instruction stringlengths 12 30k |
|---|
If x is a p-value, what do we call x*(1-x)? |
It depends on your population and the confidence that you need in your results. Typically, you should know something about what you are trying to achieve. If you know your confidence level, confidence interval, and population size, you can choose a sample size that will give you the properly confidence.
Your confide... |
I know that many types graphs have unique names, ie. a directed graph where each node has exactly one outbound edge is known as a `functional` graph. Do the graphs with exactly 2 inbound/outbound edges have a name?
**Bonus**: If the 2 inbound/outbound graph is named, is the nth inbound/outbound graph named? |
Here is a standard lemma you could use. If $M$ is a closed proper subspace of a normed linear space $E$, then for all $\epsilon\gt0$ there is an $x\in E$ of norm 1 whose distance to $M$ is greater than $1-\epsilon$. (E.g., [here's a proof][1] in Lemma 3-6.10 of Tsoy-Wo Ma's *Classical analysis on normed spaces*.)
H... |
Here is a standard lemma you could use. If $M$ is a closed proper subspace of a normed linear space $E$, then for all $\epsilon\gt0$ there is an $x\in E$ of norm 1 whose distance to $M$ is greater than $1-\epsilon$. (E.g., [here's a proof][1] in Lemma 3-6.10 of Tsoy-Wo Ma's *Classical analysis on normed spaces*.)
H... |
The statement is called the [isoperimetric inequality][1]. [Here][2] is a proof that relies on the divergence theorem on the plane and Cauchy-Schwarz and Wirtinger's inequalities.
[1]: http://en.wikipedia.org/wiki/Isoperimetric_inequality
[2]: http://books.google.es/books?id=JTjHeTnZGD8C&pg=PA4&dq=%22isoperim... |
If we know the rational homology of X is 0, can we get some information about the rational homology of X/G, where G is a finite group? Thank you very much for the answers! |
I've somewhat got this question down but I'm only half way.
Q: How many 7-digit even numbers less than 3 000 000 can be formed using all the digits 1,2,2,3,5,5,6?
So I figured that there's about 4 possible approaches
1 _ _ _ _ _ 6
1 _ _ _ _ _ 2
2 _ _ _ _ _ 6
2 _ _ _ _ _ 2
How do I fill in the middle? I trie... |
Here's the setup: Let $U,V$ open $\subset \mathbb{R}^n$, and let $u:V\rightarrow \mathbb{R}$ be harmonic, and $v:U\rightarrow V$ be conformal, i.e. $v$ is $C^1$ and the Jacobian $J_v(x)$ is a scalar multiple of an orthogonal transformation for all [; x\in U ;].
I'm trying to prove $u\circ v$ is harmonic. [I've see... |
I came across the two congruences that for $p$ an odd prime,
$$1^2\cdot 3^2\cdot 5^2\cdots (p-2)^2\equiv (-1)^{(p+1)/(2)}(mod\ p)$$
and
$$2^2\cdot 4^2\cdot 6^2\cdots (p-1)^2\equiv (-1)^{(p+1)/(2)}(mod\ p)$$
What is the reason for this? It seems to me to be closely related to Wilson's Theorem, and possibl... |
Why is the square of all odds less than an odd prime $p$ congruent to $(-1)^{(p+1)/(2)}(mod\ p)$? |
I forget for a while, we don't need the compactness condition here right? |
Does noncompact manifold or orbifold have the homotopy type,of CW complex? |
If I want to find the number of all possible groupings of a set of numbers, including all groupings with only a single number and groupings will all numbers is there a better method than this. For this use I don't want to count empty groups and order is important.
(err... can't get math symbols working so I'll just ... |
Is there an alternative method for this? |
Let $M$ be a 2n dimensional manifold, and let $A$ be a n-dimensional cycle on M. I want to compute the self-intersection $(A.A)$ of A with itself. Let $\eta_A$ be the form in $H^n(M, \mathbb{R})$ given by the Poincare duality isomorphism, i.e. $\int_B \eta_A=(A.B)$ for an n-cycle B. Then by the rule that intersection o... |
I have a terrain, which is represented by a lot of [meshes][1] as shown below:
![alt text][2]
This terrain will be cut by a plane at a certain level. So there are volumes of the terrains that are located above the plane ( cut volume), and volumes that are located below the plane ( fill volume).
The question... |
I'll add nothing to alext87 answer, or J.M. comments. Just "display" them. :-)
Remeber that you can compute the inverse of a matrix by reducing it to [row echelon form][1] and solving the simultaneous systems of linear equations $ (A \vert I)$, where $A$ is the matrix you want to invert and $I$ the unit matrix. ... |
You could find the proof in [this book][1]: <i>Proofs from the Book</i> by Martin Aigner, Günter M. Ziegler, Karl H. Hofmann, pages 40-41
[1]: http://books.google.com/books?id=CuTpgZaLahkC&pg=PA40 |
Let $p : E \rightarrow B$ be a cover s.t. Aut(E) acts transitively on $p^{-1}(b)$ for some fix $b \in B$. Then $E/Aut(E)$ is homeomorphic to $B$ where $Aut(E) \subset Cov(E,E)$. The problem is that I don't even understand how the map E/Aut(E) -> B is defined, would someone please give me this definition? Apparently it ... |
Suppose I have an number of distances from an unknown location to a known location. I can use these distances and the known locations to draw a number of circles. The point where all the circles intersect is my unknown location. This is easy to solve. However, if my distance measurements have some error, I will get a c... |
I am obsessed with [the nine point circle][1]. I was thinking, is there a generalisation to aribtrary tetrahedra and spheres? What about higher dimensions? For each face of the tetrahedron, there is a nine point circle. Do these circles all lie on a sphere?
[1]: http://en.wikipedia.org/wiki/Nine-point_circle |
Does the nine point circle generalise to some theorem about n-spheres and n-simplices? |
If I want to find the number of all possible groupings of a set of numbers, including all groupings with only a single number and groupings will all numbers is there a better method than this. For this use I don't want to count empty groups and order is important.
(err... can't get math symbols working so I'll just ... |
Let $S$ be a (n-1)-submanifold of a n-manifold $M$ and that be a submanifold of (n+1)-manifold V. (All of the manifolds are assumed orientable) Let $V$ carry a metric of signature $(1,n)$. Using terminology as in Einstein's Theory let $n$ be a future directed unit normal to $M$ and $m$ be an unit normal to $S$ pointing... |
Let $S$ be a (n-1)-submanifold of a n-manifold $M$ and that be a submanifold of (n+1)-manifold V. (All of the manifolds are assumed orientable) Let $V$ carry a metric of signature $(1,n)$. Using terminology as in Einstein's Theory let $n$ be a future directed unit normal to $M$ and $m$ be an unit normal to $S$ pointing... |
Let $C$ be a small pre-additive category. Let $R(C)$ denote its category ring, that is,
$$
R(C)=\bigoplus_{a,b\in \mathrm{Ob}(C)} C(a,b)
$$
as Abelian group, where the direct sum runs over all object $a$, $b$ of $C$. The multiplication in $R(C)$ is given by composition of composable morphisms and 0 for uncomposable... |
Can we turn the functor "category ring" into a 2-functor in a natural way? |
Let $C$ be a small pre-additive category. Let $R(C)$ denote its category ring, that is,
$$
R(C)=\bigoplus_{a,b\in \mathrm{Ob}(C)} C(a,b)
$$
as Abelian group, where the direct sum runs over all object $a$, $b$ of $C$. The multiplication in $R(C)$ is given by composition of composable morphisms and 0 for uncomposable... |
Ok this is a bit of an unanswerable question, but hopefully someone will answer. As I have been going through college & high school there has been a kind of "path" through which you learn math. For example I went through basic math, prealgebra, algebra, precalc, calculus 1-3. My question is where to go after calculus t... |
Where to go after calculus? |
A polynomial continued fraction (p.c.f.) is a continued fraction (c.f.)
$b_{0}+\underset{j=1}{\overset{\infty }{\mathbb{K}}}\left( a_{j}|b_{j}\right) =b_{0}+\dfrac{a_{1}}{b_{1}+\dfrac{a_{2}}{b_{2}+\dfrac{a_{3}}{b_{3}+\cdots }}}\qquad \left( 0\right) $
whose elements $a_{n},b_{n}$ are polynomial sequences [D. Bowm... |
Let $A \subset \mathbb{C} $ be an open set containing the closed unit disc.
Let $f$ be an analytic function from $A$ to $\mathbb{C}$ s.t. $|f(z)|=1$ if |z|=1.
Does it follow that $f(z) = a z^{n} \frac{z-b}{1-z\bar{b}} $ for some $a,b \in \mathbb{C}$
s.t. $|a|=1$, $|b|<1$ and some $n\ge 0 $? |
In real analysis, we can define a line integral, but we also define (earlier) the regular definite integral.
Why is it that in complex analysis we are interested only in a line integral? |
Why is an integral of a complex function defined as a line integral? |
Let's assume X,Y are arbitrary sets, and f is a function between them. What must those sets, and f, satisfy, in order for us to be able to define an integral of f in a way that "makes sense"? |
$f:X \rightarrow Y$. What are the conditions on $f$,$X$,$Y$ in order for an integral of $f$ to be defined? |
If I want to find the number of all possible groupings of a set of numbers, including all groupings with only a single number and groupings will all numbers is there a better method than this. For this use I don't want to count empty groups and order is important.
Take the summation of the number of permutations wit... |
Given a collection of items, how many ordered subsets can I make? |
Let $A \subset \mathbb{C} $ be an open set containing the closed unit disc.
Let $f$ be an analytic function from $A$ to $\mathbb{C}$ s.t. $|f(z)|=1$ if |z|=1.
Does it follow that $f(z) = a z^{n} \frac{cz^{m}-b}{1-cz^{m}\bar{b}} $ for some $a,b,c \in \mathbb{C}$ s.t. $|c|=1$ $|a|=1$, $|b|<1$ and some $n,m\ge 0 $? |
Let $A \subset \mathbb{C} $ be an open set containing the closed unit disc.
Let $f$ be an analytic function from $A$ to $\mathbb{C}$ such that $|f(z)|=1$ if $|z|=1$.
Does it follow that $f(z) = a z^{n} \frac{cz^{m}-b}{1-cz^{m}\bar{b}} $ for some $a,b,c \in \mathbb{C}$ s.t. $|c|=1$ $|a|=1$, $|b|<1$ and some $n,m\ge ... |
I've read through the elementary proof of why there exists a solution $x$ to $x^2\equiv -1\pmod p$ iff $p\equiv 1 \pmod 4$ for $p$ an odd prime. Is there a group theory generalization for this fact as well? |
First of all, the question is not "what conditions must $X$ and $Y$ satisfy" but "what *structures* must $X$ and $Y$ have?" Second of all, there is not one most general definition of the integral; depending on what you consider an "integral," there are several definitions which generalize in different directions.
P... |
Here's the setup: Let $U,V$ open $\subset \mathbb{R}^n$, and let $u:V\rightarrow \mathbb{R}$ be harmonic, and $v:U\rightarrow V$ be conformal, i.e. $v$ is $C^1$ and the Jacobian $J_v(x)$ is a scalar multiple of an orthogonal transformation for all [; x\in U ;].
I'm trying to prove $u\circ v$ is harmonic. [I've see... |
Let $T$ be the operator from $L^2(\mathbb R^n)$ to $L^2(\mathbb R^n)$ that is given by $Tf := f * g$ where $g$ is in $L^2$.
How do I now find that the spectrum of $T$ is equal to the essential range of $\hat{g}$? How is the spectrum of $T$ related to the invertibility of the operator $G\hat{f} = \hat{f}\hat{g}$?
... |
Define a binary step-order set as a finite set S of positive integers s<sub>i</sub>, i = 0 to n-1, where s<sub>i</sub> ≡ 2<sup>i</sup> (mod 2<sup>i+1</sup>) for each i from 0 to n-1. So, i is the power of 2 appearing in the prime factorization of s<sub>i</sub>. It’s elementary that binary step-order sets have distinct ... |
What is the least upper bound on the number of pairs of disjoint subsets of a binary step-order set S whose sums differ by 1? |
In the book Introduction to Linear Optimization by Bertsimas Dimitri, a polyhedron is defined as a set $ \lbrace x \in \mathbb{R^n} | Ax \geq b \rbrace $, where A is an m x n matrix and b is a vector in $\mathbb{R^m}$. What it means is that a polyhedron is the intersection of several halfspaces.
A ball can also be v... |
Is a ball a polyhedron? |
In many elementary (and not-so-elementary) Euclidean geometry texts, a polygon is said to be <em>tangential</em> if it is convex and has an inscribed circle (i.e., a circle that intersects and is tangent to each side of the polygon). The assumption of convexity is not needed: I've come up with a rather laborious... |
Proof that every polygon with an inscribed circle is convex? |
In Stefan Rolewicz book [Metric Linear Spaces][1] there are different kinds of general and useful results regarding integration on "more abstract spaces".
For example $f:[0,1]\to X$ where $X$ is an [$F$-space][2], he discuss the Riemann-integral and if, in addition, $X$ is [locally convex][3] then the Bochner-Lebe... |
First of all, the question is not "what conditions must $X$ and $Y$ satisfy" but "what *structures* must $X$ and $Y$ have?" Second of all, there is not one most general definition of the integral; depending on what you consider an "integral," there are several definitions which generalize in different directions.
P... |
In the book Introduction to Linear Optimization by Bertsimas Dimitri, a polyhedron is defined as a set $ \lbrace x \in \mathbb{R^n} | Ax \geq b \rbrace $, where A is an m x n matrix and b is a vector in $\mathbb{R^m}$. What it means is that a polyhedron is the intersection of several halfspaces.
A ball can also be v... |
In many elementary (and not-so-elementary) Euclidean geometry texts, a (simple) polygon is said to be <em>tangential</em> if it is convex and has an inscribed circle (i.e., a circle that intersects and is tangent to each side of the polygon). The assumption of convexity is not needed: I've come up with a rather ... |
In the book Introduction to Linear Optimization by Bertsimas Dimitri, a polyhedron is defined as a set $ \lbrace x \in \mathbb{R^n} | Ax \geq b \rbrace $, where A is an m x n matrix and b is a vector in $\mathbb{R^m}$. What it means is that a polyhedron is the intersection of several halfspaces.
A ball can also be v... |
Why is saying $\frac{dy}{dx}\frac{dy}{dx}=y\frac{d^2y}{(dx)^2 }$ not valid? Does Leibniz notation (and thinking of it as an infinitesimal quotient) not work for higher-order derivatives? |
Can Leibniz Notation Be Treated As a Quotient? |
It is not quite intuitive , at least from its origin. Could any one can give me an intuitive explanation?Thank you! |
Say we have a undirected, connected graph that represents a network of streets. How would you prove that there always exists a tour of the network, where one ``drives'' in both directions on every street exactly once? |
A polynomial continued fraction (p.c.f.) is a continued fraction (c.f.)
$b_{0}+\underset{j=1}{\overset{\infty }{\mathbb{K}}}\left( a_{j}|b_{j}\right) =b_{0}+\dfrac{a_{1}}{b_{1}+\dfrac{a_{2}}{b_{2}+\dfrac{a_{3}}{b_{3}+\cdots }}}\qquad \left( 0\right) $
whose elements $a_{n},b_{n}$ are polynomial sequences [D. Bowm... |
The following seems to hold in numerical simulations, is it true?
$$\lim_{n\to \infty} \int_0^1 dx \frac{n! 2^{-n} n}{(n x)!(n-n x)!\sqrt{x(1-x)}}=2$$
It's a combination of two known integrals
$$\lim_{n\to \infty} \int_0^1 dx \frac{n! 2^{-n} n}{(n x)!(n-n x)!\}=1$$
$$\int_0^1 dx \frac{1}{\sqrt{x(1-x)}}=\pi$$
... |
The following seems to hold in numerical simulations, is it true?
$$\lim_{n\to \infty} \int_0^1 dx \frac{n! 2^{-n} n}{(n x)!(n-n x)!\sqrt{x(1-x)}}=2$$
It's a combination of two known integrals
$$\lim_{n\to \infty} \int_0^1 dx \frac{n! 2^{-n} n}{(n x)!(n-n x)!\}=1$$
$$\int_0^1 dx \frac{1}{\sqrt{x(1-x)}}=\pi$$
... |
A polynomial continued fraction (p.c.f.) is a continued fraction (c.f.)
$b_{0}+\underset{j=1}{\overset{\infty }{\mathbb{K}}}\left( a_{j}|b_{j}\right) =b_{0}+\dfrac{a_{1}}{b_{1}+\dfrac{a_{2}}{b_{2}+\dfrac{a_{3}}{b_{3}+\cdots }}}\qquad \left( 0\right) $
whose elements $a_{n},b_{n}$ are polynomial sequences [D. Bowm... |
Your intuition is correct that the derivative OPERATOR can be multiplied together in such a manner i.e. d/dx*d/dx = d^2/(dx)^2. However, the expression dy/dx is an evaluated derivative on the function y. When you evaluate a derivative, it no longer behaves like the derivative operator and is now simply a function. I.e.... |
Your intuition is correct that the derivative OPERATOR can be multiplied together in such a manner i.e. d/dx*d/dx = d^2/(dx)^2. However, the expression dy/dx is an evaluated derivative on the function y. When you evaluate a derivative, it no longer behaves like the derivative operator and is now simply a function. Anot... |
Your intuition is correct that the derivative OPERATOR can be multiplied together in such a manner i.e. $\frac{d}{dx}\frac{d}{dx} = \frac{d^2}{dx^2}$. However, the expression $\frac{dy}{dx} = \frac{d}{dx}(y)$ is an evaluated derivative on the function $y$ we write the $y$ in the numerator with $d$ as a notational conve... |
Sometime back i watched the documentary of Andrew Wiles proving the Fermat's Last theorem. A truly inspiring video and i still watch it whenever i am in a depressed mood. There are certain things(infact many) which i couldn't follow and i would like it to be explained here.
The first is:
> The Taniyama-Shimura co... |
A polynomial continued fraction (p.c.f.) is a continued fraction (c.f.)
$b_{0}+\underset{j=1}{\overset{\infty }{\mathbb{K}}}\left( a_{n}|b_{n}\right) =b_{0}+\dfrac{a_{1}}{b_{1}+\dfrac{a_{2}}{b_{2}+\dfrac{a_{3}}{b_{3}+\cdots }}}\qquad \left( 0\right) $
whose elements $a_{n},b_{n}$ are polynomial sequences [D. Bowm... |
A polynomial continued fraction (p.c.f.) is a continued fraction (c.f.)
$b_{0}+\underset{n=1}{\overset{\infty }{\mathbb{K}}}\left( a_{n}|b_{n}\right) =b_{0}+\dfrac{a_{1}}{b_{1}+\dfrac{a_{2}}{b_{2}+\dfrac{a_{3}}{b_{3}+\cdots }}}\qquad \left( 0\right) $
whose elements $a_{n},b_{n}$ are polynomial sequences [D. Bowm... |
Let $S$ be a (n-1)-submanifold of a n-manifold $M$ and that be a submanifold of (n+1)-manifold V. (All of the manifolds are assumed orientable) Let $V$ carry a metric of signature $(1,n)$. Using terminology as in Einstein's Theory let $n$ be a future directed unit normal to $M$ and $m$ be an unit normal to $S$ pointing... |
This is not an answer, but an elementay question about this category ring $R({\cal C})$. Because of Martin's remark, I'll use the product notation: elements of $R({\cal C})$ are tuples
$$
f= (f_{ab}) \in \prod_{a, b \in {\mathrm ob} {\cal C}} {\cal C}(a,b) \ ,
$$
in which every $f_{ab} : a \longrightarrow b$ i... |
>Find the Cartesian equation for the perpendicular bisector of the line joining A(2,3) and B(0,6)
How do I do this?
Thank you! |
How close do you have to be?
Can you just find the center of the each triangle in space then divide the cutting plane into a 2d grid and do a series of cubic volume calculations using the length and width of the grid section and the average height of the triangle midpoint above/below that grid section? |
How close do you have to be?
Can you just find the center of the each triangle in space then divide the cutting plane into a 2d grid and do a series of rectangular volume calculations using the length and width of the grid section and the average height of the triangle midpoint above/below that grid section? |
The following seems to hold in numerical simulations, is it true?
$$\lim_{n\to \infty} \int_0^1 dx \frac{n! 2^{-n} n}{(n x)!(n-n x)!\sqrt{x(1-x)}}=2$$
It's a combination of two known integrals
$$\lim_{n\to \infty} \int_0^1 dx \frac{n! 2^{-n} n}{(n x)!(n-n x)!\}=1$$
$$\int_0^1 dx \frac{1}{\sqrt{x(1-x)}}=\pi$$
... |
Let f:C to C be a complex-valued function. If you want to do ''regular'' integration, by which I assume you mean integrating over 2-dimensional regions in the complex plane, then you would have that the integral of f(x+iy)=u(x,y)+iv(x,y) over some region A in R<sup>2</sup> equals the integral of the real component over... |
I am presuming you are talking about the last digit of a number in base 10 being 1 and 5 implying the powers of that number will also end with 1 and 5.
I will give you a hint:
Give two numbers whose last digits are 1, what does their product end with?
|
The following seems to hold in numerical simulations, is it true?
$$\lim_{n\to \infty} \int_0^1 dx \frac{n! 2^{-n} n}{(n x)!(n-n x)!\sqrt{x(1-x)}}=2$$
It's a combination of two known integrals
$$\lim_{n\to \infty} \int_0^1 dx \frac{n! 2^{-n} n}{(n x)!(n-n x)!\}=1$$
$$\int_0^1 dx \frac{1}{\sqrt{x(1-x)}}=\pi$$
... |
Let f:C to C be a complex-valued function. If you want to do ''regular'' integration, by which I assume you mean integrating over 2-dimensional regions in the complex plane, then you would have that the integral of f(x+iy)=u(x,y)+iv(x,y) over some region A in R<sup>2</sup> equals the integral of the real component over... |
Let f:C to C be a complex-valued function. If you want to do ''regular'' integration, by which I assume you mean integrating over 2-dimensional regions in the complex plane, then you would have that the integral of f(x+iy)=u(x,y)+iv(x,y) over some region A in R<sup>2</sup> equals the integral of the real component over... |
Let's suppose, imitating the real case, the we want to integrate the expression
$f(z) dz$, where $f$ is a function of the complex variable $z$. What would this mean?
Well, imitating the real case, we find that we have to form Riemann sums
$\sum_{i = 1}^{n-1} f(z_i) (z_{i+1} - z_i)$, and then let the absolute value ... |
I was reviewing my class notes and found the following:
"The name 'torsion' comes from topology and refers to spaces that are twisted, ex. Mobius band"
What is the relationship between torsion modules and twisted spaces? Was the definition of torsion module somehow motivated from topological considerations of twi... |
I was reviewing my class notes and found the following:
"The name 'torsion' comes from topology and refers to spaces that are twisted, ex. Mobius band"
In our notes we used the following definition for torsion element and torsion module:
An element m of an R-module M is called a torsion element if $rm=0$ for som... |
I have to show that any line parallel to Ax+ By + c =0 is of the form Ax + By + k =0
How do I show this without writing out multiple examples?
Thank you! |
What is wrong with my reasoning? |
I have to show that any line parallel to Ax+ By + c =0 is of the form Ax + By + k =0
How do I show this?
Thank you! |
I'm not too confident with my probability, but I'll try my hand at an explanation.
In the first part, you are considering the next 3 trials as successive events so you can have a success in either the first spot, second spot, or third spot.
In part b, you require all 3 events to be failures so there is only 1 way... |
This is not an answer, but an elementay question about this category ring $R({\cal C})$. Because of Martin's remark, I'll use the product notation: elements of $R({\cal C})$ are tuples
$$
f= (f_{ab}) \in \prod_{a, b \in {\mathrm ob} {\cal C}} {\cal C}(a,b) \ ,
$$
in which every $f_{ab} : a \longrightarrow b$ i... |
The following seems to hold in numerical simulations, is it true?
$$\lim_{n\to \infty} \int_0^1 dx \frac{n! 2^{-n} n}{(n x)!(n-n x)!\sqrt{x(1-x)}}=2$$
It's a combination of two known integrals
$$\lim_{n\to \infty} \int_0^1 dx \frac{n! 2^{-n} n}{(n x)!(n-n x)!\}=1$$
$$\int_0^1 dx \frac{1}{\sqrt{x(1-x)}}=\pi$$
... |
In the area of convex analysis and the area of optimization in their general sense, are convex subsets assumed to be in vector spaces or topological vector spaces? Are convex functions defined to be from convex subsets in vector spaces or topological vector spaces to general field or some special field such as $\mathbb... |
A polynomial continued fraction (p.c.f.) is a continued fraction (c.f.)
$b_{0}+\underset{n=1}{\overset{\infty }{\mathbb{K}}}\left( a_{n}|b_{n}\right) =b_{0}+\dfrac{a_{1}}{b_{1}+\dfrac{a_{2}}{b_{2}+\dfrac{a_{3}}{b_{3}+\cdots }}}\qquad \left( 0\right) $
whose elements $a_{n},b_{n}$ are polynomial sequences [D. Bowm... |
If you feel like picking up a good book on this topic, "Dr. Euler's Fabulous Formula" by Nahin provides many beautiful applications of complex numbers to other parts of mathematics (he is an electrical engineer).
http://www.amazon.com/Dr-Eulers-Fabulous-Formula-Mathematical/dp/0691118221/ref=sr_1_1?s=books&ie=UTF8... |
Suppose $A$ is a commutative algebra over $\mathbb{R}$ with unity. $\mathbb{R}$-linear map $\xi\colon A\to A$ is a [derivation][1] of $A$ iff $\xi(ab)=a\xi(b)+\xi(a)b$ for any $a,b\in A$.
If $\gamma\colon \mathbb{R}\to A$, $a\in A$, then we say, that $a=\frac{\partial}{\partial t}| _ {t=\tau} \gamma(t)$ iff $h(a) =\... |
In which commutative algebras does any derivation possess a flow? |
We know this is true for commutative ring, but if $S\subset R$ is a left and right Ore set, and $S^{-1}R$ its localization by this Ore set, is this always a flat $R$-module? |
Are localized rings always flat as R-modules? |
There are too many examples to count. Let me just mention one that is particularly concrete: how many subsets of an $n$-element set have a cardinality divisible by $3$ (or any positive integer $k$)? In other words, how do we evaluate
$$\sum_{k=0}^{\lfloor \frac{n}{3} \rfloor} {n \choose 3k}$$
in closed form? A... |
The answer is No. Recall the question:
Let $A\subset\mathbb{C}$ be an open set containing the closed unit disc. Let $f$ be an analytic function from $A$ to $\mathbb{C}$ such that $|f(z)|=1$ if $|z|=1$.
Does it follow that
$$f(z)=az^{n}\,\frac{cz^{m}-b}{1-\bar{b}cz^{m}}$$
for some $a,b,c\in\mathbb{C}$ s.t. $|c... |
A polynomial continued fraction (p.c.f.) is a continued fraction (c.f.)
$b_{0}+\underset{n=1}{\overset{\infty }{\mathbb{K}}}\left( a_{n}|b_{n}\right) =b_{0}+\dfrac{a_{1}}{b_{1}+\dfrac{a_{2}}{b_{2}+\dfrac{a_{3}}{b_{3}+\cdots }}}\qquad \left( 0\right) $
whose elements $a_{n},b_{n}$ are polynomial sequences [D. Bowm... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.