suppose $\Pi_{1}\text{-Th}(\mathbb{N})$ denotes ... | Yes, because $\: I\Delta_0 + \text{WPHP}\left(\Delta_0\right) \:$ proves the unboundedness of primes (see this answer),
since the assetion that a $\Delta_0$-defined relation is an injection from $\:[0\hspace{.01 in},\hspace{-0.02 in}2\hspace{-0.05 in}\cdot\hspace{-0.04 in}x]\:$ to $\:[0,\hspace{-0.01 in}x]$
can be ma... |
Ok, I obviously understand basic multiplication and understand why those don't equal. But in web colors, therr is FFFFFF hexadecimal different colors (or rather $16,777,215$ in base $10$). This amount of colors can also be described by using three sets of $255$ (RGB). However, $255 \times 255 \times 255 = 16,581,375$. ... | It is frequently useful not to think in terms of closed intervals -- i.e. that a byte is an integer in the interval $[0,255]$ -- but in terms of half-open intervals -- i.e. that a byte is an integer in the interval $[0,256)$.
Among the benefits of this is that you will make fewer off-by-one errors related to the size ... |
Let $B=\{f\in C^1[0,1]:\Vert f\Vert _\infty \le A\}$. Is $B$ a closed subset of $C([0,1],\mathbb R)$?
Here's what I tried to do:
Let $\{f_n\}$ be a sequences of functions of $B$ such that $f_n \to f$, with $f\in C([0,1],\mathbb R)$.
If I prove that $f\in B$, then I finish the proof, i.e. $B$ will be closed in $C([... | Ahh, another simple example of sequence of functions is $$ f_{n} = \sqrt{x + \frac{1}{n}} $$
which uniformly convergences to $f(x)= \sqrt x $ which is not differentiable at $x=0$ |
You are given a coin, which when tossed lands on H (heads) with probability $p$ and T (tails) with probability $1-p$. What is the expected number of flips that you must do before you get $L$ consecutive H?
I am aware how to solve the problem when $L$ is fixed.
Let $X$ denote the number of needed flips. Then the answe... | Let $S\subset\mathbb{N}$.
Let $$R_S=\{p_s^t:s\in S,t\notin S\}$$, where $p_s$ is the $s^{th}$ prime number. |
I am looking for optimization books. Can you suggest some good materials?
First, I started with Convex Optimization by Stephen Boyd & Lieven Vandenberghe, but I don't like it because they don't give examples of proofs and techniques, only theory and talking. I need some classical books, for example I like books such ... | Try these books $(1)$ N. S. Kambo, Mathematical Programming Techniques, East West Press, 1997 and
$(2)$ E.K.P. Chong and S.H. Zak, An Introduction to Optimization, 2nd Ed., Wiley, 2010. |
I use SmoothKernelDistribution (with no options and a single argument — a list of values) to get an approximate reconstruction of a continuous distribution based on a finite sample of values. It is known a priori that the actual distribution is unimodal and its PDF is exactly 0 for x ≤ 0.
Can I provide some options to... | SeedRandom[1234]
data = RandomVariate[LogNormalDistribution[1, 1], 1000];
From the documentation, "The kernel function ker can be specified to account for known bounding on the underlying density using {"Bounded", c, ker}, where c can be any real number, a list {Subscript[c, 1],Subscript[c, 2]} such that Subscript[c,... |
I want to look at the summation for both cos,sin..
$cos(x) = 1-\frac{x^2}{2!}+\frac{x^4}{4!}-...= \sum\nolimits_{i=0}^\infty \frac{(-1)^nx^{2n}}{(2n)!}$
so, $cos(x+\frac{\pi}{2})=\sum\nolimits_{i=0}^\infty \frac{(-1)^n(x+\frac{\pi}{2})^{2n}}{(2n)!}$. So how I need to expand $(x+\frac{\pi}{2})^{2n}$ using the binomial ... | Start from $$\cos(x+\pi/2) = \sum_{n\ge 0} (-1)^n \frac{(x+\pi/2)^{2n}}{(2n)!}.$$
Now ask yourself about the coefficient of $x^k$ in this series, it is given by
$$\sum_{2n\ge k} \frac{(-1)^n}{(2n)!} \binom{2n}{k} (\pi/2)^{2n-k}
= \frac{1}{k!} \sum_{2n\ge k} \frac{(-1)^n}{(2n-k)!} (\pi/2)^{2n-k}.$$
In this last series t... |
Prove that if $(G, ◦)$ is a (not necessarily finite) commutative group, and if $g$ and $g'$ are members of $G$ which have finite orders (say $ω$ and $ω'$ respectively), then $g ◦ g'$is of finite order.
What is that order?
I've been looking online for some help with this but most places say that you can't really say to... | Hint. $(g ◦ g')^n = g^n ◦ (g')^n = 1$. What does it mean? |
A cone of radius $r$ and height $h$ sits inside a cylinder, $C$, of radius $r$ and height $2h$ in such a way that the axis of the cone and the axis of cylinder $C$ coincide (call this the $z$-axis). The vertex of the cone lies exactly at the center of the bottom circular base of cylinder $C$ (so that the top circula... | First convert the equation of the slant into cylindrical coordinates.
Start with the observation that axes are arbitrary. Next we need to find the equation of the line in a cylinder with the height on the Z axis. $${Z=mX +H}$$ Where H is the Z intercept (the apex of the cone)
Change the X axis to ${\rho}$ then the sl... |
The problem is as in the question title. Only one addition - $n$ is not divisible by $5$.
I already have a solution involving permutations, but recently I read about generating functions and I was wondering if this problem can be solved with them.
A similar problem is the following:
Find the number of all subsets of... | This is a straightforward application of the Polya Enumeration
Theorem. We treat the problem of subsets with $n$ elements of the set
$\{1,2,\ldots, q\}$ whose sum is divisible by $k.$ Suppose $Z(P_n)$ is
the cycle index of the set operator $\mathfrak{P}_{=n}$ given by the
recurrence by Lovasz which is
$$Z(P... |
So I'm a little confused about how to finish of this homework question and even unsure if my attempt is correct so far...
" Find an orthonormal basis of matrix A" = $$\begin{pmatrix} 2 & 1 & 1 \\ 1 & 1 & 0 \\ 1 & 0 & 1\\ \end{pmatrix}$$
so I found the eigenvalues to be
$\mathbb c=0, 1, 3 $ and the respective norma... | The largest eigenvalue $\lambda_{1}$ of an $n \times n$ symmetric matrix $\mathbf{A}$ admits the following (variational) characterization:
\begin{align}
\lambda_{1}
&= \max_{\mathbf{x}} \frac{1}{\|\mathbf{x}\|^{2}}\mathbf{x}^{T}\mathbf{A}\mathbf{x}.
\end{align}
This is equivalent to
\begin{align}
\lambda_{1}
&= \max... |
So there are 15 cards total, 5 red, 7 orange, and 3 yellow. At random you pick 3 (no replacement).
What's the probability of picking:
1) Exactly 2 Red?
2) Not more than one yellow?
3) One of each?
So I understand since it's without replacement, you'd divide the total probability by 15C3, but after that I'm kinda lo... | $${\overline{z}}^2=z^2\implies{\overline{z}}^2z^2=z^4\implies \vert z \vert ^4 = z^4\implies z^4\in\mathbb{R}^+$$
A quick look at polar form/arguments shows that $z^4$ is a positive real iff $z$ is pure real or pure imaginary, and we are done. |
I just considered if/how one could implement Sequence in Mathematica if it were not predefined. It turned out that the following simple definition has in all my tests exactly the right behaviour:
myseq /: f_[x___, myseq[y___], z___]:= f[x, y, z]
Now my question: Does this already correctly reproduce the full behaviou... | There are functions related to Sequence: BlankSequence, BlankNullSequence and SlotSequence. These can be used to perform operations similar to Sequence but they are not identical.
As pointed out in the comments Sequence obeys the attribute SequenceHold:
SetAttributes[test, SequenceHold]
test[ Sequence[1, 2, 3] ]
t... |
The sequence looks like this:
$e_0 = 2$
$e_1 = 4(e_{1-1}) + 5 = 13$
$e_2 = 4(e_{2-1}) + 5 = 57$
$e_3 = 4(e_{3-1}) + 5 = 233$
$e_4 = 4(e_{4-1}) + 5 = 937$
How would I go about finding the explicit formula for this? For something a little simpler it's fairly easy to make a guess, and I've been told that 'guessing'... | Let $e_n=4^nu_n$, then
$$
4^{n}u_{n}=e_{n}=4e_{n-1}+5=4\cdot 4^{n-1}u_{n-1}+5=4^{n}u_{n-1}+5
$$
and thus
$$
u_{n}=u_{n-1}+\frac{5}{4^{n}}=\left(u_{n-2}+\frac{5}{4^{n-1}}\right)+\frac{5}{4^{n}}=\cdots=u_0+\frac{5}{4}+\frac{5}{4^2}+\cdots+\frac{5}{4^{n}} \\
=u_0+\frac{5}{4}\left(1+\frac{1}{4}+\cdots+\frac{1}{4^{n-1}}\ri... |
For the following graph:
g = Graph[{0 -> 1, 1 -> 2, 2 -> 3, 3 -> 0, 0 -> 0, 1 -> 1, 2 -> 2,
3 -> 3}, VertexLabels -> Automatic,
VertexLabelStyle -> Directive[Black, Bold, 16]]
I want to calculate all paths of length 6 that exist between node 0 and every other node in the graph. The FindPath[] function only all... | For the case of source-target pairs (as opposed to source to all other vertices), using a modification of Szabolcs's step to avoid vertices that, if taken, will take us too far from the target vertex (too far to reach the target within remaining number of steps) gives us all walks from source to target vertex with spec... |
Let $M \in \mathbb{C}^{n \times n}$ be a complex-symmetric $n \times n$ matrix. That is, $M$ is equal to its own transpose (without conjugation). If the real part of $M$ is positive-definite, then is $M$ necessarily diagonalizable?
Of course, if we drop the positive-definiteness requirement, then there are examples of... | The answer is no. Based on your example,
$$
\pmatrix{3&i\\i&1}
$$
will not be diagonalizable, but has a positive definite real part. |
Let $g \in C^{1}[0,1]$
Let $T:C^{1}[0,1] \to C[0,1]$ linear transformation such that $T(f) = (fg)'$
I have to calculate $\operatorname{Ker}(T)$
I know if $f \in \operatorname{Ker}(T)$ then $(fg)' = 0$ so $fg = c$ in $[0,1]$
If $c \not = 0$, then $g(x) \not = 0$ $\forall x \in [0,1]$
Thus $f = \frac{c}{g}$
But if ... | Noting as you did that
$$
fg=c
$$
for some constant $c$, we can deduce that if $g$ vanishes anywhere, $c=0$, and $f$ then is forced to vanish whenever $g$ doesn't, indeed
$$
\ker(T)=\{ f\in C^1[0,1]:\;f(x)=0 \;\forall x\;\text{s.t}\; g(x)\ne 0\}
$$
If $g$ does not vanish, we have the simpler expression $f=\frac{c}{g}$... |
I can see by manipulating the expression why $\mathbb{E}X$ works out to be $\int_0^\infty 1-F(x)\,dx$, where $F$ is the distribution function of $X$, but what is an intuitive explanation for why that is true? If at each point we sum the probability $\mathbb{P}(X>x)$, why should we end up with the expectation?
Thanks | If you are looking for intuition, the discrete case is your best bet.
Look at $\sum_0^\infty P[X > n]$ and count how many times you count the set $\{X = k\}$. You don't count $\{X=0\}$ at all. The only one which includes $\{X=1\}$ is $P[X>0]$ so it gets counted once. You will count $\{X=2\}$ twice, when $n=0$ and $n=1$... |
Determine a differentiable function $y = f(x)$ which has the properties: $f '(x)=f(x)^2$ and $f(0)= -{1\over 2}$.
Can someone solve this problem I cannot for the life of me figure it out.
It will be a great help. | \begin{align}
\dfrac{dy}{dx} &= y^2 \\
dx &= \dfrac{dy}{y^2} \\
x + c &= -\dfrac 1y \\
y &= -\dfrac{1}{x+c} \\
\hline
y(0) &= -\dfrac 12 \\
-\dfrac 1c &= - \dfrac 12 \\
c &= 2 \\
\hline
y &= -\dfrac{1}{2+x}
\end{align} |
Say that
j = \begin{pmatrix} 2 \\ 5 \\ -1 \end{pmatrix} and k = \begin{pmatrix} -6 \\ 4 \\ -3 \end{pmatrix}
There are two nonzero 3D vectors
a = \begin{pmatrix} x \\ y \\ z \end{pmatrix},
which are orthogonal to both j and k, such that vector a's entries $x$, $y$, and $z$ are integers which satisfy $\gcd(x,y,z) = 1$. F... | The orthogonal vectors have to has the same direction as the vector
$$ v = \begin{pmatrix}
\det
\begin{pmatrix}
4 & -3 \\
5 & -1
\end{pmatrix},
-\det
\begin{pmatrix}
-6 & -3 \\
2 & -1
\end{pmatrix},
\det
\begin{pmatrix}
-6 & 4 \\
2 & 5
\end{pmatri... |
How can you calculate sums such as:
$$\sum_{k=1}^∞{\frac{1}{k(k+1)}}$$
How do you best explain this to students in a rigorous or non-rigorous way? | You just do the actual work: by definition, $\sum_{k \ge 1} \frac 1{k(k+1)} = \lim_{N \to \infty} \sum_{k=1}^N \frac 1{k(k+1)}$. Computing the partial sums, you get
$$
\sum_{k=1}^N \frac 1{k(k+1)} = \sum_{k=1}^N \frac 1k - \frac 1{k+1} = 1- \frac 1{N+1}
$$
because this sum is telescopic (feel free to explicitly write t... |
Let $H\leq G$ be two groups. I'm interested in the Commensurator $$\mathrm{comm}_G(H)=\{g\in G: gHg^{-1} \cap H \text{ has finite index in both}\}.$$
Obviously, $\mathrm{comm}_G(H)\leq G$. I read on wiki, that the equality holds for any compact open group $G$. But does anyone have an easy example, where $1\ne\mathrm{c... | I'm no expert on the topic of Lie groups, so no example springs to mind in this context.
For abstract groups, however, we can construct an example as follows:
Define $G$ as the finitely presented group
$$G:= \langle a,b,c ~|~ cac^{-1} = b \rangle$$
and let $H:= \langle a \rangle \le G$. Then $c \not\in \text{comm}_G(H)... |
In book "Introduction to the Theory of Computation" by M. Sipser.
Captioned statement claimed in proof idea of theorem 2.42: "The class of DCFLs is closed under complementation."
My understanding is that DPDA means that all transitions are deterministic, how DPDA can go to both accept state & non-accept state after al... | It may help to think of an input word as an infinite sequence of letters, which is eventually a special "after the end of the input" character. Instead of $ababab$, think $ababab\#\#\#\#\#\ldots$
Depending on the specific definition of pushdown automaton you're using, the pushdown automaton is allowed to continue read... |
Question: Given that $x=\frac{a}{\frac{a}{x+b}+c+b}+c$ for all real $x\neq -b $, prove that $a\neq 0$ and that $c=-b$.
Proving that $a\neq 0$ probably involves a proof by contradiction, but I am not sure exactly what this involves. I also think this can be shown without proving that $c=-b$, but $a\neq 0$ follows dire... | Note that from
$$x=\frac{a}{\frac{a}{x+b}+c+b}+c$$
we need
$x+b \neq 0 \implies x\neq -b$
and
$\frac{a}{x+b}+c+b\neq 0$
then
$$x=\frac{a}{\frac{a}{x+b}+c+b}+c\iff \frac{ax}{x+b}+cx+bx=a+\frac{ac}{x+b}+c^2+bc\\\iff ax+cx^2+bx^2+bcx+b^2x=ax+ab+ac+c^2x+bcx+c^2b+b^2c\\\iff (b+c)x^2+(a+bc+b^2-a-c^2-bc)x-(ab+ac+c^2b+b... |
I know that this immediately follows from Holder's inequality if all the $c_i=1$, and I take $\mu$ as the lebesgue measure and define $f(x)=\sum a_i I_{[i-1,i)}(x)$, $g(x)=\sum b_i I_{[i-1,i)}(x)$. But I'm not sure how to handle the $c_i$'s.
Is there perhaps some way to show that $$\int|fg|h\leq\left(\int|f|^ph\right)... | This is just the usual Holder's inequality applied to $(a_ic^{1/p})$ and $(b_ic^{1/q})$. |
For Chi-Squared test on contingency tables there is a proof to get from:
$\sum\frac{(O_i - E_i)^2}{E_i}$ which equals $\frac{N(ad-bc)^2}{(a+b)(c+d)(a+c)(b+d)}$
Can anyone explain the steps in the proof i know how to get from one to other but not sure why certain steps happen!
Below ill put the proof if anyone wants t... | I went through the same problem as yours. I used Mathematica to ease the computations.
$\chi ^2=\sum _{i=1}^4 \frac{(O_i-E_i)^2}{E_i}=\frac{n(a d-b c)^2}{(a+c)(b+d)(a+b)(c+d)}$
$O_i$ are the observed values, $O_1=a, O_2=b,O_3=c$ and $O_4=d$
$E_i$ are the expected values, $E_1=\frac{(a+b)(a+c)}{n}, E_2=\frac{(a+b)(b+... |
Possible Duplicate:
What's the thing with $\sqrt{-1} = i$
What is wrong with the following?:
$$ \frac{1}{\text{i}}=\frac{1}{\sqrt{-1}}=\frac{\sqrt{1}}{\sqrt{-1}}=\sqrt{\frac{1}{-1}}=\sqrt{-1}=\text{i} $$
There have been a few posts here with similar queries as above saying that
$$\frac{\sqrt{1}}{\sqrt{-1}}=\sqrt{... | You are assuming that $\dfrac{\sqrt a}{\sqrt b} = \sqrt{ \dfrac{a}{b}}$ and that is indeed true when $a$ and $b$ are positive real numbers.
But you cannot hope extend this to all real numbers or to complex numbers ($b \not = 0$) without a definition of the $\sqrt{ }$ function, and if this is single valued then the e... |
For the usual $4$-dimensional Minkowski space $M$, the standard Dirac operator is given by
$$
D: C^{\infty}(M) \to C^{\infty}(M), ~~~~~ f \mapsto \sum_{i=1}^4 \gamma_i\frac{\partial f}{\partial x_i},
$$
where the $\gamma_i$ are the usual gamma matrises. As we all know, $D$ was originally constructed as a square root of... | As Sebastian and A.J. Tolland point out, the problem is that the two operators act on different bundles. However, the exterior algebra is canonically isomorphic to the Clifford algebra as vector spaces. Using this identification, you find two operators $D$ and $d+d^{\ast}$. Both operators have first order and they have... |
Question: $$\lim_{x\to 0}\frac{\tan x-\sin x}{x^3}$$
The answer, by L'Hopital's rule as well as wolfram and desmos is $\frac{1}{2}$
Here's what I did:
$$\lim_{x\to0}({\tan x \over x}\times{1\over x^2}-{\sin x \over x}\times{1\over x^2})$$
$$\lim_{x\to0}({1 \over x^2}-{1 \over x^2})=0$$
Im not sure where the mistake ... | Let’s look at the limit in two ways, using trigonometric properties on one hand and using Taylor expansions on the other. The indetermination is $0/0$; what you do is just writing $0/0=(0-0)/0=0/0-0/0$. This is still indeterminate.
Trigonometry
Write
$$\begin{align}
{\tan{x}-\sin{x}\over x^3}&={\tan{x}\left(1-\cos{x... |
Prove that the map: f: $\{0,1\}^\mathbb{N} \times \{0,1\}^\mathbb{N}$ $\to$ $[0,1] \times [0,1]$ is continuous.
I know that the map can be written as ($m_1,m_2,m_3,m_4,...$) $\times$ ($n_1,n_2,n_3,n_4,...$) $\mapsto$ ($0.m_1m_2m_3m_4...,0.n_1n_2n_3n_4...$) as binary expansions, and f is surjective, but how do I show ... | Here is how I would do it. First, note that as John Ma noted, it suffices to show continuity of
$$
F: \{0,1\}^\Bbb{N}\to [0,1], (x_n)_n \mapsto \sum_{n=1}^\infty x_n/2^n.
$$
To see this, show that each of the maps
$$
F_N: \{0,1\}^\Bbb{N}\to [0,1], (x_n)_n \mapsto \sum_{n=1}^N x_n/2^n
$$
is continuous (use that each pr... |
Due to difficulties in finding references I am not sure of what relative minimality is about for conic bundles. Suppose we're working on con. bun. over surfaces.
The definition is ok: the fiber over any irreducible component of the discriminant curve should not be reducible. But how can I handle that concretely?
My ... | You might consult
http://hal.archives-ouvertes.fr/docs/00/46/16/98/PDF/modelsgeomratcompositio.pdf
Your own definition seems a little confusing. A conic bundle is a map whose fibers are conics. Of course conics are embedded objects so this requires some kind of definition of a conic structure on the fibers. Plane ... |
how do you find the general law which simplifies the product:
$(1-1/4)(1-1/9)...(1-1/n^2)$
i have no idea how to do this the only thing i could do is take blind guesses, how are you meant to solve this kind of problem?
thanks | Just use the fact that $(3n)! \leq (3n)^{3n}$. This gives $(3n)!+n^{n} \leq 2(3n)^{3n}$. The answer is $1$. |
I've seen lots of casual claims that Riemannian manifolds (even without assuming second-countability) are metrizable. In the path-connected case, we can use arc-length to create a distance function. But how can one prove this without assuming path-connectedness (and second-countability)?
(Motivated by Problem 2-D of M... | Assume the manifold $M$ is connected. If there exists a piecewise-smooth path from $p$ to $q$ in $M$, then there exists such a path between $p$ and $q'$ for any $q'$ in a small closed neighborhood of $q$, since $M$ is locally path-connected. The set of such $q$ is thus open and closed in $M$, and so must be $M$ itself.... |
Let $P, Q, R$ be propositions. Consider:
$\big[\forall x \big(P(x)\implies Q(x)\big)\big]\lor \big[\forall x\big(P(x)\implies R(x)\big)\big]$
Given the fact"$\forall x$" and "$ P(x)$" appear twice, what is the best way to simplify this logic statement?
For example, I guess there exists $S$ such that $\big[\forall x ... | If you allow the use of bounded quantifiers, then you can simplify this in a manner of speaking.
$$ (\forall x \mathop. P(x) \to Q(x)) \lor (\forall x \mathop. P(x) \to R(x)) $$
Is equivalent to
$$ (\forall x: P \mathop. Q(x)) \lor (\forall x: P \mathop. R(x)) $$
But that's as far as you can go.
The candidate belo... |
I've been staring at this problem for over an hour, trying different things and getting approximately nowhere. Here's the problem:
Let $$w_1=\begin{bmatrix}1\\2\\0\end{bmatrix},w_2=\begin{bmatrix}2\\5\\1\end{bmatrix}, w_3=\begin{bmatrix}2\\4\\1\end{bmatrix}$$
Let $f:\mathbb{R}^3\rightarrow\mathbb{R}^3$ be the linear... | Let A is a required 3x3 matrix and from the problem we have A [w1 w2 w3] = [f(w1) f(w2) f(w3)]
Therefore A = [f(w1) f(w2) f(w3)]*[w1 w2 w3]^-1
find [f(w1) f(w2) f(w3)] = (\left[\begin{matrix}0&0&5\\1&-1&11\\0&0&2\end{matrix}\right])
and [w1 w2 w3]^- = (\left[\begin{matrix}1&0&-2\\-2&1&0\\2&-1&1\end{matrix}\right])
ther... |
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