1,720,984 research outputs found
The even Clifford structure of the fourth Severi variety
Full text linkThe Hermitian symmetric space M= EIII appears in the classification of complete simply connectedRiemannian manifolds carrying a parallel even Clifford structure [19]. This means the existence of a real oriented Euclidean vector bundle E over it together with an algebra bundle morphism φ: Cl^0(E)→End(TM) mapping Λ2 Einto skew-symmetric endomorphisms, and the existence of a metric connection on E compatible with φ. We give an explicit description of such a vector bundle E as a sub-bundle ofEnd(TM). From this we construct a canonical differential 8-form onEIII, associated with its holonomySpin(10)·U(1) ⊂ U(16), that represents a generator of its cohomology ring. We relate it with a Schubert cycle structure by looking at EIII as the smooth projective variety V_(4) ⊂ CP^26 known as the fourth Severi variety
Almost complex structures on spheres
Full text linkIn this paper we review the well-known fact that the only spheres admitting an almost complex structure are S2 and S6. The proof described here uses characteristic classes and the Bott periodicity theorem in topological K-theory. This paper originates from the talk “Almost Complex Structures on Spheres” given by the second author at the MAM1 workshop “(Non)-existence of complex structures on S6”, held in Marburg from March 27th to March 30th, 2017. It is a review paper, and as such no result is intended to be original. We tried to produce a clear, motivated and as much as possible self-contained exposition. © 2017 Elsevier B.V
Clifford systems in octonionic geometry
No full textWe give an inductive construction for irreducible Clifford systems on Euclidean vector spaces. We then discuss how this notion can be adapted to Riemannian manifolds, and outline some developments in octonionic geometry
Pseudo Random Number Generation through Reinforcement Learning and Recurrent Neural Networks
Full text linkA Pseudo-Random Number Generator (PRNG) is any algorithm generating a sequence of numbers approximating properties of random numbers. These numbers are widely employed in mid-level cryptography and in software applications. Test suites are used to evaluate the quality of PRNGs by checking statistical properties of the generated sequences. These sequences are commonly represented bit by bit. This paper proposes a Reinforcement Learning (RL) approach to the task of generating PRNGs from scratch by learning a policy to solve a partially observable Markov Decision Process (MDP), where the full state is the period of the generated sequence, and the observation at each time-step is the last sequence of bits appended to such states. We use Long-Short Term Memory (LSTM) architecture to model the temporal relationship between observations at different time-steps by tasking the LSTM memory with the extraction of significant features of the hidden portion of the MDP’s states. We show that modeling a PRNG with a partially observable MDP and an LSTM architecture largely improves the results of the fully observable feedforward RL approach introduced in previous work
The Role of Spin(9) in Octonionic Geometry
Full text linkStarting from the 2001 Thomas Friedrich’s work on Spin(9), we review some interactions
between Spin(9) and geometries related to octonions. Several topics are discussed in this respect:
explicit descriptions of the Spin(9) canonical 8-form and its analogies with quaternionic geometry
as well as the role of Spin(9) both in the classical problems of vector fields on spheres and in the
geometry of the octonionic Hopf fibration. Next, we deal with locally conformally parallel Spin(9)
manifolds in the framework of intrinsic torsion. Finally, we discuss applications of Clifford systems
and Clifford structures to Cayley–Rosenfeld planes and to three series of Grassmannians
Holomorphic submersions of locally conformally Kähler manifolds
No full textA locally conformally Kähler (LCK) manifold is a complex manifold covered by a Kähler manifold, with the covering group acting by homotheties. We show that if such a compact manifold X admits a holomorphic submersion with positive-dimensional fibers at least one of which is of Kähler type, then X is globally conformally Kähler or biholomorphic, up to finite covers, to a small deformation of a Vaisman manifold (i.e., a mapping torus over a circle, with Sasakian fiber). As a consequence, we show that the product of a compact non-Kähler LCK and a compact Kähler manifold cannot carry a LCK metric
Spheres with more than 7 vector fields: All the fault of Spin(9)
No full textWe give an interpretation of the maximal number of linearly independent vector fields on spheres in terms of the Spin(9) representation on R-16. This casts an insight on the role of Spin(9) as a subgroup of SO(16) on the existence of vector fields on spheres, parallel to the one played by complex, quaternionic and octonionic structures on R-2, R-4 and R-8. respectively. (C) 2012 Elsevier Inc. All rights reserved
Spin(9) and almost complex structures on 16-dimensional manifolds
No full textFor a Spin(9)-structure on a Riemannian manifold M (16) we write explicitly the matrix psi of its Kahler 2-forms and the canonical 8-form Phi(Spin(9)). We then prove that Phi(Spin(9)) coincides up to a constant with the fourth coefficient of the characteristic polynomial of psi. This is inspired by lower dimensional situations, related to Hopf fibrations and to Spin(7). As applications, formulas are deduced for Pontrjagin classes and integrals of Phi(Spin(9)) and Phi(Spin)(9) in the special case of holonomy Spin(9)
Signature-Based Community Detection for Time Series
No full textCommunity detection for time series without prior knowledge poses an open challenge within complex networks theory. Traditional approaches begin by assessing time series correlations and maximizing modularity under diverse null models. These methods suffer from assuming temporal stationarity and are influenced by the granularity of observation intervals. In this study, we propose an approach based on the signature matrix, a concept from path theory for studying stochastic processes. By employing a signature-derived similarity measure, our method overcomes drawbacks of traditional correlation-based techniques. Through a series of numerical experiments, we demonstrate that our method consistently yields higher modularity compared to baseline models, when tested on the Standard and Poor’s 500 dataset. Moreover, our approach showcases enhanced stability in modularity when the length of the underlying time series is manipulated. This research contributes to the field of community detection by introducing a signature-based similarity measure, offering an alternative to conventional correlation matrices
Rigidity of Oeljeklaus–Toma manifolds
Full text linkWe prove that Oeljeklaus-Toma manifolds are rigid, and that any line bundle on Oeljeklaus-Toma manifolds of simple type is flat
- …
