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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 have a terrain, which is represented by [one mesh with a lot of polygons][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).... |
*Motivation*: Assume that all $a_{n},b_{n}$ are integers. **I am looking for a c.f. transformation** that can convert $b_{0}+\underset{n=1}{\overset{\infty }{\mathbb{K}}}\left( a_{n}|b_{n}\right) $ into another p.c.f. $b_{0}^*+\underset{n=1}{\overset{\infty }{\mathbb{K}}}\left( a_{n}^*,b_{n}^*\right) $ with integer ele... |
*Motivation*: Assume that all $a_{n},b_{n}$ are integers. **I am looking for a c.f. transformation** that can convert $b_{0}+\underset{n=1}{\overset{\infty }{\mathbb{K}}}\left( a_{n}|b_{n}\right) $ into another c.f. $b_{0}^*+\underset{n=1}{\overset{\infty }{\mathbb{K}}}\left( a_{n}^*,b_{n}^*\right) $ with integer eleme... |
*Motivation*: Assume that all $a_{n},b_{n}$ are integers. **I am looking for a c.f. transformation** that can convert $b_{0}+\underset{n=1}{\overset{\infty }{\mathbb{K}}}\left( a_{n}|b_{n}\right) $ into another c.f. $b_{0}^*+\underset{n=1}{\overset{\infty }{\mathbb{K}}}\left( a_{n}^*,b_{n}^*\right) $ with integer eleme... |
*Motivation*: Assume that all $a_{n},b_{n}$ are integers. **I am looking for a c.f. transformation** that can convert $b_{0}+\underset{n=1}{\overset{\infty }{\mathbb{K}}}\left( a_{n}|b_{n}\right) $ into another c.f. $b_{0}^*+\underset{n=1}{\overset{\infty }{\mathbb{K}}}\left( a_{n}^*,b_{n}^*\right) $ with integer eleme... |
*Motivation*: Assume that all $a_{n},b_{n}$ are integers. **I am looking for a c.f. transformation** that can convert $b_{0}+\underset{n=1}{\overset{\infty }{\mathbb{K}}}\left( a_{n}/b_{n}\right) $ into another c.f. $b_{0}^*+\underset{n=1}{\overset{\infty }{\mathbb{K}}}\left( a_{n}^*/b_{n}^*\right) $ with integer eleme... |
How to compute the partial numerators and denominators of this continued fraction from its approximants? |
*Motivation*: Assume that all $a_{n},b_{n}$ are integers. **I am looking for a c.f. transformation** that can convert $b_{0}+\underset{n=1}{\overset{\infty }{\mathbb{K}}}\left( a_{n}/b_{n}\right) $ into another c.f. $b_{0}^*+\underset{n=1}{\overset{\infty }{\mathbb{K}}}\left( a_{n}^*/b_{n}^*\right) $ with integer eleme... |
My (maybe wrong) impression is that while probability is widely used in science (for example, in statistical mechanics), it is rarely seen in pure mathematics. Which leads me to the question -
**Are there some interesting application of Probability Theory in pure mathematics, outside Probability Theory itself?** |
How to compute the partial numerators and denominators of this continued fraction from its approximants? (Rewritten) |
I have a problem, which has $n$ variables. Let's call these variables as $a_1$, $a_2$, $a_3$ ... $a_n$. Each of these variables have their own range (i.e, $a_{imin} \leq a_i \leq a_{imax}$ for all $i$). And most importantly, each $a_n$ is ranked in terms of importance. Without a loss of generality one can assume that ... |
*Motivation*: Assume that all $a_{n},b_{n}$ are integers. **I am looking for a c.f. transformation** that can convert $b_{0}+\underset{n=1}{\overset{\infty }{\mathbb{K}}}\left( a_{n}/b_{n}\right) $ into another c.f. $b_{0}^*+\underset{n=1}{\overset{\infty }{\mathbb{K}}}\left( a_{n}^*/b_{n}^*\right) $ with integer eleme... |
This question may seem simple, but I have no idea where to start.
I'm in design phase of a hobby electronics project and need your help. Basically, I'm going to have two platforms with electronics on both of them. I'm planning to have a [stepper motor][1] on the bottom platform which will be connected to top platfo... |
The birational change of variables $(u,v) = (\frac{36+y}{6x},\frac{36-y}{6x})$ maps $u^3+v^3=1$ to $y^2 = x^3 - 432$ which has discriminant $-2^{12}\cdot 3^9$.
Using pari/gp we can compute the torsion subgroup:
? elltors(ellinit([0,0,0,0,-432]))
%1 = [3, [3], [[12, 36]]]
This says the torsion subgroup... |
How could I calculate the rank of this elliptic curve? |
**Q1:** Are convex subsets assumed to be in vector spaces or topological vector spaces?
Convex sets are easily defined in topological vector spaces over a subfield of $\mathbb{C}$, following the usual definition: $A$ is a convex set if $\forall x,y\in A$ we have $tx+(1-t)y\in A$ for $t\in[0,1]$. Basically means we c... |
This question may seem simple, but I have no idea where to start.
I'm in design phase of a hobby electronics project and need your help. Basically, I'm going to have two platforms with electronics on both of them. I'm planning to have a [stepper motor][1] on the bottom platform which will be connected to top platfo... |
**Q1:** Are convex subsets assumed to be in vector spaces or topological vector spaces?
Convex sets are easily defined in topological vector spaces over a subfield of $\mathbb{C}$, following the usual definition: $A$ is a convex set if $\forall x,y\in A$ we have $tx+(1-t)y\in A$ for $t\in[0,1]$. Basically means we c... |
This is a total and utter guess, but suppose you fix one endpoint of the wire and unroll the axle to make a rectangle? This is the same trick used to justify the rectangle in the surface area of a cylinder. Then the wire is the diagonal of a rectangle of width $L$ and height $r\theta$. So by the Pythagorean theorem, $l... |
*Motivation*: Assume that all $a_{n},b_{n}$ are integers. **I am looking for a c.f. transformation** that can convert $b_{0}+\underset{n=1}{\overset{\infty }{\mathbb{K}}}\left( a_{n}/b_{n}\right) $ into another c.f. $b_{0}^*+\underset{n=1}{\overset{\infty }{\mathbb{K}}}\left( a_{n}^*/b_{n}^*\right) $ with integer eleme... |
Let $I$ be a finite nonempty set, and $D$ a nonvoid compact subset of an N-dimensional Euclidean space. The functions $V:D\rightarrow\mathbb{R}$ and $u_i:D\rightarrow\mathbb{R}$ are given. They are all continuous, and, moreover, each $u_i$ satisfies $u_i(\lambda p + (1-\lambda) q) = \lambda u_i( p) + (1-\lambda) u_i(q)... |
*Motivation*: Assume that all $a_{n},b_{n}$ are integers. **I am looking for a c.f. transformation** that can convert $b_{0}+\underset{n=1}{\overset{\infty }{\mathbb{K}}}\left( a_{n}/b_{n}\right) $ into another c.f. $b_{0}^*+\underset{n=1}{\overset{\infty }{\mathbb{K}}}\left( a_{n}^*/b_{n}^*\right) $ with integer eleme... |
*Motivation*: Assume that all $a_{n},b_{n}$ are integers. **I am looking for a c.f. transformation** that can convert $b_{0}+\underset{n=1}{\overset{\infty }{\mathbb{K}}}\left( a_{n}/b_{n}\right) $ into another c.f. $b_{0}^*+\underset{n=1}{\overset{\infty }{\mathbb{K}}}\left( a_{n}^*/b_{n}^*\right) $ with integer eleme... |
I'm interested in solving equations of the form $f(f(x))=g(x)$ for $x\in\mathbb{R}$ where $g(x)$ is a known function.
For example: if $g(x)=x$ then we $f(f(x))=x$ so one solution is $f(x)=x$, $\forall x\in\mathbb{R}$. However, another is
$f(x)=x+1$ if $x\in(0,1]$, $f(x)=x-1$ if $x\in(1,2]$ and $f(x)=x$ otherw... |
In my daughter's class of 23, three students and the teacher all share the same birthday. Of course, there are 365 days in the year, and the first case of the shared birthday is not counted in the probability. But is the likelihood of this just (1/365)*3 = .0082? For my curiosity, leap years can be omitted. |
What does the following term mean?
> category of geometrically reduced schemes of finite type over some field
(I know what a category and a field is but I cannot translate any of the middle bits) |
What is the meaning of "reduced schemes of finite type"? |
*Motivation*: Assume that all $a_{n},b_{n}$ are integers. **I am looking for a c.f. transformation** that can convert $b_{0}+\underset{n=1}{\overset{\infty }{\mathbb{K}}}\left( a_{n}/b_{n}\right) $ into another c.f. $b_{0}^*+\underset{n=1}{\overset{\infty }{\mathbb{K}}}\left( a_{n}^*/b_{n}^*\right) $ with integer eleme... |
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) =\... |
The birational change of variables $(u,v) = (\frac{36+y}{6x},\frac{36-y}{6x})$ maps $u^3+v^3=1$ to $y^2 = x^3 - 432$ which has discriminant $-2^{12}\cdot 3^9$.
Using pari/gp we can compute the torsion subgroup:
? elltors(ellinit([0,0,0,0,-432]))
%1 = [3, [3], [[12, 36]]]
This says the torsion subgroup... |
You might find interesting various probabilistic plausibility arguments for the $\rm\:3n+1$ conjecture. For an introduction see [Lagarias's survey][1] excerpted below, and for more sophisticated arguments see his paper [How random are the 3x+1 function iterates?][2].
**1. Introduction.**$\;$ The $\rm\:3x + 1\:$ prob... |
Suppose $\mathcal{F}$ is a famliy of analytic functions of the unit disc. Suppose also that
$( Re(f(z)) )^2 \ne ( Im(f(z)) ) $for all $|z|<1$ and all $f \in \mathcal{F}$.
It follows from the <a href="http://en.wikipedia.org/wiki/Fundamental_normality_test">Fundamental Normality test</a> that $\mathcal{F}$ is... |
*Motivation*: Assume that all $a_{n},b_{n}$ are integers. **I am looking for a c.f. transformation** that can convert $b_{0}+\underset{n=1}{\overset{\infty }{\mathbb{K}}}\left( a_{n}/b_{n}\right) $ into another c.f. $b_{0}^*+\underset{n=1}{\overset{\infty }{\mathbb{K}}}\left( a_{n}^*/b_{n}^*\right) $ with integer eleme... |
> **Possible Duplicate:**
> [Square root of a function (in the sense of composition)](http://math.stackexchange.com/questions/3633/square-root-of-a-function-in-the-sense-of-composition)
<!-- End of automatically inserted text -->
I'm interested in solving equations of the form $f(f(x))=g(x)$ for $x\in\mathbb{R}$ ... |
Such results on continuous induction are folklore dating back over a century - much older than the references in the Kalantari paper referenced by Pete L. Clark. I once had a large file of papers on this topic but, alas, it's long lost. But I can still recall one [old paper \[1\]][1] because the author was the world-... |
How to start with mathematics? |
Why are row reductions so useful in linear algebra? It is easy to get lost in the mechanical solving of equations. I know I can get a matrix into reduced row echelon form. But what are the outcomes of this? What can this mean? |
What are we solving with row reduction? |
So you have a matrix $A$ and vectors $x,b$, to solve $Ax=b$ you apply row reductions.
So considering a concrete example, if we wanted to solve
$$\left(\begin{array}{cc} 1 & 1 \\\\ 1 & 2\end{array}\right) x = \left(\begin{array}{c} 5 \\\ 8\end{array}\right)$$
We can use the row that changes the bottom row into ... |
Pardon me if I'm understating your current knowledge, but are you familiar with geometry, elementary algebra and pre-calculus? Those are pretty standard courses for high school students, and I'm sure most standard textbooks is fine to get the basics.
If you are already familiar with that, computational calculus and ... |
Pardon me if I'm understating your current knowledge, but are you familiar with geometry, elementary algebra and pre-calculus? Those are pretty standard courses for high school students, and I'm sure most standard textbooks is fine to get the basics.
If you are already familiar with that, computational calculus and ... |
*Motivation*: Assume that all $a_{n},b_{n}$ are integers. **I am looking for a c.f. transformation** that can convert $b_{0}+\underset{n=1}{\overset{\infty }{\mathbb{K}}}\left( a_{n}/b_{n}\right) $ into another c.f. $b_{0}^*+\underset{n=1}{\overset{\infty }{\mathbb{K}}}\left( a_{n}^*/b_{n}^*\right) $ with integer eleme... |
I was reading about isomorphisms and homomorphisms on general structures, and first came across the definition of an injective homomorphism, or an embedding. This made me curious, is it possible for two structures $A$ and $B$ to be embeddable in each other, yet no isomorphism exists between them?
After some looking ... |
Can two structures be embeddable in each other, but not isomorphic? |
I have a problem, which has $n$ variables. Let's call these variables as $a_1$, $a_2$, $a_3$ ... $a_n$. Each of these variables have their own range (i.e, $a_{imin} \leq a_i \leq a_{imax}$ for all $i$). And most importantly, each $a_n$ is ranked in terms of importance. Without a loss of generality one can assume that ... |
Well, Dan and muad have already told you everything, but let me stress the two main points (in my opinion):
1. Row reduction is just the same as solving one equation for one unkown in terms of the others and then plugging the obtained expression into the remaining equations. (But without having to write down the un... |
Is there any formula for this series? |
1 + 1/2 + 1/3 + ... + 1/n = ? |
I was reading about isomorphisms and homomorphisms on general structures, and first came across the definition of an injective homomorphism, or an embedding. This made me curious, is it possible for two structures $A$ and $B$ to be embeddable in each other, yet no isomorphism exists between them?
After some looking ... |
For the definition of moduli space of Riemann surface, it seems that it is not based on the definiton in terms of functor(representable...) but rather put a topology on the set of isormophism classes. Is there any difference between these? Or if I miss something? Is there any reference ? Thank you! |
I have a problem, which has $n$ variables. Let's call these variables as $a_1$, $a_2$, $a_3$ ... $a_n$. Each of these variables have their own range (i.e, $a_{imin} \leq a_i \leq a_{imax}$ for all $i$). And most importantly, each $a_n$ is ranked in terms of importance. Without a loss of generality one can assume that ... |
**Motivation**: Assume that all $a_{n},b_{n}$ are integers. **I am looking for a c.f. transformation** that can convert $b_{0}+\underset{n=1}{\overset{\infty }{\mathbb{K}}}\left( a_{n}/b_{n}\right) $ into another c.f. $b_{0}^*+\underset{n=1}{\overset{\infty }{\mathbb{K}}}\left( a_{n}^*/b_{n}^*\right) $ with integer ele... |
I'm also an older person trying to teach myself math. I took math till high school and since then I have taught myself to beginning-graduate level in one area (analysis) and bits and pieces of other areas at an undergraduate level. My suggestions,
- **Time is limited, don't waste it on popular books.** Study high-s... |
**Motivation**: Assume that all $a_{n},b_{n}$ are integers. **I am looking for a c.f. transformation** that can convert $b_{0}+\underset{n=1}{\overset{\infty }{\mathbb{K}}}\left( a_{n}/b_{n}\right) $ into another c.f. $b_{0}^*+\underset{n=1}{\overset{\infty }{\mathbb{K}}}\left( a_{n}^*/b_{n}^*\right) $ with integer ele... |
**Motivation**: Assume that all $a_{n},b_{n}$ are integers. **I am looking for a c.f. transformation** that can convert $b_{0}+\underset{n=1}{\overset{\infty }{\mathbb{K}}}\left( a_{n}/b_{n}\right) $ into another c.f. $b_{0}^*+\underset{n=1}{\overset{\infty }{\mathbb{K}}}\left( a_{n}^*/b_{n}^*\right) $ with integer ele... |
Well, Dan and muad have already told you everything, but let me stress the two main points (in my opinion):
1. Row reduction is just the same as solving one equation for one unkown in terms of the others and then plugging the obtained expression into the remaining equations. (But without having to write down the un... |
**Motivation**: Assume that all $a_{n},b_{n}$ are integers. **I am looking for a c.f. transformation** that can convert $b_{0}+\underset{n=1}{\overset{\infty }{\mathbb{K}}}\left( a_{n}/b_{n}\right) $ into another c.f. $b_{0}^*+\underset{n=1}{\overset{\infty }{\mathbb{K}}}\left( a_{n}^*/b_{n}^*\right) $ with integer ele... |
According this problem/solution set from an MIT class (http://ocw.mit.edu/courses/mathematics/18-100c-analysis-i-spring-2006/exams/exam1_sol.pdf), the assertion:
"Every collection of disjoint intervals in R is countable."
is True, because "every interval contains a rational number", and the rations are countable.... |
**Motivation**: Assume that all $a_{n},b_{n}$ are integers. **I am looking for a c.f. transformation** that can convert $b_{0}+\underset{n=1}{\overset{\infty }{\mathbb{K}}}\left( a_{n}/b_{n}\right) $ into another c.f. $b_{0}^*+\underset{n=1}{\overset{\infty }{\mathbb{K}}}\left( a_{n}^*/b_{n}^*\right) $ with integer ele... |
I have heard that Legendre functions are important in number theory. Can any one tell me how?
The Legendre function of the first kind $P_s$ is defined by
\begin{eqnarray*}P_s(x) =& \frac{1}{2\pi}\int_{-\pi}^{\pi}\left(x+\sqrt{x^2-1}\cos\theta\right)^s d\theta
\newline
=&\frac{1}{\pi}\int_0^1\left(x+\sqrt{x^2-1}... |
**Motivation**: Assume that all $a_{n},b_{n}$ are integers. **I am looking for a c.f. transformation** that can convert $b_{0}+\underset{n=1}{\overset{\infty }{\mathbb{K}}}\left( a_{n}/b_{n}\right) $ into another c.f. $b_{0}^*+\underset{n=1}{\overset{\infty }{\mathbb{K}}}\left( a_{n}^*/b_{n}^*\right) $ with integer ele... |
The german mathematician [Martin Eichler][1] once stated that there were five fundamental operations of mathematics: addition, subtraction, multiplication, division, and modular forms. This was also mentioned in Simon Singh's book on Fermat's last theorem.
I don't know much about modular forms beyond the [Wikipedia ... |
How are modular forms fundamental operations? |
According this problem/solution set from an MIT class (http://ocw.mit.edu/courses/mathematics/18-100c-analysis-i-spring-2006/exams/exam1_sol.pdf), the assertion:
"Every collection of disjoint intervals in R is countable."
is True, because "every interval contains a rational number", and the rationals are countabl... |
**Motivation**: Assume that all $a_{n},b_{n}$ are integers. **I am looking for a c.f. transformation** that can convert $b_{0}+\underset{n=1}{\overset{\infty }{\mathbb{K}}}\left( a_{n}/b_{n}\right) $ into another c.f. $b_{0}^*+\underset{n=1}{\overset{\infty }{\mathbb{K}}}\left( a_{n}^*/b_{n}^*\right) $ with integer ele... |
How to compute the partial numerators and denominators of this (generalized) continued fraction from its approximants? (Rewritten) |
My question was inspired by [this][Harmonic] stackexchange question. For the last 90 minutes I have been trying to prove this formula due to [Gregory Fontana][GF]:
$$H_n = \gamma + \log n + {1 \over 2n} - \sum_{k=2}^\infty { (k-1)! C_k \over n(n+1)\ldots(n+k-1)},
\qquad \textrm{ for } n=1,2,3,\ldots,$$
where $H_... |
I managed to prove that if $\phi(0)$ and $\forall n(n\in\mathbb{N},\phi(n)\rightarrow\phi(n+1))$ then $\forall n(n\in\mathbb{N}\rightarrow\phi(n))$. The proof is based on the fact that $\\{m\in\mathbb{N}:\phi(m)\\}$ is an inductive set. Now I'd like to show (but I can't) that if you take a $k\in\mathbb{N}$, if $\phi(k)... |
For a continuous scalar field on a circle, there is a diameter of the circle such that the endpoints of the diameter have the same value. If you think of the scalar field as "temperature", then what this says is that there are points on opposite sides of the circle that are the same temperature: isothermic antipodes.
... |
**Motivation**: Assume that all $a_{n},b_{n}$ are integers. **I am looking for a c.f. transformation** that can convert $b_{0}+\underset{n=1}{\overset{\infty }{\mathbb{K}}}\left( a_{n}/b_{n}\right) $ into another c.f. $b_{0}^*+\underset{n=1}{\overset{\infty }{\mathbb{K}}}\left( a_{n}^*/b_{n}^*\right) $ with integer ele... |
$x^tAy$ is a scalar. So $x^tAy=(x^tAy)^t$. Can you continue from here? |
I'd like to mention another combinatorial result, this time about asymptotics. Let $a_n$ denote the number of ways that $n$ horses can finish in a race (with ties); in other words, $a_n$ is the number of "lists of (nonempty) sets" with $n$ total members. Generating function methods allow you to deduce that
$$A(z) ... |
This condition can actually be used as an equivalent definition of $A$ being symmetric. The key point is that both sides are bilinear in $x$ and $y$, so it suffices to prove the result when $x$ and $y$ are basis vectors, say $e_i$ and $e_j$. What does the condition say then? |
How do I prove that $x^TAy = y^TAx$ if A is symmetric? |
Is there any rectangle, that if you devide it into another rectangle (or any other quadrilateral), the **relation** between the two different edges is the same in both rectangles?
For example - **Orginal rectangle:**
![alt text][1]
**Devided Rectangle:**
![alt text][2]
Note that these illustrations aren... |
Is there any rectangle, that if you divide it into another rectangle (or any other quadrilateral), the **relation** between the two different edges is the same in both rectangles?
For example - **Orginal rectangle:**
![alt text][1]
**Divided Rectangle:**
![alt text][2]
Note that these illustrations aren... |
Given $f(x) = \frac{1}{2}x^TAx + b^Tx + \alpha $
where A is an nxn symmetric matrix, b is an n-dimensional vector, and alpha a scalar. Show that
$\bigtriangledown _{x}f(x) = Ax + b$
and
$H = \bigtriangledown ^{2}_{x}f(x) = A$
Is this simply a matter of taking a derivative with respect to X, how would ... |
**Motivation**: **I am looking for a c.f. transformation that can convert $b_{0}+\underset{n=1}{\overset{\infty }{\mathbb{K}}}\left( a_{n}/b_{n}\right) $ into another c.f. $b_{0}^*+\underset{n=1}{\overset{\infty }{\mathbb{K}}}\left( a_{n}^*/b_{n}^*\right) $, so that all elements $a_{n},b_{n},a_{n}^{*},b_{n}^{*}$ are in... |
**Motivation**: **I am looking for a c.f. transformation that can convert $b_{0}+\underset{n=1}{\overset{\infty }{\mathbb{K}}}\left( a_{n}/b_{n}\right) $ into another c.f. $b_{0}^*+\underset{n=1}{\overset{\infty }{\mathbb{K}}}\left( a_{n}^*/b_{n}^*\right) $, so that all elements $a_{n},b_{n},a_{n}^{*},b_{n}^{*}$ are in... |
I know how to partition given $p(x)$ using a generating function as my textbook on discrete mathematics explains it in detail. However, I want to know if it is possible to restrict the source elements of the partition? So, given I want to find the partition of $x$ is it possible to do it using only elements from $S=\{y... |
Given a set of vectors {x1,...xn} how can I find a set of orthonormal vectors {y1,...ym;M<=N} spanning the same space spanned by {xi}? |
How to compute the partial numerators and denominators of this (generalized) continued fraction from its approximants? |
Given a set of vectors $\\{x_1,\ldots,x_n\\}$, how can I find a set of orthonormal vectors $\\{y_1,\ldots,y_m\colon m\leq n\\}$ spanning the same space spanned by $\\{x_i\\}$? |
I'd like to mention another combinatorial result, this time about asymptotics. Let $a_n$ denote the number of ways that $n$ horses can finish in a race (with ties); in other words, $a_n$ is the number of "lists of (nonempty) sets" with $n$ total members. Generating function methods allow you to deduce that
$$A(z) ... |
_Just out of pure curiosity ..._
Suppose I want to connect the two points $(0|0)$ and $(1|1)$ with the graph of some continuous and differentiable function
$$f : [0; 1] \to [0; 1]$$
and let $s$ be the [arc length][1] of that function in $[0; 1]$.
Of course, the function with minimum $s$ that satisfies the ... |
**Motivation**: **I am looking for a c.f. transformation that can convert $b_{0}+\underset{n=1}{\overset{\infty }{\mathbb{K}}}\left( a_{n}/b_{n}\right) $ into another c.f. $b_{0}^*+\underset{n=1}{\overset{\infty }{\mathbb{K}}}\left( a_{n}^*/b_{n}^*\right) $, so that all elements $a_{n},b_{n},a_{n}^{*},b_{n}^{*}$ are in... |
How to compute the partial numerators and denominators of these two generalized continued fractions given their approximants? |
**Motivation**: **I am looking for a c.f. transformation that can convert $b_{0}+\underset{n=1}{\overset{\infty }{\mathbb{K}}}\left( a_{n}/b_{n}\right) $ into another c.f. $b_{0}^*+\underset{n=1}{\overset{\infty }{\mathbb{K}}}\left( a_{n}^*/b_{n}^*\right) $, so that all elements $a_{n},b_{n},a_{n}^{*},b_{n}^{*}$ are in... |
I studied applied math, so each course (except abstract algebra) was dedicated to solution of a similar problems. After those courses it seems that every branch of mathematics has a developed theory with a few unsolved problems. I believe that is wrong. I found out that there is no general approach to Diophantine equat... |
I think this is obvious and I am just missing something but:
>Given some finitely presentable algebra, can we always reduce the relations to commutator relations?
And moreover, if yes, then is there a reasonable algorithm?
As I said, I feel like I should know this, so I apologize if it is obvious.
Thanks,... |
(Apologies for putting this in the form of an answer - I haven't got enough rep yet to make it a comment)
What about elements of finite order? It seems like even a simple algebra such as the one generated by $a^m = b^n = e$ couldn't possibly have a commutator presentation, unless I'm missing something immensely obv... |
For the equation
$f(x,y) = x^2/y$
given that $x$ and $y$ are integers
What is the easiest/fastest way to find at least one tuple $(x_0,y_0)$ solution to $f(x,y)=4$ or any other number really. I know that $(2,1)$ would work, but is there a process for finding the solution to any integer that is equal to $f(x,y... |
Using the contrapositive isn't the only indirect way of proving a theorem, and I highly doubt it's what your instructors had in mind here. As a hint for part (a): what can you say about the first $n$ even numbers (individually) compared to the first $n$ odd numbers? Can you find some convenient correspondence between... |
When studying group theory, I was interested to learn that a the quotient set $G/N$ of a group $G$ is again a group with $\pi\colon G\to G/N$ a homomorphism when $N$ is a normal subgroup of $G$.
Is there a more general set of criteria for this situation outside of group theory? As in, for any set $U$ and an equival... |
I'm taking a course in Complex Analysis,
and the teacher mentioned that if we do not
restrict our attention to analytic functions,
we would just be looking at functions from $\mathbb{R}^2$ to $\mathbb{R}^2$.
What I don't understand is why this is not true
when we do restrict our attention to analytic functions.
... |
How is $\mathbb{C}$ different than $\mathbb{R}^2$? |
For a function $f : \mathbb R^2 \to \mathbb R^2$ "differentiable" at a point $x \in \mathbb R^2$ means you have a linear approximation $f'_x$ which satisfies
$$\lim_{y \to x} \frac{f(x)-f(y)-f'_x(x-y)}{|x-y|} = 0$$
Saying that $f$ is complex analytic is the constraint that $f'_x$ is a complex linear function for ... |
I have a problem, which has $n$ variables. Let's call these variables as $a_1$, $a_2$, $a_3$ ... $a_n$. Each of these variables have their own range (i.e, $a_{imin} \leq a_i \leq a_{imax}$ for all $i$). And most importantly, each $a_n$ is ranked in terms of importance. Without a loss of generality one can assume that ... |
- Mathematica is great. Mathematica is a crutch. If you depend on a calculator, you won't get the meaning of your answer and you won't know what to do with it.
- If you know nothing about the RHS, to get a 'minimal' answer by generate and test, go in rings around the origin:
{0,0}
{1,0},{0,1},{-1,0},{0,-1}... |
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