instruction stringlengths 12 30k |
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Voronoi decomposition implementation in four dimensions? |
I know that a continous function which is a BV may not be absolutley continuous. Is there an example of such a function? I was looking for a BV whose derivative is not Lebeguse integrable but I couldnt find one. |
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... |
I know that a continuous function which is a BV may not be absolutely continuous. Is there an example of such a function? I was looking for a BV whose derivative is not Lebesgue integrable but I couldn't find one. |
My frustration started after hours of searching failed to turn up a formula for the vertex of a parabola in the general form
$$Ax^2+Bxy+Cy^2+Dx+Ey+F=0$$
As is already well known, the discriminant $\Delta=B^2-4AC$ can be used to diagnose the type of conic one has if it it given in the general form. Other useful fo... |
The standard survey is [Aurenhammer and Klein](http://www.pi6.fernuni-hagen.de/publ/tr198.pdf). Have you looked at that?
[This would be a comment, but I don't have enough rep on this site yet] |
In signal detection, an observer is assigned the task of discerning the presence (or absence) of some signal with accompanying noise. There are four possible outcomes: a hit, a miss, a false alarm, or a correct rejection. An iso-sensitivity function is the function that describes the relation between correct and fals... |
Try division? 1/(1-x) = 1 + x + x^2 + x^3 ...
(edit: I'm not sure why you would try to derive it from the series for (1+x)^n ) |
While teaching the concept of vector spaces, my professor mentioned that addition and multiplication aren't necessarily what we *normally* call addition and multiplication, but any other function that complies with the eight axioms needed by the definition of a vector space (for instance, associativity, commutativity o... |
What are some alternative definitions of vector addition and scalar multiplication? |
This example is certainly not "widely used", but I think it's worth thinking about anyway. This answer comes from an MO post by John Goodrick: http://mathoverflow.net/questions/9402/pedagogical-question-about-linear-algebra
I'll quote the entirety of his post here in case you don't feel like clicking on the link (a... |
Among the smooth 1-manifolds (with or without boundary) which embed into $\mathbb{R}^2$, which ones can be represented by a single parametrization $z = (x,y) = f(t)$, for $t \in I$, where $I$ is an interval (not necessarily open or closed), and $f$ is smooth (i.e. infinitely differentiable)?
The reason I ask is that... |
Which Smooth 1-Manifolds Can Be Represented By A Single Smooth Parametrization? |
Which smooth 1-manifolds can be represented by a single smooth parametrization? |
Intuition behind the "large but finite search space" of the proof of the four colour theorem? |
Under the condition that the spaces (or maybe just the total) are connected and locally path connected, is then the a covering the same as a homeomorphism? |
What do you "normally" call addition and multiplication?
*Just* those operations with real numbers, or *all* kind of addition and multiplication "derived" from the well-known operations with real numbers, or that "look like" these operations?
Because, in the first case, you have plenty of elementary and widely us... |
If I have an arbitrary shape, I would like to fill it only with 45-45-90
triangles.
The aim is to get a Tangram look, so it's related to [this question][1].
Starting with 45-45-90 triangles would be an amazing start. After
the shape if filled I imagine I could pick adjacent triangles, either
2 or 4 and draw s... |
How can I pack 45-45-90 triangles inside an arbitrary shape ? |
If I have an arbitrary shape, I would like to fill it only with 45-45-90
triangles.
The aim is to get a Tangram look, so it's related to [this question][1].
Starting with 45-45-90 triangles would be an amazing start. After
the shape if filled I imagine I could pick adjacent triangles, either
2 or 4 and draw s... |
You seem to have two main observations.
statement $A_{n}$ (for $n \ge 1$):
$$f^{n}(x) = e^{q(x)}p_{n}(x), x \neq 0$$
where $$p_{n}(x) = \sum \frac{a_{r}}{x^r}, x \neq 0$$
and
that
statement $B_{n}$ (for $n \ge -1$):
$$\lim_{h \to 0} \frac{1}{h^{n} e^{h^{-2}}} = 0$$
The result you have is that $A_{n... |
I've just come up with this question as I'm studying for a number theory midterm. If $p$ and $q$ are different prime numbers, and it's known that $2^p \equiv 1 \bmod{q}$, then $q\equiv 1 \bmod{p}$. I've tried a few cases and it seems to be true. How can it be proved? |
Is there a reliable method for testing how invertible a Hermitian Toeplitz matrix is without going through the work of actually inverting it?
A determinant is obviously easy to compute, but I'm not sure what threshold to compare it against.
The algorithm I am using is Levinson recursion. The function can detect ... |
Some years ago I was interested in the following Markov chain
whose state space is the positive integers. The chain begins at state "1",
and from state "n" the chain next jumps to a state uniformly
selected from {n+1,n+2,...,2n}.
As time goes on, this chain goes to infinity, with occasional
large jumps. In any ... |
I'm taking a course in differential geometry, and have here been introduced to the wedge product of to vectors defined (in [Differential Geometry of Curves and Surfaces by Manfredo Perdigão do Carmo][1]) by:
`Let u, v be in R^3. u ^ v in R^3 is the unique vector that satisfies:`
`(u ^ v) · w = det(u, v, w) for al... |
Wedge product and cross product - any difference? |
I have a stocastic event (say a weighted coin toss) that produces a positive outcome (heads) according to some unknown probability P.
Given N total events (coin tosses) and n positive outcomes (heads), how can I measure the likelihood that n/N is a good approximation of P?
Obviously, as N grows, it becomes more l... |
I'm taking a course in differential geometry, and have here been introduced to the wedge product of to vectors defined (in [Differential Geometry of Curves and Surfaces by Manfredo Perdigão do Carmo][1]) by:
Let $\mathbf{u}$, $\mathbf{v}$ be in $\mathbb{R}^3$. $\mathbf{u}\wedge\mathbf{v}$ in $\mathbb{R}^3$ is the un... |
No difference at all.
I've been trying to write a little proof, but the software on this page seems to have forgotten how to write maths. :-(
Anyway: I assume that by "regular cross/vector product" you mean the definition with coordinates as in [Wikipedia][1].
Try to compute both sides of your equation $(... |
For functions on $\mathbb{R}^2$, it lets you classify extreme points. When function is log-likelihood, it gives asymptotic squared error of maximum likelihood estimator. Are there other places it comes up? |
Determinant of the Hessian, what is it good for? |
$f(x)=x sin(1/x)$ for $x \ne 0$ and $f(x)=0$ for $x=0$ is not absolutely continuous so it $\sout{cannot}$ might not satisfy the Lusin N condition. Is there a direct proof of this? I wanted to know how to construct a set of zero measure which does not satisfy Lusin's Condition for this function. |
$f(x)=x sin(1/x)$ for $x \ne 0$ and $f(x)=0$ for $x=0$ is not absolutely continuous so it might not satisfy the Lusin N condition. Is there a direct proof of this? I wanted to know how to construct a set of zero measure which does not satisfy Lusin's Condition for this function.
Edit: Should say might not sa... |
Let $f$ be defined as
$$f(x)=\begin{cases} x \sin(\frac{1}{x}) & x\ne 0 \\\\ 0 & x=0 \end{cases}$$
$f(x)$ is not absolutely continuous so it *might not* satisfy the Lusin N condition.
Is there a direct proof of that it does not? i.e. I wanted to know how to construct a set of zero measure which does not sat... |
Let $f$ be defined as
$$f(x)=\begin{cases} x \sin(\frac{1}{x}) & x\ne 0 \\\\ 0 & x=0 \end{cases}$$
$f(x)$ is not absolutely continuous so it <strike>cannot</strike> *might not* satisfy the Lusin N condition.
Is there a direct proof of that it does not? i.e. I wanted to know how to construct a set of zero me... |
According to [Do Carmo][1], exercise 22 in page 173, the Hessian gives you the second fundamental form of a surface, as John pointed out.
Precisely, let $S$ be a surface, $p\in S$ a point and $h: S \longrightarrow \mathbb{R}$ be the height function of $S$ relative to $T_p S$; that is, $h(q) = (q-p)\cdot N(p)$, where... |
Let $f$ be defined as
$$f(x)=\begin{cases} x \sin(\frac{1}{x}) & x\ne 0 \\\\ 0 & x=0 \end{cases}$$
$f(x)$ is not absolutely continuous so it *might not* satisfy the Lusin N condition.
Is there a direct proof of that it does not? i.e. I wanted to know how to construct a set of zero measure which does not sa... |
Does there exists a family of discs $\lbrace D_{n}\rbrace_{n=1}^{\infty}$ in euclidean plane such that:
-the radius of $D_{n}$ is $\frac{1}{n+1}$
-each $D_{n}$ is contained in the unit disc
and
-$D_{n}\cap D_{m} = \emptyset$ for each $n\neq m$ ?
(I'm not sure what tags are appropriate for this kind quest... |
I've got two questions:
(1). I would like to know if by definition the Sobolev space norm is defined only by differentials with respect to one variable i.e.
$\parallel f \parallel \; = \; \left[ \int_{\mathbb{R}} \left( \left| f \right|^p \; + \; \left| \frac{\partial f}{\partial x} \right|^p \; + \ldots + \; \le... |
Is this proof of $a^{1/2}$ being either integer or irrational circular/incorrect? |
Let $f$ be defined as
$$f(x)=\begin{cases} x \sin(\frac{1}{x}) & x\ne 0 \\\\ 0 & x=0 \end{cases}$$
$f(x)$ is not absolutely continuous so it <strike>cannot</strike> *might not* satisfy the Lusin N condition.
Is there a direct proof of that it does not? i.e. I wanted to know how to construct a set of zero me... |
Let $f$ be defined as
$$f(x)=\begin{cases} x \sin(\frac{1}{x}) & x\ne 0 \\\\ 0 & x=0 \end{cases}$$
$f(x)$ is not absolutely continuous so it <strike>cannot</strike> _might not_ satisfy the Lusin N condition.
Is there a direct proof of that it does not? i.e. I wanted to know how to construct a set of zero me... |
I am looking for an intuitive explanation of $\liminf$ and $\limsup$ for sets and how it corresponds to the same for real numbers. I researched online but cannot find a good comparison. Any link, reference or answer very much appreciated.
For example, what is $\liminf$ and $\limsup$ of $(-1)^n$ and $1/n$. Correspond... |
Suppose $\alpha: \mathbb{R} \rightarrow \mathbb{R}^3$ is a $C^\infty$
curve, parameterized by arc length ($\left\|\alpha'(x)\right\| = 1$),
and with $\alpha(0) = \alpha(\ell)$. Show that
there exists a $t_0 \in [0,\ell]$ such that
$
\left\| \alpha''(t_0)\right\| \geq \frac{2\pi}{\ell}.
$
If we had that $\al... |
Suppose $\alpha: \mathbb{R} \rightarrow \mathbb{R}^3$ is a $C^\infty$
curve, parameterized by arc length ($\left\|\alpha'(t)\right\| = 1$),
and with $\alpha(0) = \alpha(\ell)$. Show that
there exists a $t_0 \in [0,\ell]$ such that
$
\left\| \alpha''(t_0)\right\| \geq \frac{2\pi}{\ell}.
$
![loop][1]
If we h... |
I am looking for an intuitive explanation of $\liminf$ and $\limsup$ for sets and how it corresponds to the same for real numbers. I researched online but cannot find a good comparison. Any link, reference or answer very much appreciated.
For example, what is $\liminf$ and $\limsup$ of real number sequences $a_n=(-1... |
Given that I have two formal power series:
$$ A(x) = \sum_{k \ge 0} a_k x^k $$
$$ B(x) = \sum_{k \ge 0} b_k x^k $$
The Cauchy Product gives a series
$$ C(x) = \sum_{k \ge 0} b_k x^k $$
$$ c_k = \sum_{n=0}^k a_n b_{k-n} $$
Which comes from taking the product of the two series $C(x)=A(x)B(x)$. What then, in te... |
I am looking for an intuitive explanation of $\liminf$ and $\limsup$ for sequence of sets and how it corresponds to $\liminf$ and $\limsup$ for sets of real numbers. I researched online but cannot find a good comparison. Any link, reference or answer very much appreciated.
For example, what is $\liminf$ and $\limsup... |
Given that I have two formal power series:
$$ A(x) = \sum_{k \ge 0} a_k x^k $$
$$ B(x) = \sum_{k \ge 0} b_k x^k $$
The Cauchy Product gives a series
$$ C(x) = \sum_{k \ge 0} c_k x^k $$
$$ c_k = \sum_{n=0}^k a_n b_{k-n} $$
Which comes from taking the product of the two series $C(x)=A(x)B(x)$. What then, in te... |
This is the answer from [Wolframalpha][1]: note that you have to use the substitution
$$u=\sqrt{x}$$
first.
[1]: http://www.wolframalpha.com/input/?i=integrate+of+sqrt(1+%2B+sqrt(x)) |
Let's deal with the examples first.
The $\liminf$ of the real sequence $\\{a_n\\}$ with $a_n = (-1)^n$ is $-1$; the $\limsup$ of the same sequence is $1$. For the sequence $\\{b_n\\}$, with $b_n=\frac{1}{n}$, since the sequence converges to $0$, every subsequence converges to $0$ so both $\liminf$ and $\limsup$ are ... |
Let's deal with the examples first.
The $\liminf$ of the real sequence $\\{a_n\\}$ with $a_n = (-1)^n$ is $-1$; the $\limsup$ of the same sequence is $1$. For the sequence $\\{b_n\\}$, with $b_n=\frac{1}{n}$, since the sequence converges to $0$, every subsequence converges to $0$ so both $\liminf$ and $\limsup$ are ... |
I first noticed this is true for the integers of the sequence $9, 99, 999, 9999,\dots$, since for some term $a_n=10^n-1$ in the sequence and $p$ a prime other than $2$ or $5$, we have $a_n\equiv 0 (mod\ p)\Leftrightarrow 10^n\equiv 1 (mod\ p)$ and we can find infinitely many $n=k(p-1)$ so that $10^n=(10^{p-1})^k\equiv ... |
Why do primes other than 2 and 5 divide infinitely many repunits? |
I wasnt sure where to ask this but since this is an algorithmic question here it goes. I've come face to face with a math problem and can't seem to get over it for the last couple of days. It goes like this:
> You are given an adding machine that sums a set of N+1 digits
> consisting of positive integers 1 to N
> ... |
I have a grid made up of overlapping 3x3 squares like so:
![Grid][1]
The numbers on the grid indicate the number of overlapping squares. Given that we know the maximum number of overlapping squares (9 at the middle), and the size of the squares (3x3), is there a simple way to calculate the rest of the number of o... |
This is the answer from [Wolframalpha][1]: note that you have to use the substitution
$$u=\sqrt{x}$$
first. Note that Wolframalpha able to give you the full step by step solution.
[1]: http://www.wolframalpha.com/input/?i=integrate+of+sqrt(1+%2B+sqrt(x)) |
A sequence of sets is defined as $A_n=\\{x \in [0,1] : |\sum_{i=0}^{n-1} 1_{[\frac{i}{2n},\frac{2i+1}{4n})} - 1_{[\frac{2i+1}{4n},\frac{i+1}{2n})}| \geq p\\}$ for some positive $p\geq0$
What is $\limsup_{n\to\infty}A_n(p)$ and $\liminf_{n\to\infty}A_n(p)$?
For any $x$, the function $f_n(x) = \sum_{i=0}^{n-1} 1_{[... |
A sequence of sets is defined as $A_n=\\{x \in [0,1] : |\sum_{i=0}^{n-1} 1_{[\frac{i}{2n},\frac{2i+1}{4n})} - 1_{[\frac{2i+1}{4n},\frac{i+1}{2n})}| \geq p\\}$ for some positive $p\geq0$. What is $\limsup_{n\to\infty}A_n(p)$ and $\liminf_{n\to\infty}A_n(p)$?
For any $x$, the function $f_n(x) = \sum_{i=0}^{n-1} 1_{[\f... |
Assume $M$ is a set, in which all axioms of $ZF - P + (V=L)$ hold. Why does $M$ believe that there exists an uncountable ordinal? I mean, why should the class of all countable ordinal numbers be a set?
I think this is used in the proof of Diamond in $L$ in Kunen's set theory. |
A sequence of sets is defined as $A_n=\\{x \in [0,1] : |\sum_{i=0}^{n-1} 1_{[\frac{i}{2n},\frac{2i+1}{4n})} - 1_{[\frac{2i+1}{4n},\frac{i+1}{2n})}| \geq p\\}$ for some positive $p\geq0$. What is $\limsup_{n\to\infty}A_n(p)$ and $\liminf_{n\to\infty}A_n(p)$?
For any $x$, the function $f_n(x) = \sum_{i=0}^{n-1} 1_{[\f... |
I'm going to claim something that seems to fly in the face of what's accepted. So tell me if I've done something wrong (which is quite possible, whence I will delete this answer), but I believe all three proofs of this claim on [wikipedia][1] are invalid.
Let's begin with the observation that, assuming that identit... |
Assume $M$ is a set, in which all axioms of $ZF - P + (V=L)$ hold. Does then $M$ believe that there exists an uncountable ordinal? I mean, why should the class of all countable ordinal numbers be a set? |
The complexity class RL is described at the complexity zoo as: Has the same relation to L as RP does to P. The randomized machine must halt with probability 1 on any input. It must also run in polynomial time (since otherwise we would just get NL).
The question is - how can we get NL when omitting the demand for pol... |
Complexity - why is RL=NL when ommitting the demand for polynomial run-time? |
Complexity - why is RL=NL when omitting the demand for polynomial run-time? |
We know that $$ 2^{n}= (1+1)^{n} = \sum\limits_{k=0}^{n} {n \choose k}$$ I was asked to solve this limit, $$\lim_{n \to \infty} \ \sum\limits_{k=0}^{n} {n \choose k}^{-1}=? \quad \text{for} \ n \geq 1$$ |
I claim all three "proofs" of this result (currently) on [wikipedia][1] are incomplete. Let's begin with the observation that, assuming that identity permutation is not an odd permutation, then the result follows fairly easily.
**Theorem**: Assuming the identity permutation is not an odd permutation, then all permu... |
For a sequence of positive integers $a_1, \ldots, a_n$ and a base ring $R$ there is a graded ring $R[x_1,\ldots, x_n]$ where $x_i$ is in degree $a_i$. We can then apply Proj and get a scheme, and this is usually called a weighted projective space; if all of the $a_i$ are 1, then the resulting scheme really is projecti... |
Suppose $X$ is a metric space. A real valued function $f$ defined on a metric space $X$ is said to be of first baire class, if $f$ is a pointwise, limit of a sequence of continuous functions on $X$.
A famous well know theorem asserts that if *$f$ is in the first baire class, then the set of points of discontinuity ... |
Let's take the numbers 0-10. Their mean is 5, and the individual deviations from 5 are
`-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5`
And so the average (magnitude of) deviation from the mean is $30/11 \approx 2.72$.
However, this is not the standard deviation. The standard deviation is $\sqrt{10} \approx 3.16$.
... |
Motivation behind standard deviation? |
Let's take the numbers 0-10. Their mean is 5, and the individual deviations from 5 are
`-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5`
And so the average (magnitude of) deviation from the mean is $30/11 \approx 2.72$.
However, this is not the standard deviation. The standard deviation is $\sqrt{10} \approx 3.16$.
... |
Despite its name, its often claimed that the fundamental theorem of algebra (which shows that the Complex numbers are algebraically closed - this is not to be confused with the claim that a polynomial of degree n has **at most** n roots) is not considered fundamental by algebraists as its not needed for the development... |
How fundamental is the fundemental theorem of algebra? |
If I have an arbitrary shape, I would like to fill it only with 45-45-90
triangles.
The aim is to get a Tangram look, so it's related to [this question][1].
Starting with 45-45-90 triangles would be an amazing start. After
the shape if filled I imagine I could pick adjacent triangles, either
2 or 4 and draw s... |
If I have an arbitrary shape, I would like to fill it only with 45-45-90
triangles.
The aim is to get a Tangram look, so it's related to [this question][1].
Starting with 45-45-90 triangles would be an amazing start. After
the shape if filled I imagine I could pick adjacent triangles, either
2 or 4 and draw s... |
Well, there is something called "Algebraic Geometry". You may check the first pages of most of the books on that subject and you'll see something like: "Let $k$ be an algebraically closed field..." [for instance, $\mathbb{C}$]. Indeed, this is the beginning of [Hartshorne's book][1].
The reason? It's much harder to ... |
Despite its name, its often claimed that the fundamental theorem of algebra (which shows that the Complex numbers are algebraically closed - this is not to be confused with the claim that a polynomial of degree n has **at most** n roots) is not considered fundamental by algebraists as it's not needed for the developmen... |
How fundamental is the fundamental theorem of algebra? |
Suppose $X$ is a metric space. A real valued function $f$ defined on a metric space $X$ is said to be of first baire class, if $f$ is a pointwise, limit of a sequence of continuous functions on $X$.
A famous well know theorem asserts that if *$f$ is in the first baire class, then the set of points of discontinuity ... |
Well, there is something called "Algebraic Geometry". You may check the first pages of most of the books on that subject and you'll see something like: "Let $k$ be an algebraically closed field..." [for instance, $\mathbb{C}$]. Indeed, this is the beginning of [Hartshorne's book][1].
The reason? It's much harder to ... |
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... |
Suppose A is a <a href="http://en.wikipedia.org/wiki/Hadamard_matrix">Hadamard matrix</a> of size $d$. Let A be in normalized in a sense that first row and first column are all ones. What is the sum of rows? I tried random Hadamard matrices and seem to get $d,0,0,0,0,\ldots$ |
Well, there is something called "Algebraic Geometry". You may check the first pages of most of the books on that subject and you'll see something like: "Let $k$ be an algebraically closed field..." [for instance, $\mathbb{C}$]. Indeed, this is the beginning of [Hartshorne's book][1].
The reason? It's much harder to ... |
A sequence of sets is defined as $A_n=\\{x \in [0,1] : |\sum_{i=0}^{n-1} 1_{[\frac{i}{2n},\frac{2i+1}{4n})} - 1_{[\frac{2i+1}{4n},\frac{i+1}{2n})}| \geq p\\}$ for some positive $p\geq0$. What is $\limsup_{n\to\infty}A_n(p)$ and $\liminf_{n\to\infty}A_n(p)$?
For any $x$, the function $f_n(x) = \sum_{i=0}^{n-1} 1_{[\f... |
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... |
If $n$ is an integer, is there a nice way to write the partial fraction expansion of $\frac{1}{x^n - 1}$? I figure that if $\zeta$ is the $n$-th root of unity, then for some coefficients $a_0, a_1, \ldots, a_{n-1}$ we may write
$$
\frac{1}{x^n - 1} = \frac{a_0}{x - 1} + \frac{a_1}{x - \zeta} + \frac{a_2}{x - \zeta... |
If $n$ is an integer, is there a nice way to write the partial fraction expansion of $\frac{1}{x^n - 1}$? I figure that if $\zeta$ is the $n$-th root of unity, then for some coefficients $a_0, a_1, \ldots, a_{n-1}$ we may write
$$
\frac{1}{x^n - 1} = \frac{a_0}{x - 1} + \frac{a_1}{x - \zeta} + \frac{a_2}{x - \zeta... |
What do you call this function?
f(x) = x * (1.0 - x)
Does it have a name? It’s cropped up in at least two statistical formulas I've been asked to program so far. (In both cases, `x` was something called a *p-value* (for what it's worth).)
|
What do you call this statistics function? |
Because the second derivative exists everywhere, you also know that $f'$ is continuous everywhere. Now think about the function $f'(x)$; it is positive to the left and to the right of $c$, and is $0$ at $c$, and it is continuous. Does that tell you something about $f''$, and hence about concavity? |
In this wikipedia, article http://en.wikipedia.org/wiki/Circle#Area_enclosed its stated that the circle is the closed curve which has the maximum area for a given arc length. First, of all i would like to see different proofs, for this result. (If there are any elementary ones!)
One, interesting observation, which o... |
Why Circle encloses largest Area? |
For functions on $\mathbb{R}^2$, it lets you classify extreme points. When function is log-likelihood, it gives volume of the region where maximum likelihood estimates tend to fall. Are there other places it comes up? |
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?
If the 2 inbound/outbound graph is named, is the nth inbound/outbound graph named? |
What are the names of graphs with 2 in/outbound edges? |
Given an infinite dimensional normed linear space, how would one show that it is possible to fit an infinite collection of non-overlapping balls of radius 1/4 in the unit ball? |
Given an infinite dimensional normed linear space, how would one show that it is possible to fit an infinite collection of non-overlapping balls of radius 1/4 in the unit ball?
I guess one can immediately reduce the problem to a normed linear space of countably infinite dimensions. The solution seems clear if the co... |
Just to enlarge on Harry's answer:
Your symbol denotes one of two specified notions of implication in formal logic
$\vdash$ -the **turnstile** symbol denotes **syntactic** implication (syntactic here means related to syntax, the structure of a sentence), where the 'algebra' of the logical system in play (for ex... |
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