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I am a soon-to-be high school graduate with inclination towards mathematics. I enjoy doing math, and am relatively good at it, but I dislike the way I am being taught. I feel like I am being taught methods for solving a lot of problems, but I don't see how any of this fits together.
I was considering pursuing a career ... |
Suppose $f(x,y) = c$ for $0\lt y\lt x\lt 1$ and $0$ outside. What is $P(X+Y \leq 1)$? What is $P(X^2+Y^2 \leq 1)$?
So
\begin{equation*}
P(X+Y \leq 1) = \int_{0}^{1} \int_{0}^{1-x} 2 \ dy \ dx?
\end{equation*}
Likewise,
\begin{equation*}
$P(X^2+Y^2 \leq 1) = \int_{0}^{1} \int_{-\sqrt{1-x^2}}^{\sqrt{1-x^2}} 2 \ dy \ d... |
Many young, and not so young, mathematicians struggle with how to spend their time. Perhaps this is due to the 90%-10% rule for mathematical insight: 90 pages of work yield only 10 pages of useful ideas. A venerable mathematician once described his career to me as constantly stumbling around in the dark. Of course, thi... |
Suppose $R=S/I$ is the homomorphic image of a local ring $(S,m)$.
I know the completion of $R$ as an $S$-module is just
$R \otimes \hat S$, where $\hat S$ is the $m$-adic completion of $S$.
Is this also the completion of $R$ as a ring?
(I think this comes down to showing that the inverse limit of $S/(m^i +I)$
is the sa... |
I've wasted a few days trying to prove Thue's lemma with no success! The proof is absolutely easy if you just count. What 'red flags' are useful to give you a warning that counting might be more productive than other approaches?
I can start with one point, but it's more of a conceptual one and requires you to see the t... |
$$\sum_{n\geq0} ({\frac{2}{3}})^{2n-1}$$
I managed to find that this is a geometric series with common ratio of $4/9$, therefore convergent. That means it's possible to calculate the sum of the series. Can I get any tip of how to do the sum of any kind of geometric series that converge? I've been researching, but all a... |
There is a 5 by 5 matrix of points on a plane. How many triangles can be formed using points on this matrix?
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I've recently come upon the following (seemingly) simple observation:
Claim: A positive integer $k = d d^{\prime} > 1$ has the opposite parity of $\text{gcd}(d+1,d^{\prime}+1)$ for any pair of distinct divisors $d$ and $d^{\prime}$ of $k$.
There are substantially simple proofs (see comments), and one proof follows from... |
The definition of a Cumulative Distribution Function $(CDF)$ says that
$$P(X \le x) = F(x)$$
This is all good.
Then my text book gives the following theorem without proof:
$$P(X \lt x) = F(x-)$$
The book says that the proof is easy, but with my rusted calculus skills, I have trouble to even intuitively understand this ... |
I am interested in a system of differential equations that is non-linear, but it doesn't seem to be too crazy. I'm not very good at non-linear stuff, so I thought I'd throw it out there.
The actual equations I'm looking at have several parameters that'd I'd like to tweak eventually.
q' = k - m / r
r' = i - n r - j q
i... |
If I have a stochastic matrix $X$- the sum of each row is equal to $1$ and all elements are non-negative.
Given this property, how can I show that:
$x'X=x'$ , $x\geq 0$
Has a non-zero solution?
I'm assuming this has something to do with proving a feasible dual, but not may be wrong..
|
$$\int_1^{10}x^xdx$$
I figured it with a relative error of 1% and I have a response $$≈0.3*10^{10}$$
But I don´t know how to accurately calculate it ...
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I'm trying to use inverse cosine, in the cosine rule to find an angle of a triangle when you know 3 sides. I know this formula and have it written down. However I've left my calculator and I'm having to use Google:
https://www.google.com/search?q=arccos((5.9%5E2+%2B+4.1%5E2+-+3.2%5E2)+%2F+(2++5.9++4.1))++in+degrees
but... |
This is the first exercise from Sierpinski's Elementary Theory of Numbers. He gives a proof using induction and I was wondering if this approach was correct as well:
$a!b!|(a+b)! \iff \exists c \in \mathbb{N} \text{ such that } (a+b)! = c(a!b!)$
Assuming without loss of generality that $a \leq b$:
$a!b! = \displaystyle... |
Okay, so I'm trying to find $ \int \frac1{\cos x}\mathrm{d}x$ using the substitution $t = \tan\left(\frac{x}{2}\right)$.
I sub in the trig identity for $\sec$ as $\frac{1+t^2}{1-t^2}$ and then rearrange and substitute $\frac{\mathrm{d}t}{\mathrm{d}x} = \frac12 \left(1+ \tan^2\left(\frac{x}{2}\right)\right)$ so I am lef... |
So suppose I want a path from 0 to $c>0$ on the real line, and I am going to use the function $S(t)$ to get there in (discrete) time $T$. That is, my position at time 0 is 0, my position at time $T$ is $c$, and my position $P_t$ changes in the following way:
$$
P_t = (1-\gamma) P_{t-1} + \gamma S(t)
$$
where $\gamma<... |
Let $n$ be a natural number $k$ digits. Show that the quantity $Q$ of digits required to write the natural numbers from $1$ to $n$ is:
$Q = k(n+1) - \underbrace{111\ldots11}_{k\textrm{ digits}}$
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I'm having a hard time. I want to generate an equation system with two equations from a second degree equation.
For example, get two first degree equations from :
$$y''=- \frac k m|y'|y'-g$$
(k, m and g are constants)
And yes, it is a homework, I just can't figure out how to do it.
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Forgive the total beginner question, it's been a few years since I needed to do this stuff.
What is the expansion of $(1 + x)^n$?
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I would really appreciate if someone could just write down for me one example of a second order, or higher, differential equation for which it is known that there are no smooth solutions; and it's fine if it's a partial differential equation.
At first I thought it would be easy to either come up with an example or else... |
I have the following formula:
$\exists y \forall x ((x \geq y) \wedge \neg(y \geq x))$
This essentially boils down to:
$\exists y \forall x (x > y)$
I have to check whether this applies to certain universes. But my result depends on whether I can assume that x can also equal y, because in that case the condition will n... |
I would like to understand why polynomial multiplication works the way it is defined. For example, we know that $(x+1)^2 = x^2+2x+1$, but how can we prove that this actually works?
More generally, how did we came up with the method used to multiply polynomials?
Any insight would be appreciated.
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Is category theory worth learning for the sake of learning it? Can it be used in applied mathematics/probability? I am currently perusing Categories for the Working Mathematician by Mac Lane.
|
A group $G$ is is abelian if
$$ ab = ba $$
for all pairs of elements $a,b \in G$, i.e., if multiplication commutes. Another way of stating this condition is that the identity map $\mathrm{id}$ gives an isomorphism between left- and right- multiplication. In other words, we can think of left-and right- multiplication ... |
Let $f$ be a continuous but nowhere differentiable function. Is $f$ convolved with mollifier, a smooth function?
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Prove that the Markov-Hurwitz equation $x^2+y^2+z^2=dxyz$ is solvable in positive integers iff d= 1 or 3. Of course the reverse direction is easy, just set x=y=z=1, d=3. But I really have no idea how to attack the forward direction.
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A divisor $d$ of $k = p_{1}^{r_{1}} \cdots p_{n}^{r_{n}}$ is unitary if and only if $d = p_{1}^{\varepsilon_{1}} \cdots p_{n}^{\varepsilon_{n}}$, where each exponent $\varepsilon_{i}$ is either $0$ or $r_{i}$. Let $D_{k} = ${ $d$ } be the subset of unitary divisors of an integer $k > 1$ satisfying $\omega(d) = \omega(k... |
Prove that the continued fraction of $\tan(1)=[1;1,1,3,1,5,1,7,1,9,1,11,...]$. I tried using the same sort of trick used for finding continued fractions of quadratic irrationals and trying to find a recurrence relation, but that didn't seem to work.
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How can I compute the number of common divisors of two naturals ?
For example,if we consider (12,24) the answer is 6 i.e {1,12,2,6,3,4}.
EDIT : I got an answer here.The solution boils down to finding the number of divisor of the GCD of the two numbers.
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The standard definition of a ring is an abelian group that is a monoid under multiplication (with distributivity). However, there are some books that have a weaker definition implying that a ring only has to be closed under multiplication (no identity).
There is a problem in my algebra book asking me to prove that if a... |
Interesting problem... I tried solving it through isosceles triangles, and I've proved it doesn't work on equilateral triangles. Can anyone give me some hints towards a geometric proof?
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Question: Find the distinct number of permutations of four letters of the word MATHEMATICS.
The answer is $\dfrac{11P4}{2!^3} = 990$. Is this correct?
This question along with the answer is from a module from my course, so this is all the details that I am able to provide.
My understanding of this question is how many ... |
Given two integers A and B , how to efficiently compute the sum of all the digits of every number int the set $\{N | A \le N \le B \} $.
For example: A = 100 and B = 777, then the required answer is 8655.
I am interested in deriving an formula/efficient algorithm for the same.
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Fair warning: I am not a math expert (that's why I'm here).
I would like to be able to calculate the probability of rolling a certain side on a die with n sides where any number of those sides has an unfair advantage over the others.
For example: On a 6-sided die, we could assume that the 1 - 6 sides are longer than th... |
What is the probability to get $n$ times the same element $k$ choosing randomly from a set $A$ knowing the cardinality $|A| = c$?
Thank you in advance,
rubik
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what is the number of permutations of a set say {1, 2, 3, 4, 5} that keep at least one integer fixed ?
|
The 2nd part of the "Fundamental Theorem of Calculus" has never seemed as earth shaking or as fundamental as the first to me. Why is it "fundamental" -- I mean, the mean value theorem, and the intermediate value theorems are both pretty exciting by comparison. And after the joyful union of integration and the derivativ... |
I'm a senior undergrad at a top-ish(say, top 15) math school. I'm a solid, not stellar, student. This year I'm taking the qualifying exam grad courses in algebra and analysis and have been taken aback by the "pressure cooker" atmosphere among grad students here. That is, even moreso than in the undergraduate program.
I... |
Please let me summarize the method by which L. Euler solved the Basel Problem and how he found the exact value of $\zeta(2n)$ up to $n=13$. Euler used the infinite product
$$
\displaystyle f(x) = \frac{\sin(x)}{x} = \prod_{n=1}^{\infty} \Big(1-\frac{x^2}{n^2\pi^2}\Big) ,
$$
Newton's identities and the (Taylor) S... |
$f(x) = \frac{1}{\cos x}$
$f'(x) = \frac{\sin(x)}{\cos^2(x)}$
$f''(x) = \frac{2\sin^2(x)+\cos^2(x)}{\cos^3(x)}$
$f^{(3)}(x) = \frac{6\sin^3(x)+5\cos^2(x)\sin(x)}{cos^4(x)}$
$\vdots$
$f^{(n)}(x) = \frac{ ?}{cos^{n+1}(x)}$
Some of these are easy: http://darkwing.uoregon.edu/~jcomes/251exn.pdf
Others are not. Why?
|
Suppose $X$ is a space and $A_1\subseteq A_2\subseteq A_3\subseteq ...\subset X$ is a sequence of subspaces each of which is closed in $X$ and such that $X\cong \varinjlim_{n}A_n$ (i.e. $U$ is open in $X$ if and only if $U\cap A_n$ is open in $A_n$ for each $n$). This topology on $X$ has many names (direct limit, induc... |
Let's say I have a $2 \times 2$ matrix (actually the structure tensor of a discrete image - I):
$$ \begin{bmatrix}
\frac{\partial I}{\partial x}\frac{\partial I}{\partial x} & \frac{\partial I}{\partial x}\frac{\partial I}{\partial y} \\\
\frac{\partial I}{\partial y}\frac{\partial I}{\partial x} & \frac{\partial... |
Consider a situation with a bag with infinity number of balls. Each ball is of some color. Number of colors is finite but it is not known. Balls are drawn from the bag one by one and checked for the color.
We want to stop drawing balls when there is small probability that we will find color that is not already drawn.
O... |
A closed curve on a surface can be represented by the boundary it intersects. Suppose the original surface is created by glueing $a$ to $A$, $b$ to $B$ etc. Pick an orientation, each time it intersects a boundary, add the boundary to the representation. For example in the image, it's
The black dot is the starting poi... |
I know how to generate solutions to $x^2-Dy^2=\pm 1$ from the fundamental solution to $x^2-Dy^2=\pm 4$. But how do we know that all the solutions are generated from this fundamental solution?
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I am taking my first real-analysis course this year and have a midterm on friday. I have NO idea what is going on in this class.
So if anyone could clarify what the following means.
The book starts off with talking about Archimedes and his principle which states:
The value of an infinite series, if it
exists, is th... |
I encountered a problem today to prove that $(X_n)$ with $X_n = \cos(n!)$ does not have a limit (when $n$ approaches infinity). I have no idea how to do it formally. Could someone help? The simpler the proof (by that I mean less complex theorems are used) the better. Thanks
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If $A$ is a stochastic matrix, then $A$ is entry-wise nonnegative and $Ae = e$, i.e., $(1,e)$ is a right eigenpair for $A$.
Is it true that there exists a vector $b$ such that
$$(A - I)x \geq b$$
has no solutions in $x$? If so, is there a simple proof?
Motivation: I've been trying to construct an answer to anothe... |
What is the smallest square which contains 11 non-overlapping (except boundary) unit squares?
This question is open but I would like to know a method to verify the best known answer at the moment.
I'm reading paper at here . In figure 2, there has been said that the best known packing has side length about 3.8772 and t... |
I need to figure out the Probability that $Y > 23$ for a chi-square distribution with degrees of freedom $= 7$.
The book says to do this with some applet. I don't know what it's talking about.
Can I do this on my Ti-89 (preferable) or on some website somewhere?
EDIT/UPDATE:
I found a program on my ti-89 for chi-square ... |
Prove or disprove that closed 3-manifolds which are not simply connected cannot be embedded in three-dimensional Euclidean spaces. I am not a mathematics major and I am taking introductory topology this semester. But I need to apply this result for my research. Any help is much appreciated.
Thanks,
Srikanth.
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Can a derivative operation commute over an integral operation irrespective of the properties of the function under the integral ?
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This operation is similar to discrete convolution and cross-correlation, but has binomial coefficients:
$$f(n)\star g(n)=\sum_{k=0}^n \binom{n}{k}f(n-k)g(k) $$
Particularly,
$$a^n\star b^n=(a+b)^n$$
following binomial theorem.
I just wonder if there is a name for such operation and where I can read about its properties... |
Most of what I've read defines the support of a distribution topologically if it defines support precisely; e.g the smallest closed subset that has full measure (i.e. the intersection of all closed subsets of full measure). I've seen allusions to purely measure-theoretic definitions, and I think I even came across one ... |
Let $P \in \mathbb{Z}[x], P(x) = \displaystyle\sum\limits_{j=0}^n a_j x^j, a_n \neq 0$ and $a_0 \neq 0$; if $p/q$ is a root of P (with p and q coprimes) then $p|a_0$ and $q|a_n$
I've managed to prove the first part ($p|a_0$) and I suppose I'm not far from proving the second, though I'd really like some feedback since I... |
I am looking for a simple proof to show that given a countable set of natural numbers $C$ that is closed under addition and whose gcd is 1, there exists two elements $c_1, c_2 \in C$ such that $\gcd(c_1, c_2)=1$.
I used the Chinese remainder theorem, but my proof is about half a page and I was wondering if there is a s... |
The Riemann Zeta Function is defined as $ \displaystyle \zeta(s) = \sum\limits_{n=1}^{\infty} \frac{1}{n^s}$. It is not absolutely convergent or conditionally convergent for $\text{Re}(s) \leq 1$. Using analytic continuation, one can derive the fact that $\displaystyle \zeta(-s) = -\frac{B_{s+1}}{s+1}$ where $B_{s+1}$ ... |
How would I start off proving that the set $S$, of symmetric $n\times n$ matrices, is a manifold.
I tried using the definition directly by saying
$M_n =$ the space of all $n\times n$ matrices.
For every $A\in M_n$ there exists open sets $U=V=M_n$ and a bijection $F: U\to V$ by $F(A)= A-A^T$
Therefore we have $F(U \cap... |
I presume that it is because one needs to take a limit to define the derivative, but is there another reason?
|
What are some algorithms used for finding the shortest path between 2 points on a (riemannian) manifold (the manifold may have a smooth boundary)?
So far I've had 3 ideas, none of which seem that good:
1) (Rubber band principle) Find any path between the two points, then draw the path taut like a stretched rubber band ... |
The following two identities comes from my trigonometry module without any sort of proof,
If $A + B + C = \pi $ then,
$$\tan A + \tan B + \tan C = tan A \cdot tan B \cdot tan C$$
and,
$$ \tan \frac{A}{2} \cdot \tan \frac{B}{2} + \tan \frac{B}{2} \cdot \tan \frac{C}{2} + \tan \frac{C}{2} \cdot \tan \frac{A}{2} = 1 $$
PS... |
After learning about the binary number system (only 2 symbols, i.e. 0 and 1), I just thought why did we adopt the decimal number system (10 symbols) after all?
I mean if you go to see, it's rather inefficient when compared to the octal (8 symbols) and the hexadecimal (16 symbols)?
|
While reading about general topology, I started looking at the metric topology. At one point, the interior of a topological space $X$ is defined as such:
$${E}^{\circ}=\{p\in E\ | \ B_p(\epsilon)\subseteq E, \epsilon>0\}$$
Here, $B_p(\epsilon)$ is a ball with center $p$ and radius $\epsilon$. I tried verifying that thi... |
It's a really simple question. However I didn't see it in books and I tried to find the answer on the web but failed.
If I have two independent random variables, $X_1$ and $X_2$, then I define two other random variables $Y_1$ and $Y_2$, where $Y_1$ = $f_1(X_1)$ and $Y_2$ = $f_2(X_2)$.
Intuitively, $Y_1$ and $Y_2$ shoul... |
Consider a cube, with standard topological faces: 6 faces, 12 edges and 8 vertices.
Define a cube part as set of faces (possible of different dimension) so that no 2 set elements share same vertex.
E.g. there is no cube part consisting of 3 2-dimensional faces, because there have to be 2 faces that share a vertex.
Note... |
The Reuleaux polygons are analogs of the regular polygons, except that the "sides" are composed of circle arcs instead of lines. It is known that for an odd number of sides, e.g. the Reuleaux triangle, the polygon has constant width.
After reading the paper Roads and Wheels by Stan Wagon and Leon Hall, I got curious on... |
Four Color Theorem is equivalent to the statement: "Every cubic planar bridgeless graphs is 3-edge colorable". There is computer assisted proof given by Appel and Haken. Dick Lipton in of his beautiful blogs posed the following open problem:
Are there non-computer based proofs of the Four Color Theorem?
Surprisingly,... |
Let $$\nu(x) = \frac{1}{2} \log{2\pi x} - x + \int\limits_{1}^{x} \frac{\lfloor{t\rfloor}}{t} \ dt$$ then what is $\displaystyle \lim_{x \to \infty} x \cdot \nu(x)$?
I really don't know to proceed. Tried something using the expansions of $\log$ and seeing how the integral behaves, but couldn't figure out.
|
Working on the generalized birthday problem, where you draw with replacement from $\{1,2,3, \ldots,d\}$ and look for the number of draws $n$ for which you have greater than $1/2$ chance of a match I said
$\frac{d!}{d^d(d-n)!} \lt \frac{1}{2}$
Using Stirling's approximation
$\frac{d^d \exp (d-n)} {\exp(d) d^n (d-n)^... |
What are the three cube roots of -1?
Not sure if this is a trick question, But I have been asked this. one of the ansers is -1, what are the other 2?
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This is a strange question, it might be easy for all you math wizards or it even may be impossible. If you don't understand what I mean let me know so I can change the way I post the question.
Product A can be built using Product B.
To get Product B I have to transform product C or Product D.
I have a 50% chance to get... |
I'm working to understand some noise on some of our analog-to-digital converter signals. I've analyzed a log of $2^{18}$ samples in MATLAB, and can make the following observations:
the noise consists of large single-sample spikes on top of my expected signal
the FFT spectrum of the noise looks white (hard to say since... |
$$
((n+1)H_n - (n+1)) + H_{n+1} = (n+2)H_{n+1} - (n+2)
$$
I need to prove the above statement.
|
I want to find all the roots of a polynomial and decided to compute the eigenvalues of its companion matrix.
How do I do that?
For example, if I have this polynomial: $4x^3 - 3x^2 + 9x - 1$, I compute the companion matrix:
$$\begin{bmatrix} 0&0&\frac{3}{4} \\ 1&0&-\frac{9}{4} \\ 0&1&\frac{1}{4} \end{bmatrix}$$
Now how ... |
Start with the identity
$\sum_{i=1}^n i^3 = \left( \sum_{i = 1}^n i \right)^2 = \left(\frac{n(n+1)}{2}\right)^2$.
Differentiate the left-most term with respect to $i$ to get
$\frac{d}{di} \sum_{i=1}^n i^3 = 3 \sum_{i = 1}^n i^2$.
Differentiate the right-most term with respect to $n$ to get
$\frac{d}{dn} \left(\frac{n(n... |
Prove that $f_n\to f$ in measure on $E$ if and only if given $\varepsilon>0$, there exists $K$ such that |{$x\in E : |f(x)-f_k(x)|>\varepsilon$}|$<\varepsilon$ for $k\ge K$.
The "only if" direction of this is immediate from the definition of convergence in measure, but the other direction is less obvious to me.
Conve... |
I am wondering if there is a general coordinate-independent way to define a Partial Differential Equation on a Smooth manifold.
It is definitely true that in each coordinate neighborhood you could define a function to satisfy a differential equation, but when you change coordinates then the differential equation will m... |
I'm interested in finding a recursion or simple representation for a "Hadamard product" of two power series.
The Hadamard Product
The Hadamard product is defined on generating functions $f(x)$ and $f(y)$ as $f(z)$ below:
$f(x) = \sum_{i=0}^\infty{c_i x^i}$
$f(y) = \sum_{i=0}^\infty{d_i y^i}$
$f(z) = \sum_{i=0}^\infty... |
I got this problem from one of my mates, and i rearranged them, and got it in a summable form. From here could anyone tell me as to how i can sum up this interesting series.
How does one sum the given series: $$S = 1 + \Bigl(1 + \frac{1}{3}\Bigr) \cdot \frac{1}{5} \cdot \frac{1}{3} + \Bigl(1 + \frac{1}{3} + \frac{1}{5}... |
Let $B \subset [0,2\pi]$ be a Lebesgue measurable set. Prove that:
$\displaystyle \lim_{n \to \infty} \int_{B} \cos(nx) dx = 0$
OK I did this assuming B is an open interval, this is pretty easy using the fact that the sine function is bounded by 1. Now I'm stuck in the general case, I'm somewhat confused by "Lebesgue m... |
Motivation: In odd dimensions, solutions to the wave equation: $$u_{tt}(x,t)=\nabla^2 u(x,t), \qquad u_t(x,0)=0, \qquad u(x,0)=f(x)$$ where $t \geq 0$ and $x \in \mathbb{R}^n$, have the nice property that the value of $u(x,t)$ only depends on the values $f(y)$ with $|y-x|=t$. For even dimensions, the value $u(x,t)$ dep... |
The topological $K$-theory of a complex compact manifold $X$ is the commutative monoid $K(X)$ of isomorphism classes of complex vector bundles. Two classes $[E]$ and $[F]$ are equivalent in $K$-theory if they are stably equivalent:
there exists a trivial bundle $G$ with
$$
(E \oplus G) \cong (F \oplus G).
$$
In what... |
I have a semigroup $G$ (a set with associative binary operation) such that for all $a,b\in G$ there exists $x,y\in G$ such that $ax=ya=b$. Is this property enough to show that $G$ is a group, and if so, how?
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A lot of mathematics as far as I know is interested in the study of Euclidean and locally Euclidean spaces (manifolds).
What is the special feature of Euclidean spaces that makes them interesting?
Is there a field that studies spaces that are neither globally nor locally Euclidean spaces?
If such a field exists, are t... |
Américo Tavares pointed out in his answer to this question that finding the ratio of a geometric progression only from knowledge of the sum of its first $n+1$ terms $S = 1+x+x^2+\cdots+x^n$ amounts to solving a polynomial of degree $n$. This suggested to me that there might be up to $n$ real solutions of $x$ for a give... |
Here is a funny exercise
$$\sin(x - y) \sin(x + y) = (\sin x - \sin y)(\sin x + \sin y).$$
(If you prove it don't publish it here please).
Do you have similar examples?
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I am working in the field of machine learning, and I have come across a few papers that show relationships between Gröbner bases and discrete probability. So I come here for help.
Can you please explain how can Gröbner bases be used to describe discrete probability?
I have looked at Gröbner bases and I understand the g... |
A basketball player throws 100 times. On the first throw the ball is always in, on the second - always out. For all consequent throws the probability of hitting is equal to percentage of all successful throws divided by total number of throws done.
What is the probability the he will score total 50 balls out of 100?
Th... |
I recall seeing a hand-drawn picture of spec of a ring (maybe of $\mathbb Z$?) that had been passed around in the early days of the Zariski topology. Does anyone know where I can find a copy?
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Is solving problems quickly an important trait for a mathematician to have? Is solving textbook/olympiad style problems quickly necessary to succeed in math? To make an analogy, is it better to be a sprinter or a marathon runner in mathematics?
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I have a branching process of the form $p_0=0.1$, $p_1 = 0.6$, $p_2 = 0.3$. (any other $p_n = 0$). $Z_0$, the original population is $1$. $Z_1$ is the population after $1$ timestep, $Z_2$ is the population after $2$ timesteps, etc.
How can I compute the probability mass function of $Z_3$? It would suffice to have an ex... |
Given that $X_i$ is a partition of the probability space, and $Y,Z$ are events in the probability space such that $P[Y \cap X_i] > 0$ for all $i$, how can I prove that $$P[\,Z\;|\;Y\,] = \sum_{i=1}^{\infty} P[\,Z\;|\;Y \cap X_i\,] \cdot P[\,X_i\; |\; Y\,]\quad?$$
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The classic text of Deligne and Rapoport giving a model for the mod p reduction of the modular curve is found in a "Lecture Notes in Mathematics" volume.
Has it been translated from the original french and/or LaTeX-ed so that it can be searched more easily?
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My class had a good time redeveloping the unit circle under the taxicab metric. Now some of them want to do it again with another similar metric. I want to give this to some of my "honors" high-school* calculus I students to work on independently. As such, it can't be too difficult or require too many advanced concepts... |
This is somewhat of a follow-up question to Elliptic integrals with parameter outside $0<m<1$.
I have an equation that I'm attempting to simplify that has terms that look something like this:
$\Re\bigl( -i (c-1) \mathrm K(c) -2i \mathrm E(c) \bigr)$
where $c>1$. I.e., they're being evaluated outside their standard para... |
My number theory library of choice doesn't implement the logarithmic integral for complex values. I thought that I might take a crack at coding it, but I thought I'd ask here first for algorithmic advice and/or references. I'm sure there are better methods than naively calculating the integral.
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I'm a bit curious as to how "real" mathematicians would solve this problem.
"Find the least prime number greater than 2000."
Of course, I can always go brute force:
IF N IS PRIME, OUTPUT N
ELSE INCREASE N BY 1
But that's no fun. (Especially when the number is really high.)
Are there any algorithms/tricks/etc. that can... |
I came across an interesting problem but unable to see how to approach it. How do I use the dominated convergence theorem (LDCT), to show that first derivative of the characteristic function of the probability distribution at $t = 0$, $\phi^′(0)=iE[X]$? Any ideas?
References:
http://en.wikipedia.org/wiki/Characteristic... |
What does this expression mean - $\lim_{n\rightarrow\infty} E|X_n-X|=0$? $X_n$ is a sequence of random variables and $X$ is a random variable. What does this expression imply? Can I say that the sequence $X_n$ converges to $X$ in probability and almost sure convergence?
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In this post, David Speyer, actually, gave an expression for $\displaystyle \frac{t}{e^{t}-1}$.
The question is can we sum the given series, using that expression, if not how does one sum this series. $$\sum\limits_{n=1}^{\infty} \frac{n}{e^{2\pi n}-1}=\frac{1}{e^{2\pi}-1} + \frac{2}{e^{4\pi}-1} + \frac{3}{e^{6\pi}-1}... |
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