An Efficient Group-Based Replica Placement Policy for Large-Scale Geospatial 3D Raster Data on Hadoop

Geospatial three-dimensional (3D) raster data have been widely used for simple representations and analysis, such as geological models, spatio-temporal satellite data, hyperspectral images, and climate data. With the increasing requirements of resolution and accuracy, the amount of geospatial 3D raster data has grown exponentially. In recent years, the processing of large raster data using Hadoop has gained popularity. However, data uploaded to Hadoop are randomly distributed onto datanodes without consideration of the spatial characteristics. As a result, the direct processing of geospatial 3D raster data produces a massive network data exchange among the datanodes and degrades the performance of the cluster. To address this problem, we propose an efficient group-based replica placement policy for large-scale geospatial 3D raster data, aiming to optimize the locations of the replicas in the cluster to reduce the network overhead. An overlapped group scheme was designed for three replicas of each file. The data in each group were placed in the same datanode, and different colocation patterns for three replicas were implemented to further reduce the communication between groups. The experimental results show that our approach significantly reduces the network overhead during data acquisition for 3D raster data in the Hadoop cluster, and maintains the Hadoop replica placement requirements.

Equivariant connective 𝐾-theory

For separated schemes of finite type over a field with an action of an affine group scheme of finite type, we construct the bi-graded equivariant connective K K -theory mapping to the equivariant K K -homology of Guillot and the equivariant algebraic K K -theory of Thomason. It has all the standard basic properties as the homotopy invariance and localization. We also get the equivariant version of the Brown-Gersten-Quillen spectral sequence and study its convergence.

Scaling Limits of Lattice Quantum Fields by Wavelets

We present a rigorous renormalization group scheme for lattice quantum field theories in terms of operator algebras. The renormalization group is considered as an inductive system of scaling maps between lattice field algebras. We construct scaling maps for scalar lattice fields using Daubechies’ wavelets, and show that the inductive limit of free lattice ground states exists and the limit state extends to the familiar massive continuum free field, with the continuum action of spacetime translations. In particular, lattice fields are identified with the continuum field smeared with Daubechies’ scaling functions. We compare our scaling maps with other renormalization schemes and their features, such as the momentum shell method or block-spin transformations.

The Fundamental Group of a Noncommutative Space

We introduce and analyse a general notion of fundamental group for noncommutative spaces, described by differential graded algebras. For this we consider connections on finitely generated projective bimodules over differential graded algebras and show that the category of flat connections on such modules forms a Tannakian category. As such this category can be realised as the category of representations of an affine group scheme G, which in the classical case is (the pro-algebraic completion of) the usual fundamental group. This motivates us to define G to be the fundamental group of the noncommutative space under consideration. The needed assumptions on the differential graded algebra are rather mild and completely natural in the context of noncommutative differential geometry. We establish the appropriate functorial properties, homotopy and Morita invariance of this fundamental group. As an example we find that the fundamental group of the noncommutative torus can be described as the algebraic hull of the topological group $(\mathbb Z+\theta \mathbb Z)^{2}$ ( ℤ + 𝜃 ℤ ) 2 .

RECIPROCITY SHEAVES AND THEIR RAMIFICATION FILTRATIONS

We define a motivic conductor for any presheaf with transfers F using the categorical framework developed for the theory of motives with modulus by Kahn, Miyazaki, Saito and Yamazaki. If F is a reciprocity sheaf, this conductor yields an increasing and exhaustive filtration on $F(L)$ , where L is any henselian discrete valuation field of geometric type over the perfect ground field. We show that if F is a smooth group scheme, then the motivic conductor extends the Rosenlicht–Serre conductor; if F assigns to X the group of finite characters on the abelianised étale fundamental group of X, then the motivic conductor agrees with the Artin conductor defined by Kato and Matsuda; and if F assigns to X the group of integrable rank $1$ connections (in characteristic $0$ ), then it agrees with the irregularity. We also show that this machinery gives rise to a conductor for torsors under finite flat group schemes over the base field, which we believe to be new. We introduce a general notion of conductors on presheaves with transfers and show that on a reciprocity sheaf, the motivic conductor is minimal and any conductor which is defined only for henselian discrete valuation fields of geometric type with perfect residue field can be uniquely extended to all such fields without any restriction on the residue field. For example, the Kato–Matsuda Artin conductor is characterised as the canonical extension of the classical Artin conductor defined in the case of a perfect residue field.

The ℓ-modular representation of reductive groups over finite local rings of length two

Let O 2 \mathcal{O}_{2} and O 2 ′ \mathcal{O}^{\prime}_{2} be two distinct finite local rings of length two with residue field of characteristic 𝑝. Let G ⁢ ( O 2 ) \mathbb{G}(\mathcal{O}_{2}) and G ⁢ ( O 2 ′ ) \mathbb{G}(\mathcal{O}^{\prime}_{2}) be the groups of points of any reductive group scheme 𝔾 over ℤ such that 𝑝 is very good for G × F q \mathbb{G}\times\mathbb{F}_{q} or G = GL n \mathbb{G}=\operatorname{GL}_{n} . We prove that there exists an isomorphism of group algebras K ⁢ G ⁢ ( O 2 ) ≅ K ⁢ G ⁢ ( O 2 ′ ) K\mathbb{G}(\mathcal{O}_{2})\cong K\mathbb{G}(\mathcal{O}^{\prime}_{2}) , where 𝐾 is a sufficiently large field of characteristic different from 𝑝.

Instituting village savings and loan associations scheme through action research in Zimbabwe

lage savings and loan associations schemes have become one of the critical survival strategies amidst poverty, inequality and financial exclusion thus helping low income communities to ride out poverty and make their savings and to eke out a living. However, the use of scientific procedures by ordinary people when establishing such initiatives in their villages is under-reported. This study, therefore, sought to test the utility of the action research (AR) methodology in establishing a low cost village savings and loan associations scheme intervention, with a view to assessing the extent to which the scheme can improve the livelihoods of members of the scheme, and draw lessons for future interventions. The process involved planning, designing, establishing and evaluating a village savings and loan association scheme initiative involving 15 individual members (inclusive of the researcher) in ward 8 of Seke district, Zimbabwe. Results showed that creating village savings and loan associations is possible using action research, as community participation in the design, implementation and day-to-day operations of such initiatives guaranteeing ownership and control of the initiative by the host group scheme are almost natural to action research. One of the comparative advantages of using action research is that it creates spaces for ordinary people to share their experiences, reflect, and come up with context-specific solutions, as they take responsibility for their financial wellbeing, thus helping to meet their socio-economic needs and aspirations. The strength of village savings and loan associations is that they can be replicated. The study recommends that in the era of COVID-19, social distancing rules and regulations introduced to contain the virus should be observed.

Dieudonné theory via cohomology of classifying stacks

We prove that if G is a finite flat group scheme of p-power rank over a perfect field of characteristic p, then the second crystalline cohomology of its classifying stack $H^2_{\text {crys}}(BG)$ recovers the Dieudonné module of G. We also provide a calculation of the crystalline cohomology of the classifying stack of an abelian variety. We use this to prove that the crystalline cohomology of the classifying stack of a p-divisible group is a symmetric algebra (in degree $2$ ) on its Dieudonné module. We also prove mixed-characteristic analogues of some of these results using prismatic cohomology.

Almost Homogeneous Varieties of Albanese Codimension One

We classify almost homogeneous normal varieties of Albanese codimension $1$, defined over an arbitrary field. We prove that such a variety has a unique normal equivariant completion. Over a perfect field, the group scheme of automorphisms of this completion is smooth, except in one case in characteristic $2$, and we determine its (reduced) neutral component.