Narrow Results

In the framework of conformal field theory, the mapping from (unbounded) wedge regions of Minkowski spacetime to (bounded) double-cone regions is extended to the Unruh temperature associated to a uniformly accelerated observer. The link between a previous result, the diamond's temperature, and the conformal factor (Weyl rescaling of the metric) is worked out. One thus explains from a mathematical point of view why an observer with finite lifetime experiences the vacuum as a thermal state whatever his acceleration, even vanishing.

Fractons are anyons classified into equivalence classes and they obey specific fractal statistics. The equivalence classes are labeled by a fractal parameter or Hausdorff dimension h. We consider this approach in the context of the fractional quantum Hall effect (FQHE) and the concept of duality between such classes, defined by shows us that the filling factors for which the FQHE were observed just appear into these classes. A connection between equivalence classes h and the modular group for the quantum phase transitions of the FQHE is also obtained. A β-function is defined for a complex conductivity which embodies the classes h. The thermodynamics is also considered for a gas of fractons (h,ν) with a constant density of states and an exact equation of state is obtained at low-temperature and low-density limits. We also prove that the Farey sequences for rational numbers can be expressed in terms of the equivalence classes h.

In this article we study percolation on the Cayley graph of a free product of groups.

The critical probability p_c of a free product G₁ * G₂ * ⋯ * G_n of groups is found as a solution of an equation involving only the expected subcritical cluster size of factor groups G₁, G₂, …, G_n. For finite groups this equation is polynomial and can be explicitly written down. The expected subcritical cluster size of the free product is also found in terms of the subcritical cluster sizes of the factors. In particular, we prove that p_c for the Cayley graph of the modular group PSL₂(ℤ) (with the standard generators) is 0.5199…, the unique root of the polynomial 2p⁵ - 6p⁴ + 2p³ + 4p² - 1 in the interval (0, 1).

In the case when groups G_i can be "well approximated" by a sequence of quotient groups, we show that the critical probabilities of the free product of these approximations converge to the critical probability of G₁ * G₂ * ⋯ * G_n and the speed of convergence is exponential. Thus for residually finite groups, for example, one can restrict oneself to the case when each free factor is finite.

We show that the critical point, introduced by Schonmann, p_exp of the free product is just the minimum of p_exp for the factors.

In this note we apply the general multifractal analysis for growth rates derived in [10], and show that this leads to some new results in ergodic theory and the theory of multifractals of numbers. Namely, we consider Stern–Brocot growth rates and introduce the Stern–Brocot pressure P. We then obtain the results that P is differentiable everywhere and that its Legendre transformation governs the multifractal spectra arising from level sets of Stern–Brocot rates.

As the 3-string braid group B₃ and the modular group Γ are both of wild representation type one cannot expect a full classification of all their finite dimensional simple representations. Still, one can aim to describe 'most' irreducible representations by constructing for each d-dimensional irreducible component X of the variety iss_n(Γ) classifying the isomorphism classes of semi-simple n-dimensional representations of Γ an explicit minimal étale rational map 𝔸^d → X having a Zariski dense image. Such rational dense parametrizations were obtained for all components when n < 12 in [5]. The aim of the present paper is to establish such parametrizations for all finite dimensions n.

We show that the image of the representation of the modular group $\mathrm{\text{SL}}(2,\mathbb{Z})$ arising from the representation category $\mathrm{\text{Rep}}(D(G))$ of the Drinfeld double $D(G)$ of a finite abelian group $G$ of exponent $n$ is isomorphic to the special linear group $\mathrm{\text{SL}}(2,\mathbb{Z}/n\mathbb{Z})$, where $\mathbb{Z}/n\mathbb{Z}$ denotes the ring of integers modulo $n$. As a consequence, we establish that the kernel of the representation in question is the principal congruence subgroup of level $n$.

In this paper, we show that the coset diagram for the action of a linear-fractional group generated by and on the rational projective line is transitive and the coset diagram for the action is connected. Using the coset diagram, we show that are defining relations for the group.

We show that the matrix A(g), representing the element g = ((xy)²(xy²)²)^m (m ≥ 1) of the modular group PSL(2,Z) = 〈x,y : x² = y³ = 1〉, where and , is a 2 × 2 symmetric matrix whose entries are Pell numbers and whose trace is a Pell–Lucas number. If g fixes elements of , where d is a square-free positive number, on the circuit of the coset diagram, then d = 2 and there are only four pairs of ambiguous numbers on the circuit.

A k-nacci (k-step Fibonacci) sequence in a finite group is a sequence of group elements x₀, x₁, x₂, …, x_n, … for which, given an initial (seed) set x₀, x₁, x₂, …, x_j-1, each element is defined by

This paper studies modular forms of rank four and level one. There are two possibilities for the isomorphism type of the space of modular forms that can arise from an irreducible representation of the modular group of rank four, and we describe when each case occurs for general choices of exponents for the $T$-matrix. In the remaining sections we describe how to write down the corresponding differential equations satisfied by minimal weight forms, and how to use these minimal weight forms to describe the entire graded module of holomorphic modular forms. Unfortunately, the differential equations that arise can only be solved recursively in general. We conclude the paper by studying the cases of tensor products of two-dimensional representations, symmetric cubes of two-dimensional representations, and inductions of two-dimensional representations of the subgroup of the modular group of index two. In these cases, the differential equations satisfied by minimal weight forms can be solved exactly.

In this paper we have developed a graphical technique by which a suitably created fragment of a coset diagram for the action of PSL(2, ℤ) on projective lines over Galois fields can be used to obtain a complete coset diagram depicting finite homomorphic images of groups Δ(2, 3, k) = 〈x, y : x² = y³ = (xy)^k = 1〉.