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Why are the only numbers $m$ for which $n^{m+1}\equiv n \pmod{m}$ is true also unique for $\displaystyle\sum_{n=1}^{m}{n^m}\equiv 1 \bmod m$? |
Now that you've gotten the answer, here is a little bit of a post mortem clarifying the essential points, as may come up again in future problems. (In general, once you've solved a problem, it's a really good idea to ask for the "moral(s)". If you can't identify one, how are you better off now than before you solved ... |
Interesting Taxicab Problem? |
I don't know the answer (<b>I should have! -- see below</b>), but in my opinion a lot of people are misinterpreting the question, so perhaps it is worth an answer to try to set this straight.
Here is an analogy: trigonometric series is to Fourier series as power series is to Taylor series. In other words, a trigono... |
This problem isn't well defined as it relies on infinity to actually exist. There are two ways of viewing this.
If we want the limit of the number of people on the train as we approach the infinitith stop, this series is +2-1+2-1..., so a number of people with the same cardinality as the integers.
The second and ... |
I was wondering if you could help...
I have Math homework, I was hoping if you could check my answer.
Find the equation of the line perpendicular to 3x+2y-4=0 going through point (2,-3)
y = ( -3x + 4 ) /2
y = 2
x = - ( 2 ( y - 2 ) / 3 )
x = -3
Therefore my equation is correct?
th... |
I am a statistician who wishes to learn real analysis in order to better understand the foundations of statistics. With that aim in mind I plan to go through Rudin's classic on "Principles of Mathematical Analysis".
Given the above context can I skip chapter 1? It seems to me that the material in chapter 1 is not a... |
Can I skip the first chapter in Rudin's Principles of Mathematical Analysis? |
Is there any theory (analogous to Galois theory) for solving equations with irrational exponents like:
$ x^{\sqrt{2}}+x^{\sqrt{3}}=1$
? |
I recently learned that on Riemannian manifolds, one can define the _sectional curvature_ (<http://en.wikipedia.org/wiki/Sectional_curvature>) of a (2-dimensional) plane section. I was wondering if a similar concept exists for higher dimensional "space sections."
Here is what got me thinking about this: For 2-dimen... |
I would like to know something like what's the first PDE etc.
Could you recommend book on the history of PDE/ODE?
thanks. |
Book recommendation on the history of PDE/ODE? |
Now that you've gotten the answer, here is a little bit of a post mortem clarifying the essential points, as may come up again in future problems. (In general, once you've solved a problem, it's a really good idea to ask for the "moral(s)". If you can't identify one, how are you better off now than before you solved ... |
I have a matrix F = X * D * X', where D is m x m and diagonal and X is n x m.
Now, I compute F^-1.
Is there an efficient method to update F^-1 if D is updated by D = D + G, where G is a sparse diagonal matrix? |
I'm practicing some problems from past exams and found this one: Find nth derivative of this function:
$f(x)=\frac {x} {x^2-1}$.
I have no idea how to start solving this problems. Is there any theorem for finding nth derivative? |
There are some math quizzes like:<blockquote>
find a function $\phi:\mathbb{R}\rightarrow\mathbb{R}$
such that $\phi(\phi(x)) = f(x) \equiv x^2 + 1.$
</blockquote>
If such $\phi$ exists (it does in this example), $\phi$ can be viewed as a "square root" of $f$ in the sense of function composition because $\phi\circ... |
Fundamental group of GL(n,C) is isomorphic to Z. How to learn to prove facts like this? |
Being motivated by [this post](http://math.stackexchange.com/questions/3618/solving-an-equation-with-irrational-exponents), I was wondering if there is a proof (analogous to the case of Diophantine equations) that there is no general method for solving [transcendental equations](http://en.wikipedia.org/wiki/Transcenden... |
Is there a proof that there is no general method to solve transcendental equations? |
I have a matrix $F=XDX^T$, where $D$ is $m\times m$ and diagonal and $X$ is $n\times m$.
Now, I compute $F^{-1}$.
Is there an efficient method to update $F^{-1}$ if $D$ is updated by $D'=D+G$, where $G$ is a sparse diagonal matrix? |
Let $S$ be some finite set. Let $Sym(S)$ be the symmetric group on $S$. Let $A$ be the subset of permutations $\sigma$ that are such that there exist $a,b,c \in S$ such that $\sigma(a) = b, \sigma (b) = c$ and $\sigma (c) =a$ and for every other element of $S$, the action of $\sigma$ is the identity. In short, let $A$ ... |
How do you prove cycles of length 3 don't generate the whole symmetric group? |
In one of my classes we discussed the ring of 2x2 matrices $M_{2}(\mathbb{Z})$. We said that its group of units was $GL_{2}(\mathbb{Z})$ which means that it is the set of 2x2 with determinant equal to $\pm$1.
Why can't we have a 2x2 matrix with entries a,b,c, and d such that $\frac{d}{ad-bc}$,$\frac{-b}{ad-bc}$,$\frac... |
Maybe we can add some more help -just in case you didn't succeed to find the answer yourself yet. Let
$$
\frac{x}{x^2 - 1} = \frac{A}{x-1} + \frac{B}{x+1}
$$
be the splitting into partial fractions. (I'm too lazy to compute the coeffitients $A$ and $B$.) Then
$$
\frac{d}{dx} \frac{x}{x^2 - 1} = -\frac{A}{(... |
Here's a start to figuring out the formulas. The first thing is to understand the map $SU(2)\times SU(2)\to SO(4)$. Let $std$ denote the standard 2-dimensional complex representation of $SU(2)$. Then the external tensor product $V=std\otimes std$ is a natural 4-dimensional complex representation of $G=SU(2)\times SU(2)... |
I can think of the set of bounded, continuous functions from R -> R as a group, with composition as addition of functions. Since this set is path connected, the group is continuous, i.e. it is a lie group.
What is the lie algebra of this group?
I am trying to use this approach to find a relationship between Li... |
I can think of the set of bounded, continuous functions from R -> R as a group, with composition as addition of functions. {In other words, "group multiplication" of two bounded continuous functions f(x) and g(x) is the sum f(x) + g(x)} Since this set is path connected, the group is continuous, i.e. it is a lie group... |
<a href="http://math.stackexchange.com/questions/3649/how-do-you-prove-cycles-of-length-3-dont-generate-the-whole-symmetric-group">Why don't 3-cycles generate the symmetric group?</a> was asked earlier today. The proof is essentially that 3-cycles are even permutations, and products of even permutations are even.
So... |
What do all the $k$-cycles in $S_n$ generate? |
I can think of the set of bounded, continuous functions from R -> R as a group, with composition as addition of functions. {In other words, this group has rule that the composition of two elements in the group, f(x) and g(x), is their point-wise sum f(x) + g(x)} Since this set is path connected, the group is continuo... |
What is the value of 1^i? |
What is the value of 1^i? |
I can think of the set of bounded, continuous functions from R -> R as a group, with composition as addition of functions. {In other words, this group has the rule that the composition of two elements in the group, f(x) and g(x), is their point-wise sum f(x) + g(x)} Since this set is path connected, the group is cont... |
Let there be given some conditionally convergent infinite series S. Then let R be some real number, and Q<sub>k</sub> a rearrangement of S such that the sum is equal to R.
Is Q<sub>k</sub> unique? In other words, is there some other rearrangement W<sub>k</sub> for which the sum of S is also R?
I believe the ans... |
Here's a start to figuring out the formulas. The first thing is to understand the map $SU(2)\times SU(2)\to SO(4)$. Let $std$ denote the standard 2-dimensional complex representation of $SU(2)$. Then the external tensor product $V=std\otimes std$ is a natural 4-dimensional complex representation of $G=SU(2)\times SU(2)... |
To be more precise, I am interested in knowing **if the intuition that *a Euclidean zero vector does not have a particular direction* is actually correct, and if there is a rigorous formulation that would back it up.**
[Wikipedia's entries][1] on the [zero vector][2] seems to agree with that intuition, but as always... |
What is the Direction of a Zero (Null) Vector? |
Consider the following data points: 1, 1, 2, 3, 4
I understand that...
the average is the total of the numbers divided by the count of numbers in the set, the median is the central value based on location in the set, the mode is the value occurring most often in the set, and the midrange is highest and lowest value... |
Is there a name (and use) for an average based on the unique values of a set of data? |
In one of my classes we discussed the ring of 2x2 matrices $M_{2}(\mathbb{Z})$. We said that its group of units was $SL_{2}(\mathbb{Z})$ which means that it is the set of 2x2 with determinant equal to $\pm$1.
Why can't we have a 2x2 matrix with entries a,b,c, and d such that $\frac{a}{ad-bc}$,$\frac{-b}{ad-bc}$,$\frac... |
I'm relatively new to ring theory so this is probably a simple question.
Its easy to see that a finite ring that is commutative and has no-zero divisors (i.e. an integral domain) must have multiplicative inverses.
I am wondering if we can rearrange these properties and always get implication or produce finite rin... |
When does a finite ring become a finite field? |
I'm relatively new to ring theory so this is probably a simple question.
Its easy to see that a finite ring that is commutative and has no-zero divisors (i.e. an integral domain) must have multiplicative inverses.
I am wondering if we can rearrange these properties and always get implication or produce finite rin... |
I'm relatively new to ring theory so this is probably a simple question.
Its easy to see that a finite ring that is commutative and has no-zero divisors (i.e. an integral domain) must have multiplicative inverses.
I am wondering if we can rearrange these properties and always get implication or produce finite rin... |
How can I find the number of *k*-permutations of *n* objects, where there are *x* types of objects, and r<sub>1</sub>, r<sub>2</sub>, r<sub>3</sub> ... r<sub>x</sub> give the number of each type of object?
Example:
> I have 20 letters from the alphabet. There are some duplicates - 4 of them are *a*, 5 of them are... |
Is anything known about these integrals? Textbook suggestions are welcome
$f(n,p)=\int_{x=-0.5}^p \frac{n!}{n!(n-x)!} dx$
$n>0, p\le n$
For instance, as n grows $f(n,n+1/2)-2^n$ seems to be positive and monotonically decreasing...how would I find the limit?
<pre>
ListPlot[Table[2^n - NIntegrate[n!/(x! (n -... |
Yes. The problem is that the term "transcendental function" is absurdly general. For example, pick a computable bijection between the integers and the set of <a href="http://en.wikipedia.org/wiki/Turing_machine">Turing machines</a> and consider the function $f : \mathbb{R} \to \mathbb{R}$ which is $0$ on the integers... |
As Chandru1 stated tentatively, the same arguments apply for $0< p<1$ as for $1\leq p<\infty$, so it is no surprise that a separate proof is hard to find. For example, an introduction to $L^p$ spaces on subsets of $\mathbb{R}^n$ can be found in Chapter 8 of *Measure and integral* by Wheeden and Zygmund. A proof of s... |
Is anything known about these integrals? Textbook suggestions are welcome
$f(n,p)=\int_{x=-0.5}^p \frac{n!}{n!(n-x)!} dx$
$n>0, p\le n$
For instance, as n grows $2^n-f(n,n+1/2)$ seems to be positive and monotonically decreasing...how would I find the limit?
<pre>
ListPlot[Table[2^n - NIntegrate[n!/(x! (n -... |
i. If it rains next Sunday, then Andy will only go to the football game if Bob does.
ii. If Andy will only go to the foot ball game on Sunday if it rains, then Bob will definitely go to the Sunday football game regardless of the weather.
My gut feeling is they are not equivalent. I chose the following three predi... |
Are these statements logically equivalent? |
it's 1. 1 to the power of anything is 1.
**Edit:** I'll elaborate. In defining $a^b$, we know intuitively what to do in certain cases. When $a$ is a positive integer and $b$ is an integer, for example. This definition is easily extended to when $a$ is real and positive, and $b$ is a real number. In the case where $a... |
it's 1. 1 to the power of anything is 1.
**Edit:** I'll elaborate. In defining $a^b$, we know intuitively what to do in certain cases. When $a$ is a positive integer and $b$ is an integer, for example. This definition is easily extended to when $a$ is real and positive, and $b$ is a real number. In the case where $a... |
I tried to find some reference on the net but couldn't find a good one. What is the advantage of pivoting in LU decomposition over regular LU decomposition? Is it something to do with matrix singular or not? |
I know that 1 is not a prime number because $1\cdot\mathbb Z=\mathbb Z$ is, by convention, not a prime ideal in the ring $\mathbb Z$.
However, since $\mathbb Z$ is a domain, $0\cdot\mathbb Z=\\{0\\}$ *is* a prime ideal in $\mathbb Z$. Shouldn't this imply that 0 is a prime number?
(I know that this would violate ... |
Shouldn't 0 be a prime number? At least 0 is a prime ideal in Z. |
I know that 1 is not a prime number because $1\cdot\mathbb Z=\mathbb Z$ is, by convention, not a prime ideal in the ring $\mathbb Z$.
However, since $\mathbb Z$ is a domain, $0\cdot\mathbb Z=\\{0\\}$ *is* a prime ideal in $\mathbb Z$. Isn't $(p)$ being a prime ideal the very definition of $p$ being a prime element?
... |
Why doesn't 0 being a prime ideal in Z imply that 0 is a prime number? |
I know that 1 is not a prime number because $1\cdot\mathbb Z=\mathbb Z$ is, by convention, not a prime ideal in the ring $\mathbb Z$.
However, since $\mathbb Z$ is a domain, $0\cdot\mathbb Z=\\{0\\}$ *is* a prime ideal in $\mathbb Z$. Isn't $(p)$ being a prime ideal the very definition of $p$ being a prime element?
... |
I know that 1 is not a prime number because $1\cdot\mathbb Z=\mathbb Z$ is, by convention, not a prime ideal in the ring $\mathbb Z$.
However, since $\mathbb Z$ is a domain, $0\cdot\mathbb Z=\\{0\\}$ *is* a prime ideal in $\mathbb Z$. Isn't $(p)$ being a prime ideal the very definition of $p$ being a prime element?
... |
Let's take a look at the following integrals :
1) $\displaystyle \int\limits_{0}^{1} \frac{\log{x}}{1+x} \ dx = -\frac{\pi^{2}}{12} = -2 \sum\limits_{n=1}^{\infty} \frac{1}{n^2}= \zeta(2)$
2) For $c<1$ $\displaystyle \int\limits_{0}^{\frac{\pi}{2}} \arcsin(c \cos{x}) \ dx = \frac{c}{1^2} + \frac{c}{3^2} + \frac... |
Say we have a set of $n \times n$ matrices $M_i$ for $1 \leq i \leq k$, elements in $\mathbb{R}$. Suppose that for every $\boldsymbol{\lambda} = (\lambda_1, \dots , \lambda_k) \in \mathbb{R}^k$ we have that the kernel of $M_{\boldsymbol{\lambda}} = \sum_i \lambda_i M_i$ is nontrivial. Does it follow that there exists... |
Let's take a look at the following integrals :
1) $\displaystyle \int\limits_{0}^{1} \frac{\log{x}}{1+x} \ dx = -\frac{\pi^{2}}{12} = -2 \sum\limits_{n=1}^{\infty} \frac{1}{n^2}= \zeta(2)$
2) For $c<1$ $\displaystyle \int\limits_{0}^{\frac{\pi}{2}} \arcsin(c \cos{x}) \ dx = \frac{c}{1^2} + \frac{c}{3^2} + \frac... |
Say we have a set of symmetric $n \times n$ matrices $M_i$ for $1 \leq i \leq k$, elements in $\mathbb{R}$. Suppose that for every $\boldsymbol{\lambda} = (\lambda_1, \dots , \lambda_k) \in \mathbb{R}^k$ we have that the kernel of $M_{\boldsymbol{\lambda}} = \sum_i \lambda_i M_i$ is nontrivial. Does it follow that th... |
First, if you're talking about a six-digit number, the order of the digits matters, so it's generally not referred to as a combination.
Consider how many possibilities there are for the first digit. As you said, it cannot be zero, or you wouldn't have a six-digit number. Once there is a first digit, how many possi... |
The Wikipedia article gives a
<a href="http://en.wikipedia.org/wiki/Riemann_zeta_function#Mellin_transform"> Mellin transform </a>
$$\Gamma(s)\zeta(s) =\int_0^\infty\frac{x^{s-1}}{e^x-1} dx.$$
The <a href="http://en.wikipedia.org/wiki/Riemann_zeta_function#Reciprocal">Dirichlet series</a> over the Möbius functi... |
I know that 1 is not a prime number because $1\cdot\mathbb Z=\mathbb Z$ is, by convention, not a prime ideal in the ring $\mathbb Z$.
However, since $\mathbb Z$ is a domain, $0\cdot\mathbb Z=\\{0\\}$ *is* a prime ideal in $\mathbb Z$. Isn't $(p)$ being a prime ideal the very definition of $p$ being a prime element?
... |
The reason one permits zero as a prime ideal is because it facilitates useful reductions. For example, in many ring theoretic problems involving an ideal $\rm\; I\;$, one can reduce to the case $\rm\;I = P\;$ prime, and then reduce to $\rm\;R/P\;$, thus reducing to the case when the ring is a domain. In this case one s... |
Consider the piecewise constant function $\psi:I=[-1,1] \rightarrow \mathbb{R}$ given by
$$\psi(x) = \begin{cases} \psi_1 & x \leq 0, \\ \psi_2 & x > 0 \end{cases}$$
for some constants $\psi_1, \psi_2 \in \mathbb{R}$. I would like to evaluate the integral
$$\int_I \delta(x) \psi(x) dx$$
where $\delta$ is ... |
Integrate product of Dirac delta and discontinuous function? |
Suppose you have 100 coins. 96 of them are heavy and 4 of them are light. Nothing is known regarding the proportion of their weights. You want to find at least one genuine (heavy) coin. You are allowed to use a weight balance twice. How do you find it? |
Consider the piecewise constant function $\psi:I=[-1,1] \rightarrow \mathbb{R}$ given by
$$\psi(x) = \begin{cases} \psi_1 & x \leq 0, \\ \psi_2 & x > 0 \end{cases}$$
for some constants $\psi_1, \psi_2 \in \mathbb{R}$. I would like to evaluate the integral
$$\int_I \delta(x) \psi(x) dx$$
where $\delta$ is ... |
To avoid wasting readers' time when posting puzzles, it is useful (and will improve the upvote/downvote ratio) to provide:
1. *an indication of the difficulty level*. Was this puzzle in a math-is-fun book for children, or is it a generalization of a problem from the final round of a national olympiad? Having school... |
To avoid wasting readers' time when posting puzzles, it is useful (and will improve the upvote/downvote ratio) to provide:
1. *an indication of the difficulty level*. Was this puzzle in a math-is-fun book for children, or is it a generalization of a problem from the final round of a national olympiad? Having school... |
Consider the piecewise constant function $\psi:I=[-1,1] \rightarrow \mathbb{R}$ given by
$$\psi(x) = \begin{cases} \psi_1 & x \leq 0, \\ \psi_2 & x > 0 \end{cases}$$
for some constants $\psi_1, \psi_2 \in \mathbb{R}$. I would like to evaluate the integral
$$\int_I \delta(x) \psi(x) dx$$
where $\delta$ is ... |
Suppose you have 100 coins. 96 of them are heavy and 4 of them are light. Nothing is known regarding the proportion of their weights. You want to find at least one genuine (heavy) coin. You are allowed to use a weight balance twice. How do you find it?
Assumptions:
Heavy coins all have the same weight; same for... |
Suppose $\Omega$ is a domain of the complex plane (i.e. an open and connected subset of the plane). Suppose $f$ is holomorphic on $\Omega$, and $f$ is not identically zero.
Suppose $f$ has a holomorphic logarithm on $\Omega$, which means that there is a function $g$ holomorphic on $\Omega$ such that $e^g=f$. Then it... |
Is anything known about these integrals? Textbook suggestions are welcome
$f(n,p)=\int_{x=-0.5}^p \frac{n!}{n!(n-x)!} dx$
$n>0, p\le n+0.5$
For instance, as n grows $2^n-f(n,n+1/2)$ seems to be positive and monotonically decreasing...how would I find the limit?
<pre>
ListPlot[Table[2^n - NIntegrate[n!/(x! ... |
Let me elaborate on [whuber's answer][1] by putting it in a slightly different form.
Let's randomly choose 5 coins out of the 100. Let's denote this set of coins by $C$. Since there are only 4 light coins, **we are absolutely sure that $C$ contains at least one heavy coin.** Now, the problem is reduced to simply fi... |
Do you have any favorite open problem?
Let me mention one of my favorites. Let $A(\mathbb{T})$ be the Wiener algebra, that is, the linear space of absolutely convergent Fourier series on the unit circle $\mathbb{T}$ $(=\mathbb{R}/2\pi\mathbb{Z}=(-\pi,\pi])$ carrying the norm
$$ f\mapsto \|f\|=\sum_{n\in\mathbb{Z}}... |
Favourite open problem? |
Just a remark: there is some relevant basic algebraic topology here.
As Matt E points out in his answer, the "direction" of a vector $v \in \mathbb{R}^n$ is not a number. What then is it? A natural answer is that a direction is an element of the unit sphere $S^{n-1} = \{(x_1,\ldots,x_n) \in \mathbb{R}^n \ | \ x_1^... |
Just a remark: there is some relevant basic algebraic topology here.
As Matt E points out in his answer, the "direction" of a vector $v \in \mathbb{R}^n$ is not a number. What then is it? A natural answer is that a direction is an element of the unit sphere $S^{n-1} = \{(x_1,\ldots,x_n) \in \mathbb{R}^n \ | \ x_1^... |
When I first took linear algebra, we never learned about dual spaces. Today in lecture we discussed them and I understand what they are and everything, but I don't really understand why we want to study them.
I was wondering if anyone knew a nice intuitive motivation for the study of dual spaces and whether or not t... |
Why do we care about dual spaces? |
This problem first came to me in high school, and a couple times since, and I even assigned it for extra credit in one of my calculus classes after I became a teacher. So I know the solution. What I am looking for is other WAYS to obtain the solution. I've been told there exists a solution using only arithmetic, but... |
Does favorite allow me two - both about convex 3-dimensional polyhedra?
(Shephard's Conjecture)
Is it possible to cut along the edges of some spanning tree of any convex 3-dimensional polyhedron and unfold the cut open polyhedron so that the faces do not overlap (and one gets a simple polygon where the uncut edge... |
**Update:** It seems that [Moron][1] has a valid objection (*see the comments*) that renders my solution incorrect:
> Two sides are equal: You could have 2
> heavy and 2 light. Either all light or
> all heavy is incorrect.
In other words:
> Yaser's answer misses one case. Where
> we get H+L vs H+L giving us... |
When I first took linear algebra, we never learned about dual spaces. Today in lecture we discussed them and I understand what they are and everything, but I don't really understand why we want to study them within linear algebra.
I was wondering if anyone knew a nice intuitive motivation for the study of dual space... |
When I first took linear algebra, we never learned about dual spaces. Today in lecture we discussed them and I understand what they are and everything, but I don't really understand why we want to study them within linear algebra.
I was wondering if anyone knew a nice intuitive motivation for the study of dual space... |
Let $V$ be a vector space (over any field, but we can take it to be $\mathbb R$ if you like,
and for concreteness I will take the field to be $\mathbb R$ from now on;
everything is just as interesting in that case). Certainly one of the interesting concepts
in linear algebra is that of a *hyperplane* in $V$.
... |
When implicitly finding the derivative of:
>xy^3 - xy^3*sin(x) = 1
How do you find the implicit derivative of:
>xy^3*sin(x)
Is it using a *triple* product rule of sorts? |
Implicit Diferentiation... the triple product rule? |
I have been fascinated by this problem since I first heard about it in high school. From the Wikipedia article http://en.wikipedia.org/wiki/Collatz_problem:
>Take any natural number n. If n is even, divide it by 2 to get n / 2, if n is odd multiply
>it by 3 and add 1 to obtain 3n + 1. Repeat the process indefinite... |
It is possible to derive a 'triple product rule.' In fact, you can generalize it to an arbitrary number of products. You may ask why these rules aren't presented in these general forms. One reason for this is that writing the general formula out is a more complicated to memorize and more intimidating towards students j... |
It is possible to derive a 'triple product rule.' In fact, you can generalize it to an arbitrary number of products. You may ask why these rules aren't presented in these general forms. One reason for this is that writing the general formula out is a more complicated to memorize and more intimidating towards students j... |
I think the idea of convexity makes more sense in higher dimensions. I think you would agree that a sphere or any other convex shape fits your intuitive idea of what "convex" is (it bends outwards instead of inwards, or lines between two points in the shape are totally inside the shape).
An interesting way to think ... |
I think the idea of convexity makes more sense in higher dimensions. I think you would agree that a sphere or any other convex shape fits your intuitive idea of what "convex" is (it bends outwards instead of inwards, or lines between two points in the shape are totally inside the shape).
An interesting way to think ... |
I think the idea of convexity makes more sense in higher dimensions. I think you would agree that a sphere or any other convex shape fits your intuitive idea of what "convex" is (it bends outwards instead of inwards, or lines between two points in the shape are totally inside the shape).
An interesting way to think ... |
I think the idea of convexity makes more sense in higher dimensions. I think you would agree that a sphere or any other convex shape fits your intuitive idea of what "convex" is (it bends outwards instead of inwards, or lines between two points in the shape are totally inside the shape).
An interesting way to think ... |
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