This article broadens terminology and approaches that continue to advance time modelling within a relationalist framework. Time is modeled as a single dimension, flowing continuously through independent privileged poi...This article broadens terminology and approaches that continue to advance time modelling within a relationalist framework. Time is modeled as a single dimension, flowing continuously through independent privileged points. Introduced as absolute point-time, abstract continuous time is a backdrop for concrete relational-based time that is finite and discrete, bound to the limits of a real-world system. We discuss how discrete signals at a point are used to temporally anchor zero-temporal points [t = 0] in linear time. Object-oriented temporal line elements, flanked by temporal point elements, have a proportional geometric identity quantifiable by a standard unit system and can be mapped on a natural number line. Durations, line elements, are divisible into ordered unit ratio elements using ancient timekeeping formulas. The divisional structure provides temporal classes for rotational (Rt24t) and orbital (Rt18) sample periods, as well as a more general temporal class (Rt12) applicable to either sample or frame periods. We introduce notation for additive cyclic counts of sample periods, including divisional units, for calendar-like formatting. For system modeling, unit structures with dihedral symmetry, group order, and numerical order are shown to be applicable to Euclidean modelling. We introduce new functions for bijective and non-bijective mapping, modular arithmetic for cyclic-based time counts, and a novel formula relating to a subgroup of Pythagorean triples, preserving dihedral n-polygon symmetries. This article presents a new approach to model time in a relationalistic framework.展开更多
Proteins are the workhorse molecules of the cell, which are obtained by folding long chains of amino acids. Since not all shapes are obtained as a folded chain of amino acids, there should be global geometrical constr...Proteins are the workhorse molecules of the cell, which are obtained by folding long chains of amino acids. Since not all shapes are obtained as a folded chain of amino acids, there should be global geometrical constraints on the shape. Moreover, since the function of a protein is largely determined by its shape, constraints on the shape should have some influence on its interaction with other proteins. In this paper, we consider global geometrical constraints on the shape of proteins. Using a mathematical toy model, in which proteins are represented as closed chains of tetrahedrons, we have identified not only global geometrical constraints on the shape of proteins, but also their influence on protein interactions. As an example, we show that a garlic-bulb like structure appears as a result of the constraints. Regarding the influence of global geometrical constraints on interactions, we consider their influence on the structural coupling of two distal sites in allosteric regulation. We then show the inseparable relationship between global geometrical constraints and protein interactions;i.e. they are different sides of the same coin. This finding could be important for the understanding of the basic mechanisms of allosteric regulation of protein functions.展开更多
This paper presents a method for segmenting a 3D point cloud into planar surfaces using recently obtained discretegeometry results. In discrete geometry, a discrete plane is defined as a set of grid points lying betwe...This paper presents a method for segmenting a 3D point cloud into planar surfaces using recently obtained discretegeometry results. In discrete geometry, a discrete plane is defined as a set of grid points lying between two parallel planes with a small distance, called thickness. In contrast to the continuous case, there exist a finite number of local geometric patterns （LGPs） appearing on discrete planes. Moreover, such an LGP does not possess the unique normal vector but a set of normal vectors. By using those LGP properties, we first reject non-linear points from a point cloud, and then classify non-rejected points whose LGPs have common normal vectors into a planar-surface-point set. From each segmented point set, we also estimate the values of parameters of a discrete plane by minimizing its thickness.展开更多
In polar regions, floating ice exhibits distinct characteristics across a range of spatial scales. It is well recognized that the irregular geometry of these ice formations markedly influences their dynamic behavior. ...In polar regions, floating ice exhibits distinct characteristics across a range of spatial scales. It is well recognized that the irregular geometry of these ice formations markedly influences their dynamic behavior. This study introduces a polyhedral Discrete Element Method (DEM) tailored for polar ice, incorporating the Gilbert-Johnson-Keerthi (GJK) and Expanding Polytope Algorithm (EPA) for contact detection. This approach facilitates the simulation of the drift and collision processes of floating ice, effectively capturing its freezing and fragmentation. Subsequently, the stability and reli ability of this model are validated by uniaxial compression on level ice fields, focusing specifically on the influence of compression strength on deformation resistance. Additionally, clusters of ice floes nav igating through narrow channels are simulated. These studies have qualitatively assessed the effects of Floe Size Distribution (FSD), initial concentration, and circularity on their flow dynamics. The higher power-law exponent values in the FSD, increased circularity, and decreased concentration are each as sociated with accelerated flow in ice floe fields. The simulation results distinctly demonstrate the con siderable impact of sea ice geometry on the movement of clusters, offering valuable insights into the complexities of polar ice dynamics.展开更多
This study proposes a virtual globe-based vector data model named the quaternary quadrangle vector tile model(QQVTM)in order to better manage,visualize,and analyze massive amounts of global multi-scale vector data.The...This study proposes a virtual globe-based vector data model named the quaternary quadrangle vector tile model(QQVTM)in order to better manage,visualize,and analyze massive amounts of global multi-scale vector data.The model integrates the quaternary quadrangle mesh(a discrete global grid system)and global image,terrain,and vector data.A QQVTM-based organization method is presented to organize global multi-scale vector data,including linear and polygonal vector data.In addition,tilebased reconstruction algorithms are designed to search and stitch the vector fragments scattered in tiles to reconstruct and store the entire vector geometries to support vector query and 3D analysis of global datasets.These organized vector data are in turn visualized and queried using a geometry-based approach.Our experimental results demonstrate that the QQVTM can satisfy the requirements for global vector data organization,visualization,and querying.Moreover,the QQVTM performs better than unorganized 2D vectors regarding rendering efficiency and better than the latitude–longitude-based approach regarding data redundancy.展开更多
An F-polygon is a simple polygon whose vertices are F-points, which are points of the set of vertices of a tiling of R~2 by regular triangles and regular hexagons of unit edge. Let f(v) denote the least possible numbe...An F-polygon is a simple polygon whose vertices are F-points, which are points of the set of vertices of a tiling of R~2 by regular triangles and regular hexagons of unit edge. Let f(v) denote the least possible number of F-points in the interior of a convex F-polygon K with v vertices. In this paper we prove that f(10) = 10, f(11) = 12,f(12) = 12.展开更多
In this paper, we discuss fuzzy simplex and fuzzy convex hull, and give several representation theorems for fuzzy simplex and fuzzy convex hull. In addition, by giving a new characterization theorem of fuzzy convex hu...In this paper, we discuss fuzzy simplex and fuzzy convex hull, and give several representation theorems for fuzzy simplex and fuzzy convex hull. In addition, by giving a new characterization theorem of fuzzy convex hull, we improve some known results about fuzzy convex hull.展开更多
The theory of relativity links space and time to account for observed events in four-dimensional space. In this article we describe an alternative static state causal discrete time modeling system using an omniscient ...The theory of relativity links space and time to account for observed events in four-dimensional space. In this article we describe an alternative static state causal discrete time modeling system using an omniscient viewpoint of dynamical systems that can express object relations in the moment(s) they are observed. To do this, three key components are required, including the introduction of independent object-relative dimensional metrics, a zero-dimensional frame of reference, and application of Euclidean geometry for modeling. Procedures separate planes of matter, extensions of space (relational distance) and time (duration) using object-oriented dimensional quantities. Quantities are converted into base units using symmetry for space (Dihedral<sub>360</sub>), time (Dihedral<sub>12</sub>), rotation (Dihedral<sub>24</sub>), and scale (Dihedral<sub>10</sub>). Geometric elements construct static state outputs in discrete time models rather than continuous time using calculus, thereby using dimensional and positional natural number numerals that can visually encode complex data instead of using abstraction and irrationals. Static state Euclidean geometric models of object relations are both measured and expressed in the state they are observed in zero-time as defined by a signal. The frame can include multiple observer frames of reference where each origin, point, is the location of a distinct privileged point of reference. Two broad and diverse applications are presented: a one-dimensional spatiotemporal orbital model, and a thought experiment related to a physical theory beyond Planck limits. We suggest that expanding methodologies and continued formalization, novel tools for physics can be considered along with applications for computational discrete geometric modeling.展开更多
This paper proposes a novel discrete differential geometry of n-simplices. It was originally developed for protein structure analysis. Unlike previous works, we consider connection between space-filling n-simplices. U...This paper proposes a novel discrete differential geometry of n-simplices. It was originally developed for protein structure analysis. Unlike previous works, we consider connection between space-filling n-simplices. Using cones of an integer lattice, we introduce tangent bundle-like structure on a collection of n-simplices naturally. We have applied the mathematical framework to analysis of protein structures. In this paper, we propose a simple encoding method which translates the conformation of a protein backbone into a 16-valued sequence.展开更多
This paper proposes a novel category theoretic approach to describe protein’s shape, <i>i.e.</i>, a description of their shape by a set of algebraic equations. The focus of the approach is on the relation...This paper proposes a novel category theoretic approach to describe protein’s shape, <i>i.e.</i>, a description of their shape by a set of algebraic equations. The focus of the approach is on the relations between proteins, rather than on the proteins themselves. Knowledge of category theory is not required as mathematical notions are defined concretely. In this paper, proteins are represented as closed trajectories (<i>i.e.</i>, loops) of flows of triangles. The relations between proteins are defined using the fusion and fission of loops of triangles, where allostery occurs naturally. The shape of a protein is then described with quantities that are measurable with unity elements called “unit loops”. That is, protein’s shape is described with the loops that are obtained by the fusion of unit loops. Measurable loops are called “integral”. In the approach, the unit loops play a role similar to the role “1” plays in the set Z of integers. In particular, the author considers two categories of loops, the “integral” loops and the “rational” loops. Rational loops are then defined using algebraic equations with “integral loop” coefficients. Because of the approach, our theory has some similarities to quantum mechanics, where only observable quantities are admitted in physical theory. The author believes that this paper not only provides a new perspective on protein engineering, but also promotes further collaboration between biology and other disciplines.展开更多
文摘This article broadens terminology and approaches that continue to advance time modelling within a relationalist framework. Time is modeled as a single dimension, flowing continuously through independent privileged points. Introduced as absolute point-time, abstract continuous time is a backdrop for concrete relational-based time that is finite and discrete, bound to the limits of a real-world system. We discuss how discrete signals at a point are used to temporally anchor zero-temporal points [t = 0] in linear time. Object-oriented temporal line elements, flanked by temporal point elements, have a proportional geometric identity quantifiable by a standard unit system and can be mapped on a natural number line. Durations, line elements, are divisible into ordered unit ratio elements using ancient timekeeping formulas. The divisional structure provides temporal classes for rotational (Rt24t) and orbital (Rt18) sample periods, as well as a more general temporal class (Rt12) applicable to either sample or frame periods. We introduce notation for additive cyclic counts of sample periods, including divisional units, for calendar-like formatting. For system modeling, unit structures with dihedral symmetry, group order, and numerical order are shown to be applicable to Euclidean modelling. We introduce new functions for bijective and non-bijective mapping, modular arithmetic for cyclic-based time counts, and a novel formula relating to a subgroup of Pythagorean triples, preserving dihedral n-polygon symmetries. This article presents a new approach to model time in a relationalistic framework.
文摘Proteins are the workhorse molecules of the cell, which are obtained by folding long chains of amino acids. Since not all shapes are obtained as a folded chain of amino acids, there should be global geometrical constraints on the shape. Moreover, since the function of a protein is largely determined by its shape, constraints on the shape should have some influence on its interaction with other proteins. In this paper, we consider global geometrical constraints on the shape of proteins. Using a mathematical toy model, in which proteins are represented as closed chains of tetrahedrons, we have identified not only global geometrical constraints on the shape of proteins, but also their influence on protein interactions. As an example, we show that a garlic-bulb like structure appears as a result of the constraints. Regarding the influence of global geometrical constraints on interactions, we consider their influence on the structural coupling of two distal sites in allosteric regulation. We then show the inseparable relationship between global geometrical constraints and protein interactions;i.e. they are different sides of the same coin. This finding could be important for the understanding of the basic mechanisms of allosteric regulation of protein functions.
文摘This paper presents a method for segmenting a 3D point cloud into planar surfaces using recently obtained discretegeometry results. In discrete geometry, a discrete plane is defined as a set of grid points lying between two parallel planes with a small distance, called thickness. In contrast to the continuous case, there exist a finite number of local geometric patterns （LGPs） appearing on discrete planes. Moreover, such an LGP does not possess the unique normal vector but a set of normal vectors. By using those LGP properties, we first reject non-linear points from a point cloud, and then classify non-rejected points whose LGPs have common normal vectors into a planar-surface-point set. From each segmented point set, we also estimate the values of parameters of a discrete plane by minimizing its thickness.
文摘In polar regions, floating ice exhibits distinct characteristics across a range of spatial scales. It is well recognized that the irregular geometry of these ice formations markedly influences their dynamic behavior. This study introduces a polyhedral Discrete Element Method (DEM) tailored for polar ice, incorporating the Gilbert-Johnson-Keerthi (GJK) and Expanding Polytope Algorithm (EPA) for contact detection. This approach facilitates the simulation of the drift and collision processes of floating ice, effectively capturing its freezing and fragmentation. Subsequently, the stability and reli ability of this model are validated by uniaxial compression on level ice fields, focusing specifically on the influence of compression strength on deformation resistance. Additionally, clusters of ice floes nav igating through narrow channels are simulated. These studies have qualitatively assessed the effects of Floe Size Distribution (FSD), initial concentration, and circularity on their flow dynamics. The higher power-law exponent values in the FSD, increased circularity, and decreased concentration are each as sociated with accelerated flow in ice floe fields. The simulation results distinctly demonstrate the con siderable impact of sea ice geometry on the movement of clusters, offering valuable insights into the complexities of polar ice dynamics.
基金the National Natural Science Foundation of China[grant number 41171314],[grant number 41023001]the Fundamental Research Funds for the Central Universities[grant number 2014619020203].Comments from the anonymous reviewers and editor are appreciated.
文摘This study proposes a virtual globe-based vector data model named the quaternary quadrangle vector tile model(QQVTM)in order to better manage,visualize,and analyze massive amounts of global multi-scale vector data.The model integrates the quaternary quadrangle mesh(a discrete global grid system)and global image,terrain,and vector data.A QQVTM-based organization method is presented to organize global multi-scale vector data,including linear and polygonal vector data.In addition,tilebased reconstruction algorithms are designed to search and stitch the vector fragments scattered in tiles to reconstruct and store the entire vector geometries to support vector query and 3D analysis of global datasets.These organized vector data are in turn visualized and queried using a geometry-based approach.Our experimental results demonstrate that the QQVTM can satisfy the requirements for global vector data organization,visualization,and querying.Moreover,the QQVTM performs better than unorganized 2D vectors regarding rendering efficiency and better than the latitude–longitude-based approach regarding data redundancy.
基金Supported by National Natural Science Foundation of China(Grant No.12271139)。
文摘An F-polygon is a simple polygon whose vertices are F-points, which are points of the set of vertices of a tiling of R~2 by regular triangles and regular hexagons of unit edge. Let f(v) denote the least possible number of F-points in the interior of a convex F-polygon K with v vertices. In this paper we prove that f(10) = 10, f(11) = 12,f(12) = 12.
基金Supported by the Science and Technology Research Program of Chongqing Municipal Educational Committee(Grant No.KJ100518)the Foundation of Chongqing University of Posts and Telecommunications for Scholars with Doctorate (Grant No.A2009-14)
文摘In this paper, we discuss fuzzy simplex and fuzzy convex hull, and give several representation theorems for fuzzy simplex and fuzzy convex hull. In addition, by giving a new characterization theorem of fuzzy convex hull, we improve some known results about fuzzy convex hull.
文摘The theory of relativity links space and time to account for observed events in four-dimensional space. In this article we describe an alternative static state causal discrete time modeling system using an omniscient viewpoint of dynamical systems that can express object relations in the moment(s) they are observed. To do this, three key components are required, including the introduction of independent object-relative dimensional metrics, a zero-dimensional frame of reference, and application of Euclidean geometry for modeling. Procedures separate planes of matter, extensions of space (relational distance) and time (duration) using object-oriented dimensional quantities. Quantities are converted into base units using symmetry for space (Dihedral<sub>360</sub>), time (Dihedral<sub>12</sub>), rotation (Dihedral<sub>24</sub>), and scale (Dihedral<sub>10</sub>). Geometric elements construct static state outputs in discrete time models rather than continuous time using calculus, thereby using dimensional and positional natural number numerals that can visually encode complex data instead of using abstraction and irrationals. Static state Euclidean geometric models of object relations are both measured and expressed in the state they are observed in zero-time as defined by a signal. The frame can include multiple observer frames of reference where each origin, point, is the location of a distinct privileged point of reference. Two broad and diverse applications are presented: a one-dimensional spatiotemporal orbital model, and a thought experiment related to a physical theory beyond Planck limits. We suggest that expanding methodologies and continued formalization, novel tools for physics can be considered along with applications for computational discrete geometric modeling.
文摘This paper proposes a novel discrete differential geometry of n-simplices. It was originally developed for protein structure analysis. Unlike previous works, we consider connection between space-filling n-simplices. Using cones of an integer lattice, we introduce tangent bundle-like structure on a collection of n-simplices naturally. We have applied the mathematical framework to analysis of protein structures. In this paper, we propose a simple encoding method which translates the conformation of a protein backbone into a 16-valued sequence.
文摘This paper proposes a novel category theoretic approach to describe protein’s shape, <i>i.e.</i>, a description of their shape by a set of algebraic equations. The focus of the approach is on the relations between proteins, rather than on the proteins themselves. Knowledge of category theory is not required as mathematical notions are defined concretely. In this paper, proteins are represented as closed trajectories (<i>i.e.</i>, loops) of flows of triangles. The relations between proteins are defined using the fusion and fission of loops of triangles, where allostery occurs naturally. The shape of a protein is then described with quantities that are measurable with unity elements called “unit loops”. That is, protein’s shape is described with the loops that are obtained by the fusion of unit loops. Measurable loops are called “integral”. In the approach, the unit loops play a role similar to the role “1” plays in the set Z of integers. In particular, the author considers two categories of loops, the “integral” loops and the “rational” loops. Rational loops are then defined using algebraic equations with “integral loop” coefficients. Because of the approach, our theory has some similarities to quantum mechanics, where only observable quantities are admitted in physical theory. The author believes that this paper not only provides a new perspective on protein engineering, but also promotes further collaboration between biology and other disciplines.
