Hyperspace of finite unions of convergent sequences

The symbol S(X) denotes the hyperspace of finite unions of convergent sequences in a Hausdor˛ space X. This hyper-space is endowed with the Vietoris topology. First of all, we give a characterization of convergent sequence in S(X). Then we consider some cardinal invariants on S(X), and compare the character, the pseudocharacter, the sn-character, the so-character, the network weight and cs-network weight of S(X) with the corresponding cardinal function of X. Moreover, we consider rank k-diagonal on S(X), and give a space X with a rank 2-diagonal such that S(X) does not Gδ -diagonal. Further, we study the relations of some generalized metric properties of X and its hyperspace S(X). Finally, we pose some questions about the hyperspace S(X).

Exploring Geometric Feature Hyper-Space in Data to Learn Representations of Abstract Concepts

The term Concept has been a prominent part of investigations in psychology and neurobiology where, mostly, it is mathematically or theoretically represented. The Concepts are also studied computationally through their symbolic, distributed and hybrid representations. The majority of these approaches focused on addressing concrete concepts notion, but the view of the abstract concept is rarely explored. Moreover, most computational approaches have a predefined structure or configurations. The proposed method, Regulated Activation Network (RAN), has an evolving topology and learns representations of Abstract Concepts by exploiting the geometrical view of Concepts, without supervision. In the article, the IRIS data was used to demonstrate: the RAN's modeling; flexibility in concept identifier choice; and deep hierarchy generation. Data from IoT's Human Activity Recognition problem is used to show automatic identification of alike classes as abstract concepts. The evaluation of RAN with 8 UCI benchmarks and the comparisons with 5 Machine Learning models establishes the RANs credibility as a classifier. The classification operation also proved the RAN's hypothesis of abstract concept representation. The experiments demonstrate the RANs ability to simulate psychological processes (like concept creation and learning) and carry out effective classification irrespective of training data size.

Gravity and the curvature of the phase hyper-space

Gravity and the curvature of the phase hyper-space

On Bent Manifolds and Deformed Spaces

Riemannian geometry is the geometry of bent manifolds. However, as this paper shows, it is also the geometry of deformed spaces. General Relativity (GR), based on Riemannian geometry, relates to space around the Sun and other masses as a bent 3D manifold.   Although a bent 3D space manifold in a 4D hyper-space is unimaginable, physicists accept this constraint. Our geometry of deformed spaces removes this constraint and shows that the Sun and other masses simply contract the 3D space around them. Thus, we are able to understand General Relativity (GR) almost intuitively - an intuition that inspires our imagination. Space in GR is considered a continuous manifold, bent (curved) by energy/momentum. Both Einstein (1933) and Feynman (1963), considered the option of space being a deformed continuum rather than a bent (curved) continuous manifold. We, however, consider space to be a 3D deformed lattice rather than a bent continuous manifold. The geometry presented in this paper is the geometry of this kind of space.

Hyperunified field theory and Taiji program in space for GWD

In this paper, I present the recently established hyperunified field theory (HUFT)[Formula: see text] for all basic forces and elementary particles within the framework of gravitational quantum field theory (GQFT)[Formula: see text] in hyper-space–time. GQFT treats gravity as a gauge theory in the framework of quantum field theory to avoid the long term obstacle between general relativity and quantum mechanics. HUFT is built based on the guiding principle: the dimension of hyper-space–time correlates to intrinsic quantum numbers of basic building blocks of nature, and the action describing the laws of nature obeys the gauge invariance and coordinate independence, which is more fundamental than that proposed by Einstein for general relativity. The basic gravitational field is defined in biframe hyper-space–time as a bicovariant vector field, it is a gauge-type hyper-gravifield rather than a metric field. HUFT is characterized by a bimaximal Poincaré and hyper-spin gauge symmetry [Formula: see text] with a global and local conformal scaling invariance in biframe hyper-space–time. The gravitational origin of gauge symmetry is revealed through the hyper-gravifield that plays an essential role as a Goldstone-like field, which enables us to demonstrate the gauge-gravity and gravity-geometry correspondences and to corroborate the gravitational gauge-geometry duality with an emergent hidden general linear group symmetry [Formula: see text]. The Taiji Program in Space for the gravitational wave detection in China[Formula: see text] is briefly outlined.

An Extension of the VSM Documents Representation

In this paper we will present a new approach regarding the documents representation in order to be used in classification and/or clustering algorithms. In our new representation we will start from the classical "bag-of-words" representation but we will augment each word with its correspondent part-of-speech. Thus we will introduce a new concept called hyper-vectors where each document is represented in a hyper-space where each dimension is a different part-of-speech component. For each dimension the document is represented using the Vector Space Model (VSM). In this work we will use only five different parts of speech: noun, verb, adverb, adjective and others. In the hyper-space each dimension has a different weight. To compute the similarity between two documents we have developed a new hyper-cosine formula. Some interesting classification experiments are presented as validation cases.