Topological Concepts Research Articles

Coupled Fixed Point Theorems in Topological Spaces with Size Function Topology

Regarding the concept of size function topology, this allows us to view the topological space X as a metric-like space. We prove the existence and uniqueness for a coupled fixed point of the map F : X × X ⟶ X satisfying some certain contractive conditions.

Viscoelastic and Viscoplastic Glucose Theory (VGT #17): Selecting 12 Cases and Applying Theories of Viscoelasticity and Viscoplasticity from Engineering and Perturbation Theory from Quantum Mechanics to Predict the Behavior of an Output Biomarker as the Strain (ε or Behavior, Symptom) by Using the Strain Rate (dε/dt) Multiplying with a Correlated Input Biomarker of Stress Value (σ or Stressor, Cause) as

The author has collected ~3 million data regarding his health condition and lifestyle details over the past 12 years. He spent the entire year of 2014 to develop a metabolism index (MI) model using topology concept, nonlinear algebra, algebraic geometry, and finite element method. This MI model contains various measured biomarkers and recorded lifestyle details along with their induced new biomedical variables for an additional ~1.5 million data. Body weight, glucose, blood pressure, heart rate, lipids, body temperature, and blood oxygen level, along with important lifestyle details, including diet, exercise, sleep, stress, water intake, and daily life routines are included in his MI database. His developed MI model has a total of 10 categories covering approximately 500 detailed elements that constitute his defined “metabolism model” which are the building blocks or root causes for diabetes and other chronic disease complications, including but not limited to cardiovascular disease (CVD), chronic heart disease (CHD), chronic kidney disease (CKD), retinopathy, neuropathy, foot ulcer, and hypothyroidism. The end result of the MI development work is a combined MI value within any selected time period with 73.5% as its dividing line between a healthy and unhealthy state. The MI serves as the foundation to many of his follow-up medical research work. During the period from 2015 to 2017, he focused his research on type 2 diabetes (T2D), especially glucoses, including fasting plasma glucose (FPG), postprandial plasma glucose (PPG), estimated average glucose (eAG), and hemoglobin A1C (HbA1C). During the following period from 2018 to 2022, he concentrated on researching medical complications resulting from diabetes, chronic diseases, and metabolic disorders which include heart problems, stroke, kidney problems, retinopathy, neuropathy, foot ulcer, diabetic skin fungal infection, hypothyroidism, and diabetic constipation, cancer, and dementia. He also developed a few mathematical risk models to calculate the probability percentages of developing various diabetic complications. Recently, he has applied theories of elasticity, plasticity, viscoelasticity, and viscoplasticity from engineering along with the theories of wave and energy from physics to conduct his research on output biomarkers (symptoms or behaviors) resulting from a suspected or identified input biomarkers (causes or stressors). As a result, he has written 16 different articles in this research area. In this article, he utilizes the perturbation theory of quantum mechanics in modern physics to develop a set of predicted output biomarkers using the identified input biomarkers from his previous research work. Next, he compares his predicted biomarkers against his measured biomarkers using their calculated correlation coefficients and prediction accuracies.

Viscoelastic and Viscoplastic Glucose Theory (VGT #18): Applying Theories of Viscoelasticity and Viscoplasticity from Engineering and Perturbation Theory from Quantum Mechanics from 9 Cases to Predict the Behavior or Symptom of an Output Biomarker as the Strain (ε), by Selecting a Highly Correlated Input Biomarker as the Stress (σ), and using the Stress Rate (dσ/dt) Multiplied with the Stress (σ)

The author has collected ~3 million data regarding his health condition and lifestyle details over the past 12 years. He spent the entire year of 2014 to develop a metabolism index (MI) model using topology concept, nonlinear algebra, algebraic geometry, and finite element method. This model contains various measured biomarkers and recorded lifestyle details along with their induced new biomedical variables for an additional ~1.5 million data. Body weight, glucose, blood pressure, heart rate, lipids, body temperature, and blood oxygen level, along with important lifestyle details, including diet, exercise, sleep, stress, water intake, and daily life routines are included in his MI database. The developed MI model has a total of 10 categories covering approximately 500 detailed elements that constitute his defined “metabolism model” which are the building blocks or root causes for diabetes and other chronic disease complications, including but not limited to cardiovascular disease (CVD), chronic heart disease (CHD), chronic kidney disease (CKD), retinopathy, neuropathy, foot ulcer, and hypothyroidism. The end result of the MI development work is a combined MI value over a selected time period with 73.5% as its dividing line between a healthy and unhealthy state. The MI serves as the foundation to many of his follow-up medical research work. During the period from 2015 to 2017, he focused his research on type 2 diabetes (T2D), especially glucoses, including fasting plasma glucose (FPG), postprandial plasma glucose (PPG), estimated average glucose (eAG), and hemoglobin A1C (HbA1C). During the following period from 2018 to 2022, he concentrated on researching medical complications resulting from diabetes, chronic diseases, and metabolic disorders which include heart problems, stroke, kidney problems, retinopathy, neuropathy, foot ulcer, diabetic skin fungal infection, hypothyroidism, diabetic constipation, cancer, and dementia. He also developed a few mathematical risk models to calculate the probability percentages of developing various diabetic complications. Recently, he applied the theories of elasticity, plasticity, viscoelasticity, and viscoplasticity from engineering along with the theories of wave and energy from physics to conduct his research on output biomarkers (symptoms or behaviors) resulting from a suspected or identified input biomarkers (causes or stressors). As a result, he has written 17 different articles in this research area.

Nested-sphere description of the N -level Chern number and the generalized Bloch hypersphere

The geometric interpretation of (pseudo)spin 1/2 systems on the Bloch sphere has been appreciated across different areas ranging from condensed matter to quantum information and high energy physics. Although similar notions for larger Hilbert spaces are established in mathematics, they have been so far less explored beyond the two-level case for practical usage in condensed matter settings, or have involved restrictions to sub manifolds within the full Hilbert space. We here employ a coherence vector description to theoretically characterize a general N-level system on the higher dimensional generalized Bloch (hyper)sphere by respecting the structure of the underlying SU(N) algebra and construct physically intuitive geometric pictures for topological concepts. Focusing on two spatial dimensions, we reveal a geometric interpretation for the Chern number in larger Hilbert spaces in terms of a nested structure comprising N-1 two-spheres. We demonstrate that for the N-level case, there is an exterior two-sphere that provides a useful characterization of the system, notably by playing a primary role in determining the Chern number. The external sphere can be directly measured in ultracold atoms via well-established band mapping techniques, thereby imparting knowledge of the topological nature of state. We also investigate how the time evolution of the coherence vector defined on the generalized Bloch hypersphere can be utilized to extract the full state vector in experiments, allowing us to develop a tomography scheme involving quenches for three-level systems. Our geometric description opens up a new avenue for the interpretation of the topological classification and the dynamical illustration of multi-level systems, which in turn is anticipated to help in the design of new experimental probes.

The social stone: a story of transformation from ableist to accessible topology

ABSTRACT Using a case study of a kerb – the social stone – found at a tram traffic Island, the text demonstrates the re/production of the ableist city and analyses processes of transforming such a city into an accessible space for people with disabilities. The authors apply actor-network theory to analyse the mutual construction of space and bodies. To understand this construction, the concept of topology is used, which the article develops by introducing ‘ableist’ and ‘accessible’ topology. Unlike topography, which is in concord with able-bodied people, topology is more suitable for the analysis of space when also including non-standard bodies. Moreover, it allows the authors to analyse the mechanisms that reproduce and control the currently inaccessible city on the one hand, and it provide some room for change on the other. However, to make this change happen, a specific network must be established. To trace the associations that create such a network, the study uses in-depth interviews with involved actors collected over a ten-year period.

The introduction of topology into analytic philosophy: two movements and a coda

Both early analytic philosophy and the branch of mathematics now known as topology were gestated and born in the early part of the 20th century. It is not well recognized that there was early interaction between the communities practicing and developing these fields. We trace the history of how topological ideas entered into analytic philosophy through two migrations, an earlier one conceiving of topology geometrically and a later one conceiving of topology algebraically. This allows us to reassess the influence and significance of topological methods for philosophy, including the possible fruitfulness of a third conception of topology as a structure determining similarity.

Physical realization of topological Roman surface by spin-induced ferroelectric polarization in cubic lattice

Topology, an important branch of mathematics, is an ideal theoretical tool to describe topological states and phase transitions. Many topological concepts have found their physical entities in real or reciprocal spaces identified by topological invariants, which are usually defined on orientable surfaces, such as torus and sphere. It is natural to investigate the possible physical realization of more intriguing non-orientable surfaces. Herein, we show that the set of spin-induced ferroelectric polarizations in cubic perovskite oxides AMn3Cr4O12 (A = La and Tb) reside on the topological Roman surface—a non-orientable two-dimensional manifold formed by sewing a Möbius strip edge to that of a disc. The induced polarization may travel in a loop along the non-orientable Möbius strip or orientable disc, depending on the spin evolution as controlled by an external magnetic field. Experimentally, the periodicity of polarization can be the same or twice that of the rotating magnetic field, which is consistent with the orientability of the disc and the Möbius strip, respectively. This path-dependent topological magnetoelectric effect presents a way to detect the global geometry of a surface and deepens our understanding of topology in both mathematics and physics.

Topological phase transitions and Weyl semimetal phases in chiral photonic metamaterials

Recently, topologically nontrivial phases in chiral metamaterials have been proposed. However, a comprehensive description of topological phase diagrams and transitions in chiral metamaterials has not been presented. In this work, we demonstrate several forms of topological phase transitions and study the existence of edge states in different phases. In the local/lossless chiral media system, the topological phase transitions are associated with Weyl points. Along with the transitions, the edge state and Fermi arc exhibit a series of changes. When the nonlocal effect is introduced, the system shows phase transition between type-I/II Weyl semimetal phase and trivial phase. Moreover, the dissipative system also undergoes topological phase transitions owing to the annihilation of the topological charges. Our work could be helpful for the application of topological concepts and rich the topological wave physics in metamaterials.

A Fusion Topology of Higher Efficiency and Lower Capacity Hybrid Parallel Multi-Converters for Power Quality Compensation

In high-power applications, different multilevel voltage source converter (VSC) topologies are usually applied. The hybrid parallel multiconverters, which is a combination of multiactive and hybrid VSCs connecting in parallel, can be a promising solution in a high-power grid. In this article, the fusion VSC topology concept is proposed in which fusion VSC takes the advantages of both active (inductive coupling) VSC and hybrid (capacitive coupling) VSC. It corresponds to overcoming the insufficient-capacity problems when active VSCs are employed alone and vice versa. This article proposes fusion active power filter topology by combing active and hybrid power VSC through their equivalent model for power quality compensation. With the proposed VSC's equivalent model, it can reduce the converter rating by 17%–80% compared with the conventional approaches. Additionally, the corresponding control algorithm is proposed for the FAPF for power quality compensation. Finally, the simulation and real-time digital simulator hardware-in-the-loop experimental results of the proposed FAPF topology are given to demonstrate the validity and effectiveness for reducing its capacity and power loss compared with the conventional multi-VSC in power quality compensation.

Generating Soft Topologies via Soft Set Operators

As daily problems involve a great deal of data and ambiguity, it has become vital to build new mathematical ways to cope with them, and soft set theory is the greatest tool for doing so. As a result, we study methods of generating soft topologies through several soft set operators. A soft topology is known to be determined by the system of special soft sets, which are called soft open (dually soft closed) sets. The relationship between specific types of soft topologies and their classical topologies (known as parametric topologies) is linked to the idea of symmetry. Under this symmetry, we can study the behaviors and properties of classical topological concepts via soft settings and vice versa. In this paper, we show that soft topological spaces can be characterized by soft closure, soft interior, soft boundary, soft exterior, soft derived set, or co-derived set operators. All of the soft topologies that result from such operators are equivalent, as well as being identical to their classical counterparts under enriched (extended) conditions. Moreover, some of the soft topologies are the systems of all fixed points of specific soft operators. Multiple examples are presented to show the implementation of these operators. Some of the examples show that, by removing any axiom, we will miss the uniqueness of the resulting soft topology.

Spherical Fuzzy Soft Topology and Its Application in Group Decision-Making Problems

The spherical fuzzy soft set is a generalized soft set model, which is more realistic, practical, and accurate. It is an extended version of existing fuzzy soft set models that can be used to describe imprecise data in real-world scenarios. The paper seeks to introduce the new concept of spherical fuzzy soft topology defined on spherical fuzzy soft sets. In this work, we define some basic concepts including spherical fuzzy soft basis, spherical fuzzy soft subspace, spherical fuzzy soft interior, spherical fuzzy soft closure, and spherical fuzzy soft boundary. The properties of these defined set are also discussed and explained with an appropriate examples. Also, we establish certain important results and introduce spherical fuzzy soft separation axioms, spherical fuzzy soft regular space, and spherical fuzzy soft normal space. Furthermore, as an application, a group decision-making algorithm is presented based on the TOPSIS (Technique of Order Preference by Similarity to an Ideal Solution) method for solving the decision-making problems. The applicability of the proposed method is demonstrated through a numerical example. The comprehensive advantages of the proposed work have been stated over the existing methods.

Integrated Three-Level Dual-Phase Inverter

In view of reducing the number of inverter legs that provide dual-phase, three-level output voltages (as may be needed in an uninterruptible power supply), and that also provide a wide range of output frequencies (as needed in an advanced motor drive system with wide speed ranges), a three-level, dual-phase inverter topology is presented in this paper. Its three-level attribute was based on the F-type inverter topological concept, and its dual-output feature was based on the common representation of the inverter-leg concept. The proposed inverter could deliver single- and three-phase voltages to corresponding one- and three-phase loads, in common or different frequency modes of operation. A boundary between these modes of operation was established for the proposed inverter. An additional possibility of either operation in the one-phase or the three-phase system was offered by the inverter configuration. A modified carrier-based sinusoidal pulse-width modulation scheme is presented for the control of the inverter topology. The performances of the dual-phase inverter are given in the simulation results and demonstrated with a hardware prototype.

A Novel Fuzzy Structure: Infra-Fuzzy Topological Spaces

Obtaining a weaker condition that preserves some inspired topological properties is always desirable. As a result, we introduce the concept of infra-fuzzy topology, which is a subset family that degrades the concept of fuzzy topology by omitting the condition of closedness under arbitrary unions. Fundamental properties of infra-fuzzy topological spaces are investigated, including infra-fuzzy open and infra-fuzzy closed sets, infra-fuzzy interior and infra-fuzzy closure operators, and the infra-fuzzy boundary of a fuzzy set. It is not possible to expect the latter concepts to have properties identical to those in ordinary fuzzy topological spaces. More precisely, the infra-fuzzy interior of a set need not be infra-fuzzy open, and the infra-fuzzy closure and boundary of a set may not be infra-fuzzy closed. Then, employing infra-fuzzy neighborhood systems, infra-fuzzy Q-neighborhood systems, the basis of infra-fuzzy topology, and infra-fuzzy relative topology, we propose several approaches for generating infra-fuzzy topologies. Finally, we define the notions of continuity, openness, closedness, and homeomorphism of mappings in the context of infra fuzziness and investigate some of their properties and characterizations. We show that the usual characterization of earlier notions in the infra-fuzzy structure is incorrect. We demonstrate that the family of all infra-fuzzy homeomorphisms on an infra-fuzzy topological space forms a group under mappings composition. We finish this work by proving that each infra-fuzzy homeomorphism between two infra-fuzzy topological spaces produces an isomorphism on groups of infra-fuzzy homeomorphisms of the corresponding spaces.

STUDY OF CONVEXITY IN SETS AND FUZZY SETS

In this paper, we present an overview of both classical and fuzzy convexity, particularly, in conjunction with continuity5 and some topological concepts, and provide proofs of some of their algebraic properties along with suitable illustrations. We have extended the work of (Ammar E. E., 1999) by proving that if and are two convex fuzzy sets, then , , , and are also convex fuzzy sets. We have also shown that the convexity of a fuzzy set implies semistrict quasi-convexity

Triboinformatics: machine learning algorithms and data topology methods for tribology

Friction and wear are very common phenomena found virtually everywhere. However, it is very difficult to predict tribological (i.e. related to friction and wear) structure–property relationships from fundamental physical principles. Consequently, tribology remains a data-driven, mostly empirical discipline. With the advent of new machine learning (ML) and artificial intelligence methods, it becomes possible to establish new correlations in tribological data to predict and control better the tribological behavior of novel materials. Hence, the new area of triboinformatics has emerged combining tribology with data science. This paper reviews ML algorithms used to establish correlations between the structures of metallic alloys and composite materials, tribological test conditions, friction and wear. This paper also discusses novel methods of surface roughness analysis involving the concept of data topology in multidimensional data space, as applied to macro- and nanoscale roughness. Other triboinformatic approaches are considered as well.

Velocity Field Level Set Method Incorporating Topological Derivatives for Topology Optimization

Abstract The velocity field level set method constructs the velocity field by velocity design variables and basis functions, and thus facilitates the use of general optimizers while still retaining the level set-based implicit topological representation. This paper incorporates the topological derivative concept into the velocity field level set method to enable automatic nucleation of interior holes. In each design iteration, a specified volume fraction of new holes is inserted at locations with smaller values of topological derivatives. Thus, the method provides a way to directly change the structural topology during the boundary evolution using the velocity field based on the shape sensitivity. Compared with the original velocity field level set method, the current implementation can further accelerate the topological and shape evolution during the optimization process. More importantly, the capability of hole nucleation eliminates the need of prescribing initial holes and thus alleviates the dependency of the optimized design on the initial design. Several numerical examples in both 2D and 3D design domains are presented to demonstrate the validity and efficiency of the proposed method.

Viscoelastic and Viscoplastic Glucose Theory (VGT #18): Applying Theories of Viscoelasticity and Viscoplasticity from Engineering and Perturbation Theory from Quantum Mechanics from 9 Cases to Predict the Behavior or Symptom of an Output Biomarker as the Strain (ε), by Selecting a Highly Correlated Input Biomarker as the Stress (σ), and using the Stress Rate (dσ/dt) Multiplied with the Stress (σ)

The author has collected ~3 million data regarding his health condition and lifestyle details over the past 12 years. He spent the entire year of 2014 to develop a metabolism index (MI) model using topology concept, nonlinear algebra, algebraic geometry, and finite element method. This model contains various measured biomarkers and recorded lifestyle details along with their induced new biomedical variables for an additional ~1.5 million data. Body weight, glucose, blood pressure, heart rate, lipids, body temperature, and blood oxygen level, along with important lifestyle details, including diet, exercise, sleep, stress, water intake, and daily life routines are included in his MI database. The developed MI model has a total of 10 categories covering approximately 500 detailed elements that constitute his defined “metabolism model” which are the building blocks or root causes for diabetes and other chronic disease complications, including but not limited to cardiovascular disease (CVD), chronic heart disease (CHD), chronic kidney disease (CKD), retinopathy, neuropathy, foot ulcer, and hypothyroidism. The end result of the MI development work is a combined MI value over a selected time period with 73.5% as its dividing line between a healthy and unhealthy state. The MI serves as the foundation to many of his follow-up medical research work. During the period from 2015 to 2017, he focused his research on type 2 diabetes (T2D), especially glucoses, including fasting plasma glucose (FPG), postprandial plasma glucose (PPG), estimated average glucose (eAG), and hemoglobin A1C (HbA1C). During the following period from 2018 to 2022, he concentrated on researching medical complications resulting from diabetes, chronic diseases, and metabolic disorders which include heart problems, stroke, kidney problems, retinopathy, neuropathy, foot ulcer, diabetic skin fungal infection, hypothyroidism, diabetic constipation, cancer, and dementia. He also developed a few mathematical risk models to calculate the probability percentages of developing various diabetic complications.

Viscoelastic or Viscoplastic Glucose Theory (VGT #13): Applying the Theories of Elasticity, Plasticity, Viscoelasticity, and Viscoplasticity to Study the Metabolism Index (MI) and Cardiovascular Disease Risk Probability Percentage Over a 10+ Year Period from Y2012 to Y2022 Based on the GH-Method: Math-Physical Medicine (No. 592)

The author has collected ~1.5 million data of his health condition and lifestyle details over the past 12 years. In 2014, he developed a metabolism index (MI) model using a topology concept, nonlinear algebra, algebraic geometry, and finite element method. The MI model contains key biomarker data, such as body weight, glucose, blood pressure, heart rate, lipids, body temperature, and blood oxygen level, along with important lifestyle details, including diet, exercise, sleep, stress, water intake, and daily life routines. The 10 categories cover about 500 detailed elements that constitute the defined metabolism model which are the root-causes for diabetes complications, including cardiovascular disease (CVD), chronic heart disease (CHD), chronic kidney disease (CKD), retinopathy, neuropathy, foot ulcer, etc. The end result of the MI development is a combined MI value within any selected time period with 73.5% as the dividing line between a healthy state and unhealthy state. That is why he applied the MI value to develop the risk probability model of having CVD or CHD. In this article, the author conducts additional research work to discover some hidden relationship between the CVD risk probability (outcomes or strain) and MI values (inputs or stress) by applying the viscoelasticity and viscoplasticity theories. The behaviors and hidden relationships between the two biomarkers, MI values and CVD risks, are time-dependent, which are changing from time to time.

Viscoelastic and Viscoplastic Glucose Theory (VGT #17): Selecting 12 Cases and Applying Theories of Viscoelasticity and Viscoplasticity from Engineering and Perturbation Theory from Quantum Mechanics to Predict the Behavior of an Output Biomarker as the Strain (ε or Behavior, Symptom) by Using the Strain Rate (dε/dt) Multiplying with a Correlated Input Biomarker of Stress Value (σ or Stressor, Cause) as

The author has collected ~3 million data regarding his health condition and lifestyle details over the past 12 years. He spent the entire year of 2014 to develop a metabolism index (MI) model using topology concept, nonlinear algebra, algebraic geometry, and finite element method. This MI model contains various measured biomarkers and recorded lifestyle details along with their induced new biomedical variables for an additional ~1.5 million data. Body weight, glucose, blood pressure, heart rate, lipids, body temperature, and blood oxygen level, along with important lifestyle details, including diet, exercise, sleep, stress, water intake, and daily life routines are included in his MI database. His developed MI model has a total of 10 categories covering approximately 500 detailed elements that constitute his defined “metabolism model” which are the building blocks or root causes for diabetes and other chronic disease complications, including but not limited to cardiovascular disease (CVD), chronic heart disease (CHD), chronic kidney disease (CKD), retinopathy, neuropathy, foot ulcer, and hypothyroidism. The end result of the MI development work is a combined MI value within any selected time period with 73.5% as its dividing line between a healthy and unhealthy state. The MI serves as the foundation to many of his follow-up medical research work. During the period from 2015 to 2017, he focused his research on type 2 diabetes (T2D), especially glucoses, including fasting plasma glucose (FPG), postprandial plasma glucose (PPG), estimated average glucose (eAG), and hemoglobin A1C (HbA1C). During the following period from 2018 to 2022, he concentrated on researching medical complications resulting from diabetes, chronic diseases, and metabolic disorders which include heart problems, stroke, kidney problems, retinopathy, neuropathy, foot ulcer, diabetic skin fungal infection, hypothyroidism, and diabetic constipation, cancer, and dementia. He also developed a few mathematical risk models to calculate the probability percentages of developing various diabetic complications.

An elementary proof of the Brouwer’s fixed point theorem

We provide an elementary proof of the Brouwer’s fixed point theorem based solely on introductory topological concepts such as compactness and connectedness both covered in a first undergraduate course in point-set topology.