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If I toss a fair coin $n$ times, calculate the probability that no pattern HHTHTHH occurs.
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I'm reading through the first chapter of Ahlfors's Complex Analysis book, and during the section on stereographic projections, he says that we can map any $z = x+iy \in \mathbb{C}$ onto the unit sphere in three dimensions injectively using the equation $z= \frac{x_1+ix_2}{1-x_3}$.
He then says that $x:y:-1= x_1:x_2:x_... |
You have 4 machines that are supposed to create 100mg chocolate bars. One of the four machines is not working correctly and is making 90mg chocolate bars. You only have 1 chance to weigh the chocolate bars of 2 of the machines. What is the mathematical equation capable of pulling this off?
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I'm glancing over Milnor's paper on exotic 7-spheres, and one of the first few lines says, `every closed 7-manifold $M^7$ is the boundary of an 8-manifold $B^8$.
Here's what I don't understand: The unoriented cobordism ring is isomorphic to a polynomial ring over $\mathbb{Z}_2$ with a generator in every degree except ... |
I'm trying to understand Serre's proof (in Local Fields) of the fact that a fractional ideal $\mathfrak{a}$ of a Dedekind domain factors uniquely as $\mathfrak{a} = \prod_{\mathfrak{p}}\mathfrak{p}^{v_{p}(\mathfrak{a})}$.
The approach he took was that since $v_{\mathfrak{p}}(\mathfrak{a}) = 0$ for all but finitely many... |
I am working with granular materials (seeds). I am looking for a way to correctly scale the amount of different particles in one batch using weight only. I have worked with the problem a bit and instead of going in detail with the applied problem, I will try to present the problem more generally.
Let's say I have the f... |
This is an applied problem, which arises from the problem of reorienting of a sensor axes according to particle displacement directions:
Consider a sensor which is located inside the solid substance. This sensor is capable of detecting the substance oscillations along each of the three axes (usually orthogonal, but gen... |
1 can be divided by 1
2 can be divided by 1, 2
6 can be divided by 1, 2, 3
12 can be divided by 1, 2, 3, 4
60 can be divided by 1, 2, 3, 4, 5
60 can be divided by 1, 2, 3, 4, 5, 6
420 can be divided by 1, 2, 3, 4, 5, 6, 7
840 can be divided by 1, 2, 3, 4, 5, 6, 7, 8
I just wonder whether there is a general function to... |
I've been playing with Fourier transform a little and discovered the identity quoted in the title. More precisely, writing the matrix for the Fourier transform in ${\mathbb Z} / N {\mathbb Z}$ as
$$A = {1 \over \sqrt{N}} \pmatrix{
\zeta^0& \zeta^0 & \cdots & \zeta^0 \cr
\zeta^0& \zeta^1 & \cdots & \zeta^{N-1} \cr
\v... |
Suppose there are vending machine that sells its goods for $3$. It's known that a third of the buyers use three coins of $1$, a third of the buyers use $2$ and $1$, and the last third use $5$. The machine start empty (without money). What it is the probability that it could give change after $n$ buyers?
Thank you!
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This is my very first question, so don't be too hard on me :)
I am a beginner programmer, and I am working on the typical helicopter game, working with a one-button push:
When the button is pushed, the helicopter ascends, starting ascending slowly, accelerating upwards.
When the button is released, the upward accelerat... |
What is the limit as $x\to\infty$ of $\cos x$?
Thanks in advance.
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I have several tasks to solve where a set of inequalities is used to describe a region. I should then calculate the area or volume of that region.
Let's say we have the following inequality (for $x,y,z \geq 0$):
$x+2y+3z \leq 1$
Now I need to find out the boundaries for the triple integral. From a few examples I have h... |
In course of a particular research I ran into these two inequalities which I would like to have some help with.
$r,R>0$ for both the questions.
Is there a function $M(r)$ for which this inequality is satisfied?
$$\cos^{-1}\left (\frac{4M(r)}{r}-1 \right ) + \sqrt{1- \left ( \frac{4M(r)}{r}-1 \right )^2} > \pi$$
Let ... |
You are and your friend are given a list of N distinct integers and are told this:
Six distinct integers from the list are selected at random and placed one at each side of a cube. The cube is placed in the middle of a rectangular room in front of its only door, with one face touching the floor, its 6 sides parallel to... |
Let $E = \{E_k\}_{k \in \mathbb{N}}$ be an infinite sequence of sets. Then, the following inclusion holds:
$\bigcup_{n=1}^{\infty} \bigcap_{k=n}^{\infty} E_k \quad\subseteq\quad \bigcap_{n=1}^{\infty} \bigcup_{k=n}^{\infty} E_k$
I know the left-hand side (LHS) represents elements which belong to all but finitely many s... |
A box contains $n$ number of coins, $m$ of which are fair and the rest are biased. The probability of getting a head when a fair coin is tossed is $1/2$, while it is $2/3$ when a biased coin is tossed. A coin is drawn from the box at random and is tossed twice. The first time it shows a head and the second time it show... |
I haven't done very much statistics but in a Finnish programming forum I found a some exercises where answers are integers. In every exercise the best result $r$ gets 100 points. Suppose you get the result $s$. If in a given problem the best score is as high as possible, you will be awarded by $100(s/r)^2$ points round... |
On Mathworld it says some diophantine equations can be solved using algebraic numbers. I know one example which is factor $x^2 + y^2$ in $\mathbb{Z}[\sqrt{-1}]$ to find the Pythagorean triples.
I would be very interested in finding some examples of harder equations (not quadratic) which are also solved easily using alg... |
I am a little stuck at the moment with the Fourier transform of a function $w(z)$ that is defined piecewise and discontinuous at $z = 0$:
$$w\left(z\right) = \begin{cases} -\gamma_1 e^{-\gamma_{1}z} & z > 0 \\ \gamma_2 e^{\gamma_{2}z} & z < 0\end{cases}$$
with $\gamma_i \geq 0$.
In a naive attempt, I calculated the fol... |
This was an exam question of ours:
Let $\chi$ be the set, $\chi = \left \{ a, b, c, d \right \}$. Create a topology $\tau$ on $\chi$ so that $\left ( \chi ,\tau \right )$ is regular but not normal.
I couldn't solve the question but I'm wondering what the answer is. Is it possible at all? Can anyone help me with thi... |
Since computers work off formulas, without greater knowledge one would assume that it would come up with the same answer for a set formula. However, you're able to tell it to generate a random number. IE: php rand(). What's the formula that's used to get a random number? How can there be anything "random" in an algebra... |
We have:
$U = 150$
$|M| = 90$
$|E| = 75$
$|F| = 80$
$|M ∩ E| = 45$
$|M ∩ F| = 35$
$|M ∩ F ∩ E| = 10$
We want:
$|E ∩ F ∩ M^c| = ?$
I know I have to use the exclusion-inclusion principle. I have to take the last statement and transform it so that it is similar to:
$|A' ∪ B'|$
But I don't know how transform it; t... |
Possible Duplicate:
Solving the functional equation $f(x+1) - f(x-1) = g(x)$
How do I approach this problem $x[f(x+1)-f(x-1)]=1$.
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Given that $x^3[f(x+1)-f(x-1)]=1$, determine $\lim_{x\rightarrow \infty}(f(x)-f(x-1))$ explicitly.
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Given that f is continuous and $x^3[f(x+1)-f(x)]=1$, determine $\lim_{x\rightarrow \infty}f(x)$.
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While trying some problems along with my friends we had difficulty in this question.
True or False: The value of the infinite product $$\prod\limits_{n=2}^{\infty} \biggl(1-\frac{1}{n^{2}}\biggr)$$ is $1$.
I couldn't do it and my friend justified it by saying that since the terms in the product have values less than ... |
The other day, I tried to answer someone's question, you can read the question and the answer here:
Numerical Solutions of ordinary differential equations
The problem is, I got stuck towards the end of the calculation. Now my question is: does anyone know how to get order 2?
Many thanks for your help!
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Category Observed Expected (O-E) (O-E)^2 (O-E)^2 / E
0 6 5 1 1 0.5
1 6 5 1 1 0.5
2 3 5 2 4 2
3 3 5 2 4 2
4 4 5 1 1 ... |
I want to show that any two disjoint compact sets in a Hausdorff space $X$ can be separated by disjoint open sets. Can you please let me know if the following is correct? Not for homework, just studying for a midterm. I'm trying to improve my writing too.
My work:
Let $C$,$D$ be disjoint compact sets in a Hausdorff spa... |
I believe the problem of trying to find the Jacobian of the following function highlights a lack of understanding of some concept on my part. I was hoping someone could either provide specific advice about solving this problem, or computing Jacobians in general.
Consider the mapping $h : \mathbb{R}^n \rightarrow \math... |
When talking about continuous random variables, with a particular probability distribution, what are the underlying sample spaces?
Additionally, why these sample spaces are omitted oftentimes, and one simply says r.v. $X$ follows a uniform distribution on the interval $[0,1]$? Isn't the sample space critically importa... |
This question is motivated by a little anecdote. I was at home teaching some secondary school math to a relative. At some relax time, he glanced at a book I had over the table - it was some text about analytical number theory that I had recently bought, second hand- and I explained him that that was an area of mathema... |
Let $\mathscr{N}(R)$ denote the set of all nilpotent elements in a ring $R$.
I have actually done an exercise which states that if $x \in \mathscr{N}(R)$, then $x$ is contained in every prime ideal of $R$.
The converse of this statement is: if $x \notin \mathscr{N}(R)$, then there is a prime ideal which does not cont... |
I don't know much (any?) graph theory, but I'm interested in learning a bit, and doing some calculations for a game. Is there a tool online where I could construct a graph (this one has 30-40 vertices, maybe 100 edges), and play around to explore its properties? Useful things to do would be describing paths, finding re... |
Given two invertible sheaves $\mathcal{F}$ and $\mathcal{G}$, one can define their tensor product, but in this definition $\mathcal{F} \otimes \mathcal{G} (U)$ is (apparently) not simply equal to $\mathcal{F} (U) \otimes \mathcal{G}(U)$ for an open set $U$. This latter structure is a presheaf; can someone give me an ex... |
If I were to say that A intersect B was the empty set, does that imply that B intersect A is also the empty set?
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Let $p$ be a permutation of $\mathbb{N}$. We say that $p$ is bounded if there exists $k$ so that $|p(i)-i| \le k$ for all $i$. If $p$ is bounded, must there exist $M>0$ such that
$p(\{1,2,\ldots, M\}) = \{1,2,\ldots, M\}$? If so, can we bound $M$ by a function of $k$?
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Sorry to ask yet another "how do I do this integral" question! But I've really been having a hard time with this one.
$$\int_0^\infty \frac{J_0(x)}{|C^2 - x^2|}\mathrm{d}x$$
I've been through a lot of identities giving results for expressions of the form
$\int Z_\nu(x) x^p \mathrm{d}x$,
where $Z_\nu$ is a Bessel functi... |
If $b_n = \frac{1}{2n+1}$ does the notation $b_{n+1}$ make $b_n = \frac{1}{2(n+1)+1}$ or $b_n = \frac{1}{2n+2}$?
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I have no idea how to solve this. Could anyone give a hand? Thanks!
$$2x^{5a} + 6x^{3a} - x^{2a} - 3 = 0$$
PS: I'm guessing this doesn't make sense unless it's also "solve for x" and the equation is = 0, but I'm not certain of that.
PPS: My intuition was to add 3 to both sides, then multiply both sides by x, then divid... |
I tried proving the formula presented here by integrating the circumferences of cross-sections of a right circular cone:
$$\int_{0}^{h}2\pi sdt, \qquad\qquad s = \frac{r}{h}t$$
so
$$\int_{0}^{h}2\pi \frac{r}{h}tdt.$$
Integrating it got me $\pi h r$, which can't be right because $h$ isn't the slant height. So adding up ... |
I am an undergraduate Math student, i find the courses that I am in are too easy (calculus, linear algebra, Discrete math, logic).
I want to become a successful mathematician. I want to know what material i should study by myself? Should I just do all the extra problems in the book, or should I start reading about gen... |
I'm stuck on the following problem :
Does there exist a real borel measure $\mu$ on $[0,1]$ such that
$$\int_{0}^{1}x^n d\mu = e^{-n^2}$$
for all $n \geq 1$ ?
Does anyone have any hint?
Thank you
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My undergraduate University does not offer advanced courses on logic, I know truth tables, Boolean algebra, propositional calculus. However I want to pursue Mathematical Logic on the long term as a mathematician.
Can anyone suggest a study-map of Mathematical Logic.
such as
(1) Learn The following topics : a,b,c,etc..
... |
Can you calculate rigorously the limit $\lim\limits_{n \to \infty} {(\sin n)^{\frac{1}{n}}}$
|
I have the following theorem on my notes of a course on complex analysis I'm taking:
If f(z) is not bounded in a neighborhood of its isolated singular point z0, then z0 is a pole.
I think this cannot be right, for example sin(1/z) at z0=0 is not bounded, but has an essential singularity, not a pole. Just need some conf... |
$$ \lim\limits_{n\to\infty}\frac{2^n}{3^{\frac{n}{2}}}$$
I used wolfram to get the limit as follows:
"lim n tends infinity 2^n/3^(n/2)".
And using L'Hospital's rule the result was:
$\displaystyle\frac{2^{n+6}}{3^{\frac{n}{2}}}$, which tends to infinity.
My question is why does
$\displaystyle\frac{2^{n+6}}{3^{\frac{n}{... |
For a given analytic germ $f(z)=\sum_{n=0}^{\infty} a_n (z-z_0)^n$, and a simple curve $\gamma: [0,1]\to \mathbb{C}$ such that $\gamma(0)=z_0$, suppose one may analytically continue $f(z)$ along $\gamma$. So for each point $\tilde{z} =\gamma(t)$ there is a corresponding power series $f(z;\tilde{z})=\sum_{n=0}^{\infty} ... |
Problem edited since i missed a line from the problem,
A monkey wants to climb up a pole of 50 metre height. he first climbs up 1 m but he falls back by same height. Again he climbs up 2 m but he falls back by 1 m. Again he climbs up 3 m and falls back by 1 m. Again he climbs up 4 metre but falls back by 1 m. In this w... |
Naively I would have thought that a manifold becomes geodesically incomplete if there are missing points in it or if the geodesics are hitting a boundary.
But I am not sure how to think of geodesic completeness or not for manifolds with a boundary. Like how to think for the closed disk on $\mathbb{R}^2$
I would like... |
This is a problem I'm confused with:
Definition:
We say two subsets $A$ and $B$ of a topological space $X$ are completely separated if there exists a continuous map $f: X \rightarrow \mathbb{R}$ such that $f(A) \subseteq \{0\}$ and $f(B) \subseteq \{1\}$.
i.e we are not given that $A=f^{-1}(\{0\})$ but just that $f(A) ... |
Possible Duplicate:
How come the number $N!$ can terminate in exactly $1,2,3,4,$ or $6$ zeroes but never $5$ zeroes?
The number of zeros which are not possible at the end of the $n!$ is:
$a) 82 \quad\quad\quad b) 73 \quad\quad\quad c) 156 \quad\quad\quad d) \text{ none of these }$
I was trying to solve this probl... |
I'm aware that there are an infinite number of integrals that do not have a closed-form solution. But how does one know if their particular integral is one of these? For example, a particular integral I have at the moment is
Integrate[ Cos[x]^2 Log[ a + b Cos[x] + Sqrt[ c + d Cos[x] ] ] , x]
or
$$ \int \cos^2 x \,\... |
This is a follow-up question to About the limit of the coefficient ratio for a power series over complex numbers.
Cauchy's estimation in complex analysis is a consequence of the Cauchy's integral formula, which can be stated as follow:
Theorem. If $f$ is holomorphic in an open set $\Omega$ that contains a closed disc... |
Let $n$ be a number of people. At least two of them may be born on the same day of the year with probability: $$1-\prod_{i=0}^{n-1} \frac{365-i}{365}$$
But what is the probability that at least two of them are born on two consecutive days of the year (considering December 31st and January 1st also consecutive)? It seem... |
what is the 444th term of the series $1,2,3,10,11,12,13,20,21,22,23,30,31, \dots$
|
Which conditions must the matrix entries satisfy, and what would be an interpretation of the row and column sums of the matrix?
|
The question is :
6 points are located on a circle and lines are drawn connecting these points, each pair of points connected by a single line. What can be the maximum number of regions into which the circle is divided?
My answer is 32 . But the actual answer is 31 how ??
|
I was asked to sum the given series:
$\displaystyle \sum\limits_{n=1}^{\infty} \frac{1}{n\cdot (2n-1)}=\frac{1}{1} + \frac{1}{2 \cdot 3} + \frac{1}{3 \cdot 5} + \cdots \infty$
Here i workout the details.
\begin{align*}
\sum\limits_{n=1}^{\infty} \frac{1}{n \cdot (2n-1)} &= \frac{1}{1} + \frac{1}{2 \cdot 3} + \frac{1}... |
I am trying to solve the matrix equation of the form $XA = XB$. $A$, $B$ and the solution sought $X$ are $4 \times 4$ homegeneous matrices which are composed of a rotation matrix and translation vector, like in
$$ \begin{bmatrix} r_1 & r_2 & r_3 & tx\\ r_4 & r_5 & r_6 &ty\\ r_7 & r_8 & r_9 & tz\\ 0 & 0 & 0 & 1 \end... |
I am trying to find the $n$'th derivative of $f(x)=e^{1/x}$. When looking at the first few derivatives I noticed a pattern and eventually found the following formula
$$\frac{\mathrm d^n}{\mathrm dx^n}f(x)=(-1)^n e^{1/x} \cdot \sum _{k=0}^{n-1} k! \binom{n}{k} \binom{n-1}{k} x^{-2 n+k}$$
I tested it for the first $20$ d... |
$$\limsup_{i \to \infty} \; E_i = \bigcap_{k=1}^{\infty} \; \bigcup_{i=k}^{\infty} \; E_i$$
So
$$ \bigcup_{i=k}^{\infty} \; E_{i} = \{x \in E_{i} \mid i \in I\}= S $$
where $I$ is some index set. When I try try to understand next part, I get confused, seems too trivial:
$$\bigcap_{k=1}^{\infty}S = S$$
since there is on... |
Does there exist an analytic everywhere, piecewise-defined function $f$ such that:
$f(x) = g(x)$ for $x < k$
$f(x) = h(x)$ for $x>k$
$f(x) = r$ for $x=k$
With $g \ne h $ ($g$ not the same function as $h$)
If it exists, what is such an example?
|
Suppose I have a group $G$ with a finite presentation, and I know that $G$ is polycyclic. So I input the group into GAP via generators and relators. Now I want to find a presentation of an infinite index subgroup; I can do this by hand (usually), by doing a Reidemeister-Schreier rewriting, and then simplifying the pr... |
I've created a type of alphabet cipher in which each character is defined based on how distant it is (in the alphabet) from the preceding character.
So, assuming that $A=0$, $B=1$, $C=2$, etc., and you have a text input of "GOOSE," you would construct the output like this:
G $(6)$ is $2$ letters in front of E $(4)$, so... |
I am stuck up with this simple problem.
If $\alpha \cdot \beta = \pi$, then show that $$\sqrt{\alpha}\int\limits_{0}^{\infty} \frac{e^{-x^{2}}}{\cosh{\alpha{x}}} \ \textrm{dx} = \sqrt{\beta} \int\limits_{0}^{\infty} \frac{e^{-x^{2}}}{\cosh{\beta{x}}} \ \textrm{dx}$$
I tried replacing $\cosh{x} = \frac{e^{x}+e^{-x}}{2}... |
Maybe this is a dumb question, but I have to admit that it is not really clear to me what the theory is, in which incompleteness of PA and stronger theories is proved. The texts I did study so far are often not really explicit about that.
What's the (minimal) theory in which
incompleteness of PA and stronger theor... |
What is the measure of one of larger angles of a
parallelogram in the $xy$-plane that has vertices with
coordinates $(2, 1)$, $(5, 1)$, $(3, 5)$ and $(6, 5)$?
(A): $93.4$
(B): $96.8$
(C): $104.0$
(D): $108.3$
(E): $119.0$
Now the answer strip says (C), but it doesn't explain how to find the angle.
So how would I go abo... |
Prove that the the function $f(x) = |x-1|+|x-2|$ is not differentiable at $x = 1$ and $x = 2$.
This is actually an objective-type question where there are $3$ other options $(\{0\},\{1\},\{3\},\{1,2\})$, now the problem is graph is not really much of a choice, hence I guess I have to break the functions into ranges an... |
I have the following system of equations:
$$\alpha f_1(x) = \int_\mathbb{R} g(k) h_1(k) e^{\mathrm{i}kx} dk$$
$$\beta f_2(x) = \int_\mathbb{R} g(k) h_2(k) e^{\mathrm{i}kx} dk$$
with known functions $f_i(x)$ and $h_i(x)$ and unknown $\alpha$, $\beta$ and $g(k)$.
Before outlining my ansatz, here directly the question:
... |
In this very nice book, author has defined vector space as set of functions $f : S \rightarrow F$ where $S$ is a finite set and $F$ is a field. It turns out that this definition is closely resemble definition of Free Vector Space.
There are one or two threads on other online community regarding this. While I try to und... |
I want to solve this limit without using L'Hopital's rule:
$$\lim_{x\to 5}\frac{2^x-2^5}{x-5}.$$
And thanks.
|
One of my former students asked me how to go from one presentation of the Hermite Polynomial to another. And I'm embarassed to say, I've been trying and failing miserably. (I'm guessing this is a homework problem that he is having trouble with.)
http://functions.wolfram.com/Polynomials/HermiteH/07/ShowAll.html
So he ha... |
There are several equivalent ways of defining a function. We know that a differentiable function $f : \mathbb{R} \to \mathbb{R}$ is uniquely defined when its values are specified at every point in $\mathbb{R}$. Now the question is : Is the derivative of such a function $f$ always unique ?
PS: Pardon me if its a very t... |
Let $f:X\to Y$ be a finite map of varieties and let $BL_Z(Y)$ be the blow-up of a subscheme $Z\subset Y$. Is there a map $$\phi:BL_{f^{-1}(Z)}(X)\to BL_{Z}(Y)?$$ If so, what can be said about $\phi$? Is it also finite?
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Source link: http://romvf.wordpress.com/2011/01/19/open-letter/ (Referred from slashdot)
The fact of existence of the polynomial algorithm for 3-SAT problem leads to a conclusion that P=NP.
Is there validity to this argument that there is a P=NP solution? How long till this can be verified? Will this take years to ve... |
I found on Mathworld ( http://mathworld.wolfram.com/Ellipse.html ) that the quadratic equation:
$$ax^2 + 2bxy + cy^2 + 2dx + 2fy + g = 0$$
represent an ellipse only when, after defining:
$$\Delta = \left|\begin{array}{ccc} a&b&d\\b&c&f\\d&f&g\end{array}\right|$$
$$J = ac-b^2$$
$$I = a+c$$
the following conditions are m... |
I have been reading a bit about the Bochner integral and now I'm wondering the following:
For the theory to be "nice", one would expect that
$$L^1([0, \tau], L^1([0, \tau])) \cong L^1([0,\tau] \times [0, \tau]).$$
Is this the case? How do we prove this (or where can I find more about this), since if we take $u$ in the ... |
Edge-attaching many hexagons results in a plane. Edge-attaching pentagons yields a dodecahedron.
Is there some insight into why the alternation of pentagons and hexagons yields an approximated sphere? Is this special, or are there an arbitrary number of assorted n-gons sets that may be joined together to create regular... |
Melvyn Nathanson, in his book Methods in Number Theory (Chapter 8: Prime Number's) states the following:
A conjecture of Schinzel and Sierpinski asserts that every positive rational number $x$ can be represented as a quotient of shifted primes, that $x=\frac{p+1}{q+1}$ for primes $p$ and $q$. It is known that the set ... |
Let $\tau$ be the usual topology on the real line $\mathbb{R}$. Does there exists a topology $\tau_{0} \subset \tau$ such that $(\mathbb{R},\tau_{0})$ is homeomorphic to the
figure eight? Also, is it possible to find a topology $\tau_{0} \subset \tau$ and a quotient space with this topology such that this space is hom... |
As a physics student, I have occasionally run across the gamma function
$$\Gamma(n) \equiv \int_0^{\infty}t^{n-1}e^{-t} \textrm{d}t = (n-1)!$$
when we want to generalize the concept of a factorial. Why not define the gamma function so that
$$\Gamma(n) = n!$$
instead?
I realize either definition is equally good, but i... |
I have a collections of "systems" that I want to test their performance, $1,\ldots,k$.
I have a small set of samples, for which I can test whether each condition does better or not than the other, so I can get a binary result $I(i,j)$ for each $1 \le i,j \le k$ that tests whether $i$ or $j$ do better on this small samp... |
Since $\Gamma$ is not one to one over the complex domain, Is it possible to define some principal values ( analogues to Principal Roots for the Root function ) so we can have a $\Gamma^{-1} (z)$ (inverse $\Gamma$ function)?
|
How are graphs plotted or sketched?
If you have a graph plotting software for example mathematica or matlab, or you want to see what the graph of $e^{2x}$ looks like, how do you plot/sketch such a graph?
If you proceed by using a finite number of points, and "extrapolating" and "interpolating", then how do you know t... |
Given n points drawn randomly on the circumference of a circle, what is the probability they will all be within any common semicircle?
|
Drawing a parallel to linear Algebra.
Premise:
The maximum of a determinate is when all the columns are orthogonal, given that the vectors are normalized.
Conclusion:
The orthogonal trajectory should have a similar effect maximizing the wronskian. However, this does not seem to be true.
Given dy/dx = x the orthogonal t... |
How to simplify this?
$\displaystyle\frac{n^{\log m}}{m^{\log n}}$
|
Greetings All
Thanks to James and Chas for helping me get this far
btw Chas the language I wrote it in is in matlab. I tried to respond
to your post but wasn't able to do it..I guess the gremlins
are acting up on my computer. Also since I'm only able
to post one link I'll post a link with to the text file
which will ... |
Suppose you want to line up $2^n$ balls of which $x$ are black the rest are white. Find a general method to do this so that the black balls are as dispersed as possible, assuming that the pattern will repeat itself ad infinitum. The solution can be in closed form, iterative, or algorithmic.
For example, if $n=3$, whe... |
I've been given the task of computing two series programmatically one is of the form:
$ 1 + \frac{1}{2} +\frac{1}{3}+\frac{1}{4}...+ \frac{1}{n} $
The other one is an estimation for $\pi$ which is of the form:
$\pi = 4( 1 - \frac{1}{3} + \frac{1}{5} -\frac{1}{2i -1})$
I can write this program any way I see fit.
My qu... |
Since semigroups don't need to have an identity element, I was wondering if there's any kind of short exact sequence for semigroups.
|
I am wondering if there are some other examples of Riemann non-integrable but Lebesgue integrable, besides the well-known Dirichlet function.
Thanks.
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I'm currently stuck on an exercise problem from Joseph Gallian's book, "Contemporary Abstract Algebra." The question is from Chapter $2$, Exercise $12$. It says: "For $n>2$, show that there are at least two elements in $U(n)$ that satisfy $x^2=1$"
Here $U(n)$ is the set of positive integers less than $n$ and co-prime ... |
We know that if a real function is in $L^2$ then it is in $L^1$, but the reverse is not necessarily true. So what are the examples of functions that are $L^1$ but not $L^2$, especially those intrinsically unbounded ones if any? Welcome more examples, or classes of examples. Thanks.
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This is related to the content in the book by Grimmett and Stirzaker "Probability and random processes" 3rd ed. On page 111, it calculated, as an example, the conditional density function of $X_1+X_2$ given $X_1=X_2$ for two i.i.d. exponential r.v. with parameter $\lambda$. They used two methods,
1) using $Y_1=X_1+X_2... |
If I have a two-sided coin with probability $p$ showing head. I repeatedly toss it until either HTHTH or HTHH appears. Can you calculate
1) the probability when I got HTHTH, and
2) the expected value of the number of tosses before I stop?
Thanks.
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Let us suppose we have slots $n$ slots $1, \ldots, n$ and $k$ pebbles, each of which is initially placed in some slot. Now the pebbles want to space themselves out as evenly as possible, and so they do the following. At each time step $t$, each pebble moves to the slot closest to the halfway point between its neighbori... |
Given the positive sequence $a_{n+2} = \sqrt{a_{n+1}}+ \sqrt{a_n}$,
I want to prove these.
1) $|a_{n+2}| > 1 $ for sufficiently large $n \ge N$.
2) Let $b_{n} = |a_{n} - 4|$. Show that $b_{n+2} < (b_{n+1} + b_{n})/3$ for $n \ge N$.
3) Prove that the sequence converges.
How should I proceed? Is there a recurrence formul... |
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