EULER CHARACTERISTIC OF TANGO BUNDLES Nguyen, Hong Cong; Dang, Tuan Hiep; Nguyen, Thi Mai Van
Tạp chí Khoa học Đại học Đà Lạt,
01/2022
Journal Article
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We are interested in a vector bundle constructed by Tango (1976). The Tango bundle is an indecomposable vector bundle of rank \(n-1\) on the complex projective space \(\mathbb{P}^n\). In particular, ...we show that the Euler characteristic of the Tango bundle on \(\mathbb{P}^n\) is equal to \(2n-1\).
•Investigate pore heterogeneity on the dissolution pattern and permeability evolution.•Use the Euler characteristic to characterize the pore heterogeneity in porous media.•Influence of pore ...heterogeneity on permeability change presents non-linear behavior.•Dissolution is dependent on pore heterogeneity at high Péclet and Damköhler numbers.•Permeability-porosity curve is influenced by heterogeneity at high Damköhler number.
The influence of pore space heterogeneity on mineral dissolution and permeability evolution in porous media was investigated using a numerical approach. Artificial porous media were generated by the linear Boolean model, and pore heterogeneity was evaluated using the Euler–Poincaré characteristic (i.e., Euler number). We applied the lattice Boltzmann method with dual particle distribution functions to simulate mineral dissolution under the combined effect of fluid flow and a diffusion process. Simulations were conducted to investigate dissolution patterns for a wide range of Péclet (Pe) and Damköhler (Da) numbers and various pore geometries. Six dissolution regimes were observed, and two types of transition phenomena between these dissolution regimes could be characterized. At high Pe and Da, the dissolution patterns strongly depended on the pore heterogeneity. In addition, four types of porosity–permeability relationship were observed. These relationships were influenced by the pore heterogeneity at high Da numbers.
A multi-relaxation time lattice-Boltzmann model is employed to investigate the dynamic evolution of two immiscible phases through an artificial, randomly generated, porous structure. The flow is ...driven by a constant pressure gradient, in the absence of gravitational effects. Constraining attention to two dimensions, the impact of the morphological properties of the porous structure, generated using constant radius circular solid grains, on a water-wet, oil–water two-phase flow is studied. Variations in the pore space connectivity and topology are quantified by the Euler characteristic. It is found that the wetting phase saturation and the degree of pore network homogeneity have a significant impact on the dynamic evolution of the non-wetting phase topology, which is governed by a series of coalescence and snap-off events. It is also observed that the phenomenal macro-scale steady-state based on the velocity field does not also imply a temporal topological invariance of the displaced phases. The impact of the pore space morphology on the transient dynamics of the two-phase flow is monitored and quantified through a series of hydrodynamic and topological parameters that signify the underlying flow transport processes.
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•The impact of pore space connectivity on transient two-phase flows is investigated.•Macroscopic steady-state conditions occur under evolving fluid phase topology.•Transient phase fragmentation patterns highly depend on saturation levels.•Porous medium inhomogeneity promotes flow channelisation patterns.•Porous media inhomogeneity increased interface length for drainage and imbibition.
Let G be a finite group and X be a subgroup of G. We investigate the topological properties of the poset CX(G) of cosets Hx in G with X≤H<G. We show that CX(G) is non-contractible if G is solvable or ...NG(X) contains a Sylow 2-subgroup and a Sylow 3-subgroup of G. This result follows J. Shareshian and R. Woodroofe's work in 8 (2016). We also give some divisibility properties of the Euler characteristic of CX(G) when X is a p-group, which follows K. S. Brown's classical result in 3 (2000).
Glioblastoma multiforme (GBM) is an aggressive form of human brain cancer that is under active study in the field of cancer biology. Its rapid progression and the relative time cost of obtaining ...molecular data make other readily available forms of data, such as images, an important resource for actionable measures in patients. Our goal is to use information given by medical images taken from GBM patients in statistical settings. To do this, we design a novel statistic-the smooth Euler characteristic transform (SECT)-that quantifies magnetic resonance images of tumors. Due to its well-defined inner product structure, the SECT can be used in a wider range of functional and nonparametric modeling approaches than other previously proposed topological summary statistics. When applied to a cohort of GBM patients, we find that the SECT is a better predictor of clinical outcomes than both existing tumor shape quantifications and common molecular assays. Specifically, we demonstrate that SECT features alone explain more of the variance in GBM patient survival than gene expression, volumetric features, and morphometric features. The main takeaways from our findings are thus 2-fold. First, they suggest that images contain valuable information that can play an important role in clinical prognosis and other medical decisions. Second, they show that the SECT is a viable tool for the broader study of medical imaging informatics.
Supplementary materials
for this article, including a standardized description of the materials available for reproducing the work, are available as an online supplement.
Immiscible displacements in porous media are unquestionably of great significance in numerous natural and industrial processes. It has been well established and accepted that in addition to the ...immiscible fluid properties, the morphology of the porous medium also plays a significant role in the final volumetric throughput of a particular fluid. Topological features of the porous medium have been found to exert a strong influence on the hydrodynamic behaviour of single and two-phase flows as they express a measure of pore space and consequently flow path connectivity and availability. The current study investigates the effect of the pore space connectivity, expressed through the Euler characteristic, on the hydrodynamic behaviour of a water-wet, oil–water two-phase system at steady state. Two-dimensional simulations are conducted in artificially generated porous media with constant diameter circular solid grains using a multi-relaxation time lattice-Boltzmann model. It is shown that topological features of the porous medium can significantly affect the macro-scale capillary pressure and relative permeability curves for drainage and imbibition but in different ways. It is also demonstrated that pore space connectivity has a strong influence on the fluid phase distribution and fragmentation patterns in the porous structure depending on the displacement process.
•The impact of pore space connectivity on immiscible two-phase flows is investigated.•Porous media with varying topology are simulated in drainage and imbibition.•The pore space Euler Characteristic impacts the hydrodynamic features of the flow.•The pore space Euler Characteristic affects the topological features of the flow.
In this paper, we settle an open conjecture regarding the assertion that the Euler-characteristic of G/NG(T) for a split reductive group scheme G and the normalizer of a split maximal torus NG(T) ...over a field is 1 in the Grothendieck-Witt ring with the characteristic exponent of the field inverted, under the assumption that the base field contains a −1. Numerous applications of this to splittings in the motivic stable homotopy category and to Algebraic K-Theory are worked out in several related papers by Gunnar Carlsson and the authors.
This paper studies Gaussian random fields with Matérn covariance functions with smooth parameter ν>2. Two cases of parameter spaces, the Euclidean space and N-dimensional sphere, are considered. For ...such smooth Gaussian fields, we have derived the explicit formulae for the expected Euler characteristic of the excursion set, the expected number and height distribution of critical points. The results are valuable for approximating the excursion probability in family-wise error control and for computing p-values in peak inference.
We begin by describing where and when Euler obtained the famous formula V + F = E + 2, which relates the number of vertices, edges and faces of a polyhedron that satisfies certain conditions. A few ...considerations are made about the relation of this formula with other problems and some difficulties of the original proof given by Euler. Then we move to the end of the 19th and beginning of the 20th century when the Euler haracteristic and its generalization were linked to new topics in topology. Finally we present some of the generalizations of Euler characteristic which are used in recent (in the past 50 years) developments of topology.
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