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Explain why calculating divergent series could cause paradox? |
Explain why calculation divergent series could cause paradox? |
Explain why calculating divergent series could cause paradox? |
Consider the closed interval $[0,1]$, there is $\frac{2}{3} \in [0,1]$ where $p=2$ and $q=3$. Similarly consider $[2,3]$, one can have $\frac{5}{2} \in [2,3]$ where $p=5$ and $q=2$. Does every interval of the form $[a,b]$, where $a,b \in \mathbb{R}$ contain a rational of this kind. If yes, how can we prove it? |
Claim if our hypothesis holds, $a \equiv b \ (\text{mod}\ p)$ for any prime $p$.
Proof:
Find $n$ so that $n \equiv -a \ (\text{mod}\ p)$ and $n \equiv 1 (\text{mod}\ p-1)$ ( we can do this by **Chinese Remainder Theorem**). Then
$$a^{n} + n \equiv a^1 + n = a - a = 0 (\text{mod} \ p)$$
Therefore since $a^n... |
As Chandru1 pointed out, a beautiful particular case of Cauchy's integral formula gives us a very particular *winding number*. For the sake of simplicity, let us look only at closed paths on the boundary of the unit disc, $\gamma : [0,1] \longrightarrow S^1$. Then, the *winding number* of $\gamma$ is the number of time... |
Say you have a set of points and joining lines as show below:
![t junctions are at D, F, and E][1]
In this diagram of triangles, T-(or Y-like) junctions are at D, F, and E.
At vertex F for example, there is a triangle FDE, but no line FJ. If there was a line FJ, then there would be no T junction at F. Two tri... |
How do you find T-Junctions in an array of triangles? |
<a href="http://math.stackexchange.com/questions/4084/how-do-you-find-t-junctions-in-an-array-of-triangles">Follow up to</a>
Say I have triangles as shown:
![triangles][1]
Its easy to see that ABC is adjacent to ACJ: ABC and ACJ share 2 vertices (A and C).
However FEC and ACJ are __also__ adjacent, but onl... |
How do I find if two triangles are adjacent, when they differ in size? |
How do you show that $l_p \subset l_q$ for $p \leq q$? |
I'm trying to show that all graphs with 5 vertices, each of degree 2, are isomorphic to each other. Is there a more clever way than simply listing them all out? |
I'm assuming you do not allow multi-edges, as otherwise there is a trivial counterexample (the cycle of length $5$, versus a disconnected graph consisting of a triangle and two vertices joined by two edges).
Pick one vertex $v_0$; it must be joined to two other distinct vertices, which we may call $v_{-1}$ and $v_1$... |
Given sample data $x_1, \ldots, x_n$ generated from a probability distribution $f(x|\theta)$ ($\theta$ being an unknown parameter), a statistic $T(x_1, \ldots, x_n)$ of the sample data is called sufficient if $f(x|\theta, t) = f(x|t)$.
However, I'm always kinda confused by this definition, since I think of a suffici... |
What you apparently want is a monotonic interpolation scheme. As mentioned already, fitting a single polynomial to your data will result in an interpolant that does not necessarily satisfy your constraints. Using a piecewise polynomial function (a function that has a different polynomial representation at each interval... |
What's a good way of solving the Diffie-Hellman problem when those exchanging the message have chosen a low primitive root $g$?
Of course you could brute force it but I'm interested in knowing whether there is a formula for solving it when you know $g^a \pmod{p}$ and $g^b \pmod{p}$ as well as $p$ and $g$ of course. |
Suppose I have a non-negative real-valued function over $d$-dimensional real vectors as follows
$$f(\mathbf{x})=\exp(-\mathbf{x} A \mathbf{x})$$
Where $A$ is some positive definite $d\times d$ matrix. What is the normalization factor to turn this into a valid density over the following set?
$$S_d=\\{(x_1,\ldot... |
What are some examples of theories stronger than Presburger Arithmetic but weaker than Peano Arithmetic? Are all such theories decidable? If not, by what methods other than Gödelization can undecidability be established? |
What are some examples of theories stronger than Presburger Arithmetic but weaker than Peano Arithmetic? |
What's a good way of solving the Diffie–Hellman problem when those exchanging the message have chosen a low primitive root $g$?
Of course you could brute force it but I'm interested in knowing whether there is a formula for solving it when you know $g^a \pmod{p}$ and $g^b \pmod{p}$ as well as $p$ and $g$ of course.
... |
What's a good way of solving the Diffie–Hellman problem when those exchanging the message have chosen a low primitive root $g$?
Of course you could brute force it but I'm interested in knowing whether there is a formula for solving it when you know $g^a \pmod{p}$ and $g^b \pmod{p}$ as well as $p$ and $g$ of course.
... |
What you apparently want is a monotonic interpolation scheme. As mentioned already, fitting a single polynomial to your data will result in an interpolant that does not necessarily satisfy your constraints. Using a piecewise polynomial function (a function that has a different polynomial representation at each interval... |
In the definition of topological space, we require the intersection of a finite number of open sets to be open while we require the arbitrary union of open sets to be open. why is this?
I'm assuming this has something to do with the following observation: $\cap_{n=1}^{\infty} [0,\frac{1}{n}] = {0}$ and there is some... |
Why do we require a topological space to be closed under finite intersection? |
Minor correction: $[0,1/n]$ is not open in the usual topology of the real line. But you could get the same thing with $\cap_{n=1}^\infty (-1/n,1/n)=${0}. And yes, points are not open in the usual topology on the line. If they were, then every set, being the union of one-point sets, would be open. This is called the... |
In the definition of topological space, we require the intersection of a finite number of open sets to be open while we require the arbitrary union of open sets to be open. why is this?
I'm assuming this has something to do with the following observation: $\cap_{n=1}^{\infty} (0,\frac{1}{n}) = {0}$ and there is some... |
You need to think about what the intuition behind open sets are. One way to think about it is through neighborhoods: an open set is a set which is a neighborhood of each of its points. What is a neighborhood of a point? A neighborhood of a point $x$ is a set that contains of all points that are "sufficiently close" to ... |
In the definition of topological space, we require the intersection of a finite number of open sets to be open while we require the arbitrary union of open sets to be open. why is this?
I'm assuming this has something to do with the following observation: $\cap_{n=1}^{\infty} (-\frac{1}{n},\frac{1}{n}) = {0}$ and th... |
<strike>Minor correction: $[0,1/n]$ is not open in the usual topology of the real line. But you could get the same thing with</strike> The example $\cap_{n=1}^\infty (-1/n,1/n)=${0} is a good one. And yes, points are not open in the usual topology on the line. If they were, then every set, being the union of one-poi... |
In the definition of topological space, we require the intersection of a finite number of open sets to be open while we require the arbitrary union of open sets to be open. why is this?
I'm assuming this has something to do with the following observation: $\cap_{n=1}^{\infty} (-\frac{1}{n},\frac{1}{n}) = \\{0\\}$ an... |
Suppose I have a non-negative real-valued function over $d$-dimensional real vectors as follows
$$f(\mathbf{x})=\exp(-\mathbf{x}' A \mathbf{x})$$
Where $A$ is some positive definite $d\times d$ matrix. What is the normalization factor to turn this into a valid density over the following set?
$$S_d=\\{(x_1,\ldo... |
Given $C=XF$, and only data for $C$, solving for $X$ and $F$ has been described as [not tractable][1] (which I'm content to believe, but others aren't). If that's the case, where does the following argument fall apart?
Suppose $C=XF$ is written:
$\begin{bmatrix}
C_1 \\\
\vdots \\\
C_m
\end{bmatrix} = \b... |
Given $C=XF$, and only data for $C$, solving for $X$ and $F$ has been described as [not tractable][1] (which I'm content to believe, but others aren't). If that's the case, where does the following argument fall apart?
Suppose $C^i=XF^i$ is written:
$\begin{bmatrix}
C_1^i \\\
\vdots \\\
C_m^i
\end{bmatr... |
In the comments to the question: [$a^n+n|b^n+n \Longrightarrow a=b$][1], there was a claim that $5^n+n$ is never prime (for integer $n>0$).
It does not look obvious to prove, nor have I found a counter example.
Is this really true?
[1]: http://math.stackexchange.com/questions/3978/ann-mid-bnn-longrightarro... |
$5^n+n$ is never prime? |
The division is always possible, which can be proved by complete induction.
The case n=1 is trivial - one stick of length 1. Then we suppose the assumption holds for n and consider the case n+1. I.e. given a number of sticks of integral length >= n+1 whose lengths add to (n+1)(n+2)/2 - can these be divided into stic... |
What's a good way of solving the Diffie–Hellman problem when those exchanging the message have chosen a low primitive root $g$ (e.g. $g=3$)?
Of course you could brute force it but I'm interested in knowing whether there is a formula for solving it when you know $g^a \pmod{p}$ and $g^b \pmod{p}$ as well as $p$ and $g... |
A general rule-of-thumb for "is there a prime of the form f(n)?" questions is, unless there exists a set of small divisors D, called a *covering set,* that divide every number of the form f(n), then there will eventually be a prime. See, e.g. [Sierpinski numbers][1].
Running WinPFGW (it should be available from the... |
Relatively simple, but still fun...
Unknown source, other than my friend telling me yesterday
There is a room with 50 doors in it, each door leading to a cell that is completely sound-proof and light-proof, and each cell can hold a maximum of 1 person. There are 50 prisoners. There also exists a light in the mi... |
The following is the start of basic corollary in my logic text:
For any set $\Gamma$ of sentences, $\Gamma\subseteq\text{Th}(\text{Mod}(\Gamma))$.
What happens when $\text{Mod}(\Gamma)$ is empty? Suppose $\Gamma=\{p_1,\neg p_1\}$, so that there is no truth value $V$ which models $\Gamma$, or otherwise $V(p_1)=\t... |
Does this proposition hold if $\text{Mod}(\Gamma)=\emptyset$? |
Quinn and Hannan's [The estimation and tracking of frequency][1] is dedicated to this topic. I can highly recommend it.
[1]: http://books.google.com.au/books?id=mTluO2TZ8V8C&dq=estimation+and+tracking+of+frequency&printsec=frontcover&source=bn&hl=en&ei=4s2ETMuYIYqHcYa-pNAL&sa=X&oi=book_result&ct=result&resnum=4&... |
According to the definitions of [algebra over a ring][1] and [algebra over a field][2], are [Elementary Algebra][3] and [Boolean Algebra][4] examples of algebra over a ring or algebra over a field? How can one show that? What are the ring or field that the algebras are with respect to?
Thanks and regards!
[1]... |
Are Elementary Algebra and Boolean Algebra Algebras over a Ring (or Field)? |
Suppose I have a non-negative real-valued function over $d$-dimensional real vectors as follows
$$f(\mathbf{x})=\exp(-\mathbf{x}' A \mathbf{x})$$
Where $A$ is some symmetric positive definite $d\times d$ matrix. What is the normalization factor to turn this into a valid density over the following set?
$$S_d=\\... |
for somebody having a quite strong background in Mathematics, which are some good books for the domain of Operations research? I guess there are textbooks covering topics like linear and nonlinear optimization, convex optimization and quadratic programming, dynamic programming, multicriterial optimizations (did I miss ... |
I have a set containing numbers 1 to 49{1,2,3...49}. Now, i want to devide the set into 7
subsets such that each subset should contain 7 elements and sum of the elements
of each subset should be 175. Is it possible to prove that such subsets exist?
|
I have a set containing numbers $1$ to $49$ $\\{1,2,3, \cdots, 49\\}$.
Now, I want to divide the set into $7$ subsets such that each subset should contain $7$ elements and sum of the elements of each subset should be $175$.
Is it possible to prove that such subsets exist?
|
Suppose I have a non-negative real-valued function over $d$-dimensional real vectors as follows
$$f(\mathbf{x})=\exp(-\mathbf{x}' A \mathbf{x})$$
Where $A$ is some symmetric positive definite $d\times d$ matrix. What is the normalization factor to turn this into a valid density over the following set?
$$S_d=\\... |
First count the pairs with $\gcd(a,b)=n$ counting $(a,b)$ and $(b,a)$ as
distinct. You have $a=\prod p_i^{r_i}$ and $b=\prod p_i^{s_i}$
and need $\max(r_i,s_i)=e_i$ for all $i$. So, given a positive integer $e$,
how many pairs $(r,s)$ of nonnegative integers are there with $\max(r,s)=e$? |
Each of n persons draws a card from a shuffled deck of k cards numbered from 1 to k. There are at least as many cards as persons. The winner is the person who is holding the largest card. If everyone is honest, how can they mutually determine the identity of the winner, without any person learning additional informa... |
<sup>(This question is inspired by the [StackOverflow question *Converting prime numbers*](http://stackoverflow.com/questions/3653475/converting-prime-numbers) by [@Manas](http://stackoverflow.com/users/158163/manas), which turns out to be the [problem `PPATH` of SPOJ](http://www.spoj.pl/problems/PPATH/).)</sup>
Let... |
This question isn't about the actual substance of math so much as it is about style. I'm not going to school, so my feedback on these sorts of problems is limited. How would you improve this proof's readability? And if this isn't the sort of question that's suitable here, where would be a more appropriate venue? Th... |
How can I write this proof about partial derivatives better? |
If F and G are functors between two arbitrary categories C and D, does a [natural transformation][1] η from F to G always exists? What is the condition for its existence?
Thanks and regards!
[1]: http://en.wikipedia.org/wiki/Natural_transformation |
I've marked this as "community wiki" for myself...
Your argument is a bit vague in the first paragraph; you have an indefinite integral, but what you actually want is the function; that is, you want $f(x,y_1)=C$. It would be better to use the Fundamental Theorem of Calculus, and write something along these lines:
... |
I've marked this as "community wiki" for myself...
Your argument is a bit vague in the first paragraph; you have an indefinite integral, but what you actually want is the function; that is, you want $f(x,y_1)=C$. **Edit:** The problem with this is that, if you recall, the indefinite integral does not produce a funct... |
I just started to learn mathematical logic. I'm a graduate student. I need a book with relatively more examples. Any recommendation? |
Good books on mathematical logic? |
I asked this question on mathoverflow, but it was deemed too simple, so I'm posting here instead --
Is there a nice way to characterize an orthonormal basis of eigenvectors of the following $d\times d$ matrix?
$$\mathbf{I}-\frac{1}{d} \mathbf{v}\mathbf{v}'$$
Where $\mathbf{v}$ is a $d\times 1$ vector of 1's. ... |
Posting of random "tasks", with no sources to indicate difficulty or correctness of the problem, can waste a lot of users' time, and is subject to downvotes.
This one is a problem from a journal or from competitions at the level of the Putnam contest (see reference below).
Hint: $g(x) = x + Af(x)$ satisfies ... |
I'm trying to construct a simple proof of the Propositional Interpolation Theorem. For the following, let $At(\phi)$ be the set of sentence symbols that occur in a sentence $\phi$. Suppose that $\varphi$ is a tautological consequence of $\phi$, but neither $\neg\phi$ nor $\varphi$ is a tautology, and so $\varphi$ is no... |
I asked this question on mathoverflow, but it was deemed too simple, so I'm posting here instead --
Is there a nice way to characterize an orthonormal basis of eigenvectors of the following $d\times d$ matrix?
$$\mathbf{I}-\frac{1}{d} \mathbf{v}\mathbf{v}'$$
Where $\mathbf{v}$ is a $d\times 1$ vector of 1's. ... |
I'm trying to construct a simple proof of the Propositional Interpolation Theorem. For the following, let $At(\phi)$ be the set of sentence symbols that occur in a sentence $\phi$. Suppose that $\psi$ is a tautological consequence of $\phi$, but neither $\neg\phi$ nor $\psi$ is a tautology, and so $\psi$ is not a tauto... |
I was going to ask this on Stack Overflow, but finally decided this was more math than programming. I may still turn out to be wrong about that, but...
Given a number represented in binary, it's fairly easy to derive a decimal representation using integer division and remainders to extract digits in reverse order. Y... |
I asked this question on mathoverflow, but it was deemed too simple, so I'm posting here instead --
Is there a nice way to characterize an orthonormal basis of eigenvectors of the following $d\times d$ matrix?
$$\mathbf{I}-\frac{1}{d} \mathbf{v}\mathbf{v}'$$
Where $\mathbf{v}$ is a $d\times 1$ vector of 1's. ... |
I asked this question on mathoverflow, but it was deemed too simple, so I'm posting here instead --
Is there a nice way to characterize an orthonormal basis of eigenvectors of the following $d\times d$ matrix?
$$\mathbf{I}-\frac{1}{d} \mathbf{v}\mathbf{v}'$$
Where $\mathbf{v}$ is a $d\times 1$ vector of 1's. ... |
I asked this question on mathoverflow, but it was deemed too simple, so I'm posting here instead --
Is there a nice way to characterize an orthonormal basis of eigenvectors of the following $d\times d$ matrix?
$$\mathbf{I}-\frac{1}{d} \mathbf{v}\mathbf{v}'$$
Where $\mathbf{v}$ is a $d\times 1$ vector of 1's. ... |
consider the following game: 10 dice are tossed and those showing 3 are more are retained. [those showing 2 or less are discarded.] the remaining dice are tossed again and those showing 4 or more are retained. finally, the remaining dice are tossed once more and those showing 5 or more are retained.
if $Z$ denotes th... |
a slight retouching of unkz's idea is the following orthogonal basis for $v^\perp$:
$$u_1 = (1,-1,0,\dots,0),\\
u_2 = (1,1,-2,0,\dots,0),\\
\dots, \\
u_{d-1} = (1,\dots,1,-d+1)$$
in this form it is pretty easy to see that the $\\{u_i\\}$ are in $v^\perp$ and are mutually orthogonal.
i don't quite unde... |
You should take a look at [D. Van Dalen: Logic and Structure][1].
[1]: http://books.google.com/books?id=4u9gQ6pctuIC&printsec=frontcover&dq=van+dalen+logic&source=bl&ots=koMWzUiwUR&sig=Noea7aYSfL54r63KwnunQOB1qRE&hl=en&ei=Vc-FTNL1EMjNswbKq-yaBQ&sa=X&oi=book_result&ct=result&resnum=5&ved=0CCoQ6AEwBA#v=onepage&q=... |
Given complete fan $\Delta$ defining a projective toric variety (so that $\Delta$ is the normal fan of some polytope). How do one go on to find a defining ideal of the toric variety in projective space?
Or, going the other way, given binomial equations defining a projective toric variety, is there some way to recover... |
In an introductory algebra course, one proves the Structure theorem for finitely generated modules over a principal ideal domain.
http://en.wikipedia.org/wiki/Structure_theorem_for_finitely_generated_modules_over_a_principal_ideal_domain
In every book that I have looked in, the uniqueness part of the proof is a m... |
I am reading Jech's Set Theory, in particular the chapter about large cardinals. After discussing measurable cardinals he moves on to weakly-compact cardinals, which have been discussed far earlier in the book.
I went back to the chapter dealing with weakly-compact cardinals and began retracing the definition.
Ev... |
Usually, applying the conversion formulae
$r^2=x^2+y^2$
$\cos\;\theta=\frac{x}{r}$
$\sin\;\theta=\frac{y}{r}$
to transform an equation in polar coordinates to an implicit Cartesian equation is pretty straightforward for an equation of the form
$r=f(\cos(n\theta),\sin(n\theta))$
with $n$ an integer, th... |
Let X = complex manifold of dimension n. Thus, it's a real manifold of dimension 2n.
Now cohomology is a topological concept so it should not depend upon the structure given on a topological space.
We know that k'th Singular cohomology of X is 0 for k > 2n. We can also define a sheaf cohomology on that space usi... |
Let X = complex manifold of dimension n. Thus, it's a real manifold of dimension 2n.
Now cohomology is a topological concept so it should not depend upon the structure given on a topological space.
We know that k'th Singular cohomology of X is 0 for k > 2n. We can also define a sheaf cohomology on that space usi... |
One of my Students asked me whether total induction is applicable to real numbers, too ( or at least all real numbers >= 0) . We only used that for natural numbers so far.
Of course you have to change some things in the inductive step, when you want to use it on real numbers.
I guess that using induction on real n... |
Okay, I can't resist: here is a quick answer.
I am construing the question in the following way: "Is there some criterion for a subset of $[0,\infty)$ to be all of $[0,\infty)$ which is (a) analogous to the principle of mathematical induction on $\mathbb{N}$ and (b) useful for something?"
The answer is yes, at leas... |
How to expand and simplify (p^y * (pq)^o) / (p^(2y+o) * q^(o-2))? |
Okay, I can't resist: here is a quick answer.
I am construing the question in the following way: "Is there some criterion for a subset of $[0,\infty)$ to be all of $[0,\infty)$ which is (a) analogous to the principle of mathematical induction on $\mathbb{N}$ and (b) useful for something?"
The answer is yes, at leas... |
The cohomological dimension of an *real* $n$-manifold $M$ is $n$: this means that $H^i(M,\mathscr F)=0$ for each sheaf $\mathscr F$ of abelian groups on $M$ if $i>n$, and that there exist sheaves $\mathcal F$ on $M$ with $H^n(M,\mathscr F)\neq0$. You'll find this proved in Bredon's book on *Sheaf theory*, §II.16.
I... |
One of my Students asked me whether total induction is applicable to real numbers, too ( or at least all real numbers ≥ 0) . We only used that for natural numbers so far.
Of course you have to change some things in the inductive step, when you want to use it on real numbers.
I guess that using induction on real nu... |
Okay, I can't resist: here is a quick answer.
I am construing the question in the following way: "Is there some criterion for a subset of $[0,\infty)$ to be all of $[0,\infty)$ which is (a) analogous to the principle of mathematical induction on $\mathbb{N}$ and (b) useful for something?"
The answer is yes, at leas... |
Explain why calculating this series could cause paradox? |
Okay, I can't resist: here is a quick answer.
I am construing the question in the following way: "Is there some criterion for a subset of $[0,\infty)$ to be all of $[0,\infty)$ which is (a) analogous to the principle of mathematical induction on $\mathbb{N}$ and (b) useful for something?"
The answer is yes, at leas... |
Motivation: In S.J. Patterson's *An introduction to the theory of the Riemann
Zeta-Function* it is proved (p.132) that
$\displaystyle -\Gamma ^{\prime }(t)/\Gamma (t)=\gamma +t^{-1}+\underset{j\geq 1}{\sum }[(j+t)^{-1}-j^{-1}]$.
In Exercise A4.1 (p.135) one is asked to show that if $|t|<1$
$\displaystyle -\Ga... |
The cohomological dimension of a *real* $n$-manifold $M$ is $n$: this means that $H^i(M,\mathscr F)=0$ for each sheaf $\mathscr F$ of abelian groups on $M$ if $i>n$, and that there exist sheaves $\mathcal F$ on $M$ with $H^n(M,\mathscr F)\neq0$. You'll find this proved in Bredon's book on *Sheaf theory*, §II.16.
It... |
Okay, I can't resist: here is a quick answer.
I am construing the question in the following way: "Is there some criterion for a subset of $[0,\infty)$ to be all of $[0,\infty)$ which is (a) analogous to the principle of mathematical induction on $\mathbb{N}$ and (b) useful for something?"
The answer is yes, at leas... |
In b<sup>a</sup> = x, b is the base, a is the exponent and x is the result of the operation. But in its logarithm counterpart, log<sub>b</sub>(x) = a, b is still the base, and a is now the result. What is x called here? The exponent? |
What is the x in log b (x) called? |
I suspect it is impossible to split a (any) 3d solid into two, such that each of the pieces is identical in shape (but not volume) to the original. How can I prove this? |
Given complete fan $\Delta$ defining a projective toric variety (so that $\Delta$ is the normal fan of some polytope). How do one go on to find a defining ideal of the toric variety in projective space?
Or, going the other way, given binomial equations defining a projective toric variety, is there some way to recover... |
Can we slice an object into two pieces similar to the original? |
What is the x in $\log_b x$ called? |
One of my Fellows asked me whether total induction is applicable to real numbers, too ( or at least all real numbers ≥ 0) . We only used that for natural numbers so far.
Of course you have to change some things in the inductive step, when you want to use it on real numbers.
I guess that using induction on real num... |
In the course of analyzing a certain Markov chain, I once had to prove the following algebraic identity.
**Is there a slick or known proof?**
For $n$-tuples $(x_1,x_2,\dots, x_n)$ of positive real numbers define
$$\mu(x_1,x_2,\dots, x_n)=\prod_{j=1}^n {x_j\over x_j+x_{j+1}+\cdots+x_n}.$$
Then if $x^\ast$ is ... |
Is this a known algebraic identity? |
What are some examples of theories stronger than Presburger Arithmetic but weaker than Peano Arithmetic? Are all such theories decidable? If not, by what methods other than Gödelization can undecidability be established?
EDIT: Both answers have indicated Robinson Arithmetic as an intermediate theory, but I don't u... |
Say P is a convex Euclidean polytope, none of whose facets contain the origin, with n facets given by f ◦ x = 1 and m vertices v. (That is, each facet equation has been multiplied through as needed to obtain 1 on the right side.)
Is it true that the polytope P’ with m facets given by v ◦ x = 1 and n vertices f is comb... |
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