#
Symmetry, Probabiliy, Entropy: Synopsis of the Lecture at MAXENT 2014^{ †}

^{†}

## Abstract

**:**

- The symmetry group of a single round of gambling with three dice has order 288 = 6 × 6 × 8: it is a semidirect product of the permutation group S
_{3}of order 6 and the symmetry group of the 3d cube, that is, in turn, is a semidirect product of S_{3}and {±1}^{3}. - The Bernoulli spaces${({\u25a0}_{p},\phantom{\rule{0.2em}{0ex}}{\u2666}_{1-p})}^{\mathrm{\mathbb{Z}}}$, 0 < p < 1, of (■, ♦)-sequences indexed by integers $z\in \mathrm{\mathbb{Z}}=\{\cdots ,-2,-1,0,1,2,\cdots \}$. are acted upon by a semidirect product of the infinite permutation group$${S}_{\infty =\mathrm{\mathbb{Z}}}\supset \mathrm{\mathbb{Z}}=\{\cdots ,-2,-1,0,1,2,\cdots \}$$
- The system of identical point-particles •
_{i}in the Euclidean 3-space${\mathrm{\mathbb{R}}}^{3}$, that are indexed by a countable set I ∋ i, is acted upon by the isometry group of ${\mathrm{\mathbb{R}}}^{3}$ times the infinite permutation group S_{∞=I}. - Buffon’s probabilistic needle formula for π 3.141592653589793⋯ relies on the invariance of the Haar measure on the circle.

- What happens if the symmetry is enhanced, e.g., from the permutation group S
_{∞}_{=I}to the group $G{L}_{\mathbb{F}}(\infty )$ of liner transformations of the vector space ${\mathbb{F}}^{I}$ (formally) spanned by symbols [i], i ∈ I, regarded as (linearly independent) vectors over a filed $\mathbb{F}$? - What could you do if your system is inherently heterogeneous, such as a folding polypeptide chain or a natural language, for instance?

_{1}, x

_{2}, x

_{3}, x

_{4}) be a 4-linear function (form) over some field (where the variables x

_{i}run over some vector spaces X

_{i}). Then the ranks of the following four bilinear forms Φ(x

_{1}, x

_{2}⊗ x

_{3}⊗ x

_{4}), Φ(x

_{1}⊗ x

_{2}, x

_{3}⊗ x

_{4}), Φ (x

_{1}⊗ x

_{3,}x

_{2}⊗ x

_{4}) and Φ(x

_{1}⊗ x

_{4,}x

_{2}⊗ x

_{3,}) satisfy

_{∗}(X) = ⊕

_{i}H

_{i}(X) of topological spaces X and natural subgroups in H

_{∗}are graded Abelian groups: their ranks are properly represented not by individual numbers r

_{i}, but by Poincaré polynomials P

_{X}(t) = ∑

_{i}r

_{i}· t

^{i}.

_{U}, U ⊂ X, has some measure/entropy-like properties that become more pronounced for the ideal valued function that assigns the kernels

**additive**for the sum-of-subsets in the group H*(X; A) and, if A is a commutative ring, then μ* is

**super-multiplicative**for the the ◡-product of ideals:

_{1}and U

_{2}in A, and

_{1}, U

_{2}⊂ A.

_{Θ}(U) over this field by |μ

_{Θ}(U)| = |μ

_{Θ}(U)|

_{A}.

_{1}, U

_{2}⊂ ${U}_{2}\subset {\mathbb{T}}^{N}$ be non-intersecting (closed or open) subsets and let

_{i}≤ N/2, i = 1,2 and some field A. Then

_{1}n

_{2}/N

^{2}and where, observe, $\left|{\mathrm{\Theta}}_{i}={\wedge}^{{n}_{i}}A\right|=\left({}_{{n}_{i}}^{N}\right)$.

_{Θ}U may be interpreted as

_{∞}=

_{I}[11], that is expanded/corrected in [12].

_{i}}, p

_{i}> 0, ∑

_{i}p

_{i}= 1, comes as the class [P]

_{Gro}of P in the Grothendieck group$Gro(\mathcal{P})$ of the topological category $\mathcal{P}$ of finite probability spaces P and probability/measure preserving maps P → Q with a properly defined topological structure in $\mathcal{P}$.

_{Gro}can be identified with exp ent(P).

## Acknowledgments

## Conflicts of Interest

## References and Notes

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**MDPI and ACS Style**

Gromov, M.
Symmetry, Probabiliy, Entropy: Synopsis of the Lecture at MAXENT 2014. *Entropy* **2015**, *17*, 1273-1277.
https://doi.org/10.3390/e17031273

**AMA Style**

Gromov M.
Symmetry, Probabiliy, Entropy: Synopsis of the Lecture at MAXENT 2014. *Entropy*. 2015; 17(3):1273-1277.
https://doi.org/10.3390/e17031273

**Chicago/Turabian Style**

Gromov, Misha.
2015. "Symmetry, Probabiliy, Entropy: Synopsis of the Lecture at MAXENT 2014" *Entropy* 17, no. 3: 1273-1277.
https://doi.org/10.3390/e17031273