Geometric and topological approaches to shape variation in Ginkgo leaves

Leaf shape is a key plant trait that varies enormously. The range of applications for data on this trait requires frequent methodological development so that researchers have an up-to-date toolkit with which to quantify leaf shape. We generated a dataset of 468 leaves produced by Ginkgo biloba , and 24 fossil leaves produced by evolutionary relatives of extant Ginkgo . We quantified the shape of each leaf by developing a geometric method based on elastic curves and a topological method based on persistent homology. Our geometric method indicates that shape variation in modern leaves is dominated by leaf size, furrow depth and the angle of the two lobes at the leaf base that is also related to leaf width. Our topological method indicates that shape variation in modern leaves is dominated by leaf size and furrow depth. We have applied both methods to modern and fossil material: the methods are complementary, identifying similar primary patterns of variation, but also revealing different aspects of morphological variation. Our topological approach distinguishes long-shoot leaves from short-shoot leaves, both methods indicate that leaf shape influences or is at least related to leaf area, and both could be applied in palaeoclimatic and evolutionary studies of leaf shape.

Topological Method for a Study of Discriminating Three Categories of Banks and its use in Attributes Reduction

This work approaches the problem of knowledge extraction within the banking domain using rough set, rough set theory can be considered as a topological method. Our main goal is to separate of the accounting attributes to discriminate between Islamic, mixed, and conventional banks. To this end, we have used the positive region in the rough set framework is traditional uncertainty measurements, used usually as in attribute reduction. Attributes banks will be separated and we are classified with a given decision, then we theoretically analyze the variance of the rough set. In the actual application, we used the financial semantics based on the domain expertise of experts to determine between the competing approaches. The results show the value of shared financial information for distinguishing between the three types of banks with certain attributes. These results are helping us offer a new view of attribute reduction in knowledge. We used MATLAB for our applications in computing.

Topological Method for Assessing the Sensitivity to the Detection of Cybersecurity in Electrical Networks

Розглянуто теоретичні питання побудови топологічного методу оцінки чутливості до виявлення кібернетичних загроз в електричних мережах енергосистем за допомогою моделювання режимів роботи окремих (виділених) підсистем. Описано основні етапи побудови моделей топології енергосистем, запропоновано та реалізовано методи форму­вання інформаційних моделей об'єктів енергосистем. Досліджено методи візуалізації ре­зультатів моделювання умов виникнення кіберзагроз. Визначено способи використання запропонованого підходу до створення системи протидії кіберзагрозам в електричних мережах енергосистем і побудови сценаріїв їх ліквідації за допомогою навчального дис­танційного режимного тренажеру.

Multiple positive solutions of second-order nonlinear difference equations with discrete singular ϕ-Laplacian

We consider the existence and multiplicity of positive solutions of the Dirichlet problem for the quasilinear difference equation $$ \textstyle\begin{cases} -\nabla [\phi (\triangle u(t))]=\lambda a(t,u(t))+\mu b(t,u(t)), \quad t\in \mathbb{T}, \\ u(1)=u(N)=0, \end{cases} $$ { − ∇ [ ϕ ( △ u ( t ) ) ] = λ a ( t , u ( t ) ) + μ b ( t , u ( t ) ) , t ∈ T , u ( 1 ) = u ( N ) = 0 , where $\lambda ,\mu \geq 0$ λ , μ ≥ 0 , $\mathbb{T}=\{2,\ldots ,N-1\}$ T = { 2 , … , N − 1 } with $N>3$ N > 3 , $\phi (s)=s/\sqrt{1-s^{2}}$ ϕ ( s ) = s / 1 − s 2 . The function $f:=\lambda a(t,s)+\mu b(t,s)$ f : = λ a ( t , s ) + μ b ( t , s ) is either sublinear, or superlinear, or sub-superlinear near $s=0$ s = 0 . Applying the topological method, we prove the existence of either one or two, or three positive solutions.

Geometric and Topological Approaches to Shape Variation in Ginkgo Leaves

Leaf shape is a key plant trait that varies enormously. The diversity of leaf shape, and the range of applications for data on this trait, requires frequent methodological developments so that researchers have an up-to-date toolkit with which to quantify leaf shape. We generated a dataset of 468 leaves produced by Ginkgo biloba, and 24 fossil leaves produced by evolutionary relatives of extant Ginkgo. We quantified the shape of each leaf by developing a geometric method based on elastic curves and a topological method based on persistent homology. Our geometric method indicates that shape variation in modern leaves is dominated by leaf size, furrow depth, and the angle of the two lobes at the base of the leaf that is also related to leaf width. Our topological method indicates that shape variation in modern leaves is dominated by leaf size and furrow depth. Both methods indicate that there is greater diversity in the shape of fossil leaves compared to modern leaves. The two approaches we have described can be applied to modern and fossil material, and are complementary: identifying similar primary patterns of variation, but revealing some different aspects of morphological variation.