A strongly minimal structure D is called unimodular if any two finite-to-one maps with the same domain and range have the same degree; that is if/4: (/-»• V is everywhere fc4-to-l, then kx = kc,. THEOREM. Unimodular strongly minimal structures are locally modular. This extends Zil'ber's theorem on locally finite strongly minimal sets, Urbanik's theorem on free algebras with the Steinitz propert...

In this paper, binary extremal singly even self-dual codes of length 40 and extremal odd unimodular lattices in dimension 40 are studied. We give a classification of extremal singly even self-dual codes of length 40. We also give a classification of extremal odd unimodular lattices in dimension 40 with shadows having 80 vectors of norm 2 through their relationships with extremal doubly even sel...

The known facts about solvability of equations over groups are considered from a more general point of view. A generalized version of the theorem about solvability of unimodular equations over torsion-free groups is proved. In a special case, this generalized version become a multivariable variant of this theorem. For unimodular equations over torsion free groups, we prove an analogue of Magnus...

Unimodular gravity (UG) is an important theory which may explain the smallness of cosmological constant. To get insight into covariant quantization UG, we discuss BRST General Relativity (GR) with a constant in unimodular gauge. We develop novel way to gauge fix transverse diffeomorphism (TDiff) and then further fulfill This process requires introduction additional pair doublets decouple from p...

Unimodular relativity is a theory of gravity and space-time with a fixed absolute space-time volume element, the modulus, which we suppose is proportional to the number of microscopic modules in that volume element. In general relativity an arbitrary fixed measure can be imposed as a gauge condition, while in unimodular relativity it is determined by the events in the volume. Since this seems t...

We give a global picture of the Ricci flow on the space of three-dimensional, unimodular, nonabelian metric Lie algebras considered up to isometry and scaling. The Ricci flow is viewed as a two-dimensional dynamical system for the evolution of structure constants of the metric Lie algebra with respect to an evolving orthonormal frame. This system is amenable to direct phase plane analysis, and ...

A polynomial is called unimodular if each of its coefficients is a complex number of modulus 1. A polynomial P of the form P (z) = ∑n j=0 ajz j is called conjugate reciprocal if an−j = aj , aj ∈ C for each j = 0, 1, . . . , n. Let ∂D be the unit circle of the complex plane. We prove that there is an absolute constant ε > 0 such that max z∈∂D |f(z)| ≥ (1 + ε) √ 4/3m , for every conjugate recipro...

We compute the contribution of various gravitational instantons to the path integral in the standard formulation of unimodular gravity, an alternative theory of gravity where the determinant of the metric is not dynamical. Following similar computations in General Relativity, we derive the entropy/area ratio for cosmological and black hole horizons, finding in general disagreement with General ...

We construct integral bases for the SO(3)-TQFT-modules of surfaces in genus one and two at roots of unity of prime order and show that the corresponding mapping class group representations preserve a unimodular Hermitian form over a ring of algebraic integers. For higher genus surfaces the Hermitian form sometimes must be non-unimodular. In one such case, genus 3 and p = 5, we still give an exp...

In this work it is shown that a necessary condition for the completeness of the geodesics of left invariant pseudo-Riemannian metrics on Lie groups is also sufficient in the case of 3-dimensional unimodular Lie groups, and not sufficient for 3-dimensional non-unimodular Lie groups. As a consequence it is possible to identify, amongst the compact locally homogeneous Lorentzian 3-manifolds with n...