Distinguished Boundary Research Articles

Rational tetra-inner functions and the special variety of the tetrablock

The setE‾={x∈C3:1−x1z−x2w+x3zw≠0 whenever |z|<1,|w|<1} is called the tetrablock and has intriguing complex-geometric properties. It is polynomially convex, nonconvex and starlike about 0. It has a group of automorphisms parametrised by AutD×AutD×Z2 and its distinguished boundary bE‾ is homeomorphic to the solid torus D‾×T. It has a special subvarietyRE‾={(x1,x2,x3)∈E‾:x1x2=x3}, called the royal variety of E‾. RE‾ is a complex geodesic of E and it is invariant under all automorphisms of E. We make use of these geometric properties of E‾ to develop an explicit structure theory for the rational maps from the unit disc D to E‾ that map the unit circle T to the distinguished boundary bE‾ of E‾. Such maps are called rational E‾-inner functions. We call the points λ∈D‾ such that x(λ)∈RE‾ the royal nodes of x. We describe the construction of rational E‾-inner functions of prescribed degree from the zeros of x1 and x2 and the royal nodes of x. The proof of this theorem is constructive: it gives an algorithm for the construction of a 3-parameter family of such functions x subject to the computation of Fejér-Riesz factorizations of certain non-negative functions on the circle. We show that, for each nonconstant rational E‾-inner function x, either x(D‾)⊆RE‾∩E‾ or x(D‾) meets RE‾ exactly deg⁡(x) times. We study convex subsets of the set J of all rational E‾-inner functions and extreme points of J. We show that whether a rational E‾-inner function x is an extreme point of J depends on how many royal nodes of x lie on T.

'Being Chinese' in Mauritius and Madagascar: Comparing Chinese diasporic communities in the western Indian Ocean

Chinese migration to the Western Indian Ocean since the 1800s was part of an earlier historical trend that saw European colonial powers setting up plantation economies that required foreign laborers. Migrants from Southern China arrived in Mauritius and Madagascar first as indentured laborers, and later as free merchants. Despite many similarities between the two diasporas, they differed in terms of their cultural and linguistic propensities. Furthermore, since the 1990s, both Mauritius and Madagascar have been experiencing rising influences of Mandarin-speaking Chinese immigrants working in infrastructure construction, commercial and educational sectors. Based on ethnographic fieldwork conducted in these two Western Indian Ocean countries between 2015 and 2020, this paper applies the theoretical lens of ‘diaspora-for-others,’ featured in this special issue, to explore the similarities and differences between Chinese migration trajectories to Mauritius and Madagascar, and their respective diasporic identity formations. Local socio-historical contexts in Mauritius, Madagascar, and China influence the transnational experiences of Mauritian and Malagasy Chinese communities, which further contributes to their heterogeneous, fluid and changing cultural identities. In addition, the People’s Republic of China’s increasing engagement in Western Indian Ocean countries as a gateway to Africa in the past two decades has also created more nuances in the distinguishable boundaries within the Chinese diaspora communities in the region.

Domain generalization via optimal transport with metric similarity learning

Generalizing knowledge to unseen domains, where data and labels are unavailable, is crucial for machine learning models. We tackle the domain generalization problem to learn from multiple source domains and generalize to a target domain with unknown statistics. The crucial idea is to extract the underlying invariant features across all the domains. Previous domain generalization approaches mainly focused on learning invariant features and stacking the learned features from each source domain to generalize to a new target domain while ignoring the label information, which will lead to indistinguishable features with an ambiguous classification boundary. One possible solution is to constrain the label-similarity when extracting the invariant features and take advantage of the label similarities for class-specific cohesion and separation of features across domains. Therefore we adopt optimal transport with Wasserstein distance, which could constrain the class label similarity, for adversarial training and also further deploy a metric learning objective to leverage the label information for achieving distinguishable classification boundary. Empirical results show that our proposed method could outperform most of the baselines. Furthermore, ablation studies also demonstrate the effectiveness of each component of our method.

Interpolation by holomorphic maps from the disc to the tetrablock

The tetrablock is the setE={x∈C3:1−x1z−x2w+x3zw≠0whenever|z|≤1,|w|≤1}. The closure of E is denoted by E‾. A tetra-inner function is an analytic map x from the unit disc D to E‾ such that, for almost all points λ of the unit circle T,limr↑1⁡x(rλ) exists and lies in bE‾, where bE‾ denotes the distinguished boundary of E‾. There is a natural notion of degree of a rational tetra-inner function x; it is simply the topological degree of the continuous map x|T from T to bE‾. In this paper we give a prescription for the construction of a general rational tetra-inner function of degree n. The prescription exploits a known construction of the finite Blaschke products of given degree which satisfy some interpolation conditions with the aid of a Pick matrix formed from the interpolation data. It is known that if x=(x1,x2,x3) is a rational tetra-inner function of degree n, then x1x2−x3 either is identically 0 or has precisely n zeros in the closed unit disc D‾, counted with multiplicity. It turns out that a natural choice of data for the construction of a rational tetra-inner function x=(x1,x2,x3) consists of the points in D‾ for which x1x2−x3=0 and the values of x at these points.

Level curve portraits of rational inner functions

We analyze the behavior of rational inner functions on the unit bidisk near singularities on the distinguished boundary $\mathbb{T}^2$ using level sets. We show that the unimodular level sets of a rational inner function $\phi$ can be parametrized with analytic curves and connect the behavior of these analytic curves to that of the zero set of $\phi$. We apply these results to obtain a detailed description of the fine numerical stability of $\phi$: for instance, we show that $\frac{\partial \phi}{\partial z_1}$ and $\frac{\partial \phi}{\partial z_2}$ always possess the same $L^{\mathfrak{p}}$-integrability on $\mathbb{T}^2$, and we obtain combinatorial relations between intersection multiplicities at singularities and vanishing orders for branches of level sets. We also present several new methods of constructing rational inner functions that allow us to prescribe properties of their zero sets, unimodular level sets, and singularities.

Identification of high grade and low grade tumors in MR Brain Image using Modified Monkey Search Algorithm

Detection of tumors present in the Magnetic Resonance brain image is a challenging task in the research field of medical imaging processing. The tumors with distinguished boundaries are difficult to find in the MR brain images, while performing the manual segmentation process. There is a necessity of an automated segmentation technique for performing better segmentation in terms of tumors with distinguished boundaries. The automated modified monkey search technique is used to find the optimized cluster position, and a random search operation is performed is locate all the pixels present in the image and then finally the location of the tumor region is exactly segmented/predicted by using the suggested monkey search algorithm. The suggested technique will support the radiologist for finding the tumors with distinguishing boundaries and accuracy of prediction of tumors is also improved lot with this approach. Based on the early prediction of tumors, diagnosing procedures will save the lives of many human beings.

Influence of current injection scheme on electrical impedance tomography for monitoring of the respiratory function of obese subjects

Electrical impedance tomography (EIT) has been proposed as a noninvasive radiation-free functional imaging modality for bedside monitoring of lung ventilation. Due to the intrinsic low sensitivity of measured boundary voltages with respect to internal conductivity changes, EIT image reconstructions are more challenging for the regions that are located deeper from the attached surface electrodes. Considering the worldwide prevalence of obesity, it is highly desirable for an EIT system to produce high-quality images from obese as well as lean subjects. In EIT, the internal current density distribution and the sensitivity of a boundary voltage to a change of internal conductivity vary with respect to the current injection scheme. We quantified the range of measured boundary voltages subject to a chosen current injection scheme and investigated the influence of various obesity levels on the number of distinguishable boundary voltages given a fixed amount of system noise. We found that the 0-skip current injection scheme, where neighboring electrode pairs are used for current injections, could be most favorable to improve spatial resolution but produces the smallest number of distinguishable boundary voltages. To obtain an enough number of distinguishable boundary voltages from an obese subject, the gap between current injection electrodes should be increased depending on the circumference of a chosen imaging plane. We suggested a guideline for current injection schemes in EIT to produce images with better quality from obese as well as lean subjects.

Influence of synthesis strategy on the formation of microspheres of self-assembled CuO rectangular nanorods and hierarchical structures of self-assembled Cu2O nanospheres from single precursor (copper (II) acetate monohydrate) and their structural, optical, thermal and magnetic properties

Here, microspheres of self-assembled CuO rectangular nanorods and hierarchical structures of self-assembled Cu2O nanospheres have been synthesized from single precursor (copper (II) acetate monohydrate) using hydrothermal method. The utility of NH3 and N2H4·H2O as reducing agents for the formation of phases and morphology of CuO and Cu2O has been investigated. The XRD patterns and Raman spectra indicate that the synthesized CuO and Cu2O nanostructures are single phase and high crystalline. The SEM images clearly show that the rectangular shaped CuO nanorods are densely packed and self-organized into microspheres whereas self-assembled Cu2O nanospheres makes hierarchical Cu2O structures. The TEM images reveal that the individual CuO rectangular nanorod is having ends with a small faceted nano-tip-like structure while the hierarchical nanostructures of Cu2O consist of agglomerated nanospheres with distinguishable grain boundaries. The presence of vibrational modes, associated with the relative motion of both copper and oxygen atoms, in the FTIR spectra also supports the formation of CuO and Cu2O phases, respectively. The optical properties of the synthesized materials were revealed using absorption spectra. The thermal behaviour was also investigated through DSC measurements. The magnetic properties have also been investigated in order to explore new insights responsible for specific magnetic behaviour.

On the Derivatives of Cauchy-Type Integrals in the Polydisk

In the paper the formulas for derivatives of the Cauchy type integral K[f] of smooth functions f on the distinguished boundary of a polydisc are given. These formulas express the derivatives of orderm from K[f] through the derivatives of lower order (Theorem 1). We use them for the estimates of smoothness of derivatives of Cauchy type integral in terms of the Holder’s ordinary scale (Theorem 2) and in terms of anisotropic Holder’s classes (Theorem 5).

DeepSeg: deep neural network framework for automatic brain tumor segmentation using magnetic resonance FLAIR images

PurposeGliomas are the most common and aggressive type of brain tumors due to their infiltrative nature and rapid progression. The process of distinguishing tumor boundaries from healthy cells is still a challenging task in the clinical routine. Fluid-attenuated inversion recovery (FLAIR) MRI modality can provide the physician with information about tumor infiltration. Therefore, this paper proposes a new generic deep learning architecture, namely DeepSeg, for fully automated detection and segmentation of the brain lesion using FLAIR MRI data.MethodsThe developed DeepSeg is a modular decoupling framework. It consists of two connected core parts based on an encoding and decoding relationship. The encoder part is a convolutional neural network (CNN) responsible for spatial information extraction. The resulting semantic map is inserted into the decoder part to get the full-resolution probability map. Based on modified U-Net architecture, different CNN models such as residual neural network (ResNet), dense convolutional network (DenseNet), and NASNet have been utilized in this study.ResultsThe proposed deep learning architectures have been successfully tested and evaluated on-line based on MRI datasets of brain tumor segmentation (BraTS 2019) challenge, including s336 cases as training data and 125 cases for validation data. The dice and Hausdorff distance scores of obtained segmentation results are about 0.81 to 0.84 and 9.8 to 19.7 correspondingly.ConclusionThis study showed successful feasibility and comparative performance of applying different deep learning models in a new DeepSeg framework for automated brain tumor segmentation in FLAIR MR images. The proposed DeepSeg is open source and freely available at https://github.com/razeineldin/DeepSeg/.

Kinetics of Phase Transformations in FeAl(Cr,Si) / Fe&lt;sub&gt;3&lt;/sub&gt;Al(Cr,Si) / Fe(Al,Cr,Si) Laminated Coating on the Cr15Al5 Alloy

The influence of thermal action at 1150 °C on the kinetics of phase transformations in a layered coating FeAl (Cr,Si) / Fe3Al (Cr,Si) / Fe (Al,Cr,Si), obtained using a technology involving aluminizing of an Cr15Al5 alloy by hot dipping in silumin melt AK12M2 and heat treatment. It was shown that after 100 hours of exposure, the visually distinguishable boundary between the coating and the alloy disappears. The aluminum content in the Cr15Al5 alloy increased to 17 at. %, silicon up to 3 at. %, and chromium is reduced to 14 at. %. The microhardness of the alloy increases from 1.6 to 2.4 GPa and survivability at a test temperature of 1150 °C increases from 120 to at least 920 hours.

Controllable Synthesis of Leaf‐Like CuO Nanosheets for Selective CO2 Electroreduction to Ethylene

AbstractThe carbon dioxide reduction reaction (CO2RR) driven by renewable electricity is a promising way to tackle the CO2 emission woes and recycle use of CO2. The synthesis of electrocatalysts with high activity and selectivity for CO2RR to ethylene remains a great challenge. Herein, leaf‐like CuO nanosheets are fabricated in situ on nitrogen‐doped graphene (NG) by using a novel reduction‐oxidation‐reconstruction process. When used as a catalyst for the CO2RR in 0.1 M KHCO3, a high faradaic efficiency of approximately 30 % for ethylene with an ultra‐high ethylene/methane ratio of 190 was achieved at −1.3 V vs. the reversible hydrogen electrode. The SEM and TEM images confirm the leaf‐like CuO nanosheets display high‐curvature structures, while multiple distinguished grain boundaries constructed by CuO(110) and CuO(111) planes are verified by HRTEM. For the first time, we present a facile method to combine the high‐curvature structure and the grain boundary to enhance the selectivity of the CO2RR to ethylene over a CuO catalyst.

Shadow Detection in Still Road Images Using Chrominance Properties of Shadows and Spectral Power Distribution of the Illumination.

A well-known challenge in vision-based driver assistance systems is cast shadows on the road, which makes fundamental tasks such as road and lane detections difficult. In as much as shadow detection relies on shadow features, in this paper, we propose a set of new chrominance properties of shadows based on the skylight and sunlight contributions to the road surface chromaticity. Six constraints on shadow and non-shadowed regions are derived from these properties. The chrominance properties and the associated constraints are used as shadow features in an effective shadow detection method intended to be integrated on an onboard road detection system where the identification of cast shadows on the road is a determinant stage. Onboard systems deal with still outdoor images; thus, the approach focuses on distinguishing shadow boundaries from material changes by considering two illumination sources: sky and sun. A non-shadowed road region is illuminated by both skylight and sunlight, whereas a shadowed one is illuminated by skylight only; thus, their chromaticity varies. The shadow edge detection strategy consists of the identification of image edges separating shadowed and non-shadowed road regions. The classification is achieved by verifying whether the pixel chrominance values of regions on both sides of the image edges satisfy the six constraints. Experiments on real traffic scenes demonstrated the effectiveness of our shadow detection system in detecting shadow edges on the road and material-change edges, outperforming previous shadow detection methods based on physical features, and showing the high potential of the new chrominance properties.

An extremal composition operator on the Hardy space of the bidisk with small approximation numbers

We construct an analytic self-map Φ of the bidisk D2 whose image touches the distinguished boundary, but whose approximation numbers of the associated composition operator on H2(D2) are small in the sense that lim supn→∞[an2(CΦ)]1∕n<1.

Agent-based modeling and genetics reveal the Limfjorden, Denmark, as a well-connected system for mussel larvae

Fishery of blue mussels Mytilus edulis constitutes a very important economic activity in Denmark, whereas mussel farming on long-lines or nets is a new, growing sector. Spawning from natural mussel beds takes place during early summer, and larvae disperse via water currents before settling on the bottom or on spat collectors in the farms. In the present study, we coupled a 3D physical model system (FlexSem) with an agent-based model in order to examine the connectivity of this marine system in terms of mussel larval dispersal and settling potential. To address this question, we (1) estimated the dispersal and connectivity between 17 areas in the Limfjorden, (2) identified the main donor and receiver areas of mussel larvae and (3) identified possible dispersal barriers. The results show that the central narrow strait in the Limfjorden was the main donor area in all the studied years, and that the adjacent eastern areas were the main receiver areas. Towards the inner basins of the Limfjorden, isolation increased and limited connectivity was observed. The results from the cluster analysis grouped the Limfjorden into 3 to 5 clusters, but there was still some exchange of simulated larvae observed among these clusters. Analysis of molecular markers revealed no genetic differentiation between areas and supports the model results, indicating that despite distinguishable hydrographic boundaries, the mussel populations in the Limfjorden are well connected. This study demonstrates how connectivity modeling can be used to support site selection processes in aquaculture.

Domain-filling circle packings

According to a classical theorem of Constantin Caratheodory, any Jordan domain G admits a unique conformal mapping onto the unit disk $$\mathbb {D}$$ such that three distinguished boundary points of G have prescribed images on $$\partial \mathbb {D}$$ . This result can be extended to general domains when the role of boundary points is taken over by prime ends. In this paper we prove a discrete counterpart of Caratheodory’s theorem in the framework of circle packing. A trilateral is a domain whose (intrinsic) boundary is decomposed into three arcs $$\alpha $$ , $$\beta $$ , and $$\gamma $$ (of prime ends). Using Sperner’s lemma, we prove existence of circle packings and more general circle agglomerations filling arbitrary (bounded) simply connected trilaterals. In order to get complete results we have to admit (in some exceptional cases) that the packings contain degenerate circles. We study in detail which properties of the domain G and the underlying complex of the packing guarantee uniqueness and exclude degeneracy.

Geochemical Features of Annual Layers of Bottom Sediments of Freshwater Lakes, Studied via Synchrotron Radiation–Induced XRF Microanalysis

The possibility of distinguishing layer boundaries through lithological and geochemical indicators (Rb/Sr ratio) is shown by a case study of bottom sediments from Lake Kucherlinskoe (Altai). Based on the data from scanning synchrotron radiation–induced XRF microanalysis and isotope studies, a record of the Tunguska event of 1908 is found in the bottom sediments of Lake Zapovednoye, forty kilometers from the explosion’s epicenter. An anomalous layer in the sediments is associated with an increase in the terrigenous drift in the catchment area due to massive forest outfall and subsequent fires. The possibility of a targeted search for microparticles of extraterrestrial origin in lake sediments is demonstrated.

Boundary Value Problems for Non-linear First Order Systems of Partial Differential Equations in Higher Dimensions, Especially in Three Dimensions

Starting from distinguishing boundaries for monogenic functions (see Tutschke in Adv. Appl. Clifford Algebr 25:441–451, 2015), one can solve boundary value problems for monogenic functions (see Dao in Boundary value problems for monogenic functions in higher dimensions. Ph.D Thesis, Hanoi University of Science and Technology, Vietnam, 2019). A boundary value problem for monogenic function in $${\mathbf {R}}^{3}$$ is the following: prescribe the vector components $$u_1$$ and $$u_2$$ on the whole boundary, the bi-vector component $$u_{12}$$ on a (one-dimensional) curve on the boundary and the real part $$u_0$$ at one point. The goal of the present paper is to reduce the same boundary value problems for more general (linear or fully non-linear) first order systems to fixed-point problems for an operator containing the monogenic solution of the boundary value problem under consideration (provided the given system is written in its Clifford-analytic normal form). In case of $${\mathbf {R}}^{3}$$ the existence of a uniquely determined solution of the boundary value problem is proved as fixed-point of the corresponding fixed-point problem in case the right hand side is Lipschitz-continuous with sufficiently small Lipschitz constants.

Influence of individual phases on the magnetoelectric coupling and electromechanical response in (1 − x)Ba0.83Ca0.10Sr0.07TiO3-xBiFeO3 (x = 0–0.30) multiferroic composites

BaTiO3 based composites have been widely utilized to achieve better ferroelectric and magnetoelectric characteristics. BiFeO3 is one of materials that exhibit superior magnetoelectric characteristics in single phase. In this work, (1 − x)Ba0.83Ca0.10Sr0.07TiO3-xBiFeO3 (x = 0–0.30) multiferroic composites have been synthesized using mechano-chemical activation method. The strategy is to achieve optimum ferroelectric and ferromagnetic properties without hampering the inherent dielectric and ferroelectric characteristics of Ba0.83Ca0.10Sr0.07TiO3. The synthesized composites have shown diphasic nature with homogeneous grains and clearly distinguishable grain boundaries. The composition with x = 0.10 exhibits superior electrical properties, which are attributed to optimum resistivity and increased tetragonal lattice distortion introduced by BiFeO3 addition. The modification in bond length and bond angle leads to a significant enhancement in ferromagnetic characteristics of prepared composites. The composites have shown a gradual increase in magnetoelectric characteristics and substantial DC magnetic field dependence. This work provides an optimum composition and feasible route to prepare high-performance magnetoelectric composites.

Rational dilation of tetrablock contractions revisited

A classical result of Sz.-Nagy asserts that a Hilbert-space contraction operator T can be lifted to an isometry V. A more general multivariable setting of recent interest for these ideas is the case where (i) the unit disk is replaced by a certain domain contained in C3 (called the tetrablock), (ii) the contraction operator T is replaced by a commutative triple (T1,T2,T) of Hilbert-space operators having E as a spectral set (a tetrablock contraction). The rational dilation question for this setting is whether a tetrablock contraction (T1,T2,T) can be lifted to a tetrablock isometry (V1,V2,V) (a commutative operator tuple which extends to a tetrablock-unitary tuple (U1,U2,U)—a commutative tuple of normal operators with joint spectrum contained in the distinguished boundary of the tetrablock). We discuss necessary conditions for a tetrablock contraction to have a tetrablock-isometric lift. We present an example of a tetrablock contraction which does have a tetrablock-isometric lift but violates a condition previously thought to be necessary for the existence of such a lift. Thus the question of whether a tetrablock contraction always has a tetrablock-isometric lift appears to be unresolved at this time.