On the Local Case in the Aschbacher Theorem for Symplectic and Orthogonal Groups

N. Yang; A. A. Galt

doi:10.1134/s0037446621020178

Low-dimensional representations of finite orthogonal groups

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004121000037 ◽

2021 ◽

pp. 1-22

Author(s):

KAY MAGAARD ◽

GUNTER MALLE

Keyword(s):

Orthogonal Groups ◽

Brauer Characters ◽

Low Dimensional

Abstract We determine the smallest irreducible Brauer characters for finite quasi-simple orthogonal type groups in non-defining characteristic. Under some restrictions on the characteristic we also prove a gap result showing that the next larger irreducible Brauer characters have a degree roughly the square of those of the smallest non-trivial characters.

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STEENROD OPERATIONS ON THE DE RHAM COHOMOLOGY OF ALGEBRAIC STACKS

Journal of the Institute of Mathematics of Jussieu ◽

10.1017/s1474748021000177 ◽

2021 ◽

pp. 1-48

Author(s):

Federico Scavia

Keyword(s):

Finite Groups ◽

Reductive Groups ◽

Orthogonal Groups ◽

De Rham Cohomology ◽

Algebraic Stacks ◽

Steenrod Operations ◽

De Rham

Abstract Building upon work of Epstein, May and Drury, we define and investigate the mod p Steenrod operations on the de Rham cohomology of smooth algebraic stacks over a field of characteristic $p>0$ . We then compute the action of the operations on the de Rham cohomology of classifying stacks for finite groups, connected reductive groups for which p is not a torsion prime and (special) orthogonal groups when $p=2$ .

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Higher pullbacks of modular forms on orthogonal groups

Forum Mathematicum ◽

10.1515/forum-2020-0066 ◽

2021 ◽

Vol 0 (0) ◽

Author(s):

Brandon Williams

Keyword(s):

Modular Forms ◽

Differential Operators ◽

Jacobi Form ◽

Siegel Modular Forms ◽

Orthogonal Groups

Abstract We apply differential operators to modular forms on orthogonal groups O ⁢ ( 2 , ℓ ) {\mathrm{O}(2,\ell)} to construct infinite families of modular forms on special cycles. These operators generalize the quasi-pullback. The subspaces of theta lifts are preserved; in particular, the higher pullbacks of the lift of a (lattice-index) Jacobi form ϕ are theta lifts of partial development coefficients of ϕ. For certain lattices of signature ( 2 , 2 ) {(2,2)} and ( 2 , 3 ) {(2,3)} , for which there are interpretations as Hilbert–Siegel modular forms, we observe that the higher pullbacks coincide with differential operators introduced by Cohen and Ibukiyama.

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Assessing Urbanization Dynamics in Turkey’s Marmara Region Using CORINE Data between 2006 and 2018

Remote Sensing ◽

10.3390/rs13040664 ◽

2021 ◽

Vol 13 (4) ◽

pp. 664

Author(s):

Özlem Altınkaya Genel ◽

ChengHe Guan

Keyword(s):

Land Cover ◽

Land Use Planning ◽

Complex Structure ◽

Spatial Dynamics ◽

Growth Dynamics ◽

Theory And Practice ◽

Marmara Region ◽

Local Case ◽

Sustainable Regional Development ◽

Linking Theory And Practice

This study investigated the urban growth dynamics of urban regions. The study area was the Marmara Region, one of the most densely populated and ecologically diverse areas in Turkey. Using CORINE land cover data for 2006, 2012, and 2018, the study utilized multiple correspondence analyses and cluster analyses, to analyze land cover changes. The resulting maps, visualized in GIS, revealed the rapid urban transformation of the regional structure, formerly comprised of four distinct areas, into a more complex structure, in which densification and sprawl occur simultaneously. Our findings demonstrated a dissonance between the spatial dynamics of the Marmara Region during the study period, and the capacity and scope of the simultaneously initiated regional policies and mega-projects. This uncoordinated approach has endangered the region’s sustainable development. The paper, therefore, discusses the importance of land use planning and transboundary collaboration for sustainable regional development. Beyond the local case, the results contribute to critical theories in regional planning by linking theory and practice.

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The use of diversity indices for local assessment of marine sediment quality

Scientific Reports ◽

10.1038/s41598-021-94636-0 ◽

2021 ◽

Vol 11 (1) ◽

Author(s):

Shinya Hosokawa ◽

Kyosuke Momota ◽

Anthony A. Chariton ◽

Ryoji Naito ◽

Yoshiyuki Nakamura

Keyword(s):

Community Structure ◽

Sample Size ◽

Small Sample Size ◽

Benthic Communities ◽

Sediment Quality ◽

Diversity Indices ◽

Small Sample ◽

Species Density ◽

Local Case ◽

Two Indices

AbstractDiversity indices are commonly used to measure changes in marine benthic communities. However, the reliability (and therefore suitability) of these indices for detecting environmental change is often unclear because of small sample size and the inappropriate choice of communities for analysis. This study explored uncertainties in taxonomic density and two indices of community structure in our target region, Japan, and in two local areas within this region, and explored potential solutions. Our analysis of the Japanese regional dataset showed a decrease in family density and a dominance of a few species as sediment conditions become degraded. Local case studies showed that species density is affected by sediment degradation at sites where multiple communities coexist. However, two indices of community structure could become insensitive because of masking by community variability, and small sample size sometimes caused misleading or inaccurate estimates of these indices. We conclude that species density is a sensitive indicator of change in marine benthic communities, and emphasise that indices of community structure should only be used when the community structure of the target community is distinguishable from other coexisting communities and there is sufficient sample size.

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An asymptotically compatible approach for Neumann-type boundary condition on nonlocal problems

ESAIM Mathematical Modelling and Numerical Analysis ◽

10.1051/m2an/2019089 ◽

2020 ◽

Vol 54 (4) ◽

pp. 1373-1413 ◽

Cited By ~ 1

Author(s):

Huaiqian You ◽

XinYang Lu ◽

Nathaniel Task ◽

Yue Yu

Keyword(s):

Boundary Condition ◽

Boundary Conditions ◽

Nonlocal Boundary ◽

Nonlocal Diffusion ◽

Order Convergence ◽

Flux Boundary Condition ◽

Nonlocal Interactions ◽

Local Case ◽

Neumann Type ◽

Type Boundary

In this paper we consider 2D nonlocal diffusion models with a finite nonlocal horizon parameter δ characterizing the range of nonlocal interactions, and consider the treatment of Neumann-like boundary conditions that have proven challenging for discretizations of nonlocal models. We propose a new generalization of classical local Neumann conditions by converting the local flux to a correction term in the nonlocal model, which provides an estimate for the nonlocal interactions of each point with points outside the domain. While existing 2D nonlocal flux boundary conditions have been shown to exhibit at most first order convergence to the local counter part as δ → 0, the proposed Neumann-type boundary formulation recovers the local case as O(δ2) in the L∞ (Ω) norm, which is optimal considering the O(δ2) convergence of the nonlocal equation to its local limit away from the boundary. We analyze the application of this new boundary treatment to the nonlocal diffusion problem, and present conditions under which the solution of the nonlocal boundary value problem converges to the solution of the corresponding local Neumann problem as the horizon is reduced. To demonstrate the applicability of this nonlocal flux boundary condition to more complicated scenarios, we extend the approach to less regular domains, numerically verifying that we preserve second-order convergence for non-convex domains with corners. Based on the new formulation for nonlocal boundary condition, we develop an asymptotically compatible meshfree discretization, obtaining a solution to the nonlocal diffusion equation with mixed boundary conditions that converges with O(δ2) convergence.

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