Rank 2 local systems and abelian varieties II

LetX/Fq be a smooth, geometrically connected, quasi projective scheme. Let Ebe a semisimple over convergent F-isocrystal on X. Suppose that irreducible summands Ei of E have rank 2, determinant ̄Qp (−1), and infinite monodromy at∞. Suppose further that for each closed point x of X, the characteristic polynomial of E at x is in Q[t]⊂Qp[t]. Then there exists a non-trivial open set U⊂X such that E|U comes from a family of abelian varieties on U. As an application, let L1 be an irreducible lisse ̄Ql sheaf on X that has rank 2, determinant ̄Ql(−1), and infinite monodromy at∞. Then all crystalline companions to L1 exist (as predicted by Deligne’s crystalline companions conjecture) if and only if there exists a non-trivial open set U⊂X and an abelian scheme πU: AU→U such that L1|U is a summand of R1(πU)∗ ̄Ql

Variation in the Biomolecular Interactions of Nickel(Ii) Hydrazone Complexes Upon Tuning the Hydrazide Fragment

Three new bivalent nickel hydrazone complexes have been synthesised from the reactions of [NiCl2(PPh3)(2)] with H2L {L = dianion of the hydrazones derived from the condensation of o-hydroxynaphthaldehyde with furoic acid hydrazide (H2L1) (1)/thiophene-2-acid hydrazide (H2L2) (2)/isonicotinic acid hydrazide (H2L3) (3)} and formulated as [Ni(L-1)(PPh3)] (4), [Ni(L-2)(PPh3)] (5) and [Ni(L-3)(PPh3)] (6). Structural characterization of these compounds 4-6 were accomplished by using various physico-chemical techniques. Single crystal X-ray diffraction data of complexes 4 and 5 proved their distorted square planar geometry. In order to ascertain the potential of the above synthesised compounds towards biomolecular interactions, additional experiments involving interaction with calf thymus DNA (CT DNA) and bovine serum albumin (BSA) were carried out. All the ligands and corresponding nickel(II) chelates have been screened for their scavenging effect towards O-2(-), OH and NO radicals. The efficiency of complexes 4-6 to arrest the growth of HeLa, HepG-2 and A431 tumour cell lines has been studied along with the cell viability test against the non-cancerous NIH 3T3 cells under in vitro conditions.University Grants Commission, New Delhi under the UGC-SAP-DRSRobert A. Welch Foundation F-0003Chemistr

A collage of results on the divisibility and indivisibility of class numbers of quadratic fields

The investigation of the ideal class group $Cl_K$ of an algebraic number field $K$ is one of the key subjects of inquiry in algebraic number theory since it encodes a lot of arithmetic information about K. There is a considerable amount of research on many topics linked to quadratic field class groups notably intriguing aspect is the divisibility of the class numbers. This article discusses a few recent results on the divisibility of class numbers and the Izuka conjecture. We also discuss the quantitative aspect of the Izuka conjecture.Comment: Comments are welcome.11 page

Flat Norm Decomposition of Integral Currents

Currents represent generalized surfaces studied in geometric measure theory. They range from relatively tame integral currents representing oriented compact manifolds with boundary and integer multiplicities, to arbitrary elements of the dual space of differential forms. The flat norm provides a natural distance in the space of currents, and works by decomposing a $d$-dimensional current into $d$- and (the boundary of) $(d+1)$-dimensional pieces in an optimal way. Given an integral current, can we expect its flat norm decomposition to be integral as well? This is not known in general, except in the case of $d$-currents that are boundaries of $(d+1)$-currents in $\mathbb{R}^{d+1}$ (following results from a corresponding problem on the $L^1$ total variation ($L^1$TV) of functionals). On the other hand, for a discretized flat norm on a finite simplicial complex, the analogous statement holds even when the inputs are not boundaries. This simplicial version relies on the total unimodularity of the boundary matrix of the simplicial complex -- a result distinct from the $L^1$TV approach. We develop an analysis framework that extends the result in the simplicial setting to one for $d$-currents in $\mathbb{R}^{d+1}$, provided a suitable triangulation result holds. In $\mathbb{R}^2$, we use a triangulation result of Shewchuk (bounding both the size and location of small angles), and apply the framework to show that the discrete result implies the continuous result for $1$-currents in $\mathbb{R}^2$.Comment: 17 pages, adds some related work and application

Simplicial Flat Norm with Scale

We study the multiscale simplicial flat norm (MSFN) problem, which computes flat norm at various scales of sets defined as oriented subcomplexes of finite simplicial complexes in arbitrary dimensions. We show that the multiscale simplicial flat norm is NP-complete when homology is defined over integers. We cast the multiscale simplicial flat norm as an instance of integer linear optimization. Following recent results on related problems, the multiscale simplicial flat norm integer program can be solved in polynomial time by solving its linear programming relaxation, when the simplicial complex satisfies a simple topological condition (absence of relative torsion). Our most significant contribution is the simplicial deformation theorem, which states that one may approximate a general current with a simplicial current while bounding the expansion of its mass. We present explicit bounds on the quality of this approximation, which indicate that the simplicial current gets closer to the original current as we make the simplicial complex finer. The multiscale simplicial flat norm opens up the possibilities of using flat norm to denoise or extract scale information of large data sets in arbitrary dimensions. On the other hand, it allows one to employ the large body of algorithmic results on simplicial complexes to address more general problems related to currents.Comment: To appear in the Journal of Computational Geometry. Since the last version, the section comparing our bounds to Sullivan's has been expanded. In particular, we show that our bounds are uniformly better in the case of boundaries and less sensitive to simplicial irregularit

Optimisation of hood panels of a passenger car for pedestrian protection

This thesis presents a pro–active research work motivated by the prospect of the imminent implementation of the regulatory requirement for pedestrian protection, Global Technical Regulation–9 (GTR–9) (United Nations Economic Commission for Europe 1998) in the near future. To meet the performance criteria for pedestrian protection head impact, it is vital to incorporate the required design parameters into the hood design process at an early stage. These main design parameters are architectural and changing them late in the vehicle design process is very expensive and difficult to implement. The main design parameters are the inner and outer hood thickness, inner and outer hood material, inner hood structure and the available deformation space to hard components such as the engine. The main objective of this work is to develop a methodology for optimising hood panels of passenger cars to ensure that the pedestrian Head Injury Criterion (HIC) falls below the threshold values specified by both the GTR–9 and the consumer metric, the Australasian New Car Assessment Program (ANCAP). This study investigated the development of a hood configuration that provides robust and homogeneous HIC for different impact positions in the central area of the hood of a large sedan, taking into consideration of the limited space available for deformation. An extensive series of Computer Aided Engineering (CAE) simulations has been carried out to collect the acceleration data and vertical intrusion data required to validate the proposed methodology and the optimal hood configuration. These impact simulations include a stationary vehicle set up and a moving head impactor as per GTR–9. The Design of Experiments (DOE) has been set up with the control factors as inputs to the Kriging response surface and Monte Carlo methods to output the responses. The variables considered for the control factors are the inner hood structure, inner hood thickness and material, outer hood thickness and material, and the impact positions. The results from the numerical tests have been utilised to map the response surfaces in order to identify the important variables and to visualise the relationships between the inputs and the outputs. The proposed optimisation methodology is described in detail and the outcomes provide clear recommendation of the optimal configuration of passenger car hood panels. In conclusion, if the vehicle design team’s main objective were to reduce the deformation space, the preferred choice for hood material would be steel rather than aluminium. The benefits of minimising the deformation space are significant. They include the freedom of styling, improved aerodynamics, and hence improvements in vehicle stability and fuel economy. The trade–off will be a higher mass than the equivalent aluminium system. On the other hand, if the vehicle design and program team’s main objective was to reduce the system mass, then the preferred choice for hood material would be aluminium. The trade–off would be a higher deformation space than that is required for the steel system