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So for a writing assignment in one of my classes we are asked to discuss and prove some basic results about compact sets in general topological spaces. I like proving these things, but they dont help me understand what a compact( locally compact, paracompact,...) set in a topological space "looks like." That said I'm a... |
For example:
the limit of $e^t$ as $t$ approaches infinity is simply infinity.
but the limit of $e^{t^2}$ cannot be found because it explodes? Is this due to the fact that there are differing types of infinity and the second limit produces a non-"countable" infinity?
see referenced question
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Let $f(x) \in L^p(\mathbb{R})$ and $K \in C^m(\mathbb{R})$. Can I then say that $(f \ast K) (x) = \int_{\mathbb{R}} f(t) K(x-t) dt$ is in $C^m$?
I know that this is true if $K$ has compact support, but I was wondering if it is possible to have a stronger result (perhaps $K$ vanishing at $\infty$?).
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I'm attempting to come up with a first order language $L$ that is able to describe vector spaces over fields. I came up with a few sets of nonlogical symbols.
$Rs_L=\{Scal,Vec\}$ where $Scal,Vec$ are unary relation symbols which I interpret to be true or false depending on whether $x$ is a scalar or vector.
Also, $Fs_L... |
It's kind of an infamous problem in differential equations to find the correct road surface so that a car with square wheels (and an axle located in the center) keeps its axle level as it drives along. I hope I won't offend anybody by saying that one smooth piece of the solution (for a wheel with sides of length 2) is... |
Trying to remember my high school formulas, and coming up dry.
Say I have two choices: $A$ and $B$.
$P(A) = 0.25$ ; $P(B) = 0.75$
There are no conditional probabilities or anything. Each choice is independent of the last.
How do I go about calculating the probability that I will choose A exactly twice, having chosen t... |
Is there an expansion for the following summation?
$$ (a_1 + a_2 + \cdots + a_k)^n $$
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Suppose that a rational generating function is given representing the function
$f(x) = \sum_{i=0}^{2n}{c_i x^i}$ where $c_i \in \mathbb{N}$
The goal is to determine if the coefficient in the middle, $c_n$, is nonzero or not.
For example, we could be given $f(x) = \frac{1 - 10x^9 + 9x^{10}}{(x - 1)^2}$, which represents... |
Take any number $x$ (edit: x should be positive, heh)
Add 1 to it $x+1$
Find its reciprocal $1/(x+1)$
Repeat from 2
So, taking $x = 1$ to start:
1
2 (the + 1)
0.5 (the reciprocal)
1.5 (the + 1)
0.666... (the reciprocal)
1.666... (the + 1)
0.6 (the reciprocal)
1.6 (the + 1)
0.625
1.625
0.61584...
1.61584...
0.619047.... |
Thats it! Thats what I want to know.
If θ is the angle subtended by an arc of length L at its center with radius R.
We know, θ = L/R.
How did we get this?
Please don't say we got it from generalizing the idea of perimeter of circle.
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For a probability distribution, its quantile function is defined in terms of its distribution function as
$$ Q(p)=F^{-1}(p) = \inf \{ x\in R : p \le F(x) \} $$
I was wondering if, conversely, a quantile function can uniquely determine a distribution and therefore fully describe the probability distribution just as ... |
I'm trying to show that if $f$ and $1/f$ are in $H^2$ then $f$ is an outer function. I think I somewhat have a proof (if it is correct) but it doesn't feel like it is the easiest one.
We know that if $f$ is in $H^2$ and non-zero and $B$ is the Blanschke product of its zeros, then there exists a constant $K$ of modulus ... |
In $\mathbb{R}^2$ specify a set of vectors, that has many elements as possible, so that no vector is a constant multiple of another vector in the set.
What do such sets have in common?
In $\mathbb{R}^3$ do the same
I asked both of these earlier, but people are acting like it is a HW question (giving me "hints" etc.) b... |
It is well known that the differential calculus has a nice algebraization in terms of the differential rings but what about integral calculus? Of course, one sometimes defines an integral in a differential ring $R$ with a derivation $\partial$ as a projection $\pi: R\rightarrow \tilde R$, where $\tilde R$ is a quotient... |
I have some geometry in 3d (like road surface or terrain surface) and there could be several not connected parts of a road.
So I'd like to know if I may unite these two not connected pieces of road in a term "surface".
And if I couldn't, then what is the best mathematical term for this?
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Is there some place (on the internet or elsewhere) where I can find the number and preferably a list of all (isomorphism classes of) finite connected $T_0$-spaces with, say, 5 points?
In know that a $T_0$-topology on a finite set is equivalent to a partial ordering, and wikipedia tells me that there are, up to isomorph... |
I've seen recently for the first time in Special Functions (by G. Andrews, R. Askey and R. Roy) the definitions of fractional integral
$$(I_{\alpha }f)(x)=\frac{1}{\Gamma (\alpha )}\int_{a}^{x}(x-t)^{\alpha -1}f(t)dt\qquad \text{Re}\alpha >0$$
and fractional derivative
$$\frac{d^{\nu }w^{\mu }}{dw^{\nu }}=\frac{\Gamm... |
Both Fourier transform and Taylor series are means to represent functions in a different form.
What is the connection between these two? Is there a way to get from one to the other (and back again)? Is there an overall, connecting (geometric?) intuition?
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The Wronskian lets us determine if a set of functions (possibly the solutions to a differential equation) are linearly dependent or not. But, for every example in the book, it is very obvious if one of the functions is a linear combination of the others. The examples in the book use 3-5 functions. What would be an exam... |
After some searching I found:
Triangulation
Using theta and "b" to find out the height of a building.
Thanks in advance.
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Perhaps this has a simple answer, but I don't know (I wouldn't be asking if I did). Every set of outer measure zero is Lebesgue measurable with Lebesgue measure zero. Is the converse true? That is, if a set has Lebesgue measure zero, does it necessarily have outer measure zero?
And I suppose I should specify that I'm t... |
So here is the problem:
Solved for a in terms of x:
$$a^{x} = 10^{2x + 1}$$
I tried:
$\displaystyle x \cdot \log(a) = (2x+1) \cdot \log\;10 $
$\displaystyle \frac{x}{2x + 1} = \frac{\log\;10} {\log\;a} $
But this is not going in the right direction, the answer according to the book is:
$$ \frac{1} {\log\;a - 2} $$
Exc... |
Assume I have only basic math knowledge, what specific areas of math would I need to learn in order to understand the following webpage:
Black-Scholes
Many thanks.
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How to show that $$\gcd \Biggl( {m \choose k} , {m+1 \choose k } , {m+2 \choose k}, \cdots, {m+k \choose k} \Biggr)=1$$
where $m,k \in \mathbb{N}$ and $m \geq k $.
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We can easily see that if a sequence of inner functions converges then certainly the limit is still inner (using the fact that a countable union of measure zero sets still has measure zero). This implies that the set of inner functions is closed in $H^2$.
Now, is the set of the singular inner functions closed?
We know ... |
Suppose $f_{X}(x) = xe^{-x^2/2}$ for $x>0$ and $Y = \ln X$. Find the density function for $Y$. So we want to find $P(Y \leq y)$. This is the same thing as $P(\ln X \leq y)$ or $P(X \leq e^{y})$. Thus $f_{Y}(y) = f_{X}(e^y)$? Or does $f_{Y}(y) = F_{X}(e^y)$ since $P(X \leq x) = F_{X}(x)$?
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Let R be a ring with identity. An R-module M is Artinian if it satisfies the descending chain condition on submodules. What is an example of an Artinian module with a proper submodule that is not finitely generated?
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Two players put a dollar in a pot. They decide to throw a pair of dice alternatively. The first one who throws a total of $ 5$ on both dice wins the pot. How much should the player who starts add to the pot to make this a fair game?
So my interpretation of this problem is that I first throw a dice and then the other p... |
Maple tells me that $\lim_{n \to \infty} n^2 \log^n(1 - \frac{c \log n}{n}) = 0$ for any constant $c$, but I can't find a way to prove it. Any suggestions?
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I know that in general, $H_{n}(X)$ counts the number of $n$-cycles that are not $n$-boundaries of a simplicial complex $X$. So for the sphere, $H_{0}(X) \cong \mathbb{Z}$ since it is connected. Also $H_{n}(X) = 0$ for all $n>0$ (e.g. all $1$-cycles are $1$-boundaries, etc..). How do you use this geometric interpretatio... |
Each person from a group of 3 people can choose his dish from a menu of 5 options. Knowing that each person eats only 1 dish what are the number of different orders the waiter can ask the chef?
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I would like to solve:
$$
\frac{d^2y}{dx^2} - \frac{2}{y^2}=0$$
with $y(0)=a$ and $y'(0)=0$
Where $a$ is a known constant.
Thanks in advance.
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i couldn't do the following question for hours
minimize $\sum_{i=1}^{n}x_{i}^{3}$
s.t. $\sum_{i=1}^{n}x_{i}=0$
and $\sum_{i=1}^{n}x_{i}^{2}=n$.
by Lagrange multiplier rule
?
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Reading "Random number generation and Monte Carlo methods" by James E. Gentle, I encountered an example concerning the Bays-Durham Shuffling algorithm whose result I can't reproduce.
Basically, such an example is written in Random Number Generation - Durham Shuffling of Uniform Deviates. For the full text you can go to... |
What is a good introduction in gradient flows in metric spaces? I know the book Gradient flows: in metric spaces and in the space of probability measures by Luigi Ambrosio, Nicola Gigli and Giuseppe Savaré, but is too hard for an introduction.
Can someone make this a community wiki? I cannot find how you have to do tha... |
So, I thought of something nerdy...I was making an algebra problem to disguise it so that to see the joke, you'd have to a bit of work, but I think that I might be doing something wrong, because neither Wolfram|Alpha nor MyAlgebra can solve it...I fixed a few things that I had wrong initially, but I don't see anything ... |
I believe I'm right in saying that an isometry of a convex figure will be convex. There are other automorphisms that have this property, for example "stretching along one axis" (e.g. $f(x,y) = (2x,y)$) won't transform a convex shape into a non-convex one or vice versa.
What other maps have this property?
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Determine whether this series converges or diverges: $$\sum\limits_{n=0}^{\infty} \frac{1}{(2n+1)!}$$
Thought about using the limit theorem or by comparison but am so stuck. any pointers would be appreciated guys
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Given the coordinates of two rectangles on a coordinate plane, what would be the easiest way to find the coordinates of the intersecting rectangle of the two?
I am trying to do this programatically.
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Let $I$ be a finitely generated ideal of a commutative ring $R$. Assume every element of $I$ is a zero divisor. Does then exist a $x \neq 0$ in $R$ with $xI=0$?
This is true if $0$ is a decomposable ideal, for example if $R$ is noetherian. I wonder if we actually need this. Doesn't it sound plausible? The problem is th... |
This question and the described solution are copied from a test-paper :
For the equation $x^2$ + |x| - 6 = 0 analyze the four statements below for correctness.
there is only one root
sum of the roots is + 1
sum of the roots is zero
the product of the roots is +4
Answer : (3)
Answer Explanation :
If x > 0 |x| = x.
Gi... |
Let $C$ be a convex subset of $\mathbb{R}^n$. I've been trying for hours to prove that $\dot{\overline{C}}=\dot{C}$. Somehow my intuition completely fails me. I found a proof in a textbook, but just got stuck on another statement the author considered obvious. Could someone please give a proof that uses little more tha... |
Firstly, you guys are awesome, and I learn quite a bit just from reading the questions of others.
Secondly, a friend asked me recently why large primes are important for data security, and I was unable to give him an answer with which I myself was satisfied. Various wikipedia articles have mostly pointed out an embarr... |
Page 13 of Tropical Algebraic Geometry by Itenberg, Mikhalkin, and Shustin mentions 1-valent vertices, but I haven't been able to find a source that actually defines this term or managed to guess the definition myself. Either a definition or reference would be greatly appreciated. Thanks!
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How to solve $$\int_0^{\pi} \sin{2x}\sin{x} dx$$
Edit: Sorry! I should have described more. This is not a homework. Recently, Out of the blue I got interest in physics and started reading and solving problems. This is part of a physics problem where I got stuck (because I forgot all high school formulae.). Thanks all o... |
Question 1: Does such a ring exist?
Note: The definition of a noetherian topological space is similar to that in rings or sets. Every descending chain of closed subsets stops after a finite number of steps
(Question 2: is this equivalent to saying that every descending chain of opens stops?)
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Suppose there are 9 events, that have a probability of 10%, 20%, 30%, ..., 90% of being a success.
How would I find the probability of exactly n number of these events succeeding?
For n = 1, I'm thinking it is something like
(first succeeds) -> (1-0.9) * 0.8 * 0.7 * 0.6 ... 0.1
(second succeeds) -> 0.9 * (1-0.8) * 0.7 ... |
A quantile function Q is defined in terms of its distribution function F as:
$Q(p) = \inf \{ x\in R : p \le F(x) \}, p \in (0,1).$
I was wondering:
(1) if the definition of Q can be extended to $[0,1]$ by letting $ x \in R \cup \{\infty, -\infty \}$? if not, why?
(2) if it is correct that a distribution function can ... |
I was playing around with Sierpinski carpets (see pretty GPU-produced picture here), and came up with a variation that I have been unable to find mentioned elsewhere. I'm wondering if anyone can tell me an existing name for it, and help me determine where it fits into existing classifications of fractals.
As you probab... |
I am currently learning about the concept of convolution between two functions in my university course. The course notes are vague about what convolution is, so I was wondering if anyone could give me a good explanation. I can't seem to grasp other than the fact that it is just a particular integral of two functions. ... |
I'm trying to solve this non-linear differential equation using substitution $\dfrac{y}{x}$ to $t$. However, I can't solve this equation.
$$
\text{xy$\prime $} = \left(\frac{y}{x}\right)^3+y
$$
How to solve this equation and what's the general solution?
Thanks.
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Choose numbers from $1$ to $2n$ uniformly at random. How many numbers must be chosen, on average, before at least $n$ numbers have been picked?
This is similar to the coupon-collector problem, but looking for only partial completion.
Note: Choosing an appropriate meaning of 'random' is part of the question.
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I have a probability space $\mathcal{X} \times \mathcal{Y}$. $\mathcal{X}$ and $\mathcal{Y}$ could be infinite, but they are at worst case countable.
There is a matrix $A_{xy}$ ranging over $\mathcal{X}$ for rows $\mathcal{Y}$ for columns, which is either 0 or 1 for each $x,y$.
We have the information that
$P(${$ (x,y... |
The pdf of the geomtric distribution is the following: $f(x) = (1-p)^{x}p$. Also $E[X] = \frac{1-p}{p}$ and $\text{Var}[X] = \frac{1-p}{p^2}$. The pdf of the negative binomial distribution is: $f(x) = \binom{r+x-1}{x}p^{r}(1-p)^{x}$. Also $E[X] = \frac{r(1-p)}{p}$ and $\text{Var}[X] = \frac{r(1-p)}{p^2}$. So is the geo... |
In the past couple years, I've read many words pertaining to D-branes without feeling I have really comprehended them. In broad terms, I think I get what they're about: They're supposed to serve as habitats for the ends of open strings and can be viewed as submanifolds (of the target manifold in a sigma model), possibl... |
I'm having trouble finding a good explanation of the Lebesgue integral. As per the definition, it is the expectation of a random variable. Then how does it model the area under the curve? Let's take for example a function $f(x) = x^2$. How do we find the integral of $f(x)$ under $[0,1]$ using the Lebesgue integral?
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Prove $(A∩B)’=A’∪B’$.
Let $x ∈ (A ∩ B)’$
$∴ (x ∈ A ∩ B)’$
$∴ (x ∈ A ∧ x ∈ B)’$
$∴ (x ∈ A)’ ∨ (x ∈ B)’$
$∴ x ∈ A’ ∨ x ∈ B’$
$∴ x ∈ A’ ∪ B’$
$∴ (A ∩ B)’ ⊆ A’ ∪ B’$
The above is the solution provided. Sorry if it seems trivial but I don't understand how
$∴ (x ∈ A ∧ x ∈ B)’$
leads to
$∴ (x ∈ A)’ ∨ (x ∈ B)’$
and not
$∴ (x ∈... |
For $x>0$ and $y>x+1$, how do we prove that $$\sum\limits_{n=1}^{\infty} \frac{x(x+1) \cdots (x+n-1)}{y(y+1) \cdots (y+n-1)} = \frac{x}{y-x-1}$$
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In this answer on meta, Pete L. Clark said:
I think the question concerns the idea that a differentiable curve becomes more and more like a straight line segment the closer one zooms in on its graph. (And I must say that I regard part of this confusion as an artifact of badly written recent calculus books who describe... |
Given that a population (eg. of bacteria) grows exponentially with the function
$$p 2^{\frac{t}{10}}$$
where $p$ is the initial population and $t$ is the time in minutes. Given that the population consumes (eg. food) at a steady rate, how would you calculate how much the population consumes?
Assume that each bacterium ... |
A cohort in a school consists of 75 students who study for 6 years. Each year, the students are randomly distributed into 3 classrooms of 25 students each. What is the probability that, after 6 years, each student has at some point been in a classroom with every other student?
More generally: Starting with an edgele... |
I'm trying to remember a particular theorem of ZF but unfortunately my memory is quite incomplete.
The theorem is of the form (some set operation) is either (expected answer) or the empty set.
If the Axiom of Choice holds, (some set operation) is (expected answer).
I believe that if (some set operation) is (expected a... |
I have a polygon P, that may or may not be convex. Is there an algorithm that will enable me to find the collection of points A that are at a distance less than d from P?
Is A in turn always a polygon?
Does the solution change materially if we try to solve the problem on the surface of a sphere instead of on a Euclidea... |
The question what-is-the-optimum-angle-of-projection-when-throwing-a-stone-off-a-cliff was asked and answered a while back. This one has a much cleaner answer. Now you are on a uniform slope, a line through the origin that is not horizontal and you want to throw a stone as far as possible. As a function of the angle... |
List all the elements of the alternating group $A_3$ written in cyclic notation.
I come up with
Identity $(1)$
Obviously $(123)$
|
Denoted as $\zeta(s,a)$ for a > 0
Where do I find topics on the Hurwitz zeta function for a < 0?
Any links or resources would be appreciated.
(Please dont mention wiki or mathworld)
Thanks
|
Consider the group of permutation matrices $G =\{I_3, P_1, P_2, P_3, P_4, P_5\}$
For $n=3$ the permutation matrices are $I_3$ and the five matrices are:
\begin{equation*}
P_1 = [1,0,0;0,0,1;0,1,0] \\
P_2 = [0,1,0;1,0,0;0,0,1] \\
P_3 = [0,1,0;0,0,1;1,0,0] \\
P_4 = [0,0,1;0,1,0;1,0,0] \\
P_5 = [0,0,1;1,0,0;0,1,0]
\end{eq... |
I have a homework question "Show the following is true using theorems. State which theorem you use at each step." This is just one of many problems I have! So, if you can help me with this one problem than I can apply what I learn to finish the rest. I don't want a handout; I just don't know where to start or what to d... |
For all $a, m, n \in \mathbb{Z}^+$,
$$\gcd(a^n - 1, a^m - 1) = a^{\gcd(n, m)} - 1$$
|
Let $\mathbb{T}^n = { (z_1,\ldots,z_n) \in \mathbb{C}^n : |z_l| = 1, \; 1 \leq l \leq n }$ denote the $n$-torus, and let $t_1, \ldots, t_n$ be arbitrary real numbers. Then it can be shown that the topological closure $H$ in $\mathbb{T}^n$ of the one-parameter subgroup
$$H' = {(e^{2\pi i t_1 y}, \ldots, e^{2\pi i t_n y}... |
Find an isomorphism from the octic group $G$ to the group $G'$:
$G = \{e, s, s^2, s^3, b, g, d, t\}$
$$e = (1)\quad s = (1234)\quad s^2= (13)(24)\quad s^3= (1432)\quad b = (14)(23)\quad g = (24)\quad d = (12)(34)\quad t = (13)$$
$G' = \{I_2, R, R^2, R^3, H, D, V, T\}$
$$\begin{align*}I_2&=\begin{pmatrix}1 & 0\\... |
The question is:
Give an example of a function $f$, continuous on $S=(0,1) \times (0,1)$, such that $f(S)=\mathbb{R}^2$.
I'm getting stuck on $f(x)=\tan\left(\pi \left(x-\frac{1}{2}\right)\right)$,
which covers all of $\mathbb{R}$ but not all of $\mathbb{R}^2$.
|
I am trying to make a truth table from an SOP boolean algebra expression. I understand AND, OR, NOT truth tables. I just don't understand these types of tables and their outputs.
This is the expression:
$$A'BD' + BCD + ABC' + AB'D = A'BD' + BCD + ABC' + AB'D + BC'D' + A'BC + ABD.$$
I can use either side whichever is e... |
If $X\sim \Gamma(a,\sigma_x^2)$ and $Y\sim \Gamma(b,\sigma_y^2)$. What will be the probability density function of R? Where $R=\frac{X+C}{X+Y}$, here $C$ is a positive constant, $\Gamma(.,.)$ denotes standard gamma probability density function and '$\sim$' represents 'distributed as'. X and Y are independent random var... |
Let $S \subset L^2(\mathbb{R})$, and let us define $S^{\perp} =\{f \in L^2(\mathbb R) | \left< f,g\right> =0, g \in S\}$. Show that $S^{\perp}$ is a vector subspace of $L^2$.
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Can you solve Problem 19 from Chapter 8 of Rudin's Principles of Mathematical Analysis, I'm having a lot of difficulty with it
I've proven the first part, namely
$$\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^N \exp(ik(x+n\alpha))=\frac{1}{2\pi}\int_{-\pi}^\pi(\cdots) = \begin{cases} 1\text{ if }k=0\\0\text{ otherwise}\end... |
Compute the Fourier transform of $f(t) = \sin t$. Does that converge?
|
Hi I have just found the proof that 90 equals 95 and was wondering if I have made some mistake. If so, which step in my proof is not true?
Definitions:
1. $\angle ABC=90^{\circ}$
2. $\angle BCD=95^{\circ}$
3. $|AB|=|CD|$
4. $M:=$ the center of $BC$
5. $N:=$ the center of $AD$
6. $l:=$ a line perpendicular to $BC$... |
I have a question pertaining to linear separability with hyperplanes in a support vector machine.
According to Wikipedia:
...formally, a support vector machine
constructs a hyperplane or set of
hyperplanes in a high or infinite
dimensional space, which can be used
for classification, regression or
other task... |
Let $A$ be an abelian category. There are various types of full subcategories. I often wonder if it is assumed that these are nonempty, since in most proofs this is used implicitely, but also the empty case then works as a special case. To be sure, I ask you:
$S \subseteq A$ is called thick iff $S$ is closed under sub... |
Consider (non-trivial) functions that are their own nth derivatives. For instance
$\frac{\mathrm{d}}{\mathrm{d}x} e^x = e^x$
$\frac{\mathrm{d}^2}{\mathrm{d}x^2} e^{-x} = e^{-x}$
$\frac{\mathrm{d}^3}{\mathrm{d}x^3} e^{\frac{-x}{2}}\sin(\frac{\sqrt{3}x}{2}) = e^{\frac{-x}{2}}\sin(\frac{\sqrt{3}x}{2})$
$\frac{\mathrm{d}^... |
I am a EE grad. student who has had one undergraduate course in real analysis (which was pretty much the only pure math course that I have ever done). I would like to do a self study of some basic functional analysis so that I can be better prepared to take a graduate course in that material in my university. I plan to... |
UPDATED: 1/15/2014
I originally wrote this post in 2010, when I was looking for alternative ways to store and transport papers. I had my laptop, but due to its weight, limited battery life, and the LCD screen, an e-reader such as Kindle seemed like a good idea at the time. (Also, at the time of the original post, I had... |
For example, I have heard of a topological one wherein negation means the interior of the complement (but still would like a reference).
|
I am a graduate student of math right now but I was not able to get a topology subject in my undergrad... I just would like to know if you guys know the best one..
|
I am working with $G_{n,p}$ random graphs. Let $p$ (the probability of an edge) be $c/n$, so $c$ is the expected number of neighbors for a vertex when $n$ is large.
Now let $N$ be the number of components with more edges than vertices in the graph. I am trying to find the largest $c$ such that $\mathbb{E}(N)$ tends to ... |
I have attempting to solve this using the infinite ramsey theorem, with colouring based on whether the sum of two vertices has an even or odd number of distinct prime factors.
This is leading to an infinite recursion.
Is this ok? At the end of all time I will be done.
|
Given a continuous function $f$ of period $2\pi$ and $\alpha$ in $\mathbb{R}$
can you show for any real $x$ that $\lim_{ N\to \infty} \sum_{n=1}^N f(x+n\alpha) = \lim_{ N\to\infty} \sum_{n=1}^N f(x_n)$
that is, the 'average value' of $f$ diluted by $N$ is the same as the average value of the sequence $x+n\alpha$ sinc... |
Actually, I posted this long ago in MO but did not get a reply as it was unfit.
Now this is an exercise in some textbook (I think Apostol), and I would be happy to receive some answers.
Let $P(n)$ be the product of positive integers which are $\leq n$ and relatively prime to $n$. Prove that $$ \displaystyle P(n) = n^... |
Suppose one is given an arbitrary moment generating function $M_{X}(t)$. How would you determine $P(X=k)$ from this? We know that $M_{X}(t) = E[e^{tX}]$ and $M_{X}(0) = 1$.
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Suppose $E$ is a set of edges on graph $G$ over $n$ vertices, $\mathbf{x} \in \{-1,1\}^n$. The following seems to hold if G is not Eulerian, can anyone see a proof?
$$\sum_\mathbf{x} \prod_{ij\in E} x_i x_j=0$$
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first time poster.
I'm trying to determine the length of a repeating sequence, with noise.
I.e. given the sequence 4, 12, 3, 4, 12, 3, 4, 12, 3, ... the period is apparently 3.
Now, what if the sequence was more complicated. What if each value had added noise?
I was thinking of something with autocorrelation, or Fouri... |
Given a large univariate polynomial, say of degree 200 or more, is there a procedural way of finding all the complex roots? By "roots", I mean complex decimal approximations to the roots, though the multiplicity of the root is important. I have access to MAPLE and the closest function I've seen is:
with(RootFinding):
A... |
Let’s define the codicil of the Geometric Mean – Arithmetic Mean Inequality to be the statement that if the means are equal, then all the terms are equal. Then: I conjecture that most of the GM-AM Inequality is actually in its codicil, in the sense that if you can use the codicil (that is, assume the codicil as part of... |
Is there an algorithm that will allow me to numerically compute the limit of a function f(x) in a principled way?
The most naive algorithm would be to continue to compute the function for larger values of x. The first problem is how to figure out the 'large' values for x to compute the function for. How do I know when... |
It's been a while since my last statistics class...
I have $404$ files that went through some automated generation process. I would like to manually verify some of them to make sure that their data is indeed correct. I want to use probability to help me out so that I don't need to check every single file.
How would I c... |
Is there a way to determine the probability of winning a particular sweepstakes with only the following information:
the estimated odds of winning are $1$ in $13,000$
contestants must be owners (property holders) within the corporation running the sweepstakes and the total number of owners is $222,600$
each owner can ... |
Are there known sharp upper bounds for sums of the form $\sum_{p \mid k} \frac{1}{p+1}$ for $k > 1$ subject to the constraint $\sum_{p \mid k} \frac{1}{p+1} < 1$? (The factor of +1 in the denominator and the constraint of the sum of inverse "shifted prime divisors" is bounded by 1 are both necessary.)
A related questio... |
Given a smooth vector field on a closed smooth manifold, does the flow of that vector field exists for all time $t \in R$? This may be a well-known and elementary conclusion, but I just wnat to confirm it to me.
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I'm looking for an example of two metrics that induce the same topology, but so that one metric is complete and the other is not (Since it is known that completeness isn't a topological invariant).
Thanks in advance for any hints or ideas.
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