Search Results (10)

In this paper, we introduce a new generalized inverse called $\mathfrak{m}$-CCE inverse which presents a generalization of the CCE inverse in Minkowski space by using the $\mathfrak{m}$-core-EP decomposition and the Minkowski inverse. We first show the existence and the uniqueness of the generalized inverse. Then, a number of basic properties and diverse characterizations are derived for the $\mathfrak{m}$-CCE inverse as well as its limit and integral expressions. Additionally, applications of the $\mathfrak{m}$-CCE inverse are given in solving a system of linear equations. Applying the generalized inverse, we introduce a binary relation based on the $\mathfrak{m}$-CCE inverse. Full article

This paper mainly focuses on in-depth research on inequalities on symmetric cones. We will further analyze and discuss the inequalities we developed on the second-order cone and develop more inequalities. According to our past research in dealing with second-order cone inequalities, we derive more inequalities concerning the eigenvalue version of Young’s inequality and trace a version of an inverse Young inequality and its applications. These conclusions align with the results established for the positive semidefinite cone, which is also a symmetric cone. It is of considerable help to the establishment of inequalities on symmetric cones and the analysis of their derivative algorithms. Full article

While there are well-known synthetic methods in the literature for finding the image of a point under circular inversion in $l_2$-normed geometry (Euclidean geometry), there is no similar synthetic method in Minkowski geometry, also known as the geometry of finite-dimensional Banach spaces. In this study, we have succeeded in creating a synthetic construction of the circular inversion in $l_1$-normed spaces, which is one of the most fundamental examples of Minkowski geometry. Moreover, this synthetic construction has been given using the Euclidean circle, independently of the $l_1$-norm. Full article

In 1978, Friedberg and Lee introduced the phenomenological soliton bag model of hadrons, generalizing the MIT bag model developed in 1974 shortly after the formulation of QCD. In this model, quarks and gluons are confined due to coupling with a real scalar field $\rho$, which tends to zero outside some compact region $S\subset {\mathbb{R}}^3$ determined dynamically from the equations of motion. The gauge coupling in the soliton bag model runs as the inverse power of $\rho$, already at the semiclassical level. We show that this model arises naturally as a consequence of introducing the warped product metric $\mathrm{d}{s}_M^2+{\rho }^2\mathrm{d}{s}_G^2$ on the principal G-bundle $P(M,G)\cong M\times G$ with a non-Abelian group G over Minkowski space $M={\mathbb{R}}^{3,1}$. Confinement of quarks and gluons in a compact domain $S\subset {\mathbb{R}}^3$ is a consequence of the collapse of the bundle manifold $M\times G$ to M outside S due to shrinking of the group manifold G to a point. We describe the formation of such regions S as a dynamical process controlled by the order parameter field $\rho$. Full article

We write the Kerr–Schild tetrads in terms of the flat space–time tetrads and of a (1, 1) tensor $S_{\mu }^{\lambda }$. This tensor can be considered as a projection operator, since it transforms (i) flat space–time tetrads into non-flat tetrads, and vice-versa, and (ii) the Minkowski space–time metric tensor into a non-flat metric tensor, and vice-versa. The $S_{\mu }^{\lambda }$ tensor and its inverse are constructed in terms of the standard null vector field $l_{\mu }$ that defines the Kerr–Schild form of the metric tensor in general relativity, and that yields black holes and non-linear gravitational waves as solutions of the vacuum Einstein’s field equations. We demonstrate that the condition for the vanishing of the Ricci tensor obtained by Kerr and Schild, in empty space–time, is also a condition for the vanishing of the Nijenhuis tensor constructed out of $S_{\mu }^{\lambda }$. Thus, a theory based on the Nijenhuis tensor yields an important class of solutions of the Einstein’s field equations, namely, black holes and non-linear gravitational waves. We also demonstrate that the present mathematical framework can easily admit modifications of the Newtonian potential that may explain the long range gravitational effects related to galaxy rotation curves. Full article

We propose a multi-layer data mining architecture for web services discovery using word embedding and clustering techniques to improve the web service discovery process. The proposed architecture consists of five layers: web services description and data preprocessing; word embedding and representation; syntactic similarity; semantic similarity; and clustering. In the first layer, we identify the steps to parse and preprocess the web services documents. In the second layer, Bag of Words with Term Frequency–Inverse Document Frequency and three word-embedding models are employed for web services representation. In the third layer, four distance measures, namely, Cosine, Euclidean, Minkowski, and Word Mover, are considered to find the similarities between Web services documents. In layer four, WordNet and Normalized Google Distance are employed to represent and find the similarity between web services documents. Finally, in the fifth layer, three clustering algorithms, namely, affinity propagation, K-means, and hierarchical agglomerative clustering, are investigated for clustering of web services based on observed similarities in documents. We demonstrate how each component of the five layers is employed in web services clustering using randomly selected web services documents. We conduct experimental analysis to cluster web services using a collected dataset consisting of web services documents and evaluate their clustering performances. Using a ground truth for evaluation purposes, we observe that clusters built based on the word embedding models performed better than those built using the Bag of Words with Term Frequency–Inverse Document Frequency model. Among the three word embedding models, the pre-trained Word2Vec’s skip-gram model reported higher performance in clustering web services. Among the three semantic similarity measures, path-based WordNet similarity reported higher clustering performance. By considering the different word representations models and syntactic and semantic similarity measures, we found that the affinity propagation clustering technique performed better in discovering similarities among Web services. Full article

In this paper, we analyzed the fermionic condensate (FC) associated with a massive fermionic field on a five-dimensional anti-de Sitter (AdS) spacetime in the presence of a cosmic string taking into account a magnetic flux running along its core. In addition, a compactified dimension was considered. Due to this compactification, the FC is expressed in terms of two distinct contributions: the first one corresponds to the geometry without compactification, and the second one is induced by the compactification. Depending on the values of the physical parameters, the total FC can be positive or negative. As a limiting case, the expression for the FC on locally Minkowski spacetime was derived. It vanishes for a massless fermionic field, and the nonzero FC on the AdS background space in the massless case is an effect induced by gravitation. This shows that the gravitational field may essentially influence the parameter space for phase transitions. For a massive field, the FC diverges on the string as the inverse cube of the proper distance from the string. In the case of a massless field, depending on the magnetic flux along the string and planar angle deficit, the limiting value of the FC on the string can be either finite or infinite. At large distances, the decay of the FC as a function of the distance from the string is a power law for both cases of massive and massless fields. For a cosmic string on the Minkowski bulk and for a massive field, the decay is exponential. The topological part in the FC vanishes on the AdS boundary. We show that the FCs coincide for the fields realizing two inequivalent irreducible representations of the Clifford algebra. In the special case of the zero planar angle deficit, the results presented in this paper describe Aharonov–Bohm-type effects induced by magnetic fluxes in curved spacetime. Full article

Here, we study a quantum fermion field in rigid rotation at finite temperature on anti-de Sitter space. We assume that the rotation rate $\Omega$ is smaller than the inverse radius of curvature ${\ell }^{-1}$, so that there is no speed of light surface and the static (maximally-symmetric) and rotating vacua coincide. This assumption enables us to follow a geometric approach employing a closed-form expression for the vacuum two-point function, which can then be used to compute thermal expectation values (t.e.v.s). In the high temperature regime, we find a perfect analogy with known results on Minkowski space-time, uncovering curvature effects in the form of extra terms involving the Ricci scalar R. The axial vortical effect is validated and the axial flux through two-dimensional slices is found to escape to infinity for massless fermions, while for massive fermions, it is completely converted into the pseudoscalar density $-i\overline{\psi }{\gamma }^5\psi$. Finally, we discuss volumetric properties such as the total scalar condensate and the total energy within the space-time and show that they diverge as ${[1-{\ell }^2{\Omega }^2]}^{-1}$ in the limit $\Omega \to {\ell }^{-1}$. Full article

In this paper, we show that Lorentz boosts are direct isometries according to the recent mathematical definitions of direct and indirect isometries and of chirality, working for any metric space. Here, these definitions are extended to the Minkowski spacetime. We also show that the composition of parity inversion and time reversal is an indirect isometry, which is the opposite of what could be expected in Euclidean spaces. It is expected that the extended mathematical definition of chirality presented here can contribute to the unification of several definitions of chirality in space and in spacetime, and that it helps clarify the ubiquitous concept of chirality. Full article

In this paper, a novel meshless method for the transient modeling of subsurface flow in unsaturated soils was developed. A linearization process for the nonlinear Richards equation using the Gardner exponential model to analyze the transient flow in the unsaturated zone was adopted. For the transient modeling, we proposed a pioneering work using the collocation Trefftz method and utilized the coordinate system in Minkowski spacetime instead of that in the original Euclidean space. The initial value problem for transient modeling of subsurface flow in unsaturated soils can then be transformed into the inverse boundary value problem. A numerical solution obtained in the spacetime coordinate system was approximated by superpositioning Trefftz basis functions satisfying the governing equation for boundary collocation points on partial problem domain boundary in the spacetime coordinate system. As a result, the transient problems can be solved without using the traditional time-marching scheme. The validity of the proposed method is established for several test problems. Numerical results demonstrate that the proposed method is highly accurate and computationally efficient. The results also reveal that it has great numerical stability for the transient modeling of subsurface flow in unsaturated soils. Full article