Tits polygons

We introduce the notion of a Tits polygon, a generalization of the notion of a Moufang polygon, and show that Tits polygons arise in a natural way from certain configurations of parabolic subgroups in an arbitrary spherical buildings satisfying the Moufang condition. We establish numerous basic properties of Tits polygons and characterize a large class of Tits hexagons in terms of Jordan algebras. We apply this classification to give a “rank 2 2 ” presentation for the group of F F -rational points of an arbitrary exceptional simple group of F F -rank at least 4 4 and to determine defining relations for the group of F F -rational points of an an arbitrary group of F F -rank 1 1 and absolute type D 4 D_4 , E 6 E_6 , E 7 E_7 or E 8 E_8 associated to the unique vertex of the Dynkin diagram that is not orthogonal to the highest root. All of these results are over a field of arbitrary characteristic.

Application of Electrocardiogram Localization during Peripherally Inserted Central Catheter Line Insertion into the Persistent Left Superior Vena Cava of Neonates

Background Most neonates with persistent left superior vena cava (PLSVC) have no clinical symptoms or hemodynamic changes, and this anomaly is only found during cardiac catheterization, pacemaker implantation, or central venous catheterization. Electrocardiogram (ECG) localization is helpful for the application of the peripherally inserted central catheter (PICC) technique in neonates with PLSVC. Objective To explore the characteristic waveforms of the P wave when a PICC under ECG localization is applied in neonates with PLSVC. Study Design The observation and management strategies for the P wave changes during catheter insertion (CI) of two neonates with PLSVC admitted to our institution between January and July 2020, who underwent PICC line insertion, were summarized. Results The characteristic P wave changes in two children with a PICC line inserted via the PLSVC were observed. When a wide inverted P wave appeared on ECG, the catheter was immediately withdrawn by 0.5 cm, a bidirectional P wave gradually appeared and then disappeared. After that, the catheter was further withdrawn by 0.5 cm. After catheterization, the optimal position of the PICC was confirmed by X-ray photography and bedside B-ultrasound. The PICC line was removed as scheduled after indwelling for 18 and 29 days, respectively, in the two cases, and no PICC-related complications occurred during indwelling. Conclusion The characteristic P wave changes on ECG during CI provide important clinical reference values for the application of the PICC technique under ECG localization in neonates with PLSVC. Key Points

GEOMETRIC WEIGHT-SHIFTING OPERATORS ON HILBERT MODULAR FORMS IN CHARACTERISTIC p

We carry out a thorough study of weight-shifting operators on Hilbert modular forms in characteristic p, generalising the author’s prior work with Sasaki to the case where p is ramified in the totally real field. In particular, we use the partial Hasse invariants and Kodaira–Spencer filtrations defined by Reduzzi and Xiao to improve on Andreatta and Goren’s construction of partial $\Theta $ -operators, obtaining ones whose effect on weights is optimal from the point of view of geometric Serre weight conjectures. Furthermore, we describe the kernels of partial $\Theta $ -operators in terms of images of geometrically constructed partial Frobenius operators. Finally, we apply our results to prove a partial positivity result for minimal weights of mod p Hilbert modular forms.

Constacyclic Codes over Finite Chain Rings of Characteristic p

Let R be a finite commutative chain ring of characteristic p with invariants p,r, and k. In this paper, we study λ-constacyclic codes of an arbitrary length N over R, where λ is a unit of R. We first reduce this to investigate constacyclic codes of length ps (N=n1ps,p∤n1) over a certain finite chain ring CR(uk,rb) of characteristic p, which is an extension of R. Then we use discrete Fourier transform (DFT) to construct an isomorphism γ between R[x]/<xN−λ> and a direct sum ⊕b∈IS(rb) of certain local rings, where I is the complete set of representatives of p-cyclotomic cosets modulo n1. By this isomorphism, all codes over R and their dual codes are obtained from the ideals of S(rb). In addition, we determine explicitly the inverse of γ so that the unique polynomial representations of λ-constacyclic codes may be calculated. Finally, for k=2 the exact number of such codes is provided.

Unit groups of group algebras of Abelian groups of order 32

Let F be a finite field of characteristic p > 0 with q = pn elements. In this paper, a complete characterization of the unit groups U(F G) of group algebras F G for the abelian groups of order 32, over finite field of characteristic p > 0 has been obtained.

Cohomology of simple modules for sl3(k) in characteristic 3

In this paper we calculate cohomology of a classical Lie algebra of type A2 over an algebraically field k of characteristic p = 3 with coefficients in simple modules. To describe their structure we will consider them as modules over an algebraic group SL3(k). In the case of characteristic p = 3, there are only two peculiar simple modules: a simple that module isomorphic to the quotient module of the adjoint module by the center, and a one-dimensional trivial module. The results on the cohomology of simple nontrivial module are used for calculating the cohomology of the adjoint module. We also calculate cohomology of the simple quotient algebra Lie of A2 by the center.