User alex 'qubeat' - MathOverflow

Kolmogorov probability axioms without non-negativity condition

2011-12-01T13:43:15Z

What is a minimal consistent modification of probability axioms to include negative values? Is it enough to use a minimal modification of axioms obtained by formal exclusion of non-negativity requirement, i.e.:

There are sample space $\Omega$, event space $F$ ($\sigma$-algebra of subsets of $\Omega$) and a function $P$ satisfying axioms:

(A1) $P(E) \in {\mathbb R}$, $\forall E \in F$

(A2) $P(\Omega) = 1$

(A3) Any countable sequence of pairwise disjoint elements $E_k \in F$ satisfies

$$P(\bigcup_k E_k) = \sum_k P(E_k).$$

So, the only difference with standard probability axioms is lack of condition $P(E) \geq 0$ in A1.

I am not quite sure, is it necessary to add yet another axiom?:

(A0) $P(\emptyset) = 0$

From the one hand, the axiom A0 together with A1 and A3 define a signed measure and so known to be consistent. Definition above could be shortly rewritten as: Extended ("negative") probability space $\Omega =(\Omega,F,P)$ is signed measure space with $P(\Omega)=1$.

On the other hand in usual probability theory A0 is consequence of other axioms and I suppose that here it also follows from application of A3 to formal expression $P(E \cup \emptyset) = P(E)$ with arbitrary $E$.

It is possible to prove that $P(A \cup B) = P(A) + P(B) - P(A \cap B)$ and for complementary event $P(\overline{E}) \equiv P(\Omega \setminus E) = 1 - P(E)$.

Yet, monotonicity does not follow from the axioms, i.e., for $A \subseteq B$ it is not necessary $P(A) \leq P(B)$.

The question: if axioms A1, A2, A3 (maybe, together with A0) define minimal logically consistent model of "extended" probabilities? If yes, that is the possible caveats (e.g. some essential theorems are not valid or useful tools do not work - cf lack of monotonicity).

Note: My interest was inspired by an application to geometrical probability, but below is suggested a more elementary example (it may be omitted, I wrote that due to a reasonable question about interpretation of extended probabilities).

There is a family with father, mother, son and daughter. The family is poor and may only buy Xmas gift for single person. So each year son or daughter may have a present with equal probabilities and (in average) we get a distribution of gifts: father: 0, mother: 0, son: 0.5 and daughter 0.5

But let's suppose, that parents do not want to upset both children and after buying a gift they also search for gifts received during own childhood to present one to second child. So both children have gifts, but one of parents lost his own old gift. Now distribution of gifts may be formally written: father: -0.5, mother: -0.5, son: 1 and daughter: 1

In fact, the example shows, that $P > 1$ may cause even more objections against probabilistic interpretation, than $P < 0$. Here we have distribution of gifts: parents: -1, children: 2. Indeed, $P(\overline{E}) > 1$ is inevitable consequence of $P(E) < 0$ due to axiom A2.

[EDIT 2-Dec-2011] With taking into account comments of Andreas Blass and Emil Jeřáb,

I could suggest some clarification:

The question: if axioms A1, A2, A3 (maybe, together with A0) - are logically consistent system for "extended" probabilities? If yes, which are the possible caveats (e.g. some essential theorems are not valid or useful tools do not work - cf lack of monotonicity).

A construction of generators of discrete subgroups of SL(2,R)

2011-05-15T20:46:05Z

I know about geometrical method of construction of discrete subgroups of $SL(2,\mathbb{R})$ using Lobachevsky plane (e.g. B.A. Dubrovin, A.T. Fomenko, S.P. Novikov, Modern Geometry --- Methods and Applications, Springer) via fundamental polygon. Such construction has many applications and some relevant themes were already discussed in MO.

I do not know, if my question is appropriate here, but I would like to know rather opposite thing: how to construct explicitly the matrices itself. In book mentioned above it was demonstrated only for simple case of $4g$-polygon with sum of angles $2\pi$. I mean, if there is some analytical equations or an algorithm of calculation of parameters of matrices for discrete subgroups of $SL(2,\mathbb{R})$.

Answer by Alex 'qubeat' for Suitable references for the the Stone-von Neumann Theorem

2011-05-15T12:45:56Z

I would also recommend M. Reed and B. Simon, Methods of modern mathematical physics, vol I, chapter VIII (and maybe vol II, chapter X). The theorem itself is VIII.14 with Corollary after that. The reference is useful also due to counterexample, demonstrating that the idea about “one-to-one correspondence” used in Wikipedia and some other places is not very good one.

Answer by Alex 'qubeat' for Linking to Code in a Paper

2011-05-14T13:05:04Z

Recently I had a similar problem, because I wanted to use some code for illustration of a paper. I have found some table with comparison of different sites in Wikipedia. Yet, finally I decided, that such approach may be not very convenient for scientific paper and upload packed archive to Windows Live SkyDrive.

Answer by Alex 'qubeat' for Pythagorean 5-tuples

2011-05-01T14:03:54Z

The construction below shows that such tuples really hardly could be written in unique way. Maybe some exclusion from general case is due to the Hurwitz theorem about sum of squares and noncommutativity of quaternions. This construction is modification of Pietro Majer and Geoff Robinson suggestions.

Let us consider quaternion $q = t + x i + y j + z k$ and a constant quaternion $c$. I am not sure after reading of the question, if norm should be a square of some integer or not. It may be Hurwitz quaternion and simple nontrivial example is $c=(1+i+j+k)/2$. Now let us consider product $s = q c q$ as polynomial of $t, x, y, z$.

We have $|s| = |c| |q|^2 = |c| (t^2+x^2+y^2+z^2)^2 $, but $|s|=s_0^2+s_1^2+s_2^2+s_3^2$ for $s = s_0 + s_1 i + s_2 j +s_3 k$, where $s_0, s_1, s_2, s_3$ by definition are second order polynomial of $t, x, y, z$.

So different $c$ produces different tuples and $c=1$ produces solution mentioned by Pietro Majer and Geoff Robinson and in Mordell book.

[edit] To produce more general solution for second order polynomials it is possible to use some modification. Let's consider three constant quaternions $a, b, c$ with modules are squares of some integers. It may be done using Kac's method or something else. Then new solution is $a q c q b$.

Answer by Alex 'qubeat' for A puzzling remark of Manin (ICM 1978)

2011-04-27T16:34:31Z

From Introduction to “Scattering Theory for Automorphic Functions” by Lax and Phillips (1976) and some other sources I could suppose, that proper reference would be the Gelfand presentation on 1962 ICM, but I do not have access to it, and so not quite certain.

A non-associative three-valued logic

2011-04-22T14:04:47Z

There are three elements: x, y, z and a relation C:

x C y, y C z, z C x, x C x, y C y, z C z.

Let us introduce two binary operations with respect to the C: "the leftmost" (L) and "the rightmost" (R), i.e.

x L x = x L y = y L x = x, y L y = y L z = z L y = y, z L z = z L x = x L z = z

x R x = x R z = z R x = x, y R y = x R y = y R x = y, z R z = z R y = y R z = z.

Similar construction produces a multi-valued logic, if to use a linear order instead of the C, but this non-associative "logic" also has some applications. Yet, I failed to find any notes about that in a book about multi-valued logic. I would be glad to know, if described construction was used somewhere earlier to provide correct references in my works.

Answer by Alex 'qubeat' for A non-associative three-valued logic

2011-04-27T11:28:04Z

Just few hours ago I found, that the construction was used in talks of J. B. Nation “How aliens do math”, and “Logic on other planets” (here). Despite of such titles, the works look quite instructive.

Comment by Alex 'qubeat'

2012-02-14T13:35:59Z

Let us consider sequence of elements $S_k$, $k = 1,\ldots,\infty$ with positive $P(S_k) = 1/(\ln(2) 2k (2k-1))$, $\sum_k P(S_k) = 1$. Now let us split each element on pair $S_k = E_{2k-1} \cup E_{2k}$, $P(E_k) = (-1)^{k+1}/(\ln(2) k)$, $P(S_k) = P(E_{2k-1})+P(E_{2k})$ using the gifts" model above. Then $\sum_k |P(E_k)| = \infty$, but why we should exclude such a model from consideration?

Comment by Alex 'qubeat'

2012-02-14T11:16:29Z

Yes, I read that before post the question

Comment by Alex 'qubeat'

2012-02-14T11:13:00Z

Thank you for reminder about that. @Jochen Wengenroth: If the problem may be really resolved in such a simple way? What about possibility of some analogue of Banach-Tarski paradox?

Comment by Alex 'qubeat'

2011-12-02T19:51:14Z

@Andreas Blass: I added simpler formulation of the question to the end

Comment by Alex 'qubeat'

2011-12-02T16:19:28Z

Thank you for the note, but the problem for me - is using of $p$-adic framework there.

Comment by Alex 'qubeat'

2011-12-01T17:33:39Z

@Andreas Blass: I am not sure, that such modification may be considered minimal, because it is suggestion of infinite family of axiom systems instead of single one. After all, there are already many works about extended probability and I not sure that modification with fixed $\epsilon$ would be appropriate.

Comment by Alex 'qubeat'

2011-12-01T16:36:13Z

@Andreas Blass: Yes, I meant "minimal modification", not minimal axiom system. Yet, seems, due to A3 $\epsilon+\epsilon = \epsilon$, so either $\epsilon = 0$ or $\epsilon = \infty$.

Comment by Alex 'qubeat'

2011-12-01T14:00:27Z

@Andreas Blass: I agree, it may be not very clear - I added a note about "caveats" partially due to that...

Comment by Alex 'qubeat'

2011-06-01T22:42:09Z

Yes, the Clifford algebra are isomorphic with algebra of $2^n \times 2^n$ complex matixes, but it is not always necessary to write down this matrices. Expressions with products of few generators correspond to some subspaces of the exponential space, e.g. products of two generators -- is simply Lie algebra so(2n).

Comment by Alex 'qubeat'

2011-06-01T19:12:09Z

Canonical anticommutation relations with $n$ generators has a standard description using complex Clifford algebras with $2n$ generators, i.e. $a_n = (e_{2n}+ i e_{2n+1})/2$, $a^*_n = (e_{2n}- i e_{2n+1})/2$. Could it help?

Comment by Alex 'qubeat'

2011-05-23T12:26:33Z

Despite of finite number of such groups it is really instructive.

Comment by Alex 'qubeat'

2011-05-19T09:36:30Z

@algori: I may give an analogue example for spherical case, SO(3) or SU(2). You have polygon with n sides and m polygon meet in each vertex. Using n,m we may calculate ratio between side and sphere radius, e.g., icosahedron for n=3 and m=5, but we may not say, that we have coordinate of points for initial triangle. After all, even the icosahedron defined up to SO(3) action and we still need to make additional assumptions if we want all vertexes expressed as an analytical expression with square roots.

Comment by Alex 'qubeat'

2011-05-16T17:38:56Z

Thanks for answer and comments. I got the "Indra's pearls" and Ford's book.

Comment by Alex 'qubeat'

2011-05-16T15:00:49Z

@algori: Maybe I missed something, it seems to me, I do not have the points explicitly - in general case I only had a proof that such points exist.

Comment by Alex 'qubeat'

2011-05-16T13:38:52Z

@algori: The geometric proof I read do not produce precise coordinate. It is rather demonstration of possibility to construct a polygon using some geometric illustration with Lobachevsky plane. There is only one simple example I have mentioned. In such a case the matrices are really constructed directly.