On λ-ideal statistical convergence in fuzzy cone normed spaces

In this paper, we discuss several aspects of Julia sets, as well as those of the Mandelbrot set. We are interested in topological properties such as connectivity and local connectivity, geometric properties such as Hausdorff dimension and Lebesgue measure, and complex analytic properties such as holomorphic removability.

The prognosis of cancer patients suffering from solid tumors significantly depends on the developmental stage of the tumor. For cervix carcinoma the prognosis is better for compact shapes than for diffusive shapes since the latter may already indicate invasion, the stage in tumor progression that precedes the formation of metastases. In this paper, we present methods for describing and evaluating tumor objects and their surfaces based on topological and geometric properties. For geometry, statistics of the binary object's distance transform are used to evaluate the tumor's invasion front. In addition, a simple compactness measure is adapted to 3D images and presented to compare different types of tumor samples. As a topological measure, the Betti numbers are calculated of voxelized tumor objects based on a medial axis transform. We further illustrate how these geometric and topological properties can be used for a quantitative comparison of histological material and single-cell-based tumor growth simulations.

Previous studies have revealed that the visual system is highly sensitive to topological differences. Topological properties, which are first represented and processed, influence ongoing visual information processing. In addition, as a fundamental human ability, perception of duration can be affected by non-temporal information, including the magnitude, motion, and spatial frequency. Topological properties are another type of important non-temporal information, but little is known whether these can also affect duration perception. Recently, researchers have found that when an unexpected "oddball" stimulus is embedded in a train of repeated standard stimuli, its duration typically seems to be longer. This phenomenon is termed the oddball effect, and illustrates how duration perception can be affected by stimulus novelty. According to the topological approach to perceptual organization, the core intuitive notion of an object may be characterized precisely as topological invariance, and changes in topological properties will be regarded as the emergence of new objects by the vision system. Therefore, the impact of topological properties on duration perception of oddball stimuli was investigated in this research. In Experiment 1a 18 participants(7 men, 11 women) were asked to judge whether oddball stimuli(i.e., disks with one or two holes) were longer or shorter in duration than the standard stimuli(i.e., squares with one or two holes). In Experiment 1b, 14 participants(7 men, 7women) rated whether oddballs(i.e., disks with two holes, squares with one hole or squares with three holes)were longer or shorter in duration than standard stimuli(i.e., squares with two holes). Experiment 2 included 40 participants(15men, 25 women) and was designed to compare effects on duration perception of oddball stimuli between topological properties and other geometric properties. Participants were required to judge whether the oddballs(i.e., parallelograms, trapezoids, circles or rings) were longer or shorter in duration than standard stimuli(i.e., either squares or beeps of 1000 Hz). Results of Experiment 1 indicated that when there were topological differences between standard stimuli and the oddballs, the magnitude of the oddball effect was significantly larger than that of non-topological differences. Findings from Experiment 2 showed the magnitude of the oddball effect increased monotonically with increasing levels of stability of structural differences between standard stimuli and the oddballs. Compared with other geometrical properties, changes of topological properties induced the largest magnitude of oddball effect.These results suggest that topological properties are one type of important non-temporal information that affect duration perception of oddball stimuli. Furthermore, the current study supports the topological definition of perceptual objects.

Derivation of topological, geometrical, and correlational properties of porous media from pore-chart analysis of serial section data

The concept of electric and magnetic field lines is intrinsically non-relativistic. Nonetheless, for certain types of fields satisfying certain geometric properties, field lines can be defined covariantly. More precisely, two Lorentz-invariant 2D surfaces in spacetime can be defined such that magnetic and electric field lines are determined, for any observer, by the intersection of those surfaces with spacelike hyperplanes. An instance of this type of field is constituted by the so-called Hopf–Rañada solutions of the source-free Maxwell equations, which have been studied because of their interesting topological properties, namely, linkage of their field lines. In order to describe both geometric and topological properties in a succinct manner, we employ the tools of geometric algebra (aka Clifford algebra) and use the Clebsch representation for the vector potential as well as the Euler representation for both magnetic and electric fields. This description is easily made covariant, thus allowing us to define electric and magnetic field lines covariantly in a compact geometric language. The definitions of field lines can be phrased in terms of 2D surfaces in space. We display those surfaces in different reference frames, showing how those surfaces change under Lorentz transformations while keeping their topological properties. As a byproduct we also obtain relations between optical helicity, optical chirality and generalizations thereof, and their conservation laws.

The Möbius strip, a long sheet of paper whose ends are glued together after a 180° twist, has remarkable geometric and topological properties. Here, we consider dielectric Möbius strips of finite width and investigate the interplay between geometric properties and resonant light propagation. We show how the polarization dynamics of the electromagnetic wave depends on the topological properties, and demonstrate how the geometric phase can be manipulated between 0 and π through the system geometry. The loss of the Möbius character in thick cavities and for small twist segment lengths allows one to manipulate the polarization dynamics and the far-field emission, and opens the venue for applications.

Surface depressions are abundant in topographically complex landscapes, and they exert significant influences on hydrological, ecological, and biogeochemical processes at local and regional scales. The increasing availability of high-resolution topographical data makes it possible to resolve small surface depressions. By analogy with the reasoning process of a human interpreter to visually recognize surface depressions from a topographic map, we developed a localized contour tree method that is able to fully exploit high-resolution topographical data for detecting, delineating, and characterizing surface depressions across scales with a multitude of geometric and topological properties. In this research, we introduce a new concept ‘pour contour’ and a graph theory-based contour tree representation for the first time to tackle the surface depression detection and delineation problem. Beyond the depression detection and filling addressed in the previous raster-based methods, our localized contour tree method derives the location, perimeter, surface area, depth, spill elevation, storage volume, shape index, and other geometric properties for all individual surface depressions, as well as the nested topological structures for complex surface depressions. The combination of various geometric properties and nested topological descriptions provides comprehensive and essential information about surface depressions across scales for various environmental applications, such as fine-scale ecohydrological modeling, limnological analyses, and wetland studies. Our application example demonstrated that our localized contour tree method is functionally effective and computationally efficient.

Individual tree crown delineation using localized contour tree method and airborne LiDAR data in coniferous forests

A comprehensive study on geometric, topological and fractal characterizations of pore systems in low-permeability reservoirs based on SEM, MICP, NMR, and X-ray CT experiments

Every topological property can be associated with its relative version in such a way that when smaller space coincides with larger space, then this relative property coincides with the absolute one. This notion of relative topological properties was introduced by Arhangel’skii and Ganedi in 1989. Singal and Arya introduced the concepts of almost regular spaces in 1969 and almost completely regular spaces in 1970. In this paper, we have studied various relative versions of almost regularity, complete regularity, and almost complete regularity. We investigated some of their properties and established relationships of these spaces with each other and with the existing relative properties.

This thesis presents computational topology algorithms for discrete 2-manifolds. Although it is straightforward to compute the genus of a discrete 2-manifold, this topological invariant does not tell us enough for most computer graphics applications where we would like to know: what does the topology look like? Genus is a scalar value with no associated geometric appearance. We can, however, isolate geometric regions of the surface that are topologically interesting. The simplest topologically interesting, and perhaps most intuitive, regions to consider are those with genus equal to one. By isolating and examining such regions we can compute measures to better describe the appearance of relevant surface topology. Thus, this work focuses on isolating handles, regions with genus equal to one, in discrete 2-manifolds. In this thesis, we present novel algorithms guaranteed to identify and isolate handles for various discrete surface representations. Additionally, we present robust techniques to measure the geometric extent of handles by identifying two locally minimal-length non-separating cycles for each handle. We also present algorithms to retain or simplify the topology of a reconstructed surface as desired. Finally, the value of these algorithms is demonstrated through specific applications to computer graphics. For example, we demonstrate how geometric models can be greatly improved through topology simplification both for models represented by volume data or by triangle meshes. Contributions. The contributions of this work include: (1) A robust and efficient method for identifying and isolating handles for discrete 2-manifolds. (2) A method to robustly represent the topology of the surface with an augmented Reeb graph. (3) A robust method to find two locally minimal-length non-separating cycles for each handle. (4) A simple method to simplify the topology for volume data and triangle meshes which preserves the local geometry as much as possible. (5) An out-of-core method for topology simplification for volume data.

In the present paper, we introduce the sequence space using a new regular matrix of Fibonacci numbers, where . We investigate its topological properties such as Schauder basis, and duals and its geometric properties like uniformly convex, strictly convex and super-reflexive. We also characterize some matrix classes from the space to classical sequence spaces.

A number of strategies have been used to include spatial and topological properties in the image segmentation stage. It is generally accepted that grouping of nearby pixels by modelling neighbourhood relationships as (a, b) connected graphs may lead to meaningful image objects. In such approach, however, topological concepts may suffer from ambiguity since image elements (pixels) are two dimensional entities. This paper evaluates whether an alternative representation of digital images based both on Cartesian complexes and oriented matroids may improve multispectral image segmentation by enforcing topological and geometric properties and then be used in the classification stage. A conceptual model is defined, using Cartesian complexes, in order to link combinatorial properties of axiomatic locally finite spaces and their associated oriented matroids for involving topological properties. The proposed approach uses a layered architecture going from a physical level, going next through logical geospatial abstraction level and then through the Cartesian complex logical level. Additionally, there is a layer of oriented matroids composed by conceptual elements in terms of combinatorics for encoding relevant features to multispectral image segmentation. First, it is conducted an edge detection task, next an probability contour map using a Cartesian complex space rather than the conventional image space and finally, an image classification using random forest method. A computational solution including several components was developed using a framework for parallel computing. The performance of this solution was assessed using a small subset of GEOBIA2016 benchmark dataset. It is shown that the usage of a partial implementation of Cartesian complexes and associated oriented matroids is computationally but does not increase classification accuracy.

The measurement of Friedel oscillations (FOs) is conventionally used to recover the energy dispersion of electronic structure. Besides the energy dispersion, the modern electronic structure also embodies other key ingredients such as the geometrical and topological properties; it is one promising direction to explore the potential of FOs for the relevant measurement. Here, we present a comprehensive study of FOs in substrate-supported graphene under off-resonant circularly polarized light, in which a valley-contrasting feature and topological phase transition occur due to the combined breaking of inversion (Ƥ) and time reversal (T) symmetries. Depending on the position of the Fermi level, FOs may be contributed by electronic backscattering in one single valley or two valleys. In the single-valley regime, the oscillation periods of FOs can be used to determine the topological phase boundary of electronic structure, while the amplitudes of FOs distinguish trivial insulators and topological insulators in a quantitative way. In the two-valley regime, the unequal Fermi surfaces lead to a beating pattern (robust two-wave-front dislocations) of FOs contributed by intravalley (intervalley) scattering. This study implies the great potential of FOs in characterizing topological and geometrical properties of the electronic structure of two-dimensional materials.

Rapid object recognition has survival significance. The extraction of topological properties (TP) is proposed as the starting point of object perception. Behavioral evidence shows that TP processing takes precedence over other geometric properties and can accelerate object recognition. However, the mechanism of the fast TP processing remains unclear. The magnocellular (M) pathway is well known as a fast route to convey “coarse” information, compared with the slow parvocellular (P) pathway. Here, we hypothesize that the fast processing of TP occurs in a subcortical M pathway. We applied single‐pulse transcranial magnetic stimulation (TMS) over the primary visual cortex to temporarily disrupt cortical processing. Besides, stimuli were designed to preferentially engage M or P pathways (M‐ or P‐biased conditions). We found that, when TMS disrupted cortical function at the early stages of stimulus processing, non‐TP shape discrimination was strongly impaired in both M‐ and P‐biased conditions, whereas TP discrimination was not affected in the M‐biased condition, suggesting that early M processing of TP is independent of the visual cortex, but probably occurs in a subcortical M pathway. Using an unconscious priming paradigm, we further found that early M processing of TP can accelerate object recognition by speeding up the processing of other properties, e.g., orientation. Our findings suggest that the human visual system achieves efficient object recognition by rapidly processing TP in the subcortical M pathway.