Geometrical Modeling of Crystal Structures with Use of Space of Elliptic Riemannian Geometry

The space of internal geometry of a model of a real crystal is supposed to be finite, closed, and with a constant Gaussian curvature equal to unity, permitting the realization of lattice systems in accordance with Fedorov groups of transformations. For visualizing computations, the interpretation of geometrical objects on a Clifford surface (SK) in Riemannian geometry with the help of a 2D torus in a Euclidean space is used. The F-algorithm ensures a computation of 2D sections of models of point systems arranged perpendicularly to the symmetry axes l3, l4, and l6. The results of modeling can be used for calculations of geometrical sizes of crystal structures, nanostructures, parameters of the cluster organization of oxides, as well as for the development of practical applications connected with improving the structural characteristics of crystalline materials.

Interconverting Models of Gray Matrix and Geographic Coordinates Based on Space Analytic Geometry

With the progress and development of science, the synchronous satellite as the high-tech product is taken something more and more seriously by all countries. Converting gray value matrix into geographic coordinates becomes more and more important. This paper establishes the models for interconverting row and column of gray value matrix which detected by the infrared detectors on synchronous satellites and geographic coordinates based on the related theorems and properties of space analytic geometry. We draw satellite cloud image with the data of gray matrix in MATLAB, convert coastline’s latitude and longitude coordinates into row and column coordinates of a gray matrix according via these models when the sub-satellite point’s row and column coordinates and geographic coordinates are given, then add coast-lines to the original satellite cloud image with the transformation data. We find that coast-lines completely match with the bump characteristics of satellite cloud image, and that verify the correctness of the models.

Alignments, Topologies, Convex Geometries and Higgs Spaces

Aligned spaces generalize topological spaces and generate Higgs spaces. We give a necessary and sufficient condition for a finite aligned space to be a topological space, we prove the existence of two kinds of convex geometries, and we compare several concepts and results for arbitrary （that is, not necessarily finite） aligned, topological and Higgs spaces.

The Relativistic Geometry of the Complex Matter Space

Duality behavior of photons in wave-particle property has posed challenges and opportunities to discover other frontiers of fundamental particles leading to the relativistic and quantum description of matter. The speed of particles faster than the speed of light could not be recognized, and matter was always described as a real number. A new fundamental view on matter as a complex value has been introduced by many authors who present a paradigm that is shifted from real or pure imaginary particles to Complex Matter Space. A new assumption will be imposed that matter has two intrinsic components: i) mass, and ii) charge. The mass will be measured by real number systems and charged by an imaginary unit. The relativistic concept of Complex Matter Space on energy and momentum is investigated and we can conclude that the new Complex Matter Space (CMS) theory will help get one step closer to a better understanding toward: 1) Un-Euclidean description of Minkowski Geometry in the context of the Complex Matter Space, 2) transformation from Euclidean to Minkowski space and its relativistic interpretation. Finally, geometrical foundations are essential to have a real picture of space, matter, and the universe.

Physical Space Was Not Expanding

Plurality of characteristic peaks observed in number density distribution of galaxy redshift reveals that extent of physical space has been finite. Significant portion of observed celestial objects is found pair-wise associated, i.e., the observed lights were emitted from one and same luminescent source but seen at different sky directions of observer, which is a unique phenomenon that can occur but only in finite space. Cosmic microwave radiation has always been interpreted as afterglow of Big Bang event. However, such radiation is shown unobservable to current observer if Hubble-Lemaître Correlation is interpreted as caused by receding motion of celestial objects. On the other hand, cosmic radiation can be understood as a common and ordinary phenomenon due to space lens, a unique property only of finite space. From Sloan Digital Sky Survey data, internal diameter of physical space is measured as 2.0 billion light years. If celestial objects were receding, hence physical space was expanding, then characteristic peaks of finite physical space should not appear evenly in number density distribution of redshift of the objects but more sparsely with respect to redshift increase. However, as revealed by the data, locations of the characteristic peaks in the distributions are rather even that do not match the locations as required by receding motion of object. Therefore, as evidenced by the data, physical space was not expanding, at least during the recent 18 billion years. In addition, considerable portion of observed quasars is found sharing a common factor of ~1/2 for their respective gravitation redshifts.

Physical Space Is Finite

Metrological analysis shows that any clock in inertial motion in infinite space shall not have time dilation, due to relativity of such motion in such space. On the other hand, atomic clock in inertial motion in finite space shall exhibit time dilation, due to alteration of momentum of clock-defining particle caused by nonzero curvature of trajectory of such motion in such space. Therefore, time dilation experiment of atomic clock in inertial motion in physical space provides a direct and decisive way of determining geometry of physical space in real-time. Phenomenon of time dilation of atomic clock in inertial motion in physical space has long been observed and confirmed experimentally. Therefore, extent of physical space has to be finite, consistent with result of high precision experiment of free particle in high-speed motion conducted a decade ago.Keywords Geometry of Physical Space, Time Dilation, Atomic Clock, Special Relativity Theory.

ATOMIC DECOATOMIC DECOMPOSITIONS AND DUALS OF WEAK HARDY SPACES OF B-VALUED MARTINGALES

In this article, several weak Hardy spaces of Banach-space-valued martingales are introduced, some atomic decomposition theorems for them are established and their duals are investigated. The results closely depend on the geometrical properties of the Banach space in which the martingales take values.

Quantum Dark Energy from the Hyperbolic Transfinite Cantorian Geometry of the Cosmos

The quintessence of hyperbolic geometry is transferred to a transfinite Cantorian-fractal setting in the present work. Starting from the building block of E-infinity Cantorian spacetime theory, namely a quantum pre-particle zero set as a core and a quantum pre-wave empty set as cobordism or surface of the core, we connect the interaction of two such self similar units to a compact four dimensional manifold and a corresponding holographic boundary akin to the compactified Klein modular curve with SL(2,7) symmetry. Based on this model in conjunction with a 4D compact hy- perbolic manifold M(4) and the associated general theory, the so obtained ordinary and dark en- ergy density of the cosmos is found to be in complete agreement with previous analysis as well as cosmic measurements and observations such as WMAP and Type 1a supernova.

Real Interpolation Between B-valued Martingale Hardy Spaces

The interpolation spaces between Banach space valued martingale Hardy spaces, between Hardy and BMO spaces are identified respectively. Some results obtained here are connected closely with the convexity and smoothness of the Banach space which the martingales take values in.

ATOMIC DECOMPOSITION FOR B-VALUED R.V. SEQUENCE SPACES

In this article,atomic decompositions and the duals of some B-valued r.v.se- quence spaces are investigated.The results show that it closely depends on the geometrical properties of the sequence that take values in.

Two New BMO Spaces and Sharp Operators of B-Valued Martingale

For Banach space-valued martingale, two new BMO spaces, namely BMO (X), resp-BMO (X), and two new sharp operators, namely, resp. generated by the condition-al p-mean-square resp. p-mean-square operator are introduced, and then, the connections betweenBMO (X) and BMO;, BMO(X) and BMO and and are investigated. The resultsobtained here yield a new charactrization of the convexity and smoothness of Banach space.

THE HOMOTHETIC MOTIONS IN THE LORENTZ 3-SPACE

In this article, the properties of the homothetic motions in three-dimensional Lorentz space are investigated. Also, some geometric results between velocity and acceleration vectors of a point in a spatial motion are obtained.

Coarsely Invariant Hilbert Spaces over Infinite Connected Graphs

We study in this paper a Hilbert space HV associated with the coarse geometry of an infinite connected graph X(V, E) with vertex set V and edge set E. We show that X(V,E) is uniformly expanding if and only ifl2(V)can be continuously included in HV as a closed subspace,and that the inner product structure of HV is topologically invariant under uniform coarsening of the graph. We also discuss the functorial properties of these Hilbert spaces.

NEW SCALAR-TYPE EQUILIBRIUM EQUATIONS OF THE GENERAL SPACE FORCE SYSTEM

Using the principle of analytical geometry, several properties of the space straight lille are proved. Based on these properties, the equilibrium of general space force system is considered and its four new scalar-type equilibrium equations are derived which are equivalent to the vector-type necessary and sufficient conditions far equilibrium.

Computational Modeling of Dual-Phase Ceramics with Finsler-Geometric Phase Field Mechanics

A theory invoking concepts from differential geometry of generalized Finsler space in conjunction with diffuse interface modeling is described and implemented in finite element(FE)simulations of dual-phase polycrystalline ceramic microstructures.Order parameters accounting for fracture and other structural transformations,notably partial dislocation slip,twinning,or phase changes,are dimensionless entries of an internal state vector of generalized pseudo-Finsler space.Ceramics investigated in computations are a boron carbide-titanium diboride(B4C-TiB2)composite and a diamond-silicon carbide(C-SiC)composite.Deformation mechanisms-in addition to elasticity and cleavage fracture in grains of any phase-include restricted dislocation glide(TiB2 phase),deformation twinning(B4C and-SiC phases),and stress-induced amorphization(B4C phase).The metric tensor of generalized Finsler space is scaled conformally according to dilatation induced by cavitation or other fracture modes and densification induced by phase changes.Simulations of pure shear consider various morphologies and lattice orientations.Effects of microstructure on overall strength of each composite are reported.In B4C-TiB2,minor improvements in shear strength and ductility are observed with an increase in the second phase from 10 to 18%by volume,suggesting that residual stresses or larger-scale crack inhibition may be responsible for toughness gains reported experimentally.In diamond-SiC,a composite consisting of diamond crystals encapsulated in a nano-crystalline SiC matrix shows improved strength and ductility relative to a two-phase composite with isolated bulk SiC grains.

Fast,Exact and Robust Set Operations on Polyhedrons Using Localized Constructive Solid Geometry Trees

Regularized Boolean operations have been widely used in 3D modeling systems. However, evaluating Boolean operations may be quite numerically unstable and time consuming, especially for iterated set operations. A novel and unified technique is proposed in this paper for computing single and iterated set operations efficiently, robustly and exactly. An adaptive octree is combined with a nested constructive solid geometry （CSG） tree by this technique. The intersection handling is restricted to the cells in the octree where intersection actually occurs. Within those cells, a CSG tree template is instanced by the surfaces and the tree is converted to planebased binary space partitioning （BSP） for set evaluation; Moreover, the surface classification is restricted to the ceils in the octree where the surfaces only come from a model and are within the bounding-boxes of other polyhedrons. These two ways bring about the efficiency and scalability of the operations, in terms of runtime and memory. As all surfaces in such a cell have the same classification relation, they are classified as a whole. Robustness and exactness are achieved by integrating plane-based geometry representation with adaptive geometry predicate technique in intersection handling, and by applying divide-and-conquer arithmetic on surface classification. Experimental results demonstrate that the proposed approach can guarantee the robustness of Boolean computations and runs faster than other existing approaches.

On Possibility of Geometrical Interpretation of Time

The low-dimensional model of the space-time is considered where time is a real coordinate with dimensionality of length. The inertia law appears within this model as a consequence of the geometrical structure of the space.

Generalized Uncertainty Relations, Curved Phase-Spaces and Quantum Gravity

Modifications of the Weyl-Heisenberg algebra are proposed where the classical limit corresponds to a metric in (curved) momentum spaces. In the simplest scenario, the 2D de Sitter metric of constant curvature in momentum space furnishes a hierarchy of modified uncertainty relations leading to a minimum value for the position uncertainty . The first uncertainty relation of this hierarchy has the same functional form as the stringy modified uncertainty relation with a Planck scale minimum value for at . We proceed with a discussion of the most general curved phase space scenario (cotangent bundle of spacetime) and provide the noncommuting phase space coordinates algebra in terms of the symmetric and nonsymmetric metric components of a Hermitian complex metric , such . Yang’s noncommuting phase-space coordinates algebra, combined with the Schrodinger-Robertson inequalities involving angular momentum eigenstates, reveals how a quantized area operator in units of emerges like it occurs in Loop Quantum Gravity (LQG). Some final comments are made about Fedosov deformation quantization, Noncommutative and Nonassociative gravity.

Means of Choice for Interactive Management of Dynamic Geometry Problems Based on Instrumented Behaviour

Our paper presents a project that involves two research questions: does the choice of a related problem by the tutorial system allow the problem solving process which is blocked for the student to be restarted? What information about learning do related problems returned by the system provide us? We answer the first question according to the didactic engineering, whose mode of validation is internal and based on the confrontation between an a priori analysis and an a posteriori analysis that relies on data from experiments in schools. We consider the student as a subject whose adaptation processes are conditioned by the problem and the possible interactions with the computer environment, and also by his knowledge, usually implicit, of the institutional norms that condition his relationship with geometry. Choosing a set of good problems within the system is therefore an essential element of the learning model. Since the source of a problem depends on the student’s actions with the computer tool, it is necessary to wait and see what are the related to problems that are returned to him before being able to identify patterns and assess the learning. With the simultaneity of collecting and analysing interactions in each class, we answer the second question according to a grounded theory analysis. By approaching the problems posed by the system and the designs in play at learning blockages, our analysis links the characteristics of problems to the design components in order to theorize on the decisional, epistemological, representational, didactic and instrumental aspects of the subject-milieu system in interaction.

OCDMA SystemSignature Code Design Theory Via Finite Projective Geometry

Optical orthogonal code is the main signature code employed by optical CDMA system. Starting from modern mathematics theory, finite projective geometry and Galois theory, the essential connection between optical orthogonal code designing and finite geometry theory were discussed; find out the corresponding relationship between the parameter of OOC and that of finite geometry space. In this article, the systematic theory of OOC designing based on projective geometry is established in detail. The designing process and results of OOC on projective plane PG(2,q) and on m-dimension projective space are given respectively. Furthermore, the analytical theory for the corresponding relation between OOC with high cross-correlation and k-D manifold of projective space is set up. The OOC designing results given in this article have excellent performance, whose maximum cross-correlation is 1, and the cardinality reaches the Johnson upper bound, i.e. it realizes the optimization in both MUI and system capacity.