Polynomial representations of GL(m|n)

International audience We develop a modular version of a super analogue of Schur's duality by means of supergroups, rather than Lie superalgebras, in preparation for a geometric analogue.

The influence of plate strength on dynamic topography estimates

<p>A common method used to evaluate dynamic topography amplitudes begins with an estimate of Moho depth, usually from seismic data but sometimes - or also - from the inversion of gravity data. Then the principles of Airy isostasy are applied: surface topography is assumed to be in isostatic equilibrium, buoyantly supported by the displacement of high-density mantle material by the low-density crustal ‘root’ that compensates the surface topographic mass. Hence, the actual relief of the Moho yields an ‘isostatic topography’ which will depart from the actual, observed topography by a component that, in theory, must arise from convective support or subsidence. Notwithstanding the fact that the errors on the seismic Moho may be larger than the topography itself, there is another source of uncertainty, that of the flexural rigidity of the lithosphere. Airy isostasy is essentially an end-member of plate flexure models, one in which the flexural rigidity is zero. However there are very few places on Earth where the flexural rigidity, usually represented by its geometric analogue the effective elastic thickness (Te), is indeed zero. In most environments, the rigidity of the plate will act to resist flexure, with the implication that the ‘Airy isostatic topography’ and therefore the dynamic topography will be in error. Here several scenarios will be presented illustrating these issues, and paths for remediation recommended.</p>

Toward the Unification of Physics and Number Theory

This paper introduces the notion of simplex-integers and shows how, in contrast to digital numbers, they are the most powerful numerical symbols that implicitly express the information of an integer and its set theoretic substructure. A geometric analogue to the primality test is introduced: when [Formula: see text] is prime, it divides [Formula: see text] for all [Formula: see text]. The geometric form provokes a novel hypothesis about the distribution of prime-simplexes that, if solved, may lead to a proof of the Riemann hypothesis. Specifically, if a geometric algorithm predicting the number of prime simplexes within any bound [Formula: see text]-simplex or associated [Formula: see text] lattice is discovered, a deep understanding of the error factor of the prime number theorem would be realized — the error factor corresponding to the distribution of the non-trivial zeta zeros, which might be the mysterious link between physics and the Riemann hypothesis [D. Schumayer and D. A. W. Hutchinson, Colloquium: Physics of the Riemann hypothesis, Rev. Mod. Phys. 83 (2011) 307]. It suggests how quantum gravity and particle physicists might benefit from a simplex-integer-based quasicrystal code formalism. An argument is put forth that the unifying idea between number theory and physics is code theory, where reality is information theoretic and 3-simplex integers form physically realistic aperiodic dynamic patterns from which space, time and particles emerge from the evolution of the code syntax.

Elementary equivalence in Artin groups of finite type

Irreducible Artin groups of finite type can be parametrized via their associated Coxeter diagrams into six sporadic examples and four infinite families, each of which is further parametrized by the natural numbers. Within each of these four infinite families, we investigate the relationship between elementary equivalence and isomorphism. For three out of the four families, we show that two groups in the same family are equivalent if and only if they are isomorphic; a positive, but weaker, result is also attained for the fourth family. In particular, we show that two braid groups are elementarily equivalent if and only if they are isomorphic. The [Formula: see text] fragment suffices to distinguish the elementary theories of the groups in question. As a consequence of our work, we prove that there are infinitely many elementary equivalence classes of irreducible Artin groups of finite type. We also show that mapping class groups of closed surfaces — a geometric analogue of braid groups — are elementarily equivalent if and only if they are isomorphic.

The approximate method of determining the integral physical characteristics in some problems of technical theory of plates

The paper considers the dimensionless geometric characteristic R/r (R is the maximum radius of a circle inscribed in a given region, r is the minimum radius of a circle circumscribed around it), which is used as a geometric analogue of integral characteristics in problems of the technical theory of plates and theory of elasticity. By numerical experiment the authors showed that using this characteristic as the only argument made it possible to approximate with one elementary function all the solutions for short plates, which satisfied the condition (0.5 < R/r < 1) in problems of trans-verse bending, free oscillations and stability of plates.

On the Notion of Conductor in the Local Geometric Langlands Correspondence

Under the local Langlands correspondence, the conductor of an irreducible representation of Gln(F) is greater than the Swan conductor of the corresponding Galois representation. In this paper, we establish the geometric analogue of this statement by showing that the conductor of a categorical representation of the loop group is greater than the irregularity of the corresponding meromorphic connection.